# Tetrahedron equation and Schur functions

Shinsuke Iwao, Kohei Motegi, and Ryo Ohkawa Email address<EMAIL_ADDRESS>Faculty of Business and Commerce, Keio University,Hiyosi 4–1–1, Kohoku-ku,
Yokohama-si, Kanagawa 223-8521, Japan, E-mail<EMAIL_ADDRESS>Faculty
of Marine Technology, Tokyo University of Marine Science and
Technology,Etchujima 2-1-6, Koto-Ku, Tokyo, 135-8533, Japan, E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Osaka Central Advanced Mathematical
Institute, Osaka Metropolitan University, 558-8585, Osaka, Japan, Research
Institute for Mathematical Sciences, Kyoto University, 606-8502, Kyoto, Japan.

###### Abstract

The tetrahedron equation introduced by Zamolodchikov is a three-dimensional
generalization of the Yang-Baxter equation. Several types of solutions to the
tetrahedron equation that have connections to quantum groups can be viewed as
$q$-oscillator valued vertex models with matrix elements of the $L$-operators
given by generators of the $q$-oscillator algebra acting on the Fock space.
Using one of the $q=0$-oscillator valued vertex models introduced by Bazhanov-
Sergeev, we introduce a family of partition functions that admits an explicit
algebraic presentation using Schur functions. Our construction is based on the
three-dimensional realization of the Zamolodchikov-Faddeev algebra provided by
Kuniba-Okado-Maruyama. Furthermore, we investigate an inhomogeneous
generalization of the three-dimensional lattice model. We show that the
inhomogeneous analog of (a certain subclass of) partition functions can be
expressed as loop elementary symmetric functions.

##  1 Introduction

In [32, 33], Zamolodchikov introduced the tetrahedron equation as a three-
dimensional generalization of the Yang-Baxter equation [1, 31]. Various
seminal works such as [2, 3, 6, 13, 14, 16] have been done to construct
solutions to the tetrahedron equation. See [4, 5, 17, 18, 19, 21, 24], as well
as more recently [20], for examples of studies that investigate solutions to
the tetrahedron equation through the viewpoint of quantum groups [8, 9, 10,
12].

Solutions to the tetrahedron equation which we call as three-dimensional
$R$-matrices are regarded as “local Boltzmann weights” in three-dimensional
statistical physics. In this paper, we study a class of partition functions
using one of the solutions found by Bazhanov-Sergeev [5], where they used the
three-dimensional $R$-matrix and gave an unconventional realization of the
$U_{q}(\widehat{sl_{n}})$ $R$-matrix by tracing out one dimension. Our main
idea is to use a degeneration of the three-dimensional $R$-matrix, which was
studied by Kuniba-Maruyama-Okado [18, 19] from the perspective of its relation
to a multispecies generalization of a totally asymmetric symmetric exclusion
process (TASEP) [7, 29]. They showed that the matrix product form of the
steady state of the multispecies periodic TASEP introduced by Ferrari-Martin
[11] can be transformed into a certain type of partition functions constructed
from Bazhanov-Sergeev’s three-dimensional $R$-matrix. This result was given by
bypassing the crystal basis theory [15, 26]. Furthermore, they showed that
layers of the three-dimensional partition functions satisfy the Zamolodchikov-
Faddeev algebra relations, i.e. they gave a realization of generators of the
(degenerated) Zamolodchikov-Faddeev algebra. The partition functions
corresponding to the steady state were constructed by taking the traces of the
product of the Zamolodchikov-Faddeev operators acting on the tensor product of
Fock spaces and specializing spectral parameters to a specific value.

In this paper, we study a closely related but different class of three-
dimensional partition functions. Instead of taking the traces, our partition
functions are given by taking the vacuum expectation values of the
Zamolodchikov-Faddeev operators. In addition, we do not specialize spectral
parameters to a specific value, as they will play the role of symmetric
variables. We show that this class of partition functions is represented as
Schur polynomials up to an overall factor. This result is derived from
multiple commutation relations of the Zamolodchikov-Faddeev algebra.

