Incommensurate Potential Research Articles

Fate of topological states and mobility edges in one-dimensional slowly varying incommensurate potentials

We investigate the interplay between disorder and superconducting pairing for a one-dimensional $p$-wave superconductor subject to slowly varying incommensurate potentials with mobility edges. With amplitude increments of the incommensurate potentials, the system can undergo a transition from a topological phase to a topologically trivial localized phase. Interestingly, we find that there are four mobility edges in the spectrum when the strength of the incommensurate potential is below a critical threshold, and a novel topologically nontrivial localized phase emerges in a certain region. We reveal this energy-dependent metal-insulator transition by applying several numerical diagnostic techniques, including the inverse participation ratio, the density of states and the Lyapunov exponent. Nowadays, precise control of the background potential and the $p$-wave superfluid can be realized in the ultracold atomic systems, we believe that these novel mobility edges can be observed experimentally.

Almost mobility edges and the existence of critical regions in one-dimensional quasiperiodic lattices

We study a one-dimensional quasiperiodic system described by the Aubry-Andr\'e 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-Andr\'e 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 $\alpha$ of the incommensurate potential is small, and for the system with $V<V_c$, there exist almost mobility edges at the energy $E_{c_{\pm}}$, which separate the robustly delocalized states from "almost localized" states. We find that, besides $E_{c_{\pm}}$, there exist additionally another energy edges $E_{c'_{\pm}}$, at which abrupt change of inverse participation ratio occurs. By using the inverse participation ratio and carrying out multifractal analyses, we identify the existence of critical regions among $|E_{c_{\pm}}| \leq |E| \leq |E_{c'_{\pm}}|$ with the almost mobility edges $E_{c_{\pm}}$ and $E_{c'_{\pm}}$ separating the critical region from the extended and localized regions, respectively. We also study the system with $V>V_c$, for which all eigenstates are localized states, but can be divided into extended, critical and localized states in their dual space by utilizing the self-duality property of the Aubry-Andr\'e model.

Mobility edges in one-dimensional bichromatic incommensurate potentials

We theoretically study a one-dimensional (1D) mutually incommensurate bichromatic lattice system which has been implemented in ultracold atoms to study quantum localization. It has been universally believed that the tight-binding version of this bichromatic incommensurate system is represented by the well-known Aubry-Andre model. Here we establish that this belief is incorrect and that the Aubry-Andre model description, which applies only in the extreme tight-binding limit of very deep primary lattice potential, generically breaks down near the localization transition due to the unavoidable appearance of single-particle mobility edges (SPME). In fact, we show that the 1D bichromatic incommensurate potential system manifests generic mobility edges which disappear in the tight-binding limit, leading to the well-studied Aubry-Andre physics. We carry out an extensive study of the localization properties of the 1D incommensurate optical lattice without making any tight-binding approximation. We find that, for the full lattice system, an intermediate phase between completely localized and completely delocalized regions appears due to the existence of the SPME, making the system qualitatively distinct from the Aubry-Andre prediction. Using the Wegner flow approach, we show that the SPME in the real lattice system can be attributed to significant corrections of higher-order harmonics in the lattice potential which are absent in the strict tight-binding limit. We calculate the dynamical consequences of the intermediate phase in detail to guide future experimental investigations for the observation of 1D SPME and the associated intermediate phase. We consider effects of interaction numerically, and conjecture the stability of SPME to weak interaction effects, thus leading to the exciting possibility of an experimentally viable nonergodic extended phase in interacting 1D optical lattices.

Bosons with incommensurate potential and spin-orbit coupling

We chart out the phase diagram of ultracold `spin-half' bosons in a one-dimensional optical lattice in the presence of Aubry-Andr\'e (AA) potential and with spin-orbit (SO) and Raman couplings investigating the transition from superfluid (SF) to localized phases and the existence of density wave phase for nearest-neighbor interaction (NNI). We show that the presence of SO coupling and AA potential leads to a novel spin-split momentum distribution of the bosons in the localized phase near the boundary with the SF phase, which can act as a signature of such a transition. We also obtain the level statistics of the bosons in the superfluid phase with finite NNI and demonstrate its change from Gaussian Unitary Ensemble (GUE) to Gaussian Orthogonal Ensemble (GOE) as a function of the Raman coupling. We discuss experiments which can test our theory.

