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Physical review. B, Condensed matter

Sensitivity to boundary conditions of Ising spin-glasses

Marek Cieplak, Jayanth R. Banavar

Published Jan 1, 1983• 19 citations
Publisher

Abstract

Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions in two random two-dimensional Ising systems at $T=0$ are presented. The spin-glass with a Gaussian distribution of exchange couplings is characterized by an algebraic-size dependence of this sensitivity. The results may be interpreted to yield a lower critical dimensionality of 4 for this system. For the system with a $\ifmmode\pm\else\textpm\fi{}J$ probability distribution for the exchange couplings we expect an exponential-size dependence of the sensitivity but our data are inconclusive.

DOI
10.1103/physrevb.27.293

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Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions in two random two-dimensional Ising systems at T=0T=0T=0 are presented. The spin-glass with a Gaussian distribution of exchange couplings is characterized by an algebraic-size dependence of this sensitivity. The results may be interpreted to yield a lower critical dimensionality of 4 for this system. For the system with a \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ probability distribution for the exchange couplings we expect an exponential-size dependence of the sensitivity but our data are inconclusive.
Results of numerical investigations of the scaling properties of the equilibrium sensitivity to changes in the boundary conditions for two-dimensional and four-dimensional Heisenberg spin-glasses at T=0T=0T=0 are presented. The results are consistent with an algebraic size dependence of this sensitivity. They suggest that the two-dimensional system has a zero-temperature phase transition and that the four-dimensional system behaves as if it were at its lower critical dimensionality.
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.
A new equilibrium characterization of the spin-glass-paramagnet transition is presented. It is based on the size dependence of the sensitivity of the free energy to the boundary conditions. Results of numerical studies of a three-dimensional Heisenberg spin-glass at T=0T=0T=0 are consistent with an algebraic decay of this sensitivity and suggest a zero-temperature phase transition.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2{J}^{2}J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for nonfree boundary conditions is governed by the stiffness exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282$ describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.
We use finite size scaling to study Ising spin glasses in two spatial dimensions. The issue of universality is addressed by comparing discrete and continuous probability distributions for the quenched random couplings. The sophisticated temperature dependency of the scaling fields is identified as the major obstacle that has impeded a complete analysis. Once temperature is relinquished in favor of the correlation length as the basic variable, we obtain a reliable estimation of the anomalous dimension and of the thermal critical exponent. Universality among binary and Gaussian couplings is confirmed to a high numerical accuracy.