algebraic formulation of duality transformations for abelian lattice models

ANNALS OF PHYSICS 141, 225-253 (1982)

Algebraic Formulation of Duality Transformations for Abelian Lattice Models

K. DR~~HL* AND H. WAGNER

Sektion Physik der Universitiit MCnchen. D-8000 Miinchen, Germany

Received October 26. 1981; revised February 1. 1982

We derive duality relations for the generating functional of correlation-functions for disor dered (“frustrated”) lattice Higgs and gauge models with arbitrary locally compact abelian symmetry groups. We examine the effects on duality of nontrivial lattice topology and of configurational boundary conditions. Technically, we employ some elementary concepts borrowed from algebraic topology. Combined with Fourier expansions on groups. they provide tools to handle functions on lattices in a very efficient way. For convenience, these concepts are described in some detail in the first part of the paper.

1. INTRODUCTION

About forty years ago, Kramers and Wannier [ 1 ] discovered a peculiar symmetry relation between the low and high temperature expansions for the partition function of the square lattice Ising model. With the assumption that the model has a single phase transition, this relation enabled them to locate the exact transition temperature. Onsager 121 gave a topological interpretation of the Kramers-Wannier symmetry in terms of dual lattices and showed how to extend it to a wider class of two- dimensional models [3]. McKean [4] has pointed out that the Kramers-Wannier relation arises from a Fourier transformation on the group L,, which is represented by the Ising spins. This recognition opened the way for the application of duality transformations to models with other abelian symmetry groups.

As long as one confines attention to models with their spins residing on lattice sites only, duality transformations are restricted to two dimensions. In higher dimensions the dual of a “site’‘-model is a model with local gauge symmetry, where the basic variables are attached to higher dimensional lattice cells (bonds, plaquettes, etc.). The introduction of lattice gauge models by Wegner ]5 ] and Wilson [6] has led to an upsurge of interest in duality methods, which provide qualitative insights into the apparently complicated phase structure of these systems [7].

In this paper we present a coherent formulation of duality transformations which combines their topological and group-theoretical aspects in a natural way by using the concept of cell-complexes. The efficiency of the cell-complex formalism has

* Present address: Institute of Modern Optics, University of New Mexico, Albuquerque, N.M. 87106.

225 0003 4’) 16/82/080225 -2Y505.00/0

CopyrIght c 1982 by Academic Press. Inc. All rights of reproduction m any form rewrx’t4

226 DRiiHLANDWAGNER

already been noticed by Drouffe [8]. It has also been employed by Ukawa, et al. [9] in their study of the phase structure of Z, lattice gauge models.

For convenience we first review some notions of algebraic topology and Fourier analysis on groups. We then derive the duality relation for the generating functional of correlation-functions in disordered (“frustrated”) systems. This provides a kind of master expression which described duality in a compact form. In particular, we obtain the general relationships between order and disorder variables, originally introduced by Kadanoff and Ceva [lo] for the two-dimensional Ising model and extended to the three-dimensional Ising and XY-model by Fradkin et al. [ 111. We also consider the effect of boundary conditions arising from the geometry of the lattice or from constraints on the physical states. It has been noticed by Guth [ 121 that the cell-complex method accounts automatically for nontrivial boundary conditions.

During the preparation of the manuscript we came accross a recent paper by Zinov’ev [ 131. His approach to duality is similar to ours, but he considers neither disordered systems nor configurational boundary conditions. Furthermore we believe that our less abstract formulation lends itself more readily to practical applications.

2. CELL-COMPLEXES

Consider a d-dimensional regular lattice A,, with basis vectors n’,..., nd. The geometrical elements of A, are its sites (x), bonds (x; n’), plaquettes (x; n’, n’), etc. We introduce an orientation and write

so = *(x)7

s, = f (x; n’), (x + n’; 4) E -(x; n’),

s, = (x; dl,..., n’q,

= sign 71 . (x; n’=(l),..., nincr)),

(2.1)

with i, < i, < -.a < i,, r < d, and where z denotes a permutation of (l,..., r). These oriented elements will be called r-cells. The boundary of an r-cell, 0 < r < d,

is defined by

as, = 0,

as, = i (-l)"+'[(x + n'"; nil,..., AiD,..., n") (2.2) "=*

- (x; nil )...) n '̂L, )..., n'r)],

where $0 is omitted.

DUALITY TRANSFORMATIONS 227

d[“ f -TJ] q g + L-+2+2

x nl

FIG. 1. Boundary of the plaquette (x: n’. n’).

Thus we have, for instance (see Fig. l), 8(x; n’, n’) = (x + n’; n2) - (x; n’) - (x + n*; n’) + (x; n’). It is easily verified that the boundary of a boundary vanishes identically :

aas, = 0, I = O,..., d. (2.3)

In writing (2.2) we have added cells. In order to give a precise meaning to this operation, we introduce formal linear combinations of r-cells with integer coefficients,

c, = 0 0 all msr = 0.

c, + CL= 1 (m,,+ ml,) s,. S?

These linear forms of r-cells are called r-chains. The boundary operation is extended to r-chains by

iYc=z m,& qc + c') = ac + Bc'.

Here and in the following we occasionally employ an abbreviated notation whenever its specification should be clear from the context.

The boundary of an r-cell is an (r - 1)chain,

as,= x [s,:s,-I]s,.-,. sr-I

(2.4)

The coefficients [s,: s,- ,] E Z are the incidence numbers. They take the values

1 s,: s,-,] = fl ifs,-, b e ongs 1 to the boundary of s, with the sign depending on the relative orientation of s, and s,- r, and [s,: s,- i] = 0, otherwise. From (2.3) it follows that

s is,+ I : s,][s,: s,-11 = 0. (2.5 1

In summary: The r-chains on the lattice A, form a linear space K, over the ring Z, with the set of r-cells as a basis. This algebraic structure is known as a free Z-

228 DRiiHL AND WAGNER

module. In this terminology the boundary operation 8 is a Z-linear map

a: K,+K,.-l,

aoa=o.

