Mobility edge in a model one-dimensional potential

Dimerized Hofstadter model in two-leg ladder quasi-crystals

We theoretically study topological features, band structure, and localization properties of a dimerized two-leg ladder with an oscillating on-site potential. The periodicity of the on-site potential can take either rational or irrational values. We consider two types of dimerized configurations; symmetric and asymmetric models. For rational values of the periodicity as long as inversion symmetry is preserved both symmetric and asymmetric ladders can host topological phases. Additionally, the energy spectrum of the models exhibits a fractal structure known as the Hofstadter butterfly spectrum, dependent on the dimerization of the hopping and the strength of the on-site potential. In the case of irrational values for the periodicity, a metal-insulator phase transition occurs with small values of the critical strength of the on-site potential in the dimerized cases. Our models incorporate the effects of lattice configuration and quasi-periodicity, paving the way for establishing platforms that host both topological and non-topological phase transitions.

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree d. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around W∼20 for a random regular graph d=4. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around W∼5, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.

Exact New Mobility Edges between Critical and Localized States

The disorder systems host three types of fundamental quantum states, known as the extended, localized, and critical states, of which the critical states remain being much less explored. Here we propose a class of exactly solvable models which host a novel type of exact mobility edges (MEs) separating localized states from robust critical states, and propose experimental realization. Here the robustness refers to the stability against both single-particle perturbation and interactions in the few-body regime. The exactly solvable one-dimensional models are featured by a quasiperiodic mosaic type of both hopping terms and on-site potentials. The analytic results enable us to unambiguously obtain the critical states which otherwise require arduous numerical verification including the careful finite size scalings. The critical states and new MEs are shown to be robust, illustrating a generic mechanism unveiled here that the critical states are protected by zeros of quasiperiodic hopping terms in the thermodynamic limit. Further, we propose a novel experimental scheme to realize the exactly solvable model and the new MEs in an incommensurate Rydberg Raman superarray. This Letter may pave a way to precisely explore the critical states and new ME physics with experimental feasibility.

Absence of Mobility Edge in Short-Range Uncorrelated Disordered Model: Coexistence of Localized and Extended States

Unlike the well-known Mott's argument that extended and localized states should not coexist at the same energy in a generic random potential, we formulate the main principles and provide an example of a nearest-neighbor tight-binding disordered model which carries both localized and extended states without forming the mobility edge. Unexpectedly, this example appears to be given by a well-studied β ensemble with independently distributed random diagonal potential and inhomogeneous kinetic hopping terms. In order to analytically tackle the problem, we locally map the above model to the 1D Anderson model with matrix-size- and position-dependent hopping and confirm the coexistence of localized and extended states, which is shown to be robust to the perturbations of both potential and kinetic terms due to the separation of the above states in space. In addition, the mapping shows that the extended states are nonergodic and allows one to analytically estimate their fractal dimensions.

Universal Subdiffusive Behavior at Band Edges from Transfer Matrix Exceptional Points

We discover a deep connection between parity-time symmetric optical systems and quantum transport in one-dimensional fermionic chains in a two-terminal open system setting. The spectrum of one dimensional tight-binding chain with periodic on-site potential can be obtained by casting the problem in terms of 2×2 transfer matrices. We find that these non-Hermitian matrices have a symmetry exactly analogous to the parity-time symmetry of balanced-gain-loss optical systems, and hence show analogous transitions across exceptional points. We show that the exceptional points of the transfer matrix of a unit cell correspond to the band edges of the spectrum. When connected to two zero temperature baths at two ends, this consequently leads to subdiffusive scaling of conductance with system size, with an exponent 2, if the chemical potential of the baths are equal to the band edges. We further demonstrate the existence of a dissipative quantum phase transition as the chemical potential is tuned across any band edge. Remarkably, this feature is analogous to transition across a mobility edge in quasiperiodic systems. This behavior is universal, irrespective of the details of the periodic potential and the number of bands of the underlying lattice. It, however, has no analog in absence of the baths.

