Area laws in a many-body localized state and its implications for topological order

Abstract

The question whether Anderson insulators can persist to finite-strength interactions -a scenario dubbed manybody localization -has recently received a great deal of interest. The origin of such a many-body localized phase has been described as localization in Fock space, a picture we examine numerically. We then formulate a precise sense in which a single energy eigenstate of a Hamiltonian can be adiabatically connected to a state of a non-interacting Anderson insulator. We call such a state a many-body localized state and define a manybody localized phase as one in which almost all states are many-body localized states. We explore the possible consequences of this; the most striking is an area law for the entanglement entropy of almost all excited states in a many-body localized phase. We present the results of numerical calculations for a one-dimensional system of spinless fermions. Our results are consistent with an area law and, by implication, many-body localization for almost all states and almost all regions for weak enough interactions and strong disorder. However, there are rare regions and rare states with much larger entanglement entropies. Furthermore, we study the implications that many-body localization may have for topological phases and self-correcting quantum memories. We find that there are scenarios in which many-body localization can help to stabilize topological order at non-zero energy density, and we propose potentially useful criteria to confirm these scenarios.

Rare‐region effects and dynamics near the many‐body localization transition

“…This phase supports an extensive set of localized integrals of motion [28][29][30][31][32][33] (termed LIOMs or "l-bits"), and certain quantum correlations can retain memory of their initial state even at infinitely late times [34]. The MBL phase resembles noninteracting Anderson insulators in some ways (e.g., spatial correlations decay exponentially, and eigenstates have area-law entanglement [35]). However, there are also important distinctions in entanglement dynamics [36,37], dephasing [38][39][40], linear [41] and nonlinear [42][43][44][45][46][47][48] response, and the entanglement spectrum [49,50].…”

“…Eigenstate entanglement.-The reasoning above also has implications for the distribution of entanglement entropy [35] for regions of size L in the MBL phase. The tail of this distribution is sensitive to rare-region effects.…”

Low-frequency conductivity in many-body localized systems

“…Instead, an isolated system in the MBL phase is a "quantum memory", retaining some local memory of its local initial conditions at arbitrarily late times [9][10][11][12][13][14][15][16][17][18]. The existence of the MBL phase can be proved with minimal assumptions [20]; many of its properties are phenomenologically understood [10,11,16], and some cases can be explored using strong-randomness renormalization group methods [9,[21][22][23].…”

Multipoint entanglement in disordered systems

“…Naively, one would think this is just another way to look at the entanglement area laws found for disordered eigesntates 65 . The advantage of this approach is that the deficit information is directly connected with the correlation length of the system, the main challenge here being the numerical study of Hamiltonians with a larger number of spins, leading to a law of the type (12) which would provide the scaling law of the correlation length, an approach which was explored recently in 49 .…”