One-dimensional Lattice Systems Research Articles

Anomalous symmetry-protected blockade of the skin effect in one-dimensional non-Hermitian lattice systems

Anomalous symmetry-protected blockade of the skin effect in one-dimensional non-Hermitian lattice systems

Continuous extension of the discrete shift translations on one-dimensional quantum lattice systems

We study the continuous extension of discrete shift translations on one-dimensional quantum lattice systems. We explore a specific example provided by a quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system, comparing it with its original discrete shift-translations. Specifically, we demonstrate its explicit dynamical formula, which reveals violations of causality and locality. Furthermore, we prove that this quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system cannot be extended to the one-dimensional quantum spin lattice system via the Jordan-Wigner transformation.

Quantum current and holographic categorical symmetry

We establish the formulation for quantum current. Given a symmetry group G, let \mathcal{C}𝒞:=Rep G be its representation category. Physically, symmetry charges are objects of \mathcal{C}𝒞 and symmetric operators are morphisms in \mathcal{C}𝒞. The addition of charges is given by the tensor product of representations. For any symmetric operator O crossing two subsystems, the exact symmetry charge transported by O can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center Z_1(\mathcal{C})Z1(𝒞). The condition for quantum currents to be is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in Z_1(\mathcal{C})Z1(𝒞) and the boundary-bulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.

Topological phase in one-dimensional momentum space lattice of ultracold atoms without chiral symmetry

Symmetry plays a crucial role in understanding topological phases in materials. In one-dimensional systems, such as the Su-Schrieffer-Heeger (SSH) model, chiral symmetry is thought to ensure the quantization of the Zak phase and the nontrivial topological phase. However, our work demonstrates that the one-dimensional lattice system with broken chiral symmetry can still possess quantized Zak phase and nontrivial topological phase. Specifically, we use a Bose-Einstein condensate of <sup>87</sup>Rb atoms in a momentum space lattice of ultracold atoms to effectively simulate a one-dimensional Zigzag model of 26 sites, which intrinsically breaks the chiral symmetry by additional next-nearest-neighbor coupling. To ensure the existence of the nontrivial topological phase, where the Zak phase can be measured from the time-averaged displacement during the system’s evolution, we need to preserve the inversion symmetry by modulating laser power so that all next-nearest-neighbor coupling strengths are equal. Furthermore, by changing the ratio of nearest-neighbor coupling strengths, we observe a topological phase transition from a nontrivial topological phase to a trivial topological phase at the point where the ratio equals 1. Our work demonstrates that the ultracold atom system provides a controllable platform for studying the symmetrical phase and topological phase, with the potential to explore nonlinear topological phenomena by increasing the interactions among atoms. In addition, our system can be used to investigate other interesting topological phenomena with more complex models, such as critical phenomena at the phase transitions and flat band structures in the extended SSH model with long-range coupling. By controlling the coupling strengths, we can also explore the influence of different symmetries on the topological properties of extended SSH models in the future. Moreover, our platform makes it possible to studythe models with more energy bands, such as the Aharonov-Bohm caging model with a three-level structure, which shows peculiar flat-band properties. This work provides opportunities for various studies in the fields of symmetry, topology, and the interaction of controllable quantum systems.

Observation of a Topological Edge State Stabilized by Dissipation.

Robust states emerging at the boundary of a system constitute a hallmark for topological band structures. Other than in closed systems, topologically protected states can occur even in systems with a trivial band structure, if exposed to suitably modulated losses. Here, we study the dissipation-induced emergence of a topological band structure in a non-Hermitian one-dimensional lattice system, realized by arrays of plasmonic waveguides with tailored loss. We obtain direct evidence for a topological edge state that resides in the center of the band gap. By tuning dissipation and hopping, the formation and breakdown of an interface state between topologically distinct regions is demonstrated.

Super-diffusion and crossover from diffusive to anomalous transport in a one-dimensional system

We study transport in a one-dimensional lattice system with two conserved quantities - “volume” and energy. Considering a slowly evolving local equilibrium state that is slightly deviated from an underlying global equilibrium, we estimate the correction to the local equilibrium distribution. This correction arises mainly through the space-time correlations of some local currents. In the continuum limit, we show that the local equilibrium distribution along with the correction yields drift-diffusion equation for the “volume” and super-diffusion equation for the energy in the linear response regime as macroscopic hydrodynamics as one would obtain from non-linear fluctuating hydrodynamic theory. We find explicit expression of the super-diffusion equation. Further, we find diffusive correction to the super-diffusive evolution. Such a correction allows us to study a crossover from diffusive to anomalous transport. We demonstrate this crossover numerically through the spreading of an initially localized heat pulse in equilibrium as well as through the system size scaling of the stationary current in non-equilibrium steady state.

