Single-particle Mobility Edge Research Articles

Phenomenon of multiple reentrant localization in a double-stranded helix with transverse electric field

The present work explores the potential for observing multiple reentrant localization behavior in a double-stranded helical (DSH) system, extending beyond the conventional nearest-neighbor hopping (NNH) interaction. The DSH system is considered to have hopping dimerization in each strand, while also being subjected to a transverse electric field. The inclusion of an electric field serves the dual purpose of inducing quasi-periodic disorder and strand-wise staggered site energies. Two reentrant localization regions are identified: one exhibiting true extended behavior in the thermodynamic limit, while the second region shows quasi-extended characteristics with partial spreading within the helix. The DSH system exhibits three distinct single-particle mobility edges linked to localization transitions present in the system. The analysis in this study involves examining various parameters such as the single-particle energy spectrum, inverse participation ratio, local probability amplitude, and more. Our proposal, combining achievable hopping dimerization and induced correlated disorder, presents a unique opportunity to study phenomenon of reentrant localization, generating significant research interest.

Incommensurate many-body localization in the presence of long-range hopping and single-particle mobility edge

We study many-body localization (MBL) in the quasiperiodic ${t}_{1}\text{\ensuremath{-}}{t}_{2}$ model, focusing on the role of next-nearest-neighbor (NNN) hopping ${t}_{2}$, which introduces a single-particle mobility edge. The calculated phase diagram can be divided into three distinct regimes, depending on the strength of the short-range interaction $U$. For weak interactions ($U\ensuremath{\ll}{t}_{1}$), this model is always nonthermal. For intermediate interactions ($U\ensuremath{\sim}{t}_{1}$), the thermal-MBL phase transition in this model is qualitatively the same as that of the Aubry-Andre (AA) model, which is consistent with existing experimental observations. For strong interactions $(U\ensuremath{\gg}{t}_{1})$, the NNN hopping produces qualitatively new physics because it breaks down the Hilbert space fragmentation present in the AA model. The NNN hopping is thus irrelevant when the interaction is intermediate but relevant for strong interactions.

Exact mobility edges in the non-Hermitian t1−t2 model: Theory and possible experimental realizations

Quantum localization in 1D non-Hermitian systems, especially the search for exact single-particle mobility edges, has attracted considerable interest recently. While much progress has been made, the available methods to determine the ME in such models are still limited. In this work, we use a new method to find a new class of exact mobility edges in 1D non-Hermitian quasiperiodic models with parity-time $(\mathcal{PT})$ symmetry. We illustrate our method by studying a specific model. We first use our method to determine the energy-dependent mobility edge as well as the spectrum for localized eigenstates in this model. We then demonstrate that the metal-insulator transition must occur simultaneously with the spontaneous $\mathcal{PT}$-symmetry breaking transition in this model. Finally, we propose an experimental protocol based on a 1D photonic lattice to distinguish the extended and localized single-particle states in our model. The results in our work can be applied to studying other non-Hermitian quasiperiodic models.

Environment-assisted quantum transport and mobility edges

Environment-assisted quantum transport (ENAQT) is a unique situation where environmental noise can, counterintuitively, enhance the transport of an open quantum system. In this paper, we investigate how the presence of a one-dimensional single-particle mobility edge can generate strong ENAQT. For this purpose, we study the energy current of a generalized Aubry-Andr\'e-Harper tight binding model coupled at its edges to spin baths of differing temperature and dephasing noise along the system. We find that the ENAQT increases by orders of magnitude and depends on the number of localized eigenstates and disorder strength nonmonotonically. We show that this enhancement is the result of the cooperation between population uniformization and localization.

Learning single-particle mobility edges by a neural network based on data compression

The breaking of the well-known Aubry-Andr\'e model with a single-particle mobility edge (SPME) is a intriguing subject that has not yet been fully understood. In particular, how to accurately and efficiently recognize a SPME in an optical lattice is currently under active debate. In this work, we develop a data compression-based neural network (DCNN) approach to identify SPMEs in one-dimensional quasiperiodic optical lattice using eigenstates as the sole diagnostic. We find that such method can successfully identify SPMEs of a large system only using a small network trained by the small system data, without onerously and repetitively training a new and large-scale network by massive data of a large system. Furthermore, we show that this method is also applicable to recognize more complex phase transitions, such as many-body localization. Our DCNN approach first paves the way for the development of a generic tool for identifying unexplored phase transitions in large systems.

Reentrant Localization Transition in a Quasiperiodic Chain.

