Session B13: Many-Body Localized Quantum Systems

Equilibration Dynamics of Strongly Interacting Bosons in 2D Lattices with Disorder

Ultracold atoms in optical lattices can, in principle, be used to probe many-body localization, a quantum entangled regime potentially arising from the interplay of strong interaction and disorder. Motivated by recent experiments [Choi et al., Science 352, 1547 (2016)] we study the dynamics of strongly interacting bosons in the presence of disorder in two dimensions. We show that Gutzwiller mean-field theory (GMFT) captures the main experimental observations, which are a result of the competition between disorder and interactions. Our findings highlight the difficulty in distinguishing glassy dynamics, which can be captured by GMFT, and many-body localization, which cannot be captured by GMFT, and indicate the need for further experimental studies of this system.   

Constructively determining the MBL spectrum using Tensor Networks

All the eigenstates of a many-body localized phase can be compactly represented in the tensor-network language. Current algorithms to find these states often only target single states and/or require difficult optimization to find. In this talk we will show how to generate every eigenstate in the spectrum constructively and discuss its implication for the properties of the MBL phase.   

Exploring one particle orbitals in Many-Body Localized systems with SIMPS

A disordered interacting quantum system can give rise to what is known as a Many-Body Localized (MBL) phase. We study the properties of the natural single particle orbitals given by the eigenvectors of the one particle density matrix of single MBL eigenstates of a system of interacting spinless fermions in one dimension, subject to a random potential. Using a recently proposed matrix product state method, SIMPS~[X.~Yu,~et~al.,~2015], to target highly excited many-body states, we are able to obtain accurate results for large system sizes.   

Scaling Theory of Entanglement at the Many-Body Localization Transition

We study the universal properties of eigenstate entanglement entropy across the transition between many-body localized and thermal phases. Using a real space renormalization group approach, we can numerically simulate large system sizes and systematically extrapolate to the infinite system size limit. Unlike conventional critical points, the short interval entanglement depends non-locally on system size. In the infinite size limit, the entanglement only shows scaling on the localized side of the transition and jumps to its fully thermal value immediately upon entering the thermal phase. On the many-body localized side, the entanglement shows universal scaling intermediate between area and volume law behavior and consistent with a logarithmic dependence on interval size. The full distributions of scaling quantities show bimodal structure and infinite randomness behavior.   

Statistical Bubble Localization with Random Interactions

We study one-dimensional spinless fermions with random interactions, but without any on-site disorder. We find that random interactions generically stabilize a many-body localized phase, in spite of the completely extended single-particle degrees of freedom. In the large randomness limit, we construct "bubble-neck" eigenstates having a universal area-law entanglement entropy on average, with the number of volume-law states being exponentially suppressed. We argue that this statistical localization is beyond the phenomenological local-integrals-of-motion description of many-body localization. With exact diagonalization, we confirm the robustness of the many-body localized phase at finite randomness by investigating eigenstate properties such as level statistics, entanglement/participation entropies, and nonergodic quantum dynamics. At weak random interactions, the system develops a thermalization transition when the single-particle hopping becomes dominant.   

Out of time order correlation in Marginal Many-Body Localized systems

In many-body localized (MBL) systems, energy, charge, and other local conserved quantities can not defuse due to the localization of excitations in the presence of strong disorder. Nevertheless, the quantum information can still propagate. The out-of-time-order correlation (OTOC) was recently proposed to quantify the quantum information scrambling and the butterfly effect in quantum many-body dynamics. We show that the out-of-time-order correlation $\langle W(t)^\dagger V(0)^\dagger W(t)V(0)\rangle$ in many-body localized and marginal MBL systems can be efficiently calculated by the spectral bifurcation renormalization group (SBRG). Previous results show that MBL system has a very slow information scrambling behavior compared to a non-integrable chaotic system. For instance, the scrambling time follows an exponential scaling with the distance. In our work, we demonstrate, in marginal MBL systems, the scrambling time $t_\text{scr}$ follows a stretched exponential scaling with the distance $d_{WV}$ between the operators $W$ and $V$: $t_\text{scr}\sim\exp(\sqrt{d_{WV}/l_0})$, which demonstrates Sinai diffusion of quantum information and the enhanced scrambling by the quantum criticality in non-chaotic systems.   

Entanglement entropy at the MBL transition: evidence for a discontinuous change

The many-body localization phase transition is further investigated in the random-field Heisenberg chain using the numerical linked cluster (NLC) expansion technique. Following a recently proposed method of series analysis, an analysis is performed on the NLC coefficients of the entanglement entropy. We find evidence for a weak singularity at a critical value $h_c\approx 4.5$ that is inconsistent with the Harris criterion and the scaling behavior expected for a continuous transition. We construct a series for the cumulative eigenstate-entanglement probability distribution, which we show is related to the structure of the local integrals of motion of the system. A natural interpretation of our results is that, in the critical regime, while the majority of the integrals of motion remain highly localized, an increasing number of very delocalized integrals of motion show up as the order is increased. We argue that our results are indicative of a very sparse interconnected thermal subregion that is able to thermalize the entire system only in the thermodynamic limit, leading to a discontinuous change in the entanglement entropy.   

