Phenomenology of fully many-body-localized systems

Abstract

We consider isolated quantum systems with all of their many-body eigenstates localized. We define a sense in which such systems are integrable, and discuss a method for finding their localized conserved quantum numbers ("constants of motion"). These localized operators are interacting pseudospins and are subject to dephasing but not to dissipation, so any quantum states of these pseudospins can in principle be recovered via (spin) echo procedures. We also discuss the spreading of entanglement in many-body localized systems, which is another aspect of the dephasing due to interactions between these localized conserved operators. PACS numbers:Isolated quantum many-body systems with shortrange interactions and static randomness may be in a many-body localized phase where they do not thermally equilibrate under their own dynamics. While this possibility was pointed out long ago by Anderson [1], such localization of highly-excited states in systems with interactions did not receive a lot of attention until after Basko, et al.[2] forcefully brought the subject into focus. Isolated systems in the many-body localized phase have strictly zero thermal conductivity [2], so if some energy is added to the system locally, it only excites localized degrees of freedom and does not diffuse, even when the system's energy density corresponds to a nonzero (even infinite [3]) temperature.We expect that the many-body eigenstates of a system's Hamiltonian in the localized phase are product states of localized degrees of freedom, with some shortrange "area-law" entanglement between the "bare" local degrees of freedom. One goal of this paper is to explore how one can define suitably "dressed" localized pseudospin operators in terms of which the many-body eigenstates within the localized phase are indeed precisely product states with zero entanglement. When the Hamiltonian is then expressed in terms of these dressed localized pseudospins it has exponentially decaying long-range interactions, and it is these long-range interactions that cause decoherence and dephasing of local observables in the many-body insulator. These interactions also cause the spreading of entanglement for a nonentangled initial product state of the bare spins, as has been seen and explored in Refs. [4][5][6][7][8].To be concrete, assume we have a system of N spin-1/2's {σ i } on some lattice (say, in one, two or three dimensions). For an example, see, e.g., Ref. [9]. Our system has a specific random Hamiltonian H that contains only short-range interactions and strong enough static random fields on each spin so that, with probability one in the limit of large N , all 2 N many-body eigenstates of this H are localized. The construction we present below should be readily generalizable to local operators with more than two states. It should also be generalizable to systems where the dominant strong randomness is instead the spin-spin interactions rather than random fields. In those cases, the pseudospins we will construct may instead be localized domain wall operators [...

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].…”

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]. 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].…”

Probing many-body localization with neural networks

“…According to the ETH, local observables in a typical many-body eigenstate should take the values that pertain to the observables in a thermal ensemble, with the whole system acting as a heat bath for its subsystems in the thermodynamic limit. A well-studied class of systems that violate the ETH are those exhibiting many-body localization (MBL) [18][19][20][21][22][23][24][25], meaning that partial memory of initial conditions is preserved for infinite times. Due to this property, which is intimately related to the emergence of an extensive number of integrals of motion [23,[26][27][28], MBL systems have been envisioned as particularly robust quantum memories [29].…”