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arXiv:2204.02177

A note on adiabatic time evolution and quasi-static processes in translation-invariant quantum systems

Jak\v{s}i\'c, Vojkan, Claude‐Alain Pillet, Clément Tauber

Published Apr 5, 2022• 0 citations
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Abstract

We study the slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice $\mathbb Z^d$. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in [Jak\v{s}i\'c, Pillet, Tauber, arXiv:2204.00440], we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.

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We study the slowly varying, non-autonomous quantum dynamics of a translation invariant spin or fermion system on the lattice Zd\mathbb Z^dZd. This system is assumed to be initially in thermal equilibrium, and we consider realizations of quasi-static processes in the adiabatic limit. By combining the Gibbs variational principle with the notion of quantum weak Gibbs states introduced in [Jak\v{s}i'c, Pillet, Tauber, arXiv:2204.00440], we establish a number of general structural results regarding such realizations. In particular, we show that such a quasi-static process is incompatible with the property of approach to equilibrium studied in this previous work.
We continue the development, started in [AHP 25 (2023), 715-749], of a structural theory of approach to equilibrium in translation-invariant quantum spin systems. Dynamical Steady States (DSS) are defined as w*-limit points of the time-averaged evolution of an initial state. We show that if a DSS is an equilibrium state of a sufficiently regular interaction, then the specific energy of this interaction is necessarily a constant of the motion. Two distinct mechanisms lead to this conclusion, one based on dynamical surface energies and the other on conservation of specific relative entropy. We conjecture that these properties hold for a natural class of interactions. For a DSS admitting a physical equilibrium description, the associated equilibrium states are characterized by a dynamical Gibbs variational principle: they are precisely the minimizers of the specific relative entropy from the initial state over the physical equilibrium states. The achieved minimum equals the increase of specific entropy. The structural theory also yields characterizations of approach to thermal equilibrium, based on two model-dependent questions: restoration of a sufficiently local thermodynamic description at large times and classification of the corresponding local constants of motion. Once locality is restored, the constants of motion determine the possible asymptotic equilibrium states. In particular, if finite-range locality is restored and the energy is the only finite-range constant of the motion, then every DSS is a thermal equilibrium state at the temperature fixed by energy conservation. The finite-time dynamical input underlying this theory is established in a companion paper.
The assumption that quasi-static transformations do not quantitatively alter the equilibrium expectation of observables is at the heart of thermodynamics and, in the quantum realm, its validity may be confirmed by the application of adiabatic perturbation theory. Yet, this scenario does not straightforwardly apply to Bosonic systems whose excitation energy is slowly driven through the zero. Here, we prove that the universal slow dynamics of such systems is always non-adiabatic and the quantum corrections to the equilibrium observables become rate independent for any dynamical protocol in the slow drive limit. These findings overturn the common expectation for quasi-static processes as they demonstrate that a system as simple and general as the quantum harmonic oscillator, does not allow for a slow-drive limit, but it always displays sudden quench dynamics.
Abstract The assumption that quasi-static transformations do not quantitatively alter the equilibrium expectation of observables is at the heart of thermodynamics and, in the quantum realm, its validity may be confirmed by the application of adiabatic perturbation theory. Yet, this scenario does not straightforwardly apply to Bosonic systems whose excitation energy is slowly driven through the zero. Here, we prove that the universal slow dynamics of such systems is always non-adiabatic and the quantum corrections to the equilibrium observables become rate independent for any dynamical protocol in the slow drive limit. These findings overturn the common expectation for quasi-static processes as they demonstrate that a system as simple and general as the quantum harmonic oscillator, does not allow for a slow-drive limit, but it always displays sudden quench dynamics.
The quasi-static evolution of steady states far from equilibrium is investigated from the point of view of quantum statistical mechanics. As a concrete example of a thermodynamic system, a two-level quantum dot coupled to several reservoirs of free fermions at different temperatures is considered. A novel adiabatic theorem for unbounded and nonnormal generators of evolution is proven and applied to study the quasi-static evolution of the nonequilibrium steady state (NESS) of the coupled system. © Birkhäuser Verlag Basel/Switzerland 2007.
We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.