Finite size effects with 2-D ZN lattice gauge models

Physica 145A (1987) 96-104 North-Holland, Amsterdam

FINITE SIZE EFFECTS W I T H 2-D

ZN

LATTICE GAUGE MODELS

R.L. GIBBS, P.B. S T E P H E N S O N

Department of Physics, Louisiana Tech University, Tech Station, Louisiana 71272, USA and Kelly A. F A R R A R Jr

Department of Physics, University of Kansas, Lawrence, Kansas 66045, USA

Received 3 December 1986 Revised manuscript received 9 February 1987

T h e 2-D Z 2 lattice gauge model with zeros or ones on the links is solved by directly producing all possible configurations, By doing this for larger and larger lattices, the proper combinatorial relations are deduced for any size lattice. T h e results of course are the same as the 1-D Ising model. A n interesting boundary value problem is studied which seems to yield infinite volume results for finite lattices. Identical results for other Z u groups were found using the M o n t e Carlo

technique.

1. Introduction The Z N group with elements U n = exp (2nrci/N) where n = 0, 1, 2 . . . . . N 1 has been used for m a n y purposes in physics, Today much interest centers around the use of groups such as Z N in conjunction with lattices1). W h e n considering spin problems, the group elements are associated with the lattice points while gauge field problems use the lines linking the points. The Z 2 model finds application in the one-, two-, and three-dimensional Ising models. We illustrate the exact solution to the 1-D Ising chain below. The exact solution to the 2-D Ising model is known and a numerical solution exists for the 3-D model. The corresponding two-dimensional Z 2 gauge model exact solution is given below. Higher-dimensional Z 2 models and their numerical solutions have been extensively studied1). The notion of an exact solution for a model centers around the possibility of determining in closed f o r m the partition function 0378-4371/87 / $03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

FINITE SIZE EFFECTS WITH 2-D Z N LATTICE GAUGE MODELS

Z(/3) = E g(c) e -t~E~ .

97 (1)

c

g(c) is the number of configurations which give energy E c and/3 is a parameter we can think of as ( k T ) -1. Average values of quantities can be found by

(Q) =

z-1E

Q c g ( c ) e -~Ec .

(2)

c

The computational problems enter when one tries to find g(c) from all the possible configurations which are formed by an arrangement of the Un on the lattice sites for spin problems or on the links for gauge field problems. The 1-D spin chain with Z 2 is the easiest model to illustrate the above method2). Consider a line of N zeros and ones . . . 0 0 1 1 0 1 0 1 1 . . . . Let 0 represent spin down and 1 represent spin up. To obtain the energy for this set, one usually considers the energy to be - J for pairs with spins aligned and + J for pairs with opposing spins. By counting up the opposing spin pairs and subtracting the n u m b e r of aligned spin pairs one gets the energy in units of J. The 2 u possible configurations are sorted into equal energy sets and g(c) identified by the numbers in each set. There are other ways of doing this and certainly the 0, 1 structure of this representation suggests a computer technique using bit structures. The result is that a simple solution is found3), namely that the sets are found to be grouped so that the number of configurations for the kth energy is simply the binomial coefficient (U--~), where k = 1, 2 , . . . , N. Using this distribution of configurations, one finds

(3)

Z = 2 N cosh N 1(/3J) .

When periodic boundary conditions are imposed (i.e. S(1) = S ( N ) where S(k) represents the value 0 or 1 at point k) one finds 2) Z = 2 N- I ( c o s h N - I ( ~ J )

+

sinhU-'(/3J))

(4)

for the cyclic or closed chain of spins2). The specific heat or energy fluctuation is usually of great interest in these computations. As usual C/k =/32((E-

~ E ) ) 2) ,

(5)

( E ) ~- - Z - 1 0 Z / O / 3 ,

(6)

( E 2 ) = Z - ' 02Z/0/3 2 .

(7)

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R.L. GIBBS et al.

Now using eq. (3) we find for C / k ( N - 1), the specific heat per unit length (N - 1 = length in lattice units)

C/k(N - 1) = (/3J/cosh(/3J))2.

(8)

The significant point here is that C / k ( N - 1) is independent of N. However, using the Z from eq. (4) leads to a complicated dependence on N for C / k ( N - 1). In section 3, we show exactly the same thing for the lattice gauge Z z model when certain boundary conditions are imposed. We also show through Monte Carlo computations that the other Z u groups exhibit the same behavior. In section 2 the method of directly determining g(c) for the lattice gauge Z 2 in 2-D is described.

2. Z 2 o n a t w o - d i m e n s i o n a l

lattice

The partition function is given as

Z(/3) = ~ g(E,) e -¢E' ,

(9)

l

w h e r e / 3 = (J/kT) -~, E~ is one of the allowed energy values, and g(El) is the degeneracy factor. The values of g(Et) are the required quantities for the energies which are easily found. The energy is given as ~)

E=Z(1 -cos(0.),

(10)

P

where reference to fig. 1 is necessary to understand how Op is determined. On the lattice an elementary plaquette, p, is a unit square with the perimeter defined as shown. Links are defined between the lattice points and the perimeter is defined as positive in a counter clockwise direction. The plaquette angle is defined as 1)

Op = 2 ~ r ( N 1 + N 2 - N 3 - N 4 ) / N , where N N=2so Once Zp = B)

(11)

determines the Z u model and N k = 0, 1 , . . . , N - 1. In the Z 2 model theN k~{0,2}. all possible combinations (2 2B for periodic boundary conditions, are found, the g(Et) can be found.

FINITE SIZE EFFECTS WITH 2-D Z N LATTICE GAUGE MODELS

99

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Fig. 1. A lattice with B = 49. N~, N 2, N 3, N4 are the link variables required to define

Op.