Fate of Weyl semimetals in the presence of incommensurate potentials

We investigate the effect of the incommensurate potential on Weyl semimetal, which is proposed to be realized in ultracold atomic systems trapped in three-dimensional optical lattices. For the system without the Fermi arc, we find that the Weyl points are robust against the incommensurate potential and the system enters into a metallic phase only when the incommensurate potential strength exceeds a critical value. We unveil the trastition by analysing the properties of wave functions and the density of states as a function of the incommensurate potential strength. We also study the system with Fermi arcs and find the Fermi arcs are sensitive against the incommensurate potential and can be destoryed by a weak incommensurate potential.

Topological superconductivity in quantum Hall–superconductor hybrid systems

We develop a scenario to engineer a topological phase with Majorana edge states based on an integer quantum Hall (QH) system proximity coupled to a superconductor (SC). Due to the vortices in the SC order parameter, the SC--QH hybrid system is described by a Bloch problem with ten unpaired momenta, corresponding to the maxima and saddle points of the SC order parameter. For external potentials respecting the symmetry of the vortex lattice, the states with unpaired momenta have degeneracies such that the system always is in a trivial phase. However, an incommensurate potential can lift the degeneracies and drive the system into a topologically nontrivial phase.

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 an incommensurate optical lattice, which changes from an extended phase to a nonergoic many-body localized phase by increasing the strength of the incommensurate potential. We identify that there exists an intermediate regime before the system enters the many-body localized phase, in which both the localized and extended many-body states coexist, thus the system is divided into three different phases, which can be characterized by normalized participation ratios of the many-body eigenstates and distributions of natural orbitals of the corresponding one-particle density matrix. This is very different from its noninterating limit, in which all eigenstaes undergo a delocaliztion-localization transtion when the strength of the incommensurate potential exceeds a critical value.

Localization in the Ground State of an Interacting Quasi-Periodic Fermionic Chain

We consider a many body fermionic system with an incommensurate external potential and a short range interaction in one dimension. We prove that, for certain densities and weak interactions, the zero temperature thermodynamical correlations are exponentially decaying for large distances, a property indicating persistence of localization in the interacting ground state. The analysis is based on Renormalization Group, and convergence of the renormalized expansion is achieved using fermionic cancellations and controlling the small divisor problem assuming a Diophantine condition for the frequency.

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.

Enhancement of long-range correlations in a 2D vortex lattice by an incommensurate 1D disorder potential

Long range correlations in two-dimensional (2D) systems are significantly altered by disorder potentials. Theory has predicted the existence of disorder induced phenomena such as Anderson localization and the emergence of novel glass and insulating phases as the Bose glass. More recently, it has been shown that disorder breaking the 2D continuous symmetry, such as a one dimensional (1D) modulation, can enhance long range correlations. Experimentally, developments in quantum gases have allowed the observation of a wealth of phenomena induced by the competition between interaction and disorder. However, there are no experiments exploring the effect of symmetry-breaking disorder. Here, we create a 2D vortex lattice at 0.1 K in a superconducting thin film with a well-defined 1D thickness modulation and track the field induced modification using scanning tunneling microscopy. We find that the 1D modulation becomes incommensurate to the vortex lattice and drives an order-disorder transition, behaving as a scale-invariant disorder potential. We show that the transition occurs in two steps and is mediated by the proliferation of topological defects. We find that critical exponents determining the loss of positional and orientational order are far above theoretical expectations for scale-invariant disorder and follow instead the critical behaviour which describes dislocation unbinding melting. Our data show for the first time that randomness disorders a 2D crystal, and evidence enhanced long range correlations in presence of a 1D modulation demonstrating the transformation induced by symmetry breaking disorder in interactions and the critical behaviour of the transition.

