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Journal of Physics A Mathematical and General

The thermodynamics of site-random mean-field quantum spin systems

Nick G. Duffield, Reimer Kühn

Published Nov 7, 1989• 5 citations
Publisher

Abstract

The authors provide a general scheme for the treatment of the thermodynamics of mean-field site-random quantum spin systems, including systems where bond randomness is expressed as an underlying site randomness. They use the method to find the phase structure of a mean-field Heisenberg model in a random field, and of a mean-field Heisenberg spin glass model.

DOI
10.1088/0305-4470/22/21/025

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The authors provide a general scheme for the treatment of the thermodynamics of mean-field site-random quantum spin systems, including systems where bond randomness is expressed as an underlying site randomness. They use the method to find the phase structure of a mean-field Heisenberg model in a random field, and of a mean-field Heisenberg spin glass model.
In realistic spin glasses a fraction c of spins is located at random positions. Their couplings are determined by the relative spin positions. For such systems a field theory is formulated. In a certain mean-field limit it generalizes the Sherrington-Kirkpatrick free energy of a random bond spin glass. In general, an infinity of multistate overlaps occurs, as in the diluted random bond spin glass of Viana and Bray. For small c there occurs a percolation transition for short-range exchange and the "concentration scaling" T G ∼ c for RKKY couplings.
Ground-state properties of the two-dimensional S=1/2S=1/2S=1/2 random Heisenberg models are investigated by the exact-diagonalization method. The phase diagram of the bond-random model (the ±J\pm J±J model) is the same as that of the corresponding Ising spin glass model, in spite of quantum fluctuation. In the site-random model, the spin glass phase exists at least in the ferro-rich case. In comparison with the corresponding Ising model, the Néel phase of this model becomes much narrower in the ferro-rich case, while it is comparable in the antiferro-rich case. KEYWORDS: quantum spin glass, Heisenberg model, ground-state phase transition, site-random model, exact diagonalization
We develop a mean-field theory for random quantum spin systems using the spin coherent state path integral representation. After the model is reduced to the mean-field one-body Hamiltonian, the integral is analyzed with the aid of several methods such as the semiclassical method and the gauge transformation. As an application we consider the Sherrington-Kirkpatrick model in a transverse field. Using the Landau expansion and its improved versions, we give a detailed analysis of the imaginary-time dependence of the order parameters. Integrating out the quantum part of the order parameters, we obtain the effective renormalized free energy written in terms of the classically defined order parameters. Our method allows us to obtain the spin glass-paramagnetic phase transition point $\ensuremath{\Gamma}∕J\ensuremath{\sim}1.62$ at T=0T=0T=0.
Ground-state properties of the two-dimensional S = 1/2 random Heisenberg models are investigated by the exact-diagonalization method. The phase diagram of the bondrandom model (the ±J model) is the same as that of the corresponding Ising spin glass model, in spite of quantum fluctuation. In the site-random model, the spin glass phase exists at least in the ferro-rich case. In comparison with the corresponding Ising model, the Néel phase of this model becomes much narrower in the ferro-rich case, while it is comparable in the antiferro-rich case.