Universal Slow Growth of Entanglement in Interacting Strongly Disordered Systems

Abstract

Recent numerical work by Bardarson et. al.[1] revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interactioninduced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξ ln(V t/ ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase. Introduction. While it is well-known that arbitrarily weak disorder localizes all single-particle quantummechanical states in 1 and 2 dimensions, the effect of disorder potential on the states of interacting systems largely remains an open problem. Works [2,3] conjectured that localization in a many-body system survives in the presence of weak interactions. When the strength of the interactions is increased, at some critical value a transition to the delocalized phase -a "many-body localization" transition -takes place, as observed in the numerical simulations [4][5][6][7][8][9][10][11][12][13][14].

Low-frequency conductivity in many-body localized systems

“…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]. While the eigenstate properties of MBL systems are in some respects similar to those of noninteracting Anderson insulators, there are important differences in the dynamics, such as the logarithmic spreading of entanglement in the MBL phase [6,9,11,18,19,24,25].…”

“…These mechanisms are absent in noninteracting systems: thus, the differences in transport between single-particle and many-body localization can be traced to the much larger connectivity of the many-body Hilbert space. The two mechanisms we discuss involve dissipative dynamics, and are thus distinct from the "pure dephasing" processes that cause the slow growth of entanglement within the MBL phase [11,25].…”

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]).…”

Quantum Entanglement Growth under Random Unitary Dynamics

“…This irreversible growth of entanglementquantified by the growth of the von Neumman entropyis important for several reasons. It is an essential part of thermalization, and as a result has been addressed in diverse contexts ranging from conformal field theory [1][2][3][4] and holography [5][6][7][8][9][10][11][12] to integrable [13][14][15][16][17][18][19], nonintegrable [20][21][22][23], and strongly disordered spin chains [24][25][26][27][28][29][30]. Entanglement growth is also of practical importance as the crucial obstacle to simulating quantum dynamics numerically, for example, using matrix product states or the density matrix renormalization group [31].…”