Statistics of Linear Polymers in Disordered Media

Statistics of Linear Polymers in Disordered Media

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Statistics of Linear Polymers in Disordered Media

Edited by

Bikas K. Chakrabarti Saha Institute of Nuclear Physics Kolkata, India

2005

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PREFACE With the mapping of the partition function graphs of the n-vector magnetic model in the n —t 0 limit as the self-avoiding walks, the conformational statistics of linear polymers was clearly understood in early 1970s. Various models of disordered solids, percolation model in particular, were also established by late seventies. Subsequently, investigations on the statistics of linear polymers or of self-avoiding walks in, say, porous medium or disordered lattices were started in early 1980s. Inspite of the brilliant ideas forwarded and extensive studies made, the problem is not yet completely solved in its generality. This intriguing and important problem has remained since a topic of vigorous and active research. This book intends to offer the readers a first hand and extensive review of the various aspects of the problem, written by the experts in the respective fields. S. M. Bhattacharjee has reviewed the success in dealing with the directed polymers in random medium and has also discussed the problem of unzipping of a pair of directed chains where disorder appears along the chains. A. J. Guttmann has reviewed the series studies of self-avoiding walk statistics for various constrained and random geometries, including the problem of unzipping of the DNA chains. V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch has reviewed extensively the field theoretic and real space renormalization group studies for the problem, including the effects of correlated disorder. D. Dhar and Y. Singh have reviewed most of the exact results for self-avoiding walks on different non-random fractals, including various simplex lattices, employing the real space renormalization group technique. A. Ordemann, M. Porto and H. E. Roman have reviewed the extensive numerical studies on self-avoiding walks on various deterministic and random fractals; percolation clusters in particular. In absence of strict self-avoiding restriction, the analogy of the problem with that of quantum particles in disorder, and the consequent localization of the polymers in random media, have been reviewed by Y. Y. Goldschmidt and Y. Shiferaw. P. Bhattacharyya and A. Chatterjee have reviewed the properties of various optimal and most probable (self-avoiding in general) paths on randomly disordered lattices, including the statistics of the Travelling Salesman Problem on dilute lattices. Finally, G. D. J. Phillies has given an extensive overview of the experimental studies on polymer diffusion in random environments of the solutions. We earnestly hope, the contents of the book will provide a valuable guide for researchers in statistical physics of polymers and will surely induce further research and advances towards a complete understanding of the problem. I am grateful to all the contributors for their wonderful contributions and cooperations. I am thankful to Arnab Chatterjee for his help in the compilation of the book.

Bikas K. Chakrabarti Theoretical Condensed Matter Physics Division, and Centre for Applied Mathematics & Computational Science Saha Institute of Nuclear Physics, Kolkata 700064, India. January 2005.

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Contents Polymers in random media: An introduction B. K. Chakrabarti

1

Directed polymers and randomness S. M. Bhattacharjee

9

Self-avoiding walks in constrained and random geometries: Series studies A. J. Guttmann

59

Renormalization group approaches to polymers in disordered media V. Blavats 'ka, C. von Ferber, R. Folk and Yu. Holovatch

103

Linear and branched polymers on fractals D. Dhar and Y. Singh

149

Self-avoiding walks on deterministic and random fractals: Numerical results A. Ordemann, M. Porto and H. E. Roman

195

Localization of polymers in random media: Analogy with quantum particles in disorder Y. Y. Goldschmidt and Y. Shiferaw 235 Geometric properties of optimal and most probable paths on randomly disordered lattices P. Bhattacharyya and A. Chatterjee 271 Phenomenology of polymer single-chain diffusion in solution G. D. J. Phillies

305

Index

357

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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.

Polymers in random media: An introduction Bikas K. Chakrabarti

a

a

Theoretical Condensed Matter Physics Division, and Centre for Applied Mathematics & Computational Science, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India. email: [email protected] We introduce first the (lattice) Self-Avoiding Walk (SAW) model of polymer chains, their critical statistics and the criteria indicating effects of lattice disoder on the critical behaviour. Prominent indications for the effect of disorder on the SAW statistics are then discussed. Next, some mean field and scaling arguments are discussed for the SAW statistics in disordered medium; percolating lattice in particular. 1. POLYMER STATISTICS AND SAW MODEL Linear polymers are long flexible molecular chains [1] whose building blocks are the monomers. The chain flexibility arises when the chains are dissolved in solvents. The chains are completely flexible in good solvent. To study the conformational properties of polymer in good solvent, e.g., to estimate the variation of average radius of gyration or the end-to-end distance with the chain length, we use a lattice model of linear polymers where the polymer is viewed as a walk on a lattice: the monomer size is represented by the lattice constant and the size of the polymer chain by the walk length. A random walk can easily capture the flexibility of the chain. However, such a walk can cross itself or may trace back the same path. For polymer chains, the steric hindrence induces monomermonomer excluded volume restriction which invalidates this model. Hence a realistic model of a polymer chain in a good solvent is a random self-avioding walk (SAW) on a (translationally invariant) lattice. These SAWs are random walks without self-intersection or crossing; as such they are the self-avoiding subset of random walks. The statistics of the SAW model [2] of linear polymers is quite well studied. The generator of the self-avoiding walk statistics or the distribution function G^(r), which represents the number of IV-stepped SAW configurations with end-to-end distance r, is not Gaussian as in the case of random walk (for random walk Gjv(r) ~ exp[—r2/N]). From the distribution function Gx(r), one can obtain the asymptotic behaviour of various moments. A brief summary is as follows: The statistics of SAWs are characterised by the connectivity constant [i (GM = X)r Gjv(r) ~ ^N1'1), which is nonuniversal and depends on the lattice types, and the universal exponents like the radius of gyration exponent v (R2N = 52r r2Gff(r)/GN ~ N2l/), which depend only on the lattice dimension d. Extensive numerical studies give, fi = / / ~ 2.638, 4.151 and 4.684 for square, triangular and simple 1

2

B. K. Chakrabarti

cubic lattices respectively [3]. Various theoretical and numerical studies give the value of v = z/° = 3/4, ~ 0.592 and 1/2 (and 7 = 7° = 43/32, ~ 1.17 and 1) for d = 2, 3 and 4 respectively; here the superscript 0 stands for pure lattice [4]. For d > 4, the (statistical) effect of the excluded volume fluctuation disappear and self-avoiding walk and random walk belongs to the same universality class; the upper critical dimensionality (for the SAW statistics) is four [2]. In the limit of high temperature the effective interactions between the monomers arise mostly from the excluded volume considerations and the random SAW model, discussed above, is quite successful in capturing the universal behaviour of the conformational statistics. With lowering of the temperature, or in poor solvents, the effect of monomermonomer attraction grows and the polymer radius shrinks. The changes in the conformational statistics of linear polymers, with lowering of temperature from a high temperature limit, have also been studied extensively [1,2]. At the 0-point temperature, the two body excluded volume term is exactly cancelled by the growing attractive interactions and the statistics is governed by the higher order excluded volume terms. This point has been identified [2] as the tricritical point. The ^-points for the lattice SAW model have been estimated and (like the connectivity constant ji) they depend on the lattices. At this particular temperature, a crossover occurs from the high temperature SAW statistics to a tricritical (0-point) statistics. The size exponent is then given by the 0-point exponent value v = Vg = 4/7 [4] and 1/2 in d = 2 and 3 respectively: with the upper critical dimension for #-point statistics equal to three [2]. Below this tricritical temperature the attractive force dominates over the repulsive (entropic) term and induces the chain to collapse (with collapsed polymer size exponent v = v\ = l/d). We summarised so far the statistics of SAWs on regular lattices. 2. SAW STATISTICS IN DISORDERED MEDIUM We consider in this book the problem of polymer chain statistics in a disordered (say, porous) medium. If the porous medium is modelled by a percolating lattice [5], we can consider the following problem: let the bonds (sites) of a lattice be randomly occupied with concentration p (> pc; the percolation threshold); the SAWs are then allowed to have their steps only on the occupied bonds (through the occupied sites). We address the following questions [6,7]: does the lattice irregularity (of the dilute lattice) affect the SAW statistics? We expect /x or 9 to vary with the lattice occupation concentration p: /j,(p) < fi° and 9(p) < 6° for p < 1 where superscript 0 refers to that for pure lattice (p = 1). In particular, the values of (J,(pc) and 8(pc) on various lattices are very significant lattice statistical quantities: n(pc) (> 1) and 9(pc) (> 0) values signify the nature of ramification [5] of the percolation clusters (see e.g., [7]). We will discuss if the size exponents (i/s or vg) are affected by the lattice (configurational) fluctuations: if vs(p) is different from v°s or if i/g is different from ifg for p < 1. This question arises naturally from the application of the Harris criterion [8] to the n-vector model in the n — 0 limit [2], when the partition function graphs are all SAWs. A naive application of the criterion to the SAWs suggested [6] a possible disorder induced crossover in the critical behaviour of SAW statistics for any amount of disorder (p < 1). A modified

Introduction

3

analysis [9] of course indicated that a tricky cancellation of the disorder induced crossover occurs at n —¥ 0 limit and the SAW statistics remains unchanged for 1 > p > pc. However, since at pc the dimensionality of the fractal percolating lattice is different from that of the Euclidean lattice, and as the size exponents are determined by the dimensionality, we expect vvsc and vvcc to be different from those (v° and v\ respectively) of pure lattice. Let us now look into the initial indication of the effect of lattice disorders on SAW statistics. Harris [8] and Fisher [10] gave heuristic arguments which suggested that the critical behaviour of a system would be affected by the presence of disorder (quenched and annealed respectively), if the internal energy fluctuation or specific heat of the (pure) system diverges (with positive specific heat exponent a). For quenched disorder, the Harris criterion indicates only the possibility of a crossover for systems with diverging specific heat but it cannot be extended to indicate the new critical behaviour. For annealed disorder, the arguments by Fisher gives also the nature of the new (Fisher renormalised) critical behaviour. These findings, using these arguments, had later been supported using renormalisation group techniques. 2.1. Quenched impurity: Harris criterion Here, as in the percolating systems discussed earlier, the impurities are not in the same thermal bath as the ordering system; rather they are quencehed to zero temperature. The mean square fluctuation in the disorder concentration in a typical volume element £d (£ denoting the thermal correlation length of the ordering system) is then (APr = Y/I-
2] ^ f P (i - P).

(i)

i

Here p is not critical; p » pc. The change in internal energy due to this fluctuation in interaction strength corresponds to a fluctuation ATC in local transition temperature Tc given by f*ATc ~ Ap ~ £ d/2 , o r A T c ~ r d / 2 -

(2) llu

However, since £ ~ | AT\~", where AT = \T - Tc\/Tc, we can express AT as ~ £,~ . For a sharp transition the local fluctuations in Tc is required to be much smaller than this critical temperature interval AT, and consequently we need ATC v"1, or (2 — dv) < 0. Using the hyperscaling relation a = 2 — dv, the above condition becomes a < 0; indicating nontrivial effect (a new different sharp transition with negative value of a, or a smeared transition) due to quenched disorder for (pure) systems with a > 0 [8]. Using the Flory estimate v = v\ = 3/(2 + d), as discussed in the next section, and the hyperscaling relation a°s = 2 — dv°, we get a°s = (4 — d)/(d+2) > 0 for d < 4 for SAWs on pure lattices. This would indicate [6] a disorder induced crossover in the SAW statistics for any amount of disorder at p > pc. However, this indication is not quite correct for the n-vector model in the n —> 0 limit [9]. For a general n-vector model, a similar estimate of transition temperature fluctuation would give £~dATc ~ nAp, since the Hamiltonian contains a sum over all the n components. This gives ATC ~ n£~d/2, compared to the usual AT ~ ^"1/", mentioned above. Hence, in the n -> 0 (SAW) limit, ATC « AT always, irrespective of the a = 2 — dv exponent value. This modified Harris criterion, thus, does not indicate any effect of lattice disorder on SAW statistics (for p > pc) as the

4

B. K. Chakraharti

critical temperature fluctuation never exceeds the temperature interval of the correlation length in an n-vector model in the n —> 0 limit, although the specific heat exponent of the model is positive. 2.2. Annealed impurity: Fisher renormalisation Since the impurities here are in the same thermal bath as the ordering system, one can consider a total Hamiltonian of the system including the impurity. Let the impurity coupling be denoted by A. For such annealed systems, Fisher assumed [10] F(AT, A) ~ F(AT*(AT, A)),

(3) 2 Q

where F denotes the free energy of the system with F(AT*) ~ |AT*[ ~ , a denoting the specific heat of the pure system, and the renormalised temperature interval AT* is an analytic function of its argument AT and A. The impurity concentration p is then given by p ~ dF/dX ~ (dF/dT*)(dT* /dX). Assuming now analytic constraints over p, both sides of the above expression (3) can be expanded and the lowest order terms in AT and AT* on both sides may be compared to give (since both AT and AT* are much less than unity): |AT| ~ |AT*|(1~Q), or |AT*| ~ lATI^ 1 "^ for a > 0

(4)

and | AT* | = |AT| for a < 0. Usual critical behaviour of the annealed system, expressed in terms of AT*, gives in effect the Fisher renormalised exponents QR = — a/(l — a), /3R = (3/(1 — a), 7# = 7/(1 — a) and VR/(1 — a) for the impure system when a > 0 for the pure system. These renormalisation of exponents keeps the scaling and hyperscaling relation a + 2/3 + 7 = 2 and a = 2 — dv unchanged. Using a° = (4 - d)/(2 + d) for SAWs, using the Flory value for u° = 3/(2 + d), one gets yR = i/R ~ 3/2(d— 1) for the Fisher renormalised polymer size exponent (in the annealed random medium), which is often unacceptable (e.g., vf > 1 for d = 2!) [7]. 3. ESTIMATES OF SAW SIZE EXPONENT ON PERCOLATION CLUSTER The above considerations indicate that SAW statistics may have different critical behaviours in disoderd media. We now try to estimate the SAW size exponent v using some approximate theories. 3.1. Flory theory The Flory theory [1,2] estimates the functional form of the free energy F(R) of a chain of size TV and average radius of gyration or end-to-end distance R. For this, it proceeds as follows: For N monomers in a volume of average radius R, the average monomer concentration c = N/Rd. Now the repulsive energy due to monomer-monomer interaction is proportional to c2, and the total repulsive energy Frep ~ c2Rd ~ N2/Rd. From entropic consideration, the elastic energy Fei ~ R2/N. Hence, total energy F(R) ~ (N2/Rd) + (R2/N). Minimising the total energy F with respect to R, we get Rd+2 ~ TV3. This gives R ~ Nv> with vs = u° = 3/(d + 2). This estimate for v\ (= 3/4, 3/5 and 1/2 for d = 2, 3 and 4 respectively) is remarkably accurate [2] compared to the exact

Introduction

5

value (for d = 2) and best estimates [4,5]. This also gives dc = 4 for the upper critical dimension of SAW statistics, where the entropy term Fei wins over the excluded volume term Frep. One can easily see that even a naive replacement of d in this Flory formula by the appropriate fractal dimension (< d) of the percolation clusters indicate a significant change in the SAW statistics; namely the polymer size exponent vvsc on the percolation cluster clearly changes (v^c > u°) [11]. The above method can be extended to the 0-point, where the above two-body repulsion (O(c2)) is exactly balanced by the attractive interaction and three-body interaction (O(c3)) comes into play. Hence, here F(R) ~ {N3/R2d) + (R2/N). Again from energy minimum condition we get R ~ Nve, ug = v*jj = 2/(d+ 1). This correctly gives the upper critical dimension dc = 3 (where Vg = 1/2) for the 0-point statistics. However, the value v% = 2/3 ~ 0.67 in d = 2 is far from the real value (i$ = 4/7 ~ 0.57 [4]). There have been several attempts to extend the above Flory theory to correct for its failure for the vg estimate (in d = 2) and for estimating vs and vg on fractals. We give here a simple and elegant one (cf. [7]) which starts with a functional estimate of the radius of gyration distribution P(R) ~ exp[—F(R)], instead of that for the free energy F(R). The form of the distribution function P{R) of the polymer radius of gyration is given by P{R) ~ exp[-N(Na/R)e

+ (R/Nb)s],

(5)

where /3,5 > 0. The proportionality factors are omitted for simplicity. This form of P(R) ensures that the probability of getting the polymer size R outside the bound Na < R < Nb is exponetially small. In fact, the above form of P(R) assumes that it decays exponentially to zero as R crosses the above bounds. The distribution function is maximum at the most probable size RN when dP(R)/dR\n~RN = 0. If we express the most probable size RN ~ N", then we get, v = (a + K6)/(1 + K), where re = 5//3. Let us first consider the regular lattices: (i) For SAWs in a pure lattice in the high temperature limit (random SAW limit), Nlld < R < N. In this case the two-body interaction term in the Flory free eneregy (F(R) ~ lnP(R)) is ensured by choosing /? = d. Similarly the elastic term in F(R) will come by choosing 5 = 2. In the SAW limit, therefore, we get v = v°s = 3/(rf + 2), the normal Flory estimate for SAW size exponent. (ii) At the #-point, the appropriate bound was suggested to be (cf. [7]) Nxld < R < N"°. Here the three-body term in F(R) is ensured by the choice /3 = 2d and the elastic term again comes from the choice 6 = 2. One thus gets v = i>g = (d + 5)/[(d + 2)(d + 1)]. It gives v% = 7/12 ~ 0.58, compared to the exact value 4/7 ~ 0.57 in d = 2. This d = 2 result for v\ (and also the F(R)) is the same as that obtained from various screening considerations. Also, Vg = 1/2 when three-body term vanishes (becomes independent of N) at d > dc ~ 2.4 in this approxmation [7]. We now consider the percolation clusters: (i) Here in the high temperature limit, the bound on R is JVx/dfl < R < Nlldmi", where ds is the percolation backbone dimension and dmin is the shortest chemical path dimension [5]. K (= 6/p) should incorporate here the spectral (random walk) dimension of the percolation cluster for the elastic energy term and is given by re = dwdmin/dB{dm — dmin), where dw is the random walk dimension on the percolation cluster. One then gets v = vvsc = (dmin + KdB)IdBdmin(\ + K). The same expression has also been reproduced by

6

B. K. Chakrabarti

others using quite different methods (see [7]). This gives the values of vvsc ~ 0.77, 0.66 and 1/2 in d = 2, d = 3 and d > 6 respectively. (ii) For SAWs at the 0-point on the percolation cluster, the appropriate bound is assumed to be (cf. [7]) N1^3 < R < Nv'°. The values of n corresponding to the three-body interaction gives v = vvec — (1 + Kdeffc)/'dg(2 + K). This gives Vgc{>: Vg) ~ 0.68, 0.61 and 1/2 in d = 2, d = 3 and d > 6 respectively. It is to be noted that here also the upper critical dimensionality for the #-point on the percolation cluster shifts to dc = 6.

3.2. Scaling theory We consider now a scaling theory of SAW statistics on percolation cluster. This indeed gives the (exact) result that v\c > v® for d < 6 and that v^c = 1/2 at d > dc = 6. Since there is no new fixed point for SAWs on p > pc cluster [7], we can write R^ ~ A(p)Nv', where the enhancement factor A(p) = 1 for p = 1 and A(p) diverges as p approaches pc. This factor comes because of the swelling of the SAW with lattce dilution (keeping vs unchanged at the pure lattice value UQ). SO, for p > pc, we assume RN^(p-pc)-aNv-

(6)

with a > 0. For p just below pc the SAWs are restricted on the 'incipient infinite cluster' [5], which will force the SAW to move on the cluster within its boundary. Hence there should be no N dependence on R^ and RM will be of the order of the average size of the cluster for such N values. So, for p < pc, SAWs are possible until RN^^^(PC-P)-"P,

(7)

where fp is the percolation correlation length and vv is the corresponding exponent. One can now combine the above limiting results for different regions of p into one consistent theory by constructing a suitable scaling function of the general form (cf. [7]) RN~N»*cY(Nx(p-pc)).

