One-dimensional Tight-binding Model Research Articles

Influence of the disorder correlation length on localization and quantum state transference in tight-binding channels

We investigate in detail the transference of quantum states in a disordered channel. We consider a one-dimensional tight-binding model consisting of a source S connected to a receiver R throughout a disordered channel. The disorder distribution contains a single tunable spatial correlation length. We demonstrate that the disorder correlation length plays a relevant role within the localization properties of the channel. The hopping parameter between the sites S and R and the channel are also adjustable parameters. We investigate the possibility of transference of quantum states along this quantum channel model and describe the optimal conditions for the occurrence of a high fidelity process.

Solvable non-Hermitian skin effects and real-space exceptional points: non-Hermitian generalized Bloch theorem

Non-Hermitian systems can exhibit extraordinary boundary behaviors, known as the non-Hermitian skin effects, where all the eigenstates are localized exponentially at one side of lattice model. To give a full understanding and control of non-Hermitian skin effects, we have developed the non-Hermitian generalized Bloch theorem to provide the analytical expression for all solvable eigenvalues and eigenstates, in which translation symmetry is broken due to the open boundary condition. By introducing the Vieta’s theorem for any polynomial equation with arbitrary degree, our approach is widely applicable for one-dimensional non-Hermitian tight-binding models. With the non-Hermitian generalized Bloch theorem, we can analyze the condition of existence or non-existence of the non-Hermitian skin effects at a mathematically rigorous level. Additionally, the non-Hermitian generalized Bloch theorem allows us to explore the real-space exceptional points. We also establish the connection between our approach and the generalized Brillouin zone method. To illustrate our main results, we examine two concrete examples including the Su–Schrieffer–Heeger chain model with long-range couplings, and the ladder model with non-reciprocal interaction. Our non-Hermitian generalized Bloch theorem provides an efficient way to analytically study various non-Hermitian phenomena in more general cases.

Possibility of massless Dirac fermions in an Aubry–André–Harper potential

In this study, we present a one-dimensional tight-binding model designed to explore the impact of electric fields on an incommensurate quantum system. We specifically focus on the Aubry–André–Harper model, a quasiperiodic model known to exhibit a metal–insulator transition at a critical potential value of λc = 2. This model combines Anderson and Aubry–André–Harper localization phenomena in a quantum system, leading to intriguing effects on the lattice band structure upon the application of an electric field F to the Aubry–André–Harper potential. Our investigation reveals that by choosing a specific value for the applied electric field, it becomes feasible to generate effective massless Dirac fermions within our Aubry–André–Harper system. Furthermore, we note that the extension or localization of the massless particle wave function is contingent upon the potential strength value λ within our incommensurate model. Importantly, our findings highlight the potential for detecting this intriguing phenomenon through experimental means.

Exploring entanglement characteristics in disordered free fermion systems through random bi-partitioning

This study investigates the entanglement properties of disordered free fermion systems undergoing an Anderson phase transition from a delocalized to a localized phase. The entanglement entropy is employed to quantify the degree of entanglement, with the system randomly divided into two subsystems. To explore this phenomenon, one-dimensional tight-binding fermion models and Anderson models in one, two, and three dimensions are utilized. Comprehensive numerical calculations reveal that the entanglement entropy, determined using random bi-partitioning, follows a volume-law scaling in both the delocalized and localized phases, expressed as [Formula: see text], where D represents the dimension of the system. Furthermore, the role of short and long-range correlations in the entanglement entropy and the impact of the distribution of subsystem sites are analyzed.

Effect of electron-electron interaction on polarization process of exciton and biexciton in conjugated polymers

Abstract By using the one-dimensional tight-binding model modified to include electron-electric field interaction and electron-electron interaction, we theoretically explore the polarization process of exciton and biexciton in cis-polyacetylene. The dynamical simulation is performed by adopting the non-adiabatic evolution approach. The results show that under the effect of moderate electric field, when the strength of electron-electron interaction is weak, the singlet exciton is stable but its polarization presents obvious oscillation. With the enhancement of interaction, it is dissociated into polaron pair, the spin-flip of which could be observed through modulating the interaction strength. For the triplet exciton, the strong electron-electron interaction restrains its normal polarization, but it is still stable. In case of biexciton, the strong electron-electron interaction not only dissociate it, but also flip its charge distribution. We also calculate the yield of the possible states formed after the dissociation of exciton and biexciton.

