One-dimensional Tight-binding Model Research Articles

Anomalous Edge State in a Non-Hermitian Lattice.

We show that the bulk-boundary correspondence for topological insulators can be modified in the presence of non-Hermiticity. We consider a one-dimensional tight-binding model with gain and loss as well as long-range hopping. The system is described by a non-Hermitian Hamiltonian that encircles an exceptional point in momentum space. The winding number has a fractional value of 1/2. There is only one dynamically stable zero-energy edge state due to the defectiveness of the Hamiltonian. This edge state is robust to disorder due to protection by a chiral symmetry. We also discuss experimental realization with arrays of coupled resonator optical waveguides.

PT-symmetry breaking for the scattering problem in a one-dimensional non-Hermitian lattice model

We study the $\mathcal{PT} $-symmetry breaking for the scattering problem in a one-dimensional (1D) non-Hermitian tight-binding lattice model with balanced gain and loss distributed on two adjacent sites. In the scattering process the system undergoes a transition from the exact $\mathcal{PT} $-symmetry phase to the phase with spontaneously breaking $\mathcal{PT} $-symmetry as the amplitude of complex potentials increases. Using the S-matrix method, we derive an exact discriminant, which can be used to distinguish different symmetry phases, and analytically determine the exceptional point for the symmetry breaking. In the $\mathcal{PT} $-symmetry breaking region, we also confirm the appearance of the unique feature, i.e., the coherent perfect absorption Laser, in this simple non-Hermitian lattice model. The study of the scattering problem of such a simple model provides an additional way to unveil the physical effect of non-Hermitian $\mathcal{PT} $-symmetric potentials.

Trap-assisted charge transport at conjugated polymer interfaces

The trap-assisted charge transport in conjugated polymers is numerically investigated in the framework of a one-dimensional tight-binding model. Our findings show that a polaron trapped within an interchain region can be released migrating between the chains at low temperature regimes. Conversely, a trapped bipolaron cannot be released even considering high temperature values. Interestingly, for systems containing more than one polaron, the formation of trapped states is avoided and there is no charge transfer between the chains. Considering more than one bipolaron, the generation of trapped states occurs and a bipolaron can migrates between the chains.

Delocalization in one-dimensional tight-binding models with fractal disorder

In the present work, we investigated the correlation-induced localization-delocalization transition in the one-dimensional tight-binding model with fractal disorder. We obtained a phase transition diagram from localized to extended states based on the normalized localization length by controlling the correlation and the disorder strength of the potential. In addition, the transition of the diffusive property of wavepacket dynamics is shown around the critical point.

Maximum group velocity in a one-dimensional model with a sinusoidally varying staggered potential

We use Floquet theory to study the maximum value of the stroboscopic group velocity in a one-dimensional tight-binding model subjected to an on-site staggered potential varying sinusoidally in time. The results obtained by numerically diagonalizing the Floquet operator are analyzed using a variety of analytical schemes. In the low frequency limit we use adiabatic theory, while in the high frequency limit the Magnus expansion of the Floquet Hamiltonian turns out to be appropriate. When the magnitude of the staggered potential is much greater or much less than the hopping, we use degenerate Floquet perturbation theory; we find that dynamical localization occurs in the former case when the maximum group velocity vanishes. Finally, starting from an "engineered" initial state where the particles (taken to be hard core bosons) are localized in one part of the chain, we demonstrate that the existence of a maximum stroboscopic group velocity manifests in a light cone like spreading of the particles in real space.

