arXiv:1509.01451 · Journal of Physics A Mathematical and Theoretical

The elliptic Gaudin model: a numerical study

C Esebbag, J. Dukelsky

Published Oct 28, 2015• 5 citations

Abstract

The elliptic Gaudin model describes completely anisotropic spin systems with long range interactions. The model was proven to be quantum integrable by Gaudin and latter the exact solution was found by means of the algebraic Bethe ansatz. In spite of the appealing properties of the model, it has not yet been applied to any physical problem. We here generalize the exact solution to systems with arbitrary spins, and study numerically the behavior of the Bethe roots for a system with three different spins. Then, we propose an integrable anisotropic central spin model that we study numerically for very large systems.

DOI: 10.1088/1751-8113/48/47/475303

Related papers

‘Bethe-ansatz-free’ eigenstates for spin-1/2 Richardson–Gaudin integrable models

Alexandre Faribault, Claude Dimo

Abstract In this work we construct the eigenstates of the most general spin-1/2 Richardson–Gaudin model integrable in an external magnetic field. This includes the possibility for fully anisotropic XYZ coupling such that the S i x S j x , S i y S j y and S i z S j z terms all have distinct coupling strengths. While insuring that integrability is maintained in the presence of an external field excludes the elliptic XYZ model which is only integrable at zero field, this work still covers a wide class of fully anisotropic (XYZ) models associated with non skew-symmetric r -matrices. The eigenstates, as constructed here, do not require any usable Bethe ansatz and therefore: no proper pseudo-vacuum, Bethe roots, or generalised spin raising (Gaudin) operators have to be defined. Indeed, the eigenstates are generically built only through the conserved charges which define the model of interest and the specification of the set of eigenvalues defining the particular eigenstate. Since these eigenvalues are, in general, solutions to a simple set of quadratic equations, the proposed approach is simpler to implement than any Bethe ansatz and, moreover, it remains completely identical independently of the symmetries of the model. Indeed, the construction removes any distinction between XYZ models and XXZ/XXX models and, generically, that between models with or without U (1) symmetry so that any difficulties associated with the use of a Bethe ansatz in any of these cases are avoided.

arXiv:1812.06428Journal of Physics A Mathematical and TheoreticalSep 16, 20221 citation

Exact solution of the Izergin-Korepin Gaudin model with periodic and open boundaries

Xiaotian Xu, Pei Sun, Xin H. H. Zhang +2 more

We study the Izergin-Korepin Gaudin models with both periodic and open integrable boundary conditions, which describe quantum systems exhibiting novel long-range interactions. Using the Bethe Ansatz approach, we derive the eigenvalues of the Gaudin operators and the corresponding Bethe Ansatz equations.

arXiv:2412.04761SciPost Physics CoreMay 21, 20250 citations

The dynamical elliptic quantum Gaudin models and their solutions

Bo-Yu Hou, Kangjie Shi, Rui‐Hong Yue +1 more

In this paper, we construct the Hamiltonians of both periodic and open elliptic quantum Gaudin models and show their relations with the elliptic quantum group, and the boundary elliptic quantum group, respectively. We define the eigenstates of these two models to be the Bethe vectors with η=0 of the elliptic quantum group and the boundary elliptic quantum group, respectively. Then, the Hamiltonians are exactly diagonalized by using the algebraic Bethe ansatz method.

Journal of Mathematical PhysicsSep 26, 20020 citations

The Open Boundary Dynamical Elliptic Quantum Gaudin Model and Its Solution

Shao-you Zhao A, Rui‐Hong Yue

We construct the Hamiltonians of open elliptic quantum Gaudin model and show its relation with the open boundary elliptic quantum group. We define eigenstates of the model to be Bethe vectors with of the boundary elliptic quantum group. Then, the Hamiltonian is exactly diagonalized by using the algebraic Bethe ansatz method.

Communications in Theoretical PhysicsSep 15, 20010 citations

Integrable spin- 1 2 Richardson–Gaudin XYZ models in an arbitrary magnetic field

Pieter W. Claeys, Claude Dimo, Stijn De Baerdemacker +1 more

Abstract We establish the most general class of spin- integrable Richardson–Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin- relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Richardson–Gaudin models. The full set of conserved charges are constructed explicitly and shown to satisfy a set of quadratic equations, allowing for the numerical treatment of a fully anisotropic central spin in an external magnetic field. While this approach does not provide expressions for the exact eigenstates, it allows their eigenvalues to be obtained, and expectation values of local observables can then be calculated from the Hellmann–Feynman theorem.

arXiv:1810.06059Journal of Physics A Mathematical and TheoreticalJan 15, 201929 citations

Separation of Variables in the Elliptic Gaudin Model

Evgeny Konstantinovich Sklyanin, St. Petersburg Branch, Takashi Takebe

. For the elliptic Gaudin model (a degenerate case of XYZ integrable spin chain) a separation of variables is constructed in the classical case. The corresponding separated coordinates are obtained as the poles of a suitably normalized Baker-Akhiezer function. The classical results are generalized to the quantum case where the kernel of separating integral operator is constructed. The simplest onedegree -of-freedom case is studied in detail. 1 Introduction The quantum elliptic (or XYZ) Gaudin model was introduced in [1], see also [2], as a limiting case of the integrable XYZ spin chain [3]. The commuting Hamiltonians H n of the model are expressed as quadratic combinations of sl 2 spin operators. Determining the spectrum of H n turned out to be a difficult problem like the original XYZ spin chain. Let us list the known facts related to this problem. ffl A solution by means of Algebraic Bethe Ansatz has been obtained only recently [4]. See also [5] ffl As shown in [6], in the SU(2...

Jan 1, 199821 citations