Mathematical Methods of Many-Body Quantum Field Theory

Mathematical Methods of Many-Body Quantum Field Theory

CHAPMAN & HALL/CRC Research Notes in Mathematics Series Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor) Editorial Board R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware D. Jerison, Massachusetts Institute of Technology B. Lawson, State University of New York at Stony Brook

B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

Submission of proposals for consideration Suggestions for publication, in the form of outlines and representative samples, are invited by the Editorial Board for assessment. Intending authors should approach one of the main editors or another member of the Editorial Board, citing the relevant AMS subject classifications. Alternatively, outlines may be sent directly to the publisher’s offices. Refereeing is by members of the board and other mathematical authorities in the topic concerned, throughout the world. Preparation of accepted manuscripts On acceptance of a proposal, the publisher will supply full instructions for the preparation of manuscripts in a form suitable for direct photo-lithographic reproduction. Specially printed grid sheets can be provided. Word processor output, subject to the publisher’s approval, is also acceptable. Illustrations should be prepared by the authors, ready for direct reproduction without further improvement. The use of hand-drawn symbols should be avoided wherever possible, in order to obtain maximum clarity of the text. The publisher will be pleased to give guidance necessary during the preparation of a typescript and will be happy to answer any queries. Important note In order to avoid later retyping, intending authors are strongly urged not to begin final preparation of a typescript before receiving the publisher’s guidelines. In this way we hope to preserve the uniform appearance of the series. CRC Press UK Chapman & Hall/CRC Statistics and Mathematics 23 Blades Court Deodar Road London SW15 2NU Tel: 020 8875 4370

Detlef Lehmann

Mathematical Methods of Many-Body Quantum Field Theory

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data Lehmann, Detlef. Mathematical methods of many-body quantum field theory / Detlef Lehmann. p. cm. -- (Chapman & Hall/CRC research notes in mathematics series ; 436) ISBN 1-58488-490-8 (alk. paper) 1. Many-body problem. 2. Quantum field theory--Mathematical models. I. Title. II. Series. QC174.17.P7L44 2004 530.14'3--dc22

2004056042

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2005 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-490-8 Library of Congress Card Number 2004056042 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

 


 

    

38 "   ' #  33  '+  #      3; %   "  ?
@ % '  ?
 ,     0 is attractive and λl0 > λl for all l = l0 . Then lim 1 d ψ¯kσ ψkσ L→∞ βL

ik0 +ek = − k2 +e 2 +|∆ l 0

k

0

where |∆l0 |2 is a solution of the BCS equation
tanh( β √e2 +|∆l |2 ) λl0 0 √2 2 k =1 2 Ld k |ek |≤ωD

2

ek +|∆l0 |

|2

(6.75)

(6.76)

94 c) Let d = 3 and suppose that the electron-electron interaction in (6.66) is given by a single, even  term, 


ˆ−p ˆ ) = λ V (k

ˆ m (ˆ Y¯m (k)Y p)

(6.77)

m=−

Then, if eRk = ek for all R ∈ SO(3), one has  ik0 +ek 1 ¯ lim βLd ψkσ ψkσ β,L = k2 +e2 +λ ρ2 |Σm α0 L→∞

S2

0

k

m Ym (p)|

0

2

dΩ(p) 4π

(6.78)

where ρ0 ≥ 0 and α0 ∈ C2+1 , Σm |α0m |2 = 1, are values at the global minimum (which is degenerate) of & '2  √2 ˆ 2) cosh( β ek +λ ρ2 |Σm αm Ym (k)| 2 dd k 1 2 W (ρ, α) = ρ − log (6.79) (2π)d β cosh β e 2

M

k

In particular, the momentum distribution is given by    √2 ek +|∆(p)|2 ) tanh( β dΩ(p) + 1 1 2 √ lim Ld akσ akσ β,L = 2 2 1 − ek 4π 2 L→∞

ek +|∆(p)|

S2

(6.80)

1

and has SO(3) symmetry. Here ∆(p) = λ2 ρ0 Σm α0m Ym (p). Proof: a) The Grassmann integral representation of Z = T r e−βH /T r e−βH0 , H given by (6.70), is given by Z = Z(skσ = 0) where (recall that κ := βLd )  
1
¯ ˆ l (ˆ Z({skσ }) = exp − 3 λl y¯l (k)y p) × ψk0 ,k,↑ ψ¯q0 −k0 ,−k,↓ κ k,p,q0 l even + ψp0 ,p,↑ ψq0 −p0 ,−p,↓ × P

¯ ψ −1 (ik −e −s )ψ Π ik κ−ek e κ kσ 0 k kσ kσ kσ Π dψkσ dψ¯kσ kσ 0 kσ  


√ 1 ˆ ψ¯k ,k,↑ ψ¯q −k ,−k,↓ × = exp − 3 λl y¯l (k) 0 0 0 κ l even q0 k0 ,k + √
λl yl (ˆ p)ψp0 ,p,↑ ψq0 −p0 ,−p,↓ × p0 ,p

Π

kσ

κ ik0 −ek

e− κ 1

P

¯

kσ (ik0 −ek −skσ )ψkσ ψkσ

Π dψkσ dψ¯kσ (6.81)

kσ

Here k = (k0 , k), k0 , p0 ∈ πβ (2Z + 1), q0 ∈ 2π β Z. By making the HubbardStratonovich transformation  P P P 2 ¯ ¯ dφ dφ − l,q al,q0 bl,q0 0 e = ei l,q0 (al,q0 φl,q0 +bl,q0 φl,q0 ) e− l,q0 |φl,q0 | Π l,q0π l,q0 l,q0

BCS Theory and Spontaneous Symmetry Breaking

95

with al,q0 := bl,q0 :=

 λ 12  l κ3

p)ψp0 ,p,↑ ψq0 −p0 ,−p,↓ p0 ,p yl (ˆ

κ3

¯l (k)ψk0 ,k,↑ ψq0 −k0 ,−k,↓ k0 ,k y

 λ 12  l

,

¯

ˆ ¯

we arrive at  Z({skσ }) =

det det

! P 2 ··· ! e− l,q0 |φl,q0 | Π l,q0 ·

¯l,q dφl,q0 dφ 0 π

(6.82)

where the quotient of determinants is given by (

)  igl ¯ ˆ p,k √ (ak − sk↑ )δk,p ¯l (k)δ l κ φl,p0 −k0 y det  igl ˆ √ (a−k − s−k↓ )δk,p l κ φl,k0 −p0 y (k)δk,p & ' = (6.83) ak δk,p 0 det 0 a−k δk,p ( ) ˆ ¯ √i Φ (ak0 ,k − sk0 ,k,↑ )δk0 ,p0 (k) κ p0 −k0 det ˆ √i Φk −p (k)  (a−k0 ,−k − s−k0 ,−k,↓ )δk0 ,p0 0 0 κ & ' a δ 0 k ,k k ,p 0 0 0 k det 0 a−k0 ,−k δk0 ,p0 and we abbreviated ˆ := Φq0 (k)




ˆ , gl φl,q0 yl (k)

ˆ := ¯ q0 (k) Φ

l




ˆ gl φ¯l,q0 y¯l (k)

(6.84)

l

Observe that in the first line of (6.83) the matrices are labelled by k, p = (k0 , k), (p0 , p) whereas in the second line of (6.83) the matrices are only labelled by k0 , p0 and there is a product over spatial momenta k since the matrices in the first line of (6.83) are diagonal in the spatial momenta k, p. As in Theorem 5.1.2, the correlation functions are obtained from (6.82) by differentiating with respect to stσ :  ( 1 κ

ψ¯tσ ψtσ =

¯ p0 −k0 (t) ak0 ,t δk0 ,p0 √iβ Φ √i Φk −p (t) a−k ,−t δk ,p 0 0 0 0 0 β

where, making the substitution of variables

)−1 dP ({φl,q0 }) (6.85) t0 σ,t0 σ

√1 φl,q 0 Ld

→ φl,q0 ,

d e−L V ({φl,q0 }) Πl,q dφl,q0 dφ¯l,q0 dP ({φl,q0 }) = * −Ld V ({φ }) 0 l,q0 e Πl,q0 dφl,q0 dφ¯l,q0

(6.86)

96 with an effective potential (

¯ p0 −k0 (k) ak0 ,k δk0 ,p0 √iβ Φ det √i Φ (k) a−k0 ,−k δk0 ,p0
1
β k0 −p0 & ' |φl,q0 |2 − d log V ({φl,q0 }) = L ak0 ,k δk0 ,p0 0 l,q0 k det 0 a−k0 ,−k δk0 ,p0

)

(6.87) * dd k  If we substitute the Riemannian sum L1d k in (6.87) by an integral (2π) d, then the only place where the volume shows up in the above formulae is the prefactor in the exponent of (6.86). Thus, in the infinite volume limit we can evaluate the integral by evaluating the integrand at the global minimum of the effective potential, or, if this is not unique, by averaging the integrand over all minimum configurations. Since the global minimum of (6.87) is at Φq0 (k) = 0 for q0 = 0 (compare the proof of Theorem 6.2.1) we arrive at (Φq0 =0 (k) ≡ Φ(k))  ( 1 κ

ψ¯t↑ ψt↑ =  =

¯ ak0 ,t δk0 ,p0 √iβ Φ(t)δ k0 ,p0 i √ Φ(t)δk ,p a−k ,−t δk ,p 0 0 0 0 0 β

|at

|2

)−1 dP ({φl }) t0 ↑,t0 ↑

a−t dP ({φl }) ¯ + Φ(t)Φ(t)

(6.88)

where d e−βL V ({φl }) Πl dφl dφ¯l dP ({φl }) = * −Ld V ({φ }) l e Πl dφl dφ¯l

(6.89)

with an effective potential V ({φl }) =


l

|φl |2 −

1 βLd




 log

¯ |ak |2 +Φ(k)Φ(k) |ak |2

 (6.90)

k0 ,k

which coincides with (6.72). b) This part is due to A. Sch¨ utte [58]. We have to prove that the global minimum of the real part of the effective potential (6.74) is given by φl = 0 for l = l0 . To this end we first prove that for e ∈ R and z, w ∈ C , ,2   , , z + iw) ¯ , ≤ cosh2 e2 + |z|2 ,cosh e2 + (z + iw)(¯ and , ,2   , , z + iw) ¯ , = cosh2 e2 + |z|2 ,cosh e2 + (z + iw)(¯

⇔

w=0

(6.91)

(6.92)

BCS Theory and Spontaneous Symmetry Breaking

97

√

a2 + b2 , a + ib = r eiϕ one has   √ - 1+cos ϕ √ √  1−cos ϕ a + ib = r cos ϕ2 + i sin ϕ2 = r ± i 2 2 r−a = r+a 2 ±i 2

Namely, for real a and b, r =

and | cosh(a + ib)|2 = cosh2 a − sin2 b which gives ,2 ,  , , z + iw) ¯ , = ,cosh e2 + (z + iw)(¯ 2 2 2 cosh2 R2 + e +|z|2 −|w| − sin2 R 2 −

e2 +|z|2 −|w|2 2

where R=

 (e2 + |z|2 − |w|2 )2 + 4(Re w¯ z )2

Since e2 +|z|2 −|w|2 2

≤ e2 + |z|2  ⇔ (e2 + |z|2 − |w|2 )2 + 4(Re w¯ z )2 ≤ e2 + |z|2 + |w|2 ⇔ (e2 + |z|2 − |w|2 )2 + 4(Re w¯ z )2 ≤ (e2 + |z|2 + |w|2 )2 R 2

+

⇔

(Re w¯ z )2 ≤ (e2 + |z|2 )|w|2

and because of (Re w¯ z )2 ≤ |w|2 |z|2 (6.91) and (6.92) follow. Now suppose that {λl } = {λ } ∪ {λm } where the λ are attractive couplings, λ > 0, and the λm are repulsive couplings, λm ≤ 0. Then ¯ k = (zk + iwk )(¯ Φk Φ zk + iw ¯k )  √  √ ¯k are where zk =  λ φ y (k), wk = m −λm φm ym (k) and z¯k and w the complex conjugate of zk and wk . Thus, using (6.91) and (6.92) , ,  
, cosh( β2 √e2k +Φ¯ k Φk ) ,2  2 d2 k 1 , , min Re VBCS ({φl }) = min |φl | − (2π)2 β log , , cosh β e φl

φl

= min φ




 |φ |2 −



l d2 k 1 (2π)2 β

2

, , , cosh( β2 √e2k +|zk |2 ) ,2  , , log , , cosh β 2 ek

Using polar coordinates k = k(cos α, sin α), φl = ρl eiϕl , we have
 g g ρ ρ ei[ϕ −ϕ +(− )α] |zk |2 = ,

k

(6.93)

98 , , , cosh( β2 √e2k +x) ,2 , is a concave , |zk | = Since W (x) := log , cosh β e and 0 , 2 k function, we can use Jensen’s inequality to obtain * 2π



dα 2π

2

d2 k 1 (2π)2 β



1 β

2  λ ρ .

, ,  , cosh( β2 √e2k +|zk |2 ) ,2 , log ,, ≤ , cosh β e 2

k

dk k 1 2π β

 dk k 1 2π β

=

, , cosh( β √e2 +R 2π k 2 0 log ,, cosh β e 2

dα 2 2π |zk | )

k

, , , cosh( β2 √e2k +P λ ρ2 ) ,2 , , log , , cosh β 2 ek

,2 , , ,

and  
2 min Re VBCS ({φl }) = min ρ − ρ

φl

dk k 1 2π β



, , , cosh( β2 √e2k +P λ ρ2 ) ,2  , , log , , cosh β 2 ek

By assumption, we had λl0  ≡ λ0 > λ for  all  = 0 . Thus, if there is some ρ > 0 for  = 0 , we have  λ ρ2 < λ0  ρ2 and, since W (x) is strictly monotone increasing,  dk k 1 2π β

, , , cosh( β2 √e2k +P λ ρ2 ) ,2 * , < , log , , cosh β 2 ek

dk k 1 2π β

, , , cosh( β2 √e2k +λ0 P ρ2 ) ,2 , , log , , cosh β 2 ek

Hence, we arrive at   min Re VBCS ({φl }) = min ρ20 − ρ0

φl

dk k 1 2π β

, , , cosh( β qe2 +λ ρ2 ) ,2  k 2 0 0 , , (6.94) log , , cosh β , , 2 ek

which proves part (b). c) We have to compute the infinite volume limit of * a−p −κV (φ)  Πm=− dum dvm R4+2 ap a−p +|Φp |2 e 1 ¯ * κ ψpσ ψpσ = −  e−κV (φ) Πm=− dum dvm R4+2 where V (φ) =


m=−

&

 |φm | − 2

M

dd k 2 (2π)d β

log

cosh( β 2

√

e2k +|Φk |2 )

cosh

β 2 ek

and (vm → −vm ) 1

Φk = λ2




ˆ φ¯m Ym (k)

m=−

Let U (R) be the unitary representation of SO(3) given by
ˆ = ˆ Ym (Rk) U (R)mm Ym (k) m

'

(6.95)

BCS Theory and Spontaneous Symmetry Breaking  ˆ =: (U Φ)k . and let m (U φ)m Ym (k)  U (R)Φ k = ΦR−1 k and  
2 V U (R)φ = |[U (R)φ]m | − m

=




 |φm |2 − M

m

=




M

 |φm |2 −

M

m

99

Then for all R ∈ SO(3) one has

dd k 2 (2π)d β

&

dd k 2 (2π)d β

log

dd k 2 (2π)d β

log

& log

cosh( β 2

cosh( β 2

cosh( β 2

e2k +|[UΦ]k |2 )

cosh

β 2 ek

√

e2k +|ΦR−1 k |2 )

cosh

&

√

β 2 ek

√

e2k +|Φk |2 )

cosh

'

'

'

β 2 ek

= V (φ)

(6.96)

 Let S 4+1 = {φ ∈ C2+1 | m |φm |2 = 1}. Since U (R) leaves S 4+1 invariant, S 4+1 can be written as the union of disjoint orbits, S 4+1 = ∪[α]∈O [α] where [α] = {U (R)α|R ∈ SO(3)} is the orbit of α ∈ S 4+1 under the action of U (R) and O is the set of all orbits. If one chooses a fixed representative α in each orbit [α], that is, if one chooses a fixed section σ : O → S 4+1 , [α] → σ[α] with [σ[α] ] = [α], every φ ∈ C2+1 can be uniquely written as ρ = φ ≥ 0, α =

φ = ρ U (R)σ[α] ,

φ
φ
,

[α] ∈ O and R ∈ SO(3)/I[α]

σ where I[α] = I[α] = {S ∈ SO(3) | U (S)σ[α] = σ[α] } is the isotropy subgroup of σ[α] . Let

* R4+2

Πm dum dvm f (φ) =

* R+

Dρ

* O

D[α]

* [α]

 DR f ρ U (R)σ[α]

be the integral in (6.95) over R4+2 in the new coordinates. That is, for example, Dρ = ρ4+1 dρ. In the new coordinates ,
 ,
,2  ,,2  , , , ˆ , U (R)σ[α] Ym (ˆ p), = λ ρ2 , σ ¯[α],m Ym R−1 p |Φp |2 = λ ρ2 , m

m

m

such that −a−p ap a−p +|Φp |2

=

−a−p ˆ )|2 ap a−p +λ ρ2 |Σm σ ¯[α],m Ym (R−1 p

≡ f (ρ, [α], R−1 p)

100 Since V (φ) = V (ρ, [α]) is independent of R, one obtains * 1 ¯ κ ψpσ ψpσ

=



−a−p ap a−p +|Φp |2

e−κV (φ) Π dum dvm m=−

(6.97)  e−κV (φ) Π dum dvm m=− * * * Dρ O D[α] [α] DR f (ρ, [α], R−1 p) e−κV (ρ,[α]) R+ * * * = −κV (ρ,[α]) R+ Dρ O D[α] [α] DR e R * * DRf (ρ,[α],R−1 p) −κV (ρ,[α]) [α] R e R+ Dρ O D[α] vol([α]) DR [α] * * = Dρ O D[α] vol([α]) e−κV (ρ,[α]) R+ *

It is plausible to assume that at the global minimum of V (ρ, [α]) ρ is uniquely determined, say ρ0 . Let Omin ⊂ O be the set of all orbits at which V (ρ0 , [α]) takes its global minimum. Then in the infinite volume limit (6.97) becomes R

lim 1 ψ¯pσ ψpσ κ→∞ κ

=

Omin

R

DR f (ρ,[α],R−1 p)

R D[α] vol([α]) [α] [α] DR R D[α] vol([α]) O

(6.98)

min

Consider the quotient of integrals in the numerator of (6.98). Since  ,  ,2  ˆ , f (ρ, [α], R−1 p) = f ρ2 ,Σm σ ¯[α],m Ym R−1 p  ,   ,2  = f ρ2 ,Σm U (S)σ[α] m Ym R−1 p ,  , ,2   ˆ , = f (ρ, [α], (RS)−1 p) ¯[α],m Ym (RS)−1 p = f ρ2 ,Σm σ for all S ∈ I[α] , one has, since [α]  SO(3)/I[α] * * * −1 −1 p) I[α] DS p) SO(3)/I[α] DR f (ρ, [α], R [α] DR f (ρ, [α], R * * * = DR DR I[α] DS [α] SO(3)/I[α] * * −1 p) SO(3)/I[α] DR I[α] DS f (ρ, [α], (RS) * * = DR DS SO(3)/I[α] I[α] * −1 p) SO(3) DR f (ρ, [α], R * = SO(3) DR * * −1 p) S 2 dΩ(t) SO(3)t→p DR f (ρ, [α], R * * = S 2 dΩ(t) SO(3)t→p DR * * S 2 dΩ(t) f (ρ, [α], t) SO(3)t→p DR * * = (6.99) S 2 dΩ(t) SO(3)t→p DR that DR has the where SO(3)t→p = {R ∈ SO(3) | Rt = p}. If one assumes * usual invariance properties of the Haar measure, then SO(3)t→p DR does not

BCS Theory and Spontaneous Symmetry Breaking

101

depend on t such that it cancels out in (6.99). Then (6.98) gives R

*

dΩ(t) f (ρ,[α],t)

D[α] vol([α]) S2 R 2 dΩ(t) Omin S * lim κ1 ψ¯pσ ψpσ = κ→∞ D[α] vol([α]) Omin

(6.100)

Now, since the effective potential, which is constant on Omin , may be written as   , ,2  dΩ(t) 2, V (ρ, [α]) = ¯[α],m Ym (t), 4π G ρ Σm σ S2

with G(X) = ρ − 2

also

* S2

*

&

dk k2 2π 2

log

cosh( β 2

dΩ(t) f (ρ, [α], t) * = S 2 dΩ(t)

√

e2k +λ X)

cosh

β 2 ek

' , it is plausible to assume that

 S2

ip0 +ep dΩ(t) 4π p20 +e2p +λ ρ2 |Σm σ ¯[α],m Ym (t)|2

is constant on Omin . In that case also the integrals over Omin in (6.100) cancel out and the theorem is proven.


We now compare the results of the above theorem to the predictions of the quadratic Anderson-Brinkman Balian-Werthamer [3, 5] mean field theory. This formalism gives  √2 ∗    tanh( β ek +∆k ∆k ) + 1 1 2 √ lim Ld akσ akσ = 2 1 − ek (6.101) 2 ∗ ek +∆k ∆k

L→∞

σσ

where the 2 × 2 matrix ∆k , ∆Tk = −∆−k , is a solution of the gap equation  √2 ∗ ek +∆k ∆k ) tanh( β dd k 2 √ ∆p = U (p − k) ∆ (6.102) k d 2 ∗ (2π) Mω

2

ek +∆k ∆k

Consider first the case d = 2. The electron-electron interaction in (6.102) is given by  U (p − k) = l λl eil(ϕp −ϕk ) (6.103) Usually it is argued that the interaction flows to an effective interaction which is dominated by a single attractive angular momentum sector λl0 > 0. In other words, one approximates U (p − k) ≈ λl0 eil0 (ϕp −ϕk )

(6.104)

where l0 is chosen as in the theorem above. The proof of part (b) of the theorem gives a rigorous justification for that, the global minimum of the effective potential for an interaction of the form (6.103) is identical to the global minimum of the effective potential for the interaction (6.104) if λl0 > λl

102 for all l = l0 . Once U (k − p) is approximated by (6.104), one can solve the equation (6.102). In two dimensions there are the unitary isotropic solutions   cos lϕk sin lϕk ∆(k) = d (6.105) sin lϕk − cos lϕk for odd l and

 ∆(k) = d

0 eilϕk −eilϕk 0

 (6.106)

for even l which gives ∆(k)+ ∆(k) = |d|2 Id such that (6.101) is consistent with (6.75). In three dimensions, for an interaction (6.77), it has been proven [21] that for all  ≥ 2 (6.102) does not have unitary isotropic (∆∗k ∆k = const Id) solutions. That is, the gap in (6.102) is angle dependent but part (c) of the above theorem states that a+ kσ akσ has SO(3) symmetry. For SO(3) symmetric ek also the effective potential has SO(3) symmetry which means that also the global minimum has SO(3) symmetry. Since in the infinite volume limit the integration variables are forced to take values at the global minimum, the integral over the sphere in (6.80) is the averaging over all global minima. However, it may very well be that in the physically relevant case there is SO(3) symmetry breaking. That is, instead of the Hamiltonian H of (6.70) one should consider a Hamiltonian H + HB where HB is an SO(3)-symmetry breaking term which vanishes if the external parameter B (say, a magnetic field) goes to zero. Then one has to compute the correlations lim lim 1d a+ kσ akσ B→0 L→∞ L

(6.107)

which are likely to have no SO(3) symmetry. However, the question to what extent the results of the quadratic mean field formalism for this model relate to the exact result for the quartic Hamiltonian H + HB (in the limit B → 0) needs some further investigation. Several authors [10, 12, 7, 36] have investigated the relation between the reduced quartic BCS Hamiltonian

+ HBCS = H0 + L13d U (k − p) a+ (6.108) kσ a−kτ apσ a−pτ σ,τ ∈{↑,↓} k,p

and the quadratic mean field Hamiltonian 

+ HMF = H0 + L13d U (k − p) a+ kσ a−kτ apσ a−pτ σ,τ ∈{↑,↓} k,p

(6.109)

 + + + + a+ kσ a−kτ apσ a−pτ − akσ a−kτ apσ a−pτ

BCS Theory and Spontaneous Symmetry Breaking

103

where the numbers apσ a−pτ are to be determined according to the relation apσ a−pτ = T r e−βHMF apσ a−pτ T r e−βHMF . The idea is that H  = HBCS − HMF  

+ + + = L13d U (k − p) a+ kσ a−kτ − akσ a−kτ × σ,τ ∈{↑,↓} k,p

(6.110)

  apσ a−pτ − apσ a−pτ

is only a small perturbation which should vanish in the infinite volume limit. It is argued that, in the infinite volume limit, the correlation functions of the models (6.108) and (6.110) should coincide. More precisely, it is claimed that 1 d L→∞ L

lim

−βHBCS

log TTrree−βHMF

(6.111)

vanishes. To this end it is argued that each order of perturbation theory, with  respect to H  , of T r e−β(HMF +H ) /T r e−βHMF is finite as the volume goes to infinity. The Haag paper argues that spatial averages of field operators * like L1d [0,L]d dd xψ↑+ (x)ψ↓+ (x) may be substituted by numbers in the infinite volume limit, but there is no rigorous control of the error. However, in part (c) of Theorem 6.3.1 we have shown for the model (6.70), that the correlation functions of the quartic model do not necessarily have to coincide with those of the quadratic mean field model. Thus, in view of Theorem 6.3.1, it is questionable whether the above reasoning is actually correct.

Chapter 7 The Many-Electron System in a Magnetic Field

This chapter provides a nice application of the formalism of second quantization to the fractional quantum Hall effect. Throughout this chapter we will stay in the operator formalism and make no use of functional integrals. Whereas in the previous chapters we worked in the grand canonical ensemble, the fractional quantum Hall effect is usually discussed in the canonical ensemble which is mainly due to the huge degeneracy of the noninteracting system. For filling factors less than one, a suitable approximation is to consider only the contributions in the lowest Landau level. This projection of the many-body Hamiltonian onto the lowest Landau level can be done very easily with the help of annihilation and creation operators. The model which will be discussed in this chapter contains a certain long range approximation. This approximation makes the model explicitly solvable but, as it is argued below, this approximation also has to be considered as unphysical. Nevertheless, we think it is worthwhile considering this model because, besides providing a nice illustration of the formalism, it has an, in finite volume, explicitly given eigenvalue spectrum, which, in the infinite volume limit, most likely has a gap for rational fillings and no one for irrational filling factors. This is interesting since a similar behavior one would like to prove for the unapproximated fractional quantum Hall Hamiltonian.

