Dynamics of many-body localization

Abstract

Following the field theoretic approach of Basko et al., Ann. Phys. 321, 1126(2006, we study in detail the real-time dynamics of a system expected to exhibit many-body localization. In particular, for time scales inaccessible to exact methods, we demonstrate that within the second-Born approximation that the temporal decay of the density-density correlation function is non-exponential and is consistent with a finite value for t → ∞, as expected in a non-ergodic state. This behavior persists over a wide range of disorder and interaction strengths. We discuss the implications of our findings with respect to dynamical phase boundaries based both on exact diagonalization studies and as well as those established by the methods of Ref. 1.It has been known for more than 50 years that noninteracting particles in a one-dimensional disordered system exhibit Anderson localization [2], namely the exponential suppression of transport. While a localized system is non-ergodic and thus does not thermalize, coupling the system to other degrees of freedom with a continuous spectrum, such as a heat bath, allows thermalization to occur via processes such as variable-range hopping [3]. For an isolated many-body system, only interactions between the particles may lead to thermalization. The question of whether or not localization is stable in the presence of interactions was first considered by Fleishman and Anderson [4], who concluded that short-ranged interactions cannot destabilize the insulating phase. A similar and still open question also exists for Bose-Einstein condensates, treated in the framework of the time-dependent Gross-Pitaevskii (or nonlinear Schrödinger) equation [5]. In this case numerics, as well as analytical arguments, suggest a temporally sub-diffusive or even logarithmic thermalization behavior for not very strong interactions [5].Using a diagrammatic approach, Basko et al. argued that for a general class of isolated, disordered and interacting systems, a many-body mobility edge exists, similarly to the Anderson mobility edge in a threedimensional non-interacting system [1]. Namely, a critical energy separates "insulating" and "metallic" eigenstates, which can be distinguished by evaluating the spatial correlations of any local operator. "Metallic" eigenstates will have non-vanishing or slowly decaying correlations, while "insulating" states will have exponentially decaying correlations. By changing the energy (or the micro canonical temperature) of the system across the mobilityedge, the system will undergo an insulator-metal transition. Similar to the Anderson transition, the many-body localization (MBL) transition is a dynamical and not a thermodynamic phenomena [1]. However, the MBL transition is also not a conventional quantum phase transition since the critical energy, which depends on the parameters of the system, may be very far from the ground state. In fact, for systems of bounded energy density (e.g., finite number of states per site), Oganesyan and Huse suggested that the transition will persist up t...

Multifractal metal in a disordered Josephson junctions array

“…In this region, the volume occupied by a typical wavefunction scales as anomalous power, D, of the full Hilbert space volume that continuously changes from D = 0 in the insulator to D = 1 in a fully delocalized state. In a qualitative agreement several groups have found that the dynamics in this region is often described by non-trivial power laws that are neither diffusive nor localized [6][7][8][9].…”

Theory of the Many-Body Localization Transition in One-Dimensional Systems

“…Grover [22] showed that the eigenstates at this critical point have volume-law entanglement of small subsystems. Recent numerical work, which focused on the vicinity of the transition, has identified and explored the subdiffusive regime in the vicinity of the transition [17,23]. Finally, a very recent work explores a different RG approach to the MBL transition based on an assumed hierarchical structure of the relevant many-body resonances, arriving at similar conclusions to ours [24].…”

Localization in a random XY model with long-range interactions: Intermediate case between single-particle and many-body problems

“…In the thermodynamic limit of infinite system this parameter is defined as the local spin-spin correlation function (29) taken in the infinite time limit [43] and averaged over the system eigenstates having zero energy. The latter choices correspond to the infinite temperature thermodynamic limit [27,28]. In that limit the average ergodicity parameter should approach zero in the delocalized state where correlations are subject to decay, while in the localized state it should be finite (unity in an infinitely strong disorder limit).…”

“…[26]). We consider the infinite temperature limit similarly to the previous work [27,28] because it is more convenient for analytical and numerical considerations while its generalization to the finite non-zero temperature is straightforward [29].…”

“…The critical strength of disorder W c for the given number of spins can be found resolving Eqs. (26), (27), (28) with respect to the disorder W at fixed system size R or number of spins N . The dependencies W c (N ) are also shown in Table II.…”