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Ising fermions (II). Three dimensions
Nuclear Physics B210[FS6] (1982) 477-498 © North-Holland Publishing Company
ISING FERMIONS (II). Three dimensions C. ITZYKSON
CEN-Saclav, 91191 Gif-sur-Yvette, Cedex, France Received 21 May 1982
We present a simple derivation of a grassmannian integral for the 3-dimensional Ising model partition function and study some elementary consequences.
1. Introduction This is the second part of a discussion of fermionic representations for Ising-like systems. We have previously reviewed the two-dimensional case with emphasis on the relation between the transfer matrix (operator) formalism and the use of integrals over anticommuting variables. Here we concentrate on the three-dimensional Ising model with spin variables, + 1, interacting by neighboring pairs. Its dual is the Z 2 three-dimensional gauge model with similar variables assigned to links and interacting on elementary plaquettes. For future convenience we write the partition functions 1
exp(
e NFI(~) ,
~
°site
Z ~ ( B ) =2- ~1
Y'~ exp ( /3 Olink
E
ooo~ ) = eNFcdB)"
(1.1)
plaquettes
Here N is the total number of sites on the cubic lattice and a thermodynamic limit N ~ oo is understood. The suffixes I and G refer to the Ising and gauge models. When/3 and/~ are related by duality ~
t=tanhfl=e
28,
(1.2)
the free energies obey F o ( f l ) = ~ In sinh2fl - ½ln2 + F~ (/~). Observables of either model can be translated into the other one's language. 477
(1.3)
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Data on the three-dimensional model are numerous. Its second-order transition is well understood [1]. Series analysis and renormalization group calculations have given precise values for the critical exponents. We quote from a recent work [2]* a value of t c = 0.642 on a simple cubic lattice as the one used here, with a specific heat singularity in It - tcl-~, a -- 0.11. It has been suggested that the 3-dimensional Ising model is related to random surfaces as is the 2-dimensional one to random paths. An elegant model for these surfaces has been presented by Polyakov [3]. On the other hand Samuel [4] and Fradkin, Srednicki and Susskind [5] have derived a grassmannian representation some time ago, while recently Casher, Foerster and Windey have analyzed the corresponding equations of motion [6]. The low-temperature expansion of the Ising model, identical to the high-temperature one of the gauge model, introduces surfaces or Bloch walls which separate plus from minus spins, and extend to three dimensions the ideas of Peierls contours familiar in the two-dimensional case. The expansion is originally for the partition function and one has to appeal to a thermodynamic argument to extract the free energy or any correlation function by retaining only terms proportional to the volume. In this way disconnected surfaces play a role due to geometrical obstructions. The fermionic formalism enables one to perform a step similar to that in two dimensions, i.e. to obtain a precise set of rules directly for the free energy in terms of suitably weighted connected surfaces. This weight includes a sign and a symmetry factor to be discussed below. The purpose of this paper is simply to extend the techniques of part I [7], to give a very simple derivation of the grassmannian integral and to draw some conclusions. We will prove the expansion of the free energy in terms of connected closed surfaces and obtain their weights. As a by product we find a property which has perhaps already been proved, and certainly noticed, to the effect that the internal energy per link has a low-temperature expansion with integer coefficients, or similarly for the high-temperature expansion of the average plaquette in the dual model. We are able to compare to a very modest order our series with the results of Domb [1] and (fortunately) find agreement. For completeness, our discussion starts with the transfer matrix formalism and the corresponding field equations (sect. 2). In sect. 3 we obtain the grassmannian integral using a decoupling trick which enables us to use results from I. We also write an alternative expression which might be useful for a high-temperature study. In sect. 4 we examine the expansion of the free energy and we enumerate the first contributions in sect. 5. The final section summarizes our conclusions and speculations.
* Further references on the Ising transition are to be found in this paper.
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2. Transfer matrix Let us discuss a three-dimensional Ising model enclosed in a periodic box with sides l x, ly, l z. We call/3 the coupling between neighboring spins. We introduce Pauli matrices o (1~, o (2), o (3) at each site with the eigenvalues of o (~ considered as the original spin variables. At fixed z we write a transfer matrix T in such a way that (2s)N/2ZN([~)=
T r U~.
(2.1)
T h e total n u m b e r of sites N equals the product lxl.J z, with all lengths assumed to tend to infinity. As in I, the transfer matrix T can be factored into a product T = 0/J, 0 = exp(/3
~
~'x~(')=(')]"xi_/''
t~ = exp( fl ~20x1(3)) .
neighbors
(2.2)
sites
Here x± corresponds to a point in the ( x y ) plane. In this plane for each site s with coordinates x± let us set os = Ox(3] and for each link of neighboring points ( x ± x ~ ) let Ol= O(l±)O(xl). W e have a m o n g these operators the following identities: Os = o, = l ,
[ o , , o,,1 =
[o,, o,,1 = o,
(2.4)
ora l = e,lol%,
es/= 1,
(2.3)
if l is not incident on s, - 1 if l is incident on s.
Incident clearly means that s is one of the end points of the link. To characterize the model completely we have to add the fact that the product of o~'s around any plaquette p is unity: 17 °l = 1.
(2.5)
/cp
As in I a fermionisation of the model is achieved if we can express the operators as and o t in terms of Clifford variables in such a way that (2.3) and (2.4) become identities. For this purpose we assume a set of such variables (F) labelled by a pair consisting of a site s and a link L of the dual lattice in such a way that s belongs to a plaquette of the dual lattice incident on L. Equivalently L is dual to a link l incident on s (fig. 1). The geometrical duality relations refer here to the two-dimensional plane (xy) on which the o ' s are defined. The above construction requires, therefore, a Clifford algebra of 41xly variables while on account of (2.5) the n u m b e r of independent o ' s is
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480
i
I
:
L
i
s
t
i
Fig. 1. A neighboring pair (sL) as a label for the Clifford variables F.
21xly. At this point we cannot expect to be able to express the F ' s in terms of o ' s without further restrictions in contradistinction with the situation in the two-dimensional model. Nevertheless with l and L dual to each other, we set
o/= -iI-[Fsl, s
(2.6)
as=I"IFsL. L
Thus o/is quadratic in the F ' s while os is quartic. A specific ordering is chosen on the right-hand sides in (2.6). One can easily check that (2.3) and (2.4) become identities. This representation reveals naturally that the Ising and gauge models are dual to each other. However, because condition (2.5) is not yet satisfied our F ' s would have a representation in a vector space of dimension 2 number of V's/2 221,l, instead of 2/`/` which is the dimension of the vector space in which the o ' s operate. If S on the dual lattice is conjugate to the plaquette p let us introduce the operators =
Os = I-I o/= 1-I FsL, lop
s,L
L incident on S,
(2.7)
which involve eight F ' s (fig. 2). They commute with all o / a n d o, in the form (2.6) and have a square equal to unity. We can require a restriction to the subspace spanned by their + 1 eigenvectors (for an appropriate ordering of the F's),
Os = 1,
(2.8)
O
Fig. 2. The product of eight F ' s to build p.
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thereby reducing the dimensionality of the vector space to 2ix6, as required. In the gauge version (2.8) is the analog of the C o u l o m b gauge condition to characterize the neutral sector. Once (2.8) is enforced we have in operator form a fermionization of the Ising model with the transfer matrix expressed as a product of exponentials of quadratic and quartic form symbolized in fig. 3. Physical observables c o m m u t e with all the Ps, a condition which means that through each point S of the dual lattice are attached an even n u m b e r of F operators representing the observable. N o t e that we do not attempt to solve for the F ' s in terms of o ' s (generalized Jordan-Wigner transformation) because we do not need such an expression. But this could be done with a large arbitrariness. For instance an individual of I) could be thought of as a limit of a product of%f) ) when s' recesses to infinity. Such a product can be written as a p r o d u c t of o(I)o °) for neighboring spins along an arbitrary path joining s to s ' and therefore expressed as a string of F ' s using (2.6). Having a transcription of the transfer matrix in terms of fermionic variables we can obtain field equations for the F ' s for instance. These include linear and cubic terms. Turning the crank as we did in I we can also derive a path integral where care has to be taken to enforce the remaining gauge invariance. As we shall proceed in a more symmetric fashion we shall not pursue this matter further here.
3. Grassmannian integral Let us use as a starting point the gauge model on the dual lattice with coupling ft. This was, incidentally, the reason for inverting fl and D in the notations of sect. 1. With t defined as in (1.2) we therefore rewrite
Z G ( f l ) = [½coshfl] 3N Y'. I - i ( 1 + t o o o o ) . 0£= +1 P
(3.1)
We observe that on each plaquette p two pairs of link variables interact. For each pair the links are parallel to one of the coordinate axes. It is convenient for some time to index links according to their direction, thus o~, 02, 03 refer to link variables parallel to the x, y, z axes, respectively. For instance a plaquette parallel to the ( x y ) plane will contribute a factor (1 + tOlO~O2o~) to (3.1). Next, for each plaquette p we
°xtx Fig. 3. Symbolic representation of the quadratic and quartic forms occurring in the transfer matrix.
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482
introduce a new variable (l +
7p = + 1
and notice that
tOlO~020~)= ½ E
(1 +
tl/2ypOlO~)(1 + tl/2yp020~).