(8)

The scaling function Y(z) (~ z~a for z > 0 and ~ z~Vv for z < 0; z = Nx(p — pc) ) is assumed to be asymptotically defined with power law variations. This gives R^ ~ N"'c at p = pc. From the fitting of the scaling form with the assumed forms for the abovementioned limits, we get x = vvscfvp and a = (y^ — v°)/x. Since a > 0, vpsc > v°s. Of course, since the percolation cluster backbone dimension ds is less than two in d = 4 (de = 2 for d > 6 only [5]), where the fractal dimension ((^s) 1 ) °f t n e ordinary SAW becomes equal to two, they cannot be accomodated on the percolation fractal until they swell, suggesting v\c > vas at least for d = 4 and indicating generally the same for d < 6. An independent estimate of a can be made by node-link-blob model [5] of the percolating cluster and this in turn can give the estimate of vpsc (from the above relation). For p > Pc, in the 'nodes and links' model the infinite percolation cluster (without dangling ends) is a super-lattice where one dimensional channels or 'links' (with occasional blobs in it) meet at the crossing points or 'nodes'. The nodes of the superlattice are separated by a 'crow flying' distance (~ O(£p), fp ~ \p — pcY"v denoting the percolation correlaion length) and also by a chemical distance L (the length of the quasi-one dimensional links), where it is assumed that L (> £p) diverges near pc i.e., L ~ (p — p c )~^. Hence, with

Introduction

7

an appropriate scaling of the variable -RJV as RN/£,P a n d number of steps N as N/L, the relation R^ ~ Nv° becomes R^ ~ {£,PL~"')N~V'. This relation, when recast in the form (6) appropriate for p > pc, gives a = vp — £pi/° F° r d < 6, when the blob exists, Cp depends on the appropriate choice of chemical length L. At d > 6, blobs do not exist and CP = 1 [5]. Also vp = 1/2 for d > 6 where z/f = u° = 1/2. Hence i/J" > i/° for d < 6 and i/Vc = iP = 1/2 for d > 6. This gives the upper critical dimension for SAW statistics on percolation cluster to be six. 4. OUTLOOK Here we have very briefly introduced the (lattice) SAW model of linear polymers, their configurational statistics and the (lattice) percolation model of disodered media. Approximate mean field-like and scaling arguments have been forwarded to indicate that the SAW critical behaviour on disordered lattices, percolating lattice in particular, could be significantly different from those of SAWs on pure lattices. More careful analysis, as we will see in the following chapters, show even more subtle effects of disoder on the polymer conformation statistics. Also, as we will see, such effects are not necessarily confined only to the cases of extreme disorder like percolating fractals. REFERENCES 1. P. Flory, Statistics of Chain Molecules, Cornell Univ. Press, Ithaca, NY, 1969. 2. P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. 3. See e.g., A. J. Guttmann, in Phase Transition and Critical Phenomena, vol. 13, Eds. C. Domb and J. L. Lebowitz, Academic Press, London, 1989, p. 1. 4. B. Nienhuis, Phys. Rev. Lett. 49 (1982) 1062; B. Duplantier and H. Saleur, Phys. Rev. Lett. 59 (1987) 539. 5. See e.g., D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd edition, Taylor and Francis, London, 1992. 6. B. K. Chakrabarti and J. Kertesz, Z. Phys B 44 (1981) 221. 7. K. Barat and B. K. Chakrabarti, Phys. Rep. 258 (1995) 377. 8. A. B. Harris, J. Phys. C 7 (1974) 1671. 9. A. B. Harris, Z. Phys. B. 49 (1983) 347. 10. M. E. Fisher, Phys. Rev. 176 (1968) 257. 11. K. Kremer, Z. Phys. B 45 (1981) 149.

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Statistics of Linear Polymers in Disordered Media Edited by Bikas K. Chakrabarti © 2005 Elsevier B.V. All rights reserved.

Directed polymers and randomness Somendra M. Bhattacharjeea institute of Physics, Bhubaneswar 751 005, India email: [email protected] The effects of two types of randomness on the behaviour of directed polymers are discussed in this chapter. The first part deals with the effect of randomness in medium so that a directed polymer feels a random external potential. The second part deals with the RANI model of two directed polymers with heterogeneity along the chain such that the interaction is random. The random medium problem is better understood compared to the RANI model.

1. DIRECTED POLYMERS A long flexible elastic string, to be called a polymer, has several features of a critical system, showing power law behaviours without much fine tuning [1,2]. An important quantity for a polymer is its size or the spatial extent as the length N becomes large. For a translationally invariant system with one end [z = 0) fixed at origin, the average position at z = N is zero but the size is given by the rms value

=0, (riyi2~N»,

(1)

with v = 1/2, for the free case. Similar power laws can be defined in other properties also. In general, such exponents like the size exponent v define the polymer universality class and these depend only on a few basic elements of the polymer. In addition to the geometric properties, the usual thermodynamic quantities, e.g. free energy (or energy at temperature T = 0), entropy etc., are also important, especially if one wants to study phase transitions. Power laws generally imply absence of any length scale in the problem. Given a microscopic Hamiltonian with its own small length scales like the bond length, interaction range etc, power laws occur only in the long distance limit (large N) for thermally averaged quantities which require summing over all possible configurations. As a result, in the long distance limit, these power laws become insensitive to minute details at the microscopic level, a feature that may be exploited to choose appropriate simplified models to describe a polymer. In thermal equilibrium, the Boltzmann distribution ultimately determines the macroscopic behaviour. In most cases one may avoid the issue of probability distribution but instead may concentrate only on the first few, may be the first two, moments or cumulants. For example, one needs to know the average energy, entropy etc and the various response 9

10

S, M. Bhattacharjee

functions which come from the width of the distribution. Thermodynamic descriptions do not generally go beyond that. In random physical systems, one encounters an extra problem. If the randomness is not thermalizable ("quenched"), any quantity of interest becomes realization dependent. As a result, an additional disorder averaging has to be done over and above the usual thermal averaging for each realization. It is therefore necessary to know if and how this extra averaging alters the behaviour of the system. Critical-like systems [3] become a natural choice for studying the effect of quenched randomness because it is hoped that at least some of the properties would be insensitive to the minute details of the randomness. Since for a critical system, the influence of the randomness on a long scale is what matters, it transpires that the critical behaviour will be affected if the disorder is a relevant variable. In the renormalization group language, a coupling is classified as relevant, irrelevant or marginal, if, with increasing length scale, it grows, decays or remains invariant, because the contribution of a relevant quantity cannot be ignored at long distances even if it is numerically small to start with. With relevant disorder, there is the obvious possibility of a change in the critical properties (e.g. new set of critical exponents). More complex situations may include emergence of new or extra length scales. One may recall that approach to criticality, most often, is described by a diverging length scale. Developing a description of the system in terms of this large length scale only goes by the name of scaling theory. Emergence of any new or extra length scale would then alter the corresponding scaling description. In case it is possible to change the nature of disorder from relevant to irrelevant (say by changing temperature), then a phase transition would occur which will have no counterpart in the pure problem. For the disorder-dominated phase, on a large scale, there are possibilities of rare events (see Appendix A) which necessitates a distinction between the average value and the typical (e.g. most probable) value. In such situations higher moments of the quantity concerned become important. These are some of the aspects that make disorder problems important, interesting and difficult.

Figure 1. (a) A random walk in d dimensions with z as the variable along the contour of the polymer i.e. giving the location of the monomers, (b) Directed polymer on a square lattice. A polymer as of (a) can be drawn in d + 1 dimensions. This is like a path of a quantum particle in nonrelativistic quantum mechanics, (c) A situation where both the transverse space (r) and z are continuous, (d) The directed polymers on a hierarchical lattice. Three generations are shown for 4 bonds, (e) A general motif of 26 bonds.

Directed polymers and randomness

11

The problem of a polymer in a random medium was initiated by Chakrabarti and Kertesz [4,5] with the application of the Harris criterion. This problem has enriched our overall understanding of polymers and random systems in general, but still a complete understanding remains elusive. Not surprisingly, the look out for simpler problems that capture the basic essence of the original complex system gained momentum. In this context, directed polymers played a very crucial role. Let us define the problem here. Consider a polymer where each monomer sees a different, independent, identically distributed random potentials. Geometrically this can be achieved if the monomers live in separate spaces. One way to get that is to consider the polymer to be a d + 1 dimensional string with the monomers in d dimensional planes but connected together in the extra dimension. As shown in Fig. 1, this is a polymer which is directed in one particular direction. Hence the name directed polymer [6-9]. For a directed polymer, the size would now refer to the size in the transverse d-directions and so Eq. (1) refers to the transverse size as the length in the special z-direction increases For long enough chains, it is this size that matters and enters the scaling description. The significance of directed polymer lies in the fact that the pure system is very well understood and exactly solvable in all dimensions while the random problem can be attacked in several different ways, a luxury not affordable in most situations. Two types of randomness can arise in the context of directed polymers. One type would involve imposition of a random external potential (random medium problem). In the second type, the interaction (between say two chains) is random (RANI model). In the random medium problem, the random potential would like to have a realization dependent ground state which may not match with the zero-field state. In the RANI model, the randomness in the interaction may lead to a change in the phase transition behaviour shown by the polymers. These two classes are discussed separately. 2. OUTLINE We first consider the random medium problem and then the random interaction (RANI) model. In both cases, disorder turns out to be marginally relevant though at two different dimensions. The quantities of interest in a disordered system are introduced in Sec. 3. For the random medium problem, the question of relevance of disorder, the size exponent v and the free energy fluctuation exponent 9 are discussed in Sees. 4 and 5. Sec 4 deals with the renormalization group (RG) for the moments of the partition function and also the special Bethe ansatz results for the 1 + 1 dimensional problem. A Flory approach and scaling ideas are also summarized there. Sec. 5 deals with the RG approach for the free energy via the Kardar-Parisi-Zhang equation. The behaviour of the overlap especially near the transition to strong disorder phase may be found in Sec. 6 . We briefly mention the unzipping behaviour in presence of a force and summarize some recent results for the pure case in Sec. 7. More on unzipping of a two chain system may be found in Sec. 9.4. These results and scaling arguments are then used to visualize the nature of the ground state in Sec. 8. Various aspects of the RANI model can be found in Sees. 9 and 10. The question of relevance, and annealed versus quenched disorder in multi chain system are analyzed in Sec. 9. The two different types of randomness or heterogeneity on hierarchical lattices are studied in Sec. 10. In the last part of this section, one may

12

S. M. Bhattacharjee

find the general validity and extension of the Harris criterion as applied to this polymer problem. Various technical issues are delegated to the Appendixes. An example of rare events is given in App. A. A short review of the pure polymer behaviour can be found in App. B. The issue of self-averaging and some recent results about it for disordered systems are surveyed in App. C. The renormalization group approach to polymers as used in Sec. 4.2.1 is spelt out in App. D in the minimal subtraction scheme with dimensional regularization. The momentum shell RG approach for the free energy is dealt with in App. E. All the possible flow diagrams are reviewed in App. F. A short introduction to the transfer matrix approach for the directed polymer problem is given in App. G. The exact RG for the RANI model can be found in App. H. On Notation and convention: To avoid proliferation of symbols, we reserve the symbol f to denote an arbitrary or unspecified function, not necessarily same everywhere. By a sample or a realization we would mean one particular arrangement of the random parameters over the whole system. For a thermodynamic (infinitely large) system the sample space is also infinite. Sample averaging is denoted by [...]av while thermal averaging is denoted by (...). The Boltzmann constant is set, most often, to one, k& = 1. "Disorder" and "randomness" will be used interchangeably. 3. HAMILTONIAN AND RANDOMNESS By definition, a directed polymer is defined in D dimensions out of which one direction is special that represents the sense of direction of the polymer. It is then useful to show that explicitly by writing D = d + 1 where d is the dimension of the transverse space. One may consider various possible models of the underlying space as shown in Fig. 1. One may consider a lattice (square lattice in the Fig. lb) with the polymer as a random walk on the lattice with a bias in the diagonal ^-direction, never taking a step in the —z direction. The length of the polymer is then the number of steps on the lattice. Instead of a lattice model, one may consider continuum where both the space and the z-direction are continuous as shown in Fig. lc. The polymer itself may consist of small rods or bonds whose length provides us with a small scale cut-off. In many situations, it helps to consider polymers without any intrinsic small scale cutoff. Quite often it is useful to consider very special lattices, e.g. hierarchical lattices as shown in Figs. Id, and le, because of the possibility of exact analysis. Here one starts with a bond and then replaces the bond iteratively by a predetermined motif ("diamond" of 1b bonds) and the process can be iterated ad infinitum. This is a lattice with a well-defined dimension (see Sec. 10) but unfortunately without any proper Euclidean distance. Consequently geometric properties are not useful here.

Directed polymers and randomness

13

The effective dimension of the lattice is deg = (In 26)/In 2, if there are 26 bonds per motif. A directed polymer can be taken as a random walk on this lattice starting from say the bottom point, going up, and ending at the top end.

3.1. Pure case Taking the polymer as an elastic string, one may define a Hamiltonian

(21

*-!*/*(£)'

which gives a normalized probability distribution of the position vector r at length z from the end at (0,0) p ( r > z) =

Wzyii

(Kd/k*T = i)-

e rV2z

"

(3)

Here k-a is the Boltzmann constant and T is the temperature. One can even write down the distribution for any two intermediate points (r.;,;Zj) and (TJ,ZJ) as G(F

" 2 ' | r i ' Z i ) = [2,(zf - *)] 2, there is a fixed point at u* = |e| where e = 2 — d. For u < u*, the disorder strength goes to zero and one recovers a "pure"-like behaviour. This is a weak disorder limit. But, if u > u* the disorder is relevant. Based on the fixed point analysis, we conclude, as already mentioned, that disorder can be relevant depending on the dimensions we are in (i.e. the value of d) and temperature or strength of disorder. In particular, one finds

Directed polymers and randomness

23

1. A disorder-dominated or strong disorder phase for all temperatures for d < 2. 2. A disorder dominated or strong disorder phase at low temperatures for d > 2. For d > 2, one sees a phase transition by changing the strength of the disorder or equivalently temperature for a given A. This is an example of a phase transition induced by disorder which cannot exist in a pure case. It is to be noted that the phase transition (the unstable O(\e\) fixed point) occurs because of the positive u2 term in the flow equation of Eq. (30). At d = 2, u is marginal (no L dependence) but renormalization effects lead to an eventual growth. Such parameters are called marginally relevant. Any marginally relevant variable will produce an unstable fixed point, and hence a phase transition, in dimensions higher than the dimension in which it is marginal. A general statement can then be made: Disorder is expected to produce a new phase transition if it is marginally relevant at its critical dimension. The new phase transition (a critical point) is to be characterized by its own set of exponents. An important quantity is the length scale behaviour. The flow equation around the fixed point for d > 2 shows that one may define a diverging "length-scale" associated with the critical point as £ ~ \u-u*\-(,

with C = l / | 2 - d | .

(32)

In the critical dimension d = 2, there are log corrections. In the weak disorder phase where the disorder is irrelevant, [lnZ] av « ln[Z] av , and therefore one may put a bound on the transition temperature Tc for a lattice model as Tc > TA as defined below Eq. (25). Attempts were made to determine £ by numerical methods and verify the RG prediction. However, the results from specific heat [15] and size calculations [16] agree neither with each other nor with the RG result of Eq. (32). This remains an open problem. We come back to the strong disorder phase in Sec. 5.6.

4.3. Bethe ansatz and 9 For the directed polymer problem, a mapping to a quantum problem helps in the evaluation of \Zn\w at least in d = 1. For a Gaussian distributed, delta-correlated disorder, [Zn]av corresponds to the partition function of an n-polymer system with the Hamiltonian given by Eq. (26). Noting the similarity with the quantum Hamiltonian with z playing the role of imaginary time, finding TV"1 In [Zn]m for N —> oo is equivalent to finding the ground state energy £ of a quantum system of n particles. This problem can be solved exactly only in one dimension (d = 1) using the Bethe ansatz [23,24]. This gives the ground state energy as E = -K(n-n3)

in d = 1,

(33)

which gives 8 = \,

(34)

from Eq. (21) As we shall see below, this implies v = 2/3 so that the polymer has swollen far beyond the random walk or Gaussian behaviour. What looks surprising in

24

S. M. Bhattacharjee

this approach is that there is no "variance" (2nd cumulant) contribution. It is just not possible to have a probability distribution whose variance vanishes identically. This is a conspiracy of the N —> oo limit inherent in the quantum mapping and the value of the exponent 9 that suppressed the second cumulant contribution (see Eq. (21)). 4.3.1. Flory approach Using the quantum analogy, we may try to estimate the ground state energy in a simple minded dimensionally correct calculation based on the assumption of only one length scale. Such an approach generally goes by the generic name of "Flory approach". The elastic energy is like the kinetic energy of quantum particles which try to delocalize the polymers (random walk) while the attractive potential tries to keep the polymer together. For n polymers there are n(n — l)/2 interactions. We take the large n limit so that if the particles are bound in a region of size R, the energy is (using dimensionally correct form with R as the only length scale)

*-£-=£

W

which on minimization gives E ~ n3 consistent with the Bethe ansatz solution. At this point we see the problem of the replica approach if the limit is taken too soon. Since our interest is eventually in n —> 0, we could have used in this argument the linear term of the combinatorics. That would have made energy "extensive" with respect to the number of particle and replaced the disorder-induced attraction by a repulsion (note the negative sign). The end result would however have no n 3 dependence. This is a real danger and any replica calculation has to watch out of these pitfalls. Quite strangely we see that the correct answer comes by taking n —> oo first and then n 4 0 or, probably better to say, by staying along the "attractive part" of the interaction only. 4.3.2. Confinement energy Suppose we confine the polymer in a tube of diameter D. This is like the localization length argument used to justify the energy in the quantum formulation. The polymer in the random medium won't feel the wall until its size is comparable to that, D ~ Nfi which gives the length at which the polymer feels the wall. Elastic energy of a blob is D2/N0. But because of the tube, the polymer will be stretched in the tube direction. One may then consider the polymer as consisting of free N/No blobs aligning with the force, so that the energy is 2

N

=N

wowr {i3 D^-

(36)

This gives the known form 1/D2 used in the quantum analogy, Eq. (35), ( and consistent with dimensional analysis) but for v = 2/3, this gives 1/D. A cross-check of this comes from the energy of a blob. Each blob has the fluctuation energy N$ and so free energy per unit length F/N ~ NQ/N0 ~ D~2{-l~v^v.

Directed polymers and randomness

25

5. ANALYSIS OF FREE ENERGY: SPECIALTY OF DIRECTED POLYMER Another unique feature of this directed polymer problem is that there is a way to study the average free energy and implement RG directly for the free energy bypassing the n —> 0 problem of the replica analysis completely, giving an independent way of checking the results of replica approach. 5.1.

Free energy and the KPZ equation For a polymer, the partition function satisfies a diffusion or Schrodinger-like equation. This equation can be transformed to an equation for the free energy F(T,Z) = —Tin Z(T, Z). This is the free energy of a polymer whose end point at z is fixed at r. To maintain the distance fixed at r a force is required which is given by g = — VF. If we want to increase the length of a polymer by one unit, we need to release the constraint at the previous layer (think of a lattice). The change in free energy would then depend on the force at that point, and of course the random energy at the new occupied site. The change dF{r)/dz being a scalar can then depend only on the two scalars V g and g2. A direct derivation of the differential equation for the free energy shows that these are the three terms required. The differential equation, now known as the Kardar-Parisi-Zhang equation [7], is ^

= 2K^F--{VF)>

+ V{r,z).

(37)

If we can solve this exact equation and average over the random energy rj, we get all the results we want. One may also write down the equation for the force in this "fixed distance" ensemble as

vag

v +v

i = ^ -i«- « ^'^

(38)

This equation is known as the Burgers equation. 5.2.

Free energy of extension: pure like We want to know the free energy cost in pulling a polymer of length N from origin (where the other end is fixed) to a position r. For the pure case, the free energy follows from Eq. (3) (with K inserted) as F(v, N) - F(0, N) = \K ^ . (pure)

(39)

The probability distribution for the free energy can be obtained by choosing g = Kdrj^/N in Eq. (13) as. P ( F M ) = P(F(0) + ^ ) .

