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On the emergence of U(1) invariances in the fermionisation of Baxter type models
Volume 139B, number 1,2
PHYSICS LETTERS
3 May 1984
ON THE EMERGENCE OF U(1) INVARIANCES IN THE FERMIONISATION OF BAXTER TYPE MODELS D. F O E R S T E R Max Planck-lnstitut fhr Physik und Astrophysik - Werner Heisenberg-Institut fiir Physik -, D-8000 Munich 40, Fed. Rep. Germany
Received 10 December 1983
Recently, it has been observed that the d = 3 Ising model fermionises into a locally U(1) invariant theory of Grassmann variables. Here we notice the analogy with the situation in d = 2 where the Baxter model fermionises into the globally U (1) invariant Thirring model. This suggests that a suitable generalisation of the Baxter model to d = 3 should fermionise into a locally U(1) invariant Grassmann theory which may have ~ many commuting charges.
1. In this letter I want to point out the possibility that d = 3 B a x t e r - Z a m o l o d c h i k o v type models might be twin brothers of locally U(1) invariant Grassmann gauge theories with an infinity of conserved charges. The d = 2 Baxter model is a Z2 valued exactly integrable model which has been generalised to d = 3 by Zamolodchikov [1,2]. A Grassmann gauge theory [3] has a local (gauge) invariance but its link variables are Grassmann valued and differ in their structure from those of Wilson's theory [4] ,1. My argument that d = 3 Baxter models and U(1) Grassmann gauge theories are related is based on the following observations: (i) The d = 3 ( d = 2) Ising model fermionises into a locally U(1) invariant Grassmann gauge theory (globally U ( I ) invariant free Dirac fermions). (ii) The d = 2 Baxter model fermionises into a globally U(1) invariant Thirring model (with ~ many conserved charges). The U(1) invariances in (i) and (ii) are on a different footing since the conventional fermionisation of ,1 Grassmann gauge theories axe also theories of interacting fermionic surfaces just as Thirring models are theories of interacting fermionic paths. The passage from global invariance to local invariance parallels the replacement of paths by surfaces while the relation between Grassmann and bosonic gauge theories is presumably analogous to that of Thirring and sine-Gordon theory [5]. 62
the d = 2 Ising model via a J o r d a n - W i g n e r transformation leads to Majorana fermions, while a euclidean treatment leads more naturally to charged Dirac fermions, By contrast, the U(1) invariance ( o f the Thirring model) in (ii) appears already at the level of the conventional J o r d a n - W i g n e r fermionisation [6] (in the continuum limit) ,2. F r o m d = 2 we learn that the U(1) invariance of the fermionic partner survives the extension of the Ising model to a Baxter model. In d = 3 we expect persistence of the local U(1) invariance in an analogous extension of the d = 3 Ising model. In d = 2, the Baxter model has an infinity of conserved charges [7] which reappear in its fermionic image [6] and if the d = 3 Baxter model should have an infinity of conserved charges then there should exist similar charges also in the corresponding Grassmann gauge theory. Zamolodchikov has found an "integrable" extension of the Baxter model from d = 2 to d = 3 based on the tetrahedron equations but his model has negative statistical weights [ 1,2,8 ]. Finally, to counter an often heard criticism, nontriviality o f the d = 3 theory (in spite of oo many conserved charges) is guaranteed if it contains the d = 3 ,2 As shown below, this U(1) invariance persists in the euclidean fermionisation. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 139B, number 1,2
PHYSICS LETTERS
Ising model (along with its fermionisation) as a special case since the latter is known to be a nontrivial field theory. Ultimately we are, of course, interested in SU(N) Yang-Mills theory in d = 4. In this connection it is encouraging that the entire SU(N) series of Thirring models is integrable [91j. The existence of oo many conserved charges appears absolutely crucial to gain leverage in analysing Grassmann gauge theories since a perturbation approach for such theories is not feasible. 2. We shall now reexamine the fermionisation of the d = 2 Baxter model as an exercise for d = 3 and to motivate our expectations there. The fermionisation of the d = 2 Baxter model that is known in the literature proceeds in two steps: d = 2 Baxter
anisotropic . limit ¢ a = 1 x y z quantum chain
Jordan • ~ massive Thirring model transformation Wlgner
In other words, one first determines the hamiltonian of the Baxter model before fermionising it [6,10]. Hamiltonian language is not convenient for going beyond d = 2 because: (i) Already the d = 3 Ising model contains loop variables for the analysis of which hamiltonian methods are inappropriate. (ii) The local U(1) invariance had been missed in a previous hamiltonian fermionisation of the d = 3 Ising model [ 11 ]. Because of this, we shall give here a new euclidean fermionisation of the d = 2 Baxter model that should generalise to d = 3. Following refs. [3,12] we shall proceed in 3 steps: (a) Identify fermions as order-disorder pairs. (b) Derive equations of motion for them. (c) Find a Grassmann action that reproduces the equations of motion. The Baxter model is obtained by decorating with arrows the links of a two-dimensional lattice, such that they are conserved mod 4 at each vertex and with the vertex statistical weights a, b, c, d invariant under reversal of all 4 associated arrows (see fig. 1). A disorder variable is generated by drawing an open path of even length on the dual lattice with two oppositely pointing arrows at every cut link (see fig. 2). A similar
3 May 1984
B'
+ + + + + + + + 1
2
3
a
b
I.
