Many-Body Localization Characterized from a One-Particle Perspective

Abstract

We show that the one-particle density matrix ρ can be used to characterize the interaction-driven many-body localization transition in closed fermionic systems. The natural orbitals (the eigenstates of ρ) are localized in the many-body localized phase and spread out when one enters the delocalized phase, while the occupation spectrum (the set of eigenvalues of ρ) reveals the distinctive Fockspace structure of the many-body eigenstates, exhibiting a step-like discontinuity in the localized phase. The associated one-particle occupation entropy is small in the localized phase and large in the delocalized phase, with diverging fluctuations at the transition. We analyze the inverse participation ratio of the natural orbitals and find that it is independent of system size in the localized phase. Introduction. While the theory of noninteracting disordered systems is well developed [1,2], the possibility of a localization transition in closed interacting systems has only recently been firmly established . This manybody localization (MBL) transition occurs at finite energy densities and is not a conventional thermodynamic transition [24,25]. Instead, it can be understood as a dynamical phase transition, associated with the emergence of a complete set of local conserved quantities in the localized phase, which thus behaves as an integrable system [26][27][28][29][30]. This restricts the entanglement entropy of the eigenstates to an area law [31], in contrast to the volume law predicted by the eigenstate thermalization hypothesis for the ergodic delocalized phase [32][33][34]. At the localization transition, the fluctuations of the entanglement entropy diverge [16,35]. The effects of MBL are also observed in the dynamics following, for example, a global quench from a product state, wherein dephasing between the effective degrees of freedom leads to a characteristic logarithmic growth of the entanglement entropy [6,10,12]. These features comprise a much richer set of signatures than in the context of noninteracting systems, for which, in the spirit of one-parameter scaling, the notion of a localization length based on single-particle wave functions generally suffices [1,2].

Criterion for Many-Body Localization-Delocalization Phase Transition

“…The delocalization transition takes place at some critical disorder strength W c . Several numerical studies [5,21,[23][24][25][26][27]41,42] have estimated the location of the transition in this model using various probes, including the statistics of eigenenergies, entanglement entropy and its fluctuations, participation ratios, and different dynamical probes. In particular, state-of-the-art exact diagonalization performed on systems up to L ¼ 22 spins [24] has determined the location of the transition at W c ≈ 3.6.…”

Early Breakdown of Area-Law Entanglement at the Many-Body Delocalization Transition

“…Most studies have focused on exact diagonalization or Lanczos methods which are able to address both sides of the transition in small systems [32][33][34][35][36][37][38][39][40][41][42][43][44][45]. However, since much about this phase transition is still not well understood, extension of finite size results to the thermodynamic limit can prove difficult.…”

Multipoint entanglement in disordered systems

“…Using the same Hamiltonian and family of initial states, as in the previous section, we compute the steady state magnetic imbalance (32) in Fig. 6.…”

“…Latter behavior is usually related to "eigenstate thermalization hypothesis" 15,[27][28][29][30] . On the contrary, MBL phases display a -possibly partially-localized spectrum and a correspondent mobility edge [31][32][33][34][35] . The need for a full knowledge of the entire spectrum makes the study of the MBL phases numerically extremely challenging 36,37 .…”