7.1

Solution of the Single Body Problem

In this section we solve the single body problem for one electron in a constant magnetic field in two dimensions. We consider a disc geometry and a rectangular geometry. The many-body problem is considered in the next section in a finite volume rectangular geometry.

105

106

7.1.1

Disk Geometry

We start by considering one electron in two dimensions in a not necessarily constant but radial symmetric magnetic field
= (0, 0, B(r)) B Let H(r) =

1 r2


r 0

(7.1)

B(s) s ds

(7.2)

then the vector potential A(x, y) = H(r) (−y, x) = rH(r) eϕ

(7.3)

rotA = (0, 0, B(r))

(7.4)

satisfies

and the Hamiltonian is given by  2 H =
i ∇ − eA   ∂ =
2 −∆ + 2ih(r) ∂ϕ + r2 h(r)2 ≡
2 K

(7.5)

if we define h(r) =
e H(r)

(7.6)

∂ K = −∆ + 2ih(r) ∂ϕ + r2 h(r)2

(7.7)

and

As for the harmonic oscillator, we can solve the eigenvalue problem by introducing annihilation and creation operators. To this end we introduce complex variables

∂ ∂z

=

such that ∂ ∂z z

=

1 2



1 2



∂x ∂x

z = x + iy,  ∂ ∂ ∂x − i ∂y ,

− i2 ∂y ∂y

 =1=

z¯ = x − iy   ∂ 1 ∂ ∂ ∂ z¯ = 2 ∂x + i ∂y

∂ ¯, ∂ z¯ z

∂ ¯ ∂z z

=0=

∂ ∂ z¯ z

Now define the following operators a = a(x, y, h) and b = b(x, y, h): √ √ ∂ a = 2 ∂∂z¯ + √12 h z, a+ = − 2 ∂z + √12 h z¯ √ √ ∂ b+ = 2 ∂∂z¯ − √12 h z b = − 2 ∂z − √12 h z¯, Note that b(h) = a+ (−h) and b+ (h) = a(−h). There is the following

(7.8) (7.9)

The Many-Electron System in a Magnetic Field

107

Lemma 7.1.1 a) There are the following commutators: [a, a+ ] =
e B(r) = [b, b+ ]

(7.10)

and all other commutators [a(+) , b(+) ] = 0. b) The Hamiltonian is given by H =
2 K where K = a+ a + aa+ = 2a+ a +
e B(r) Proof: We have   ∂ [a, a+ ] = ∂∂z¯ , h¯ = z − hz, ∂z

∂(h¯ z) ∂ z¯

+

∂(hz) ∂z

(7.11)

∂h = h + z¯ ∂h ∂ z¯ + h + z ∂z

Since h = h(r) depends only on r, one obtains with ρ = r2 = z z¯ ∂(z z¯) dh dh z ∂h ¯ ∂h ∂z = z ∂z dρ = ρ dρ = z ∂ z¯

which results in

Using H(r) =

1 r2

(7.12)

    e dH [a, a+ ] = 2 h + ρ dh dρ = 2
H + ρ dρ

(7.13)


r

B(s) s ds, one finds 
r 2 H + ρ dH − r14 0 B(s)s ds + dρ = H + r 0

1 = = B(r)r 2r

1 dr r 2 B(r)r d(r 2 )



B(r) 2

(7.14)

Thus one ends up with [a, a+ ] =
e B(r) = [b, b+ ] Furthermore [b+ , a+ ] = [a(−h), a+ (h)] = =









∂ ∂ z + hz, ∂z ∂ z¯ , h¯ ∂(h¯ z) ∂(hz) ¯ ∂h ∂ z¯ − ∂z = z ∂ z¯

− z ∂h ∂z = 0

because of (7.12). This proves part (a). To obtain part (b), observe that  2  ∂ ∂ 1 ∂ ∂2 = + (7.15) = 14 ∆ 2 2 ∂z ∂ z¯ 4 ∂x ∂y and ∂ ∂ϕ

= = =

∂y ∂ ∂x ∂ ∂ ∂ ∂ϕ ∂x + ∂ϕ ∂y = −y ∂x + x ∂y   1 ∂ − 2i (z − z¯) ∂z + ∂∂z¯ + 12 (z +  ∂  i z ∂z − z¯ ∂∂z¯

z¯) 1i



∂ ∂ z¯

−

∂ ∂z

 (7.16)

108 which gives ∂ + r2 h(r)2 K = −∆ + 2ih(r) ∂ϕ   ∂ ∂ ∂ = −4 ∂z ¯ ∂∂z¯ + h2 z z¯ ∂ z¯ − 2h z ∂z − z

(7.17)

Since ∂ ∂ a+ a = −2 ∂z ∂ z¯ −

= =

∂ z ∂∂z¯ + 12 h2 z z¯ ∂z hz + h¯   ∂ ∂ ∂ −2 ∂z ¯ ∂∂z¯ + 12 h2 z z¯ − ∂(hz) ∂ z¯ − h z ∂z − z ∂z 1 e B(r) 2K −
2

the lemma follows.
We now focus on the case of a constant magnetic field B(r) = B = const

(7.18)

eB 2


(7.19)

In that case h(r) = where

=

B =

1 22B


eB

(7.20)

denotes the magnetic length. We change variables w=

√1 z 2 B

(7.21)

∂ c+ = B a+ = − ∂w + 12 w ¯

(7.22)

− 12 w

(7.23)

and introduce the rescaled operators c = B a =

∂ ∂w ¯

+ 12 w,

∂ − 12 w, ¯ d = B b = − ∂w

d+ = B b+ =

∂ ∂w ¯

which fulfill the commutation relations [c, c+ ] = [d, d+ ] = 1,

[c(+) , d(+) ] = 0

Then the Hamiltonian becomes 2 
 +  1 1 =
eB 2m i ∇ − eA m c c+ 2

(7.24)

(7.25)

and the orthonormalized eigenfunctions are given by ¯ = φnm (w, w)

√ 1 n!m!

n

m

(c+ ) (d+ ) φ00

(7.26)

The Many-Electron System in a Magnetic Field

109

where 2

e−

1

φ00 (w, w) ¯ =

(2π2B )1/2

ww ¯ 2

=

1 (2π2B )1/2

e

− |z|2 4


(7.27)

B

The eigenfunctions can be expressed in terms of the generalized Laguerre polynomials Lα n . One obtains φnm (w, w) ¯ = =

1 (2π2B )1/2 1 (2π2B )1/2

 n!  12

wm−n Lm−n (ww) ¯ e− n

m!

 n!  12  m!

√z 2 B

m−n

Lm−n n

ww ¯ 2



|z|2 22B



2

e

− |z|2 4


B

(7.28)

We summarize some properties of the eigenfunctions in the following lemma. By a slight abuse of notation, we write φnm (z) = φnm (z, z¯) in the lemma z z¯ . below instead of φnm √2 , √2 B

B

Lemma 7.1.2 Let φnm (z) be given by the second line of (7.28). Then (i) φnm (z) = (−1)n−m φmn (z)

(7.29)

(ii) φˆnm (k = k1 + ik2 ) =


R2

dx dx e−i(k1 x+k2 y) φnm (x + iy)

= 4π2B (−1)n φnm (−2i2B k)

(7.30)

(iii) ∞ 

φn1 m (z1 )φn2 m (z2 ) =

m=0

1 (2π2B )1/2

φn1 n2 (z1 − z2 ) e

i

Im(z1 z ¯2 ) 2
2 B

(7.31)

Proof: Part (i) is obtained by using the relation L−l n (x) =

(n−l)! l l n! (−x) Ln−l (x)

for l = n − m. To obtain the second part, one can use the integral ∞  2  y2 n ax2 y l l e− 2a , a > 0 xl+1 Lln (ax2 ) e− 2 Jl (xy) dx = (−1) y L l+1 n a a 0

110 where Jl is the l’th Bessel function. One may look in [43] for the details. The third part can be proven by using a generating function for the Laguerre polynomials, ∞ 

tk Lm−k (x)Lk−n (y) = tn (1 + t)m−n e−tx Lm−n n n k

 1+t t

 (tx + y) ,

|t| < 1

k=0

but probably it is more instructive to give a proof in terms of annihilation and creation operators. To this end we reintroduce the rescaled variables √ w = z/( 2B . Then we have to show ∞ 

φn1 m (w1 )φn2 m (w2 ) =

m=0

1 (2π2B )1/2

φn1 n2 (w1 − w2 ) eiIm(w1 w¯2 )

Since φ¯jm = (−1)j−m φmj we have to compute (c+ 1 denotes the creation operator with respect to the w1 variable) (−1)j j (c+ )n (d+ 2) (n!j!)1/2 1

∞ 

+ m (−d+ 1 c2 ) φ00 (w1 )φ00 (w2 ) m!

m=0

=

j

(−1) j (c+ )n (d+ 2) (n!j!)1/2 1

∞ 

¯ 2 )m (w1 w φ00 (w1 )φ00 (w2 ) m!

m=0

=

(−1)j j w1 w ¯2 1 (c+ )n (d+ 2) e 2π2B (n!j!)1/2 1

=

(−1)j 1 j 12 (w1 w ¯ 2 −w ¯ 1 w2 ) (c+ )n (d+ φ00 (w1 2) e (2π2B )1/2 (n!j!)1/2 1

e− 2 (w1 w¯1 +w2 w¯2 ) 1

− w2 )

Now, because of A

e Be

−A

=

∞ 

1 k!

  A, A, · · · [A, B] · · ·

k=0

and

 1 + w1 ∂ = (w w ¯ − w ¯ w ), d (w w ¯ − w ¯ w ), 1 2 1 2 1 2 1 2 2 2 2 ∂w ¯2 = − 2

1 one gets

¯ 2 −w ¯1 w2 ) 2 (w1 w e− 2 (w1 w¯2 −w¯1 w2 ) d+ = d+ 2e 2 − 1

1

=

∂ ∂w ¯2

1

−

and j 2 (w1 w ¯ 2 −w ¯1 w2 ) j e− 2 (w1 w¯2 −w¯1 w2 ) (d+ = (−d+ w1 −w2 ) 2) e 1

1



¯2 − w ¯1 w2 ), d+ 2 2 (w1 w + 1 1 2 w2 + 2 w1 = −dw1 −w2

The Many-Electron System in a Magnetic Field

111

Similarly one finds n 2 (w1 w ¯ 2 −w ¯1 w2 ) n = (c+ e− 2 (w1 w¯2 −w¯1 w2 ) (c+ w1 −w2 ) 1) e 1

1

and we end up with 1 + 1 1 n + j e 2 (w1 w¯2 −w¯1 w2 ) (n!j!) 1/2 (cw1 −w2 ) (dw1 −w2 ) φ00 (w1 (2π2B )1/2

=

1 (2π2B )1/2

− w2 )

eiIm(w1 w¯2 ) φnj (w1 − w2 )

which proves the third part of the lemma.


7.1.2

Rectangular Geometry

In this section we solve the eigenvalue problem for a single electron in a constant magnetic field in two dimensions for a rectangular geometry. We start with the half infinite case. That is, coordinate space is given by [0, Lx ] × R. The following gauge is suitable A(x, y) = (−By, 0, 0)

(7.32)

Then H=




i∇

2 + eA

 ∂ =
2 −∆ + 2i eB
y ∂x +

e2 B 2 2
2 y



≡
2 K

(7.33)

where, recalling that B = {
/(eB)}1/2 denotes the magnetic length, K = −∆ +

2i y∂ 2B ∂x

+

1 2 y 4B

(7.34)

We make the ansatz ψ(x, y) = eikx ϕ(y)

(7.35)

2π Imposing periodic boundary conditions on [0, Lx ] gives k = L m with m ∈ Z. x The eigenvalue problem Hψ = εψ is equivalent to   2 2ky ∂2 2 + y4 ϕ(y) = ε ϕ(y)
2 − ∂y 2 + k − 2 B B   ∂2 1 2 2 ⇔
− ∂(y−2 k)2 + 4 (y − B k)2 ϕ(y) = ε ϕ(y) B

B

This is the eigenvalue equation for the harmonic oscillator shifted by 2B k. Thus, if hn denotes the normalized Hermite function, then the normalized eigenfunctions of H read   (7.36) ψnk (x, y) = √L1  eikx hn (y − 2B k)/B x B

112 with eigenvalues 1/(2m)Hψnk = εn ψnk , εn =
eB m (n + 1/2). As in the last section, the eigenvalues have infinite degeneracy. This is due to the fact that, as in the last section, we computed in infinite volume. If one turns to finite volume, the degeneracy is reduced to a finite value. Namely, the degeneracy is equal to the number of flux quanta flowing through the sample. A flux quantum is given by
= 4, 14 · 10−11 T cm2 (7.37) e and there is flux quantization which means that the magnetic flux through a given sample has to be an integer multiple of φ0 . Thus, for finite volume [0, Lx ] × [0, Ly ] the number of flux quanta is equal to φ0 = 2π

M :=

Lx Ly ! BLx Ly Φ = = ∈N φ0 2π
/e 2π2B

(7.38)

This is the case we consider next. So let coordinate space be [0, Lx ] × [0, Ly ]. The following boundary conditions are suitable (‘magnetic boundary conditions’): 2

ψ(x, y + Ly ) = eixLy /B ψ(x, y)

ψ(x + Lx , y) = ψ(x, y) ,

(7.39)

The finite volume eigenvalue problem can be solved by periodizing the states (7.36). This in turn can be obtained by a suitable superposition. To this end observe that in the y-direction the wavefunctions (7.36) are centered at 2π 2π m by K := L M , then the center yk+K yk = 2B k. Now, if we shift k = L x x 2π is shifted by 2B L M = 2B L2πx x eigenfunctions:

Lx Ly 2π2B

= Ly . As a result, one finds the following

Lemma 7.1.3 Let H be given by (7.33) on finite volume [0, Lx ] × [0, Ly ] with the Landau gauge (7.34) and with magnetic boundary conditions (7.37). Then a complete orthonormal set of eigenfunctions of H is given by ψn,k (x, y) =

√ 1 L x B

∞ 

ei(k+jK)x hn,k (y − jLy )

(7.40)

j=−∞

n = 0, 1, 2, ... where K :=

2π Lx M

=

Ly 2B

k=

2π Lx m,

m = 1, 2, ..., M

  y2 and hn,k (y) = hn (y−2B k)/B , hn (y) = cn Hn (y) e− 2

the normalized Hermite function, cn = π − 4 (2n n!)− 2 . The eigenvalues are given by   1 1 (7.41) εn =
eB 2m Hψn,k = εn ψn,k , m n+ 2 1

1

For the details of the computation one may look, for example, in [42].

The Many-Electron System in a Magnetic Field

7.2

113

Diagonalization of the Fractional Quantum Hall Hamiltonian in a Long Range Limit

Having discussed the single body problem in the previous two sections, we now turn to the N -body problem. We consider the many-electron system in two dimensions in a finite, rectangular volume [0, Lx ] × [0, Ly ] in a constant
= (0, 0, B). The noninteracting Hamiltonian is magnetic field B H0,N =

N  


i ∇i

2 − eA(xi )

(7.42)

i=1

with A given by the Landau gauge (7.32). Apparently the eigenstates of (7.42) are given by wedge products of single body eigenfunctions (7.40), Ψn1 k1 ,··· ,nN kN = ψn1 k1 ∧ · · · ∧ ψnN kN

(7.43)

with eigenvalues En1 ···nN = εn1 +· · ·+εnN . Recall that N denotes the number of electrons and M (see (7.38) and (7.40)) is the degeneracy per Landau level which is equal to the number of flux quanta flowing through the sample. The quotient ν=

N M

(7.44)

is the filling factor of the system. For integer filling, there is a unique ground state, namely the wedge product of the N = νM single body states of the lowest ν ∈ N Landau levels. This state is separated by a gap of size
eB m from the other states. The existence of this gap leads to the integer quantum Hall effect. If the filling is not an integer, the ground state of the noninteracting N -body system is highly degenerate. Suppose that ν = 1/3. Then M = 3N 2π m with m, the ground states are given by Ψ0m1 ,··· ,0mN and, identifying k = L x where m1 , ..., mN ∈ {0, 1, ...M − 1} and m1 < · · · < mN . Apparently, there are   3 N    √ (3N )! M 3 √ 3 (7.45) = 3N = (2N )! N ! ≈ 22 N N 4πN √ such choices. Here we used Stirling’s formula, n! ≈ 2πn (n/e)n to evaluate the factorials. Since 33 /22 = 6 43 , this is an extremely large number for macroscopic values of N like 1011 or 1012 which is the number of conduction electrons in Ga-As samples where the fractional quantum Hall effect is observed. Now, if the electron-electron interaction is turned on, some of these states are energetically more favorable and others less, but, due to the huge degeneracy of

114 11

about (6.75)(10 ) , one would expect a continuum of energies. However, the discovery of the fractional quantum Hall effect in 1982 demonstrated that this is not true. For certain rational values of ν with odd denominators, mainly n , p, n ∈ N, the interacting Hamiltonian ν = 2pn±1 HN =

N  


i ∇i

 2 − eA(xi ) + V (xi − xj )

i=1

(7.46)

i,j=1 i
=j

with V being the Coulomb interaction, should have a gap. Since then, a lot of work has been done on the system (7.46) (see [34, 39, 13] for an overview). Numerical data for small system size are available which give evidence for a gap. However, so far it has not been possible to give a rigorous mathematical proof that the Hamiltonian (7.46) has a gap for certain rational values of the filling factor. In this section we consider the Hamiltonian (7.46) in a certain approximation in which it becomes explicitly solvable. We will argue below that this approximation has to be considered as unphysical. However it is interesting since it gives a model with an, in finite volume, explicitly given energy spectrum which, in the infinite volume limit, most likely has a gap for ν ∈ Q and no gap for ν ∈ / Q. For that reason we think it is worth discussing that model. We consider the complete spin polarized case and neglect the Zeemann energy. We take a Gaussian as the electron-electron interaction, V (x, y) = λ e−

x2 +y2 2r2

(7.47)

Opposite to Coulomb, this has no singularity at small distances. We assume that it is long range in the sense that r >> B , B being the magnetic length. This length is of the order 10−8 m for typical FQH magnetic fields which are about B ≈ 10T . The long range condition is used to make the approximation (see (7.68, 7.70) below)



2 B

 2

ds ds hn (s) hn (s ) e− 2r2 (s−s )

≈




ds ds hn (s) hn (s )

(7.48)

s2

where hn (s) = cn Hn (s) e− 2 denotes the normalized Hermite function. With this approximation, the Hamiltonian PLL HN PLL , PLL being the projection onto the lowest Landau level, can be explicitly diagonalized. There is the following Theorem 7.2.1 Let HN be the Hamiltonian (7.46) in finite volume [0, Lx ] × [0, Ly ] with magnetic boundary conditions (7.39), let A(x, y) = (−By, 0, 0) and let the interaction be Gaussian with long range, V (x, y) = λ e−

x2 +y2 2r2

,

r >> B

(7.49)

The Many-Electron System in a Magnetic Field

115

LL be the projection onto the lowest Landau level, where Let PLL : FN → FN LL FN is the antisymmetric N -particle Fock space and FN is the antisymmetric Fock space spanned by the eigenfunctions of the lowest Landau level. Then, with the approximation (7.48), the Hamiltonian HN,LL = PLL HN PLL becomes exactly diagonalizable. Let M be the number of flux quanta flowing through [0, Lx ] × [0, Ly ] such that ν = N/M is the filling factor. Then the eigenstates and eigenvalues are labelled by N -tuples (n1 , · · · , nN ), n1 < · · · < nN and ni ∈ {1, 2, · · · , M } for all i,

HN,LL Ψn1 ···nN = (ε0 N + En1 ···nN )Ψn1 ···nN

(7.50)

where ε0 =
eB/(2m) and En1 ···nN =

N 

W (ni − nj ) ,

W (n) = λ

i,j=1 i
=j



2

e− 2r2 (Lx M −jLx ) 1

n

(7.51)

j∈Z

and the normalized eigenstates are given by Ψn1 ···nN = φn1 ∧ · · · ∧ φnN where φn (x, y) = √π

−1 4

B L y

∞ 

−

1 2
2 B

2 n Lx −sLx ) (x− M

ei(x− M Lx −sLx )y/B n

2

(7.52)

s=−∞ 1

=

e

− √1 √π 4 M B L x

∞ 

e

−

1 2
2 B

n r (y− Mr Ly )2 i(x− M Lx ) M Ly /2B e

(7.53)

r=−∞

Before we start with the proof we make some comments. The approximation (7.48) looks quite innocent. However, by reviewing the computations in the proof one finds that it is actually equivalent to the approximation 2 2 2 2 2 V (x, y) = λ e−(x +y )/(2r ) ≈ λ e−x /(2r ) . That is, if we write in (7.49) 2 2 V (x, y) = λ e−x /(2r ) then the above Theorem is an exact statement. Recall that the single body eigenfunctions are localized in the y-direction and are given by plane waves in the x-direction. Since the eigenstates are given by wedge products, we can explicitely compute the expectation value of the energy. One may speculate that for fillings ν = 1/q the ground statesare la belled by the N -tuples (n1 , · · · , nN ) = j, j +q, j +2q, · · · , j +(N −1)q which have a q-fold degeneracy, 1 ≤ j ≤ q. A q-fold degeneracy for fillings ν = p/q, p, q without common divisor, follows already from general symmetry considerations (see [42] or [61]). In particular, for ν = 1/3, one may speculate that there are three ground states labelled by (3, 6, 9, ..., 3N ), (2, 5, 8, ..., 3N − 1) and (1, 4, 7, ..., 3N −2). Below Lemma 7.2.3 we compute the expectation value of the energy of these states with respect to the Coulomb interaction. It is about as twice as big as the energy of the Laughlin wavefunction. This is not too surprising since a wedge product vanishes only linearly in the differences xi − xj whereas the Laughlin wavefunction, given by P √ 2 1 wi = zi /( 2B ) (7.54) ψ(w1 , ...wN ) = Π (wi − wj )3 e− 2 j |wj | , i<j

116 vanishes like |xi − xj |3 which results in a lower contribution of the Coulomb energy 1/|xi − xj |. Thus we have to conclude that the approximation (7.48), implemented in (7.68) and (7.70) below, has to be considered as unphysical. Nevertheless, the energy spectrum (7.51) seems to have the property that it has a gap for rational fillings in the infinite volume limit. That is, we expect   > 0 if ν ∈ Q  (7.55) ∆(ν) := lim E1 (N, M ) − E0 (N, M ) N,M →∞ = 0 if ν ∈ /Q N/M =ν E0 being the lowest and E1 the second lowest eigenvalue in finite volume. This is interesting since a similar behavior one would like to prove for the original Hamiltonian (7.46). However, it seems that the energy (7.51) does not distinguish between even and odd denominators q. That is, it looks like the approximate model does not select the observed fractional quantum Hall fillings (see [34, 39, 13] for an overview). This should be due to the unphysical nature of the approximation (7.48).

