Modular Invariance, Lattice Field Theories, and Finite Size Corrections

Annals of Physics 273, 72�98 (1999)

Modular Invariance, Lattice Field Theories,and Finite Size Corrections

Charles Nash

Department of Mathematical Physics, St. Patrick's College, Maynooth, Ireland,and School of Theoretical Physics, Dublin Institute for Advanced Studies,

10 Burlington Road, Dublin 4, Ireland

and

Denjoe O' Connor

Departamento de F@� sica, CINVESTAV, Apdo. Postal 14-740, 07000 Me� xico, D.F., Mexico

Received July 7, 1998

We obtain the exact partition function for a lattice Gaussian model where the site degreesof freedom are sections of a U(1) bundle over a triangular lattice which globally forms a torus,with three independent nearest neighbour interactions in the different lattice directions. Wefind that in the scaling limit, even off criticality, the finite size contribution is invariant underthe double cover of the modular group. Demanding that the singular part of the bulk con-tribution be similarly invariant provides a natural method of identifying this contribution. Theorigin of this symmetry is shown to be coordinate invariance of the continuum microscopicenergy functional together with the discrete symmetries of parity and global space inversion.We similarly find the exact scaling function for the two dimensional Ising model and byworking with three independent lattice couplings access the full range of the modularparameter which we identify in terms of the underlying lattice couplings. � 1999 Academic Press

1. INTRODUCTION

In both laboratory experiments and the computer simulation of theoreticalmodels a finite lattice plays an essential role. The finite lattice gives rise to manyadditional effects which provide corrections to the bulk system behaviour. Thetheory of finite size scaling, formulated in the early 1970s [1] has become a power-ful tool for the investigation of the vicinity of a critical point (for more recentreviews see [2]). Finite size and lattice effects have a complex interplay with theother features of these systems giving rise to phenomena such as crossovers from

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one characteristic behaviour to another [3]. Though considerable progress hasbeen made since the inception of the subject, much still remains to be done andunfortunately only very few lattice models admit a complete analytical solution ona lattice of arbitrary size and where a detailed comparison of numerical and exactresults is feasible. Two dimensional physics has provided a framework where sub-stantial progress has been made in understanding such problems and considerableinterest has been focused on them. Some of these calculations use generalisedGaussian models [4] and this paper can be considered in a similar spirit.

Pure finite size effects can be isolated from those due to the presence of a bound-ary by ensuring that the connectivity of the underlying lattice is such that it has noboundary. One can avoid other complications, such as local curvature of the lattice,while retaining the finite size effects, by considering a flat torus. In particular westudy a model where the basic degree of freedom is a complex continuous spinvariable .(x), however, a crucial feature of our spin field will be that, though theunderlying lattice is periodic, the fields will not be periodic functions; instead theywill be sections of appropriate bundles which possess a non-trivial holonomy whenrotated round the non-trivial loop of the base space. We begin by considering a onedimension version of the model; this serves to set our notation and illustrate themethods used.

In two dimensions the model is considered on a triangular lattice which forms aflat torus. The flat torus has the advantage of being the only zero curvature finitevolume orientable manifold without boundary in two dimensions. The model pos-sesses two holonomy phases (cf. Section 3) u0 and u1 associated with the differentnon-contractible loops of the torus. When the phases are zero we recover simpleperiodic boundary conditions, while when one or both are 1�2 we have anti-peri-odic boundary conditions in that direction. Anti-periodic boundary conditions werestudied in order to access the helicity modulus in superfluidity by [5] and moregeneral twisted boundary conditions emerged naturally in universal conductancefluctuations of mesoscopic systems [6]. This motivated Bre� zin et al. [7] to studyan O(N ) extension of where the twisting occurs between two components of theO(N ) field. General twisted boundary conditions have also been studied in the con-text of the spherical model by Allen and Pathria [8�10]. However in this contextthey are considered an artificial construct as it is necessary to extend the model tocomplex fields when the ui are not integers or half integers.

For our two dimensional theory we obtain an exact expression for the partitionfunction on a general finite triangular lattice with three different nearest neighbourcouplings associated with the lattice bonds. From this we obtain the exact finite sizefinite lattice corrections to the free energy of our lattice model. When we take thecontinuum or scaling limit these corrections form a modular invariant expression,irrespective of whether we are at the bulk critical point or not. At the bulk criticalpoint, where the mass m=0, this modular invariant function is expressible in termsof elliptic theta functions and the associated partition function holomorphically fac-torises in the modular parameter {. The modular invariant expression also possessessome interesting non-commuting limits.

73LATTICE FIELD THEORIES

The modular invariance of the finite size [11] corrections,1 in the continuumlimit, is argued to be a consequence of the general covariance of the continuumenergy functional. It is in fact the residual ability to perform constant coordinatetransformations and leaves the continuum part of the free energy unaffected. Thisallows the singular part of the free energy to be isolated��as the modular invariantpart of the free energy.

By using the fact that the Ising partition function for a toroidal lattice is the sumof four Pfaffians we are able to use our results for the Gaussian model to obtainthe same data for the Ising model. Hence we obtain the modular parameters of thismodel in terms of the underlying couplings. In order to span all possible modularparameters three independent couplings on a triangular lattice are necessary.A brief summary of some of these results has been presented in [12].

The principal new results contained in this work are:

(i) The determination of the exact partition function for a massive Gaussianmodel where the spin variables are sections over a generic triangulated torus.

(ii) The demonstration that in the scaling limit the finite size corrections tothe bulk partition function are invariant under the double cover of the modulargroup SL

t(2, Z).

(iii) The demonstration that the origin of modular invariance in the scalingor continuum limit is a consequence of general covariance at the level of thecontinuum energy functional. Hence it is as natural for the massive theory as it isfor the massless one.

(iv) The determination of the exact lattice partition function for the twodimensional Ising model on a triangular lattice with three independent nearestneighbour couplings and the extraction of its scaling limit, the modular invariantpart of which gives the universal free energy scaling function on a generic torus inthe absence of a magnetic field.

(v) The determination of the modular parameter for the Ising partitionfunction in terms of the underlying Ising couplings.

