Trigonometric Limit Research Articles

Analytical studies of two-component Bose–Einstein condensates in two-dimensional optical lattices

The two-component Bose–Einstein condensates (BECs) trapped in 2D optical lattice potential is studied analytically. A new family of stationary exact solutions of the coupled Gross–Pitaevskii (GP) equations with 2D periodic potential are obtained. In particular, the phase diagram of the system in the trigonometric limit is determined analytically according to the nontrivial phase macroscopic wave functions of the condensates.

Gravity duals for the Coulomb branch of marginally deformed Script N = 4 Yang-Mills

Supergravity backgrounds dual to a class of exactly marginal deformations of N supersymmetric Yang-Mills can be constructed through an SL(2,R) sequence of T-dualities and coordinate shifts. We apply this transformation to multicenter solutions and derive supergravity backgrounds describing the Coulomb branch of N=1 theories at strong 't Hooft coupling as marginal deformations of N=4 Yang-Mills. For concreteness we concentrate to cases with an SO(4)xSO(2) symmetry preserved by continuous distributions of D3-branes on a disc and on a three-dimensional spherical shell. We compute the expectation value of the Wilson loop operator and confirm the Coulombic behaviour of the heavy quark-antiquark potential in the conformal case. When the vev is turned on we find situations where a complete screening of the potential arises, as well as a confining regime where a linear or a logarithmic potential prevails depending on the ratio of the quark-antiquark separation to the typical vev scale. The spectra of massless excitations on these backgrounds are analyzed by turning the associated differential equations into Schrodinger problems. We find explicit solutions taking into account the entire tower of states related to the reduction of type-IIB supergravity to five dimensions, and hence we go beyond the s-wave approximation that has been considered before for the undeformed case. Arbitrary values of the deformation parameter give rise to the Heun differential equation and the related Inozemtsev integrable system, via a non-standard trigonometric limit as we explicitly demonstrate.

Central elements of the elliptic [formula omitted] monodromy matrix algebra at roots of unity

The central elements of the algebra of monodromy matrices associated with the Z n R-matrix are studied. When the crossing parameter w takes a special rational value w = n N , where N and n are positive coprime integers, the center is substantially larger than that in the generic case for which the “quantum determinant” provides the center. In the trigonometric limit, the situation corresponds to the quantum group at roots of unity. This is a higher rank generalization of the recent results by Belavin and Jimbo.

A remark on solutions of reflection equation for the critical ZN-symmetric vertex model

We consider solutions to the reflection equation for the critical Z N -symmetric vertex model, which is the trigonometric limit of the elliptic Z N -symmetric R-matrix of Belavin. These critical R-matrices have two parameters u and η. The transfer matrices T(u, η) constructed from this R-matrix R(u, η) under the cyclic boundary condition are commutative among different u when η is in common, [T(u, η), T(v, η)] = 0. We prove that an arbitrary solution to the reflection equation is independent of the parameter η.

THE 8V CSOS MODEL AND THE sl2 LOOP ALGEBRA SYMMETRY OF THE SIX-VERTEX MODEL AT ROOTS OF UNITY

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group Eτ,η(sl2) at the discrete coupling constants: 2N η = m1 + im2τ, where N, m1 and m2 are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector SZ ≡ 0 ( mod N) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector SZ ≡ 0 ( mod N) of the six-vertex model at Nth roots of unity is given by [Formula: see text], where [Formula: see text] is the maximal value of the total spin operator SZ in the degenerate eigenspace.

An elliptic analogue of the multiple gamma function

A hierarchy of functions including the elliptic gamma function is introduced. It can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q-analogue of the multiple gamma function. Some properties of the functions are considered.

