Finite One-dimensional Lattice Research Articles

Exit times of totally asymmetric simple exclusion processes.

We address the question of the time needed by N particles, initially located on the first sites of a finite one-dimensional lattice of size L, to exit that lattice when they move according to a TASEP transport model. Using analytical calculations and numerical simulations, we show that when N≪L, the mean exit time of the particles is asymptotically given by T_{N}(L)∼L+β_{N}sqrt[L] for large lattices. Building upon exact results obtained for two particles, we devise an approximate continuous space and time description of the random motion of the particles that provides an analytical recursive relation for the coefficients β_{N}. The results are shown to be in very good agreement with numerical results. This approach sheds some light on the exit dynamics of N particles in the regime where N is finite while the lattice size L→∞. This complements previous asymptotic results obtained by Johansson [Commun. Math. Phys. 209, 437 (2000)0010-361610.1007/s002200050027] in the limit where both N and L tend to infinity while keeping the particle density N/L finite.

Nonreciprocal energy transmission in short discrete systems with strong spatiotemporal modulations

Introducing spatiotemporally varying properties in a phononic lattice can enable nonreciprocal transmission of energy. The hallmark of this phenomenon is the unidirectional energy transmission in infinitely long systems. In short lattices, however, identifying nonreciprocity through differences in transmitted energies (norm bias) becomes challenging because the primary contributor to nonreciprocity is the transmitted phases. Although stronger modulation can achieve a higher norm bias, it can also result in parametric instabilities and trigger large-amplitude oscillations that lead to device failure. To better understand the tradeoff between stability and norm bias in strongly modulated systems, we investigate the parametric stability of a finite one-dimensional lattice with spatiotemporally modulated elasticity. We use Floquet theory to compute the stability charts for strongly modulated systems. Thus, we can identify operating conditions that allow for a relatively large norm bias while maintaining a stable, bounded response.

Microscopic dynamics and an effective Landau-Zener transition in the quasiadiabatic preparation of spatially ordered states of Rydberg excitations

We examine the adiabatic preparation of spatially-ordered Rydberg excitations of atoms in finite one-dimensional lattices by frequency-chirped laser pulses, as realized in a number of recent experiments simulating quantum Ising model. Our aims are to unravel the microscopic mechanism of the phase transition from the unexcited state of atoms to the antiferromagnetic-like state of Rydberg excitations by traversing an extended gapless phase, and to estimate the preparation fidelity of the target state in a moderately sized system amenable to detailed numerical analysis. We find that, in the basis of the bare atomic states, the system climbs the ladder of Rydberg excitations predominantly along the strongest-amplitude paths towards the final ordered state. We show that, despite its complexity, the interacting many-body system can be described as an effective two-level system involving a pair of lowest-energy instantaneous collective eigenstates of the time-dependent Hamiltonian. The final preparation fidelity of the target state can then be well approximated by the Landau-Zener formula, while the nonadiabatic population leakage during the passage can be estimated using a perturbative approach applied to the instantaneous collective eigenstates.

Vanishing and Nonvanishing Persistent Currents of Various Conserved Quantities.

For every conserved quantity written as a sum of local terms, there exists a corresponding current operator that satisfies the continuity equation. The expectation values of current operators at equilibrium define the persistent currents that characterize spontaneous flows in the system. In this Letter, we consider quantum many-body systems on a finite one-dimensional lattice and discuss the scaling of the persistent currents as a function of the system size. We show that, when the conserved quantities are given as the Noether charges associated with internal symmetries or the Hamiltonian itself, the corresponding persistent currents can be bounded by a correlation function of two operators at a distance proportional to the system size, implying that they decay at least algebraically as the system size increases. In contrast, the persistent currents of accidentally conserved quantities can be nonzero even in the thermodynamic limit and even in the presence of the time-reversal symmetry. We discuss "the current of energy current" in S=1/2 XXZ spin chain as an example and obtain an analytic expression of the persistent current.

