Hermitian zero modes protected by nonnormality: Application of pseudospectra

Abstract

Recently, it was established that there exists a direct relation between the non-Hermitian skin effects, -strong dependence of spectra on boundary conditions for non-Hermitian Hamiltonians-, and boundary zero modes for Hermitian topological insulators. On the other hand, in terms of the spectral theory, the skin effects can also be interpreted as instability of spectra for nonnormal (non-Hermitian) Hamiltonians. Applying the latter interpretation to the former relation, we develop a theory of zero modes with quantum anomaly for general Hermitian lattice systems. Our theory is applicable to a wide range of systems: Majorana chains, non-periodic lattices, and long-range hopping systems. We relate exact zero modes and quasi-zero modes of a Hermitian system to spectra and pseudospectra of a non-Hermitian system, respectively. These zero and quasi-zero modes of a Hermitian system are robust against a class of perturbations even if there is no topological protection. The robustness is measured by nonnormality of the corresponding non-Hermitian system. We also present explicit construction of such zero modes by using a graphical representation of lattice systems. Our theory reveals the presence of nonnormality-protected zero modes, as well as the usefulness of the nonnormality and pseudospectra as tools for topological and/or non-Hermitian physics.

11 pages, 4 figures

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• Quantum anomaly, non-Hermitian skin effects, and entanglement entropy in open systems
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• Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics
• Interaction-induced non-Hermitian topological phases from a dynamical gauge field
• Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification
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• Quantum Simulation of the Bosonic Kitaev Chain
• Quantum-Squeezing-Induced Point-Gap Topology and Skin Effect
• R-DMFT study of a non-Hermitian skin effect for correlated systems: analysis based on a pseudo-spectrum
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• Solvable non-Hermitian skin effect in many-body unitary dynamics
• Non-Hermitian skin effect edge
• Anomalously large relaxation times in dissipative lattice models beyond the non-Hermitian skin effect
• Exponentially growing bulk Green functions as signature of nontrivial non-Hermitian winding number in one dimension
• Topological enhancement of non-normality in non-Hermitian skin effects
• General criterion for non-Hermitian skin effects and Application: Fock space skin effects in many body systems
• Non-normal Hamiltonian dynamics in quantum systems and its realization on quantum computers
• Infernal and Exceptional Edge Modes: Non-Hermitian Topology Beyond the Skin Effect
• Phantom relaxation rate of the average purity evolution in random circuits due to Jordan non-Hermitian skin effect and magic sums
• Hidden zero modes and topology of multiband non-Hermitian systems
• Evolution of spectral topology in one-dimensional long-range nonreciprocal lattices
• Transient amplification in stable Floquet media
• Hermitian Topologies originating from non-Hermitian braidings
• The Interplay of Finite and Infinite Size Stability in Quadratic Bosonic Lindbladians
• Purity decay rate in random circuits with different configurations of gates
• Numerical instability of non-Hermitian Hamiltonian evolutions
• Magnetic Flux Response of Non-Hermitian Topological Phases
• Boundary-dependent dynamical instability of bosonic Green's function: Dissipative Bogoliubov-de Gennes Hamiltonian and its application to non-Hermitian skin effect
• Coexistence of non-Hermitian skin effect and extended states in one-dimensional nonreciprocal lattices
• Real spectra in one-dimensional single-band non-Hermitian Hamiltonians
• Dissolution of the non-Hermitian skin effect in one-dimensional lattices with linearly varying nonreciprocal hopping
• Gaussian eigenstate pinning in non-Hermitian quantum mechanics
• Scaling of pseudospectra in exponentially sensitive lattices
• Boom and bust cycles due to pseudospectra of matrices with unimodular spectra
• Real spectra and phase transition of skin effect in nonreciprocal systems
• Wiener-Hopf factorization and non-Hermitian topology for Amoeba formulation in one-dimensional multiband systems