(3.2)
yp=±l This splitting is only one among three possible choices and we could also have partitioned the factor t differently. Our choice is to respect the permutation symmetry among the three directions x, y, z. For shortness we denote the average over the ~p's as ( )p. Consequently,
p
In (3.3) the product involves terms referring to link variables parallel to the same coordinate axis, belonging to the same plaquette with an effective coupling tl/2yp. One realizes that, for fixed yp'S, we have decoupled the 3-dimensional gauge model into three families of independent 2-dimensional Ising models, with couplings y p t 1/2 for each interaction. Each family pertains to one direction of coordinate planes (fig. 4). A grassmannian integral representation is already available for these 2-dimensional models [7]. This representation contains four variables per link. These grassmannian variables are naturally in correspondence with the F-matrices introduced in the operator formalism of sect. 2. In this formulation the symmetry between all directions is, however, respected, so that we have four such variables for any link no matter whether it lies in a "space" or "time" direction. We recall that the lattice here is the one pertaining to the gauge model. Fig. 5 represents these four variables. Each variable points from the link to one of the four plaquettes to which it belongs. The notation for a z-type link is also shown in fig. 5 and is in agreement with the corresponding one in I. From I we find that ZG(fl) = (cosh fl)3u( f o b [ (~]exp(S, +
tl/2S~})p.
× Fig. 4. A member of one of the three families of 2-dimensional Ising models.
(3.4)
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Fig. 5. Four grassmannian variables per link.
Here S~ is a sum over link contributions S, = I2 SL. links
(3.5)
In turn Sz, for a z-link, for instance, is expressed as a sum of six quadratic combinations
SL=~3,I~g3,,-t-~3,2~3,2 q-~3,2~3,1 q-~3,2~3,1 q- ~3,1~3,2 q- ~3,2~3,1 "
(3.6)
For an x or y link, S L is given by the appropriate cyclic permutation of the indices. The quantity S~ in the action is a sum over plaquette contributions, S~=
Y'~ S~. plaquettes
(3.7)
For a specific plaquette, parallel to the xy plane say, with the notations of fig. 6, S; = "/p(~1,2~1,2 "}" ~2 ~'2,,),
(3.s)
and similar expressions for other plaquettes with corresponding permutations of indices. The average over the extra variables yp is now easily performed. A given pair ~(' commutes with any other and has a vanishing square. Consequently (exp{t'/2yp[~,
--
l
. 2 ~ ., 2 q .- ~ 2.
1~
,])
p=
-
!
-
t
1 q-t~l . 2~1 . 2~2 . 1~2 . I
= exp{ t~,.2(;.2(2.,~i., }.
(3.9)
Thus the final expression of the partition function has an action built out of a quadratic part corresponding to links and a quartic part corresponding to plaquettes as expected from sect. 2 [4, 5]. It reads Z o ( ~ ) = (cosh ~)3uf0> [g~]exp(S, + tS2),
(3.10)
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I l --X----
I i
~'I,2
Fig. 6. The variablesin Sp. with S t as above given by (3.5) and (3.6) and S2=Y'~S p,
(3.11)
P
Sp=(/ ,j ( 'i,j ( j,i~j,i" ~'
(3.12)
The notation in (3.12) was, of course, for a plaquette parallel to the (i, j ) plane. The average over the ,{p'S has recoupled the 2-dimensional integrals and reintroduced the variable t. The anticommuting variables ~ are to be thought as products of orderdisorder variables in both the Ising and the gauge model. The presence in (3.10) of a quartic piece in the action means that a closed form solution can hardly be expected. To recognize the usual low-temperature expansion (t << 1) of the partition function notice that for a z link, for instance, exp S L = (1 + S L + ~3,1~3,2~3, 1(3,2 )' and for an
(3.13)
xy plaquette, for instance, exp tSp = 1 + t~'l,Z~,t~2,1~l, 2.
(3.14)
To obtain the required low-temperature expansion, insert these expressions in the integral (3.10) for every term in the action and collect the coefficients of the product of all ~'s and ~ in a definite order. We can start from the quartic contribution from the links. This gives the leading term 1 in Z(fl)/(cosh fl)au. Then we introduce the plaquettes terms perturbatively. Careful considerations [4, 5] show that one retrieves the usual small-t expansion involving closed surfaces. Each term in this expansion appears several times with a +_ sign, but their sum conspires to give a unique term with a positive combinatorial factor arising from the various positions these surfaces can occupy on the three-dimensional lattice.
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On the other hand it is far less obvious how the usual high-temperature expansion (1 - t << 1) for the partition function arises from (3.10). To derive it we need an alternative expression for the grassmannian integral, a sort of dual form, which does not seem obtainable by a simple change of integration variables. The steps involved a m o u n t (i) first to return from (3.10) to (3.3) and (ii) then to use a grassmannian representation for each two-dimensional model in its dual form. For shortness we will leave the details to the reader. The result is the following. The conversion factor from Zo(/3) to ZI(/~ ) appears naturally and we get
2
U(cosh )'Nfo)[ nlexp(Z, +
(3.15)
There are 12 variables attached to each site of the Ising lattice. Through each vertex of the latter we have four variables per lattice direction as indicated in fig. 7 for a z direction and again the labelling is in accordance with I. The two terms St and N2 in (3.15) are reminiscent of S t and S 2 in (3.10). St involves, at each site, and for each direction, six terms like S t. For a given site and a given direction (say the z direction) ~1 = ~3,1T13,1 q-~3,2~3,2-}-~3,2T]3,1 + ~3,2T13.1 q- ~3,1T13,2 q- ~3,2T13,1 "
(3.16)
However ~2 is of slightly different structure than S 2 having both quadratic and quartic terms. It involves four variables pertaining to neighboring sites and two directions orthogonal to the link joining these sites, the remaining indices being relative to the link direction as shown in fig. 8. The corresponding term in N2 is =
B3,1 ,7'3 , 1 + ~ 2 , 1 n'2 , 1 ) + (
, 1 - t - 2 -)~3. J~3, - 1B2,1~2,1-
(3.17)
Notice that for a high-temperature expansion the small parameter is
(3.18)
7 = tanh/~, and that the exponential of a given contribution in ~2 is e x p ~ 2 = 1 + t(~3,1 7/'3, i mr- ~2, ~ 31~2, .t 11)- q--
Tit3, 1~2, 112,1.
J
'q 3,2
73,1
~3,2
Fig. 7. Four of the 12 variables r~ attached to a site.
(3.19)
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C. Itzykson / Ising fermions (II)
_
T(2,1
_
~3,1
/
U
Fig. 8. The four variables involvedin a typical term of -~2-
Insertion of these terms in (3.15) will allow an expansion in powers of t which reduces to the usual high-temperature expansion of the Ising model in terms of closed paths. The structure of (3.19) is such that at the level of the partition function we can ignore links described more than once. However, the details are hard to work out. Comparison of the two integrals obtained for the partition function suggests why it is simpler, from a "field theoretic" point of view, to pursue perturbation theory in terms of t (closed surfaces, low temperature) rather than in terms of t (closed paths, high temperature). In the former case the " u n p e r t u r b e d " form is quadratic and the perturbation quartic while in the latter we have to perturb an already quartic action. But no doubt it would be worthwhile to investigate the small 7 expansion more closely. We therefore return now to the low-temperature expansion and its diagrammatic rules.
4. The free energy in terms of closed surfaces
Since S 1 in (3.10) is quadratic and t independent, it can be used as a free action, an integral over a free lagrangian. We therefore only expand e x p t S 2= ~
tA ),4 ~. ( S z
(4.1)
A=O and find perturbative rules even though the " p r o p a g a t o r " actually only shifts the indices of link variables while it is the interaction which displaces these variables in real space by one unit spacing at a time. The Feynman rules are the familiar ones in such a way that Fermi statistics is implemented by sums of diagrams but not generally by individual ones. In particular, antisymmetrization can relate diagrams with different connectivity. When dealing with anticommuting variables we have indeed the choice to write, for instance, (expt,X~,~2~3~3) = (w(1 + t~,~2~2~3)),
(4.2)
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which makes full use of the statistics but for which exponentiation is not obvious. Or we may compute the same mean value using (4.1), i.e.
(exp tZ~,~2~3~3) =
~
~.. (X~1~2~3~4)
.
(4.3)
A=0
In this second form Wick's theorem takes care of antisymmetrization and exponentiation of connected diagrams follows from standard arguments. It is clear, however, when comparing (4.2) and (4.3) that the added terms have to cancel. Trivial enough as this point may be, it is at the basis of the successful treatment of anticommuting integrals. Consequently, the perturbative rules yield the free energy as a sum over connected diagrams: 1 F ~ ( f l ) = lim ~ In N---~oo
f(r)
1 = ~ •
Zc,(fl ) =
31ncoshfl
+f(fl),
connected vacuum diagrams.