Proof of Theorem 7.2.1: We proceed in three steps: Projection onto the lowest Landau level using fermionic annihilation and creation operators, implementation of the approximation (7.48) and finally diagonalization. (i) Projection onto the Lowest Landau Level To project HN onto the lowest Landau level, we rewrite HN in terms of fermionic annihilation and creation operators   2 d2 x ψ + (x)
i ∇ − eA(x) ψ(x) HN =   (7.56) + d2 x d2 x ψ + (x)ψ + (x  )V (x − x  )ψ(x  )ψ(x)  FN

where FN is the antisymmetric N -particle Fock space. We consider the complete spin polarized case in which only one spin direction (say ψ = ψ↑ ) contributes and we neglect the Zeeman energy. Let ψ and ψ + be the fermionic annihilation and creation operators in coordinate space. In order to avoid confusion with the single body eigenfunctions ψn,k (7.40), we denote the latter ones as ϕn,k . Introducing an,k , a+ n,k according to ψ(x) = an,k =






ϕn,k (x)an,k ,

n,k

d2 x ϕ¯n,k (x)ψ(x) ,

a+ n,k =






ϕ¯n,k (x)a+ n,k

(7.57)

d2 x ϕn,k (x)ψ + (x)

(7.58)

ψ + (x) =

n,k

the an,k obey the canonical anticommutation relations {an,k , a+ n ,k } = δn,n δk,k

(7.59)

The Many-Electron System in a Magnetic Field

117

and (7.56) becomes, if H = ⊕N HN , H = Hkin + Hint

(7.60)

where Hkin =



ε n a+ n,k an,k

(7.61)

n,k

The interacting part becomes 
Hint = d2 x d2 x ψ + (x)ψ + (x  )ϕ¯n,k (x) nk|V |n k  ϕn ,k (x  )ψ(x  )ψ(x) n,k n ,k

=





(n1 l1 ; n2 l2 ; nk) nk|V |n k  ×

n,k n1 ,··· ,n4 n ,k l1 ,··· ,l4

+ (n k  ; n3 l3 ; n4 l4 ) a+ n1 ,l1 an3 ,l3 an4 ,l4 an2 ,l2

(7.62)

where we used the notation

nk|V |n k  := d2 x d2 x ϕn,k (x) V (x − x  ) ϕ¯n ,k (x  )
(n1 l1 ; n2 l2 ; nk) := d2 x ϕ¯n1 ,l1 (x) ϕn2 ,l2 (x) ϕ¯n,k (x)

(7.63) (7.64)

Now we consider systems with fillings ν =

N < 1 M

(7.65)

and restrict the electrons to the lowest Landau level. Since the kinetic energy is constant, we consider only the interacting part, HLL := PLL Hint PLL =

  n,k n ,k

(7.66)

+ (0l1 ; 0l2 ; nk) nk|V |n k  (n k  ; 0l3 ; 0l4 ) a+ l1 a l3 a l4 a l2

l1 ,··· ,l4

where we abbreviated al := a0,l ,

+ a+ l := a0,l

(7.67)

(ii) The Approximation The matrix element nk|V |n k  is computed in part (a) of Lemma 7.2.2 below. For a gaussian interaction (7.47) the exact result is (7.68) n, k|V |n , k  =
2 √

 2 r2 2 B 2πB δk,k λ r [e− 2 k ]M ds ds hn (s) hn (s ) e− 2r2 (s−s )

118 where, if k = 2πm/Lx , 2

[e

− r2 k2

]M :=

∞ 

2

e

− r2 (k−jK)2

j=−∞

=

∞ 

2

2 r 2π e− 2 [ Lx (m−jM)]

(7.69)

j=−∞

is an M -periodic function (as a function of m). For a long range interaction r >> B , we may approximate this by √

r2 2 n, k|V |n , k  ≈ 2πB δk,k λ r [e− 2 k ]M ds hn (s) ds hn (s )

=: δk,k vk ds hn (s) ds hn (s ) (7.70) Then HLL becomes HLL =   n,n k

=

(0l1 ; 0l2 ; nk) vk




hn (s)ds




+ hn (s )ds (n k; 0l3 ; 0l4 ) a+ l1 a l3 a l4 a l2

l1 ,··· ,l4

 

+ (0l1 ; 0l2 ; 1y k) vk (1y k; 0l3 ; 0l4 ) a+ l1 a l3 a l4 a l2

(7.71)

k l1 ,··· ,l4

Here we used that ∞ 

hn (y)




hn (s)ds = 1

(7.72)

n=0

which is a consequence of ∞ 

ϕ¯n,k (x, y)




∞ n=0

hn (y) hn (s) = δ(y − s). Thus

hn (s)ds

n=0

=

=

√ 1 L x B

√ 1 L x B

∞  j=−∞ ∞ 

e−i(k+jK)x

∞ 

 
hn (s)ds hn (y − yk − jLy )/B

n=0

e−i(k+jK)x

(7.73)

j=−∞

and (7.71) follows if we define (0l1 ; 0l2 ; 1y k) :=

dxdy ϕ¯0,l1 (x, y) ϕ0,l2 (x, y) √L1 

∞ 

x B

e−i(k+jK)x

j=−∞

(7.74) These matrix elements are computed in part (b) of Lemma 7.2.2 below and the result is M √ 1 (0l1 ; 0l2 ; 1y k) = δm,m [e− 2 −m1 L  x B


2 B 4

(l1 −l2 )2

]M

(7.75)

The Many-Electron System in a Magnetic Field

119

2π M m, lj = L2πx mj and δm = 1 iff m1 = m2 mod M . In the if k = L 1 ,m2 x 2π M following we write, by a slight abuse of notation, also δl,l  if l = Lx m. Then the Hamiltonian (7.71) becomes

HLL =

1 L x B

 


2 B 4

M δk,l [e− 2 −l1

(l1 −l2 )2

M ]M vk δk,l × 3 −l4

k l1 ,··· ,l4

=

[e−



1 Lx


2 B 4

(l3 −l4 )2

+ + ]M a+ l1 a l3 a l4 a l2

+ δlM wl2 −l1 a+ l1 a l3 a l4 a l2 2 −l1 ,l3 −l4

(7.76)

l1 ,··· ,l4

where the interaction is given by wk :=

√

2π λ r [e−

r2 2

k2

]M [e−


2 B 4

k2 2 ]M

(7.77)

(iii) Diagonalization Apparently (7.76) looks like a usual one-dimensional many-body Hamiltonian in momentum space. Thus, since the kinetic energy is constant, we can easily diagonalize it by taking the discrete Fourier transform. For 1 ≤ n ≤ M let ψn :=

√1 M

M 

nm

e2πi M am ,

ψn+ =

M 

√1 M

m=1

e−2πi M a+ m nm

(7.78)

m=1

or am =

√1 M

M 

e−2πi M ψn , nm

a+ m =

√1 M

n=1

M 

nm

e2πi M ψn+

(7.79)

n=1

Substituting this in (7.76), we get HLL =



ψn+ ψn+ W (n − n ) ψn ψn

(7.80)

n,n

with an interaction W (n) =

1 Lx

M 

nm

e2πi M wm

(7.81)

m=1

where wm ≡ wk is given by (7.77), k = 2πm/Lx . The N -particle eigenstates of (7.80) are labelled by N -tuples (n1 , · · · , nN ) where 1 ≤ nj ≤ M and

120 n1 < n2 < · · · < nN and are given by Ψn1 ···nN = ψn+1 ψn+2 · · · ψn+N |1  2πi + 1 = M N/2 e− M (n1 j1 +···nN jN ) a+ j1 · · · ajN |1 j1 ···jN

=



1 M N/2

e− M

2πi

(n1 j1 +···nN jN )

ϕ0j1 ∧ · · · ∧ ϕ0jN

j1 ···jN

= φn1 ∧ · · · ∧ φnN

(7.82)

if we define φn (x, y) :=

M 

√1 M

nj

e−2πi M ϕ0j (x, y)

(7.83)

j=1

The energy eigenvalues are HLL Ψn1 ···nN = En1 ···nN Ψn1 ···nN

(7.84)

where N 

En1 ···nN =

W (ni − nj )

(7.85)

i,j=1 i
=j

The Fourier sums in (7.81) and (7.83) can be performed with the Poisson summation formula. This is done in part (c) and (d) of Lemma 7.2.2. If we √ r2 2 approximate wk ≈ 2π λ r [e− 2 k ]M , since by assumption r >> B , we find for this wk W (n) = λ



e− 2r2 (Lx M −jLx ) 1

j∈Z

=λ



e

n

2

“ ”2 − 2r12 2B 2πn Ly −jLx

(7.86)

j∈Z

Thus the theorem is proven.


Lemma 7.2.2 a) For the matrix element in (7.63) one has nk|V |n k  = (7.87)
2 √

 2 r2 2 B 2πB δk,k λ r [e− 2 k ]M ds ds hn (s) hn (s ) e− 2r2 (s−s )

The Many-Electron System in a Magnetic Field

121

where, if k = 2πm/Lx , [e−

r2 2

k2

∞ 

]M :=

e−

r2 2

(k−jK)2

∞ 

=

j=−∞

2

r 2π e− 2 [ Lx (m−jM)]

2

(7.88)

j=−∞

is an M -periodic function (as a function of m). b) The matrix elements (7.74) are given by M √ 1 (0, l1 ; 0, l2 ; 1y , k) = δm,m [e− 2 −m1 L 


2 B 4

(l1 −l2 )2

x B

if k =

2π Lx m, lj

=

2π L x mj

]M

(7.89)

M and δm = 1 iff m1 = m2 mod M . 1 ,m2

c) For m ∈ Z let vm = [e−

r2 2

k2

]M =



e−

r2 2

2π 2 (L ) (m−jM)2 x

(7.90)

j∈Z

and let V (n) =

1 Lx

M

nm

e2πi M vm . Then  − 1 L n −jL 2 1 x) e 2r2 ( x M V (n) = √2π r m=1

(7.91)

j∈Z

d) Let k = 2πm/Lx and let ϕ0,m ≡ ϕ0,k be the single-body eigenfunction (7.40). Then M 

√1 M

e−2πi M ϕ0,m (x, y) nm

m=1

= √π

−1 4

B L y

∞ 

−

1 2
2 B

2 n Lx −sLx ) (x− M

e

i

(x− n Lx −sLx )y M
2 B

(7.92)

s=−∞ 1

=

e

− √1 √π 4 M B L x

∞ 

e

−

1 2
2 B

n r (y− Mr Ly )2 i(x− M Lx ) M Ly /2B e

(7.93)

r=−∞

Proof: a) We have
n, k|V |n , k  = d2 x d2 x ϕn,k (x) V (x − x ) ϕ¯n ,k (x ) =

1 L x B






dxdx dydy  ei(k−jK)x−i(k −j



K)x

(7.94)

×

j,j 

hn,k (y − jLy ) hn ,k (y  − j  Ly ) V (x − x ) =

1 L x B








dxdx dydy  ei(k−jK)(x−x ) ei(k−jK−k +j



K)x

×

j,j 

    (x−x )2 (y−y )2 hn (y − yk − jLy )/B hn (y  − yk − j  Ly )/B λ e− 2r2 e− 2r2

122 The x -integral gives Lx δm−jM,m −j  M = Lx δm,m δj,j  if k = 2πm/Lx, k  = 2πm /Lx , 0 ≤ m, m ≤ M − 1. Thus we get n, k|V |n , k  = 
  x2
λ  dx ei(k−jK)x e− 2r2 dydy  hn (y − yk − jLy )/B × B δk,k j

  (y−y )2 hn (y  − yk − jLy )/B e− 2r2 =

 r2
2 √  2 2
B 2πB λr δk,k e− 2 (k−jK) dsds hn (s) hn (s ) e− 2r2 (s−s ) j


2 √
 2 B r2 2 = 2πB λr δk,k [e− 2 k ]M dsds hn (s) hn (s ) e− 2r2 (s−s )

(7.95)

and part (a) follows. b) One has (0, l1 ; 0, l2 ; 1y , k) =  √ 1 3 L x B

(7.96) dxdy e−i(l1 +j1 K)x ei(l2 +j2 K)x e−i(k+jK)x h0,l1 (y) h0,l2 (y)

j1 ,j2 ,j

The plane waves combine to

 2π exp i L (m + j M − m − j M − m − jM )x 2 2 1 1 x

(7.97)

and the x-integral gives a volume factor Lx times a Kroenecker delta which is one iff m2 + j2 M − m1 − j1 M − m − jM = 0 or m = m2 − m1 ∧ j = j2 − j1

if m2 ≥ m1

m = m2 − m1 + M ∧ j = j2 − j1 − 1

if m2 < m1

(7.98)

Thus (7.96) becomes (0, l1 ; 0, l2 ; 1y , k) = M δm,m 2 −m1

√ 1 3 √1 Lx B

M √1 = δm,m 2 −m1

B

3

√1 Lx

 j1 ,j2

0

 j1 ,j2

Ly

0

  dy h0 (y − yl1 − j1 Ly )/B ×   h0 (y − yl2 − j2 Ly )/B

Ly

  dy h0 (y − yl1 − j1 Ly )/B ×   h0 (y − yl2 − j1 Ly + (j1 − j2 )Ly )/B

The Many-Electron System in a Magnetic Field  Ly   M 1 √1 √ = δm,m dy h0 (y − yl1 − j1 Ly )/B × 2 −m1 3 Lx

B

M √1 = δm,m 2 −m1

B

0

j1 ,j

3



√1 Lx



M √ 1 = δm,m 2 −m1 L 

x B

−∞

j

e

∞

− 12 4
B

123

  h0 (y − yl2 − j1 Ly + jLy )/B     dy h0 (y − yl1 )/B h0 (y − yl2 + jLy )/B

(yl1 −yl2 +jLy )2

j



M √ 1 = δm,m 2 −m1 L 

x B

e−


2 B 4

(l1 −l2 +jK)2

j M √ 1 = δm,m [e− 2 −m1 L 


2 B 4

(l1 −l2 )2

x B

]M

(7.99)

M  where δm,m  equals one iff m = m mod M and equals zero otherwise.

c) It is [e−

r2 2

k2



]M =

e

2 M2 L2 x

− r2

m −2πj ) (2π M

2

(7.100)

j∈Z

We use the following formula which is obtained from the Poisson summation theorem   t 2  2 1 t e− 2t (x−2πj) = 2π e− 2 j eijx (7.101) j∈Z

j∈Z

with m x = 2π M ,

t=

L2x r2 M 2

(7.102)

Then vm =

Lx √1 2π r M



L2 x

e− 2r2 M 2 j e−2πi M 1

2

jm

(7.103)

j∈Z

and V (n) becomes V (n) =

1 Lx

M 

nm

e2πi M

Lx √1 2π r M

m=1

=

1 √1 2π r M

1 √1 2π r M

L2 x

e− 2r2 M 2 j e−2πi M 1

2

jm

j∈Z



e

L2 − 2r12 Mx2

j2

e

L2 − 2r12 Mx2

2



M 

e2πi

(n−j)m M

m=1

j∈Z

=



j

M M δn,j

j∈Z

=

√1 2π r

 s∈Z

L2 x

e− 2r2 M 2 (n−sM) 1

2

(7.104)

124 which proves part (c). d) According to (7.40) we have

√1 M

M 

e−2πi M ϕ0,m (x, y) = nm

m=1 √1 M

M 

e−2πi M

nm

m=1

=

∞ 

√ 1 L x B

2π

ei Lx (m+jM)x h0,k (y − jLy )

j=−∞ ∞ 

−1

√π 4 MLx B

M 

2π

n

2π

n

2π

ei Lx (x− M Lx )m ei Lx jMx e

−

1 2
2 B

(y−2B L2πx m−jLy )2

j=−∞ m=1

=

M ∞  

−1

√π 4 MLx B

ei Lx (x− M Lx )m e

m=1 j=−∞

√1 2π

1

=

− √π 4 √1 MLx B 2π



dq e−

q2 2

e

iq
y

i

Ly
2 B

dq e−

M 

B

jx

q2 2

× “

e

iq

y
B

L

2π −B L m−j
y x

”

B

ei Lx (x− M Lx −qB )m × 2π

n

m=1 ∞ 

e

“ ”L i
x −q
y j B

B

(7.105)

j=−∞ 1

=

− √π 4 √1 MLx B 2π



dq e−

q2 2

e

iq
y

M 

B

ei Lx (x− M Lx −qB )m × 2π

n

m=1

  L 2π δ ( xB − q) By − 2πr

∞  r=−∞

The delta function forces q to take values q=

x B

B − 2πr L y

which gives x−

n M Lx

2

n − qB = − M Lx + 2πr LBy =

r−n M

Lx

Therefore the m-sum in (7.105) becomes M  m=1

ei Lx (x− M Lx −qB )m = 2π

n

M  m=1

e2πi

(r−n)m M

M = M δr,n

(7.106)

The Many-Electron System in a Magnetic Field

125

and we get √1 M

M 

e−2πi M ϕ0,m (x, y) = nm

m=1 ∞ 

√ √ −1 4 B π√ M 2π L L x B y

= √π

∞ 

−1 4

B L y

= √π

−1 4

B L y

s=−∞ ∞ 

e e

e

−

1 2
2 B

„ «2 “ ” 2π
2
x−r LyB i
x −2πr LBy
y

e

B

B

M δr,n

r=−∞ „ «2 “ ” 2π
2
− 12 x−(n+sM) LyB i
x −2π(n+sM) LBy
y 2


e

B

−

1 2
2 B

2 n Lx −sLx ) (x− M

e

i

B

(x− n Lx −sLx )y M
2 B

B

(7.107)

s=−∞

This proves (7.92). (7.93) is obtained directly from (7.40) by putting r = m + jM .
Since the eigenstates of the approximate model are given by pure wedge products, we cannot expect that their energies, for a Coulomb interaction, are close to those of the Laughlin or Jain wavefunctions. The reason is that a wedge product only vanishes linearly in xi − xj while the Laughlin wavefunction vanishes like (xi − xj )3 if xi goes to xj . This gives a lower contribution to the Coulomb repulsion 1/|xi − xj |. In general it is not possible to make an exact analytical computation of the expectation value of the energy if the wavefunction is not given by a pure wedge product, like the Laughlin or composite fermion wavefunction. One has to rely on numerical and analytical approximations or exact numerical results for small system size. For a pure wedge product, the exact result can be written down and it looks as follows. Lemma 7.2.3 a) Let ψ(x1 , ..., xN ) be a normalized antisymmetric wavefunc
N tion and let W (x1 , ..., xN ) = i,j=1 V (xi − xj ). Let A = d2 x be the sample i<j

size, let nν = N/A = ν/(2π2B ) be the density
and let ρ be the density of the constant N -particle
wavefunction such that dx1 · · · dxN ρ = 1, that is, ρ = 1/AN . Let W = d2N x W (|ψ|2 − ρ). Then 
W  1 1 dx1 dx2 V (x1 − x2 ) g(x1 , x2 ) − 1 (7.108) N = 2 nν A where g(x1 , x2 ) =

N (N −1) n2ν




dx3 · · · dxN |ψ(x1 , ..., xN )|2

(7.109)

b) Suppose that ψ(x1 , ..., xN ) = φm1 ∧ · · · ∧ φmN (x1 , ..., xN ) = √1N ! det [φmi (xj )1≤i,j≤N ]

(7.110)

126 Then 1 n2ν

g(x1 , x2 ) =

  P (x1 , x1 )P (x2 , x2 ) − |P (x1 , x2 )|2

(7.111)

where P is the kernel of the projector onto the space spanned by {φm1 , ..., φmN }. N That is, P (x, x ) = j=1 φmj (x)φ¯mj (x ). Proof: a) One has W =

d2N x

N 

  V (xi − xj ) |ψ(x1 , ..., xn )|2 − ρ

i,j=1 i<j

=

N (N −1) 2

=

1 2





dx1 dx2 V (x1 − x2 )



dx1 dx2 V (x1 − x2 ) × 

= N NA−1 12 A1



N (N − 1)

  dx3 · · · dxN |ψ(x1 , ..., xn )|2 − ρ

dx3 · · · dxN |ψ(x1 , ..., xn )|2 −

N (N −1) A2



dx1 dx2 V (x1 − x2 ) ×   N (N −1) dx3 · · · dxN |ψ(x1 , ..., xn )|2 − 1 n2 ν

where we used nν = N/A ≈ (N − 1)/A. This proves part (a). b) Let ψ = φm1 ∧ · · · ∧ φmN . Then dx3 · · · dxN |ψ(x1 , ..., xN )|2  = N1 ! επ εσ dx3 · · · dxN φmπ1 (x1 )φ¯mσ1 (x1 ) · · · φmπN (xN )φ¯mσN (xN ) π,σ∈SN

=

1 N!

   |φmπ1 (x1 )|2 |φmπ2 (x2 )|2 − φmπ1 (x1 )φ¯mπ1 (x2 )φmπ2 (x2 )φ¯mπ2 (x1 )

π∈SN

=

1 N (N −1)

N    |φmi (x1 )|2 |φmj (x2 )|2 − φmi (x1 )φ¯mi (x2 )φmj (x2 )φ¯mj (x1 ) i,j=1 i
=j

=

1 N (N −1)

N 

{· · · }

i,j=1

=

1 N (N −1)

  Pm (x1 , x1 )Pm (x2 , x2 ) − Pm (x1 , x2 )Pm (x2 , x1 )

where Pm (x1 , x2 ) =

N j=1

φmj (x1 )φ¯mj (x2 ).


(7.112)

The Many-Electron System in a Magnetic Field

127

Now consider the eigenvalues En1 ...nN given by (7.51). For −M/2 ≤ n ≤ M/2, the dominant contribution from the periodizing j-sum for W (n) is L2 n 2 the j = 0 term which is exp{− 2rx2 ( M ) }. This is small if n = ni − nj is large. Thus, it seems that those configurations (n1 , ..., nN ) have low energy for which ni − nj in average is large. Hence, one may speculate that, for ν = 1/3, the minimizing configurations are (3, 6, 9, ..., 3N ), (2, 5, 8, ..., 3N − 1) and (1, 4, 7, ..., 3N − 2) and the lowest excited states should be obtained from these states by just changing one ni or a whole group of neighboring ni ’s by one each. The projection P{3k} onto the space spanned by φ3 , φ6 , ..., φ3N can be explicitly computed. In the infinite volume limit, one simply obtains P{3k} = 13 Pν=1 where Pν=1 is the projection onto the whole lowest Landau level, spanned by all the ϕn ’s. Thus, the energy per particle U{3k} for the  2 2 wavefunction φ3 ∧ φ6 ∧ · · · ∧ φ3N is 13 Uν=1 = − 31 π8 eB ≈ −0, 21 eB which is much bigger than the energy of the Laughlin wavefunction which is about 2 Uν=1/3 = −0, 42 eB .

Chapter 8 Feynman Diagrams

8.1

The Typical Behavior of Field Theoretical Perturbation Series

In chapters 3 and 4 we wrote down the perturbation series for the partition function and for some correlation functions. We found that the coefficients of λn were given by a sum (d + 1)n-dimensional integrals if the space dimension is d. Typically, some of these integrals diverge if the cutoffs of the theory are removed. This does not mean that something is wrong with the model, but merely means first of all that the function which has been expanded is not analytic if the cutoffs are removed. To this end we consider a small example. Let
∞
1 1 e−x (8.1) Gδ (λ) := 0 dx 0 dk √k+λx+δ where δ > 0 is some cutoff and the coupling λ is positive. One may think of δ = T , the temperature, or δ = 1/L if Ld is the volume of the system. By explicit computation, using Lebesgue’s theorem of dominated convergence to interchange the limit with the integrals, √ √
∞ G0 (λ) = lim Gδ (λ) = 0 dx 2( 1 + λx − λx) e−x δ→0 √ = 2 + O(λ) − O( λ) (8.2) Thus, the δ → 0 limit is well defined but it is not analytic. This fact has to show up in the Taylor expansion. It reads Gδ (λ) =

n    −1
∞ 2

0

j j=0

dx


1 0

dk

xj e−x 1

(k+δ)j+ 2

λj + rn+1

(8.3)

Apparently, all integrals over k diverge for j ≥ 1 in the limit δ → 0. Now, very roughly speaking, renormalization is the passage from the expansion (8.3) to the expansion G0 (λ) =

n   
∞ 1 2



0

dx 2x e−x λ − c

√ λ + Rn+1

(8.4)

=0

129

130 √ where the last one is obtained from (8.2) by expanding the 1 + λx term. One would √ say ‘the diverging integrals have been resumed to the nonanalytic term c λ’. In the final expansion (8.4) all coefficients are finite and, for small λ, the lowest order terms are a good approximation since (θλ ∈ [0, λ])  
 1    ∞ 1 2 xn+1 λn+1 e−x  |Rn+1 | = 2 0 dx n+1 n+ 1 2 (1+θλ x)  1 
∞  2 dx xn+1 e−x λn+1 ≤ 2 n+1 0 √  n n n+1 1 (2n)! n+1 n→∞ = 22n ∼ 2 e λ (8.5) n! λ Here we used the Lagrange representation√of the n + 1’st Taylor remainder in the first line and Stirling’s formula, n! ∼ 2πn(n/e)n , in the last line. An estimate of the form (8.5) is typical for renormalized field theoretic perturbation series. The lowest order terms are a good approximation for weak coupling, but the renormalized expansion is only asymptotic, the radius of convergence of the whole series is zero. The approximation becomes more accurate if n approaches 1/λ, but then quickly diverges if n > e/λ. Or, for fixed n, the n lowest order terms are a good approximation as long as λ < 1/n. In this small example we went from ‘the unrenormalized’ or ‘naive’ perturbation expansion (8.3) to the ‘renormalized’ perturbation expansion (8.4) by going through the exact answer (8.2). Of course, for the models we are interested in, we do not know the exact answer. Thus, for weak coupling, the whole problem is to find this rearrangement which transforms a naive perturbation expansion into a renormalized expansion which is (at least) asymptotic. In the next section we prove a combinatorial formula which rewrites the perturbation series in terms of n’th order diagrams whose connected components are at least of order m, which, for m = n, results in a proof of the linked cluster theorem. This reordering has nothing to do with the rearrangements considered above, it simply states that the logarithm of the partition function is still given by a sum of diagrams which is not obvious. In section 8.3 we start with estimates on Feynman diagrams. That section is basic for an understanding of renormalization, since it identifies the divergent contributions in a sum of diagrams.

Feynman Diagrams

8.2

131

Connected Diagrams and the Linked Cluster Theorem

The perturbation series for the partition function reads Z(λ) =

∞ 

(−λ)n n!




dξ1 · · · dξ2n U (ξ1 − ξ2 ) × · · ·

(8.6)

n=0

· · · × U (ξ2n−1 − ξ2n ) det [C(ξi , ξj )]1≤i,j≤2n If we expand the 2n × 2n determinant and interchange the sum over permutations with the ξ-integrals, we obtain the expansion into Feynman diagrams: Z(λ) =

∞ 

(−λ)n n!

n=0



signπ G(π)

(8.7)

π∈S2n

where the graph or the value of the graph defined by the permutation π is given by  n (8.8) G(π) = dξ1 · · · dξ2n Π U (ξ2i−1 − ξ2i ) C(ξ1 , ξπ1 ) · · · C(ξ2n , ξπ2n ) i=1

In general the above integral factorizes into several connected components. The number of U ’s in each component defines the order of that component. The goal of this subsection is to prove the linked cluster theorem which states that the logarithm of the partition function is given by the sum of all connected diagrams. A standard proof of this fact can be found in many books on field theory or statistical mechanics [40, 56]. In the following we give a slightly more general proof which reorders the perturbation series in terms of n’th order diagrams whose connected components are at least of order m where 1 ≤ m ≤ n is an arbitrary given number. See (8.17) below. Linked Cluster Theorem: The logarithm of the partition function is given by the sum of all connected diagrams, log Z(λ) =

∞  n=0

We use the following

(−λ)n n!

 π∈S2n G(π) connected

signπ G(π)

(8.9)

132 Lemma 8.2.1 Let {wn }n∈N be a sequence with wi wj = wj wi ∀i, j ∈ N and let a be given with awi = wi a ∀i ∈ N (for example wi , a ∈ C or even elements of a Grassmann algebra). For fixed m ∈ N define the sequence {vn }n∈N by (n = mk + l, 0 ≤ l ≤ m − 1, k ∈ N)  aj wmk−mj+l vmk+l = . (−1)j (mk + l)! j=0 j! (mk − mj + l)! k

(8.10)

Then the wn ’s can be computed from the vn ’s by  aj vmk−mj+l wmk+l = (mk + l)! j=0 j! (mk − mj + l)!

(8.11)

∞ ∞   wn vn a = e . n! n! n=0 n=0

(8.12)

k

and one has

Proof: One has k  vmk−mj+l aj j! (mk − mj + l)! j=0

=

r=i+j

=

k−j k  wm(k−j)−mi+l aj  ai (−1)i j! i! (m(k − j) − mi + l)! j=0 i=0

r=0

=

wmk−mr+l   r  (−1)i r! (mk − mr + l)! i=0 i

k  ar

r

k  ar r=0

wmk−mr+l wmk+l δr,0 = r! (mk − mr + l)! (mk + l)!

which proves the first formula. The second formula is obtained as follows ∞ ∞ m−1 ∞ m−1 k    wmk+l   vmk−mj+l wn aj = = n! (mk + l)! j! (mk − mj + l)! n=0 j=0 k=0 l=0

=

k=0 l=0

∞ ∞ m−1  aj   vmk−mj+l j! (mk − mj + l)! j=0 k=j l=0

= ea

∞  vn n! n=0

which proves the lemma.


r=k−j

=

∞ ∞ m−1  aj   vmr+l j! r=0 (mr + l)! j=0 l=0

(8.13)

Feynman Diagrams

133

of n’th order diagrams whose connected Now we define the sum Det(m) n components are at least of order m inductively by (1) Det(1) n = Detn (C, U )  n := dξ1 · · · dξ2n Π U (ξ2i−1 − ξ2i ) det [C(ξi , ξj )]1≤i,j≤2n

(8.14)

i=1

and for m ≥ 1, n = mk + l , 0 ≤ l ≤ m − 1 1 j (m) (m+1) k  Detmk−mj+l Detmk+l Det(m) m = (−1)j m! (mk + l)! j=0 j! (mk − mj + l)!