Systems with quasiperiodic disorder are known to exhibit a localization transition in low dimensions. After a critical strength of disorder, all the states of the system become localized, thereby ceasing the particle motion in the system. However, in our analysis, we show that in a one-dimensional dimerized lattice with staggered quasiperiodic disorder, after the localization transition, some of the localized eigenstates become extended for a range of intermediate disorder strengths. Eventually, the system undergoes a second localization transition at a higher disorder strength, leading to all states being localized. We also show that the two localization transitions are associated with the mobility regions hosting the single-particle mobility edges. We establish this reentrant localization transition by analyzing the eigenspectra, participation ratios, and the density of states of the system.

Plane wave study on the localized-extended transition in the one-dimensional incommensurate systems

Based on our recently proposed plane wave framework, we theoretically study the localized-extended transition in the one dimensional incommensurate systems with cosine type of potentials, which are in close connection to many recent experiments of the ultracold atoms and photonic crystals. With the aid of numerical calculations, we formulate a propagator based scattering picture to explain the transition at the ground state and single particle mobility edge, in which deeper connections among the incommensurate potentials, eigenstate compositions and transition mechanism are revealed. We further show that there exists an upper limit of localization length for all localized eigenstates, leading to a fundamental difference to the Anderson localization. The scattering picture provides unique insights on the localization mechanism in the incommensurate systems, and can be generalized to study the more complex phenomena in higher dimensions, with different potentials or beyond the single particle regime.

Transport in the nonergodic extended phase of interacting quasiperiodic systems

We study the transport properties and the spectral statistics of a one-dimensional closed quantum system of interacting spinless fermions in a quasiperiodic potential which produces a single-particle mobility edge in the absence of interaction. For such systems, it has been shown that the many-body eigenstates can be of three different kinds: extended and eigenstate thermalization hypothesis (ETH) obeying (thermal), localized and ETH violating (many body localized), and extended and ETH violating (nonergodic extended). Here we investigate the nonergodic extended phase from the point of view of level spacing statistics and charge transport. We calculate the dc conductivity and the low-frequency conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and show that both are consistent with subdiffusive transport. This is contrasted with diffusive transport in the thermal phase and blocked transport in the MBL phase.

Mobility edge and intermediate phase in one-dimensional incommensurate lattice potentials

We study theoretically the localization properties of two distinct one-dimensional quasiperiodic lattice models with a single-particle mobility edge (SPME) separating extended and localized states in the energy spectrum. The first one is the familiar Soukoulis-Economou trichromatic potential model with two incommensurate potentials, and the second is a system consisting of two coupled 1D Aubry-Andre chains each containing one incommensurate potential. We show that as a function of the Hamiltonian model parameters, both models have a wide single-particle intermediate phase (SPIP), defined as the regime where localized and extended single-particle states coexist in the spectrum, leading to a behavior intermediate between purely extended or purely localized when the system is dynamically quenched from a generic initial state. Our results thus suggest that both systems could serve as interesting experimental platforms for studying the interplay between localized and extended states, and may provide insight into the role of the coupling of small baths to localized systems. We also calculate the Lyapunov (or localization) exponent for several incommensurate 1D models exhibiting SPME, finding that such localization critical exponents for quasiperiodic potential induced localization are nonuniversal and depend on the microscopic details of the Hamiltonian.

Quasiperiodic quantum heat engines with a mobility edge

Steady-state thermoelectric machines convert heat into work by driving a thermally-generated charge current against a voltage gradient. In this work, we propose a new class of steady-state heat engines operating in the quantum regime, where a quasi-periodic tight-binding model that features a mobility edge forms the working medium. In particular, we focus on a generalization of the paradigmatic Aubrey-Andr\'e-Harper (AAH) model, known to display a single-particle mobility edge that separates the energy spectrum into regions of completely delocalized and localized eigenstates. Remarkably, these two regions can be exploited in the context of steady-state heat engines as they correspond to ballistic and insulating transport regimes. This model also presents the advantage that the position of the mobility edge can be controlled via a single parameter in the Hamiltonian. We exploit this highly tunable energy filter, along with the peculiar spectral structure of quasiperiodic systems, to demonstrate large thermoelectric effects, exceeding existing predictions by several orders of magnitude. This opens the route to a new class of highly efficient and versatile quasi-periodic steady-state heat engines, with a possible implementation using ultracold neutral atoms in bichromatic optical lattices.

Energy Current Rectification and Mobility Edges.

We investigate how the presence of a single-particle mobility edge in a system can generate strong energy current rectification. Specifically, we study a quadratic bosonic chain subject to a quasiperiodic potential and coupled at its boundaries to spin baths of differing temperature. We find that rectification increases by orders of magnitude depending on the spatial position in the chain of localized eigenstates above the mobility edge. The largest enhancements occur when the coupling of one bath to the system is dominated by a localized eigenstate, while the other bath couples to numerous delocalized eigenstates. By tuning the parameters of the quasiperiodic potential it is thus possible to vary the amplitude, and even invert the direction, of the rectification.