Exact mobility edges in quasiperiodic systems without self-duality

Mobility edge (ME), a critical energy separating localized and extended states in spectrum, is a central concept in understanding localization physics. However, there are few models with exact MEs, and their presences are fragile against perturbations. In the paper, we generalize the Aubry-André-Harper model proposed in (Ganeshanet al2015Phys. Rev. Lett.114146601) and recently realized in (Anet al2021Phys. Rev. Lett.126040603), by introducing a relative phase in the quasiperiodic potential. Applying Avila's global theory, we analytically compute localization lengths of all single-particle states and determine the exact expression of ME, which both significantly depend on the relative phase. They are verified by numerical simulations, and physical perception of the exact expression is also provided. We show that old exact MEs, guaranteed by the delicate self-duality which is broken by the relative phase, are special ones in a series controlled by the phase. Furthermore, we demonstrate that out of expectation, exact MEs are invariant against a shift in the quasiperiodic potential, although the shift changes the spectrum and localization properties. Finally, we show that the exact ME is related to the one in the dual model which has long-range hoppings.

Quantum Information Entropy of Hyperbolic Potentials in Fractional Schrödinger Equation

In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number (0<n≤2) by calculating position and momentum entropy. We find that the wave function will move towards the origin as the fractional derivative number n decreases and the position entropy density becomes more severely localized in more fractional system, i.e., for smaller values of n, but the momentum probability density becomes more delocalized. And then we study the Beckner Bialynicki-Birula−Mycieslki (BBM) inequality and notice that the Shannon entropies still satisfy this inequality for different depth u even though this inequality decreases (or increases) gradually as the depth u of the hyperbolic potential U1 (or U2) increases. Finally, we also carry out the Fisher entropy and observe that the Fisher entropy increases as the depth u of the potential wells increases, while the fractional derivative number n decreases.

Observation of Interaction-Induced Mobility Edge in an Atomic Aubry-André Wire

A mobility edge, a critical energy separating localized and extended excitations, is a key concept for understanding quantum localization. The Aubry-André (AA) model, a paradigm for exploring quantum localization, does not naturally allow mobility edges due to self-duality. Using the momentum-state lattice of quantum gas of Cs atoms to synthesize a nonlinear AA model, we provide experimental evidence for a mobility edge induced by interactions. By identifying the extended-to-localized transition of different energy eigenstates, we construct a mobility-edge phase diagram. The location of a mobility edge in the low- or high-energy region is tunable via repulsive or attractive interactions. Our observation is in good agreement with the theory and supports an interpretation of such interaction-induced mobility edge via a generalized AA model. Our Letter also offers new possibilities to engineer quantum transport and phase transitions in disordered systems.

Generation of circular spin current in an AB magnetic ring with vanishing net magnetization: a new prescription

In this work we report for the first time the appearance of non-decaying circular spin current in a magnetic ring with vanishing net magnetization, even in absence of any spin chirality. Breaking the symmetry in hopping integrals we can misalign up and down spin electronic energy levels which yields a net spin current in the magnetic quantum ring, threaded by an Aharonov-Bohm flux. Along with spin current, a net charge current also appears, and we compute both these currents using the second quantized approach. A tight-binding framework is employed to describe the magnetic ring where each site of the ring contains a finite magnetic moment. Itinerant electrons get scattered from the localized magnetic moments at different lattice sites, and the moments are arranged in such a way that the net magnetization vanishes. The interplay between magnetic moments and asymmetric hopping integrals leads to several atypical features in energy spectra, especially the existence of vanishing current carrying energy eigenstates together with the current carrying ones. The formation of such states those do not contribute any current is the artifact of different kinds of on-site energies and/or hopping integrals in different segments of the magnetic ring. The atypical signatures of energy levels are directly reflected into the charge and spin currents, and here we critically investigate the behaviors of circular currents as functions of electron filling, hopping integrals, strength of spin-moment interaction and ring size. Finally, we discuss briefly the possible experimental realization to implement our proposed magnetic system. The present analysis may provide a new route of generating persistent spin current in magnetic quantum rings with vanishing net magnetization, circumventing the use of spin-orbit coupled systems.

Localization transitions and mobility edges in quasiperiodic ladder

We investigate localization properties of two-coupled uniform chains (ladder) with quasiperiodic modulation on interchain coupling strength. We demonstrate that this ladder is equivalent to two Aubry-André chains when two legs are symmetric. Analytical and numerical results indicate the appearance of mobility edges in asymmetric ladder systems. We propose an easy-to-engineer quasiperiodic Moiré superlattice ladder system comprising two-coupled uniform chains. An irrational lattice constant difference results in a quasiperiodic structure. Numerical simulations indicate that such a system supports the existence of mobility edges. Furthermore, we demonstrate that the mobility edges can be detected through a dynamical method, that is based on the measurement of survival probability in the presence of a single imaginary negative potential. The results provide insights into localization transitions and mobility edges in experiments.

