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

The issue of completeness has been addressed in section 6. Indeed, in https://scipost.org/submissions/1603.03542v4/ the completeness of the Quadratic Bethe equations in the usual (z-oriented field) XXX case has been demonstrate. The simple one-one correspondence between eigenstates in the generic field and the z-oriented case which was presented in this section (simple shift of the eigenvalues) consequently leads to a “trivial” homotopy which preserves the completeness in the case treated here.

Following a referee’s suggestion, we have vastly expanded the discussion concerning other approaches (and consequently the list of references as well) which have been developed to deal with similar situations in a variety of integrable models. Instead of simply mentioning separation of variables, we present a broader picture of the many techniques, which should give the interested reader much more material to look into. From the sheer number of such approaches, it is in our opinion well beyond the scope of this work, to present explicitly the numerous formal possible links to our work.

We have also added references to Claeys’ work on the p+ip superconductor, which has also made use of the same technique in this explicit XXZ-Gaudin case as well as that of Lukayenko on the same subject which, despite not discussing explicitly the eigenstates of the system has also derived similar conserved charges, Bethe equations and eigenvalues.

Section 5 has been slightly modified to make the final result appear at the very end of the section while section 4 has been reworked in order to regroup the results at beginning and leave the rest of the section as proof.

We also added a short discussion of the “triangular boundary” case, where the proposed ansatz using as many Bethe roots as we have spins, would no longer be valid. In this case, eigenstates are defined by a smaller number of Bethe roots as the pseudovacuum is now a properly defined eigenstate of the transfer matrix.

As it was explicitly mentioned in one of the reports, we added subsection 7.1 to give explicit determinant formulas the scalar products between generic off-the-shell Bethe states and generic off-the-shell dual states, for a given fixed (but generic) orientation of the field. This has lead to a slight shortening of 7.2, since the formulas for the canonical basis projections are obtained in an more or less identical fashion as those in 7.1

The various typos, notational errors and small precisions requested in the various reports have also all been explicitly addressed.