The translational symmetry of/i, entered into the definition of the cells, (2.1), and of the boundary operation, (2.2). It is easily seen, however, that the regularity of/i, is irrelevant for the concepts introduced to remain meaningful. Let us start, for instance, with the “lattice” depicted in Fig. 2. As cells we take the enumerated vertices (sites), bonds, and elementary polygonal faces with an arbitrary but fixed orientation. The incidence numbers have an obvious intuitive meaning and their values can be read off by inspection. In the present example, they also obey Eq. (2.5). We now take Eq. (2.4) as a definition of the boundary operation. Then aa = 0 also holds. In defining the cell basis we even have the option to omit, for instance some or all of the polygonal faces. This allows us to introduce holes into the lattice.

The above informal considerations motivate the following definition of an abstract cell-complex [ 141. A d-dimensional cell-complex (K, a) consists of

(1) a family of free Z-modules K,, 0 < r < d, with bases (s,), and

(2) a family of Z-linear maps

~,:K,+K,-,, a,-, oa,=o.

The basis elements S, are r-cells and c,. E K, are r-chains. In order to avoid repeated mentioning of exceptional cases it is convenient to introduce Kd+, = {0} = K-, .

FIG. 2. A two-dimensional cell-complex and its dual. The orientation of the cells (which can be chosen arbitrarily) is not shown.


A cell-complex is finite if the number of r-cells is finite for all 0 ,< r < d. In a complex there exists a natural scalar product given by

(s,s’)= 1 if s=s’,

=o otherwise,

which reads for chains c(‘) = ES mj”s,

(c. c’) = 2 m,mj. s

(2.6)

The incidence numbers may now be defined by

1 s,: ST-,] = (as,, s,_,).

If the incidence numbers are given, then the abstract complex can be realized geometrically by embedding in an Euclidean space of sufficiently high dimension.

Each K, has two special subspaces: the space of cycles Z, = (c, 1 ac, = 0) and the space of boundaries B, = {c, 1 c, = k,, , ). Since 83 = 0, every boundary is a cycle. B, c Z,. The converse is not true in general. Consider, for example, the two- dimensional complex shown in Fig. 3, with the hatched square omitted from K,. We have ac, = 0 = ac;, but neither c, nor c; is a boundary. However, c{ = c, + iic:, where c2 is the sum of squares spanned between cl and c,.

Two r-cycles are called homologous if they differ by a boundary,

The hole in the complex of Fig. 3 is characterized by the equivalence class of l- cycles homologous to c,.

The equivalence classes of homologous cycles are the elements of the quotient space IH, = Z,/B, which is known as the rth homology group of (K, a). If K is finite. then the number r,. of generators in IH, is finite (dim lh, = rc,). 71, is called the rth Betti number and is a topological invariant.

FIG. 3. Homologous nonbounding cycles. The hatched square is omitted from the complex. Squares are oriented clockwise.


x4 I3 x3

X1 X2 FIG. 4. A complex which acquires torsion after suitable identification of cells; see text.

Torsion coefficients constitute a second class of topological invariants. Without entering into details, let us illustrate the concept of torsion by an example, Fig. 4. If we identify the sites, x‘ =x2 =x3 =x4 =x, and the bonds I’ = I3 = I, 1’ = -I4 = I’, then we obtain a complex with the topology of Klein’s bottle, with 8p = 21, CY = 0. Hence the cycle 1 becomes a boundary only if we run through it twice. The factor 2 is an example of a torsion coefficient.

In this paper, we only deal with finite, torsion-free complexes. By definition, each module Z$r has a cell basis. From the latter we may construct

other bases by forming linear combinations. A standard result of algebraic topology asserts that each K, admits a so-called canonical or &basis [ 141. It consists of three families of r-chains {Q: E K, ] a = l,..., d,}, {bf E K, 1 p = l,..., .4,+ i), and {h; E K, 1 y = l,..., z,}, with the properties

i3aF = bF_, ,

6’b: = 0,

ah; = 0,

a = l,..., d,,

P = L..., A,, I,

y = l,..., 72,.

Here, A, = rank(8,) and A, + A,+ 1 + n, = N, = dim K,. (In a complex with torsion, one has aa; = OF-_, bF-_, with positive integers OF-, . Those OF-_, 2 2 are the torsion coefficients.)

With respect to a a-basis, K, can be written as a direct sum,

K,=A,OB,OH,,

where (a;}, {bf}, {hjf} are bases in A,, B,, H,, respectively. Each h,YE H, is a representative of an equivalence class of cycles in IH,. Obviously, A, is isomorphic to B r-1.

DUALITYTRANSFORMATIONS 231

A a-basis will be written compactly in the form (cr ) ,U = l,..., N,} with

an = cm I r) a = l,..., A,,

b; = ,;r+o, B = I,..., A,, , >

h; = ctr+4tl+~, y = l,.... Xi,.

Then

N,

M;” = (c;, s:‘) E {0, 1 }, det M,= 1. (2.7)

As we shall see, a-bases are useful in connection with duality transformation and for the discussion of gauge invariance of lattice gauge models. They allow us to manipulate functions on a complex in a simple and efficient way.

Here are two examples of a-bases:

1. EXAMPLE. Consider again the two-dimensional complex in Fig. 4. One choice of a ii-basis is:

r= 2: ai =p;

r= 1: af = I”, a = 1, 2, 3; 6; = 1’ + 1’ + I’ + 1’;

r=O: b:=x4+‘-x5, p= 1,2,3; h:,=x’.

Hence, K, = A 2, Az=l, x,=0; K,=A,@B,, A,=3, z,=O; K,=B,@H,,. A,=O, 7co= 1.

a-bases are not unique. Another choice is a :̂ = ai; 6: = I’, 6: = 1’ + I’, ~~~,‘+1’+1”.~~~b~;~~~~~-~~,~~~x’-x’,~~~~~-~~;jl~~h~~

2. EXAMPLE. N x N square lattice with one plaquette (p”) omitted; see Fig. 5.

FIG. 5. The collection of heavy solid bonds (orientation suppressed) form a basis for the space A,. The plaquette p” is omitted.

