Equilibrium behaviour of quantum Ising spin glass

A phenomenological theory of the ordered phase of short-range quantum Ising spin glass is developed in terms of droplet excitations, and presented in detail. These excitations have free energies that provene from an interplay between the classical excitation energies ε{lunate}_L, with a broad distribution whose characteristic magnitude grows with length scale L as L^θ, and quantum tunneling rates Γ_L, decreasing faster than exponentially with L. At temperature T = 0, the equal-time spatial correlations due to quantum fluctuations do not show the power-law decay that occurs for T > 0 due to thermal fluctuations, but ather an exponential decay of Ornstein-Zernicke type. At finite, but still very small T, there exists a crossover length scale L^*(T), set by h{stroke}Γ _L^*(T){reversed tilde equals} k_BT, below which the droplets behave quantum-mechanically and above which the behaviour is essentially classical. As for the classical spin glass, only a small fraction of droplets are thermally active at low T; it is shown that many of the low-T static properties are dominated by the thermally active droplets at the crossover length L^*(T). The uniform static linear susceptibility is found to diverge at T = 0 below the lower critical dimension, d_l, and to be finite above d_l. The static nonlinear susceptibility diverges in all dimensions, d. The zero temperature linear ac susceptibility χ(ω) is dominated by droplets at a length scale such that ω is of order the characteristic frequency of the quantum system. The behaviour near to the quantum critical point is discussed within a conventional scaling framework if it is approached at strictly zero temperature as well as from finite T. Implications of the existence of Griffiths singularities at the critical point and the disordered phase are pointed out: In the disordered phase, Griffiths singularities dominate the low-T specific heat and the long-time correlations.