Uniqueness and statistical properties of the Gibbs state on general one-dimensional lattice systems with Markovian structure

Let [Formula: see text] be a compact metric space and [Formula: see text], we consider a set of admissible sequences [Formula: see text] determined by a continuous admissibility function [Formula: see text] and a compact set [Formula: see text]. Given a Lipschitz continuous potential [Formula: see text], we prove uniqueness of the Gibbs state [Formula: see text] and we show that it is a Gibbs–Bowen measure and satisfies a central limit theorem.

Localization of matter waves in lattice systems with moving disorder

We study the localization phenomena in a one-dimensional lattice system with a uniformly moving disordered potential. At a low-moving velocity, we find a sliding localized phase in which the initially localized matter wave adiabatically follows the moving potential without diffusion, thus resulting in an initial state memory in the many-body dynamics. Such an intriguing localized phase distinguishes itself from the standard Anderson localization in two aspects: it is not robust against interaction, but persists in the presence of slowly varying perturbations. Such a sliding localized phase can be understood as a consequence of interference between the wave-packet paths under moving quasiperiodic potentials with various periods that are incommensurate with the lattice constant. The experimental realization and detection are also discussed.

Dissipation-induced recurrence of non-Hermitian edge burst

In quantum mechanics, the Hermitian Hamiltonian is generally used to describe the ideal closed quantum system, but in reality, the physical system is closely related to the environment, and the open quantum system coupled to the environment can be described by the equivalent non-Hermitian Hamiltonian to a certain extent. Among them, the dissipation intensity is closely related to the dynamic properties of non-Hermitian quantum systems. Therefore, it is of great practical significance to study how dissipation affects particle loss. In this paper, the dynamic law related to dissipation intensity in a one-dimensional non-Hermitian system under open boundary condition is studied, and it is found that dissipation can induce the recurrence of edge burst. After the time-dependent evolution of the particles in the one-dimensional non-Hermitian dissipative lattice system with open boundary condition, there is an edge burst in the system, that is, there is a large probability of particle loss at the edge, and the edge burst disappears after increasing the intracell hopping. It is found that if the dissipation intensity increases or decreases, the edge burst will reappear. This kind of reappearance is different from the original edge burst, which is mainly manifested in the loss probability distribution of particles from the edge distribution to the bulk distribution, which is due to the difference in probability of particle motion direction between the two cases. Under the re-induced edge burst, the particles move leftward and rightward from their initial positions, and rebound from the left after having reached the boundary, forming a more obvious loss probability at the edge and gradually decreasing to the body area. In the original edge burst, the particles only move to the left with a greater probability, and are ‘trapped’ at the edge to completely dissipated, forming a distribution with an independent loss peak at the edge. The movement to the left is due to the non-Hermitian skin effect. The deeper reason for different movement directions is related to parity-time symmetry. Under the parameter near the parity-time symmetry breaking point, the loss probability of the particle is of unilateral distribution, and the loss probability of the particle moving to both sides is of bilateral distribution when it is far away. This is the description of the dissipation-induced edge burst recurrence phenomenon and its characteristics. In addition, this paper also studies the influence of impurity barrier on the probability distribution of particle loss in non-Hermitian dynamics. The results show that placing a small barrier on the non-dissipative <i>A</i>-site can obviously hinder the particle motion, and when the barrier increases to a certain height, its influence on the particle motion tends to be unchanged. And the barrier at the dissipative <i>B</i> lattice has little effect on the dynamics.

Tensor-network approach to work statistics for one-dimensional quantum lattice systems

We introduce a tensor-network approach to calculate the statistics of work done on one-dimensional quantum lattice systems initially prepared in thermal equilibrium states. In this approach, the dynamics is simulated with time-evolving block decimation (TEBD), and the initial thermal equilibrium state is prepared either directly with TEBD or with minimally entangled typical thermal states, which generates a set of typical states representing the Gibbs canonical ensemble. As an illustrative example, we apply this approach to the Ising chain in mixed transverse and longitudinal fields. Under a prescribed protocol, the moment generating function for work distribution can be calculated, from which the quantum Jarzynski equality and the generalized quantum work relation involving a functional of an arbitrary observable are tested.