Hopping tunneling through a quasiperiodic potential

In this work, we have developed a one-dimensional tight-binding model to investigate the conductivity of tunneling through a quasiperiodic or slowly-varying potential. Our study focuses on the hopping tunneling properties of a tight-binding model in which the on-site potential is replaced by both quasiperiodic and periodic functions. By comparing the hopping transport in scenarios involving a periodic potential and an slowly-varying model, the latter being the only known case of a 1D potential with a mobility edge and a metal-insulator transition, we gain valuable insights. In our analysis, we consider a proposed device situated between two quantum reservoirs in both quasiperiodic and periodic cases. Remarkably, we find that the tunnel transport through the lattice completely diminishes at the band center. This intriguing phenomenon is likely to hold fundamental importance in the interpretation of experimental data.

Mobility edges in one-dimensional finite-sized models with large quasi-periodic disorders

We study the one-dimensional tight-binding model with quasi-periodic disorders, where the quasi-period is tuned to be large compared to the system size. It is found that this type of model with large quasi-periodic disorders can also support the mobility edges, which is very similar to the models with slowly varying quasi-periodic disorders. The energy-matching method is employed to determine the locations of mobility edges in both types of models. These results of mobility edges are verified by numerical calculations in various examples. We also provide qualitative arguments to support the fact that large quasi-periodic disorders will lead to the existence of mobility edges.

Drude weights in one-dimensional systems with a single defect

Ballistic transport of a quantum system can be characterized by Drude weight, which quantifies the response of the system to a uniform electric field in the infinitely long timescale. The Drude weight is often discussed in terms of the Kohn formula, which gives the Drude weight by the derivative of the energy eigenvalue of a finite-size system with the periodic boundary condition in terms of the Aharonov-Bohm flux. Recently, the Kohn formula is generalized to nonlinear responses. However, the nonlinear Drude weight determined by the Kohn formula often diverges in the thermodynamic limit. In order to elucidate the issue, in this work we examine a simple example of a one-dimensional tight-binding model in the presence of a single defect at zero temperature. We find that its linear and non-linear Drude weights given by the Kohn formula (i) depend on the Aharonov-Bohm flux and (ii) diverge proportionally to a power of the system size. We argue that the problem can be attributed to different order of limits. The Drude weight according to the Kohn formula (``Kohn--Drude weight'') indicates the response of a finite-size system to an adiabatic insertion of the Aharonov-Bohm flux. While it is a well-defined physical quantity for a finite-size system, its thermodynamic limit does not always describe the ballistic transport of the bulk. The latter should be rather characterized by a ``bulk Drude weight'' defined by taking the thermodynamic limit first before the zero-frequency limit. While the potential issue of the order of limits has been sometimes discussed within the linear response, the discrepancy between the two limits is amplified in nonlinear Drude weights. We demonstrate the importance of the low-energy excitations of $O(1/L)$, which are excluded from the Kohn--Drude weight, in regularizing the bulk Drude weight.

Time-based Chern number in periodically driven systems in the adiabatic limit

To define the topology of driven systems, recent works have proposed synthetic dimensions as a way to uncover the underlying parameter space of topological invariants. Using time as a synthetic dimension, together with a momentum dimension, gives access to a synthetic two-dimensional ($2\mathrm{D}$) Chern number. It is, however, still unclear how the synthetic 2D Chern number is related to the Chern number that is defined from a parametric variable that evolves with time. Here we show that in periodically driven systems in the adiabatic limit, the synthetic $2\mathrm{D}$ Chern number is a multiple of the Chern number defined from the parametric variable. The synthetic 2D Chern number can thus be engineered via how the parametric variable evolves in its own space. We justify our claims by investigating Thouless pumping in two one-dimensional ($1\mathrm{D}$) tight-binding models, a three-site chain model, and a two-$1\mathrm{D}$-sliding-chains model. The present findings could be extended to higher dimensions and other periodically driven configurations.

