Single-particle Localization Research Articles

From Bloch oscillations to many-body localization in clean interacting systems

In this work we demonstrate that nonrandom mechanisms that lead to single-particle localization may also lead to many-body localization, even in the absence of disorder. In particular, we consider interacting spins and fermions in the presence of a linear potential. In the noninteracting limit, these models show the well-known Wannier-Stark localization. We analyze the fate of this localization in the presence of interactions. Remarkably, we find that beyond a critical value of the potential gradient these models exhibit nonergodic behavior as indicated by their spectral and dynamical properties. These models, therefore, constitute a class of generic nonrandom models that fail to thermalize. As such, they suggest new directions for experimentally exploring and understanding the phenomena of many-body localization. We supplement our work by showing that by using machine-learning techniques the level statistics of a system may be calculated without generating and diagonalizing the Hamiltonian, which allows a generation of large statistics.

Single-particle tracking localization microscopy reveals nonaxonemal dynamics of intraflagellar transport proteins at the base of mammalian primary cilia.

Primary cilia play a vital role in cellular sensing and signaling. An essential component of ciliogenesis is intraflagellar transport (IFT), which is involved in IFT protein recruitment, axonemal engagement of IFT protein complexes, and so on. The mechanistic understanding of these processes at the ciliary base was largely missing, because it is challenging to observe the motion of IFT proteins in this crowded region using conventional microscopy. Here, we report short-trajectory tracking of IFT proteins at the base of mammalian primary cilia by optimizing single-particle tracking photoactivated localization microscopy for IFT88-mEOS4b in live human retinal pigment epithelial cells. Intriguingly, we found that mobile IFT proteins “switched gears” multiple times from the distal appendages (DAPs) to the ciliary compartment (CC), moving slowly in the DAPs, relatively fast in the proximal transition zone (TZ), slowly again in the distal TZ, and then much faster in the CC. They could travel through the space between the DAPs and the axoneme without following DAP structures. We further revealed that BBS2 and IFT88 were highly populated at the distal TZ, a potential assembly site. Together, our live-cell single-particle tracking revealed region-dependent slowdown of IFT proteins at the ciliary base, shedding light on staged control of ciliary homeostasis.

Slow Growth of Out-of-Time-Order Correlators and Entanglement Entropy in Integrable Disordered Systems.

We investigate how information spreads in three paradigmatic one-dimensional models with spatial disorder. The models we consider are unitarily related to a system of free fermions and, thus, are manifestly integrable. We demonstrate that out-of-time-order correlators can spread slowly beyond the single-particle localization length, despite the absence of many-body interactions. This phenomenon is shown to be due to the nonlocal relationship between elementary excitations and the physical degrees of freedom. We argue that this nonlocality becomes relevant for time-dependent correlation functions. In addition, a slow logarithmic-in-time growth of the entanglement entropy is observed following a quench from an unentangled initial state. We attribute this growth to the presence of strong zero modes, which gives rise to an exponential hierarchy of time scales upon ensemble averaging. Our work on disordered integrable systems complements the rich phenomenology of information spreading and we discuss broader implications for general systems with nonlocal correlations.

Single-particle localization in dynamical potentials

Single particle localization of an ultra-cold atom is studied in one dimension when the atom is confined by an optical lattice and by the incommensurate potential of a high-finesse optical cavity. In the strong coupling regime the atom is a dynamical refractive medium, the cavity resonance depends on the atomic position within the standing-wave mode and nonlinearly determines the depth and form of the incommensurate potential. We show that the particular form of the quasi-random cavity potential leads to the appearance of mobility edges, even in presence of nearest-neighbour hopping. We provide a detailed characterization of the system as a function of its parameters and in particular of the strength of the atom-cavity coupling, which controls the functional form of the cavity potential. For strong atom-photon coupling the properties of the mobility edges significantly depend on the ratio between the periodicities of the confining optical lattice and of the cavity field.

Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

We develop a novel analytical approach to the problem of single particle localization in infinite dimensional spaces such as Bethe lattice and random regular graph models. The key ingredient of the approach is the notion of the inverted order thermodynamic limit (IOTL) in which the coupling to the environment goes to zero before the system size goes to infinity. Using IOTL and Replica Symmetry Breaking (RSB) formalism we derive analytical expressions for the fractal dimension D1 that distinguishes between the extended ergodic, D1=1, and extended non-ergodic (multifractal), 0<D1<1 states on the Bethe lattice and random regular graphs with the branching number K. We also employ RSB formalism to derive the analytical expression lnStyp−1=−〈lnS〉∼(Wc−W)−1 for the typical imaginary part of self-energy Styp in the non-ergodic phase close to the Anderson transition in the conventional thermodynamic limit. We prove the existence of an extended non-ergodic phase in a broad range of disorder strength and energy and establish the phase diagrams of the models as a function of disorder and energy. The results of the analytical theory are compared with large-scale population dynamics and with the exact diagonalization of Anderson model on random regular graphs. We discuss the consequences of these results for the many body localization.

Control of a single-particle localization in open quantum systems

We investigate the possibility to control localization properties of the asymptotic state of an open quantum system with a tunable synthetic dissipation. The control mechanism relies on the matching between properties of dissipative operators, acting on neighboring sites and specified by a single control parameter, and the spatial phase structure of eigenstates of the system Hamiltonian. As a result, the latter coincide (or near coincide) with the dark states of the operators. In a disorder-free Hamiltonian with a flat band, one can either obtain a dominating localized asymptotic state or populate whole flat and/or dispersive bands, depending on the value of the control parameter. In a disordered Anderson system, the asymptotic state can be localized anywhere in the spectrum of the Hamiltonian. The dissipative control is robust with respect to an additional local dephasing.

Characterizing Time Irreversibility in Disordered Fermionic Systems by the Effect of Local Perturbations.

We study the effects of local perturbations on the dynamics of disordered fermionic systems in order to characterize time irreversibility. We focus on three different systems: the noninteracting Anderson and Aubry-André-Harper (AAH) models and the interacting spinless disordered t-V chain. First, we consider the effect on the full many-body wave functions by measuring the Loschmidt echo (LE). We show that in the extended or ergodic phase the LE decays exponentially fast with time, while in the localized phase the decay is algebraic. We demonstrate that the exponent of the decay of the LE in the localized phase diverges proportionally to the single-particle localization length as we approach the metal-insulator transition in the AAH model. Second, we probe different phases of disordered systems by studying the time expectation value of local observables evolved with two Hamiltonians that differ by a spatially local perturbation. Remarkably, we find that many-body localized systems could lose memory of the initial state in the long-time limit, in contrast to the noninteracting localized phase where some memory is always preserved.

Universality of Schmidt decomposition and particle identity

Schmidt decomposition is a widely employed tool of quantum theory which plays a key role for distinguishable particles in scenarios such as entanglement characterization, theory of measurement and state purification. Yet, its formulation for identical particles remains controversial, jeopardizing its application to analyze general many-body quantum systems. Here we prove, using a newly developed approach, a universal Schmidt decomposition which allows faithful quantification of the physical entanglement due to the identity of particles. We find that it is affected by single-particle measurement localization and state overlap. We study paradigmatic two-particle systems where identical qubits and qutrits are located in the same place or in separated places. For the case of two qutrits in the same place, we show that their entanglement behavior, whose physical interpretation is given, differs from that obtained before by different methods. Our results are generalizable to multiparticle systems and open the way for further developments in quantum information processing exploiting particle identity as a resource.

Conduction in quasiperiodic and quasirandom lattices: Fibonacci, Riemann, and Anderson models

We study the ground state conduction properties of noninteracting electrons in aperiodic but non-random one-dimensional models with chiral symmetry, and make comparisons against Anderson models with non-deterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's many-particle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all single-particle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be a conductor at most electron densities, including the half-filled case; however, at various specific fillings $\rho$, including the values $\rho = 1/g^n$, where $g$ is the golden ratio and $n$ is any integer, the gas turns into an insulator due to spectral gaps. Metallic behaviour is found at half-filling in the Riemann lattice as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to single-particle eigenstate localization, likely at all other fillings. Its behaviour turns out to be alike that of the off-diagonal Anderson model, albeit with different system-size scaling of the band-centre anomalies. The advantages of analysing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted.