232 DRijHL AND WAGNER

r = 2: B, = (0} = H,. The plaquettes p”, a = l,..., (N- 1)’ - 1, form a basis in A,.HenceK,=A,withA,=N’-2N,a,=O.

r = 1: For {a:} we take all horizontal bonds and the vertical bonds on the first row, thus A, = N2 - 1. We have {bf] = {$#,/I = l,..., A2}. H, has a single element /I: = ape, which represents the equivalence class of l-cycles going around the holep’. ThusK,=A,OB,OH,,K,=l.

r=O:Ko=Bo@Ho, {b{}={&.zf;/3= l,...,A,},h;=xl.

The adjoint of 8 with respect to the scalar product (2.6) defines the coboundary operator 8 *,

In particular, we have (see Fig. 6)

a,*,,%= c Sr+lbr+l: s,l. Sr+l

(2.9)

a* can be extended to a Z-linear map,

Given a &basis, we may construct a reciprocal a*-basis {E;} = {ZF, @, h”;} as follows. We set

;; = 2 s,“(jkf;‘)“‘. “El

Thus, by (2.7)

(c,u, q) = BP”, 1, v = l,..., N,, (2.10)

FIG. 6. Illustration of coboundary operation: 8*x = I’ + 1’ + I’ + 14. a*/ =p’ +p2.

DUALITYTRANSFORMATIONS

and each c, E K, may be written as

233

(2.11)

In particular, taking c,, , = a*EF and using (2.8) yields

With (2.10) it follows that

a*(j=zzI) r 1 a = l,..., d,,

a*E4 = a’4 r r+1, p = I,..., A,, 1,

a*p = 0 I 3 y = l,..., 7T,.

The subbases {c$}, {@}, {ZY,} g enerate the subspaces A”,, g,., fi?,, and we have

K,=&@&@fi,.

An example of a a*-basis for the complex in Fig. 4 is i$ =p; 2; = 1” - 1”. a = 1,2,3; b: = Z4; bi = x2 + x3 + x4, 8: = x3 + x4, &i =x4; i;A = x’ + x2 + x3 + .y4.

3. PHYSICAL STATES ON A COMPLEX

In the Ising model, a spin variable S taking the values fl is attached to each lattice site x. We write S, = exp[inm(x)] with m(x) = 0, 1. Since S: = 1, the addition of the m-variables is defined modulo 2. In other words, m(x) represents an element of the group L,. An IV,-tuple (m(x’),..., m(xNO)) E Z, X ... X Z, (N, times) represents a spin configuration or physical state on the N, sites of the lattice. This notion of physical states will be generalized as follows. Let G denote an abelian group, written additively.

A map

pyc, + c:> = #(c,> + rp’(C>.

4f(-c,) = -v’(c,)~

is called a G-valued r-cochain on K,.


If we define

(P’ + V)(c) = P’(C) + q”(C),

the r-cochains form a Z-linear space K’(G). By construction, we have the correspondence

K’(G) ++ { (p’(s;),..., cf(s~)} = G x . -- x G (N’ times).

The cochains are now interpreted as physical states on the complex (K, a). We define a “differential” operator d by

(dq’)(c,+ I> = Wcr+ 11, (3.1)

which extends to a Z-linear map,

d’: K’(G) --f K’+‘(G),

d ‘+lod’=o.

dd = 0 is a consequence of &Y = 0. A a-basis in K’ induces a canonical decomposition of K’(G) in the following way:

First, we assert that dcp’ = 0 o qf(B’) = 0, (3.2)

cp’ = dqf-’ o qf(Z’) = 0. (3.3)

Equation (3.2) holds, since df(K’+ J = q’(i?K’+ ,) = #(B’).

Proof of (3.3). =-: ((Z,) = dqf-‘(Z’) = rp’-‘(8Z’) = q’-‘(O) = 0.

e: Define p’-’ by

p’- ‘(b;- 1) = p’(a;), a = l,..., d’,

qqC’- 1) = 0, c,-1 @B’-,.

Then,

dqf-‘(a;) = p’- ‘(b;- 1) = @(a;).

Secondly, for each cochain I$’ E K’(G) we set

(3.4)

where {c; 1 v = l,..., N’} is a &basis. Since the boundary of any chain c’+, is of the form


we find

a = l,..., d,,

p = 1 )...I d,+,,

y = l...., Tc,.

As a result, we obtain the decomposition

A, A ,+I

9r= c 9:,+ c p;+ 5 9;=9;+9;+9;, a=1 /3=1 y= I

whereby

9; = &p;-’ = dcp-‘,

d9; = d9’,

d9; = 0.

(3.5)

(3.6)

We denote the spaces of 9>, 9;, and 9; by A’(G), B’(G), and H’(G), respectively. Then

K’(G) = B’(G) @A’(G) @ H’(G)

E B’(G) @ C’(G).

A’(G) = {9’ ( 9’ = d9’-’ } is the group of exact cochains; C’(G) = (9’ 1 d9’ = 0} 1 A’(G) is the group of closed cochains. IH’ = Cr/Ar denotes the so- called rth cohomology group of the complex (K, a).

Finally, we introduce the “codifferential” operator d* by

(4 9’)(c,- d = 9’@, cr- I),

which induces the Z-linear map

d;, : K’(G) -+ K’- ‘(G),

S,- ’ o d; = 0.

A a*-basis ($‘} in K, generates a decomposition of K’(G) of the form

9’=9~+9g+9~=9>+9~,

with

236

and

DRiiHL AND WAGNER

etc. Thus

K’(G) = p(G) @l?(G) @ I?‘(G)

= K’(G) 0 Z’(G).

One also finds that

q+&.p~+‘=d.&+‘,

d,q$=O.

The structure of the decompositions is conveniently summarized by

(3.7)

K’ =A: @iI’ @H’

dI \ K’-’ =A’-] @B’-‘@H’-1

and

K’t1 =K’+1

d*I \

d*T =Pyq* @P

K'-'=R'-1 @#PI BE?'-'.