The organisation of the material in the rest of this paper is as follows. InSection 2 we discuss the one dimensional lattice theory. Section 3 discusses the twodimensional lattice theory on a generically skewed torus T2. Section 4 describes thescaling or continuum limit and modular invariance is discussed in Section 5. InSection 6 we focus on the universal scaling function that describes the scaling regionof critical point, a function that has various non-commuting limits. The Sectioncloses by specializing to the geometric limit of a cylinder geometry. Section 7 brieflydescribes the corresponding results for the two dimensional Ising model. The paperends with conclusions and additional comments in Section 8.

74 NASH AND O' CONNOR

1 For an early exploitation of modular invariance at the critical point see Cardy and Cappelli et al.[11].

2. A TWISTED LATTICE THEORY IN ONE DIMENSION

We take as our underlying lattice a polygon with N vertices, which is topologi-cally a circle S1. We take the spacing between the vertices to be uniform and oflength a and hence the circumference is of length L=Na. At each lattice site weplace a complex field (or spin) . that can take any value in the complex plane.However, though the underlying lattice is periodic we do not require the field to beso, rather we require that

.(x+mL)=e2?imu.(x) (2.1)

where u is a real number between zero and one. This means that the . is actuallya Section of a complex line bundle L over the circle which carries a representationof the fundamental group of the circle labeled by u. We take the spins to have onlynearest neighbour interactions, in particular the interactions are given by the dis-cretised laplacian so that the energy of a configuration is given by the discretisedGaussian

EL[S1, .*, .]= 12 :

k, k$

- g .*(k)(2(k, k$)+m2$k, k$) .(k$) (2.2)

where - g=a and the discrete Laplacian 2 is an N_N symmetric matrix all thenon-zero entries of which can be deduced from

2(k, k)=2a2 , 2(k, k+1)=&

1a2 . (2.3)

The corresponding lattice partition function we denote by ZN(S 1, L, m) and isgiven by

ZN(S 1, L, m)=| `k

d,1(k) d,2(k) exp[&EL[S 1, .*, .], (2.4)

where we have taken ,=,1+i,2 and .=- a ,, which means that . correspondsto the usual continuum field and the lattice field , is dimensionless as is thepartition function. From (2.4) we have

ZN(S 1, L, m)={detN(2+m2) a2

2? =&1

, (2.5)

where we use the notation detN (M) to denote the determinant of the N dimen-sional matrix M.

The discretised basis of eigensections is then given by

[eixnk : k=1, ..., N], where xn=2?(n+u)

N, n=0, ..., N&1, (2.6)


and so we find

detN [2+m2]=`n

*n , where *n=1a2 sin(xn)2+m2 (2.7)

are eigenvalues of 2+m2 with eigensections (2.6).Defining W=&ln ZN we find that W is given by the sum

W= :N&1

n=0

ln*na2

2?

=&N ln ?+ :N&1

n=0

ln _1+m2a2

2&cos(xn)& . (2.8)

The sum can be performed using (see Appendix A for a derivation) the basicidentity

:N&1

n=0

ln[z&cos(xn)]=N ln _z+

2 &+ln |1&zN&e2?iu|2, (2.9)

where z\=z\- z2&1.On using (2.9) we find that W splits in the form

W=NWB+WF ,

where the extensive term is known as the bulk contribution. The quantity WB is thecontribution to W per lattice site in the thermodynamic limit, i.e.,

WB= limN � �

WN

and is given by

WB=ln _1+m2a2

2+�m2a2+m4a4

4 &&ln 2?. (2.10)

The remaining term WF captures the effects of a finite lattice and is given by

WF=ln } 1&\1+m2a2

2&�m2a2+

m4a4

4 +N

e2?iu }2

(2.11)

The continuum or scaling limit is a constrained thermodynamic limit andcorresponds to taking N � � and a � 0 while keeping Na=L and m fixed. In thislimit we have

NWB=&N ln 2?+1B , where 1B=mL (2.12)


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and

1F=ln |1&e&mL+2?iu| 2 (2.13)

3. A TWISTED LATTICE THEORY ON THE TORUS

We now replace the polygon S 1 by a discrete lattice forming a parallelogram asfollows: the parallelogram is composed of a triangular lattice composed of similartriangles, pairs of which form parallelograms. The basic triangles have two sides oflengths a0 and a1 with an angle % between them and are fitted together to form theglobal parallelogram which consists of K0K1 sites and 2K0K1 triangles, forming aparallelogram of sides L0=K0a0 and L1=K1a1 . The resulting lattice is depicted inFig. 1. The geometry is such that a point z # C is identified with points of the formz+mL0+nL1 exp[i%] where m, n # Z. Hence if z=(x, y) then

(x, y)#(x+mL1 cos %+nL0 , y+mL1 sin %), m, n # Z (3.1)

and the complete lattice therefore forms our torus, T2.At each lattice site we again place a complex field (or spin) . which forms a

section of a complex line bundle L over the torus. The field section . is again notperiodic but satisfies the condition

.(x+mL1 cos %+nL0 , y+mL1 sin %)=e2?i(mu1+nu0).(x, y), m, n # Z, (3.1)

where the real numbers ui , i=1, 2, with 0�ui�1 characterize the bundle.We label the lattice sites by (k0 , k1)#k=k0+(k1&1)K0 with ki=1, ..., Ki , and

the energy of a lattice configuration is given by

EL[T2, .*, .]= 12 :

kk$

- g .*(k)(2(k, k$)+m2$k, k$) .(k$), (3.3)

where - g=a0 a1 sin %. Since the interactions are given by the lattice Laplacian 2,the model has again only nearest neighbour interactions.

FIG. 1. The triangulated torus.


