Non-Hermitian quasicrystal in dimerized lattices

Non-Hermitian quasicrystals possess PT and metal–insulator transitions induced by gain and loss or nonreciprocal effects. In this work, we uncover the nature of localization transitions in a generalized Aubry–André–Harper model with dimerized hopping amplitudes and complex onsite potential. By investigating the spectrum, adjacent gap ratios and inverse participation ratios, we find an extended phase, a localized phase and a mobility edge phase, which are originated from the interplay between hopping dimerizations and non-Hermitian onsite potential. The lower and upper bounds of the mobility edge are further characterized by a pair of topological winding numbers, which undergo quantized jumps at the boundaries between different phases. Our discoveries thus unveil the richness of topological and transport phenomena in dimerized non-Hermitian quasicrystals.     

Majorana zero modes,unconventional real-complex transition,and mobility edges in a one-dimensional non-Hermitian quasi-periodic lattice

A one-dimensional non-Hermitian quasiperiodic p-wave superconductor without PT-symmetry is studied.By analyzing the spectrum,we discovered that there still exists real-complex energy transition even if the inexistence of PT-symmetry breaking.By the inverse participation ratio,we constructed such a correspondence that pure real energies correspond to the extended states and complex energies correspond to the localized states,and this correspondence is precise and effective to detect the mobility edges.After investigating the topological properties,we arrived at a fact that the Majorana zero modes in this system are immune to the non-Hermiticity.     

Topological phase transition in the quasiperiodic disordered Su–Schriffer–Heeger chain

We study the stability of the topological phase in one-dimensional Su–Schrieffer–Heeger chain subject to the quasiperiodic hopping disorder. Two different hopping disorder configurations are investigated, one is the Aubry–André quasiperiodic disorder without mobility edges and the other is the slowly varying quasiperiodic disorder with mobility edges. Interestingly, we find topological phase transitions occur at the critical quasiperiodic disorder strengths which have an exact linear relation with the dimerization strengths for both disorder configurations. We further investigate the localized property of the Su–Schrieffer–Heeger chain with the slowly varying quasiperiodic disorder, and identify that there exist mobility edges in the spectrum when the dimerization strength is unequal to 1. These interesting features of models will shed light on the study of interplay between topological and disordered systems.     

具有p波超流的一维非公度晶格中迁移率边研究

研究了具有p波超流的一维非公度晶格中迁移率边的性质. 发现适当的p波超流可以增加体系中的迁移率边的数目, 并且通过多分形分析确定了迁移率边所在的位置.     

Non-Hermitian Kitaev chain with complex periodic and quasiperiodic potentials

We study the topological properties of the one-dimensional non-Hermitian Kitaev model with complex either periodic or quasiperiodic potentials. We obtain the energy spectrum and the phase diagrams of the system by using the transfer matrix method as well as the topological invariant. The phase transition points are given analytically. The Majorana zero modes in the topological nontrivial regimes are obtained. Focusing on the quasiperiodic potential, we obtain the phase transition from the topological superconducting phase to the Anderson localization, which is accompanied with the Anderson localization–delocalization transition in this non-Hermitian system. We also find that the topological regime can be reduced by increasing the non-Hermiticity.     

Localization attractors in active quasiperiodic arrays

In dissipationless linear lattices, spatial disorder or quasiperiodic modulations in on-site potentials induce localization of the eigenstates and block the spreading of wave packets. Quasiperiodic inhomogeneities allow for the metal–insulator transition at a finite modulation amplitude already in one dimension. We go beyond the dissipationless limit and consider nonlinear quasi-periodic arrays that are additionally subjected to dissipative losses and energy pumping. We find finite excitation thresholds for oscillatory phases in both metallic and insulating regimes. In contrast to disordered arrays, the transition in the metallic and weakly insulating regimes display features of the second order phase transition accompanied by a large-scale cluster synchronization. In the limit of strong localization, we find the existence of globally stable asymptotic states consisting of several localized modes. These localization attractors and chaotic synchronization effects can be potentially implemented with polariton condensate lattices and cavity-QED arrays.     

