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

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.