Switch Operators for the Six-Vertex Model

The six-vertex model is a widely studied statistical mechanics model due to its connections to various areas of mathematics, including representation theory, combinatorics, and integrable systems. It was first introduced by Pauling in [Pau35].

In this paper, we view the six-vertex model as a combinatorial system of paths on a lattice model under prescribed boundary conditions. Each combination of paths forms an admissible state of the model, in which we assign vertex weights to the vertices in each state. The partition function, which is a weighted sum over all possible configurations, encodes the statistical properties of the model.

A recurrent problem in the combinatorics of integrable lattice models is to find appropriate Boltzmann weights that lead to meaningful and useful values in the partition function. If these weights satisfy the Yang-Baxter equation (see [Bax82]), then the partition function satisfies functional equations and can be both computed and identified with special functions from literature. For multiple instances, see [BBF11, BMN14, ABPW21, MS15, Mot17a, Mot17b, Nap23] and references therein.

One of the key challenges in studying the six-vertex model is understanding the dependence of the partition function on the boundary conditions, which are the states assigned to the vertices on the boundary of the lattice. Functional relations for the partition functions with different boundary conditions exist by the Yang-Baxter equation. In this paper, we introduce a novel method for analyzing this dependence: the switch operators. These operators, derived from the Yang-Baxter equation, allow us to express the partition function with arbitrary boundary conditions in terms of a base case with simple domain wall boundary conditions.

The main result is the following:

As an application, we compute the explicit expression for the factorial Schur functions and their generalizations. We also prove that these functions are asymptotically symmetric in column parameters. These results extend and generalize those from Section 7 of [BMN14].

Acknowledgements. This paper was created through the 2022 Stanford Undergraduate Research in Mathematics (SURIM) program. We would like to thank everyone involved in organizing SURIM, and in particular we thank Lernik Asserian for directing the program.

In this section, we review the six-vertex model from statistical mechanics and introduce notation. For a treatment from the point of view of statistical mechanics, see [Bax82]. Here, we represent the model as a rectangular lattice with paths traveling from northwest to southeast. That is, paths enter the model from the top or the left and leave the model on the right and bottom. Paths may intersect, but their movement is restricted to the right and downward directions. Thus, there are only six admissible states for a vertex, illustrated in Figure 1.

Defining a six-vertex model requires specifying the size of the grid, as well as the boundary conditions (at what positions the paths enter or leave the grid). Given such a model, we have a system of admissible configurations of paths. Each configuration is called a state, and each state consists only of the six mentioned vertices. The weight of a given type of vertex at a certain position in the lattice is represented using the following weight functions:

$$a_{1},a_{2},b_{1},b_{2},c_{1},c_{2}\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}.$$

For each state $s$, its weight $w(s)$ is given by computing the product of weights over all vertices in the state. Then the partition function $Z(\mathfrak{S})$ of a six-vertex model $\mathfrak{S}$ is the sum of weights of all admissible states:

$$Z(\mathfrak{S})=\sum_{\text{states}}\prod_{\text{vertices}}W\text{(vertex)}.$$

More generally, we consider the six-vertex model with row labels $I=(I_{1},I_{2},\dots,I_{n})$, column labels $J=(J_{1},J_{2},\dots,J_{m})$, and boundary conditions $\beta=(\beta^{l},\beta^{t},\beta^{r},\beta^{b})$ for the left, top, right, and bottom boundaries, respectively. We denote such a model by

$$\operatorname{\mathfrak{S}}(I;J;\beta)=\operatorname{\mathfrak{S}}(I_{1},\dots,I_{n};J_{1},\dots,J_{m};\beta^{l},\beta^{t},\beta^{r},\beta^{b}).$$

For brevity, we denote the partition function of such a model by

$$Z(I;J;\beta^{l},\beta^{t},\beta^{r},\beta^{b})=Z(\mathfrak{S}(I;J;\beta^{l},\beta^{t},\beta^{r},\beta^{b})).$$

Lastly, we use the notation $[n]=(1,2,\dots,n)$.