5 c
'
6
7
d
8
Fig. 1.
construction has been suggested to me by Dotsenko [13]. In fig. 1 we may, by invariance under reversal of arrows at P, deform the path from C to C' etc. The dual of this disorder-disorder correlation is an o r d e r order correlation defined by multiplying along a path C the arrow values of the links across C. An arrow value is +1 ( - I ) if the arrow points along a positive (negative) coordinate direction. The loop C associated with an order-order correlation may be deformed using conservation of arrows (mod 4). Because of arrow reversal symmetry respectively flux conservation (rood 4) order-order and disorder-disorder correlations are non-zero only if A and B are of the same checkerboard parity. The fermion operators are given by multiplying order with nearby (next neighbour) disorder operators (see fig. 2). We shall refer to o r d e r disorder pairs as even or odd according to the parity of the order operator. In the following, we shall write identities for products of order or disorder operators located at centres of squares labeUed (m, n), (m + 1, n), (m + 1, n + 1) and (m, n + 1). These points will be denoted as (: :), with e.g. (. ") denoting a product of order variables at (m, n) and (m + 1, n + 1) (dots denote order variables and crosses disorder variables, as in fig. 2). We have the following identities:
I
i Fig. 2.
63
Volume 139B, number 1,2
PHYSICS LETTERS
1 =el(~, : ) + f l ( . e o ) + g l ( :
:)+hl(:
3 May 1984
gl = (sinh 2')' sinh 2/3)/[cosh 2a(sinh 2t3 cosh 2a cosh 23,)
:),
+ sinh 2')' sinh 2a cosh 2/3] , 1 =e2(: .)+f2(o
. ° ) + g 2 ( o• ~ ) + h 2 ( .* *, ) ,
(1) h 1 = (sinh 23, sinh 2a)/[cosh 2a(cosh 2')' cosh 2a sinh 2t3
These identities (1) are conveniently derived by passing from "arrow language" to spin language with a partition function Z=
(5 cont'd)
(e2f2g2h2) are given by the same expressions but with
a ,e,/3 interchanged. Now multiply, for example, the operator (~ *) by the identity (1) and use that the squares o f • and * are 1 [it is implicit that (~ *) is part of a correlation function] :
~ exp(-S) (o(m,n)}
S = ~_1 (aOm,n+ 10m+l,n + [JOm,nOm+l,n+l m,n + 3`Om,n+l Om, n Om+l,n Om+l, n+l)
+ sinh 27 cosh 2/3 cosh 2a sinh 2a)] .
(m,n+l)
(2)
(! :)=el(~
")+fl(: ;)+gl(o
7)
with a = exp(a + j3 + 3`),
b = e x p ( - a -/3 + 3`),
c = e x p ( - a +/3 - 3`),
d = exp(a - 13- 3`),
(3)
establishing the connection with the vertex weights. The spins may be thought of as sitting on the centres of the previous lattice squares and products of neighbouring spins determine the arrow values (see ref. [7]). In spin language, the order variables are just spins and the disorder variables are given by open paths of fixed checkerboard parity across which one flips two couplings in each square. Calling (m, n + 1), (m + 1, n + 1), (m + 1, n), (m, n) points 1, 2, 3, 4 respectively, disorder-disorder correlations can be shown to be given by: p(1) ~(3) = exp(-2/3o2o4) exp(--27OlO2O304), p(2) p(4) = exp(-2aO103) exp(-23`O1020304).