(40)

One then gets the surprising result,

[F(T, N) - F(0, JV)L = iKJf-

(disorder)

(41)

26

S. M. Bhattacharjee

Therefore, on the average the stretching of a chain is pure-like (elastic) with the same elastic constant though the fluctuation is anomalous (9 / 0). This is analogous to the pure-like result for the correlation function, Eq. (16). These results have a far reaching consequence that in a renormalization group procedure, the elastic constant must remain an invariant (independent of length scale). As we shall see, this invariance condition puts a constraint on v and 9, making only one independent. 5.3. RG of the KPZ equation To analyze the nonlinear KPZ equation, an RG procedure may be adopted. This RG is based on treating the nonlinear term in an iterative manner by starting from the linear equation. This is a bit unusual because here we are not starting with a "Gaussian" polymer problem, rather, a formal linear equation[25] that does not necessarily represent a polymer. Leaving aside such peculiarity, one may implement the coarse-graining of RG to see how the couplings change with length scale. 5.3.1. Scale transformation and an important relation Under a scale transformation x —¥ bx, z —¥ bxl"z, and F —$ belvF, the randomness transforms like A —» &-<J-(1/") A. This transformation done on Eq. (41) shows that for K to be an invariant (no 6-dependence) we must have, 9 + \ = 2v.

(42)

This is trivially valid for the Gaussian pure polymer problem but gives a relation between the free energy fluctuation and the size of the polymer. This is borne out by the intuitive picture we develop below. This relation gives the size exponent v = 2/3 in d = 1. The equation in terms of the transformed variables is then ^ = ^&(i- 2 "V" V2F - l-b^-2"^!" uz 2K zK

(VF)2 + &(i-*-2»)/«' ^

z)

(43)

The 6-dependent factors can be absorbed to define new parameters, except for K. The temperature however gets renormalized as T —> Tb^1'2"^". Its flow is described by the flow equation L—- OLj

~

T (to leading order)

(44)

V

For v > 1/2, T(L) — 0. The disorder dominated phase is therefore equivalent to a zero temperature problem. In other words, the fluctuation in the ground state energy and ground state configurations dominate the behaviour at low temperatures in situations with v > 1/2. It is this renormalization that was missing in Sec 4.2.1. 5.4. RG flow equation The nonlinearity (or g2) contributes further to the renormalization of the temperature through the appropriate dimensionless variable u = (KA/T3)L2~d (same as in Eq. (30). As in the previous section, the important flow equation is for this parameter u (upto constant factors). The renormalization of temperature acquires subtle d-dependence that

Directed polymers and randomness

27

introduces a new element in the flow equation. Some details may be found in Appendix E. We quote the flow equation here as T du

L

,„

,.

Id — 3 ,

= {2 d)u+

dL

-

u

^r -

,,_. (45)

General cases of such a flow equation are discussed in Appendix F. One immediately sees a major difference with the flow equation Eq. (31) for d = 1. Because of the change of sign of the quadratic term in Eq. (45), there is now a fixed point for d = 1 as shown in Fig. 6(b). This flow equation does not behave properly in a range d € [1.5, 2) but that is more of a problem of implementation of RG than directed polymer per se, and so, may be ignored here. Note also that no extra information can be obtained from Eq. (45) for d > 2 other than what we have obtained so far in Sec. 4, namely the existence of a critical point. This approach however has the advantage of getting the renormalization of temperature by u. Some details of this RG is given in Appendix E. Eq. (44) then gets modified to dT (\-2v , 2-d \ dL = ("TT- + ^d-u) T-

L

(46)

If we now demand scale invariance at a fixed point of u, we can determine the exponent v. At d = l , the stable fixed point u* — 2 from Eq. (45) then gives the exact exponents at d=l

"=l>0=\>

(47)

in agreement with the Bethe ansatz results mentioned above. To get 9 the exponent relation Eq. (42) (from invariance of K) has been used. The RG results for d = 1 are expected to be exact. 5.5. Critical point for d > 2 For d > 2, the unstable fixed point is u* = O(\e\), (d = 2 — e). This gives at the critical point v = 1/2 + O(e2) indicating the possibility of v = 1/2, and 9 = 0

(48)

to be exact. One may argue [26] for 9 = 0 in the following way. At T = Tc, thermal fluctuation enables the polymer to get out of the trap set by the random potential ("ground state"). Just above the critical point, on a scale determined by the correlation length of the critical point, the random potential scaling is set by be with the value of 9 at Tc. As T —¥ Tc+, the length scale diverges and therefore the relevant energy scale would also grow with the same exponent. However a critical point implies the energy scale to be O(TC) which is finite. These can be reconciled if 9 = 0 at the critical point. One then gets v = 1/2. Though this is the same as that of a Gaussian polymer, we shall see later that the polymer has extra non-Gaussian features.

28

S. M. Bhattacharjee

5.6. Strong disorder phase for d > 2 Unlike the strong disorder phase at d=l, the absence of any fixed point for the strong disorder phase for d > 2 in this approach forbids quantitative results about the phase itself. In addition, the behaviour of the strong disorder phase in d = 1 can be obtained by various methods. Unfortunately, there are few concrete results in higher dimensions d > 2. Most reliable values of the exponents come from various numerical approaches based on the KPZ equation. A recent estimate for d = 2 is [17] 1/v = 1.67

0.025, and 9 = 0.229

0.05.

(49)

Numerical studies indicate that v decreases as d increases. A question arises about the existence of an upper critical dimension d = ducD such that v = 1/2 for d > rfucD- For example, higher loop contributions in the RG of Sec. 4.2.5 show singularities at d = 4 which could indicate d = 4 as another critical dimension. Various analytical approaches [6,18,19] suggest ducD = 4, or even nonintegral dimension [20]. But numerical simulations and other arguments [21,22,9] suggest rfucD = oo. This issue is yet to be resolved. The fact that the size exponent v (often called wandering exponent) is different from 1/2 has important implications in various applications, especially for flux lines in superconductors. For example, confinement of a flux line in presence of many other flux lines would lead to a steric repulsion[27] (similar to the confinement energy in Sec. 4.3.2) and the interaction of the vortices may lead to an attractive fluctuation induced (van der Waals type) interaction[28]. 6. OVERLAP In a replica approach, occupancy of different ground states may be achieved by "replica symmetry breaking" (i.e. various replicas occupying various states) but the difficulty arises from the n —> 0 limit. In the case of directed polymer, we have argued that the degenerate states occur only rarely and therefore the effect of "replica symmetry breaking" if any has to be very small. This is why the Bethe ansatz gave correct results without invoking replica symmetry breaking. The method to compute the overlap was developed by Mukherji [29]. By introducing a repulsive potential v J dz5(ri(z) — ^(z)) for the two polymers in the same random medium,

pN

N

/

dzr](T2{z),z)+

dz v0 5{T12(Z)). Jo The free energy F(ri(z),i2(z), z,v) satisfies a modified KPZ type equation

(50)

-Q-Z = E \^2iF

- ^( v ^) 2 +"(*> z )) +v5^ -r^)-

( 51 )

which can be studied by RG. The mutual interaction has no effect on the single chain behaviour but the interaction gets renormalized. The flow equation for the dimensionless

Directed polymers and randomness

29

parameter u of Eq. (45) remains the same. The exponent relation of Eq. (42) also remains valid. The interaction gets renormalized as

where v is in a dimensionless form. For the pure problem (0 — 0, v = 1/2) this reduces to the expected flow equation of Eq. (31) for repulsive interaction {u —> —u). For overlap one needs only the first order term because we need v —> 0. The overlap can be written in a polymer-type scaling form q = Ns f(vN~'i"/), where E = 9 — 4>f — 1. The above RG equation for v shows that the exponent E = 0 at the stable fixed point for u of Fig 6(b) at d = 1. However, E < 0 at the transition point for d > 2. This means that the overlap vanishes at the transition point from the strong disorder side as q ~ \T - Tc\™. This approach to overlap can be extended to m-chain overlaps also, which show a nonlinear dependence on m at the transition point [30]. This suggests that eventhough the size exponent is v — 1/2 Gaussian like, there is more intricate structure than the pure Gaussian chain. Overlaps of directed polymers on trees have been considered in Ref. [31]. A case of cross-correlation of randomness (each polymer seeing a different noise) has been considered by Basu in Ref. [32].

7. UNZIPPING: PURE CASE Unzipping was first considered in the context of DNA [10]. However the same ideas play a role here. Let us consider a pure case of a directed polymer with one end fixed at origin and with an attractive interaction with a line at T(Z) = 0 (instead of being in a random medium). The Hamiltonian for a delta-function interaction can be written as

H=d K dz

2 [ ^)-v[dz

s

^

(53)

very similar in form with Eq. (13) except here we have an attractive interaction instead of a random medium. For the zero force case, there is a critical unbinding transition at v = vc. For d < 2, vc = 0. The pulling force would like to align the polymer in the direction of the force while the interaction would like to keep the polymer attached to the rod. At zero temperature the unzipping transition takes place at a force where the binding energy is compensated by the force term. Upto a geometric factor a, this is given by Nv = agN. At nonzero temperature, the entropic effects are to be taken into account, which may be done by using the quantum analogy. The problem can be mapped on to a quantum Hamiltonian, albeit non-hermitian, for a particle of co-ordinate r Hq(g) = ~(p + ig)2 + V(T),

(54)

in units of h(= ISBT) = 1 and mass = 1, with p as momentum. For long chains ./V —> oo the free energy is the ground-state energy of this non-hermitian Hamiltonian. A phase

30

S. M. Bhattacharjee

transition takes place whenever the ground state is degenerate. The analysis done in Ref. [10] shows that if the ground state energy ( i.e. the binding energy of the polymer per unit length) is EQ, then the critical force is given by 5c

= 2v / ^H^-^l 1 / | 2 - d | .

(55)

where the v-dependences of Eo close to vc, for general d, is used. In fact if the bound state has extensive entropy, then there is a possibility of a re-entrance at low temperatures (see Sec. 9.4. This however is not possible in this case in hand. 8. NATURE OF GROUND STATES AND EXCITATIONS Powered by the quantitative estimates of the free energy fluctuation and size exponents, we now try to generate a physical picture. 8.1. Rare events We have seen that there is a low temperature region (in lower dimensions for all T) where randomness results in a new phase but the response to an unzipping force is the same as for the pure system. For the pure case as N —» oo the width of P(rjv, N) increases. Hence the increase of CT with N. With randomness, for T —> 0 we need to look for the minimum energy path. Let us suppose that there is a unique ground state, i.e. E(TN) or F(TN) is a minimum for a particular path. This tells us that as the temperature is changed, T still low, the polymer explores the nearby region so that the probability distribution gains some width which is determined by the thermal length. Susceptibility would be the width of the distribution and this is independent of N. This cannot satisfy the relation given by Eq. (42). If we invoke the the unzipping argument, then we need to exert a force exceeding the critical force to take the polymer out of the bound state and so the response to a small force (g —> 0) would be insignificant. The situation will not be any better even on averaging over the random samples if every sample has a unique ground state. However, it may happen that most of the samples have unique ground-states but once in a while (rare samples) there is more than one ground state which happens to be far away from each other. Suppose there are such rare samples, whose probabilities decay as N~K, where the paths are separated by TV", then the contribution to the fluctuation from these samples would be N2v~K. In case K = 2v — 1, we get back the exact result. The relation of Eq. (42) tells us K = 6. The rare events control the free energy fluctuation. From the unzipping point of view, the threshold in such rare cases is at zero force because a small force can take a polymer from one ground state to another one, gaining energy in the process. Following Ref. [33], one may argue that the gain in energy from the force should be similar to the energy fluctuation. Assuming a scaling of the force, g ~ gN°, then gNaNv ~ Ne which gives a = Q - v = 1/3 in d = 1. This argument implies that if the average stretching is proportional to g and to N, i.e., [ < r > ] av ~ gN, then one should get a linear plot if [ < r > ]W/N11 is plotted against gNv~e. The surprising feature is the sample dependence of such a plot. For a directed polymer, these quantities can be obtained by a transfer matrix calculation which is exact for a given sample and finite N. Some details on the transfer matrix approach are given in Appendix G. As shown in Fig.

Directed polymers and randomness

31

7, one sees steps with an overall linear dependence. The susceptibility within a flat step is zero as seen in the plot of the fluctuations.

Figure 7. (a) Plot of < r > /N" vs gNv~B for d = 1. Two different values of N and in different realizations of disorder, (b) Corresponding fluctuation in position. From Ref. [34]. What we see here is that though the average behaviour is the same as that of the pure system, the underlying phenomenon is completely different; the average thermal response is determined by the rare samples that have widely separated degenerate ground state and the probability of such states also decays as a power law of the chain length. This picture also shows the ensemble dependence. What we have discussed is the fixed force ensemble. In the fixed distance ensemble, if we keep the end point fixed at r and try to determine the force to maintain it at that point, then by definition, the force comes from a small displacement around r. Such small displacements will never lead to the big jumps that ultimately contribute to the average susceptibility. This difference in behaviour in the two ensembles is one of the important features of quenched randomness. In a given sample elastic energy ~ r2/N ~ N2"'1. The pinning energy would also grow with length say as Ne. We see, 9 = 2v — 1 = 6. One way to say this is that the sample to sample fluctuation and the energy scale for a given sample are the same. These results can now be combined for an image of the minimum energy paths. If the end points at z = N are separated by r JV", the two paths explore independent regions and they need not meet. This picture (Fig.8) is often alluded to as the river-basin network.

8.2. Probability distribution For a pure polymer, the probability distribution of the end point is Gaussian but it need not be so for the disordered case. One way to explore the probability distribution is to study the response of the polymer as we take it out of its optimal or average position, e.g.

32

S. M. Bhattacharjee

Figure 8. Various paths for various locations of the end point.

by applying a force. In a previous section we used the fixed distance ensemble where the end point was kept fixed and we looked at the force g required to maintain that distance (see Eq. (38)). Here we consider the conjugate fixed force ensemble. 8.2.1. Response to a force Let us apply a force that tries to pull the end of the polymer beyond the equilibrium value r ~ RQ. In equilibrium, the average size R or extension by the force can be expressed in a scaling form R = Rof(^).

(56)

This is because for zero force one should get back the unperturbed size while the force term may enter only in a dimensionless form in the above equation where the quantities available are the size RQ and the thermal energy. For small g, linearity in g is expected. This requirement gives R = R0Y9

(*B = 1),

(57)

R is not proportional to N if v ^ 1/2 (_R0 ~ N"). The polymer acts as a spring with T/R% as the effective spring constant.

8.2.2. Scaling approach Let us try to develop a physical picture and the corresponding algebraic description (called a scaling theory). The polymer in absence of any force has some shape of characteristic size RQ. The force stretches it in a way that it breaks up into blobs of size £g = T/g. For size < £9 the polymer looks like a chain without any force and these blobs, connected linearly by geometry, act as a "new" polymer to respond to the force by aligning along it. We now have two scales RQ and £s = T/g. A dimensionless form is then

R=R0f(^y^g

(58)

Directed polymers and randomness

33

Now each blob is of length Ng so that £9 = Ng and there are iV/iV9 blobs. We therefore expect ll i

R=%-gt, = Ntl-V =

"' "

(59)

This gives a susceptibility \ = dR/dg ~ g^-2")!" . 8.2.3. Probability Distribution Let us try to get the susceptibility of Eq. (59) in another way. Let us assume that the probability distribution for large R is P(r)~exp(-(r/flo)«).

(60)

The entropy is given by S(r) = — In P. The free energy in presence of a force which stretches the polymer to the tail region is given by F = T(r/R0)s - gr.

(61)

This on minimization gives

By equating this form with Eq. (59), we get

d

(63)

=lb

and

P(i?)~exp - ^ — j

.

(64)

For v = 1/2 we do get back the Gaussian distribution. The above analysis, done routinely for polymers, relies on the fact that there is only one length scale in the problem, namely, the size of the polymer. If we are entitled to do the same for the disorder problem, namely only one scale, Ro ~ N", matters, then the blob picture goes through in toto. The chain breaks up into "blobs" and the blobs align as dictated by the force. Each blob is independent and the polymer inside a blob is exploring its environment like a directed polymer pinned at one end. The probability distribution is therefore given by Eq. (64) which for d = 1 is P{r) ~ exp(-|r| 3 /iV 2 )

(d = 1).

(65) 2

If we use the relation AF = F(x,N) - F(0,N) ~ x /N, then the above probability distribution can be mapped to the distribution of the free energy as

P (f) .« p [_(^) iw -"]. exp (_^!)

(d = 1).

This has been tested numerically [6]. See, e.g. Ref. [35] for more recent work.

m

34

S. M. Bhattacharjee

9. RANDOM INTERACTION - RANI MODEL So far we have been considering the problem of random medium. A different situation arises if there is randomness in the interaction of polymers. This is the RANI model [36,37]. Consider the problem of two directed polymers interacting with a short range interaction as in Eq. (51) but take v to be random. Such problems are of interest, especially in the context of DNA where the base sequence provides heterogeneity along the chain. In this DNA context, the randomness can be taken to be dependent only on the z coordinate and not on others like the transverse position r. It can be written as #int = [ dz v0 [1 + b{z)] 6{ri{z) - r2{z)), (67) Jo where the randomness in introduced through b(z). We take uncorrelated disorder with a Gaussian distribution [b(z)]n = 0, [b(Zl)b(z2)lv = A6(Zl - z2).

(68)

This would correspond to uncorrelated base sequence of a DNA problem. The full Hamiltonian can be written as

where Tt(z) is the d-dimensional position vector of a point of chain i at a contour length z, and 1:12(2) = ri(z) —1:2(2). Though written for a general potential V(r), we shall consider only short-range interaction that, when convenient, can be replaced by a 5-function.

9.1. Annealed case: two chain As expected, the average partition function (annealed averaging) would correspond to a pure-type problem. This however is not the case always as we see in Sec. 9.3 for more than two chains. A straightforward averaging of Z — J Dr\ DT-I exp(—H) using the probability distribution of Eq. (68) defines an effective Hamiltonian ?4ff such that

\Z\W = j Dr, Dr2 expi-Ha),

(70)

and it is given by

ft* = \ [ dzj^ {^>j

+ v0 j " dz V(v12(z)) - ^ j " dz V\vu{z)).

It appears from the above expression of the effective Hamiltonian that an attraction is generated between the two chains. Now, since any short range potential under renormalization maps onto a 8 function potential, we can take the "minimal" effective Hamiltonian for (Z) as

(71)

Directed polymers and randomness

35

where v0 is the reduced coupling constant which takes care of the attraction described earlier. We believe that the large length scale properties as described by Eq. (72) is same as that of Eq. (71). If necessary, we can restrict the strength of the disorder so that VQ, which represents the effective coupling between the two chains, is positive (i.e. repulsive interaction). Now the problem reduces to a relatively simple situation where the two chains interact with a pure nc-,

(81)

with r = 0.71 0.02. This exponent is independent of temperature but depends on b. As mentioned, in the site version there are special sites with large degree. Whether such vertices lead to a Griffiths' type singularity needs to explored. Griffiths singularity in the context of random interactions has recently been discussed in Ref. [49].