The discrete eigensections of 2 are En0 n1(k0 , k1) where

En0 n1(k0 , k1)=exp _2?i {(n0+u0)

k0

K0

+(n1+u1)k1

K1=& (3.4)

and the discrete version of the Laplacian 2 is a K0K1 _K0K1 symmetric matrix2(K0 , K1) a general element of which we denote by 2[(k0 , k1), ($0 , k$1)] so that wehave

2(K0 , K1)=[2[(k0 , k1), (k$, l$)]]K0 K1_K0 K1with 1�k0 , k$0�K0 ,

and 1�k1 , k$1�K1. (3.5)

All the non-zero entries of 2(K0 , K1) can be deduced by its symmetry and byrecording explicitly that

2[(k0 , k1), (k0+1, k1)]=&:=&1

sin2% \1a2

0

&cos %a0a1 +

2[(k0 , k1), (k0 , k1+1)]=&;=&1

sin2% \1

a21

&cos %a0a1 + (3.6)

2[(k0+1, k1), (k0 , k1+1)]=&#=&1

sin2% \cos %a0a1 +

2[(k0 , k1), (k0 , k1)]=2_, where _=:+;+#

the remaining elements being zero. The lattice eigenvalues are then

2n0 n1=2(_&: cos(xn0

)&; cos(xn1)&# cos(xn0

&xn1))

where xni=

2?(ni+ui)Ki

, i=0, 1. (3.7)

In contrast to the one dimensional case the lattice field .=.1+i.2 is nowdimensionless. So the partition function that interests us is

ZK0 K1(T 2, L, m)=| _` d.1 d.2& e&EL[T2, .*, .]

={detK0K1 _- g (2+m2)2? &=

&1

. (3.8)

The problem is therefore the computation of this lattice determinant. This is mostconveniently computed by focusing on W=&ln ZK0 K1

, i.e., on


W=ln detK0 K1 _- g (2+m2)2? &

= :K0&1

n0=0

:K1&1

n1=0

ln _- g (2n0 n1

+m2)

2? & . (3.9)

Since the results of this computation will be of use to us more generally thansimply for the Gaussian model we have focused on so far, we shall perform theeigenvalue summations in a more general setting. Let us define $, by $=_+m2�2,which implies that, together with (3.7), the eigenvalues of K0_K1 dimensionalmatrix 1

2 (2+m2) are given by

*n0 n1=$&: cos(xn0


&xn1). (3.10)

The quantity of interest to us therefore is

W= :K0&1

n0=0

:K1&1

n1=0

ln _- g ($&: cos(xn0


&xn1)

? & . (3.11)

We will in fact solve the model by performing the sums for general positive :, ;, #and $. A key step in the following analysis is that we do one of the sums in (3.9)completely, as in the one dimensional model of Section 2 by repeated use of (A.1).

To bring the (3.11) into a form where A.1 can be applied define the newquantities, ;n0

, %n0by writing

;n0=(;+# exp[ixn0

])# |;n0| exp[i%n0

],

where

|;n0|=- ;2+#2+2;# cos(xn0

)

and

%n0=tan&1 \ # sin(xn0

);+# cos(xn0

)+ (3.12)

O *n0 n1=$&: cos(xn0

)&|;n0| cos(xn1

&%n0)

Hence using (A.1) we are in a position to completely perform the summation overn1 . On doing this we have

:K1&1

n1=0

ln *n0 n1=K1 ln _

|;n0|

2(zn0

+- z2n0

&1)&+ln |1&e&K1vn0e2?iu1| 2,


where

zn0=

$&: cos(xn0)

|;n0|

and vn0=&ln[zn0

&- z2n0

&1]+i%n0(3.13)

and we have entirely done one of the summations.By using the integral identity

|?�2

0

d&?

ln[`2&*2 cos2 (&)]=ln _`+- `2&*2

2 & (3.14)

we find that the first term on the r.h.s. of (3.13) takes the form

ln _|;n0

| (zn0+- z2

n0&1)

2 &=|

?�2

0

d&?

ln[($&: cos(xn0))2&|;n0

|2 cos2 (&)]. (3.15)

But the argument of the log on the r.h.s. of (3.15) is a quadratic in cos(xn0), thus

if we factor it we can use (A.1) on the individual terms and perform this summationcompletely. In factorised form we write

($&: cos(xn0))2&|;n0

|2 cos2(&)=:2( p&2q cos(xn0)+cos2(xn0

))

=:2(q+&cos(xn0))(q&&cos(xn0

))

with

q\=q\- q2& p, where {q=

:$+;# cos2(&):2 and

p=$2&(;2+#2) cos2(&)

:2 .(3.16)

If we now employ (A.1) we find that we can do the sum over n0 in (3.15) leavingus with a trivial two term sum originating in the roots of the quadratic. We get

:K0&1

n0=0

ln _- g |;n0

| (zn0+- z2

n0&1)

2? &=K0WB+WC , (3.17)

where

WB=|?�2

0

d&?

ln _g:2

4?2 (q++- q2+&1)(q&+- q2

&&1)&


and

WC = :==\

|?�2

0

d&?

ln |1&(q=&- q=2&1)K0 e2?iu0|2. (3.18)

Thus we have accomplished our summation goal and have found for W=&ln ZK0 K1

the exact lattice expression

W=K0K1 WB+WF,

where

WF =K1 WC+ :K0&1

n0=0

1&e&K1vn0e2?iu1| 2. (3.19)

The complete finite size and finite lattice corrections are contained in the term WF .The coefficient of K0K1 above is the bulk term, and can be isolated as

WB= limK0 , K1 � �

WK0K1

(3.20)

and is easily checked to be given by

WB=|?

&?

d&1 d&2

(2?)2 ln _- g?

[$&: cos(&1)&; cos(&2)&# cos(&1&&2)]& . (3.21)

The coefficient of K1 corresponds to the finite size finite lattice corrections to acylinder limit wherein K1 is taken to infinity but K0 is retained finite. It can beisolated as

WC= limK1 � �

WF

K1

. (3.22)

We could have taken the alternative limit in which K1 is retained finite while K0 issent to infinity. The resulting expression would be most easily obtained by perform-ing the summation over n0 first and then over n1 . This would lead to an alternativebut completely equivalent expression for WF where the roles of the set of variables[K0 , u0 , :] and the set [K1 , u1 , ;] are interchanged.

4. THE SCALING OR CONTINUUM LIMIT

We now turn to the scaling or continuum limit for our Gaussian model. This isa constrained thermodynamic limit achieved by taking K0 and K1 to infinity whilesimultaneously approaching the critical point. For us this amounts to takingKi � � while keeping the geometry and distance from the critical point fixed, i.e.,we hold Li=Kiai , %, m2 and the ratio k=K1 �K0 fixed.