(4.4)
(4.5)
In the sequel we shall refer to f ( f l ) as the free energy. Let us set S l = - ½ Y', ~,,K~,b~¢,,, ,
(4.6)
a,b
where a is a composite index which includes the location of a link on the lattice and the type of link variable. It runs, therefore, over 12N values. The matrix K is antisymmetric and invariant under translations and cyclic permutations of the axis. For a specific link location it is identical to the corresponding one described in I and denoted there by - K 0. From (4.7)
o =
it follows in the free theory that
=
(4.8)
We recall from I that the antisymmetric matrices K and K - l are block diagonal, in four by four blocks (per link), have unit determinant and matrix elements equal to 0,+1. Wick's theorem states ....
)o=
E pairings
(4.9)
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with the sum running over distinct pairings ((2n)!/2"n! in number), the sign being that of the corresponding permutation. We can also write S 2 in condensed form as 1
(4.10)
• abcd
with Vabcd totally antisymmetric, equal to 0, +_ 1 and non-vanishing when abcd refer to four link variables around a plaquette. The diagrams arise from
Z ~ ( B ) = (cosh3B) 3N ~ tA vo,~c,~... E A=o A!(4!) A ~, ...dA
V~,A~,~A{~,...~ ) (4.11)
upon application of (4.9). The abstract structure of unlabelled vertices and connecting lines corresponding to a diagram from (4.11) will be referred to as a graph. Here we are concerned with vacuum connected diagrams to compute In Z. The corresponding graphs are simple, connected with four lines (or edges) incident on each vertex. The qualification "simple" arises from the antisymmetric structure of V and K - 1 in such a way that no line emerging from a vertex returns to the same one (no tadpole) and two vertices cannot be connected by more than one line as depicted in fig. 9. From the concrete expressions for S~ and S2 we find that a given graph prescribes the connectivity relations between a set of plaquettes, some of which can possibly share the same location on the lattice, and their common edges (or links) on the lattice. To be precise let us call such a set closed connected surface if the following requirements are satisfied: (i) The corresponding graph is connected, or equivalently one can join two plaquettes of the surface by a continuous path across plaquettes sharing common links• (ii) Two plaquettes share at most a link and when they do, they do not occupy the same location on the lattice. Nevertheless, a closed surface can (and generally will) have some of its plaquettes occupy the same location on the lattice. (iii) Any plaquette of the set has no free links, each of its four links being identified with links from another plaquette of the set. It is important to distinguish the abstract graph with unlabeled vertices and edges from its concrete realizations on the lattice which correspond to the computation of the corresponding terms according to the Feynman rules. In such a concrete realization prescriptions (i), (ii), (iii) amount to identifying links and their end vertices for connected neighboring plaquettes. The final result is thus a well-defined connected surface (two-dimensional complex) with faces (plaquettes) links and
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Fig. 9. Forbidden graphs.
vertices and a corresponding Euler characteristic X = plaquettes - links + vertices = 2 - 2 H , H = genus of the surface ( n u m b e r of handles).
(4.12)
To be sure the graph contains part of the information, and in fact is a piece of the dual complex of the surface. Missing information is that pertaining to the vertices of the surface or equivalently the faces of the dual. This would follow if we could draw the propagators with double lines to follow the end points of the corresponding link variables. We would then have analogs of the graphs of the large-N matrix field theories which carry complete topological information [8]. We shall return to this point in sect. 6. A n y closed connected surface on the lattice will contribute to the free energy a quantity which is the p r o d u c t of three factors: (a) t A, A -= area = n u m b e r of plaquettes; (b) 1 / k , k a symmetry factor and, as such, a positive integer; (c) e, a sign. To obtain (a), (b), (c) we have to decipher (4.11) given the values for K and V given in (3.5), (3.6), (3.11) and (3.12). The factor t A from (a) is obvious. Let us discuss (b) and (c). F r o m the original integral (3.10) it is first clear that a term in t A arising from the coupling of distinct plaquettes will contribute _+tA; thus k = 1 in this case. We have therefore to look at circumstances where a given location is occupied several times by the plaquettes of a surface. Such a connected surface will arise, in general,
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from several sets of contractions in
/
P]'--~'P~'---~"" " Pr"---~/ ' nl! n2! n~! 0
where each Pi stands for the product of the four corresponding ~'s and some n~ are larger than one. The situation is the same as if we had assigned different couplings to distinct plaquette contributions in the action S 2. For each i (i = 1.... , r) let us introduce an extra index ranging from 1 to n i in order to distinguish the different plaquettes at the same location. We want to classify the Fl~nfi permutations of these A = E~ni labels. Call this group of permutations 6). We start from a given surface corresponding to a set of contractions in the above average and label the vertices of the corresponding graph p~)...p}"r). Let ~rl,2 ~ 6) correspond to one and the same set of contractions; then ~r~ ~r2 is a permutation of the labels of the graph corresponding to the original set of contractions. In other words, let G c 6) be the subgroup of order k of those permutations of the labels which leave invariant one set of contractions corresponding to a given surface (little group); then 6) can be subdivided into fight equivalence classes which are the cosets 6 ) / G all with the same number of elements n i ! . . . n r ! / k . Consequently, the symmetry factor k is the order of G, made of those permutations which correspond to a unique term in Wick's theorem. The crucial property of G is the following. Let ~r ~ G be different from the identity. Let us subdivide each set of p, plaquettes into the disjoint union of the smallest possible invariant non-empty subsets. According to the definition of G each plaquette in such a subset is connected to four other plaquettes belonging to four other invariant subsets in such a way that two connected pairs are interchanged with their link or edge. If we identify the plaquettes in the subsets and the edges connecting equivalent plaquettes, we therefore obtain a contracted, connected simple graph of the same category as the original one but with fewer vertices and edges. The important fact here is that this reduced graph is connected. Indeed two original vertices v and v' could be joined by a string of edges, and if ~r ~ G interchanges two vertices it also permutes the nearest neighbors to which they were joined. Step by step we see that the subsets to which v and v' belong have to be connected. It is also clear that two distinct 7r's, each one different from the identity, will lead to the same partition which therefore establishes an equivalence relation on the original graph between vertices and edges. Finally, each set contains the same number of elements and this number is the order k of G. Indeed if w and qr' both map P~(~) on /'1~/3) (a ~/3), ~r' ~Tr leaves Ps(~) invariant and therefore contradicts the assumption that we had a smallest possible invariant subset. In short the closed surface is a k-covering of a simpler one obtained by identifying equivalent plaquettes and their connections. The latter has obviously a symmetry factor one. It may be remarked that k divides nl, n 2 n r and therefore their sum A = ~r= 1n~. .....
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The above discussion justifies calling k the periodicity of the imbedded closed surface. Perusal of D o m b ' s results [1] collected in table 2 enables one to identify this periodicity factor as ½ to order 12 (double covering of a cube), ½ to order 18 (triple covering of a cube), ½ to order 20 (double covering of the union of two cubes with an intermediate plaquette omitted) and so on. Finally we come to point (c), i.e. the question of sign. This sign arises from a combination of the signs in (4.9) as well as those arising from the propagators K 1 and vertices V. Stated in this way it would seem hopeless to keep track of them. Fortunately we can use our derivation of the integral representation as well as known properties of the two-dimensional model. For this purpose we use the intermediate representation (3.4) and recall that graphs in the two-dimensional model correspond to closed curves with sign equal to their number of self-intersections. Thus we cut any closed surface by three families of planes parallel to the coordinate planes at half-integer ordinates, say. In each of these planes the intersection is a set of closed curves each one with a factor ( - 1) n, where n is its number of self-intersections. The overall sign e is the product of all the signatures in the three families of planes. We call it ( - 1) L, where L is the total length of self-intersection. As remarked by Fradkin, Srednicki and Susskind, ( - 1 ) c is not a topological invariant, prima facie, and awaits a clever interpretation, part of which may be contained in the paper by Casher, Foerster and Windey [6]. The final outcome of our rather cumbersome discussion is therefore the following. The free energy Nf(~) is a sum over all possible connected closed surfaces in the precise sense introduced above, each one weighted by a factor ( - 1)t'tA/k: 1
(--1)Lt A surfaces
A total area, k periodicity, L length of self-intersection.
(4.13)
An interesting consequence arises from the fact that the periodicity k divides the area A. Consider in the gauge model the average plaquette denoted U= (n).
(4.14)
From (1.1) and (4.4) it is given by 1 d
U = ~ 5-BF~;(/~) = t + (1 taR
tz)~f'(t).
(4.15)
In turn lf'(t) is the average of four ~'s around a plaquette and can be symbolically represented as a decorated plaquette
>.
(4.16)
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Then we find that U and ½f'(t) are both series in t with integer coefficients. This can either be established directly by an analysis of the contributions of surfaces bounded by the plaquette, or by differentiation from (4.13). Indeed for those closed surfaces which are not invariant in a permutation of the coordinate axis the number of realizations is a multiple of three. For those which are invariant their area A and even A/k, i.e. the area of the corresponding non-periodic closed surface, is also a multiple of three. The stated property follows. We can translate this result in terms of the Ising model free energy
1± r u= -3 3D
(4.17)
This can be thought as the average of a link Coo') denoted symbolically ~.---.). Then u = (.---.) = ~ [([3) - cotanhZfl] cl/s = 1 - 2t < ~ ]
>.