Many-body localization in the presence of long-range interactions and long-range hopping

We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of long-range interactions or long-range hopping. Based on perturbative arguments there is a common belief that MBL can exist only in systems with short-range interactions and short-range hopping. We analyze effects of power-law interactions and power-law hopping, separately, on a system which has all the single particle states localized in the absence of interactions. Since delocalization is driven by proliferation of resonances in the Fock space, we mapped this model to an effective Anderson model on a complex graph in the Fock space, and calculated the probability distribution of the number of resonances up to third order. Though the most-probable value of the number of resonances diverge for the system with long-range hopping ($t(r) \sim t_0/r^\alpha$ with $\alpha < 2$), there is no enhancement of the number of resonances as the range of power-law interactions increases. This indicates that the long-range hopping delocalizes the many-body localized system but in contrast to this, there is no signature of delocalization in the presence of long-range interactions. We further provide support in favor of this analysis based on dynamics of the system after a quench starting from a charge density wave ordered state, level spacing statistics, return probability, participation ratio and Shannon entropy in the Fock space. We demonstrate that MBL persists in the presence of long-range interactions though long-range hopping with $1<\alpha <2$ delocalizes the system partially, with all the states extended for $\alpha <1$. Even in a system which has single-particle mobility edges in the non-interacting limit, turning on long-range interactions does not cause delocalization.

Observation of Many-Body Localization in a One-Dimensional System with a Single-Particle Mobility Edge.

We experimentally study many-body localization (MBL) with ultracold atoms in a weak one-dimensional quasiperiodic potential, which in the noninteracting limit exhibits an intermediate phase that is characterized by a mobility edge. We measure the time evolution of an initial charge density wave after a quench and analyze the corresponding relaxation exponents. We find clear signatures of MBL when the corresponding noninteracting model is deep in the localized phase. We also critically compare and contrast our results with those from a tight-binding Aubry-André model, which does not exhibit a single-particle intermediate phase, in order to identify signatures of a potential many-body intermediate phase.

Investigating many-body mobility edges in isolated quantum systems

The existence of many-body mobility edges in closed quantum systems has been the focus of intense debate after the emergence of the description of the many-body localization phenomenon. Here we propose that this issue can be settled in experiments by investigating the time evolution of local degrees of freedom, tailored for specific energies and initial states. An interacting model of spinless fermions with exponentially long-ranged tunneling amplitudes, whose non-interacting version known to display single-particle mobility edges, is used as the starting point upon which nearest-neighbor interactions are included. We verify the manifestation of many-body mobility edges by using numerous probes, suggesting that one cannot explain their appearance as merely being a result of finite-size effects.

Machine Learning Many-Body Localization: Search for the Elusive Nonergodic Metal.

The breaking of ergodicity in isolated quantum systems with a single-particle mobility edge is an intriguing subject that has not yet been fully understood. In particular, whether a nonergodic but metallic phase exists or not in the presence of a one-dimensional quasiperiodic potential is currently under active debate. In this Letter, we develop a neural-network-based approach to investigate the existence of this nonergodic metallic phase in a prototype model using many-body entanglement spectra as the sole diagnostic. We find that such a method identifies with high confidence the existence of a nonergodic metallic phase in the midspectrum at an intermediate quasiperiodic potential strength. Our neural-network-based approach shows how supervised machine learning can be applied not only in locating phase boundaries but also in providing a way to definitively examine the existence or not of a novel phase.

Configuration-controlled many-body localization and the mobility emulsion

We uncover a new non-ergodic phase, distinct from the many-body localized (MBL) phase, in a disordered two-leg ladder of interacting hardcore bosons. The dynamics of this emergent phase, which has no single-particle analog and exists only for strong disorder and finite interaction, is determined by the many-body configuration of the initial state. Remarkably, this phase features the $\textit{coexistence}$ of localized and extended many-body states at fixed energy density and thus does not exhibit a many-body mobility edge, nor does it reduce to a model with a single-particle mobility edge in the noninteracting limit. We show that eigenstates in this phase can be described in terms of interacting emergent Ising spin degrees of freedom ("singlons") suspended in a mixture with inert charge degrees of freedom ("doublons" and "holons"), and thus dub it a $\textit{mobility emulsion}$ (ME). We argue that grouping eigenstates by their doublon/holon density reveals a transition between localized and extended states that is invisible as a function of energy density. We further demonstrate that the dynamics of the system following a quench may exhibit either thermalizing or localized behavior depending on the doublon/holon density of the initial product state. Intriguingly, the ergodicity of the ME is thus tuned by the initial state of the many-body system. These results establish a new paradigm for using many-body configurations as a tool to study and control the MBL transition. The ME phase may be observable in suitably prepared cold atom optical lattices.

