Imaginary-time replica formalism study of a quantum sphericalp-spin-glass model

Abstract

We study a quantum extension of the spherical $p$-spin-glass model using the imaginary-time replica formalism. We solve the model numerically and we discuss two analytical approximation schemes that capture most of the features of the solution. The phase diagram and the physical properties of the system are determined in two ways: by imposing the usual conditions of thermodynamic equilibrium and by using the condition of marginal stability. In both cases, the phase diagram consists of two qualitatively different regions. If the transition temperature is higher than a critical value $T^{\star}$, quantum effects are qualitatively irrelevant and the phase transition is {\it second} order, as in the classical case. However, when quantum fluctuations depress the transition temperature below $T^{\star}$, the transition becomes {\it first order}. The susceptibility is discontinuous and shows hysteresis across the first order line, a behavior reminiscent of that observed in the dipolar Ising spin-glass LiHo$_x$Y$_{1-x}$F$_4$ in an external transverse magnetic field. We discuss in detail the thermodynamics and the stationary dynamics of both states. The spectrum of magnetic excitations of the equilibrium spin-glass state is gaped, leading to an exponentially small specific heat at low temperatures. That of the marginally stable state is gapless and its specific heat varies linearly with temperature, as generally observed in glasses at low temperature. We show that the properties of the marginally stable state are closely related to those obtained in studies of the real-time dynamics of the system weakly coupled to a quantum thermal bath. Finally, we discuss a possible application of our results to the problem of polymers in random media.