(4)
Identity (1) then follows by expanding 5 operators over a basis of only 4 linearly independent ones, i.e. 1,
OlO 3 ,
a2o 4 ,
01020304 .
we may also multiply the same operator by the second identity (1) translated one unit in the +y direction:
(i
:)+4"-')
(m,n+l) + h2 *o
"
(7)
So we see that there are always two equations per operator which is due to our symmetric treatment of fermions. We resolve this difficulty as in refs. [1,3,12] by distinguishing between right- and left-handed equations according to whether * rotates around • in a clockwise fashion or vice versa. We also stipulate that the equations of motion pertaining to a given square may not be used twice in succession. With a notation
The coefficients el, fl, gl, hl are given by e 1 = (sinh 2/3)/(sinh 213 cosh 23' cosh 2a = ffl(m,n), + sinh 2a sinh 2')' cosh 213), fl = tanh 2a, 64
(5)
~2(m,n),
q)3(m,n),
~4(rn, n)
for order-disorder pairs subject to right-handed equations and ~ ~+ ~ for left-handed equations we obtain
Volume 139B, number 1,2
PHYSICS LETTERS
equations of motion that can be integrated in terms of a Grassmann action only for a =/3 which implies (elflglhl) = (e2f2g2h2). In this special case, the equations of motion may be derived from the following Grassmann action: S = S (2) + S (int) ,
3 May 1984
where
l(--~xp(-ioyrr/2) U=-~
1
and
7x=
O1
'
7Y= 0
s (2) = ~A ¢ ,
A=
0
_0
1
e
-J~y_
;rx+y
1
0
1 0 0-1
In terms of ff = U~b one may rewrite S (int) ofeq. (10) to obtain (g = h to lowest order in 7)
- l T x - y -e
- f r - x - y --e
/ '
(lO)
s(int) = ~g[~-e( 1 + T5) ~be~o(1 + T5) qSo
0 + ~-e( 1 -- V5) {be~-o(1 -- V5) •o] ,
_ -
~03 ~02 '
where
~4
q~e=(~b3~ ~ba]e'
Ty ~(rn, n) = t~(m, n+ 1),
Tx t~(m, n) = qJ(m+ 1,n), s(int) = g
2
m,n [ f 3 ( m + 1, n) ~4(m, n+ 1)
+ f l ( r n , n+ 1) ff2(rn+ 1,n)] X [~b2(m, n) f 4 ( m + l , n + l ) - f f 3 ( m + l , n + l ) h ~
[~l(m,n+l)
~2(m,n)]
t~4(m,n+l)
+ "2 m, n
- ~ 3 ( m + l , n ) ff2(m+ 1,n)] X [ff3(m+l,n+l) ~ 2 ( m , n ) + ~ l ( m , n ) ~ 4 ( m + l , n + l ) ] . The continuum limit is (for 7 = 0 and in the isotropic case) at detA Co)[p=O = (1 - 2el) 2 + (e 2 - f2)2 = 0 ~+e=f
(13)
'
(9)
where ,3
1
)
exp(+ioyrr/2)
= 1/V~.
(11)
In this case A may be diagonalised A = - i U-ruU - 1 +
O(p2),
(12)
,3 The convention taken for disorder paths was to run in the +x - y direction [this generates - signs not indicated in eqs. (6), (7)].
(q~3) , ~b°= ~b4 o
75=TxTy.
