In this work, we construct an alternative formulation to the traditional Algebraic Bethe ansatz for quantum integrable models derived from a generalised rational Gaudin algebra realised in terms of a collection of spins 1/2 coupled to a single bosonic mode. The ensemble of resulting models which we call Dicke-Jaynes-Cummings- Gaudin models are particularly relevant for the description of light-matter interaction in the context of quantum optics. Having two distinct ways to write any eigenstate of these models we then combine them in order to write overlaps and form factors of local operators in terms of partition functions with domain wall boundary conditions. We also demonstrate that they can all be written in terms of determinants of matrices whose entries only depend on the eigenvalues of the conserved charges. Since these eigenvalues obey a much simpler set of quadratic Bethe equations, the resulting expressions could then offer important simplifications for the numerical treatment of these models.
We solve the XXZ Gaudin model with generic boundary using the modified algebraic Bethe ansatz. The diagonal and triangular cases have been recovered in this general framework. We show that the model for odd or even lengths has two different behaviors. The corresponding Bethe equations are computed for all the cases. For the chain with even length, inhomogeneous Bethe equations are necessary. The higher spin Gaudin models with generic boundary is also treated.
We study the Gaudin models associated with $\mathfrak{gl}(1|1)$. We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}(1|1)[t]$-modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. Therefore, we conf…
In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield st…
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 $S^x_iS^x_j$, $S^y_iS^y_j$ and $S^z_iS^z_j$ 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 co…
We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where $p$ is a prime number. We define the Bethe ansatz equations and show that if $(t^0_1,\dots,t^0_k)$ is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the Gaudin Hamiltonians. We characterize solutions $(t^0_1,\dots,t^0_k)$ of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials $F_p[x]$. We consider the case when the number of parameters $k$ equals 1. In …
This thesis presents an introduction to the class of Richardson-Gaudin integrable models, with special focus on the Bethe ansatz wave function, and investigates ways of applying the properties of Richardson-Gaudin models both in and out of integrability. A framework is outlined for the numerical and theoretical treatment of these systems, exposing a duality allowing the Bethe equations to be solved numerically. This is extended to the calculation of inner products and correlation functions. Using this framework, the influence of particle exchange on the Bethe ansatz is discussed, after which i…
The generic Heun operator of Lie type is identified as a certain $BC$-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We show also that these Bethe roots are intimately associated to the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of $O(3)$ and the computation of the entanglement …
We describe a reproduction procedure which, given a solution of the $\mathfrak{gl}_{M|N}$ Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family $P$ of other solutions called the population. To a population we associate a rational pseudodifferential operator $R$ and a superspace $W$ of rational functions. We show that if at least one module is typical then the population $P$ is canonically identified with the set of minimal factorizations of $R$ and with the space of full superflags in $W$. We conjecture that the singular eigenvectors (up to rescal…
We show that the Lipkin-Meshkov-Glick $2N$-fermion model is a particular case of one-spin Gaudin-type model in an external magnetic field corresponding to a limiting case of non-skew-symmetric elliptic $r$-matrix and to an external magnetic field directed along one axis. We propose an exactly-solvable generalization of the Lipkin-Meshkov-Glick fermion model based on the Gaudin-type model corresponding to the same $r$-matrix but arbitrary external magnetic field. This model coincides with the quantization of the classical Zhukovsky-Volterra gyrostat. We diagonalize the corresponding quantum Ham…
In this article we study the thermodynamic limit of the form factors of the XXX Heisenberg spin chain using the algebraic Bethe ansatz approach. Our main goal is to express the form factors for the low-lying excited states as determinants of matrices that remain finite dimensional in the thermodynamic limit. We show how to treat all types of the complex roots of the Bethe equations within this framework. In particular we demonstrate that the Gaudin determinant for the higher level Bethe equations arises naturally from the algebraic Bethe ansatz.
We study quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(m|n)$-invariant $R$-matrix. We compute the norm of the Hamiltonian eigenstates. Using the notion of a generalized model we show that the square of the norm obeys a number of properties that uniquely fix it. We also show that a Jacobian of the system of Bethe equations obeys the same properties. In this way we prove a generalized Gaudin hypothesis for the norm of the Hamiltonian eigenstates.
We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$. We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, whose factors are characterized by two highest coefficients. We provide different recursions for these highest coefficients. In addition, we show that when the Bethe vectors are on-shell, their norm takes the form of a Gaudin…
A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for $SL(N)$. We introduce a difference equation version of opers called $q$-opers and prove a $q$-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with regular singularities on the projective line. We show that the quant…