We note that d and d* induce one-one correspondences Art ’ t) B’ and K’ ++ @‘, respectively.

There is a remarkable analogy between the concepts described here and those of integration on manifolds. A complex corresponds to a discretized (“triangulated”) manifold; chains and cochains correspond to integration domains and (G-valued) differential forms. p(c) is the algebraic analogue of the integral I,cp, and (3.1) is nothing but the algebraic version of the Stokes formula lacy = SC dp.


For a three-dimensional complex, do, d’, dZ correspond to grad, rot, div. d rt ’ o d’ = 0 combines the identities rot grad = 0 (I = 0) and div rot = 0 (r = 1).

A r-cochain rp’ may be visualized by giving its support, i.e., the set of those r-cells on which cpr is nonvanishing,

supp rp’ - {s;“, #I E I, 1 ff(s9) # O}, (3.8)

with the index set I, c (l,..., IV,}. One easily sees that supp dp’ c (a, s;‘, 1 E I,}.

4. DUAL COMPLEX

A d-dimensional complex (K*, a) is said to be dual to the d-dimensional complex (K, a) if there is a family of maps

D,: K,-+ K$-,, r = O,..., d,

c, t+ D,(c,) = cd*- ,. 7

with D,-, o D, = identity map, and

[s&+*: sjYrl = (s,: q-,1. (4.1)

This definition formalizes the well-known intuitive construction of dual lattices: see Fig. 2.

Relation (4.1) also fixes the (relative) orientation of the dual cells s,*_, = D,(s,) in terms of the orientation of the cells s,. From (4.1) and since D, is invertible, one immediately finds that the incidence numbers of K* satisfy the dual form of (2.5). Hence, 8 = 0 in K* is consistent with (4.1). Furthermore, it is easily verified that

Doa=a*oD,

Doa*=aoD. (4.2)

An example of dual complexes is shown in Fig. 7. Note that each 2-cell p’* = D(x’) is represented by an infinite quadrant, with the boundary in K* of p’*, for instance, consisting of the half-lines I’ * and l4 *, 8~’ * = -1’ * + l4 *.

FIG. 7. Dual complexes with orientations related by Eq. (4.1).


To each complex there exists a dual one. This can be proven constructively, by employing the method of subdivision [lo]. Given a a*-basis in (K, a), we can construct a a-basis in (K*, 3) by using (4.2):

Therefore, if we define

a 0 D@) = D(a*q) = D(d),

a 0 D(6:) = D(a*q) = 0,

a 0 D(Q) = D(a*i,y) = 0.

;1-$-, = D@), B= L...,A,+l (-A$-,), r;,u-, z D(qT), a = l,..., A r wd*-r+ 117

ij;+. = D(Q), y = I,..., 7r, (=Q- ,>,

we have a &basis in (K*, a). To each cochain (p’ on K, we associate a dual cochain q$-’ on K$+ by

or

The transformation under D, of the differential operators can be read Off from

d&‘(c$-,+I) = &-“(k,*_,+,> = P$-YW*c,-,)I

=~r(a*Cr--l)=d*(P’(C,-*)= (d,P’),(&,,)~

which yields

dcpd,-r 0 D,-, =d*p’.

The dual relationships are summarized in the diagram

K’ a , K’-’

D D

(4.3)

(4.4)

KL 4 a Kd*-r+,


5. FOURIER TRANSFORMATION ON ABELIAN GROUPS

In this section we collect a few elementary results from harmonic analysis on an abelian group 6 [ 151. In the applications, 0 will be the group of r-cochains K’(G) which is isomorphic to G x . . . x G (N, times).

Let f be a real or complex valued function on 6, 8 3 g t-+ f (g) E R, C; we write f( g + g,) E f go( g) with arbitrary but fixed g, E G. For the groups considered here, there exists an invariant measure &(g) = &( g + go). The invariant integral off over 6 will be denoted by

A character of 8 is a complex-valued function 1: 0 -+ C, with the properties

x+(g)Xw = 1 (t: complex conjugation), (5.1)

x(g + g’> =x(g)xW)*

Obviously, x is a one-dimensional unitary representation of 8. We set

where

x(g) E &g) 3 .&J .y,

&: 8 + R/2n

gti&(g)=(3 .g.

Because of (5.1), we have

6. (g+g’)=&.g+h.g) (mod 271).

With the definition

(cj+&‘).g=h*g+&‘.g,

(5.2)

the &‘s form an abelian group O*. The invariant integral on 0” will be denoted by C *. Examples are given in Table I.

Functions f on 6 from a suitable space can be expanded in a Fourier series (or integral),

f(g) = I f*(G) exp(ih - g), v* j (4) = x f (g) exp(-i& . g). * w R

We shall also employ the invariant &functions,

x*exp[i&.(g-g’)]=a(g-g’), w

C exp[-i(& - &‘) . g] 3 6(& - &‘), K


TABLE I

SOME ABELIAN GROUPS, THEIR DUAL CHARACTER GROUPS,

AND INVARIANT SUMS

Note. For the definition of w . g see Eq. (5.2)

which obey

Cf(d 4&T-g’) =f(g’)v

&)S(&&‘)=h(d’). iI

Note that 6(O) (6*(O)) is infinite if 6 (6*) is noncompact. If (s is taken to be the group of r-cochains K’(G) N {(q’(si),..., (p’(s>))}, the

characters of K’(G) are N,-fold products of characters of G,

XW) = @fiI x,(cpW)) = exp [i 2 f$ . (I’]. #El

We set G* 3 CAL 3 Or@!) and extend ~6’ via linearity to a G*-valued cochain on K,, b, E K’(G*). We write

g1 &yS:) - q’(sf!) = &’ * q’. (5.3)

Instead of evaluating expression (5.3) in a cell basis we may insert a a*-basis. Using (2.1 l), we then find

NT ;r.,r= c ;‘v;) * [cc:, $1 ~‘(~,“)I

*=I

(5.4)


In applications, terms of the form &’ . d9’- * arise. Using can be rewritten as

(2.4) and (2.9), these terms

= x 2 uY(sf!). [[s~:s~~,J9’-‘(sf-,)] p=l u=l

N,-I = 2 a(a*s;-,)~ rprp’(s;-,),

L’= 1

or

&i’ . d9’-’ = de&‘. 9r-‘. (5.5)

Since dtp-’ = d&’ = (dp)> and d*cif = d*cci: = (d,&);l-’ (see (3.6) and (3.7)), we find, with (5.5)

(jr . &f- ’ = ;; . dq;- ’ = (d, ;);- ’ . 9;- ‘.

