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.
Highlights
The representation is simple to define, it lacks a highest-weight state which would serve as the reference state in the Algebraic Bethe Ansatz (ABA). Despite this fact, we show that a common pseudovacuum can be used to define a common framework for arbitrarily oriented fields, leading to a construction which is as closely related to the traditional ABA as can be
Even with a single spin involved (N = 1), the known non-degenerate eigenstates of Sz(u), namely the |↑〉 and |↓〉, are shown to not be eigenstates of S2(u) in the presence of the in-plane field terms B0±. While it allows for a straightforward inclusion of additional in-plane magnetic field components, the chosen representation does not provide us with a proper highest weight state to be used as a pseudovacuum
This lack of a proper vacuum reference state is typical of integrable models without U(1) symmetry and a wide variety of approaches have been built in order to address this specific issue: from the diagonalisation of the XXZ open spin chain from the representation theory of the q-Onsager algebra [12], to the generalisation of the coordinate Bethe ansatz used in the XXZ chain and the antisymmetric simple exclusion process (ASEP) models in [13] as well as the functional Bethe ansatz used to treat various boundaries problems in XXX and XXZ chains [14,15,16]
Summary
The rational Richardson-Gaudin [1,2,3,4] models define a class of integrable quantum spin models in which the coupling between spins is isotropic. In the fourth section we will demonstrate that, by relaxing a usual ABA constraint on the pseudovacuum, one can still apply the QISM and find proper Bethe equations for this system, for which, working with a common reference state, defines this common framework leading to -built representations of the eigenstates for any magnetic field. These eigenvalue-based variables allow, in section 6, to define the simple transformation which, for any given eigenstate, links its rotated basis representation to this newly proposed common framework representation and gives us, in section 7 a simple determinant representation of certain scalar products
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