Non-Equilibrium Dynamics and Weakly Broken Integrability

Scientists still lack a fundamental understanding of how and when complex many-body systems out of equilibrium will relax back to thermal equilibrium. Empirically, many systems do relax readily, justifying many of the core approximations of thermodynamics. In classical physics, however, exceptions include the famous one-dimensional Fermi-Past-Ulam system of oscillators with nonlinear interactions, which persists out of equilibrium indefinitely for many initial conditions. This reflects its status as an integrable system, prevented from relaxing to thermal equilibrium by the existence of many conserved quantities. In the quantum setting, the quantum Newton’s cradle experiment offers a similar example – the absence of thermalization over long timescales in a simple one-dimensional Bose gas with point-like collisional interactions. This is also an integrable system, and here conservation laws lead to conserved quantities associated with distinct effective temperatures, with slow, anomalous thermalization the result. This understanding has stimulated a wealth of recent theoretical activity.
Despite recent advances in the understanding of integrable systems, real physical systems always contain imperfections, which appear as perturbations to ideal models. These perturbations may influence and destabilize the integrable dynamics, but their effect is hard to quantify. In the classical domain, the well-known Kolmogorov-Arnold-Moser (KAM) theorem guarantees the persistence of some quasi-periodic orbits under small perturbations. A useful concept in the quantum many-body domain is that of “pre-thermalization,” as some systems initially relax to a quasi-stationary disequilibrium state before moving on toward thermal equilibrium. Yet insights analogous to KAM theory have been hard to establish, leaving many open questions: To what extent does quantum integrability survive in the presence of weak perturbations? How can we quantify and organize the dynamical effects of integrability-destroying interactions? What are the relevant timescales?
In a new paper, LML External Fellow Joe Bhaseen and colleagues address these questions by developing a quantum kinetic approach to weakly perturbed integrable models out of equilibrium. They show that the dynamics of physical observables from short to long timescales can be described using the exact matrix elements of the underlying integrable model. They also illustrate their findings using numerical evaluation of the key formulae, including the time-evolution of the average densities, quasiparticle distributions, and effective temperatures. Developing their kinetic approach further into a general theory, the researchers also identify dynamical response functions which encode the timescales of thermalization, and describe a family of integrability-breaking, KAM-like perturbations which do not lead to thermalization in the weakly coupled regime. Overall, the researchers believe these results should be useful in helping to clarify the role of perturbations in dynamical quantum systems, with applications to experiments on cold atomic gases and condensed matter.
The paper is available at

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