Furthermore, we also investigate an inhomogeneous generalization. Since this
generalization does not respect integrability, techniques based on integrable
systems such as the Zamolodchikov-Faddeev algebra cannot be applicable here.
However, we show that a certain subclass of inhomogeneous partition functions
can be expressed as loop elementary symmetric functions, which was originally
introduced by Yamada [30] in the context of the discrete Toda lattice. See
Lam-Pylyavskyy [22, 23] for more details. We also show that several other
partition functions are given by loop elementary symmetric functions.

This paper is organized as follows. Section 2 first reviews some basic facts
about the three-dimensional $R$-matrix and its equivalent description as an
operator-valued two-dimensional $R$-matrix. We next recall a realization of
the Zamolodchikov-Faddeev operators, and introduce partition functions. We
derive the correspondence with the Schur functions in Section 3 and also
discuss some corollaries/applications. In section 4, we introduce an
inhomogeneous version for a subclass and several other related partition
functions, and prove the correspondence with the loop elementary symmetric
functions.

##  2 Three-dimensional $R$-matrix and partition functions

We first recall the three-dimensional $R$-matrix [5, 18, 19] and operators
that are used in this paper, and introduce the partition functions which we
investigate.

Let $V={\mathbb{C}}v_{0}\oplus{\mathbb{C}}v_{1}$ be the complex two-
dimensional space spanned by the standard basis $\\{v_{0},v_{1}\\}$, and let
$\mathcal{F}=\displaystyle\oplus_{m=0}^{\infty}\mathbb{C}|m\rangle$ be the
bosonic Fock space spanned by the basis $\\{|m\rangle\;;\;(m=0,1,\dots)\\}$.
We introduce the linear operators $\mathbf{t},\mathbf{b}^{+},\mathbf{b}^{-}$
acting on $\mathcal{F}$ as

$\displaystyle\mathbf{t}|m\rangle=\delta_{m,0}|m\rangle,\ \ \
\mathbf{b}^{+}|m\rangle=|m+1\rangle,\ \ \
\mathbf{b}^{-}|m\rangle=|m-1\rangle.$ (2.1)

Here, $|-1\rangle:=0$. These operators satisfy the relations

$\displaystyle\mathbf{t}\mathbf{b}^{+}=\mathbf{b}^{-}\mathbf{t}=0,\ \ \
\mathbf{b}^{+}\mathbf{b}^{-}=1-\mathbf{t},\ \ \
\mathbf{b}^{-}\mathbf{b}^{+}=1.$ (2.2)

We refer to $\mathbf{t},\mathbf{b}^{+},\mathbf{b}^{-}$ as the zero-number
projection operator, the creation operator, and the annihilation operator,
respectively. The bosonic Fock space and the algebra associated can be
regarded as the $q=0$ limit of the $q$-bosonic Fock space and the
$q$-oscillator algebra.

Let $V^{\ast}=\mathbb{C}v^{\ast}_{0}\oplus\mathbb{C}v^{\ast}_{1}$ denote the
dual space of $V^{\ast}$, which is spanned by
$\\{v_{0}^{\ast},v_{1}^{\ast}\\}$ such that $v_{j}^{\ast}(v_{i})=\delta_{ij}$.
The dual Fock space $\mathcal{F}^{\ast}=\oplus_{m=0}^{\infty}\mathbb{C}\langle
m|$ is the vector space spanned by $\\{\langle m|\;|\;m=0,1,\dots\\}$ such
that $\langle m^{\prime}\;|\;m\rangle=\delta_{m^{\prime},m}$, where $(\langle
m^{\prime}|,|m\rangle)\mapsto\langle m^{\prime}\;|\;m\rangle$ is the standard
pairing.