Non-Hermitian topological Anderson insulators

Non-Hermitian systems can exhibit exotic topological and localization properties.Here we elucidate the non-Hermitian effects on disordered topological systems using a nonreciprocal disordered Su-Schrieffer-Heeger model.We show that the non-Hermiticity can enhance the topological phase against disorders by increasing bulk gaps.Moreover,we uncover a topological phase which emerges under both moderate non-Hermiticity and disorders,and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge modes.Such topological phases induced by the combination of non-Hermiticity and disorders are dubbed non-Hermitian topological Anderson insulators.We reveal that the system has unique non-monotonous localization behavior and the topological transition is accompanied by an Anderson transition.These properties are general in other non-Hermitian models.     

Bloch-like oscillations in a one-dimensional lattice with long-range correlated disorder

We study the dynamics of an electron subjected to a uniform electric field within a tight-binding model with long-range-correlated diagonal disorder. The random distribution of site energies is assumed to have a power spectrum S(k) approximately 1/k(alpha) with alpha>0. de Moura and Lyra [Phys. Rev. Lett. 81, 3735 (1998)]] predicted that this model supports a phase of delocalized states at the band center, separated from localized states by two mobility edges, provided alpha>2. We find clear signatures of Bloch-like oscillations of an initial Gaussian wave packet between the two mobility edges and determine the bandwidth of extended states, in perfect agreement with the zero-field prediction.     

Multifractal analysis of electronic states on random Voronoi-Delaunay lattices

We consider the transport of non-interacting electrons on two- and three-dimensional random Voronoi-Delaunay lattices. It was recently shown that these topologically disordered lattices feature strong disorder anticorrelations between the coordination numbers that qualitatively change the properties of continuous and first-order phase transitions. To determine whether or not these unusual features also influence Anderson localization, we study the electronic wave functions by multifractal analysis and finite-size scaling. We observe only localized states for all energies in the two-dimensional system. In three dimensions, we find two Anderson transitions between localized and extended states very close to the band edges. The critical exponent of the localization length is about 1.6. All these results agree with the usual orthogonal universality class. Additional generic energetic randomness introduced via random potentials does not lead to qualitative changes but allows us to obtain a phase diagram by varying the strength of these potentials.     

Phase diagram of a family of one-dimensional nearest-neighbor tight-binding models with an exact mobility edge

Recently, an interesting family of quasiperiodic models with exact mobility edges(MEs) has been proposed(Phys.Rev. Lett. 114 146601(2015)). It is self-dual under a generalized duality transformation. However, such transformation is not obvious to map extended(localized) states in the real space to localized(extended) ones in the Fourier space. Therefore,it needs more convictive evidences to confirm the existence of MEs. We use the second moment of wave functions, Shannon information entropies, and Lypanunov exponents to characterize the localization properties of the eigenstates, respectively.Furthermore, we obtain the phase diagram of the model. Our numerical results support the existing analytical findings.     

Aggregation and intermediate phases in dilute spin systems

We study a variety of dilute annealed lattice spin systems. For site diluted problems with many internal spin states, we uncover a new phase characterized by the occupation and vacancy of staggered sublattices. In cases where the uniform system has a low temperature phase, the staggered states represent an intermediate phase. Furthermore, in many of these cases, we show that (at least part of) the phase boundary separating the low-temperature and staggered phases is a line of phase coexistence-i.e. the transition is first order. We also study the phenomenon of aggregation (phase separation) in bond diluted models. Such transitions are known, trivially, to occur in the large-q Potts models. However, it turns out that phase separation is typical in bond diluted spin systems with many internal states. (In particular, a bond aggregation transition is not tied to a discontinuous transition in the uniform system.) Along the portions of the phase boundary where any of these phenomena occur, the prospects for a Fisher renormalization effect are deemed to be highly unlikely or are ruled out altogether.Partly supported by the NSF grant DMS-93-02023 (L.C.), the grants GAR 202/93/0449 and GAUK 376 (R.K.), and the NSF grant DMS-92-08029 and the Russian Fund of Fundamental Investigations grant 93-01-01470 (S.B.S.).     

