Geometry of quantum observables and thermodynamics of small systems

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.

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Cited by in corpus (10)

• Quantum Thermodynamics
• Quench dynamics and relaxation in isolated integrable quantum spin chains
• Quantum heat engines and refrigerators: Continuous devices
• Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One Dimensional Bose Gases
• Achieving the Landau bound to precision of quantum thermometry in systems with vanishing gap
• Local and Quasilocal Conserved Quantities in Integrable Systems
• Statistical mechanics of an integrable system
• Fluctuations of the heat exchanged between two quantum spin chains
• Role of topology in determining the precision of a finite thermometer
• Signatures of Anderson localization and delocalized random quantum states