Physics Reports vol.394

Available online at www.sciencedirect.com Physics Reports 394 (2004) 1 – 40 www.elsevier.com/locate/physrep Magnetic q...

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Physics Reports 394 (2004) 1 – 40 www.elsevier.com/locate/physrep

Magnetic quantum dots and magnetic edge states S.J. Leea; b;∗ , S. Soumab , G. Ihmc , K.J. Changd a Department of Physics, State University of New York, Bualo, NY 14260, USA Quantum-Functional Semiconductor Research Center, Dongguk University, Seoul 100-715, South Korea c Department of Physics, Chungnum National University, Taejon 305-764, South Korea d Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701, South Korea b

Accepted 1 November 2003 editor: A.A. Maradudin

Abstract Starting with de/ning the magnetic edge state in a magnetic quantum dot, which becomes quite popular nowadays conjunction with a possible candidate for a high density memory device or spintronic materials, various magnetic nano-quantum structures are reviewed in detail. We study the magnetic edge states of the two dimensional electron gas in strong perpendicular magnetic /elds. We /nd that magnetic edge states are formed along the boundary of the magnetic dot, which is formed by a nonuniform distribution of magnetic /elds. These magnetic edge states circulate either clockwise or counterclockwise, depending on the number of missing 6ux quanta, and exhibit quite di8erent properties, as compared to the conventional ones which are induced by electrostatic con/nements in the quantum Hall system. We also /nd that a close relation between the quantum mechanical eigenstates and the classical trajectories in the magnetic dot. When a magnetic dot is located inside a quantum wire, the edge-channel scattering mechanism by the magnetic quantum dot is very di8erent from that by electrostatic dots. Here, the magnetic dot is formed by two di8erent magnetic /elds inside and outside the dot. We study the ballistic edge-channel transport and magnetic edge states in this situation. When the inner /eld is parallel to the outer one, the two-terminal conductance is quantized and shows the features of a transmission barrier and a resonator. On the other hand, when the inner /eld is reversed, the conductance is not quantized and all channels can be completely re6ected in some energy ranges. The di8erence between the above two cases results from the distinct magnetic con/nements. We also describe successfully the edge states of magnetic quantum rings and others in detail. c 2003 Elsevier B.V. All rights reserved. PACS: 73.23.Ad; 73.20.Dx Keywords: Two-dimensional electron gases; Magnetic edge states; Magnetic quantum dots; Ballistic transport

∗

Corresponding author. Tel.: +82-2-2260-3953; fax: +82-2-2260-3945. E-mail address: [email protected] (S.J. Lee).

c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.11.004

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S.J. Lee et al. / Physics Reports 394 (2004) 1 – 40

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Chronological survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Magnetic edge states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Two di8erent magnetic domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Energy levels of the magnetic quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Correspondence between the quantum mechanical eigenstates and the classical trajectories in the magnetic quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The resonant tunneling through the magnetic quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Quantum wires with magnetic quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Edge state transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Magnetic edge states of magnetic quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Edge-channel scattering by magnetic quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Conductance and local density of states of quantum wire with two magnetic quantum dots . . . . . . . . . . . . . . 5. Modi/ed magnetic quantum dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Formulation of modi/cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Angular momentum transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Magnetic quantum ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Electronic structures of magnetic quantum ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Modi/ed magnetic quantum ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 6 6 6 9 10 12 12 13 15 18 23 23 25 28 28 34 37 38 38

1. Introduction In past decades, advances in semiconductor nano-technology have enabled researchers to fabricate low dimensional nano-scale structures with great control [1–16]. This has made an enormous interest in the study of the mesoscopic systems for novel phenomena, especially those relating to electron transport [17]. The two-dimensional electron gas (2DEG) created in the interface of semiconductor heterostructures such as GaAs=Alx Ga1−x As is one of the important source of the low dimensional systems. Quantum dots, rings, wires and antidots are the typical examples obtained through the additional con/nements on 2DEG. In these systems, the quantum mechanical e8ects such as level quantization due to the con/nement potential and the magnetic /eld, quantum interference, strong electron–electron interaction, as well as single-electron charging e8ects in6uence the electron transport. The Landauer–BLuttiker formalism [18,19] is one of the central tool to understand electron transport in these mesoscopic systems. The conductance of large samples is well known to obey an ohmic scaling low. But as we go to smaller dimension, the conductance does not decrease linearly with the width of the sample, instead, it depends on the number of transverse modes in the conductor. Moreover, there is an interface resistance independent of the length of the sample. In the Landauer–BLuttiker formalism, these features are incorporated. First, we brie6y study the chronological survey. Second, we study the nature of magnetic edge states in magnetic quantum dot, which is formed by spatially nonuniform magnetic /elds on 2DEG.


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Magnetic quantum structures such as magnetic quantum dots have attracted much attention recently. With the application of spatially inhomogeneous magnetic /eld, a number of alternative magnetic structures were proposed on the 2DEG, such as magnetic quantum dots using a scanning tunneling microscope lithographic technique [3], magnetic superlattices by the patterning of ferromagnetic materials integrated by semiconductors [4], type-II superconducting materials deposited on conventional hetero-structures [1], and non-planer 2DEG systems grown by a molecular beam epitaxy [20]. The electron transport features through these magnetic quantum structures are very di8erent from the electrostatic quantum structures, thus, a variety of new phenomena associated with the magnetic structures are expected. Motivated from these features, we calculate exactly the single electron eigenstates and energies of a magnetic quantum dot as a function of magnetic /eld. We propose magnetic edge states, which are the states along the boundary between two di8erent magnetic domains, and /nd two types of magnetic edge states which circulate in opposite directions to each other along the boundary of the magnetic dot. The properties of magnetic edge states largely depend on the missing magnetic 6ux quanta inside the dot and the angular momentum of the states. We also /nd a close relation between the quantum mechanical eigenstates and the classical trajectories in the magnetic quantum dot. Very recently, the e8ects of magnetic edge states on magnetoresistance have been reported experimentally in transverse magnetic steps [21]. Third, we study the ballistic electrons transport and magnetic edge states in quantum wires with a magnetic quantum dot. It is easily expected from classical trajectories that the scattering of conventional edge channels [22] by magnetic quantum structures is quite di8erent from those by electrostatic quantum structures. The study of such a scattering mechanism is important to understand electron transport in magnetic structures and to suggest future device application. The magnetic dot, considered here, is formed by two di8erent magnetic /elds inside and outside the dot. When the inner /eld is parallel to the outer one, we /nd that the two-terminal conductance is quantized like the electrostatic quantum point contact [23] and shows the features of a transmission barrier and a resonator. This feature results from the harmonic potential-like magnetic con/nements and is similar to those of electrostatic dots or antidots. On the other hand, when the inner /eld is reversed and conventional edge states interact with magnetic edge states, the conductance is not quantized and all incident channels can be completely re6ected by the dot in some energy ranges, so that the conductance G oscillates between 0 and 2e2 =h with G = 0 plateaus. In this case, the magnetic con/nements are the types of double wells and merged single wells, which are caused by the /eld reversal at the dot boundary. The di8erence of the conductance and magnetic edge states between the two cases of parallel and reversed /elds results from the distinct magnetic con/nements. The calculation method to evaluate the conductance of quantum wires with magnetic quantum dot is improved and generalized to take into account the presence of two or more magnetic quantum dots, where the use of the region-dependent functional form of the vector potential and the gauge transformation is essential. Applying the proposed calculation method, we found the presence of magnetic edge states extended spatially over two magnetic quantum dots. Fourth, we study the modi/ed magnetic quantum dot. In modi/ed magnetic quantum dots, electrons are magnetically con/ned to the plane where the magnetic /elds inside and outside the dot are di8erent from each other. The energy spectrum exhibits quite di8erent features depending on the directions of the magnetic /elds inside and outside the dot. In particular, the case of opposite directions of the /elds is more interesting than that of the same direction. An electrostatic potential

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is introduced to the system to study the e8ects of an electric con/ning potential on the eigenenergy of a single electron in the modi/ed magnetic quantum dot. The additional potential raises the whole energy spectrum and changes its shape. The ground-state angular momentum transitions occurring in a bare modi/ed magnetic quantum dot disappear on introduction of the additional parabolic potential. Finally, we present the model of a magnetic quantum ring, where electrons are con/ned to a plane, and the magnetic /elds are zero inside the ring and constant elsewhere. The energy states that deviate from the Landau levels are found to form the magnetic edge states along the boundary regions of the magnetic quantum ring. The probability densities of these magnetic edge states are found to be well corresponded to the circulating classical trajectories. In contrast to magnetic or conventional quantum dots, the eigenstates of the magnetic quantum ring show angular momentum transitions in the ground state as the magnetic /eld increases, even without including electron–electron interactions. For a modi/ed magnetic quantum ring with the distribution of nonzero magnetic /elds inside the ring and di8erent /elds outside it, we also /nd similar behaviors such as the angular momentum transitions in the ground state with increasing the magnetic /eld. Our review does not include the conventional magnetic thin /lm, stripes, and nano-particle, e.g. Fe nano-particle on Cu(GaAs) substrate, which can be used for the high density recording media. Some people use the terminology of “magnetic quantum dot (QD)” for this magnetic nano-particle but we do not treat this subject in our paper. We are concentrated in magnetic quantum dot made by the circular dot of two dimensional electron gas (2DEG) due to boundary of the two di8erent region of the inhomogeneous magnetic /eld Bin = B0 (r ¡ r0 ) and Bout = B (r ¿ r0 ). 2. Chronological survey In this section, we survey chronologically the various approaches for the magnetic quantum dot. As previously mentioned, we do not include the magnetic nano-particle for which some people use the terminology of the magnetic quantum dot. In 1990 von-Klitzing group [1] made thin gates of type II superconducting materials on top of the 2DEG in a GaAs/AlGaAs heterostructure to /nd the e8ect of the modulation of an applied magnetic /eld. They found a weak-localization magnetoconductance for small /elds proportional to the magnetic /eld B, in contrast to the B2 homogeneous result. Later on they used similar structure of 2DEG for a 6ux detector of a superconducting /lm [2]. In the same year McCord and Awschalom [3] of IBM found the method of direct deposition of magnetic dots using a scanning tunneling microscope. They used Fe(CO)5 as the source gas for the deposits and gold /lm and SQUID pick up coil as the substrates and got magnetic dots with diameters ranging from 10 to 30 nm and heights from 30 to 100 nm. They estimated that the dots were composed of about 50% iron, the remainder being primarily carbon contamination along with a small amount of oxygen. They claimed that measurements on the particles at low temperatures showed them to be magnetic and reveal macroscopic spin properties. They used the nomenclature of “magnetic dots” or “magnetic particles” instead of “magnetic quantum dots”. After that many di8erent groups made various kinds of structures to study the behavior of electrons in inhomogeneous magnetic /elds [4–6]. Leadbeater et al. [4] studied single crystal /lms of -MnAl grown by MBE on GaAs substrates and showed a large hysteresis loop by the extraordinary


5

Hall e8ect. Krishnan et al. [5] reported, in 1992, the structure and properties of the thermodynamically stable -phase Mn1−x Gax single crystal thin /lms grown on GaAs and suggested that this set of materials is a very promising one for magneto-optic recording with the additional potential of integrating semiconductor/magnetic devices. The /rst realization of a spatially modulated periodic magnetic /eld was performed by Carmona et al. [7] by putting superconducting stripes on the surface of the heterostructure with a 2DEG. They observed oscillatory magnetoresistance due to a commensurability e8ect between the classical cyclotron diameter and the period of magnetic modulation. Other realization was through magnetic superlattices or periodic magnetic modulations by the patterning of ferromagnetic materials integrated by semiconductors [8]. Authors of Ref. [7] investigated the e8ect of the parallel ferromagnetic stripes of Dy on 2DEG of AlGaAs/GaAs heterostructure and found that the longitudinal resistance of the 2DEG displays, as a function of the externally applied /eld, the magnetic commensurability oscillations which result from the interplay between the two characteristic length scales of the system, the classical cyclotron radius Rc of the electrons and the period a of the magnetic /eld modulation, 2Rc = ( + 1=4)a, where = 0; 1; : : : is an integer oscillation index. Motivated by these experiments a number of theoretical investigations was followed pursuing the transport properties of these magnetic structures [9–14]. Peeter’s group [9,10,13] studied systems of magnetic quantum steps, barriers, and magnetic wells and found the energy spectrum and the nature of the bound and/or scattered states. They showed the interesting features of inhomogeneous magnetic-/eld can bind the electrons. This is essentially di8erent from elastic potential steps, which always act repulsive. In 1994 Chang and Niu [11] studied the energy spectrum of a 2DEG in a 2D periodic magnetic /eld. Both a square magnetic lattice and a triangular one were considered. They found that a general feature of the band structure was bandwidth oscillation as a function of the Landau index. This theory could be applied to a triangular magnetic lattice on a 2DEG which was realized by the vortex lattice of a superconductor /lm coated on top of a heterojunction. Later on You et al. [12,14] also investigated the transport properties of the nanostructures consisting of magnetic barriers produced by the deposition of ferromagnetic stripes on heterostructures and found that the electron tunneling through multiple-barrier magnetic structures exhibits complicated resonant features. For this study they took two types of magnetic barriers, which were produced by the deposition, on top of a heterostructure, of a ferromagnetic stripe with magnetization (a) perpendicular and (b) parallel to the 2DEG located below the stripe. They used a logarithmic function and a arctangent function for the vector potential, for cases (a) and (b), respectively. In 1997, Nogaret et al. [15] studied similar subject as what von Klitzing’s group did [8], but they put ferromagnets di8erent way and got a periodic magnetic /eld that alternates in sign. They observed a giant low-/eld magnetoresistance due to electrons propagating in open orbits along lines of zero magnetic /eld. The main contribution came from the open orbits among the three types of electron orbits, open, intermediate, and closed. They could explain the observed form and magnitude of the magnetoresistance in a semiclassical model. However, most of the previous investigations overlooked the importance of magnetic edge states which can be formed at the boundary of di8erent magnetic domains in close analogy with conventional edge states formed by the electrostatic con/nements. Some of present authors [16] presented a model of a magnetic quantum dot and explained the magnetoresistance in terms of the magnetic edge state which will be explained in detail in next section.

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3. Magnetic edge states 3.1. Two dierent magnetic domains For the two-dimensional electron gas (2DEG) applied by inhomogeneous magnetic /elds, which provide two di8erent magnetic domains, as shown schematically in Fig. 3.1, the current-carrying states (hereafter referred to the magnetic edge states in close analogy with electrostatically induced conventional ones) exist near the boundary between the two domains [24]. These magnetic edge states have quite di8erent properties from the conventional ones, thus, a variety of new phenomena associated with the magnetic structures are expected in the electron transport. However, to our knowledge, only a little attention has been paid to this problem [25]. In this chapter, we investigate the nature of magnetic edge states in a magnetic quantum dot which is formed by inhomogeneous magnetic /elds; electrons are apparently con/ned to a plane and within that plane the magnetic /eld is zero within a circular disc and constant B outside it [26]. We calculate exactly the single electron eigenstates and energies of a magnetic quantum dot as a function of magnetic /eld, using a single scaled parameter s=r02 B=0 , which represents the number of missing magnetic 6ux quanta within the dot, where r0 is the radius of the quantum dot and 0 (=h=e) is the 6ux quantum. We /nd two types of edge states which circulate in opposite directions to each other along the boundary of the magnetic dot and exhibit quite di8erent energy dependences on angular momentum. We /nd a close relation between the quantum mechanical eigenstates and the classical trajectories in the magnetic quantum dot; the quantum mechanical eigenstate corresponds to a certain ensemble average of the classical motions which consist of straight line paths in the dot region and cyclotron orbits with a quantized radius in the outside region. These radius and central positions of the cyclotron orbits critically depend on the value of s. For a narrow two-dimensional conductor with a magnetic quantum dot at the center, the calculated magnetoconductances show aperiodic oscillations instead of the Aharonov–Bohm type of periodic oscillations [27], and this behavior is attributed to the characteristics of the magnetic edge states, which is absent in the conventional ones. 3.2. Energy levels of the magnetic quantum dot The single particle SchrLodinger equation for a two-dimensional magnetic quantum dot is (˜ p+ 2 ∗ ∗ ˜ eA) =(2m ) (˜r) = E (˜r), where m is the e8ective mass of electron and e is the absolute value of

Fig. 3.1. Schematic diagram of classical trajectories of electrons for the magnetic edge states on the magnetic domain boundary.


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the electron charge. In polar coordinates (r; ) on the plane, the vector potential Ã can be chosen as 0 for r ¡ r0 and (r 2 − r02 )B=(2r)ˆ for r ¿ r0 , so that B = 0 for r ¡ r0 and nonzero Bzˆ otherwise. The wave functions and the energies are easily determined by the continuity of the wave functions and their derivatives at the boundary of the dot. Since the wave functions are separable, i.e., nm (˜r) = Rnm (r)eim ; where m is the angular momentum quantum number and n (=0; 1; 2; : : :) is the radial quantum number (the number of nodes in the radial wave function), the equation for the radial part is written as 2 m2 d 1 d − 2 + 2E Rnm (r) = 0 (r ¡ r0 ) ; + (3.1) dr 2 r dr r 2 (m − s)2 d 1 d 2 − + − r + 2[E − (m − s)] Rnm (r) = 0 (r ¿ r0 ) : (3.2) dr 2 r dr r2 √ 2 Here Rnm (r) = C1 J|m| ( 2Er) for r ¡ r0 and Rnm (r) = C2 r |m−s| e−r =2 U (a; b; r 2 ) for r ¿ r0 . It is convenient to express all quantities in dimensionless units by letting ˝!L [ =√ ˝eB=(2m∗ )] and the inverse 2 ∗ 2 ∗ length $ = m !L =˝ be 1. Then, since ˝ =m = ˝!L =$ → 1 and r0 → s, s = Br02 e=h is only the relevant parameter. The function Jm is the Bessel function of the order m, and U is the con6uent hypergeometric function with a = −(E − me8 − |me8 | − 1)=2, b = |me8 | + 1, and me8 = m − s. It is noted that Eq. (3.2) has the same form as that of the uniform magnetic /eld case, except that the angular momentum m is replaced by the e8ective angular momentum me8 . The meaning of this replacement of m to me8 will be discussed in the next section. In the magnetic quantum dot, the Landau level degeneracy is lifted for the states near the dot. From Eqs. (3.1) and (3.2), if the e8ective potential Ve8 (r) is de/ned as  2 m   (r ¡ r0 ) ;  2 2r (3.3) Ve8 (r) = 2 2    me8 + r + me8 (r ¿ r0 ) ; 2r 2 2 √ the minimum of Ve8 (r) always occurs at r = r0 (= s) for the states with |me8 | ¡ s, i.e., 0 ¡ m ¡ 2s, which correspond to the magnetic edge states circulating counterclockwise, as we will see below. The m = 0 state is widely distributed over the dot due to the lack of the centrifugal force, and the minimum of Ve8 (r) for the states with |me8 | ¿ s, i.e., m ¡ 0 or m ¿ 2s, is located at r = |me8 | outside the quantum dot, similar to the case of uniform magnetic /elds. The states with m ¡ 0, which exist near the dot, give rise to the magnetic edge states circulating clockwise. Fig. 3.2 shows the energy levels of the magnetic quantum dot for di8erent values of m at s = 5, U for magnetic /elds of teslas. The lowest energy state occurs the radius of which is about 500 A at m = 0 and the degeneracy of the Landau levels are removed, as shown in Fig. 3.2. This result indicates that the inhomogeneity of magnetic /elds mostly perturbs the states near the boundary of the quantum dot, and this perturbation is caused by the missing of s 6ux quanta. From the wave functions, the probability current density Jnm carried by the state nm can be calculated as

˝˜ 1 ∗ ∇ + eÃ Jnm = ∗ Re nm ; (3.4) nm m i

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Fig. 3.2. Dependence of the energy eigenvalues Enm on the angular momentum m for s = 5. Dashed lines represent the bulk Landau levels.