(15)
(16)
This differs from the usual Thirring model in that we have even (e) and odd (o) fermions in the continuum limit due to conservation of checkerboard parity. There is no contradiction with known results because =/3 is outside the asymmetric region previously considered in fermionising the Baxter model. The general fermionisation (a ~/3) will be given elsewhere. The general result is crucial to check the conjecture that the Grassmann theory obeys Grassmann triangle equations that replace the triangle equations of the original Baxter model [7]. The above technique of euclidean fermionisation should be sufficient to treat the d = 3 Baxter model, whichever will turn out to be the correct generalisation of the Baxter model from d = 2 tod=3. 3. Finally, we should write down a d = 3 Baxter model that is a viable candidate for the generalisation from d = 2 to d = 3. The first generalisation due to Zamolodchikov [1,2] was inspired by the d = 2 relation between S matrices in systems without particle production and the vertex weights o f d = 2 integrable systems (see ref. [ 1 ]). To generalise this relation to d = 3, Zamolodchikov had to discuss the scattering of infinitely straight strings. As Zamolodchikov states himself, such objects probably do not exist except to motivate the solution of the tetrahedron equations. It was the intention of the present note to obtain 65
Volume 139B, number 1,2
PHYSICS LETTERS
alternative guiding principles in the construction of the d = 3 Baxter model. Our guiding principle shall be that the d -- 3 Baxter model should be a generalisation of the d = 3 Ising model and that it should allow a fermionisation into a U(1) Grassmann .gauge theory. One possible definition of such a model involves Z 2 valued spins located on the squares of a simple cubic lattice. The spins on the boundary squares of each cube should satisfy the constraint
P 1P2P3P4P5P6 = + 1 ,
(17)
which is conveniently solved in terms of Z2 valued link variables associated with each square p=
1-I {p) cubes
pC, C),
C where w(PC --" pC, C) is the probability of a cube C having boundary spins pC ... pC (subject to PC" P~" .... pC = +1). This is clearly a model with 32 possible configurations at each vertex and which contains Z2 gauge theory as a special case. The ferrnionisations of d = 3 Z2 gauge theory in refs. [3,12] involved disordered Wilson loops i.e. Wilson loops with a pointlike disorder variable associated with each link of a loop, see refs. [3,12] for further details. The present model contains both Wilson loops and pointlike disorder operators (the definition of which is straightforward) and we may, therefore, expect the present model to fermionise into a locally U(1) invariant Grassmann theory. The present model has been previously considered by Maillard who also wrote down the associated tetrahedron equations [14]. We should also mention an implicit assumption in the present letter - i.e. the assumption that the d = 3 Ising model is not integrable. If it were "integrable", we would try to analyse the structure of its conserved charges and of the conserved charges of the fermionic theory and there would be no motivation to generalise
66
the d = 3 Ising model to a Baxter model. In fact, it has not yet been shown (as far as I know) that the d = 3 Ising model is not integrable or that it does not satisfy the tetrahedron equations but I intend to show this elsewhere.
4. Summary and conclusion. In this letter we have seen that two difficult problems are very likely to be strongly interrelated: the problem of finding d > 2 Baxter models and the problem of understanding Yang-Mills theory outside of the perturbative regime. Clearly much further work is required on the generalisation of the Baxter model from d = 2 to d = 3 and on the fermionisation of such a model.
(18)
I-1 £. ~E ~p
The partition function is given by
z--
3 May 1984
I am particularly indebted to S. Rouhani for discussions on the tetrahedron equations. I have also had useful discussions with A. Luther, J.M. Maillard, D. Maison, H. Sharatchandra and U. Wolff.
References [1] A.B. Zamolodchikov, JETP 52 (1980) 325. [2] A.B. Zamolodchikov, Commun. Math. Phys. 75 (1981) 489. [3] D. FiSrster, Phys. Lett. 120B (1983) 150; Proc. XVI Ahrenshoop Meeting. [4] K. Wilson, Phys. Rev. D10 (1974) 2445. [5] A. Luther and V.J. Emery, Phys. Rev. Lett. 33 (1974) 589; S. Coleman, Phys. Rev. Dll (1975) 1088; S. Mandelstam, Phys. Rev. Dll (1975) 3076. [6] M. Liischer, Nucl. Phy~ Bl17 (1976) 475. [7] R.J. Baxter, Exactly solved models in statistical mechanics (Academic Press, New York, 1982). [8] R.J. Baxter, Commun. Math. Phys. 88 (1983) 185. [9] See e.g.H. Bergknoff and H.B. Thacker, Phys. Rev. D19 (1979) 3666. [10] A. Luther, Phys. Rev. B14 (1976) 2153. [11] E. Fradkin, M. Srednicki and L. Susskind, Phys. Rev. D21 (1980) 2885; C. Itzykson, Nucl. Phys. B210 [FS6] (1982) 477. [12] A. Casher, D. F~Srster and P. Windey, CERN preprint TH 3200 (1981); see also V. Dotsenko, Thesis [in Russian]. [13] V. Dotsenko, private communication. [14] J.M. MaiUard, CEA report (April 1983) [in French].
