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

摘要

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 |ψ〉 of the system. Using this basis, we compute D_q and other indices quantifying the Fock-space localization, such as the local purity 5, which is derived from the occupation spectrum {n_α } (eigenvalues of the OPDM). We highlight the statistical distribution of Hamming distance x_(μv) occurring in the pairwise coefficients |a_μ|~2|a_v|~2 in S. and compare this with a related quantity considered in the literature.