(4.18) ~
Thus, also the low-temperature expansion of u in powers of t = e 2~ involves only integer coefficients. More general averages can be studied. For instance the connected correlation function C(1,2) =<~-r~, Frq2> _ < ~ , q , > <
FA2>
(4.19)
referring to two distinct plaquettes p~ and P2 will be computed in terms of connected surfaces bounded by p] and P2. We can also consider a generalisation of Wilson's average by taking a product of ~'s along a closed loop C with 4 Icl extra degrees of freedom corresponding to the individual ~ 's. Such an average will again involve connected surfaces bounded by the curve C with obvious constraints in the manner in which they are incident on C. The relation of such quantities for large C with the Wilson average is not entirely clear. Using a different language, even though we expect these quantities to decrease exponentially with the area enclosed by C for t < to, the coefficient of this area law does not seem directly related to the surface tension as defined conventionally.
5. E n u m e r a t i o n
of closed surfaces
It must be admitted that the procedure to generate the low-temperature series for the free energy in terms of closed surfaces is not the most efficient one. Large cancellations occur due to the signs ( - 1 ) L. A side effect of these cancellations is indeed the absolute convergence of the series around the origin in the complex t
C. lt~vkson / Isingfermions (1I)
493
plane. A very poor upper bound on the coefficients indicates a radius of absolute convergence larger than ½ [9] and, as we saw in the introduction, t c is almost twice as large (but, of course, closer singularities in the complex t plane may exist, see below). Nevertheless it is comforting to check that the enumeration according to (4.13) agrees with known results, as they should of course. In table 1 we summarize such an enumeration up to A = 16, a very modest order indeed. To increase the table would certainly only expose the weaknesses of the present author by increasing the probability of mistakes without adding much to the proof of consistency. Table 1 contains the abstract connectivity graph and shows its realizations on the lattice. It also includes the sign ( - 1 ) L multiplying the reduced number of realizations r.n.r. (i.e. number of realizations per site), its periodicity and topological characteristic X of the surface. To show the topological parallel with the graph the latter is drawn on a torus when its corresponding genus is one, the highest value encountered up to order sixteen. The details for certain surfaces, the contributions of which cancel, have been omitted. Up to A ~< 16 only periodicity 2 is encountered and as we just said X >/0. But to order 18 we already expect surfaces with genus 2 (X = - 2 ) , period 3. Due to the possibility of using a plaquette location several times, non-orientable surfaces also occur at higher orders (at least already at order 22). The meager results from table 1 can be compared with the impressive series quoted by D o m b [1] giving the series f o r f ( t ) up to order 40 and reproduced in table 2. Table 3 giving the decorated plaquette ½f'(t) is simply obtained from the previous one and simply illustrates the property about integer coefficients. An intriguing feature is noticeable in these tables. They suggest that the series for f(t) alternate in sign with the coefficients of powers 0 mod 4 negative and those of powers 2 mod 4 positive. The fact that only even powers of t contribute is on the other hand obvious. We do not find any simple proof of this fact, nor do we know whether it persists to higher order, which is very likely. If so, on the radius of convergence of the series some singularities necessarily lie on the imaginary t axis!
6. Conclusion
The fermionization methods developed at length in the specific examples of the 2and 3-dimensional models can, of course, be extended to other cases, like the 4-dimensional gauge or Ising model. More interestingly in 2 dimensions we can also derive such representations for vertex models. It remains to translate the Bethe ansatz into a path integral language in 2 dimensions. Returning to the 3-dimensional model we do not know, of course, any algorithm which would enable one to sum the set of vacuum diagrams in closed form. Short of this one can speculate on the asymptotics of the perturbation series. The connected surfaces appearing in this expansion are hardly constrained. They can have arbitrary Euler characteristics, may or may not be orientable. In this sense they are essentially
C. ltzykson- / Ising fermions ( H)
494
TABLE l Closed surfaces up to A = 16
Order
Graph
6
~
_
Realization
~
r.n.r,
+ 1
Period. k
1
Charact. X
Contrib.
2
t6
Total
t6
,o@
1
3,,0 3t ]o
12
[ ~ ) Eance|
. ~ ~ ~
.
~ ~
- 12
1
2
- 12t 12
-4
2
2
- 2 t h2
+6
1
2
6t 12
+2
2
2
t 12
+3
1
0
3t 12
+1
2
0
i 12 ~t
--3E2t12
495
C. Itzykson / Ising fermions (II) TABLE 1,(continued) Order
Graph
Realization
r.n.r,
Period. k
Charact. X
Contrib.
Totol
14
(
/ ~ ' ~
+3
1
2
3t TM
+ 12
l
2
12t 14 15t ~4
16
~
Cancel
/
I U ~ ~
~ ~ ~
~
(
~
L20
1
2
- 120fl 6
- 24
1
2
24t 16
+ 60
1
2
60t 16
+ 12
1
2
12t 16
+30
1
0
30t t6
+6
1
0
6t 16
1
2
3t ~6
+3
-33fl 6
*The two cubes are next to each other; identified links are shown. **Similar drawing, the two cubes are one into the other. tThe double and single cube are next to each other or one on top of the other. tiThe cube is inside the double cube.
C. Itzykson / Isingfermions ( II)
496
TABLE 2 Series f o r f ( t ) in even powers of t from D o m b [1] Order
Coefficient
6 8 l0
1 0 3
12 14 16
- 3½ 15 33
18
104½
20 22
- 280½ 849
24 26
- 2461~ 7485
28
- 22534½
30
693935
32 34
213754½ 666750
36 38
2086 734~ 6583341
40
- 20 8523634
[
I
TABLE 3 Series in odd powers of t for the decorated p l a q u e t t e ~3f'(t) = ~ [ ] Order
Coefficient
5 7 9 II 13 15 17 19 21 23 25 27 29 31 33 35 37 39
2 0 10 -14 70 -176 626 -1870 6226 - 12 694 64 870 - 210322 693 932 2280048 7556 500 -25040810 83 388986 -278031510
C. Itzvkson / lsingfermions (II)
497
random and the anticommuting integral, cumbersome as it is to evaluate, may be considered as a generating functional. It would be interesting to know if the topological character is coded in their connectivity graph as would be suggested by our low order study. We have mentioned that this graph is only a piece of the dual of the 2-dimensional complex attached to the quadrangulation of the surface. What are missing are the faces of the graph corresponding to the vertices of the complex. It is known that any graph admits a genus which is the smallest one of an orientable surface on which it can be drawn with its edges crossing only at its vertices. Poor bounds can be obtained for this genus but in the words of the author of a recent text book on graph theory, R.J. Wilson, "little is known about the problem of finding the genus of an arbitrary graph" [10]. So much more work is still required to extract useful information from this series and understand the embedding of surfaces in 3-space. The sign question is also intriguing, in view of the comments of ref. [6] where the authors would like to relate it to reference frames on the surface since the expansion contains non-orientable surfaces. By regrouping the terms in the series one could try to mimic in discrete form Polyakov's conjecture [3], possibly valid in the critical region since large classes of graphs correspond to imbedding the same abstract surface in real space. This still looks like a formidable task. We have also encountered a mysterious alternation of sign in the free energy. Intuitively this might be related to the fact that for /~ imaginary, or t 2 < 0, the partition function Zo(/3 ) expressed as in (1.1) is obviously still real and of modulus smaller than one, so that F~ is negative under such circumstances. If true the corresponding singularity on the negative t 2 axis has to be clarified. Let us make a very exotic suggestion. We have seen that one can introduce intermediate variables "fp = + 1 in such a way as to decouple three families of non-interacting Ising models for fixed "fp. If we write the average over yp as
"E yp =
~ 1
Yp =
~ I
with A to be set equal to one, we can develop a perturbation theory in A, the starting point of which is an expression which has the critical properties of the two-dimensional model. Investigation of this series might be of some interest. Finally we recall that we have obtained in sect. 3 an alternative integral adapted to the study of high-temperature series. It too might be worth working out. The Ising models remain fascinating and challenging. Discussions with P. Windey and J.B. Zuber are gratefully acknowledged.
498
C. ltzykson / lsing fermions ( H )
References [1] C. Domb, in Phase transitions and critical phenomena, ed. C. Domb and M.C. Green, vol. 3 (Academic Press, New York); D.S. Gaunt, ibid.; A.J. Guttmann, ibid. [2] J. Zinn-Justin, J. de Phys. 40 (1979) 969 [3] A. Polyakov, Phys. Lett. 103B (1981) 207, 211 [4] S. Samuel, J. Math. Phys. 21 (1980) 2806, 2815, 2820 [5] E. Fradkin, M. Srednicki and L. Susskind, Phys. Rev. D21 (1980) 2885 [6] A. Casher, D. Foerster and P. Windey, CERN preprint TH 3200 (1981) [7] C. Itzykson, Nucl. Phys. B210[FS6] (1982) 448 [8] E. Brbzin, C. Itzykson, G. Parisi and J.B. Zuber, Comm. Math. Phys. 59 (1978) 35 [9] R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D l l (1975) 2098 [10] R.J. Wilson, Introduction to graph theory (Longman, London, 1979)
ISING FERMIONS (II). Three dimensions C. ITZYKSON
CEN-Saclav, 91191 Gif-sur-Yvette, Cedex, France Received 21 May 1982
We present a simple derivation of a grassmannian integral for the 3-dimensional Ising model partition function and study some elementary consequences.