Directed polymers and randomness

39

10.2. Bond randomness: Problem with Harris criterion

According to the Harris criterion, disorder is irrelevant at a critical point if the pure specific heat exponent a < 0. We show that a more general criterion is required for the directed polymer problem [46]. We place two interacting polymers on a hierarchical lattice. They start from one end (bottom) and meet at the other end (top). See Fig. 1. There is an attractive interaction —v(v > 0) if a bond of the lattice is shared by two polymers. As in real space, here also the polymers undergo a binding unbinding transition for b > 2. Randomness is introduced by allowing the interaction energy to be random on each and every bond. The first model, model A, has independent random energy on all the 26 bonds. The randomness in the second model, model B, is taken only along the longitudinal direction so that equivalent bonds on all directed paths have identical random energy. Model B is a hierarchical lattice version of the continuum RANI model. The pure problem can be solved easily by a Real space renormalization approach where one needs only the renormalization of the Boltzmann factor y = exp(v/T). Let yn,Zn, and En be the renormalized weight, partition function and energy at the nth generation. By decimating the diamonds the recursion relations are given by yn_i = (y*+b-l)/b, Zn+1

(82)

2

= (Z n + b-l)/b,

(83)

2 ZnEn -fc-n+i

=

(°4J

T -^ o Ai+i

The two fixed points of the quadratic recursion relation, Eq. (82), are 1 and 6 — 1 of which the larger one is the unstable fixed point representing the transition point. Since y > 1 there is a transition at yc = b— 1 only for b > 2. The other fixed point y* = 1 corresponds to the high temperature limit. The length scale exponent £, and the specific heat exponent, a, can be obtained from Eqs. (82)-(84) as C=

W-2l)/ft]'anda

= 2

-C-

(85)

Note that a < 0 for b < 2 + \/2. It is clear that hyperscaling holds good with d = 1 and not the effective dimension deg of the lattice. It is also gratifying to observe that the same u and a describe the finite size scaling form of En. For the disordered case, the recursion relation for the Boltzmann weight can be written as

»„_! = r V u ) i £ 2 ) + v(2l)yT] + ...y(61)yf») + (& -1)&-\

(86)

where y£ is the Boltzmann weight in the nth generation for the disorder on the upper (j = 1) or lower (j = 2) part of the ith branch as in Fig. 1. To understand the effect of the disorder at the pure critical point we introduce a small disorder yn = yc + £n The average of the disorder, [e]av, acts like the temperature as it measures the deviation from the pure critical point. The second moment is the measure of disorder. In principle, one

S. M. Bhattacharjee

40

should look at the variance of e , but at the pure critical point the variance would be the same as the second moment. Since our motivation is to find the flow of the disorder at the pure fixed point, we need only study the first two moments, starting from a finite and small variance. The crossover exponent for the disorder is defined through the homogeneity of the singular part of the free energy in terms of the scaling fields fi-i (temperature) and /J.2 (disorder). Under decimation, the free energy behaves /sing(/ii,M2)=M?"af(M2/Mf)

(87)

defining <j> which can be obtained from the RG recursion relations for the first two moments. This crossover exponent determines the relevance of disorder at the critical point and can be computed exactly for both models A and B. One finds a striking difference between the two models as

For model A, <j> is negative for all b > 2, implying irrelevance of disorder and / a but it is equal to 2 — desv, while for model B, <j> — a and not 2 — degv. Since the randomness in model B is highly correlated, the Harris criterion is less expected to be valid here as opposed to model A. but it turns out to be so. In order to construct a general framework for checking the validity of the Harris criterion, we start with the Taylor expansion of the recursion relation of Eq. (86), e = gs{b){e1+e2 + ...eM) + Q{e2) + ...,

{N = 26)

(89)

which defines gs(b). Simple arguments show that gs{b) determines a whose positivity is guaranteed if gs(b) > \Z2J\f~1 Now, suppose that the bonds are grouped in n sets with Ni bonds in the ith group such that the members of a set have the same randomness. Obviously ]T) iVj = J\f. Starting with a narrow distribution, the relevance of the disorder at the pure transition then requires g.(b) > (N* + Nl + .... + Hi)-1'2.

(90)

Hence, the Harris criterion holds good if either 5 s (6)>max

—r,—==

, or gs(b) < min — — = =

.

(91)

For model A, M = 26 and Nt = l Mi, while for model B, n = 2 with Nt = b. If the disordered models are classified by according to the sign of a, and I or R according to irrelevance or relevance of disorder, then the Harris criterion suggests the existence of only two classes (+R) and (-1). On the other hand, the above inequalities allow special classes like (+1) and (-R) where the Harris criterion fails. Model A is in the (+1) class for b > 2 + \/2. Model B is in either the (+R) or (-1) class depending on 6. It is possible to construct models that would belong to any of the four possible classes, especially (-R) [46].

Directed polymers and randomness

41

11. SUMMARY The behaviour of a directed polymer in a random medium in 1+1 dimensions seems to be well understood. There is a strong disordered phase at all temperatures for d < 2. For d > 2 renormalization group analysis shows a phase transition from a low temperature strong disordered phase to a weak disorder, pure-like phase. There are rare configurations with degenerate widely separated ground states, giving a contribution to "overlap", and strong sample dependent response to an unzipping force. The RANI model remains less understood compared to the random medium problem. Exact renormalization analysis establish the marginal relevance of the disorder at d = 1, indicating a disorder dominated unbinding transition in d > 1. Several features including a generalization of the Harris criterion for this criticality via relevant disorder and aspects of unzipping have been discussed. In both cases of random medium and random interaction, many issues still remain open. Acknowledgments: The author thanks Rajeev Kapri and Soumen Roy for many useful comments on the manuscript. APPENDIX A. Typical vs. average Consider a random variable x that takes two values Xx

= eaVW and X2=e?N,/3>l,

(92a)

with probabilities Pl

= 1 - e~N, and p2 = e~N.

(92b) 1

In the limit N —> oo, the average value [x]av —> e^" )" while the typical or most probable value is x = X\ with probability 1. On the other hand [lnx] av —> ayN in the same limit, showing that [ln:r]av is determined by the typical value of the variable while the moments are controlled by the rare events. Note that this peculiarity disappears if x has a smooth probability distribution in the sense of no special or rare events. B. Pure polymers The universality of the "Gaussian" polymer is actually a consequence of the Central limit theorem. Suppose we construct a flexible polymer from bonds with independent probability distribution tjj(r) for a bond vector r. The end-to-end distance is given by R = Yli ri so that the probability density of R is

P(R) = /"ndr»^wi>-R) i

=

i

Y[ > ^( >) P ( * " Ti) exp(-ik R)rfk dr

r

ex

(93)

i

=

A^(k)] iv exp(-ik R)dk,

(94)

42

S. M. Bhattacharjee

where V>(k) is the Fourier transform of ip{r)- For a symmetric distribution with finite variance, In V>(k) ~ 1 —Ak2/2..., which on integration over k leads to a Gaussian distribution. This is valid for a lattice model also. With the probability distribution of Eq. (3), the entropy in a fixed distance ensemble can be written as 1 r2

S(T) = 5(0) - -

w

1 Tr2

, which gives F(v) = F(0) + - - , - .

Z Ho

Z Ko

(95)

This identifies an effective spring constant for the polymer, namely 3T/2.RQ. This spring like behaviour is purely an entropic effect. An important and general point is to be noted. The macroscopic quantity involves an "external" parameter like r which is scaled by Ro the characteristic long-length scale size of the polymer. That the long distance behaviour can be described by a single length scale is the basis of "scaling" approach to polymers. Another approach to scaling is to study the changes in the properties of a polymer as the microscopic variables are scaled. E.g., if we make a scale change, r —¥ br and z —> b1!"'z, the Hamiltonian of Eq. (2) remains invariant if v = 1/2. Under such a transformation, the size behaves as R0(N) = b-'Roib^N),

(96)

so that by choosing b = N~", Ro ~ N", i.e., the size exponent becomes v = 1/2. In presence of interaction or disorder, v may not be obtained so simply and then renormalization group methods become useful.

B.I. Scaling approach in presence of a force If the polymer is now pulled with a force g, keeping the end at z = 0 fixed, the polymer would align on the average with the force. The polymer can be thought of as consisting of blobs within each of which a polymer can be considered as free from the force whereas the blobs as unit form a still coarse grained model that shows stretching. This is shown schematically in Fig. 3 and is used in Sec. 8.2.1. This description is called a "blob picture". This picture essentially depends on the scaling idea that Ro is the only relevant scale for the macroscopic description. This can definitely be justified at the Gaussian level. The partition function with the unzipping force can be written as

Z = JdrP(r,N)exp{(3g-r),

(97)

where P(r, N) is Eq. (3). The Gaussian integral can be done (keeping T explicitly in Eq. (3)) to obtain l < r > | = ^ff,

(98)

which is consistent with the idea of an effective spring constant of the polymer derived after Eq. (95). A scaling approach to derive Eq. (98) would as follows. Let R9 be the characteristic size of the polymer in presence of the force. Then, from a dimensional analysis point of

Directed polymers and randomness

43

view, this is similar to the zero force size RQ. From the nature of the force term, g is dimensionally like T divided by a length scale. Only lengthscale in the problem is RQ. Hence a dimensionally correct form is Rg ~ #0 f(gRo/T).

(99)

Note the absence of any microscopic scale in the above form. All microscopies go in Ro. For a linear law at small force we require f(x) ~ x (x —» 0) giving back Eq. (98). One may rephrase this by saying that the force has a characteristic size £9 ~ T/g. If the polymer is confined in a tube of diameter D then the dimensionless variable is Ro/D. This will appear in the form for change in entropy or in "confinement energy". This is used in Sec. 4.3.2. C. Self-averaging Let us build a large system by adding blocks A, B, C, D, ... systematically so that an extensive quantity is a sum over its values on individual blocks. In case this averaging over blocks leads to a very sharp probability distribution, then no further disorder averaging is warranted, i.e., any large sample would show the average behaviour. A quantity with this property is often called self-averaging. This may not be the case if the distribution is broad especially in the sense discussed in App. A. A self-averaging quantity has the advantage that one may study one realization of a large enough system without any need of further disorder or sample averaging. For numerical simulations, the statistics of a non-self-averaging quantity cannot be improved by increasing the number of realizations. To be quantitative, let us choose a quantity M which is extensive meaning M = Nm where m is the "density" or per particle value. This is based on the additive property over subsystems M = J2^i- For a random system we better write M = M(N,{Q}), with {Q} representing all the random variables. To recover thermodynamics, we want [M]av to be proportional to N for N —> oo. Now, if it so happens that for large JV M{N, {Q}) -> Nmd,

(100)

with rrid independent of the explicit random variables, then M is said to have the self averaging property. Note that no averaging has been done in Eq. (100). One way to guarantee this self-averaging is to have a probability distribution P(M/N) ^ °

6{md).

(101)

This is equivalent to the statement that the sum over a large number of subsystems gives the average value, something that would be expected in case the central limit theorem (CLT) is applicable. This generally is the case if quantities like M for the sub-blocks are independent and uncorrelated random variables. For many critical systems CLT may not be applicable and self-averaging is not selfevident. A practical procedure for testing self-averaging behaviour of a quantity X is to study the fluctuations a\ = [X 2 ] av — [X]^v and then check if 2

RX,N

= jy^

-> 0, as N -> oo.

(102)

44

S. M. Bhattacharjee

A quantity is not self-averaging if the corresponding R does not decay to zero. The central limit theorem would suggest RX,N ~ N~x, while a decay of Rx slower than this would be termed as '"weakly" self-averaging. We may then classify a quantity X, based on the large N behaviour, as follows: RX,N

~ constant => non self-averaging ~ N"1 => strongly self-averaging ~ N~p with 0 < p < 1 => weakly self-averaging.

Recent renormalization group arguments seem to suggest that if disorder is relevant then at the new (disorder-dominated) critical point thermodynamic quantities are not self-averaging [50]. The arguments leading to this extremely significant prediction of non self-averaging nature of critical quantities can be summarized as follows. Let it = \T — Tc(i, N)\/Tc be a sample dependent reduced temperature where Tc(i, N) is a pseudo-critical temperature of sample i of N sites with Tc as the ensemble averaged critical temperature in the N —I oo limit. In terms of this temperature, a critical quantity X is expected to show a sample dependent finite size scaling form Xi{T, N) = Np Q{UN1/d 0, i.e., if d < 2. This means a is a relevant variable at the noninteracting point. For the disordered system it translates to relevance of disorder at the pure fixed point. One gets a nontrivial fixed point at u* = — 2ire where j5(u) = 0 which is an unstable fixed point if e < 0. This unstable fixed point for d > 2 represents the disorder induced critical point. For the disorder problem, u < 0 is not meaningful, but for the interaction problem as in the RANI model, full range of u is allowed. The nontrivial fixed point for d < 2 is a stable fixed point and it describes the phase of repulsive polymers (fermion like). There is a

48

S. M. Bhattacharjee

critical binding-unbinding transition for d > 2 for pure attractive short range interaction. The unbinding transition is at zero potential for d < 2 For the transition one can define a characteristic length £ so that at scales> £, the critical effects can be ignored. If we start with an initial value UQ = u* + Ao, then one may say this crossover happens at some arbitrarily chosen value of AM = 1 say where Au is the renormalized deviation from the fixed point. One may determine this by linearization as

L^p

(120)

=| e | Au,

dL which gives AM = A 0 (L/L 0 )' e '. If we take Ao < < 1 as a measure of the deviation from the critical point (like T — Tc), then a small deviation grows and goes over either to the stable f.p. for the unbound phase or to — oo for the bound phase depending on the starting initial sign. Setting L = f, we then get

e = | A o |"c, and C = y ^ - .

(121)

Since the beta function is exact, we have obtained the exact correlation length scale exponent for the binding-unbinding critical transition. At the critical dimension, d = 2, the flow equation is = (A.) 2 ,

L ^

(122)

which gives £ ~ exp(l/Ao). This exponential dependence of £ on Ao accounts for the divergence of v at e = 0.

E. RG of the KPZ equation: Momentum shell technique We discuss the momentum-shell RG approach to the KPZ equation P)fprp

^

i =

2KV2F~2K{VF)2

+ 11 Z)

(123)

^ -

which is Eq. (37). More details on this equation may be found in Ref. [55]. The idea is to get the behaviour of the parameters of the differential equation in the long distance limit. The three parameters are T, K and A, of which K remains invariant so that 2v — 6 + 1, Eq. (42), is satisfied. Note that if the nonlinear term (i.e. the force term in the fixed distance ensemble) is absent, the differential equation becomes linear which can be solved exactly[25]. The RG scheme uses this exact solution for an iterative approach to tackle the nonlinear term. In contrast to the RG in the polymer approach of App. D, here the starting point is not a Gaussian polymer but a linear equation which need not represent any polymer. It is convenient to work in the Fourier space

F(x,z)=rpf ./-oo 27T Jk oo) is apparent in the integral of Eq. (128). If in the residual integration in Eq. (128), we perform a thin-shell integration between A(l — 51) < q < A and set A = 1, the effective temperature for the left-over long wavelength part is given by T bx, we have k —> (1 — . n->oo

(1)

n

where we have assumed isotropy, so that the choice of co-ordinate direction is irrelevant.

2. SELF-AVOIDING WALKS IN RANDOM GEOMETRIES Consider a SAW on a lattice in which some sites are missing. One can remove sites at random, such that a fraction p of sites is removed, and study the behaviour of SAW averaged over an ensemble of such lattices. Thus one obtains the properties of the SAW averaged over a uniform distribution of such lattices. This is the quenched situation.

SAW in constrained and random geometries

61

Brief reflection allows one to conjecture the effect of this site dilution. Firstly, it is clear that the problem now becomes one of SAW on a percolation cluster. For p > pc there is no infinite cluster, and hence all SAW are of finite length. For p < pc, the average number of neighbours of a lattice site is modified from q, the coordination number of the undiluted lattice, to q(l —p). Thus one would expect the growth constant n also to be reduced, and /x(p) to be a monotone decreasing function of p for 0 < p < pc. What is less clear is how the critical exponents behave. As discussed above, for the undiluted lattice one has (R2)n ~ const.n2" in the limit as n becomes large. The exponent v is given phenomenologically by the Flory formula, v = 3/(d+2) which seems to be exact for d = 2 and d = dc = 4 (ignoring logarithmic corrections at d = dc = 4). The Flory formula prediction is about 2% high at d = 3. Above four dimensions the exponent "sticks" at the value 1/2, as the SAW constraint becomes increasingly irrelevant. Field theory, again for the undiluted lattice [7], gives v = 1/2 + e/16 + 15e2/512 + ..., where e = 4 - d. Naive application of the Harris criterion [8] would suggest that for pc > p > 0 the exponents would be modified. That is to say, the slightest level of disorder would be sufficient to change the exponent. Loosely speaking, the Harris criterion says that the universality class of a system is affected by the presence of disorder if the specific heat exponent a of the pure system is positive. For both two- and three-dimensional SAW this is the case. For d = 2, a = 1/2, while for d = 3, a « 0.236. However, as SAW are given by the JV —» 0 limit of the O(N) model, a modified Harris criterion applies [8]. In the usual Harris criterion argument, one can relate the concentration induced change in transition temperature to the correlation length, and then using the hyperscaling relation dv — 2 — a deduce that if a > 0, the exponent v will change with impurity concentration p > 0, no matter how small. However, in the JV —» 0 limit, the transition temperature change does not depend on a, but only on the dimensionality of the system. Thus weak disorder is not expected to modify the universality class. However, at p = pc we are in a regime of strong disorder, and one expects a change in universality class. This situation was first discussed within the framework of the epsilon expansion by Meir and Harris [9] in 1989, and demonstrated numerically by Grassberger [10] in 1993. The epsilon expansion has recently been extended by von Ferber et al. [11], who found that at p = pc one has vc = 1/2 + e/42 + 110e2/213 + ... where now e = 6 — d. The appropriate Flory formula [12] is now vc = 3/(d p +2), where dp = d~Pp/vp and j3p and vp are the appropriate percolation exponents. In two dimensions we have j3p = 5/36 and up = 4/3, hence dp = 91/48 and uc = 144/187 = 0.7700.... In [9] the values 0.76 0.08, 0.67 0.04, 0.63 0.02 and 0.54 0.02 were obtained for d = 2,3,4,5 respectively, based on the analysis of short series. Grassberger [10] obtained v = 0.783 0.003 by Monte Carlo analysis, while Lam [13] estimated vc = 0.81 0.03. The most recent series work, [14] finds uc = 0.778 0.015 and uc = 0.787 0.010 from two different analyses of the two-dimensional case, and vc = 0.662 0.006 in the three-dimensional case. These are consistent with recent Monte Carlo analyses, and also with the epsilon-expansion given above, which evaluates to vc — 0.785.... and vc — 0.678... for two- and three-dimensions respectively. Thus all numerical work and theory is in reasonable agreement, and this situation can be considered to be well understood.

62

A. J. Guttmann

3. RANDOM COPOLYMERS For the problem of random copolymers, unlike the situation discussed in the previous section, the randomness is now not in the medium, but in the SAW itself. One of the simpler cases is when there are two types of monomer, say A and B. Typically one has a fraction p of A-type monomers, and a fraction (1 — p) of B-type monomers. One usually assumes that the monomers are randomly distributed and constrained only by the value of p. An excellent contemporary review of this topic can be found in [2]. More generally, one can consider the case with k types of monomer, denoted m i , . . . , m j , where the state of the polymer, modelled by an n-step SAW, is given by an n-tuple a = a 1 ; . . . , an, where «j £ {mi,... ,rrik}. The values of a.^ are taken from some distribution, appropriately chosen to model the problem at hand. In [2] three representative situations are discussed. The first is adsorption of a copolymer onto a surface, the second is the localization of a copolymer at an interface between two immiscible liquids, and the third is the temperature- or solventinduced coil-ball collapse of a copolymer. We discuss these three representative problems below. We will consider the case of an isolated SAW, of length n. Two types of averaging are usually considered, corresponding to the quenched and annealed cases respectively. In the quenched case, the average is taken of the logarithm of the partition function over the entire distribution, so that the free energy is Tq:=

Urn -{logZn),

(2)

while in the annealed case Ta:=

lim -\og{Zn).

(3)

n->oo n

The averages are taken over all a. It is noteworthy that the annealed free energy is an upper bound on the quenched free energy. This is useful as the former is usually easier to calculate. Better approximations are often provided by the Morita [15] approximation and the replica method. We refer readers to [2] and references therein for further details. We will now consider the three problems referred to above. Random copolymer adsorption refers to the situation in which a copolymer (most simply described as one with two types of monomer, say A and B) is adsorbed onto a surface. Consider an n-step SAW, with a proportion p of A type monomers, and a proportion 1 — p of B type monomers in a half-space. In two dimensions we are therefore referring to that section of Z 2 with y > 0. Let the origin of the SAW lie in the line y — 0. Finally, the energy comes from the A monomers in the line y = 0. That is to say, only the A monomers are attracted to the surface. There is no surface interaction with B monomers. For those SAW, with cardinality c+(v^|a) with VA vertices in y — 0, and monomer distribution as described above given by a, where a* = 1 if the ith vertex is A and zero otherwise, the partition function is given by

^n («l«) = J2cn(vA\a)

exp{nvA).