PHYSICS LETTERS
3 May 1984
ON THE EMERGENCE OF U(1) INVARIANCES IN THE FERMIONISATION OF BAXTER TYPE MODELS D. F O E R S T E R Max Planck-lnstitut fhr Physik und Astrophysik - Werner Heisenberg-Institut fiir Physik -, D-8000 Munich 40, Fed. Rep. Germany
Received 10 December 1983
Recently, it has been observed that the d = 3 Ising model fermionises into a locally U(1) invariant theory of Grassmann variables. Here we notice the analogy with the situation in d = 2 where the Baxter model fermionises into the globally U (1) invariant Thirring model. This suggests that a suitable generalisation of the Baxter model to d = 3 should fermionise into a locally U(1) invariant Grassmann theory which may have ~ many commuting charges.
1. In this letter I want to point out the possibility that d = 3 B a x t e r - Z a m o l o d c h i k o v type models might be twin brothers of locally U(1) invariant Grassmann gauge theories with an infinity of conserved charges. The d = 2 Baxter model is a Z2 valued exactly integrable model which has been generalised to d = 3 by Zamolodchikov [1,2]. A Grassmann gauge theory [3] has a local (gauge) invariance but its link variables are Grassmann valued and differ in their structure from those of Wilson's theory [4] ,1. My argument that d = 3 Baxter models and U(1) Grassmann gauge theories are related is based on the following observations: (i) The d = 3 ( d = 2) Ising model fermionises into a locally U(1) invariant Grassmann gauge theory (globally U ( I ) invariant free Dirac fermions). (ii) The d = 2 Baxter model fermionises into a globally U(1) invariant Thirring model (with ~ many conserved charges). The U(1) invariances in (i) and (ii) are on a different footing since the conventional fermionisation of ,1 Grassmann gauge theories axe also theories of interacting fermionic surfaces just as Thirring models are theories of interacting fermionic paths. The passage from global invariance to local invariance parallels the replacement of paths by surfaces while the relation between Grassmann and bosonic gauge theories is presumably analogous to that of Thirring and sine-Gordon theory [5]. 62
the d = 2 Ising model via a J o r d a n - W i g n e r transformation leads to Majorana fermions, while a euclidean treatment leads more naturally to charged Dirac fermions, By contrast, the U(1) invariance ( o f the Thirring model) in (ii) appears already at the level of the conventional J o r d a n - W i g n e r fermionisation [6] (in the continuum limit) ,2. F r o m d = 2 we learn that the U(1) invariance of the fermionic partner survives the extension of the Ising model to a Baxter model. In d = 3 we expect persistence of the local U(1) invariance in an analogous extension of the d = 3 Ising model. In d = 2, the Baxter model has an infinity of conserved charges [7] which reappear in its fermionic image [6] and if the d = 3 Baxter model should have an infinity of conserved charges then there should exist similar charges also in the corresponding Grassmann gauge theory. Zamolodchikov has found an "integrable" extension of the Baxter model from d = 2 to d = 3 based on the tetrahedron equations but his model has negative statistical weights [ 1,2,8 ]. Finally, to counter an often heard criticism, nontriviality o f the d = 3 theory (in spite of oo many conserved charges) is guaranteed if it contains the d = 3 ,2 As shown below, this U(1) invariance persists in the euclidean fermionisation. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 139B, number 1,2
PHYSICS LETTERS
Ising model (along with its fermionisation) as a special case since the latter is known to be a nontrivial field theory. Ultimately we are, of course, interested in SU(N) Yang-Mills theory in d = 4. In this connection it is encouraging that the entire SU(N) series of Thirring models is integrable [91j. The existence of oo many conserved charges appears absolutely crucial to gain leverage in analysing Grassmann gauge theories since a perturbation approach for such theories is not feasible. 2. We shall now reexamine the fermionisation of the d = 2 Baxter model as an exercise for d = 3 and to motivate our expectations there. The fermionisation of the d = 2 Baxter model that is known in the literature proceeds in two steps: d = 2 Baxter
anisotropic . limit ¢ a = 1 x y z quantum chain
Jordan • ~ massive Thirring model transformation Wlgner
In other words, one first determines the hamiltonian of the Baxter model before fermionising it [6,10]. Hamiltonian language is not convenient for going beyond d = 2 because: (i) Already the d = 3 Ising model contains loop variables for the analysis of which hamiltonian methods are inappropriate. (ii) The local U(1) invariance had been missed in a previous hamiltonian fermionisation of the d = 3 Ising model [ 11 ]. Because of this, we shall give here a new euclidean fermionisation of the d = 2 Baxter model that should generalise to d = 3. Following refs. [3,12] we shall proceed in 3 steps: (a) Identify fermions as order-disorder pairs. (b) Derive equations of motion for them. (c) Find a Grassmann action that reproduces the equations of motion. The Baxter model is obtained by decorating with arrows the links of a two-dimensional lattice, such that they are conserved mod 4 at each vertex and with the vertex statistical weights a, b, c, d invariant under reversal of all 4 associated arrows (see fig. 1). A disorder variable is generated by drawing an open path of even length on the dual lattice with two oppositely pointing arrows at every cut link (see fig. 2). A similar
3 May 1984
B'
+ + + + + + + + 1
2
3
a
b
I.
