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Journal of Physics A Mathematical and General

Dynamical phase diagram of the van Hemmen model of spin glass

Francisco A. Tamarit, Evaldo M. F. Curado

Published Feb 7, 1994• 8 citations
Publisher

Abstract

The dynamical behaviour of the van Hemmen model of spin glass is studied by comparing the time evolution of two configurations subjected to the same thermal noise. It was found that the system undergoes several dynamical phase transitions among the paramagnetic, ferromagnetic, spin-glass and mixed phases, The dynamical phase diagram seems to be in good agreement with the static one. In the spin glass phase, the system presents a quite different dynamical behaviour from that found in short-range spin glass models as well as in the Sherrington-Kirkpatrick model.

DOI
10.1088/0305-4470/27/3/013

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The dynamical behaviour of the van Hemmen model of spin glass is studied by comparing the time evolution of two configurations subjected to the same thermal noise. It was found that the system undergoes several dynamical phase transitions among the paramagnetic, ferromagnetic, spin-glass and mixed phases, The dynamical phase diagram seems to be in good agreement with the static one. In the spin glass phase, the system presents a quite different dynamical behaviour from that found in short-range spin glass models as well as in the Sherrington-Kirkpatrick model.
The van Hemmen model with a transverse field is studied to describe the quantum Ising spin glass. The free energy and phase diagrams (T(T(T versus $\ensuremath{\Omega},$ and T versus Jo/J,{J}_{o}/J,Jo​/J, where $\ensuremath{\Omega}$ is the transverse field, Jo{J}_{o}Jo​ and J are the ferromagnetic and random exchange interactions, respectively) are calculated for the model with two-peaked and Gaussian exchange distribution. The system presents three ordered phases, namely, spin glass, mixed, and ferromagnetic phases, besides the paramagnetic disordered phase. The influence of the transverse field $\ensuremath{\Omega}$ is to destroy the ordered phases. In the $(T,\ensuremath{\Omega})$ plane our results are compared with those obtained by the replica-symmetry-breaking solution and the same qualitative behavior is observed for the spin-glass transition.
We study the time evolution of two configurations submitted to the same thermal noise for several two dimensional models (Ising ferromagnet, symmetric spin glass, non symmetric spin glass). For all these models, we find a non zero critical temperature above which the two configurations always meet. Using finite size scaling ideas, we determine for these three models this dynamical phase transition and some of the critical exponents. For the ferromagnet, the transition Tc ≃ 2.25 coincides with the Curie temperature whereas for the two spin glass models ( ± J distribution of bonds) we obtain Tc ≃ 1.5-1.7.
We study the competition between ferromagnetic and spin glass phases using a system of coupled infinite-range Ising and Sherrington-Kirkpatrick models. We obtain the replica-symmetric solution for the free energy of this system in terms of magnetization and Edwards-Anderson order parameters in both subsystems. Using these order parameters we are able to identify different phases in the system and determine which phase is dominant for different strengths of the coupling between two subsystems. We observe that both subsystems are in the same phase although with different order parameters. The phase boundary between paramagnetic-ferromagnetic and paramagnetic--spin glass phases is more or less similar to that in the Sherrington-Kirkpatrick phase diagram. But the boundary between ferromagnetic--spin glass phases becomes qualitatively different as the coupling between the two subsystems changes. Remarkably, for intermediate values of the coupling, this phase boundary is such that there could be a reentrant transition between spin glass and ferromagnetic phases by increasing the temperature. We found that for some range of the coupling, the second-order transition between these phases turns into first order at a tricritical point. Further, we carry out the stability analysis by considering deviations from the replica-symmetric solution.
We study the time evolution of the distance between two configurations submitted to the same thermal noise for the 3 d ± J Ising spin glass. We observe three temperature regimes: a high-temperature regime where the distances vanishes in the long-time limit. An intermediate-temperature regime where the distance has a nonzero limit independent of the initial distance. A low-temperature regime where the distance in the long time limit seems to depend upon the initial distance. For the sake of comparison, we have repeated our simulations for the ferromagnetic case.