Dynamical Confinement-Deconfinement Transitions in Many-body Localized Topological Phases

Many-body localization (MBL) is a dynamical quantum phase of matter in which the properties of quantum many-body ground states can be extended to highly excited states in the presence of strong disorder. While the existence and the properties of such many-body localized phases are rather well understood in one dimension, much less is known in higher dimensions in which more exotic quantum orders can be realized. In this talk, I will discuss the role of disorder in topological-to-trivial quantum phase transitions in order to provide explicit examples of MBL-protected topological order in two dimensions. In particular, I will give a universal real-space renormalization picture of such phase transitions both at zero temperature and in highly excited states.   

Many-body localization of fermions with attractive interactions

We consider spinless fermions in one dimension with attractive interactions in the presence of uncorrelated diagonal disorder. This system is known to have a delocalized Luttinger-liquid phase in its ground state for weak disorder [1]. We provide numerical evidence that this zero-temperature delocalized phase smoothly evolves into an ergodic phase at finite temperatures. Moreover, the finite-energy density transition between the ergodic and the MBL phase terminates at the zero-temperature transition between the delocalized and localized ground-state phases. Our work is based on analyzing the finite-size scaling of the von-Neumann entropy, the level spacing distribution and properties of the one-particle density matrix [2]. As a result we obtain the energy-density versus interaction-strength phase diagram at weak disorder. [1] Schmitteckert et al. Phys. Rev. Lett. 80, 560 (1998) [2] Bera et al. Phys. Rev. Lett. 115, 046603 (2015)   

Many-body delocalization with random vector potentials

In this talk we present the ergodic properties of excited states in a model of interacting fermions in quasi-one dimensional chains subjected to a random vector potential. In the non-interacting limit, we show that arbitrarily small values of this complex off-diagonal disorder triggers localization for the whole spectrum; the divergence of the localization length in the single particle basis is characterized by a critical exponent $\nu$ which depends on the energy density being investigated. However, when short-ranged interactions are included, the localization is lost and the system is ergodic regardless of the magnitude of disorder in finite chains. Our numerical results suggest a delocalization scheme for arbitrary small values of interactions. This finding indicates that the standard scenario of the many-body localization cannot be obtained in a model with random gauge fields. \\ Reference [1] C. Cheng, and R. Mondaini, \textit{Many-body delocalization with random vector potentials}, arXiv:1508.06992.   

Many-body delocalization: Keldysh sigma model approach

Disordered, interacting quantum systems can exhibit many-body localization (MBL), a remarkable interference phenomenon that can preserve quantum mechanical coherence across a macroscopic sample at finite, even large energy densities. An isolated MBL system is non-ergodic and cannot thermalize, i.e., it cannot serve as its own heat bath and can act as a quantum memory. Although much has been recently clarified about the MBL phase, the nature (or even the existence) of the transition between MBL and the ergodic phases remains unclear, especially in dimensions higher than one. In this work, we reformulate the Keldysh approach to interacting non-linear sigma models for Anderson localization in order to approach the transition from the metallic (ergodic) side in two spatial dimensions. We study a system that can undergo a metal-insulator transition at zero temperature. Our goal is to explore the MBL-ergodic transition across a many-body mobility edge by deforming the quantum critical point to finite temperature. We will discuss the prospects for a dephasing catastrophe that signals the onset of MBL, as encountered by approaching from the ergodic side.   

Density propagator for many-body localization: finite size effects, transient subdiffusion, (stretched-) exponentials

We investigate charge relaxation in the spin-less disordered fermionic Hubbard chain. Our observable is the time-dependent density propagator, $\Pi_{\varepsilon}(x,t)$, calculated in windows of different energy density, $\varepsilon$, of the many-body Hamiltonian and at different disorder strengths, $W$, not exceeding the critical value $W_\text{c}$. The width $\dex_\varepsilon(t)$ of $\Pie(x,t)$ exhibits a behavior $d\ln \dex_\varepsilon(t) / d\ln t {=} \beta_\varepsilon(t)$, where $\beta_\varepsilon(t){\lesssim}1/2$ is seen to depend strongly on $L$ at all investigated parameter combinations. (i) We do not find a region in phase space that exhibits subdiffusive dynamics in the sense that $\beta{<}1/2$ in the thermodynamic limit. Instead, subdiffusion may be transient, giving way eventually to conventional diffusive behavior, $\beta{=}1/2$. (ii) (Transient) subdiffusion $0{<}\beta_\varepsilon(t)\lesssim 1/2$, coexists with an enhanced probability for returning to the origin, $\Pie(0,t)$, decaying much slower than $1/\dex_\varepsilon (t)$. Correspondingly, the spatial decay of $\Pie(x,t)$ is far from Gaussian, i.e. exponential or even slower. On a phenomenological level, our findings are broadly consistent with effects of strong disorder and Griffiths regions.