Finite Lattice Research Articles

Quantized Axial Charge of Staggered Fermions and the Chiral Anomaly

In the 1+1D ultralocal lattice Hamiltonian for staggered fermions with a finite-dimensional Hilbert space, there are two conserved, integer-valued charges that flow in the continuum limit to the vector and axial charges of a massless Dirac fermion with a perturbative anomaly. Each of the two lattice charges generates an ordinary U(1) global symmetry that acts locally on operators and can be gauged individually. Interestingly, they do not commute on a finite lattice and generate the Onsager algebra, but their commutator goes to zero in the continuum limit. The chiral anomaly is matched by this non-Abelian algebra, which is consistent with the Nielsen-Ninomiya theorem. We further prove that the presence of these two conserved lattice charges forces the low-energy phase to be gapless, reminiscent of the consequence from perturbative anomalies of continuous global symmetries in continuum field theory. Upon bosonization, these two charges lead to two exact U(1) symmetries in the XX model that flow to the momentum and winding symmetries in the free boson conformal field theory. Published by the American Physical Society 2025

Correlation decay for finite lattice gauge theories at weak coupling

Correlation decay for finite lattice gauge theories at weak coupling

A class of stable nonlinear non-Hermitian skin modes

Abstract The non-Hermitian skin effect (NHSE) is a well-known phenomenon in open topological systems that causes a large number of eigenstates to become localized at the boundary. Although many aspects of its theory have been investigated in linear systems, this phenomenon remains novel in nonlinear models. In the first step of this paper, we look at the conditions for the presence of quasi-skin modes in a semi-infinite, one-dimensional, nonlinear, nonreciprocal lattice. In the following phase, we explore the survival time of the quasi-skin mode in a finite nonlinear lattice with open edges. We study the dependency of the survival time on the system's parameters and demonstrate how the nonreciprocity of the system affects the survival time. This study introduces a method for achieving a stable localized state in a nonlinear finite lattice.

Basic Cells Special Features and Their Influence on Global Transport Properties of Long Periodic Structures.

In this contribution, we address quantum transport in long periodic arrays whose basic cells, localized potentials U(x), display certain particular features. We investigate under which conditions these "local" special characteristics can influence the tunneling behavior through the full structure. As the building blocks, we consider two types of U(x)s: combinations of either Pöschl-Teller, U0/cosh2[αx], potentials (for which the reflection and transmission coefficients are known analytically) or Gaussian-shaped potentials. For the latter, we employ an improved potential slicing procedure using basic barriers, like rectangular, triangular and trapezoidal, to approximate U(x) and thus obtain its scattering amplitudes. By means of a recently derived method, we discuss scattering along lattices composed of a number, N, of these U(x)s. We find that near-resonance energies of an isolated U(x) do impact the corresponding energy bands in the limit of very large Ns, but only when the cell is spatially asymmetric. Then, there is a very narrow opening (defect or rip) in the system conduction quasi-band, corresponding to the energy of the U(x) quasi-state. Also, for specific U0's of a single Pöschl-Teller well, one has 100% transmission for any incident E>0. For the U(x) parameters rather close to such a condition, the associated array leads to a kind of "reflection comb" for large Ns; |TN(k)|2 is not close to one only at very specific values of k, when |TN|2≈0. Finally, the examples here-illustrating how the anomalous transport comportment in finite but long lattices can be inherited from certain singular aspects of the U(x)s-are briefly discussed in the context of known effects in the literature, notably for lattices with asymmetric cells.

Solving an internal problem for finite regular two-dimensional lattice spiral elements, excitable plate electromagnetic wave

