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

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.

10 pages; v2: couple of refs. and a comment on the brick-wall added

References in corpus (11)

• Making Sense of Non-Hermitian Hamiltonians
• The physics of exceptional points
• Topological Origin of Non-Hermitian Skin Effects
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Scrambling in Random Unitary Circuits: Exact Results
• Quantum anomaly, non-Hermitian skin effects, and entanglement entropy in open systems
• Exact convergence times for generation of random bipartite entanglement
• Emergence of typical entanglement in two-party random processes
• Hermitian zero modes protected by nonnormality: Application of pseudospectra
• R-DMFT study of a non-Hermitian skin effect for correlated systems: analysis based on a pseudo-spectrum
• Two-step phantom relaxation of out-of-time-ordered correlations in random circuits

Cited by in corpus (18)

• Topological Non-Hermitian skin effect
• Does provable absence of barren plateaus imply classical simulability?
• Non-Hermitian strongly interacting Dirac fermions: a quantum Monte-Carlo study
• Anomalously large relaxation times in dissipative lattice models beyond the non-Hermitian skin effect
• Topological zero modes and edge symmetries of metastable Markovian bosonic systems
• Momentum dependent quantum Ruelle-Pollicott resonances in translationally invariant many-body systems
• Quantum criticality in interacting bosonic Kitaev-Hubbard models
• Phantom relaxation rate of the average purity evolution in random circuits due to Jordan non-Hermitian skin effect and magic sums
• The Interplay of Finite and Infinite Size Stability in Quadratic Bosonic Lindbladians
• Numerical instability of non-Hermitian Hamiltonian evolutions
• Purity decay rate in random circuits with different configurations of gates
• Two-step relaxation in local many-body Floquet systems
• Theory of Irreversibility in Quantum Many-Body Systems
• Non-thermal eigenstates and slow relaxation in quantum Fredkin spin chains
• Mixedness timescale in non-Hermitian quantum systems
• Absorption to Fluctuating Bunching States in Non-Unitary Boson Dynamics
• Boom and bust cycles due to pseudospectra of matrices with unimodular spectra
• Recurrence method in Non-Hermitian Systems