Quasi-periodic Boundary Conditions Research Articles

A contour integral-based method for nonlinear eigenvalue problems for semi-infinite photonic crystals

In this study, we introduce an efficient method for determining isolated singular points of two-dimensional semi-infinite and bi-infinite photonic crystals, equipped with perfect electric conductor and quasi-periodic mixed boundary conditions. This specific problem can be modeled by a Helmholtz equation and is recast as a generalized eigenvalue problem involving an infinite-dimensional block quasi-Toeplitz matrix. Through an intelligent implementation of cyclic structure-preserving matrix transformations, the contour integral method is elegantly employed to calculate the isolated eigenvalue and to extract a component of the associated eigenvector. Moreover, a propagation formula for electromagnetic fields is derived. This formulation enables rapid computation of field distributions across the expansive semi-infinite and bi-infinite domains, thus highlighting the attributes of edge states. The preliminary MATLAB implementation is available at https://github.com/FAME-GPU/2D_Semi-infinite_PhC.

Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin - [formula omitted] chain

It is known that for the Heisenberg XXZ spin - 12 chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schrödinger equation for the anharmonic oscillator.

An Adaptive and Quasi-periodic HDG Method for Maxwell’s Equations in Heterogeneous Media

With the aim to continue developing a hybridizable discontinuous Galerkin (HDG) method for problems arisen from photovoltaic cells modeling, in this manuscript we consider the time harmonic Maxwell’s equations in an inhomogeneous bounded bi-periodic domain with quasi-periodic conditions on part of the boundary. We propose an HDG scheme where quasi-periodic boundary conditions are imposed on the numerical trace space. Under regularity assumptions and a proper choice of the stabilization parameter, we prove that the approximations of the electric and magnetic fields converge, in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2$$\\end{document}-norm, to the exact solution with order hk+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1}$$\\end{document} and hk+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1/2}$$\\end{document}, resp., where h is the meshsize and k the polynomial degree of the discrete spaces. Although, numerical evidence suggests optimal order of convergence for both variables. An a posteriori error estimator for an energy norm is also proposed. We show that it is reliable and locally efficient under certain conditions. Numerical examples are provided to illustrate the performance of the quasi-periodic HDG method and the adaptive scheme based on the proposed error indicator.

Three types of boundary conditions in a four-fermion interacting model for quarks

In this note, we study the phase transition of an effective four-fermion model describing interactions between quarks, taking into account finite size and chemical potential effects on the system, with periodic, quasi-periodic, and anti-periodic boundary conditions in D - Euclidean dimensions through Jacobi theta functions. Strictly speaking, this is performed by the generalized Matsubara formalism and the Schwinger proper-time representation. We consider the system at temperature T and the spatial coordinates compacted along the direction j with length Lj. The gap equation of the system is solved numerically and we investigate its behavior when the thermodynamic parameters of the model are changed.

A fast method for imposing periodic boundary conditions on arbitrarily-shaped lattices in two dimensions

A new scheme is presented for imposing periodic boundary conditions on unit cells with arbitrary source distributions. We restrict our attention here to the Poisson, modified Helmholtz, Stokes and modified Stokes equations. The approach extends to the oscillatory equations of mathematical physics, including the Helmholtz and Maxwell equations, but we will address these in a companion paper, since the nature of the problem is somewhat different and includes the consideration of quasiperiodic boundary conditions and resonances. Unlike lattice sum-based methods, the scheme is insensitive to the unit cell's aspect ratio and is easily coupled to adaptive fast multipole methods (FMMs). Our analysis relies on classical “plane-wave” representations of the fundamental solution, and yields an explicit low-rank representation of the field due to all image sources beyond the first layer of neighboring unit cells. When the aspect ratio of the unit cell is large, our scheme can be coupled with the nonuniform fast Fourier transform (NUFFT) to accelerate the evaluation of the induced field. Its performance is illustrated with several numerical examples.

Quasi-periodic boundary conditions for hierarchical algorithms used for the calculation of inter-particle electrostatic interactions

In the present work, a simplified way to apply tri-periodic Boundary Conditions (BCs) in hierarchical algorithms has been devised. The application of these BCs is demonstrated using an in-house algorithm that is used for the calculation of electrostatic interactions in a system of charged particles. The proposed quasi-periodic BCs entail a truncation of the infinite periodic domain with a reasonable cut-off error. For a correct representation of the physics, two properties have to be ensured: the convergence of the electrostatic forces and the isotropy of the electric field. The developed algorithm allows for a rather efficient and precise calculation of the former in a tri-periodic computational domain by separating them in short- and long-range parts, which are calculated exactly and approximately, respectively. The approximation error, computational cost and performance of the proposed algorithm are documented and thoroughly analyzed. Then, an application of the method is presented for dry mono-charged (all particles bear equal charge) particle flows where the fundamentals physics of particle-particle electrostatic interactions are investigated via characteristic length and time scales. It is shown that the underlying mechanism of these interactions is the Coulomb collision, a concept that allows to interpret these interactions in a rather intuitive way. In addition, an attempt is made to provide analytical estimations for several statistical quantities via dimensional analysis based on measured simulation data. Finally, the particle-induced electric field is presented and its characteristics are related to particle motion.