Interaction-induced analog of a non-Hermitian skin effect in a lattice two-body problem

We present a theoretical study of the quantum states of two repelling spinless particles in a one-dimensional tight-binding model with a simple periodic lattice and open boundary conditions. We demonstrate that, when the particles are not identical, their interaction drives nontrivial correlated two-particle states, such as bound states and edge states, and induces interaction-induced flat bands. We show that the localization of the center of mass of the two particles enforces the localization of their relative motion, which means formation of the bound states. While the considered system is Hermitian, an insight into the bound states is provided by an approximate effective non-Hermitian model for the relative motion that features the non-Hermitian skin effect.

Statistical properties related to angle variables in Hamiltonian map approach for one-dimensional tight-binding models with localization

Statistical properties related to angle variables in Hamiltonian map approach for one-dimensional tight-binding models with localization

Topological energy braiding of non-Bloch bands

The non-Hermitian skin effect, as a unique feature of non-Hermitian systems, will break the topological energy braiding of Bloch bands in open boundary systems. Going beyond the Bloch band theory, we unveil the energy braiding of non-Bloch bands by introducing a one-dimensional non-Hermitian tight-binding model. We find a generic class of topological non-Bloch bands such as the Hopf link, which is generally generated by the non-Hermitian skin effect. The energy braiding is topologically robust against any perturbations without gap closing. Furthermore, non-Bloch topological invariants are proposed based on the generalized Brillouin zone to characterize the topology of these non-Bloch bands. The topological phase transition between the distinct phases occurs with non-Bloch bands touching at exceptional points. We hope that our work can shed light on the topological energy braiding of non-Bloch bands for non-Hermitian systems.

Wave-function extreme value statistics in Anderson localization

We consider a disordered one-dimensional tight-binding model with power-law decaying hopping amplitudes to disclose wavefunction maximum distributions related to the Anderson localization phenomenon. Deeply in the regime of extended states, the wavefunction intensities follow the Porter-Thomas distribution while their maxima assume the Gumbel distribution. At the critical point, distinct scaling laws govern the regimes of small and large wavefunction intensities with a multifractal singularity spectrum. The distribution of maxima deviates from the usual Gumbel form and some characteristic finite-size scaling exponents are reported. Well within the localization regime, the wavefunction intensity distribution is shown to develop a sequence of pre-power-law, power-law, exponential and anomalous localized regimes. Their values are strongly correlated, which significantly affects the emerging extreme values distribution.

Coexistence of extended and localized states in the one-dimensional non-Hermitian Anderson model

In one-dimensional Hermitian tight-binding models, mobility edges separating extended and localized states can appear in the presence of properly engineered quasi-periodical potentials and coupling constants. On the other hand, mobility edges don't exist in a one-dimensional Anderson lattice since localization occurs whenever a diagonal disorder through random numbers is introduced. Here, we consider a nonreciprocal non-Hermitian lattice and show that the coexistence of extended and localized states appears with or without diagonal disorder in the topologically nontrivial region. We discuss that the mobility edges appear basically due to the boundary condition sensitivity of the nonreciprocal non-Hermitian lattice.

Local orbital formulation of the Floquet theory of projectile electronic stopping

A recently proposed theoretical framework for the description of electronic quantum friction for constant-velocity nuclear projectiles traversing periodic crystals is here implemented using a local basis representation. The theory requires a change of reference frame to that of the projectile, and a basis set transformation for the target basis functions to a gliding basis is presented, which is time-periodic but does not displace in space with respect to the projectile, allowing a local-basis Floquet impurity-scattering formalism to be used. It is illustrated for a one-dimensional single-band tight-binding model, as the simplest paradigmatic example, displaying the qualitative behaviour of the formalism. The time-dependent non-orthogonality of the gliding basis requires care in the proper (simplest) definition of a local projectile perturbation. The Fermi level is tilted with a slope given by the projectile velocity, which complicates integration over occupied states. It is solved by a recurrent application of the Lippmann-Schwinger equation, in analogy with previous non-equilibrium treatment of electron ballistic transport. Aiming towards a first-principles mean-field-like implementation, the final result is the time-periodic particle density in the region around the projectile, describing the stroboscopically stationary perturbation cloud around the projectile, out of which other quantities can be obtained, such as the electronic stopping power.

