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arXiv:cond-mat/0509280 · Europhysics Letters (EPL)

Numerical studies of the compressible Ising spin glass

Adam H. Marshall, Bulbul Chakraborty, Sidney R. Nagel

Published Apr 4, 2006• 4 citations
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Abstract

We study a two-dimensional compressible Ising spin glass at constant volume. The spin interactions are coupled to the distance between neighboring particles in the Edwards-Anderson model with ± J interactions. We find that the energy of a given spin configuration is shifted from its incompressible value, E 0 , by an amount quadratic in E 0 and proportional to the coupling strength. We then construct a simple model expressed only in terms of spin variables that predicts the existence of a critical value of the coupling above which the spin-glass transition cannot exist.

DOI
10.1209/epl/i2005-10564-5

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We study a two-dimensional compressible Ising spin glass at constant volume. The spin interactions are coupled to the distance between neighboring particles in the Edwards-Anderson model with ± J interactions. We find that the energy of a given spin configuration is shifted from its incompressible value, E 0 , by an amount quadratic in E 0 and proportional to the coupling strength. We then construct a simple model expressed only in terms of spin variables that predicts the existence of a critical value of the coupling above which the spin-glass transition cannot exist.
Numerical studies of a compressible version of the Ising spin glass in two dimensions are reported. Compressibility is introduced by adding a term that couples the spin-spin interactions and local lattice deformations to the standard Edwards-Anderson model. The relative strength of this coupling is controlled by a single dimensionless parameter $\ensuremath{\mu}$. The time scale associated with the dynamics of the system grows exponentially as $\ensuremath{\mu}$ is increased, and the energy of the compressible system is shifted downward by an amount proportional to $\ensuremath{\mu}$ times the square of the uncoupled energy. This result leads to the formulation of a simplified model that depends solely on spin variables; analysis and numerical simulations of the simplified model predict a critical value of the coupling strength above which the spin-glass transition cannot exist at any temperature.
We present numerical simulations of the four-dimensional Edwards Anderson Ising spin glass with binary couplings. Our results, in the midst of strong finite-size effects, suggest the existence of a spin-glass phase transition. We present a preliminar determination of critical exponents. We discuss spin-glass susceptibilities, cumulants of the overlap and energy overlap probability distributions, finite-size effects, and the behaviour of the disorder-dependent and integrated probability distributions.
By mapping the Edwards-Anderson model onto a nonrandom Ising model we argue that the lower critical dimension d∗{d}^{*}d∗ for the Edwards-Anderson spin-glass is close to 3 and may depend on the probability distribution of the bond strengths. For d>d∗d>{d}^{*}d>d∗, the approach to spin-glass order from the paramagnetic phase is characterized by Ising exponents. At d=d∗d={d}^{*}d=d∗, however, the critical exponents defined for T>TcT>{T}_{c}T>Tc​ are twice the corresponding Ising ones. These results are consistent with recent computer simulations.
The Edwards and Anderson model (1975) for spin glasses is investigated by a real space rescaling transformation on an S= 1 / 2 Ising Hamiltonian with nearest-neighbour-only interactions in three dimensions. A transition to a spin glass ordered phase is found at about half the mean transition temperature, in contrast to earlier work on a two-dimensional model where no transition was found. The results suggest that the lower critical dimensionality, d c , for spin glass order is slightly greater than 2, although possibly d c =2 exactly. The specific heat is calculated and found to exhibit a broad maximum above T c , with no discernable singularity at the transition since the exponent alpha is large and negative.
We characterize numerically the properties of the phase transition of the three-dimensional Ising spin glass with Gaussian couplings and of the low-temperature phase. We compute critical exponents on large lattices. We study in detail the overlap probability distribution and the equilibrium overlap-overlap correlation functions. We find a clear agreement with off-equilibrium results from previous work. These results strongly support the existence of a continuous spontaneous replica symmetry breaking in three-dimensional spin glasses.