Topological Properties of Ultracold Bosons in One-Dimensional Quasiperiodic Optical Lattice

We analyze the topological properties of the one-dimensional Bose–Hubbard model with a quasiperiodic superlattice potential. This system can be realized in interacting ultracold bosons in an optical lattice in the presence of an incommensurate superlattice potential. We first analyze the quasiperiodic superlattice formed by the cosine function, which we call the Harper-like Bose–Hubbard model. We compute the Chern number and observe gap-closing behavior as the interaction strength U is changed. Also, we discuss the bulk-edge correspondence in our system. Furthermore, we explore the phase diagram as a function of U and a continuous deformation parameter β between the Harper-like model and another important quasiperiodic lattice, the Fibonacci model. We numerically confirm that the incommensurate charge density wave (ICDW) phase is topologically nontrivial and that it is topologically equivalent in the whole ICDW region.

Dynamical Anderson transition in one-dimensional periodically kicked incommensurate lattices

We study the dynamical localization transition in a one-dimensional periodically kicked incommensurate lattice, which is created by perturbing a primary optical lattice periodically with a pulsed weaker incommensurate lattice. The diffusion of wave packets in the pulsed optical lattice exhibits either extended or localized behaviors, which can be well characterized by the mean square displacement and the spatial correlation function. We show that the dynamical localization transition is relevant to both the strength of incommensurate potential and the kicked period, and the transition point can be revealed by the information entropy of eigenfunctions of the Floquet propagator.

Effect of incommensurate potential on the resonant tunneling through Majorana bound states on the topological superconductor chains

We study the transport through the Kitaev chain with incommensurate potentials coupled to two normal leads by the numerical operator method. We find a quantized linear conductance of e 2 / h, which is independent to the disorder strength and the gate voltage in a wide range, signaling the Majorana bound states. While the incommensurate potential suppresses the current at finite voltage bias, and then narrows the linear response regime of the I-V curve which exhibits two plateaus corresponding to the superconducting gap and the band edge, respectively. The linear conductance abruptly drops to zero as the disorder strength reaches the critical value 2g s + 2Δ with Δ the p-wave pairing amplitude and g s the hopping between neighbor sites, corresponding to the transition from the topological superconducting phase to the Anderson localized phase. Changing the gate voltage also causes an abrupt drop of the linear conductance by driving the chain into the topologically trivial superconducting phase, whose I-V curve exhibits an exponential shape.

Quantum phase transitions and phase diagram for a one-dimensional p-wave superconductor with an incommensurate potential

The effect of the incommensurate potential is studied for the one-dimensional p-wave superconductor. It is determined by analyzing various properties, such as the superconducting gap, the long-range order of the correlation function, the inverse participation ratio and the Z2 topological invariant, etc. In particular, two important aspects of the effect are investigated: (1) as disorder, the incommensurate potential destroys the superconductivity and drives the system into the Anderson localized phase; (2) as a quasi-periodic potential, the incommensurate potential causes band splitting and turns the system with certain chemical potential into the band insulator phase. A full phase diagram is also presented in the chemical potential–incommensurate potential strength plane.

Topological Superconductor to Anderson Localization Transition in One-Dimensional Incommensurate Lattices

We study the competition of disorder and superconductivity for a one-dimensional p-wave superconductor in incommensurate potentials. With the increase in the strength of the incommensurate potential, the system undergoes a transition from a topological superconducting phase to a topologically trivial localized phase. The phase boundary is determined both numerically and analytically from various aspects and the topological superconducting phase is characterized by the presence of Majorana edge fermions in the system with open boundary conditions. We also calculate the topological Z2 invariant of the bulk system and find it can be used to distinguish the different topological phases even for a disordered system.

Finite-temperature properties of one-dimensional hard-core bosons in a quasiperiodic optical lattice

We investigate the properties of impenetrable bosons confined in a one-dimensional lattice at finite temperature in the presence of an additional incommensurate periodic potential. Relying on the exact Fermi-Bose mapping, we study the effects of temperature on the one-particle density matrix and related quantities such as the momentum distribution function and the natural orbitals. We found evidence of a finite-temperature crossover related to the zero-temperature superfluid-to-Bose-glass transition that induces a delocalization of the lowest natural orbitals.