Fig. 3.3. Dependence of the probability current Inm (in units of !L ) on the angular momentum m for s = 5 and n = 0 case.

and the probability current Inm of the state nm is related to the derivative of Enm with respect to m as follows [9]: ∞ 1 9Enm Inm = : (3.5) Jnm dr = h 9m 0 In Fig. 3.3, the probability current Inm for s=5 and n=0 is drawn as a function of m. The probability currents for the perturbed states are found to have nonzero, resulting in the magnetic edge states. For m ¿ 0, Inm have positive values for counterclockwise circulations whereas for m ¡ 0, Inm have negative values for clockwise ones. In Fig. 3.4, the energy levels are plotted as a function of magnetic /eld s for di8erent values of (n; m), with the energy ˝!L set to one at s = 5 and the radius r0 /xed. As the magnetic /eld


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Fig. 3.4. Energy spectra as a function of s. The energy unit of ˝!L = 1 at s = 5 is used. Dotted lines represent the Landau levels.

increases, the deviations of energies from the bulk Landau levels become signi/cant, which lead to the magnetic edge states near the boundary of the quantum dot. In the limit of B → ∞, we /nd that the energies approach to those for the conventional circular dot which is electrostatically con/ned by hard walls without magnetic /elds. 3.3. Correspondence between the quantum mechanical eigenstates and the classical trajectories in the magnetic quantum dot To see the signi/cance of me8 , let us consider momentarily the 2DEG in a uniform magnetic /eld, in which the eigenstates are described by the degenerate Landau levels, Ei = ˝!c (i + 1=2), where !c = eB=m∗ . When the symmetric gauge is chosen, n and m remain good quantum numbers and the probability density of the eigenstate (n = 0; m) has a maximum at r = |m| in dimensionless units. In this case, the quantum mechanical eigenstate (n; m) with the eigenvalue Enm corresponds to the ensemble average of the classical cyclotron motions [28] with the radius ri and its center located at rj from the origin, which satisfy the following relations from the conservations of energy and angular momentum;

Enm 2n + |m| + m + 1 ri = = ; rj = ri2 − m : (3.6) 2 2 However, because of the uncertainty principle, the central position of the cyclotron orbit cannot be determined quantum mechanically. From the analysis used for the uniform /eld case, we can also show that the (n; m) state exactly corresponds to the ensemble average of the classical motions which consist of the straight line paths in the dot region and the cyclotron orbits with the radius ri and the center located at rj outside the dot. These straight lines and cyclotron orbits intersect each other at the dot boundary. In this

10


case, the relations between the (n; m) states and the corresponding ri and rj values are determined from the conservations of energy and angular momentum for the magnetic quantum dot and are written as

Enm ; rj = ri2 − me8 : (3.7) ri = 2 Eq. (3.7) has the same form as Eq. (3.6) except that Enm calculated from Eqs. (3.1) and (3.2) is lifted from the bulk Landau level in Eq. (3.6) and m is replaced by me8 due to the inhomogeneity of magnetic /elds. The replacement of me8 from m is due to the missing magnetic 6ux quanta s. For example, the (0; m ¡ 0) state locates at rj ∼ |m| in the uniform /eld case and encloses |m| magnetic 6ux quanta. When the s 6ux quanta are missed inside the dot, this state moves outside to keep enclosing |m| magnetic 6ux quanta and locates at rj ∼ |m| + s in result. The classical trajectories for the (0; 0), (0; −1), and (0; 1) states are drawn in Fig. 3.5, showing a clear correspondence between the quantum eigenstates and the classical motions; the probability densities |Rnm (r)|2 and the directions of the probability currents Inm correspond to the classical motions. The classical trajectory corresponding to the (0; 0) state carries no current because it always passes through the origin, and the classical motions of the (0; −1) and (0; 1) states correspond to the probability currents of the states in the clockwise and counterclockwise directions, respectively. We /nd that our correspondence analysis may answer to the important question whether the classical motions corresponding to the quantum eigenstates are periodic or not. In the magnetic quantum dot, periodic motions occur if the angle ) made by two lines connected from the origin to the centers of two successive orbits [see Fig. 3.5(c)] is 2p=q, where p and q are integers. From a simple geometrical argument, ) is found to satisfy the relation cos()=2) = (ri2 + rj2 − r02 )=(2ri rj ). However, at this moment, it is diVcult to make a de/nite answer because of the numerical errors for evaluating Enm . 3.4. The resonant tunneling through the magnetic quantum dot In this section, we present a phenomenological discussion of the tunneling of an electron through the magnetic quantum dot placed in the quantum wire (narrow two-dimensional conductor). More accurate numerical calculations of such structure will be presented in the next section. We consider a narrow two-dimensional conductor with a magnetic quantum dot at the center. Under the applied strong magnetic /elds which give the quantum Hall plateaus, the transport along the boundary of the sample, which is usually promoted by conventional edge states, can be backscattered by the resonant tunneling into the magnetic edge states along the boundary of the dot, because of the impurity e8ect in the narrow region between two boundaries. In usual quantum dots or ring structures, the resonant tunneling e8ect in magnetoresistance measurements gives rise to the Aharonov–Bohm oscillations [27,29], which are periodic with magnetic /eld. In the magnetic quantum dot considered here, we do not see such periodic oscillations. We calculate the two-terminal conductance, which is the inverse of the sum of magnetoresistance and Hall resistance, taking into account the resonant backscattering via the magnetic edge channels as follows; 2e2 +2 G(B) = 1− ; (3.8) h (EF − Enm (B))2 + +2 n; m


11

Fig. 3.5. Classical trajectories of electrons and corresponding probability densities for the eigenstates (a) (0; 0), (b) (0; −1), and (c) (0; 1).

where + is the elastic resonance width and a constant value of + = 0:005 is used for simplicity. The calculated conductance is plotted as a function of magnetic /eld in Fig. 3.6, with the Fermi energy of EF = 2 in units of Fig. 3.4. In this case, the magnetic /elds represented by s are in the , = 2 quantum Hall plateau region, where , is the Landau level /lling factor. We /nd that the oscillations are not periodic, in contrast to the Aharonov–Bohm type of oscillations. The /rst dip in the conductance around s = 3:7 is due to the resonant backscattering via the (1; −3) magnetic edge state. The other dips are found to be associated with the (0; 3), (1; 1), (1; −2), and (1; −1) states in the increasing order of s. In the narrow ring structure of Jain [27], the intervals between the dips were shown to be periodic, which indicates the subsequent change of one 6ux quantum passing through the inner boundary. In our magnetic dot structure, the resonances occur via the two di8erent magnetic edge states circulating in di8erent directions, depending on the sign of m. Since there is

12


Fig. 3.6. Magnetoconductance as a function of s.

no magnetic /eld inside the magnetic dot, the magnetic edge states may not enclose the magnetic 6ux, resulting in the missing of 6ux quanta, which is absent in the edge states formed by electrostatic con/nements. 4. Quantum wires with magnetic quantum dots 4.1. Edge state transport Transport properties of two-dimensional electron gas (2DEG) in spatially nonuniform magnetic /elds have also attracted much attention recently. As a counterpart of electrostatic structures, various magnetic structures have been realized experimentally [3,30,31,21], patterning of ferromagnetic or superconducting materials on 2DEG or using nonplanar 2DEG. Theoretically, it was shown that nonuniform magnetic /elds can cause electron drifts [25,32], transmission barriers [33], commensurability e8ects [34], and electron con/nements [35,36]. Magnetic edge states, which exist along the boundary between two di8erent magnetic domains, were also proposed [35,36] in the same analogy with the conventional edge states [22,19] in quantum Hall systems. Recently, the e8ects of them on magnetoresistance were reported in transverse magnetic steps [21]. In the edge state transport regime, the conductance of quantum wires with a local electrostatic modulation is quantized except for resonant re6ections [37] and exhibits Aharonov–Bohm oscillations [38]. These interesting features can be modi/ed when such a modulation is replaced by a magnetic one such as a magnetic quantum dot (or magnetic antidot) [35,36] which is formed in 2DEG by nonuniform perpendicular magnetic /elds; ˜B = B∗ zˆ within a circular disk with radius r0 , while ˜B = B0 zˆ outside it. Classical electron trajectories (see Fig. 4.1) scattered by a magnetic dot with -[ = B∗ =B0 ] ¡ 0 are very di8erent from those for - ¿ 0 and those by an electrostatic dot (or antidot). This indicates that the edge-channel scattering by local magnetic modulations can be quite di8erent from that by electrostatic ones. The study of such a scattering mechanism is important to understand electron transport in magnetic structures and to suggest future device application. However, to our knowledge, little attention has been paid to it [33]. In this chapter, we study the ballistic transport of conventional edge channels through quantum wires with a magnetic quantum dot [35]. The magnetic edge states near the dot and two-terminal


13

Fig. 4.1. Schematic diagram of a quantum wire with a magnetic quantum dot. The solid (dotted) arrows represent classical electron trajectories for - ¿ 0 (- ¡ 0), where - = B∗ =B0 .

conductance G(EF ) of the wires in the limit of zero bias are found to exhibit distinct features between two cases of - ¿ 0 and - ¡ 0, where EF is the Fermi energy. For - ¿ 0, G(EF ) is quantized except for resonances and shows the behavior of a transmission barrier and a resonator, depending on the value of -, when the magnetic length inside the dot is smaller than r0 . This feature results from the harmonic-potential-like magnetic con/nements and is similar to those of electrostatic dots (or antidots). On the other hand, for - ¡ 0, G(EF ) is not quantized when incident edge channels are scattered by the dot. Moreover, for - ¡ − 1, incident channels can be completely re6ected by the dot in some ranges of EF , resulting in the plateaus of G(EF ) = 0. This interesting feature is due to the double-well and merged-well magnetic con/nements caused by the /eld reversal at the dot boundary. We also propose a calculation method for conductance, based on the Green’s function along with the lattice-Hamiltonian and the symmetric gauge. In the /nal section of this chapter, the method of conductance calculation based on the Green’s function is generalized to include many magnetic quantum dots. 4.2. Magnetic edge states of magnetic quantum dots As shown in Fig. 4.1, the dot is assumed to be located at the center of the wire. For simplicity, the wire potential along the transverse (y) ˆ direction is assumed to be an in/nite square well with width Ly . The magnitudes of the magnetic /eld determine the characteristic length andthe energy scales; the magnetic energy inside (outside) the dot is lB∗ (j) = (2j + 1)˝=(e|B∗ |) length and the Landau (lB0 (j)= (2j + 1)˝=(eB0 )) and E ∗ (j)=(j+1=2)˝e|B∗ |=m∗ (E0 (j)=(j+1=2)˝eB0 =m∗ ), respectively, where j = 0; 1; 2; : : : and m∗ is the e8ective mass. We focus on the edge state transport regime [i.e., Ly lB0 (N − 1)] and ignore the e8ects of spin and disorder, where N is the number of Landau levels below EF far away from the dot. Note that in previous studies [16] for magnetic dots, B∗ is /xed to be zero. We /rst consider the case 1 ¿ lB0 (N − 1), where 1 is the distance between the dot and wire edge. In this case, current-carrying conventional edge channels do not interact with the dot, thus, G(EF ) = NG0 (including the spin degeneracy) except for backward scatterings of channels by the resonant tunneling into the magnetic edge states of the dot, where G0 =2e2 =h. To study these magnetic edge states, one can neglect the wire con/nement.

14


For a magnetic dot in an in/nite 2DEG, the SchrLodinger’s equation is given by [(˜ p +eÃ)2 =2m∗ ] = ˜ E where the vector potential A can be chosen in the form of the symmetric gauge: 1  (r ¡ r0 ) ;  -B0 r Ã = ˆ 2 (4.1)   1 r 2 B (- − 1) + 1 B r (r ¿ r ) : 0 0 0 2r 0 2 Here we used the polar coordinates (r; ) and assumed that the dot center is located at r = 0. Then the eigenstates can be written as nm (˜r) = Rnm (r)eim , where m is the angular momentum quantum number and n(=0; 1; 2; : : :) is the number of nodes in R(r). The states can be classi/ed by their radial locations. A (n; m ¡ 0) state located far away from the dot interacts with B0 . From the gauge invariance [39], its radial wave function is found to be the same as that of the (n; me8 ) state in the uniform /eld B0 . Here, me8 = m − s and s[ = (1 − -)r 2 B0 =0 ] is the number of removed magnetic 6ux quanta (or additional ones for s ¡ 0) to form the magnetic dot in 2DEG where uniform B0 is already applied. Then, nm¡0 is located at rp (me8 ; B0 ), encloses |m| 6ux quanta, and its energy Enm is E0 (n), where rp (m; B) = 2|m|h=(eB). On the other hand, nm ’s near the dot interact with both of B0 and B∗ , thus, Enm ’s deviate from E0 (n). They are magnetic edge states, carry nonzero probability current Inm (˙ 9Enm =9m), and result in resonances when they interact with conventional edge channels. When r0 lB∗ (n − 1) and |m| is small, nm ’s are located at rp (m; B∗ ) inside the dot and Enm = E ∗ (n). Interestingly, for - ¡ 0, nm ’s with small m ¡ 0 can be located also at rp (me8 ; B0 ) outside the dot. √ The above features are clearly shown in Fig. 4.2. In dimensionless units of E0 (0) → 1 and 2lB (0) → 1, Enm is calculated from the radial part of the single particle SchrLodinger equation, 2 d 1 d + (4.2) + 2(Enm − Ve8 (r)) Rnm (r) = 0 ; dr 2 r dr where

 1 m 2  (r ¡ r0 ) ; + -r  2 r Ve8 (r) = (4.3)   1 me8 + r 2 (r ¿ r ) : 0 2 r The magnetic con/nement can be de/ned as the e8ective potential Ve8 and becomes the harmonic potential in uniform /elds (i.e., Ve8 with - = 1). For - ¿ 0, magnetic con/nements are similar to the harmonic potential. Thus, Enm ’s vary monotonously from E0 (j) at large |m| [i.e., rp (me8 ; B0 )r0 ] to E ∗ (j) at small |m| [i.e., rp (m; B∗ )r0 ], where j = n + (m + |m|)=2. For - ¿ 1, magnetic edge states circulate counterclockwise around the dot, while either clockwise or counterclockwise for 0 ¡ - ¡ 1. These magnetic edge states are similar to edge states around electrostatic dots or antidots. For - ¡ 0, magnetic con/nements are very di8erent from the harmonic potential. For |m| ¡ |-|s0 , Ve8 is a double-well potential, where s0 = r02 B0 =0 . The barrier in this potential is enough high to con/ne nm only in one of the wells, if r0 lB∗ . Then, for small m ¡ 0, the inner well allows energies E ∗ (j1 ) with j1 = |m|; |m| + 1; |m| + 2; : : : ; while the outer well E0 (j2 ) with j2 = 0; 1; 2; : : : Thus, nm with small m ¡ 0 can be located either inside or outside the dot, depending on n, as discussed before. This feature results in abrupt changes of Enm ’s from E0 to E ∗ [see Fig. 4.2(d)].


15

Fig. 4.2. (a) – (d) Enm ’s and (e) Ve8 (r; m)’s for s0 (=r02 B0 =0 ) = 5 and some -’s. In (e), m = −1 (solid), 4 (dashed), 15 (dotted) are chosen. The energy unit is E0 (0).

Note that the abrupt change appears only in the n = 0 level in Fig. 4.2(c), since lB∗ (0) ≈ r0 . For |-|s0 6 m 6 s + s0 , the two wells in Ve8 merge into a single well [see dotted line in Fig. 4.2(e)], which minimum occurs at r0 . Magnetic edge states in this merged well circulate counterclockwise along r = r0 with snake-like classical motions. 4.3. Edge-channel scattering by magnetic quantum dots Next, we study the scattering of incident conventional edge channels by the magnetic dot when WlB0 (N − 1). We calculate a transmission probability T ()) [or G()) = T ())G0 ] of incident channels with dimensionless energy )[ = EF =(2E0 (0))] in a quantum mechanical way based on the lattice Green’s function [38], where a continuous 2DEG is approximated by a tight-binding square lattice with lattice constant a. The vector potential is included as the Peierls’ phase factor [exp(−ie=h l Ã · d˜l)] in hopping matrix element. The symmetric gauge is essential to study the scattering by the magnetic dot [40], however, to our knowledge, it has never been used so far to calculate T in a quantum mechanical way. The behavior in T ()) can be classi/ed by - (see Fig. 4.3). For - ¿ 0, T ()) is quantized when lB∗ ¡ r0 . In this case, magnetic con/nements are similar to the harmonic potential. Thus, when edge channels pass the constriction between the dot and wire edge, they are still well con/ned near edges and do not interact with those in the opposite edge, resulting in the quantization of T ()). For - ¿ 1, T ()) is smaller than that of the uniform-/eld case (-=1). This feature results from that some of incident edge channels are re6ected by the dot due to the large magnetic energy E ∗ (¿ E0 ). Thus, the magnetic dot with - ¿ 1 is similar to an electrostatic antidot [38]. A transition energy Et (j), where T ()) changes from j − 1 to j, is approximately determined by min {E ∗ (j); (˝j)2 =(2m2 12 )}.

16


Fig. 4.3. Dependence of T on ) for some -’s and r0 ’s. For all cases, Ly and lB (0) are /xed as 35a and 5a.

As 1 decreases, Et (j) approaches to E ∗ (j) and the number of resonances decreases [see Figs. 4.3(a) and (c)], because magnetic edge states are con/ned in a narrower region. For 0 ¡ - ¡ 1, T ()) is the same as that for - = 1 except for resonant dips. In this case, the small magnetic energy E ∗ (¡ E0 ) does not re6ect any incident edge channels and binds electrons like as electrostatic dots, thus, the magnetic dot behaves as a resonator. The number of resonances increases as - decreases from 1 and r0 increases. The features of the magnetic dot with - ¡ 0 are very di8erent from those for - ¿ 0 and those by electrostatic dot or antidots. For −1 ¡ - ¡ 0, T ()) is not quantized and smaller than that of the uniform-/eld case, although E ∗ ¡ E0 , in contrast to the case of 0 ¡ - ¡ 1. For - ¡ − 1, T ()) is not quantized. Moreover, when 1 ≈ lB (0), incident edge channels are completely re6ected, except for resonances, in some ranges of ), so that G()) oscillates between 0 and G0 with G = 0 plateaus. It contrasts with the barrier with same 1 for - ¿ 1. The features for - ¡ 0 result from the double-well and merged-well magnetic con/nements, which are caused by the /eld reversal. To understand these features, we imitate the region near the dot as a magnetic step [32], which is con/ned by an in/nite square well U (y) with width Ly and is divided into three strips by di8erent magnetic /elds; in the middle strip (|y| ¡ r0 ), B = B∗ , while in the upper (y ¿ r0 ) and lower ones (y ¡ − r0 ), B = B0 . Eigenstates can be written as eikx Yk (y) and Ve8 (y; k) is de/ned in a similar way to that for the dot; Ve8 (y; k) = ˝2 {k + F(y)=l2B0 }2 =(2m∗ ) + U (y), where F(y) is y + r0 (- − 1) for y ¿ r0 , -y for |y| ¡ r0 , and y − r0 (- − 1) for y ¡ − r0 . In Fig. 4.4,


17

Fig. 4.4. (a) – (b) Ve8 (y; k)’s and (c) – (d) E(k)’s for magnetic steps. The energy unit is 2E0 (0) while the length unit is arbitrary. For all cases, lB (0) = 2:47 and r0 = 9.