Interacting spinning fermions with quasi‐random disorder

Interacting spinning fermions with strong quasi‐random disorder are analyzed via rigorous Renormalization Group (RG) methods combined with KAM techniques. The correlations are written in terms of an expansion whose convergence follows from number‐theoretical properties of the frequency and cancellations due to Pauli principle. A striking difference appears between spinless and spinning fermions; in the first case there are no relevant effective interactions while in presence of spin an additional relevant quartic term is present in the RG flow. The large distance exponential decay of the correlations present in the non interacting case, consequence of the single particle localization, is shown to persist in the spinning case only for temperatures greater than a power of the many body interaction, while in the spinless case this happens up to zero temperature. image

Microscopic Movement of Slow-Diffusing Nanoparticles in Cylindrical Nanopores Studied with Three-Dimensional Tracking.

To study slow mass transport in confined environments, we developed a three-dimensional (3D) single-particle localization technique to track their microscopic movements in cylindrical nanopores. Under two model conditions, particles are retained much longer inside the pores: (1) increased solvent viscosity, which slows down the particle throughout the whole pore, and (2) increased pore wall affinity, which slows down the particle only at the wall. In viscous solvents, the particle steps decrease proportionally to the increment of the viscosity, leading to macroscopically slow diffusion. As a contrast, the particles in sticky pores are microscopically active by showing limited reduction of step sizes. A restricted diffusion mode, possibly caused by the heterogeneous environment in sticky pores, is the main reason for macroscopically slow diffusion. This study shows that it is possible to differentiate slow diffusion in confined environments caused by different mechanisms.

Criterion for Many-Body Localization-Delocalization Phase Transition

We propose a new approach to probing ergodicity and its breakdown in quantum many-body systems based on their response to a local perturbation. We study the distribution of matrix elements of a local operator between the system's eigenstates, finding a qualitatively different behaviour in the many-body localized (MBL) and ergodic phases. To characterize how strongly a local perturbation modifies the eigenstates, we introduce the parameter ${\cal G}(L)=\langle \ln (V_{nm}/\delta) \rangle$, which represents a disorder-averaged ratio of a typical matrix element of a local operator $V$ to the energy level spacing, $\delta$; this parameter is reminiscent of the Thouless conductance in the single-particle localization. We show that the parameter ${\cal G}(L)$ decreases with system size $L$ in the MBL phase, and grows in the ergodic phase. We surmise that the delocalization transition occurs when ${\cal G}(L)$ is independent of system size, ${\cal G}(L)={\cal G}_c\sim 1$. We illustrate our approach by studying the many-body localization transition and resolving the many-body mobility edge in a disordered 1D XXZ spin-1/2 chain using exact diagonalization and time-evolving block decimation methods. Our criterion for the MBL transition gives insights into microscopic details of transition. Its direct physical consequences, in particular logarithmically slow transport at the transition, and extensive entanglement entropy of the eigenstates, are consistent with recent renormalization group predictions.

Fast weighted centroid algorithm for single particle localization near the information limit.

A simple weighting scheme that enhances the localization precision of center of mass calculations for radially symmetric intensity distributions is presented. The algorithm effectively removes the biasing that is common in such center of mass calculations. Localization precision compares favorably with other localization algorithms used in super-resolution microscopy and particle tracking, while significantly reducing the processing time and memory usage. We expect that the algorithm presented will be of significant utility when fast computationally lightweight particle localization or tracking is desired.

Lattice bosons in a quasi-disordered environment: The effects of a superlattice potential on single particle and many particle properties

In this paper we present a theoretical investigation of the effect of a superlattice potential on some properties of non-interacting bosons in one dimensional lattices with Aubry–Andŕe disorder potential. In the first part, we investigate the single particle localization properties. We find a re-entrant localization–delocalization transition and the development of multiple mobility edges for a range of superlattice potential strengths. In the second part, we study the Bose–Einstein condensation with an additional harmonic trapping potential. We find that an increase in the superlattice potential leads to an increase in the depletion of the condensate in the low temperature limit.