1. Introduction This is the second part of a discussion of fermionic representations for Ising-like systems. We have previously reviewed the two-dimensional case with emphasis on the relation between the transfer matrix (operator) formalism and the use of integrals over anticommuting variables. Here we concentrate on the three-dimensional Ising model with spin variables, + 1, interacting by neighboring pairs. Its dual is the Z 2 three-dimensional gauge model with similar variables assigned to links and interacting on elementary plaquettes. For future convenience we write the partition functions 1
exp(
e NFI(~) ,
~
°site
Z ~ ( B ) =2- ~1
Y'~ exp ( /3 Olink
E
ooo~ ) = eNFcdB)"
(1.1)
plaquettes
Here N is the total number of sites on the cubic lattice and a thermodynamic limit N ~ oo is understood. The suffixes I and G refer to the Ising and gauge models. When/3 and/~ are related by duality ~
t=tanhfl=e
28,
(1.2)
the free energies obey F o ( f l ) = ~ In sinh2fl - ½ln2 + F~ (/~). Observables of either model can be translated into the other one's language. 477
(1.3)
478
C. ltzykson / Isingfermions ( H)
Data on the three-dimensional model are numerous. Its second-order transition is well understood [1]. Series analysis and renormalization group calculations have given precise values for the critical exponents. We quote from a recent work [2]* a value of t c = 0.642 on a simple cubic lattice as the one used here, with a specific heat singularity in It - tcl-~, a -- 0.11. It has been suggested that the 3-dimensional Ising model is related to random surfaces as is the 2-dimensional one to random paths. An elegant model for these surfaces has been presented by Polyakov [3]. On the other hand Samuel [4] and Fradkin, Srednicki and Susskind [5] have derived a grassmannian representation some time ago, while recently Casher, Foerster and Windey have analyzed the corresponding equations of motion [6]. The low-temperature expansion of the Ising model, identical to the high-temperature one of the gauge model, introduces surfaces or Bloch walls which separate plus from minus spins, and extend to three dimensions the ideas of Peierls contours familiar in the two-dimensional case. The expansion is originally for the partition function and one has to appeal to a thermodynamic argument to extract the free energy or any correlation function by retaining only terms proportional to the volume. In this way disconnected surfaces play a role due to geometrical obstructions. The fermionic formalism enables one to perform a step similar to that in two dimensions, i.e. to obtain a precise set of rules directly for the free energy in terms of suitably weighted connected surfaces. This weight includes a sign and a symmetry factor to be discussed below. The purpose of this paper is simply to extend the techniques of part I [7], to give a very simple derivation of the grassmannian integral and to draw some conclusions. We will prove the expansion of the free energy in terms of connected closed surfaces and obtain their weights. As a by product we find a property which has perhaps already been proved, and certainly noticed, to the effect that the internal energy per link has a low-temperature expansion with integer coefficients, or similarly for the high-temperature expansion of the average plaquette in the dual model. We are able to compare to a very modest order our series with the results of Domb [1] and (fortunately) find agreement. For completeness, our discussion starts with the transfer matrix formalism and the corresponding field equations (sect. 2). In sect. 3 we obtain the grassmannian integral using a decoupling trick which enables us to use results from I. We also write an alternative expression which might be useful for a high-temperature study. In sect. 4 we examine the expansion of the free energy and we enumerate the first contributions in sect. 5. The final section summarizes our conclusions and speculations.
* Further references on the Ising transition are to be found in this paper.
C. ltzykson / lsingfermions ( II)
479
2. Transfer matrix Let us discuss a three-dimensional Ising model enclosed in a periodic box with sides l x, ly, l z. We call/3 the coupling between neighboring spins. We introduce Pauli matrices o (1~, o (2), o (3) at each site with the eigenvalues of o (~ considered as the original spin variables. At fixed z we write a transfer matrix T in such a way that (2s)N/2ZN([~)=
T r U~.
(2.1)
T h e total n u m b e r of sites N equals the product lxl.J z, with all lengths assumed to tend to infinity. As in I, the transfer matrix T can be factored into a product T = 0/J, 0 = exp(/3
~
~'x~(')=(')]"xi_/''
t~ = exp( fl ~20x1(3)) .
neighbors
(2.2)
sites
Here x± corresponds to a point in the ( x y ) plane. In this plane for each site s with coordinates x± let us set os = Ox(3] and for each link of neighboring points ( x ± x ~ ) let Ol= O(l±)O(xl). W e have a m o n g these operators the following identities: Os = o, = l ,
[ o , , o,,1 =
[o,, o,,1 = o,
(2.4)
ora l = e,lol%,
es/= 1,
(2.3)
if l is not incident on s, - 1 if l is incident on s.
Incident clearly means that s is one of the end points of the link. To characterize the model completely we have to add the fact that the product of o~'s around any plaquette p is unity: 17 °l = 1.
(2.5)
/cp
As in I a fermionisation of the model is achieved if we can express the operators as and o t in terms of Clifford variables in such a way that (2.3) and (2.4) become identities. For this purpose we assume a set of such variables (F) labelled by a pair consisting of a site s and a link L of the dual lattice in such a way that s belongs to a plaquette of the dual lattice incident on L. Equivalently L is dual to a link l incident on s (fig. 1). The geometrical duality relations refer here to the two-dimensional plane (xy) on which the o ' s are defined. The above construction requires, therefore, a Clifford algebra of 41xly variables while on account of (2.5) the n u m b e r of independent o ' s is
C. ltzykso n / Ising fermions (II)
480
i
I
:
L
i
s
t
i
Fig. 1. A neighboring pair (sL) as a label for the Clifford variables F.
21xly. At this point we cannot expect to be able to express the F ' s in terms of o ' s without further restrictions in contradistinction with the situation in the two-dimensional model. Nevertheless with l and L dual to each other, we set
o/= -iI-[Fsl, s
(2.6)
as=I"IFsL. L
Thus o/is quadratic in the F ' s while os is quartic. A specific ordering is chosen on the right-hand sides in (2.6). One can easily check that (2.3) and (2.4) become identities. This representation reveals naturally that the Ising and gauge models are dual to each other. However, because condition (2.5) is not yet satisfied our F ' s would have a representation in a vector space of dimension 2 number of V's/2 221,l, instead of 2/`/` which is the dimension of the vector space in which the o ' s operate. If S on the dual lattice is conjugate to the plaquette p let us introduce the operators =
Os = I-I o/= 1-I FsL, lop
s,L
L incident on S,
(2.7)
which involve eight F ' s (fig. 2). They commute with all o / a n d o, in the form (2.6) and have a square equal to unity. We can require a restriction to the subspace spanned by their + 1 eigenvectors (for an appropriate ordering of the F's),
Os = 1,
(2.8)
O
Fig. 2. The product of eight F ' s to build p.
c. Itzykson / Isingfermions ( II)
481
thereby reducing the dimensionality of the vector space to 2ix6, as required. In the gauge version (2.8) is the analog of the C o u l o m b gauge condition to characterize the neutral sector. Once (2.8) is enforced we have in operator form a fermionization of the Ising model with the transfer matrix expressed as a product of exponentials of quadratic and quartic form symbolized in fig. 3. Physical observables c o m m u t e with all the Ps, a condition which means that through each point S of the dual lattice are attached an even n u m b e r of F operators representing the observable. N o t e that we do not attempt to solve for the F ' s in terms of o ' s (generalized Jordan-Wigner transformation) because we do not need such an expression. But this could be done with a large arbitrariness. For instance an individual of I) could be thought of as a limit of a product of%f) ) when s' recesses to infinity. Such a product can be written as a p r o d u c t of o(I)o °) for neighboring spins along an arbitrary path joining s to s ' and therefore expressed as a string of F ' s using (2.6). Having a transcription of the transfer matrix in terms of fermionic variables we can obtain field equations for the F ' s for instance. These include linear and cubic terms. Turning the crank as we did in I we can also derive a path integral where care has to be taken to enforce the remaining gauge invariance. As we shall proceed in a more symmetric fashion we shall not pursue this matter further here.
3. Grassmannian integral Let us use as a starting point the gauge model on the dual lattice with coupling ft. This was, incidentally, the reason for inverting fl and D in the notations of sect. 1. With t defined as in (1.2) we therefore rewrite
Z G ( f l ) = [½coshfl] 3N Y'. I - i ( 1 + t o o o o ) . 0£= +1 P
(3.1)
We observe that on each plaquette p two pairs of link variables interact. For each pair the links are parallel to one of the coordinate axes. It is convenient for some time to index links according to their direction, thus o~, 02, 03 refer to link variables parallel to the x, y, z axes, respectively. For instance a plaquette parallel to the ( x y ) plane will contribute a factor (1 + tOlO~O2o~) to (3.1). Next, for each plaquette p we
°xtx Fig. 3. Symbolic representation of the quadratic and quartic forms occurring in the transfer matrix.
C. Itzykson / lsingferrnions(II)
482
introduce a new variable (l +
7p = + 1
and notice that
tOlO~020~)= ½ E
(1 +
tl/2ypOlO~)(1 + tl/2yp020~).
