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

• Published in: Physical review. B 2021-06, Vol.103 (21), Article 214206
• Main Authors: Orito, Takahiro, Imura, Ken-Ichiro
• Format: Article
• Language: English
• Subjects: Eigenvalues; Eigenvectors; Fractal geometry; Localization; Many body interactions; Multiplexing; Particle density (concentration); Thermalization (energy absorption)
• ISSN: 2469-9950; 2469-9969
• DOI: 10.1103/PhysRevB.103.214206

Many-body localization (MBL) is well characterized in Fock space. To quantify the degree of this Fock-space localization, the multifractal dimension Dq is employed; it has been claimed that Dq shows a jump from the delocalized value Dq = 1 in the eigenstate thermalization hypothesis (ETH) phase to a smaller value 0 < Dq < 1 at the ETH-MBL transition, yet exhibiting a conspicuous discrepancy from the fully localized value Dq = 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 Dq and other indices quantifying the Fock-space localization, such as the local purity S, which is derived from the occupation spectrum {nα} (eigenvalues of the OPDM). We highlight the statistical distribution of Hamming distance xμν occurring in the pairwise coefficients ∣ ∣ aμ∣∣2|aν|2 in S, and compare this with a related quantity considered in the literature.