The introduction of cross vertices to facilitate the Yang-Baxter equation, as illustrated in the above figure, gives rise to horizontal and vertical cross vertices. We provide the six allowable horizontal cross vertices of the six-vertex model in Figure 4. Similarly, the six allowable vertical cross vertices of the six-vertex model are shown in Figure 5.

We now provide an extension of the Yang-Baxter equation, called the train argument. This argument is the basis of our construction of the switch operators.

In this section we define the generalized Demazure operators, which we call the switch operators, that can be applied to both horizontal and vertical boundaries. We then axiomatically develop the algebra of partition functions under the switch operators.

Let

	$\displaystyle a_{1},a_{2},b_{1},b_{2},c_{1},c_{2}$	$\displaystyle\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{C},$
	$\displaystyle a_{1}^{\operatorname{H}},a_{2}^{\operatorname{H}},b_{1}^{\operatorname{H}},b_{2}^{\operatorname{H}},c_{1}^{\operatorname{H}},c_{2}^{\operatorname{H}}$	$\displaystyle\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{C},$
	$\displaystyle a_{1}^{\operatorname{V}},a_{2}^{\operatorname{V}},b_{1}^{\operatorname{V}},b_{2}^{\operatorname{V}},c_{1}^{\operatorname{V}},c_{2}^{\operatorname{V}}$	$\displaystyle\colon\mathbb{Z}\times\mathbb{Z}\to\mathbb{C},$

be the integrable weight functions, where $b_{1}^{\operatorname{H}},b_{2}^{\operatorname{H}},b_{1}^{\operatorname{V}},b_{2}^{\operatorname{V}}$ are non-zero, and where

	$\displaystyle a_{1}^{\operatorname{H}}(i+1,i)a_{1}^{\operatorname{H}}(i,i+1)=b_{1}^{\operatorname{H}}(i+1,i)b_{2}^{\operatorname{H}}(i+1,i)+c_{1}^{\operatorname{H}}(i+1,i)c_{2}^{\operatorname{H}}(i+1,i)$
	$\displaystyle a_{1}^{\operatorname{V}}(i+1,i)a_{1}^{\operatorname{V}}(i,i+1)=b_{1}^{\operatorname{V}}(i+1,i)b_{2}^{\operatorname{V}}(i+1,i)+c_{1}^{\operatorname{V}}(i+1,i)c_{2}^{\operatorname{V}}(i+1,i).$

Note that the last condition is equivalent to assuming that the $R$-matrix of the cross-vertex is invertible. This is demonstrated in detail in [Nap22].

Let $\alpha=(\alpha_{1},\dots,\alpha_{d})\in\mathbb{N}^{d}$ be a strictly decreasing signature of length $d$. Let $\delta_{d}=(d,d-1,\dots,1)$ and let $n,m\geq d$.

We consider the six-vertex model $\mathfrak{S}^{n,m}(\emptyset,\delta_{d},\alpha,\emptyset)$, that is, a model with $n$ rows and $m$ columns. The lattice consists of an empty left boundary, a dense top boundary, a boundary of $\alpha$ on the right, and an empty bottom boundary.

Let $Z_{\alpha}$ denote the partition function of this model. The main aim of this section is to give connection between the partition functions $Z_{\alpha}$ for different $\alpha$’s.

The partition functions $Z_{\alpha}$ depend on the spectral parameters $I=(i_{1},\dots,i_{n})$ and $J=(j_{1},\dots,j_{m})$. Let the permutation group $S_{\infty}$ act on the spectral parameters as follows:

$$\pi I=(i_{\pi(1)},\dots,i_{\pi(n)}).$$

We also define the action of the simple transposition $s_{i}$ on the rightmost boundary as follows:

1. (1)

   if $i,i+1\in\alpha$, then $s_{i}\alpha=\alpha$;

2. (2)

   if $i,i+1\not\in\alpha$, then $s_{i}\alpha=\alpha$;

3. (3)

   if $i\in\alpha$ but $i+1\not\in\alpha$, then $s_{i}\alpha=\alpha^{\prime}$, where $\alpha^{\prime}$ has the value $i+1$ replaced by $i$;

4. (4)

   if $i+1\in\alpha$ but $i\not\in\alpha$, then $s_{i}\alpha=\alpha^{\prime}$, where $\alpha^{\prime}$ has the value $i$ replaced by $i+1$.

