One-dimensional Periodic Lattice Research Articles

Time-dependent properties of run-and-tumble particles. II. Current fluctuations.

We investigate steady-state current fluctuations in two models of hardcore run-and-tumble particles (RTPs) on a periodic one-dimensional lattice of L sites, for arbitrary tumbling rate γ=τ_{p}^{-1} and density ρ; model I consists of standard hardcore RTPs, while model II is an analytically tractable variant of model I, called a long-ranged lattice gas (LLG). We show that, in the limit of L large, the fluctuation of cumulative current Q_{i}(T,L) across the ith bond in a time interval T≫1/D grows first subdiffusively and then diffusively (linearly) with T: 〈Q_{i}^{2}〉∼T^{α} with α=1/2 for 1/D≪T≪L^{2}/D and α=1 for T≫L^{2}/D, where D(ρ,γ) is the collective- or bulk-diffusion coefficient; at small times T≪1/D, exponent α depends on the details. Remarkably, regardless of the model details, the scaled bond-current fluctuations D〈Q_{i}^{2}(T,L)〉/2χL≡W(y) as a function of scaled variable y=DT/L^{2} collapse onto a universal scaling curve W(y), where χ(ρ,γ) is the collective particle mobility. In the limit of small density and tumbling rate, ρ,γ→0, with ψ=ρ/γ fixed, there exists a scaling law: The scaled mobility γ^{a}χ(ρ,γ)/χ^{(0)}≡H(ψ) as a function of ψ collapses onto a scaling curve H(ψ), where a=1 and 2 in models I and II, respectively, and χ^{(0)} is the mobility in the limiting case of a symmetric simple exclusion process; notably, the scaling function H(ψ) is model dependent. For model II (LLG), we calculate exactly, within a truncation scheme, both the scaling functions, W(y) and H(ψ). We also calculate spatial correlation functions for the current and compare our theory with simulation results of model I; for both models, the correlation functions decay exponentially, with correlation length ξ∼τ_{p}^{1/2} diverging with persistence time τ_{p}≫1. Overall, our theory is in excellent agreement with simulations and complements the prior findings [T. Chakraborty and P. Pradhan, Phys. Rev. E 109, 024124 (2024)1539-375510.1103/PhysRevE.109.024124].

Compact gap solitons and compact edge states of exciton–polariton condensates with spin–orbit coupling in a one-dimensional flatband lattice

We propose that the compact gap solitons and compact edge states can be excited in a flatband of the incoherently-pumped exciton–polariton condensate under a one-dimensional periodic potential lattice. The combined effects of spin–orbit coupling and periodic potential depth on the flatband structures are investigated. Then how the compact gap solitons and edge states are localized and extended inside only a fraction of a single lattice site will be studied with varying pump strengths, pump spot-sizes as well as energy detuning.

Catalog of noninteracting tight-binding models with two energy bands in one dimension

We classify Hermitian tight-binding models describing noninteracting electrons on a one-dimensional periodic lattice with two energy bands. To do this, we write a generalized Rice-Mele model with two orbitals per unit cell, including all possible complex-valued long-range hoppings consistent with Hermicity. We then apply different forms of time-reversal, charge-conjugation and chiral symmetry in order to constrain the parameters, resulting in an array of possible models in different symmetry classes. For each symmetry class, we define a single, canonical form of the Hamiltonian and identify models that are related to the canonical form by an off-diagonal unitary transformation in the atomic basis. The models have either symmorphic or nonsymmorphic nonspatial symmetries (time $T$, chiral and charge-conjugation). The nonsymmorphic category separates into two types of state of matter: an insulator with a $\mathbb{Z}_2$ topological index in the absence of nonsymmorphic time-reversal symmetry or, in the presence of nonsymmorphic time-reversal symmetry, a metallic state. The latter is an instance of Kramer's degeneracy with one degeneracy point in the Brillouin zone as opposed to no degeneracy points in symmorphic systems with $T^2 = 1$ and two in symmorphic systems with $T^2 = - 1$.

Source-driven homogenization theory for electro-momentum coupled scatterers

Willis materials are metamaterials whose subwavelength asymmetry couples the macroscopic pressure-strain and momentum-velocity relations. Recently, the design space of these metamaterials has been expanded to consider asymmetric piezoelectric scatterers and thereby couple the electric field-electric displacement relation to the constitutive equations in the Willis form. The existence of this so-called electro-momentum coupling was first predicted using dynamic homogenization of heterogeneous piezoelectric media [J. Mech. Phys. Solids 134, 103770 (2020)] but it can also be understood through a generalized polarizability tensor α that calculates the electro-acoustic field scattered from a single asymmetric piezoelectric inhomogeneity as a function of local fields [Proc. Mtgs. Acoust. 46, 065002 (2022)]. We present a multiple-scattering dynamic homogenization method that extends the work of Sieck et al. [Phys. Rev. B 96, 104303 (2017)] using the generalized polarizability tensor. The model considers the microscale scattered pressure, velocity, and electric fields in a one-dimensional periodic lattice of identical scatterers to find analytical expressions for the effective macroscale fields. The macroscale fields are then used to find the effective electro-momentum coupling constants in terms of the polarizability of the individual scatterers and the concentration of scatterers in a background medium. [Work supported by DARPA and ARL:UT McKinney Fellowship in Acoustics.]

