Real spectra in one-dimensional single-band non-Hermitian Hamiltonians

Abstract

In general, the energy spectrum of a non-Hermitian system turns out to be complex, which is not so satisfactory since the time evolution of eigenstates with complex eigenvalues is either exponentially growing or decaying. Here we provide a sufficient and necessary condition of the real spectrum under open boundary conditions for one-dimensional non-Hermitian tight-binding Hamiltonians. The necessity is directly related to the fact that the generalized Brillouin zone in one dimension is a closed loop with the origin in its interior. For some simple models, we demonstrate the sufficiency by analytically determining when the preimage of the characteristic polynomial contains a loop and showing that this loop is just the generalized Brillouin zone itself. For more complex models, the numerical results are shown. Our results indicate that real spectra are more common than one may have expected in non-Hermitian systems and are helpful for designing non-Hermitian models with real spectra.

References in corpus (50)

• Topological Insulators
• Topological insulators and superconductors
• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Edge states and topological invariants of non-Hermitian systems
• Exceptional Topology of Non-Hermitian Systems
• Non-Hermitian Physics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Localization transitions in non-Hermitian quantum mechanics
• Topological phases of non-Hermitian systems
• Symmetry and Topology in Non-Hermitian Physics
• Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
• Topological Origin of Non-Hermitian Skin Effects
• Topological Band Theory for Non-Hermitian Hamiltonians
• Anomalous edge state in a non-Hermitian lattice
• Non-Bloch Band Theory of Non-Hermitian Systems
• Correspondence between winding numbers and skin modes in non-hermitian systems
• Pseudo-Hermitian Representation of Quantum Mechanics
• Non-Hermitian Boundary Modes
• Observation of non-Hermitian bulk-boundary correspondence in quantum dynamics
• Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems
• Vortex Pinning and Non-Hermitian Quantum Mechanics
• Non-Hermitian skin effect and chiral damping in open quantum systems
• Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory
• Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas
• Second-Order Topological Phases in Non-Hermitian Systems
• Non-Hermitian Topological Invariants in Real Space
• Periodic Table for Topological Bands with Non-Hermitian Bernard-LeClair Symmetries
• Universal non-Hermitian skin effect in two and higher dimensions
• Classification of Exceptional Points and Non-Hermitian Topological Semimetals
• Non-Hermitian topological phenomena: A review
• Higher-order non-Hermitian skin effect
• Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics
• Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence
• Exact solution of non-Hermitian systems with generalized boundary conditions: size-dependent boundary effect and fragility of skin effect
• Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition
• Non-Hermitian Hopf-Link Exceptional Line Semimetals
• Skin effect and winding number in disordered non-Hermitian systems
• Non-Hermitian Edge Burst
• Knotted Non-Hermitian Metals
• Continuous Phase Transition without Gap Closing in Non-Hermitian Quantum Many-Body Systems
• Many-body topology of non-Hermitian systems
• Symmetry breaking and spectral structure of the interacting Hatano-Nelson model
• Non-Hermitian Skin Effects in Hermitian Correlated/Disordered Systems: Boundary-Sensitive/Insensitive Quantities and Pseudo Quantum Number
• Non-Hermitian skin effect beyond the tight-binding models
• Hermitian zero modes protected by nonnormality: Application of pseudospectra
• Localization in non-Hermitian quantum mechanics and flux-line pinning in superconductors
• Real spectra in non-Hermitian topological insulators
• Non-Hermitian spatial symmetries and their stabilized normal and exceptional topological semimetals
• Chiral symmetry in non-Hermitian systems: product rule and Clifford algebra

Cited by in corpus (5)

• Anderson localization versus hopping asymmetry in a disordered lattice
• Nontrivial worldline winding in non-Hermitian quantum systems
• Multifractal statistics of non-Hermitian skin effect on the Cayley tree
• Topology of the generalized Brillouin zone of one-dimensional models
• Recurrence method in Non-Hermitian Systems