Geometry of quantum observables and thermodynamics of small systems

TL;DR

This paper explores the geometric structure of quantum observables related to ergodicity and thermodynamics in small systems, revealing a conserved quantity that links classical and quantum behaviors during the integrability-ergodicity transition.

Contribution

It introduces a geometric framework for quantum observables that captures ergodic phenomena and optimizes conserved quantities for thermodynamic descriptions of small systems.

Findings

The sum of ensemble variance of temporal average and temporal variance remains constant across the transition.

The geometry encodes eigenstate thermalization and the disappearance of integrals of motion.

Application to optimize conserved quantities in various thermodynamic regimes.

Abstract

The concept of ergodicity---the convergence of the temporal averages of observables to their ensemble averages---is the cornerstone of thermodynamics. The transition from a predictable, integrable behavior to ergodicity is one of the most difficult physical phenomena to treat; the celebrated KAM theorem is the prime example. This Letter is founded on the observation that for many classical and quantum observables, the sum of the ensemble variance of the temporal average and the ensemble average of temporal variance remains constant across the integrability-ergodicity transition. We show that this property induces a particular geometry of quantum observables---Frobenius (also known as Hilbert-Schmidt) one---that naturally encodes all the phenomena associated with the emergence of ergodicity: the Eigenstate Thermalization effect, the decrease in the inverse participation ratio, and the disappearance of the integrals of motion. As an application, we use this geometry to solve a known problem of optimization of the set of conserved quantities---regardless of whether it comes from symmetries or from finite-size effects---to be incorporated in an extended thermodynamical theory of integrable, near-integrable, or mesoscopic systems.