Antiperiodic Boundary Conditions Research Articles

A representation basis for the quantum integrable spin chain associated with the su(3) algebra

An orthogonal basis of the Hilbert space for the quantum spin chain associated with the su(3) algebra is introduced. Such kind of basis could be treated as a nested generalization of separation of variables (SoV) basis for high-rank quantum integrable models. It is found that all the monodromy-matrix elements acting on a basis vector take simple forms. With the help of the basis, we construct eigenstates of the su(3) inhomogeneous spin torus (the trigonometric su(3) spin chain with antiperiodic boundary condition) from its spectrum obtained via the off-diagonal Bethe Ansatz (ODBA). Based on small sites (i.e. N=2) check, it is conjectured that the homogeneous limit of the eigenstates exists, which gives rise to the corresponding eigenstates of the homogenous model.

On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators

The paper is concerned with the Riesz basis property of a boundary value problem associated in L2[0,1]⊗C2 with the following 2×2 Dirac type equation(0.1)Ly=−iB−1y′+Q(x)y=λy,B=(b100b2),y=col(y1,y2), with a summable potential matrix Q∈L1[0,1]⊗C2×2 and b1<0<b2. If b2=−b1=1 this equation is equivalent to one dimensional Dirac equation. It is proved that the system of root functions of a linear boundary value problem constitutes a Riesz basis in L2[0,1]⊗C2 provided that the boundary conditions are strictly regular. By analogy with the case of ordinary differential equations, boundary conditions are called strictly regular if the eigenvalues of the corresponding unperturbed (Q=0) operator are asymptotically simple and separated. In opposite to the Dirac case there is no simple algebraic criterion of the strict regularity whenever b1+b2≠0. However under certain restrictions on coefficients of the boundary linear forms we present certain algebraic criteria of the strict regularity in the latter case. In particular, it is shown that regular separated boundary conditions are always strictly regular while antiperiodic boundary conditions are strictly regular if b1, b2 are coprime integers of different parity. The proof of the main result is based on existence of triangular transformation operators for system (0.1). Their existence is also established here in the case of a summable Q. In the case of regular (but not strictly regular) boundary conditions we prove the Riesz basis property with parentheses. The main results are applied to establish the Riesz basis property of the dynamic generator of spatially non-homogeneous damped Timoshenko beam model.

Deconfinement on ℝ 2 × S L 1 × S β 1 $$ {\mathrm{\mathbb{R}}}^2\times {S}_L^1\times {S}_{\beta}^1 $$ for all gauge groups and duality to double Coulomb gas

I study finite-temperature $\mathcal N=1$ super Yang-Mills for any gauge group $G=A_N, B_N, C_N, D_N, E_{6,7,8},F_4,G_2$, compactified from four dimensions on a torus, $\mathbb R^2\times S^1_L\times S^1_{\beta}$. I examine in particular the low temperature regime $L\ll\beta=1/T$, where $L$ is the length of the spatial circle with periodic boundary conditions and with anti-periodic boundary conditions for the adjoint gauginos along the thermal cycle $S^1_{\beta}$. For small such $L$ we are in a regime were semiclassical calculations can be performed and a transition occurs at $T_c$ much smaller than $1/NL$. The transition is mediated by the competition between non-perturbative objects including 'exotic' topological molecules: neutral and magnetic bions composed of BPS and KK monopole constituents, with $r=rank(G)$ different charges in the co-root lattice of the gauge group $G$, and the perturbative electrically charged W-bosons (along with their wino superpartners). I determine a duality to a double Coulomb gas of neutral and magnetic bions of different charges of their constituent monopole-instantons, and W-bosons of both scalar and electric charges. Aharanov-Bohm interactions exist between magnetic bions and W-bosons, and scalar charges of W-bosons and neutral bions attract like charges, as opposed to the magnetic and electric charges where like charges repel. It is hoped in the future that lattice studies of this Coulomb gas can be done as in [1] for all gauge groups. It is hoped that a dual lattice 'affine' XY model with symmetry breaking perturbations can also be found in future studies of general gauge group as done in [1] for $SU(2)$.

Periodic Boundary Conditions in the Natural Element Method

The purpose of this paper is to introduce periodic and antiperiodic boundary conditions in the natural element method (NEM). It is shown that as the NEM shape functions verify the Kronecker delta property, the imposition of such boundary conditions can be done in an easy way as in the finite-element method (FEM). These boundary conditions are important because they allow exploitation of the inherent symmetry of the electromagnetic devices. Therefore, with these techniques the NEM can now easily take advantage of the symmetry of electrical machines. The proposed approach is evaluated, and its results are compared with those obtained by the traditional FEM.

