Extreme statistics of the excitations in the random transverse Ising chain

TL;DR

This paper investigates the distribution of low-energy excitations in the disordered Griffiths phase of the random transverse Ising chain, revealing that weak correlations between excitations are not entirely negligible, challenging assumptions of their irrelevance.

Contribution

The study provides high-precision numerical analysis showing that correlations between excitations influence their distribution, contrasting with the behavior expected for independent random variables.

Findings

Excitation energies follow the Fréchet distribution asymptotically.

Finite size corrections indicate non-negligible correlations.

Weak correlations affect the extreme value statistics of excitations.

Abstract

In random quantum magnets, like the random transverse Ising chain, the low energy excitations are localized in rare regions and there are only weak correlations between them. It is a fascinating question whether these correlations are completely irrelevant in the sense of the renormalization group. To answer this question, we calculate the distribution of the excitation energy of the random transverse Ising chain in the disordered Griffiths phase with high numerical precision by the strong disorder renormalization group method and - for shorter chains - by free-fermion techniques. Asymptotically, the two methods give identical results, which are well fitted by the Fr\'echet limit law of the extremes of independent and identically distributed random numbers. Given the finite size corrections, the two numerical methods give very similar results, but they differ from the correction term for uncorrelated random variables. This fact shows that the weak correlations between low-energy excitations in random quantum magnets are not entirely irrelevant.