The three-dimensional $R$-matrix $R$ acting on $V\otimes V\otimes\mathcal{F}$
is defined by acting on the basis $\\{v_{i}\otimes
v_{j}\otimes|k\rangle\;;\;(i,j=0,1,k=0,1,2,\dots)\\}$ as

$\displaystyle\displaystyle R(v_{i}\otimes
v_{j}\otimes|k\rangle)=\sum_{a=0}^{1}\sum_{b=0}^{1}\sum_{\ell=0}^{\infty}v_{a}\otimes
v_{b}\otimes|\ell\rangle[R]_{ijk}^{ab\ell},$ (2.3)

where

$\displaystyle[R]_{00k}^{00\ell}$ $\displaystyle=\delta_{k\ell},\ \ \
[R]_{11k}^{11\ell}=\delta_{k\ell},\ \ \ [R]_{10k}^{01\ell}=\delta_{k+1,\ell},$
(2.4) $\displaystyle[R]_{01k}^{10\ell}$ $\displaystyle=\delta_{k,\ell+1},\ \ \
[R]_{01k}^{01\ell}=\delta_{k0}\delta_{\ell 0},$ (2.5)

and $[R]_{ijk}^{ab\ell}=0$ otherwise. This is equivalent to the operator-
valued two-dimensional $L$-operator $\mathcal{R}$ acting on $V\otimes V$ as

$\displaystyle\mathcal{R}(v_{i}\otimes
v_{j})=\sum_{a=0}^{1}\sum_{b=0}^{1}v_{a}\otimes v_{b}[\mathcal{R}]_{ij}^{ab},$
(2.6)

where

$\displaystyle[\mathcal{R}]_{00}^{00}$ $\displaystyle=1,\ \ \
[\mathcal{R}]_{11}^{11}=1,\ \ \ [\mathcal{R}]_{10}^{01}=\mathbf{b}^{+},$ (2.7)
$\displaystyle[\mathcal{R}]_{01}^{10}$ $\displaystyle=\mathbf{b}^{-},\ \ \
[\mathcal{R}]_{01}^{01}=\mathbf{t},$ (2.8)

and $[\mathcal{R}]_{ij}^{ab}=0$ otherwise.

Figure 1: Three-dimensional $R$-matrix. The two solid lines represent two two-
dimensional spaces, and the dashed line represents the bosonic Fock space. To
this configuration, the ”Boltzmann weight” $[R]_{ijk}^{ab\ell}$ is assigned.
One can also regard this is as an ”operator-valued Boltzmann weight”
$[\mathcal{R}]_{ij}^{ab}$ if not specifying the basis in the bosonic Fock
space.

Figure 1 presents a graphical description of the three-dimensional $R$-matrix
$R$ in (2.3) (or equivalently, the operator-valued two-dimensional $R$-matrix
$\mathcal{R}$ in (2.6)), which we use in this paper. The matrix elements of
$\mathcal{R}$ are shown in Figure 2.

Let
$\mathbb{Z}^{3}=\mathbb{Z}\bm{e}_{1}\oplus\mathbb{Z}\bm{e}_{2}\oplus\mathbb{Z}\bm{e}_{3}$
be the three-dimensional lattice spanned by the standard basis
$\\{\bm{e}_{1},\bm{e}_{2},\bm{e}_{3}\\}$. An element
$\alpha\bm{e}_{1}+\beta\bm{e}_{2}+\gamma\bm{e}_{3}\in\mathbb{Z}^{3}$ is often
identified with the vector $(\alpha,\beta,\gamma)$. We will refer to the
planes $H^{1}_{k}=\\{(\alpha,\beta,\gamma)\;|\;\alpha=k\\}$,
$H^{2}_{\ell}=\\{(\alpha,\beta,\gamma)\;|\;\beta=\ell\\}$, and
$H^{3}_{m}=\\{(\alpha,\beta,\gamma)\;|\;\gamma=m\\}$ as the $k$-th row, the
$\ell$-th column, and the $m$-th slice, respectively.

Let $D_{n}:=\\{(k,\ell)\in\mathbb{Z}^{2}\;|\;k\geq 1,\ell\geq 1,k+\ell\leq
n\\}$ be the set consisting of $\frac{n(n-1)}{2}$ points. We assign a copy of
the Fock space $\mathcal{F}$ to each line $L_{k\ell}:=H^{1}_{k}\cap
H^{2}_{\ell}$ for $(k,\ell)\in D_{n}$. Hereinafter, $\mathcal{F}_{k\ell}$
denotes the Fock space assigned to $L_{k\ell}$.