4.1. The Bulk Term

We begin by discussing the scaling limit of the bulk term WB . For this purposeit is most convenient to re-express WB in terms of a single integral in the form

WB=|2?

0

d&2?

ln _- g?

[$&: cos &+- Q($, :, ;, #, &)]& , (4.1)

where

Q($, :, ;, #, &)=($&: cos &)2&(;2+#2+2;# cos &)

so that with m2=2($&:&;&#)

dWB

dm2=W (0, 1)

B =|2?

0

d&

4?

1

- Q(:, ;, #, &). (4.2)

This latter integral can be reduced to a complete elliptic integral (see Fan and Wu[13]) and we find

W (0, 1)B =

- 2

?

K(k)

- s2+r2,

where K(k) is the complete elliptic integral

K(k)=|?�2

0

d&

- 1&k2 sin2 &(4.4)

with modulus k given by

k2=s2&r2

s2+r2 , (4.5)

where

s2=$2&:2&;2&#2

and

r2=- ($&:&;&#)($+:&;&#)($&:+;&#)($&:&;+#). (4.6)

At the critical point r=0 and therefore k=1, hence in the neighbourhood of thecritical point with k2=1&k$2 we can use the expansion [14, 8.113.4]

K(k)=ln[4�k$]+k$2

4(ln[4�k$]&1)+ } } } . (4.7)


The critical point value of s2=2�g together with (4.7) yields

W (0, 1)B =&

- g4? {ln _m2

- g4?2 &&2\= ,

where

\=ln[4�?]& 34 ln g& 1

2 ln[(:+;)(;+#)(#+:)] (4.8)

Hence with V=- g K0K1 the scaling limit of WB is given by

limscaling

WB=K0K1 4B+m2V4?

[ln[K0K1]+2\]&m2V4? {ln _m2V

4?2 &&1= ,

where

4B=|2?

0

d&1 d&2

(2?)2 ln _- g?

[_&: cos(&1)&; cos(&2)&# cos(&1&&2)]& (4.9)

and is the value of WB at the critical point, m=0.

4.2. The Cylinder Term

In order to extract the scaling limit of the term WC we need the following expan-sions with m2 and & infinitesimal

q+ t1+2

:2g&&2 \2;#

:2 +g(;+#)2

2 ++m2

: \1&g:(;+#)

2 + (4.10)

q& t1+&2 g(;+#)2

2+m2 g(;+#)

2. (4.11)

Thus for m=0 and &=0 we have q&=1 but q+>1 hence in the limit of K0 � �only q& contributes to WC . For fixed infinitesimal &0 and asymptotically large K0

we have

WC t|&0

0

d&?

ln |1&e&K0- (- g (;+#) &)2+ gm2(;+#)e2?iu0|2. (4.12)

If we change variables to p=K0 - g(;+#) & the scaling limit of K1 WC is welldefined and we find

limscaling

K1 WC #1C={1 |�

&�

dp2?

ln |1&e&- p2+m2V�{1+2?iu0|2


and where

{1=k

- g (;+#)(4.13)

which we assume to be positive.

4.3. The Pure Toroidal Contribution

To obtain the scaling limit for the remaining term in WF we observe that unlesszn0

approaches one sufficiently rapidly all of the individual terms in the summationgo exponentially to zero as K1 � �. But zn0 relabeling � 1 only when $ approaches:+;+# and cos(xn0

) � 1, the latter occurring for n0 near zero or near K0&1.Therefore due to the periodicity of the eigenvalues it is convenient to relabel thosen0 from [(K0&1)�2] (where [x] means the integer part of x) up to K0&1 by then0 [ n0&K0 so that we have

WT= :K0&1

n0=0

|1&e&K1 vn0e2?iu1|2= :[(K0&1)�2]

n0=&[K0 �2]

|1&e&K1 vn0e2?iu1|2. (4.14)

The small m and small xn0expansion of vn0

is given by

vn0t� m2

;+#+

x2n0

g(;+#)2+i#xn0

;+#. (4.15)

So the limiting form of WT is given by

limscaling

WT #1T= :�

n=&�

ln |1&e&2?{1- (n+u0)2+(m2V�4?2{1)+2?i[u1&{0(n+u0)]|2, (4.16)

where

{0=k#

;+#. (4.17)

Hence we have the scaling limit of the entire finite size contribution given by1F=1C+1T or explicitly

1F =&?{1

6c \u0 ,

m2V{1 +

+ :�

n=&�

ln |1&e&2?{1- (n+u0)2+(m2V�4?2{1)+2?i[u1&{0(n+u0)]|2, (4.18)


where c(u0 , m2V�{1) which we refer to as the ``cylinder charge'' is given by

|�

&�

dp2?

ln |1&e&- p2+m2V�{1+2?iu0|2=&?6

c \u0 ,m2V{1 + . (4.19)

Observe that the dependence of WF on the five variables [:, ;, #, $, K0 , K1] reducesin the scaling limit to a dependence of 1F on only three combinations of these {0 ,{1 and m2V. In terms of the original model (3.3) and the lattice spacings (3.6) withL0=K0a0 and L1=K1a1 we have

{0 #k#

;+#=

L1 cos %

L0

, {1 #k

- g (;+#)=

L1 sin %

L0

and

V#- g K0K1=L0L1 sin %. (4.20)

5. MODULAR INVARIANCE IN THE SCALING LIMIT

In this section we show that the scaling limit of the finite size corrections hasacquired additional symmetries, over and above the symmetries that existed in theoriginal lattice model, the most important of these is modular invariance. By usingthese symmetries we will be able to split the bulk term into a modular invariantpart, which will have a natural interpretation as the ``singular part'' of the freeenergy, and an additional part which is non-modular invariant and which we inter-pret as the non-universal regular part of the free energy. We finish the section byestablishing that the origin of modular invariance lies in the freedom to performconstant coordinate transformations of the continuum energy functional.