(3.2)
yp=±l This splitting is only one among three possible choices and we could also have partitioned the factor t differently. Our choice is to respect the permutation symmetry among the three directions x, y, z. For shortness we denote the average over the ~p's as ( )p. Consequently,
p
In (3.3) the product involves terms referring to link variables parallel to the same coordinate axis, belonging to the same plaquette with an effective coupling tl/2yp. One realizes that, for fixed yp'S, we have decoupled the 3-dimensional gauge model into three families of independent 2-dimensional Ising models, with couplings y p t 1/2 for each interaction. Each family pertains to one direction of coordinate planes (fig. 4). A grassmannian integral representation is already available for these 2-dimensional models [7]. This representation contains four variables per link. These grassmannian variables are naturally in correspondence with the F-matrices introduced in the operator formalism of sect. 2. In this formulation the symmetry between all directions is, however, respected, so that we have four such variables for any link no matter whether it lies in a "space" or "time" direction. We recall that the lattice here is the one pertaining to the gauge model. Fig. 5 represents these four variables. Each variable points from the link to one of the four plaquettes to which it belongs. The notation for a z-type link is also shown in fig. 5 and is in agreement with the corresponding one in I. From I we find that ZG(fl) = (cosh fl)3u( f o b [ (~]exp(S, +
tl/2S~})p.
× Fig. 4. A member of one of the three families of 2-dimensional Ising models.
(3.4)
C. Itzykson / lsingfermions (II)
483
Fig. 5. Four grassmannian variables per link.
Here S~ is a sum over link contributions S, = I2 SL. links
(3.5)
In turn Sz, for a z-link, for instance, is expressed as a sum of six quadratic combinations
SL=~3,I~g3,,-t-~3,2~3,2 q-~3,2~3,1 q-~3,2~3,1 q- ~3,1~3,2 q- ~3,2~3,1 "
(3.6)
For an x or y link, S L is given by the appropriate cyclic permutation of the indices. The quantity S~ in the action is a sum over plaquette contributions, S~=
Y'~ S~. plaquettes
(3.7)
For a specific plaquette, parallel to the xy plane say, with the notations of fig. 6, S; = "/p(~1,2~1,2 "}" ~2 ~'2,,),
(3.s)
and similar expressions for other plaquettes with corresponding permutations of indices. The average over the extra variables yp is now easily performed. A given pair ~(' commutes with any other and has a vanishing square. Consequently (exp{t'/2yp[~,
--
l
. 2 ~ ., 2 q .- ~ 2.
1~
,])
p=
-
!
-
t
1 q-t~l . 2~1 . 2~2 . 1~2 . I
= exp{ t~,.2(;.2(2.,~i., }.
(3.9)
Thus the final expression of the partition function has an action built out of a quadratic part corresponding to links and a quartic part corresponding to plaquettes as expected from sect. 2 [4, 5]. It reads Z o ( ~ ) = (cosh ~)3uf0> [g~]exp(S, + tS2),
(3.10)
484
C. ltzykson / Ising ferrnions (II)
I l --X----
I i
~'I,2
Fig. 6. The variablesin Sp. with S t as above given by (3.5) and (3.6) and S2=Y'~S p,
(3.11)
P
Sp=(/ ,j ( 'i,j ( j,i~j,i" ~'
(3.12)
The notation in (3.12) was, of course, for a plaquette parallel to the (i, j ) plane. The average over the ,{p'S has recoupled the 2-dimensional integrals and reintroduced the variable t. The anticommuting variables ~ are to be thought as products of orderdisorder variables in both the Ising and the gauge model. The presence in (3.10) of a quartic piece in the action means that a closed form solution can hardly be expected. To recognize the usual low-temperature expansion (t << 1) of the partition function notice that for a z link, for instance, exp S L = (1 + S L + ~3,1~3,2~3, 1(3,2 )' and for an
(3.13)
xy plaquette, for instance, exp tSp = 1 + t~'l,Z~,t~2,1~l, 2.
(3.14)
To obtain the required low-temperature expansion, insert these expressions in the integral (3.10) for every term in the action and collect the coefficients of the product of all ~'s and ~ in a definite order. We can start from the quartic contribution from the links. This gives the leading term 1 in Z(fl)/(cosh fl)au. Then we introduce the plaquettes terms perturbatively. Careful considerations [4, 5] show that one retrieves the usual small-t expansion involving closed surfaces. Each term in this expansion appears several times with a +_ sign, but their sum conspires to give a unique term with a positive combinatorial factor arising from the various positions these surfaces can occupy on the three-dimensional lattice.
C. ltzvkson / Isingfermions (1I)
485
On the other hand it is far less obvious how the usual high-temperature expansion (1 - t << 1) for the partition function arises from (3.10). To derive it we need an alternative expression for the grassmannian integral, a sort of dual form, which does not seem obtainable by a simple change of integration variables. The steps involved a m o u n t (i) first to return from (3.10) to (3.3) and (ii) then to use a grassmannian representation for each two-dimensional model in its dual form. For shortness we will leave the details to the reader. The result is the following. The conversion factor from Zo(/3) to ZI(/~ ) appears naturally and we get
2
U(cosh )'Nfo)[ nlexp(Z, +
(3.15)
There are 12 variables attached to each site of the Ising lattice. Through each vertex of the latter we have four variables per lattice direction as indicated in fig. 7 for a z direction and again the labelling is in accordance with I. The two terms St and N2 in (3.15) are reminiscent of S t and S 2 in (3.10). St involves, at each site, and for each direction, six terms like S t. For a given site and a given direction (say the z direction) ~1 = ~3,1T13,1 q-~3,2~3,2-}-~3,2T]3,1 + ~3,2T13.1 q- ~3,1T13,2 q- ~3,2T13,1 "
(3.16)
However ~2 is of slightly different structure than S 2 having both quadratic and quartic terms. It involves four variables pertaining to neighboring sites and two directions orthogonal to the link joining these sites, the remaining indices being relative to the link direction as shown in fig. 8. The corresponding term in N2 is =
B3,1 ,7'3 , 1 + ~ 2 , 1 n'2 , 1 ) + (
, 1 - t - 2 -)~3. J~3, - 1B2,1~2,1-
(3.17)
Notice that for a high-temperature expansion the small parameter is
(3.18)
7 = tanh/~, and that the exponential of a given contribution in ~2 is e x p ~ 2 = 1 + t(~3,1 7/'3, i mr- ~2, ~ 31~2, .t 11)- q--
Tit3, 1~2, 112,1.
J
'q 3,2
73,1
~3,2
Fig. 7. Four of the 12 variables r~ attached to a site.
(3.19)
486
C. Itzykson / Ising fermions (II)
_
T(2,1
_
~3,1
/
U
Fig. 8. The four variables involvedin a typical term of -~2-
Insertion of these terms in (3.15) will allow an expansion in powers of t which reduces to the usual high-temperature expansion of the Ising model in terms of closed paths. The structure of (3.19) is such that at the level of the partition function we can ignore links described more than once. However, the details are hard to work out. Comparison of the two integrals obtained for the partition function suggests why it is simpler, from a "field theoretic" point of view, to pursue perturbation theory in terms of t (closed surfaces, low temperature) rather than in terms of t (closed paths, high temperature). In the former case the " u n p e r t u r b e d " form is quadratic and the perturbation quartic while in the latter we have to perturb an already quartic action. But no doubt it would be worthwhile to investigate the small 7 expansion more closely. We therefore return now to the low-temperature expansion and its diagrammatic rules.
4. The free energy in terms of closed surfaces
Since S 1 in (3.10) is quadratic and t independent, it can be used as a free action, an integral over a free lagrangian. We therefore only expand e x p t S 2= ~
tA ),4 ~. ( S z
(4.1)
A=O and find perturbative rules even though the " p r o p a g a t o r " actually only shifts the indices of link variables while it is the interaction which displaces these variables in real space by one unit spacing at a time. The Feynman rules are the familiar ones in such a way that Fermi statistics is implemented by sums of diagrams but not generally by individual ones. In particular, antisymmetrization can relate diagrams with different connectivity. When dealing with anticommuting variables we have indeed the choice to write, for instance, (expt,X~,~2~3~3) = (w(1 + t~,~2~2~3)),
(4.2)
C. Itzykson / Ising fermions ( II)
487
which makes full use of the statistics but for which exponentiation is not obvious. Or we may compute the same mean value using (4.1), i.e.
(exp tZ~,~2~3~3) =
~
~.. (X~1~2~3~4)
.
(4.3)
A=0
In this second form Wick's theorem takes care of antisymmetrization and exponentiation of connected diagrams follows from standard arguments. It is clear, however, when comparing (4.2) and (4.3) that the added terms have to cancel. Trivial enough as this point may be, it is at the basis of the successful treatment of anticommuting integrals. Consequently, the perturbative rules yield the free energy as a sum over connected diagrams: 1 F ~ ( f l ) = lim ~ In N---~oo
f(r)
1 = ~ •
Zc,(fl ) =
31ncoshfl
+f(fl),
connected vacuum diagrams.
