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Communications in Theoretical Physics

Pair Approximation Method Applied to a Quantum Transverse Ising Spin-Glass Model

Yu‐qiang Ma, Zhen‐Ya Li

Published Mar 1, 1991• 0 citations
Publisher

Abstract

The quantum version of the Sherrington-Kirkpatrick model in a transverse field is investigated by combining the pair approximation with the discrete path-integral representation. The free energy and the field-dependent freezing temperature of the model are obtained, and the classical result for the vanishing of the transverse field can be reproduced.

DOI
10.1088/0253-6102/15/2/251

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The quantum version of the Sherrington-Kirkpatrick model in a transverse field is investigated by combining the pair approximation with the discrete path-integral representation. The free energy and the field-dependent freezing temperature of the model are obtained, and the classical result for the vanishing of the transverse field can be reproduced.
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.
The effect of a transverse field on the spin-glass freezing transition in the Sherrington-Kirkpatrick version of the quantum Ising model is studied using the thermofield dynamics (the real-time, finite-temperature quantum field theory) due to Takahashi and Umezawa (1975) as a substitute for the 'n-replica trick'. The field-dependent transition line to the spin-glass phase is found and the averaged local static metastable susceptibility in the transverse direction is evaluated in the simple mean-field approximation for the effective dynamic Lagrangian.
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 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.