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Journal of Physics A Mathematical and General

Two-fluid picture of the SK model of a spin glass

M. Gabay, Thomas Garel

Published Dec 1, 1981• 4 citations
Publisher

Abstract

The authors show how the known properties of the Sherrington-Kirkpatrick (SK) model of a spin glass fit naturally into a two-fluid description. The normal fluid is responsible for thermal excitations, whereas the condensed part yields a temperature independent zero field susceptibility at all temperatures below T c . The relevance of the overblocking effect to the SK model is also pointed out.

DOI
10.1088/0305-4470/14/12/032

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The authors show how the known properties of the Sherrington-Kirkpatrick (SK) model of a spin glass fit naturally into a two-fluid description. The normal fluid is responsible for thermal excitations, whereas the condensed part yields a temperature independent zero field susceptibility at all temperatures below T c . The relevance of the overblocking effect to the SK model is also pointed out.
Using Sherrington-Kirkpatric （SK） model we investigated the nonequilibrum properties of spin glass by dynamical Monte Carlo simulation. Our results revealed that the susceptibility in the weak field shows a cusp near the glass transition, while the specific heat shows a more broad transition. The aging and zero-field colding memory effect is also reproduced in our simulation, and the phenomena can be explained by energy relaxation. Our simulation also reveals that homogeneous SK model can not reproduce the exchange bias effect observed in the experiment.
In the Sherrington-Kirkpatrick (SK) model the effect of a transverse field Delta on the spin glass freezing is studied on the basis of the theory of Thouless, Anderson and Palmer (1977). It is shown from the logarithmic singular point of the free energy in the high-temperature region that T c decreases with increasing Delta . To be consistent with this result the mean-field equation is set up for T<T c . Due to the correlation effect spin flips associated with the transverse field are unlikely to occur at low temperatures and the system eventually undergoes the spin glass transition irrespective of the magnitude of Delta , which is characteristic of the SK model.
We consider the quantum Sherrington–Kirkpatrick (SK) spin-glass model with a transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures β > 0. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with Rd-valued spins. This enables us to use the results of Panchenko who generalized [Ann. Probab. 46(2), 829–864 (2018); ibid, 46(2), 865–896 (2018)] the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynamic limit of the free energy of the quantum SK model as the d → ∞ limit of the free energies of the d-dimensional approximations of the model.
We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.
The microscopic probability distribution function of the Sherrington-Kirkpatrick (SK) model of spin glasses is calculated explicitly as a function of time by a high-temperature expansion. The resulting formula to the third order of the inverse temperature shows that an assumption made by Coolen, Laughton and Sherrington in their recent theory of dynamics is violated. Deviations of their theory from exact results are estimated quantitatively. Our formula also yields explicit expressions of the time dependence of various macroscopic physical quantities when the temperature is suddenly changed within the high-temperature region.