Gaudin models solver based on the correspondence between Bethe ansatz and ordinary differential equations

Abstract

We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a new set of variables, the canceling divergences which occur for certain values of the coupling strength no longer appear explicitly. The problem is thus reduced to a set of quadratic algebraic equations. The required inverse transformation can then be realized using only linear operations and a standard polynomial root finding algorithm. The method is applied to Richardson's fermionic pairing model, the central spin model and generalized Dicke model.The correspondence between Bethe Ansatz / Ordinary Differential Equation (ODE) has been found some time ago 1 on the basis of similarity between T-Q system of the Bethe Ansatz and functional relations including spectral determinants on the ODE side. For extensive review we refer to 2 . The correspondence is related to the Langlands correspondence where the ODE part has to do with socalled Miura opers 3 . It has many interesting links with conformal field theory, quasi-exact solvability, supersymmetry and PT-symmetric quantum mechanics. It was also used it in the context of physics of cold atoms and quantum impurity problem 4 . Here we develop further this remarkable correspondence to facilitate the solving of otherwise difficult-to-solve Gaudin systems and implement it to several physically interesting situations.Indeed, numerous integrable models derived from a generalized Gaudin algebra have been used to describe the properties of physical systems. The Richardson fermionic pairing Hamiltonian 5 has explained most properties of superconducting nanograins 6,7 , numerous studies of the decoherence of a single electron spin trapped in a quantum dot rely on the central spin model 8 while the inhomogeneous Dicke models 9 have been used to study light-matter interaction in many cavity-based systems.The quantum integrability of such systems brings major simplifications to the structure of their eigenstates. In a system containing N degrees of freedom, they can be fully described by a number M of complex parameters we call rapidities. M is here a number of the same order as N despite the exponentially large Hilbert space. The possible values of this set of parameters can be obtained by finding solutions to a set of M coupled non-linear algebraic equation: the Bethe equations. However, analytically finding solutions to the Bethe equations remains impossible except in very specific cases. The problem is still numerically approachable but solving non-linear equations remains a challenge that only iterative methods can tackle.While previous efforts in designing algorithms have shown promising results [10][11][12][13][14], methods which are sufficiently fast and stable for systematically finding a large number of eigenstates remained elusive. As was shown by one of the authors, computing efficiently only a small fraction of the complete Hilbert space can be sufficient to access static 15 , dynamical 16 and even non-equilibrium dyna...

Eigenvalue-based method and form-factor determinant representations for integrable XXZ Richardson-Gaudin models

“…circumventing the singular points in the Richardson-Gaudin equations [20]. A set of equations equivalent to the RG equations can be found for these variables; however, these equations are void of singular behavior.…”

“…(22) to the set of equations found for the rational model [20], it is straightforward to extend the solution method for the rational model to our equations. General sets of nonlinear equations have to be solved by an iterative approach starting from an initial guess, such as the Newton-Raphson method.…”

“…As in [20], we start from an approximation of the solution in the weak-coupling limit (small |g|). A solution at any value of the coupling constant can then be found by adiabatically varying g starting from the weak-coupling limit and using the solution at the previous step as the starting point for an iterative solution at the current step.…”

“…As an illustration, the set of RG equations for the doubly degenerate XXX model [Eq. (14)] is isomorphic to the set of quadratic equations [20] …”

“…* pieterw.claeys@ugent.be Our method consists of a generalization of the numerical method first proposed by Faribault et al [20] for a set of nondegenerate rational RG models and later extended to degenerate rational models by El Araby et al [21]. The equations for the Dicke model were independently presented by Babelon and Talalaev [22].…”