2020/03/31 by Akshay Krishna, R. N. Bhatt · 10 citations
Mathematics · Physics and Astronomy · #Lambda #Lattice (music) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Physics of Superconductivity and Magnetism #Quantum and electron transport phenomena #Quantum many-body systems #Quantum mechanics #Realization (probability) #Singularity #Statistics #cond-mat.dis-nn #cond-mat.stat-mech #k-nearest neighbors algorithm
paper · pdf · doi:10.1103/physrevb.101.224203
published in Physical review. B./Physical review. B 101(22) (American Physical Society) · 14 pages, 7 figures
arxiv created 2020/03/31 · openalex publication_date 2020/06/15 · arxiv updated 2020/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the one-dimensional nearest-neighbor tight-binding model of electrons with independently distributed random hopping and no on-site potential (i.e., off-diagonal disorder with particle-hole symmetry, leading to sublattice symmetry, for each realization). For nonsingular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states \ensuremathρ(E)\ensuremath∼1/|Eln3|E|| and of the localization length \ensuremathξ(E)\ensuremath∼|ln|E||, near the band center E=0. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain.) Simultaneously, the state at E=0 shows a universal, subexponential decay at large distances \ensuremath∼exp[\ensuremath-√r/r0]. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small t is of the form \ensuremath∼1/[tln^\ensuremathλ+1(1/t)], characterized by a single, continuously tunable parameter \ensuremathλ>0. We find, using a combination of analytic and numerical methods, that while the universal result applies for \ensuremathλ>2, it no longer holds in the interval 0<\ensuremathλ<2. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the nonuniversal parameter \ensuremathλ; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below \ensuremathλ=1. For \ensuremathλ<2, the fall-off of the E=0 state at large distances also deviates from the universal result and is of the form \ensuremath∼exp[\ensuremath-(r/r0)^1/\ensuremathλ], which decays faster than an exponential for \ensuremathλ<1.