We introduce a self-consistent theory of mobility edges in nearest-neighbour tight-binding chains with quasiperiodic potentials. Demarcating boundaries between localised and extended states in the space of system parameters and energy, mobility edges are generic in quasiperiodic systems which lack the energy-independent self-duality of the commonly studied Aubry-André-Harper model. The potentials in such systems are strongly and infinite-range correlated, reflecting their deterministic nature and rendering the problem distinct from that of disordered systems. Importantly, the underlying theoretical framework introduced is model-independent, thus allowing analytical extraction of mobility edge trajectories for arbitrary quasiperiodic systems. We exemplify the theory using two families of models, and show the results to be in very good agreement with the exactly known mobility edges as well numerical results obtained from exact diagonalisation.
Mobility edges, separating localized from extended states, are known to arise in the single-particle energy spectrum of disordered systems in dimension strictly higher than two and certain quasiperiodic models in one dimension. Here we unveil a different class of mobility edges, dubbed anomalous mobility edges, that separate bands of localized states from bands of critical states in diagonal and off-diagonal quasiperiodic models. We first introduce an exactly solvable quasi-periodic diagonal model and analytically demonstrate the existence of anomalous mobility edges. Moreover, numerical multi…
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, if correlations are present in the disorder potential, the localization transition can occur at a finite disorder strength and SPMEs become possible. In this work, we find experimental evidence for the existence of such a SPME in a one-dimensional quasi-periodic optical lattice. Specifically, we find a re…
We investigate localization properties in a two-coupled uniform chains with quasiperiodic modulation on interchain coupling strength. We demonstrate that this ladder is equivalent to a Aubry-Andre (AA) chain when two legs are symmetric. Analytical and numerical results indicate the appearance of mobility edges for asymmetric ladder. We also propose an easily engineered quasiperiodic ladder system which is a moiré superlattice system consisting of two-coupled uniform chains. An irrational lattice constant difference results in quasiperiodic structure. Numerical simulations show that such a syst…
We aim to study a one-dimensional $p$-wave superconductor with quasiperiodic on-site potentials. A modified real-space-Pfaffian method is applied to calculate the topological invariants. We confirm that the Majorana zero mode is protected by the nontrivial topology the topological phase transition is accompanied by the energy gap closing and reopening. In addition, we numerically find that there are mobility edges which originate from the competition between the extended $p$-wave pairing and the localized quasi-disorder. We qualitatively analyze the influence of superconducting pairing paramet…
Engineering exact mobility edges in quasiperiodic Aharonov-Bohm chains
Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences（中国科学院精密测量科学与技术创新研究院）；University of Chinese Academy of Sciences（中国科学院大学）
Within the framework of the Aubry-Andre model, one kind of self-dual quasiperiodic lattice, it is known that a sharp transition occurs from \emph{all} eigenstates being extended to \emph{all} being localized. The common perception for this type of quasiperiodic lattice is that the self-duality excludes the appearance of the mobility edge separating localized from extended states. In this work, we propose a multi-chromatic quasiperiodic lattice model retaining the self-duality identical to the Aubry-Andre model, and demonstrate numerically the occurrence of the localization-delocalization trans…
Anomalous mobility edges and extended-localized transition in a quasiperiodic emitter-cavity array
School of Physics and Optoelectronic Engineering & Institute of Theoretical Physics, Ludong University（鲁东大学物理与光电工程学院暨理论物理研究所）；Center for Quantum Sciences, Northeast Normal University（东北师范大学量子科学中心）
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-André-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 t…
We study the stability of the topological phase in one-dimensional Su-Schrieffer-Heeger chain subject to the quasiperiodic hopping disorder. We investigate two different hopping disorder configurations, one is the Aubry-André quasiperiodic disorder without mobility edges and the other is the slowly varying quasiperiodic disorder with mobility edges. With the increment of the quasiperiodic disorder strength, the topological phase of the system transitions to a topologically trivial phase. Interestingly, we find the occurrence of the topological phase transition at the critical disorder strength…
