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.
Highlights
IntroductionIntegrable models have proven to be a powerful tool in the study of many-body physics
The existence of two distinct representations for each on-shell Bethe state allows such inner products to be recast as domain wall boundary partition functions (DWPF), and we show here how the Slavnov determinant follows as a corollary
Integrable Richardson-Gaudin models can be investigated in two distinct ways - either by solving the Bethe equations for the rapidities, or by solving a set of quadratic Bethe equations for the eigenvalues of the conserved charges
Summary
Integrable models have proven to be a powerful tool in the study of many-body physics. One of the key expressions is the well-known Slavnov formula for the inner product between two Bethe states, one with rapidities satisfying the Bethe equations (an on-shell state), and one with arbitrary rapidities (an off-shell state) [5] Such determinant expressions provide a basic building block for the calculation of correlation coefficients from the Bethe states, which has allowed for massive simplifications in the calculations of correlation coefficients in these models [6,7,8,9,10].
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