Non-Hermitian spectra and Anderson localization

Abstract

The spectrum of exponents of the transfer matrix provides the localization lengths of Anderson's model for a particle in a lattice with disordered potential. I show that a duality identity for determinants and Jensen's identity for subharmonic functions, give a formula for the spectrum in terms of eigenvalues of the Hamiltonian with non-Hermitian boundary conditions. The formula is exact; it involves an average over a Bloch phase, rather than disorder. A preliminary investigation of non-Hermitian spectra of Anderson's model in D=1,2 and on the smallest exponent is presented.

8 pages, 10 figures

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• Exact non-Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice
• Non-Hermitian Topological Phases: Principles and Prospects
• Transport and spectral features in non-Hermitian open systems
• Non-Hermitian skin effect beyond the tight-binding models
• Dimerization induced mobility edges and multiple reentrant localization transitions in non-Hermitian quasicrystals
• Spontaneous PT symmetry breaking in non-Hermitian coupled cavities array
• Non-Hermitian Floquet Topological Matter -- A Review
• Dynamical evolution in a one-dimensional incommensurate lattice with $\mathcal{P}\mathcal{T}$ symmetry
• Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase
• Driving-induced multiple $\mathcal{P}\mathcal{T}$-symmetry breaking transitions and reentrant localization transitions in non-Hermitian Floquet quasicrystals
• Non-Abelian generalization of non-Hermitian quasicrystal: PT-symmetry breaking, localization, entanglement and topological transitions
• Erratic non-Hermitian skin localization
• Non-Hermiticity induces localization: good and bad resonances in power-law random banded matrices
• Tight-binding lattices with an oscillating imaginary gauge field
• Transport properties in a Non-Hermitian triple-quantum-dot structure
• Interplay between Fano resonance and $\mathcal{P}\mathcal{T}$ symmetry in non-Hermitian discrete systems
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• $\mathcal{P}\mathcal{T}$ symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model
• Modulational Instability and Dynamical Growth Blockade in the Nonlinear Hatano-Nelson Model
• Exact complex mobility edges and flagellate spectra for non-Hermitian quasicrystals with exponential hoppings
• Scale-free localization versus Anderson localization in unidirectional quasiperiodic lattices
• Lifshitz tail states in non-Hermitian disordered photonic lattices
• Spectral Curves and Localization in Random Non-Hermitian Tridiagonal Matrices
• Phase transitions in non-Hermitian superlattices
• Sparse Random Block Matrices
• Edge states and persistent current in a PT-symmetric extended Su-Schrieffer-Heeger model with generic boundary conditions
• Counting the exponents of single transfer matrices