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arXiv:cond-mat/0607089 · Physical Review B

Compressible Ising spin glass: Simulation results

Adam H. Marshall

Published Feb 20, 2007• 2 citations

Abstract

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.

DOI: 10.1103/physrevb.75.054414

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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 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.

We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to

\ensuremath{\mu}

using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain

{T}_{\mathrm{c}}=1.1019(29)

for the critical temperature, $\ensuremath{\nu}=2.562(42)$ for the thermal exponent, $\ensuremath{\eta}=\ensuremath{-}0.3900(36)$ for the anomalous dimension, and $\ensuremath{\omega}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $\ensuremath{\alpha}=\ensuremath{-}5.69(13)$, $\ensuremath{\beta}=0.782(10)$, and $\ensuremath{\gamma}=6.13(11)$. We also compute several universal quantities at

{T}_{\mathrm{c}}

.

We performed numerical simulations of 2D and 3D Edwards-Anderson spin glass models by using the recently developed multicanonical ensemble. Our ergodicity times increase with the lattice size approximately as

V^3

. The energy, entropy and other physical quantities are easily calculable at all temperatures from a single simulation. Their finite size scalings and the zero temperature limits are also explored.

We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size

L

: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter

p

,

p=0.5

and

p=0.7

(up to

L=28

and

L=20

, respectively), and the bond-diluted bimodal model for bond-occupation probability

{p}_{b}=0.45

(up to

L=16

). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at

{p}_{b}=0.7

and

{p}_{b}=0.35

(up to

L=10

) and of the Ising spin glass model with Gaussian bond distribution (up to

L=8

) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.

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.