One-Dimensional Quasiperiodic Mosaic Lattice with Exact Mobility Edges

Exactly Solvable Mobility Edges for Phonons in One-Dimensional Quasiperiodic Chains

Mobility edges, which demarcate the boundary between extended and localized states, are fundamental to understanding the physics of localization in condensed matter systems. Systems exhibiting exact mobility edges are rare, and the localization properties of phonons have received limited prior investigation. In this work, we reveal analytical mobility edges in one-dimensional quasiperiodic-modulated spring-mass chains. The mobility edges are exactly solved and numerically validated through the eigenfrequency spectra, inverse/normalized participation ratios, and lattice wave dynamics. Our research demonstrates the Anderson localization transition in phonon systems, paving the way for experimental observations of phonon localization.

Observation of Photonic Mobility Edge Phases

Inverse Anderson localizations in lower dimensions predict that, as the hopping rates increase, all localized eigenmodes transition to extended states. Here, through the implementation of a mosaic quasiperiodic photonic waveguide lattice, we experimentally demonstrate a distinctive scenario, where the intermediate-energy eigenmodes become extended, while the low- or high-energy eigenmodes remain localized, leading to the emergence of energy-dependent Anderson localization transitions and mobility edge phases. Our experiment is enabled by developing an adiabatic procedure to prepare the photonic lattice into the zero-energy, lower and upper middle-energy, and ground and highest excited eigenmodes and subsequently measuring their localization properties. Moreover, we also experimentally investigate nonequilibrium quench dynamics for photons and show that photonic Loschmidt echoes can identify the appearance of mobility edge phases. Our study thus opens new avenues for investigating energy-dependent photonic Anderson localizations and harnessing photons to explore intriguing nonequilibrium physics.

Probing multi-mobility edges in quasiperiodic mosaic lattices

The mobility edge (ME) is a crucial concept in understanding localization physics, marking the critical transition between extended and localized states in the energy spectrum. Anderson localization scaling theory predicts the absence of ME in lower dimensional systems. Hence, the search for exact MEs, particularly for single particles in lower dimensions, has recently garnered significant interest in both theoretical and experimental studies, resulting in notable progress. However, several open questions remain, including the possibility of a single system exhibiting multiple MEs and the continual existence of extended states, even within the strong disorder domain. Here, we provide experimental evidence to address these questions by utilizing a quasiperiodic mosaic lattice with meticulously designed nanophotonic circuits. Our observations demonstrate the coexistence of both extended and localized states in lattices with broken duality symmetry and varying modulation periods. By single-site injection and scanning the disorder level, we could approximately probe the ME of the modulated lattice. These results corroborate recent theoretical predictions, introduce a new avenue for investigating ME physics, and offer inspiration for further exploration of ME physics in the quantum regime using hybrid integrated photonic devices.

Quasiperiodic disorder induced critical phases in a periodically driven dimerized p-wave Kitaev chain

The intricate relationship between topology and disorder in non-equilibrium quantum systems presents a captivating avenue for exploring localization phenomenon. Here, we look for a suitable platform that enables an in-depth investigation of the topic. To this end, we delve into the nuanced analysis of the topological and localization characteristics exhibited by a one-dimensional dimerized Kitaev chain under periodic driving and perform detailed analyses of the Floquet Majorana modes. Such a non-equilibrium scenario is made further interesting by including a spatially varying quasiperiodic potential with a temporally modulated amplitude. Apriori, the motivation is to explore an interplay between dimerization and a quasiperiodic disorder in a topological setting which is also known to demonstrate unique (re-entrant) localization properties. While the topological properties of the driven system confirm the presence of zero and π Majorana modes, the phase diagram obtained by constructing a pair of topological invariants ( Z × Z ), also referred to as the real space winding numbers, at different driving frequencies reveal intriguing features that are distinct from the static scenario. In particular, at either low or intermediate frequency regimes, the phase diagram concerning the zero mode involves two distinct phase transitions, one from a topologically trivial to a non-trivial phase, and another from a topological phase to an Anderson localized phase. On the other hand, the study of the Majorana π mode unveils the emergence of a unique topological phase, characterized by complete localization of both the bulk and the edge modes, which may be called as the Floquet topological Anderson phase. Moreover, different frequency regimes showcase distinct localization features which can be examined via the localization toolbox, namely, the inverse and the normalized participation ratios. Specifically, the low and high-frequency regimes demonstrate the existence of completely extended and localized phases, respectively. While at intermediate frequencies, we observe the critical (multifractal) phase of the model which is further investigated via a finite-size scaling analysis of the fractal dimension. Finally, to add depth into our study, we have performed a mean level spacing analyses and computed the Hausdorff dimension which yields specific characteristics inherent to the critical phase, offering profound insights into its underlying properties.