The transformation from a cell- to a a*-basis is unimodular. Hence

and

\‘ o’EYw,

exp(icj’ . 9’) = fi 6*(G’(s:)) bl=l

= &(c$ = 6;(G) &(ci) &CA).

It is well known that Fourier expansions can also be performed for class functions on non-abelian groups. However, in this case the characters do not form a group. This is the main reason why duality transformations as described here cannot be applied to models with non-abelian symmetry [ 161.

6. LATTICE MODELS

In this section we introduce the class of abelian lattice models on which the subse- quent discussion of duality transformations is based. The models are characterized by their Hamiltonians or actions. A Hamiltonian is an observable, i.e., a map associating a real number with every physical state. The general definition of an observable F is


ST: K’(G) --t R,

S(qq = sT[fp’(s:),..., yl’(sF)].

This includes the special class of local observables,

The Fourier transform of an observable reads

y(f) = C* $(c5’) exp(i&’ . q’), l&r

(6.1)

where the sum on b’ runs over K’(G*) and .@+(A) =3(-k). In the notation of (5.2), we interprete exp[i&(s) . o(s)] as the value of the variable

exp[i&(s)] on the state C+A The collection

{ V;,(s:)} = {exp[i&‘(s;)] 1~1 = l,..., N,}, (6.2)

constitutes a set of basic local variables from which observables can be constructed according to (6.1). Examples are the Z,-spins S,(s, m) = exp[i(2n/N) n(s) m(s)], n, m = o,..., N - 1, and the spins of the XY-model Sn(s, 6) = exp(in(s) 6(s)], n E H, 6 E [0, 27r]. Usually one takes n = 1 (“fundamental representation”) in both cases. In general, according to its definition, V;(s) is a character of G in the irreducible representation specified by &.

The generic form of our model Hamiltonians is

(6.3)

with qr-’ E K’- l(G), w’-* E K’-*(G). This type of Hamiltonian includes a number of lattice models of current interest [5, 71 which differ in the specific form of &‘,, p;-, and in the choice of the symmetry group G. For instance, if we take r = 2 (<d) and G = Z,, then (6.3) reads in conventional notation

(6.4)

where A, = exp[irtp’(l)] = f 1 and s, = exp[iaw”(x)] = fl denote b,-variables attached to bonds and sites, respectively. L, and M, are coupling coefficients; the


sum on p is over elementary plaquettes of the lattice. Equation (6.4) is easily seen to be of the form (6.3) since

and similarly for the second term in (6.4). The Hamiltonian (6.4) describes a gauge invariant (see below) Ising model, with

the spins S, coupled “minimally” to gauge field variables A, [ 17, IS]. In particle physics, models like (6.4) are called Higgs-type models.

For r = 1, G = Z,, (6.3) denotes the usual Ising model with external magnetic field.

Because of dd = 0. the Hamiltonian (6.3) with r > 2 is invariant under the substitution

w r-2 ‘l/Y r-2 2 -Er- ,

v, r-1 (6.5)

+v, r- 1 + d&‘- ‘,

where srP2 E K’-‘(G) is arbitrary. Equation (6.5) is a local gauge transformation. In example (6.4), (6.5) corresponds to

with t, = exp[ins’(x)] = fl. In the gauge c = w, .P in (6.3) takes the form

-;i” = Pr(dqo) + T’;- ,(a,).

For reasons which become obvious when we consider duality transformations, we modify models (6.3) by introducing a special kind of disorder through

-x,=~~(dyl+p)+7;_,(dli/+cp), (6.6)

with arbitrary but fixed p’E K’(G). In example (6.4), p has the effect L, + L, exp[iw(p)] = fL,. Since p(p) is arbitrary, the coupling coefficient now varies randomly over the lattice.

The partition function for models (6.6) on a finite complex is

S@;K,G;Sr,;r;p,)

=Q-’ x y exp[~(da,+p)+‘~;-,(dW+yl)l. o’-l *‘-~I

(6.7)

The temperature is absorbed in %r and ?‘;- r. Q is a normalization factor. Because of gauge invariance, the sum on y/‘-’ in (6.7) is redundant. Hence the obvious choice


for Q is Q = C, 1 =8;-‘(O). This eliminates the possibly infinite constant &-‘(0) from %. Therefore

In the special case of pure gauge models with Y,-i = 0 we employ decomposition (3%

According to (3.6), p: is exact, i.e., it can always be written as pi = d$-’ with some r- ‘, and can thus be absorbed in I&-‘.

g Since the Hamiltonian in (6.7), with

= 0 (and the sum on w dropped), depends on cp only via (Pi, the sums on oA and &$e redundant and we set Q = J>(O) C&O). Thus,

(6.8)

where the configurational sum is over B'-' (G), and & =pL + p;l. Equation (6.8) is free of possibly infinite constants due to pure gauge degrees of freedom.

p; describes the relevant disorder which cannot be gauged away. Its topological part, p;, arises from the r-dimensional holes in the lattice. p; is related to the concept of frustration [ 11, 181: (r + 1)-cells on which dp; # 0 are called frustrated. For instance, if r = 1 and p;(l) # 0 for the single bond I, then the plaquettes in a*1 are frustrated; dpj,(a*l) # 0.

