de Gennes–Suzuki–Kubo Quantum Ising Mean-Field Dynamics: Applications to Quantum Hysteresis, Heat Engines, and Annealing

We briefly review the early development of the mean-field dynamics for cooperatively interacting quantum many-body systems, mapped to pseudo-spin (Ising-like) systems. We start with (Anderson, 1958) pseudo-spin mapping the BCS (1957) Hamiltonian of superconductivity, reducing it to a mean-field Hamiltonian of the XY (or effectively Ising) model in a transverse field. Then, we obtain the mean-field estimate for the equilibrium gap in the ground-state energy at different temperatures (gap disappearing at the transition temperature), which fits Landau’s (1949) phenomenological theory of superfluidity. We then present in detail a general dynamical extension (for non-equilibrium cases) of the mean-field theory of quantum Ising systems (in a transverse field), following de Gennes’ (1963) decomposition of the mean field into the orthogonal classical cooperative (longitudinal) component and the quantum (transverse) component, with each of the component following Suzuki–Kubo (1968) mean-field dynamics. Next, we discuss its applications to quantum hysteresis in Ising magnets (in the presence of an oscillating transverse field), to quantum heat engines (employing the transverse Ising model as a working fluid), and to the quantum annealing of the Sherrington–Kirkpatrick (1975) spin glass by tuning down (to zero) the transverse field, which provides us with a very fast computational algorithm, leading to ground-state energy values converging to the best-known analytic estimate for the model. Finally, we summarize the main results obtained and draw conclusions about the effectiveness of the de Gennes–Suzuki–Kubo mean-field equations for the study of various dynamical aspects of quantum condensed matter systems.