(4)

"A

For technical reasons one usually insists that the last step of the SAW also lies in the line y = 0, so that we are studying self-avoiding loops, rather than SAW. The reason for this

SAW in constrained and random geometries

63

restriction is that it allows concatenation of such walks, and hence application of standard tools like sub-additivity, which are so ubiquitous in existence proofs of free-energies and the like. What makes this problem so difficult—and indeed most copolymer problems difficult— is the necessity to take the average over all possible 2" distributions of monomers on the polymer. Nevertheless, some exact enumeration studies of this problem have been conducted. Martin [16] estimated the location of the transition using exact enumeration methods. An open question is the value of the crossover exponent, <j>, which describes the shape of the free-energy near the adsorption critical temperature. The conclusion from the above study, and others, is that the difference between this exponent and its homopolymer counterpart, if it exists, is too small to be detectable by any current numerical studies. The second problem of considerable interest is that of localization of a copolymer. Consider a mixture of two immiscible liquids such as oil and water. Then consider a copolymer, modelled as usual by a SAW, with two kinds of monomer, distributed uniformly and independently, and with origin at the interface between the two liquids. Let A monomers be energetically attracted to the water phase and B monomers to the oil phase. At high temperatures, the SAW will sit entirely in the liquid that gives rise to the lowest energy configuration. At low temperatures however, the SAW will try to orient itself with A monomers in the water and B monomers in oil. It will thus cross the interface frequently. This is the so called localized phase, as in it the SAW is largely constrained to the vicinity of the interface. As the temperature is raised, one expects a phase transition to a delocalized phase, heralded by a singularity in the quenched free energy. The model can also be generalized to include an interaction energy for monomers in the interface. James et al. [17] and Martin et al. [18] studied the localization problem for the simplecubic lattice Z3, where the plane z = 0 defines the interface. Most of their studies are devoted to establishing rigorous results, of the existence and convexity type, and determining qualitatively the shape of the phase diagram. The third problem we will consider is the collapse of copolymers. In the homopolymer case (where there is only one type of monomer), the collapse transition is known to take place at the 9 temperature, brought about by sufficiently strong attractive interactions between nearest neighbour monomers that are not joined by a path of the SAW. In that case, the exponent v that characterises metric properties such as the mean-square endto-end distance and the radius of gyration, which are characterised by the exponent u, changes abruptly to u$ at the 8 temperature, and then to vc = \/d at lower temperatures, when the SAW collapses into a d-dimensional ball. In the copolymer case, the simplest realistic model is as usual to have A monomers with probability p and B monomers with probability 1 — p. Then instead of just considering interactions between nearest neighbour contacts, as in the homopolymer case, we associate different energies to AA, BB and AB contacts. Typically, one allows like monomers to attract and unlike monomers to repel, or vice versa. In the former case we have a model of hydrophilic and hydrophobic monomers, and in the latter case we have a model of a highly screened Coulomb system. Golding and Kantor [19] and Kantor and Kardar [20] studied a version of this problem by both exact enumeration and Monte Carlo methods. In the former situation described above, they find a collapse transition much like the transition discussed above in the

64

A. J. Guttmann

homopolymer case, which holds irrespective of the ratio of type A to type B monomers. In the latter situation, modelling a screened Coulomb system, the behaviour depends on the ratio of the two types of monomers. If this is around 1, so that the two types of monomer have similar cardinality, the system has no nett charge, and there is a collapse transition. But if the monomer ratio is substantially different from 1, the repulsive interactions between the monomers that are in excess prevent any collapse, and the SAW remains in its coiled state at all temperatures. A later enumeration study by Monari and Stella [21] considered the system with equal numbers of A and B monomers and a Coulombic interaction energy. They found that at the 6 temperature, the radius of gyration exponent v and the crossover exponent (f> were indistinguishable from their homopolymer values in both two- and three-dimensions. Many features of this fascinating problem have not been mentioned here, but can be found in the comprehensive and up-to-date review of random copolymers in [2]. In particular, many of the rigorous results and the numerical results obtained both by other approximation methods and by Monte Carlo methods which have not been discussed here can be found there 4. SAW IN WEDGES In this section we consider the effect of confining a SAW to a wedge, formed by the intersection of two (non-parallel) d — 1-dimensional planar surfaces in a d-dimensional hypercubic lattice, with co-ordinates labelled by {xg\5 = 1,2,..., d}. The surfaces intersect in a d — 2 dimensional surface perpendicular to both X\ and x2, with wedge angle a. It is not difficult to show that the growth constant fi remains unchanged for most "obvious" geometries, such as the semi-infinite lattice, or indeed any positive wedge angle. A much more subtle result was proved by Hammersley and Whittington [22] who considered SAW confined to a subset Z d (/) of the d-dimensional hypercubic lattice, such that the co-ordinates (xi,x2, ,xa) of each walk vertex satisfy Xi > 0 and 0 < x^ < fk(xi) for k = 2 , 3 , . . . , d. They showed that the growth constant remains unchanged provided that fk{x) - > o o a s i - > oo. In this problem, either end of a walk can be in the bulk, at a surface, or at the wedge "edge". This gives rise to a number of possible generating functions, and we now develop the theory in terms of the general iV-vector model, and take the limit N —> 0 in order to recover the SAW situation. Expressed as a problem in the framework of the O(N) model, the Hamiltonian is

H = -K'£<Ti.oi-L0Yi^-L1Yio?>-L2Yi*V

i

i

(5) i

where ai is an iV-dimensional unit vector with components (a\ , /3 = 1 , 2 , . . . , TV) and LQ, L\ and L2 are bulk, surface and edge fields respectively, all parallel to a\ . The first sum is over nearest-neighbour pairs, the second is over all spins, the third is over all spins in a surface, and the fourth is over all edge spins. The edge magnetization is the expectation value of an edge spin:

m2{K;L0,L1,L2)

= {af\Xl = 0 - x2)).

(6)

SAW in constrained and random geometries

65

Differentiation with respect to the three fields Lo, L\ and Li allows for three susceptibilities to be naturally defined, viz: X2 ~ const.r72,

(7)

X21 ~consi.r 7 2 1 ,

(8)

22

X22 ~ const.r~< ,

(9)

where t = T/Tc — 1. If we now write the free energy in the form

T = Vfb + Af. + Lfe + ...

(10)

where V is the 'volume' of the system, A is the 'area' of the surface, and L is the 'length' of the edge, then /{,, /„ and fe denote the bulk, surface and edge free energies respectively. Standard scaling theory informs us that in this case, with three fields, the singular part of the free-energy will scale as: fstn9 = A-d/(Aw5o, A^si, \y2g2)

(11)

where yo, 2/1 and 2/2 are the renormalization group eigenvalues associated with the three fields. The three free energies referred to above can be written in terms of the canonical scaling form as fh ~ tt-a*h{ht-*>v),

(12)

fs^t2-a'^b(hr^,hlr^), 2 ae

V0I/

fe ~ t - ^b(hr

,

(13) 1

y2

M " " " , h2t- "),

(14)

where yo, y\ and y2 are the bulk, surface and edge scaling indices, and (2 - a) - dv, (2 - as) = (d - l)v, and (2 - a e ) = (d- 2)v.

(15)

All susceptibilities now follow by taking the appropriate derivatives, so that

X=y^~ const*-!,

i=

co

r71

* Hr ^- ' d2f Xn =-gjp ~ const*-*1,

X2 = ^rk-^const.r^,

(16)

(17) (18) (19)

ahahi

X2i= or-fe-~ c o n s *- r 7 a i '

(2°)

a/iid/i2

X22 = ^ ~ c o n s U - w ,

(21)

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A. J. Guttmann

and by performing these differentiations, we immediately find 7 = u(2yo - d)

(22)

71 = "(yo + ! / i - d + l )

(23)

7n = K 2 2 / i - d + l )

(24)

72 = Ks/o + 2/2 - d + 2)

(25)

721 = i/(yi +y2-d

(26)

+ 2)

722 = K22/2-d + 2).

(27)

These results can be combined to produce the scaling law 271 - 7n = 7 + v,

(28)

which was first derived by Barber [23], and the additional scaling laws, 272 - 722 = 7 + 2i/,

(29)

and 2721 - 722 = 7n + v,

(30)

first given by Guttmann and Torrie [24]. The susceptibilities can also be defined in terms of correlation functions, which will be useful in taking the N —> 0 limit of the O(iV) model. Define the correlation function C{r' = 0;p,xux2)

(31)

between a spin at the edge (r' = 0) and a spin at (p, xi, x2) where p is a (d—2)-dimensional vector with components x3, X4,..., X4. First, note that the correlation function will depend on the orientation of the vector (p, Xi,x2) even near Tc. Denote its magnitude by r, its orientation within the (xi, 22) plane by 6 (clearly 0 < 6 < a), where a is the wedge angle, and its orientation within the surface plane by <j>. For T > Tc we have C(r, 6,4) »„«, /(r, 9, ) exp[-r/6, 0 (i)],

(32)

where / decays more slowly than the exponential term. At T = Tc, C(r, 9, ) ~ M O O A(9, 4>)/rd-2-^,

(33)

which leads to the special cases 772 = f] for (9 > 0, <j> > 0), 772,1 for (6 = 0, > 0), and 772,2 for {6 = 0,4> = 0). Clearly 772 < 7721 < 7722 as there are more possible paths leading t o 772 than the others, and similarly for the last inequality. A typical susceptibility is 00

00

p xi— 0 £2=0

SAW in constrained and random geometries

67

with a similar expression holding for the other susceptibilities. (The prime is to remind one of the restriction n < a). Following an argument of Sarma, given in the appendix to a paper by Daoud et al. [25], we see that \2 is the generating function for SAW in the wedge, with one end tethered to the edge, and no restrictions on the other end. X21 is given by the generating function for SAW with one edge tethered to the edge, and the other end in the surface, while X22 is given by the generating function for SAW with both edges tethered to the edge. Replacing sums by integrals in the above expression, it is straightforward to show that 72 = v{2 - 772),

(35)

712 = v(l - »7i2),

(36)

722 = -VTI22-

(37)

Defining 7700 = r], and 77^0 — Vi these expressions can be combined to give Vpq = (Vpp + »7 M )/2,

a result first obtained by Cardy [26] from field-theoretical arguments. Based on extensive enumerations in wedges of various angles, Guttmann and Torrie [24] found compelling numerical evidence for SAW in two-dimensional wedges of angle a, that 2/2 = —57r/8a. If one recalls that 2/0 = 91/48 and 1/1 = 3/8 were already known, it follows that all exponents for the edge problem follow. Cardy subsequently verified this conjecture for 2/2(0), again from field theoretical arguments. For three-dimensional SAW, the enumerations in [24] led to the conjecture 2/2(0;) — 1/2 — 0.85(2)/a. Both these results can be compared to the mean-field theory result, y?fF = 1 — d/2 — n/a. Note that the mean-field result agrees with the enumeration results in both the constant term and the form of the angular dependence. An interesting and physically significant generalisation of this problem can be achieved by associating a fugacity with those monomers lying in the surface, or surfaces, in the case of a wedge geometry. If the fugacity is sufficiently attractive then at some critical temperature the SAW becomes adsorbed onto the surface. In an interesting paper, Hammersley, Torrie and Whittington [27] considered the simpler case of a single surface (corresponding to a = IT) on the hypercubic lattice and showed that there must be a phase transition. In [28], Batchelor et al. studied the problem of two-dimensional SAW, and more general SAW networks in which the surface is allowed to assume a variety of boundary conditions on either side of the wedge. That is to say, there is a different fugacity associated with SAW contacts in one surface to that in the other surface. As a result of this study, they were able to give conjectured exponents for an arbitrary polymer network—that is to say, not just SAW, but stars—connected to the surface of an arbitrary wedge (in two dimensions) where the surface is allowed to have general mixed boundary conditions. 5. SELF-AVOIDING WALKS IN STRIPS AND SLABS Consider SAW in a strip of width L on the square lattice. This is essentially a onedimensional problem. Indeed, the column-to-column transfer matrix is finite, and hence by a well-known theorem [29] the generating function is rational. The "critical exponent"

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A. J. Guttmann

is 1, corresponding to a simple pole, and the number of SAW of length n, which we denote cn(L) ~ const./j,(L)n, where the growth constant (sometimes called the connective constant, though this latter term is also applied to the logarithm of // by some authors) is a strictly monotone increasing function of L [22], and lim,,.^ n(L) = fi, the growth constant in the (two-dimensional) bulk case. The generating functions have been investigated for small values of L in [30]. Daoud and de Gennes [31] has given a scaling argument for the L dependence of /x(L), which leads to log/x — log/z(L) ~ DwL~ll"', where v = 3/4 as usual denotes the exponent that characterises the mean-square end-to-end distance of an n-step SAW, and Dw is an amplitude. An interesting point occurs if we consider self-avoiding polygons, similarly constrained. An n-step self-avoiding polygon (SAP) is a subset of (n — l)-step SAW; those with adjacent end-points. Clearly, joining the end-points results in an n-step SAP. By identical arguments we see that the generating function of SAP in a strip is rational. The "critical exponent" is 1, corresponding to a simple pole, and the number of SAP of perimeter n, which we denote pn(L) ~ const. nP(L)n, where the growth constant is a monotone increasing function of L, and lim/,_>0O/ip(L) = n. However, [32] nP(L) < fi(L), even though these two quantities have the same limit as L becomes infinite. This follows from Kesten's pattern theorem [32], which, crudely speaking, says that if a SAW pattern in one geometry can occur at least three times, and it cannot occur in a different geometry, then SAW in the former class are exponentially more numerous than those in the latter. Soteros and Whittington [32] displayed such a pattern for SAW in strips of finite widths, hence the above result follows. As a similar scaling Ansatz applies for SAP, viz: log// - log//(L) ~ DvL-ll", it follows that Dw / Dp. We can make use of existing results in the literature to quantify this effect. In [30] the growth constants for SAW and SAP in a strip of width L are given, for L < 6. Using these results, and Daoud and de Gennes's scaling form above, we find Dw/Dp « 0.52. For a slab geometry, corresponding to SAW constrained between two parallel planes, which is essentially a two-dimensional problem, there is no similar pattern, and one has the same growth constant for SAW and SAP confined between two planes separated by a distance L [33]. The reason for this difference is, loosely speaking, that it is easy to devise "blocking" patterns for SAW in strips, but not in slabs, where the extra dimension allows the SAW to avoid the blockage. The generating function is no longer rational in this geometry, but the strictly monotone increasing property of the growth constant with separation L still prevails [22].

6. SELF-AVOIDING WALKS IN SQUARES AND CUBES Another interesting constrained SAW problem is that of SAW confined to lie entirely within a square, or rectangle. We will considering in detail the problem of self-avoiding walks on a subset of the square lattice Z2, though several of the theorems we give apply to the hypercubic lattice, and so hold for Zd, d>2. We are interested in a restricted class of square lattice SAW which start at the origin (0,0), end at (L,L), and are entirely contained in the square [0,L] x [0,L]. A fugacity x is associated with each step of the walk. Historically, this problem seems to have led two largely independent lives. One as a problem in combinatorics (in which case the fugacity

SAW in constrained and random geometries

69

has been implicitly set to x = 1), and one in the statistical mechanics literature where the behaviour as a function of fugacity x has been of considerable interest, as there is a fugacity dependent phase transition. The problem seems to have first been discussed by Abbott and Hanson [34] in 1978, many of whose results and methods are still the most powerful today. A key question considered both then and now, is the number of distinct SAW on the constrained lattice, and their growth as a function of the size of the lattice. Let cn(L) denote the number of n-step SAW which start at the origin (0,0), end at (L, L) and are entirely contained in the square [0,L] x [0,L]. Further, let CL(x) := ^2nCn(L)xn. Then CL(\) is the number of distinct walks from the origin to the diagonally opposite corner of an L x L lattice. In [34], and independently in [35] it was proved that Ci(l) ~ const.\L . The value of A is not known, though bounds and estimates have been given in [34-36]. In the statistical mechanics literature, the problem appears to have been introduced by Whittington and Guttmann [35] in 1990, who were particularly interested in the phase transition that takes place as one varies the fugacity associated with the walk length. At a critical value, xc the average walk length of a path on an L x L lattice changes from growing as L to growing as L2. In [35] the critical fugacity was proved to be > l//z, and conjectured to be xc = 1/fi, and in [37] the conjecture was proved. In [34] the more general problem of SAW constrained to an L x M lattice was considered, where the analogous question was asked: how many non-self-intersecting paths are there from (0,0) to (L, M). If one denotes the number of such paths by CL,M, it is clear that, for M finite, the paths can be generated by a finite dimensional transfer matrix, and hence that the generating function is rational. Indeed, in [34] it was proved that G

'(*>~gg^~ 1 - 4 x ^ - 2 ^ '

(38)

L>0

(where here we have corrected a typographical error). It follows that CL,2 ~ const. A^, where A2 = J^zr3 = 1.81735 .... In [36] two further problems which can be seen as generalisations of the stated problem were considered. Firstly, they considered the problem in which SAWs are allowed to start anywhere on the left edge of the square and terminate anywhere on the right edge; so these are walks spanning the rectangle from left to right. Secondly, they considered the problem in which there may be several independent SAW, each SAW starting and ending on the perimeter of the square. The SAW are not allowed to take steps along the edges of the perimeter. Such walks partition the rectangle into distinct regions and by colouring the regions alternately black and white one gets a cow-patch pattern. Each problem is illustrated infig.1. Following the work in [35], Madras in [37] proved a number of theorems. In fact, most of Madras's results were proved for the more general d-dimensional hypercubic lattice, but here we will quote them in the more restricted two-dimensional setting. Let Ai(z) := lim CL(x)1/L L-K5O

(39)

A. J. Guttmann

70

Figure 1. An example of a SAW configuration crossing a square (left panel), spanning a square from left to right (middle panel) and a cow-patch (right panel).

X2(x) := lim CL{xflL2

(40)

L—foo

Theorem 1. (%) The limit (39) exists and is finite for 0 < x < 1/jU, and is infinite for x > l//i. We have 0 < Ai(x) < 1 for 0 < x < l//i and Ai(l/mw) = 1.

fiij Tfte Zz'mzi (40) exists and is finite for all x > 0. We have \2{x) = 1 for 0 < x < 1//J, and \2{x) > 1 for x > l//i. The average length of a walk is defined to be

(n(x))L := J2 ncn(L)xn/ £ n

Cn (L)^.

(41)

n

In the next theorem, we use the notation a « b to mean that there exist two positive constants Ci and C2 such that Cx& < a < C2b. Theorem 2. As L —> oo, we have {n{x))i ss L /or 0 < x < 1/fj, and (n(x))i « L2 /or x > l/fj,.

The situation at x = \j\i is unknown. Compelling numerical evidence is given in [36] that in fact (n(l/fj,))i is proportional to L1/", in accordance with an intuitive suggestion in [37]. Theorem 3. For x > 0, define fi{x) = log Ai(a:) and ^(z) = log A2(a;). (i) The function f\ is a strictly increasing, negative-valued convex function of log x for 0 < x < 1/n, and fi(x) fa — m as x —» l/fi~. (m is the mass, defined above.) (ii) The function f2 is a strictly increasing, convex function of log x for x > l//i, and satisfies 0 < /^(x) < log/U + logx. Some, but not all of the above results were previously proved in [35], but these three theorems elegantly capture all that is rigorously known.