5 c
'
6
7
d
8
Fig. 1.
construction has been suggested to me by Dotsenko [13]. In fig. 1 we may, by invariance under reversal of arrows at P, deform the path from C to C' etc. The dual of this disorder-disorder correlation is an o r d e r order correlation defined by multiplying along a path C the arrow values of the links across C. An arrow value is +1 ( - I ) if the arrow points along a positive (negative) coordinate direction. The loop C associated with an order-order correlation may be deformed using conservation of arrows (mod 4). Because of arrow reversal symmetry respectively flux conservation (rood 4) order-order and disorder-disorder correlations are non-zero only if A and B are of the same checkerboard parity. The fermion operators are given by multiplying order with nearby (next neighbour) disorder operators (see fig. 2). We shall refer to o r d e r disorder pairs as even or odd according to the parity of the order operator. In the following, we shall write identities for products of order or disorder operators located at centres of squares labeUed (m, n), (m + 1, n), (m + 1, n + 1) and (m, n + 1). These points will be denoted as (: :), with e.g. (. ") denoting a product of order variables at (m, n) and (m + 1, n + 1) (dots denote order variables and crosses disorder variables, as in fig. 2). We have the following identities:
I
i Fig. 2.
63
Volume 139B, number 1,2
PHYSICS LETTERS
1 =el(~, : ) + f l ( . e o ) + g l ( :
:)+hl(:
3 May 1984
gl = (sinh 2')' sinh 2/3)/[cosh 2a(sinh 2t3 cosh 2a cosh 23,)
:),
+ sinh 2')' sinh 2a cosh 2/3] , 1 =e2(: .)+f2(o
. ° ) + g 2 ( o• ~ ) + h 2 ( .* *, ) ,
(1) h 1 = (sinh 23, sinh 2a)/[cosh 2a(cosh 2')' cosh 2a sinh 2t3
These identities (1) are conveniently derived by passing from "arrow language" to spin language with a partition function Z=
(5 cont'd)
(e2f2g2h2) are given by the same expressions but with
a ,e,/3 interchanged. Now multiply, for example, the operator (~ *) by the identity (1) and use that the squares o f • and * are 1 [it is implicit that (~ *) is part of a correlation function] :
~ exp(-S) (o(m,n)}
S = ~_1 (aOm,n+ 10m+l,n + [JOm,nOm+l,n+l m,n + 3`Om,n+l Om, n Om+l,n Om+l, n+l)
+ sinh 27 cosh 2/3 cosh 2a sinh 2a)] .
(m,n+l)
(2)
(! :)=el(~
")+fl(: ;)+gl(o
7)
with a = exp(a + j3 + 3`),
b = e x p ( - a -/3 + 3`),
c = e x p ( - a +/3 - 3`),
d = exp(a - 13- 3`),
(3)
establishing the connection with the vertex weights. The spins may be thought of as sitting on the centres of the previous lattice squares and products of neighbouring spins determine the arrow values (see ref. [7]). In spin language, the order variables are just spins and the disorder variables are given by open paths of fixed checkerboard parity across which one flips two couplings in each square. Calling (m, n + 1), (m + 1, n + 1), (m + 1, n), (m, n) points 1, 2, 3, 4 respectively, disorder-disorder correlations can be shown to be given by: p(1) ~(3) = exp(-2/3o2o4) exp(--27OlO2O304), p(2) p(4) = exp(-2aO103) exp(-23`O1020304).