The effect of graph connectivity on metastability in a stochastic system of spiking neurons

We consider a continuous-time stochastic model of spiking neurons originally introduced by Ferrari et al. in Ferrari et al. (2018). In this model, we have a finite or countable number of neurons which are vertices in some graph G where the edges indicate the synaptic connection between them. We focus on metastability, understood as the property for the time of extinction of the network to be asymptotically memory-less, and we prove that this model exhibits two different behaviors depending on the nature of the specific underlying graph of interaction G that is chosen. In this model the spiking activity of any given neuron is represented by a point process, whose rate fluctuates between 1 and 0 over time depending on whether the membrane potential is positive or null. The membrane potential of each neuron evolves in time by integrating all the spikes of its adjacent neurons up to the last spike of the said neuron, so that when a neuron spikes, its membrane potential is reset to 0 while the membrane potential of each of its adjacent neurons is increased by one unit. Moreover, each neuron is exposed to a leakage effect, modeled as an abrupt loss of membrane potential which occurs at random times driven by a Poisson process of some fixed rate γ. It was previously proven that when the graph G is the infinite one-dimensional lattice, this model presents a phase transition with respect to the parameter γ. It was also proven that, when γ is small enough, the renormalized time of extinction (the first time at which all neurons have a null membrane potential) of a finite version of the system converges in law toward an exponential random variable when the number of neurons goes to infinity. The present article is divided into two parts. First we prove that, in the finite one-dimensional lattice, this last result does not hold anymore if γ is large enough, and in fact we prove that for γ>1 the renormalized time of extinction is asymptotically deterministic. Then we prove that conversely, if G is the complete graph, the result of metastability holds for any positive γ.

The Many Roles of Binding Cooperativity in the Control of DNA Replication

The Many Roles of Binding Cooperativity in the Control of DNA Replication

Stark Many-Body Localization.

We consider spinless fermions on a finite one-dimensional lattice, interacting via nearest-neighbor repulsion and subject to a strong electric field. In the noninteracting case, due to Wannier-Stark localization, the single-particle wave functions are exponentially localized even though the model has no quenched disorder. We show that this system remains localized in the presence of interactions and exhibits physics analogous to models of conventional many-body localization (MBL). In particular, the entanglement entropy grows logarithmically with time after a quench, albeit with a slightly different functional form from the MBL case, and the level statistics of the many-body energy spectrum are Poissonian. We moreover predict that a quench experiment starting from a charge-density wave state would show results similar to those of Schreiber etal. [Science 349, 842 (2015)SCIEAS0036-807510.1126/science.aaa7432].

Characterization of the binding interactions between EvaGreen dye and dsDNA.

Understanding the dsDNA·EG binding interaction is important because the EvaGreen (EG) dye is increasingly used in real-time quantitative polymerase chain reaction, high resolution melting analysis, and routine quantification of DNA. In this work, a binding isotherm for the interactions of EG with duplex DNA (poly-dA17·poly-dT17) has been determined from the absorption and fluorescence spectra of the EG and dsDNA·EG complex. The isotherm has a sigmoidal shape and can be modeled with the Hill equation, indicating positive cooperativity for the binding interaction. A Scatchard plot of the binding data yields a concave-down curve in agreement with the Hill analysis of the binding isotherm for a positive cooperative binding interaction. Analysis of the Scatchard plot with the modified McGhee and von Hippel model for a finite one-dimensional homogeneous lattice and nonspecific binding of ligands to duplex DNA yields the intrinsic binding constant, the number of lattice sites occluded by a bound ligand, and the cooperativity parameter of 3.6 × 105 M-1, 4.0, and 8.1, respectively. The occluded site size of 4 indicates that moieties of the EG intercalate into the adjacent base pairs of the duplex DNA with a gap of 1 intercalation site between EG binding sites, as expected for a bifunctional molecule. Interestingly, at high [EG]/[base pair], the intercalation is disrupted. A model is proposed based on the fluorescence spectrum where the formation of anti-parallel stacked chains of EGs bound externally to the duplex DNA occur at these high ratios.

