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

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 eigenstate thermalization hypothesis (ETH) phase 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 indicates 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 $|\ensuremath{\psi}\ensuremath{\rangle}$ 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 ${{n}_{\ensuremath{\alpha}}}$ (eigenvalues of the OPDM). We highlight the statistical distribution of Hamming distance ${x}_{\ensuremath{\mu}\ensuremath{\nu}}$ occurring in the pairwise coefficients $|{a}_{\ensuremath{\mu}}{|}^{2}{|{a}_{\ensuremath{\nu}}|}^{2}$ in $S$, and compare this with a related quantity considered in the literature.