Strong-disorder renormalization-group study of the one-dimensional tight-binding model

We formulate a strong-disorder renormalization-group (SDRG) approach to study the beta function of the tight-binding model in one dimension with both diagonal and off-diagonal disorder for states at the band center. We show that the SDRG method, when used to compute transport properties, yields exact results since it is identical to the transfer matrix method. The beta function is shown to be universal when only off-diagonal disorder is present even though single-parameter scaling is known to be violated. A different single-parameter scaling theory is formulated for this particular (particle-hole symmetric) case. Upon breaking particle-hole symmetry (by adding diagonal disorder), the beta function is shown to crossover from the universal behavior of the particle-hole symmetric case to the conventional non-universal one in agreement with the two-parameter scaling theory. We finally draw an analogy with the random transverse-field Ising chain in the paramagnetic phase. The particle-hole symmetric case corresponds to the critical point of the quantum Ising model while the generic case corresponds to the Griffiths paramagnetic phase.

Analyticity of quantum states in one-dimensional tight-binding model

Analytical complexity of quantum wavefunction whose argument is extended into the complex plane provides an important information about the potentiality of manifesting complex quantum dynamics such as time-irreversibility, dissipation and so on. We examine Pade approximation and some complementary methods to investigate the complex-analytical properties of some complex quantum states such as impurity states, Anderson-localized states and localized states of Harper model. The impurity states can be characterized by simple poles of the Pade approximation, and the localized states of Anderson model and Harper model can be characterized by an accumulation of poles and zeros of the Pade approximated function along a critical border, which implies a natural boundary (NB). A complementary method based on shifting the expansion-center is used to confirm the existence of the NB numerically, and it is strongly suggested that the both Anderson-localized state and localized states of Harper model have NBs in the complex extension. Moreover, we discuss an interesting relationship between our research and the natural boundary problem of the potential function whose close connection to the localization problem was discovered quite recently by some mathematicians. In addition, we also examine the usefulness of the Pade approximation for numerically predicting the existence of NB by means of two typical examples, lacunary power series and random power series.

Critical integer quantum Hall topology and the integrable Maryland model as a topological quantum critical point

One-dimensional tight binding models such as the Aubry-Andr\'e-Harper (AAH) model (with an on-site cosine potential) and the integrable Maryland model (with an on-site tangent potential) have been the subject of extensive theoretical research in localization studies. AAH can be directly mapped onto the two-dimensional (2D) Hofstadter model which manifests the integer quantum Hall topology on a lattice. However, such a connection needs to be made for the Maryland model (MM). Here we describe a generalized model that contains AAH and MM as the limiting cases with the MM lying precisely at a topological quantum phase transition (TQPT) point. A remarkable feature of this critical point is that the one-dimensional MM retains well defined energy gaps whereas the equivalent 2D model becomes gapless, signifying the 2D nature of the TQPT.

Gate-modulated thermopower in disordered nanowires: I. Low temperature coherent regime

Using a one-dimensional tight-binding Anderson model, we study a disordered nanowire in the presence of an external gate which can be used for depleting its carrier density (field effect transistor device configuration). In this first paper, we consider the low temperature coherent regime where the electron transmission through the nanowire remains elastic. In the limit where the nanowire length exceeds the electron localization length, we derive three analytical expressions for the typical value of the thermopower as a function of the gate potential, in the cases where the electron transport takes place (i) inside the impurity band of the nanowire, (ii) around its band edges and eventually (iii) outside its band. We obtain a very large enhancement of the typical thermopower at the band edges, while the sample to sample fluctuations around the typical value exhibit a sharp crossover from a Lorentzian distribution inside the impurity band towards a Gaussian distribution as the band edges are approached.

Dynamics of one-dimensional tight-binding models with arbitrary time-dependent external homogeneous fields

The exact propagators of two one-dimensional systems with time-dependent external fields are presented by following the path-integral method. It is shown that the Bloch acceleration theorem can be generalized to the impulse-momentum theorem in quantum version. We demonstrate that an evolved Gaussian wave packet always keeps its shape in an arbitrary time-dependent homogeneous driven field. Moreover, that stopping and accelerating of a wave packet can be achieved by the pulsed field in a diabatic way.