Charge transfer and anderson localization in one-dimensional finite-size disordered systems

The metal–insulator transition in finite one-dimensional lattices with diagonal disorder is considered. Although the true phase transition does not occur in such systems, a conditional criterion can be introduced to distinguish the regions of localization and delocalization of eigenvectors and wave functions, depending on the degree of disorder W. A relationship between the degree of disorder W and the characteristic length separating the localized and delocalized states is obtained: with high accuracy, ξcr ~ W –2. The calculations are performed by integrating the evolutional Schrodinger equations and through analyzing the spectral properties of the corresponding random matrices. The probability of charge transfer is estimated within the framework of a simple model of DNA.

Combinatorial game associated to the one dimensional Schelling’s model of social segregation

In this paper we consider a finite one-dimensional lattice with \(N = 2n + 1\) sites such that one of them is empty and the others have a black or white token. There are two players (one for each color), such that step by step alternately they move one of their tokens to the empty site trying to obtain a connected configuration. This game is related with the Schelling’s social segregation model, where colors represent two different populations such that each one tries to take up a position with more neighbors as itself (same color). In this work we study strategies to play the game as well as their relation with the associated Schelling’s one-dimensional case (line and cycle graphs).

Integrability of a deterministic cellular automaton driven by stochastic boundaries

We propose an interacting many-body space–time-discrete Markov chain model, which is composed of an integrable deterministic and reversible cellular automaton (rule 54 of Bobenko et al 1993 Commun. Math. Phys. 158 127) on a finite one-dimensional lattice , and local stochastic Markov chains at the two lattice boundaries which provide chemical baths for absorbing or emitting the solitons. Ergodicity and mixing of this many-body Markov chain is proven for generic values of bath parameters, implying the existence of a unique nonequilibrium steady state. The latter is constructed exactly and explicitly in terms of a particularly simple form of matrix product ansatz which is termed a patch ansatz. This gives rise to an explicit computation of observables and k-point correlations in the steady state as well as the construction of a nontrivial set of local conservation laws. The feasibility of an exact solution for the full spectrum and eigenvectors (decay modes) of the Markov matrix is suggested as well. We conjecture that our ideas can pave the road towards a theory of integrability of boundary driven classical deterministic lattice systems.

Effective stochastic generators for conditioned dynamics at an atypical reaction-diffusion current.

We consider the fluctuations of a time-integrated particle current around an atypical value in a generic stochastic Markov process involving classical particles with two-site interaction and hard-core repulsion on a finite one-dimensional lattice with open boundaries. We address the question of which interactions one has to impose on such a process to make the atypical value of the current typical. It is known that a corresponding effective stochastic Markov process might exist whose typical current value is equal to the atypical current value in the original process within a time-translational invariant regime. This effective process has, in principle, nonlocal transition rates. Nevertheless, it turns out that under some conditions, the stochastic generator of the effective process has the same dynamical rules as the stochastic generator of the original process. We find these conditions and show that our approach can be generalized to any time-integrated observable.

Spin Chains with Dynamical Lattice Supersymmetry

Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the N=(2,2) superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension admits a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjectured.

Velocity correlations of a discrete-time totally asymmetric simple-exclusion process in stationary state on a circle

The discrete-time version of totally asymmetric simple-exclusion process (TASEP) on a finite one-dimensional lattice is studied with the periodic boundary condition. Each particle at a site hops to the next site with probability 0≤p≤1 if the next site is empty. This condition can be rephrased by the condition that the number n of vacant sites between the particle and the next particle is positive. Then the average velocity is given by a product of the hopping probability p and the probability that n≥1. By mapping the TASEP to another driven diffusive system called the zero-range process, it is proved that the distribution function of vacant sites in the stationary state is exactly given by a factorized form. We define k-particle velocity correlation function as the expectation value of a product of velocities of k particles in the stationary distribution. It is shown that it does not depend on positions of k particles on a circle but depends only on the number k. We give explicit expressions for all velocity correlation functions using the Gauss hypergeometric functions. Covariance of velocities of two particles is studied in detail, and we show that velocities become independent asymptotically in the thermodynamic limit.