Existence of Integral and Anti-periodic Boundary Valued Problem of Fractional Order $$0&lt;\alpha \le 3$$ 0 &lt; α ≤ 3

We are concerned with the existence of solutions of a class of fractional differential equations with anti-periodic and integral boundary conditions involving the Caputo fractional derivative with order $$\alpha \in (0,3]$$ . We give three results based on Banach fixed-point theorem, and Schauder fixed-point theorems.

On determinant representations of scalar products and form factors in the SoV approach: the XXX case

In the present article we study the form factors of quantum integrable lattice models solvable by the separation of variables (SoVs) method. It was recently shown that these models admit universal determinant representations for the scalar products of the so-called separate states (a class which includes in particular all the eigenstates of the transfer matrix). These results permit to obtain simple expressions for the matrix elements of local operators (form factors). However, these representations have been obtained up to now only for the completely inhomogeneous versions of the lattice models considered. In this article we give a simple algebraic procedure to rewrite the scalar products (and hence the form factors) for the SoV related models as Izergin or Slavnov type determinants. This new form leads to simple expressions for the form factors in the homogeneous and thermodynamic limits. To make the presentation of our method clear, we have chosen to explain it first for the simple case of the XXX Heisenberg chain with anti-periodic boundary conditions. We would nevertheless like to stress that the approach presented in this article applies as well to a wide range of models solved in the SoV framework.

Existence results for impulsive fractional q-difference equations with anti-periodic boundary conditions

This paper studies a Caputo type anti-periodic boundary value problem of impulsive fractional q-difference equations involving a q-shifting operator of the form ${}_{a}\Phi_{q}(m) = qm + (1-q)a$ . Concerning the existence of solutions for the given problem, two theorems are proved via Schauder’s fixed point theorem and the Leray-Schauder nonlinear alternative, while the uniqueness of solutions is established by means of Banach’s contraction mapping principle. Finally, we discuss some examples illustrating the main results.

The existence of solution for a k-dimensional system of fractional differential inclusions with anti-periodic boundary value conditions

We investigate the existence of solutions for a k-dimensional systems of fractional differential inclusions with anti-periodic boundary conditions. We provide two results via different conditions for obtaining solutions of the k-dimensional inclusion problem. We provide some examples to illustrate our results.

Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions

Abstract We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated) and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration but also yield some new special cases for specific choices of parameters involved in the problems.

The eight-vertex model with quasi-periodic boundary conditions

We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T–Q functional equations, where the Q-functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.

FRACTAL DIMENSION OF SPIN-GLASSES INTERFACES IN DIMENSION d = 2 AND d = 3 VIA STRONG DISORDER RENORMALIZATION AT ZERO TEMPERATURE

For Gaussian Spin-Glasses in low dimensions, we introduce a simple Strong Disorder renormalization at zero temperature in order to construct ground states for Periodic and Anti-Periodic boundary conditions. The numerical study in dimensions [Formula: see text] (up to sizes [Formula: see text]) and [Formula: see text] (up to sizes [Formula: see text]) yields that Domain Walls are fractal of dimensions [Formula: see text] and [Formula: see text], respectively.

Asymptotically flat structure of hypergravity in three spacetime dimensions

The asymptotic structure of three-dimensional hypergravity without cosmological constant is analyzed. In the case of gravity minimally coupled to a spin-$5/2$ field, a consistent set of boundary conditions is proposed, being wide enough so as to accommodate a generic choice of chemical potentials associated to the global charges. The algebra of the canonical generators of the asymptotic symmetries is given by a hypersymmetric nonlinear extension of BMS$_{3}$. It is shown that the asymptotic symmetry algebra can be recovered from a subset of a suitable limit of the direct sum of the W$_{\left(2,4\right)}$ algebra with its hypersymmetric extension. The presence of hypersymmetry generators allows to construct bounds for the energy, which turn out to be nonlinear and saturate for spacetimes that admit globally-defined "Killing vector-spinors". The null orbifold or Minkowski spacetime can then be seen as the corresponding ground state in the case of fermions that fulfill periodic or anti-periodic boundary conditions, respectively. The hypergravity theory is also explicitly extended so as to admit parity-odd terms in the action. It is then shown that the asymptotic symmetry algebra includes an additional central charge, being proportional to the coupling of the Lorentz-Chern-Simons form. The generalization of these results in the case of gravity minimally coupled to arbitrary half-integer spin fields is also carried out. The hypersymmetry bounds are found to be given by a suitable polynomial of degree $s+\frac{1}{2}$ in the energy, where $s$ is the spin of the fermionic generators.