Many-body localization in a quantum gas with long-range interactions and linear external potential

We study theoretically transitions between the localized and chaotic many-body regimes in one-dimensional quantum lattice systems with long-range couplings between particles and linear external potential. In terms of established criteria characterizing localization, we construct effective phase diagrams for several types of lattice systems with variable amplitude of the external linear tilt and interaction strength. By means of exact diagonalization and time-dependent variational principle numerical approaches we analyze system dynamics after quenches. Our results reveal that the Stark localization without any artificial source of disorder remains stable upon inclusion of long-range interactions.

Spread of correlations in strongly disordered lattice systems with long-range coupling

We investigate the spread of correlations in a one-dimensional lattice system with high on-site energy disorder and long-range couplings with a power-law dependence on the distance ($\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\mu}}$). The increase in correlation between the initially quenched node and a given node exhibits three phases: quadratic in time, linear in time, and saturation. No further evolution is observed in the long time regime. We find an approximate solution of the model valid in the limit of strong disorder and reproduce the results of numerical simulations with analytical formulas. We also find the time needed to reach a given correlation value as a measure of the propagation speed. Because of the triple-phase evolution of the correlation function, the propagation changes its time dependence. In the particular case of $\ensuremath{\mu}=1$, the propagation starts as a ballistic motion and then, at a certain crossover time, turns into standard diffusion.

Exact solution of the “Rule 150” reversible cellular automaton

We study the dynamics and statistics of the Rule 150 reversible cellular automaton (RCA). This is a one-dimensional lattice system of binary variables with synchronous (Floquet) dynamics that corresponds to a bulk deterministic and reversible discretized version of the kinetically constrained "exclusive one-spin facilitated" (XOR) Fredrickson-Andersen (FA) model, where the local dynamics is restricted: A site flips if and only if its adjacent sites are in different states from each other. Similar to other RCA that have been recently studied, such as Rule 54 and Rule 201, the Rule 150 RCA is integrable, however, in contrast is noninteracting: The emergent quasiparticles, which are identified by the domain walls, behave as free fermions. This property allows us to solve the model by means of matrix product ansatz. In particular, we find the exact equilibrium and nonequilibrium stationary states for systems with closed (periodic) and open (stochastic) boundaries, respectively, resolve the full spectrum of the time evolution operator and, therefore, gain access to the relaxation dynamics, and obtain the exact large deviation statistics of dynamical observables in the long-time limit.

Product Type Potential on the One-Dimensional Lattice Systems: Selection of Maximizing Probability and a Large Deviation Principle

Product Type Potential on the One-Dimensional Lattice Systems: Selection of Maximizing Probability and a Large Deviation Principle

Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure

Consider a compact metric space $(M, d_M)$ and $X = M^{\mathbb{N}}$. We prove a Ruelle's Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in [Bull. Braz. Math. Soc. 45 (2014), pp. 53-72] which are defined from a continuous function $A : M \times M \to \mathbb{R}$ that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle $S^1$. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associated to its maximal eigenvalue and present an extension of its corresponding Gibbs state to the bilateral approach. From these results, we prove existence of equilibrium states and accumulation points at zero temperature in a particular class of countable Markov shifts.

Critical exponents of block-block mutual information in one-dimensional infinite lattice systems.

We study the mutual information between two lattice blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems. Quantum q-state Potts model and transverse-field spin-1/2XY model are considered numerically by using the infinite matrix product state approach. As a system parameter varies, block-block mutual information exhibit singular behaviors that enable us to identify the critical points for the quantum phase transitions. As happens with von Neumann entanglement entropy of single block, at critical points, block-block mutual information for two adjacent blocks show a logarithmic leading behavior with increasing the size of the blocks, which yields the central charge c of the underlying conformal field theory, as it should be. It seems that two disjoint blocks show a similar logarithmic growth of the mutual information as a characteristic property of critical systems but the proportional coefficients of the logarithmic term are very different from the central charges. As the separation between the two lattice blocks increases, the mutual information reveals a consistent power-law decaying behavior for various truncation dimensions and lattice-block sizes. The critical exponent of block-block mutual information in the thermodynamic limit is estimated by extrapolating the exponents of power-law decaying regions for finite truncation dimensions. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. Whereas the critical exponents have different values to a degree of distinction for the different universality classes. Asthelattice-block size becomes bigger, the critical exponent becomes smaller. We discuss a relation between the exponents of block-block mutual information and correlation with the Shatten one-norm of block-block correlation.