Influence of weak nonlinearity on the 1D Anderson model with long-range correlated disorder

We investigate theoretically the nature of the states and the localization properties in a one-dimensional Anderson model with long-range correlated disorder and weak nonlinearity. Using the stationary discrete nonlinear Schrödinger equation, we calculate the disorder-averaged logarithm of the transmittance and the localization length in the fixed input case in a numerically exact manner. Unlike in many previous studies, we strictly fix the intensity of the incident wave and calculate the localization length as a function of other parameters. We also calculate the wave functions in a given disorder configuration. In the linear case, flat phased localized states appear near the bottom of the band and staggered localized states appear near the top of the band, while a continuum of extended states appears near the band center. We find that the focusing Kerr-type nonlinearity enhances the Anderson localization of flat phased states and suppresses that of staggered states. We observe that there exists a perfect symmetry relationship for the localization length between focusing and defocusing nonlinearities. Above a critical value of the strength of nonlinearity, delocalization due to the long-range correlations of disorder is destroyed and all states become localized.     

Delocalization and Heisenberg's uncertainty relation

In the one-dimensional Anderson model the eigenstates are localized for arbitrarily small amounts of disorder. In contrast, the Aubry-André model with its quasiperiodic potential shows a transition from extended to localized states. The difference between the two models becomes particularly apparent in phase space where Heisenberg's uncertainty relation imposes a finite resolution. Our analysis points to the relevance of the coupling between momentum eigenstates at weak potential strength for the delocalization of a quantum particle. Received 3 May 2002 / Received in final form 2 October 2002 Published online 29 November 2002     

Many-body ground state localization and coexistence of localized and extended states in an interacting quasiperiodic system

We study the localization problem of one-dimensional interacting spinless fermions in anincommensurate optical lattice, which changes from an extended phase to a non-ergoicmany-body localized phase by increasing the strength of the incommensurate potential. Weidentify that there exists an intermediate regime before the system enters the many-bodylocalized phase, in which both the localized and extended many-body states coexist, thusthe system is divided into three different phases, which can be characterized bynormalized participation ratios of the many-body eigenstates and distributions of naturalorbitals of the corresponding one-particle density matrix. This is very different from itsnoninteracting limit, in which all eigenstates undergo a delocalization-localizationtransition when the strength of the incommensurate potential exceeds a critical value.     

Dynamical localization and eigenstate localization in trap models

The one-dimensional random trap model with a power-law distribution of mean sojourn times exhibits a phenomenon of dynamical localization in the case where diffusion is anomalous: the probability to find two independent walkers at the same site, as given by the participation ratio, stays constant and high in a broad domain of intermediate times. This phenomenon is absent in dimensions two and higher. In finite lattices of all dimensions the participation ratio finally equilibrates to a different final value. We numerically investigate two-particle properties in a random trap model in one and in three dimensions, using a method based on spectral decomposition of the transition rate matrix. The method delivers a very effective computational scheme producing numerically exact results for the averages over thermal histories and initial conditions in a given landscape realization. Only a single averaging procedure over disorder realizations is necessary. The behavior of the participation ratio is compared to other measures of localization, as for example to the states’ gyration radius, according to which the dynamically localized states are extended. This means that although the particles are found at the same site with a high probability, the typical distance between them grows. Moreover the final equilibrium state is extended both with respect to its gyration radius and to its Lyapunov exponent. In addition, we show that the phenomenon of dynamical localization is only marginally connected with the spectrum of the transition rate matrix, and is dominated by the properties of its eigenfunctions which differ significantly in dimensions one and three.     