Background. The work is aimed at developing and researching rigorous methods for solving internal problem of electrodynamics for multi-element structures (metastructures) consisting from the final number of elements, as well as to study the physical processes occurring in them. A special case of such structures are two-dimensional lattices with a fixed interelement distance, consisting of identical elements having the same spatial orientation (regular lattices). Aim. In this work, based on an iterative approach, the internal solution is solved. problems of electrodynamics for a finite regular two-dimensional lattice of spiral elements. In order to obtain a priori information about the electrodynamic characteristics of elements lattice and justification for the choice of projection function systems are analyzed spectral characteristics of the integral operator of the internal problem for a single spiral element. Then the currents on the structure elements are calculated, their spectral characteristics are determined. The results of spectral analysis allow increase the efficiency of solving an internal problem. Methods. The research is based on a strict electrodynamic approach, within the framework of which, for the specified structure in the thin-wire approximation, an integral representation of the electromagnetic field is formed, which, when considered on the surface of conductors together with boundary conditions, is reduced to a system of Fredholm integral equations of the second kind, written relative to unknown current distributions on conductors (internal task). The solution of the internal problem within the framework of the method of moments is reduced to solving a SLAE with a block matrix. Results. A mathematical model of a finite two-dimensional lattice of spiral elements is proposed radiating structure. For the specified structure, in the case of its excitation by a flat electromagnetic wave, based on the iterative approach, the internal problem of electrodynamics was solved. The following were carried out in a wide frequency range: analysis of the convergence of the iterative process, spectral analysis of the integral operator of the internal problem for a single spiral element, as well as spectral analysis of external field and current functions functions on lattice elements. Conclusion. The feasibility of determining the spectral characteristics of integral operators is shown internal task for the elements forming the metastructure. A relationship has been identified between the frequency dependence eigenvalues of the integral operator of the internal problem of single elements, forming a metastructure, with resonance phenomena arising in the metastructure, the influence of resonances on the convergence of the iterative process was confirmed. The feasibility of considering averaged amplitude current spectra is shown. It was revealed that the averaged spectrum of current functions is close to degenerate, especially near resonant frequencies. This allows for use as projection functions a compact set of eigenfunctions that have significant amplitudes in the vicinity of the frequency under study, which significantly simplifies the solution of the internal problem.

Numerical Simulation and Experimental Verification of Rotor Airflow Field Based on Finite Volume Method and Lattice Boltzmann Method

The primary focus of research in agricultural unmanned aerial vehicle (UAV) pesticide application technology is the investigation of droplet drift and deposition. The influence of the rotor airflow on droplets is particularly significant, making numerical simulations a crucial tool for airflow field analysis. Among existing numerical simulation methods, the Finite Volume Method (FVM) and the Lattice Boltzmann Method (LBM) are commonly used, but there is limited research that compares the two approaches. Therefore, this paper conducts numerical simulations of the rotor airflow of an agricultural UAV using Fluent, representing the FVM, and XFlow, representing the LBM. This research aims to reveal the distribution patterns of airflow field numerical simulations under different theoretical methods, validate them through practical experiments, and select the optimal method for simulating rotor airflow. The ultimate goal is to establish an effective airflow field model to enhance the precision of pesticide application by an agricultural UAV. The results indicate that the lift error calculated by XFlow in this paper is 2.57% smaller than that by Fluent. The wind field of Fluent entered the “stable state” earlier than XFlow, and the speed value of Fluent was smaller than that of XFlow. The difference between the two speed values became larger and larger as the distance from the rotor was longer. Compared with XFlow, Fluent changes more obviously in the core region, and the center region gradually disappears with the distance from the rotor. However, in the velocity field calculated by XFlow, there are still more turbulent flows outside the core region, indicating that the transient calculation method based on the LBM can better show the details of fluid flow than the steady-state calculation method based on the FVM. Through comparison with the actual test data, it is found that the relative error of the velocity value of XFlow at 0.2 m and 0.4 m is small, while that of Fluent at 0 and 0.2 m is small. It shows that XFlow simulation has higher accuracy for external turbulent flow, while Fluent simulation has higher accuracy for steady laminar flow. The research results provide data comparison and a basis for further analysis of the wind field model of the rotor wing of the plant protection UAV, and they lay a foundation for further research on precision application technology.

Congruences of finite semidistributive lattices

Congruences of finite semidistributive lattices

Exploring run-and-tumble movement in confined settings through simulation.

Motion in bounded domains is a fundamental concept in various fields, including billiard dynamics and random walks on finite lattices, and has important applications in physics, ecology, and biology. An important universal property related to the average return time to the boundary, the Mean Path Length Theorem (MPLT), has been proposed theoretically and experimentally confirmed in various contexts. We investigated a wide range of mechanisms that lead to deviations from this universal behavior, such as boundary effects, reorientation, and memory processes. This study investigates the dynamics of run-and-tumble particles within a confined two-dimensional circular domain. Through a combination of theoretical approaches and numerical simulations, we validate the MPLT under uniform and isotropic particle inflow conditions. This research demonstrates that although the MPLT is generally applicable for different step length distributions, deviations occur for non-uniform angular distributions, non-elastic boundary conditions, or memory processes. These results underline the crucial influence of boundary interactions and angular dynamics on the behavior of particles in confined spaces. Our results provide new insights into the geometry and dynamics of motion in confined spaces and contribute to a better understanding of a broad spectrum of phenomena ranging from the motion of bacteria to neutron transport. This type of analysis is crucial in situations where inhomogeneity occurs, such as multiple real-world scenarios within a limited domain.