On the spectrum of non-self-adjoint Dirac operators with quasi-periodic boundary conditions

In this paper, we consider non-self-adjoint Dirac operators on a finite interval with complex-valued potentials and quasi-periodic boundary conditions. Necessary and sufficient conditions for a set of complex numbers to be the spectrum of the indicated problem are established.

Thermal Casimir effect in a classical liquid in a quasiperiodically identified conical spacetime

In this paper, we study the finite-temperature quantum fluctuation of a classical liquid induced by the topology of an effective conical spacetime, as well as by a quasiperiodic boundary condition. The conical spacetime could be either a disclination or a cosmic string. In this context, we consider a phonon field representing quantum excitations of the liquid density, which obeys an effective Klein-Gordon equation with the sound velocity replaced by the light velocity. We obtain closed analytic expressions for the thermal Hadamard function, and consequently, the renormalized mean square density fluctuation of the liquid along with thermodynamics quantities such as internal energy, free energy, total energy, and entropy densities. We also discuss the limiting cases, including low- and high-temperature regimes, and the situations in which there is only either the conical spacetime or quasiperiodicity.

Casimir wormholes in 2+1 dimensions with applications to the graphene

In this paper we show that wormholes in (2+1) dimensions (3-D) cannot be sourced solely by both Casimir energy density and tension, differently from what happens in a 4-D scenario, in which case it has been shown recently, by the direct computation of the exact shape and redshift functions of a wormhole solution, that this is possible. We show that in a 3-D spacetime the same is not true since the arising of at least an event horizon is inevitable. We do the analysis for massive and massless fermions, as well as for scalar fields, considering quasi-periodic boundary conditions and find that a possibility to circumvent such a restriction is to introduce, besides the 3-D Casimir energy density and tension, a cosmological constant, embedding the surface in a 4-D manifold and applying a perpendicular weak magnetic field. This causes an additional tension on it, which contributes to the formation of the wormhole. Finally, we discuss the possibility of producing the condensed matter analogous of this wormhole in a graphene sheet and analyze the electronic transport through it.

Dirac particles on periodic quantum graphs.

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasiperiodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum graphs is obtained. Band spectra of the periodic quantum graphs of different topologies are calculated. Universality of the probability to be in the spectrum for certain graph topologies is observed.

Unfitted Nitsche’s method for computing band structures of phononic crystals with periodic inclusions

In this paper, we propose an unfitted Nitsche’s method to compute the band structures of phononic crystal with periodic inclusions of general geometry. The proposed method does not require the background mesh to fit the interfaces of periodic inclusions, and thus avoids the expensive cost of generating body-fitted meshes and simplifies the inclusion of interface conditions in the formulation. The quasi-periodic boundary conditions are handled by the Floquet–Bloch transform, which converts the computation of band structures into an eigenvalue problem with periodic boundary conditions. More importantly, we show the well-posedness of the proposed method using a delicate argument based on the trace inequality, and further prove the convergence by the Babuška–Osborn theory. We achieve the optimal convergence rate at the presence of the periodic inclusions of general geometry. We demonstrate the theoretical results by two numerical examples, and show the capability of the proposed methods for computing the band structures without fitting the interfaces of periodic inclusions.

Correlation functions by separation of variables: the XXX spin chain

We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.

Perfect Integrability and Gaudin Models

We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite types, with periodic and regular quasi-periodic boundary conditions.

