Spectral Curves of Non-Hermitean Hamiltonians

Abstract

Recent analytical and numerical work have shown that the spectrum of the random non-hermitean Hamiltonian on a ring which models the physics of vortex line pinning in superconductors is one dimensional. In the maximally non-hermitean limit, we give a simple "one-line" proof of this feature. We then study the spectral curves for various distributions of the random site energies. We find that a critical transition occurs when the average of the logarithm of the random site energy squared vanishes. For a large class of probability distributions of the site energies, we find that as the randomness increases the energy at which the localization-delocalization transition occurs increases, reaches a maximum, and then decreases. The Cauchy distribution studied previously in the literature does not have this generic behavior. We determine the critical value of the randomness at which "wings" first appear in the energy spectrum. For distributions, such as Cauchy, with infinitely long tails, we show that this critical value is infinitesimally above zero. We determine the density of eigenvalues on the wings for any probability distribution. We show that the localization length on the wings diverges linearly as the energy approaches the energy at which the localization-delocalization transition occurs. These results are all obtained in the maximally non-hermitean limit but for a generic class of probability distributions of the random site energies.

36 pages, 5 figures (.ps), LaTex

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• "Single Ring Theorem" and the Disk-Annulus Phase Transition
• Localization-delocalization transition in disordered systems with a direction
• Extracting many body localization lengths with an imaginary vector potential
• Critical statistics for non-Hermitian matrices
• Random Fibonacci Sequences
• Topological delocalization transitions and mobility edges in the nonreciprocal Maryland model
• Spectral duality and distribution of exponents for transfer matrices of block tridiagonal Hamiltonians
• Eigenvalue Repulsion and Eigenfunction Localization in Sparse Non-Hermitian Random Matrices
• Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms
• The Eigenvalues of Tridiagonal Sign Matrices are Dense in the Spectra of Periodic Tridiagonal Sign Operators
• Hatano-Nelson model with a periodic potential
• Correlation-induced phase transitions and mobility edges in an interacting non-Hermitian quasicrystal
• "Lifshitz tails" and extended states in an Imaginary random potential
• Spectrum of a Feinberg-Zee Random Hopping Matrix
• Symmetries of the Feinberg-Zee Random Hopping Matrix
• Probing localization properties of many-body Hamiltonians via an imaginary vector potential
• A class of tridiagonal operators associated to some subshifts
• The numerical range of a class of periodic tridiagonal operators
• The numerical range of a periodic tridiagonal operator reduces to the numerical range of a finite matrix
• The numerical range of some periodic tridiagonal operators is the convex hull of the numerical ranges of two finite matrices
• The numerical range of periodic banded Toeplitz operators