Finally, let us consider correlation functions. The thermal average of an observable jr is given by

In the case FT,-, = 0, the configurational sum is restricted to B'-'(G). If we insert the Fourier transform (6.1), we see that thermal averaging reduces to the computation of

(6.9)

By suitable choices of ci’ we generate correlation functions for basic local variables (6.2). The “generating functional” r, in (6.9) is normalized such that r,(O, p) = Z,@). As an example, we set r = q - 1 and take (iq-’ = d, iq, with some iq E Kq(G*). (We shall see that for models with %’ = 0, r,- ,((;, p) is nonvanishing only if &q-’ is a codifferential.)


To specify iq, we assume it to take the constant value g* E G* on all cells st in supp iq, I. E I,; see (3.8). We then fmd

(6.10)

where c,(Z) s CIElt si. For q = 1, this denotes an n-point correlation function, n being the number of

lattice sites in a~,(?). If q = 2, (6.10) is the expectation value of a Wegner-Wilson loop variable [5,6, 171.

7. DUALITY TRANSFORMATIONS

We start with the generating functional

and consider the cases 7 ;-, # 0 and Y’;-, = 0 separately.

(a) Higgs Models (?‘;-, # 0)

The duality transformation proceeds in two steps,

1. Step. The local weight factors are Fourier transformed.

exp[W(s>>l = x * exp[L$(&(s)) - i&(s) . a(s)], 05(S)

expl. 4OiWl = x * expl, %.(v+)) + ids) . v-(s)], G(S)

(7.1)

(7.2)

with .D = l,..., N, and v = l,..., N,- , . The weight factors exp[2i;(s)] and exp 1, <,&)I are assumed to be positive definite functions on G. Then, by Bochner’s theorem 1 19 I. Y@ and (4, are real functions on G. In general they contain constant (& and I,& independent) terms. With (7.2) inserted into (7.1) the sum on cp’-’ yields a factor 6;-‘(~5 + II, + d,&). The sum on I+P’ then results in

(7.31

2. Step. Expression (7.3) has a structure rather similar to (7.1), except that d.+ instead of d appears in (7.3). However, by passing over to the dual complex, d, is transformed into d according to (4.4). Therefore we make the replacements


a -+ (x-’ E K”*-‘(G), d* (jr + d&d,-‘,

pr+pd*-r E K”*-‘(G),

u *r--l+(jd--r+l EKd,-r+](G).

Since D: s,tt s,*_, is 9,-l ++ q-r+l.

a one-one mapping we also have $r t+ gd-,, and

Thus, expression (7.3) becomes

rr(dv P) = c * exp[“tj&-,+ ,(d&, + 6,) + F&&-,(&) + ip* . CA*], *d-r W*

or, more explicitly,

This is already the final result of the duality transformation. Notice the interchange in the roles of ?/ and F-, and of p and 8. Special cases:

(i) p= O=ci. We obtain

B(K, G; 22,, T-J = B(K*, G*; F’--r+, , c&,) = 3-&+ 1.

In general, Zr and BJer+i denote partition functions of different systems. For Higgs- type models, necessary conditions for self-duality are K = K*, G = G*, and 2r=d+ 1.

(ii) ci = 0. From (7.1) it follows that r,.(O, p) = Z@) is the partition function of a disordered model. Equation (7.4) yields

which expresses ZP@) in terms of correlation-functions in the dual model.

(b) Gauge Models (ST,- 1 = 0)

We use decompositions (3.5), (5.4), and dqf-’ = d r-1 r-’ in (7.1) with Y-r =0 results in &-l(&) 6”

;1-‘. Then the summation on a)A ,V)H h--‘(d). This implies by (3.7) that Tr(d, p) is nonvanishing only if ci’-’ = d* i’. Therefore,

T,(d, z’, p) = c exp[F&:(y, + p) + id* 5’ . v’-‘I. (7.6) r-1 QB

After insertion of the Fourier transformation (7.2), the sum on &’ gives


&-‘(dmi+d&)=&(i+ *) h o , w ere the equality holds since A”‘(G*) and @-‘(G*) are isomorphic. We find

r,(d, i, p) = C* exp[S&+ - ?) + i&z . pz - if . p 1 ;‘- z

= exp(--i . p) I* 2 * exp [gr(d, 6 + &* - 5) fir;’ I .4 &;i

+ 4 1 dp + 6, . pH]. (7.7 1

We used A”‘+’ ‘v B”’ by writing &Is = de $+I. The transition to the dual complex proceeds as before. In addition, we only have to remember that under D one has x,+ , H BzmrP, and fi, H HTmr, cf. (4.3). As a result we obtain

~h&i,p)exp(ii .P)

zz y * fdur(d*p*, Lj,,, - iy;) exp(ipA, . 3,.). (7.8) &drr H

This duality relation differs in two essential aspects from (7.4). First, the topology of the lattice enters explicitly via homology. Secondly, a phase factor exp(ii . p) appears, which will be seen to be intimately connected with gauge invariance.

A discussion of the effects on nontrivial homology is deferred to the next section. Here we assume zr=O for the rest of this section Then Eq. (7.8) gives in particular

Jr@, = f’&d*p*, 0). (7.9)

In contrast to the Higgs case (7.5!, P, in (7.9) only depends on dp; see also (7.7). For p = 0, (7.9) reduces to Zr = &,, and a necessary condition for self-duality is now 2r = d.

From (7.6) it follows that Tr(d* i, p) changes under the gauge transformation

p’ -+ pr + dir- ‘, (7.10)

according to

T,(d, i, p + dC) = exp(--it . d<) T,(d, i, p).

However, the product

I’,.(d, 5, p) exp(ii . p) = f,(f. p), (7.11)

is invariant under (7.10) and generates gauge invariant correlation functions. Note that this holds also in the case 7t, # 0, since pH is not affected by (7.10). For r, = 0, r,. depends on p only via dp and (7.8) can be put into the symmetric form

(7.12)

248 DRijHLANDWAGNER

Under the transformation i’ -P i’ + d* Pi, F,, changes as

I;,(? + d* E*, dp) = exp(i2 . dp) r,(f, dp). (7.13)

The phase factors appearing in (7.11) and (7.13) are interpreted as follows: We take i’, p’, and gr+’ to be constant, with values i,, gP, and d, on the cells of their supports. With the notation used in (6.10), we then have

This has a group-theoretical part and a geometrical part, For a Z,-model, for instance, the group-theoretical part B, . g, = 27rn, m,/N, with n,, mp E { l,..., N - 1).