To get definite expressions for Z in eq. (1), the boundary conditions on the lattice must be stated. Usually, periodic boundary conditions are prescribed to minimize the finite size effects (case (a)). However, other conditions can also be studied as shown below. Once the boundary conditions are settled upon, then the g(Et)'s are found by generating all possible configurations by putting a one or zero on each link (maintaining the boundary conditions). It is easy to demonstrate that if there are B elementary plaquettes, then there are 228 possible configurations. The lowest energy possible for a configuration is zero and the largest is 2B. We wrote a machine language program for a Commodore 64 to generate these configurations. By generating configurations for different lattice sizes, a process of induction allows one to find g(Et). This is done by counting the number of times each Ez(0 ~< E ~<2B) occurs in a configuration and then doing this for each of the 22B configurations. The result is

(~) is the binomial coefficient, a I / b ( a - b ) ! , 1 = O, 1 . . . . . B/2. Using this result, Z=

~t (2/)2B+1

e-4t°=

B = n u m b e r of plaquettes and

2B((1 + e-2t~)B + ( 1 -

e-2B)~) "

(13)

The specific heat determined using eqs. (5), (6), (7) is shown in figs. 2 and 3 for various B.

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R . L . G I B B S et al.

JB.6 SPECIFIC HEAT

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FINITE SIZE EFFECTS WITH 2-D Z u LATTICE GAUGE MODELS

101

T h r e e other cases can be easily treated which correspond to the bottom edge (b), bottom and top edge (c) and all edges (d) of the lattice having the link variables set equal to zero. The corresponding g(Et)'s are Case (b): g(El) = ( f ) , where l = 0, 1, 2 , . . . , B, see fig. 4, Z = ~ ( B ) e-2~¢ = (1 + e-2~)B ;

Case (c): same as case (a); Case (d): same as case (a). Finally, the relationship of this model to the linear spin model can be explicitly shown. If the energy in eq. (10) is renormalized by an additive term equal to B then eq. (13) becomes Z = 22B{coshBfl + sinh~fl} which is within an unimportant constant the same Z found for the cyclic one-dimensional Ising model 2) and is of course essentially the same as eq. (4). This connection has been studied in detail for generalized Ising models 4) and in regard to gauge-invariant Ising modelsS). It has been shown here explicitly the correspondence between B plaquettes on the Z 2 lattice and B spin sites of the 0.6

SPECIFIC HEAT

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Fig. 4. Specificheat vs. ft. Two opposite sides have periodic boundary conditions, one side has all link values set to zero, and the fourth side has a free variation of link values. The curve is independent of lattice size.

102

R,L. GIBBS et al.

one-dimensional cyclic Ising model. This relationship between gauge and spin lattice problems has been discussed in the context of phase transitions6).

3. R e s u l t s f o r Z N The previous section showed how the Z 2 lattice gauge partition function could be directly determined. W h e n the boundary conditions (BC = 1) were periodic, definite size effects were quite obvious as seen in figs. 2 and 3. As the n u m b e r of plaquettes, B, increases the curves approach that of fig. 4 which has boundary conditions (BC = 2) specified as one side free, one side set to zero, and two sides with periodic assignments of the link variables. In analogy with the results for the 1-D Ising model discussed in section 1 we call the results for Z 2 with B C = 2 the infinite size result and B C = 1 the finite size results. Monte Carlo results are shown in fig. 5 for Z 2 with BC = 2. The solid line represents the exact results. It is quite obvious the fit is excellent. The question naturally arises whether this is a result somehow connected to Z 2 alone due to the simplicity of zero or one as a link variable. Figs. 6 and 7 show that for Z 3 and Z 4 the same result is confirmed with MC calculations. In 8.6

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104

R.L. GIBBS et al.

each figure we have also included the exact results for Z 2 with BC = 1 (solid curve) and BC = 2 (dashed curve). The points designated by crosses show that size effects occur for the Z 3 and Z 4 exactly as with the Z 2 model. The dots form the curves we determine (BC = 2) to be the infinite size results. We have in fact verified these results for Z N up to Z 8. The curves formed from dots (BC = 2) are found to be the same for any B, hence the designation "infinite size result" even though we do not have exact analytic expressions as we do for Z 2. Finally, what is responsible for this behavior and are the results for the BC = 2 models truly infinite size effects? We feel that these effects are caused by the gauge invariance of the model as explained by Creutz7). In other words when BC = 2 is used, this amounts to gauge fixing and since the energy is gauge invariant we must have r e m o v e d the finite size effects by a proper choice of gauge. Does BC = 2 truly represent infinite size results? Obviously, if the variable requires a path that incloses an area larger than our lattice, the results do not agree with the true infinite lattice results. H o w e v e r , we have tried Z 2 models with actions different f r o m eq. (10) utilizing techniques described in section 2. We find interesting results for BC = 2 but have not been able to identify the g(c) with simple combinatorial factors such as binomial coefficients.

References 1) C. Rebbi, Lattice Gauge Theories and Monte Carlo Simulations (World Scientific, Singapore, 1983). Also, see L.P. Kadanoff, Rev. Mod. Phys. 49 (1967) 267 for an excellent overview of the relationship between quantum field theory and lattice models. 2) H.W. Graben, Am. J. Phys. 45 (1977) 211. 3) K. Huang, Statistical Mechanics (Wiley, New York, 1963), p. 346. 4) F.J. Wegner, J. Math. Phys. 12 (1971) 2259. 5) R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D 11 (1975) 2098. 6) A.A. Migdal, Soy. Phys. JETP 42 (1976) 743. 7) M. Creutz, Phys. Rev. 15 (1977) 1130; Quarks, Gluons, and Lattices (Cambridge Univ. Press, Cambridge, 1983).