Figure 2: The operator-valued matrix elements of $\mathcal{R}$. We color the
half-edges which are connected to 1s in the two-dimensional vector spaces with
red, and color 0s with blue.

We denote the basis of $\mathcal{F}_{k\ell}$ by
$\\{|m_{k\ell}\rangle_{k\ell}\;;\;(m_{k\ell}=0,1,2,\dots)\\}$. The standard
bases of $\mathcal{F}^{\otimes n(n-1)/2}$ and its dual $(\mathcal{F}^{\otimes
n(n-1)/2})^{*}$ are given by $\\{\otimes_{(k,\ell)\in
D_{n}}|m_{k\ell}\rangle_{k\ell}\;;\;(m_{k\ell}=0,1,2,\dots)\\}$ and
$\\{\otimes_{(k,\ell)\in D_{n}}\ {}_{k\ell}\langle
m_{k\ell}|\;;\;(m_{k\ell}=0,1,2,\dots)\\}$, respectively. We introduce
notations for the vacuum state and its dual as
$|\Omega\rangle:=|0\rangle^{\otimes n(n-1)/2}\in\mathcal{F}^{\otimes
n(n-1)/2}$ and $\langle\Omega|:=(|\Omega\rangle)^{*}\in(\mathcal{F}^{\otimes
n(n-1)/2})^{*}$.

We introduce the linear operator $X_{i}^{(n)}(z)$ $(i=0,\dots,n)$ [17, 18, 19]
acting on the tensor space $\mathcal{F}^{\otimes
n(n-1)/2}\otimes\mathbb{C}[z]$, which is graphically represented as Figure 3.
Here, to each of $\frac{n(n-1)}{2}$ intersections, a matrix element of
$\mathcal{R}$ is assigned, which is determined according to the configuration
of the red and blue segments. $X_{i}^{(n)}(z)$ is a weighted sum (see Figure
3) of all matrix elements of $\mathcal{R}^{\otimes n(n-1)/2}$ determined
according to the possible configuration in which (i) the segments at the
bottom of $1,2,\dots,i$-th columns are colored by red, and (ii) the segments
at the bottom of $i+1,i+2,\dots,n$-th columns are colored by blue.
Graphically, the operator $X^{(n)}_{i}(z)$ sends a vector in
$\mathcal{F}^{\otimes n(n-1)/2}\otimes\mathbb{C}[z]$ to the next slice. We
will abbreviate $X_{i}^{(n)}(z)$ as $X_{i}(z)$ if there is no confusion.

Figure 3: The operator $X_{i}(z)=X_{i}^{(n)}(z)$. We take the sum over all
configurations except one side of the boundary is fixed. The fixed boundary
condition on one side is labeled by the subscript $i$ of the operator, which
corresponds to the number of consecutive number of integers which is fixed as
$1$ on the fixed boundary. The sum is weighted, and the power
$\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$ of $z$ is the number of $1$s at the
top boundary.

In this paper, we investigate the following three-dimensional partition
function

$\displaystyle\langle\Omega|X_{i_{1}}(z_{1})X_{i_{2}}(z_{2})\cdots
X_{i_{m-1}}(z_{m-1})X_{i_{m}}(z_{m})|\Omega\rangle,$ (2.9)

which is graphically represented as Figure 4.

Figure 4: The three-dimensional partition functions
$\langle\Omega|X_{i_{1}}(z_{1})X_{i_{2}}(z_{2})\cdots
X_{i_{m-1}}(z_{m-1})X_{i_{m}}(z_{m})|\Omega\rangle$.

We mainly investigate the case $n\geq i_{1}\geq i_{2}\geq\dots\geq i_{m-1}\geq
i_{m}\geq 0$ in this paper.