5.1. Symmetries of the Partition Function

In the derivation of (3.19) we performed the summation over n1 first and thenthat over n0 . We could of course have reversed this order. The only difference thatwould have arisen is that K0 would be replaced by K1 , u0 by u1 and : by ;. Thebulk term WB is unaffected by this substitution, but it leads to a different expressionfor the finite size finite lattice term, WF ; both expressions, of course, are completelyequivalent. The first (3.19) is a sum of the K0 cylinder term WC and a toroidal termWT that contains the K1 cylinder contribution, while the second (obtained byperforming the summations in the opposite order) is a sum of the K1 cylinder termand a toroidal term that now contains the K0 cylinder contribution. Naturally, thisinvariance remains in the scaling limit and corresponds to the replacements

{0 [1

k

#

:+#=

{0

|{|2, {1 [

1

k

1

- g (:+#)=

{1

|{|2(5.1)

u0 [ u1 , u1 [ u0 ,


where we find it convenient to define {={0+i{1 . The lattice partition function isfurther invariant under (u0 , u1) [ (&u0 , &u1) as, of course, is the scaling limit.

In the scaling limit there are in fact two further transformations that leave thefinite size contribution invariant. So in all we have the following four transforma-tions that leave the 1F invariant. [eqlabel][fulltransforms]

(a) {0 [ {0+1, {1 [ {1 , u0 [ u0 , u1 [ u0+u1

(5.2)(b) {0 [

{0

|{| 2 , {1 [{1

|{|2 , u0 [ u1 , u1 [ u0

(c) {0 [ {0 , {1 [ {1 , u0 [ &u0 , u1 [ &u1

(d) {0 [ &{0 , {1 [ {1 , u0 [ &u0 , u1 [ u1 ,

where (a) and (d) are the new invariances. In the scaling limit (a) appears to berather trivial in nature; however, it corresponds to the transformation of thecouplings

: [ :&;&#

k+

;+#k2 , ; [ ;&

;+#k

, # [ #+;+#

k. (5.3)

which is evidently not a symmetry of the underlying lattice system. The invariances,(a)�(d), for the scaling limit can be regrouped and further simplified, by definingw=u1&{u0 (with {={0+i{1), to read

(i) { [ {+1, w [ w

(5.4)(ii) { [ &

1{

, w [w{

(iii) { [ {, w [ &w

(iv) { [ &{*, w [ w*.

Transformations (i) and (ii) correspond to the non-trivial action of the generatorsof the modular group with (i) recording the action of the T generator and (ii) thatof the S generator. Together they generate the entire group SL(2, Z), a generalelement corresponding to a ``word'' formed by S and T.

It is useful at this point to spell out some of the details of the invariance proper-ties of our expressions in a short discussion. We recall some basic facts. If Mdenotes an element of the modular group then in matrix form

M=\ac

bd+ # SL(2, Z)�Z2 , a, b, c, d # Z, ad&cb=1 (5.5)

and matrices M and &M are identified. When such transformations M act on thetorus they act on the cycles and hence act both on the modular parameter { and


on the generators of the fundamental group ?1(T 2); or equivalently on theholonomy phases u0 and u1 which specify the bundle L. However, we note that asecond action of the generator S is such that { [ { but w [ &w, but this can becompensated for by the transformation (iii) above. Hence we can indeed factor bythe Z2 corresponding to M being minus the identity to reduce from SL(2, Z) to themodular group SL(2, Z)�Z2 . For completeness the action of a general element Mof the modular group is given by

under M : u0 [ cu1+du0 , u1 [ au1+bu0

and

{ [a{+bc{+d

with M # SL(2, Z)�Z2 . (5.6)

It is possible to check numerically the invariance of (4.18) for general M. This weindeed have done. Note in particular that none of these transformations changesthe sign of {1 .

From the above discussion we see that the full invariance of the finite size con-tributions in the scaling limit can be characterized as (1) invariance under themodular group SL(2, Z)�Z2 , (2) under the group Z2 generated by M=&I, andfinally (3) invariance under parity which corresponds to the generator

P=\10

0&1+ (5.7)

with determinant &1. This full invariance is the double cover of SL(2, Z) which wedenote SL

t(2, Z). This group leaves both the mass (or measure of the deviation from

the critical point) m and the symmetric product of the couplings - g and the latticevolume K0K1 and hence V=- g K0K1 invariant.

We finally observe that the two functions 4B and \, which enter the scaling limitof the bulk contribution, are not modular invariant. Therefore the bulk contribu-tion (4.9) can split into a part that is modular invariant, 1B , and a portion whichis not 1 reg; however, this decomposition is not unique, therefore we introduce a freeparameter, *, into the decomposition where * is simply a numerical scale that canbe chosen at out convenience. The two terms of the decomposition are then

1 reg=K0 K14B+m2V4?

[ln *+2\] (5.8)

and

1B=&m2V4? {ln _m2* - g

4?2 &&1= . (5.9)


The sum

1=1B+1F , (5.10)

where 1B is given by (5.9) and 1F is given by (4.18) is then invariant under thedouble cover of the modular group SL

t(2, Z) and can be identified as the universal

scaling function that the describes the neighbourhood of the critical point.

5.2. The Origins of Modular Invariance

We now argue that the origin of the above symmetries of the scaling limit lie inthe general covariance of the continuum energy functional. In the continuum limit(3.3) becomes the energy functional,

I[.]=12 |

T 2- h [�+.*h+&�&.+m2.*.], (5.11)

where h+& is an arbitrary metric and we have purposely written the expression in acoordinate free form. If we choose rectangular coordinates where y runs from 0 toL1 sin %, where % is the angle of the parallelogram as in Fig. 1, then the metric isdiagonal with unit entries. We are of course free to choose a different coordinatesystem by for example making the coordinate transformation

z=L0(x0+{x1),

where {={0+i{1 with {0=(L1 �L0) cos % and {1=(L1 �L0) sin %, in which x0 and x1

both run between 0 and 1. In this coordinate system the metric g takes the form

ds2=h+& dx+ dx&

=L20 |dx0+{ dx1| 2 (5.12)

and we have chosen our unit of length to be L0 . We could of course equally wellhave chosen our unit of length to be L1 and used the coordinate system defined byz={L0[x~ 0+(&1�{) x~ 1] which corresponds to x~ 0=&x1 and x~ 1=x0 or a rotationby ?�2 of the (x0 , x1) coordinate system. In these new coordinates the metric isgiven by

ds2=L21 }dx~ 0+\&

1{+ dx~ 1 }

2

and we see that aside from the change in our unit of measure from L0 to L1 wehave replaced { by &1�{. Now by studying the implications of this coordinatechange on the phase dependence of . we see that it is equivalent to interpreting thenew phases in the new coordinate system according to the transformation u~ 0=u1

and u~ 1=&u0 . But with these transformations, which clearly do not affect theenergy functional (5.11), we have precisely implemented the transformation (ii) of


