Phantom relaxation rate of the average purity evolution in random circuits due to Jordan non-Hermitian skin effect and magic sums

Abstract

Phantom relaxation is relaxation with a rate that is not given by a finite spectral gap. Studying the average purity dynamics in a staircase random Haar circuit and the spectral decomposition of a non-symmetric matrix describing the underlying Markovian evolution, we explain how that can arise out of an ordinary-looking spectrum. Crucial are alternating expansion coefficients that diverge in the thermodynamic limit due to the non-Hermitian skin effect in the matrix describing the average purity dynamics under an overall unitary evolution. The mysterious phantom relaxation emerges out of localized generalized eigenvectors describing the Jordan normal form kernel, and, independently, also out of interesting trigonometric sums due to localized true eigenvectors. All this shows that when dealing with non-Hermitian matrices it can happen that the spectrum is not the relevant object; rather, it is the pseudospectrum, or, equivalently, a delicate cancellation enabled by localized eigenvectors.

arXiv commentv2: slight title change

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Cited by in corpus (8)

• Momentum dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems
• The Interplay of Finite and Infinite Size Stability in Quadratic Bosonic Lindbladians
• Two-step relaxation in local many-body Floquet systems
• Numerical instability of non-Hermitian Hamiltonian evolutions
• Theory of Irreversibility in Quantum Many-Body Systems
• Delocalization of skin steady states
• Absorption to Fluctuating Bunching States in Non-Unitary Boson Dynamics
• Boom and bust cycles due to pseudospectra of matrices with unimodular spectra