Ve8 (y)’s and the calculated energy levels E(k ¿ 0)’s are shown. Note that E(k ¡ 0) is the same as E(|k|). The states near y = ±Ly =2 correspond to the current-carrying states near the magnetic dot, while those near y = ±r0 to the magnetic edge states circulating around the dot. And, the triple wells [solid and dashed lines in Fig. 4.4(a) – (b)] correspond to the double-well magnetic con/nements of the magnetic dot, while the double wells (dotted lines) to the merged-well magnetic con/nements. For −1 ¡ - ¡ 0, as 1(=Ly =2 − r0 ) decreases, the edge states near y = −Ly =2 are determined by Ve8 with smaller k ¿ 0, which has a smaller barrier at y = −r0 , due to the wire con/nement [see Fig. 4.4(a)]. When 1 ≈ lB0 , the barrier is so small that the states near y = −Ly =2 can be extended to the center or upper strip. Then, the states in the lower strip can easily interact with those in the upper one. The same behavior arises in the case of the magnetic dot: When 1 ≈ lB0 , conventional edge channels can interact with the doublewell magnetic con/nement with small barrier, so that they are extended in the transverse direction. Then, the left-going channels easily interact with the right-going ones, thus, the conductance is not quantized. This behavior well corresponds to the classical trajectories in Fig. 4.1. For - ¡ − 1 and k ¿ 0, when lB0 1 ¡ |-|r0 , states con/ned in the local minimum at y = Ly =2 of triple wells are the conventional edge states. Their energies are much larger than E0 at k = 0 and meet the relation dE=d k ¡ 0. As 1 decreases, the well near y = Ly =2 becomes narrower, so that the conventional edge states have larger energies and begin to be mixed with the magnetic edge states near r = r0 , resulting in the level splitting. The number of the pure conventional edge channels near y = Ly =2 is, ∼ M , where M is the largest number satisfying 2lB (M − 1) ¡ 1. In Fig. 4.4(c), two energy levels of the pure conventional edge channels are shown; note that the levels of channels near y = −Ly =2 do not appear because of their very large energies. Thus, when 1 ≈ lB (0), there exist no pure conventional edge states. In this case, eigenstates are classi/ed into those with dE=d k = 0 inside the middle strip, those with dE=d k ¿ 0 caused by the merged wells at y = r0 , and those with dE=d k ¡ 0 which result from the triple wells with small barrier at r0 [see Fig. 4.4(d)]. The third states are the mixed ones of the magnetic and conventional edge states and their energies are smaller

18


than E ∗ ’s. Then, in some energy ranges above E ∗ ’s, no states with dE=d k ¡ 0 are allowed in the upper strip. This feature indicates that all conventional edge channels cannot pass the constriction between the magnetic dot and wire edge, thus, G(EF ) = 0 in some ranges above E ∗ ’s. The plateaus of G(EF ) = 0 appear in longer energy ranges for larger |-|, smaller EF , and smaller 1. The resonant peaks in the ranges of T ()) = 0 in Fig. 4.3(b) result from the snake magnetic edge states in the merged-well magnetic con/nement. Finally, the shapes of magnetic con/nements for - ¡ 0 (- ¿ 0) are still the double wells or merged wells (the harmonic-like potentials) in realistic situations, where magnetic /elds slowly vary near the boundary of a magnetic dot or a step, thus, our /ndings can be observed experimentally. We propose that the constriction between the magnetic dot and wire edge can be considered as a magnetic quantum point contact. The conductance in this geometry with - ¿ 1 is similar to that in electrostatic quantum point contacts [23], while it can be very di8erent for - ¡ 1, showing a switching behavior with the plateaus of G(EF ) = 0. 4.4. Conductance and local density of states of quantum wire with two magnetic quantum dots In this section, we present the theoretical formulation to calculate the conductance of quantum wires in which many magnetic quantum dots are placed in series [41]. The proposed calculation method is applied to calculate the conductance and the local density of states (LDOS) of quantum wire in which two magnetic quantum dots are placed in series. In order to calculate the conductance of such systems, the method of calculation given in the last section (single magnetic dot problem) has to be modi/ed so as to allow us to treat more complicated distribution of the magnetic /elds. To be speci/c, we consider a quantum wire with width Lz (narrow two-dimensional system) de/ned by the con/nement potential Uconf (y) = 0 for |y| 6 Ly =2 and ∞ otherwise. The presence of the magnetic quantum dots is de/ned by the following magnetic /eld pro/le: ˜ r) = B(˜

N

Bi (Ri −

(x − xi )2 + y2 )zˆ ;

(4.4)

i=1

where (x) is the step function, zˆ the unit vector along the z-direction, ˜r i = (xi ; 0) and Ri are the center position and the radius of the each ith magnetic quantum dot, respectively. Here we restrict our attention to the case xi+1 − xi ¿ Ri + Ri+1 (i = 1; 2; : : : ; N − 1), so that those magnetic dots are separated with each other and are arranged in the form of an array. In the presence of an applied perpendicular (static) magnetic /eld ˜B0 = B0 z, ˆ the total magnetic /eld felt by electrons is given by ˜ r) ; ˜B(˜r) = ˜B0 + B(˜

(4.5)

which can be rewritten as ˜B(˜r) = Bi∗ z(≡ ˆ (B0 + Bi )z) ˆ if ˜r is in the ith dot region while ˜B(˜r) = B0 if ˜r is outside dot regions. In order to attack the quantum mechanical scattering problem of electrons in such system, it is convenient to model the considered system by a square lattice with lattice spacing a in the x–y plane. Let us consider M lattice sites along the y-direction, so that the width of the quantum wire is given by Ly = (M + 1)a. The lattice coordinates in the x- and the y-directions are speci/ed by the lattice indices l and m, respectively: (la; (m − (M + 1)=2)a). Assuming that the functional form of the vector potential Ã(˜r) which satis/es ˜B(˜r) = ∇ × Ã(˜r) is known, the total Hamiltonian can be


19

written as H=

∞ M

[4t|l; ml; m|

l=−∞ m=1

− t{Px (l; m)|l + 1; ml; m| + Py (l; m)|l; m + 1l; m| + h:c:}] 2

(4.6)

2

Here, t ≡ ˝ =2ma is the hopping integral between the nearest-neighbor sites within the 2D plane, with m∗ being the e8ective mass of an electron. In Eq. (4.6), the position dependent perpendicular magnetic /eld ˜B(˜r) has been implemented in the Peierls phase factor ie Ax ((˜r l; m + ˜r l+1; m )=2)a ; (4.7) Px (l; m) = exp h ie Ay ((˜r l; m + ˜r l; m+1 )=2)a : Py (l; m) = exp (4.8) h However, the complicated form of the magnetic /eld pro/le given in Eq. (4.5) makes it diVcult to /nd the functional form of Ã(˜r). Therefore, we make use of di8erent vector potentials depending on regions, along with appropriate gauge transformations for wavefunctions to take into account the uni/cation of the gauge. As seen in Eq. (4.1), the vector potential in a region including only the ith magnetic quantum dot (i = 1; 2; : : : ; N ) can be chosen in the form of symmetric gauge:  - i B0   (r(i) ¡ Ri ) ;  2 (−y; x(i); 0) (i) Ã (˜r) = (4.9) B0 R2i    (-i − 1) + 1 (−y; x(i); 0) (r(i) ¿ Ri ) ; 2 r 2 (i) where -i ≡ (B0 + Bi )=B0 . Here we have introduced a new coordinate system: (x(i); y) ≡ (x − xi ; y) which has the origin at the center of the each ith magnetic quantum dot, and r(i) ≡ (x2 (i) + y2 )1=2 the distance from the ith origin. The each vector potential Ã(i) (˜r) gives rise to the magnetic /eld ˜B(˜r) = (B0 + Bi )zˆ for r(i) ¡ Ri and ˜B(˜r) = B0 zˆ otherwise, as expected. On the other hands, in the left (i = 0) and the right (i = N + 1) lead regions where magnetic quantum dot does not exist, it is natural to employ the Landau gauge for vector potential: Ã(0) (˜r) = Ã(N +1) (˜r) = B0 (−y; 0; 0) :

(4.10)

Here we note that the boundary between the region where we use Ã(i) and the region where we use Ã(i+1) is appropriate, as long as the magnetic /eld at that boundary is ˜B0 . Suppose that the magnetic /eld in the region including lattice columns l − 1, l, and l + 1 can be described by the vector potential Ã(i) . Then the SchrLodinger equation (Hˆ − E Iˆ)|: = 0 can be reduced to the equation ˜ (i) ˜ (i) ˆ (i)∗ ˆ (i) ˜ (i) (E Iˆ − hˆ(i) l )C l + t P x (l − 1)C l−1 + t P x (l)C l+1 = 0 ;

(4.11)

where hˆ(i) l is the M × M -matrix which describes the Hamiltonian for an isolated lth column chain (i)∗ ˆ(i) (slice), and the (m; m ) element of hˆ(i) (l; m − 1) m; m +1 + l is given by {hl }mm = 4t mm − t(P (i) (i) ˆ P (l; m) m ; m+1 ). The M × M -matrix P x (l) couples the nearest neighbor column chains, and the (i) (i) (i) (m; m ) element of which is given by {Pˆ (i) x (l)}mm = Px (l; m) mm . Here Px (l; m) and Py (l; m) are the Peierls phase factors (Eqs. (4.7) and (4.8)) de/ned using the vector potential Ã(i) . In Eq. (4.11),

20


˜ (i) is a M -dimensional vector which describes the wavefunction in the lth column chain, such that C l ˜ (i) }m = l; m| (i) . Let l be a lattice column in the boundary region between the ith region and the {C l i + 1th region (i = 0; 1; : : : ; N ), such that ˜B(xl ; y) = ˜B0 . Then, noting the fact that ∇ × Ã(i) (x; y)|x=xl = ∇ × Ã(i+1) (x; y)|x=xl = ˜B0 , the wavefunction vector in the column l expressed using the Ã(i) is related to that expressed using the Ã(i+1) by the following gauge transformation: ˜ (i) ; ˜ (i+1) = Pˆ l (i + 1; i)C C l l where Pˆ l (i + 1; i) is the M × M gauge function matrix de/ned by

ie ˆ ;i+1; i (l; m) mm ; {Pl (i + 1; i)}m; m = exp h

(4.12)

(4.13)

and is introduced to account for the required uni/cation of the gauge [42]. The functional form of the gauge function ;i+1; i is de/ned by the relation (;i+1; i (l; m + 1) − ;i+1; i (l; m))=a = A(i) r l; m + ˜r l; m+1 )=2) − A(i+1) ((˜r l; m + ˜r l; m+1 )=2) ; y ((˜ y

(4.14)

with arbitrary boundary condition (e.g., ;i; i+1 (xl ; ym=0 )=0). Eq. (4.14) is the /nite di8erence version of the following di8erence equation: ∇;i; i+1 (˜r)|x=xl = (Ã(i) (˜r) − Ã(i+1) (˜r))|x=xl . Combining a set of Eqs. (4.9)–(4.14) with the recursive Green’s function method [43] generalized to the strong magnetic /eld case [44], one can calculate the transmission/re6ection probability in our system. Then the Landauer–BLuttiker formalism gives us the conductance G [18,19]. Here we note that the calculation method presented in this section has an advantage over the previous method given in the last section even in the case of single magnetic dot; the method given in this section is eVcient because it does not require to introduce the /ctitious gradation of the magnetic /eld, which is necessary in the previous method. First in order to con/rm the validity of the calculation method presented here, we carried out the numerical calculation for the case of single magnetic dot. Fig. 4.5 shows the calculated conductance (single dot case) as a function of the Fermi energy for the quantum limit [EF =(˝!0 ) ¡ 1:5] with !0 = eB0 =m∗ . Here the total magnetic /eld in the dot region (r ¡ R1 ) is chosen to be zero such that $1 = −B0 (i.e., B1∗ ≡ $1 + B0 = 0, -1 = B1∗ =B0 = 0:0). The ratio between the dot radius and the wire width is chosen asR1 = 0:28Ly , and the ratio between the wire width and the magnetic length is chosen as Ly = 7:2 ˝=(eB0 ). Therefore the number of additional magnetic 6ux quanta threading the dot region is given by s1 = R21 (B1∗ − B0 )=0 = −2:0 (the number of missing 6ux quanta is 2.0), with 0 = h=e being the magnetic 6ux quantum. As seen in Fig. 4.5, the calculated conductance is almost quantized except for the appearance of aperiodic dips. By comparing the positions of those dips with the energy spectrum of magnetic quantum dot given in Fig. 3.4 (at s = 2), one can recognize that four dips seen in Fig. 4.5 correspond to the magnetic dot eigenstates (a) (n; m) = (0; 1), (b) (1; 0), (c) (0; 2), and (d) (1; −1), respectively. Note that the energy axis in Fig. 4.5 is scaled by eB=(2m∗ ) with B satisfying R21 B=0 = 5:0. In order to understand the formation of the magnetic quantum dot more clearly, in Fig. 4.6 we show the two-dimensional distribution of the local density of states (LDOS) at those four dip positions [(a) – (d)] and at an o8-resonance point EF = 0:8˝!c [(e)]. Here the LDOS at a particular position ˜r is calculated by using the diagonal element of the Green’s function as cation The electronic properties of a modi/ed magnetic quantum dot are also studied. The modi/ed magnetic quantum dot is a quantum structure that is formed by spatially inhomogeneous distributions of magnetic /elds. Electrons are magnetically con/ned to the plane where the magnetic /elds inside and outside the dot are di8erent from each other. The energy spectrum exhibits quite di8erent features depending on the directions of the magnetic /elds inside and outside the dot. In particular, the case of opposite directions of the /elds is more interesting than that of the same direction. An electrostatic potential is introduced to the system to study the e8ects of an electric con/ning potential on the eigenenergy of a single electron in the modi/ed magnetic quantum dot. The additional potential raises the whole energy spectrum and changes its shape. The ground-state angular momentum transitions occurring in a bare modi/ed magnetic quantum dot disappear on introduction of the additional parabolic potential. The modi/ed magnetic quantum dot is formed by spatially inhomogeneous distributions of magnetic /elds [45]. Electrons are magnetically con/ned to the plane where the magnetic /elds inside and outside the dot are di8erent, i.e., (0; 0; B1 ) for r ¡ r0 and (0; 0; B2 ) for r ¿ r0 . This is a more complicated system than a magnetic quantum dot (B1 = 0 for r ¡ r0 ). Here, r0 is the radius of the dot. This kind of magnetic /eld pro/le may be obtained by the combination of a ferromagnetic disk with a homogeneous magnetic /eld as shown in Ref. [46]. An electrostatic potential is introduced to the system to study the e8ects of an electric con/ning potential on the eigenenergy of a single electron in the modi/ed magnetic quantum dot. The electrostatic potentials that we have considered are V (r) = ar 2 + b=r 2 and describe a quantum dot, an antidot, and a quantum ring, depending on the values of a and b. The exact single-electron eigenstates and energies of a modi/ed magnetic quantum dot are calculated by using a general form of the single-particle SchrLodinger equation without electron–electron interactions, i.e.,

2 1 ˜ p ˜ + eA + V (r) (˜r) = E (˜r) ; 2m∗

(5.1)

24


where e is the absolute value of the electron charge. In plane polar coordinates (r; ), the vector potential Ã can be chosen in the symmetric gauge as 1  (r ¡ r0 ) ;  - 1 B0 r Ã = ˆ 2 (5.2)   1 r 2 B (- − - ) + 1 - B r (r ¿ r ) ; 0 1 2 2 0 0 2r 0 2 where -1 = B1 =B0 and -2 = B2 =B0 are the ratios of the applied external magnetic /eld to the standard magnetic /eld B0 . The additional electrostatic potential V (r) is expressed as V (r) =

∗ 2 4 1 ∗ 2 2 2 m !0 r0 + m ) !0 r : 2r 2 2

(5.3)

Here m∗ is the e8ective mass of the electron and d and a are parameters that decide the strength of the antidot and the parabolic potential, respectively. The standard cyclotron frequency is !0 = eB0 =2m∗ . The wave functions are separable, i.e., nm (˜r) = Rnm (r)eim , where m is the angular momentum quantum number and n (=0; 1; 2; : : :) is a radial quantum number which gives the number of nodes in the radial wave function. All quantities are expressed in dimensionless units by setting ˝!0 (=˝eB0 =2m∗ ) and the inverse length $ = m∗ !0 =˝ equal to 1. In these units, ˝2 =m∗ = ˝!0 =$2 → 1, the radius of √ the magnetic quantum dot r0 → s0 where s0 = B0 r02 =0 is the number of magnetic 6ux quanta enclosed by a circle of radius r0 with magnetic /eld B0 and magnetic 6ux quantum 0 (=h=e). The SchrLodinger equation of the radial part is written as 2 d 1 d + (5.4) + 2(E − Ve8 ) Rnm (r) = 0 : dr 2 r dr Here the e8ective potential Ve8 is expressed as  2 m + s02 1 2   + (-1 + )2 )r 2 + m-1  2r 2 2 Ve8 = 2 2    me8 + s0 + 1 (-2 + )2 )r 2 + me8 -2 2r 2 2 2

(r ¡ r0 ) ;

(5.5)

(r ¿ r0 ) ;

where me8 = m + (-1 − -2 )s0 . The wave functions can be solved using Eq. (5.4) and the energy eigenvalues are determined by the continuity of the wave functions and their derivatives at the dot boundary r = r0 . The wave functions are expressed in con6uent hypergeometric functions M and U as

√m2 + s02 √ 4 − -21 +)2 r 2 =2 2 2 2 2 2 (r ¡ r0 ) ; Rnm (r) = C1 -1 + ) r e M a; b; -1 + ) r E m-1 1 − 2 − m2 + s02 − 1 ; a=− 2 -21 + )2 -1 + ) 2 b = m2 + s02 + 1 ;


25

Fig. 5.1. Eigenenergies for √ the states (n; m) (n = 0; 1 and m = −2; −1; 0; 1) of a modi/ed magnetic quantum dot as a function of -1 with r0 = 5 and -2 = 1.

4

√m2e8 + s02

−

√

-22 +)2 r 2 =2

Rnm (r) = C2 + e M (c; d; -22 + )2 r 2 ) (r ¿ r0 ) ; E 1 me8 -2 2 2 c=− − 2 − me8 + s0 − 1 ; 2 -22 + )2 -2 + ) 2 (5.6) d = m2e8 + s02 + 1 : √ U for magnetic The radius of the modi/ed magnetic quantum dot is r0 = 5, which is about 1500 A /elds of 0:1 T. For calculations at the special point = 0, the radial wave function Rnm (r) = 1 √ C1 J|m| ( 2Er) is used in the region of r ¡ r0 just as in a regular magnetic quantum dot [16]. Here the function J|m| is the Bessel function of order m. -22

)2 r

5.2. Angular momentum transitions The low lying energy levels for the states (n; m) (n = 0; 1 and m = −2; −1; 0; 1) of a modi/ed magnetic quantum dot () = 0; = 0) as a function of -1 are shown in Fig. 5.1. Here, the magnetic /eld outside the dot is /xed (-2 = 1) and the magnetic /eld inside the dot is varied. The energy spectrum exhibits quite di8erent features depending on the directions of the magnetic /elds inside and outside the dot. In particular, when the /elds inside and outside the dot are opposite to each other in direction the spectrum shows more interesting features. These peculiar behaviors of eigenstates (n; m) can be understood from Ve8 in Eq. (5.5) without detailed calculations. Ve8 (r) for several cases are shown in Fig. 5.2. When r1 (= |m=-1 |) ¡ r0 and r2 (= |me8 =-2 ) ¿ r0 , Ve8 has a double well structure with two local minima, Ve8 (r1 ) = |-1 m| + -1 m and Ve8 (r2 ) = |-2 me8 | + -2 me8 . However, when r1 ¿ r0 and r2 ¡ r0 , Ve8 has one minimum at r = r0 . For states with m ¡ 0, there are |m| 6ux quanta inside a circle of radius rmin , where Ve8 (rmin ) = 0. In fact, this is manifest in Eq. (5.5) and

26


Fig. 5.2. E8ective potential Ve8 (r). (a) The cases of a double well (m = −1, -1 = −1, and -2 = 1) and a single well (m = −3, -1 = 0:5, and -2 = 1). (b) The cases having a minimum inside the dot (m = −5, -1 = 3, and -2 = 1) and a minimum outside the dot (m = −5, -1 = −3, and -2 = 1).

is explained in detail in Ref. [16]. These behaviors are clearly shown in Fig. 5.2. Thus, as -1 (¿ 0) increases, (n; m ¡ 0) states are located in the deeper region of the dot, resulting in the Landau levels (2n + 1)|-1 |˝!0 , as shown in the region of -1 ¿ 1 in Fig. 5.1. For -1 ¡ 0, when |-1 | is increased, (n; m ¡ 0) states are located farther away from the dot to enclose |m| 6ux quanta and approach the Landau levels (2n + 1)|-2 |˝!0 . [See the dotted line in Fig. 5.2(b).] States for m ¿ 0 show distinctive variations as also predicted from the shape of Ve8 . Besides convergence to the Landau levels, there is also breaking of Landau level degeneracy except at -1 = 1 where the magnetic /eld is homogeneous. There are also ground-state angular momentum transitions as |-1 | increases in contrast to the regular magnetic quantum dot [16]. These are mainly caused by change of magnetic 6ux quanta due to the inhomogeneous magnetic /elds in the dot. This type of angular momentum transition can occur in conventional quantum rings with /nite width, or in magnetic quantum rings, or in conventional quantum dots including electron–electron interactions. When -1 = 0, the ground state occurs at m = 0 [16]. As |-1 | increases, the ground state occurs at the state m(¿ 0) for negative -1 as in Fig. 5.1 and at m(¡ 0) for positive -1 , and the values m are determined by the given parameters. The main reason is that for -1 6 0 the ground state corresponds to nearby states from the dot while for -1 ¿ 1 it corresponds to states away from the dot as we see from Ve8 in Eq. (5.5).


Fig. 5.3. Probability densities |Rnm (r)|2 of states (a) (0; 2) and (b) (0; −10) for di8erent -1 ’s with r0 =

27

√

5 and -2 = 1.