Critical regions in a one-dimensional flat band lattice with a quasi-periodic potential

In our previous work, the concept of critical region in a generalized Aubry-André model (Ganeshan-Pixley-Das Sarma's model) has been established. In this work, we find that the critical region can be realized in a one-dimensional flat band lattice with a quasi-periodic potential. It is found that the above flat band lattice model can be reduced to an effective Ganeshan-Pixley-Das Sarma's model. Depending on various parameter ranges, the effective quasi-periodic potential may be bounded or unbounded. In these two cases, the Lyapunov exponent, mobility edge, and critical indices of localized length are obtained exactly. In this flat band model, the localized-extended, localized-critical and critical-extended transitions can coexist. Furthermore, we find that near the transitions between the bound and unbounded cases, the derivative of Lyapunov exponent of localized states with respect to energy is discontinuous. At the end, the localized states in bounded and unbounded cases can be distinguished from each other by Avila's acceleration.

Dephasing-Induced Mobility Edges in Quasicrystals

Mobility edges (ME), separating Anderson-localized states from extended states, are known to arise in the single-particle energy spectrum of certain one-dimensional lattices with aperiodic order. Dephasing and decoherence effects are widely acknowledged to spoil Anderson localization and to enhance transport, suggesting that ME and localization are unlikely to be observable in the presence of dephasing. Here it is shown that, contrary to such a wisdom, ME can be created by pure dephasing effects in quasicrystals in which all states are delocalized under coherent dynamics. Since the lifetimes of localized states induced by dephasing effects can be extremely long, rather counterintuitively decoherence can enhance localization of excitation in the lattice. The results are illustrated by considering photonic quantum walks in synthetic mesh lattices.

Observation of reentrant metal-insulator transition in a random-dimer disordered SSH lattice

The interrelationship between localization, quantum transport, and disorder has remained a fascinating focus in scientific research. Traditionally, it has been widely accepted in the physics community that in one-dimensional systems, as disorder increases, localization intensifies, triggering a metal-insulator transition. However, a recent theoretical investigation [Phys. Rev. Lett. 126, 106803] has revealed that the interplay between dimerization and disorder leads to a reentrant localization transition, constituting a remarkable theoretical advancement in the field. Here, we present the first experimental observation of reentrant localization using an experimentally friendly model, a photonic SSH lattice with random-dimer disorder, achieved by incrementally adjusting synthetic potentials. In the presence of correlated on-site potentials, certain eigenstates exhibit extended behavior following the localization transition as the disorder continues to increase. We directly probe the wave function in disordered lattices by exciting specific lattice sites and recording the light distribution. This reentrant phenomenon is further verified by observing an anomalous peak in the normalized participation ratio. Our study enriches the understanding of transport in disordered mediums and accentuates the substantial potential of integrated photonics for the simulation of intricate condensed matter physics phenomena.

Dissipation-Induced Extended-Localized Transition

A mobility edge (ME), representing the critical energy that distinguishes between extended and localized states, is a key concept in understanding the transition between extended (metallic) and localized (insulating) states in disordered and quasiperiodic systems. Here we explore the impact of dissipation on a quasiperiodic system featuring MEs by calculating steady-state density matrix and analyzing quench dynamics with sudden introduction of dissipation. We demonstrate that dissipation can lead the system into specific states predominantly characterized by either extended or localized states, irrespective of the initial state. Our results establish the use of dissipation as a new avenue for inducing transitions between extended and localized states and for manipulating dynamic behaviors of particles.

Quantum criticality of generalized Aubry-André models with exact mobility edges using fidelity susceptibility

In this study, we explore the quantum critical phenomena in generalized Aubry-André models, with a particular focus on the scaling behavior at various filling states. Our approach involves using quantum fidelity susceptibility to precisely identify the mobility edges in these systems. Through a finite-size scaling analysis of the fidelity susceptibility, we are able to determine both the correlation-length critical exponent and the dynamical critical exponent at the critical point of the generalized Aubry-André model. Based on the Diophantine equation conjecture, we can determines the number of subsequences of the Fibonacci sequence and the corresponding scaling functions for a specific filling fraction, as well as the universality class. Our findings demonstrate the effectiveness of employing the generalized fidelity susceptibility for the analysis of unconventional quantum criticality and the associated universal information of quasiperiodic systems in cutting-edge quantum simulation experiments.

Realization of Gapped and Ungapped Photonic Topological Anderson Insulators

It is commonly believed that topologically nontrivial one-dimensional systems support edge states rather than bulk states at zero energy. In this work, we find an unanticipated case of topological Anderson insulator (TAI) phase where two bulk modes are degenerate at zero energy, in addition to degenerate edge modes. We term this "ungapped TAI" to distinguish it from the previously known gapped TAIs. Our experimental realization of both gapped and ungapped TAIs relies on coupled photonic resonators, in which the disorder in coupling is judiciously engineered by adjusting the spacing between the resonators. By measuring the local density of states both in the bulk and at the edges, we demonstrate the existence of these two types of TAIs, together forming a TAI plateau in the phase diagram. Our experimental findings are well supported by theoretical analysis. In the ungapped TAI phase, we observe stable coexistence of topological edge states and localized bulk states at zero energy, highlighting the distinction between TAIs and traditional topological insulators.

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.

