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Bulletin of the American Physical Society

On thermalization in a nonintegrable quantum system: who thermalizes?

HyungWon Kim, Mari Carmen Bañuls, David A. Huse, J. I. Cirac

Published Mar 7, 2014• 0 citations
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Thermalization Processes in Quantum Mechanics

Van A. Ngo, Stephan Haas

Bulletin of the American Physical Society•Mar 21, 2013•0 citations
We study quantum quenches between integrable and nonintegrable hard-core boson models in the thermodynamic limit with numerical linked cluster expansions. We show that while quenches in which the initial state is a thermal equilibrium state of an integrable model and the final Hamiltonian is nonintegrable (quantum chaotic) lead to thermalization, the reverse is not true. While this might appear counterintuitive given the fact that the eigenstates of both Hamiltonians are related by a unitary transformation, we argue that it is generic. Hence, the lack of thermalization of integrable systems is robust against quenches starting from stationary states of nonintegrable ones. Nonintegrable systems thermalize independently of the nature of the initial Hamiltonian.
Understanding how isolated quantum systems thermalize has recently gathered renewed interest almost 100 years after the first work by von Neumann, thanks to the experimental realizations of such systems. Experimental and numerical pieces of evidence imply that nonintegrability of the system plays an important role in thermalization. Nonintegrable systems that conserve energy alone are expected to be effectively described by the (micro)canonical ensemble due to the so-called eigenstate thermalization hypothesis (ETH) in the thermodynamic limit. In contrast, it is expected that stationary states in integrable systems are described not by the canonical ensemble but by the generalized Gibbs ensemble (GGE) due to the existence of many nontrivial conserved quantities. In this thesis, we study thermalization and its mechanism in nonintegrable systems from two perspectives. We first study how well the ETH and its finite-size corrections can be predicted by random matrix theory (RMT). Next, we present our study on the emergence of the GGE in a nonintegrable system with an extensive number of local symmetries.
The eigenstate thermalization hypothesis (ETH) plays a major role in explaining thermalization of isolated quantum many-body systems. However, there has been no proof of the ETH in realistic systems due to the difficulty in the theoretical treatment of thermal energy eigenstates of nonintegrable systems. Here, we write down analytically thermal eigenstates of nonintegrable spin chains. We consider a class of theoretically tractable volume-law states, which we call entangled antipodal pair (EAP) states. These states are thermal, in the most fundamental sense that they are indistinguishable from the Gibbs state with respect to all local observables, with infinite temperature. We then identify Hamiltonians having the EAP state as an eigenstate and rigorously show that some of these Hamiltonians are nonintegrable. Furthermore, a thermal pure state at an arbitrary temperature is obtained by the imaginary time evolution of an EAP state. Our results offer a potential avenue for providing a provable example of the ETH.