(4.4)
(4.5)
In the sequel we shall refer to f ( f l ) as the free energy. Let us set S l = - ½ Y', ~,,K~,b~¢,,, ,
(4.6)
a,b
where a is a composite index which includes the location of a link on the lattice and the type of link variable. It runs, therefore, over 12N values. The matrix K is antisymmetric and invariant under translations and cyclic permutations of the axis. For a specific link location it is identical to the corresponding one described in I and denoted there by - K 0. From (4.7)
o =
it follows in the free theory that
=
(4.8)
We recall from I that the antisymmetric matrices K and K - l are block diagonal, in four by four blocks (per link), have unit determinant and matrix elements equal to 0,+1. Wick's theorem states ....
)o=
E pairings
(4.9)
488
C. ltzvkson / Isingfermions ( I I )
with the sum running over distinct pairings ((2n)!/2"n! in number), the sign being that of the corresponding permutation. We can also write S 2 in condensed form as 1
(4.10)
• abcd
with Vabcd totally antisymmetric, equal to 0, +_ 1 and non-vanishing when abcd refer to four link variables around a plaquette. The diagrams arise from
Z ~ ( B ) = (cosh3B) 3N ~ tA vo,~c,~... E A=o A!(4!) A ~, ...dA
V~,A~,~A{~,...~ ) (4.11)
upon application of (4.9). The abstract structure of unlabelled vertices and connecting lines corresponding to a diagram from (4.11) will be referred to as a graph. Here we are concerned with vacuum connected diagrams to compute In Z. The corresponding graphs are simple, connected with four lines (or edges) incident on each vertex. The qualification "simple" arises from the antisymmetric structure of V and K - 1 in such a way that no line emerging from a vertex returns to the same one (no tadpole) and two vertices cannot be connected by more than one line as depicted in fig. 9. From the concrete expressions for S~ and S2 we find that a given graph prescribes the connectivity relations between a set of plaquettes, some of which can possibly share the same location on the lattice, and their common edges (or links) on the lattice. To be precise let us call such a set closed connected surface if the following requirements are satisfied: (i) The corresponding graph is connected, or equivalently one can join two plaquettes of the surface by a continuous path across plaquettes sharing common links• (ii) Two plaquettes share at most a link and when they do, they do not occupy the same location on the lattice. Nevertheless, a closed surface can (and generally will) have some of its plaquettes occupy the same location on the lattice. (iii) Any plaquette of the set has no free links, each of its four links being identified with links from another plaquette of the set. It is important to distinguish the abstract graph with unlabeled vertices and edges from its concrete realizations on the lattice which correspond to the computation of the corresponding terms according to the Feynman rules. In such a concrete realization prescriptions (i), (ii), (iii) amount to identifying links and their end vertices for connected neighboring plaquettes. The final result is thus a well-defined connected surface (two-dimensional complex) with faces (plaquettes) links and
C. Itzvkson / Isingfermions ( II)
489
Fig. 9. Forbidden graphs.
vertices and a corresponding Euler characteristic X = plaquettes - links + vertices = 2 - 2 H , H = genus of the surface ( n u m b e r of handles).
(4.12)
To be sure the graph contains part of the information, and in fact is a piece of the dual complex of the surface. Missing information is that pertaining to the vertices of the surface or equivalently the faces of the dual. This would follow if we could draw the propagators with double lines to follow the end points of the corresponding link variables. We would then have analogs of the graphs of the large-N matrix field theories which carry complete topological information [8]. We shall return to this point in sect. 6. A n y closed connected surface on the lattice will contribute to the free energy a quantity which is the p r o d u c t of three factors: (a) t A, A -= area = n u m b e r of plaquettes; (b) 1 / k , k a symmetry factor and, as such, a positive integer; (c) e, a sign. To obtain (a), (b), (c) we have to decipher (4.11) given the values for K and V given in (3.5), (3.6), (3.11) and (3.12). The factor t A from (a) is obvious. Let us discuss (b) and (c). F r o m the original integral (3.10) it is first clear that a term in t A arising from the coupling of distinct plaquettes will contribute _+tA; thus k = 1 in this case. We have therefore to look at circumstances where a given location is occupied several times by the plaquettes of a surface. Such a connected surface will arise, in general,
C. ltzykson / Ising fermions ( H)
490
from several sets of contractions in
/
P]'--~'P~'---~"" " Pr"---~/ ' nl! n2! n~! 0
where each Pi stands for the product of the four corresponding ~'s and some n~ are larger than one. The situation is the same as if we had assigned different couplings to distinct plaquette contributions in the action S 2. For each i (i = 1.... , r) let us introduce an extra index ranging from 1 to n i in order to distinguish the different plaquettes at the same location. We want to classify the Fl~nfi permutations of these A = E~ni labels. Call this group of permutations 6). We start from a given surface corresponding to a set of contractions in the above average and label the vertices of the corresponding graph p~)...p}"r). Let ~rl,2 ~ 6) correspond to one and the same set of contractions; then ~r~ ~r2 is a permutation of the labels of the graph corresponding to the original set of contractions. In other words, let G c 6) be the subgroup of order k of those permutations of the labels which leave invariant one set of contractions corresponding to a given surface (little group); then 6) can be subdivided into fight equivalence classes which are the cosets 6 ) / G all with the same number of elements n i ! . . . n r ! / k . Consequently, the symmetry factor k is the order of G, made of those permutations which correspond to a unique term in Wick's theorem. The crucial property of G is the following. Let ~r ~ G be different from the identity. Let us subdivide each set of p, plaquettes into the disjoint union of the smallest possible invariant non-empty subsets. According to the definition of G each plaquette in such a subset is connected to four other plaquettes belonging to four other invariant subsets in such a way that two connected pairs are interchanged with their link or edge. If we identify the plaquettes in the subsets and the edges connecting equivalent plaquettes, we therefore obtain a contracted, connected simple graph of the same category as the original one but with fewer vertices and edges. The important fact here is that this reduced graph is connected. Indeed two original vertices v and v' could be joined by a string of edges, and if ~r ~ G interchanges two vertices it also permutes the nearest neighbors to which they were joined. Step by step we see that the subsets to which v and v' belong have to be connected. It is also clear that two distinct 7r's, each one different from the identity, will lead to the same partition which therefore establishes an equivalence relation on the original graph between vertices and edges. Finally, each set contains the same number of elements and this number is the order k of G. Indeed if w and qr' both map P~(~) on /'1~/3) (a ~/3), ~r' ~Tr leaves Ps(~) invariant and therefore contradicts the assumption that we had a smallest possible invariant subset. In short the closed surface is a k-covering of a simpler one obtained by identifying equivalent plaquettes and their connections. The latter has obviously a symmetry factor one. It may be remarked that k divides nl, n 2 n r and therefore their sum A = ~r= 1n~. .....
C. Itzykson / lsingfermions (II)
491
The above discussion justifies calling k the periodicity of the imbedded closed surface. Perusal of D o m b ' s results [1] collected in table 2 enables one to identify this periodicity factor as ½ to order 12 (double covering of a cube), ½ to order 18 (triple covering of a cube), ½ to order 20 (double covering of the union of two cubes with an intermediate plaquette omitted) and so on. Finally we come to point (c), i.e. the question of sign. This sign arises from a combination of the signs in (4.9) as well as those arising from the propagators K 1 and vertices V. Stated in this way it would seem hopeless to keep track of them. Fortunately we can use our derivation of the integral representation as well as known properties of the two-dimensional model. For this purpose we use the intermediate representation (3.4) and recall that graphs in the two-dimensional model correspond to closed curves with sign equal to their number of self-intersections. Thus we cut any closed surface by three families of planes parallel to the coordinate planes at half-integer ordinates, say. In each of these planes the intersection is a set of closed curves each one with a factor ( - 1) n, where n is its number of self-intersections. The overall sign e is the product of all the signatures in the three families of planes. We call it ( - 1) L, where L is the total length of self-intersection. As remarked by Fradkin, Srednicki and Susskind, ( - 1 ) c is not a topological invariant, prima facie, and awaits a clever interpretation, part of which may be contained in the paper by Casher, Foerster and Windey [6]. The final outcome of our rather cumbersome discussion is therefore the following. The free energy Nf(~) is a sum over all possible connected closed surfaces in the precise sense introduced above, each one weighted by a factor ( - 1)t'tA/k: 1
(--1)Lt A surfaces
A total area, k periodicity, L length of self-intersection.
(4.13)
An interesting consequence arises from the fact that the periodicity k divides the area A. Consider in the gauge model the average plaquette denoted U= (n).
(4.14)
From (1.1) and (4.4) it is given by 1 d
U = ~ 5-BF~;(/~) = t + (1 taR
tz)~f'(t).
(4.15)
In turn lf'(t) is the average of four ~'s around a plaquette and can be symbolically represented as a decorated plaquette
>.
(4.16)
492
C. Itzvkson / lsingfermions (11)
Then we find that U and ½f'(t) are both series in t with integer coefficients. This can either be established directly by an analysis of the contributions of surfaces bounded by the plaquette, or by differentiation from (4.13). Indeed for those closed surfaces which are not invariant in a permutation of the coordinate axis the number of realizations is a multiple of three. For those which are invariant their area A and even A/k, i.e. the area of the corresponding non-periodic closed surface, is also a multiple of three. The stated property follows. We can translate this result in terms of the Ising model free energy
1± r u= -3 3D
(4.17)
This can be thought as the average of a link Coo') denoted symbolically ~.---.). Then u = (.---.) = ~ [([3) - cotanhZfl] cl/s = 1 - 2t < ~ ]
>.