(8.15)

Then the fact that the logarithm gives only the connected diagrams may be formulated as follows. Theorem 8.2.2 The logarithm of the partition function is given by ∞  λn log Z(λ) = Det(n) n (C, U ) n! n=1

(8.16)

and Det(n) n (C, U ) is the sum of all connected n’th order diagrams. Proof: We claim that for arbitrary m Z(λ) =



∞ m  λn λs (s) Det(m+1) Det exp n s n! s! n=0 s=1

(8.17)

For m = 0, (8.17) is obviously correct. Suppose (8.17) is true for m− 1. Then, because of (m+1)

λn

Detmk+l Det(m+1) n = λmk+l n! (mk + l)!  λm  (m) k (m) j  Detmk−mj+l j m! Detm mk−mj+l λ (−1) = j! (mk − mj + l)! j=0

(8.18)

and the lemma, one obtains Z(λ) =

m−1

∞   λs λn (s) Det(m) Det exp n s n! s! n=0 s=1

m−1

∞   λs m λn (m+1) λm! Det(m) (s) m Detn Dets = e exp n! s! n=0 s=1 =



∞ m  λn λs (s) Det(m+1) Det exp n s n! s! n=0 s=1

(8.19)

134 Furthermore we claim that for a given m ∈ N (m+1)

Det1

(m+1)

= Det2

= · · · = Det(m+1) =0 m

(8.20)

Namely, for n = mk + l < m one has k = 0, n = l such that  Det(m+1) n = (−1)j n! j=0 0



 (m) j 1 m! Detm j!

Det(m) Det(m) n n = n! n!

(8.21)

and it follows Det(m+1) = Det(m) = · · · = Det(n+1) n n n

(8.22)

But, for n = m = 1m + 0, by definition  Det(m+1) m = (−1)j m! j=0 1



Det(m) m = − m!

 (m) j 1 m! Detm j! 

 (m) 1 1 m! Detm 1!

(m)

Detm−mj (m − mj)! = 0

(8.23)

which proves (8.20). Thus one obtains  

∞ m  λn λs (m+1) (s) Z(λ) = 1 + Detn Dets exp n! s! n=m+1 s=1

(8.24)

By taking the limit m → ∞ one gets log Z(λ) =

∞  λs s=1

s!

Det(s) s

(8.25)

which proves (8.16). It remains to prove that Det(n) n is the sum of all connected n’th order diagrams. To this end we make the following definitions. Let π ∈ S2n be given. We say that π is of type t(π) = 1b1 2b2 · · · nbn

(8.26)

iff Gπ consists of precisely b1 first order connected components, b2 second order connected components, · · · , bn n’th order connected components, where Gπ is the graph produced by the permutation π. Observe that, contrary to section (1.3.1), the br are not the number of r-cycles of the permutation π, but, as defined above, the number of r’th order connected components of the diagram (8.8) given by the permutation π. Let (b ,··· ,bn )

S2n1

= {π ∈ S2n | t(π) = 1b1 2b2 · · · nbn }

(8.27)

Feynman Diagrams

135

Then S2n is the disjoint union

S2n =

(b ,··· ,bn )

S2n1

(8.28)

(0,··· ,0,bm ,··· ,bn )

(8.29)

0≤b1 ,··· ,bn ≤n 1b1 +···+nbn =n

Let

(m)

S2n =

S2n

0≤bm ,··· ,bn ≤n mbm +···+nbn =n

and (n)

(0,··· ,0,1)

c S2n = S2n = S2n

(8.30)

One has (b ,··· ,bn )

|S2n1

|=

(1!)b1

n! c bn |S c |b1 · · · |S2n | , · · · (n!)bn b1 ! · · · bn ! 2

(8.31)

in particular (0,··· ,0,bm ,··· ,bn )

|S2n

|=

n! (0,··· ,0,bm+1 ,··· ,bn−mbm ) |S c |bm |S2n−2mbm | (n − mbm )! (m!)bm bm ! 2m (8.32)

We now prove by induction on m that  Det(m) n

dξ1 dξ2 · · · dξ2n

=

n 

U (ξ2i−1 − ξ2i ) det(m) [C(ξj , ξk )j,k=1,··· ,2n ]

i=1

(8.33) where det(m) [C(ξj , ξk )j,k=1,··· ,2n ] =



C(ξ1 , ξπ1 ) · · · C(ξ2n , ξπ(2n) ) (8.34)

(m)

π∈S2n

It follows from (8.33) that Det(n) n is the sum of all connected diagrams. Obviously (8.33) is correct for m = 1. Suppose (8.33) is true for m. Let (m+1)

 Det n

=

 dξ1 dξ2 · · · dξ2n

n 

U (ξ2i−1 − ξ2i ) det(m+1) [C(ξj , ξk )j,k=1,··· ,2n ]

i=1

(8.35)

136 Then, for n = mk + l with k ∈ N and 0 ≤ l ≤ m − 1 2n   

Det(m) = n

i=1

(m) π∈S2n

  2n

bm ,bm+1 ,··· ,bn =0 mbm +···+nbn =n

(0,··· ,0,bm ,bm+1 ,··· ,bn ) π∈S2n

bm =0

c π∈S2m

i=1





dξi

m 

dξi · · ·

i=1

U (ξ2i−1 − ξ2i )



bm C(ξr , ξπr ) ×

2m  r=1

i=1

2(n−mbm )

(m+1) π∈S2n−2mbm

bm =0

C(ξr , ξπr )

n! × (n − mbm )! (m!)bm bm !

   2m

k 

2n  r=1

i=1



k 

=

U (ξ2i−1 − ξ2i )

n 

=

=

n 

dξi

dξi

i=1

n−mb m



2(n−mbm )

U (ξ2i−1 − ξ2i )

C(ξr , ξπr )

r=1

i=1

bm  (m+1) n! (m)  Det Det m n−mbm b m (n − mbm )! (m!) bm !

(8.36)

or (m) k  Detmk+l = (mk + l)!



(m) 1 m! Detm

bm

bm !

bm =0

(m+1)

 Det mk−mbm +l (mk − mbm + l)!

(8.37)

(m+1)

Then, by the lemma and the definition of Detmk+l , (m+1)

k  mk+l  Det = (−1)bm (mk + l)! bm =0



(m) 1 m! Detm

bm !

bm

(m)

(m+1)

Detmk−mbm +l Detmk+l = (mk − mbm + l)! (mk + l)!

which proves (8.33).


8.3 8.3.1

Estimates on Feynman Diagrams Elementary Bounds

In this section we identify the large contributions which are typically contained in a sum of Feynman diagrams. These large contributions have the

Feynman Diagrams

137

effect that the lowest order terms of the naive perturbation expansion are not a good approximation. The elimination of these large contributions is called renormalization. We find that the size of a graph is determined by its subgraph structure. For the many-electron system with short range interaction, that is, for V (x) ∈ L1 , the dangerous subgraphs are the two- and four-legged ones. Indeed, in Theorem 8.3.4 below we show that an n’th order diagram without two- and four-legged subgraphs is bounded by constn which is basically the best case which can happen. A sum of diagrams where each diagram is bounded by constn can be expected to be asymptotic. That is, the lowest order terms in this expansion would be indeed a good approximation if the coupling is not too big. However usually there are certain subgraphs which produce anomalously large contributions which prevent the lowest order terms in the perturbation series from being a good approximation. For the many-electron system with short range interaction these are two- and four-legged subdiagrams. Fourlegged subgraphs produce factorials, that is, an n’th order bound of the form constn n!, the constant being independent of the cutoffs. Diagrams which contain two-legged subgraphs in general diverge if the cutoffs are removed. The goal of this section is to prove these assertions. We start with a lemma which sets up all the graph theoretical notation and gives the basic bound in coordinate space. The diagrams in Lemma 8.3.1 consist of generalized vertices or subgraphs which are represented by some functions I2qv (x1 , · · · , x2qv ) and lines to which are assigned propagators C(x − x ). A picture may be helpful. In figure 8.1 below, G = G(x1 , x2 ) and there is an integral over the remaining variables x3 , · · · , x10 .

C(x6-x3)

I(x1,x3,x4,x5) x1

5

3 4

C(x5-x9)

C(x7-x4)

I(x9,x10) 9

10

Figure 8.1

I(x2,x6,x7,x8) 6 7

8

C(x10-x8)

x2

138 Lemma 8.3.1 (Coordinate Space Bound) Let I2q = I2q (x1 , · · · , x2q ), xi ∈ Rd , be a generalized vertex (or subgraph) with 2q legs obeying

  2q I2q ∅ := sup sup dxj |I2q (x1 , · · · , x2q )| < ∞ xi

i

(8.38)

j=1 j=i

Let G be a connected graph built up from vertices I2qv , v ∈ VG , the set of all vertices of G, by pairing some of their legs. Two paired legs are by definition a line  ∈ LG , the set of all lines of G. To each line, assign a propagator v v C (xvi , xi ) = C (xvi − xi ). Suppose 2q legs remain unpaired. The value of   G is by definition G(x1 , · · · , x2q ) =

  v∈VG

2qv 

dxvi

i=1 xv int. i



I2qv (xv1 , · · · , xv2qv )

v∈VG



v

C (xvi , xi ) 

∈LG

(8.39) v v where x1 , · · · , x2q ∈ ∪v∈VG ∪2q i=1 {xi } are by definition the variables of the

v

v v unpaired legs and xvi , xi ∈ ∪v∈VG ∪2q i=1 {xi } are the variables of the legs  connected by the line . Let f1 , · · · , f2q be some test functions. Fix s = |S| legs of G where S ⊂ {1, · · · , 2q}. These s legs, which will be integrated against test functions, are by definition the external legs of G, and the other legs, which are integrated over Rd , are called internal. For S = ∅, define the norm       G S := (8.40) dxk |fk (xk )| dxi |G(x1 , · · · , x2p )|

i∈S c

k∈S

Then there are the following bounds: a) G ∅ ≤



C L1

∈T



C ∞



I2qv ∅

(8.41)

v∈V

∈L\T

where T is a spanning tree for G which is a collection of lines which connects all vertices of G such that no loops are formed. b) G S ≤

 ∈T¯

C L1

 ∈L\T¯

C ∞



I2qv Sv

(8.42)

v∈V

w where now T¯ = i=1 Ti is a union of w trees which spans G and w is the number of vertices to which at least one external leg is hooked which, by definition, is the number of external vertices. Each Ti contains precisely one external vertex. Finally, Sv is the set of external legs at I2qv .

Feynman Diagrams

139

c) Let G be a vacuum diagram, that is, a diagram without unpaired legs (q = 0). Then    |G| ≤ Ld C L1 C ∞ I2qv ∅ (8.43) ∈T

∈L\T

v∈V

where |G| is the usual modulus of G and Ld =




dx 1.

Proof: a) Choose a spanning tree T for G, that is, choose a set of lines T ⊂ L = LG which connects all vertices such that no loops are formed. For all lines not in T take the L∞ -norm. Let xi ∈ {x1 , · · · , x2q } be the variable where the supremum is taken over. We get G ∅ ≤

  v∈VG

≤

  v∈VG

2qv 

dxvi

i=1 xv i =xi

2qv  i=1 xv i =xi



|I2qv (xv1 , · · · , xv2qv )|

v∈VG

dxvi





v

|C (xvi − xi )| 

∈LG

|I2qv (xv1 , · · · , xv2qv )|

v∈VG



v

|C (xvi − xi )| × 

∈T



C ∞

∈LG \T

The vertex to which the variable xi belongs we define as the root of the tree. To perform the integrations, we start at the extremities of the tree, that is, at those vertices I2qv which are not the root and which are connected to the tree only by one line. To be specific, choose one of these vertices, I2qv1 (xv11 , · · · , xv2q1v ). Let xvr1 be the variable which belongs to the tree. This 1 variable also shows up in the propagator for the corresponding line , C (xvr1 −  xvr ) where v  is a vertex necessarily different from v since  is on the tree. Now we bound as follows    dxv11 · · · dxv2q1v |I2qv1 (xv11 , · · · , xv2q1v )||C (xvr1 − xvr )| |I2qv ({xvj })| 1 1  2q  v1   dxvi 1 |I2qv1 (xv11 , · · · , xv2q1v )||C (xvr1 − xvr )| |I2qv ({xvj })| = dxvr1 Π i=1  ≤

i=r

  dxvr1 sup v

xr 1

  dxvi 1 |I2qv1 (xv11 , · · · , xv2q1v )| |C (xvr1 − xvr )| × Π i=1

2qv1 i=r

  |I2qv ({xvj })|  2qv1 
  v v v = sup Π dxi 1 |I2qv1 (x11 , · · · , x2q1v )| dxvr1 |C (xvr1 − xvr )| |I2qv ({xvj })| v

xr 1

i=1 i=r



≤ I2qv1 ∅ C L1 |I2qv ({xvj })|

(8.44)

140 Now we repeat this step until we have reached the root of the tree. To obtain an estimate in this way we refer in the following to as ‘we apply the tree identity’. For each line on the tree we get the L1 norm, lines not on the tree give the L∞ norm and each vertex is bounded by the · ∅ norm which results in (8.41). b) Choose w trees T1 , · · · , Tw with the properties stated in the lemma. For each line not in T¯ take the L∞ -norm. Then for each Ti apply the tree identity with the external vertex as root. Suppose this vertex is I2qv (y1 , · · · , y2qv ) and y1 , · · · , ypv are the external variables. Then, instead of I2qv ∅ as in case (a) one ends up with pv  

2qv     dyk fjk (yk ) dyi |I2qv (y1 , · · · , y2qv )|

(8.45)

i=pv +1

k=1

which by definition is I2qv Sv . c) Here we proceed as in (a), however we can choose an arbitrary vertex I2qv to be the root of the tree, since we do not have to take a supremum over some xi . We apply the tree identity, and the integrations at the last vertex, the root I2qv , are bounded by   2qv 2qv v v v v v v v Π dxj |I2qv (x1 , · · · , x2qv )| ≤ dxi sup Π dxj |I2qv (x1 , · · · , x2qv )| xv i

j=1

j=1 j=i

≤ Ld I2qv ∅ which proves the lemma


For an interacting many body system the basic vertex is  ¯  )ψ(ξ  ) ¯ (ξ − ξ  )ψ(ξ dξdξ  ψ(ξ)ψ(ξ)U

(8.46)

which corresponds to the diagram ψ(ξ) ξ ψ(ξ)

ψ(ξ’) U(ξ-ξ’)

ξ’ ψ(ξ’)

In order to represent this by some generalized vertex I4 (ξ1 , ξ2 , ξ3 , ξ4 ) which corresponds to the diagram

Feynman Diagrams ψ(ξ3)

141

ψ(ξ4)

I4(ξ1,ξ2,ξ3,ξ4)

ψ(ξ1)

ψ(ξ2)

we rewrite (8.46) as  ¯ 3 )ψ(ξ1 )ψ(ξ ¯ 4 )ψ(ξ2 )I4 (ξ1 , ξ2 , ξ3 , ξ4 ) dξ1 dξ2 dξ3 dξ4 ψ(ξ

(8.47)

which coincides with (8.46) if we choose I4 (ξ1 , ξ2 , ξ3 , ξ4 ) = δ(ξ3 − ξ1 )δ(ξ4 − ξ2 )U (ξ1 − ξ2 )

(8.48)

I4 ∅ = U L1 (Rd+1 ) = V L1 (Rd ) < ∞

(8.49)

Then

if we assume a short range potential. The next lemma is the momentum space version of Lemma 8.3.1. We use the same letters for the Fourier transformed quantities, hats will be omitted. For translation invariant I4 (ξ1 , ξ2 , ξ3 , ξ4 ) = I4 (ξ1 + ξ  , ξ2 + ξ  , ξ3 + ξ  , ξ4 + ξ  ) we have  ¯ 4 )ψ(ξ2 )I4 (ξ1 , ξ2 , ξ3 , ξ4 ) = ¯ 3 )ψ(ξ1 )ψ(ξ dξ1 dξ2 dξ3 dξ4 ψ(ξ  d-k1 d-k2 d-k3 d-k4 (2π)d δ(k1 + k2 − k3 − k4 ) I4 (k1 , k2 , k3 , k4 )ψ¯k ψk ψ¯k ψk 3

1

4

2

dd k where we abbreviated d-k := (2π) d . For example,  dx1 dx2 dx3 dx4 ei(k1 x1 +k2 x2 −k3 x3 −k4 x4 ) δ(x3 − x1 )δ(x4 − x2 )V (x1 − x2 )  = dx1 dx2 ei(k1 −k3 )x1 +i(k2 −k4 )x2 V (x1 − x2 )  = dx1 dx2 ei(k1 −k3 )(x1 −x2 ) V (x1 − x2 ) ei(k1 +k2 −k3 −k4 )x2

= (2π)d δ(k1 + k2 − k3 − k4 ) V (k1 − k3 )

(8.50)

The value of the graph G defined in the above Lemma 8.3.1 reads in momentum space (2π)d δ(k1 + · · · + k2q ) G(k1 , · · · , k2q ) = (8.51)     v v dk C (k ) (2π)d δ(k1v + · · · + k2q )I2qv (k1v , · · · , k2q ) v v ∈LG

∈LG

v∈VG

142 where kiv ∈ ∪∈L {k } is the momentum ±k if  is the line to which the i’th leg of I2qv is paired. Observe that G(k1 , · · · , k2q ) is the value of the Fourier transformed diagram after the conservation of momentum delta function has been removed. However, for a vacuum diagram, we do not explicitly remove the factor δ(0) = Ld . In (8.51) we have |LG | =: |L| integrals and |VG | =: |V | constraints. The |V | delta functions produce one overall delta function which is explicitly written in (8.51) and |V | − 1 constraints on the |L| integration variables. Let T be a tree for G, that is, a collection of lines which connect all the vertices such that no loops are formed. Since |T | = |V | − 1, we can use the momenta k for lines  ∈ L \ T which are not on the tree as independent integration variables. We get     d-k C (K ) I2qv (Kv1 , · · · , Kv2qv ) (8.52) G(k1 , · · · , k2q ) = ∈L\T

∈L

v∈V

Here the sum of all momenta K flowing through the line  is defined as follows. Each line  not on the tree defines a unique loop which contains only lines on the tree with the exception of  itself. To each line in this loop assign the loop momentum k . The sum of the external variables k1 + · · · + k2q vanishes, thus we may write G(k1 , · · · , k2q ) = G(k1 , · · · , k2q−1 , −k1 − · · · − k2q−1 )

(8.53)

There are 2q −1 unique paths γi on G, containing only lines on the tree, which connect the i’th leg, to which the external momentum ki is assigned, to the 2q’th leg, to which the momentum k2q = −k1 − · · · − k2q−1 is assigned. To each line on γi assign the momentum ki . Then K is the sum of all assigned momenta. In particular, K = k for all  ∈ L \ T . The Ki appearing in I2qv (K1 , · · · , K2qv ) are the sum of all momenta flowing through the leg of I2qv labelled by i .

Lemma 8.3.2 (Momentum Space Bound) Define the following norms   I2q (k1 , · · · , k2q ) I2q ∞ = sup (8.54) k1 ,··· ,k2q ∈Rd k1 +···+k2q =0

and for S ⊂ {1, · · · , 2q}, S = ∅,     d-ks δ(k1 + · · · + k2q ) I2q (k1 , · · · , k2q ) I S = sup ks ∈Rd s∈S /

(8.55)

s∈S

Recall the notation of Lemma 8.3.1. Then there are the following bounds

Feynman Diagrams

143

a) G ∞ ≤





C ∞

∈T

C 1



I2qv ∞

(8.56)

v∈V

∈L\T

b) G S ≤



C ∞

∈T¯



C 1

∈L\T¯



I2qv Sv

v∈V Sv =∅



I2qv ∞ (8.57)

v∈V Sv =∅

c) Let G be a vacuum diagram, that is, G has no unpaired legs (q = 0). Then    |G| ≤ Ld C ∞ C 1 I2qv ∅ (8.58) ∈T

v∈V

∈L\T

where Ld is the volume of the system. Remark: Observe that the · S -norms in coordinate space depend on the choice of testfunctions whereas in momentum space the · S -norms are defined independent of testfunctions. Proof: a) We have   G(k1 , · · · , k2q ) ≤          C (K ) C (K ) I2qv (K1 , · · · , K2qv ) d-k ∈L\T

≤ =



C ∞

∈T

v∈V





C ∞

∈T

=

∈T





∈T



I2qv ∞  I2qv ∞

v∈V

C ∞



v∈V

v∈V

∈L\T



d-k

∈L\T



∈L\T

d-k

∈L\T

I2qv ∞



   C (K )    C (k ) ∈L\T

C 1

(8.59)

∈L\T

b) Let ki1 , · · · , ki|S| be the external momenta of G with respect to S . Observe that exactly |Svi | , i = 1, · · · , w , of these momenta are external to I2qvi and |Sv1 | + · · · + |Svw | = |S|

(8.60)

For each j = 1, · · · , w , let kj∗ , be one of the momenta external to Ivj . That is, kj∗ = kij∗ for some 1 ≤ j ∗ ≤ |S| . Let k˜1 , · · · , k˜w be the complementary external momenta. Here, w = |S| − w . We have  ∗    ∗ ˜ k1 , · · · , kw (8.61) , k1 , · · · , k˜w = ki1 , · · · , ki|S|

144 By definition, G S = sup

 

ks ∈Rd s∈S /

= sup ks ∈Rd s∈S /

d-k˜j

j=1



(8.62)

s∈S

  w

= sup

  d-ks δ(k1 + · · · + k2q ) G(k1 , · · · , k2q )

w 

  d-kj∗ δ(k1 + · · · + k2q ) G(k1 , · · · , k2q )

j=1 

w 

d-k0

ks ∈Rd s∈S /

d-k˜j

j=1

w 

∗ d-kj∗ δ(k0 + k1∗ + · · · + kw )×

j=1

  δ(k1 + · · · + k2q ) G(k1 , · · · , k2q )

Observe that   δ(k1 + · · · + k2q ) G(k1 , · · · , k2q ) ≤    d-k |C (k )| |I2qv (k1 , · · · , k2qv )| δ(k1 + · · · + k2qv )

(8.63)

v∈V

∈LG

Thus,  G S ≤ sup



dk0

ks ∈Rd s∈S /

w 

d-k˜j H(k0 , k˜1 , · · · , k˜w , ks ; s ∈ / S)

(8.64)

j=1

where / S) = H(k0 , k˜1 , · · · , k˜w , ks ; s ∈   w  ∗ d-kj∗ d-k |C (k )| δ(k0 + k1∗ + · · · + kw ) j=1

(8.65)

∈LG

×



|I2qv (k1 , · · · , k2qv )| δ(k1 + · · · + k2qv )

v∈V

Let G∗ be the graph obtained from G and the vertex I∗ = δ(k0 + k1∗ + ∗ ) with w + 1 legs by joining the leg of Ivj with momentum kj∗ · · · + kw to one leg of I∗ other than the leg labeled by k0 . Let Ti∗ be the tree obtained from Ti by adjoining the line that connects the external vertex at the end of Ti to I∗ . Observe that T ∗ = T1∗ ∪ · · · ∪ Tw∗ is a spanning tree for G∗ . For each  ∈ LG∗ \ T ∗ there is a momentum cycle obtained by attaching  to the unique path in T ∗ that joins the ends of  . Let p∗ be the momentum flowing around that cycle. The external legs of G∗ are labeled by the momenta k˜1 , · · · , k˜w , ks ; s ∈ / S , and k0 . Connect each of the external / S , to k0 by the unique path in T ∗ that joins them legs k˜1 , · · · , k˜w , ks ; s ∈ and let the external momenta flow along these paths. We have   / S) = δ · · · G∗ (k0 , k˜1 , · · · , k˜w , ks ; s ∈ / S) (8.66) H(k0 , k˜1 , · · · , k˜w , ks ; s ∈

Feynman Diagrams

145

where      ks δ · · · = δ k0 + k˜1 + · · · + k˜w +

(8.67)

s∈S /

and G∗ (k0 , k˜1 , · · · , k˜w , ks ; s ∈ / S) =     d-p∗ |C (K∗ )| |I2qv (K∗1 , · · · , K∗qv )| ∈LG∗ \T ∗

v∈V

∈LG

Here, K∗ is the sum of all momenta flowing through  and K∗i appearing in I2qv is the sum of all momenta flowing through the leg of I2qv labelled by i . Observe that, by construction, G∗ (k0 , k˜1, · · · , k˜w , ks ; s ∈ / S) is  independent of k0 . Now split LG = T1 ∪ · · · ∪ Tw ∪ LG \ (T1 ∪ · · · ∪ Tw ) . Then 

∗



G =



=







=



d-p∗ w 

, K∗2qv )| 

|C (K∗ )|

∈T1 ∪···∪Tw

∈LG∗ ) \T ∗

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

|I2qv (K∗1 , · · ·

v∈V



, K∗qv )| 

|C (K∗ )|

∈T1 ∪···∪Tw

∈LG∗ \T ∗

|C (K∗ )| ×

∈L\(T1 ∪···∪Tw )

|I2qv (K∗1 , · · ·

v∈V



d-p∗



|C (K∗ )|

∈T1 ∪···∪Tw

∈LG∗ \T ∗





d-p∗

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

|I2qvi (K∗1 , · · · , K∗2qv )|



i

i=1

|I2qv (K∗1 , · · · , K∗2qv )|

v=vi i=1,··· ,w

Thus we get  G S ≤ sup



dk0

ks ∈Rd s∈S /

w 

   d-k˜j δ k0 + k˜1 + · · · + k˜w + ks × s∈S /

j=1 ∗

= sup ks ∈Rd s∈S /

  w j=1

G (k0 , k˜1 , · · · , k˜w , ks ; s ∈ / S) d-k˜j G∗ (k˜1 , · · · , k˜w , ks ; s ∈ / S)

146   w

≤ sup ks ∈Rd s∈S /



d-k˜j



w 

≤

C ∞

ks ∈Rd s∈S /

I2qv ∞

v=vi i=1,··· ,w

  w

× sup

|I2qv (K∗1 , · · · , K∗2qv )|

v=vi i=1,··· ,w



∈T1 ∪···∪Tw

|C (p∗ )|

∈L\(T1 ∪···∪Tw )



|I2qvi (K∗1 , · · · , K∗2qvi )|

i=1





|C (K∗ )|

∈T1 ∪···∪Tw

∈LG∗ \T ∗

j=1



d-p∗



d-k˜j

 ∈LG∗ \T ∗

j=1 w 



d-p∗

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

|I2qvi (K∗1 , · · · , K∗2qvi )|

(8.68)

i=1

Exchanging integrals, we obtain   w

d-k˜j



 ∈LG∗ \T ∗

j=1

=

|I2qvi (K∗1 , · · · , K∗2qvi )|

i=1





d-p∗

∈LG∗ \T ∗



 ∈LG∗ \T ∗

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

  w

=

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

w 





d-p∗

j=1

d-p∗

d-k˜j

w 

|I2qvi (K∗1 , · · · , K∗2qv )| i

i=1



|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

w  



d-k˜j |I2qvi (K∗1 , · · · , K∗2qv )| i

i=1

(8.69)

˜ ∈Sv k j i j=1,··· ,w

For each i = 1, · · · , w , exactly one of the arguments, K∗1 , · · · , K∗2qv , say for i convenience the first, is the momentum flowing through the single line that connects I2qvi to I∗ . It is the sum of all external momenta flowing into I2qvi and some loop momenta. Furthermore, exactly |Svi | − 1 of the arguments K∗1 , · · · , K∗qv appearing in I2qvi are equal to external momenta in the set i {k˜1 , · · · , k˜w } . By construction, no other momenta flow through these legs. For convenience, suppose that K∗2 = k˜2 , · · · , K∗|Sv | = k˜|Svi | . The remaining i arguments on the list K∗1 , · · · , K∗qv are sums of loop momenta only. Recall i

Feynman Diagrams

147

that K∗1 = −K∗2 − · · · − K∗2qv . Thus, i   d-k˜j |I2qvi (K∗1 , · · · , K∗2qv )| = i

˜ ∈Sv k j i j=1,··· ,w



dp



  d-k˜j δ p + K∗2 + · · · + K∗2qvi |I2qvi (p, K∗2 , · · · , K∗2qvi )|

˜ ∈Sv k j i j=1,··· ,w

 =

 d-k˜j δ p + k˜2 + · · · + k˜|Svi | + K∗|Sv



dp

|+1 i

 + · · · + K∗2qvi ×

˜ ∈Sv k j i j=1,··· ,w

|I2qvi (p, k˜2 , · · · , K∗2qv )| i

 ≤

K∗ |S

sup vi |+1

,··· ,K∗ 2qv

 d-k˜j δ p + k˜2 + · · · + k˜|Svi | +



dp

˜ ∈Sv k j i j=1,··· ,w

i

+ K∗|Sv

|+1 i

+ · · · + K∗2qv

 i

|I2qvi (p, k˜2 , · · · , K∗2qv )| i

= I2qvi Svi

(8.70)

Combining (8.69) and (8.70), we arrive at   w

d-k˜j

j=1



 ∈LG∗ \T ∗

≤



d-p∗

w 

∈L\(T1 ∪···∪Tw )



I2qvi Svi

=



|I2qvi (K∗1 , · · · , K∗2qvi )| 

d-p∗



I2qvi Svi

w  i=1

∈LG∗ \T ∗

i=1 w 

|C (p∗ )|

|C (p∗ )|

∈L\(T1 ∪···∪Tw )

C 1

(8.71)

∈L\(T1 ∪···∪Tw )

i=1

Finally, G S ≤



C ∞

∈T1 ∪···∪Tw

 v=vi i=1,··· ,w

I2qv ∞

w  i=1

I2qvi Svi



C 1

∈L\(T1 ∪···∪Tw )

which proves the lemma.