(5.4) corresponding to the S generator of the modular group. The T generatorcorresponds to implementing a coordinate transformation to the (x$0 , x$1) coor-dinate system given by z=L0[x$0+({+1) x$1] where the x$1 axis lies along thediagonal of the (x0 , x1) coordinate square. Tracking the phase dependence in thisnew coordinate system implies that the full mapping of parameters correspondsprecisely to (i) of (5.4) or the action of the T generator of the modular group. Sincethe functional (5.11) is invariant under the discrete symmetries of both parity andspace inversion we have accounted for all the symmetries of (5.4).

If we now proceed formally and calculate the partition function by a continuumregularisation method, for example, `-function regularisation where the continuumeigenvalues are summed over raised to a negative power which is continued into thecomplex plane to yield the generalized `-function,

`2+m2(&)= :�

n0 , n1=&�

#&&n0 n1

,

with in our case

#n0 n1=\ 2?

L0{1+2

|(n1+u1)&{(n0+u0)|2+m2, (5.13)

then &ln Z is given by

1=&`$2+m2(0)&`2+m2(0) ln +2, (5.14)

where + is an arbitrary scale that enters in the regularisation process to keep thepartition function dimensionless. This `-function (5.13) can be easily calculated byrepeated use of the Plana sum formula [15] which can be cast in the form

:�

n=&�

1((n+u)2+x)s

=?1�21(s&1�2)

1(s)x&s+1�2+

sin(?s)? |

�

0dqq&2s d

dqln |1&e&2? - q2+x+2?iu|2

(5.15)

and on computation of (5.14) yields precisely the finite size contribution 1F of(4.18) but the bulk term is given by &(Vm2�4?)(ln[(m�2?+)2]&1) where + is thearbitrary undetermined scale that arose in the regularisation process. If we interpret+2=1�* - g we see that we have recovered precisely the singular part of the parti-tion function in the scaling limit. Furthermore since we were able to regulate in afashion that preserved all of the symmetries discussed above we see that theinvariance of the scaling function under SL

t(2, Z) is precisely a consequence of the

diffeomorphism invariance of the continuum action functional together with itsdiscrete symmetries.


6. THE APPROACH TO THE CRITICAL POINT ANDNON-COMMUTING LIMITS

We now focus on the universal scaling function 1, which exhibits the property,characteristic of all non-trivial scaling functions, of having non-commuting limits.We will first focus on the bulk critical point where $=:+;+# and then examinethe approach to this point. Finally we will consider the geometrical limit where thetorus reduces to a cylinder.

6.6. Shape and Size Dependent Effects at the Critical Point

At the bulk critical point the mass m is zero (i.e., $=:+;+#) and the integral(4.13) can be performed. We find

1C =&{1

2?

:�

l=1

cos(2?u0 l )l2

=&2?{1 \u20&u0+

16+ (6.1)

which we see from the first representation is a periodic of u0 with period 1; howeverit is non-analytic at u=0, having a cusp at this point. Setting m=0 in (4.16) andafter a little rearranging we have

1T = :�

n=&�

ln |1&exp[2?i[ |n| {+=n(u1&u0{)]]| 2,

where

=n={1,&1,

n�0,n<0,

{={0+i{1 . (6.2)

The two terms (6.1) and (6.2) can in fact be combined to give

1F=ln } exp[?iu20{]

%1(u1&{u0 , {)'({) }

2

. (6.3)

The modular invariance of this expression (and more generally invariance underSLt

(2, Z)) can be verified directly using the known transformation properties of theclassical functions from which it is built.

Observe that as far as the variable { is concerned 1F is the sum of a holomorphicand anti-holomorphic part, and so the finite size contribution to the partitionfunction exhibits holomorphic factorization in the complex variable {,

exp[&1F]#ZF =FF�


with

F=exp[?iu20 {]

%1(w, {)'({)

. (6.4)

Furthermore observe that if the phase factor exp[?iu20{] were absent from F above

then ZF would also exhibit holomorphic factorisation in the Picard variety labelw=u1&{u0 . If we re-express 1F in terms of the complex variable w we find

1F=ln } %1(w, {)'({) }

2

+?

2{1

(w&w� )2

but this latter expression is proportional to the Greens function with the zero moderemoved (see [16]) of the Laplacian �w�w� . Hence we find

�w�w� 1F=&4?$(w, w� )+?{1

. (6.5)

For w{0 from (6.5) we see that it is the absence of the zero mode that is respon-sible for the breakdown of holomorphic factorization. Actually the presence of thephase exp[?iu2

0{] is a consequence of the central charge c being non-zero��lack ofholomorphic factorisation in this sense is the existence of a holomorphic anomaly[17] of the �� L determinant bundle over the Picard variety.

6.7. Non-commuting Limits

A further rich feature of this model is that it possesses non-commuting limits,namely the limits u0 , u1 � 0 and m � 0. To establish this we can expand 1F , thefinite size contribution to (4.18), for small ui and m obtaining

1F=ln[(2?)2 |u1&{u0|2+{1 m2V ]+2 ln |'({)|2+ } } } . (6.6)

Now the r.h.s. of (6.6) can tend to the two distinct (logarithmically singular)expressions

ln |u1&{u0|2+2 ln |'({)|2

and

ln[{1m2V]+2 ln |'({)|2 (6.7)

depending on the order in which the limits are taken. Nevertheless both limits andindeed (6.6) itself are modular invariant.