Mobility edges in one-dimensional models with quasi-periodic disorder

We study the mobility edges in a variety of one-dimensional tight binding models with slowly varying quasi-periodic disorders. It is found that the quasi-periodic disordered models can be approximated by an ensemble of periodic models. The mobility edges can be determined by the overlaps of the energy bands of these periodic models. We demonstrate that this method provides an efficient way to find out the precise location of mobility edge in quasi-periodic disordered models. Based on this approximate method, we also propose an index to indicate the degree of localization of each eigenstate.

Interactions and Mobility Edges: Observing the Generalized Aubry-André Model

Using synthetic lattices of laser-coupled atomic momentum modes, we experimentally realize a recently proposed family of nearest-neighbor tight-binding models having quasiperiodic site energy modulation that host an exact mobility edge protected by a duality symmetry. These one-dimensional tight-binding models can be viewed as a generalization of the well-known Aubry-André model, with an energy-dependent self-duality condition that constitutes an analytical mobility edge relation. By adiabatically preparing low and high energy eigenstates of this model system and performing microscopic measurements of their participation ratio, we track the evolution of the mobility edge as the energy-dependent density of states is modified by the model's tuning parameter. Our results show strong deviations from single-particle predictions, consistent with attractive interactions causing both enhanced localization of the lowest energy state due to self-trapping and inhibited localization of high energy states due to screening. This study paves the way for quantitative studies of interaction effects on self-duality induced mobility edges.

Moiré versus Mott: Incommensuration and Interaction in One-Dimensional Bichromatic Lattices

Inspired by the rich physics of twisted 2D bilayer moiré systems, we study Coulomb interacting systems subjected to two overlapping finite 1D lattice potentials of unequal periods through exact numerical diagonalization. Unmatching underlying lattice periods lead to a 1D bichromatic "moiré" superlattice with a large unit cell and consequently a strongly flattened band, exponentially enhancing the effective dimensionless electron-electron interaction strength and manifesting clear signatures of enhanced Mott gaps at discrete fillings. An important nonperturbative finding is a remarkable fine-tuning effect of the precise lattice commensuration, where slight variations in the relative lattice periods may lead to a suppression of the correlated insulating phase, in qualitative agreement with the observed fragility of the correlated insulating phase in twisted bilayer graphene. Our predictions, which should be directly verifiable in bichromatic optical lattices, establish that the competition between interaction and incommensuration is a key element of the physics of moiré superlattices.

One-Dimensional Quasiperiodic Mosaic Lattice with Exact Mobility Edges

The mobility edges (MEs) in energy that separate extended and localized states are a central concept in understanding the localization physics. In one-dimensional (1D) quasiperiodic systems, while MEs may exist for certain cases, the analytic results that allow for an exact understanding are rare. Here we uncover a class of exactly solvable 1D models with MEs in the spectra, where quasiperiodic on-site potentials are inlaid in the lattice with equally spaced sites. The analytical solutions provide the exact results not only for the MEs, but also for the localization and extended features of all states in the spectra, as derived through computing the Lyapunov exponents from Avila's global theory and also numerically verified by calculating the fractal dimension. We further propose a novel scheme with experimental feasibility to realize our model based on an optical Raman lattice, which paves the way for experimental exploration of the predicted exact ME physics.

Engineering topological phase transition and Aharonov-Bohm caging in a flux-staggered lattice

A tight binding network of diamond shaped unit cells trapping a staggered magnetic flux distribution is shown to exhibit a topological phase transition under a controlled variation of the flux trapped in a cell. A simple real space decimation technique maps a binary flux staggered network into an equivalent Su-Shrieffer-Heeger (SSH) model. In this way, dealing with a subspace of the full degrees of freedom, we show that a topological phase transition can be initiated by tuning the applied magnetic field that eventually simulates an engineering of the numerical values of the overlap integrals in the paradigmatic SSH model. Thus one can use an external agent, rather than monitoring the intrinsic property of a lattice to control the topological properties. This is advantageous from an experimental point of view. We also provide an in-depth description and analysis of the topologically protected edge states, and discuss how, by tuning the flux from outside one can enhance the spatial extent of the Aharonov-Bohm caging of single particle states for any arbitrary period of staggering. This feature can be useful for the study of transport of quantum information. Our results are exact.