In each case, we use the train argument to derive information about the relations that must hold between different horizontal cross vertex weights.

In the first case, we have

$$a_{1}^{\operatorname{H}}(i+1,i)Z_{\alpha}(s_{i}\,x;y)=a_{2}^{\operatorname{H}}(i+1,i)Z_{\alpha}(x;y).$$

In the second case,

$$Z_{\alpha}(s_{i}\,x;y)=Z_{\alpha}(x;y).$$

That is, swapping empty rows has no effect on the partition function. The final two cases provide the most significant insights. In the third case, we have

(3.1)

$$a_{1}^{\operatorname{H}}(i+1,i)Z_{\alpha}(s_{i}\,x;y)=b_{2}^{\operatorname{H}}(i+1,i)Z_{s_{i}\alpha}(x;y)+c_{1}^{\operatorname{H}}(i+1,i)Z_{\alpha}(x;y).$$

Lastly, the fourth case gives us the following:

(3.2)

$$a_{1}^{\operatorname{H}}(i+1,i)Z_{\alpha}(s_{i}\,x;y)=b_{1}^{\operatorname{H}}(i+1,i)Z_{s_{i}\alpha}(x;y)+c_{2}^{\operatorname{H}}(i+1,i)Z_{\alpha}(x;y).$$

If we look at the third case in particular, we notice that we can rewrite the terms and define a Demazure-like operator:

$$\partial_{i}^{\operatorname{H}}=\frac{a_{1}^{\operatorname{H}}(i+1,i)s_{i}^{\operatorname{H}}-c_{1}^{\operatorname{H}}(i+1,i)}{b_{2}^{\operatorname{H}}(i+1,i)},$$

such that

$$Z_{s_{i}\alpha}(x;y)=\partial_{i}^{\operatorname{H}}(Z_{\alpha}(x;y))=\left(\frac{a_{1}^{\operatorname{H}}(i+1,i)s_{i}^{\operatorname{H}}-c_{1}^{\operatorname{H}}(i+1,i)}{b_{2}^{\operatorname{H}}(i+1,i)}\right)Z_{\alpha}(x;y).$$

If $\partial_{i}^{\operatorname{H}}$ is defined as above, then the inverse $\overline{\partial}_{i}^{\operatorname{H}}=(\partial_{i}^{\operatorname{H}})^{-1}$ is defined by the fourth case:

$$\overline{\partial}_{i}^{\operatorname{H}}=\frac{a_{1}^{\operatorname{H}}(i+1,i)s_{i}^{\operatorname{H}}-c_{2}^{\operatorname{H}}(i+1,i)}{b_{1}^{\operatorname{H}}(i+1,i)}.$$

By [Nap22], we have the following relations between the weights:

The property that these two operators are inverses of each other follows from the relations on the weights. By definition,

	$\displaystyle\partial_{i}^{\operatorname{H}}\overline{\partial}_{i}^{\operatorname{H}}$	$\displaystyle=\frac{a_{1}(i+1,i)s_{i}^{\operatorname{H}}-c_{1}^{\operatorname{H}}(i+1,i)}{b_{2}^{\operatorname{H}}(i+1,i)}\frac{a_{1}^{\operatorname{H}}(i+1,i)s_{i}^{\operatorname{H}}-c_{2}^{\operatorname{H}}(i+1,i)}{b_{1}^{\operatorname{H}}(i+1,i)}$
		$\displaystyle=\frac{c_{1}^{\operatorname{H}}(i+1,i)c_{2}^{\operatorname{H}}(i+1,i)-a_{1}^{\operatorname{H}}(i+1,i)a_{1}^{\operatorname{H}}(i,i+1)}{b_{1}^{\operatorname{H}}(i+1,i)b_{2}^{\operatorname{H}}(i+1,i)}$
		$\displaystyle=1.$

Throughout the computation, we use the properties from Lemma 3.1.