A Gauge-Invariant Geometric Phase for Electrons in a One-Dimensional Periodic Lattice

Here the notion of geometric phase acquired by an electron in a one-dimensional periodic lattice as it traverses the Bloch band is carefully studied. Such a geometric phase is useful in characterizing the topological properties and the electric polarization of the periodic system. An expression for this geometric phase was first provided by Zak, in a celebrated work three decades ago. Unfortunately, Zak’s expression suffers from two shortcomings: its value depends upon the choice of origin of the unit cell, and is gauge dependent. Upon careful investigation of the time evolution of the system, here we find that the system displays cyclicity in a generalized sense wherein the physical observables return in the course of evolution, rather than the density matrix. Recognition of this generalized cyclicity paves the way for a correct and consistent expression for the geometric phase in this system, christened as Pancharatnam-Zak phase. Pancharatnam-Zak geometric phase does not suffer from the shortcomings of Zak’s expression, and correctly classifies the Bloch bands of the lattice. A naturally filled band extension of the Pancharatnam-Zak phase is also constructed and studied.

A simple symmetric exclusion process driven by an asymmetric tracer particle

We consider an exclusion process on a periodic one-dimensional lattice where all particles perform simple symmetric exclusion at rate $1$ except for a single tracer particle, which performs partially simple asymmetric exclusion with rate $p$ to the right and rate $q$ to the left. This model was first considered by Ferrari, Goldstein and Lebowitz (1985) as a test for the validity of the Einstein relation in microscopic systems. The main thrust of this work is an exact solution for the steady state of this exclusion process. We show that the stationary probabilities factorize and give an exact formula for the nonequilibrium partition function. We provide formulas for the current and two-point correlations. When the tracer particle performs totally asymmetric exclusion ($q=0$), the results are shown to simplify significantly and we find an unexpected connection with the combinatorics of set partitions. Finally, we study the system from the point of view of the tracer particle, the so-called environment process. In the environment process, we show that the density of particles decays with the scaled position in front of the tracer particle in the thermodynamic limit.

High-quality resonances in quasi-periodic clusters of scatterers for flexural waves

Multiple scattering theory is applied to the study of clusters of point-like scatterers attached to a thin elastic plate and arranged in quasi-periodic distributions. Two types of structures are specifically considered: the twisted bilayer and the quasi-periodic line. The former consists in a couple of two-dimensional lattices rotated a relative angle, so that the cluster forms a moiré pattern. The latter can be seen as a periodic one-dimensional lattice where an incommensurate modulation is superimposed. Multiple scattering theory allows for the fast and efficient calculation of the resonant modes of these structures as well as for their quality factor, which is thoroughly analyzed in this work. The results show that quasi-periodic structures present a large density of states with high quality factors, being therefore a promising way for the design of high quality wave-localization devices.

The silicon model photonic structure for a full-function thermal photodetector

The out-of-plane optical properties of a combined one-two-dimensional comb-type photonic structure based on macroporous silicon under illumination with polarized normally incident light have been experimentally investigated. Technologically simple quasi-periodic structure with parallel air grooves in the form of mutually overlapping macropores (i.e., one-dimensional air/Si type structure) together with macropores arbitrarily distributed over the crystal surface (two-dimensional air/Si structure) was studied in comparison with the calculated one-dimensional periodic metal lattice on silicon. It is shown that this combined air/Si photonic structure has significant polarization selectivity and can serve as a basis for developing the full-function out-of-plane thermal photodetector on macroporous silicon.

Quantum random walk and tight-binding model subject to projective measurements at random times

What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N. For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m. By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, and with δ > 1. These scales are such that (i) for m large and satisfying , the decay of the survival probability is as m −2, (ii) for m satisfying , the decay is as m −3/2, while (iii) for , the decay is as an exponential. The results for the dynamics with the leftover component, already known for the case of measurements at regular intervals, are being extended here to the case of measurements at random intervals. We find that our results hold independently of the choice of the distribution of times between successive measurements, as have been corroborated by our results for a wide range of distributions including exponential and power-law distributions as well as for the case of measurements at regular intervals. This fact hints at robustness and ubiquity of our derived results.

