Solvable non-Hermitian skin effect in many-body unitary dynamics

TL;DR

This paper investigates the boundary-localized skin effect in many-body quantum dynamics, revealing a transition in purity relaxation rates caused by non-Hermitian effects and the spectrum's discontinuity.

Contribution

It provides a full solution to the dynamics of purity in a bipartite system with random gates, highlighting the role of non-Hermitian skin effects and pseudospectrum in relaxation behavior.

Findings

Discovery of a sudden transition in purity relaxation rate.

Identification of boundary localized eigenmodes as the skin effect.

Demonstration of the importance of pseudospectrum over spectrum for relaxation rates.

Abstract

We study unitary evolution of bipartite entanglement in a circuit with nearest-neighbor random gates. Deriving a compact non-unitary description of purity dynamics on qudits we find a sudden transition in the purity relaxation rate the origin of which is in the underlying boundary localized eigenmodes -- the skin effect. We provide the full solution of the problem, being one of the simplest iterations of two-site matrices, namely, that each is a sum of only two projectors. This leads to rich dynamics influenced by the Jordan normal form of the kernel and, most importantly, a spectrum that is completely discontinuous in the thermodynamic limit. It provides a simple example of how a seemingly innocuous many-body unitary evolution can harbor interesting mathematical effects: an effective non-symmetric Toeplitz transfer matrix description causes a phantom relaxation, such that the correct relaxation rate is not given by the matrix spectrum, but rather by its pseudospectrum.