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arXiv:1704.01873 · SciPost Physics

Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

Alexandre Faribault, Hugo Tschirhart

Published Aug 4, 2017• 13 citations
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Abstract

In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure differs vastly from the most natural approach which would be to simply orient the spin quantisation axis in the same direction as the magnetic field through an appropriate rotation. Instead, we define a modified realisation of the rational Gaudin algebra and use the quantum inverse scattering method which allows us, within a slightly modified implementation, to build an algebraic Bethe ansatz using the same unrotated reference state (pseudovacuum) for any external field. This common framework allows us to easily write determinant expressions for certain scalar products which would be highly non-trivial in the rotated system approach.

DOI
10.21468/scipostphys.3.2.009

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In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure differs vastly from the most natural approach which would be to simply orient the spin quantisation axis in the same direction as the magnetic field through an appropriate rotation. Instead, we define a modified realisation of the rational Gaudin algebra and use the quantum inverse scattering method which allows us, within a slightly modified implementation, to build an algebraic Bethe ansatz using the same unrotated reference state (pseudovacuum) for any external field. This common framework allows us to easily write determinant expressions for certain scalar products which would be highly non-trivial in the rotated system approach.
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 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.
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.
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 Hamiltonian by means of the modified algebraic Bethe ansatz. We explicitly solve the corresponding Bethe-type equations for the case of small fermion number N=1,2.
We present a numerical approach which allows the solving of Bethe equations whose solutions define the eigenstates of Gaudin models. By focusing on a different 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 the generalized Dicke model.
We consider the elliptic Gaudin-type model in an external magnetic field [11], [12], [13], [16], [15], [22] associated with non-skew-symmetric elliptic r-matrix [11] defined on 4:1 unramified covering of the Weierstrass cubic y2=(u+j1)(u+j2)(u+j3). We develop a modified algebraic Bethe ansatz for the considered elliptic r-matrix and for the algebra generated by the entries of the corresponding Lax operator with the aim of obtaining the spectra of the relevant Gaudin-type Hamiltonians in terms of solutions of modified Bethe equations. The applications of the obtained result to the diagonalization of the anisotropic quantum Euler top, quantum Zhukovsky-Volterrra top, quantum Steklov and Rubanovsky tops are given.