Sample-size dependence of the ground-state energy in a one-dimensional localization problem

TL;DR

This paper investigates how the average ground-state energy in a one-dimensional localization model depends on sample size, revealing a stretched-exponential decay characterized by an exponent linked to the potential's correlations.

Contribution

It provides bounds and exact results for the sample-size dependence of ground-state energy in a 1D localization problem with Gaussian random potentials, including the white noise case.

Findings

Ground-state energy decreases as a stretched-exponential with sample size.

For Gaussian white noise, the exponent z equals 1/3.

The exponent z varies with the correlation structure of the potential.

Abstract

We study the sample-size dependence of the ground-state energy in a one-dimensional localization problem, based on a supersymmetric quantum mechanical Hamiltonian with random Gaussian potential. We determine, in the form of bounds, the precise form of this dependence and show that the disorder-average ground-state energy decreases with an increase of the size $R$ of the sample as a stretched-exponential function, $\exp( - R^{z})$, where the characteristic exponent $z$ depends merely on the nature of correlations in the random potential. In the particular case where the potential is distributed as a Gaussian white noise we prove that $z = 1/3$. We also predict the value of $z$ in the general case of Gaussian random potentials with correlations.