Polaron formation: Ehrenfest dynamics vs. exact results

We use a one-dimensional tight binding model with an impurity site characterized by electron-vibration coupling, to describe electron transfer and localization at zero temperature, aiming to examine the process of polaron formation in this system. In particular we focus on comparing a semiclassical approach that describes nuclear motion in this many vibronic-states system on the Ehrenfest dynamics level to a numerically exact fully quantum calculation based on the Bonca-Trugman method [J. Bonča and S. A. Trugman, Phys. Rev. Lett. 75, 2566 (1995)]. In both approaches, thermal relaxation in the nuclear subspace is implemented in equivalent approximate ways: In the Ehrenfest calculation the uncoupled (to the electronic subsystem) motion of the classical (harmonic) oscillator is simply damped as would be implied by coupling to a Markovian zero temperature bath. In the quantum calculation, thermal relaxation is implemented by augmenting the Liouville equation for the oscillator density matrix with kinetic terms that account for the same relaxation. In both cases we calculate the probability to trap the electron by forming a polaron and the probability that it escapes to infinity. Comparing these calculations, we find that while both result in similar long time yields for these processes, the Ehrenfest-dynamics based calculation fails to account for the correct time scale for the polaron formation. This failure results, as usual, from the fact that at the early stage of polaron formation the classical nuclear dynamics takes place on an unphysical average potential surface that reflects the distributed electronic population in the system, while the quantum calculation accounts fully for correlations between the electronic and vibrational subsystems.

Shockley model description of surface states in topological insulators

Surface states in topological insulators can be understood based on the well-known Shockley model, a one-dimensional tight-binding model with two atoms per elementary cell, connected via alternating tunneling amplitudes. We generalize the one-dimensional model to the three-dimensional case representing a sequence of layers connected via tunneling amplitudes $t$, which depend on the in-plane momentum $\mathbit{p}=({p}_{x},{p}_{y})$. The Hamiltonian of the model is a $2\ifmmode\times\else\texttimes\fi{}2$ matrix with the off-diagonal element $t(k,\mathbit{p})$ depending also on the out-of-plane momentum $k$. We show that the existence of the surface states depends on the complex function $t(k,\mathbit{p})$. The surface states exist for those in-plane momenta $\mathbit{p}$ where the winding number of the function $t(k,\mathbit{p})$ is nonzero when $k$ is changed from 0 to $2\ensuremath{\pi}$. The sign of the winding number determines the sublattice on which the surface states are localized. The equation $t(k,\mathbit{p})=0$ defines a vortex line in the three-dimensional momentum space. Projection of the vortex line onto the space of the two-dimensional momentum $\mathbit{p}$ encircles the domain where the surface states exist. We illustrate how this approach works for a well-known model of a topological insulator on the diamond lattice. We find that different configurations of the vortex lines are responsible for the ``weak'' and ``strong'' topological insulator phases. A topological transition occurs when the vortex lines reconnect from spiral to circular form. We apply the Shockley model to Bi${}_{2}$Se${}_{3}$ and discuss applicability of a continuous approximation for the description of the surface states. We conclude that the tight-binding model gives a better description of the surface states.

Investigation of polarons in perovskite manganites

Perovskite manganites have aroused a great interest in their outstanding electrical and magnetic properties, but the characteristics of carriers in these materials are still under debate. According to the Mn-O chain, we build a one-dimensional tight-binding model to study the characteristics of charge carriers in manganites. It is obtained that at doping concentration x=0.5, the system shows a ferromagnetic state and the energy bands of spin up and spin down are completely splitted. A gap exists between valence band and conduction band, and all the electronic states are extended. With further doping, a localized electronic state appears, which we call a polaron. Accompanied with the electronic state, local distortions of the lattice and deep levels appear in the gap. The depth of the polaron increases with the doping quantity of electrons. It is also found that the polaron is spin polarized and has a maximum electronic charge of 0.621 e in the present parameters, beyond which the polaron will be divided into two separate states called solitons.