Polynomial crisp-minimization algorithm for fuzzy deterministic automata

Designing automata minimization algorithms is a significant topic in Automata Theory and Languages with practical applications. In this paper, we develop an efficient minimization algorithm for deterministic fuzzy finite automata over locally finite lattices. More precisely, the algorithm outputs an equivalent minimal crisp-deterministic fuzzy finite automaton for an input fuzzy deterministic finite automaton (FDfA). The running time of the proposed algorithm is polynomial for particular types of locally finite lattices, specifically for max-min-based complete residuated lattices. The intuition behind the proposed algorithm relies on the polynomial minimization algorithm's intuition for ordinary deterministic automata developed by Vazquez de Parga, Garcia, and Lopez (2013) [35]. The motivation for this study comes from the fact that the original algorithm's notions and meanings are lost in the context of fuzzy automata. Thus, we establish a new theoretical foundation that provides the correctness and the polynomial-time nature of this new crisp-minimization algorithm for FDfAs.

Cauchy universality and random billiards

Motion in bounded domains represents a paradigm in several settings: from billiard dynamics, to random walks in a finite lattice, with applications to relevant physical, ecological, and biological problems. A remarkable universal property, involving the average of return times to the boundary, has been theoretically proposed and experimentally verified in quite different contexts. We discuss here mechanisms that lead to violations of universality, induced by boundary effects and we also emphasize the role played by replacing straight lines with random walks in this framework. We suggest that our analysis should be relevant where nonhomogeneity appears in the stationary probability distribution in bounded domain. Published by the American Physical Society 2024

Dual digraphs of finite meet-distributive and modular lattices

We describe the digraphs that are dual representations of finite lattices satisfying conditions related to meet-distributivity and modularity. This is done using the dual digraph representation of finite lattices by Craig, Gouveia and Haviar (2015). These digraphs, known as TiRS digraphs, have their origins in the dual representations of lattices by Urquhart (1978) and Ploščica (1995). We describe two properties of finite lattices which are weakenings of (upper) semimodularity and lower semimodularity respectively, and then show how these properties have a simple description in the dual digraphs. Combined with previous work in this journal on dual digraphs of semidistributive lattices (2022), it leads to a dual representation of finite meet-distributive lattices. This provides a natural link to finite convex geometries. In addition, we present two sufficient conditions on a finite TiRS digraph for its dual lattice to be modular. We close by posing three open problems.

Time-Dependent Variational Principle with Controlled Bond Expansion for Matrix Product States.

We present a controlled bond expansion (CBE) approach to simulate quantum dynamics based on the time-dependent variational principle (TDVP) for matrix product states. Our method alleviates the numerical difficulties of the standard, fixed-rank one-site TDVP integrator by increasing bond dimensions on the fly to reduce the projection error. This is achieved in an economical, local fashion, requiring only minor modifications of standard one-site TDVP implementations. We illustrate the performance and accuracy of CBE-TDVP with several numerical examples on finite quantum lattices, including new results on bipolaron formation in the Peierls-Hubbard model and spin pumping via adiabatic flux insertion in a chiral spin liquid.

Tailoring band gap properties of curved hexagonal lattices with nodal cantilevers

Metamaterials find applications across diverse domains such as electromagnetics, elasticity, and acoustics by creating band gaps. Lattice-based metamaterials also exhibit band gaps, which have a great potential to influence engineering design in vibration and noise reduction problems. The geometry of the repetitive unit cell in the lattice plays a crucial role in diversifying the location and number of stop bands across the frequency range. One of the key hurdles is devising unit cell architectures that can effectively suppress vibrations across diverse frequency ranges. This work proposes an innovative two-dimensional hexagonal lattice with tailored band gap characteristics through curved beam members and auxiliary cantilever beams at the nodes. We have thoroughly explored the impact of various design parameters on dispersion characteristics, wave directionality through iso-frequency contours of dispersion surfaces, and the transmission loss considering finite lattice. The investigation demonstrates an improvement in band gap characteristics, indicating the generation of more band gaps across the entire frequency range and the widening of the same. This study has the potential to serve as a future benchmark in the development of lattice-based elastic/acoustic metamaterials, particularly for addressing vibration reduction challenges at user-defined frequencies.

