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arXiv:2107.13358 · Nuclear Physics B

Six-vertex model on a finite lattice: Integral representations for nonlocal correlation functions

Filippo Colomo, Giuseppe Di Giulio, A. G. Pronko

Published Sep 15, 2021• 10 citations

Abstract

We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably combining such representations, and using certain antisymmetrization relation in two sets of variables, it is possible to derive integral representations for various correlation functions. In particular, focusing on the emptiness formation probability, besides reproducing the known result, obtained by other means elsewhere, we provide a new one. By construction, the two representations differ in the number of integrations and their equivalence is related to a hierarchy of highly nontrivial identities.

DOI: 10.1016/j.nuclphysb.2021.115535

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We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably combining such representations, and using certain antisymmetrization relation in two sets of variables, it is possible to derive integral representations for various correlation functions. In particular, focusing on the emptiness formation probability, besides reproducing the known result, obtained by other means elsewhere, we provide a new one. By construction, the two representations differ in the number of integrations and their equivalence is related to a hierarchy of highly nontrivial identities.

We consider the six-vertex model with domain wall boundary conditions. We choose the inhomogeneities as solutions of the Bethe Ansatz equations. The Bethe Ansatz equations have many solutions, so we can consider a wide variety of inhomogeneities. For certain choices of the inhomogeneities we study arrow correlation functions on the horizontal line going through the centre. In particular we obtain a multiple integral representation for the emptiness formation probability that generalizes the known formulæ for XXZ antiferromagnets. 1

We review our recent work on the boundary correlation functions of integrable vertex models on an N × N lattice with domain wall boundary conditions. Particularly considered is the six vertex model. The general expression of the boundary correlation functions is obtained for the six vertex model by use of the quantum inverse scattering method. We also comment on the potential application of the boundary correlation functions, and the relation between the boundary correlation functions for the nineteen vertex model.

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