The probability densities |Rnm (r)|2 for (0; 2) and (0; −10) states are plotted in Fig. 5.3 for several di8erent -1 ’s. In Fig. 5.3(a), the (0; 2) state can be localized inside or outside the dot but not on the boundary since the condition of r1 ¿ r0 and r2 ¡ r0 is never satis/ed. These values of r1 and r2 are the positions of the minima in Ve8 inside and outside the dot, respectively. Otherwise, states are distributed away from the boundary. In Fig. 5.3(b), states with m = −10 are located farther away from the dot to enclose 10 magnetic 6ux quanta as -1 goes to negative values. When -1 = 2, the probability density has a maximum peak at r0 because the modi/ed magnetic quantum dot includes 10 magnetic 6ux quanta inside the dot exactly at -1 = 2 and s0 = 5. The energy levels for the states (n; m) (n = 0; 1 and m = −2; −1; 0; 1) of the modi/ed magnetic quantum dot with an additional electrostatic potential ()=1, =0) as a function of -1 are shown in Fig. 5.4. The additional potential raises the whole energy spectrum and changes its shape, and breaks the Landau level degeneracy at -1 = 1. The ground state angular momentum transitions that occur in a bare modi/ed magnetic quantum dot disappear due to the additional parabolic potential. When -1 ¿ 0, states gain more energy by virtue of the parabolic potential. These states show similar energy dispersion to that of a conventional quantum dot with a homogeneous external magnetic /eld (see the inset in Fig. 5.4). However, when -1 ¡ 0, the energy spectrum becomes more complicated. Most of the low lying energy states that are localized outside the dot without the parabolic potential are con/ned in the dot by the parabolic potential. As |-1 | increases, i.e., the e8ect of magnetic nonuniformity gets stronger,

28


Fig. 5.4. Eigenenergies for the states (n; m) (n = 0; 1 and m = −2; −1; 0; 1) of a modi/ed magnetic quantum dot as a √ function of -1 with additional parabolic potential () = 1 and = 0), -2 = 1, and r0 = 5.

the higher energy states are less a8ected by the parabolic potential and tend to stay where they were despite the electric con/nement. But those states are no longer Landau levels because of gaining additional energy due to the electric con/nement. If we choose a big enough ) to neglect magnetic /eld nonuniformity, we can get a similar energy dispersion to that of the inset. Figs. 5.5(a) and (b) show the energy dependence Enm on di8erent angular momentum quantum number m for ) = 0 and ) = 1, respectively. These show that all states for ) = 1 are increased in energy and more states (m ¿ 0 → m ¿ − 3) are localized inside the dot compared to the case of ) = 0. In Fig. 5.5(a), the states m ¡ 0 are located outside the dot and are just Landau levels. The state (1; 0) represents the localized state inside the dot having doubled energy E1; 0 compared to E0; 0 . This simply re6ects the /eld ratio of -1 =-2 = 2. The ground state of the system occurs at m ¿ 0 (in fact m = 7), which is associated with a double well situation of Ve8 as already discussed. The states m ¿ 10 represent the edge states, which are localized at the dot boundary and show rapid increase with increase of m. 6. Magnetic quantum ring 6.1. Electronic structures of magnetic quantum ring Now we consider the electronic structure of a magnetic quantum ring formed by inhomogeneous magnetic /elds [47], where electrons are con/ned to a plane, and the magnetic /elds are zero inside the ring and constant elsewhere. The energy states that deviate from the Landau levels are found to form the magnetic edge states along the boundary regions of the magnetic quantum ring. The probability densities of these magnetic edge states are found to be well corresponded to the circulating classical trajectories. In contrast to magnetic or conventional quantum dots, the eigenstates of the magnetic quantum ring show angular momentum transitions in the ground state as the magnetic /eld increases, even without including electron–electron interactions. For a modi/ed magnetic quantum ring with the distribution of nonzero magnetic /elds inside the ring and di8erent /elds outside it,


29

Fig. 5.5. Dependence of the energy eigenvalues Enm of a modi/ed magnetic quantum dot on the angular momentum quantum number m for (a) ) = 0 and (b) ) = 1 at -1 = −2, -2 = 1, and s0 = 5.

we also /nd similar behaviors such as the angular momentum transitions in the ground state with increasing the magnetic /eld. In the discussions of the magnetic quantum dots given in Sections 3–5, the magnetic edge state was shown to have quite di8erent properties from the conventional electrostatic edge state; a notable feature is that for a small conductor with a magnetic quantum dot at the center, magnetoconductances have aperiodic oscillations instead of the well-known Aharonov–Bohm-type periodic oscillations. Therefore it is interesting to see the formation of magnetic edge states in other magnetic quantum structures like magnetic quantum rings, which have di8erent magnetic /elds in the ring region and outside it. Since the magnetic edge states are related to the characteristics of the electronic-structure, the related physical properties are expected to be di8erent from those of the magnetic quantum dot. In this chapter, we investigate the electronic structure and the magnetic edge states of magnetic quantum rings. We calculate exactly single electron energies by neglecting electron–electron interactions, and /nd that the energy spectra critically depend on the number of missing magnetic 6ux quanta rather than the geometry of the structure or the /eld abruptness. In contrast to the magnetic quantum dot [16], the angular momentum transitions in the ground state are found to occur as the magnetic /eld varies. The classical trajectories of the quantum states are obtained by using the general rules, which are derived from the energy and angular momentum conservation laws.

30


We /rst consider the 2DEG con/ned in a magnetic quantum ring formed by inhomogeneous magnetic /elds; the magnetic /eld perpendicular to the plane is zero within a circular ring and constant B outside it. When electron–electron interactions are neglected, the single-particle SchrLodinger equation for the magnetic quantum ring is 1 (˜ p + eÃ)2 (˜r) = E (˜r) ; (6.1) 2m∗ where m∗ is the e8ective mass of an electron and e is the absolute value of the electron charge. In polar coordinates (r; ) on the plane, the vector potential Ã is chosen in a symmetric gauge such as  1   Br (r ¡ r1 ) ;   2    1 Ã = ˆ (6.2) Br12 (r1 ¡ r ¡ r2 ) ;  2r       1 Br − 1 (r22 + r12 ) (r2 ¡ r) : 2 2r Then, the wave functions are separable, i.e., Rnm (˜r) = Rnm (r)eim , where m is the angular momentum quantum number and n(=0; 1; 2; : : :) is the number of nodes in Rnm (r), and the equation for the radial part is written as 2 d 1 d + (6.3) + 2(E − Ve8 ) Rnm (r) = 0 ; dr 2 r dr  1 2 m2e8 ; 1    + me8 ; 1 (r ¡ r1 ) ; r +   2 2r 2    2 me8 ; 2 Ve8 = (6.4) (r1 ¡ r ¡ r2 ) ; 2  2r     2    1 r 2 + me8 ; 3 + me8 ; 3 (r ¿ r2 ) ; 2 2r 2 where me8 ; 1 , me8 ; 2 , and me8 ; 3 are de/ned as m, m + s1 , and m − S, respectively. All quantities are ∗ ∗ expressed in dimensionless units by letting ˝!L [=˝eB=(2m )] and the inverse length $= m !L =˝ to be 1. The dimensionless parameter S(=s2 − s1 ) represents the number of missing 6ux quanta, where s1 = r12 B=0 , s2 = r22 B=0 , and 0 (=h=e) is the 6ux quantum. In these units ˝2 =m∗ = ˝!L =$2 → 1, √ √ r1 → s1 , and r2 → s2 , where r1 and r2 are the inner and outer radii of the ring, respectively. The solutions for Rnm (r) are found to be  2 (r ¡ r1 ) ; C1 r |me8 ; 1 | e−r =2 M (a1 ; b1 ; r 2 )     √ √ 2Er + C3 N|me8 ; 2 | 2Er (r1 ¡ r ¡ r2 ) ; Rnm (r) = C2 J|me8 ; 2 | (6.5)    2  C4 r |me8 ; 3 | e−r =2 U (a2 ; b2 ; r 2 ) (r ¿ r2 ) ; where a1 = (|me8 ; 1 | + 1 − E + me8 ; 1 )=2, b1 = |me8 ; 1 | + 1, a2 = (|me8 ; 3 | + 1 − E + me8 ; 3 )=2, b2 = |me8 ; 3 | + 1. Here, J and N denote the Bessel functions, and M and U are the con6uent hypergeometric functions. The eigenenergies are determined by the continuity of the wave functions and their derivatives at the boundaries of the inner and outer circles. From the e8ective potential Ve8 in Eq. (6.4), we can obtain


31

Fig. 6.1. E8ective potentials for the states (m = 0; −1; 1) in the magnetic quantum ring with s1 = 2 and s2 = 6.

very useful information on the properties of the (n; m) eigenstates without detailed calculations. For the states with m 6 0, the minimum value of Ve8 is always 0, and |m| magnetic 6ux quanta are enclosedat the minimum. For m ¿ 0 and |me8 ; 3 | ¿ s2 , Ve8 has the minimum value of |me8 ; 3 | + me8 ; 3 at r = |me8 ; 3 |, which is located outside the outer circle of the magnetic ring. For m ¿ 0 and |me8 ; 3 | 6 s2 , the minimum of Ve8 is always located at r = r2 , and the corresponding states are localized near the ring boundaries. The e8ective potentials for m=0; −1, and 1 are drawn√in Fig. 6.1. √ √ In this case, we choose the parameters, s1 = 2 and s2 = 6, i.e., r1 = s1 and r2 = s2 = 3r1 , which U for magnetic /elds of few teslas. As the magnetic give the ring size of about several hundreds A /eld S increases, more states are populated in the ring region, with the energies deviated from the Landau levels. This deviation results in the formation of magnetic edge states, similar to the case of magnetic quantum dots [16].

32


Fig. 6.2. Energy spectra of the magnetic quantum ring as a function of S. Dotted lines represent the Landau levels, and Enm represents the energies of the (n; m) states, which are normalized by that for the lowest Landau level at S = 4, where n(=0; 1; 2; : : :) and m denote the radial and angular momentum quantum numbers, respectively.

For the magnetic ring considered here, the calculated energies Enm are plotted as a function of the magnetic /eld S in Fig. 6.2. For weak-magnetic /elds, since the density of magnetic 6ux is low over the ring, the (0; m ¡ 0) states must be localized in the region very far from the ring to enclose |m| 6ux quanta, if the magnitude of m is very large. In this case, the states resemble the Landau levels, which are normally formed by the uniform distribution of magnetic /elds over the whole region. As B increases, the localized region of the states get closer to the ring, and these states feel the absence of magnetic /elds inside the ring. Then, these states start to deviate from the Landau levels, with lower energies. When B is strong enough for the (0; m ¡ 0) states to be prominent inside the inner circle, where uniform magnetic /elds are present, the energies return to the Landau levels. The larger the magnitude of m, the faster the recovery of the Landau levels takes place with increasing the magnetic /eld. Thus, the magnitude of m in the ground state continuously increases as B increases, i.e., the high-energy states with large values of m (m ¡ 0) turn into the ground state. Such angular momentum transitions are mainly caused by the missing of 6ux quanta in the ring area, while these transitions were not seen in the magnetic quantum dot [16]. In conventional quantum dots con/ned by electrostatic potentials, this type of angular momentum transitions in the ground state can occur only if electron–electron interactions are included [48]. For a conventional quantum ring with a /nite width con/ned by electrostatic potentials, the angular momentum transition was also observed with increasing magnetic /eld [49]. When the (0; m ¡ 0) states have the quantum number of m = −s1 , Ve8 is zero over the ring region. Thus, for S = 4, the ground state is the (0; −2) state instead of the (0; 0) state, as clearly shown in Fig. 6.2. In the limit of B → ∞, the whole energy spectra of the magnetic quantum ring become identical to those of the conventional quantum ring with the angular momentum quantum number shifted from m to me8 ; 2 . This is because very high-magnetic /elds outside the ring area act as an in/nite barrier for electrons in the ring. In the case of m ¿ 0,


33

Fig. 6.3. Dependence of Enm on m in the magnetic quantum ring for S = 4, i.e., s1 = 2 and s2 = 6.

as expected from the behavior of Ve8 , the states with |me8 ; 3 | 6 s2 are mainly localized near the outer circle, while for |me8 ; 3 | ¿ s2 they are distributed outside the ring. Then, once the (0; m ¿ 0) states deviate from the Landau levels, they never turn to the Landau levels again even for very high-magnetic /elds, as shown in Fig. 6.2. For a magnetic /eld given by s1 = 2 and s2 = 6, the dependence of Enm on m is drawn in Fig. 6.3. If me8 ; 3 in Ve8 satis/es the condition |me8 ; 3 |s2 , which gives m10 or m − 2, the (0; m ¡ 0) states are distributed very far from the ring, and their energies becomes the Landau levels. As mentioned earlier, the (0; m ¡ 0) states near m = −2, where the ground state occurs, are perturbed by the missing of S 6ux quanta in the ring region. Since these states have lower energies than the Landau level, they have nonzero probability currents Inm [22], where Inm = 1=h9Enm =9m, forming magnetic edge states. Depending on the sign of 9Enm =9m, the magnetic edge states carry currents circulating either clockwise or counterclockwise, while the degenerated Landau levels of m ¡ 0 carry no probability currents. In Fig. 6.4, the classical trajectories of electrons and their corresponding probability densities |Rnm (r)|2 are drawn for the (0; 0), (0; −1), and (0; 1) states, which exhibit clearly the classical behavior of the magnetic edge states formed near the ring boundaries. In fact, these states represent the ensemble average of trajectories, which consist of straight line paths in the ring region and cyclotron orbits with the radius ri = Enm =2 and the center located at rj = ri2 − me8 outside the ring. Here, the value of me8 depends on the region, as shown in Eq. (6.4). Thus, for given n and m, the value of ri is /xed whereas that for rj varies with region. The general rules for ri and rj are derived from the conservation of both energy and angular momentum [16]. Since the (0; −2) state is the ground state, the (0; 0), (0; −1), and the (0; 1) states have the probability currents drifting along the counterclockwise direction, i.e., Inm =1=h9Enm =9m is positive (see Fig. 6.3). Besides the direction of classical motions, we /nd other interesting feature that although no tunneling is allowed between the classical trajectories which consist of separate sets of motions in general, the corresponding probability densities between the trajectories are connected smoothly due to quantum mechanical tunneling.

34


Fig. 6.4. Classical trajectories of electrons and corresponding probability densities for the (a) (0; 0), (b) (0; −1), and (c) (0; 1) states in the magnetic quantum ring.

6.2. Modi>ed magnetic quantum ring We extend our study to the modi/ed structure of the magnetic quantum ring, which has nonzero magnetic /eld B∗ in the ring area (r1 ¡ r ¡ r2 ) and B(= B∗ ) elsewhere. The vector potential for this new structure is chosen to be  1   Br (r ¡ r1 ) ;   2    Ã = ˆ 1 B∗ r + 1 (B − B∗ )r 2 (6.6) (r1 ¡ r ¡ r2 ) ; 1  2 2r       1 Br + 1 B∗ (r22 − r12 ) − 1 B(r22 − r12 ) (r2 ¡ r) : 2 2r 2r


35

Fig. 6.5. Dependence of Enm on m in the modi/ed magnetic quantum ring with s1 = 2 and s2 = 6.

Then, the e8ective angular momentum quantum numbers me8 are modi/ed such as me8 ; 1 = m

(r ¡ r1 ) ; ∗

me8 ; 2 = m + s1 − s1

(r1 ¡ r ¡ r2 ) ;

me8 ; 3 = m + s2∗ − s1∗ − (s2 − s1 )

(r2 ¡ r) ;

(6.7)

where s1 = r12 B=0 , s2 = r22 B=0 , s1∗ = r12 B∗ =0 , and s2∗ = r22 B∗ =0 . Using the same dimensionless √ √ units as those in the last section, we can express r1 = s1 and r2 = s2 . The radial function in the ring region is written as the combination of the con6uent hypergeometric functions M and U , 2 2

R(r) = -(-r)|me8 ; 2 | e−1=2- r [C2 M (a3 ; b3 ; -2 r 2 ) + C3 U (a3 ; b3 ; -2 r 2 )]

(r1 ¡ r ¡ r2 ) ;

(6.8)

where a3 =[|me8 ; 2 |+1−-−2 E +sign(B∗ )me8 ; 2 ]=2, b3 =|me8 ; 2 |+1, and -2 =|B∗ =B|. The wave functions outside the ring have the same forms as those of the magnetic quantum ring with the modi/ed me8 ’s in Eq. (6.7). Here, we only consider a special case of B∗ = −B, which gives the e8ective potential Ve8 = m2e8 =(2r 2 ) + r 2 =2 ± me8 . The positive and negative signs of the last term corresponds to the regionsof r ¿ r2 and r ¡ r1 , respectively. In each region, Ve8 has a minimum value of |me8 | ± me8 at r = |me8 |. The calculated energies of the modi/ed magnetic quantum ring are plotted as function of m for s1 = 2 and s2 = 6 in Fig. 6.5. Both the ground and excited states are found to deviate more severely from the Landau levels, as compared to the regular magnetic quantum ring considered in the last section. The results indicate that the energy levels and the angular momentum transitions can be modulated for both the ground and excited states by varying the ratio B∗ =B, i.e., the number of missing 6ux quanta in the ring region. For the states localized far outside the ring, which satisfy the condition |me8 ; 3 |s2 , i.e., m2 or m14, their energies approach to the Landau levels under uniform magnetic /elds with the quantum number shifted from m to me8 ; 3 . For the (n; m) eigenstates, the corresponding

36


Fig. 6.6. Classical trajectories of electrons and corresponding probability densities for the (a) (0; 0), (b) (0; −1), and (c) (0; 1) states in the modi/ed magnetic quantum ring.

classical trajectories can also be constructed usingthe energy and angular momentum conservation ∗ | E =2 and r = |B=B∗ | E =2 − B∗ =Bm laws such as r = |B=B i nm j nm e8 inside the ring whereas ri = Enm =2 and rj = Enm =2 − me8 elsewhere, where me8 ’s are given in Eq. (6.7). In each region, the trajectories consist of circular orbits centered at rj with the radius ri , and the resulting magnetic edge states circulate clockwise for r1 ¡ r ¡ r2 while counterclockwise elsewhere. For s1 = 2, s2 = 6, and B∗ =B = −1, the classical trajectories and the corresponding probability densities |Rnm (r)|2 are drawn for the (0; 0), (0; −1), and (0; 1) states in Fig. 6.6. These states exhibit wavelike trajectories unlike the magnetic quantum dot in the last section. Since the tunnelings between electrons are prohibited classically, some trajectories may consist of separated orbit sets. The radial distribution of the eigenstates obtained quantum mechanically shows rather smooth variations and agrees excellently with the corresponding trajectories. Although the (0; 0) and (0; −1) states are almost degenerate as shown in Fig. 6.5, they have di8erent probability currents, opposite to each other. The (0; 0) state circulates counter-clockwise along the outer boundary of the ring, while the (0; −1) state does


37

clockwise along the inner boundary (see Fig. 6.6). For the m = 0 states, the e8ective potential has three local minima of Ve8 = 0, in contrast to the magnetic quantum ring in Section 2, where a minimum of Ve8 occurs only at r = 0. This behavior illustrates why the probability density of the (0; 0) state in Fig. 6.6(a) is more reduced for r ¡ r1 than that for the regular magnetic ring in Fig. 6.4(a). The (0; 1) state is found to have no trajectories for r ¡ r1 and show a wavelike circulation along the outer boundary of r = r2 , which is also manifested in the radial distribution in Fig. 6.6(c). 7. Summary We have investigated the electronic structure and the electronic transport in various magnetic nano-structures which are de/ned by various non-uniform distributions of magnetic /elds; the magnetic quantum dots, the modi/ed magnetic quantum dots, and the magnetic quantum ring. Through the theoretical study of such magnetic nano-structures, we found that the magnetic edge states are formed along the boundaries of those magnetic quantum dots or rings. Such magnetic edge states are essentially di8erent from conventional (electrostatic) edge states formed due to the electrostatic modulations, and are interpreted to correspond to the complicated classical electron trajectories formed due to the non-uniform magnetic /elds. The formation of magnetic edge states are essentially determined by the number of additional (or missing) magnetic 6ux quanta threading the systems, and are directly related to the discrete energy eigenvalues of such systems. As a simplest case, if the distribution of magnetic /elds is given by ˜Bin = 0 and ˜Bout = B0 zˆ inside and outside the circular region with radius r0 , respectively, the number of missing magnetic 6ux quanta s threading the circular region importantly determines the energy spectrum, and the e8ective angular momentum quantum number me8 = m − s serves as an important factor which determines the e8ective potential, the e8ective radius of the electron’s wavefunction, and so on. When such a magnetic quantum dot (˜Bin = 0 and ˜Bout = B0 z) ˆ is placed at the center of quantum wire (narrow 2DEG), the formation of the magnetic edge states is found to give rise to quite distinctive aperiodic oscillations in the magnetoconductance. When a magnetic quantum dot, de/ned by ˜Bin = B∗ zˆ and ˜Bout = B0 z, ˆ is located at the center of quantum wire (narrow 2DEG), more varieties of phenomena can be found. The magnetic quantum dot is found to be a characteristic scattering center which results in a transmission barrier and a resonator. The /eld reversal at the dot boundary gives rise to distinct magnetic edge states and transport properties, such as non-quantized conductance and the conductance plateaus. The scattering mechanism based on the magnetic con/nement can be useful to understand electron transport in other magnetic structures combined with electrostatic con/nements. Moreover, the electronic transport in quantum wires with two magnetic quantum dots were also investigated, by using the calculation method based on the gauge transformations. The calculations of the local density of states have shown the presence of magnetic edge states extended over the two magnetic dots. Studies on the electronic structures of the magnetic quantum dot is generalized by considering a modi/ed magnetic quantum dot [˜Bin = -1 B0 zˆ and ˜Bout = -2 B0 z] ˆ and the e8ects of an electric con/ning potential on the eigenenergy of a single electron in a modi/ed magnetic quantum dot have been studied. The energy spectrum exhibits quite interesting features depending on the direction of the magnetic /eld inside the dot when the direction of the magnetic /eld outside the dot is