Separation of variables bases for integrable $gl_{\mathcal{M}|\mathcal{N}}$ and Hubbard models

We construct quantum Separation of Variables (SoV) bases for both the fundamental inhomogeneous % gl_{\mathcal{M}|\mathcal{N}}glℳ|𝒩 supersymmetric integrable models and for the inhomogeneous Hubbard model both defined with quasi-periodic twisted boundary conditions given by twist matrices having simple spectrum. The SoV bases are obtained by using the integrable structure of these quantum models, i.e. the associated commuting transfer matrices, following the general scheme introduced in [1]; namely, they are given by set of states generated by the multiple actions of the transfer matrices on a generic co-vector. The existence of such SoV bases implies that the corresponding transfer matrices have non-degenerate spectrum and that they are diagonalizable with simple spectrum if the twist matrices defining the quasi-periodic boundary conditions have that property. Moreover, in these SoV bases the resolution of the transfer matrix eigenvalue problem leads to the resolution of the full spectral problem, i.e. both eigenvalues and eigenvectors. Indeed, to any eigenvalue is associated the unique (up to a trivial overall normalization) eigenvector whose wave-function in the SoV bases is factorized into products of the corresponding transfer matrix eigenvalue computed on the spectrum of the separated variables. As an application, we characterize completely the transfer matrix spectrum in our SoV framework for the fundamental gl_{1|2}gl1|2 supersymmetric integrable model associated to a special class of twist matrices. From these results we also prove the completeness of the Bethe Ansatz for that case. The complete solution of the spectral problem for fundamental inhomogeneous gl_{\mathcal{M}|\mathcal{N}}glℳ|𝒩 supersymmetric integrable models and for the inhomogeneous Hubbard model under the general twisted boundary conditions will be addressed in a future publication.

Numerical method for the projected Gross-Pitaevskii equation in an infinite rotating two-dimensional Bose gas.

We present a method for evolving the projected Gross-Pitaevskii equation in an infinite rotating Bose-Einstein condensate, the ground state of which is a vortex lattice. We use quasiperiodic boundary conditions to investigate the behavior of the bulk superfluid in this system, in the absence of boundaries and edge effects. We also give the Landau gauge expression for the phase of a BEC subjected to these boundary conditions. Our spectral representation uses the eigenfunctions of the one-body Hamiltonian as basis functions. Since there is no known exact quadrature rule for these basis functions we approximately implement the projection associated with the energy cutoff, but we show that by choosing a suitably fine spatial grid the resulting error can be made negligible. We show how the convergence of this model is affected by simulation parameters such as the size of the spatial grid and the number of Landau levels. Adding dissipation, we use our method to find the lattice ground state for N vortices. We can then perturb the ground-state, to investigate the melting of the lattice.

Stationary solutions to cubic nonlinear Schrödinger equations with quasi-periodic boundary conditions

In this paper we give the quantization rules to determine the normalized stationary solutions to the cubic nonlinear Schrödinger equation with quasi-periodic conditions on a given interval. Similarly to what happen in the Floquet’s theory for linear periodic operators, also in this case some kind of band functions there exist.

Spectral Estimates for the Fourth-Order Operator with Matrix Coefficients

The fourth-order differential operator with matrix coefficients with the domain determined by quasi-periodic boundary conditions is considered. For this operator, the asymptotics of the arithmetic mean of the eigenvalues is found. Moreover, for various special cases, the asymptotics of the eigenvalues is also obtained. The spectral characteristics in the case of periodic and antiperiodic boundary conditions are studied separately. The results are better than those known before.

Inverse spectral problems for differential operators with non-separated boundary conditions

Abstract We give a short review of results on inverse spectral problems for second-order differential operators on an interval with non-separated boundary conditions. We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential operators from given spectral characteristics. In the first part of the review, we provide the main results and methods related to inverse problems for Sturm–Liouville operators with non-separated boundary conditions: periodic, quasi-periodic and Robin-type boundary conditions. At the end, we present the main results on inverse problems for differential pencils with non-separated boundary conditions.

Computational analysis of transport in three-dimensional heterogeneous materials

Porous and heterogeneous materials are found in many applications from composites, membranes, chemical reactors, and other engineered materials to biological matter and natural subsurface structures. In this work we propose an integrated approach to generate, study and upscale transport equations in random and periodic porous structures. The geometry generation is based on random algorithms or ballistic deposition. In particular, a new algorithm is proposed to generate random packings of ellipsoids with random orientation and tunable porosity and connectivity. The porous structure is then meshed using locally refined Cartesian-based or unstructured strategies. Transport equations are thus solved in a finite-volume formulation with quasi-periodic boundary conditions to simplify the upscaling problem by solving simple closure problems consistent with the classical theory of homogenisation for linear advection–diffusion–reaction operators. Existing simulation codes are extended with novel developments and integrated to produce a fully open-source simulation pipeline. A showcase of a few interesting three-dimensional applications of these computational approaches is then presented. Firstly, convergence properties and the transport and dispersion properties of a periodic arrangement of spheres are studied. Then, heat transfer problems are considered in a pipe with layers of deposited particles of different heights, and in heterogeneous anisotropic materials.

On the graphene Hamiltonian operator

We solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by Kuchment and Post (Commun Math Phys 275(3):805–826, 2007) to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies.