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Physical review. B, Condensed matter

Domain-wall renormalization-group study of the random Heisenberg model

W. L. McMillan

Published Jan 1, 1985• 99 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.31.342

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Physical review. B, Condensed matter•Sep 1, 1989•5 citations
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.
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 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.
For the Ising model with Gaussian random coupling of average J0{J}_{0}J0​ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter J0{J}_{0}J0​ can be studied via the size-LLL-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension d=2,3,4,5,6d=2,3,4,5,6d=2,3,4,5,6. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.
The XY\mathrm{XY}XY model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.
The quantum renormalization-group technique recently developed by Chakravarty and the present author is used to study weak random couplings in quantum-mechanical and classical nearest-neighbor Heisenberg ferromagnets in D=2+\ensuremath{\epsilon} dimensions and at low temperatures T. Momentum-shell recursion relations are obtained by considering the renormalization-group transformations of the cumulants of the probability distribution of the random exchange couplings. From a one-loop calculation it is found that, if the second cumulants ${\ensuremath{\sigma}}^{2}$ of the random couplings are smaller than a critical value ${\ensuremath{\sigma}}_{c}^{2}$, the only modification due to randomness in the low-temperature behavior of the order-parameter correlation length \ensuremath{\xi} in two dimensions is a renormalization of the spin stiffness ${\ensuremath{\rho}}_{s}$ according to ${\ensuremath{\rho}}_{s}$(X)=(1-X${)}^{\ensuremath{\varphi}}$${\ensuremath{\rho}}_{s}$(0), where X=${\ensuremath{\sigma}}^{2}$/${\ensuremath{\sigma}}_{c}^{2}$ and \ensuremath{\varphi}=(1/3 within the one-loop approximation. At zero temperature \ensuremath{\xi} diverges for with ${\ensuremath{\nu}}_{x}$=1/D. Precisely at X=1 the critical behavior of the system is characterized by new exponents; the dynamical critical exponent z is given by z=2+\ensuremath{\varphi}/${\ensuremath{\nu}}_{x}$. For D(1-\ensuremath{\varphi})>2 a new phase appears in the T-X phase diagram, and it is argued that this phase is characterized by spin-glass order.