38


/xed. As -1 (¿ 0) increases, (n; m ¡ 0) states are located in the deeper region of the dot, resulting in the Landau levels (2n + 1)|-1 |˝!0 . For -1 ¡ 0, when |-1 | is increased, (n; m ¡ 0) states are located farther away from the dot to enclose |m| 6ux quanta and approach to the Landau levels (2n + 1)|-2 |˝!0 . Additional electrostatic potentials lift up the whole energy spectrum, break the Landau level degeneracy at -1 = 1, and change the shape of the electronic structure of the modi/ed magnetic quantum dot. Low lying energy levels, which are a8ected strongly by added electrostatic potentials, no longer have ground state angular momentum transitions. Finally, the electronic structure of the magnetic quantum ring have been investigate. The eigenstates are found to deviate from the Landau levels due to the missing of magnetic 6ux quanta and form the magnetic edge states. These edge states carry nonzero probability currents and depend sensitively on the number of the enclosed magnetic 6ux quanta. We /nd that the magnetic quantum ring exhibits the angular momentum transitions in the ground state as the magnetic /eld increases. For extremely high-magnetic /elds, the energy spectra resemble those for a conventional quantum ring without magnetic /elds. For the modi/ed magnetic quantum ring with nonzero but di8erent magnetic /elds inside the ring, the angular momentum transitions are also found in the ground state, which are enhanced by modifying the geometry of the quantum structure and varying the strength of magnetic /eld. Acknowledgements This work was supported by the Korea Science and Engineering Foundation through the QuantumFunctional Semiconductor Research Center at Dongguk University and the authors are grateful for the discussions with Dr. H.S. Sim, Dr. N. Kim, and Dr. J.W. Kim. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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[40]

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39

H.-S. Sim, K.-H. Ahn, K.J. Chang, G. Ihm, N. Kim, S.J. Lee, Phys. Rev. Lett. 80 (1998) 1501. L.L. Sohn (Ed.), Mesoscopic Electron Transport, Kluwer Academic Publishers, Dordrecht, 1997. R. Landauer, IBM J. Res. Dev. 32 (1998) 306. M. BLuttiker, Phys. Rev. B 38 (1988) 9375. M.L. Leadbeater, C.L. Foden, T.M. Burke, J.H. Burroughes, M.P. Grimshaw, D.A. Ritchie, L.L. Wang, M. Pepper, J. Phys.: Condens. Matter 7 (1995) L307. A. Nogaret, S.J. Bending, M. Henini, Phys. Rev. Lett. 84 (2000) 2231. B.I. Halperin, Phys. Rev. B 25 (1982) 2185. B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven, D. van der Marel, C.T. Foxon, Phys. Rev. Lett. 60 (1988) 848; D.A. Wharam, T.J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie, G.A.C. Jones, J. Phys. C 21 (1988) L209. These conventional channels are often referred to the magnetic edge channels to distinguish from the narrow channels in the absence of magnetic /elds. In our work this is not the case. J.E. MLuller, Phys. Rev. Lett. 68 (1992) 385. L. Solimany, B. Kramer, Solid State Commun. 96 (1995) 471. J.K. Jain, Phys. Rev. Lett. 60 (1988) 2074. C.S. Lent, Phys. Rev. B 43 (1991) 4179. B.J. van Wees, L.P. Kouwenhoven, C.J.P.M. Harmans, J.G. Williamson, C.E. Timmering, M.E.I. Broekaart, C.T. Foxon, J.J. Harris, Phys. Rev. Lett. 62 (1989) 2523; W.-C. Tan, J.C. Inkson, Phys. Rev. B 53 (1996) 6947. H.A. Carmona, A.K. Geim, A. Nogaret, P.C. Main, T.J. Foster, M. Henini, S.P. Beaumont, M.G. Blamire, Phys. Rev. Lett. 74 (1995) 3009; P.D. Ye, D. Weiss, R.R. Gerhardts, M. Seeger, K. von Klitzing, K. Eberl, H. Nickel, Phys. Rev. Lett. 74 (1995) 3013; A. Nogaret, S. Carlton, B.L. Gallagher, P.C. Main, M. Henini, R. Wirtz, R. Newbury, M.A. Howson, S.P. Beaumont, Phys. Rev. B 55 (1999) R16037. M.L. Leadbeater, C.L. Foden, J.H. Burroughes, M. Pepper, T.M. Burke, L.L. Wang, M.P. Grimshaw, D.A. Ritchie, Phys. Rev. B 52 (1995) R8629. B.-Y. Gu, W.-D. Sheng, X.-H. Wang, J. Wang, Phys. Rev. B 56 (1997) 13434; J. Reijniers, F.M. Peeters, arXiv:cond-mat/0009303 (2000). A. Matulis, F.M. Peeters, P. Vasilopoulos, Phys. Rev. Lett. 72 (1994) 1518; J. Reijniers, F.M. Peeters, A. Matulis, Physica E 6 (2000) 759. F.M. Peeters, P. Vasilopoulos, Phys. Rev. B 47 (1994) 1466. H.-S. Sim, G. Ihm, N. Kim, K.J. Chang, Phys. Rev. Lett. 87 (2001) 146601. J. Reijniers, F.M. Peeters, A. Matulis, Phys. Rev. B 59 (1999) 2817. J.K. Jain, S.A. Kivelson, Phys. Rev. Lett. 60 (1988) 1542. Y. Takagaki, D.K. Ferry, Phys. Rev. B 48 (1993) 8152. Let us consider a state with m ¡ 0 [ m = eim Rm (r)] in the uniform /eld B0 . This state is located at rp (m ; B0 ) and encloses |m | 6ux quanta. If s 6ux quanta are removed inside r0 (rp ), m can be written as ei(m +s) Rm (r) due to the gauge invariance and then rewritten as eim Rme8 (r). In the calculations given in this section, to make Ã = 0 in leads (i.e., wire regions of |x| ¿ Lx =2), we choose that B = 0 for r ¿ r1 (r0 ) and Ã = 0 for r ¿ r2 (∼ Lx =2) (see Ref. [38]). Here, r1 and Lx are chosen as 38a and 4000a, respectively. This arti/cial choice for Ã does not a8ect our results only when Ly =Lx 1; otherwise, the Peierls’ phase is counted incorrectly. Also, to imitate the continuum limit by the lattice Hamiltonian, the condition |2Ba2 =0 1| should be satis/ed. S. Souma, S.J. Lee, N. Kim, T.W. Kang, G. Ihm, J.C. Woo, A. Suzuki, in: J.H. Davies, A.R. Long (Eds.), Proc. 26th Int. Conf. Physics of Semiconductors, Edinburgh, 2002, IOP Conf. Ser. 171, H187, IOP, Bristol, 2003; S. Souma, S.J. Lee, N. Kim, T.W. Kang, G. Ihm, K.S. Yi, A. Suzuki, J. Superconductivity: Incorporating Novel Magnetism 16 (2003) 339. J.J. Palacios, C. Tejedor, Phys. Rev. B 48 (1993) 5386. D.S. Fisher, P.A. Lee, Phys. Rev. B 23 (1991) 6851.

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[44] T. Ando, Phys. Rev. B 44 (1991) 8017; S. Souma, A. Suzuki, J. Korean Phys. Soc. 39 (2001) 553. [45] N. Kim, G. Ihm, H.-S. Sim, T.W. Kang, Phys. Rev. B 63 (2001) 235317. [46] J.A.K. Freire, A. Matulis, F.M. Peeters, V.N. Freire, G.A. Farias, Phys. Rev. B 61 (2000) 2895. [47] N. Kim, G. Ihm, H.-S. Sim, K.J. Chang, Phys. Rev. B 60 (1999) 8767. [48] U. Merkt, J. Huser, M. Wagner, Phys. Rev. B 43 (1991) 7320; M. Wagner, U. Merkt, A.V. Chaplik, ibid. 45 (1992) 1951; J.H. Oh, K.J. Chang, G. Ihm, S.J. Lee, ibid. 50 (1994) 15397. [49] G. Kirczenow, Superlatt. Microstruct. 14 (1994) 237; W.-C. Tan, J.C. Inkson, Phys. Rev. B 53 (1996) 6947.

Available online at www.sciencedirect.com

Physics Reports 394 (2004) 41 – 156 www.elsevier.com/locate/physrep

Random matrix theory and symmetric spaces M. Casellea;∗ , U. Magneab a

Department of Theoretical Physics, University of Torino and INFN, Sez. di Torino Via P. Giuria 1, I-10125 Torino, Italy b Department of Mathematics, University of Torino, Via Carlo Alberto 10, I-10123 Torino, Italy Accepted 30 December 2003 editor: C.W.J. Beenakker

Abstract In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important results can be obtained from this identi4cation. In particular the Cartan classi4cation of triplets of symmetric spaces with positive, zero and negative curvature gives rise to a new classi4cation of random matrix ensembles. The review is organized into two main parts. In Part I the theory of symmetric spaces is reviewed with particular emphasis on the ideas relevant for appreciating the correspondence with random matrix theories. In Part II we discuss various applications of symmetric spaces to random matrix theories and in particular the new classi4cation of disordered systems derived from the classi4cation of symmetric spaces. We also review how the mapping from integrable Calogero–Sutherland models to symmetric spaces can be used in the theory of random matrices, with particular consequences for quantum transport problems. We conclude indicating some interesting new directions of research based on these identi4cations. c 2003 Elsevier B.V. All rights reserved. PACS: 02.10.−v

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. Lie groups and root spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.1. Lie groups and manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ∗

Corresponding author. Tel.: +39-011-6707205; fax: +39-011-6707214. E-mail addresses: [email protected] (M. Caselle), [email protected] (U. Magnea).

c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.12.004

42

M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156 2.2. The tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The Lie algebra and the adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Semisimple algebras and root spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. The Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. The simple root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Involutive automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The action of the group on the symmetric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Radial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The metric on a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The algebraic structure of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Real forms of semisimple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The real forms of a complex algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The classi4cation machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The classi4cation of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The curvature tensor and triplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Restricted root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Real forms of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operators on symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Casimir operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Laplace operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Zonal spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. The analog of Fourier transforms on symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrable models related to root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. The root lattice structure of the CS models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Mapping to symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 48 50 51 55 56 57 57 59 59 62 64 64 65 68 71 72 75 79 81 81 83 88 91 94 95 96

Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Random matrix theories and symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Introduction to the theory of random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1. What is random matrix theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2. Some of the applications of random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3. Why are random matrix models successful? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. The basics of matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Identi4cation of the random matrix integration manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1. Circular ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Gaussian ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3. Chiral ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4. Transfer matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5. The DMPK equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6. BdG and p-wave ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.7. S-matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Identi4cation of the random matrix eigenvalues and universality indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1. Discussion of the Jacobians of various types of matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Fokker–Planck equation and the Coulomb gas analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1. The Coulomb gas analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2. Connection with the Laplace–Beltrami operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3. Random matrix theory description of parametric correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6. A dictionary between random matrix ensembles and symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. On the use of symmetric spaces in random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 98 98 98 99 100 101 104 104 106 111 114 117 119 120 121 122 124 125 126 127 127 128

3.

4. 5.

6.

7.

M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156 9.1. 9.2. 9.3. 9.4.

Towards a classi4cation of random matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetries of random matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of symmetric spaces in quantum transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. Exact solvability of the DMPK equation in the = 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Asymptotic solutions in the = 1; 4 cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Magnetic dependence of the conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4. Density of states in disordered quantum wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Beyond symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. NonCartan parametrization of symmetric spaces and S-matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1. NonCartan parametrization of SU (N )=SO(N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Clustered solutions of the DMPK equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Triplicity of the Weierstrass potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Zonal spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. The Itzykson–Zuber–Harish–Chandra integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. The Duistermaat–Heckman theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 128 131 131 133 134 137 137 138 139 140 141 143 145 147 149 150 151 152 152

1. Introduction The study of symmetric spaces has recently attracted interest in various branches of physics, ranging from condensed matter physics to lattice QCD. This is mainly due to the gradual understanding during the past few years of the deep connection between random matrix theories and symmetric spaces. Indeed, this connection is a rather old intuition, which traces back to Dyson [1] and has subsequently been pursued by several authors, notably by HNuOmann [2]. Recently it has led to several interesting results, like for instance a tentative classi4cation of the universality classes of disordered systems. The latter topic is the main subject of this review. The connection between random matrix theories and symmetric spaces is obtained simply through the coset spaces de4ning the symmetry classes of the random matrix ensembles. Although Dyson was the 4rst to recognize that these coset spaces are symmetric spaces, the subsequent emergence of new random matrix symmetry classes and their classi4cation in terms of Cartan’s symmetric spaces is relatively recent [3–7]. Since symmetric spaces are rather well understood mathematical objects, the main outcome of such an identi4cation is that several non-trivial results concerning the behavior of the random matrix models, as well as the physical systems that these models are expected to describe, can be obtained. In this context an important tool, that will be discussed in the following, is a class of integrable models named Calogero–Sutherland models [8]. In the early 1980s, Olshanetsky and Perelomov showed that also these models are in one-to-one correspondence with symmetric spaces through the reduced root systems of the latter [9]. Thanks to this chain of identi4cations (random matrix ensemble–symmetric space–Calogero–Sutherland model) several of the results obtained in the last 20 years within the framework of Calogero–Sutherland models can also be applied to random matrix theories. The aim of this review is to allow the reader to follow this chain of correspondences. To this end we will devote the 4rst half of the paper (Sections 2–7) to the necessary mathematical background

44

M. Caselle, U. Magnea / Physics Reports 394 (2004) 41 – 156

and the second part (Sections 8–10) to the applications in random matrix theory. In particular, in the last section we discuss some open directions of research. The reader who is not interested in the mathematical background could skip the 4rst part and go directly to the later sections where we list and discuss the main results. This review is organized as follows: The 4rst 4ve sections of Part I (Sections 2–6) are devoted to an elementary introduction to symmetric spaces. As mentioned in the Abstract, these sections consist of the material presented in [10], which is a self-contained introductory review of symmetric spaces from a mathematical point of view. The material on symmetric spaces should be accessible to physicists with only elementary background in the theory of Lie groups. We have included quite a few examples to illustrate all aspects of the material. In the last section of Part I, Section 7, we brieQy introduce the Calogero– Sutherland models with particular emphasis on their connection with symmetric spaces. After this introductory material we then move on in Part II to random matrix theories and their connection with symmetric spaces (Section 8). Let us stress that this paper is not intended as an introduction to random matrix theory, for which very good and thorough references already exist [11,26–29]. In this review we will assume that the reader is already acquainted with the topic, and we will only recall some basic information (de4nitions of the various ensembles, main properties, and main physical applications). The main goal of this section is instead to discuss the identi4cations that give rise to the close relationship between random matrix ensembles and symmetric spaces. Section 9 is devoted to a discussion of some of the consequences of the above mentioned identi4cations. In particular we will deduce, starting from the Cartan classi4cation of symmetric spaces, the analogous classi4cation of random matrix ensembles. We discuss the symmetries of the ensembles in terms of the underlying restricted root system, and see how the orthogonal polynomials belonging to a certain ensemble are determined by the root multiplicities. In this section we also give some examples of how the connection between random matrix ensembles on the one hand, and symmetric spaces and Calogero–Sutherland models on the other hand, can be used to obtain new results in the theoretical description of physical systems, more precisely in the theory of quantum transport. The last section of the paper is devoted to some new results that show that the mathematical tools discussed in this paper (or suitable generalizations of these) can be useful for going beyond the symmetric space paradigm, and to explore some new connections between random matrix theory, group theory, and diOerential geometry. Here we discuss clustered solutions of the Dorokhov– Mello–Pereyra–Kumar equation, and then we go on to discuss the most general Calogero–Sutherland potential, given by the Weierstrass P-function, and show that it covers the three cases of symmetric spaces of positive, zero and negative curvature. Finally, in the appendix we discuss some intriguing exact results for the so called zonal spherical functions, which not only play an important role in our discussion, but are also of great relevance in several other branches of physics. There are some important and interesting topics that we will not review because of lack of space and competence. For these we refer the reader to the existing literature. In particular we shall not discuss: • the supersymmetric approach to random matrix theories and in particular their classi4cation in terms of supersymmetric spaces. Here we refer the reader to the original paper by Zirnbauer [4], while a good introduction to the use of supersymmetry in random matrix theory and a complete set of the relevant references can be found in [12];


45

• the very interesting topic of phase transitions. For this we refer to the recent and thorough review by Cicuta [13]; • the extension to two-dimensional models of the classi4cation of symmetric spaces, and more generally the methods of symmetric space analysis [14]; • the generalization of the classi4cation of symmetric spaces to non-hermitean random matrices [15] (see however a discussion in Section 11); • the so-called q-ensembles [16]; • the two-matrix models [17] and multi-matrix models [18] and their continuum limit generalization. The last item in the list given above is a very interesting topic, which has several physical applications and would indeed deserve a separate review. The common feature of these two- and multi-matrix models which is of relevance for the present review, is that they all can be mapped onto suitably chosen Calogero–Sutherland systems. These models represent a natural link to two classes of matrix theories which are of great importance in high energy physics: on the one hand, the matrix models describing two-dimensional quantum gravity (possibly coupled to matter) [19], and on the other hand, the matrix models pertaining to large N QCD, which trace back to the original seminal works of ’t Hooft [20]. In particular, a direct and explicit connection exists between multi-matrix models (the so called Kazakov–Migdal models) for large N QCD [21] and the exactly solvable models of two-dimensional QCD on the lattice [22]. The mapping of these models to Calogero–Sutherland systems of the type discussed in this review can be found for instance in [23]. The relevance of these models, and in particular of their Calogero– Sutherland mappings, for the condensed matter systems like those discussed in the second part of this review, was 4rst discussed in [24]. A recent review on this aspect, and more generally on the use of Calogero–Sutherland models for low-dimensional models, can be found in [25]. We will necessarily be rather sketchy in discussing the many important physical applications of the random matrix ensembles to be described in Section 8. We refer the reader to some excellent reviews that have appeared in the literature during the last few years: the review by Beenakker [26] for the solid state physics applications, the review by Verbaarschot [27] for QCD-related applications, and [28,29] for extensive reviews including a historical outline. Part I The theory of symmetric spaces has a long history in mathematics. In this 4rst part of the paper we will introduce the reader to some of the most fundamental concepts in the theory of symmetric spaces. We have tried to keep the discussion as simple as possible without assuming any previous familiarity of the reader with symmetric spaces. The review should be particularly accessible to physicists. In the hope of addressing a wider audience, we have almost completely avoided using concepts from diOerential geometry, and we have presented the subject mostly from an algebraic point of view. In addition we have inserted a large number of simple examples in the text, that will hopefully help the reader visualize the ideas. Since our aim in Part II will be to introduce the reader to the application of symmetric spaces in physical integrable systems and random matrix models, we have chosen the background material presented here with this in mind. Therefore we have put emphasis not only on fundamental issues but on subjects that will be relevant in these applications as well. Our treatment