8.3.2

Single Scale Bounds

The Lemmata 8.3.1 and 8.3.2 estimate a diagram in terms of the L1 - and 1 L - norms of its propagators. An L∞ bound of the type, say, 1+x 2 ≤ 1 is ∞

148 of course very crude since the information is lost that there is decay for large x. In order to get sharp bounds, one introduces a scale decomposition of the covariance which isolates the singularity and puts it at a certain scale. To this end let M be some constant bigger than one and write C(k) = =

=:

χ(|ik0 −ek |≤1) 1 k |>1) + χ(|ikik00−e ik0 −ek = ik0 −ek −ek ∞  χ(M −j−1 1) + χ(|ikik00−e ik0 −ek −ek j=0 ∞  j UV

C (k) + C

(k)

(8.72)

j=0

For each C j (k) we have the momentum space bounds C j ∞ ≤ M j ,

C j 1 ≤ c M −j

(8.73)

 since vol{(k0 , k) | k02 + e2k ≤ M −j } ≤ c M −2j . Here c is some j independent constant. Then the strategy to bound a diagram is the following one. Substitute each covariance C of the diagram in (8.51) by its scale decomposition. Interchange the scale-sums with the momentum space integrals. Then one has to bound a diagram where each propagator has a fixed scale. The L1 and L∞ -bounds on these propagators are now sharp bounds. Thus we apply Lemma 8.3.2. The bounds of Lemma 8.3.2 depend on the choice of a tree for the diagram. Propagators on the tree are bounded by their L∞ -norm which is large whereas propagators not on the tree are bounded by their L1 -norm which is small. Thus the tree should be chosen in such a way such that propagators with small scales are on the tree and those with large scales are not on the tree. This is the basic idea of the Gallavotti Nicolo tree expansion [33] which has been applied to the diagrams of the many-electron system in [29],[8]. An application to QED can be found in [17]. Instead of completely multiplying out all scales one can also apply an inductive treatment which is more in the spirit of renormalization group ideas which are discussed in the next section. Let C

≤j

(k) :=

j 

C i (k)

(8.74)

i=0

such that C ≤j+1(k) = C ≤j (k) + C j+1 (k)

(8.75)

We consider a diagram up to a fixed scale j, that is, each covariance is given by C ≤j . Then we see how the bounds change if we go from scale j to scale j +1 by using (8.75). If the diagram has L lines, application of (8.75) produces 2L terms. Each term can be considered as a diagram, consisting of subdiagrams

Feynman Diagrams

149

which have propagators C ≤j , and lines which carry propagators of scale j + 1. Then we apply Lemma 8.3.2 where the generalized vertices I2q are given by the subdiagrams of scale ≤ j which we can bound by a suitable induction hypothesis and all propagators are given by C j+1 . The next lemma specifies Lemmata 8.3.1, 8.3.2 for the case that all propagators have the same scale and satisfy the bounds (8.73) and in Theorem 8.3.4 diagrams are bounded using the induction ∞ outlined above. In the following we will consider only the infrared part j=0 C j of the covariance which contains the physically relevant region around the singularity at the Fermi surface ek = 0 and k0 = 0 and neglect the ultraviolet part C UV (k) in (8.72). The proof that all n’th order diagrams with C UV (k) as propagator are bounded by constn can be found in [29]. Lemma 8.3.3 Let G be a graph as in Lemma 8.3.1 or 8.3.2 and C be some covariance. a) Coordinate Space: Suppose that each C satisfies the estimates C (x) ∞ ≤ c M −j ,

C (x) 1 ≤ c M αj

(8.76)

Then there are the bounds   G ∅ ≤ c|LG | I2qv ∅ M −(qv −1−α)j M (q−1−α)j v∈VG

G S ≤ c|LG |

(8.77)

  I2qv ∅ M −(qv −1−α)j ×



v∈VG,int.





I2qv Sv M −(qv −

|Sv | 2 )j

(8.78) 

M (q−

|S| 2 )j

v∈VG,ext.

and if G has no unpaired legs   I2qv ∅ M −(qv −1−α)j M −(1+α)j |G| ≤ βLd c|LG |

(8.79)

v∈VG

b) Momentum Space: Suppose that each C (k) satisfies the estimates C (k) 1 ≤ c M −j

(8.80)

Then there are the bounds   I2qv ∞ M −(qv −2)j M (q−2)j G ∞ ≤ c|LG |

(8.81)

C (k) ∞ ≤ c M j

v∈VG

G S ≤ c|LG |

  I2qv ∞ M −(qv −2)j ×



v∈VG,int.



v∈VG,ext.



I2qv Sv M −(qv −

|Sv | 2 )j

(8.82) 

M (q−

|S| 2 )j

150 and if G has no unpaired legs   I2qv ∞ M −(qv −2)j M −2j |G| ≤ βLd c|LG |

(8.83)

v∈VG

Proof: a) Apply Lemma 8.3.1. Choose a spanning tree T for G. Suppose G is made of n = v 1 vertices. Then  1  |T | = n − 1, |L| = 2qv − 2q (8.84) 2 v∈V   qv − q − n + 1 = (qv − 1) − q + 1 (8.85) |L \ T | = v∈V

v∈V

Thus  n−1  P G ∅ ≤ c M αj c M −j v∈V = c|L| M −j ( = c|L| M −j



qv −q−n+1

I2qv ∅

v∈V

P v∈V



qv −(1+α)n+1+α−q)

I2qv ∅

v∈V

P v∈V

(qv −1−α)

M j(q−1−α)



I2qv ∅

v∈V

= c|L|

  I2qv ∅ M −(qv −1−α)j M (q−1−α)j

(8.86)

v∈V

which proves (8.77). To wobtain (8.78), we use (8.42). Let w be |Vext |. Choose a union of trees T¯ = i=1 Ti such that each Ti contains one external vertex and T¯ spans G. Then |T¯ | = n − 1 − (w − 1),   |L \ T¯ | = qv − q − (n − 1 − (w − 1)) = qv − p − n + w v∈V

v∈V

Recall that |Sv | is the number of external legs of I2qv and |S| the number of external legs of G, so v∈Vext |Sv | = |S|. Thus one gets n−w  P  c M −j v∈V G S ≤ c M αj = c|L| M −j = c|L|

I2qv Sv

v∈V

P v∈V

(qv −1−α)

M −j((1+α)w−q)

   I2qv ∅ M −(qv −1−α)j × v∈Vint



qv −q−n+w



I2qv Sv

v∈V

   I2qv Sv M −j(qv −1−α) M −(1+α)j M qj

v∈Vext

= c|L|

Feynman Diagrams    I2qv ∅ M −(qv −1−α)j × v∈Vint

 

 P |S| |S| I2qv Sv M −j ( v∈Vext qv − 2 ) M j(q− 2 )

v∈Vext

= c|L|

  v∈Vint

I2qv ∅ M −(qv −1−α)j

 

151



I2qv Sv M −j(qv −

× |Sv | 2 )



M j(q−

|S| 2 )

(8.87)

v∈Vext

The bounds of part (b) are obtained in the same way by using Lemma 8.3.2. The L1 - and L∞ -norms are interchanged on the tree and not on the tree but also the covariance bounds for the L1 - and L∞ -norm are interchanged (now α = 1) which results in the same graph bounds.


8.3.3

Multiscale Bounds

Theorem 8.3.4 Let G be a 2q-legged diagram made from vertices I2qv as in Lemma 8.3.1. Suppose that each line  ∈ LG , the set of all lines of G, carries ∞ the covariance C = j=0 C j where C j satisfies the momentum space bounds (8.73), C j 1 ≤ cM M −j

C j ∞ ≤ cM M j ,

a) Suppose that q ≥ 3, qv ≥ 3 for all vertices and that G has no two- and four-legged subgraphs (which is formalized by (8.101) below). Then one has  |L | I2qv Sv , (8.88) G {1,··· ,2q} ≤ aM G v∈VG

where aM = cM

∞

M−3 . j

j=0

b) Suppose that q ≥ 2, qv ≥ 2 for all vertices and that G has no two-legged subgraphs (which is formalized by (8.115) below). Then one has |L |

G {1,··· ,2q} ≤ rM,|LG | cM G



I2qv Sv ,

(8.89)

v∈VG

where rM,|LG | = 1 +

∞  j=1

j

|L |

j |LG | M − 2 (1 − M − 2 ) ≤ constM G |LG |! (8.90) 1

152 Proof: Let C ≤j =

j i=0

C i and

(2π)d δ(k1 + · · · + k2q ) G ≤j (k1 , · · · , k2q ) =     ≤j v v dk C (k ) (2π)d δ(k1v + · · · + k2q )I2qv (k1v , · · · , k2q ) v v ∈LG

v∈VG

∈LG

We write C ≤j+1 = C ≤j + C j+1 and multiply out: (2π)d δ(k1 + · · · + k2q ) G ≤j+1(k1 , · · · , k2q )      ≤j C + C j+1 (k ) × = dk ∈LG

=

   A⊂LG

=

∈LG

v v (2π)d δ(k1v + · · · + k2q )I2qv (k1v , · · · , k2q ) v v

v∈VG

dk



C ≤j (k )

∈LG \A

∈LG

   A⊂LG A=∅





∈A

v (2π) δ(· · · )I2qv (k1v , · · · , k2q ) v



∈LG \A

∈LG



C j+1 (k ) ×

d

v∈VG

dk



C ≤j (k )



C j+1 (k ) ×

∈A

v (2π)d δ(· · · )I2qv (k1v , · · · , k2q ) v

v∈VG

+ (2π) δ(k1 + · · · + k2q ) G ≤j (k1 , · · · , k2q ) d

(8.91)

The lines in A have scale j + 1 propagators while the lines in LG \ A have C ≤j as propagators. Generalized vertices which are connected by ≤ j lines we consider as a single subdiagram such that, for given A ⊂ LG , every term in (8.91) can be considered as a diagram consisting of certain subgraphs H2qw and only scale j + 1 lines. See the following figure 8.2 for some examples. Thus, let WG,A be the set of connected components which consist of vertices I2qv which are connected by lines in LG \ A. Each connected component is labelled by w ∈ WG,A and is itself a connected amputated diagram with, say, 2qw unpaired legs. These legs may be external legs of G or come from scale j + 1 lines. We write the connected components as (2π)d δ(k1w + · · · + w w )H2qw (k1w , · · · , k2q ) such that we obtain k2q w w (2π)d δ(k1 + · · · + k2q ) G ≤j+1(k1 , · · · , k2q )      w = dk C j+1 (k ) (2π)d δ(· · · )H2qw (k1w , · · · , k2q ) w A⊂LG A=∅

∈A

∈A

w∈WG,A

+ (2π)d δ(k1 + · · · + k2q ) G ≤j (k1 , · · · , k2q )

(8.92)

Feynman Diagrams

153

j I4

I4

j+1

H6

j+1

⇒

j+1

j+1 j+1

I2

j+1 H2=I2

j+1 I4

I4

j+1

j

j+1 j+1

⇒

j+1

H4

I2

H4=I4 j+1

Figure 8.2 w where the delta function δ(· · · ) = δ(k1w + · · · + k2q ). The above expression w can be considered as the value of a diagram with generalized vertices H2qw and lines  ∈ A. Let TG,A ⊂ A be a tree for that diagram. Then we can eliminate all the momentum-conserving delta functions by a choice of loop momenta:

G ≤j+1(k1 , · · · , k2q ) =     dk C j+1 (K ) A⊂LG A=∅

∈A

∈A\TG,A



w H2qw (Kw 1 , · · · , K2qw )

w∈WG,A

+ G ≤j (k1 , · · · , k2q )

(8.93)

where, as in (8.52), K is the sum of momenta flowing through the line . Then we get with the momentum space bound of Lemma 8.3.3 G ≤j+1 − G ≤j ∞ ≤



w∈WG,A

A⊂LG A=∅

G ≤j+1 − G ≤j S ≤

   M −(j+1)(qw −2) H2qw ∞ M (j+1)(q−2)

c|A|



 

c|A|

A⊂LG A=∅



(8.94)

M −(j+1)(qw −2) H2qw ∞ ×

w∈WG,A w int.

   |Sw | |S| M −(j+1)(qw − 2 ) H2qw Sw M (j+1)(q− 2 )

w∈WG,A w ext.

(8.95)

154 Part a) We verify the following bounds by induction on j: For each connected ≤j

amputated 2q-legged diagram G2q with q ≥ 3 one has ≤j

G2q ∞ ≤ c|LG | s(j)|LG | M −j 3 M j(q−2) q



I2qv ∞

(8.96)

v∈VG

and for all S = ∅, {1, · · · , 2q} : ≤j

j

G2q S ≤ c|LG | s(j)|LG | M − 3 (q− 

≤j

G2q {1,··· ,2q} ≤ c|LG | s(j)|LG |

|S| 2 )

M j(q−

|S| 1 2 −3)



I2qv Sv (8.97)

v∈VG

I2qv Sv

(8.98)

v∈VG

j j where s(j) = i=0 M − 3 . Then part (a) is a consequence of (8.98). For j = 0 one has C = C 0 for all lines and one obtains with Lemma 8.3.3: G ∞ ≤ c|LG |

  I2qv ∞ M −(qv −2)0 M (q−2)0 v∈VG

=c

|LG |



s(0)|LG |

I2qv ∞

v∈VG

G S ≤ c|LG |



  I2qv ∞ M −(qv −2)0 ×

v∈VG,int.





I2qv Sv M −(qv −

|Sv | 2 )0



M (q−

|S| 2 )0

v∈VG,ext.

= c|LG | s(0)|LG |



I2qv Sv

(8.99)

v∈VG

Suppose (8.96) through (8.98) are correct for j. Then, to verify (8.96) for j + 1, observe that by (8.93) and the induction hypothesis (8.96) ≤j+1

G2q ≤



c|A|

A⊂LG A=∅

≤



A⊂LG A=∅

≤j+1

∞ ≤ G2q

≤j

≤j

− G2q ∞ + G2q ∞

   ≤j M −(j+1)(qw −2) H2qw ∞ M (j+1)(q−2) + G2q ∞ w∈WG,A

c|A|

  qw M −(j+1)(qw −2) cLw s(j)Lw M −j 3 M j(qw −2) ×

w∈WG,A

 v∈Vw

 I2qv ∞ M (j+1)(q−2) +

Feynman Diagrams  q I2qv ∞ + c|LG | s(j)|LG | M −j 3 M j(q−2) 

=

 

c|LG | s(j)LG \A

v∈VG

M −(qw −2) M −j

qw 3

 

w∈WG,A

A⊂LG A=∅

155

I2qv ∞ M (j+1)(q−2)

v∈VG

q 3

+ c|LG | s(j)|LG | M −j M j(q−2)



I2qv ∞

(8.100)

v∈VG

Now the assumption that G has no two- and four-legged subgraphs means qw ≥ 3

(8.101)

for all possible connected components H2qw in (8.93). Thus we have  

M −(qw −2) M −j

qw 3



≤

w∈WG,A



M −(j+1)

qw 3

|A|+q  j+1 = M− 3

w∈WG,A

and (8.100) is bounded by  |A|+q  j+1 c|LG | s(j)|LG \A| M − 3 I2qv ∞ M (j+1)(q−2)



v∈VG

A⊂LG A=∞ q

q

+ c|LG | s(j)|LG | M −j 3 M (j+1)(q−2) M − 3



I2qv ∞

v∈VG q

= c|LG | M −(j+1) 3

|LG | 



|LG | k

  k  j+1 s(j)|LG |−k M − 3 I2qv ∞ M (j+1)(q−2)

k=1

+c

|LG |

|LG |

s(j)

M

−(j+1) q3

M

(j+1)(q−2)



v∈VG

I2qv ∞

v∈VG q

= c|LG | M −(j+1) 3

|LG | 



|LG | k

  k  j+1 s(j)|LG |−k M − 3 I2qv ∞ M (j+1)(q−2) v∈VG

k=0

q

= c|LG | M −(j+1) 3 =c

|LG |

M

−(j+1) q3

 |LG |  j+1 I2qv ∞ M (j+1)(q−2) s(j) + M − 3 |LG |

s(j + 1)



v∈VG

I2qv ∞ M (j+1)(q−2)

(8.102)

v∈VG

which verifies (8.96). To verify (8.97) and (8.98) for scale j + 1, observe that

156 by (8.95) and the induction hypothesis (8.97,8.98) ≤j+1

G2q

≤j+1

≤j

S ≤ G2q

≤ M (j+1)(q−



|S| 2 )

≤j

− G2q S + G2q S  

c|A|

A⊂LG A=∅

 M −(j+1)(qw −2) H2qw ∞ ×

w∈WG,A w int.

   |Sw | M −(j+1)(qw − 2 ) H2qw Sw + G≤j 2q S w∈WG,A w ext.

≤ M (j+1)(q−



|S| 2 )

 

c|A|

A⊂LG A=∅

M j(qw −2)



j

|Sw | 2 )

I2qv ∞



=M

 

×

  |Sw | M −(j+1)(qw − 2 ) c|Lw | s(j)Lw ×

w∈WG,A w ext.

M j(qw −



|Sw | 1 2 −3)

I2qv Sv

v∈Vw (j+1)(q− |S| 2 )

qw 3

w∈WG,A w int.

v∈Vw

M − 3 (qw −

M −(j+1)(qw −2) c|Lw | s(j)Lw M −j



c|LG |

A⊂LG A=∅

M −(qw −

 

s(j)|LG \A|

|Sw | 2 )

j

|Sw | 2 )

+ G≤j 2q S

M −(qw −2) M −j

w∈WG,A w int.

M − 3 (qw −



M −j 3 1

 

qw 3



×

I2qv Sv

v∈VG

w∈WG,A w ext.

+ G≤j 2q S

(8.103)

Now, for S = ∅ but S = {1, · · · , 2q} may be allowed, one has  

M −(qw −2) M −j

w∈WG,A w int.

≤ M

− j+1 3

qw 3



 

M −(qw −

w∈WG,A w ext.

P

w∈WG,A w int.

M

− 23

qw

P

M

− j+1 3

w∈WG,A w ext.

− 23

|Sw | 2 )

P w∈WG,A w ext.

(qw − |S2w | )

P

= M−

|S| j+1 3 (|A|+q− 2 )

≤ M−

|S| j+1 3 (|A|+q− 2 )

M−3×2 M−3

= M−

|S| j+1 3 (|A|+q− 2 )

M−

M

2

j+1 3

w∈WG,A w ext.

1

j

M − 3 (qw −

M

|Sw | 2 )

(qw − |S2w | )

M −j 3 1



×

− j3 |Vext. |

(qw − |S2w | )

j

M − 3 |Vext. |

j

(8.104)

Feynman Diagrams

157

Therefore it follows from (8.103) for S = ∅ ≤j+1

G2q

S ≤ c|LG | M −

|S| j+1 3 (q− 2 )



M (j+1)(q−



|S| 1 2 −3)

I2qv Sv ×

v∈VG

s(j)|LG \A| M −

j+1 3 |A|

≤j

+ G2q S

(8.105)

A⊂LG A=∅

Now suppose first that S = {1, · · · , 2q}. By the induction hypothesis (8.97) one has  |S| |S| j 1 ≤j I2qv Sv G2q S ≤ c|LG | s(j)|LG | M − 3 (q− 2 ) M j(q− 2 − 3 ) v∈VG

= c

|LG |

|LG |

s(j)

M

− 23 (q− |S| 2 )

≤ c|LG | s(j)|LG | M −

1 3

M M

|S| j+1 3 (q− 2 )

|S| − j+1 3 (q− 2 )

M (j+1)(q−

M

1 (j+1)(q− |S| 2 −3)



|S| 1 2 −3)



I2qv Sv

v∈VG

I2qv Sv

(8.106)

v∈VG

Substituting this in (8.105), one obtains ≤j+1

G2q

S ≤ c|LG | M − 

|S| j+1 3 (q− 2 )

I2qv Sv

v∈VG

|S|

M (j+1)(q− 2 − 3 ) ×  j+1 s(j)|LG \A| M − 3 |A| 1

A⊂LG A=∅

+ c|LG | s(j)|LG | M −

|S| j+1 3 (q− 2 )

M (j+1)(q−

|S| 1 2 −3)



I2qv Sv

v∈VG

= c|LG | M −  v∈VG

= c|LG | M −

|S| j+1 3 (q− 2 )

I2qv Sv

|S|

M (j+1)(q− 2 − 3 ) ×  j+1 s(j)|LG \A| M − 3 |A| 1

A⊂LG |S| j+1 3 (q− 2 )

M (j+1)(q−



|S| 1 2 −3)

I2qv Sv s(j + 1)|LG |

v∈VG

which verifies (8.97) for j + 1. Now let S = {1, · · · , 2q}. Then by the induction hypothesis (8.98) one has ≤j

G2q {1,··· ,2q} ≤ c|LG | s(j)|LG |

 v∈VG

I2qv Sv

(8.107)

and (8.105) becomes ≤j+1

G2q

{1,··· ,2q} ≤ c|LG | M −(j+1) 3 1



I2qv Sv

v∈VG

+ c|LG | s(j)|LG |

 v∈VG

 A⊂LG A=∅

I2qv Sv

s(j)|LG \A| M −

j+1 3 |A|

158

≤ c|LG |



I2qv Sv

v∈VG



s(j)|LG \A| M −

j+1 3 |A|

A⊂LG



= c|LG | s(j + 1)|LG |

I2qv Sv

(8.108)

v∈VG

which verifies the induction hypothesis (8.98) for j + 1. Part b) We verify the following bounds by induction on j: For each connected ≤j

amputated 2q-legged diagram G2q with q ≥ 2 one has 

|L |

≤j

G2q ∞ ≤ cM G j |LG | M j(q−2)

I2qv ∞

(8.109)

v∈VG

and for all S = ∅, {1, · · · , 2q}: |L |

≤j

G2q S ≤ cM G j |LG | M j(q−



|S| 1 2 −2)

I2qv Sv

(8.110)

v∈VG |L |

≤j

G2q {1,··· ,2q} ≤ rM (j) cM G



I2qv Sv

(8.111)

v∈VG

where rM (j) = 1 +

j−1 

i|LG | (M − 2 − M − i

i+1 2

j

) + j |LG | M − 2

(8.112)

i=1

Then part (b) is a consequence of (8.111). For j = 0 one has C = C 0 for all lines and one obtains with Lemma 8.3.3: |L |

G ∞ ≤ cM G



  |L | I2qv ∞ M −(qv −2)0 M (q−2)0 = cM G I2qv ∞

v∈VG |L |

G S ≤ cM G



v∈VG,int.

=



|L | cM G v∈VG

  I2qv ∞ M −(qv −2)0 × 

v∈VG

  |Sv | |S| I2qv Sv M −(qv − 2 )0 M (q− 2 )0

v∈VG,ext.