A new tunable elastic metamaterial structure for manipulating band gaps/wave propagation

A one-dimensional mechanical lattice system with local resonators is proposed as an elastic metamaterial model, which shows negative mass and negative modulus under specific frequency ranges. The proposed representative units, consisting of accurately arranged rigid components, can generate controllable translational resonance and achieve negative mass and negative modulus by adjusting the local structural parameters. A shape memory polymer is adopted as a spring component, whose Young’s modulus is obviously affected by temperature, and the proposed metamaterials’ tunable ability is achieved by adjusting temperature. The effect of the shape memory polymer’s stiffness variation on the band gaps is investigated detailedly, and the special phenomenon of intersecting dispersion curves is discussed, which can be designed and controlled by adjusting temperature. The dispersion relationship of the continuum metamaterial model affected by temperature is obtained, which shows great tunable ability to manipulate wave propagation.

Characterization of topological insulators based on the electronic polarization with spiral boundary conditions

We introduce the electronic polarization originally defined in one-dimensional lattice systems to characterize two-dimensional topological insulators. The main idea is to use spiral boundary conditions which sweep all lattice sites in one-dimensional order. We find that the sign of the polarization changes at topological transition points of the two-dimensional Wilson-Dirac model (the lattice version of the Bernevig-Hughes-Zhang model) in the same way as in one-dimensional systems. Thus the polarization plays the role of "order parameter" to characterize the topological insulating state and enables us to study topological phases in different dimensions in a unified way.

Approximations based on density-matrix embedding theory for density-functional theories

Recently a novel approach to find approximate exchange–correlation functionals in density-functional theory was presented (Mordovina et al 2019 J. Chem. Theory Comput. 15 5209), which relies on approximations to the interacting wave function using density-matrix embedding theory (DMET). This approximate interacting wave function is constructed by using a projection determined by an iterative procedure that makes parts of the reduced density matrix of an auxiliary system the same as the approximate interacting density matrix. If only the diagonal of both systems are connected this leads to an approximation of the interacting-to-non-interacting mapping of the Kohn–Sham approach to density-functional theory. Yet other choices are possible and allow to connect DMET with other density-functional theories such as kinetic-energy density functional theory or reduced density-matrix functional theory. In this work we give a detailed review of the basics of the DMET procedure from a density-functional perspective and show how both approaches can be used to supplement each other. We do not present a specific realization of combining density-functional methods with DMET but rather provide common grounds to facilitate future developments that encompass both approaches. We do so explicitly for the case of a one-dimensional lattice system, as this is the simplest setting where we can apply DMET and the one that was originally presented. Among others we highlight how the mappings of density-functional theories can be used to identify uniquely defined auxiliary systems and projections in DMET and how to construct approximations for different density-functional theories using DMET inspired projections. Such alternative approximation strategies become especially important for density-functional theories that are based on non-linearly coupled observables such as kinetic-energy density-functional theory, where the Kohn–Sham fields are no longer obtainable by functional differentiation of an energy expression, or for reduced density-matrix functional theories, where a straightforward Kohn–Sham construction is not feasible.

Quantized transport induced by topology transfer between coupled one-dimensional lattice systems

We show that a topological pump in a one-dimensional (1D) insulator can induce a strictly quantized transport in an auxiliary chain of non-interacting fermions weakly coupled to the first. The transported charge is determined by an integer topological invariant of the ficticious Hamiltonian of the insulator, given by the covariance matrix of single-particle correlations. If the original system consists of non-interacting fermions, this number is identical to the TKNN (Thouless, Kohmoto, Nightinghale, den Nijs) invariant of the original system and thus the coupling induces a transfer of topology to the auxiliary chain. When extended to particles with interactions, for which the TKNN number does not exist, the transported charge in the auxiliary chain defines a topological invariant for the interacting system. In certain cases this invariant agrees with the many-body generalization of the TKNN number introduced by Niu, Thouless, and Wu (NTW). We illustrate the topology transfer to the auxiliary system for the Rice-Mele model of non-interacting fermions at half filling and the extended superlattice Bose-Hubbard model at quarter filling. In the latter case the induced charge pump is fractional.