The fractional nonlinear impurity: A Green function approach

We use a lattice Green function approach to study the stationary modes of a linear/nonlinear (Kerr) impurity embedded in a periodic one-dimensional lattice where we replace the standard discrete Laplacian by a fractional one. The energies and the mode profiles are computed in closed form, for different fractional exponents and different impurity strengths. The energies of the impurity mode lie outside the linear band whose bandwidth decreases steadily as the fractional exponent decreases. For any fractional exponent values, there is always a single bound state for the linear impurity while for the nonlinear (Kerr) case, up to two bound states are possible, for impurity strengths above certain threshold. The energy of the linear mode (or that of the upper energy nonlinear one), becomes directly proportional to the impurity strength at large impurity strengths. The transmission of plane waves is also computed in closed form for several fractional exponents, and various impurity strengths. We observe that fractionality tends to increase the overall transmission. The selftrapping transition for the nonlinear impurity shifts to lower nonlinearity values as the fractional exponent is decreased. In both cases, linear and nonlinear, we observe a form of trapping at zero impurity strength, which can be explained by the near-degeneracy of the spectrum in the limit of a small fractional exponent.

Bloch's band structures of a pair of interacting electrons in simple one- and two-dimensional lattices

We have applied the Bloch's theorem to a pair of interacting electrons in periodic simple one-dimensional lattice and two-dimensional square lattice with both intra-site and nearest neighbor inter-site interactions. The electronic structures can be classified as binding bands and unbinding continuum states of a pair of electrons. In addition to the usual intra-site binding band, we found the nearest-neighbor inter-site binding bands in both spin singlet and triplet states irrespective of interaction nature. The binding bands are either above or below the unbinding continuum states separated by gaps for a repulsive or attractive interaction potential.

Polarization-differentiated band dynamics of resonant leaky modes at the lattice Γ point.

In the physical description of photonic lattices, leaky-mode resonance and bound states in the continuum are central concepts. Understanding of their existence conditions and dependence on lattice parameters is of fundamental interest. Primary leaky-wave effects are associated with the second stop band at the photonic lattice Γ point. The pertinent band gap is defined by the frequency difference between the leaky-mode band edge and the bound-state edge. This paper address the polarization properties of the band gaps resident in laterally periodic one-dimensional photonic lattices. We show that the band gaps pertinent to TM and TE leaky modes exhibit significantly differentiated evolution as the lattice parameters vary. This is because the TM band gap is governed by a surface effect due to the discontinuity of the dielectric constant at the interfaces of the photonic lattice as well as by a Bragg effect due to the periodic in-plane dielectric constant modulation. We find that when the lattice is thin (thick), the surface (Bragg) effect dominates the Bragg (surface) effect in the formation of the TM band. This leads to complex TM band dynamics with multiple band closures possible under parametric variation. In complete contrast, the TE band gap is governed only by the Bragg effect thus exhibiting simpler band dynamics. This research elucidates the important effect of polarization on resonant leaky-mode band dynamics whose explanation has heretofore not been available.

Asymptotic behavior of correlation functions of one-dimensional polar-molecules on optical lattices

We combine a slave-spin approach with a mean-field theory to develop an approximate theoretical scheme to study the density, spin, and, pairing correlation functions of fermionic polar molecules. We model the polar molecules subjected to a one-dimensional periodic optical lattice potential using a generalized t–J model, where the long-range part of the interaction is included through the exchange interaction parameter. For this model, we derive a set of self-consistent equations for the correlation functions, and evaluate them numerically for the long-distance behavior. We find that the pairing correlations are related to spin correlations through the density and the slave-spin correlations. Further, our calculations indicates that the long-range character of the interaction can be probed through these correlation functions. In the absence of exact solutions for the one-dimensional t–J model, our approximate theoretical treatment can be treated as a useful tool to study one dimensional long-range correlated fermions.

Driven tracers in a one-dimensional periodic hard-core lattice gas

Totally asymmetric tracer particles in an environment of symmetric hard-core particles on a ring are studied. Stationary state properties, including the environment density profile and tracer velocity are derived explicitly for a single tracer. Systems with more than one tracer are shown to factorise into single-tracer subsystems, allowing the single tracer results to be extended to an arbitrary number of tracers. We demonstrate the existence of a cooperative effect, where many tracers move with a higher velocity than a single tracer in an environment of the same size and density. Analytic calculations are verified by simulations. Results are compared to established results in related systems.

