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