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Journal de physique

Low temperature behaviour of 3-D spin glasses in a magnetic field

Sergio Caracciolo A, Giorgio Parisi A, Stefano Patarnello, Nicolas Sourlas

Published Jan 1, 1990• 55 citations
Publisher

Abstract

We present the results of two sets of numerical simulations of 3-d Ising spin-glasses in the presence of an uniform magnetic field. In the first set, among other quantities, we compute the spin-spin overlap probability distribution P (q ), the link-link overlap probability distribution P e(qe) and the spin-glass susceptibility χ SG for different volumes and temperatures. The results are in good agreement with mean-field behaviour : P(q) and P e(qe) are non trivial and non self-averaging and χSG shows an increase with the linear size L of the system as χSG ∼ Lω. Our estimate is ω = 1,8 ± 0.25 at T = 0.83. In the second set of simulations a small coupling ε is introduced between two copies of the system and the copy overlap Q(ε) is computed as a function of ε. Q(ε) becomes steeper around ε ∼ 0 as L increases, in agreement with the previous set of simulations. Our data seem anyhow incompatible with the alternatives to mean-field theory proposed so far.

DOI
10.1051/jphys:0199000510170187700

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We present the results of two sets of numerical simulations of 3-d Ising spin-glasses in the presence of an uniform magnetic field. In the first set, among other quantities, we compute the spin-spin overlap probability distribution P (q ), the link-link overlap probability distribution P e(qe) and the spin-glass susceptibility χ SG for different volumes and temperatures. The results are in good agreement with mean-field behaviour : P(q) and P e(qe) are non trivial and non self-averaging and χSG shows an increase with the linear size L of the system as χSG ∼ Lω. Our estimate is ω = 1,8 ± 0.25 at T = 0.83. In the second set of simulations a small coupling ε is introduced between two copies of the system and the copy overlap Q(ε) is computed as a function of ε. Q(ε) becomes steeper around ε ∼ 0 as L increases, in agreement with the previous set of simulations. Our data seem anyhow incompatible with the alternatives to mean-field theory proposed so far.
In order to determine if the effects of an overlap length are visible in an Ising Edwards-Anderson spin-glass model, we have performed Monte Carlo simulations that mimic experimental procedures. The spin system was quenched from a random initial condition to a temperature T1{\mathit{T}}_{1}T1​, equilibrated in zero field for a time tw1{\mathit{t}}_{\mathit{w}1}tw1​, whereupon the system was subjected to a perturbation in field \ensuremath{\Delta}H, temperature \ensuremath{\Delta}T, or bond \ensuremath{\Delta}J with a duration tw2{\mathit{t}}_{\mathit{w}2}tw2​. After the perturbation of the original field, the temperature or bonds were restored. After waiting for a time tw3{\mathit{t}}_{\mathit{w}3}tw3​, a weak magnetic field H was applied. The susceptibility \ensuremath{\chi}(t)=[M+{\mathit{M}}_{+}M+​(t)-${\mathit{M}}_{\mathrm{\ensuremath{-}}}$(t)]/2H, where ${\mathit{M}}_{\ifmmode\pm\else\textpm\fi{}}$(t) is the magnetization due to positive or negative field, and the spin autocorrelation function q(t) were studied as functions of time. For small perturbations, \ensuremath{\Delta}HH0{\mathit{H}}_{0}H0​, \ensuremath{\Delta}TT0{\mathit{T}}_{0}T0​, or \ensuremath{\Delta}JJ0{\mathit{J}}_{0}J0​, respectively, the system is undisturbed. For larger perturbations, the correlations built up during tw1{\mathit{t}}_{\mathit{w}1}tw1​ are gradually destroyed. Similar results are found in all three types of simulations and the qualitative features are in agreement with experimental results. These features are found for both two-dimensional and three-dimensional spin-glass systems. The results can be interpreted in the domain models for spin glasses. \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ is found to be strongly dependent on the duration tw2{\mathit{t}}_{\mathit{w}2}tw2​ of the perturbation. If tw1{\mathit{t}}_{\mathit{w}1}tw1​ is constant and tw2{\mathit{t}}_{\mathit{w}2}tw2​ increases, \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ decreases. If tw1{\mathit{t}}_{\mathit{w}1}tw1​ is set equal to tw2{\mathit{t}}_{\mathit{w}2}tw2​ and tw1{\mathit{t}}_{\mathit{w}1}tw1​=tw2{\mathit{t}}_{\mathit{w}2}tw2​ is increased, \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ decreases, in agreement with the expected behavior. \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ was found to increase with increasing temperature T, which is in agreement with experimental results, but contrary to the theoretical expectations.
Excitations of three-dimensional spin glasses are computed numerically. We find that one can flip a finite fraction of an LxLxL lattice with an O(1) energy cost, confirming the mean field picture of a non-trivial spin overlap distribution P(q). These low energy excitations are not domain-wall-like, rather they are topologically non-trivial and they reach out to the boundaries of the lattice. Their surface to volume ratios decrease as L increases and may asymptotically go to zero. If so, link and window overlaps between the ground state and these excited states become ``trivial\'\'.
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.
We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of L3{L}^{3}L3 spins. After averaging over large sets of EA system samples of $3\ensuremath{\le}L\ensuremath{\le}10$, we obtain accurate numbers for distributions p(q)p(q)p(q) of the overlap parameter qqq at very low-temperature TTT. We find $p(0)/T\ensuremath{\rightarrow}0.233(4)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links---between replica pairs---that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of $q\ensuremath{\sim}0$ excitations in the EA model ${d}_{s}\ensuremath{\rightarrow}2.59(3)$ as $T\ensuremath{\rightarrow}0$. This is in contrast with ${d}_{s}\ensuremath{\rightarrow}3$ as $T\ensuremath{\rightarrow}0$ that is predicted by mean-field theory for the macroscopic limit.
Zero-temperature computer simulations are reported for the Sherrington-Kirkpatrick random Ising model of a spin glass, including an external field. Results are presented for the internal field distribution P(H), and for the ground state energy and magnetisation as functions of field. P(H) has a linear rise from H=0 for all external fields. The zero-temperature susceptibility chi (0) is close to unity when equilibrium states are examined, in agreement with Parisi's replica symmetry breaking theory and in conflict with linear response theory. The linear response result chi (0)=0 can be obtained by searching for metastable local energy minima close to the zero-field ground state in configuration space.
Zero-temperature computer simulations are reported for the Sherrington-Kirkpatrick random Ising model of a spin glass, including an external field. Results are presented for the internal field distribution P(H), and for the ground state energy and magnetisation as functions of field. P(H) has a linear rise from H=0 for all external fields. The zero-temperature susceptibility chi (0) is close to unity when equilibrium states are examined, in agreement with Parisi's replica symmetry breaking theory and in conflict with linear response theory. The linear response result chi (0)=0 can be obtained by searching for metastable local energy minima close to the zero-field ground state in configuration space.