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Journal of Physics C Solid State Physics

Transverse freezing in the quantum Ising spin glass: a thermofield dynamic approach

T. K. Kopeć

Published Jan 20, 1988• 50 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/21/2/013

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

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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.

Journal of Physics C Solid State Physics•Nov 30, 1985•88 citations
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 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.
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.
The infinite-range quantum Ising spin glass in transverse ($\ensuremath{\Gamma}$) and longitudinal (hhh) fields is studied by means of thermo-field-dynamics as a substitute for the nnn-replica trick. With the use of a one-loop approximation for the dynamic self-interaction the stability of the mean-field-type solution against the action of fluctuations is investigated in the parameter space (${k}_{B}T, \ensuremath{\Gamma}, h$).
The p-component quantum Potts spin-glass model is considered in the mean-field approximation via the thermofield-dynamics (TFD) method. Quantum effects appear as the action of a transverse field \ensuremath{\Gamma} which represents the tunneling frequency between different discrete Potts states. Applying the Sherrington-Kirkpatrick infinite-interaction-range scheme in the TFD formalism, we find a continuous spin-glass transition for p\ensuremath{\le}6 and discontinuous for p>6 at the field-dependent critical line Tc{\mathit{T}}_{\mathit{c}}Tc​(\ensuremath{\Gamma}), as demonstrated in the jump of the temperature dependence of the spin-glass order parameter. Interestingly, in contrast to the statics the dynamical theory leads to a continuous slowing down as Tc{\mathit{T}}_{\mathit{c}}Tc​(\ensuremath{\Gamma}) is approached from the ergodic side, indicating the existence of the infinitely-long-lived metastable states.
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.
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.