Multifractality and Fock-space localization in many-body localized states: one-particle density matrix perspective

TL;DR

This paper investigates many-body localization using the one-particle density matrix basis, revealing persistent multifractality within the MBL phase and providing new insights into Fock space localization measures.

Contribution

It introduces the use of the OPDM eigenstates as a basis to analyze Fock space localization, offering a novel perspective on multifractality in MBL states.

Findings

Multifractality persists inside the MBL phase.

OPDM basis provides new insights into Fock space localization.

Distribution of Hamming distances reveals statistical properties of eigenstate coefficients.

Abstract

Many-body localization (MBL) is well characterized in Fock space. To quantify the degree of this Fock space localization, the multifractal dimension $D_q$ is employed; it has been claimed that $D_q$ shows a jump from the delocalized value $D_q=1$ in the ETH phase (ETH: eigenstate thermalization hypothesis) to a smaller value $0<D_q<1$ at the ETH-MBL transition, yet exhibiting a conspicuous discrepancy from the fully localized value $D_q=0$, which indicate that multifractality remains inside the MBL phase. Here, to better quantify the situation we employ, instead of the commonly used computational basis, the one-particle density matrix (OPDM) and use its eigenstates (natural orbitals) as a Fock state basis for representing many-body eigenstates $|\psi\rangle$ of the system. Using this basis, we compute $D_q$ and other indices quantifying the Fock space localization, such as the local purity $S$, which is derived from the occupation spectrum $\{n_\alpha\}$ (eigenvalues of the OPDM). We highlight the statistical distribution of Hamming distance $x_{\mu\nu}$ occurring in the pair-wise coefficients $|a_\mu|^2|a_\nu|^2$ in $S$, and compare this with a related quantity considered in the literature.