Non-Hermitian anomalous skin effect

A non-Hermitian topological insulator is fundamentally different from conventional topological insulators. The non-Hermitian skin effect arises in a nonreciprocal tight binding lattice with open edges. In this case, not only topological states but also bulk states are localized around the edges of the nonreciprocal system. We discuss that controllable switching from topological edge states into topological extended states in a chiral symmetric non-Hermitian system is possible. We show that the skin depth decreases with non-reciprocity for bulk states but increases with it for topological zero energy states.     

Topological Anderson insulator in two-dimensional non-Hermitian systems

We study the disorder-induced phase transition in two-dimensional non-Hermitian systems. First, the applicability of the noncommutative geometric method(NGM) in non-Hermitian systems is examined. By calculating the Chern number of two different systems(a square sample and a cylindrical one), the numerical results calculated by NGM are compared with the analytical one, and the phase boundary obtained by NGM is found to be in good agreement with the theoretical prediction. Then, we use NGM to investigate the evolution of the Chern number in non-Hermitian samples with the disorder effect. For the square sample, the stability of the non-Hermitian Chern insulator under disorder is confirmed. Significantly,we obtain a nontrivial topological phase induced by disorder. This phase is understood as the topological Anderson insulator in non-Hermitian systems. Finally, the disordered phase transition in the cylindrical sample is also investigated. The clean non-Hermitian cylindrical sample has three phases, and such samples show more phase transitions by varying the disorder strength:(1) the normal insulator phase to the gapless phase,(2) the normal insulator phase to the topological Anderson insulator phase, and(3) the gapless phase to the topological Anderson insulator phase.     

The geometric mean density of states and its application to one-dimensional nonuniform systems

By using the measure of the ratio R of the geometric mean of the local density of states (LDOS) and the arithmetic mean of LDOS, the localization properties can be efficiently characterized in one-dimensional nonuniform single-electron and two-interacting-particle (TIP) systems. For single-electron systems, the extended and localized states can be distinguished by the ratio R. There are sharp transitions in the ratio R at mobility edges. For TIP systems, the localization properties of particle states can also be reflected by the ratio R. These results are in accordance with what obtained by other methods. Therefore, the ratio R is a suitable quantity to characterize the localization properties of particle states for these 1D nonuniform systems.     

Anderson localization and nonlinearity in one-dimensional disordered photonic lattices

We experimentally investigate the evolution of linear and nonlinear waves in a realization of the Anderson model using disordered one-dimensional waveguide lattices. Two types of localized eigenmodes, flat-phased and staggered, are directly measured. Nonlinear perturbations enhance localization in one type and induce delocalization in the other. In a complementary approach, we study the evolution on short time scales of delta-like wave packets in the presence of disorder. A transition from ballistic wave packet expansion to exponential (Anderson) localization is observed. We also find an intermediate regime in which the ballistic and localized components coexist while diffusive dynamics is absent. Evidence is found for a faster transition into localization under nonlinear conditions.     

交错跃迁Hofstadter梯子的量子流相

为玻色Hofstadter梯子模型引入交错跃迁,来扩展模型支持的量子流相.基于精确对角化和密度矩阵重整化群计算发现,无相互作用时,系统中包含横流相、涡旋相和纵流相;横流相来自均匀跃迁时Hofstadter梯子模型的Meissner相,纵流相是交错跃迁时才可见的流相.强相互作用极限下系统的超流区也包含横流相、纵流相和涡旋相,但存在更多的相变级数;超流区的横流相、纵流相之间存在相变但Mott区的不存在,把Mott区的"横、纵流相"称为Mott-均匀相,在Mott区只存在均匀相和涡旋相.跃迁的交错会压缩涡旋相存在的区域,使Mott区最终只剩下均匀相;跃迁的交错不仅能驱动Mott-超流相变,还使磁通的改变也能够驱动系统的Mott-超流相变.对这一系统的研究丰富了磁通系统中的量子流相,同时为研究拓扑流特性提供了模型支持.