(4)
Identity (1) then follows by expanding 5 operators over a basis of only 4 linearly independent ones, i.e. 1,
OlO 3 ,
a2o 4 ,
01020304 .
we may also multiply the same operator by the second identity (1) translated one unit in the +y direction:
(i
:)+4"-')
(m,n+l) + h2 *o
"
(7)
So we see that there are always two equations per operator which is due to our symmetric treatment of fermions. We resolve this difficulty as in refs. [1,3,12] by distinguishing between right- and left-handed equations according to whether * rotates around • in a clockwise fashion or vice versa. We also stipulate that the equations of motion pertaining to a given square may not be used twice in succession. With a notation
The coefficients el, fl, gl, hl are given by e 1 = (sinh 2/3)/(sinh 213 cosh 23' cosh 2a = ffl(m,n), + sinh 2a sinh 2')' cosh 213), fl = tanh 2a, 64
(5)
~2(m,n),
q)3(m,n),
~4(rn, n)
for order-disorder pairs subject to right-handed equations and ~ ~+ ~ for left-handed equations we obtain
Volume 139B, number 1,2
PHYSICS LETTERS
equations of motion that can be integrated in terms of a Grassmann action only for a =/3 which implies (elflglhl) = (e2f2g2h2). In this special case, the equations of motion may be derived from the following Grassmann action: S = S (2) + S (int) ,
3 May 1984
where
l(--~xp(-ioyrr/2) U=-~
1
and
7x=
O1
'
7Y= 0
s (2) = ~A ¢ ,
A=
0
_0
1
e
-J~y_
;rx+y
1
0
1 0 0-1
In terms of ff = U~b one may rewrite S (int) ofeq. (10) to obtain (g = h to lowest order in 7)
- l T x - y -e
- f r - x - y --e
/ '
(lO)
s(int) = ~g[~-e( 1 + T5) ~be~o(1 + T5) qSo
0 + ~-e( 1 -- V5) {be~-o(1 -- V5) •o] ,
_ -
~03 ~02 '
where
~4
q~e=(~b3~ ~ba]e'
Ty ~(rn, n) = t~(m, n+ 1),
Tx t~(m, n) = qJ(m+ 1,n), s(int) = g
2
m,n [ f 3 ( m + 1, n) ~4(m, n+ 1)
+ f l ( r n , n+ 1) ff2(rn+ 1,n)] X [~b2(m, n) f 4 ( m + l , n + l ) - f f 3 ( m + l , n + l ) h ~
[~l(m,n+l)
~2(m,n)]
t~4(m,n+l)
+ "2 m, n
- ~ 3 ( m + l , n ) ff2(m+ 1,n)] X [ff3(m+l,n+l) ~ 2 ( m , n ) + ~ l ( m , n ) ~ 4 ( m + l , n + l ) ] . The continuum limit is (for 7 = 0 and in the isotropic case) at detA Co)[p=O = (1 - 2el) 2 + (e 2 - f2)2 = 0 ~+e=f
(13)
'
(9)
where ,3
1
)
exp(+ioyrr/2)
= 1/V~.
(11)
In this case A may be diagonalised A = - i U-ruU - 1 +
O(p2),
(12)
,3 The convention taken for disorder paths was to run in the +x - y direction [this generates - signs not indicated in eqs. (6), (7)].
(q~3) , ~b°= ~b4 o
75=TxTy.