Disordered polyhalide anion effect on the Fermi surface of the incommensurate organic superconductor (MDT-TSF)I0.77Br0.52

The effect of disorder in polyhalide anions is investigated in terms of the degree of charge transfer and the Fermi surface in the incommensurate organic superconductor (MDT-TSF)I${}_{0.77}$Br${}_{0.52}$, where MDT-TSF is methylenedithio-tetraselenafulvalene. The Raman spectra suggest that the infinite polyhalide chains include the asymmetric trihalide anion I${}_{2}$Br${}^{\ensuremath{-}}$ and that the degree of charge transfer from the donor molecule to the anion is $0.429$, estimated from the lattice constants. The Shubnikov--de Haas (SdH) oscillation is in good agreement with this charge transfer degree. The beating behavior of the SdH oscillation gives a large interlayer transfer integral, which is consistent with a corrugated conducting sheet structure. The randomness of the disordered anions reduces the periodicity of the incommensurate anion potential; this destroys the Fermi surface reconstruction by the incommensurate anion potential. The factor dominating the superconducting transition temperature is not the randomness of the polyhalide anions but the effective cyclotron mass enhancement, i.e., the many-body effect.

Electronic structures of Ga-induced incommensurate and commensurate overlayers on the Si(111) surface

Electronic band structures of Ga-induced dense overlayers on the Si(111) surface have been investigated using angle-resolved photoelectron spectroscopy and first-principle density-functional theory calculations. The well-known incommensurate 6.3 × 6.3 phase formed by the growth on the Si(111)7×7 surface and the newly found 1 × 1 phase grown on the preformed Si(111) 3 × 3 -Ga surface are characterized in detail. A highly dispersive surface state (S) is observed for the incommensurate phase but only a weakly dispersing one (S') for the 1 × 1 surface. Both surfaces are found to be nonmetallic, with their surface states fully occupied. The theoretical calculation reproduces well the S band of the 6.3 × 6.3 phase on the basis of a simple 1 × 1 Ga–Si bilayer structure formed with substitutional Ga atoms, which was proposed previously. No explicit sign of the incommensurate periodicity is found in the measured band dispersions, indicating a very weak incommensurate potential. The S band is shown to originate in the sp 2-like planar bonds within the Ga–Si bilayer. The width of the S band is sensitive to the surface strain in the calculation and about 8 % expansion of the Ga–Si bilayer lattice was needed to reproduce the measured dispersions, which is in good agreement with the previous X-ray study. On the other hand, the surface state S' of the commensurate 1 × 1 phase cannot be explained by any simple model of a substitutional or adsorbed Ga layer. Further structural studies are thus requested. The second Ga layer, which is metallic, grows two-dimensionally over this 1 × 1 layer up to a total coverage of about 5 ML.

Bosons in one-dimensional incommensurate superlattices

We investigate numerically the zero-temperature physics of the one-dimensional Bose-Hubbard model in an incommensurate cosine potential, recently realized in experiments with cold bosons in optical superlattices [L. Fallani et al., Phys. Rev. Lett. 98, 130404 (2007)]. An incommensurate cosine potential has intermediate properties between a truly periodic and a fully random potential, displaying a characteristic length scale (the quasiperiod) which is shown to set a finite lower bound to the excitation energy of the system at special incommensurate fillings. This leads to the emergence of gapped incommensurate band-insulator (IBI) phases along with gapless Bose-glass (BG) phases for strong quasiperiodic potential for both hard-core and soft-core bosons. Enriching the spatial features of the potential by the addition of a second incommensurate component appears to remove the IBI regions, stabilizing a continuous BG phase over an extended parameter range. Moreover, we discuss the validity of the local-density approximation in the presence of a parabolic trap, clarifying the notion of a local BG phase in a trapped system; we investigate the behavior of first- and second-order coherence upon increasing the strength of the quasiperiodic potential; and we discuss the ab initio derivation of the Bose-Hubbard Hamiltonian with quasiperiodic potential starting from the microscopic Hamiltonian of bosons in an incommensurate superlattice.

Disorder-induced delocalization in an incommensurate potential

The role of disorder in the localization problem under an incommensurate potential is studied theoretically for a two-dimensional symplectic model. With an incommensurate potential only, there appear wave functions localized along the incommensurate wave vector and extended perpendicular to it. Once the disorder potential is introduced, the wave functions become extended in both directions beyond the scale of the elastic mean free path, and eventually become localized at a critical strength of the disorder. A possible experimental observation of the incommensurability-induced localization is also discussed.