We use a similar approach to define the vertical switch operator. Let $s_{j}^{\operatorname{V}}$ be the vertical analog of $s_{i}^{\operatorname{H}}$, where $s_{j}^{\operatorname{V}}$ acts by transposing the spectral parameters corresponding to columns $j$ and $j+1$. We will use the notation $s_{j}$ when it is clear we are acting in the vertical case.

We may now define the operators we have derived from our applications of the train argument.

Note that we can use a composition of simple reflections $s_{i}$ to bring any signature $(n,n-1,\dots,1)$ to $\alpha$. For example, if we look at the horizontal direction:

$$(3,2,1)\xrightarrow{s_{3}}(4,2,1)\xrightarrow{s_{4}}(5,2,1)\xrightarrow{s_{2}}(5,3,1)\xrightarrow{s_{1}}(5,3,2).$$

Hence, using the operator $\partial_{i}^{\operatorname{H}}$, we can express $Z_{\alpha}$ in terms of $Z_{\delta}$, where $\delta=(n,n-1,\dots,1)$. We call $Z_{\delta}$ the base case. Note that $Z_{\delta}=Z_{n}$ from Definition 2.1.

Thus, we can express $Z_{\alpha}$ as

$$Z_{\alpha}=\partial_{i_{1}}^{\operatorname{H}}\partial_{i_{2}}^{\operatorname{H}}\dots\partial_{i_{k}}^{\operatorname{H}}(Z_{\delta}),$$

for some $\partial_{i_{m}}^{\operatorname{H}}$’s. The vertical case follows this same property, but instead prescribes a strictly decreasing signature of $\beta=(\beta_{1},\dots,\beta_{n})\in\mathbb{N}^{n}$ to the top boundary.

We express this concept with more precise notation. Let $D=(d,d-1,\cdots,1)$ where $d$ is the length of the signature $\alpha$. Then let

$$\partial_{\alpha}^{\operatorname{H}}=\prod_{k=1}^{d}\prod_{\ell=1}^{\alpha_{d-k+1}-D_{k}}\partial_{\alpha_{d-k+1}-\ell}^{\operatorname{H}}.$$

Lastly, let $M$ denote the number of columns in the lattice, and let

$$\partial_{\beta}^{V}=\prod_{k=1}^{d}\prod_{\ell=\beta_{k}}^{M-k}\partial_{\ell}^{\operatorname{V}}.$$

Then, consider the partition function being normalized in the $a_{1}$ weight, so that the addition of extra empty columns or rows does not impact the partition function.

Before we reach our theorem, we develop some useful notation regarding repeated use of the switch operators. For $i<j$, denote by $\partial_{[i,j]}$ the operator sequence $\partial_{j-1,j}\partial_{j-2,j-1}\dots\partial_{i,i+1}$ and $s_{[i,j]}$ the sequence $s_{j-1,j}s_{j-2,j-1}\dots s_{i,i+1}$. Let $\partial_{\alpha}^{\operatorname{H}}$ be the expression $\partial_{[1,\alpha_{n}]}^{\operatorname{H}}\partial_{[2,\alpha_{n-1}]}^{\operatorname{H}}\dots\partial_{[n,\alpha_{1}]}^{\operatorname{H}}$. Similarly, we denote $\partial_{\alpha}^{\operatorname{V}}$ to be $\partial_{[1,\alpha_{n}]}^{\operatorname{V}}\partial_{[2,\alpha_{n-1}]}^{\operatorname{V}}\dots\partial_{[n,\alpha_{1}]}^{\operatorname{V}}$.

Thus we have proved it is possible to express the partition function of a lattice with arbitrary boundaries in terms of the operators acting on the base case.