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.

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.

Universal Properties of Many-Body Localization Transitions in Quasiperiodic Systems

The precise nature of many-body localization (MBL) transitions in both random and quasiperiodic (QP) systems remains elusive so far. In particular, whether MBL transitions in QP and random systems belong to the same universality class or two distinct ones has not been decisively resolved. Here, we investigate MBL transitions in one-dimensional (d=1) QP systems as well as in random systems by state-of-the-art real-space renormalization group (RG) calculation. Our real-space RG shows that MBL transitions in 1D QP systems are characterized by the critical exponent ν≈2.4, which respects the Harris-Luck bound (ν>1/d) for QP systems. Note that ν≈2.4 for QP systems also satisfies the Harris-Chayes-Chayes-Fisher-Spencer bound (ν>2/d) for random systems, which implies that MBL transitions in 1D QP systems are stable against weak quenched disorder since randomness is Harris irrelevant at the transition. We shall briefly discuss experimental means to measure ν of QP-induced MBL transitions.

Phase diagram of the incommensurate off-diagonal Aubry-André model with p-wave pairing

Using exact numerical methods, we study the interplay between p-wave superconductivity and off-diagonal modulation in the incommensurate off-diagonal Aubry-André model with p-wave pairing. When chemical potential is zero, the modulation not only leads to an Anderson-like localization with all bulk states changing into critical ones, but also induces a topological phase transition accompanying the disappearing of exponentially localized zero-energy edge Majorana fermions. These two transitions happen at different places and true Anderson localization is absent. With a finite chemical potential, the quasi-periodicity of the modulation causes band splitting and drives the system into a band insulator phase. It has a fixed incommensurate particle density which is determined by the modulation frequency. Phase diagrams are identified, and can be tested with existing proposals for experimental realizations of the effective off-diagonal disorder and the p-wave pairing.

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.

The electronic and transport properties of two-dimensional conjugated polymer networks including disorder

Two-dimensional (2D) conjugated polymers exhibit electronic structures analogous to that of graphene with the peculiarity of π-π* bands which are fully symmetric and isolated. In the present letter, the suitability of these materials for electronic applications is analyzed and discussed. In particular, realistic 2D conjugated polymer networks with a structural disorder such as monomer vacancies are investigated. Indeed, during bottom-up synthesis, these irregularities are unavoidable and their impact on the electronic properties is investigated using both ab initio and tight-binding techniques. The tight-binding model is combined with a real space Kubo-Greenwood approach for the prediction of transport characteristics for monomer vacancy concentrations ranging from 0.5% to 2%. As expected, long mean free paths and high mobilities are predicted for low defect densities. At low temperatures and for high defect densities, strong localization phenomena originating from quantum interferences of multiple scattering paths are observed in the close vicinity of the Dirac energy region while the absence of localization effects is predicted away from this region suggesting a sharp mobility transition. These predictions show that 2D conjugated polymer networks are good candidates to pave the way for the ultimate scaling and performances of future molecular nanoelectronic devices.

Many-Body Localization and Quantum Nonergodicity in a Model with a Single-Particle Mobility Edge

We investigate many-body localization in the presence of a single-particle mobility edge. By considering an interacting deterministic model with an incommensurate potential in one dimension we find that the single-particle mobility edge in the noninteracting system leads to a many-body mobility edge in the corresponding interacting system for certain parameter regimes. Using exact diagonalization, we probe the mobility edge via energy resolved entanglement entropy (EE) and study the energy resolved applicability (or failure) of the eigenstate thermalization hypothesis (ETH). Our numerical results indicate that the transition separating area and volume law scaling of the EE does not coincide with the nonthermal to thermal transition. Consequently, there exists an extended nonergodic phase for an intermediate energy window where the many-body eigenstates violate the ETH while manifesting volume law EE scaling. We also establish that the model possesses an infinite temperature many-body localization transition despite the existence of a single-particle mobility edge. We propose a practical scheme to test our predictions in atomic optical lattice experiments which can directly probe the effects of the mobility edge.