Quantum search on Hanoi network

Hanoi network (HN) has a one-dimensional periodic lattice as its main structure with additional long-range edges, which allow having efficient quantum walk algorithm that can find a target state on the network faster than the exhaustive classical search. In this paper, we use regular quantum walks and lackadaisical quantum walks, respectively, to search for a target state. From the curve fitting of the numerical results for HN of degrees three and four, we find that their running time for the regular quantum walks are followed by amplitude amplification scales as [Formula: see text] and [Formula: see text], respectively. For the search by lackadaisical quantum walks, the running time scales are as [Formula: see text] and [Formula: see text], respectively.

Non-Markovian stochastic Schrödinger equation in k-space toward the calculation of carrier dynamics in organic semiconductors.

A non-Markovian stochastic Schrödinger equation developed in our former work [Y. Ke and Y. Zhao, J. Chem. Phys. 147, 184103 (2017)] is extended to the reciprocal (k-) space to calculate the carrier dynamics in organic semiconductors with both local and nonlocal carrier-phonon interactions taken into account. The validity of this approach is examined by comparing with numerically exact benchmark results. As an application, the carrier mobilities are calculated within a one-dimensional periodic lattice model. The results reveal an inversion in the magnitude of the mobility as the nonlocal carrier-phonon interaction varies from weak to strong strengths, indicating a transition of the transport mechanism. This is also demonstrated by a variation in the temperature dependence of the mobility. In addition, a transient localization diffusive behavior caused by intramolecular vibrations is also found.

Energy and momentum diffusion in one-dimensional periodic and asymmetric nonlinear lattices with momentum conservation.

The energy diffusion in one-dimensional (1D) momentum conserving nonlinear lattices usually exhibits anomalous superdiffusion, except the coupled rotator lattice with symmetric and periodic interacting potential which has normal energy diffusion corresponding to normal heat conduction. For nonperiodic 1D lattices with momentum conservation, it has been argued that the asymmetric potential can induce normal heat conduction. Later results indicate the observed normal behavior might be the finite size effect and the anomalous behavior will appear in the thermodynamical limit. Here we propose asymmetric and periodic 1D nonlinear lattices with momentum conservation. The energy and momentum diffusion behaviors will be investigated in detail and the same normal diffusion behaviors for both energy and momentum can be observed. These results confirm that the periodicity is the key for normal transport behavior in 1D momentum conserving lattices, whether the potential is symmetric or asymmetric.

Brewster effect when approaching exceptional points of degeneracy: Epsilon-near-zero behavior

We reveal that the phenomenon of full transmission without phase accumulation commonly associated with epsilon-near-zero (ENZ) materials for a plane-wave does not require vanishing of permittivity. We theoretically connect the phenomenon with condition of the Brewster effect satisfied at the edges of stop bands (so called exceptional points of degeneracy) and show that the full transmission without phase accumulation can be observed in various one-dimensional periodic structures. Particularly, exploiting the manifold of exceptional points of degeneracy in one-dimensional all-dielectric periodic lattices, we demonstrate that these structures not only offer a lossless and extremely simple, CMOS compatible alternative for some applications of ENZ media, but exhibit new properties of all-angle full transmission with zero phase delay.

Optimizing a jump-diffusion model of a starving forager

We analyze the movement of a starving forager on a one-dimensional periodic lattice, where each location contains one unit of food. As the forager lands on sites with food, it consumes the food, leaving the sites empty. If the forager lands consecutively on $s$ empty sites, then it will starve. The forager has two modes of movement: it can either diffuse, by moving with equal probability to adjacent sites on the lattice, or it can jump to a uniformly randomly chosen site on the lattice. We show that the lifetime $T$ of the forager in either paradigm can be approximated by the sum of the cover time $\tau_{\rm cover}$ and the starvation time $s$, when $s$ far exceeds the number $n$ of lattice sites. Our main findings focus on the hybrid model, where the forager has a probability of either jumping or diffusing. The lifetime of the forager varies non-monotonically according to $p_j$, the probability of jumping. By examining a small system, analyzing a heuristic model, and using direct numerical simulation, we explore the tradeoff between jumps and diffusion, and show that the strategy that maximizes the forager lifetime is a mixture of both modes of movement.

Modelling photon condensation in a fluorescent dye filled optical microcavity

In this paper, a mathematical model for photon condensation in an optical microcavity is considered based on Gross-Pitaevskii equation. The interaction of photons with fluorescent dye molecules in an optical microcavity resembles electron-ion interaction in a periodic lattice. The system is modelled analogous to Kronig-Penney model for electrons in a one-dimensional periodic lattice. The laser used as the light source gives rise to nonlinearity due to the deformation of electron orbits in the fluorescent dye. To accommodate the optical Kerr effect which arises due to this nonlinear interaction, the resultant Schrödinger equation is modified, which assumes the form of Gross-Pitaevskii equation. The model is validated using equation of state, whose virial coefficients can be evaluated from grand canonical ensemble scheme taking into account the interaction between photons and fluorescent dye molecules.