Quasiperiodic systems serve as fertile ground for studying localisation, due to their propensity already in one dimension to exhibit rich phase diagrams with mobility edges. The deterministic and strongly-correlated nature of the quasiperiodic potential nevertheless offers challenges distinct from disordered systems. Motivated by this, we present a theory of localisation in quasiperiodic chains with nearest-neighbour hoppings, based on the convergence of local propagators; exploiting the fact that the imaginary part of the associated self-energy acts as a probabilistic order parameter for loca…
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 of such models are still limited. In this work, we propose a new method to determine the exact mobility edge in a large class of 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 f…
We study if periodic driving of a model with a quasiperiodic potential can generate interesting Floquet phases which have no counterparts in the static model. Specifically, we consider the Aubry-André model which is a one-dimensional time-independent model with an on-site quasiperiodic potential $V_0$ and a nearest-neighbor hopping amplitude which is taken to have a staggered form. We add a uniform hopping amplitude which varies periodically in time with a frequency $ω$. Unlike the static Aubry-André model which has a simple phase diagram with only two phases (only extended or only localized s…
In this paper, we study a one-dimensional tight-binding model with tunable incommensurate potentials. Through the analysis of the inverse participation rate, we uncover that the wave functions corresponding to the energies of the system exhibit different properties. There exists a critical energy under which the wave functions corresponding to all energies are extended. On the contrary, the wave functions corresponding to all energies above the critical energy are localized. However, we are surprised to find that the critical energy is a constant independent of the potentials. We use the self-…
Entanglement growth and information capacity in a quasiperiodic system with a single-particle mobility edge
Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）
We study a one-dimensional system that includes both a commensurate off-diagonal modulation of the hopping amplitude and an incommensurate, slowly varying diagonal on-site modulation. By using asymptotic heuristic arguments, we identify four closed form expressions for the mobility edges. We further study numerically the inverse participation ratio, the density of states and the Lyapunov exponent. The numerical results are in exact agreement with our theoretical predictions. Besides a metal-insulator transition driven by the strength of the slowly varying potential, another four insulator-meta…
The mobility edges (MEs) in energy which 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 which 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 …
We uncover the relationship of topology and disorder in a one-dimensional Su-Schrieffer-Heeger chain subjected to a slowly varying quasi-periodic modulation. By numerically calculating the disorder-averaged winding number and analytically studying the localization length of the zero modes, we obtain the topological phase diagram, which implies that the topological Anderson insulator (TAI) can be induced by a slowly varying quasi-periodic modulation. Moreover, unlike the localization properties in the TAI phase caused by random disorder, mobility edges can enter into the TAI region identified b…
We construct an example of a 1$d$ quasiperiodically driven spin chain whose edge states can coherently store quantum information, protected by a combination of localization, dynamics, and topology. Unlike analogous behavior in static and periodically driven (Floquet) spin chains, this model does not rely upon microscopic symmetry protection: Instead, the edge states are protected purely by emergent dynamical symmetries. We explore the dynamical signatures of this Emergent Dynamical Symmetry-Protected Topological (EDSPT) order through exact numerics, time evolving block decimation, and analytic…
Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems as well as disordered ones. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization group transformation. These discrete sequences are therefore fixed points of a \emph{functional} renormalizat…
We investigate and map out the non-equilibrium phase diagram of a generalization of the well known Aubry-André-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 b…
In this work, we investigate the Anderson localization problems of the generalized Aubry-André model (Ganeshan-Pixley-Das Sarma's model) with an unbounded quasi-periodic potential where the parameter $|α|\geq1$. The Lyapunov exponent $γ(E)$ and the mobility edges $E_c$ are exactly obtained for the unbounded quasi-periodic potential. With the Lyapunov exponent, we find that there exists a critical region in the parameter $λ-E$ plane. The critical region consists of critical states. In comparison with localized and extended states, the fluctuation of spatial extensions of the critical states is …