Anomalous localization and dual role of correlation in one-dimensional electronic systems with long-range correlated disorder

The localization behaviors of electronic states and the role of long-range correlation (LRC) in a one-dimensional tight-binding Anderson model with long-range correlated diagonal disorder were numerically studied at the band center by using the transfer matrix method. Two different kinds of localized states with anomalous properties were found in two special localized regimes: regime I with both weak correlation and weak disorder and regime II with both strong correlation and strong disorder. Further calculations show that the LRC plays inverse role in these two regimes: by increasing the degree of LRC, the localization is suppressed in regime I while the localization is enhanced in regime II.

Disorder in materials with complex crystal structures: the Non-Local Coherent Potential Approximation for compounds with multiple sublattices

Over the last few years the Non-Local Coherent Potential Approximation (NL-CPA) has been shown to provide an effective way to describe the electronic structure and related properties of disordered systems, where short-range order (SRO) and other local environment effects are important. Here we present its generalization to materials with multi-atom per unit cell lattices. The method is described using a Green function formalism and illustrated by an implementation for a simplified one-dimensional tight-binding model with substitutional disorder. This development paves the way for a natural reimplementation of the Korringa-Kohn-Rostoker (KKR) multiple scattering solution of Kohn-Sham equations for ab-initio calculations of real materials.

Density of States and Wave Function Localization in Disordered Conjugated Polymers: A Large Scale Computational Study

We present large-scale calculations of electronic structure of strongly disordered conjugated polymers. The calculations have been performed using the density functional theory based charge patching method for the construction of single-particle Hamiltonian and the overlapping fragments method for the efficient diagonalization of that Hamiltonian. We find that the hole states are localized due to the fluctuations of the electrostatic potential and not by the breaks in the conjugation of the polymer chain. The tail of the density of hole states exhibits an exponentially decaying behavior. The main features of the electronic structure of the system can be described by an one-dimensional nearest neighbor tight-binding model with a correlated Gaussian distribution of on-site energies and constant off-site coupling elements.

Nonlinear Current Response of an Isolated System of Interacting Fermions

Nonlinear real-time response of interacting particles is studied on the example of a one-dimensional tight-binding model of spinless fermions driven by electric field. Using equations of motion and numerical methods we show that for a nonintegrable case at finite temperatures the major effect of nonlinearity can be taken into account within the linear response formalism extended by a renormalization of the kinetic energy due to the Joule heating. On the other hand, integrable systems show on constant driving a different universality with a damped oscillating current whereby the frequency is related but not equal to the Bloch oscillations.

Localization and fractality in inhomogeneous quantum walks with self-duality

We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-André model describing the one-dimensional tight-binding model with a quasiperiodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at the origin. We also numerically study the eigenvalues of the one-step time evolution operator and find the Hofstadter butterfly spectrum which indicates the fractal nature of this class of quantum walks.

A one-to-two-dimensional mapping using a partial Fast Fourier Transform

It will be shown how to map a simple one-dimensional tight binding model with a cosine potential in one dimension exactly to a two-dimensional tight binding model with periodic boundary conditions in the presence of a single flux quantum spread evenly on the torus. The mapping is achieved by a partial sequence of “Fast Fourier Transform” (FFT) steps which if completed would be an exact Fourier transform of the original model. Each step of the FFT recursively maps a tight binding model into two decoupled sublattices of half the lattice length.

Modeling excitonic line shapes in weakly disordered semiconductor nanostructures

Excitonic spectra of weakly disordered semiconductor heterostructures are simulated on the basis of a one-dimensional tight-binding model. The influence of the length scale of weak disorder in quantum wells on the redshift of the excitonic peak and its linewidth is studied. By calculating two-dimensional Fourier-transform spectra we are able to determine the contribution of disorder to inhomogeneous and also to homogeneous broadenings separately. This disorder-induced dephasing is related to a Fano-type coupling and leads to contributions to the homogeneous linewidth that depends on energy within the inhomogeneously broadened line. The model includes heavy- and light-hole excitons and yields smaller inhomogeneous broadening for the light-hole exciton if compared to the heavy-hole exciton, which agrees qualitatively with the experiment.