46


will be somewhat rigorous; however, we skip proofs that can be found in the mathematical literature and concentrate on simple examples that illustrate the concepts presented. The reader is referred to Helgason’s book [30] for a rigorous treatment; however, this book may not be immediately accessible to physicists. For the reader with little background in diOerential geometry we recommend the book by Gilmore [31] (especially Chapter 9) for an introduction to symmetric spaces of exceptional clarity. In Section 2, after reviewing the basics about Lie groups, we will present some of the most important properties of root systems. In Section 3 we de4ne symmetric spaces and discuss their main characteristics, de4ning involutive automorphisms, spherical decomposition of the group elements, and the metric on the Lie algebra. We also discuss the algebraic structure of the coset space. In Section 4 we show how to obtain all the real forms of a complex semisimple Lie algebra. The same techniques will then be used to classify the real forms of symmetric spaces in Section 5. In this section we also de4ne the curvature of a symmetric space, and discuss triplets of symmetric spaces with positive, zero and negative curvature, all corresponding to the same symmetric subgroup. We will see why curved symmetric spaces arise from semisimple groups, whereas the Qat spaces are associated to nonsemisimple groups. In addition, in Section 5 we will de4ne restricted root systems. The restricted root systems are associated to symmetric spaces, just like ordinary root systems are associated to groups. As we will discuss in detail in Part II of this paper, they are key objects when considering the integrability of Calogero–Sutherland models. In Section 6 we discuss Casimir and Laplace operators on symmetric spaces and mention some known properties of the eigenfunctions of the latter, so called zonal spherical functions. These functions play a prominent role in many physical applications. The introduction to symmetric spaces we present contains the basis for understanding the developments to be discussed in more detail in Part II. The reader already familiar with symmetric spaces is invited to start reading in the last section of Part I, Section 7, where we give a brief introduction to Calogero–Sutherland models. 2. Lie groups and root spaces In this introductory section we de4ne the basic concepts relating to Lie groups. We will build on the material presented here when we discuss symmetric spaces in the next section. The reader with a solid background in group theory may want to skip most or all of this section. 2.1. Lie groups and manifolds A manifold can be thought of as the generalization of a surface, but we do not in general consider it as embedded in a higher-dimensional euclidean space. A short introduction to diOerentiable manifolds can be found in Ref. [32], and a more elaborate one in Refs. [33,34, Chapter III]. The points of an N -dimensional manifold can be labelled by real coordinates (x1 ; : : : ; xN ). Suppose that we take an open set U of this manifold, and we introduce local real coordinates on it. Let be the function that attaches N real coordinates to each point in the open set U . Suppose now that the manifold is covered by overlapping open sets, with local coordinates attached to each of them. If for each pair of open sets U , U , the function ◦ −1 is diOerentiable in the overlap region


47

U ∩ U , it means that we can go smoothly from one coordinate system to another in this region. Then the manifold is diOerentiable. Consider a group G acting on a space V . We can think of G as being represented by matrices, and of V as a space of vectors on which these matrices act. A group element g ∈ G transforms the vector v ∈ V into gv = v . If G is a Lie group, it is also a diOerentiable manifold. The fact that a Lie group is a diOerentiable manifold means that for two group elements g, g ∈ G, the product (g; g ) ∈ G × G → gg ∈ G and the inverse g → g−1 are smooth (C ∞ ) mappings, that is, these mappings have continuous derivatives of all orders. Example. The space Rn is a smooth manifold and at the same time an abelian group. The “product” of elements is addition (x; x ) → x + x and the inverse of x is −x. These operations are smooth. Example. The set GL(n; R) of nonsingular real n × n matrices M , det M = 0, with matrix multiplication (M; N ) → MN and multiplicative matrix M → M −1 is a non-abelian group manifold. inverse t i Xi i Any such matrix can be represented as M = e where Xi are generators of the GL(n; R) algebra and t i are real parameters. 2.2. The tangent space In each point of a diOerentiable manifold, we can de4ne the tangent space. If a curve through a point P in the manifold is parametrized by t ∈ R xa (t) = xa (0) + a t;

a = 1; : : : ; N;

(2.1)

where P = (x1 (0); : : : ; xN (0)), then = (1 ; : : : ; N ) = (x˙1 (0); : : : ; x˙N (0)) is a tangent vector at P. Here x˙a (0)=d=dt xa (t)|t=0 . The space spanned by all tangent vectors at P is the tangent space. In particular, the tangent vectors to the coordinate curves (the curves obtained by keeping all the coordinates 4xed except one) through P are called the natural basis for the tangent space. Example. In euclidean 3-space the natural basis is {eˆ x ; eˆ y ; eˆ z }. On a patch of the unit 2-sphere parametrized by polar coordinates it is {eˆ ; eˆ ! }. For a Lie group, the tangent space at the origin is spanned by the generators, that play the role of (contravariant) vector 4elds (also called derivations), expressed in local coordinates on the group manifold as X = X a (x)9a (for an introduction to diOerential geometry see Ref. [35, Chapter 5] or [34]). Here the partial derivatives 9a = 9=9xa form a basis for the vector 4eld. That the generators span the tangent space at the origin can easily be seen from the exponential map. Suppose X is a generator of a Lie group. The exponential map then maps X onto etX , where t is a parameter. This mapping is a one-parameter subgroup, and it de4nes a curve x(t) in the group manifold. The tangent vector of this curve at the origin is then d tX e |t=0 = X : (2.2) dt All the generators together span the tangent space at the origin (also called the identity element).

48


2.3. Coset spaces The isotropy subgroup Gv0 of a group G at the point v0 ∈ V is the subset of group elements that leave v0 4xed. The set of points that can be reached by applying elements g ∈ G to v0 is the orbit of G at v0 , denoted Gv0 . If Gv0 = V for one point v0 , then this is true for every v ∈ V . We then say that G acts transitively on V . In general, a symmetric space can be represented as a coset space. Suppose H is a subgroup of a Lie group G. The coset space G=H is the set of subsets of G of the form gH , for g ∈ G. G acts on this coset space: g1 (gH ) is the coset (g1 g)H . We will refer to the elements of the coset space by g instead of by gH , when the subgroup H is understood from the context, because of the natural mapping described in the next paragraph. If g ∈ H , gH corresponds to a point on the manifold G=H away from the origin, whereas hH = H (h ∈ H ) is the identity element identi4ed with the origin of the symmetric space. This point is the north pole in the example below. If G acts transitively on V , then V = Gv for any v ∈ V . Since the isotropy subgroup Gv0 leaves a 4xed point v0 invariant, gGv0 v0 = gv0 = v ∈ V , we see that the action of the group G on V de4nes a bijective action of elements of G=Gv0 on V . Therefore the space V on which G acts transitively, can be identi4ed with G=Gv0 , since there is one-to-one correspondence between the elements of V and the elements of G=Gv0 . There is a natural mapping from the group element g onto the point gv0 on the manifold. Example. The SO(2) subgroup of SO(3) is the isotropy subgroup at the north pole of a unit 2-sphere imbedded in three-dimensional space, since it keeps the north pole 4xed. On the other hand, the north pole is mapped onto any point on the surface of the sphere by elements of the coset SO(3)=SO(2). This can be seen from the explicit form of the coset representatives. As we will see in Eq. (3.20) in Section 3.5, the general form of the elements of the coset is   0 C I2 − XX T X  ; = M = exp (2.3) T T −C T 0 −X 1−X X where C is the matrix 2 t C= t1

(2.4)

and t 1 , t 2 are real coordinates. I2 in Eq. (2.3) is the 2 × 2 unit matrix. For the coset space SO(3)=SO(2), M is equal to     0 0 0 0 0 1 2   1 1 0 0 1 0 0 0 t i Li ; L1 =  M = exp ; L2 =  (2.5)     : 2 2 i=1 0 −1 0 −1 0 0 The third SO(3) generator   0 1 0  1 −1 0 0  L3 =    2 0 0 0

(2.6)


spans the algebra of the stability subgroup SO(2), that keeps the north pole 4xed:     0 0     3    exp(t L3 )  0=0 : 1 1

49

(2.7)

The generators Li (i = 1; 2; 3) satisfy the SO(3) commutation relations [Li ; Lj ] = 12 jijk Lk . Note that since the Li and the t i are real, C † = C T . In (2.3), M is a general representative of the coset SO(3)=SO(2). By expanding the exponential we see that the explicit form of M is   1 2 2 2 1 2 2 2 1 2 2 2 2 2 cos (t ) + (t ) − 1 1 2 cos (t ) + (t ) − 1 2 sin (t ) + (t ) t t t  1 + (t ) (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2       1 2 2 2 1 2 2 2 1 2 2 2   1 2 cos (t ) + (t ) − 1 1 2 cos (t ) + (t ) − 1 1 sin (t ) + (t )  M = t t 1 + (t ) t  : 1 )2 + (t 2 )2   (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (t     1 )2 + (t 2 )2 1 )2 + (t 2 )2   sin (t sin (t 2 1 −t −t cos (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (t 1 )2 + (t 2 )2 (2.8) Thus the matrix X = yx is given in terms of the components of C by (cf. Eq. (3.21)):   1 2 2 2 2 sin (t ) + (t ) t  x  (t 1 )2 + (t 2 )2   :  X= (2.9) =  1 2 2 2 y  1 sin (t ) + (t )  t (t 1 )2 + (t 2 )2 De4ning now z = cos (t 1 )2 + (t 2 )2 , we see that the variables x, y, z satisfy the equation of the 2-sphere: x2 + y2 + z 2 = 1 : When the coset space of the 2-sphere:    0 :      M 0=: 1 :

(2.10)

representative M acts on the north pole it is easily seen that the orbit is all : : :

    0 x         y 0 = y : z 1 z x

(2.11)

This shows that there is one-to-one correspondence between the elements of the coset and the points of the 2-sphere. The coset SO(3)=SO(2) can therefore be identi4ed with a unit 2-sphere imbedded in three-dimensional space.

50


2.4. The Lie algebra and the adjoint representation A Lie algebra G is a vector space over a 4eld F. Multiplication in the Lie algebra is given by the bracket [X; Y ]. It has the following properties: (1) (2) (3) (4)

If X; Y ∈ G, then [X; Y ] ∈ G, [X; Y + Z] = [X; Y ] + [X; Z] for , ∈ F, [X; Y ] = −[Y; X ], [X; [Y; Z]] + [Y; [Z; X ]] + [Z; [X; Y ]] = 0 (the Jacobi identity).

The algebra G generates a group through the exponential mapping. A general group element is t i Xi ; t i ∈ F; Xi ∈ G : (2.12) M = exp i

We de4ne a mapping ad X from the Lie algebra to itself by ad X : Y → [X; Y ]. The mapping X → ad X is a representation of the Lie algebra called the adjoint representation. It is easy to check that it is an automorphism: it follows from the Jacobi identity that [ad Xi ; ad Xj ]=ad [Xi ; Xj ]. Suppose we choose a basis {Xi } for G. Then ad Xi (Xj ) = [Xi ; Xj ] = Cijk Xk ;

(2.13)

where we sum over k. The Cijk are called structure constants. Under a change of basis, they transform as mixed tensor components. They de4ne the matrix (Mi )jk = Cikj associated with the adjoint representation of Xi . One can show that there exists a basis for any complex semisimple algebra in which the structure constants are real. This means the adjoint representation is real. Note that the dimension of the adjoint representation is equal to the dimension of the group. Example. Let us construct the adjoint representation of SU (2). The generators in the de4ning representation are 0 1 0 −i 1 1 0 1 J3 = ; J± = ±i (2.14) 2 0 −1 2 1 0 i 0 and the commutation relations are [J3 ; J± ] = ±J± ;

[J+ ; J− ] = 2J3 :

(2.15)

− + + 3 3 The structure constants are therefore C3+ = −C+3 = −C3−− = C− 3 = 1, C+− = −C−+ = 2 and the adjoint representation is given by (M3 )++ = 1, (M3 )− − = −1, (M+ )+3 = −1, (M+ )3− = 2, (M− )−3 = 1, (M− )3+ = −2, and all other matrix elements equal to 0:       0 0 0 0 0 2 0 −2 0            M3 =  (2.16)  0 1 0  ; M+ =  −1 0 0  ; M− =  0 0 0  : 0 0 −1 0 0 0 1 0 0

These representation matrices are real, have the same dimension as the group, and satisfy the SU (2) commutation relations [M3 ; M± ] = ±M± , [M+ ; M− ] = 2M3 .


51

2.5. Semisimple algebras and root spaces In this paragraph we will brieQy recall the basic facts about root spaces and the classi4cation of complex simple Lie algebras, to set the stage for our discussion of real forms of Lie algebras and 4nally symmetric spaces. An ideal, or invariant subalgebra I is a subalgebra such that [G; I] ⊂ I. An abelian ideal also satis4es [I; I] = 0. A simple Lie algebra has no proper ideal. A semisimple Lie algebra is the direct sum of simple algebras, and has no proper abelian ideal (by proper we mean diOerent from {0}). A Lie algebra is a linear vector space over a 4eld F, with an antisymmetric product de4ned by the Lie bracket (cf. Section 2.4). If F is the 4eld of real, complex or quaternion numbers, the Lie algebra is called a real, complex or quaternion algebra. A complexi4cation of a real Lie algebra is obtained by taking linear combinations of its elements with complex coeVcients. A real Lie algebra H is a real form of the complex algebra G if G is the complexi4cation of H. In any simple algebra there are two kinds of generators: there is a maximal abelian subalgebra, called the Cartan subalgebra H0 = {H1 ; : : : ; Hr }; [Hi ; Hj ] = 0 for any two elements of the Cartan subalgebra. There are also raising and lowering operators denoted E . is an r-dimensional vector = ( 1 ; : : : ; r ) and r is the rank of the algebra. 1 The latter are eigenoperators of the Hi in the adjoint representation belonging to eigenvalue i : [Hi ; E ] = i E . For each eigenvalue, or root i , there is another eigenvalue − i and a corresponding eigenoperator E− under the action of Hi . Suppose we represent each element of the Lie algebra by an n × n matrix. Then [Hi ; Hj ] = 0 means the matrices Hi can all be diagonalized simultaneously. Their eigenvalues -i are given by Hi |- = -i |-, where the eigenvectors are labelled by the weight vectors - = (-1 ; : : : ; -r ) [36]. A weight whose 4rst non-zero component is positive is called a positive weight. Also, a weight - is greater than another weight - if - − - is positive. Thus we can de4ne the highest weight as the one which is greater than all the others. The highest weight is unique in any representation. The roots i ≡ (Hi ) of the algebra G are the weights of the adjoint representation. Recall that in the adjoint representation, the states on which the generators act are de4ned by the generators themselves, and the action is de4ned by Xa |Xb ≡ ad Xa (Xb ) ≡ [Xa ; Xb ] :

(2.17)

The roots are functionals on the Cartan subalgebra satisfying ad Hi (E ) = [Hi ; E ] = (Hi )E ;

(2.18)

where Hi is in the Cartan subalgebra. The eigenvectors E are called the root vectors. These are exactly the raising and lowering operators E± for the weight vectors -. There are canonical commutation relations de4ning the system of roots belonging to each simple rank r-algebra. These are summarized below: 2 [Hi ; Hj ] = 0;

[Hi ; E ] = i E ;

[E ; E− ] = i Hi :

(2.19)

1 The rank of an algebra is de4ned through the secular equation (see Section 6.1). For a non-semisimple algebra, the maximal number of mutually commuting generators can be greater than the rank of the algebra. 2 For the reader who wants to understand more about the origin of the structure of Lie algebras, we recommend Chapter 7 of Gilmore [31].

52


One can prove the fundamental relation [35,36] 2 · = −(p − q) ; 2

(2.20)

where is a root, - is a weight, and p, q are positive integers such that E |- + p = 0, E− |- − q = 0. 3 This relation gives rise to the strict properties of root lattices, and permits the complete classi4cation of all the complex (semi)simple algebras. Eq. (2.20) is true for any representation, but has particularly strong implications for the adjoint representation. In this case - is a root. As a consequence of Eq. (2.20), the possible angle between two root vectors of a simple Lie algebra is limited to a few values: these turn out to be multiples of /=6 and /=4 (see e.g. [36, Chapter VI]). The root lattice is invariant under reQections in the hyperplanes orthogonal to the roots (the Weyl group). As we will shortly see, this is true not only for the root lattice, but for the weight lattice of any representation. Note that the roots are real-valued linear functionals on the Cartan subalgebra. Therefore they are in the space dual to H0 . A subset of the positive roots span the root lattice. These are called simple roots. Obviously, since the roots are in the space dual to H0 , the number of simple roots is equal to the rank of the algebra. The same relation (2.20) determines the highest weights of all irreducible representations. Setting p = 0, choosing a positive integer q, and letting run through the simple roots, = i (i = 1; : : : ; r), we 4nd the highest weights -i of all the irreducible representations corresponding to the given value of q [36]. For example, for q = 1 we get the highest weights of the r fundamental representations of the group, each corresponding to a simple root i . For higher values of q we get the highest weights of higher-dimensional representations of the same group. The set of all possible simple root systems are classi4ed by means of Dynkin diagrams, each of which correspond to an equivalence class of isomorphic Lie algebras. The classical Lie algebras SU(n + 1; C), SO(2n + 1; C), Sp(2n; C) and SO(2n; C) correspond to root systems An , Bn , Cn , and Dn , respectively. In addition there are 4ve exceptional algebras corresponding to root systems E6 , E7 , E8 , F4 and G2 . Each of these complex algebras in general has several real forms associated with it (see Section 4). These real forms correspond to the same Dynkin diagram and root system as the complex algebra. Since we will not make reference to Dynkin diagrams in the following, we will not discuss them here. The interested reader can 4nd suVcient material for example in the book by Georgi [36]. 3

Here the scalar product · can be de4ned in terms of the metric on the Lie algebra. For the adjoint representation, - is a root and 2K(H ; H ) 2 · 2(H ) = ≡ ; 2 K(H ; H ) (H )

(2.21)

where K denotes the Killing form (see Section 3.4). There is always a unique element H in the algebra such that K(H; H ) = (H ) for each H ∈ H0 (see for example [35, Chapter 10]). In general for a linear form - on the Lie algebra, 2 · 2-(H ) = : 2 (H )

(2.22)

Then - is a highest weight for some representation if and only if this expression is an integer for each positive root .


The (semi)simple complex algebra G decomposes into a direct sum of root spaces [35]: G = H0 ⊕ G ;

53

(2.23)

where G is generated by {E± }. This will be evident in the example given below. Example. The root system An−1 corresponds to the complex Lie algebra SL(n; C) and all its real forms. In a later section we will see how to construct all the real forms associated with a given complex Lie algebra. Let us see here explicitly how to construct the root lattice of SU(3; C), which is one of the real forms of SL(3; C). The generators are determined by the commutation relations. In physics it is common to write the commutation relations in the form [Ti ; Tj ] = ifijk Tk

(2.24)

(an alternative form is to de4ne the generators as Xi = iTi and write the commutation relations as [Xi ; Xj ] = −fijk Xk ) where fijk are structure constants for the algebra SU(3; C). a Using the notation g = eit Ta for the group elements (with t a real and a sum over a implied), the generators Ta in the fundamental representation of this group are hermitean: 4       0 1 0 0 −i 0 1 0 0    1 1 1 1 0 0 i 0 0 0 −1 0  T1 =  ; T2 =  ; T3 =        ; 2 2 2 0 0 0 0 0 0 0 0 0       0 0 1 0 0 −i 0 0 0      1  ; T5 = 1  0 0 0  ; T6 = 1  0 0 1  ; 0 0 0 T4 =     2 2 2 1 0 0 i 0 0 0 1 0     0 0 0 1 0 0   1 1  0 1 0  :  ; T8 = √ 0 0 −i (2.25) T7 =    2 2 3 0 i 0 0 0 −2 In high energy physics the matrices 2Ta are known as Gell–Mann matrices. The generators are normalized in such a way that tr (Ta Tb ) = 12 6ab . Note that T1 , T2 , T3 form an SU(2; C) subalgebra. We take the Cartan subalgebra to be H0 = {T3 ; T8 }. The rank of this group is r = 2. 4

Note that we have written an explicit factor of i in front of the generators in the expression for the group elements. This is often done for compact groups; since the Killing form (Section 3.4) has to be negative de4nite, the coordinates of the algebra spanned by the generators must be purely imaginary. Here we use this notation because it is conventional. If we absorb the factor of i into the generators, we get antihermitean matrices Xa = iTa ; we will do this in the example in Section 3.1 to comply with Eq. (3.1). Of course, the matrices in the algebra are always antihermitean.