I2qv Sv

(8.113)

Suppose (8.109) through (8.111) are correct for j. Then, to verify (8.109) for

Feynman Diagrams

159

j + 1, observe that by (8.93) and the induction hypothesis (8.109) ≤j+1

G2q ≤

≤j+1

∞ ≤ G2q



w∈WG,A

A⊂LG A=∅

≤

  |L | M −(j+1)(qw −2) cM w j |Lw | M j(qw −2) ×

|A|

cM

A⊂LG A=∅

≤j

   ≤j M −(j+1)(qw −2) H2qw ∞ M (j+1)(q−2) + G2q ∞

c|A|



≤j

− G2q ∞ + G2q ∞

w∈WG,A

 I2qv ∞ M (j+1)(q−2)



v∈Vw



|L |

+ cM G j |LG | M j(q−2)

I2qv ∞

v∈VG

=



    M −(qw −2) I2qv ∞ M (j+1)(q−2)

|L |

cM G j |LG \A|

w∈WG,A

A⊂LG A=∅

v∈VG



|L |

+ cM G j |LG | M j(q−2)

I2qv ∞

(8.114)

v∈VG

Now the assumption that G has no two-legged subgraphs means qw ≥ 2

(8.115)

for all possible connected components H2qw in (8.93). Thus we have 

M −(qw −2) ≤ 1

w∈WG,A

and (8.114) is bounded by 

|L |

cM G j |LG \A|

A⊂LG A=∞



I2qv ∞ M (j+1)(q−2)

v∈VG |L |

+ cM G j |LG | M (j+1)(q−2)



I2qv ∞

v∈VG

=

|L | cM G

|LG | 



k=1

+

|LG | k



j |LG |−k

 v∈VG

|L | cM G j |LG | M (j+1)(q−2)

I2qv ∞ M (j+1)(q−2)  v∈VG

I2qv ∞

160 |L |

= cM G

|LG | 



|LG | k



k=0

=

|L | cM G (j

=

|L | cM G (j

|LG |

+ 1)



j |LG |−k 

I2qv ∞ M (j+1)(q−2)

v∈VG

I2qv ∞ M (j+1)(q−2)

v∈VG



|LG |

+ 1)

I2qv ∞ M (j+1)(q−2)

(8.116)

v∈VG

which verifies (8.109). To verify (8.110, 8.111) for scale j + 1, observe that by (8.95) and the induction hypothesis (8.110, 8.111) ≤j+1

G2q

≤j+1

S ≤ G2q

≤ M (j+1)(q−

|S| 2 )

≤j

≤j

− G2q S + G2q S



|A|

cM

A⊂LG A=∅

 

 M −(j+1)(qw −2) H2qw ∞ ×

w∈WG,A w int.

   |Sw | M −(j+1)(qw − 2 ) H2qw Sw + G≤j 2q S w∈WG,A w ext.

≤ M (j+1)(q−

|S| 2 )



|A|

cM

A⊂LG A=∅

M j(qw −2)



w∈WG,A w int.

I2qv ∞

|Sw | 1 2 −2)



|L |

M −(j+1)(qw −2) cM w j |Lw | ×

 



=

|Sw | 2 )

|L |

cM w j |Lw | ×

 I2qv Sv + G≤j 2q S

v∈Vw |S| |L | M (j+1)(q− 2 ) cM G

M −(j+1)(qw −

w∈WG,A w ext.

v∈Vw

M j(qw −

 





j |LG \A|

A⊂LG A=∅

M −(qw −2) ×

w∈WG,A w int.

    |Sw | 1 I2qv Sv + G≤j M −(qw − 2 ) M −j 2 2q S

(8.117)

v∈VG

w∈WG,A w ext.

Now, for S = ∅ but S = {1, · · · , 2q} may be allowed, one has  w∈WG,A w int.

M −(qw −2)

   |Sw | 1 M −(qw − 2 ) M −j 2 w∈WG,A w ext.

≤ 1 · M − 2 M −j 2 = M − 1

1

j+1 2

(8.118)

since there is at least one external vertex and at least one internal line because

Feynman Diagrams

161

of A = ∅. Therefore it follows from (8.117) for S = ∅ ≤j+1

G2q

|L |

S ≤ cM G M (j+1)(q−

|S| 1 2 −2)





I2qv Sv

≤j

j |LG \A| + G2q S

A⊂LG A=∅

v∈VG

(8.119) Now suppose first that S = {1, · · · , 2q}. By the induction hypothesis (8.110) one has  |S| 1 ≤j |L | I2qv Sv G2q S ≤ cM G j |LG | M j(q− 2 − 2 ) v∈VG

≤

|S| 1 |L | cM G j |LG | M (j+1)(q− 2 − 2 )



I2qv Sv

(8.120)

v∈VG

Substituting this in (8.119), one obtains ≤j+1

G2q

|L |

S ≤ cM G M (j+1)(q−

|S| 1 2 −2)



I2qv Sv

v∈VG |L |

|L |

|S| 1 2 −2)

|S| 1 2 −2)



|L |

|S| 1 2 −2)

I2qv Sv

v∈VG



I2qv Sv

v∈VG

= cM G M (j+1)(q−

j |LG \A|

A⊂LG A=∅

+ cM G j |LG | M (j+1)(q− = cM G M (j+1)(q−





j |LG \A|

A⊂LG



I2qv Sv (j + 1)|LG |

(8.121)

v∈VG

which verifies (8.110) for j + 1. Now let S = {1, · · · , 2q}. By the induction hypothesis (8.111) one has |L |

≤j

G2q {1,··· ,2q} ≤ rM (j) cM G

 v∈VG

I2qv Sv

(8.122)

where rM (j) = 1 +

j−1 

i|LG | (M − 2 − M − i

i+1 2

j

) + j |LG | M − 2

i=1

and (8.119) becomes ≤j+1

G2q

|L |

{1,··· ,2q} ≤ cM G M −(j+1) 2 1



I2qv Sv

v∈VG |L |

+ rM (j) cM G

 v∈VG

 A⊂LG A=∅

I2qv Sv

j |LG \A|

(8.123)

162    1 |L | I2qv Sv = cM G M −(j+1) 2 (j + 1)|LG | − j |LG | 

|L |

+ rM (j) cM G

v∈VG

I2qv Sv

v∈VG |L |



= rM (j + 1) cM G M −(j+1) 2 1

I2qv Sv

(8.124)

v∈VG

since   1 rM (j) + M −(j+1) 2 (j + 1)|LG | − j |LG | = 1+

j−1   i=1

i|LG | M − 2 − i|LG | M − i

i+1 2



+ j |LG | M − 2 j

  1 + (j + 1)|LG | − j |LG | M −(j+1) 2

= rM (j + 1)

(8.125)

which verifies the induction hypothesis (8.111) for j + 1.


8.4

Ladder Diagrams

In this section we explicitly compute n’th order ladder diagrams and show that they produce factorials. Consider the following diagram:

s+q/2

t+q/2 s

k1

k2

...

kn

t

-s+q/2

-t+q/2

Its value is given by  n Λn (s, t, q) := Π

i=1

dd+1 ki (2π)d+1

n

Π C(q/2 + ki )C(q/2 − ki ) ×

i=1

n−1

V (s − k1 ) Π V (ki − ki+1 ) V (kn − t) i=1

(8.126)

Feynman Diagrams

163

Specializing to the case of a delta function interaction in coordinate space or a constant in momentum space, V (k) = λ, one obtains Λn (s, t, q) = λn+1 Λn (q) where  n n dd+1 k Λn (q) = Π (2π)d+1i Π C(q/2 + ki )C(q/2 − ki ) i=1 i=1 
d+1 n n d k = C(q/2 + k)C(q/2 − k) = {Λ(q)} (8.127) d+1 (2π) and Λ(q) =




dd+1 k (2π)d+1

C(q/2 + k)C(q/2 − k)

(8.128)

is the value of the particle-particle bubble,

Λ(q) =

q→

k

→q

The covariance is given by C(k) = C(k0 , k) = 1/(ik0 − ek ). For small q the value of Λ(q) is computed in the following Lemma 8.4.1 Let C(k) = 1/(ik0 − k2 + 1), let d = 3 and for q = (q0 , q), |q| < 1, let  ∞  dk0 d3 k Λ(q) := (8.129) 2π (2π)3 C(q/2 + k)C(q/2 − k) −∞

|k|≤2

Then for small q Λ(q) = − 8π1 2 log[q02 + 4|q|2 ] + O(1)

(8.130)

Proof: We first compute the k0 -integral. By the residue theorem  ∞ 1 dk0 1 2π i(k + q /2) − e i(−k + q /2) − e−k+q/2 0 0 0 0 k+q/2 −∞  ∞ 1 dk0 1 = 2π k + q /2 + ie k − q /2 − ie−k+q/2 0 0 0 k+q/2 0 −∞

χ(ek+q/2 > 0)χ(e−k+q/2 > 0) 2πi χ(ek+q/2 < 0)χ(e−k+q/2 < 0) = + 2π −q0 − iek+q/2 − ie−k+q/2 q0 + iek+q/2 + ie−k+q/2 =

χ(ek+q/2 > 0)χ(ek−q/2 > 0) χ(ek+q/2 < 0)χ(ek−q/2 < 0) + iq0 + |ek+q/2 | + |ek−q/2 | −iq0 + |ek+q/2 | + |ek−q/2 |

(8.131)

164 To compute the integral over the spatial momenta, we consider the cases ek+q/2 < 0, ek−q/2 < 0 and ek+q/2 > 0, ek−q/2 > 0 separately. Case (i): ek+q/2 < 0, ek−q/2 < 0. Let p = q/2 and let k := |k|, p := |p|, and cos θ = kp/(kp). Then k 2 + 2kp cos θ + p2 < 1, k 2 − 2kp cos θ + p2 < 1 ⇔ k 2 < 1 − p2 , 2kp| cos θ| < 1 − p2 − k 2

(8.132)

then the last inequality in (8.132) gives no If 2kp < 1 − p2 − k 2 or k < 1 − p, restriction on θ. For 1 − p ≤ k < 1 − p2 one gets | cos θ|
1, k >1−p , k 2 > 1 − p2 , 2

2

k 2 > 1 − p2 ,

k 2 − 2kp cos θ + p2 > 1

⇔

±2kp cos θ > 1 − p − k ⇔ 2 2 ∓2kp cos θ < k − (1 − p ) ⇔ 2

2

2kp |cos θ| < k 2 − (1 − p2 )

(8.136)

If 2kp < k 2 − (1 − p2) or k > 1 + p (recall that  |q| < 1), then the last inequality in (8.136) gives no restriction on θ. For 1 − p2 ≤ k < 1 + p one gets | cos θ| < Thus we have 

k 2 − (1 − p2 ) 2kp

(8.137)

d3 k χ(ek+q/2 > 0)χ(ek−q/2 > 0) 3 −iq + |e 0 k+q/2 | + |ek−q/2 | |k| 0)χ(ek−p > 0) = 2 dk k 2 dθ sin θ 4π 0 −iq0 + 2(k 2 + p2 − 1) 0   k2 −(1−p2 )  1+p  π χ | cos θ| < 2kp 1 + = 2 √ dk k 2 dθ sin θ 2 + p2 − 1) 4π −iq + 2(k 2 0 0 1−p

 π  2 1 dk k 2 dθ sin θ −iq0 + 2(k 2 + p2 − 1) 1+p 0  1+p 2 2 1 2 ) + dk k 2 k −(1−p = 2 √ 2kp 2 + p2 − 1) 4π −iq + 2(k 0 1−p2

 2 2 2 dk k (8.138) −iq0 + 2(k 2 + p2 − 1) 1+p

166  Now we substitute x = k 2 − (1 − p2 ), k = x + 1 − p2 = kx to obtain  d3 k χ(ek+q/2 > 0)χ(ek−q/2 > 0) 3 −iq + |e 0 k+q/2 | + |ek−q/2 | |k| 0 we have arg[±iq0 + 4p(1 ∓ p)] ∈ (−π/2, π/2) and therefore log[iq0 + 4p(1 − p)] + log[−iq0 + 4p(1 + p)]   = 12 log[q02 + 16p2 (1 − p)2 ] + log[q02 + 16p2 (1 + p)2 ] + iO(1) = log[q02 + 4|q|2 ] + O(1)

(8.140)

which proves the lemma.
Thus the leading order behavior of Λn (q) for small q is given by  n Λn (q) = {Λ(q)}n ∼ constn log[1/(q02 + 4|q|2 )]

(8.141)

Feynman Diagrams

167

If Λn (q) is part of a larger diagram, there may be an integral over q which then leads to an n!. Namely, if ω3 denotes the surface area of S 3 ,    n ω3 n 1 3 2 2 2 dq0 d q log[1/(q0 + 4|q| )] = dρ ρ3 (log[1/ρ])n 8 0 q02 +4|q|2 ≤1  ∞ ρ=e−x ω3 n 2 = dx e−4x xn 8 0 =

constn n!

(8.142)

Chapter 9 Renormalization Group Methods

In the last section we proved that, for the many-electron system with a short range potential (that is, V (x) ∈ L1 ), an n’th order diagram Gn allows the following bounds. If Gn has no two- and four-legged subgraphs, it is bounded by constn (measured in a suitable norm), if Gn has no two-legged but may have some four-legged subgraphs it is bounded by n! constn (and it was shown that the factorial is really there by computing n’th order ladder diagrams with dispersion relation ek = k2 /2m − µ), and if Gn contains two-legged subdiagrams it is in general divergent. The large contributions from two- and four-legged subdiagrams have to be eliminated by some kind of renormalization procedure. After that, one is left with a sum of diagrams where each n’th order diagram allows a constn bound. The perturbation series for the logarithm of the partition function and similarly those series for the correlation functions are of the form (8.16) log Z(λ) =

∞
n=1




λn n!

signπ Gn (π)

(9.1)

π∈S2n Gn (π) connected

where each permutation π ∈ S2n generates a certain diagram. The sign signπ is present in a fermionic model like the many-electron system but would be absent in a bosonic model. The condition of connectedness and similar conditions like ‘Gn (π) does not contain two-legged subgraphs’ do not significantly change the number of diagrams. That is, the number of diagrams which contribute to (9.1) or to a quantity like F (λ) =

∞
n=1




λn n!

signπ Gn (π) =:

∞


gn λn

(9.2)

n=1

π∈S2n Gn (π) connected, without 2−,4−legged subgraphs

is of the order (2n)! or, ignoring a factor constn , of the order (n!)2 , even in the case that diagrams which contain two- and four-legged subgraphs are removed. If we ignore the sign in (9.2) we would get a bound |F (λ)| ≤

∞
n=0

|λ|n n!

constn (n!)2 =

∞


n! (const|λ|)n

(9.3)

n=0

169

170 and the series on the right hand side of (9.3) has radius of convergence zero. Now there are two possibilities. This also holds for the original series (9.2) or the sign in (9.2) improves the situation. In the first case, which unavoidably would be the case for a bosonic model, there are again two possibilities. Either the series is asymptotic, that is, there is the bound n  
  gj λj  ≤ n! (const|λ|)n (9.4) F (λ) − j=1

or it is not. If one is interested only in small coupling, which is the case we restrict to, an asymptotic series would already be sufficient in order to get information on the correlation functions since (9.4) implies that the lowest order terms are a good approximation if the coupling is small. For the many-electron system with short range interaction the sign in (9.2) indeed improves the situation and one obtains a small positive radius of convergence (at least in two space dimensions) for the series in (9.2). One obtains the bound n  
  gj λj  ≤ (const|λ|)n (9.5) F (λ) − j=1

Concerning the degree of information we can get about the correlation functions or partition function we are interested in (here the quantity F (λ)), there is practically no difference whether we have (9.4) or (9.5). In both cases we can say that the lowest order terms are a good approximation if the coupling is small and not more. We compute the lowest order terms for, say, n = 1, 2 and then it does not make a difference whether the error is 3! (const|λ|)3 or just (const|λ|)3 . However, from a technical point of view, it is much easier to rigorously prove a bound for a series with a small positive radius of convergence, as in (9.5), which eventually may hold for the sum of convergent diagrams for a fermionic model, than to prove a bound for an expansion which is only asymptotic, as in (9.4), which typically holds for the sum of convergent diagrams for a bosonic model. The goal of this section is to rigorously prove a bound of the form (9.5) on the sum of convergent diagrams for a typical fermionic model. By ‘convergent diagrams’ we mean diagrams which allow a constn -bound. In particular, for the many-electron system, this excludes for example ladder diagrams which, although being finite, behave like n! constn . The reason is the following. There are two sources of factorials which may influence the convergence properties of the perturbation series. The number of diagrams which is of order (n!)2 , which, including the prefactor of 1/n! in (9.2), may produce (if the sign is absent) an n! and there may be an n! due to the values of certain n’th order diagrams, like the ladder diagrams. Now, roughly one can expect the following: • A sum of convergent diagrams, that is, a sum of diagrams where each diagram allows a constn bound, is at least asymptotic. That is, its

Renormalization Group Methods

171

lowest order terms are a good approximation if the coupling is small, regardless whether the model is fermionic or bosonic. • A sum of finite diagrams which contains diagrams which behave like n! constn is usually not asymptotic. That is, its lowest order contributions are not a good approximation. Those diagrams which behave like n! constn have to be resummed, for example by the use of integral equations as described in the next chapter. For the many-electron system in two space dimensions a bound of the form (9.5) on the sum of convergent diagrams has been rigorously proven in [18, 16]. The restriction to two space dimensions comes in because in the proof one has to use conservation of momentum for momenta which are close to the Fermi surface k2 /2m − µ = 0 which is the place of the singularity of the free propagator. Then in two dimensions one gets more restrictive conditions than in three dimensions [24]. If the singularity of the covariance would be at a single point, say at k = 0, these technicalities due to the implementation of conservation of momentum are absent and the proof of (9.5) becomes more transparent. Therefore in this section we choose a model with covariance C(k) =

1 d

|k| 2

, k ∈ Rd

(9.6)

and, as before, a short range interaction V ∈ L1 . This C has the same power counting as the propagator of the many-electron system which means that the bounds on Feynman diagrams in Lemma 8.3.3 and Theorem 8.3.4 for d both propagators are the same (with M substituted by M 2 in case of (9.6)). However, as mentioned above, the proof of (9.5), that is, the proof of the Theorems 9.2.2 and 9.3.1 below will become more transparent.

9.1

Integrating Out Scales

In this subsection we set up the scale by scale formalism which allows one to compute the sum of all diagrams at scale j from the sum at scale j − 1. We start with the following

Lemma 9.1.1 Let C, C 1 , C 2 ∈ CN ×N be invertible complex matrices and 1 , dµC 2 be the corresponding Grassmann Gaussian C = C 1 + C 2 . Let dµC , dµCP ¯ −1 − N ¯ ¯ i,j=1 ψi Cij ψj measures, dµC = det C e ΠN i=1 (dψi dψi ). Let P (ψ, ψ) be some

172 polynomial of Grassmann variables. Then   ¯ = ¯ dµC (ψ, ψ) P (ψ 1 + ψ 2 , ψ¯1 + ψ¯2 ) dµC 1 (ψ 1 , ψ¯1 ) dµC 2 (ψ 2 , ψ¯2 ) P (ψ, ψ) (9.7)

Proof: Since thePintegral is linear and every monomial can be obtained from N ¯ the function e−i j=1 (¯ηj ψj +ηj ψj ) by differentiation with respect to the Grass¯ ¯ = e−i¯η,ψ−iη,ψ , mann variables η and η¯, it suffices to prove (9.7) for P (ψ, ψ) N ¯ η , ψ := j=1 η¯j ψj . We have 

¯

e−i¯η,ψ−iη,ψ dµC = e−i¯η,Cη = e−i¯η,C η e−i¯η,C η  1 2 ¯1 ¯2 = e−i¯η,ψ −iη,ψ  dµC 1 e−i¯η ,ψ −iη,ψ  dµC 2  1 2 ¯1 ¯2 = e−i¯η,ψ +ψ −iη,ψ +ψ  dµC 1 dµC 2 (9.8) 1

2

which proves the lemma.
Now let C

≤j

=

j


Ci

(9.9)

i=0

be a scale decomposition of some covariance C and let dµ ≤j := dµ

C ≤j

= det[C

≤j

]e

fi fl ≤j −1 ¯ − ψ,(C ) ψ

¯ Π(dψdψ)

(9.10)

be the Grassmann Gaussian measure with covariance C ≤j . We consider a general interaction of the form ¯ =λ λV(ψ, ψ)

m 
q=1

2q

¯ 2 ) · · · ψ(ξ2q−1 )ψ(ξ ¯ 2q ) Π dξi V2q (ξ1 , · · · ξ2q ) ψ(ξ1 )ψ(ξ

i=1

(9.11) which we assume to be short range. That is, m
q=1

V2q ∅ < ∞

(9.12)

Renormalization Group Methods

173

The connected amputated correlation functions up to scale j are generated ¯ by the functional (from now on we suppress the ψ-dependence, that is, for ¯ brevity we write F (ψ) instead of F (ψ, ψ))  ≤j ≤j 1 (9.13) eλV(ψ+ψ ) dµ ≤j (ψ ≤j ) W (ψ) := log ≤j Z  ≤j (9.14) Z ≤j := eλV(ψ ) dµ ≤j (ψ ≤j ) and the sum of all 2q-legged, connected amputated diagrams up to scale j is ≤j

given by the coefficient W2q in the expansion W

≤j

(ψ) =

∞ 


2q

≤j ¯ 2 ) · · · ψ(ξ2q−1 )ψ(ξ ¯ 2q ) (9.15) Π dξi W2q (ξ1 , · · · ξ2q ) ψ(ξ1 )ψ(ξ

i=1

q=1

W ≤j can be computed inductively from W ≤j−1 in the following way. Lemma 9.1.2 For some function F let F (ψ; η) := F (ψ) − F (η)

(9.16)

Let dµj := dµC j be the Gaussian measure with covariance C j . Define the quantities V j inductively by  P j−1 i j j 1 V (ψ) := log Yj (9.17) e( i=0 V +λV )(ψ+ψ ;ψ) dµj (ψ j )  P j−1 i j e( i=0 V +λV )(ψ ) dµj (ψ j ) (9.18) Yj = where, for j = 0,

−1 i=0

V i := 0 and λV is given by (9.11). Then

W ≤j =

j


V i + λV = W ≤j−1 + V j

(9.19)

i=0

Proof: Induction on j. For j = 0 W ≤0(ψ) = log Z 1

≤0

 eλV(ψ+ψ



= log Y10

eλV(ψ+ψ

0

≤0

;ψ)

)

dµ ≤0

dµ0 + λV(ψ)

= V 0 (ψ) + λV(ψ) since

 Z ≤0 =

0

eλV(ψ ) dµ0 = Y0

(9.20)

(9.21)

174 Furthermore W ≤0(0) = V 0 (0) = 0. Suppose (9.19) holds for j and V i (0) = 0 for all 0 ≤ i ≤ j. Then, using Lemma 9.1.1 in the second line,  ≤j+1 ≤j+1 ) W (ψ) = log eλV(ψ+ψ dµ ≤j+1 − log Z ≤j+1     λV(ψ+ψ j+1 +ψ ≤j ) = log exp log e dµ ≤j dµj+1 − log Z ≤j+1  ≤j j+1 = log eW (ψ+ψ ) dµj+1 + log Z ≤j − log Z ≤j+1  ≤j j+1 ≤j = log eW (ψ+ψ ;ψ) dµj+1 + W ≤j (ψ) + log Z≤j+1 Z    P j i j+1 ≤j V (ψ+ψ ;ψ) 1 Z ≤j i=0 = log Yj+1 dµj+1 + W (ψ) + log Yj+1 ≤j+1 e Z   ≤j = V j+1 (ψ) + W ≤j (ψ) + log Yj+1 Z≤j+1 (9.22) Z

and, since V i (0) for 0 ≤ i ≤ j, 1 V j+1 (0) = log Yj+1

 e

Pj i=0

V i (ψ j+1 )

dµj+1 = log 1 = 0

Since also by definition W ≤j+1 (0) = W ≤j (0) = 0 the constant log Yj+1 in (9.22) must vanish and the lemma is proven


9.2

(9.23) Z ≤j Z



≤j+1

A Single Scale Bound on the Sum of All Diagrams

We consider the model with generating functional  1 W(η) = log Z eλV(η+ψ) dµC (ψ)  Z = eλV(ψ) dµC (ψ) and covariance C(ξ, ξ  ) = δσ,σ C(x − x ) where  dd k C(x) = eikx χ(|k|≤1) d (2π)d |k| 2

(9.24)

(9.25)

The interaction is given by (9.11) which we assume to be short range, that is, V2q ∅ < ∞ for all 1 ≤ q ≤ m. The norm  · ∅ is defined in (8.38). The strategy to controll W2q,n , the sum of all n’th order connected amputated 2q-legged diagrams, is basically the same as those of the last section. We

Renormalization Group Methods

175

introduce a scale decomposition of the covariance and then we see how the bounds change if we go from scale j to scale j +1. Thus, if ρ ∈ C0∞ , 0 ≤ ρ ≤ 1, ρ(x) = 1 for x ≤ 1 and ρ(x) = 0 for x ≥ 2 is some ultraviolet cutoff, let C(k) :=

ρ(|k|) d

|k| 2

=

=

∞
ρ(M j |k|)−ρ(M j+1 |k|) d

j=0 ∞


|k| 2 f (M j |k|) d

j=0

|k| 2

=:

∞


C j (k)

(9.26)

j=0

1 ≤ x ≤ 2 which implies where f (x) := ρ(x) − ρ(M x) has support in M  1 supp C j ⊂ k ∈ Rd | M (9.27) M −j ≤ |k| ≤ 2M −j

Lemma 9.2.1 Let C j (k) = f (M d|k|) be given by (9.26) and let C j (x) = |k| 2

dd k ikx j e C (k). Then there are the following bounds: (2π)d j

a) Momentum space: d

C j (k)∞ ≤ cM M 2 j ,

C j (k)1 ≤ cM M − 2 j d

(9.28)

b) Coordinate space: C j (x)∞ ≤ cM M − 2 j , d

d

C j (x)1 ≤ cM M 2 j

(9.29)

where the constant cM = c(M, d, f ∞ , · · · , f (2d) ∞ ) is independent of j. Proof: The momentum space bounds are an immediate consequence of (9.27). d Furthermore C j (x)∞ ≤ (2π)− 2 C j (k)1 . To obtain the L1 bound on C j (x), observe that  d   j  d k  |(M −2j x2 )N C j (x)| =  (2π) (−M −2j ∆)N eikx f (M d|k|)  d |k| 2   d j   d k ikx (−M −2j ∆)N f (M d|k|)  =  (2π) d e |k| 2  N

dd k  ∂ 2 f (M j |k|)  d−1 ∂ ≤ (M −2j )N (2π)  d  ∂|k|2 + |k| ∂|k| d |k| 2

≤ constN sup{f ∞, · · · , f (2N ) ∞ } |k|≤2M −j dd k 1 d |k| 2

≤ constN sup{f ∞, · · · , f (2N ) ∞ } M

−d 2j

(9.30)

since each derivative either acts on f (M j |k|), producing an M j by the chain d rule, or acts on |k|− 2 , producing an additional factor of 1/|k| which can be

176 estimated against an M j on the support of the integrand. Thus, including the supremum over the derivatives of f into the constant, |[1 + (M −2j x2 )d ]C j (x)| ≤ const M − 2 j d

or M−2j |C (x)| ≤ const 1 + (M −j |x|)2d d

j

(9.31)

from which the L1 bound on C j (x) follows by integration.
Let  W ≤j (ψ) := log =

