# Entanglement entropy of finite-temperature pure quantum states and thermodynamic entropy as a Noether invariant

**Speaker:**Sho Sugiura, University of Tokyo

**When:**August 16, 2017 (Wed), 11:00AM to 12:00PM (add to my calendar)

**Location:**SCI 352

**Hosted by:**Anatoli Polkovnikov

*This event is part of the Condensed Matter Theory Seminar Series. *

Entropy is a fundamental concept in statistical mechanics. In this talk, I investigate two aspects of entropy. First, we reveal a universal behavior of entanglement entropy of the pure quantum states which are in thermal equilibrium states. Second, we characterize thermodynamic entropy as a Noether invariant under a non-uniform time translation.

In the first part, we focus on the size dependence of the entanglement entropy of the pure quantum states which can fully describe thermal equilibrium as long as one focuses on local observables. The thermodynamic entropy can also be recovered as the entanglement entropy of small subsystems. When the size of the subsystem increases, however, quantum correlations break the correspondence and mandate a correction to this simple volume-law. The elucidatione of the size dependence of the entanglement entropy is thus of essential importance in linking quantum physics with thermodynamics, and in analyzing recent experiments on ultra-cold atoms. In this talk, we derive an analytic formula of the entanglement entropy for a class of pure states called cTPQ states representing thermal equilibrium. We find that our formula applies universally to any sufficiently scrambled pure state representing thermal equilibrium, i.e., general energy eigenstates of non-integrable models and states after quantum quenches. Our universal formula can be exploited as a diagnostic tool for chaotic systems; we can distinguish integrable models from chaotic modelsand detect many-body localization with high accuracy.

In the second part, we investigate a thermally isolated quantum many-body system with an external control represented by a time-dependent parameter. We formulate a path integral in terms of thermal pure states and derive an effective action for trajectories in a thermodynamic state space, where the entropy appears with its conjugate variable. In particular, for quasi-static operations, the symmetry for the uniform translation of the conjugate variable emerges in the path integral. This leads to the entropy as a Noether invariant.