54


Let us 4rst 4nd the weight vectors of the fundamental representation. To this end we look for the eigenvalues -i of the operators in the abelian subalgebra H0 :         1 1 1 1   1      1        T3  (2.26)  0  = 2  0  ; T8  0  = 2√3  0  ; 0 0 0 0 therefore the eigenvector (1 0 0)T corresponds to the state |- where 1 1 ; √ - ≡ (-1 ; -2 ) = 2 2 3

(2.27)

is distinguished by its eigenvalues under the operators Hi of the Cartan subalgebra. In the same way we 4nd that (0 1 0)T and (0 0 1)T correspond to the states labelled by weight vectors 1 1 1 - = − ; √ ; - = 0; − √ ; (2.28) 2 2 3 3 respectively. -, - , and - are the weights of the fundamental representation 7 = D and they form an equilateral triangle in the plane. The highest weight of the representation D is - = 12 ; 2√1 3 . There is also another fundamental representation DW of the algebra SU(3; C), since it generates a group of rank 2. Indeed, from Eq. (2.20), for p = 0, q = 1, there is one highest weight -i , and one fundamental representation, for each simple root i . The highest weight -W of the representation DW is 1 1 ;− √ -W = : (2.29) 2 2 3 The highest weights of the representations corresponding to any positive integer q can be obtained as soon as we know the simple roots. Then, by operating with lowering operators on this weight, we obtain other weights, on which we can further operate with lowering operators until we have obtained all the weights in the representation. For an example of this procedure see [36, Chapter IX]. Let us see now how to obtain the roots of SU(3; C). Each root vector E corresponds to either a raising or a lowering operator: E is the eigenvector belonging to the root i ≡ (Hi ) under the adjoint representation of Hi , like in Eq. (2.32). Each raising or lowering operator is a linear combination of generators Ti that takes one state of the fundamental representation to another state of the same representation: E± |- = N± ; - |- ± . Therefore the root vectors will be diOerences of weight vectors in the fundamental representation. We 4nd the raising and lowering operators E± to be E±(1; 0) = E 1

√1

√ 3 ± 2; 2

E

2

(T1 ± iT2 ) ;

=

√ 3 1 ± −2; 2

√1

=

2

(T4 ± iT5 ) ;

√1

2

(T6 ± iT7 ) :

(2.30)

These generate the subspaces G in Eq. (2.23). In the fundamental representation, we 4nd using the Gell–Mann matrices that these are matrices with only one non-zero element. For example,


55

the raising operator E that  0 1 1  0 0 E+(1; 0) = √  2 0 0

corresponds to the root = (1; 0) is  0  0 (2.31)  : 0 This operator takes us from the state |- = − 12 ; 2√1 3 to the state |- = 12 ; 2√1 3 . The components of the root vectors of SU(3; C) are the eigenvalues i of these under the adjoint representation of the Cartan subalgebra. That is, Hi |E ≡ ad Hi (E ) ≡ [Hi ; E ] = i |E :

(2.32)

This way we easily 4nd the roots: we can either explicitly use the structure constants of SU (3) c Tc (note the explicit factor of i due to our conventions regarding the in [Ta ; Tb ] = ifabc Tc = −iCab generators) or we can use an explicit representation for Hi , E like in Eqs. (2.25), (2.30), (2.31), to calculate the commutators: ad H1 (E±(1; 0) ) = [H1 ; E±(1; 0) ] = T3 ; √12 (T1 ± iT2 ) = √12 (iT2 ± T1 ) = ±E±(1; 0) ≡ 1± E±(1; 0) ; ad H2 (E±(1; 0) ) = [H2 ; E±(1; 0) ] = T8 ; √12 (T1 ± iT2 ) = 0 ≡ 2± E±(1; 0) :

(2.33)

The root vector corresponding to the raising operator E+(1; 0) is thus = ( 1+ ; 2+ ) = (1; 0) and the − − root vector corresponding to the lowering operator E−(1; 0) is − = These root ( 1 ; 2 ) = (−1; 0). 1 √ 1 vectors are indeed the diOerences between the weight vectors - = 2 ; 2 3 and - = − 12 ; 2√1 3 of the fundamental representation. √ √ In the same way we 4nd the other root vectors ± 12 ; ± 23 , ∓ 12 ; ± 23 , and (0; 0) (with multiplicity 2), by operating with H1 and H2 on the remaining E± ’s and on the Hi ’s. The last root with multiplicity 2 has as its components the eigenvalues under H1 , H2 of the states |H1 and |H2 : Hi |Hj = [Hi ; Hj ] = 0; i, j ∈ {1;

2}. The root vectors

form a regular hexagon in the plane. The positive √

√

roots are (1; 0), 1 = 12 ; 23 and 2 = 12 ; − 23 . The latter two are simple roots. (1; 0) is not simple because it is the sum of the other positive roots. There are two simple roots, since the rank of SU (3) is 2 and the root lattice is two-dimensional. The root lattice of SU (3) is invariant under reQections in the hyperplanes orthogonal to the root vectors. This is true of any weight or root lattice; the symmetry group of reQections in hyperplanes orthogonal to the roots is called the Weyl group. It is obtained from Eq. (2.20): since for any root and any weight -, 2( · -)= 2 is the integer q − p, 2( · -) - = - − (2.34) 2 is also a weight. Eq. (2.34) is exactly the above mentioned reQection, as can easily be seen. 2.6. The Weyl chambers The roots are linear functionals on the Cartan subalgebra. We may denote the Cartan subalgebra by H0 and its dual space by H0∗ . A Weyl reQection like the one in (2.34) can be de4ned not only

56


for the weights or roots - in the space H0∗ , but for an arbitrary vector q ∈ H0∗ or, in all generality, for a vector q in an arbitrary 4nite-dimensional vector space: s (q) = q − ∗ (q) :

(2.35)

Note that q ∈ H0∗ is in the space dual to H0 and may denote a root. In (2.35) the function ∗ (q) is a linear functional on H0∗ such that ∗ ( ) = 2. We will be concerned only with the crystallographic case when ∗ (q) is integer. We denote the hyperplanes in H0∗ where the function ∗ (q) vanishes by H ( ) : H ( ) = {q ∈ H0∗ : ∗ (q) = 0} ;

(2.36)

( )

where H is orthogonal to the root , and s (q) is a reQection in this hyperplane. By identifying the dual spaces H0 and H0∗ (this is possible since they have the same dimension), we can consider hyperplanes like the ones in (2.36) in the space H0 . The role of the linear functional ∗ (q) is then played by 2q · 2q(H ) ; (2.37) ∗ (q) = = 2 (H ) where (H ) = K(H ; H ). Here K is the Killing form (a metric on the algebra to be de4ned in Section 3.4) and H is the unique element in H0 such that K(H; H ) = (H ). The open subsets of H0 where roots are nonzero are called Weyl chambers. Consequently, the walls of the Weyl chambers are the hyperplanes in H0 where the roots q(H ) are zero. 2.7. The simple root systems We have just shown by an example, in Section 2.5, how to obtain a root system of type An . In general, for any simple algebra the commutation relations determine the Cartan subalgebra and raising and lowering operators, that in turn determine a unique root system, and correspond to a given Dynkin diagram. In this way we can classify all the simple algebras according to the type of root system it possesses. The root systems for the four in4nite series of classical nonexceptional Lie groups can be characterized as follows [36] (denote the r-dimensional space spanned by the roots by V and let {e1 ; : : : ; en } be a canonical basis in Rn ): An−1 : Let V be the hyperplane in Rn that passes through the points (1; 0; 0; : : : ; 0); (0; 1; 0; : : : ; 0); : : : ; (0; 0; : : : ; 0; 1) (the endpoints of the ei , i = 1; : : : ; n). Then the root lattice contains the vectors {ei − ej ; i = j}. Bn : Let V be Rn ; then the roots are {±ei ; ±ei ± ej ; i = j}. Cn : Let V be Rn ; then the roots are {±2ei ; ±ei ± ej ; i = j}. Dn : Let V be Rn ; then the roots are {±ei ± ej ; i = j}. The root lattice BCn , that we will discuss in conjunction with restricted root systems, is the union of Bn and Cn . It is characterized as follows: BCn : Let V be Rn ; then the roots are {±ei ; ±2ei ; ±ei ± ej ; i = j}. Because this system contains both ei and 2ei , it is called non-reduced (normally the only root collinear with is − ). However, it is irreducible in the usual sense, which means it is not the direct sum of two disjoint root systems Bn and Cn . This can be seen from the root multiplicities (cf. Table 1).


57

The semisimple algebras are direct sums of simple ones. That means the simple constituent algebras commute with each other, and the root systems are direct sums of the corresponding simple root systems. Therefore, knowing the properties of the simple Lie algebras, we also know the semisimple ones.

3. Symmetric spaces In the previous section, we have reminded ourselves of some elementary facts concerning root spaces and the classi4cation of the complex semisimple algebras. In this section we will de4ne and discuss symmetric spaces. A symmetric space is associated to an involutive automorphism of a given Lie algebra. As we will see, several diOerent involutive automorphisms can act on the same algebra. Therefore we normally have several diOerent symmetric spaces deriving from the same Lie algebra. The involutive automorphism de4nes a symmetric subalgebra and a remaining complementary subspace of the algebra. Under general conditions, the complementary subspace is mapped onto a symmetric space through the exponential map. In the following subsections we make these statements more precise. We discuss how the elements of the Lie group can act as transformations on the elements of the symmetric space. This naturally leads to the de4nition of two coordinate systems on symmetric spaces: the spherical and the horospheric coordinate systems. The radial coordinates associated to each element of a symmetric space through its spherical or horospheric decomposition will be of relevance when we discuss the radial parts of diOerential operators on symmetric spaces in Section 6. In the same section we explain why these operators are important in applications to physical problems, and in Part II we will discuss some of their uses. In all of this paper we will distinguish between compact and non-compact symmetric spaces. In order to give a precise notion of compactness, we will de4ne the metric tensor on a Lie algebra in terms of the Killing form in Section 3.4. The latter is de4ned as a symmetric bilinear trace form on the adjoint representation, and is therefore expressible in terms of the structure constants. We will give several examples of Killing forms later, as we discuss the various real forms of a Lie algebra. The metric tensor serves to de4ne the curvature tensor on a symmetric space (Section 5.1). It is also needed in computing the Jacobian of the transformation to radial coordinates. This Jacobian is relevant in calculating the radial part of the Laplace–Beltrami operator (see Section 6.2). We will close this section with a discussion of the general algebraic form of coset representatives in Section 3.5. 3.1. Involutive automorphisms An automorphism of a Lie algebra G is a mapping from G onto itself such that it preserves the algebraic operations on the Lie algebra. For example, if : is an automorphism, it preserves multiplication: [:(X ); :(Y )] = :([X; Y ]), for X , Y ∈ G. Suppose that the linear automorphism : : G → G is such that :2 = 1, but : is not the identity. That means that : has eigenvalues ±1, and it splits the algebra G into orthogonal eigensubspaces corresponding to these eigenvalues. Such a mapping is called an involutive automorphism.

58


Suppose now that G is a compact simple Lie algebra, : is an involutive automorphism of G, and G = K ⊕ P where :(X ) = X

for X ∈ K;

:(X ) = −X

for X ∈ P :

(3.1)

From the properties of automorphisms mentioned above, it is easy to see that K is a subalgebra, but P is not. In fact, the commutation relations [K; K] ⊂ K;

[K; P] ⊂ P;

[P; P] ⊂ K

(3.2)

hold. A subalgebra K satisfying (3.2) is called a symmetric subalgebra. If we now multiply the elements in P by i (the “Weyl unitary trick”), we construct a new noncompact algebra G∗ = K ⊕ iP. This is called a Cartan decomposition, and K is a maximal compact subalgebra of G∗ . The coset spaces G=K and G ∗ =K are symmetric spaces. Example. Suppose G = SU (n; C), the group of unitary complex matrices with determinant +1. The algebra of this group then consists of complex antihermitean 5 matrices of zero trace (this follows by diOerentiating the identities UU † = 1 and det U = 1 with respect to t where U (t) is a curve passing a through the identity at t = 0); a group element is written as g = et Xa with t a real. Therefore any matrix X in the Lie algebra of this group can be written X =A+iB, where A is real, skew-symmetric, and traceless and B is real, symmetric and traceless. This means the algebra can be decomposed as G=K⊕P, where K is the compact connected subalgebra SO(n; R) consisting of real, skew-symmetric and traceless matrices, and P is the subspace of matrices of the form iB, where B is real, symmetric, and traceless. P is not a subalgebra. Referring to the example for SU(3; C) in Section 2.5 we see, setting Xa = iTa , that the {Xa } split into two sets under the involutive automorphism : de4ned by complex conjugation : = K. This splits the compact algebra G into K ⊕ P, since P consists of imaginary matrices:          0 1 0 0 0 1 0 0 0   1   1  1        −1 0 0 0 0 0 0 0 1 K = {X2 ; X5 ; X7 } = ; 2  ; 2   ;  2    0 0 0 −1 0 0 0 −1 0  P = {X1 ; X3 ; X4 ; X6 ; X8 }      0 1 0 1  i   i 1 0 0; 0 =  2  2   0 0 0 0 

1

i  √  0 2 3 0 5

0 1 0

     0  ;   −2 

0 −1 0

0





0

 i  0 ; 2 0 0 1

0 0 0

1





0

 i  0 ; 2 0 0 0

0

0



0

 1 ;

1

0

0

See the footnote in Section 2.5.

(3.3)


59

K spans the real subalgebra SO(3; R). Setting X2 ≡ L3 , X5 ≡ L2 , X7 ≡ L1 , the commutation relations for the subalgebra are [Li ; Lj ] = 12 jijk Lk . The Cartan subalgebra iH0 = {X3 ; X8 } is here entirely in the subspace P. Going back to the general case of G = SU(n; C), we obtain from G by the Weyl unitary trick the non-compact algebra G∗ = K ⊕ iP. iP is now the subspace of real, symmetric, and traceless matrices B. The Lie algebra G∗ = SL(n; R) is then the set of n × n real matrices of zero trace, and generates the linear group of transformations represented by real n × n matrices of unit determinant. The involutive automorphism that split the algebra G above was de4ned to be complex conjugation T −1 : =K. The involutive automorphism that splits G∗ is de4ned by :(g)=(g ˜ ) for g ∈ G ∗ , as we will now see. On the level of the algebra, :(g) ˜ = (gT )−1 means :(X ˜ ) = −X T . Suppose now g = etX ∈ G ∗ with X real and traceless and t a real parameter. If now X is an element of the subalgebra K, we then have :(X ˜ ) = +X , i.e. −X T = X and X is skew-symmetric. If instead X ∈ iP, we have T :(X ˜ ) = −X = −X , i.e. X is symmetric. The decomposition G∗ = K ⊕ iP is the usual decomposition of a SL(n; R) matrix in symmetric and skew-symmetric parts. G=K = SU (n; C)=SO(n; R) is a symmetric space of compact type, and the related symmetric space of non-compact type is G ∗ =K = SL(n; R)=SO(n; R). 3.2. The action of the group on the symmetric space Let G be a semisimple Lie group and K a compact symmetric subgroup. As we saw in the preceding paragraph, the coset spaces G=K and G ∗ =K represent symmetric spaces. Just as we have de4ned a Cartan subalgebra and the rank of a Lie algebra, we can de4ne, in an exactly analogous way, a Cartan subalgebra and the rank of a symmetric space. A Cartan subalgebra of a symmetric space is a maximal abelian subalgebra of the subspace P (see Section 5.2), and the rank of a symmetric space is the number of generators in this subalgebra. If G is connected and G = K ⊕ P where K is a compact symmetric subalgebra, then each group element can be decomposed as g = kp (right coset decomposition) or g = pk (left coset decomposition), with k ∈ K = eK , p ∈ P = eP . P is not a subgroup, unless it is abelian and coincides with its Cartan subalgebra. However, if the involutive automorphism that splits the algebra is denoted :, one can show [37, Chapter 6] that gp:(g−1 ) ∈ P. This de4nes G as a transformation group on P. Since :(k −1 ) = k −1 for k ∈ K, this means p = kpk −1 ∈ P

(3.4)

if k ∈ K, p ∈ P. Now suppose there are no other elements in G that satisfy :(g) = g than those in K. This will happen if the set of elements satisfying :(g) = g is connected. Then P is isomorphic to G=K. Also, G acts transitively on P in the manner de4ned above (cf. Section 2.3). The tangent space of G=K at the origin (identity element) is spanned by the subspace P of the algebra. 3.3. Radial coordinates In this paragraph we de4ne two coordinate systems frequently used on symmetric spaces. Let G = K ⊕ P be a Cartan decomposition of a semisimple algebra and let H0 ⊂ P be a maximal abelian

60


subalgebra in the subspace P. De4ne M to be the subgroup of elements in K such that M = {k ∈ K: kHk −1 = H; H ∈ H0 } :

(3.5)

This set is called the centralizer of H0 in K. Under conjugation by k ∈ K, each element H of the Cartan subalgebra is preserved. Further, denote M = {k ∈ K: kHk −1 = H ; H; H ∈ H0 } :

(3.6)

This is a larger subgroup than M that preserves the Cartan subalgebra as a whole, but not necessarily each element separately, and is called the normalizer of H0 in K. If K is a compact symmetric subgroup of G, one can show [37, Chapter 6] that every element p of P G=K is conjugated with some element h = eH for some H ∈ H0 by means of the adjoint representation 6 of the stationary subgroup K: p = khk −1 = kh:(k −1 ) ;

(3.8)

where k ∈ K=M and H is de4ned up to the elements in the factor group M =M . This factor group coincides with the Weyl group that was de4ned in Eq. (2.34): since the space H0 can be identi4ed with its dual space H0∗ , we can identify M =M with the Weyl group of the restricted root system (see Section 5.2). The eOect of the Weyl group is to transform the algebra H0 ⊂ P into another Cartan subalgebra H0 ⊂ P conjugate with the original one. This amounts to a permutation of the roots of the restricted root lattice corresponding to a Weyl reQection. Eq. (3.8) means that every element g ∈ G can be decomposed as g = pk = k hk −1 k = k hk , and this is very much like the Euler angle decomposition of SO(n). Thus, if x0 is the 4xed point of the subgroup K, an arbitrary point x ∈ P can be written x = khk −1 x0 = khx0 :

(3.9)

The coordinates (k(x); h(x)) are called spherical coordinates. k(x) is the angular coordinate and h(x) is the spherical radial coordinate of the point x. Eq. (3.8) de4nes the so-called spherical decomposition of the elements in the coset space. Of course, a similar reasoning is true for the space P ∗ G ∗ =K. This means every matrix p in the coset space G=K can be diagonalized by a similarity transformation by the subgroup K, and the radial coordinates are exactly the set of eigenvalues of the matrix p. These “eigenvalues” are not necessarily real numbers. This is easily seen in the example in Eq. (3.3). It can also be seen in the adjoint representation. Suppose the algebra G = K ⊕ P is

6

Note that eK H e−K = ead K H ≡

∞ (ad K)n H : n! n=0

(3.7)


compact. From Eq. (2.13), in the adjoint representation Hi ∈ H0 has the form   0 :::     .. :  .     :  0     i     ; Hi =   − i       ..   .       ;i   −;i

61

(3.10)

where the matrix is determined by the structure constants ([Hi ; Hj ] = 0, [Hi ; E± ] = ± i E± : : : and ± i ; : : : ; ±;i are the roots corresponding to Hi ). Since the Killing form must be negative (see Section 3.4) for a compact algebra, the coordinates of the Cartan subalgebra must be purely imaginary and the group elements corresponding to H0 must have the form   1 :::   ..   .  :     : 1   it · H   (3.11) = e  eit ·       ..   .   e−it · with t = (t 1 ; t 2 ; : : : t r ) and t i real parameters. In particular, if the eigenvalues are real for p ∈ P ∗ , they are complex numbers for p ∈ P. Example. In the example we gave in the preceding subsection, the coset space G ∗ =K = SL(n; R)= SO(n) P ∗ = eiP consists of real positive-de4nite symmetric matrices. Note that G = K ⊕ P implies that G can be decomposed as G =PK and G ∗ as G ∗ =P ∗ K. The decomposition G ∗ =P ∗ K in this case is the decomposition of a SL(n; R) matrix in a positive-de4nite symmetric matrix and an orthogonal one. Each positive-de4nite symmetric matrix can be further decomposed: it can be diagonalized by an SO(n) similarity transformation. This is the content of Eq. (3.8) for this case, and we know it to be true from linear algebra. Similarly, according to Eq. (3.8) the complex symmetric matrices in G=K = SU (n; C)=SO(n) P = eP can be diagonalized by the group K = SO(n) to a form where the eigenvalues are similar to those in Eq. (3.11). In terms of the subspace P of the algebra, Eq. (3.8) amounts to saying that any two Cartan subalgebras H0 , H0 of the symmetric space are conjugate under a similarity transformation by K,

62


and we can choose the Cartan subalgebra in any way we please. However, the number of elements that we can diagonalize simultaneously will always be equal to the rank of the symmetric space. There is also another coordinate system valid only for spaces of the type P ∗ ∼ G ∗ =K. This coordinate system is called horospheric and is based on the so called Iwasawa decomposition [37] of the algebra: G = N + ⊕ H0 ⊕ K :

(3.12)

Here K; H0 ; N+ are three subalgebras of G. K is a maximal compact subalgebra, H0 is a Cartan subalgebra, and N+ = G (3.13) ∈ R+

is an algebra of raising operators corresponding to the positive roots (H ) ¿ 0 with respect to H0 (G is the space generated by E ). As a consequence, the group elements can be decomposed g=nhk, in an obvious notation. This means that if x0 is the 4xed point of K, any point x ∈ G ∗ =K can be written as x = nhkx0 = nhx0 :

(3.14)

