Advances in Mathematical Physics

Volume 2017 (2017), Article ID 7563781, 11 pages

https://doi.org/10.1155/2017/7563781

## Izergin-Korepin Analysis on the Projected Wavefunctions of the Generalized Free-Fermion Model

Faculty of Marine Technology, Tokyo University of Marine Science and Technology, Etchujima 2-1-6, Koto-Ku, Tokyo 135-8533, Japan

Correspondence should be addressed to Kohei Motegi

Received 17 April 2017; Accepted 21 May 2017; Published 20 June 2017

Academic Editor: Andrei D. Mironov

Copyright © 2017 Kohei Motegi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We apply the Izergin-Korepin analysis to the study of the projected wavefunctions of the generalized free-fermion model. We introduce a generalization of the -operator of the six-vertex model by Bump-Brubaker-Friedberg and Bump-McNamara-Nakasuji. We make the Izergin-Korepin analysis to characterize the projected wavefunctions and show that they can be expressed as a product of factors and certain symmetric functions which generalizes the factorial Schur functions. This result can be seen as a generalization of the Tokuyama formula for the factorial Schur functions.

#### 1. Introduction

Integrable lattice models [1–4] are special classes of models in statistical physics in which many exact calculations are believed to be able to be done. The most local object in integrable models is called the -matrix, and its mathematical structure was revealed in the mid-1980s [5, 6]. The underlying mathematical structure was named as the quantum groups, and the investigation of the quantum groups naturally leads to immediate constructions of various -matrices.

From the point of view of statistical physics, -matrices are the most local objects, and the study on the -matrices is a starting point. The most important objects in statistical physics are partition functions. For the case of integrable models, partition functions are objects constructed from multiple -matrices and are determined by boundary conditions. One of the most famous partition functions in integrable lattice models are the domain wall boundary partition functions which was first introduced and analyzed in [7, 8]. In recent years, a more general class of partition functions which we shall call as the projected wavefunctions are attracting attention in its relation with algebraic combinatorics. The projected wavefunctions are the projection of the off-shell Bethe vector of integrable models into a class of some simple states labelled by the sequences of the particles or down spins. For the case of the free-fermion model in an external field, it was first shown by Bump-Brubaker-Friedberg [9] that the projected wavefunctions give a natural realization of the Tokuyama combinatorial formula for the Schur functions [10], which is a one-parameter deformation of the Weyl character formula (note that there are pioneering works using the free-fermion model implicitly in [11–13], and the Izergin-Korepin analysis and observation of the factorization phenomena on the domain wall boundary partition functions of the related models are called the Perk-Schultz (supersymmetric vertex) model [14] and the Felderhof free-fermion model [15] in [16, 17]. There is also an application to the correlation functions in [18]). This observation triggered studies on finding various generalizations and variations of the Tokuyama-type formula for symmetric functions [19–27] such as the factorial Schur functions and symplectic Schur functions, and an interesting notion was introduced furthermore which the number theorists call it the metaplectic ice.

In this paper, we analyze the free-fermion model using the method initiated by Izergin-Korepin [7, 8]. The method was developed by them in order to find the explicit expression of polynomials representing the domain wall boundary partition functions of the six-vertex model, from which the famous Izergin-Korepin determinant formula was found. The Izergin-Korepin analysis is the important method to study variants of the domain wall boundary partition functions. For example, it was applied to the domain wall boundary partition functions of the six-vertex model with reflecting end by Tsuchiya [28] to find its determinant formula. Extending the Izergin-Korepin analysis to more general class of partition functions is also important. Wheeler [29] invented a method to extend the Izergin-Korepin analysis on a class of partition functions called the scalar products. And in our very recent work [30], we extended the Izergin-Korepin analysis to study the projected wavefunctions of the six-vertex model. The resulting symmetric polynomials representing the projected wavefunctions contains the Grothendieck polynomials as a special case when the six-vertex model reduces to the five-vertex model [31–33]. We apply this technique to study the free-fermion model in an external field. To this end, we first introduce an ultimate generalization of the -operator by introducing the inhomogeneous parameters and factorial parameters. We use an inhomogeneous version of the generalized -operator in our forthcoming paper [34] having two types of factorial parameters, which generalizes the factorial -operator by Bump-McNamara-Nakasuji [22]. We next view the projected wavefunctions as a function of the inhomogeneous parameters and characterize its properties by using the Izergin-Korepin analysis. We then show that the product of factors and certain symmetric functions satisfies all the required properties the projected wavefunctions must satisfy. The result is a generalization of [9, 22] and hence can be viewed as a generalization of the Tokuyama for the factorial Schur functions. The Izergin-Korepin analysis views the partition functions as functions of inhomogeneous parameters in the quantum spaces, whereas the arguments initiated in [9] view the partition functions as functions of the free parameter in the auxiliary spaces. The comparison of the two different ways of arguments seems to be interesting.

We will use the results of the projected wavefunctions to the algebraic combinatorial study of the generalized Schur functions [34]. For example, two ways of evaluations of the same partition functions can lead to integrable model constructions of algebraic identities of the symmetric functions. For example, two ways of evaluations of the domain wall boundary partition functions, a direct evaluation and an indirect evaluation using the completeness relation and the projected wavefunctions, can give rise to the dual Cauchy formula of the generalized Schur functions. This idea can also be applied to partition functions of integrable models under reflecting boundary to give dual Cauchy identities of the generalized symplectic Schur functions. Further detailed Izergin-Korepin analysis on the domain wall boundary partition functions and the dual projected wavefunctions are required for the studies.

There are also studies on deriving Cauchy identities using the domain wall boundary partition functions like an intertwiner, invented in [35]. Deriving algebraic combinatorial properties of symmetric functions using their integrable model realizations is an active line of research. See [36–40] for more examples on Cauchy-type identities and more recent studies on the Littlewood-Richardson coefficients by [33, 41].

In any case, in order to conduct these studies, we first of all have to find out what are the explicit functions representing the projected wavefunctions. We think the Izergin-Korepin analysis presented in this paper is a fairly simple way to find out the explicit forms.

This paper is organized as follows. In the next section, we first list the generalized -operator and introduce the projected wavefunctions. In Section 3, we make the Izergin-Korepin analysis and list the properties needed to determine the explicit form of the projected wavefunctions. In Section 4, we show that the product of factors and certain symmetric functions satisfies all the required properties extracted from the Izergin-Korepin analysis, which means that the product is the explicit form of the projected wavefunctions. Section 5 is devoted to the conclusion of this paper.

#### 2. The Generalized Free-Fermion Model and the Projected Wavefunctions

The most fundamental objects in integrable lattice models are the -matrices and -operators. The -matrix of the free-fermion model we treat in this paper is given byacting on the tensor product of the complex two-dimensional space .

The -operator of the free-fermion model we use as bulk pieces of the projected wavefunctions in this paper is given byacting on the tensor product of the space and the two-dimensional Fock space at the th site .

The parameters , , and can be regarded as parameters associated with the quantum space . The -operators giving the Schur functions [9] and factorial Schur functions [22] are a special limit of the generalized -operator (2) given byrespectively.

The -operator (2) together with the -matrix (1) satisfies the relation:acting on .

Let us denote the orthonormal basis of and its dual as and and the orthonormal basis of and its dual as and . The matrix elements of the -operator can be written as , which we will use this form in the next section. See Figure 1 for a pictorial description of the -operator (2).