Boom and bust cycles due to pseudospectra of matrices with unimodular spectra
arXiv:2402.19201 · doi:10.1088/1751-8121/aded52
Abstract
We discuss dynamics obtained by increasing powers of non-normal matrices that are roots of the identity, and therefore have all eigenvalues on the unit circle. Naively, one would expect that the expectation value of such powers cannot grow as one increases the power. We demonstrate that, rather counterintuitively, a completely opposite behavior is possible. In the limit of infinitely large matrices one can have an exponential growth. For finite matrices this exponential growth is a part of repeating cycles of exponential growths followed by exponential decays. The effect can occur if the spectrum is different than the pseudospectrum, with the exponential growth rate being given by the pseudospectrum. We show that this effect appears in a class of transfer matrices appearing in studies of two-dimensional non-interacting systems, for a matrix describing the Ehrenfest urn, as well as in previously observed purity dynamics in a staircase random circuit.
10 pages, 5 figures
References in corpus (14)
- Liouvillian Skin Effect: Slowing Down of Relaxation Processes without Gap Closing
- Pseudospectrum and black hole quasi-normal mode (in)stability
- Resolving Discrepancy between Liouvillian Gap and Relaxation Time in Boundary-Dissipated Quantum Many-Body Systems
- Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics
- Hermitian zero modes protected by nonnormality: Application of pseudospectra
- Symmetrized Liouvillian Gap in Markovian Open Quantum Systems
- Fastest local entanglement scrambler, multistage thermalization, and a non-Hermitian phantom
- Solvable non-Hermitian skin effect in many-body unitary dynamics
- Anomalously large relaxation times in dissipative lattice models beyond the non-Hermitian skin effect
- Two-step phantom relaxation of out-of-time-ordered correlations in random circuits
- Topological zero modes and edge symmetries of metastable Markovian bosonic systems
- Phantom relaxation rate of the average purity evolution in random circuits due to Jordan non-Hermitian skin effect and magic sums
- Purity decay rate in random circuits with different configurations of gates
- Two-step relaxation in local many-body Floquet systems