(15)
(16)
This differs from the usual Thirring model in that we have even (e) and odd (o) fermions in the continuum limit due to conservation of checkerboard parity. There is no contradiction with known results because =/3 is outside the asymmetric region previously considered in fermionising the Baxter model. The general fermionisation (a ~/3) will be given elsewhere. The general result is crucial to check the conjecture that the Grassmann theory obeys Grassmann triangle equations that replace the triangle equations of the original Baxter model [7]. The above technique of euclidean fermionisation should be sufficient to treat the d = 3 Baxter model, whichever will turn out to be the correct generalisation of the Baxter model from d = 2 tod=3. 3. Finally, we should write down a d = 3 Baxter model that is a viable candidate for the generalisation from d = 2 to d = 3. The first generalisation due to Zamolodchikov [1,2] was inspired by the d = 2 relation between S matrices in systems without particle production and the vertex weights o f d = 2 integrable systems (see ref. [ 1 ]). To generalise this relation to d = 3, Zamolodchikov had to discuss the scattering of infinitely straight strings. As Zamolodchikov states himself, such objects probably do not exist except to motivate the solution of the tetrahedron equations. It was the intention of the present note to obtain 65
Volume 139B, number 1,2
PHYSICS LETTERS
alternative guiding principles in the construction of the d = 3 Baxter model. Our guiding principle shall be that the d -- 3 Baxter model should be a generalisation of the d = 3 Ising model and that it should allow a fermionisation into a U(1) Grassmann .gauge theory. One possible definition of such a model involves Z 2 valued spins located on the squares of a simple cubic lattice. The spins on the boundary squares of each cube should satisfy the constraint
P 1P2P3P4P5P6 = + 1 ,
(17)
which is conveniently solved in terms of Z2 valued link variables associated with each square p=
1-I {p) cubes
pC, C),
C where w(PC --" pC, C) is the probability of a cube C having boundary spins pC ... pC (subject to PC" P~" .... pC = +1). This is clearly a model with 32 possible configurations at each vertex and which contains Z2 gauge theory as a special case. The ferrnionisations of d = 3 Z2 gauge theory in refs. [3,12] involved disordered Wilson loops i.e. Wilson loops with a pointlike disorder variable associated with each link of a loop, see refs. [3,12] for further details. The present model contains both Wilson loops and pointlike disorder operators (the definition of which is straightforward) and we may, therefore, expect the present model to fermionise into a locally U(1) invariant Grassmann theory. The present model has been previously considered by Maillard who also wrote down the associated tetrahedron equations [14]. We should also mention an implicit assumption in the present letter - i.e. the assumption that the d = 3 Ising model is not integrable. If it were "integrable", we would try to analyse the structure of its conserved charges and of the conserved charges of the fermionic theory and there would be no motivation to generalise
66
the d = 3 Ising model to a Baxter model. In fact, it has not yet been shown (as far as I know) that the d = 3 Ising model is not integrable or that it does not satisfy the tetrahedron equations but I intend to show this elsewhere.
4. Summary and conclusion. In this letter we have seen that two difficult problems are very likely to be strongly interrelated: the problem of finding d > 2 Baxter models and the problem of understanding Yang-Mills theory outside of the perturbative regime. Clearly much further work is required on the generalisation of the Baxter model from d = 2 to d = 3 and on the fermionisation of such a model.
(18)
I-1 £. ~E ~p
The partition function is given by
z--
3 May 1984
I am particularly indebted to S. Rouhani for discussions on the tetrahedron equations. I have also had useful discussions with A. Luther, J.M. Maillard, D. Maison, H. Sharatchandra and U. Wolff.
References [1] A.B. Zamolodchikov, JETP 52 (1980) 325. [2] A.B. Zamolodchikov, Commun. Math. Phys. 75 (1981) 489. [3] D. FiSrster, Phys. Lett. 120B (1983) 150; Proc. XVI Ahrenshoop Meeting. [4] K. Wilson, Phys. Rev. D10 (1974) 2445. [5] A. Luther and V.J. Emery, Phys. Rev. Lett. 33 (1974) 589; S. Coleman, Phys. Rev. Dll (1975) 1088; S. Mandelstam, Phys. Rev. Dll (1975) 3076. [6] M. Liischer, Nucl. Phy~ Bl17 (1976) 475. [7] R.J. Baxter, Exactly solved models in statistical mechanics (Academic Press, New York, 1982). [8] R.J. Baxter, Commun. Math. Phys. 88 (1983) 185. [9] See e.g.H. Bergknoff and H.B. Thacker, Phys. Rev. D19 (1979) 3666. [10] A. Luther, Phys. Rev. B14 (1976) 2153. [11] E. Fradkin, M. Srednicki and L. Susskind, Phys. Rev. D21 (1980) 2885; C. Itzykson, Nucl. Phys. B210 [FS6] (1982) 477. [12] A. Casher, D. F~Srster and P. Windey, CERN preprint TH 3200 (1981); see also V. Dotsenko, Thesis [in Russian]. [13] V. Dotsenko, private communication. [14] J.M. MaiUard, CEA report (April 1983) [in French].