Spectral gap and the exponential localization in general one-particle systems

TL;DR

This paper establishes a refined upper bound relating the spectral gap and localization length in one-particle quantum systems, improving previous bounds and confirming tightness through specific models.

Contribution

It proves a new inequality xi ≤ const. × delta E_0^{-1/2} for one-particle systems, refining the known bound based on the spectral gap and localization length.

Findings

The new bound is quantitatively tight for nearest-neighbor hopping models.

The inequality xi ≤ const. × delta E_0^{-1/2} improves upon the previous xi ≤ const. × delta E_0^{-1}.

The proof uses an inequality related to the uncertainty principle, not the Lieb-Robinson bound.

Abstract

We investigate the relationship between the spectral gap delta E_0 and the localization length xi in general one-particle systems. A relationship for many-body systems between the spectral gap and the exponential clustering has been derived from the Lieb-Robinson bound, which reduces to the inequality xi le const. times delta E_0^{-1} for one-particle systems. This inequality, however, turned out not to be optimal qualitatively. As a refined upper bound, we here prove the inequality xi le const. times delta E_0^{-1/2} in general one-particle systems. Our proof is not based on the Lieb-Robinson bound, but on our complementary inequality related to the uncertainty principle [T. Kuwahara, J. Phys. A: Math. Theor. 46 (2013)]. We give a specific form of the upper bound and test its tightness in the tight-binding Hamiltonian with a diagonal impurity, where the localization length behaves as xi ~ delta E_0^{-1/2}. We ensure that our upper bound is quantitatively tight in the case of nearest-neighbor hopping.