Nearest neighbor tight binding models with an exact mobility edge in one dimension

We investigate localization properties in a family of deterministic (i.e., no disorder) nearest neighbor tight binding models with quasiperiodic on site modulation. We prove that this family is self-dual under a generalized duality transformation. The self-dual condition for this general model turns out to be a simple closed form function of the model parameters and energy. We introduce the typical density of states as an order parameter for localization in quasiperiodic systems. By direct calculations of the inverse participation ratio and the typical density of states we numerically verify that this self-dual line indeed defines a mobility edge in energy separating localized and extended states. Our model is a first example of a nearest neighbor tight binding model manifesting a mobility edge protected by a duality symmetry. We propose a realistic experimental scheme to realize our results in atomic optical lattices and photonic waveguides.

Universal scheme to generate metal-insulator transition in disordered systems

We propose a scheme to generate metal-insulator transition in the random binary layer (RBL) model, which is constructed by randomly assigning two types of layers along the longitudinal direction. Based on a tight-binding Hamiltonian, the localization length is calculated for a variety of RBLs with different cross section geometries by using the transfer-matrix method. Both analytical and numerical results show that a band of extended states could appear in the quasi-one-dimensional RBLs and the systems behave as metals by properly tuning the model parameters, due to the existence of a completely ordered subband, leading to a metal-insulator transition in parameter space. Furthermore, the extended states are irrespective of the diagonal and off-diagonal disorder strengths. Our results can be generalized to two- and three-dimensional disordered systems with arbitrary layer structures, and may be realized in Bose-Einstein condensates.

Topological zero-energy modes in gapless commensurate Aubry-André-Harper models

The Aubry-André or Harper (AAH) model has been the subject of extensive theoretical research in the context of quantum localization. Recently, it was shown that one-dimensional quasicrystals described by the incommensurate AAH model has a nontrivial topology. In this Letter, we show that the commensurate off-diagonal AAH model is topologically nontrivial in the gapless regime and supports zero-energy edge modes. Unlike the incommensurate case, the nontrivial topology in the off-diagonal AAH model is attributed to the topological properties of the one-dimensional Majorana chain. We discuss the feasibility of experimental observability of our predicted topological phase in the commensurate AAH model.

Comparison of Shannon information entropies in position and momentum space for an electron in one-dimensional nonuniform systems

We investigate numerically the position- and momentum-space Shannon information entropies, S(x)(β) and S(p)(β), respectively, of energy eigenstates |β} for an electron in four kinds of one-dimensional (1D) nonuniform systems, i.e., the Harper model, the slowly varying potential ones, the complex quasiperiodic potential ones, and the random-dimer potential ones. In the former three models, electronic localization properties are well-defined. For them, we find it interesting that, S(x)(β) is greater than, equal to, and less than S(p)(β) for delocalized, critical, and localized states in position-space, respectively, which can be used as signatures of the transition from a delocalized phase to a localized ones. With the criterion, we analyze the random-dimer potential model. We give another perspective and propose a consistent interpretation of discrepancies about the random-dimer potential model. Therefore, all these provide us a simple method to discern the nature of states in these 1D nonuniform systems.

Vibrational modes in a two-dimensional aperiodic harmonic lattice

We study the nature of collective excitations in classical harmonic lattices with aperiodic and pseudo-random mass distributions. Using a matrix recursive reformulation of the mass displacement equation, we compute the localization length within the band of allowed frequencies. Our numerical calculations indicate that, for aperiodic arrays of masses, a new phase of extended states appears in this model. Solving numerically the Hamilton equations for momentum and displacement along the chain, we compute the spreading of an initially localized energy excitation. We find that for sufficient aperiodicity, there is a ballistic propagation of the energy pulse.

Predicted mobility edges in one-dimensional incommensurate optical lattices: an exactly solvable model of anderson localization

Localization properties of noninteracting quantum particles in one-dimensional incommensurate lattices are investigated with an exponential short-range hopping that is beyond the minimal nearest-neighbor tight-binding model. Energy dependent mobility edges are analytically predicted in this model and verified with numerical calculations. The results are then mapped to the continuum Schrödinger equation, and an approximate analytical expression for the localization phase diagram and the energy dependent mobility edges in the ground band is obtained.

Metal-insulator transition in an aperiodic ladder network: an exact result

We prove that a tight-binding ladder network composed of atomic sites with on-site potentials distributed according to the quasiperiodic Aubry model can exhibit a metal-insulator transition at multiple values of the Fermi energy. For specific values of the first and second neighbor electron hopping, the result is obtained exactly. With a more general model, we numerically calculate the two-terminal conductance. The numerical results corroborate the analytical findings.