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Physical review. B, Condensed matter

Domain-wall renormalization-group study of the two-dimensional random Ising model

W. L. McMillan

Published Apr 1, 1984• 180 citations
Publisher

Abstract

The distribution of domain-wall free energies is calculated with the use of a transfer-matrix approach for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall free-energy distribution as well as the susceptibility when the lattice parameter is changed. The fixed points, critical exponents, and phase diagram are determined for the two-dimensional random Ising model with a Gaussian distribution of nearest-neighbor interactions.

DOI
10.1103/physrevb.29.4026

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The distribution of domain-wall free energies is calculated with the use of a transfer-matrix approach for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall free-energy distribution as well as the susceptibility when the lattice parameter is changed. The fixed points, critical exponents, and phase diagram are determined for the two-dimensional random Ising model with a Gaussian distribution of nearest-neighbor interactions.
The three-dimensional random Ising model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall free energies at finite temperature is calculated with the use of the Metropolis Monte Carlo algorithm for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall free-energy distribution when the lattice parameter is changed. The spin-glass transition temperature is found to be Tg{T}_{g}Tg​=(1.0\ifmmode\pm\else\textpm\fi{}0.2)J\ifmmode \tilde{}\else \~{}\fi{} where J\ifmmode \tilde{}\else \~{}\fi{} is the variance of the Gaussian interaction distribution. The thermal exponent is \ensuremath{\nu}=1.8\ifmmode\pm\else\textpm\fi{}0.5 and the heat-capacity exponent is \ensuremath{\alpha}=-3.4\ifmmode\pm\else\textpm\fi{}1.5. The heat capacity exhibits a rounded peak at higher temperatures.
The three-dimensional random Ising model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall energies at zero temperature is calculated using a Monte Carlo-quench algorithm to find the ground-state energy for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall energy distribution when the lattice parameter is changed. In the strong-coupling regime (zero temperature) the model iterates toward strong coupling and therefore exhibits a spin-glass phase transition at nonzero temperature. The thermal exponent is $\ensuremath{\nu}=3.0\ifmmode\pm\else\textpm\fi{}1.0$ and the heat capacity exponent is $\ensuremath{\alpha}=\ensuremath{-}7\ifmmode\pm\else\textpm\fi{}3$.
The random Heisenberg model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall energies at zero temperature is calculated using a spin-quench algorithm to find the ground-state energy for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall energy distribution when the lattice parameter is changed. In the strong-coupling regime (zero temperature) the model iterates toward weak coupling and therefore exhibits a ``phase transition at zero temperature'' in both two and three dimensions. The correlation-length exponent is \ensuremath{\nu}=0.714\ifmmode\pm\else\textpm\fi{}0.015 in two dimensions and \ensuremath{\nu}=1.54\ifmmode\pm\else\textpm\fi{}0.19 in three dimensions. The lower critical dimension is four.
Using a recently proposed new renormalization group method (tensor renormalization group), we analyze the Ising model on the two-dimensional square lattice. For the lowest-order approximation with two-domain wall states, it realizes the idea of coarse graining of domain walls. We write down explicit analytic renormalization transformation and prove that the picture of the coarse graining of the physical domain walls does hold for all physical renormalization group flows. We solve it to get the fixed point structure and obtain the critical exponents and the critical temperature. These results are very near to the exact values. We also briefly report the improvement using four-domain wall states.
Analytic phenomenological scaling is carried out for the random field Ising model in general dimensions d using a bar geometry. Domain wall configurations and their decorated profiles and associated wandering and other exponents (ζ,γ,δ,μ) are obtained by free-energy minimization. Scaling between different bar widths provides the renormalization group (RG) transformation. Its consequences are i) criticality at h = T = 0 in d ⩽ 2 with correlation length ξ( h , T ) diverging like ξ( h ,0) ∝ h −2/(2 − d ) for d < 2 and ξ( h ,0) ∝ exp [1/( c 1 γ h γ )] for d = 2, where c 1 is a decoration constant; ii) criticality in d = 2 + ε dimensions at T = 0, h * = (ε/2 c 1 ) 1/γ , where ξ ∝ [( s − s *)/ s ] −2ε/γ , s ≡ h γ . Finite-temperature generalizations are outlined. Numerical transfer matrix calculations and results from a ground-state algorithm adapted for strips in d = 2 confirm the ingredients which provide the RG description.
We represent the spin configuration of 2-dimensional Ising model by the do- main wall configuration defined on the dual link and formulate the domain wall renormalization group (DWRG) according to the tensor network renormalization technique.In this report, DWRG is extended to include the external magnetic field by introducing the oriented domain wall variables.The renormalization group transformation is obtained in 6 dimensional parameter space completely analyti- cally and its eigenvalues around the non-trivial fixed point is calculated to give the magnetic susceptibility exponent.