SAW in constrained and random geometries

71

In [36] a highly efficient algorithm for enumerating such walks is given. The time required to obtain the number of walks on L x M rectangles grows exponentially with M and linearly with L. The algorithm is easily generalised to include a step fugacity x. The generalisation to spanning walks is also quite simple. The generalisation to cow-patch patterns is more complicated. Graphs can now have many separate components. Using this algorithm, they calculated cn(L) for all n for L < 17. In addition, they computed Ci 8 (l) and Ci 9 (l). l

As noted, it has been proved-[34,35] that limM-+oo Cjjjntf = A exists. From this it is reasonable to expect, but not a logical consequence, that RM = CM+I,M+I/CM,M ~ \2M If so, the generating function 1Z(x) = J2M RMXM has a radius of convergence xc = I/A2, which they estimated using differential approximants [38]. For the crossing problem they found xc = 0.32858(5), for the spanning problem xc = 0.3282(6) and for the cow-patch problem xc = 0.328574(2). It can be proved that these three problems have the same growth constant, so taking the most precise estimate, they obtained A = 1.744550(5). As well as this estimate, it is possible to obtain rigorous bounds. In [34] it was proved that i

Theorem 4. For each fixed M, lim^oo C£"M = \M exists. i

It similarly follows that lim^-nx, C~^M = A exists, which was proved rather differently M

in [35]. In [34] the useful bound A > A^+1 is proved. The above evaluation of A2 immediately yields A > 1.4892.... Based on exact enumeration, Bousquet-Melou et al. [36] obtained the exact generating functions G3{x), G4(x) and G5{x). From these, they found the following values: A3 - 1.76331..., A4 = 1.75146..., and A5 = 1.74875 . . . from which they obtained the bound A > 1.59321

6.1. Asymptotics Bousquet-Melou et al. [36] also considered the exact and asymptotic results for walks crossing a square of length 2L + 2K. Recall that the shortest walk is of length 2L. For K = 0 the number of such SAW is just (2^). These are so-called ballot numbers. This result is obvious, as there are 1L steps in the path, of which L must be in the positive x (and of course positive y) direction. Note that this has the asymptotic expansion ^Lfi ^/L^V 1

1 I 1 I 5 , AL ^ 128L2 ^ 10241,3 T

\ ;.

(Aocnxn. This is because the possible origin vertices produce an infinite range of different series C(x). Each origin produces a different series which is representative only of that vertex's immediate neighbourhood in the tiling. As described above, in [46] Rogers et al. adopted three different approaches to enumerating SAWs on quasiperiodic tilings, viz: Fixed origin walks. Mean number of walks. Total number of walks. 7.3. Fixed origin walks Take a random selection of origin vertices x £ L (if xj_ ^ W the vertex is not a suitable choice and is ignored). For each suitable origin, generate the neighbourhood of the vertex, including all vertices up to some Euclidean distance N away. Two such neighbourhoods are shown in Figure 2 and Figure 3. Enumerate all SAWs from the origin up to length n in the neighbourhood using backtracking [61]. This takes time proportional to the number of walks cn.

Figure 4. The actual Ammann-Beenker neighbourhoods chosen for the enumerations.

If each of the series in Table 1 and Table 2 showed lattice consistent properties, it would be a good indication that these properties belong to the entire tiling.

78

A. J. Guttmann

n x± = (0,0) xx = (1,0) xx = (1 - % 4) xx = (l->/2,0) 1 1 1 1 0 4 5 1 3 8 12 2 13 16 16 42 46 34 3 48 152 4 144 108 108 292 388 374 5 448 1194 6 1088 952 976 7 3042 3412 3680 2458 8 9678 9584 8330 7746 9 24556 27218 21348 28336 10 68376 79150 61478 82960 11 217562 225408 197820 177230 12 628996 554108 495808 657536 1576464 1741464 1412152 13 1834768 14 5140752 4400920 4968606 3985706 13724682 15 12531794 14584112 11125408 34541864 39209054 40222672 16 31617786 17 87149372 107503768 98846548 114683280 18 270221012 248799302 306845714 313146848 896810944 840463852 19 773046904 680172768 20 2386875508 2109562128 2437468000 1943692238 21 6958267152 5303535884 6548653714 6011045200 22 18981078176 15086983820 16431248782 18500898140 23 53728620912 41295324398 46538635588 50883461478 24 147472084608 116624466842 127704810544 142927122532 25 413887940176 Table 1 The number of n-step fixed origin SAWs for various starting points xx in the AmmannBeenker tiling, with starting point coordinates (x, y) given in the internal space.

Figure 5. The actual rhombic Penrose neighbourhoods chosen for the enumerations.

SAW in constrained and random geometries

79

n xx = (0,0,1) xx = (0,0,0) xx = (0.5,0.5,1) xx = (0.25,0.5,0) 1 1 1 1 0 4 5 1 5 5 12 14 2 20 10 46 40 40 50 3 112 4 160 130 130 394 450 310 406 5 1140 1177 938 6 1170 3416 7 4000 2680 3316 9480 9360 9723 7866 8 27312 23150 27356 9 32910 77747 66150 10 76090 72520 215924 224102 11 262250 196980 545062 12 619460 615549 555290 1812802 1698548 13 1646990 2050500 4411293 14 4292010 5052310 4869786 13367278 15 13403280 15828550 14455725 33637420 41103090 38524509 35243859 16 105117832 106779600 121759470 114089288 17 279216083 265198150 331072990 304434061 18 825140032 840669610 937563530 894584372 19 2199738033 20 2092703550 2642381430 2399386239 6988332717 6459329037 21 7227151280 6573888100 17267339059 22 20931973090 16491425740 18844561759 50419312152 54473434666 23 55793302330 51185968460 147471723662 135162732506 24 164764171030 129673789110 Table 2 The number of fixed origin n-step SAWs for various starting points Xx in the rhombic Penrose tiling, with starting point coordinates (x, y, z) given in the internal space.

80

A. J. Guttmann

Figure 6. Examples of W° ... W7 for a particular walk on the Ammann-Beenker tiling.

7.4. Mean number of walks Given that a pair of vertices from A(W) are adjacent if and only if they are adjacent in L, one sees that the neighbours of a vertex with image X_L can be found by sequentially adding the image of all possible edges in L to Xj_ and testing if the new points lie in W. If it does the adjacent vertex exists in A(W). By recursively checking all possible neighbours of a vertex, all possible walks on the lattices will be found. Given an origin vertex x° in A(W), we know its image x° must lie somewhere in W, i.e. x° £ W° = W (Wn is the region x0^ can lie in, given our knowledge of the n steps in the walk). If one takes a step s (with projection onto to a possible adjacent vertex x1, then one knows x]_ = x° + Furthermore if x1 e A(W) is true, then x]_ € W. Hence W1 = (W n (W° + — The probability that the step s is possible from a random x° is given by the ratio between the areas of W1 and W. Extending this to a walk of length n with steps s\i = 1 . . . n, Wk = (W n (Wh~1 + 52i=is i ) ) "" £i=i s ! the probability of the walk existing is the ratio of the areas of Wn and W. For example, consider the shaded Wl in Figure 6, for the particular walk in the Ammann-Beenker tiling which steps west, south, south-west, north-west, west, north then north. For the Ammann-Beenker tiling, these probabilities are of the form a + bX, where A = 1 + \/2 and a, b e Q. Adding the self-avoiding constraint and summing the probabilities results in the expected number of SAWs beginning at a random origin. This process leads to mean numbers of the form a + br, where r = (1 + v/5)/2 is the golden ratio, and a, b G Z. The rhombic Penrose tiling also allows steps in ten directions, more than the AmmannBeenker tiling's eight. These facts combine to allow greater length series to be computed on the Ammann-Beenker tiling.

SAW in constrained and random geometries

81

Ammann-Beenker n rhombic Penrose 1 1 0 1 4 4 2 52-16A 62-30r 80-16A -4 + 28r 3 4 444-134A 914-488r 5 1280-380A -820 + 7 3 2 T 4492-1430A 13842-7894r 6 10848-3248A -17732 + 12860r 7 173876-101988r 8 60988-21700A -255784 + 173720r 9 89800-27036A 643248-237732A 10 1923078-1143988r 11 979776-324200A -3149856 + 2 0 7 3 1 9 2 T 12 5486960-2043420A 19566548-11734706r 10785736-3819788A 13 -34951044 + 22612992T 45253532-16927618A 14 192557132 -116151274T 110294592-40576780A -366912524 + 234803904r 15 375796808-141368464A 16 1058437232-398339560A 17 3259350860-1238175678A 18 9526156024-3632872284A 19 29127575440-11192322668A 20 81536068712-31337365980A 21 22 259724099656-100797073134A Table 3 The mean number of n-step SAWs for the Ammann-Beenker tiling and the rhombic Penrose tiling, where A = 1 4- -\/2 and r = (1 + v / 5)/2.

7.5. Total number of walks Investigating all possible walks as in the mean number of walks method, Rogers et al. [46] counted instead the number of non-zero contributions to the mean value. This counts the number of translationally inequivalent walks with Wn having positive area or, equivalently, the number of translationally inequivalent walks which may occur anywhere in the tiling.

A. J. Guttmann

82

Penrose n Ammann-Beenker 1 1 0 8 10 1 2 56 90 288 560 3 1280 2800 4 5344 12060 5 48520 20288 6 74192 182000 7 658300 260336 8 2282400 892800 9 7749440 2976512 10 25634920 11 9828256 83615140 31758112 12 101847216 268113660 13 322240144 850895040 14 1012048208 2668534600 15 3147031584 16 9732815728 17 29852932384 18 19 91182029360 276695822928 20 21 836719766336 22 2516664888416 Table 4 The total number of n-step S AWs for the Ammann-Beenker tiling and the rhombic Penrose tiling.

7.6. Self-avoiding polygons A self-avoiding polygon is equivalent to a self-avoiding walk in which the two degree one vertices are adjacent. In the enumeration of self-avoiding polygons, one must distinguish between polygons having, up to a translation, the same boundary but different fillings of the interior. SAPs may be enumerated in the same manner as SAWs. For the computation of occurrence probabilities of self-avoiding polygon patches, all vertices of the patch were taken into account. If the self-avoiding walk patch has n vertices x1 £ A.(W), where i = 1,..., n, the acceptance domain is Wn = f]i(W — x' ), and the occurrence probability is given by the ratio between the areas of Wn and W, see also [40]. The various sequences were analysed using standard methods of asymptotic analysis of power series expansions as described in [38]. For self-avoiding walks and polygons, it is easy to prove that the limit limn_>oo(cn)1/n exists by use of concatenation arguments [4]. We assume the usual asymptotic growth of the sequence coefficients cn, viz: Cn = Ax^n1-1

[l + O{n~£)]

{n -> oo,0 < e < 1).

(51)

SAW in constrained and random geometries

83

n mean number total number mean number total number 2 8 4 10 4 8 4 8 80 48 12A 6 108-48T 840 384 8 240-64T 800-272A 2960 6480 10 2840-880A 49760 6192-3364r 21600 12 28152-9984A 394080 170256 25584-13248r 14 179200-95340T 47712-9884A 3087140 1322048 869600-299392A 24020160 162976-5440T 16 10194720 215712+294408A 183529440 18 79960896 2704140-1067580r 14980920-3730840A 20 618248240 22 152588920-47048100A 4726263168 Table 5 The mean number of n-step SAPs and the total number of SAPs for the Ammann-Beenker tiling (first two columns) and for the rhombic Penrose tiling (last two columns), where A = 1 + %/2 and T = (1 + \/5)/2. On the square lattice, there is overwhelming evidence [52] of the above asymptotic behaviour with exponent 7 = 43/32. There is however no proof of this assumption. The above assumption results in the following asymptotic growth of the ratios rn

rn = -^- = - [l + ^—^ + O(n-1^)]

(n-K»,0<e w > 1. Note that there is no interaction between different segments in a chain. The generating function Z(x, w) can be expressed in terms of the generating functions for paths V(x) and loops U(x). These are oo

oo

V{x) = Y, Kxn,

U(x) = Y,P2nXn,

n=0

(56)

n=l

where bn is the number of paths of length n, and p2n is the corresponding number of loops. Due to the chain structure, one gets a geometric series in V(wx)U(x), v(

Z

^

=

\

i-u^mwx)

°°

=

^

w

^ -

(57)

SAW in constrained and random geometries

91

Since we want to analyze phase transitions of the model, which can only occur in the infinite system, we define the free energy of the model as f(w) = lim -logZ n (w) = -\ogxc(w),

(58)

n->oo n

where, for fixed w, xc(w) is the radius of convergence of Z(x, w). Concatenation arguments and supermultiplicative inequalities can be used to show that the free energy exists [90]. We will alternatively investigate properties of the free energy in terms of the generating functions for paths and loops. It is instructive to consider an exactly solvable model, which is provided by the following fully directed model. 9.3. Fully directed walks and loops This directed model consists of fully directed walks for the paths in the chain. These are clearly self-avoiding, and only take steps in positive directions. The corresponding loops are staircase polygons, which consist of two fully directed walks, which do not intersect or touch, but have a common starting point and end point. Paths are attached to these points. We distinguish the two strands of a loop. In d = 2, the generating functions for paths and marked loops are [90] V(x) = —-—,

U{x) = 1 - 2x - y/1 - Ax.

(59)

U(x) is twice the generating function of staircase polygons. The loop class exponent is c = 3/2. The generating functions V and U have critical points xy = 1/2, XJJ — 1/4 and wc = 1, i.e., a phase transition occurs at T = oo. (If the empty path would not be allowed, a phase transition would occur at a finite temperature). The free energy f(w) is given by 2 ^~^l h \A/1 + (W-1)2

f(w) = log (

- ) -

1

(1 < «>< oo).

(60)

/

The fraction of shared bonds follows as \ / l + (w - I) 2 - 1 - (w - 1)72 ,, ^ . ,C1. — . ' J=- (Kw< oo), (61) ~ W - l 1 + (W - I) 2 - y/\ + {W - I) 2 which approaches zero linearly in w — 1. The asymptotic behavior of Zn(w) about w = 1 is given by

a, ^ 0(w) =

Z n

2w

4«-i

^ ~ ~^r/2hVH/&(w - 1)

(n -»oo, w -> 1+),

(62)

uniformly in w, where ha(x) = -^ + axe^ax^ (1 + erf (ax)) and eri(x) is the error function. As shown in [89], this PS model of fully directed walks and loops is exactly solvable in arbitrary dimension. The phase transition is found to be first order for d > 6, to have a continuous phase transition in dimensions 2 < d < 5, and to occur at finite temperature in d > 4 only.

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A. J. Guttm&nn

Since the critical behavior of PS models is essentially determined by the properties of loops, PS [70,71], and later Fisher [72], were led to consider various loop classes (together with straight paths for the double stranded segments). Whereas PS analyzed loop classes derived from random walks, Fisher considered loop classes derived from self-avoiding walks. 9.4. Loops and walks An oriented, rooted loop of length n is a walk of length n — 1, whose starting point and end point are lattice nearest neighbors. We identify such loops if they have the same shape, i.e., if they are equal up to a translation, possibly followed by a change of orientation. These objects we call unrooted, unoriented loops, or simply loops. Each loop of length n has at most 2n corresponding walks. If the walks are self-avoiding, each loop has exactly 2n corresponding walks. The number of loops of length n is denoted by pn. For example, for self-avoiding loops on Z 2 we have P4 = 1 and pe = 2. For a given class of walks, the above description defines the corresponding (unmarked) loop class. Within a chain structure, two paths are attached to each loop. Different choices for attachment positions increase the number of (marked) loop configurations to pn > pn. If we assume the DNA condition that the two paths attached to a loop bisect it into pieces of equal length, then the number of possible attachments of two paths to a loop of length n is less than or equal to In. (We distinguish the two strands of a marked loop). PS and Fisher consider classes of oriented rooted loops. Self-avoiding oriented rooted loops can be interpreted as loops with 2n possible attachment positions of paths to a loop of length n. A similar interpretation for oriented rooted random loops is not obvious. We stress that these loop classes result in chains which are not segment-avoiding, as paths will intersect the loops. Both models cannot therefore represent real (self-avoiding) DNA. 9.5. Oriented rooted random loops The first simple example of loops, discussed by PS [71], is that of oriented rooted random loops derived from random walks. Random walks on 7Ld have the generating function Vd{x) = 1/(1 — 2dx). The asymptotic behavior of the number of oriented rooted random loops of length 2n is given [4] (Appendix A) by p2n ~ A{2d)2n{2n)-d/2

(n -> oo).

(63)

This implies no phase transition in d — 2 and a continuous phase transition in d = 3. The PS results led to the question [72] whether accounting for excluded volume effects within a loop increases the loop class exponent c, which might change the order of the phase transition. This led to considering self-avoiding loops, which are loops derived from self-avoiding walks. By definition, the self-avoiding loop class fully accounts for excluded volume interactions within a loop. In d = 3, self-avoiding loops of length n > 24 may be knotted. Fisher considered oriented rooted loops p2n = 4np2n- Their loop class exponent c > Vin ~ B/i^n^0, is related to the mean square displacement exponent v of self-avoiding walks by the hyperscaling relation c = dv. For unknotted self-avoiding loops, which is the preferable model from a biological point of view, it has been proved [32] that the exponential growth constant is strictly less than that of all self-avoiding loops, while the exponent (if it exists) is expected to coincide with that of all self-avoiding loops.

SAW in constrained and random geometries

93

Fisher concluded that the above values of the loop class exponent c imply a continuous phase transition in d = 2 and d = 3. In [89] it is shown that the phase transition condition is satisfied in both d = 2 and in d = 3 for the case when the double stranded segments are treated as straight paths. Note that in d = 2, self-avoiding walks as paths will result in no phase transition. The previously discussed PS models are not fully self-avoiding, in that the chains are not mutually avoiding. Using SAW for walks, a PS model with segment-avoiding chains may be defined as follows: Take only those walks with extremal first and last vertex. That is to say, if v(0), v(l),..., v(n) are the vertices of an n-step walk v, this walk is taken as a path iff vx(0) < vx(i) < vx(n) for all 1 < i < n. Such walks are bridges [4] (Sec. 1.2), whose last step is in the x-direction. For (unmarked) loops, take loops derived from the walks, that is, SAP. Marking of the loops may be achieved in different ways. To this end, consider for a given loop the sets A\ (AT) of vertices of smallest (largest) x-coordinate. We distinguish four different types of marking: complete marking (with DNA constraint), where we mark a loop at all vertex pairs from A\ and Ar (whenever the DNA condition is satisfied), and unique marking (with DNA constraint), where we only mark a loop at a single vertex pair, for example the bottom vertex and the top vertex in a lexicographic ordering (if they satisfy the DNA condition). Unique marking would then imply pn = 2pn (we distinguish the two strands of a marked loop) and hence increase the previous exponents by one. Hence, such a PS model displays a first order transition with c = 5/2 in d = 2 and with c = 2.7631(18) in d — 3. Unique marking with the DNA constraint results in pn < 2pn. If we assume that the exponential growth constant for marked self-avoiding loops is given by the usual SAP growth constant, this implies a critical exponent c greater or equal to the model with unique marking, i.e., a first order phase transition in d = 2 and d = 3. For complete marking, we have 2pn < Pn < 2npn, which rules out a decrease of c by more than one. However we expect that the number of possible markings is of order 1 as n —¥ oo, and so c remains unchanged, and the model again displays a first order phase transition. Similar considerations apply for the case of complete marking with DNA constraint. In summary, as we have shown in brief, and as shown in more detail in [89], a selfavoiding PS model (with unique marking, with self-avoiding bridges and (unrooted) selfavoiding loops) as defined above yields a first order phase transition in both d = 2 and d = 3. For the presumably more realistic model of pairs of interacting self-avoiding walks [79,83-85,80-82], the results of [89] suggest an interpretation of excluded volume effects, which complements the common one [83-85]. The self-avoiding PS models discussed above correctly account for excluded volume effects within a loop, but overestimate excluded volume effects between different segments of a chain, due to their directed chain structure. Since this leads to a first order phase transition in d = 2 and d = 3, one can conclude that the relaxation of excluded volume effects between different segments of the chain does not change the nature of the transition. The simpler, directed model displays a first order phase transition in d = 2, but such models seem to be of limited relevance to the biological problem due to their directed structure and other limitations. In conclusion, the question of the mechanisms applying in real DNA which are respon-

A. J. Guttmann

94

sible for the denaturation process and which explain multistep behavior as observed in melting curves, are still far from being satisfactorily answered in our opinion, though models based on SAW and SAP still seem to be the most appropriate way to study this important problem. 9.6. Vesicles collapse In modelling vesicle collapse by SAP, let pm(n) be the number of SAP per site on an infinite lattice, with perimeter m enclosing area n. In [75] it was proved that the free energy Yirn^ — \og^2pm,nqn

:= n(q)

(64)

exists and is finite for all values of the fugacity q < 1. Further, n(q) is log-convex and continuous for these values of y and is infinite for y > 1. In terms of the natural two-variable generating function P(x,q) ~^S^pmnxmqn,

(65)

x,q

it was further proved that for q < 1, P(x,q) converges for x < e~ K ^, while for q > 1, P(x, q) converges only for x = 0. The expected phase diagram is shown in figure 9 below.