Interaction-induced connectivity of disordered two-particle states

We study the interaction-induced connectivity in the Fock space of two particles in a disordered one-dimensional potential. Recent computational studies showed that the largest localization length $\xi_2$ of two interacting particles in a weakly random tight binding chain is increasing unexpectedly slow relative to the single particle localization length $\xi_1$, questioning previous scaling estimates. We show this to be a consequence of the approximate restoring of momentum conservation of weakly localized single particle eigenstates, and disorder-induced phase shifts for partially overlapping states. The leading resonant links appear among states which share the same energy and momentum. We substantiate our analytical approach by computational studies for up to $\xi_1 = 1000$. A potential nontrivial scaling regime sets in for $ \xi_1 \approx 400$, way beyond all previous numerical attacks.

Marginal Anderson localization and many-body delocalization

We consider d dimensional systems which are localized in the absence of interactions, but whose single particle (SP) localization length diverges near a discrete set of (single-particle) energies, with critical exponent \nu. This class includes disordered systems with intrinsic- or symmetry-protected- topological bands, such as disordered integer quantum Hall insulators. In the absence of interactions, such marginally localized systems exhibit anomalous properties intermediate between localized and extended including: vanishing DC conductivity but sub-diffusive dynamics, and fractal entanglement (an entanglement entropy with a scaling intermediate between area and volume law). We investigate the stability of marginal localization in the presence of interactions, and argue that arbitrarily weak short range interactions trigger delocalization for partially filled bands at non-zero energy density if \nu \ge 1/d. We use the Harris/Chayes bound \nu \ge 2/d, to conclude that marginal localization is generically unstable in the presence of interactions. Our results suggest the impossibility of stabilizing quantized Hall conductance at non-zero energy density.

Quantum chaotic subdiffusion in random potentials

Two interacting particles (TIP) in a disordered chain propagate beyond the single particle localization length $\xi_1$ up to a scale $\xi_2 > \xi_1$. An initially strongly localized TIP state expands almost ballistically up to $\xi_1$. The expansion of the TIP wave function beyond the distance $\xi_1 \gg 1$ is governed by highly connected Fock states in the space of noninteracting eigenfunctions. The resulting dynamics is subdiffusive, and the second moment grows as $m_2 \sim t^{1/2}$, precisely as in the strong chaos regime for corresponding nonlinear wave equations. This surprising outcome stems from the huge Fock connectivity and resulting quantum chaos. The TIP expansion finally slows down towards a complete halt -- in contrast to the nonlinear case.

Fast and High-Accuracy Localization for Three-Dimensional Single-Particle Tracking

We report a non-iterative localization algorithm that utilizes the scaling of a three-dimensional (3D) image in the axial direction and focuses on evaluating the radial symmetry center of the scaled image to achieve the desired single-particle localization. Using this approach, we analyzed simulated 3D particle images by wide-field microscopy and confocal microscopy respectively, and the 3D trajectory of quantum dots (QDs)-labeled influenza virus in live cells. Both applications indicate that the method can achieve 3D single-particle localization with a sub-pixel precision and sub-millisecond computation time. The precision is almost the same as that of the iterative nonlinear least-squares 3D Gaussian fitting method, but with two orders of magnitude higher computation speed. This approach can reduce considerably the time and costs for processing the large volume data of 3D images for 3D single-particle tracking, which is especially suited for 3D high-precision single-particle tracking, 3D single-molecule imaging and even new microscopy techniques.

Universal Slow Growth of Entanglement in Interacting Strongly Disordered 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.

Many-body localization in a quasiperiodic system

Recent theoretical and numerical evidence suggests that localization can survive in disordered many-body systems with very high energy density, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in extended quantum systems far from the zero-temperature limit. It separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting ("ergodic") phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present numerical evidence that many-body localization also occurs in models without disorder but rather a quasiperiodic potential. In one dimension, these systems already have a single-particle localization transition, and we show that this transition becomes a many-body localization transition upon the introduction of interactions. We also comment on possible relevance of our results to experimental studies of many-body dynamics of cold atoms and non-linear light in quasiperiodic potentials.