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

TL;DR

This work develops and applies the deGennes-Suzuki-Kubo mean-field dynamical framework for cooperatively interacting quantum Ising systems. By combining the de Gennes long-range mean-field decomposition with Suzuki–Kubo dynamics, it yields a general dynamical equation for the local magnetizations and explicit updates, enabling analysis of non-equilibrium phenomena. The authors demonstrate the approach through three applications: quantum hysteresis with dynamic scaling, enhanced efficiency in quantum Ising heat engines approaching Carnot limits, and a fast quantum annealing scheme for the Sherrington–Kirkpatrick spin glass that closely approaches the known ground-state energy. Overall, the framework provides a tractable, continuous-order-parameter method that captures key dynamical features across diverse quantum condensed-matter problems and suggests avenues for further extensions and improved annealing strategies.

Abstract

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 of the BCS (1957) Hamiltonian of superconductivity, reducing it to a mean-field Hamiltonian of XY (or effectively Ising) model in a transverse field. Then we get 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 orthogonal classical cooperative (longitudinal) component and the quantum (transverse) component, with each component following Suzuki--Kubo (1968) mean-field dynamics. Next we discuss its applications to quantum hysteresis in Ising magnets (in presence of an oscillating transverse field), to quantum heat engines (employing transverse Ising model as working fluid), and to the quantum annealing of the Sherrington--Kirkpatrick (1975) spin glass by tuning down (to zero) the transverse field, which provides 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 conclude about the effectiveness of the de Gennes--Suzuki--Kubo mean-field equations for the study of various dynamical aspects of quantum condensed matter systems.

Figures (9)

• Figure 1: A schematic of the dispersion of low energy excitations in a quantum many-body system.
• Figure 2: The variation of scaled loop area $\tilde{A}^x(\equiv A^x \Gamma_a^{-\alpha} T^{\beta})$ with the scaled frequency $\tilde{\omega}\space(\equiv{{\omega} \over {\Gamma_a^{\gamma} T^{\delta}}})$. The inset shows unscaled data ($A^x$) plotted against the frequency ($\omega$) for different values of $\Gamma_a$ and $T$. Adapted from AcharyyaJPhysA1994
• Figure 3: The phase diagram for the dynamical phase transition: below the critical $\Gamma_a^c(T)$ line, indicated by the symbols (o), the order parameter $Q$ acquires a nonzero value in the 'F' phase and $Q=0$ in the 'P' phase above the critical line. Here, for the numerical data $\omega=2\pi\times500$. The continuous curve represents the approximate phase boundary $T = {{\pi \Gamma_a/2} \over {{{\rm sinh}(\pi \Gamma_a/2)}}}$. Adopted from AcharyyaJPhysA1994.
• Figure 4: Schematic diagram of a cycle of the heat engine: It starts from A and returns to A after a clockwise rotation following the strokes AB, BC, CD and DA on the working fluid (here an Ising system) of the engine. For both classical and quantum Ising heat engine, stroke AB corresponds to fixed high temperature $T = T_H$ of the source and stroke CD corresponds to fixed low temperature $T = T_L$ of the sink. In the stroke AB, the transverse field $\Gamma$ changes from $\Gamma_L$ to $\Gamma_H$ for a quantum Ising engine (while for the classical engine the longitudinal field $h$ would change from $h_L$ to $h_H$). During the stroke BC the temperature decreases from $T_H$ to $TL$ (for both kinds engines). During the stroke CD, the transverse field $\Gamma$ changes from $\Gamma_H$ to $\Gamma_L$ for a quantum Ising engine (while the longitudinal field $h$ changes from $h_H$ to $h_L$ for the classical heat engine). In the fourth stroke DA, the temperature of the working fluid changes from $T_L$ to $T_H$ (for both classical and quantum Ising engines)
• Figure 5: Numerically estimated steady state internal energy ($U = {m^{z}}^2 + \Gamma m^x$) of the working fluid (quantum Ising magnet), plotted against time ($t$) over a full steady state cycle (for longitudinal field $h = 0$, and in the presence of transverse field $\Gamma$). Adapted from acharyya3.