File: 595J 586821 . By:SD . Date:16:03:99 . Time:13:26 LOP8M. V8.B. Page 01:01Codes: 2116 Signs: 1055 . Length: 46 pic 0 pts, 194 mm

FIG. 2. The ``cylinder charge'' function c(u, x) for various u.

6.8. The Cylinder Limit

As discussed in Section 4 if we take the limit K1 � � while keeping K0 and themass fixed we obtain the lattice cylinder limit

Wcylinder=K0 K1WB+K1 WC. (6.8)

Furthermore if we consider the scaling limit of this expression we find

limscaling

Wcylinder=1reg+1B&?{1

6c \u0 ,

m2V{1 + (6.9)

which can equally well be obtained by taking k � � of the torus expressions. Interms of our original model this corresponds to the limit L1 � �. For large L1 thequantity 1F�V tends to the finite value2 #cylinder where

#cylinder=&?

6L20

c(u0 , m2L20) (6.10)

and we refer to c(u0 , m2L20) as the ``cylinder charge''. As has been argued in [18]

this limit enables us to access the central charge c of the model. If we set m=u0=0we see that c(0, 0)=c

c=2. (6.11)

Because the cylinder charge determines the central charge at appropriate values ofits variables it might appear reasonable to expect that the cylinder charge #cylinder


2 We could equally take L0 large if we interchange (u0 , L0) with (u1 , L1).

be equal to the Zamolodchikov c-function [19]��this is not so. The c-function mustbe monotonic in m and the cylinder charge is not, cf. Fig. 2, where we plot c(u, x)versus - x�2? for various u. For large values of x we see the dependence on u dropsout and we have limx � � c(u, x) � 0 whatever the value of u.

7. THE ISING MODEL

A model closely related to the Gaussian model considered above is the dimermodel (for a review see Priezzhev [20] and the books [21, 22]) whose partitionfunction is given by a sum of Pfaffians. Furthermore the dimer model, on adecorated lattice, has been established to be equivalent to the Ising model on theoriginal lattice [23]. Hence to obtain the corresponding results for either of thesemodels all that is necessary is an analysis of the appropriate lattice determinants.Our results of Sections 3 and 4 can therefore be easily adopted to these models. Inparticular they can be applied to the Ising model on a torus��only the special caseof the Ising model on a square lattice with equal couplings, which as we will seebelow corresponds to {0=0 and {1=k, has been studied in detail previously [24].

Denoting nearest neighbour spin�spin couplings by J1 , J2 , and J3 we extract :,;, #, and $ from [25]. They take the form

:=sinh \ 2J1

kBT+ , ;=sinh \ 2J2

kBT+ , #=sinh \ 2J3

kBT+$=cosh \ 2J1

kBT+ cosh \ 2J2

kBT+ cosh \ 2J3

kBT+ (7.1)

+sinh \ 2J1

kBT+ sinh \ 2J2

kBT+ sinh \ 2J3

kBT+and displaying the phase dependence of WF by WF (u0 , u1) we find (for ferro-magnetic couplings) the Ising partition function is given by

ZIsing= 12e&W B

Ising[�e(1�2) WF(0, 0)+e(1�2) WF(0, 1�2)+e(1�2) WF(1�2, 0)+e(1�2) WF(1�2, 1�2)]

(7.2)

with + referring to T<Tc and & to T>Tc . This result (7.2) incorporates the com-plete lattice and finite size corrections for the Ising model on a triangular lattice.For the triangular lattice Ising model it is convenient to define the polynomial

p(v1 , v2 , v3)=v1v2v3&v1v2&v2v3&v3v1&v1&v2&v3+1,(7.3)

vi=tanh \ Ji

kBT+ .


We then have

1g

=1&p(v1 , v2 , v3) p(&v1 , &v2 , &v3)

(1&v21)(1&v2

2)(1&v23)

and

$&:&;&#=p(v1 , v2 , v3)2

(1&v21)(1&v2

2)(1&v23)

. (7.4)

The critical temperature is given by the root of the polynomial

p(v1 , v2 , v3)=0 (7.5)

and at the critical temperature g=1 and as previously $=:+;+#. The conditionfor the criticality (7.5) implies that on the critical surface in coupling space we canexpress one of the couplings in terms of the others, e.g.,

v1*=1&v2*&v3*&v2*v3*1+v2*+v3*&v2*v3*

(7.6)

and cyclic permutations. The asterisk indicates that the quantities are evaluated atthe critical temperature.

The scaling limit in the Ising model corresponds to the constrained ther-modynamic limit where t=((T&Tc)�Tc) - K0K1 , k=K1 �K0 , and the couplings Ji

are held fixed as the Ki are sent to infinity. When we take the scaling limitWF (u0 , u1) [ 1F (u0 , u1) and again the expression for the finite size contributionbecomes a modular invariant function. The finite size contribution to the partitionfunction is given by

ZIsingF =�e(1�2) 1F(0, 0)+e(1�2) 1F(0, 1�2)+e(1�2) 1F(1�2, 0)+e(1�2) 1F(1�2, 1�2), (7.7)

where 1F (u0 , u1) is given by (4.18). We are furthermore in a position to identify themodular parameter { and the combination m2V in terms of the Ising couplings andthe temperature. We find

{=k(1&v2*

2)v2*(1&v3*

2)+v3*(1&v2*2) _v3*+i

1&v3*2

2 & and m2V=At2,

where after some simplification we find

A=8 { J1

kBTcb(v2*, v3*)+

J2

kBTcb(v3*, v1*)+

J3

kBTcb(v1*, v2*)=

2


and

b(v1*, v2*)=�(1&v1*v2*)(v1*+v2*)(1&v1*

2)(1&v2*2)

.

When we restrict to the square lattice with symmetric couplings, i.e., J3=0 andJ1=J2 we find that the critical value of v is v*=- 2&1 and hence substitutingv3*=0 and v2*=- 2&1 we find {=ik and A=[2 ln[1+- 2]]2 in agreement withFerdinand and Fisher [24].

