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Physical review. B, Condensed matter

Hopfield spin-glass model in a transverse field

Yu‐qiang Ma, Chang-De Gong

Published Nov 1, 1993• 12 citations
Publisher

Abstract

A quantum Hopfield spin-glass model with transverse fields is presented. A set of equations for the order parameters of a large class of spin-glass models with transverse fields is obtained, and the phase transitions are analyzed with a finite number of patterns p. We find that a reentrant phenomenon is possible for a quantum Mattis model with transverse fields, contrary to the quantum transverse Sherrington-Kirkpatrick-like model. The effects of quantum fluctuations (induced by the transverse field) on the ordered phases of systems with competing interactions are also discussed.

DOI
10.1103/physrevb.48.12778

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Physical review. B, Condensed matter•Sep 1, 1994•1 citation
A quantum Hopfield spin-glass model with transverse fields is presented. A set of equations for the order parameters of a large class of spin-glass models with transverse fields is obtained, and the phase transitions are analyzed with a finite number of patterns p. We find that a reentrant phenomenon is possible for a quantum Mattis model with transverse fields, contrary to the quantum transverse Sherrington-Kirkpatrick-like model. The effects of quantum fluctuations (induced by the transverse field) on the ordered phases of systems with competing interactions are also discussed.
The p-component quantum Potts spin-glass model is considered in the mean-field approximation via the thermo-field-dynamics method. Quantum effects appear as the action of a transverse field \ensuremath{\Gamma} that represents the tunneling frequency between different discrete Potts states. Applying the Sherrington-Kirkpatrick infinite-range-interaction scheme, we estimate the spin-glass phase boundary in the temperature-field (T-\ensuremath{\Gamma}) plane. We find that the transverse field tends to decrease the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​(\ensuremath{\Gamma}). However, it turns out that with an increasing number of states p, a pronounced nonmonotonic temperature dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$(T) appears, resulting in a reentrant behavior in the low-temperature region.
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.
In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of CH/T{C}_{H}/TCH​/T near the glass transition temperature Tg,{T}_{g},Tg​, associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.
A generalization of the p-spin-interaction quantum multicomponent Mattis model in a transverse field is investigated by use of the Suzuki-Trotter approach with the thermodynamic perturbation theory. The phase diagram is obtained in the limit p\ensuremath{\rightarrow}\ensuremath{\infty}, and the effects of quantum fluctuations on the first-order phase transitions separating the disordered phase from the ordered phases are discussed. We find that paramagnetic phase is characterized by transverse ordering, whereas the ordered phases are not affected by the quantum fluctuations.