Systems with quasiperiodic disorder are known to exhibit localization transition in low dimension. 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…
We study a one-dimensional quasiperiodic system described by the Aubry-André model in the small wave vector limit and demonstrate the existence of almost mobility edges and critical regions in the system. It is well known that the eigenstates of the Aubry-André model are either extended or localized depending on the strength of incommensurate potential $V$ being less or bigger than a critical value $V_c$, and thus no mobility edge exists. However, it was shown in a recent work that this conclusion does not hold true when the wave vector $α$ of the incommensurate potential is small, and for the…
We study the many-body localization (MBL) properties of a chain of interacting fermions subject to a quasiperiodic potential such that the non-interacting chain is always delocalized and displays multifractality. Contrary to naive expectations, adding interactions in this systems does not enhance delocalization, and a MBL transition is observed. Due to the local properties of the quasiperiodic potential, the MBL phase presents specific features, such as additional peaks in the density distribution. We furthermore investigate the fate of multifractality in the ergodic phase for low potential va…
We investigate the topology and localization of one-dimensional Hermitian and non-Hermitian Su-Schrieffer-Heeger chains with quasiperiodic hopping modulations. In the Hermitian case, phase diagrams are obtained by numerically and analytically calculating various topological and localization characters. We show the presence of topological extended, intermediate, and localized phases due to the coexistence of independent topological and localization phase transitions driven by the quasiperiodic disorder. Unlike the gapless and localized TAI phase in one-dimensional random disordered systems, we …
Sufficient disorder is believed to localize static and periodically-driven interacting chains. With quasiperiodic driving by $D$ incommensurate tones, the fate of this many-body localization (MBL) is unknown. We argue that randomly disordered MBL exists for $D=2$, but not for $D\geq 3$. Specifically, a putative two-tone driven MBL chain is neither destabilized by thermal avalanches seeded by rare thermal regions, nor by the proliferation of long-range many-body resonances. For $D\geq 3$, however, sufficiently large thermal regions have continuous local spectra and slowly thermalize the entire …
Single particle states in a chain with quasiperiodic potential show a metal-insulator transition upon the change of the potential strength. We consider two particles with local interaction in the single particle insulating regime. The two particle states change from being localized to delocalized upon an increase of the interaction strength to a nonperturbative finite value. At even larger interaction strength the states become localized again. This transition of two particle bound states into a correlated metal is due to a resonant mixing of the noninteracting two particle eigenstates. In the…
In their Letter[Phys. Rev. Lett. 126, 106803 (2021)], the authors found an interesting reentrant localization phenomenon in a one-dimensional dimerized lattice with quasiperiodic disorder, i.e., the system undergoes a second localization transition at a higher disorder strength. They claimed that this reentrant feature would appear in the case of staggered disorder, but not for the uniform ones. In this Comment, we show that such reentrant feature can also appear for the uniform disorder when larger system parameters are taken into account.
We investigate localization-delocalization transition in one-dimensional non-Hermitian quasiperiodic lattices with exponential short-range hopping, which possess parity-time ($\mathcal{PT}$) symmetry. The localization transition induced by the non-Hermitian quasiperiodic potential is found to occur at the $\mathcal{PT}$-symmetry-breaking point. Our results also demonstrate the existence of energy dependent mobility edges, which separate the extended states from localized states and are only associated with the real part of eigen-energies. The level statistics and Loschmidt echo dynamics are al…
In one dimension, noninteracting particles can undergo a localization-delocalization transition in a quasiperiodic potential. Recent studies have suggested that this transition transforms into a many-body localization (MBL) transition upon the introduction of interactions. It has also been shown that mobility edges can appear in the single particle spectrum for certain types of quasiperiodic potentials. Here, we investigate the effect of interactions in two models with such mobility edges. Employing the technique of exact diagonalization for finite-sized systems, we calculate the level spacing…
