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arXiv:cond-mat/9806346 · Physical Review Letters

Numerical Study of Order in a Gauge Glass Model

J. Michael Kosterlitz, Nobuhiko Akino

Published Nov 23, 1998• 25 citations
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Abstract

The $\mathrm{XY}$ model with quenched random phase shifts is studied by a $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ finite size defect energy scaling method in 2D and 3D. The defect energy is defined by a change in the boundary conditions from those compatible with the true ground state configuration for a given realization of disorder. A numerical technique, which is exact in principle, is used to evaluate this energy and to estimate the stiffness exponent $\ensuremath{\theta}$. This method gives $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.36\ifmmode\pm\else\textpm\fi{}0.01$ in 2D and $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+0.31\ifmmode\pm\else\textpm\fi{}0.015$ in 3D, which are considerably larger than previous estimates, strongly suggesting that the lower critical dimension is less than 3. Some arguments in favor of these new estimates are given.

DOI
10.1103/physrevlett.81.4672

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The XY\mathrm{XY}XY model with quenched random phase shifts is studied by a T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 finite size defect energy scaling method in 2D and 3D. The defect energy is defined by a change in the boundary conditions from those compatible with the true ground state configuration for a given realization of disorder. A numerical technique, which is exact in principle, is used to evaluate this energy and to estimate the stiffness exponent \ensuremathθ\ensuremath{\theta}\ensuremathθ. This method gives \ensuremathθ=\ensuremath−0.36\ifmmode±\else\textpm\fi0.01\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.36\ifmmode\pm\else\textpm\fi{}0.01\ensuremathθ=\ensuremath−0.36\ifmmode±\else\textpm\fi0.01 in 2D and \ensuremathθ=+0.31\ifmmode±\else\textpm\fi0.015\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+0.31\ifmmode\pm\else\textpm\fi{}0.015\ensuremathθ=+0.31\ifmmode±\else\textpm\fi0.015 in 3D, which are considerably larger than previous estimates, strongly suggesting that the lower critical dimension is less than 3. Some arguments in favor of these new estimates are given.
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.
Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal $(\ifmmode\pm\else\textpm\fi{}J)$ distributions of the disorder are calculated using a ``matching''algorithm, which allows large system sizes of up to N=4802{N=480}^{2}N=4802 spins to be investigated. We study domain walls induced by two rather different types of boundary condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy,'' which we denote by $\ensuremath{\Delta}E.$ For Gaussian disorder, we find a power-law behavior $\ensuremath{\Delta}E\ensuremath{\sim}{L}^{\ensuremath{\theta}},$ with $\ensuremath{\theta}=\ensuremath{-}0.266(2)$ and $\ensuremath{\theta}=\ensuremath{-}0.282(2)$ for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension dc.{d}_{c}.dc​. For the $\ifmmode\pm\else\textpm\fi{}J$ model, we obtain a different result, namely that the domain-wall energy saturates at a nonzero value for $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty},$ so $\ensuremath{\theta}=0,$ indicating that the lower critical dimension for the $\ifmmode\pm\else\textpm\fi{}J$ model is exactly dc=2.{d}_{c}=2.dc​=2.
An efficient Monte Carlo method is extended to evaluate directly domain-wall free energy for randomly frustrated spin systems. Using the method, critical phenomena of spin-glass phase transition are investigated in the 4d+/-J Ising model under the replica boundary condition. Our values of the critical temperature and exponent, obtained by finite-size scaling, are in good agreement with those of the standard Monte Carlo and the series expansion studies. In addition, two exponents, the stiffness exponent and the fractal dimension of the domain wall, which characterize the ordered phase, are obtained. The latter value is larger than d-1, indicating that the domain wall is really rough in the 4d Ising spin-glass phase.
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.
We investigate the XY\mathrm{XY}XY spin-glass model in two and three dimensions using the domain-wall renormalization-group method. The results for systems of linear sizes up to L=12L\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}12L=12 (2D) and L=8L\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8L=8 (3D) strongly suggest that the lower critical dimension for spin-glass ordering may be ${d}_{\mathrm{c}}\ensuremath{\approx}3$, rather than 4 as is commonly believed. Our 3D data favor the scenario of a low but finite spin-glass ordering temperature below the chiral transition, but they are also compatible with the system being at or slightly below its lower critical dimension.
With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.