A fractional Anderson model

We examine the interplay between disorder and fractionality in a one-dimensional tight-binding Anderson model. In the absence of disorder, we observe that the two lowest energy eigenvalues detach themselves from the bottom of the band, as fractionality s is decreased, becoming completely degenerate at s=0, with a common energy equal to a half bandwidth, V. The remaining N−2 states become completely degenerate forming a flat band with energy equal to a bandwidth, 2V. Thus, a gap is formed between the ground state and the band. In the presence of disorder and for a fixed disorder width, a decrease in s reduces the width of the point spectrum while for a fixed s, an increase in disorder increases the width of the spectrum. For all disorder widths, the average participation ratio decreases with s showing a tendency towards localization. However, the average mean square displacement (MSD) shows a hump at low s values, signaling the presence of a population of extended states, in agreement with what is found in long-range hopping models.

Anomalous band center localization in the one-dimensional Anderson model with a disordered distribution of infinite variance.

We perform a detailed numerical study of the influence of distributions without a finite second moment on the Lyapunov exponent through the one-dimensional tight-binding Anderson model with diagonal disorder. Using the transfer matrix parametrization method and considering a specific distribution function, we calculate the Lyapunov exponent numerically and demonstrate its relation with the fractional lower order moments of the disorder probability density function. For the lower order of moments of disorder distribution with an infinite variance, we obtain the anomalous behavior near the band center.

Synthetic dimension band structures on a Si CMOS photonic platform.

Synthetic dimensions, which simulate spatial coordinates using nonspatial degrees of freedom, are drawing interest in topological science and other fields for modeling higher-dimensional phenomena on simple structures. We present the first realization of a synthetic frequency dimension on a silicon ring resonator integrated photonic device fabricated using a CMOS process. We confirm that its coupled modes correspond to a one-dimensional tight-binding model through acquisition of up to 280-GHz bandwidth optical frequency comb-like spectra and by measuring synthetic band structures. Furthermore, we realized two types of gauge potentials along the frequency dimension and probed their effects through the associated band structures. An electric field analog was produced via modulation detuning, whereas effective magnetic fields were induced using synchronized nearest- and second nearest–neighbor couplings. Creation of coupled mode lattices and two effective forces on a monolithic Si CMOS device represents a key step toward wider adoption of topological principles.

Illuminating the bulk-boundary correspondence of a non-Hermitian stub lattice with Majorana stars

Topological characterization of non-Hermitian band structures demands more than a straightforward generalization of the Hermitian cases. Even for one-dimensional tight-binding models with nonreciprocal hopping, the appearance of point gaps and the skin effect leads to the breakdown of the usual bulk-boundary correspondence. Luckily, the correspondence can be resurrected by introducing a winding number for the generalized Brillouin zone for systems with even number of bands and chiral symmetry. Here, we analyze the topological phases of a nonreciprocal hopping model on the stub lattice, where one of the three bands remains flat. Due to the lack of chiral symmetry, the biorthogonal Zak phase is no longer quantized, invalidating the winding number as a topological index. Instead, we show that a $Z_2$ invariant can be defined from Majorana's stellar representation of the eigenstates on the Bloch sphere. The parity of the total azimuthal winding of the entire Majorana constellation correctly predicts the appearance of edge states between the bulk gaps. We further show that the system is not a square-root topological insulator, despite the fact that its parent Hamiltonian can be block diagonalized and related to a sawtooth lattice model. The analysis presented here may be generalized to understand other non-Hermitian systems with multiple bands.

Dyson’s disordered linear chain from a random matrix theory viewpoint

The first work of Dyson relating to random matrix theory, “The dynamics of a disordered linear chain,” is reviewed. Contained in this work is an exact solution of the so-called type I chain in the case of the disorder variables being given by a gamma distribution. The exact solution exhibits a singularity in the density of states about the origin, which has since been shown to be universal for one-dimensional tight binding models with off diagonal disorder. We discuss this context and also point out some universal features of the weak disorder expansion of the exact solution near the band edge. Furthermore, a link between the exact solution and a tridiagonal formalism of anti-symmetric Gaussian β-ensembles with β proportional to 1/N is made.