Z

≤j

∞ 
q=1

eλV(ψ+ψ

1

≤j

)

dµ ≤j (ψ ≤j )

(9.32)

2q

≤j ¯ 2 ) · · · ψ(ξ2q−1 )ψ(ξ ¯ 2q ) Π dξi W2q (ξ1 , · · · ξ2q ) ψ(ξ1 )ψ(ξ

i=1

≤j

be as in (9.13), then W2q is given by the sum of all connected amputated j 2q-legged diagrams up to scale j, that is, with covariance C ≤j = i=0 . We ≤j

≤j

want to control all n’th order contributions W2q,n which are given by W2q = ∞ ≤j n n=1 λ W2q,n . According to Lemma 9.1.2 we have, for all n ≥ 2, ≤j

W2q,n =

j


i V2q,n

(9.33)

i=0

where the V i ’s are given by (9.17). The goal of this subsection is to prove the i . following bounds on the V2q,n

j Theorem 9.2.2 Let V2q,n be given by

 V j (ψ) = log Y1j =

∞
∞
q=1 n=1

e

Pj−1 i=0

 λn



V i +λV (ψ+ψ j ;ψ)

dµj (ψ j )

(9.34)

2q

j ¯ 2q ) Π dξl V2q,n (ξ1 , · · · , ξ2q ) ψ(ξ1 ) · · · ψ(ξ

l=1

where C j is given by(9.26) and let the interaction λV be given by (9.11). n j j Define V2q S, ≤n := =1 |λ| V2q, S where the norms  · ∅ and  · S are

Renormalization Group Methods

177

defined by (8.38) and (8.40). Then there are the following bounds: d

j ∅, ≤n ≤ M 2 j(q−2) V2q

sup

1≤r≤n P qv ≥1, qv ≤nm

 P P q −q 211 v qv cM v v × r  

(9.35)

≤j−1

M − 2 j(qv −2) W2qv ∅, ≤n d



v=1

and for S ⊂ {1, · · · , 2q}, S = ∅ d

j S, ≤n ≤ M 2 j(q− V2q

|S| 2 )

sup

1≤r+s≤n, s≥1 P qv ≥1, qv ≤nm |Sv | 0 with attractive delta interaction (λ > 0) given by the Hamiltonian H = H0 − λHint  2 k = L1d ( 2m − µ)a+ kσ akσ − kσ

λ L3d

 kpq

+ a+ k↑ aq−k↓ aq−p↓ ap↑

(10.82)

Our normalization conventions concerning the volume factors are such that the d canonical anticommutation relations read {akσ , a+ pτ } = L δk,p δσ,τ . The mo 2π d $  2π d ( (|ek | ≤ mentum sums range over some subset of Z , say M = k ∈ L L Z % 2 1 , ek = k /2m − µ, and q ∈ {k − p |k, p ∈ M}. We are interested in the momentum distribution ) −βH + akσ akσ ] T r e−βH (10.83)
a+ kσ akσ  = T r[e and in the expectation value of the energy 
Hint  = Λ(q) q

(10.84)

222 where Λ(q) =

λ L3d

 k,p

) + −βH T r[e−βH a+ k↑ aq−k↓ aq−p↓ ap↑ ] T r e

(10.85)

By writing down the perturbation series for the partition function, rewriting it as a Grassmann integral
P λ ¯ ψ ¯ ψ ψ ψ T r e−β(H0 −λHint ) ¯ = e (βLd )3 kpq k↑ q−k↓ q−p↓ p↑ dµC (ψ, ψ) (10.86) T r e−βH0 dµC = Π

kσ

βLd ik0 −ek

e

−

1 βLd

P

¯

kσ (ik0 −ek )ψkσ ψkσ

Π dψkσ dψ¯kσ

kσ

performing a Hubbard-Stratonovich transformation (φq = uq + ivq , dφq dφ¯q := duq dvq )
P P P 2 ¯ ¯ dφ dφ e− q aq bq = ei q (aq φq +bq φq ) e− q |φq | Π qπ q (10.87) q

with 1

aq =

λ2 3 (βLd ) 2



ψ¯k↑ ψ¯q−k↓ , bq =

k

1

λ2 3 (βLd ) 2



ψp↑ ψq−p↓

(10.88)

p

and then integrating out the ψ, ψ¯ variables, one arrives at the following representation which is the starting point for our analysis:  + 1 1 1
a+
ψkk ψkk0 σ  (10.89) kσ akσ  = βLd β Ld 0σ k0 ∈ π β (2Z+1) 1

where, abbreviating k = (k, k0 ), κ = βLd , ak = ik0 − ek , g = λ 2 , +−1
* ak δk,p √igκ φ¯p−k 1 ¯ dP (φ) ig κ
ψtσ ψtσ  = √ φ a−k δk,p κ k−p

(10.90)

tσ,tσ

and dP (φ) is the normalized measure * + ig ¯ P √ a δ φ 2 k k,p p−k κ e− q |φq | Π dφq dφ¯q dP (φ) = Z1 det ig √ φk−p a−k δk,p q κ Furthermore Λ(q) =



1 β

q0 ∈ 2π β

where Λ(q) =

λ (βLd )3



Λ(q, q0 )

(10.91)

(10.92)

Z


ψ¯k↑ ψ¯q−k↓ ψq−p↓ ψp↑ 

k,p

=
|φq |2  − 1

(10.93)

Resummation of Perturbation Series

223

and the expectation in the last line is integration with respect to dP (φ). The expectation on the ψ variables in
ψ¯kσ ψkσ  is Grassmann integration, P  λ ¯k↑ ψ ¯q−k↓ ψq−p↓ ψp↑ ψ 1 ¯ ¯ 3 k,p,q
ψkσ ψkσ  = Z ψkσ ψkσ e κ dµC , but these representations are not used in the following. The matrix and the integral in (10.90) become finite dimensional if we choose some cutoff on the k0 variables which is removed in the end. The set M for the spatial momenta is already finite since we have chosen a fixed UV-cuttoff |ek | = |k2 /2m − µ| ≤ 1 which will not be removed in the end since we are interested in the infrared properties at k2 /2m = µ. Our goal is to apply the inversion formula to the inverse matrix element in (10.90). Instead of writing the matrix in terms of four N × N blocks (ak δk,p )k,p , (φ¯p−k )k,p , (φk−p )k,p and (a−k δk,p )k,p where N is the number of the d + 1-dimensional momenta k, p, we interchange rows and columns to rewrite it in terms of N blocks of size 2 × 2 (the matrix U in the next line interchanges the rows and columns): * U

+ ak δk,p √igκ φ¯p−k U −1 = B = (Bkp )k,p ig √ φ a−k δk,p κ k−p

where the 2 × 2 blocks Bkp are given by , Bkk =

ak √igκ φ¯0 ig √ φ a−k κ 0

-

 , Bkp =

ig √ κ

0 φk−p

φ¯p−k 0

 if k = p

(10.94)

We want to compute the 2 × 2 matrix

G(k) =

G(k) dP (φ)

(10.95)

where G(k) = [B −1 ]kk

(10.96)

As for the Anderson model, we start again with the two-loop approximation which retains only the r = 2 term in the denominator of (10.7). The result will be equation (10.101) below where the quantities
σk  and
|φ0 |2  appearing in (10.101) have to satisfy the equations (10.102) and (10.105) which have to be solved in conjunction with (10.110). The solution to these equations is discussed below (10.111).

224 We first derive (10.101). In the two loop approximation,  −1  G(k) ≈ Bkk − Bkp Gk (p) Bpk , = , =:

p=k

ak √igκ φ¯0 ig √ φ0 a−k κ ak ig √ φ κ 0

ig √ κ

φ¯0

a ¯k

+ -

λ κ



φ¯p−k φk−p

p=k



 Gk (p)

−1 + Σ(k)

φ¯k−p

−1

φp−k (10.97)

where, substituting again Gk (p) by G(p) in the infinite volume limit, −1   ,  ig ¯  √ ¯ φ a p 0 φ φ¯k−p p−k κ λ Σ(k) = κ + Σ(p) ig √ φk−p φp−k φ a ¯p κ 0 p=k

(10.98) Anticipating the fact that the off-diagonal elements of
Σ(k) will be zero (for ‘zero external field’), we make the ansatz   σk Σ(k) = (10.99) σ ¯k and obtain   σk = σ ¯k λ κ

 p=k

1 2 (ap +σp )(¯ ap +¯ σp )+ λ κ |φ0 |

(10.100) , (ak + σk )|φp−k |2 − √igκ φ0 φ¯k−p φ¯p−k − √igκ φ¯0 φk−p φp−k (¯ ak + σ ¯k )|φk−p |2

As for the Anderson model, we perform the functional integral by substituting the quantities |φq |2 by their expectation values
|φq |2 . Apparently this is less obvious in this case since dP (φ) is no longer Gaussian and the |φq |2 are no longer identically, independently distributed. We will comment on this after (10.122) below and in section 10.6 by reinterpreting this procedure as a resummation of diagrams. For now, we simply continue in this way. Then , ig ¯ √ a ¯ +
¯ σ  −
φ  k k 0 κ (10.101)
G(k) = |a + σ |21+ λ |φ |2  0 k k − √igκ
φ0  ak +
σk  κ where the quantity
σk  has to satisfy the equation  σp  a ¯ p + ¯
σk  = λκ
|φp−k |2  |a + σ |2 + λ |φ |2  p

p=k

p

κ

0

(10.102)

Resummation of Perturbation Series

225

Since dP (φ) is not Gaussian, we do not know the expectations
|φq |2 . However, by partial integration, we obtain 
ig 2 √
|φq |  = 1 + κ (10.103) φq [B −1 (φ)]p↑,p+q↓ dP (φ) p

Namely,




P

2 φq φ¯q det [{Bkp (φ)}k,p ] e− q |φq | dφq dφ¯q
 P  2 1 ∂ = 1+ Z det [{B (φ)} ] e− q |φq | dφq dφ¯q φq ∂φ kp k,p q ⎤ ⎡ | | |
P  2 ∂Bkσ,pτ Bkσ,p

τ

⎦ e− q |φq | dφq dφ¯q = 1 + Z1 det ⎣ Bkσ,p
τ
∂φ φq q p,τ | | |


|φq |2  =

1 Z



Since ∂ ∂φq Bkp

we have ⎡

|

det ⎣ Bkσ,p
τ
|

|

=

ig √ κ

00 10

 δk−p,q

⎧ ⎤ if τ =↓ ⎨0 . Bkσ,p

τ

⎦ det [{Bkp }k,p ] = ⎩ √ig [B −1 ] p↑,p+q↓ if τ =↑ | κ |

∂Bkσ,pτ ∂φq

|

which results in (10.103). The inverse matrix element in (10.103) we compute again with (10.7,10.8) in the two-loop approximation. Consider first the case q = 0. Then one gets 

|φ0 |2  = 1 + √igκ φ0 G(p)↑↓ dP (φ) p

= 1+

ig √ κ


p

= 1+

λ κ


p

, φ0 |a +σ |21+ λ |φ |2 p p 0 κ

φ0

¯0 φ 2 |ap +σp |2 + λ κ |φ0 |

a ¯p + σ ¯p − √igκ φ¯0 − √igκ φ0 ap + σp

dP (φ)

dP (φ) ↑↓

(10.104)

Performing the functional integral by substitution of expectation values gives 
|φ0 |2  |a + σ |21+ λ |φ |2 
|φ0 |2  = 1 + λκ p

p

p

κ

0

or
|φ0 |2  = 1−

λ κ

 p

1 1 2 |ap + σp |2 + λ κ |φ0 | 

(10.105)

226 Before we discuss (10.105), we write down the equation for q = 0. In that case we use (10.8) to compute [B −1 (φ)]p↑,p+q↓ in the two-loop approximation. We obtain [B −1 (φ)]p↑,p+q↓ ≈ − [G(p)Bp,p+q G(p + q)]↑↓ = − |a

1

p +σp |



=

2 + λ |φ |2 0 κ

ig 1 √ 2 κ |ap+q +σp+q |2 + λ κ |φ0 |

×

ig −√ [(¯ a+σ ¯ )p+q φ¯0 φ−q + (¯ a+σ ¯ )p φ0 φ¯q ] (¯ a+σ ¯ )p (a + σ)p+q φ¯q − λ φ¯2 φ κ 0 −q κ ig 2φ ¯q ¯ ¯ √ (a + σ)p (¯ a+σ ¯ )p+q φ−q − λ φ − [(a + σ) φ + (a + σ) φ p φ0 φ−q ] p+q 0 q κ 0 κ

¯q − λ φ ¯2 (¯ a+¯ σ) (a+σ) φ κ 0 φ−q − √igκ |a +σ |2 + λp|φ |2 p+q ( p p κ 0 )(|ap+q +σp+q |2 + λκ |φ0 |2 )

 ↑↓

(10.106)

which gives
|φq |  = 1 + 2

λ κ




φq

p

= 1+

λ κ

 p

¯q − λ φ ¯2 (¯ a+¯ σ)p (a+σ)p+q φ κ 0 φ−q

(|ap +σp |2 + λκ |φ0 |2 )(|ap+q +σp+q |2 + λκ |φ0 |2 )

¯2 σp )(ap+q + σp+q ) |φq |2 − λ (¯ ap + ¯ κ φ0 φq φ−q  2 + λ |φ |2  2 + λ |φ |2  |a + σ | |a + σ | ( p p )( p+q p+q ) 0 0 κ κ

dP (φ) (10.107)

The expectation
φ¯20 φq φ−q  can be computed again by partial integration:
P 2 2 1 ¯
φ0 φq φ−q  = Z φ¯20 φq φ−q det [{Bkp (φ)}k,p ] e− q |φq | dφq dφ¯q
  P 2 = Z1 φ¯20 φq ∂ φ¯∂−q det [{Bkp (φ)}k,p ] e− q |φq | dφq dφ¯q ⎤ ⎡ | | |
 P 2 ∂B ⎦ e− q |φq | dφq dφ¯q = Z1 det ⎣ Bkσ,p
τ
∂ φkσ,pτ φ¯20 φq ¯−q Bkσ,p

τ

p,τ | | | The above determinant is multiplied and divided by det [{Bkp }k,p ] to give ⎧ ⎡ ⎤ | | | if τ =↑ ⎨0 . ∂B kσ,pτ ⎣ ⎦





det [{Bkp }k,p ] = det Bkσ,p τ ∂ φ¯−q Bkσ,p τ ⎩ √ig [B −1 ] p↓,p+q↑ if τ =↓ | | | κ Computing the inverse matrix element again in the two-loop approximation (10.106), we arrive at / (ap +σp )(¯ap+q +σp+q )φ¯2 φq φ−q − λ φ¯2 φ2 φq φ¯q 0 0 κ 0 0
φ¯20 φq φ−q  = λκ (|ap +σp |2 + λκ |φ0 |2 )(|ap+q +σp+q |2 + λκ |φ0 |2 ) p

Abbreviating gp =

ap + σp  2 , |ap + σp |2 + λ κ |φ0 | 

√λ

fp =

2 κ |φ0 |  2 |ap + σp |2 + λ κ |φ0 | 

(10.108)

Resummation of Perturbation Series this gives λ ¯2 κ
φ0 φq φ−q 

=

λ κ



gp g¯p+q κλ
φ¯20 φq φ−q  −

p

λ κ



227

fp fp+q κλ
|φ0 |2 
|φq |2 

p

or λ ¯2 κ
φ0 φq φ−q 

=

− λκ



fp fp+q κλ
|φ0 |2 
|φq |2   1 − λκ p gp g¯p+q p

Substituting this in (10.107), we finally arrive at  1 − λκ p gp g¯p+q 2
|φq |  = ( (   (1 − λ  gp g¯p+q (2 − λ  fp fp+q 2 p p κ κ

(10.109)

(10.110)

where gp , fp are given by (10.108). Observe that, since dP (φ) is complex, also
|φq |2  is in general complex. Only after summation over the q0 variables we obtain necessarily a real quantity which is given by (10.85) and (10.92).

10.4.2

Discussion

We now discuss the solutions to (10.105) and (10.110). We assume that the solution
σk  of (10.102) is sufficiently small such that the BCS equation λ κ



1 |ap + σp |2 +|∆|2

=1

(10.111)

p

has a nonzero solution ∆ = 0 (in particular this excludes large corrections like
σp  ∼ pα 0 , α ≤ 1/2, which one may expect in the case of Luttinger liquid behavior, for d = 1 one should make a separate analysis), and make the ansatz λ
|φ0 |2  = βLd |∆|2 + η

(10.112)

where η is independent of the volume. Then λ κ

 p

=

λ κ

1 2 |ap + σp |2 + λ κ |φ0 | 

 p

= 1−

1 |ap + σp |2 +|∆|2

= −

λ κ

λ κ





1 η p |ap + σp |2 +|∆|2 + κ

η/κ (|ap + σp |2 +|∆|2 )2

  + O ( κη )2

p

  + O ( κη )2 λ 

c∆ κη

(10.113)

where we put c∆ = κ p (|ap + σp |1 2 +|∆|2 )2 and used the BCS equation (10.111) in the last line. Equation (10.105) becomes κ |∆|2 + η =

c∆

η κ

λ λ  =κ  + O(1) c∆ η + O ( κη )2

(10.114)

228 and has a solution η = λ/(c∆ |∆|2 ). Now consider
|φq |2  for small but nonzero q. In the limit q → 0 the denominator in (10.110) vanishes, or more precisely, is of order O(1/κ) since 1−

λ κ



gp g¯p −

p

1−

λ κ

 p

λ κ



fp fp =

p 1 2 |ap + σp |2 + λ κ |φ0 | 

= O(1/κ)

(10.115)

because of (10.113). If we assume the second derivatives of
σk  to be integrable (which should be the case for d = 3 and
|φq |2  ∼ 1/q 2 by virtue of (10.102)), then, since the denominator in (10.110) is an even function of q, the small q behavior of
|φq |2  is 1/q 2 . This agrees with the common expectations [25, 14, 11]. Usually the behavior of
|φq |2  is inferred from the second order Taylor expansion of the effective potential Veff ({φq }) =



* |φq | − log det

δk,p

2

ig φk−p √ κ a−k

q

¯p−k ig φ √ κ ak

+

δk,p

(10.116)

around its global minimum (see section 5.2) = φmin q

1 βLd

|∆| √ λ

δq,0 eiθ0

(10.117)

where the phase θ0 of φ0 is arbitrary. If one expands Veff up to second order in 2  1 1 d |∆| √ ρ eiθ0 for q = 0 − βL |∆| iθ0 0 d λ (10.118) ξq = φq − δq,0 βL √λ e = for q = 0 ρq eiθq one obtains (see section 5.2) Veff ({φq }) = Vmin + 2β0 (ρ0 − + 12



1 βLd

|∆| 2 √ ) λ

+



(αq + iγq ) ρ2q

q=0

βq |e

−iθ0

φq + e

iθ0

φ¯−q | + O(ξ 3 ) 2

(10.119)

q=0

where for small q one has αq , γq ∼ q 2 . Hence, if Veff is substituted by the right hand side of (10.119) one obtains
|φq |2  ∼ 1/q 2 . For d = 3, this seems to be the right answer, but in lower dimensions one would expect an integrable singularity due to (10.102) and (10.84), (10.85) and (10.92). In particular, we think it would be a very interesting problem to solve the integral equations (10.102), (10.105) and (10.110) for d = 1 and to check the result for Luttinger liquid behavior. A good warm-up exercise

Resummation of Perturbation Series

229

would be to consider the 0 + 1 dimensional problem, that is, we only have the k0 , p0 , q0 -variables. In that case the ‘bare BCS equation’ 

λ β

p0 ∈ π β (2Z+1)

1 p20 +|∆|2

=1

(10.120)

still has a nonzero solution ∆ for sufficiently small T = 1/β and the question would be whether the correction
σp0  is sufficiently big to destroy the gap. That is, does the ‘renormalized BCS equation’ 

λ β

p0 ∈ π β (2Z+1)

1 |p0 + σp0 |2 +|∆|2

=1

(10.121)


σp0  being the solution to (10.102), (10.105) and (10.110), still have a nonzero solution? We remark that, if the gap vanishes (for arbitrary dimension), then also the singularity of
|φq |2  disappears. Namely, if the gap equation has no 1 1 2 solution, that is, if κ p |ap + σ 2 < ∞, then
|φ0 |  given by (10.105) is no p | longer macroscopic (for sufficiently small coupling) and λκ
|φ0 |2  vanishes in the infinite volume limit. And the denominator in (10.110) becomes for q → 0 1−

λ κ



1 |ap + σp |2

p

which would be nonzero (for sufficiently small coupling). Finally we argue why it is reasonable to substitute |φ0 |2 by its expectation value while performing the functional integral. We may write the effective potential (10.116) as Veff ({φq }) = V1 (φ0 ) + V2 ({φq })

(10.122)

where V1 (φ0 ) = |φ0 |2 −  =κ



 log 1 +

k |φ0 |2 κ

−

1 κ



2 λ |φ0 | κ k02 +e2k



|φ |2

log 1 +

k



0 λ κ k02 +e2k



≡ κ VBCS



|φ0 | √ κ

 (10.123)

and V2 ({φq }) =

⎡,  δk,p |φq |2 − log det ⎣ √ig φ0 q=0

δ κ a−k k,p

¯0 ig φ √ δ κ ak k,p

δk,p

-−1 ,

δk,p ig φk−p √ κ a−k

(10.124) -⎤ ⎦

¯ ig φ p−k √ κ ak

δk,p

230 If we ignore the φ0 -dependence of V2 , then the φ0 -integral   1 F κ |φ0 |2 e−V1 (φ0 ) dφ0 dφ¯0  = (10.125) e−V1 (φ0 ) dφ0 dφ¯0   2  −κV (ρ) BCS   F ρ e ρ dρ κ→∞  → F (ρ2min ) = F κ1
|φ0 |2  −κV (ρ) BCS e ρ dρ (10.126) simply puts |φ0 |2 at the global minimum of the (BCS) effective potential.

10.5 10.5.1

Application to Bosonic Models The ϕ4 -Model

In this section we choose the ϕ4 -model as a typical bosonic model to demonstrate our resummation method. As usual, we start in finite volume [0, L]d on a lattice with lattice spacing 1/M . The two point function is given by S(x, y) =
ϕx ϕy  (10.127) P  g2 1 P 4 2 1 − 2 x ϕx Md e− M 2d x,y (−∆+m )x,y ϕx ϕy Πx dϕx N d ϕx ϕy e := R  P P 2 4 2 − g2 1d − 12d x ϕx x,y (−∆+m )x,y ϕx ϕy M M e d e Πx dϕx N R where d    (−∆ + m2 )x,y = M d −M 2 (δx,y−ei /M + δx,y+ei /M − 2δx,y ) + m2 δx,y i=1

(10.128) 

First we have to bring this into the form [P + Q]−1 x,y dµ, P diagonal in momentum space, Q diagonal in coordinate space. This is done again by making a Hubbard-Stratonovich transformation which in this case reads
P P 2 P 2 1 dux − 12 ax x = ei x ax ux e− 2 x ux Π √ (10.129) e 2π x

with ax =

√ g ϕ2 Md x

(10.130)

Resummation of Perturbation Series

231

The result is Gaussian in the ϕx -variables and the integral over these variables gives
 −1 ig 1 2 √ (−∆ + m ) − u δ dP (u) (10.131) S(x, y) = x,y x x,y d M 2d M

RN d

x,y

where dP (u) =

1 Z

 det

1 (−∆ M 2d

+ m2 )x,y −

√ig ux δx,y Md

− 12

e− 2 1

P x

u2x

Π dux x

Since we have bosons, the determinant comes with a power of −1/2 which is the only difference compared to a fermionic system. In momentum space this reads  eik(x−y)
G(k) (10.132) S(x − y) = L1d k

where (γq = vq + iwq , γ−q = γ¯q , dγq d¯ γq := dvq dwq )
 −1 ak δk,p − √ig d γk−p dP (γ)
G(k) = L

RN d

(10.133)

kk

and dP (γ) =

1 Z

 det ak δk,p −

√ig γk−p Ld

− 12

e− 2 v0 dv0 1

2

Π+e

−|γq |2

dγq d¯ γq

q∈M

and M+ again is a set such that either q ∈ M+ or −q ∈ M+ . Furthermore ak = 4M 2

d 

sin2



ki 2M



+ m2

(10.134)

i=1

Equation (10.133) is our starting point. We apply (10.7) to the inverse matrix element in (10.133). In the two loop approximation one obtains (γ0 = v0 ∈ R)  ak δk,p −

−1 √ig γk−p d L kk

≈ =:

ak −

igv0 √ Ld

+

g2 Ld

1  p=k

Gk (p)|γk−p |2

1 ak + σk

(10.135)

where σk = − √ig d v0 + L

g2 Ld

 p=k

|γk−p |2 igv0 ap − √ + σp d L

(10.136)

232 which results in
G(k) =

1 ak +
σk 

(10.137)

where
σk  has to satisfy the equation
σk  = − √ig d
v0  + L


|γk−p |2  ap +
σp 

g2 Ld

(10.138)

p=k

=

g2 2Ld




G(p) +

p


|γk−p |2  = ap +
σp 

g2 Ld

p=k

where the last line is due to
 1
v0  = Z v0 det ak δk,p −
3

g2 Ld

√ig γk−p Ld

− 12

e− 2 v0 dv0 1


|γk−p |2  + ap +
σp 

1 2

p=k

2

e−|γq | dγq d¯ γq 2

Π

q∈M+

− 12 4 1 2  2 e− 2 v0 dv0 Π e−|γq | dγq d¯ ak δk,p − √ig d γk−p γq L + q∈M
 −1  = − 21 (− √ig d ) ak δk,p − √ig d γk−p dP (γ) (10.139) =

1 Z

∂ ∂v0 det

L

L

pp

p

As for the many-electron system, we can derive an equation for
|γq |2  by partial integration:
− 12 v02  2 e− 2 dv0 Π e−|γq | dγq d¯ γq γq γ¯q det ak δk,p − √ig d γk−p
|γq |2  = Z1 L q 
− 12  v02  2 1 = 1 + Z1 γq γq ∂γ∂ q det ak δk,p − √ig d γk−p e− 2 dv0 Π e− 2 |γq | dγq d¯ L q   ∂
√ig γk−p det a δ − k k,p d ∂γq L  dP (γ)  = 1 − 12 γq ig det ak δk,p − √ d γk−p L
 −1  −ig √ √ig γk−p a δ − dP (γ) (10.140) γ = 1 − 12 q k k,p d d L

L

p,p+q

p

Computing the inverse matrix element in (10.140) again with (10.8) in the two-loop approximation, one arrives at  g2 1
|γq |2  = 1 −
|γq |2  2L d (ap + σp )(ap+q + σp+q ) p

or
|γq |2  =

1+

g2 2

 [0,2πM]d

1 dd p 1 (2π)d (ap + σp )(ap+q + σp+q )

(10.141)

Resummation of Perturbation Series which has to be solved in conjunction with  dd p
σk  = g 2 [0,2πM]d (2π) d

233

|γk−p |2 + 12 ap + σp 

(10.142)

Introducing the rescaled quantities
σk  = M 2 s Mp ,


|γq |2  = λ Mq ,

ak = M 2 ε k , ε k =

d 

M

sin2

ki 2

+

m2 M2

i=1

(10.143) (10.141) and (10.142) read sk = M d−4 g 2 λq =

1+



1 dd p λk−p + 2 [0,2π]d (2π)d εp +sp

2

M d−4 g2

(10.144)

1



(10.145)

dd p 1 [0,2π]d (2π)d (εp +sp )(εp+q +sp+q )

Unfortunately we cannot check this result with the rigorously proven triviality theorem since
σk  and
|γq |2  only give information on the two-point function  g2  4 2 S(x, y), (10.127), and on M d x
ϕ(x)  = q Λ(q) where Λ(q) =
|γq |  − 1. However, the triviality theorem [32, 30] makes a statement on the connected four-point function S4,c (x1 , x2 , x3 , x4 ) at noncoinciding arguments, namely that this function vanishes in the continuum limit in dimension d > 4.

