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Physical review. B, Condensed matter

Model calculation for the susceptibility of a quantum spin glass

Varsha Banerjee, Sushanta Dattagupta

Published Oct 1, 1994• 9 citations
Publisher

Abstract

We have studied the dynamic susceptibility in a quantum spin glass which is described by an Ising model in a transverse field, the latter introducing quantum tunneling in an otherwise classical problem. An effective single-spin Hamiltonian is obtained on the basis of an extant thermofield dynamic approach. The coupling to a dissipative heat bath is designed to affect thermal as well as quantal fluctuations and to yield Glauber kinetics in the absence of the transverse field. A resolvent expansion of the underlying time-development operator is then set up and the dynamic susceptibility calculated to leading order in perturbation theory. It is shown that the frequency of the peak of the susceptibility shifts towards higher values of \ensuremath{\omega}, its amplitude is reduced and peaks are broadened as quantum effects strengthen. Our results are in qualitative agreement with the experimentally studied crystal-field-split system of ${\mathrm{LiHo}}_{\mathit{x}}$${\mathrm{Y}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{F}}_{4}$ by Wu et al.

DOI
10.1103/physrevb.50.9942

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We have studied the dynamic susceptibility in a quantum spin glass which is described by an Ising model in a transverse field, the latter introducing quantum tunneling in an otherwise classical problem. An effective single-spin Hamiltonian is obtained on the basis of an extant thermofield dynamic approach. The coupling to a dissipative heat bath is designed to affect thermal as well as quantal fluctuations and to yield Glauber kinetics in the absence of the transverse field. A resolvent expansion of the underlying time-development operator is then set up and the dynamic susceptibility calculated to leading order in perturbation theory. It is shown that the frequency of the peak of the susceptibility shifts towards higher values of \ensuremath{\omega}, its amplitude is reduced and peaks are broadened as quantum effects strengthen. Our results are in qualitative agreement with the experimentally studied crystal-field-split system of {\mathrm{LiHo}}_{\mathit{x}}$${\mathrm{Y}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{F}}_{4} by Wu et al.
We use quantum Monte Carlo methods and various analytic approximations to study the infinite-range Ising spin-glass model in a transverse field in the disordered phase. We focus on the behavior of the frequency dependent susceptibility of the system above and below the critical field. We establish that, at the quantum critical point, there exists an equivalence between the long-time behavior of this model and that of the single-impurity Kondo model. Our predictions for the long-time dynamics of the model are in good agreement with experimental results on LiHo0.167Y0.833F4{\mathrm{LiHo}}_{0.167}{\mathrm{Y}}_{0.833}{\mathrm{F}}_{4}LiHo0.167​Y0.833​F4​.
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.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.
We study effects of random fluctuations in quantum spin-glass systems. We show that the expression of the magnetic susceptibility is divided into the classical and quantum parts. The former can be analyzed by the standard replica method with the static approximation. The latter incorporates dynamical correlations and is determined by the energy gap distribution. By using the many-body interacting Ising spin-glass model with transverse field, we discuss possible mechanisms of the quantum phase transitions.
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.