(4.18) ~
Thus, also the low-temperature expansion of u in powers of t = e 2~ involves only integer coefficients. More general averages can be studied. For instance the connected correlation function C(1,2) =<~-r~, Frq2> _ < ~ , q , > <
FA2>
(4.19)
referring to two distinct plaquettes p~ and P2 will be computed in terms of connected surfaces bounded by p] and P2. We can also consider a generalisation of Wilson's average by taking a product of ~'s along a closed loop C with 4 Icl extra degrees of freedom corresponding to the individual ~ 's. Such an average will again involve connected surfaces bounded by the curve C with obvious constraints in the manner in which they are incident on C. The relation of such quantities for large C with the Wilson average is not entirely clear. Using a different language, even though we expect these quantities to decrease exponentially with the area enclosed by C for t < to, the coefficient of this area law does not seem directly related to the surface tension as defined conventionally.
5. E n u m e r a t i o n
of closed surfaces
It must be admitted that the procedure to generate the low-temperature series for the free energy in terms of closed surfaces is not the most efficient one. Large cancellations occur due to the signs ( - 1 ) L. A side effect of these cancellations is indeed the absolute convergence of the series around the origin in the complex t
C. lt~vkson / Isingfermions (1I)
493
plane. A very poor upper bound on the coefficients indicates a radius of absolute convergence larger than ½ [9] and, as we saw in the introduction, t c is almost twice as large (but, of course, closer singularities in the complex t plane may exist, see below). Nevertheless it is comforting to check that the enumeration according to (4.13) agrees with known results, as they should of course. In table 1 we summarize such an enumeration up to A = 16, a very modest order indeed. To increase the table would certainly only expose the weaknesses of the present author by increasing the probability of mistakes without adding much to the proof of consistency. Table 1 contains the abstract connectivity graph and shows its realizations on the lattice. It also includes the sign ( - 1 ) L multiplying the reduced number of realizations r.n.r. (i.e. number of realizations per site), its periodicity and topological characteristic X of the surface. To show the topological parallel with the graph the latter is drawn on a torus when its corresponding genus is one, the highest value encountered up to order sixteen. The details for certain surfaces, the contributions of which cancel, have been omitted. Up to A ~< 16 only periodicity 2 is encountered and as we just said X >/0. But to order 18 we already expect surfaces with genus 2 (X = - 2 ) , period 3. Due to the possibility of using a plaquette location several times, non-orientable surfaces also occur at higher orders (at least already at order 22). The meager results from table 1 can be compared with the impressive series quoted by D o m b [1] giving the series f o r f ( t ) up to order 40 and reproduced in table 2. Table 3 giving the decorated plaquette ½f'(t) is simply obtained from the previous one and simply illustrates the property about integer coefficients. An intriguing feature is noticeable in these tables. They suggest that the series for f(t) alternate in sign with the coefficients of powers 0 mod 4 negative and those of powers 2 mod 4 positive. The fact that only even powers of t contribute is on the other hand obvious. We do not find any simple proof of this fact, nor do we know whether it persists to higher order, which is very likely. If so, on the radius of convergence of the series some singularities necessarily lie on the imaginary t axis!
6. Conclusion
The fermionization methods developed at length in the specific examples of the 2and 3-dimensional models can, of course, be extended to other cases, like the 4-dimensional gauge or Ising model. More interestingly in 2 dimensions we can also derive such representations for vertex models. It remains to translate the Bethe ansatz into a path integral language in 2 dimensions. Returning to the 3-dimensional model we do not know, of course, any algorithm which would enable one to sum the set of vacuum diagrams in closed form. Short of this one can speculate on the asymptotics of the perturbation series. The connected surfaces appearing in this expansion are hardly constrained. They can have arbitrary Euler characteristics, may or may not be orientable. In this sense they are essentially
C. ltzykson- / Ising fermions ( H)
494
TABLE l Closed surfaces up to A = 16
Order
Graph
6
~
_
Realization
~
r.n.r,
+ 1
Period. k
1
Charact. X
Contrib.
2
t6
Total
t6
,o@
1
3,,0 3t ]o
12
[ ~ ) Eance|
. ~ ~ ~
.
~ ~
- 12
1
2
- 12t 12
-4
2
2
- 2 t h2
+6
1
2
6t 12
+2
2
2
t 12
+3
1
0
3t 12
+1
2
0
i 12 ~t
--3E2t12
495
C. Itzykson / Ising fermions (II) TABLE 1,(continued) Order
Graph
Realization
r.n.r,
Period. k
Charact. X
Contrib.
Totol
14
(
/ ~ ' ~
+3
1
2
3t TM
+ 12
l
2
12t 14 15t ~4
16
~
Cancel
/
I U ~ ~
~ ~ ~
~
(
~
L20
1
2
- 120fl 6
- 24
1
2
24t 16
+ 60
1
2
60t 16
+ 12
1
2
12t 16
+30
1
0
30t t6
+6
1
0
6t 16
1
2
3t ~6
+3
-33fl 6
*The two cubes are next to each other; identified links are shown. **Similar drawing, the two cubes are one into the other. tThe double and single cube are next to each other or one on top of the other. tiThe cube is inside the double cube.
C. Itzykson / Isingfermions ( II)
496
TABLE 2 Series f o r f ( t ) in even powers of t from D o m b [1] Order
Coefficient
6 8 l0
1 0 3
12 14 16
- 3½ 15 33
18
104½
20 22
- 280½ 849
24 26
- 2461~ 7485
28
- 22534½
30
693935
32 34
213754½ 666750
36 38
2086 734~ 6583341
40
- 20 8523634
[
I
TABLE 3 Series in odd powers of t for the decorated p l a q u e t t e ~3f'(t) = ~ [ ] Order
Coefficient
5 7 9 II 13 15 17 19 21 23 25 27 29 31 33 35 37 39
2 0 10 -14 70 -176 626 -1870 6226 - 12 694 64 870 - 210322 693 932 2280048 7556 500 -25040810 83 388986 -278031510
C. Itzvkson / lsingfermions (II)
497
random and the anticommuting integral, cumbersome as it is to evaluate, may be considered as a generating functional. It would be interesting to know if the topological character is coded in their connectivity graph as would be suggested by our low order study. We have mentioned that this graph is only a piece of the dual of the 2-dimensional complex attached to the quadrangulation of the surface. What are missing are the faces of the graph corresponding to the vertices of the complex. It is known that any graph admits a genus which is the smallest one of an orientable surface on which it can be drawn with its edges crossing only at its vertices. Poor bounds can be obtained for this genus but in the words of the author of a recent text book on graph theory, R.J. Wilson, "little is known about the problem of finding the genus of an arbitrary graph" [10]. So much more work is still required to extract useful information from this series and understand the embedding of surfaces in 3-space. The sign question is also intriguing, in view of the comments of ref. [6] where the authors would like to relate it to reference frames on the surface since the expansion contains non-orientable surfaces. By regrouping the terms in the series one could try to mimic in discrete form Polyakov's conjecture [3], possibly valid in the critical region since large classes of graphs correspond to imbedding the same abstract surface in real space. This still looks like a formidable task. We have also encountered a mysterious alternation of sign in the free energy. Intuitively this might be related to the fact that for /~ imaginary, or t 2 < 0, the partition function Zo(/3 ) expressed as in (1.1) is obviously still real and of modulus smaller than one, so that F~ is negative under such circumstances. If true the corresponding singularity on the negative t 2 axis has to be clarified. Let us make a very exotic suggestion. We have seen that one can introduce intermediate variables "fp = + 1 in such a way as to decouple three families of non-interacting Ising models for fixed "fp. If we write the average over yp as
"E yp =
~ 1
Yp =
~ I
with A to be set equal to one, we can develop a perturbation theory in A, the starting point of which is an expression which has the critical properties of the two-dimensional model. Investigation of this series might be of some interest. Finally we recall that we have obtained in sect. 3 an alternative integral adapted to the study of high-temperature series. It too might be worth working out. The Ising models remain fascinating and challenging. Discussions with P. Windey and J.B. Zuber are gratefully acknowledged.
498
C. ltzykson / lsing fermions ( H )
References [1] C. Domb, in Phase transitions and critical phenomena, ed. C. Domb and M.C. Green, vol. 3 (Academic Press, New York); D.S. Gaunt, ibid.; A.J. Guttmann, ibid. [2] J. Zinn-Justin, J. de Phys. 40 (1979) 969 [3] A. Polyakov, Phys. Lett. 103B (1981) 207, 211 [4] S. Samuel, J. Math. Phys. 21 (1980) 2806, 2815, 2820 [5] E. Fradkin, M. Srednicki and L. Susskind, Phys. Rev. D21 (1980) 2885 [6] A. Casher, D. Foerster and P. Windey, CERN preprint TH 3200 (1981) [7] C. Itzykson, Nucl. Phys. B210[FS6] (1982) 448 [8] E. Brbzin, C. Itzykson, G. Parisi and J.B. Zuber, Comm. Math. Phys. 59 (1978) 35 [9] R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D l l (1975) 2098 [10] R.J. Wilson, Introduction to graph theory (Longman, London, 1979)