Assigning Temperatures to Eigenstates

Abstract

In the study of thermalization in finite isolated quantum systems, an inescapable issue is the definition of temperature. We examine and compare different possible ways of assigning temperatures to energies or equivalently to eigenstates in such systems. A commonly used assignment of temperature in the context of thermalization is based on the canonical energy-temperature relationship, which depends only on energy eigenvalues and not on the structure of eigenstates. For eigenstates, we consider defining temperature by minimizing the distance between (full or reduced) eigenstate density matrices and canonical density matrices. We show that for full eigenstates, the minimizing temperature depends on the distance measure chosen and matches the canonical temperature for the trace distance; however, the two matrices are not close. With reduced density matrices, the minimizing temperature has fluctuations that scale with subsystem and system size but appears to be independent of distance measure. In particular limits, the two matrices become equivalent while the temperature tends to the canonical temperature.

v2: Updated with changes made for publication in Phys. Rev. E. 17 pages, 16 figures. v1: 15 Pages, 15 figures

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• Designing refrigerators in higher dimensions using quantum spin models
• Structure of the Hamiltonian of mean force
• Resolving the Debate Between Boltzmann and Gibbs Entropy: Relative Energy Window Eliminates Thermodynamic Inconsistencies and Allows Negative Absolute Temperatures
• Inhomogeneous Floquet thermalization
• Taking the temperature of quantum many-body scars
• Thermalization of Quantum Many-Body Scars in Kinetically Constrained Systems