TY - JOUR

T1 - Beyond universal behavior in the one-dimensional chain with random nearest-neighbor hopping

AU - Krishna, Akshay

AU - Bhatt, R. N.

N1 - Funding Information:
This project was conceived while R.N.B. was at the Aspen Center for Physics, which is supported by the NSF. It was completed during R.N.B.'s sabbatical stay at IAS, just prior to the passing away of Freeman Dyson, an IAS icon for several decades. R.N.B. acknowledges brief discussions with Kedar Damle, David Huse, Gil Refael, and Thomas Spencer. A.K. thanks Gerasimos Angelatos for useful discussions. We acknowledge support from Department of Energy BES grant DE-SC0002140.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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 ρ(E)∼1/|Eln3|E|| and of the localization length ζ(E)∼|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 ∼exp[-r/r0]. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small t is of the form ∼1/[tlnλ+1(1/t)], characterized by a single, continuously tunable parameter λ>0. We find, using a combination of analytic and numerical methods, that while the universal result applies for λ>2, it no longer holds in the interval 0<λ<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 λ; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below λ=1. For λ<2, the fall-off of the E=0 state at large distances also deviates from the universal result and is of the form ∼exp[-(r/r0)1/λ], which decays faster than an exponential for λ<1.

AB - 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 ρ(E)∼1/|Eln3|E|| and of the localization length ζ(E)∼|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 ∼exp[-r/r0]. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small t is of the form ∼1/[tlnλ+1(1/t)], characterized by a single, continuously tunable parameter λ>0. We find, using a combination of analytic and numerical methods, that while the universal result applies for λ>2, it no longer holds in the interval 0<λ<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 λ; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below λ=1. For λ<2, the fall-off of the E=0 state at large distances also deviates from the universal result and is of the form ∼exp[-(r/r0)1/λ], which decays faster than an exponential for λ<1.

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U2 - 10.1103/PhysRevB.101.224203

DO - 10.1103/PhysRevB.101.224203

M3 - Article

AN - SCOPUS:85086986616

VL - 101

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 22

ER -