The scalar product of the chains counts (with appropriate signs) the number of inter- sections of c,(f) with the dual chain c$-,@,). Similarly,

& *r+l - W = CC,+ ,G>, &CAP)) B, - g,

is a measure for the frustration content in supp Zr+ ’ [ 111. The duality relations (7.8) and (7.12) provide a compactly written generalization

to arbitrary abelian symmetry groups and lattice dimensions of the Kadanoff-Ceva concept [ 10,201 of order-disorder variables. The extension of (7.12) to SU(N)-gauge models, due to t’Hooft [9,21] plays an important role in the discussion of quark con- finement.

8. EFFECTS OF NONTRIVIAL HOMOLOGY

We noted earlier that nontrivial homology arises if there are holes in the lattice. Actually, there are different types of holes, namely, “local” ones due to omission of cells as in Figs. 3 and 4, and “global” ones introduced by identifying cells as in the case of periodic (toroidal) boundary conditions.

Homology specifies the features of topological disorder which influence duality transformations. However, it is remarkable that nontrivial homology only affects the duality relation for models with F’/^,-i = 0.

If we set p = 0 = i in (7.8), we find, with q = d - I,

(8.1)

Consider for a moment an arbitrary complex. Any configuration 9; on this complex may be evaluated in a cell basis,

Nq s4”= 2 (s;,qJc;, K = l,..., N,,

V=l


by using (3.4) and (3.5),

= 2 qqhj;)(s,“, Fi,y). (8.2) y= I

When we apply (8.2) for ~$7,~ in (Sl), we see that homology affects the interaction in the Hamiltonian on those cells which overlap with some 6fy E I?,*.

As an example of a “local hole,” we take the two-dimensional complex in Fig. 8 with Hamiltonian PI, where an internal site, its cobounding bonds and their coboundary (hatched plaquettes) are (by definition) not in (K, a). In this case n, = 1, and we choose for h (E h’) and i (E 6’) the heavy solid and dashed bonds, respectively. In the dual complex, 6* = Ci=, Z*k and the chain of heavy dashed bonds constitutes h* = D(K). The sites 2, j = 1 ,..., 4, and the hatched plaquette are not in (K*, a). Thus &,,*(s*) = &*(h*)(s*, 6*) = &,(/I*), if S = I*k, k = I,..., 8; and t

WH’ = 0, otherwise. The terms in %‘, affected by the hole in K* are

+ 2k[$*(ar*k) + 6*(h”)l kz-l

= + Gk[?j*(-jk) + &*(r;)l. k=l

Since the sites 9 do not belong to (I?, a), we have al*k = -j”. However, we may extend K* to a K* by including the sites 3 and defining

rj*(S) 5% CA(h), j = l,.... 4. (8.3)

Then fi*(-i,“) + G*(h) = {,($ -3”) = +.+(Jf*“) = ~&.+(f*~), where 2 and d denote extensions of CY and d to K* and K,(G*), respectively. We conclude that for this example, (8.1) can be rewritten as

x i,(&,*,= \‘ exP[~,Z:($h)l I\ 6(k++‘) - i&i-9), I U/I. 6 *E~(G*I * j-2

whereby the constraint derives from (8.3).

FIG. 8. Complex with a hole and its dual. For further description see text.

250 DRtiHL AND WAGNER

FIG. 9. Complex with periodic boundary conditions. Identification of cells is made such that embedding in R3 yields a two-dimensional torus. Orientations are not shown, being irrelevant for the L,- Ising model.

There are cases where nontrivial homology is irrelevant in the context of duality, even for models with Fr-i = 0. Take, for instance, an Ising model X = -Z!i(&) on a two-dimensional lattice. If we omit one plaquette (2-cell) from the lattice complex, the Hamiltonian and the partition function are obviously unaffected. On the other hand, duality tran:forms plaquettes into sites (O-cells) and the missing dual site clearly matters in Z?i. However, it is straightforward to show that the sum over 6”’ exactly substitutes for the missing site, and 3, = 8, is regained from (8.1) in this case.

We now turn to “global holes” as exemplified by periodic boundary conditions. After pairwise identification of opposite edges of the square lattice in Fig. 9 we create two classes of nonbounding cycles represented by h’ and h*. The associated noncobounding cocycles 6’ and /? are also depicted. Hence H, = {h ‘, h* } and rcl = 2. (Periodic boundary conditions for a three-dimensional lattice induce also a nontrivial ,Zhomology with 743) = 3 = 71,(3).) Obviously we have K = K*. The terms in pi modified by the sum over configurations &(h’) and &(h*) involve the interactions on the bonds in R’ and /i?. In the special case of a ferromagnetic Z,-Ising model, the dual coupling constants in &, for these bonds are multiplied by exp(i&), nf = 0, 1. The sum on 3, in (8.1) then results in

%(2YJ = Jqz?J + a@;l”) + 8(22;‘1”) + _z@y’);

Z?i” has antiferromagnetic coupling on the bonds in ff, and &;i’*” has these on the bonds in I;’ as well as in /?.

9. BOUNDARY CONDITIONS

In our preceding treatment of duality transformations we have assumed that the underlying complex is finite. Apart from mathematical complications which would occur otherwise, this assumption is clearly necessary if we want the Hamiltonians and the partition functions to be well defined. The complete description of a model on a finite complex (K, a) requires that we also specify boundary conditions on the surface of K. In Section 7 we tacitly assumed “free” boundary conditions which


means that in the generating functional we sum over states on K without constraints. We expect, of course, that boundary conditions become irrelevant for bulk model properties in the thermodynamic limit. However, there are topical problems in the study of lattice models, where boundary conditions play a crucial role. As an example we mention the interface roughening transition in the context of phase coex- istence. In this final section we show how boundary conditions, imposed on configurations, may be taken into account in duality transformations [ 9, 12 1.

