Localization in Out-Of-Equilibrium Disorderless Systems
Update: 2018-02-24 10:03:10      Author: yangjuan@csrc.ac.cn

When isolated quantum systems are taken out of equilibrium, its dynamics can be described by a time evolution which is unitary, unlike classically chaotic systems. The conditions that lead these systems to thermalize and the information regarding its initial preparations to be lost over the course of its evolution can be seen as result of the dephasing in the dynamics of the coherence between the eigenstates. Its mathematical basic grounds were based in what we call nowadays as the Eigenstate Thermalization Hypothesis (ETH) [1,2] and it became clear that interactions between the constituents of the system are essential in the thermalization mechanism. On the other hand, much attention has been also given to another aspect that may, in fact, prevent thermalization. When disorder comes into play an isolated quantum system can experience a halting of thermalization and, ultimately, localization takes place. This localization is manifest in the absence of mass transport and it can be seen as a generalization of the famous Anderson localization phenomenon when interactions are included.

Several numerical studies have shown this phenomenon in the presence of interactions [3], which we dub as many-body localization (MBL), and also a handful number of experiments [4] have also tackled it. In the latter, the analyzed quantum system is consisted of atoms trapped in a potential generated by laser beams in an optical lattice. By introducing disorder on the onsite energy levels, one can emulate the physics of the Anderson model with interactions which can be tuned in a controllable fashion. The most common way to identify this localization (and the associated lack of thermalization) is to follow the dynamical properties of a carefully prepared quantum system. Specifically, experiments in optical lattices employ high-fidelity preparation of initial states whose properties are well known, as for example, by confining the atoms in certain regions of the trapping environment. By measuring how the information of the initial prepared state is preserved for long-times after the release of this constraint one is able to identify the regimes where disorder is sufficient to lead to the MBL phenomenon.

On the other hand, much recent theoretical work has been devoted to understanding whether disorder is a necessary ingredient to generate localization in interacting systems. Pioneering works by Kagan and Maksimov on Helium mixtures have laid the foundation of localization when there are two constituents in the system, a light and a heavy one. In these cases, the heavy particles generate an effective random quasistatic potential which blocks the diffusion of the light ones, thus localizing them. More recently, some proposals tried to tackle this problem and much to the surprise found that an initial transient localization ultimately leads to diffusion.  In a recent publication [5], Rubem Mondaini from CSRC and Zi Cai (Shanghai Jiaotong University) have investigated one of the first evidences of many-body localization in a translation invariant Hamiltonian with a single species of particles. Specifically, they tackled the problem of (hardcore) bosons in a lattice which interact via a quasi-periodic infinite range interaction. Despite the model may sound rather artificial, it has been shown to be emulated experimentally for trapped atoms in a optical lattice embedded in a cavity [6]. The long (infinite)-range interaction is mediated by vacuum modes of the cavity and are independently controlled by tuning the cavity resonance. The Hamiltonian reads,

                               blob.png                          (1)

where blob.png is the nearest-neighbor tunneling amplitude, and the interactions blob.png have the functional form blob.png. By taking blob.png as an irrational number, the long-range interactions are no longer commensurate with the lattice and glassy behavior emerges when the interactions are sufficiently large. We investigate this glass-like regime at zero temperature and show it can also be obtained in an isolated quantum system at infinite temperatures. To make a closer connection with the experiments, they investigate the persistence of the information of initial preparations under a quench. In particular, they quantify the degree of inhomogeneity of an initial pure state, via blob.png.


Fig. 1: Time evolution of the integrated charge inhomogeneity averaged over states with similar number of domain walls in ergodic [nonergodic] regimes in panel (a) [panel (b)] with V/J = 101 [V/J = 10]. Panels (c) and (d) are comparisons of the diagonal ensemble prediction (infinite-time limit) and the microcanonical (thermodynamic) result for the corresponding values of interactions.

Before the quench, this observable represents the number of domain walls are present in the initial state.  Its time evolution is depicted in Fig. (1) for two values of interactions. If a system thermalizes, a time-evolved state can be thought as a featureless state that no longer preserves local information of the initial state. That is the case in Fig. 1(a) for small values of interactions. Surprisingly, if one increases its magnitude, the tine-evolved state no longer loses information and memory of the initial preparations can be retrieved for arbitrarily long times, as shown in Fig. 1(b). with this clear indication of many-body localization in a translation-invariant (disorderless) system, we expect that this work may sparkle interest in the experimental investigation of the ideas we present, ultimately settling the issue of whether disorder is a necessary mechanism to its observation. Further aspects of these problem, as the study of commensurate long-range interactions, are an ongoing research being currently investigated at CSRC.



[1]      Quantum statistical mechanics in a closed system, J. M. Deutsch, Phys. Rev. A 43 2046 (1991).

[2]      Chaos and quantum thermalization, M. Srednicki, Phys. Rev. E 50, 888 (1994).

[3]      Many-body localization and thermalization in disordered Hubbard chains, R. Mondaini and M. Rigol, Phys. Rev. A 92, 041601 (2015).

[4]      Observation of many-body localization of interacting fermions in a quasirandom optical lattice, M. Schreiber, S. S. Hodgman, P. Bordia, Henrik P. Lüschen, M. H Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Science 349, 842–845 (2015).

[5]      Many-body self-localization in a translation-invariant Hamiltonian, R. Mondaini and Zi Cai, Phys. Rev. 96 94, 035153 (2017).

Quantum phases from competing short- and long-range interactions in an optical lattice, R. Landig, L. Hruby, N. Dogra, M. Landini, R. Mottl, T. Donner, and T. Esslinger, Nature 532, 476 (2016).

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