Existence Uniqueness Theorems for Multi-Term Fractional Delay Differential Equations

Abstract In this paper we analyze non-linear multi-term fractional delay differential Equation denotes the Caputo fractional derivative of order α. The Schaefer fixed point theorem and Banach contraction principle are used to investigate the existence and uniqueness of solutions for above equation with periodic/ anti-periodic boundary conditions.

Chaos in spin glasses revealed through thermal boundary conditions

We study the fragility of spin glasses to small temperature perturbations numerically using population annealing Monte Carlo. We apply thermal boundary conditions to a three-dimensional Edwards-Anderson Ising spin glass. In thermal boundary conditions all eight combinations of periodic versus antiperiodic boundary conditions in the three spatial directions are present, each appearing in the ensemble with its respective statistical weight determined by its free energy. We show that temperature chaos is revealed in the statistics of crossings in the free energy for different boundary conditions. By studying the energy difference between boundary conditions at free-energy crossings, we determine the domain-wall fractal dimension. Similarly, by studying the number of crossings, we determine the chaos exponent. Our results also show that computational hardness in spin glasses and the presence of chaos are closely related.

Local dynamics of an equation with a large state-dependent delay

The behavior of solutions to first- and second-order differential equations with a large state-dependent delay is analyzed in a neighborhood of equilibrium. In critical cases, it is shown that their behavior is determined by the global dynamics of a nonlinear parabolic equation with periodic or antiperiodic boundary conditions, which is known as a quasi-normal form. Asymptotic formulas are presented that relate the solutions of the quasi-normal form and the original equation.

Exact solution of Heisenberg model with site-dependent exchange couplings and Dzyloshinsky–Moriya interaction**Project supported by the National Natural Science Foundation of China (Grant Nos. 11174335, 11375141, 11374334, and 11434013) and the National Program for Basic Research of China and the Fund from the Chinese Academy of Sciences.

We propose an integrable spin-1/2 Heisenberg model where the exchange couplings and Dzyloshinky–Moriya interactions are dependent on the sites. By employing the quantum inverse scattering method, we obtain the eigenvalues and the Bethe ansatz equation of the system with the periodic boundary condition. Furthermore, we obtain the exact solution and study the boundary effect of the system with the anti-periodic boundary condition via the off-diagonal Bethe ansatz. The operator identities of the transfer matrix at the inhomogeneous points are proved at the operator level. We construct the T–Q relation based on them. From which, we obtain the energy spectrum of the system. The corresponding eigenstates are also constructed. We find an interesting coherence state that is induced by the topological boundary.

AN INVERSE NODAL PROBLEM AND AMBARZUMYAN PROBLEM FOR THE PERIODIC $p$-LAPLACIAN OPERATOR WITH INTEGRABLE POTENTIALS

In this note, we solve the inverse nodal problem and Ambarzumyanproblem for the $p$-Laplacian coupled with periodic or anti-periodicboundary conditions. We also extend some results in a previous paperto $p$-Laplacian with $L^1$ potentials, and for arbitrary linearseparated boundary conditions. There we prove a generalizedRiemann-Lebesgue Lemma which is of independent interest.

Regularized trace of the Dirac operator

Formulas for the regularized trace of the one-dimensional non—self-adjoint Dirac operator with L2-potential are obtained. The cases of periodic and antiperiodic boundary conditions as well as of the Dirichlet boundary conditions are considered. The formulas are obtained by using the method of similar operators on the basis of results from the papers [1] and [2].

The entropy of the six-vertex model with a variety of different boundary conditions

We study the dependence of entropy [per lattice site] of six-vertex model on boundary conditions. We start with lattices of finite size and then proceed to thermodynamic limit. We argue that the six-vertex model with periodic, anti-periodic and mixed boundary conditions produce the same free-energy in the thermodynamic limit. We have found fixed boundary conditions such that the entropy varies continuously from zero to its value for periodic boundary condition. We have also shown that the physical quantities of the six-vertex model at the isotropic point does not change in the case of singular toroidal boundary.

Fractional Boundary Value Problems with Integral and Anti-periodic Boundary Conditions

In this paper, we consider a class of boundary value problems of fractional differential equations with integral and anti-periodic boundary conditions, which is a new type of mixed boundary condition. Using the contraction mapping principle, Krasnosel’skii fixed point theorem, and Leray-Schauder degree theory, we obtain some results of existence and uniqueness. Finally, several examples are provided for illustrating the applications of our theoretical analysis.