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Progress of Theoretical Physics

Successive Screw Approximation in Ising Lattice Gauge Theory

Hiroshi Katsumori

Published Jan 1, 1984• 0 citations
Publisher

Abstract

Kramers and Wannier's successive screw technique is applied to the three-dimensional Ising lattice gauge theory. The largest eigenvalue of the transfer matrix is directly calculated on a computer, independently of Monte Carlo simulation. Anomalous behavior of the specific heat implies the existence of a phase transition.
DOI
10.1143/ptp.71.226

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The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.
The eigenvalue problem of three-dimensional Ising model is solved in successive screw approximation on a computer. Suitable extrapolation of the results suggests that the maximum value of the specific heat of the simple cubic lattice tends to infinity, in proportion to the logarithm of the number of spins making up each layer. It does not seem unreasonable to consider that the temperature at which the specific heat has a maximum gradually approaches to the critical temperature predicted from the Padé approximant method as the number of spins in each layer increases.
We investigate the Kramers-Wannier approximation for the three-dimensional (3D) Ising model. The variational state is represented by an effective 2D Ising model, which contains two variational parameters. We numerically calculate the variational partition function using the corner transfer matrix renormalization group (CTMRG) method, and find its maximum with respect to the variational parameters. The value of the calculated transition point, Kc = 0.2184, is only 1.5% less than the true Kc. This result is better than that obtained using the corner transfer tensor renormalization group (CTTRG) approach. The calculated phase transition is mean-field like.
Using the transfer matrices constructed by Kramers and Wannier's successive screw method for the 2-d (l × 蜴) and 3-d (l ×m ×蜴) q state Potts model (2 蠆q 蠆7), we examine the extent to which the presently available supercomputers can provide us with finite size effects and pseudocritical properties of the systems. We estimate the bulk coupling parameters Kc from the specific heat anomaly in two and three dimensions and also from the correlation length anomaly in two dimensions. In two dimensions, our results Kc are in good agreement with those given by the Potts duality formula. In three dimensions, however, we need extended precision computations to obtain the numerical results with sufficient accuracy. We regard the behavior of the exponent µ, which characterizes the relation between the specific heat peak Cv0(l) and the lattice size l, as an indicator to discriminate the order of the phase transition by investigating Cv0(l) for several small values of l.
The exact analytical solution of the three-dimensional (3D) Ising model remains a central unresolved problem in statistical mechanics. While exact solutions exist in one and two dimensions, the three-dimensional case has resisted closed-form treatment. While standard Kramers-Wannier duality fails to yield a self-dual spin model in 3D, instead mapping to a gauge theory, we introduce a quasi-duality transformation by perturbing the isotropic cubic lattice with a slight, localized spatial inhomogeneity. Specifically, this is introduced via the boundary correction term A n ( ? )= δ η,nN ( s η ? s η?N+1 ? ? s η ? s η+1 ? ) , which arises from flattening the 2D planes into 1D row-continuous chains. By enforcing a condition where the introduced inhomogeneity algebraically absorbs the leading-order topological mismatch of the dual gauge fields, we recover an effective self-dual relation. This algebraic solution yields a critical coupling of K c = 1 4 ln( 1+ 2 )≈0.22033 , which remarkably deviates by only ~0.6% from the widely accepted homogeneous Monte Carlo estimation of K c ( MC ) ?0.22165 . We discuss the derivation and outline future applications of this method to study phase transitions in complex, structurally disordered systems. The method provides insight into how weak inhomogeneity restores a form of dual symmetry that approximates the true critical manifold. Implications for critical phenomena, renormalization structure, and complex biological systems operating near criticality are discussed.