Figure 9. The phase diagram showing the phase boundary xc(q). In the region below the phase boundary, the polygons are ramified objects, closely resembling branched polymers. That is to say, they are collapsed and string-like. As q

SAW in constrained and random geometries

95

approaches unity, they fill out more, and become less string-like. At q = 1 one has pure SAP. For q > 1 the polygons become "fat", and approximate squares, with their average area scaling as the square of their perimeter. In [75] rigorous upper and lower bounds to the shape of the phase boundary were found, and the locus of the actual phase boundary was found numerically from extrapolation of SAP enumerations by area and perimeter. In the extended phase q = 1, the mean area of polygons (a)m of perimeter m grows asymptotically like m 3 / 2 , whereas it grows like m in the deflated phase q < 1. It can be shown that in the limit q —> 0 the generating function is dominated by polygons of minimal area. Since for SAPs these polygons may be viewed as branched polymers, the phase q < 1 is also referred to as the branched polymer phase. This change of asymptotic behaviour is reflected in the singular behaviour of the perimeter and area generating function. Typically, the line q — 1 is a line of finite essential singularities for x < xc. The line xc(q), where P{x,q) is singular for q < 1, is typically a line of logarithmic singularities. For branched polymers in the continuum limit, the logarithmic singularity has been proved recently in [91]. Of special interest is the point (xc, 1) where these two lines of singularities meet. The behaviour of the singular part of the perimeter and area generating function about (xc, 1) is expected to take the special form P{x,q)~I*" 0? Numerous MC simulations [16-21], exact enumerations [22-28], and theoretical studies [29-39] which have been published since the early 80-ies [14], lead to the conclusion that there are the following regimes for the scaling of a SAW on a disordered lattice: (i) weak uncorrelated disorder, when the concentration p of bonds allowed for the random walker is higher than the percolation concentration pc (ii) weak, but long-range correlated disorder, and (iii) strong disorder, directly at p = pc. By further diluting the lattice to p < pc no macroscopically connected cluster, "percolation cluster", remains and the lattice becomes disconnected. In regime (i) the scaling law (1) is valid with the same exponent v for the diluted lattice independent of p, whereas in cases (ii) and (iii) the scaling law (1) holds with a new exponent vv / v. For magnetic systems, the effect of weak quenched uncorrelated point-like disorder on the critical behavior is usually predicted by the Harris criterion [40]: disorder changes the critical exponents only if the specific heat critical exponent avtlTf. of the pure (undiluted) system is positive: ctpure = 2 - dvpure > 0, v

(3)

pure being the correlation length critical exponent of the pure system. This was confirmed by numerous theoretical and experimental studies (see [41,42]). The straightforward application of the Harris criterion to the statics of SAWs indicates, that the critical exponents should be modified in the presence of any amount of lattice

106

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

disorder, as far as a for SAW on the pure lattice is positive. However, the renormalization group study [29] contradicts this prediction. The Harris criterion was then modified [43] for the quenched disordered m-vector model in the m —> 0 limit and leads to the conclusion, that the critical behavior of SAWs is not affected by weak uncorrelated lattice disorder. Since this point is of central interest in this review, let us consider it in more details. Let 7 denote a SAW configuration of N steps starting at some site i and R* the square of the end-to-end vector. Then for the mean square end-to-end distance of the trajectory one has:

(R*) = ^l~N2- {

}

^P(C)J2r2,

Er? E P(C)

~ T E l T " E W E i " E E P(C) C

f€C

{>

7 C:7£C

here P(C) is the probability of the realization of the configuration C. In case the distribution of occupied sites is uncorrelated, every SAW with ./V steps can be realized with probability pN. So, the last equality can be rewritten:

lR2)

" TF"" t r 7

(6)

7

and the expression obtained is the same as for the pure (non-diluted) lattice. The only effect of the disorder lies in the change of the possibility of formation of each SAW with TV steps, whereas the scaling properties of the realized SAW are not influenced. In the present review, we turn our attention to the three different classes of quenched (non-equilibrium) disorder, which from one side can be treated within the RG scheme and from the other side allow for a physical realization: 1) non-correlated point-like defects, 2) long-range correlated defects with a correlation function, governed by a power law ~ r~a at large distances, where o is some constant 1 , 3) the case when the concentration of dilution is exactly at the percolation threshold. While the influence of short- and longrange correlated disorder can be studied by related field-theoretical models with an upper critical dimension dupper — 4, the percolation problem leads to a different field theory 1 This type of disorder allows to describe so-called extended impurities: a = d — 2 corresponds to the presence in the system of straight lines of impurities of random direction, whereas a = d— 1 gives random planes of defects.

Renormalization group approaches to polymers in disordered media

107

with dupper — 6. Let us note, that in the case of m-component systems with long-rangecorrelated disorder the Harris criterion is modified: for a < d the disorder is relevant, if the correlation length critical exponent of the pure system obeys v < 2/a. For a > d the original Harris criterion (3) remains valid. Again, this criterion cannot be applied directly to the polymer case, due to the peculiarities of the polymer limit. The set-up of the paper is as follows. In the following section we present the fieldtheoretical description of the polymer model, introduce different types of structural disorder into this model and present an introduction to real space renormalization. Section 3 reviews different treatments of these systems by field theoretical and real space RG approaches to analyze the scaling properties and to estimate the critical exponents. In the present review we focus on static properties of polymers. RG treatments of the impact of disorder on polymer dynamics may be found e.g. in [44-48].

2. MODELING THE SYSTEM 2.1. Polymers: Field theory with zero-components Let us consider the model that we use to describe a single polymer chain in solution. In a first discrete version we describe a configuration of the polymer by a set of positions of segment endpoints rf. Configuration^, ...,rN}€

MdxN.

Its statistical weight (Boltzmann factor) with the Hamiltonian H divided by the product of Boltzmann constant &B and temperature T will be given by

exp[--l-if] = exp{--L f > - r;_02 - ^

£

5\n - r3)}.

(7)

The first term describes the chain connectivity, the parameter £o governs the mean square segment length. The second term describes the excluded volume interaction forbidding two segment end points to take the same position in space. The parameter /? = 1/kT gives the strength of this interaction. The third parameter in our model is the chain length or number of segments N. The partition function Z is then calculated as an integral over all configurations of the polymer divided by the system volume fi, thus dividing out identical configurations just translated in space: z N

( ) = ^jfld^M-^TH{n}}.

(8)

This will give us the 'number of configurations' of the polymer, which obey the scaling relation (1). We will do our investigations by mapping the polymer model to a renormalizable field theory making use of well developed formalisms (see [1,9,7] for example). To this end we make use of a continuous version of our model as proposed by Edwards [49,50]. The configuration of polymer is given by a path r(s) in d - dimensional space lRd parameterized by a variable 0 < s < S that has the dimension of a surface. The Hamiltonian H is then given by

108

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

where Mo is the excluded volume parameter with densities p(r) = Jo ds6d(r — r(s)) . In this formalism the partition function is calculated as a functional integral:

Z(S) = Jv[r(s)}eM-^fH[r}}-

(10)

Here the symbol 2?[r(s)] includes normalization such that Z(S) = 1 for u = 0. To make the exponential of 5-functions in (10) and the functional integral well-defined in the bare theory a cutoff s0 is introduced such that all simultaneous integrals of any variables s and s' on the same chain are cut off by | s — s' |> SQ. Let us note here that the continuous chain model (9) may be understood as a limit of a model of discrete self-avoiding walks, when the length of each step is decreasing £Q —> 0 while the number of steps N is increasing keeping the 'Gaussian surface' S = Nd\ fixed. The continuous chain model (10) can be mapped onto a corresponding field theory by a Laplace transform in the Gaussian surface variables 5 to conjugate chemical potential ("mass variables") \i [9]: />oo

Z(n) = / Jo

dSe-»sZ(S).

(11)

The Laplace-transformed partition function Z(fi) can be expressed as the m = 0 limit of the functional integral over vector fields $ with m components {i,..., (j>m} :

Z(M) = I V[4,{r)] eM-K[4>}} |m=o.

(12)

The Landau-Ginzburg-Wilson Hamiltonian 7i[(j)} of this ^-theory then reads

W[ = fddr [i(/^(r) + |V0(r)|2) + |(^(r)) 2 ]

(13)

here <j>2(r) = Y^ii&i)2 Note that this theory is symmetric under O(m) transformations of the m-component vectors <j>. The limit m = 0 in (13) results in the cancellation of some special types of diagrams contributing to the perturbation theory expansions, which contains closed loops and thus are proportional to m. The same field-theoretical representation may be obtained starting from the quite different type of lattice model known as one of the basic models in the theory of magnetic systems. We present it here since for what follows it serves to introduce different types of disorder in the polymer system. Let us consider a simple (hyper) cubic lattice of dimension d, and to each site prescribe a m-component vector S(r) with a fixed length (for convenience one usually sets | 5 | 2 = m). Imposing a pair interaction with the energy proportional to the scalar products between pairs of spins, this defines the Stanley model (also known as the O(m) symmetric model). The Hamiltonian of this model reads [77]:

n = - E J^

^(r')'

(14)

0 is a bare coupling constant. Note that the effective Hamiltonian (13) preserves the O(m) symmetry of the Stanley model (14). As far as the Stanley model is in this sense equivalent to the O(m) symmetric (j>4 theory, the analytic continuation m —> 0 of this model again leads to the polymer limit. 2.2. Randomness: symmetry consideration To introduce different types of disorder in the polymer system, we start from the mcomponent model (14). The presence of impurities can be modeled by a class of sitediluted models [51]. One considers some fraction of the sites to be occupied with some defects. The site-diluted Stanley model is introduced by the Hamiltonian: -H = -J2JcrCr>S(r)S(r'),

(15)

{r,r>)

where the occupation numbers cr are introduced, which equal 1 when the site r is occupied and 0 when it is empty. 2.2.1. Uncorrelated disorder Let us consider the case when the values cr in (15) are non-correlated random values distributed according to the probability distribution: V({Cr})=\{P{Crl r

£7>({c r }) = l

(16)

cr

with:

**>={(!-,$: 5*:i

(17)

Here p is the spin concentration and 1 — p the concentration of impurities. The pure non-diluted lattice corresponds to the case p = 1. To explain possible generalizations of the model (15) with site occupation distribution (16) let us note, that the first two moments of the distribution determine the critical behavior of the model. For the distribution (16) one gets: (cT)=p, g(\f-

f |) = (crcr,) - (cr>2 = p(l - p)S(r - r')

(18) (19)

where (...) means averaging with the distribution function (16), and 8(r — r') is Kronecker's delta. Let us introduce the notation VQ = —p(l — p) for the following. As far we consider the case of quenched disorder, the free energy of the system is obtained by averaging the logarithm of the partition function Z over the disorder distribution [15]; this amounts to use so-called replica trick [52] writing the logarithm in the

110

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

form of the following limit: In Z = lim Z

~1.

(20)

While the powers of Z can only be evaluated for integer values of n, analytical continuation in n is assumed to perform the limit n —> 0. As a result one ends up [53] with the effective Hamiltonian:

* = E / " ^ U [ ^ » + v4(r)2] + ^(^(r)) 2 }-f £ [d\$l(r)$j(r).(21) Here, Greek indices denote replicas. Comparing this expression with the Hamiltonian of the pure model (13), we notice an additional interaction of the order parameter field: the last term presents the effective interaction between different replicas. The coupling constant VQ is proportional to the variance of the disorder distribution. The coupling UQ must be positive, otherwise the pure system undergoes a 1st order transition. For final results the replica limit n —> 0 has to be taken. There exists a second way to obtain the effective Hamiltonian (21). A weak quenched disorder term can be introduced directly into the effective Hamiltonian (13). The presence of non-magnetic impurities in a microscopic model (15) manifests itself in fluctuations of the local temperature of the phase transition. Introducing ip = ip{r) as the field of local critical temperature fluctuations, one obtains the effective disordered Hamiltonian [53]:

%S) = f*dr{^\\V$\2 + (ti + m\2} +^\$\4}-

(22)

The Hamiltonian (22) depends on a number of macroscopic parameters that describe the specific configuration of the field ip(r). On the other hand, the observables should not depend on the specific realization of the random field tp and are to be averaged over the possible configurations of ip [15]. In particular, the singular contribution to the free energy of the diluted quenched Stanley model (15) can be written in the form of a functional integral:

FcxJDty(r)}P[iP(r)}\nZ{iP(r)},

(23)

where the configuration-dependent partition function Z[i/i(r)] is the normalizing factor of the Gibbs distribution with effective Hamiltonian (22); P[ip] defines a probability distribution of the field ip{r). Introducing n replicas of the model (22) and taking that tp obeys a Gaussian distribution: P(V0 = ^ - e x p ( - V 7 4 w 2 )

(24)

with ui2 being the dispersion parameter, one again ends up with the effective Hamiltonian (21).

Renormalization group approaches to polymers in disordered media

111

The model (21) is interesting in the polymer limit m —t 0. In this case it can be interpreted as a model for SAWs in disordered media. Note that such a limit is not trivial. As noted by Kim [62], once the double limit m, n —> 0 has been taken, both w0 and v0 terms are of the same symmetry, and an effective Hamiltonian with one coupling Uo = uo — Vo of the 0(mn = 0) symmetry results. This leads to the conclusion that weak quenched uncorrelated disorder does not change the universal critical properties of SAWs. These results were confirmed by numerical [16-19,26,63,64] and analytical methods [38,39].

2.2.2. Long-range correlated disorder In the above case of the diluted d-dimensional m-component spin model the disorder is correlated according to (16). Another type of disorder was proposed in the work of Weinrib and Halperin [65], with a pair correlation function of defects that decays at large distances \f— f1] according to a power law: g(\r - f\) = (crcr.) - (cr)2 ~ |f- r T a

(25)

where a is some constant. This model describes so called extended impurities in the system. In particular, the correlation function (25) with o = d — 1 describes randomly oriented impurity lines, while planar defects correspond to a = d — 2. In magnets, longrange-correlated disorder may be present in the form of continuously distributed dislocations and disclinations, so-called extended structural defects. These defects may have the form of lines or planes of random orientation [69] or may form some sponge-like fractal objects, which are considered as aggregation clusters [70]. The particular case of systems with dislocation lines or planes of parallel orientation, is investigated in Ref. [73]. The Fourier transform of the correlation function (25) at small k has the form [65]: g{k) ~vo + woka-d,

(26)

where Vo and u>o are some constants. Let us recall, that for the case of point-like noncorrelated defects the correlation function (19) reads g(\f — f\) = voS(\f— f \ ) , so its Fourier transform for small k behaves as:

g(k) ~ vo.

(27)

Comparing (26) and (27), we can see that the case a = d describes point-like disorder. Applying the replica method in order to average the free energy over different configurations of quenched disorder one finds the effective Hamiltonian of the m-vector model with long-range-correlated disorder [65]:

^eff=E /d d r[i(/x^ + (v4) 2 + 5(^) 2 ] - E fddxddyg(\x-y\)$i(x)^(y)(28) Here, the replica interaction vertex g(x) is the correlation function with Fourier image (26). Passing to the Fourier image in (28) and taking into account (26), an effective Hamiltonian results that contains three bare couplings U^,VQ,W^. For a > d the Wo-term is

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V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

irrelevant in the RG sense and one obtains an effective Hamiltonian of a quenched diluted (short-range correlated) m-vector model (21) with two couplings Mo,^o- For a < d we have, in addition to the momentum-independent couplings, the momentum dependent one Woka~d. Note that g(k) is positively definite being the Fourier image of the correlation function. From here one gets w0 > 0 at small k. A one-loop approximation for the model was given in Ref. [65] using an expansion in e = i — d,S = A — a. A new long-range-correlated fixed point was found with a correlationlength exponent v = 2/a and it was argued, that this new scaling relation is exact and also holds in higher order approximations. This result was questioned recently in Refs. [66,67], where the static and dynamic properties of 3d systems with long-range-correlated disorder were studied by means of the massive field-theoretical RG approach in a 2-loop approximation, for different fixed values of the correlation parameter, 2 < a < 3. The /3 and 7 functions in the two-loop approximation were calculated as an expansion series in renormalized vertices u, v and w. The study revealed the existence of a stable random fixed point with u* ^ 0, v* ^ 0, w* ^ 0 in the whole region of the parameter a. The obtained exponents v for various a however violate the supposed exact relation v = 2/a of ref. [65]. The authors assign difference to a more accurate field-theoretical description using higher order approximations for the 3d system directly together with methods of series summation. Recently, Ballesteros and Parisi [68] presented Monte-Carlo simulations of the diluted Ising model in three dimensions with extended defects in the form of lines of parallel orientation, confirming that the simulated critical exponents for the Ising model agree fairly well with theoretical predictions of Weinrib and Halperin. We are interested in the polymer limit m —> 0 of the model (28) interpreting it as a model for SAWs in disordered media. Note, that such a limit is not trivial, as explained in the last section. In Refs. [71,72] the asymptotic behavior of SAWs in long-range-correlated disorder of type (25) is investigated and found to be governed by a set of critical exponents which are different from that of the pure case. In ref. [48] an environment of a quenched configuration of a semi-dilute polymer solution is introduced as a special case of long-range correlated disorder for polymer dynamics. For the statics this environment is shown to be equivalent to an annealed one, i.e. without impact. 2.3. Geometry: Percolation system 2.3.1. Percolation clusters and the Potts model Like the SAW, the problem of percolation can also be treated as geometrical critical phenomenon [74]. To introduce the percolation problem, let us consider a regular hypercubic d-dimensional lattice, where either the sites or bonds are occupied with probability p. In percolation one asks questions concerning the connectivity of occupied bonds. Sets of mutually connected bonds form cluster. One can then ask what is the probability that there is a cluster spanning from the one end of the lattice to the opposite end; in the thermodynamic limit (the number of sites goes to infinity) this spanning cluster becomes the infinite cluster. An important quantity in percolation theory is the percolation probability P(p), which gives the probability that given site belongs to the infinite cluster. One can show that there exists a critical value pc (also called the percolation threshold) such that P(p) is

Renormalization group approaches to polymers in disordered media

113

d = 2 [99] d = 4[98] d = 3[98] d = 5 [98] d> 6 91/49 2 .51 2 3.05 4 2 5 3.69 df Table 1 Fractal dimension of percolation clusters in dimensions d = 2 , . . . , 6.

zero when p < pc. For p > pc, it obeys the following scaling law, approaching pc from above: P{p)~\p-Pc\0pc-

(29)

where /3pc is an universal critical exponent. Because of this property, we can consider P(p) is an order parameter. The correlation length of the percolation lattice (or connectedness length) is defined as follows:

X>yP« eP = ^—{P-PC)-2VPC,

(30)

3

where p^ is the probability for two sites i and j at the distance r,j = |r*j — r,| to belong to the infinite cluster; vpc is the critical exponent. Let us note, that the infinite percolation cluster itself is a fractal with fractal dimension dependent on d: d;{d) = d—Ppc{d)/i/pc(d) [74]. The estimates for the fractal dimensions of the percolation cluster are given in Table 1. Like SAWs that are related to the O(m)-model, percolation is connected to the qstate Potts model. The g-state Potts model [86] is a spin model whose high temperature expansion (when q —> 1) corresponds to diagrams that are percolation configurations. In the Potts model to the each site x of a regular lattice a spin variable ax corresponds which can be in q possible states (q = 2 corresponds to the Ising case). Interactions favor nearest neighbors which are in the same state. The effective Hamiltonian for the model reads:

H = JJ2

^r> ~ 1)

(31)

where dmiyi is the fractal dimension of the minimum path on a fractal.