Integrable Many-Body Systems via the Inosemtsev Limit

The Inozemtsev limit (IL), or the scaling limit, is known as a procedure applied to the elliptic Calogero-Moser model. It is a combination of the trigonometric limit, infinite shifts of particle coordinates, and coupling-constant rescalings. This results in an interaction of the exponential type. We show that the IL applied to the sl(N,ℂ) elliptic Euler-Calogero-Moser model and to the elliptic Gaudin model produces new Toda-like systems of N interacting particles endowed with additional degrees of freedom corresponding to a coadjoint orbit of sl(n,ℂ). The limits corresponding to the complete degeneration of the orbital degrees of freedom lead to recovering only the known periodic and nonperiodic Toda systems. We classify the systems appearing in the IL in the sl(3,ℂ) case. This classification is represented on a two-dimensional plane of parameters describing infinite shifts of particle coordinates. This space is subdivided into symmetric domains. In this picture, a mixture of the Toda and trigonometric Calogero-Sutherland potentials emerges on lower-dimensional domain walls. Because of obvious symmetries, this classification can be generalized to an arbitrary number of particles. We also apply the IL to the sl(2,ℂ) elliptic Gaudin model on a two-punctured elliptic curve and discuss the main properties of its possible limits. The limits of Lax matrices are also considered.

On the eigenstates of the elliptic Calogero-Moser model

It is known that the trigonometric Calogero-Sutherland model is obtained by the trigonometric limit (\tau \to \sqrt{-1} \infty) of the elliptic Calogero-Moser model, where (1,\tau) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero-Sutherland model, if \exp (2\pi \sqrt{-1} \tau ) is small enough then there exist square-integrable eigenstates and eigenvalues of the Hamiltonian of the elliptic Calogero-Moser model which converge to the ones of the Calogero-Sutherland model for the 2-particle and the coupling constant l is positive integer cases and the 3-particle and l=1 case. In other words, we justify the regular perturbation with respect to the parameter \exp (2\pi \sqrt{-1} \tau). With some assumptions, we show analogous results for N-particle and l is positive integer cases.

INTEGRABLE XYZ HAMILTONIAN WITH NEXT-NEAREST-NEIGHBOR INTERACTION

In this paper, we construct a Hamiltonian of the impurity model with next-nearest-neighbor interaction within the framework of the open boundary Heisenberg XYZ spin chain. This impurity model is an exactly solved one and it degenerates to the integrable XXZ impurity model under the triangular limit. It is the first approach to add the impurities and next-nearest-neighbor interaction to the integrable completely anisotropic Heisenberg spin chain. We find also that the impurity parameters in the bulk are real when the cross parameter is imaginary for the Hermitian Hamiltonian, or vice versa, when the next-nearest-neighbor interaction is introduced. The eigenvalue of the Hamiltonian and the Bethe ansatz equations for the trigonometric limit case are derived also.

On explicit parametrisation of spectral curves for Moser-Calogero particles and its applications

The system of $N$ classical particles on the line with the Weierstrass $\wp$ function as potential is known to be completely integrable. Recently D'Hoker and Phong found a beautiful parameterization by the polynomial of degree $N$ of the space of Riemann surfaces associated with this system. In the trigonometric limit of the elliptic potential these Riemann surfaces degenerate into rational curves. The D'Hoker-Phong polynomial in the limit describes the intersection points of the rational curves. We found an explicit determinant representation of the polynomial in the trigonometric case. We consider applications of this result to the theory of Toeplitz determinants and to geometry of the spectral curves. We also prove our earlier conjecture on the asymptotic behavior of the ratio of two symplectic volumes when the number of particles tends to infinity.

Diagonalization of the Elliptic Ruijsenaars Model of Type-BC

We study the elliptic Ruijsenaars model of type-BC, which is a difference analogue of the BC-type Calogero–Sutherland–Moser model with an elliptic pairwise potential. In the trigonometric limit, the difference operator of this model reduces to the Macdonald–Koornwinder operator by a gauge-transformation. We show that our difference operator acts on a finite-dimensional space of functions in a special case of the coupling constant.

Affine R-Matrix and the Generalized Elliptic Ruijsenaars Models

Commutative elliptic difference operators associated with the affine root systems are constructed in terms of affine R-matrices. These operators describe the Ruijsenaars models with elliptic potentials and reduce to the Macdonald operators in the trigonometric limit.