Engineering a Flux-Dependent Mobility Edge in Disordered Zigzag Chains

There has been great interest in realizing quantum simulators of charged particles in artificial gauge fields. Here, we perform the first quantum simulation explorations of the combination of artificial gauge fields and disorder. Using synthetic lattice techniques based on parametrically-coupled atomic momentum states, we engineer zigzag chains with a tunable homogeneous flux. The breaking of time-reversal symmetry by the applied flux leads to analogs of spin-orbit coupling and spin-momentum locking, which we observe directly through the chiral dynamics of atoms initialized to single lattice sites. We additionally introduce precisely controlled disorder in the site energy landscape, allowing us to explore the interplay of disorder and large effective magnetic fields. The combination of correlated disorder and controlled intra- and inter-row tunneling in this system naturally supports energy-dependent localization, relating to a single-particle mobility edge. We measure the localization properties of the extremal eigenstates of this system, the ground state and the most-excited state, and demonstrate clear evidence for a flux-dependent mobility edge. These measurements constitute the first direct evidence for energy-dependent localization in a lower-dimensional system, as well as the first explorations of the combined influence of artificial gauge fields and engineered disorder. Moreover, we provide direct evidence for interaction shifts of the localization transitions for both low- and high-energy eigenstates in correlated disorder, relating to the presence of a many-body mobility edge. The unique combination of strong interactions, controlled disorder, and tunable artificial gauge fields present in this synthetic lattice system should enable myriad explorations into intriguing correlated transport phenomena.

Single-Particle Mobility Edge in a One-Dimensional Quasiperiodic Optical Lattice.

A single-particle mobility edge (SPME) marks a critical energy separating extended from localized states in a quantum system. In one-dimensional systems with uncorrelated disorder, a SPME cannot exist, since all single-particle states localize for arbitrarily weak disorder strengths. However, in a quasiperiodic system, the localization transition can occur at a finite detuning strength and SPMEs become possible. In this Letter, we find experimental evidence for the existence of such a SPME in a one-dimensional quasiperiodic optical lattice. Specifically, we find a regime where extended and localized single-particle states coexist, in good agreement with theoretical simulations, which predict a SPME in this regime.

Criterion for the occurrence of many-body localization in the presence of a single-particle mobility edge

Non-interacting fermions in one dimension can undergo a localization-delocalization transition in the presence of a quasi-periodic potential as a function of that potential. In the presence of interactions, this transition transforms into a Many-Body Localization (MBL) transition. Recent studies have suggested that this type of transition can also occur in models with quasi-periodic potentials that possess single particle mobility edges. Two such models were studied in PRL 115,230401(2015) but only one was found to exhibit an MBL transition in the presence of interactions while the other one did not. In this work we investigate the occurrence of MBL in the presence of weak interactions in five different models with single particle mobility edges in one dimension with a view to obtaining a criterion for the same. We find that not all such models undergo a thermal-MBL phase transition in presence of weak interactions. We propose a criterion to determine whether MBL is likely to occur in presence of interaction based only on the properties of the non-interacting models. The relevant quantity $\epsilon$ is a measure of how localized the localized states are relative to how delocalized the delocalized states are in the non-interacting model. We also study various other features of the non-interacting models such as the divergence of the localization length at the mobility edge and the presence or absence of `ergodicity' and localization in their many-body eigenstates. However, we find that these features cannot be used to predict the occurrence of MBL upon the introduction of weak interactions.

Nonequilibrium phase diagram of a one-dimensional quasiperiodic system with a single-particle mobility edge

We investigate and map out the non-equilibrium phase diagram of a generalization of the well known Aubry-Andr\'e-Harper (AAH) model. This generalized AAH (GAAH) model is known to have a single-particle mobility edge which also has an additional self-dual property akin to that of the critical point of AAH model. By calculating the imbalance, we get hints of a rich phase diagram. We find a fascinating connection between single particle wavefunctions near the mobility edge of GAAH model and the wavefunctions of the critical AAH model. By placing this model far-from-equilibrium with the aid of two baths, we investigate the open system transport via system size scaling of non-equilibrium steady state (NESS) current. Current is calculated by fully exact non-equilibrium Green's function (NEGF) formalism. The critical point of the AAH model now generalizes to a 'critical' line separating regions of ballistic and localized transport. Like critical point of AAH model, current scales sub-diffusively with system size on the 'critical' line ($I\sim N^{-2\pm0.1}$). However, remarkably, the scaling exponent on this line is distinctly different from that obtained for the critical AAH model (where $I\sim N^{-1.4\pm0.05}$). All these results can be understood from the above-mentioned connection between states near mobility edge of GAAH model and those of critical AAH model. A very interesting high temperature non-equilibrium phase diagram of the GAAH model emerges from our calculations.