The coordinates (n(x); h(x)) are called horospheric coordinates and the element h = h(x) is called the horospheric projection of the point x or the horospheric radial coordinate. 3.4. The metric on a Lie algebra A metric tensor can be de4ned on a Lie algebra [30,31,35,37]. For our purposes, it will eventually serve to de4ne the curvature of a symmetric space and be useful in computing the Jacobian of the transformation to radial coordinates. In Sections 6 and 8 we will see the importance of this Jacobian in physical applications in connection with the radial part of the Laplace–Beltrami operator. If {Xi } form a basis for the Lie algebra G, the metric tensor is de4ned by gij = K(Xi ; Xj ) ≡ tr(ad Xi ad Xj ) = Cisr Cjrs :

(3.15)

The symmetric bilinear form K(Xi ; Xj ) is called the Killing form. It is intrinsically associated with the Lie algebra, and since the Lie bracket is invariant under automorphisms of the algebra, so is the Killing form. Example. The generators X7 ≡ L1 , X5 ≡ L2 , X2 ≡ L3 of SO(3) given in Eq. (3.3) obey the commutation relations [Li ; Lj ] = Cijk Lk = 12 jijk Lk . From Eq. (3.15), the metric for this algebra is gij = − 12 6ij . The generators and the structure constants can be normalized so that the metric takes the canonical form gij = −6ij . Just like we de4ned the Killing form K(Xi ; Xj ) for the algebra G in Eq. (3.15) using the adjoint representation, we can de4ne a similar trace form K7 and a metric tensor g7 for any representation 7 by g7; ij = K7 (Xi ; Xj ) = tr(7(Xi )7(Xj )) ;

(3.16)


63

where 7(X ) is the matrix representative of the Lie algebra element X . If 7 is an automorphism of G, K7 (Xi ; Xj ) = K(Xi ; Xj ). Suppose the Lie algebra is semisimple (this is true for all the classical Lie algebras except the Lie algebras GL(n; C), U (n; C)). According to Cartan’s criterion, the Killing form is non-degenerate for a semisimple algebra. This means that det gij = 0, so that the inverse of gij , denoted by gij , exists. Since it is also real and symmetric, it can be reduced to canonical form gij =diag(−1; : : : ; −1; 1; : : : ; 1) with p −1’s and (n − p) +1’s, where n is the dimension of the algebra. p is an invariant of the quadratic form. In fact, for any real form of a complex algebra, the trace of the metric, called the character of the particular real form (see below and in [31]) distinguishes the real forms from each other (though it can be degenerate for the classical Lie algebras [31]). The character ranges from −n, where n is the dimension of the algebra, to +r, where r is its rank. All the real forms of the algebra have a character that lies in between these values. In Section 4.1 we will see several explicit examples of Killing forms. A famous theorem by Weyl states that a simple Lie group G is compact, if and only if the Killing form on G is negative de3 = J3 ; 1 J+ + J − √ = √ =1 ; 2 2 2 i J+ − J − = √ =2 ; >2 = √ 2 2 2 >1 =

(4.10)

like in Eq. (4.3). The commutation relations then become [>1 ; >2 ] = − √12 >3 ;

[>2 ; >3 ] = − √12 >1 ;

[>3 ; >1 ] =

√1

2

>2 :

(4.11)

These commutation relations characterize the algebra SO(2; 1; R). From here we 4nd the structure 3 3 1 1 2 2 constants C12 = −C21 = C23 = −C32 = −C31 = C13 = − √12 and the diagonal metric of the normal real form with rows and columns labelled 3, 1, 2 (in order to comply with the notation in Eq. (4.4)) is   1 0 0    gij =  (4.12) 0 1 0  ; 0 0 −1 which is to be compared with Eq. (4.4). According to Eq. (4.3), the Cartan decomposition of G∗ is G∗ = K ⊕ iP where K = {>2 } and iP = {>3 ; >1 }. The Cartan subalgebra consists of >3 . Finally, we arrive at the compact real form by multiplying >3 and >1 with i. Setting i>1 = >˜ 1 , >2 = >˜ 2 , i>3 = >˜ 3 the commutation relations become those of the special orthogonal group: [>˜ 1 ; >˜ 2 ] = − √12 >˜ 3 ;

[>˜ 2 ; >˜ 3 ] = − √12 >˜ 1 ;

[>˜ 3 ; >˜ 1 ] = − √12 >˜ 2 :

(4.13)

The last commutation relation in Eq. (4.11) has changed sign whereas the others are unchanged. 2 2 C31 , C13 , and consequently g33 and g11 change sign and we get the metric for SO(3; R):   −1 0 0    : 0 −1 0 gij =  (4.14)   0 0 −1 This is the compact real form. The subspaces of the compact algebra G = K ⊕ P are K = {>˜ 2 } and P = {>˜ 3 ; >˜ 1 }. Weyl’s theorem states that a simple Lie group G is compact, if and only if the Killing form on G is negative de4nite; otherwise it is non-compact. In the present example, we see this explicitly.

68


4.2. The classi3 ; >1 } as the basis in the space iP and {>˜ 3 ; >˜ 1 } (>˜ i ≡ i>i ) as the basis in the space P, we see by comparing the signs of the entries of the metrics we computed in Eqs. (4.12) and (4.13) that the sectional curvature K at the origin has the opposite sign for the two spaces SO(2; 1)=SO(2) and SO(3)=SO(2). Actually, there is also a zero-curvature symmetric space X 0 = G 0 =K related to X + = G=K and = G ∗ =K, so that we can speak of a triplet of symmetric spaces related to the same symmetric

X−

74


subgroup K. The zero-curvature spaces were discussed in [9] and in Ch. V of Helgason’s book [30], where they are referred to as “symmetric spaces of the euclidean type”. That their curvature is zero was proved in Theorem 3.1 of [30, Chapter V]. The Qat symmetric space X 0 can be identi4ed with the subspace P of the algebra. The group G 0 is a semidirect product of the subgroup K and the invariant subspace P of the algebra, and its elements g = (k; a) act on the elements of X 0 in the following way: g(x) = kx + a;

k ∈ K; x; a ∈ X 0

(5.4)

if the x’s are vectors, and g(x) = kxk −1 + a;

k ∈ K; x; a ∈ X 0

(5.5)

if the x’s are matrices. We will see one example of each below. The elements of the algebra P now de4ne an abelian additive group, and X 0 is a vector space with euclidean geometry. In the above scenario, the subspace P contains only the operators of the Cartan subalgebra and no others: P = H0 , so that P is a subalgebra of G0 . The algebra G0 = K ⊕ P belongs to a nonsemisimple group G 0 , since it has an abelian ideal P : [K; K] ⊂ K, [K; P] ⊂ P, [P; P] = 0. Note that K and P still satisfy the commutation relations (5.1). In this case the coset space X 0 is Qat, since by (5.1), Rnijk = 0 for all the elements X ∈ P. Eq. (5.2) is valid for any space with a Riemannian structure. Indeed, it is easy to see from Eqs. (5.2), (5.3) that Rnijk = K = 0 if the generators are abelian. Even though the Killing form on nonsemisimple algebras is degenerate, it is trivial to 4nd a nondegenerate metric on the symmetric space X 0 that can be used in (5.3) to 4nd that the sectional curvature at any point is zero. For example, as we pass from the sphere to the plane, the metric becomes degenerate in the limit as [L1 ; L2 ] ∼ L3 → [P1 ; P2 ] = 0 (see the example below). Obviously, we do not inherit this degenerate metric from the tangent space on R2 like in the case of the sphere, but the usual metric for R2 , gij = 6ij provides the Riemannian structure on the plane. Examples. An example of a Qat symmetric space is E2 =K, where G 0 = E2 is the euclidean group of motions of the plane R2 : g(x) = kx + a, g = (k; a) ∈ G 0 where k ∈ K = SO(2) and a ∈ R2 . The generators of this group are translations P1 , P2 ∈ H0 = P and a rotation J ∈ K satisfying [P1 ; P2 ] = 0, [J; Pi ] = −jij Pj , [J; J ] = 0, in agreement with Eq. (5.1) de4ning a symmetric subgroup. The abelian algebra of translations 2i=1 t i Pi , t i ∈ R, is isomorphic to the plane R2 , and can be identi4ed with it. The commutation relations for E2 are a kind of limiting case of the commutation relations for SO(3) ∼ SU(2) and SO(2; 1). If in the limit of in4nite radius of the sphere S 2 we identify >˜ 1 with P1 , >˜ 2 with P2 , and >˜ 3 with J , we see that the commutation relations resemble the ones described in 3 3 Eqs. (4.11) and (4.13)—we only have to set [>˜ 1 ; >˜ 2 ] = 0, which amounts to setting C12 = −C21 → 0. From here we get the degenerate metric of the nonsemisimple algebra E2 :   −1    ; 0 gij =  (5.6)   0 where the only nonzero element is g33 . This is to be confronted with Eqs. (4.12) and (4.14) which are the metrics for SO(2; 1) and SO(3). This is an example of contraction of an algebra.


75

An example of a triplet {X + ; X 0 ; X − } corresponding to the same subgroup K = SO(n) is: (1) X + = SU (n; C)=SO(n), the set of symmetric unitary matrices with unit determinant; it is the space exp(P) where P are real, symmetric and traceless n × n matrices. (Cf. the example in Section 3.1.) (2) X 0 is the set P of real, symmetric and traceless n × n matrices and the non-semisimple group G 0 is the group whose action is de4ned by g(x)=kxk −1 +a, g=(k; a) ∈ G 0 where k ∈ K =SO(n) and x; a ∈ X 0 . The involutive automorphism maps g = (k; a) ∈ G 0 into g = (k; −a). (3) X − = SL(n; R)=SO(n) is the set of real, positive, symmetric matrices with unit determinant; it is the space exp(iP) where P are real, symmetric and traceless n × n matrices. We remark that the zero-curvature symmetric spaces correspond to the integration manifolds of many known matrix models with physical applications. The pairs of dual symmetric spaces of positive and negative curvature listed in each row of Table 1 originate in the same complex extension algebra [31] with a given root lattice. This “inherited” root lattice is listed in the 4rst column of the table. In our example in Section 4.2 this was the root lattice of the complex algebra GC = SL(n; C). The same root lattice An−1 characterizes the real forms of SL(n; C): as we saw in the example these are the algebras SU(n; C), SL(n; R), SU(p; q; C) and SU∗ (2n), and we have seen how to construct them using involutive automorphisms. However, also listed in Table 1 is the restricted root system corresponding to each symmetric space. This root system may be diOerent from the one inherited from the complex extension algebra. Below, we will de4ne the restricted root system and see an explicit example of one such system. While the original root lattice characterizes the complex extension algebra and its real forms, the restricted root lattice characterizes a particular symmetric space originating from one of its real forms. The root lattices of the classical simple algebras are the in4nite sequences An , Bn , Cn , Dn , where the index n denotes the rank of the corresponding group. The root multiplicities mo , ml , ms listed in Table 1 (where the subscripts refer to ordinary, long and short roots, respectively) are characteristic of the restricted root lattices. In general, in the root lattice of a simple algebra (or in the graphical representation of any irreducible representation), the roots (weights) may be degenerate and thus have a multiplicity greater than 1. This happens if the same weight - = (-1 ; : : : ; -r ) corresponds to diOerent states in the representation. In that case one can arrive at that particular weight using diOerent sets of lowering operators E− on the highest weight of the representation. Indeed, we saw in the example of SU (3; C) in Section 2.5, that the roots can have a multiplicity diOerent from 1. The same is true for the restricted roots. The sets of simple roots of the classical root systems (brieQy listed in Section 2.7) have been obtained for example in [31,36]. In the canonical basis in Rn , the roots of type {±ei ± ej ; i = j} are ordinary while the roots {±2ei } are long and the roots {±ei } are short. Only a few sets of root multiplicities are compatible with the strict properties characterizing root lattices in general. 5.2. Restricted root systems The restricted root systems play an important role in connection with matrix models and integrable Calogero–Sutherland models (these models will be introduced in Section 7). We will discuss this

76


in detail in Part II. In this subsection we will explain how restricted root systems are obtained and how they are related to a given symmetric space. 8 As we have repeatedly seen in the examples using the compact algebra SU(n; C) (in particular in Section 4.2), the algebra SU(p; q; C) (p + q = n) is a non-compact real form of the former. This means they share the same rank-(n − 1) root system An−1 . However, to the symmetric space SU (p; q; C)=(SU (p) ⊗ SU (q) ⊗ U (1)) one can associate another rank-r root system, where r = min(p; q) is the rank of the symmetric space. For some symmetric spaces, it is the same as the root system inherited from the complex extension algebra (see Table 1 for a list of the restricted root systems), but this need not be the case. For example, the restricted root system is, in the case of SU (p; q; C)=(SU (p) ⊗ SU (q) ⊗ U (1)), BCr . When it is the same and when it is diOerent, as well as why the rank can change, will be obvious from the example we will give below. In general the restricted root system will be diOerent from the original, inherited root system if the Cartan subalgebra is a subset of K. The procedure to 4nd the restricted root system is then to de4ne an alternative Cartan subalgebra that lies partly (or entirely) in P (or iP). To achieve this, we 4rst look for a diOerent representation of the original Cartan subalgebra, that gives the same root lattice as the original one (i.e., An−1 for the SU(p; q; C) algebra). In general, this root lattice is an automorphism of the original root lattice of the same kind, obtained by a permutation of the roots. Unless we 4nd this new representation, we will not be able to 4nd a new, alternative Cartan subalgebra that lies partly in the subspace P. Once this has been done, we take a maximal abelian subalgebra of P (the number of generators in it will be equal to the rank r of the symmetric space G=K or G ∗ =K) and 4nd the generators in K that commute with it. These generators will be among the ones that are in the new representation of the original Cartan subalgebra. These commuting generators now form our new, alternative Cartan subalgebra that lies partly in P, partly in K. Let us call it A0 . The new root system is de4ned with respect to the part of the maximal abelian subalgebra that lies in P. Therefore its rank is normally smaller than the rank of the root system inherited from the complex extension. We can de4ne raising and lowering operators E in the whole algebra G that satisfy [Xi ; E ] = i E

(Xi ∈ A0 ∩ P) :

(5.7)

The roots i de4ne the restricted root system. Example. Let us now look at a speci4c example. We will start with the by now familiar algebra SU(3; C). As before, we use the convention of regarding the Ti ’s as the generators, awithout the a factor of i (recall that the algebra consists of elements of the form t X = i a a a t Ta ; cf. the footnote in conjuction with Eq. (2.25)). In Section 2.5 we explicitly constructed its root lattice A2 . Let us write down the generators again:       0 1 0 0 −i 0 1 0 0 1 1 1    T1 =  1 0 0  ; T2 =  i 0 0  ; T3 =  0 −1 0  ; 2 2 2 0 0 0 0 0 0 0 0 0

8

The authors are indebted to Prof. Simon Salamon for explaining how the restricted root systems are obtained.




0

0

1 0 2 1  0 1 T7 =  0 2 0 T4 =

1



0

 0;

0

0

0

0



 −i  ; 0

0 i

 T5 =

0

1 0 2 i

0

−i





 0 ; 0

0

0  1 1  T8 = √  0 2 3 0

0 1 0

T6 =

0



0

1 0 2 0

0

77

0



0

 1 ;

1

0

 0  : −2

(5.8)

The splitting of the SU(3; C) algebra in terms of the subspaces K and P was given in Eq. (3.3): K = {iT2 ; iT5 ; iT7 };

P = {iT1 ; iT3 ; iT4 ; iT6 ; iT8 } :

(5.9)

The Cartan subalgebra is {iT3 ; iT8 }. The raising and lowering operators were given in (2.30) in terms of Ti : E±(1; 0) = E 1

√1

√ 3 ± 2; 2

E

2

(T1 ± iT2 ) ;

=

√ 3 1 ± −2; 2

√1

=

2

(T4 ± iT5 ) ;

√1

2

(T6 ± iT7 ) :

(5.10)

Now let us construct the Cartan decomposition of G ∗ = K ⊕ iP = SU(2; 1; C). We know from Section 4.2 that K and P are given by matrices of the form 0 B A 0 (5.11) ∈ P ; ∈ K ; 0 C −B† 0 where A and C are antihermitean and tr A + tr C = 0. Combining the generators to form this kind of block-structures (or alternatively, using the involution :2 = I2; 1 ) we need to take linear combinations of the Xi ’s, with real coeVcients, and we then see that the subspaces K and iP are spanned by            0 1 0 1 1 0 1 0   i          1 i i         √ 1 0 −1 0 0 −1 0 1 K = ; ; ;  2  2  2 3   2    0 0 0 −2  = {iT1 ; iT2 ; iT3 ; iT8 }      1  1   i   0 iP =  ; 2   2   1 0 1 = {T4 ; T5 ; T6 ; T7 } ;

−1 0





 1  0  ; 2 

0 0

1





 i  1 ; 2 

    −1     0

0

1

(5.12)

78


where the block-structure is evidenced by leaving blank the remaining zero entries. K spans the algebra of the symmetric subgroup SU (2) ⊗ U (1) and iP spans the complementary subspace corresponding to the symmetric space SU (2; 1)=(SU (2) ⊗ U (1)). iP is spanned by matrices of the form 0 B˜ : (5.13) B˜ † 0 We see that the Cartan subalgebra iH0 = {iT3 ; iT8 } lies entirely in K . It is easy to see that by using the alternative representation     1 1   1 1    ; T8 = √  0 −2 T3 =  (5.14)     2 2 3 −1 1 of the Cartan subalgebra (note that this is a valid representation of SU(3; C) generators) while the other Ti ’s are unchanged, we still get the same root lattice A2 . The eigenvectors under the adjoint representation, the E ’s, are still given by Eq. (5.10). However, their eigenvalues (roots) are permuted under the new adjoint representation of the Cartan subalgebra, so that they no longer correspond to the root subscripts in (5.10). This permutation is a Weyl reQection; more speci4cally, it is the √ 3 1 reQection in the hyperplane orthogonal to the root − 2 ; 2 . Now we choose the alternative Cartan subalgebra to consist of the generators T4 , T8 : A0 = {T4 ; T8 };

[T4 ; T8 ] = 0;

iT4 ∈ P ; iT8 ∈ K :

(5.15)

(Note that unless we 4rst take a new representation of the original Cartan subalgebra, we are not able to 4nd the alternative Cartan subalgebra that lies partly in P .) The restricted root system is now about to be revealed. We de4ne raising and lowering operators E in the whole algebra according to E± 1 ∼ (T5 ± iT3 );

E

1

±2

∼ (T6 ± iT2 ); E˜ ± 1 ∼ (T7 ± iT1 ) :

(5.16)

2

The ± subscripts are the eigenvalues of T4 ∈ iP in the adjoint representation: [T4 ; E± 1 ] = ±E±1 ;

[T4 ; E 1 ] = ± 12 E 1 ; [T4 ; E˜ ± 1 ] = ± 12 E˜ ± 1 : ±2

±2

2

2

(5.17)

These roots form a one-dimensional root system of type BC1 . We see that the multiplicity of the long roots is 1 and the multiplicity of the short roots is 2 = 2(p − q). This result is general (cf. Table 1). If we had ordinary roots, their multiplicity would be 2, but for this low-dimensional group we can have only three pairs of roots. Note that we can rescale the lengths of all the roots together by rescaling the operator T4 in (5.17), but their characters as long and short roots cannot change. The root system BC1 is with respect to the part of the Cartan subalgebra lying in iP only, thus it is called restricted. According to Eq. (3.8), every element p of P G=K is conjugated with some element h = eH (H ∈ H0 ) through p=khk −1 , where k ∈ K=M and H is de4ned up to the elements in the factor group M =M . Thus, the decomposition p=khk −1 is not unique. The factor group M =M transforms a Cartan subalgebra H0 ⊂ P into another Cartan subalgebra H0 ⊂ P conjugate with the original one. This


79

amounts to a permutation of the roots of the restricted root lattice corresponding to Weyl reQections. The factor group M =M then coincides with the Weyl group of the restricted root system. If we 4x the Weyl chamber of H , H is unique and k is de4ned up to transformations by the subgroup M . 5.3. Real forms of symmetric spaces Involutive automorphisms were used to split the algebra G into orthogonal subspaces to obtain the real forms G, G∗ , G ∗ : : : of a complex extension algebra GC . By re-applying the same involutive automorphisms to the spaces K, P, and iP, these spaces with a de4nite metric tensor can in turn be split into subspaces with eigenvalue +1 and −1 under this new involutive automorphism =. Thus, :: G → K ⊕ P;

G∗ = K ⊕ iP ;

=: K → K1 ⊕ K2 ;

H = K1 ⊕ iK2 ;

=: P → P1 ⊕ P2 ;

M = P1 ⊕ iP2 ;

=: iP → iP1 ⊕ iP2 ;

iM = iP1 ⊕ P2 :

(5.18)

As we already know, K is a compact subgroup, and exp(P) and exp(iP) de4ne symmetric spaces with a de

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