Electrical analogue of one-dimensional and quasi-one-dimensional Aubry-André-Harper lattices

This work explores the potential for achieving correlated disorder in electrical circuits by utilizing reactive elements. By establishing a direct correspondence between the tight-binding Hamiltonian and the admittance matrix of the circuit, a novel approach is presented. The localization phenomena within the circuit are investigated through the analysis of the two-port impedance. To introduce correlated disorder, the Aubry-André-Harper (AAH) model is employed. Both one-dimensional and quasi-one-dimensional AAH structures are examined and effectively mapped to their tight-binding counterparts. Notably, transitions from a high-conducting phase to a low-conducting phase are observed in these circuits, highlighting the impact of correlated disorder.

Polariton Localization and Dispersion Properties of Disordered Quantum Emitters in Multimode Microcavities

Experiments have demonstrated that the strong light-matter coupling in polaritonic microcavities significantly enhances transport. Motivated by these experiments, we have solved the disordered multimode Tavis-Cummings model in the thermodynamic limit and used this solution to analyze its dispersion and localization properties. The solution implies that wave-vector-resolved spectroscopic quantities can be described by single-mode models, but spatially resolved quantities require the multimode solution. Nondiagonal elements of the Green's function decay exponentially with distance, which defines the coherence length. The coherent length is strongly correlated with the photon weight and exhibits inverse scaling with respect to the Rabi frequency and an unusual dependence on disorder. For energies away from the average molecular energy E_{M} and above the confinement energy E_{C}, the coherence length rapidly diverges such that it exceeds the photon resonance wavelength λ_{0}. The rapid divergence allows us to differentiate the localized and delocalized regimes and identify the transition from diffusive to ballistic transport.

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.

Quantum transports in two-dimensions with long range hopping

We investigate the effects of disorder and shielding on quantum transports in a two dimensional system with all-to-all long range hopping. In the weak disorder, cooperative shielding manifests itself as perfect conducting channels identical to those of the short range model, as if the long range hopping does not exist. With increasing disorder, the average and fluctuation of conductance are larger than those in the short range model, since the shielding is effectively broken and therefore long range hopping starts to take effect. Over several orders of disorder strength (until [Formula: see text] times of nearest hopping), although the wavefunctions are not fully extended, they are also robustly prevented from being completely localized into a single site. Each wavefunction has several localization centers around the whole sample, thus leading to a fractal dimension remarkably smaller than 2 and also remarkably larger than 0, exhibiting a hybrid feature of localization and delocalization. The size scaling shows that for sufficiently large size and disorder strength, the conductance tends to saturate to a fixed value with the scaling function [Formula: see text], which is also a marginal phase between the typical metal ([Formula: see text]) and insulating phase ([Formula: see text]). The all-to-all coupling expels one isolated but extended state far out of the band, whose transport is extremely robust against disorder due to absence of backscattering. The bond current picture of this isolated state shows a quantum version of short circuit through long hopping.

Localization properties of harmonic chains with correlated mass and spring disorder: Analytical approach

We study the localization properties of normal modes in harmonic chains with mass and spring weak disorder. Using a perturbative approach, an expression for the localization length L_{loc} is obtained, which is valid for arbitrary correlations of the disorder (mass disorder correlations, spring disorder correlations, and mass-spring disorder correlations are allowed), and for practically the whole frequency band. In addition, we show how to generate effective mobility edges by the use of disorder with long range self-correlations and cross-correlations. The transport of phonons is also analyzed, showing effective transparent windows that can be manipulated through the disorder correlations even for relative short chain sizes. These results are connected to the problem of heat conduction in the harmonic chain; indeed, we discuss the size scaling of the thermal conductivity from the perturbative expression of L_{loc}. Our results may have applications in modulating thermal transport, particularly in the design of thermal filters or in manufacturing high-thermal-conductivity materials.

Numerical investigation of localization in two-dimensional quasiperiodic mosaic lattice

A one-dimensional lattice model with mosaic quasiperiodic potential is found to exhibit interesting localization properties, e.g. clear mobility edges (Wanget al2020Phys. Rev. Lett.125196604). We generalize this mosaic quasiperiodic model to a two-dimensional version, and numerically investigate its localization properties: the phase diagram from the fractal dimension of the wavefunction, the statistical and scaling properties of the conductance. Compared with disordered systems, our model shares many common features but also exhibits some different characteristics in the same dimensionality and the same universality class. For example, the sharp peak atg∼0of the critical distribution and the largeglimit of the universal scaling functionβresemble those behaviors of three-dimensional disordered systems.

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.

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.

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.

Analytical approach to Lyapunov time: Universal scaling and thermalization

Based on the geometrization of dynamics and self-consistent phonon theory, we develop an analytical approach to derive the Lyapunov time, the reciprocal of the largest Lyapunov exponent, for general nonlinear lattices of coupled oscillators. The Fermi-Pasta-Ulam-Tsingou-like lattices are exemplified by using the method, which agree well with molecular dynamical simulations for the cases of quartic and sextic interactions. A universal scaling behavior of the Lyapunov time with the nonintegrability strength is observed for the quasi-integrable regime. Interestingly, the scaling exponent of the Lyapunov time is the same as the thermalization time, which indicates a proportional relationship between the two timescales. This relation illustrates how the thermalization process is related to the intrinsic chaotic property.