An exact formula for the statistics of the current in the TASEP with open boundaries

We study the totally asymmetric exclusion process on a finite one-dimensional lattice with open boundaries, i.e. in contact with two reservoirs at different potentials. The total (time-integrated) current through the system is a random variable that scales linearly with time in the long-time limit. We conjecture a parametric representation for the generating function of the cumulants of the current, which is related to the large deviation function by Laplace transform. This formula is valid for all system sizes and for all values of the boundary coupling parameters.

Continuation and bifurcations of breathers in a finite discrete NLS equation

We present results on the continuation of breathers in the discrete cubic nonlinear Schrödinger equation in a finite one-dimensional lattice with Dirichlet boundary conditions. In the limit of small inter-site coupling the equation has a finite number of breather solutions and as we increase the coupling we see numerically that all breather branches undergo either fold or pitchfork bifurcations. We also see branches that persist for arbitrarily large coupling and converge to the linear normal modes of the system. The stability of the breathers that persist generally changes as the coupling is varied, although there are at least two branches that preserve their linear and nonlinear stability properties throughout the continuation.

Continuation of normal modes in finite NLS lattices

We study the continuation of breather solutions of the discrete NLS equation as the intersite coupling parameter is varied. Considering the case of a finite one-dimensional lattice of N sites, we show the existence of N branches of breathers that persist for arbitrary coupling, thus connecting normal modes of the linear system to breathers of the uncoupled, anticontinuous limit system. The proof is based on global bifurcation theory, applied to the continuation from the weakly nonlinear regime. As the coupling parameter varies these solutions generally change their stability, and we detect parameter regions where trajectories starting near unstable breathers appear to reach equipartition of power.

Exact results for an asymmetric annihilation process with open boundaries

We consider a nonequilibrium reaction–diffusion model on a finite one-dimensional lattice with bulk and boundary dynamics inspired by the Glauber dynamics of the Ising model. We show that the model has a rich algebraic structure that we use to calculate its properties. In particular, we show that the Markov dynamics for a system of a given size can be embedded into the dynamics of systems of higher sizes. This remark leads us to devise a technique which we call the transfer matrix Ansatz that allows us to determine the steady-state distribution and correlation functions. Furthermore, we show that the disorder variables satisfy very simple properties and we give a conjecture for the characteristic polynomial of Markov matrices. Finally, we compare the transfer matrix Ansatz used here with the matrix product representation of the steady state of one-dimensional stochastic models.

Quantum quenches and thermalization in one-dimensional fermionic systems

We study the dynamics and thermalization of strongly correlated fermions in finite one-dimensional lattices after a quantum quench. Our calculations are performed using exact diagonalization. We focus on one- and two-body observables such as the momentum distribution function [n(k)] and the density-density structure factor [N(k)], respectively, and study the effects of approaching an integrable point. We show that while the relaxation dynamics and thermalization of N(k) for fermions is very similar to the one of hardcore bosons, the behavior of n(k) is distinctively different. The latter observable exhibits a slower relaxation dynamics in fermionic systems. We identify the origin of this behavior, which is related to the off-diagonal matrix elements of n(k) in the basis of the eigenstates of the Hamiltonian. More generally, we find that thermalization occurs far away from integrability and that it breaks down as one approaches the integrable point.

Breakdown of Thermalization in Finite One-Dimensional Systems

We use quantum quenches to study the dynamics and thermalization of hard core bosons in finite one-dimensional lattices. We perform exact diagonalizations and find that, far away from integrability, few-body observables thermalize. We then study the breakdown of thermalization as one approaches an integrable point. This is found to be a smooth process in which the predictions of standard statistical mechanics continuously worsen as the system moves toward integrability. We establish a direct connection between the presence or absence of thermalization and the validity or failure of the eigenstate thermalization hypothesis, respectively.