(37)

Renoimalization group approaches to polymers in disordered media

115

Another kind of Flory-type formula is suggested in [30], where it was argued that the spectral dimension ds of the fractal percolation cluster must be an intrinsic property: (38>

-

In this problem there is one point to clarify. If one believes that the SAW can only live on the backbone of percolation cluster (otherwise it will be trapped at the dangling ends) [59,35,34,31], one should use in the expressions above the corresponding fractal characteristics (dB,df) for the backbone. A more sophisticated expression was proposed in [34]: p

" d f ( 2 d f - d f + 2)'

[

™>

where df is called the spreading or the connectivity dimension of the backbone of the percolation fractal. A more general version of Eq. (35) with the additional condition (37) was derived in [36], assuming for the probability distribution P(N,R):

p fl

w >*{-"[ c i G£f) f l + C *{N^)]}>

(4O)

representing the requirement, that for small R the free energy of a SAW is dominated by a term (N/RdB)a, which may be thought as being the repulsive energy between distant basic units, and for large R by a term (R/Nlldmin)s, which represents a configurational entropic term. Here, dmin is the fractal dimension of the shortest SAW on percolation cluster, a and 5 are unknown positive exponents. For the critical exponent vp the following expression results:

d? + kdf

"

p=

df(df + ky ™thk =s/a-

( 41 )

The mean-field result for this ratio is given in terms of the random walk dimension d^\

k= J^n_^

(42)

Another possible expression for vv was proposed in [32], using geometrical considerations for the node-link picture of the backbone of the percolation cluster. For the diluted lattice just above the percolation threshold the lattice can be viewed as a collection of nodes which are connected by links, thought of as random paths. Two important lengths enter this picture. One is the distance between nodes which is of order of the percolation correlation length (30), the other important length is the length L of the random path between nodes which is given by a separate exponent Q through: Loc|p-pc|- vAC > vpure,

d < 4,

(49)

since the fractal structure of the clusters should enhance the self-avoiding effect. The position-space renormalization-group (PSRG) approach was used in [30] to estimate i/IC in two and three dimensions.

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V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

More recently, Kim [35] claimed, based on a scaling argument, that the inequality of the type (49) is invalid and VJC = VAC should hold. It was also argued, that that the critical behavior of the IC averages is also similar to that on the backbone (BB) of an infinite cluster, suggesting: vie = vAC = VBB,

d < 4.

(50)

Both equalities are based on the following argument. The infinite cluster generated near pc in general contains many dead ends and "dangling loops" that are connected to the backbone via a single path. Since the walk that visits such bonds is eventually terminated after a finite number of steps, the walks that survive in the limit N —> oo are those confined to the backbone. With Eq. (50) and the inequality: dmin < —

< df

(51)

VBB

where dmin is the dimension of the shortest SAWs between two points on the infinite cluster and d® the fractal dimension of the backbone, Kim claimed that both VAC and vie should be larger than vpure because otherwise the last inequality breaks down for d > 4 since dB < 2. The MC simulation of Kremer [16] has been referred to as the only numerical estimate for vp for a number of years, until Lee et al. [17] performed simulations for SAWs on square and cubic lattices at dilution very close to the percolation threshold. Their results for the SAWs critical exponent VJC rather surprisingly indicate that the exponent UJC even at pc is close to the pure result. Furthermore, numerical uncertainties in the [16] were indicated; namely, it was noted, that the chains used in that simulation are probably not long enough to estimate the critical exponents. The validity of either (49) or (50) has been tested numerically for the first time in [19]. Here, the critical behavior of the AC and BB averages of SAWs on two-dimensional site-percolation for a square lattice has been studied via Monte Carlo simulations. It was found, that VAC is very close to the pure lattice value, while VBB = 0,77 0.01 which is about 3% larger than the full lattice value, concluding that the relation VBB > vic holds. In conclusion, it was claimed, that exponent vp, governing the scaling behavior of SAWs on the percolation cluster is close to the pure lattice value and, if any, the difference would be very small and virtually unobservable in Monte Carlo simulations. In Ref. [26] a detailed numerical study of enumerated SAWs on (Monte Carlo generated) random site diluted lattices in two and three dimensions was carried out. The nonsatisfactory results of MC simulations were explained in particular with difficulties in realizing the SAW trajectory on the percolation cluster. The percolation fractal contains self-similar regions connected by linearly or singly connected links, and MC diffusion of SAWs through these links is extremely difficult. This forces the MC generated SAWs on the percolation cluster to be confined (localized) practically to a some region of the percolation cluster of a size determined by the lattice size and the particular realization. As a result, this leads to incorrect estimates for the exponent of the mean-square end-toend distance of SAWs on the percolation cluster.

Renormalization group approaches to polymers in disordered media

119

2.3.4. Effective Hamiltonian The field theory for SAWs on the percolation cluster developed in Ref. [22] supports an upper critical dimension dup = 6. The calculation of vp was presented to the first order of perturbation theory, however the numerical estimates obtained from this result are in poor agreement with the numbers observed by other means. In particular, they lead to estimate that vp ~ v in d = 3. Recently this investigation has been extended to the second order in perturbation theory [101], which leads to the qualitative estimates of critical exponents in good agreement with numerical studies and Flory-like theories. To obtain the effective Hamiltonian, describing the behavior of SAWs on the percolation cluster, we make use of results from [91,22,101]. Let us consider the m-vector spin model on a diluted lattice with bond randomness, described by the Hamiltonian: H = - Y, JcrT'S(r)S(r'),

(52)



which defines an equivalent Hamil-

P

Performing the expansion of the logarithm in this expression one finds after some transformations: oo

n

m

~/^=E {0>>££^(r)S-(r') W(-9) Q

(t)

+!r E EEE^(^ (0 (w w (^(8i+s+ft)

(58)

?1,«2,«3 (t) (0 (»)

where the notations have been introduced <j)^(q) for ^'"'^{q) 771

£(...)standsfor]r; (t)

^

and

71

£

(...);

(59)

* ii,...,i( = lai,...,Qt=l Qi and L insertions of the type <j>2 as follows: 5(E^+E^)r^iV)({fc};M;^;{Ao})=/Aoe^«.+^^)x (^(n).. .02(rL)(-Ri).. .(KRrffe'/d'RL

. .ddRNddri.. AdrL.

(69)

here {p} and {k} are sets of external momenta, Ao is a cutoff and (...) stands for averaging with the corresponding effective Hamiltonian, {Ao} stands for the set of bare couplings of the effective Hamiltonian of the system: for instance, in the case of a model with point-like uncorrelated disorder, which is described by the effective Hamiltonian (21), one has two couplings: Mo and VQ. An intrinsic feature of the functions (69) is their divergence in the asymptotic limit Ao —> oo. In order to control the divergences and to map the divergent mathematical objects to convergent physical quantities one performs a rearrangement of the series for the vertex functions. Renormalization factors Z$ and Z$i for thefieldsare introduced defining the renormalized function F ^ ' . r(/lL)({*}; M; \?; {A}) = z^zN/v^N\{k}-

M; &{\Q})

(70)

Z^ and Zp are constructed such that 1^ has no divergences order by order in perturbation theory. In this procedure the mass fj,0 and couplings {Ao} are renormalized to fi and {A}. Here we describe in more detail two effective schemes, that are used in the context of the model, considered in our review: the massive and minimal subtraction schemes. 3.1.1. Massive scheme In the massive scheme [80,81], the renormalization is performed at non non-zero mass and zero external momenta, passing from the initial parameters of the model /J,Q, {AO} to the renormalized mass \i and coupling constants {A}. The regularization scheme suggested in (70) is not unique. To define the regularization scheme completely one imposes renormalization conditions for the renormalized vertex functions [4,80]. For theories, where the interaction is represented by 04-term in the Hamiltonian (e.g., (21)), these conditions read: lt'2)(fc,-£;/x2,{A})|fc=o

= /A

(71)

2)

= 1,

(72)

= M4"%,

(73)

2

^lf (*,-fc;/x ,{A})U =0 r2$(W;/* 2 .W)| {fci }=o 2)

r£' (p; £,-£;/?, {A})|^p=0 = 1.

(74)

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V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

In the case of long-range-correlated disorder we have another global parameter a along with d, and for the renormalization of the coupling Wo in Eq. (28) one imposes [67]:

0{fc};/A{A})L=o = M4-a™

(75)

with a renormalized coupling w. For the theory, where the interaction is represented by 3-term in the Hamiltonian (e.g., (58)), instead of (73) one has: rL°AJ)({fc};M2,{A})|{fci}=o = M 6 - %

(76)

The scaling properties of the system asymptotically close to the critical point are expressed by the homogeneous Callan-Symanzik equation [4,80] for the renormalized vertex function:

{%+E^({ A })^-(y- L )^(W)+ L ^({ A })} r R >JV) (M;W;M 2 ,{A}) = o, (77) with the coefficients defining the RG flow of the couplings (/3-functions) and the critical exponents (7-functions) 7^ and 7^2: pxA{x})

=

aW { A ° } '"°' ^

=

a T n V f ^ ' 7*2 ~ ~9Tny 2

2

h

_ d \f

~J (9 +i) ' -w[l

d g y-" 1

[*+*]'+iw

(107)

Note that in contrast to the usual 4>4 theory the 74, function in Eq. (106) is nonzero already at the one-loop level. This is due to the fc-dependence of the integral I4 in Eq. (107). Similarly as explained in subsection 3.2, one can proceed either by considering the polynomials in Eqs. (104), (105) for fixed a,d and look for the solution of the fixed

132

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

point equations (it is easy to check that these one-loop equations do not have any stable accessible fixed points for d < 4), or by evaluating these equations in a double expansion in e = 4 — d and 5 = 4 — a as proposed by Weinrib and Halperin [65]. Substituting the loop integrals in Eqs. (104)-(106) by their expansion in e = 4 — d and 5 = 4 — a: 1 26 £ r J / , ^ T - U l *\ I (l ~\ T -l(S-£] l2 1 h -~sV~2)' -~8{ ~2)' ~ 2^~e ^ " ~2~) ' 4 ~ 6 \~2~ ) ' one obtains the 3 fixed points given in the table 3.4. The following conclusions may be h

W* Fixed Point u* w2 —E 0 Gaussian (G) 0 -5 0 Pure SAW (P) e e/2-5 e 26* S(e-2S) h Ve2 - L[s6 + 8S2} Long-range (LR) (e-S) \{e-A5-. (e-S) Table 3 Fixed points and stability matrix eigenvalues in the first order of the e, S - expansion [72].

drawn from these first order results: Three distinct accessible fixed points are found to be stable in different regions of the a, d-plane: the Gaussian (G) fixed point, the pure (P) SAW fixed point and the long-range (LR) disorder SAW fixed point. The corresponding regions in the a, d-plane are marked by I, II and III in Fig. 6. In the region IV no stable fixed point is accessible. For the correlation length critical exponent of the SAW, one finds distinct values vpuTe for the pure fixed point and Z/LR. for the long-range fixed point. Taking into account that the accessible values of the couplings are u > 0, w > 0, one finds that the long-range stable fixed point is accessible only for 5 < e < 26, or d < a < 2 + d/2, a region where power counting shows that the disorder is irrelevant. In this sense the region III for the stability of the LR fixed point is unphysical. Formally, the first order results for d < 4 read: v_ v

- \

f *v re = l/2 + £ /16, VLR

= 1/2 + 5/8,

S<s/2, e/2 amarg) the polymer coil swells with increasing correlation of the disorder. The self avoiding path of the polymer has to take larger deviations to avoid the defects of the medium. 3.5. RG theory for SAWs on the percolation cluster In the following we elaborate the renormalization group scheme for the theory of SAWs on the percolation cluster as described by the effective Hamiltonian developed in section 2.4. Extending the ideas of Meir and Harris [22] in this respect we refer to this as the MH-model. The motivation for this model is to calculate the average of a logarithm - as usual for a quenched average-

F(p,K)=J2hj^gGij(K)}p

(113)

3

with log dj (K) the generating function of SAWs with a fugacity K per step between the sites i and j . The sites are connected if 7y = 1 (else 7y = 0). The average over bond occupation is denoted by [-}p. As usual this can be done in terms of fcth moments in the

136

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

limit k -4 0. MH derive that

G^(P,K) = X>yG%(#)]p

(H4)

3

=

Xp

+ kF(p,K) + O(k2)

(115)

where the susceptibility \v is given by the k = 0 moment. In terms of a spin model this means that the MH model seeks to calculate the scaling behavior of the moments = O7ySi(t)Si(j)>£]p (n6) 5 in the polymer limit with zero component spin fields S(i) averaging over the disorder at the critical concentration p = pc and over spin configurations (-)T at temperature T. While the 4-theory discussed in section 3.3 does not provide such averages it is essential that these can be performed in the framework of the MH model. With the effective Hamiltonian derived in section 2.4 it turns out that the moments G^ correspond to the propagators of this theory with masses T> that reflect the fact that there is a distinct critical point associated to each moment, i.e. there is no multicritical point as in the spin models with finite numbers of components and as suggested by the d > 4 interpretation of the 3fieldtheory the diagrams and integrals that appear are identical to those of the standard <j>3fieldtheory of the Q = 1-Potts model. However, we have to carefully deal with the overhead of spin component and replica indices and the algebra that results from their combinatorics. In the following we review how this is worked out in the two loop approximation in Ref. [107]. Only those 3 cubic terms are allowed in the perturbative expansion of the theory for which any pair of indices (QJ,/3J) on the contributing factors appear exactly twice. In terms of the diagrams where each line carries the indices of the corresponding field any pair of indices that "flows" into an interaction vertex must flow out on another line. Furthermore, the polymer limit of m — 0 components for the p indices implies that diagrams in which some indices /?; flow on closed loops i.e. do not appear on external lines, have vanishing contributions. As noted above we are interested in the k —t 0 limit of the algebra that is generated by replica and component index summation together with the combinatorics of the mass parameters r>. It is a somewhat tedious but straightforward matter to show that with respect to the evaluation of the standard 3 theory (see [108]) there is no change to the contributions to the vertex functions F ^ which controls the renormalization of the coupling w0 and the inverse propagator F' 2 ' in this limit. However, the contributions to the vertex functions F^1'2' with a mass insertion develops a characteristic behavior in linear and higher orders of k as we demonstrate in the following. Let us for simplicity consider the one loop contribution in detail. The one loop diagrams and their integrals contributing to F ^ and F(1>2) are labeled A\ and A[ respectively in Fig. 8. For any particular choice of the indices on the in- and outgoing lines of diagram Ai its contribution is the sum over all labellings of the inner lines that are compatible with the above rules. For k label pairs on the outer

G{k\p,T)

Renormalization group approaches to polymers in disordered media

137

Figure 8. Graphs of one- and two- loop contributions to <j>3 - theory. Graphs A,: contributions to r^2', Graphs £?;: contributions to F' 3 \ Graphs A\: contributions to F^'^.The insertion is marked by a dot. Graphs A' and B in the same column have identical contributions [108].

lines we have thus to sum over the possibilities to distribute those among the inner lines. Representing the masses r^ by an expansion r^ = Y^,eueke, this combinatorics gives for the first order contribution to F^{p, 7>) = 1 + W Q I I I ^ P , UJ) + . . .

Si,t(p, «,) = - o E 2

) /

d

^ W i

V r

-A2.^

= F^ 2 ) + kV^f + k2^

+ ...

(119)

f 1 2^

Then, with FQJ we recover the standard mass insertion while the one loop approximation to T\ I is the term considered by MH. The generalized two loop contribution to the bare vertex function may be written as

r£2> = 1 + wla^A!, + wt J2 °%A'V 9 =2

(120)

138

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

with the integrals A'q corresponding to the diagrams given in Fig. 8 and coefficients afj derived from the combinatorics of distributing indices among the lines of the diagrams. This is calculated in terms of the sums [107]

k-l

, , \ 8-1

/ \

«f = E (:)§(;>' A:—1 / , \ n (3)

_

9

s

/ \

k—s /j

\

v ^ (k\ s r I s ] Y ^ ( k ~ s ] J a 8=1 \

^ k—1

S

/T

/

r=0 ^ ^

x k—s—1

/7

(=0 ^

o

o

\

«?> = i £ ' E ( r ' ) - 1 s=l

^ '

r=l

^

n « i

'

0 else 0X = 0 control that only those terms contribute which have at least one index pair on each line. The /c-expansions of these series af' = ail + kafl + k2or^l + ... then define the coefficients afj. For our considerations we need ao,o = (-1,1,4,1,1/2,1) "i,i = (-1/2,1/3,2,2/3,1/6,1/4)

(128) (129)

to calculate the two-loop approximation of IQQ ,T\ [ . The minimal renormalization of these by Z$i-factors as explained in section 3.1.2

would the define exponents 7L of the percolation system for £ = 0 and of the polymer for £ = 1. However, it turns out that such a multiplicative renormalization does not exist for F^i . We have thus to resort to an additive renormalization procedure by taking into account the coupling between the different orders in k. i.e. the fc-linear contributions of if' 2 '. With a2>1 = (1/4, -5/18, - 1 , -5/18, -5/36, -3/8)

(131)

we find that the fc-linear term of the two-loop linear combination

(Clilr1f + c^f)

(132)

Renormalization group approaches to polymers in disordered media

139

can now be renormalized multiplicatively by a Z-factor if c^i = 1 + C2,i/2 and c2)1 = 1/2 while preserving the one-loop result [101,107]. The /3-function and the 7^exponent are determined by the minimal subtraction scheme (see sect. 3.1.2) for the standard 03-theory. With the renormalizing Z^-factors for the expressions in eqs.(130), (132) this the defines corresponding exponents 7^,2 according to eq. (87). We find the standard result for 7^' and [101] 7 $ = - e / 7 - 604e2/9261,

(133)

Together with the standard result for 74, and the scaling relation i/~l = 2 — 70 — 74,2 this brings about the correlation length exponent for the polymer in the percolation system [101] VW

=

Vp

= 1/2 + e/42 + 110 e1;/213

(134)

recovering the MH one-loop result and extending it to the second loop order. As far as the coefficients in this truncated series are small we may evaluate it without resummation. The numerical values of vp at fixed space dimension is then obtained by direct substitution Vp(d = 2) = 0.785, vp{d = 3)= 0.678, vp(d = 4) = 0.595, which is in good agreement with results of MC simulations, exact enumerations and Flory-like formulae.

Figure 9. The correlation exponent vp. Bold line: (133), thin line: one-loop result [22], filled boxes: Flory result i/p = 3/(dpc + 2) with dpc from [109]. Exponents for the shortest and longest SAW on percolation cluster [110] are shown by dotted lines.

From the physical point of view, this result for the exponent up together with the data of EE and Flory-like theories predicts a swelling of a polymer coil on the percolation

140

V. Blavats'ka, C. von Ferber, R. Folk and Yu. Holovatch

cluster with respect to the pure lattice: vp > Z^SAW for d = 2 — 5. At d = 3 the formula (133) predicts a 13% increase of vv with respect to I'SAW which is larger than at d = 2 (5%) and should be more easily observed by current state-of-art simulations. Given that even at d = 2 we are in nice agreement with MC and EE data and the reliability of the perturbative RG results increases with d, this number calls for verification in MC experiments of similar accuracy. The validity of the MH model has been questioned in Ref.[39] by an argument which in our present formulation reads that the one loop contributions to T\ \ ' for higher I indicate the instability of the fixed point with respect to each parameter U(. These authors explain that it remains to be shown that the instability with respect to ue with i > 1 does not influence the renormalization of the d/ctai-insertion. To understand how this may be resolved in the present approach it is instructive to consider the insertion corresponding to a derivative in u2. For this case one finds in the same way as before a multiplicative renormalization for the fc2-term of [107]

(c 2 l 2 r^ 2 ) + C 3 , 2 r^ 2 ) )

(135)

with coefficients C22 = 1 + C32/2 and C32 = 3/104 using the following results for the a-factors a2i2

=

(-3/4,11/18,3,17/18,11/36,5/8),

(136)

a3i2

=

(3/8,-19/36,-3/2,-11/36,-19/72,-9/16).

(137)

This way we may define a corresponding exponent [101] 7 ^ = -3