Shadowed set approximations of L-fuzzy sets

Pedrycz shadowed sets are three-way approximations of fuzzy sets by transforming the infinite levels of fuzzy set membership grades in the unit interval [0,1] into three levels. The three levels represent qualitatively the sets of the white, grey, and black members of a shadowed set. In this paper, we generalize the notion of shadowed sets to the case of L-fuzzy sets by making three new contributions. First, we consider two representations of a shadowed set. One is a three-valued L-fuzzy set and the other is three pairwise disjoint sets. Second, we introduce two methods for constructing a shadowed set. One divides a finite lattice based on the notion of a pair of a set of designated core membership grades and a set of designated null membership grades. The other uses a pair of threshold sets, which generalizes the method that uses a pair of thresholds. We study formal properties of the two methods and show that they are equivalent. Finally, based on a distance function on a lattice, we present a simple method to build the sets of designated core and null membership grades.

Hydrodynamics for Asymmetric Simple Exclusion on a Finite Segment with Glauber-Type Source

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a source term. Different from the usual boundary conditions introduced in Bardos et al (Commun Partial Differ Equ 4(9):1017–1034, https://doi.org/10.1080/03605307908820117, 1979) and Otto (C R Acad Sci Paris 322(1):729–734, 1996), discontinuity (boundary layer) does not formulate at the boundaries due to the strong relaxation scheme.

Spiral to stripe transition in the two-dimensional Hubbard model

We obtain an almost complete understanding of the mean-field phase diagram of the two-dimensional Hubbard model on a square lattice with a sizable next-nearest-neighbor hopping and a moderate interaction strength. In particular, we clarify the nature of the transition region between the spiral and the stripe phase. Complementing previous [] real-space Hartree-Fock calculations on large finite lattices, we solve the mean-field equations for coplanar unidirectional magnetic order directly in the thermodynamic limit, and we determine the nature of the magnetic states right below the mean-field critical temperature T* by a Landau free-energy analysis. While the magnetic order for filling factors n≥1 is always of Néel type, for n≤1 the following sequence of magnetic states is found as a function of increasing hole-doping: Néel, planar circular spiral, multispiral, and collinear spin-charge stripe states. Multispiral states are superpositions of several spirals with distinct wave vectors, and lead to concomitant charge order. We finally point out that nematic and charge orders inherited from the magnetic order can survive even in the presence of fluctuations, and we present a corresponding qualitative phase diagram. Published by the American Physical Society 2024

An Exploration of the Phonon Frequency Spectrum and Born–von Karman Periodic Boundary Conditions in 1D and 2D Lattice Systems Using a Computational Approach

Periodic boundary conditions (PBCs) are a pivotal concept in the treatment of ideal lattices of infinite extent as a finite lattice. Most undergraduate texts that delve into the analytical treatment of lattice dynamics solve the problem in the reciprocal space with an implicit assumption of PBCs. While the traditional approach centered in the reciprocal or [Formula: see text]-space is elegant for simple one-dimensional lattice structures, the method becomes mathematically cumbersome for more complex lattices with real-world applications, such as a honeycomb lattice involving a [Formula: see text] secular determinant. In this work, we have explored the solution of lattice dynamics in direct space numerically, by constructing unit cells through an explicit implementation of Born–von Karman PBCs and solving the lattice dynamical equations in the displacement–time domain using the fourth-order Runge–Kutta algorithm. The Fast Fourier Transform technique is then used to obtain the phonon frequency spectrum corresponding to the computed instantaneous displacements. The fidelity of the model is validated by the agreement of the computational results obtained for the phonon spectrum at high symmetry points in the Brillouin zone with the analytical values for the given problem. Initially, the dynamics and the phonon spectrum are explored for monatomic and diatomic linear lattices which serve to act as introductory cases for the approach before investigating the 2D monatomic square and honeycomb lattices using the nearest and next-nearest neighbor approximations. A short-range force constant model employing the central forces and angular forces is used for the monatomic square lattice to account for the interatomic interactions. Our computational approach is expected to be physically more intuitive for a student for visualizing the periodicity of a given lattice and understanding its related dynamics. The approach demonstrates the usage of Born–von Karman PBCs in constructing a unit cell that effectively captures the dynamics of a lattice system, complementing the traditional secular determinant formalism. The work also illustrates an instance of how the computer programming can be used as a tool to empower the students to explore abstract concepts in physics. A honing of programming and computational skills of students enables them to interact with dynamic models in physics and gain deeper insight into the real-world applications of physics principles. Target group of this work is the students and educators at undergraduate level who seek a thorough and didactic comprehension of lattice dynamics.