10.5.2

Higher Orders

We now include the higher loop terms of (10.7), (10.8) and give an interpretation in terms of diagrams. The exact equations for
G(k) and
|γq |2  are
 −1 0 / 1 ak δk,p − √ig d γk−p (10.146) dP (γ) = ak +σ
G(k) = k L

N  

kk

d

σk = − √ig d v0 + L

√ig Ld

r=2

r 

Gk (p2 ) · · · Gkp2 ···pr−1 (pr ) ×

p2 ···pr =k pi =pj

× γk−p2 γp2 −p3 · · · γpr −k and
|γq |2  = 1 +

ig √ 2 Ld




 γq ak δk,p −

p N   d

p→p2

= 1+

1 2

r=2

√ig Ld

r 

/

−1 √ig γk−p Ld p,p+q

dP (γ)

(10.147)

G(p2 )Gp2 (p3 ) × · · · × Gp2 ···pr−1 (pr ) ×

p2 ···pr =p2 +q pi =pj

× γp2 −p3 · · · γpr−1 −pr γpr −p2 −q γp2 +q−p2

0

234 For r > 2, we obtain terms
γk1 · · · γkr  whose connected contributions are, in terms of the electron or ϕ4 -lines, are at least six-legged. Since for the manyelectron system and for the ϕ4 -model [54] (for d = 4) the relevant diagrams are two- and four-legged, one may start with an approximation which ignores the connected r-loop contributions for r > 2. This is obtained by writing     γk1 · · · γkn ≈ γk1 · · · γkn 2

(10.148)

where (the index ‘2’ for ‘retaining only two-loop contributions’) 

  γk1 · · · γk2n 2 :=


γkσ1 γkσ2  · · ·
γkσ(2n−1) γkσ2n 

pairings σ




γk1 · · · γk2n dP2 (γ)

=

(10.149)

if we define

dP2 (γ) := e

−

P

|γq |2 q |γq |2

γq dγq d¯

(10.150)

Π π |γq |2  q

Substituting dP by dP2 in (10.147) and (10.148), we obtain a model which differs from the original model only by irrelevant contributions and for which we are able to write down a closed set of equations for the two-legged particle correlation function
G(k) and the two-legged interaction correlation function
|γq |2  by resumming all two-legged (particle and squiggle) subdiagrams. The exact equations of this model are
G(k) =


 ak δk,p −


|γq |2  = 1 +

ig √ 2 Ld

−1 √ig γk−p d L kk




dP2 (γ)

 γq ak δk,p −

p

(10.151)

−1 √ig γk−p Ld p,p+q

dP2 (γ)

(10.152)

and the resummation of the two-legged particle and squiggle subdiagrams is obtained by applying the inversion formula (10.7) and (10.8) to the inverse matrix elements in (10.151) and (10.152). A discussion similar to those of section 10.3 gives the following closed set of equations for the quantities
G(k) and
|γq |2 :
G(k) =

1 , ak +
σk 


|γq |2  =

1 1 +
πq 

(10.153)

Resummation of Perturbation Series

235

where
σk  =

g2 2Ld




G(p) +

p


πq  = − 12

  r=2

  r=2

√ig Ld

r−1 r  s=3

√ig Ld

r 


G(p2 ) · · ·
G(pr ) ×

p2 ···pr =k pi =pj

(10.154) ×
γk−p2 γp2 −p3 · · · γpr −k 2   δq,ps+1 −ps
G(p2 ) · · ·
G(pr ) × p2 ···pr =p2 +q pi =pj

5ps −ps+1 · · · γpr−1 −pr γpr −p2 −q 2 ×
G(p2 + q)
γp2 −p3 · · · γ

 (10.155)

In the last line we used that γq in (10.148) cannot contract to γp2 −p3 or to γpr −p2 −q . If the expectations of the γ-fields on the right hand side of (10.154) and (10.155) are computed according to (10.149), one obtains the expansion into diagrams. The graphs contributing to
σk  have exactly one string of particle lines, each line having
G as propagator, and no particle loops (up to the tadpole diagram). Each squiggle corresponds to a factor
|γ|2 . The diagrams contributing to
π have exactly one particle loop, the propagators being again the interacting two-point functions,
G for the particle lines and
|γ|2  for the squiggles. In both cases there are no two-legged subdiagrams. 1 resums ladder or bubble However, although the equation
|γq |2  = 1+ π q diagrams and more general four-legged particle subdiagrams if the terms for r ≥ 4 in (10.155) are taken into account, the right hand side of (10.154) and (10.155) still contains diagrams with four-legged particle subdiagrams. Thus, the resummation of four-legged particle subdiagrams is only partially through the complete resummation of two-legged squiggle diagrams. Also observe that, in going from (10.151), (10.152) to (10.153) to (10.155), we cut off the r-sum at some fixed order  independent of the volume since we can only expect that the expansions are asymptotic ones, compare the discussion in section 10.3.

10.5.3

The ϕ2 ψ 2 -Model

This problem was suggested to us by A. Sokal. One has two real scalar bosonic fields ϕ and ψ or ϕ1 and  ϕ2 (since ψ ‘looks fermionic’) with free action −∆ + m2i and coupling λ x ϕ21 (x)ϕ22 (x). One expects that there is mass generation in the limit m1 = m2 → 0. In the following we present a computation using (10.7) in two-loop approximation which shows this behavior. Let coordinate space be a lattice of finite volume with periodic boundary conditions, lattice spacing 1 and volume [0, L]d : Γ = {x = (n1 , · · · , nd ) | 0 ≤ ni ≤ L − 1} = Zd /(LZ)d Momentum space is given by $ %  2π d M := Γ = k = 2π /(2πZ)d L (m1 , · · · , md ) | 0 ≤ mi ≤ L − 1 = L Z

236 We consider the two-point functions (i ∈ {1, 2})
P 2 2 1 Si (x, y) = Z ϕi,x ϕi,y e−λ x∈Γ ϕ1,x ϕ2,x × R2Ld

e−

P

P2

ϕi,x (−∆+m2i )ϕi,x

x∈Γ

i=1

(10.156) Π dϕ1,x dϕ2,x

x∈Γ

where




e−λ

Z= R

P x∈Γ

ϕ21,x ϕ22,x −

e

P

P2

x∈Γ

i=1

ϕi,x (−∆+m2i )ϕi,x

2Ld

Π dϕ1,x dϕ2,x

x∈Γ

(10.157) Consider, say, S1 . We may integrate out the ϕ2 field to obtain


− 1 ϕ1,x ϕ1,y det −∆ + m22 + λϕ21,x δx,x
2 × S1 (x, y) = Z1 RL d

e−

P x

ϕ1,x (−∆+m21 )ϕ1,x

Π dϕ1,x x

(10.158)

On the other hand, integrating out the ϕ1 field gives the following representation: P 2 2 

e− x ϕ1,x (−∆+m1 +λϕ2,x )ϕ1,x 1 ϕ1,x ϕ1,y dϕ S1 (x, y) = Z Π 1,x × −1 RL d RL d det [−∆ + m21 + λϕ22 ] 2 x P

− 1 2 det −∆ + m21 + λϕ22 2 e− x ϕ2,x (−∆+m2 )ϕ2,x Π dϕ2,x x


−1 = Z1 (10.159) −∆ + m21 + λϕ22 x,y dP (ϕ2 ) RL d

where dP (ϕ2 ) =

1 Z

P

− 1 2 det −∆ + m21 + λϕ22 2 e− x ϕ2,x (−∆+m2 )ϕ2,x Π dϕ2,x x



and dP (ϕ2 ) = 1 (the Z factor above differs from its definition in (10.157) by some factors of 2π, for notational simplicity we have chosen the same symbol). By taking the Fourier transform (and dropping the ‘hats’ and the index 2 on the Fourier transformed ϕ2 field)  eik(x−y)
G1 (k) (10.160) S1 (x − y) = L1d k

where (ϕq = vq + iwq , ϕ−q = ϕ¯q , dϕq dϕ¯q := dvq dwq )
 −1 6 2) a1,k δk,p + √λ d (ϕ dP ({ϕq })
G1 (k) = k−p L k,k RL d


−1  a1,k δk,p + Lλd t ϕk−t ϕt−p k,k dP ({ϕq }) = RL d

(10.161)

Resummation of Perturbation Series

237

To keep track of the volume factors recall that ϕq is obtained from ϕx by applying the unitary matrix F = ( √1 d e−ikx )k,x of discrete Fourier transform, L that is   ϕq = √1 d e−iqx ϕx , ϕx = √1 d eiqx ϕq L

L

x

q

 2

 F ϕx δx,x
F ∗ k,p =



1 Ld

e−i(k−p)x ϕ2x =

√1 Ld

6 2) (ϕ k−p

x

and ϕ2x =

1 Ld



ei(k+p)x ϕk ϕp =

√1 Ld



eiqx

√1 Ld

q

k,p



ϕk ϕq−k

k

which gives 6 2) (ϕ k−p =

√1 Ld



ϕt ϕk−p−t =

√1 Ld

t



ϕt−p ϕk−t

(10.162)

t

Furthermore dP ({ϕq }) =

1 Z

det a1,k δk,p + ×

Π

q∈M+

− 12 λ  × t ϕk−t ϕt−p Ld −(q2 +m22 )|ϕq |2 e dϕq dϕ¯q

(10.163)

and a1,k = 4

d 

sin2

ki

2

+ m21

(10.164)

i=1

Equation (10.161) is the starting point for the application of the inversion formula (10.7). We have to compute
G1 (k) = G1 (k) dP (ϕ) where  G1 (k) = a1,k δk,p +

λ Ld



ϕk−t ϕt−p

t

−1

(10.165)

k,k

In the two-loop approximation one obtains G1 (k) =

1 a1,k + σ1,k

(10.166)

where, approximating G1,k (p) by G1 (p) in the infinite volume limit,     λ λ σ1,k = Lλd ϕk−t ϕt−k − ϕ ϕ G (p) ϕp−t ϕt−k k−s s−p 1 d d L L p p=k

t

=

λ Ld

 t

|ϕt |2 −

2

λ L2d



p,s,t p=k

s

ϕk−s ϕs−p ϕp−t ϕt−k G1 (p)

t

(10.167)

238 Thus
G1 (k) =

1 a1,k +
σ1,k 

(10.168)

where
σ1,k  =

λ Ld




|ϕt |2  −

λ2 L2d

t




ϕk−s ϕs−p ϕp−t ϕt−k 
G1 (p)

(10.169)

p,s,t p=k

To obtain a closed set of integral equations, we ignore connected three and higher loops and approximate
ϕk−s ϕs−p ϕp−t ϕt−k  ≈
ϕk−s ϕs−p 
ϕp−t ϕt−k  +
ϕk−s ϕp−t 
ϕs−p ϕt−k  +
ϕk−s ϕt−k 
ϕs−p ϕp−t  = δk,p
|ϕk−s |2 
|ϕt−k |2  + δt−p,k−s
|ϕk−s |2 
|ϕs−p |2  + δs,t
|ϕk−s |2 
|ϕs−p |2  = δt−p,k−s
|ϕk−s |2 
|ϕs−p |2  + δs,t
|ϕk−s |2 
|ϕs−p |2  (10.170) where the last line is due to the constraint k = p in (10.169) which comes out of the inversion formula (10.7). Thus
σ1,k  has to satisfy the equation
σ1,k  =

λ Ld



2


|ϕt |2  − 2 Lλ2d

t


|ϕk−s |2 
|ϕs−p |2  a1,p +
σ1,p  p,s

(10.171)

p=k

To close the system of equations, one needs another equation for
|ϕ2,q |2 . This can be done again by partial integration as we did in the last section for the ϕ4 -model. However, for the specific model at hand, one has, by virtue of (10.158) with the indices 1 and 2 interchanged,
|ϕ2,q |2  =
G2 (q)

(10.172)

Thus one ends up with
G1 (k) =

1 a1,k +
σ1,k 

(10.173)

where
σ1,k  =

λ Ld

 t

2


G2 (t) − 2 Lλ2d




G2 (k − s)
G2 (s − p)
G1 (p) (10.174)

p,s p=k

In particular, for m1 = m2 = m one has
G1  =
G2  ≡ G, G(k) =

k2

1 + m2 + σk

(10.175)

Resummation of Perturbation Series

239

and σk has to satisfy the equation σk = λ



dd p [−π,π]d (2π)d

G(p) − 2λ2



dd p dd q [−π,π]2d (2π)d (2π)d

G(p) G(q) G(k − p − q) (10.176)

To see how the solution looks like one may ignore the λ2 -term in which case σk = σ becomes of k. In the limit m2 ↓ 0 one obtains, if we d independent 2 substitute 4 i=1 sin [ki /2] by k 2 for simplicity, σ=λ



dd p 1 [−π,π]d (2π)d p2 +σ

(10.177)

which gives ⎧ ⎨ O(λ)   σ = O λ log[1/λ] 2 ⎩ O(λ 3 )

10.6

if d ≥ 3 if d = 2 if d = 1 .

(10.178)

General Structure of the Integral Equations

In the general case, without making the approximation (10.148), we expect the following picture for a generic quartic field theoretical model. Let G and G0 be the interacting and free particle Green function (one solid line goes in, one solid line goes out), and let D and D0 be the interacting and free interaction Green function (one wavy line goes in, one wavy line goes out). Then we expect the following closed set of integral equations for G and D: G=

G−1 0

1 , + σ(G, D)

D=

D0−1

1 + π(G, D)

(10.179)

where σ and π are the sum of all two-legged diagrams without two-legged (particle and wavy line) subdiagrams with propagators G and D (instead of G0 , D0 ). Thus (10.179) simply eliminates all two-legged insertions by substituting them by the full propagators. For the Anderson model D = D0 = 1 and (10.179) reduces to (10.67) and (10.76). A variant of equations (10.179) has been derived on a more heuristic level in [15] and [47]. Their integral equation (for example equation (40) of [47]) reads G=

G−1 0

1 +σ ˜ (G, D0 )

(10.180)

240 where σ ˜ is the sum of all two-legged diagrams without two-legged particle insertions, with propagators G and D0 . Thus this equation does not resum two-legged interaction subgraphs (one wavy line goes in, one wavy line goes out). However resummation of these diagrams corresponds to a partial resummation of four-legged particle subgraphs (for example the second equation in (10.183) below resums bubble diagrams), and is necessary in order to get the right behavior, in particular for the many-electron system in the BCS case. Another popular way of eliminating two-legged subdiagrams (instead of using integral equations) is the use of counterterms. The underlying combinatorial identity is the following one. Let
¯ = S(ψ, ψ)

¯ dk ψ¯k G−1 0 (k)ψk + Sint (ψ, ψ)

(10.181)

be some action of a field theoretical model and let T (k) = T (G0 )(k) be the sum of all amputated two-legged particle diagrams without two-legged par¯ ticle subdiagrams, evaluated with the bare propagator G0 . Let δS(ψ, ψ) = ¯ dk ψk T (k)ψk . Consider the model with action S − δS. Then a p-point function of that model is given by the sum of all p-legged diagrams which do not contain any two-legged particle subdiagrams, evaluated with the bare propagator G0 . In particular, by construction, the two-point function of that model is exactly given by G0 . Now, since the quadratic part of the model under consideration (given by the action S − δS) should be given by the bare Green function G−1 0 and the interacting Green function is G, one is led to the equation G−1 − T (G) = G−1 0 which coincides with (10.180). Since the quantities σ and π in (10.179) are not explicitly given but merely are given by a sum of diagrams, we have to make an approximation in order to get a concrete set of integral equations which we can deal with. That is, we substitute σ and π by their lowest order contributions which leads to the system 1 , dp D(p)G(k − p) 1  D(q) = D0 (q)−1 + dp G(p)G(p + q) G(k) =

G0 (k)−1 +



(10.182) (10.183)

This corresponds to the use of (10.7) and (10.8) retaining only the r = 2 term. Thus we assume that the expansions for σ and π are asymptotic. Roughly one may expect this if each diagram contributing to σ and π allows a constn bound (no n! and of course no divergent contributions). The equations (10.182) and (10.183) can be found in the literature. Usually they are derived from the Schwinger-Dyson equations which is the following nonclosed set of two equations for the three unknown functions G, D and Γ, Γ being the vertex

Resummation of Perturbation Series

241

function (see, for example, [1]):
dp G(p) D(k − p) Γ(p, k − p) G(k)

G(k) = G0 (k) + G0 (k)

(10.184)


D(q) = D0 (q) + D0 (q)

dp G(p) G(p + q) Γ(p + q, −q) D(q)

The function Γ(p, q) corresponds to an off-diagonal inverse matrix element as it shows up for example in (10.103). Then application of (10.8) transforms (10.184) into (10.179). One may say that although the equations (10.182) and (10.183) are known, usually they are not really taken seriously. For our opinion this is due to two reasons. First of all these equations, being highly nonlinear, are not easy to solve. In particular, for models involving condensation phenomena like superconductivity or Bose-Einstein condensation, it seems to be appropriate to write them down in finite volume since some quantities may become macroscopic. And second, since they are usually derived from (10.184) by putting Γ equal to 1 (or actually −1, by the choice of signs in (10.184)), one may feel pretty suspicious about the validity of that approximation. The equations (10.179) tell us that this is a good approximation if the expansions for σ and π are asymptotic. A rigorous proof of that, if it is true, is of course a very difficult problem.


    

     ½
       

    
     
  
      
        
   


     
 
 
  
       

          !





"

   # $ "




 


     &     !    " 







% #  $      
    

 


   
    
   
 
 
 
 







¼








¼




¼





¼








¼







  ¼ ¼



¼

¼  

Æ ¼ Æ



 

  

  

 % '  





"

 ½ ½







¼

¼

  





½ ½







½ ½

















½ ½



  


 
 

       
 (
 
      
  
 # $ 
 






# $ 
 







& 











  














 


             

 
     


   # $ "   

 





"    
   



 ) 


  

        #      * 
      
  
     
  


   

 
$" ½


       
      
 
   

+,-


 )   . 

 #$  



#


 

 $



  %
      
  
   '  


 
    
     

 .    
   . 

 
 #

 $  



  !    #
$     
           !

/#$ " 










#

$

 




#



$

 


0  
 

  +  
   /#$    Æ
     !           #!1 + $    
  

  2
    . 
 



 
    & 





   
   
 . 

 
   


      
          
        !           #!1 + 3    04         
$    
           

   '

+,,





   5 5 5
'   ! 0 6 '  7 8 49 
'
 :   ; %
  
 

 0
 0 
( -? >? B 6 @  6 6 
 0  
     % 
 
  ; !


 
   I>  I,  >> > % 5 @ 9
  :     ;
9
 5
 0  >,  I F : )  H F'
 8   
 8J
 5
  )  
 L           B  +,+B+,,I >A,  ? : 4
 
 E H
 7 
 % 
!

  %

      0  7  77 
 
     + I  +I +>   +> -,  +  A F %  (  ! H  F 2
 ;84" 5 0   H 
9

    
    - + >BB  B F % ( NG

  4 ! 8   
9 % 
!


   4


" )  
 0
    
    ,,?  E H

 >>,  > F % ( NG

  4 ! 8   
9 5 )  % 
!


 0 
  %
       6  J ! E

 >>B + F % ( NG

  8   
9  % 
!

 )  
      
   "DDDODC +  F % ( NG

  8   
9 5 H '  5
 
    
  !   ?,  ?>I?>> >>  ++ F % ( NG

  8   
9 5 H 
  % 

 )


 %
 
 
     >I  ,?I,>- >>B +- F % ( NG

  8   
9 :
 :   : @  ; %
   !  <  8( MG
 >>+ +, F % F : E H
 8   
9 5 7
 E  8. 
  : % 
 6  %
 !   ?I  ?A>A+  >>+ +I F % F : E H
 8   
9 2  7

   0  
 5
   P# $ 6   @  
  : % 
  !   ??  ,>BII >>- +? F % F : E H
 8   
9 5 7

  8. 
  :% 
  "
   +, #?$  ,-A,,+ >>-3  4

 :% 
   E

:  "
   +, #A$  I+ I+? >>- +A F % F : E H
 8   
9 % 

 : @  : 
  
    
  A :

 

+,A


 ;   7" %
    : @     F % H %  ! H  >>, +B F % :    8   
9 0  
     <  % 
  7 N
  % 
  %
.
# 
   B,  +> --? >>?3 0  
     <  % 
  77 H
   :
 % 
 " H05 ) 

 
 
      #

   I   -- +,? >>B3 H
   :
 % 
 "  %   
 
      #

   I+  +A--+, >>>3 5 7  
  
 % 
    
 
     

   I-  -I -B, + +> F % 8   
9 0  
    : % 
   !   ?-  I?+? >>3  %    8 0         
  !   ?,  + ,-IA >>  - H % 9 F % G 
 5 ' H  2' )

 0    



 ; %
   $%   

   
  >>+ -  5! %  F4 2' ;    : 0 
  :6 (
 >A  -+ F % G 
 L  


   
   5    )

 0
   & +  +B +>? >B+ -- 6 6 
 % BI -, H 8 0   : 6



 #$  ; ( 8J   

    ' $%  

  >> -I 7  6  7: H9
'   7  
  0  5
 0  >?I -? H (  :
      @  )  
J

:  

  
+I  +BA+>> >?+ -A (  8H" 5   
  0  0 
( > ,+  N    6   ; (  
 7 



 
    >  - --B  + ,- 4 ! 5 :
 
 4

   )

 :
 %

       04
 8( MG
 >>, " 
 # 
     A-  II A, >>I ,, 4 !  :8  
  %   8.  @) 5 .

 
 
     >B  ,+A,?B >>B ,I 4 !  6  :

   8J
 0 
   :8   
 47 
       +   >  +I> +AB + ,? 4 ! H
  % 4
    7  
  :

 
    -,  +B -, +  ,A F : !
  F ) 2  6   8    :% 
  77    B   I  , A ,+A >? ,B F : 6 0
 E H
 2    7

   +4 5   :   2' 4
    D>B + A3  5   :    :
. :    & #0   $ IB  ,> >>A ,> 64 : : 0 
 0
 0 0  >B 

I > I  :8 0'
 4E    5 7 
  ; %
   5
 2 >>I I+ 6 0
 : 6 K %
   5   :   2' 4
  
  7   ) J  D>A+  I- 50 0 
'  =5 @ '  L7 :
  7   
 6    @  >B? I, E H
 %  0  
  )  
 H 
9
 0
  P
 
 0  >>  II :    7  0  ) 
  % 
  H 
9
        II-IAB >>B

 

+,>

I? :    H 
9
 5 7 
 $%   

   
 >>> IA F H 
J        

 5
 2 >?, IB 5 G      6
  @) :   (


2  7 
 4
  
 8( MG 
 >>A I>      G 
 8
 
  H  0 
" 5 :
 H
 0 
   >B, ! (  

   N L   H   <  (  0
 0  >B-3     ,+  II >B,


5  @
'  
  5     >> + 


   
 A 
 
 
 -A 
 
 
 - + .

 @) B   B+ 
 ?   +B + - ++   
 . 
  - A + >

     - I +A  
 
 -? -> ?+ ?I >B  0Æ ,>    +A - -, 
   % 
  
    ? 
  % 
    B ! 
 >      

 >+ 4  
 + ?

@
2   
  @) J
 
 ?? AI > 
     B  
 @)   BB 
    
 
 >+     

 
 + 

 A  
  ?-

J
 
 I> ?I AI >  
  ( +I +A  @) B- BB >-   
  ( ? +A    + ?   
     
 + 
  
 ,> 
  -B  ?
 I 
  ,A   
  
  A?       
  
 BB .   AI BB


 
 +   
   
   A 


 
 

 

    
  

 ,+   
 ?+  
   
 A  -I +A +I

% 


   
 >I % 
  B >? %   + % 
  ,   
 -?



 ?  ?+


 + ? %  , 
  C. 
9
 + % '   > % 
    
  B  
  B  
  ( J 
 + 

   
   
 I>  

 ,+


 
   > I A  
   B+  
 @) B 
   
 
  A   
 


( 
 
  , (    
    
 ?
 % 
  +

 6  I? (  ?B
  
   >> ++  
  ?+ !

 
 > !   + ? !
  
 I 
'    
  ,+




     

 + C. +  B 
.
  
   ++ :   
 +A      
 @) B 5  @
'  @
2       A

@) B- BB >-   
  ( ? +A 6

    
 ,,  

 I 6 K

 I? 6 
  -A 6  
   5     >> +     ++   +A    @) B  - I +A ( K

 ?B (
 
  
  ( A
 % 
 

   + 
     > ++ 2
' II 

 ,,  

 
  
     
 +

,

+I+


 % 
  + 

    
   
 ?+  

 ,   
 
 ++ -   
 . 
  


 
 -   

 
 ++


   -, +> >A 0Æ ,> 
    >> +- 
  
   +-I 
9
 6  
 +-> 



 +A 
 ,>    

 ,B      

 >+

 
   -B    

     >+   BB    '
 AI BB     
 > ++ 6  
 +A   -B    .

 +B + - ++

 . 
 +,

 
 ,  
9 -A   
 @)    B!
 I    2
' 
 II   +I +A


9
 C. +   I H   I, AB

   

 
9
  +> ?> >- + A

 
9
    A

 
    
 
  - + A


    
 
  -, +> >A 
  .

 ?  G 
 
 A   
9
 I  

 ' ++    + ? 

    ++B   
  +I-