<i>q</i>-bosons, Toda lattice, Pieri rules and Baxter<i>q</i>-operator

Duval, Antoine; Pasquier, Vincent

doi:10.1088/1751-8113/49/15/154006

Cited by 5 publications

(4 citation statements)

References 35 publications

(59 reference statements)

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“…There are nevertheless strong reasons to expect that such a determinant representation for the scalar products should exist even for the most general boundary terms. Let us mention in particular [64], in which a determinant representation was obtained for the scalar products of two Bethe states in the case of a semi-infinite chain, and [26] in which such a type of formula, based on the construction of Bethe states through modified algebraic Bethe ansatz and on the description of the spectrum through an inhomogeneous T-Q equation, was conjectured for the XXX chain with generic boundaries from the study of particular lattices with one or two sites.…”

Section: Introductionmentioning

confidence: 99%

The open XXX spin chain in the SoV framework: scalar product of separate states

Kitanine¹,

Maillet²,

Niccoli³

et al. 2017

J. Phys. A: Math. Theor.

105

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We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of polynomial solutions of a usual T -Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for the periodic chain.

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Section: Introductionmentioning

confidence: 99%

The open XXX spin chain in the SoV framework: scalar product of separate states

Kitanine¹,

Maillet²,

Niccoli³

et al. 2017

J. Phys. A: Math. Theor.

105

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“…One of the interesting topics is the study on symmetric polynomials by investigating integrable boson models in half-infinite lattice initiated in [24], which resembles the q-vertex operator approach. Due to the imposition of the infinite boundary condition, great simplifications occur and several beautiful formulas are displayed (see [25,26,31,32] for further works and also for an approach from the coordinate Bethe ansatz approach [39,40] whose connections with the quantum inverse scattering method seems to not be revealed up to now).…”

Section: Introductionmentioning

confidence: 99%

“…This fact allowed us to extract various properties of the Grothendieck polynomials. This is just an example, and there are increasing interests on the studies of symmetric polynomials from the point of view of integrable lattice models today (see [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38] for examples on these subjects) . One of the interesting topics is the study on symmetric polynomials by investigating integrable boson models in half-infinite lattice initiated in [24], which resembles the q-vertex operator approach.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice

Motegi

2017

Journal of Mathematical Physics

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We introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in finite lattice. We introduce the L-operator related with the U q (sl 2 ) R-matrix, and construct the wavefunctions and their duals. We prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the Grothendieck polynomials. This is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions. As applications of the correspondence between the wavefunctions and symmetric polynomials, we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the B-operators. *

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“…New results are also presented for scalar products, form factors and correlation functions in integrable models [27,28,29,30,31,32]. Other topics included are Baxter's Q-operators [33,34], Q-colourings of the triangular lattice [35], periodic Temperley-Lieb algebras [36], the random-cluster model on isoradial graphs [37], discrete-time exclusion processes [38], susceptibility of the square lattice Ising model [39], diffusion processes [40], compressed self-avoiding walks, bridges and polygons [41], and discrete ‡ See http://baxter2015.anu.edu.au holomorphicity in the chiral Potts model [42]. An exact solution is given for three interacting friendly walks in the bulk [43] and topological defects are considered for the Ising model [44].…”

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confidence: 99%