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Journal of Physics C Solid State Physics

Effect of a transverse field on the spin glass freezing in the Sherrington-Kirkpatrick model

H Ishii, Tetsuya Yamamoto

Published Nov 30, 1985• 88 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/18/33/013

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Journal of Physics C Solid State Physics•Dec 30, 1988•31 citations
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.
In the Sherrington-Kirkpatrick model the effect of a transverse field Delta on spin glass freezing is studied. After the static approximation of the replica theory has been analysed, the nonlinear susceptibility is calculated by the perturbation method with respect to exchange coupling J ij up to the tenth order. The freezing temperature and the critical index of the nonlinear susceptibility are evaluated. The result shows that the system remains paramagnetic for Delta >1.506 J (the variance of (J ij ) is J 2 /N), which corrects the conjecture made before by the present authors. It is also shown that the static approximation in the replica theory overestimates T c for a large value of Delta .
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 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.
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).
The Sherrington-Kirkpatrick model under a transverse field is studied here employing the Suzuki-Trotter formula to map the model to an equivalent classical one. The effective Thouless-Anderson-Palmer free energy is used to study the stability of the system, and Monte Carlo computer simulations of the effective classical model are performed to obtain the phase diagram and the magnetization overlap distribution. Our results indicate a trivial overlap distribution due to quantum fluctuations. The phase diagram shows a slight initial increase in the glass transition temperature Tg{T}_{g}Tg​ as the transverse field is switched on, confirming that obtained by Yokota.
The effect of a transverse field Gamma on spin-glass freezing in the Sherrington-Kirkpatrick version of the quantum Ising model is studied by using the thermo-field-dynamic (TFD) method (the real-time finite-temperature quantum-field theory) due to Takahashi and Umezawa as a substitute for the 'n-replica trick'. An exact effective Lagrangian for the average dynamic causal and response functions is derived by using the functional-integral approach. Furthermore, the properties of the Lagrangian are studied in the mean-field limit by considering the infinite-range random exchange. In this limit the self-consistent equations for the matrix components of the TFD causal Green function are derived with the effective thermal Hamiltonian containing the dynamic spin self-interaction. The static properties of this equation are studied above and below the freezing temperature T c by extracting the time-delocalised parts of dynamic autocorrelations. The self-consistent equations for the spin-glass order parameter and local susceptibility are obtained. The classical ( Gamma to 0) and quantum (T to 0) limits are discussed. The crucial role of the choice of the thermal-state condition for the spin-glass problem and its striking similarity to the non-equilibrium formulation of the TFD are pointed out.