An important and incompletely answered question is whether a closed quantum system of many interacting particles can be localized by disorder. The time evolution of simple (unentangled) initial states is studied numerically for a system of interacting spinless fermions in one dimension described by the random-field XXZ Hamiltonian. Interactions induce a dramatic change in the propagation of entanglement and a smaller change in the propagation of particles. For even weak interactions, when the system is thought to be in a many-body localized phase, entanglement shows neither localized nor diffusive behavior but grows without limit in an infinite system: interactions act as a singular perturbation on the localized state with no interactions. The significance for proposed atomic experiments is that local measurements will show a large but nonthermal entropy in the many-body localized state. This entropy develops slowly (approximately logarithmically) over a diverging time scale as in glassy systems.
An important and incompletely answered question is whether a closed quantum system of many interacting particles can be localized by disorder. The time evolution of simple (unentangled) initial states is studied numerically for a system of interacting spinless fermions in one dimension described by the random-field XXZ Hamiltonian. Interactions induce a dramatic change in the propagation of entanglement and a smaller change in the propagation of particles. For even weak interactions, when the system is thought to be in a many-body localized phase, entanglement shows neither localized nor diffusive behavior but grows without limit in an infinite system: interactions act as a singular perturbation on the localized state with no interactions. The significance for proposed atomic experiments is that local measurements will show a large but nonthermal entropy in the many-body localized state. This entropy develops slowly (approximately logarithmically) over a diverging time scale as in glassy systems.
Recent numerical work by Bardarson, Pollmann, and Moore 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 interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), 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.
We formulate a dynamical real space renormalization group (RG) approach to describe the time evolution of a random spin-1/2 chain, or interacting fermions, initialized in a state with fixed particle positions. Within this approach we identify a many-body localized state of the chain as a dynamical infinite randomness fixed point. Near this fixed point our method becomes asymptotically exact, allowing analytic calculation of time dependent quantities. In particular, we explain the striking universal features in the growth of the entanglement seen in recent numerical simulations: unbounded logarithmic growth delayed by a time inversely proportional to the interaction strength. This is in striking contrast to the much slower entropy growth as loglogt found for noninteracting fermions with bond disorder. Nonetheless, even the interacting system does not thermalize in the long time limit. We attribute this to an infinite set of approximate integrals of motion revealed in the course of the RG flow, which become asymptotically exact conservation laws at the fixed point. Hence we identify the many-body localized state with an emergent generalized Gibbs ensemble.
We study the time evolution after a quantum quench in a family of models whose degrees of freedom are fermions coupled to spins, where quenched disorder appears neither in the Hamiltonian parameters nor in the initial state. Focusing on the behavior of entanglement, both spatial and between subsystems, we show that the model supports a state exhibiting combined area and volume-law entanglement, being characteristic of the quantum disentangled liquid. This behavior appears for one set of variables, which is related via a duality mapping to another set, where this structure is absent. Upon adding density interactions between the fermions, we identify an exact mapping to an XXZ spin chain in a random binary magnetic field, thereby establishing the existence of many-body localization with its logarithmic entanglement growth in a fully disorder-free system.
The possibility of observing many-body localization of ultracold atoms in a one-dimensional optical lattice is discussed for random interactions. In the noninteracting limit, such a system reduces to single-particle physics in the absence of disorder, i.e., to extended states. In effect, the observed localization is inherently due to interactions and is thus a genuine many-body effect. In the system studied, many-body localization manifests itself in a lack of thermalization visible in temporal propagation of a specially prepared initial state, in transport properties, in the logarithmic growth of entanglement entropy, and in statistical properties of energy levels.
We study many-body localization for a disordered chain of spin-1/2 fermions. When both down and up components are exposed to the same strong disorder, a power-law growth of the entanglement entropy was observed [P. Prelov\ifmmode \check{s}\else \v{s}\fi{}ek et al., Phys. Rev. B 94, 241104 (2016)] that suggests that many-body localization is not complete; the density (charge) degree of freedom is localized, while the spin degree of freedom is apparently delocalized. We show that this powerlike behavior is only a transient effect and that, for longer times, the growth of the entanglement entropy is logarithmic in time. At the same time, the decay of the spin correlation slows down, showing that the spin transport is strongly reduced at long times. The dynamics of the system is quite similar to the one of many-body localized systems. Because of the limited duration of the numerical experiment, it is not possible to determine whether the system is truly many-body localized or diffusing very slowly at long times.
We numerically investigate the dynamics of entanglement in a chain of spinless fermions with nonrandom but long-range hopping and interactions, and with random on-site energies. For moderate disorder in the absence of interactions, the chain hosts delocalized states at the top of the band which undergo a delocalization-localization transition with increasing disorder. We find an interesting regime in this noninteracting disordered chain where the long-time entanglement entropy scales as $S(t)\ensuremath{\sim}lnt$ and the saturated entanglement entropy scales with system size L as $S(L,t\ensuremath{\rightarrow}\ensuremath{\infty})\ensuremath{\sim}lnL$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.