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Journal of Physics Conference Series

Boundary correlation functions of integrable vertex models

Kohei Motegi

Published Feb 8, 2012• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1742-6596/343/1/012081

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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.
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.
The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a ‘reconstruction’ formula is proposed, which expresses the partition function as the trace of a suitably chosen quantum operator, in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models.
We consider the six-vertex model with domain wall boundary conditions. We choose the inhomogeneities as solutions of the Bethe Ansatz equations. These 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 formulae for XXZ antiferromagnets.
Vertical-arrow fluctuations near the boundaries in the six-vertex model on the two-dimensional NxN square lattice with the domain wall boundary conditions are considered. The one-point correlation function ("boundary polarization") is expressed via the partition function of the model on a sublattice. The partition function is represented in terms of standard objects in the theory of orthogonal polynomials. This representation is used to study the large N limit: the presence of the boundary affects the macroscopic quantities of the model even in this limit. The logarithmic terms obtained are compared with predictions from conformal field theory.