We start with a finite complex (K, a) and decompose K with respect to a cell basis in the following way:

K=I?@R, (9.1)

where each Z?‘, is generated by a subset {sf } of cells in K, such that R is “ii*-closed.”

a*k,dt,,,. (9.2)

R = K - I? denotes the complement in K of i. It is called the outer boundary (in K) of l?; see Fig. 10. From (9.1) and (9.2) we infer that R is a-closed,

aR,cR,-,. (9.3 1

Indeed, since l? is a*-closed, we have for any pair of cells s ̂in I? and t in R, (a*$ t) = 0 = (s ,̂ at). Thus at is in R.

Equations (9.2) and (9.3) imply that R and R are a*- and a-subcomplexes of (K, a), respectively. The outer boundary R defines the “surface” of K.

Any configuration rp’ E K’(G) can be written in the form

qf = (j’ on 8,

= rr (9.4)

on R.

6’ and r’ may be extended to configurations on K by setting

qi?‘=O on R,

f = 0 on I?. (9.5)

FIG. 10. (a) k + R. R consists of the open points and the dashed bonds; remaining cells are in K. (b) I?* + R*. Full points, solid bonds and hatched plaquettes constitute d*.


We also define the restriction to & of a,

&! 3x [sf : sp-,] sp-,. (9.6) i

Boundary conditions are now introduced by taking t’ to be a fixed arbitrary configuration.

For the Hamiltonian we assume

iqql) = c Y[fp@f)]. L;

Without loss of generality we have restricted the sum over cells to those in x. Since r is fixed and R is a-closed, this choice only affects the constant term in @j. The subse- quent arguments can easily be extended to cover models (6.3) with F:-, # 0.

We use (9.4), (9.5), and (9.6) to write @r as

iq(p) = 2 QqQ’-‘(c%f!) + f-y&$] i

= 2 iP[&‘-‘(sf) + d7’-‘(sf)]. r;

In the second line we introduced the restriction to R’(G) of d denoted ty d Note that dr’-‘(sf!) does no necessarily vanish, since Z? is not a-closed. There are cells in 2 which have part of their boundary in R.

The partition function with the boundary condition cp’- ’ = r’- ’ on R is given by

Since c exp[@r(&? + dz)] = %“(Z?, G; P,) = 2;“. ;<- I B

dr’-‘(St) = c [SF : t;-,] ~~-‘(t;-,) = dp(sf) +(sf), a

2: corresponds to the partition function of a disordered model with the Hamiltonian -g(d$ + p^).

Therefore, the duality relations of Section 7 can be taken over immediately. In particular, for n, = 0, Eq. (7.9) yields

2%: = 2$(p) = ~,*_,(d*p^*, 0). (9.7)

By (4.2) it follows that k* and R* are a- and a*-closed, respectively. R* contains the infinitely extended cells of K*, which necessarily arise in the duality transformation of finite complexes. However, R * does not appear in (9.7). Hence, by (9.7) the partition function with configurational boundary conditions is related to correlation functions for the dual model on R* with free boundary conditions. Both K and k* are physically reasonable lattices.


The above discussion can straightforwardly be extended to correlation functions in the presence of boundary conditions. This is left as an exercise to the interested reader.

ACKNOWLEDGMENTS

We acknowledge useful discussions with D. Kroll and R. Lipowsky

REFERENCES

1. H. A. KRAMERS AND G. H. WANNIER, Phys. Rev. 60 (1941). 252. 2. L. ONSAGER, see G. H. Wannier, Rev. Mod. Phys. 17 (1945), 50. 3. For a review on duality results for two-dimensional models see I. Syozi, in “Phase Transformations

and Critical Phenomena” (C. Domb AND M. S. Green, Eds.). Vol. I, Academic Press, New York/London, 1972.

4. H. P. MCKEAN, JR., J. Math. Phys. 5 (1964) 775. 5. F. J. WEGNER. J. Math. Phys. 12 (1971), 2259. 6. K. G. WILSON, Phys. Rev. D 10 (1974), 2445. 7. For a recent review and an extensive list of references see R. Savit. Rev. Mod. Phvs. 52 (1980), 453:

A. B. Zamolodchikov, Zh. Eksp. Tear. Fiz. 75 (1978). 1083 [Sor. Phys. JEW (Eng. Transl.) 48 (1978), 5461.

8. J. M. DROUFFE, Phys. Rev. D 18 (1978), 1174. 9. A. UKAWA, P. WINDEY, AND A. H. GUTH, Phys. Rev. D 21 (1980). 1013.

10. L. P. KADANOFF AND H. CEVA, Phvs. Rev. B 3 (1971). 3918. 11. E. FRADKIN, B. A. HUBERMAN, AND S. H. SHENKER. Phvs. Rev. B 18 (1978). 4789. 12. A. GUTH, Phys. Rev. D 21 (1980). 229 1. 13. Yu. M. ZINOV’EV, Tear. Mat. Fir. 43 (1980), 309. 14. P. ALEXANDROFF, “Combinatorial Topology,” Vol. 2, Graylock, Rochester. N. Y., 1956. 15. L. H. LOOMIS. “An Introduction to Abstract Harmonic Analysis.” Van Nostrand, Toronto. 1953. 16. Efforts to extend duality to non-abelian models are described by R. Savit in Reference 17 1, 17. R. BALIAN, J. M. DROUFFE, AND C. ITZYKSON. Phys. Rev. D I1 (1975). 2098. 18. G. TOULOUSE. Comm. Phys. 2 (1977), 115. 19. M. A. NAIMARK, “Normed Rings,” Noordhoff, Groningen, 1959. 20. E. FRADKIN AND L. P. KADANOFF. Nucl. Phys. I3 17 (FSlj (1980), 1. 21. T’HOOFT, Nucl. Phvs. B 138(1978). 1.

algebraic formulation of duality transformations for abelian lattice models

Documents