On trigonometric intertwining vectors and non-dynamical R-matrix for the Ruijsenaars model

We elaborate on the trigonometric version of intertwining vectors and factorized L-operators. The starting point is the corresponding elliptic construction with Belavin's R-matrix. The naive trigonometric limit is singular and a careful analysis is needed. It is shown that the construction admits several different trigonometric degenerations. As a by-product, a quantum Lax operator for the trigonometric Ruijsenaars model intertwined by a non-dynamical R-matrix is obtained. The latter differs from the standard trigonometric R-matrix of A n type. A connection with the dynamical R-matrix approach is discussed.

Supersymmetric sine-Gordon model and the eight-vertex free fermion model with boundary

We compute the boundary scattering amplitudes of the breathers of the supersymmetric sine-Gordon model using the fusion of the soliton-antisoliton pair scattering with the boundary with a known result of the soliton boundary scattering amplitudes. We also solve the boundary Yang-Baxter equation of the eight-vertex free fermion models to find the boundary reflection matrices. The former result is confirmed by the latter since the bulk S-matrices of the breathers can be identified with the trigonometric limit of the Boltzmann weights of the free fermion models. Our dual approach can answer a few questions on the relationships between the free parameters in the boundary potential and those in the scattering amplitudes.

Solutions of the reflection equation for face and vertex models associated with An(1), Bn(1), Cn(1), Dn(1) and An(2)

We present new diagonal solutions of the reflection equation for elliptic solutions of the star-triangle relation. The models considered are related to the affine Lie algebras A n (1), B n (1), C n (1), D n (1) and A n (2). We recover all known diagonal solutions associated with these algebras and find how these solutions are related in the elliptic regime. Furthermore, new solutions of the reflection equation follow for the associated vertex models in the trigonometric limit.

Decorated Star-Triangle Relations for the Free-Fermion Model and a New Solvable Bilayer Vertex Model

The decorated star-triangle relation (DSTR) discussed by Shastry is extended into the general free-fermion model. Using the trigonometric limit of the DSTR, we construct a new solvable bilayer vertex model. The relation between the DSTR and the tetrahedral Zamolodchikov algebra is discussed.

An elliptic quantum algebra for $$\widehat{s1}_2 $$ 2

An elliptic deformation of $\widehat{sl}_2$ is proposed. Our presentation of the algebra is based on the relation $RLL=LLR^*$, where $R$ and $R^*$ are eight-vertex $R$-matrices with the elliptic moduli chosen differently. In the trigonometric limit, this algebra reduces to a quotient of that proposed by Reshetikhin and Semenov-Tian-Shansky. Conjectures concerning highest weight modules and vertex operators are formulated, and the physical interpretation of $R^*$ is discussed.

Integrability of difference Calogero–Moser systems

A general class of n-particle difference Calogero–Moser systems with elliptic potentials is introduced. Besides the step size and two periods, the Hamiltonian depends on nine coupling constants. We prove the quantum integrability of the model for n=2 and present partial results for n≥ 3. In degenerate cases (rational, hyperbolic, or trigonometric limit), the integrability follows for arbitrary particle number from previous work connected with the multivariable q-polynomials of Koornwinder and Macdonald. Liouville integrability of the corresponding classical systems follows as a corollary. Limit transitions lead to various well-known models such as the nonrelativistic Calogero–Moser systems associated with classical root systems and the relativistic Calogero–Moser system.

The Lie algebra of the sl(2, ?)-valued automorphic functions on a torus

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, C) -valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2,C) -valued loop algebra, while the latter one - into the Lie algebra (sl(2)^)'/(centre) .

Exact Yangian symmetry in the classical Euler-Calogero-Moser model

We compute the r-matrix for the elliptic Euler-Calogero-Moser model. In the trigonometric limit we show that the model possesses an exact Yangian symmetry.