Exact counterdiabatic driving in finite topological lattice models

Adiabatic protocols are often employed in state preparation schemes but require the system to be driven by a slowly varying Hamiltonian so that transitions between instantaneous eigenstates are exponentially suppressed. Counterdiabatic driving is a technique to speed up adiabatic protocols by including additional terms calculated from the instantaneous eigenstates that counter diabatic excitations. However, this approach requires knowledge of the full eigenspectrum meaning that the exact analytical form of counterdiabatic driving is only known for a subset of problems, e.g., the harmonic oscillator and transverse field Ising model. We extend this subset of problems to include the general family of one-dimensional noninteracting lattice models with open boundary conditions and arbitrary onsite potential, tunneling terms, and lattice size. We will derive a general analytical form for the counterdiabatic term for all states of lattice models, including bound and in-gap states which appear, e.g., in topological insulators. We also derive the general analytical form of targeted counterdiabatic driving terms which are tailored to enforce the dynamical state to remain in a specific state. As an example of the use of the derived analytical forms, we consider state transfer using the topological edge states of the Su-Schrieffer-Heeger model. The derived analytical counterdiabatic driving Hamiltonian can be utilized to inform control protocols in many-body lattice models or to probe the nonequilibrium properties of lattice models. Published by the American Physical Society 2024

Homogenization of non-rigid origami metamaterials as Kirchhoff–Love plates

Origami metamaterials have gained considerable attention for their ability to control mechanical properties through folding. Consequently, there is a need to develop systematic methods for determining their effective elastic properties. This study presents an energy-based homogenization framework for non-rigid origami metamaterials, effectively linking their mechanical treatment with that of traditional materials. To account for the unique mechanics of origami systems, our framework incorporates out-of-plane curvature fields alongside the usual in-plane strain fields used for homogenizing planar lattice structures. This approach leads to a couple-stress continuum, resembling a Kirchhoff–Love plate model, to represent the homogeneous response of these lattices. We use the bar-and-hinge method to assess lattice stiffness, and validate our framework through analytical results and numerical simulations of finite lattices. Initially, we apply the framework to homogenize the well-known Miura-ori pattern. The results demonstrate the framework’s ability to capture the unconventional relationship between stretching and bending Poisson’s ratios in origami metamaterials. Subsequently, we extend the framework to origami lattices lacking centrosymmetry, revealing two distinct neutral surfaces corresponding to bending along two lattice directions, unlike in the Miura-ori pattern. Our framework enables the inverse design of metamaterials that can mimic the unique mechanics of origami tessellations using techniques like topology optimization.

Analysis of Microbubble-Blood cell system Oscillation/Cavitation influenced by ultrasound Forces: Conjugate applications of FEM and LBM

Sonoporation is a non-invasive method that uses ultrasound for drug and gene delivery for therapeutic purposes. Here, both Finite Element Method (FEM) and Lattice Boltzmann Method (LBM) are applied to study the interaction physics of microbubble oscillation and collapse near flexible tissue. After validating the Finite Element Method with the nonlinear excited lipid-coated microbubble as well as the Lattice Boltzmann Method with experimental results, we have studied the behavior of a three-dimensional compressible microbubble in the vicinity of tissue. In the FEM phase, the oscillation microbubble with a lipid shell interacts with the boundary. The range of pressure and ultrasound frequency have been considered in the field of therapeutic applications of sonoporation. The viscoelastic and interfacial tension as the coating properties of the microbubble shell have been investigated. The presence of an elastic boundary increases the resonance frequency of the microbubble compared to that of a free microbubble. The increase in pressure leads to an expansion in the range of the microbubble’s motion, the velocity induced in the fluid, and the shear stress on the boundary walls of tissue. An enhancement in the surface tension of the microbubble can influence fluid flow and reduce the shear stress on the boundary. The multi-pseudo-potential interaction LBM is used to reduce thermodynamic inconsistency and high-density ratio in a two-phase system for modeling the cavitation process. The three-dimensional shape of the microbubble during the collapse stages and the counter of pressure are displayed. There is a time difference between the occurrence of maximum velocity and pressure. All results in detail are presented in the article bodies.