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arXiv:1706.05511 · SciPost Physics

Inner products in integrable Richardson-Gaudin models

Pieter W. Claeys, Dimitri Van Neck, Stijn De Baerdemacker

Published Oct 24, 2017• 30 citations
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Abstract

We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula. Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.

DOI
10.21468/scipostphys.3.4.028

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We present the inner products of eigenstates in integrable Richardson-Gaudin models from two different perspectives and derive two classes of Gaudin-like determinant expressions for such inner products. The requirement that one of the states is on-shell arises naturally by demanding that a state has a dual representation. By implicitly combining these different representations, inner products can be recast as domain wall boundary partition functions. The structure of all involved matrices in terms of Cauchy matrices is made explicit and used to show how one of the classes returns the Slavnov determinant formula. Furthermore, this framework provides a further connection between two different approaches for integrable models, one in which everything is expressed in terms of rapidities satisfying Bethe equations, and one in which everything is expressed in terms of the eigenvalues of conserved charges, satisfying quadratic equations.
We propose alternative determinant representations of certain form factors and scalar products of states in rational Gaudin models realized in terms of compact spins. We use alternative pseudo-vacuums to write overlaps in terms of partition functions with domain wall boundary conditions. Contrarily to Slavnov's determinant formulas, this construction does not require that any of the involved states be solutions to the Bethe equations; a fact that could prove useful in certain non-equilibrium problems. Moreover, by using an atypical determinant representation of the partition functions, we propose expressions for the local spin raising and lowering operators form factors which only depend on the eigenvalues of the conserved charges. These eigenvalues define eigenstates via solutions of a system of quadratic equations instead of the usual Bethe equations. Consequently, the current work allows important simplifications to numerical procedures addressing decoherence in Gaudin models.
Abstract In this work we present a determinant expression for the domain-wall boundary condition partition function of rational (XXX) Richardson–Gaudin models which, in addition to spins , contains one arbitrarily large spin S . The proposed determinant representation is written in terms of a set of variables which, from previous work, are known to define eigenstates of the quantum integrable models belonging to this class as solutions to quadratic Bethe equations. Such a determinant can be useful numerically since systems of quadratic equations are much simpler to solve than the usual highly nonlinear Bethe equations. It can therefore offer significant gains in stability and computation speed.
In this work, we construct an alternative formulation to the traditional algebraic Bethe ansätze for quantum integrable models derived from a generalized rational Gaudin algebra realized 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.
Starting from integrable su (2) (quasi-)spin Richardson–Gaudin (RG) XXZ models we derive several properties of integrable spin models coupled to a bosonic mode. We focus on the Dicke–Jaynes–Cummings–Gaudin models and the two-channel ( p + i p )-wave pairing Hamiltonian. The pseudo-deformation of the underlying su (2) algebra is here introduced as a way to obtain these models in the contraction limit of different RG models. This allows for the construction of the full set of conserved charges, the Bethe ansatz state, and the resulting RG equations. For these models an alternative and simpler set of quadratic equations can be found in terms of the eigenvalues of the conserved charges. Furthermore, the recently proposed eigenvalue-based determinant expressions for the overlaps and form factors of local operators are extended to these models, linking the results previously presented for the Dicke–Jaynes–Cummings–Gaudin models with the general results for RG XXZ models.
In this work, we construct an alternative formulation to the traditional Algebraic Bethe ansatz for rational Gaudin models realised in terms of a collection of spins 1/2 coupled to a single bosonic mode. In doing so, we obtain two distinct ways to write any eigenstate of these models which can then be combined 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 therefore 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.
Abstract In this work we demonstrate how one can, in a generic approach, derive a set of N simple quadratic Bethe equations for integrable Richardson–Gaudin (RG) models built out of N spins-1/2. These equations depend only on the N eigenvalues of the various conserved charges so that any solution of these equations defines, indirectly through the corresponding set of eigenvalues, one particular eigenstate. The proposed construction covers the full class of integrable RG models of the XYZ (including the subclasses of XXZ and XXX models) type realised in terms of spins-1/2, coupled with one another through , , terms, including, as well, magnetic field-like terms linear in the Pauli matrices. The approach exclusively requires integrability, defined here only by the requirement that N conserved charges R i (with ) such that exist. The result is therefore valid, and equally simple, for models with or without symmetry, with or without a properly defined pseudo-vacuum as well as for models with non-skew symmetric couplings.