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Inner products in integrable Richardson-Gaudin models

Pieter Claeys (UGent) , Stijn De Baerdemacker (UGent) and Dimitri Van Neck (UGent)
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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.
Keywords
ALGEBRAIC BETHE-ANSATZ, FORM-FACTORS, METALLIC GRAINS, WAVE-FUNCTIONS, BCS MODEL, EQUATIONS, CHAIN, BOUNDARIES, FRAMEWORK, SYSTEMS

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Chicago
Claeys, Pieter, Stijn De Baerdemacker, and Dimitri Van Neck. 2017. “Inner Products in Integrable Richardson-Gaudin Models.” Scipost Physics 3 (4).
APA
Claeys, Pieter, De Baerdemacker, S., & Van Neck, D. (2017). Inner products in integrable Richardson-Gaudin models. SCIPOST PHYSICS, 3(4).
Vancouver
1.
Claeys P, De Baerdemacker S, Van Neck D. Inner products in integrable Richardson-Gaudin models. SCIPOST PHYSICS. 2017;3(4).
MLA
Claeys, Pieter, Stijn De Baerdemacker, and Dimitri Van Neck. “Inner Products in Integrable Richardson-Gaudin Models.” SCIPOST PHYSICS 3.4 (2017): n. pag. Print.
@article{8545218,
  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.},
  articleno    = {028},
  author       = {Claeys, Pieter and De Baerdemacker, Stijn and Van Neck, Dimitri},
  issn         = {2542-4653},
  journal      = {SCIPOST PHYSICS},
  language     = {eng},
  number       = {4},
  pages        = {34},
  title        = {Inner products in integrable Richardson-Gaudin models},
  url          = {http://dx.doi.org/10.21468/scipostphys.3.4.028},
  volume       = {3},
  year         = {2017},
}

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