Note that the expression (7.2) for ZIsing is a sum of four terms in which thedependence on u0 and u1 of each of the terms corresponds to a point on an orbitof SL(2, Z). The partition function ZIsing is precisely the sum of two such orbits, i.e.,of sets of values of (u0 , u1) which arise under repeated action of the transformations(5.2). It is the sum of the orbit generated by (0, 0) and that generated by (0, 1�2).This is in fact true even at the lattice level and in the scaling limit summation overan orbit of (u0 , u1) constructs a modular invariant function of the two variables {and m2V, which is independent of u0 and u1 . This construction makes it easy tounderstand that it is the summation over the ui which forms an orbit of theSL(2, Z)-action on the space of ui 's that guarantees invariance at the level ofdependence on {. Mathematically we are working with the action of SL(2, Z) onthe space of flat bundles��the so-called Picard variety��and other orbits exist whichallow one to construct many additional (phase independent) modular invariantpartition functions such as those of other conformally invariant field theories(cf. [26] and references therein) from the data given above.

The cylinder charge for the Ising model can be obtained from (7.2) with L1 � �,and it works out to be #cylinder

Ising where

#cylinderIsing =

?12L2

0

c \12

, x+ . (7.8)

This means that comparison with (6.10) above determines that the cylinder chargefor the Ising model is

&12c( 1

2 , x). (7.9)

Now if we set x=0 we should get the Ising central charge and doing this we findc= 1

2 (cf. Fig. 2) as expected [18].

8. CONCLUSION

Beginning with a lattice model where the site degrees of freedom are sections ofa U(1) bundle over a triangular lattice which globally forms a torus and where thenearest neighbour interactions are different in the three lattice directions we found


the exact partition function on a lattice with an arbitrary number of sites. Thisexpression naturally decomposed into a bulk part and a finite size finite lattice con-tribution. By taking the scaling or continuum limit we found that the finite sizefinite lattice contribution was a modular invariant function. In fact it was invariantunder the group of transformations generated by (5.4), i.e., the double cover of themodular group SL

t

(2, Z). The bulk part of the partition function was not invariantunder this symmetry but could be split into two parts one of which is invariantunder SL

t

(2, Z) and is then naturally identified as the singular part, the remainderforming the regular part. This division however is somewhat arbitrary and wetherefore allow a free parameter to specify the scale at which the division is per-formed. Thus the complete scaling function is given by (5.10) and is invariant underSLt

(2, Z). We emphasize that this full invariance (as described in Subsection 5.4,Eqs. (5.2)�(5.4)) is present for the massive theory.

The origin of this invariance is identified as the coordinate invariance of the con-tinuum energy functional together with the discrete invariances of parity and fullspace inversion. By using `-function regularisation to define the continuum parti-tion function directly we recover precisely the scaling function (5.10). The arbitraryscale associated with the division into bulk and regular portions in this instancearose by requiring that the partition function remain dimensionless.

We obtained precisely the same data as above for the two dimensional Isingmodel and in particular we find the exact free energy scaling function in the absenceof an external field. Again it is invariant under the double cover of the modulargroup. We identify the modular parameter { in terms of the lattice couplings. It wasonly by having allowed three independent couplings for the different lattice direc-tions that we were able to access all values of {.

It is clear from the discussion above that an analogue of modular invarianceexists for higher dimensional systems in the scaling limit. In fact we believe thepresence of such symmetries should provide a powerful method of checking thatnumerical simulations are accessing the scaling limit. They should also provide anatural method of separating the bulk and singular portions of the thermodynamicfunctions.

APPENDIX

The task of this appendix is to establish that for ` and * real and such that z=`�*�1, with z\=z\- z2&1 (so that z+z&=1) we have the summation formula

:K&1

n=0

ln _`&* cos \2?(n+u)K +&

=K ln _*z+

2 &+ln[1&2zK& cos(2?u)+z2K

& ]. (A.1)


The formula is established as follows. First note that

`&* cos(x)=*z+

2(1&z&eix)(1&z&e&ix).

Next note that since e2?in�K for n=0, 1, ..., K&1 are the Kth roots of unity

`K&1

n=0

(z+e2?iu�K&e2?in�K)=zK+e2?iu&1

so that

:K&1

n=0

ln[1&z&e2?i(n+u)�K]=ln[1&zK&e2?iu]. (A.2)

[27] Hence3 factoring the argument of the log in the l.h.s. of (A.1) and using thesummation (A.2) yields the right hand side of (A.1). When z\ are real (A.1)becomes

:K&1

n=0

ln _`&* cos \2?(n+u)K +&=K ln _`+- `2&*2

2 &+ln |1&zK& e2?iu|2 (A.3)

This implies the rather nice product identity:

`K&1

n=0

2 {`&* cos \2?(n+u)K +==|(`+- `2&*2)K�2&e2?iu(`&- `2&*2)K�2| 2

=*K } 2 sinh \K2

ln[z+]&i?u+}2

(A.4)

General complex ` and * lead to no additional difficulties; the product can alsobe performed and yields

`K&1

n=0

2 {`&* cos \2?(n+u)K +=

=*KzK+(1&zK

&e2?iu)(1&zK& e&2?iu)

=4*K sinh _K2

cosh&1(z)+?iu& sinh _K2

cosh&1(z)&?iu& . (A.5)


3 The formula (A.2) above could alternatively be obtained by expanding the argument of the log andusing the fact that exp[2?ir�K] are the K th roots of unity so that

:

(K&1)

r=0

exp[2?inr�K]={K,0,

if n=0 mod Kotherwise.

similar algebraic trick was used by A. Patrick in [27].

ACKNOWLEDGMENTS

Denjoe O' Connor thanks J. D. Bronkov, M. Hankle, A. Patrick, V. B. Priezzhev, and P. J. Uptonfor helpful comments and discussions and The Dublin Institute for Advanced Studies where part of thiswork was completed.

REFERENCES

1. M. E. Fisher, in ``Critical Phenomena, Proc. 51st Enrico Fermi Summer School, Varena,'' AcademicPress, New York, 1972; M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28 (1972), 1516; Y. Imryand D. Bergman, Phys. Rev. A3 (1971) 1416.

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Modular Invariance, Lattice Field Theories, and Finite Size Corrections

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