##  3 Three-dimensional partition functions and Schur functions

In this section, we study the partition function (2.9). Let $\Lambda_{i}$ be
the set of all possible configurations of the red and blue segments that fit
Figure 3. We expand $X_{i}(z)$ as

$X_{i}(z)=\sum_{p\in\Lambda_{i}}z^{\alpha(p)}\bigotimes_{(k,\ell)\in
D_{n}}x_{k\ell}(p),$ (3.1)

where $\alpha(p)$ is the number of $1$s in the topmost row in $p$ and
$x_{k\ell}(p):\mathcal{F}_{k\ell}\to\mathcal{F}_{k\ell}$. Then, $x_{k\ell}(p)$
is either $1$, $\mathbf{b}^{+}$, $\mathbf{b}^{-}$, or $\mathbf{t}$. We
sometimes use the notation $x_{k\ell}(p)=1_{b}$ (resp. $x_{k\ell}(p)=1_{r}$)
if $x_{k\ell}(p)=1$ and it arises from two blue (resp. red) lines crossing at
$(k,\ell)$.

The following is an immediate consequence of Figure 2.

######  Lemma 3.1.

We have the following:

  1. 1.

If $x_{k+1,\ell}(p)=1_{b}$ and $x_{k,\ell+1}(p)=1_{b}$, then
$x_{k,\ell}(p)=1_{b}$ or $\mathbf{b}^{+}$.

  2. 2.

If $x_{k+1,\ell}(p)=1_{r}$ and $x_{k,\ell+1}(p)=1_{r}$, then
$x_{k,\ell}(p)=1_{r}$ or $\mathbf{b}^{-}$.

  3. 3.

If $x_{k+1,\ell}(p)=\mathbf{t}$ and $x_{k,\ell+1}(p)=1_{b}$, then
$x_{k,\ell}(p)=\mathbf{t}$.

  4. 4.

If $x_{k+1,\ell}(p)=1_{r}$ and $x_{k,\ell+1}(p)=\mathbf{t}$, then
$x_{k,\ell}(p)=\mathbf{t}$.

We first show the following factorized expression for cases $n\geq i_{m}\geq
i_{m-1}\geq\dots\geq i_{2}\geq i_{1}\geq 0$.

######  Lemma 3.2.

When $n\geq i_{m}\geq i_{m-1}\geq\dots\geq i_{2}\geq i_{1}\geq 0$, we have

$\displaystyle\langle\Omega|X_{i_{1}}(z_{1})X_{i_{2}}(z_{2})\cdots
X_{i_{m-1}}(z_{m-1})X_{i_{m}}(z_{m})|\Omega\rangle=z_{1}^{i_{1}}\cdots
z_{m}^{i_{m}}.$ (3.2)

###### Proof.

We prove the lemma by showing that there is only one configuration that
contributes to (3.2). This can be checked by investigating the bosonic Fock
spaces $\mathcal{F}_{k\ell}$ assigned to
$(k,\ell)=(n-1,1)\to(n-2,2)\to(n-2,1)\to(n-3,3)\to(n-3,2)\to(n-3,1)\to\cdots\to(1,1)$.
We first see what happens to the Fock space $\mathcal{F}_{(n-1,1)}$. Let
$\mathfrak{X}:=X_{i_{1}}(z_{1})X_{i_{2}}(z_{2})\cdots
X_{i_{m-1}}(z_{m-1})X_{i_{m}}(z_{m})$. Let $\theta_{k}$ denote the number of
$p$ such that $i_{p}=k$. We let $\theta_{\leq k}$ denote the sum
$\theta_{1}+\theta_{2}+\dots+\theta_{k}$. In a similar manner, we define
$\theta_{<k}$, $\theta_{\geq k}$, and $\theta_{>k}$. By definition of
$X_{0}(z)$, the $\mathcal{F}_{(n-1,1)}$ component of each term in $X_{0}(z)$
must be $1_{b}$ or $\mathbf{b}^{+}$. From Figure 2, we also find that the
$\mathcal{F}_{(n-1,1)}$ component of a term in $X_{1}(z)$ is $\mathbf{t}$, and
that in $X_{p}(z)$ $(p>1)$ is $1_{r}$ or $\mathbf{b}^{-}$. Hence, the
$\mathcal{F}_{(n-1,1)}$ component of each term in $\mathfrak{X}$ should be a
sequence of the form $(x_{1}\dots
x_{\theta_{0}})\mathbf{t}^{\theta_{1}}(y_{1}\dots y_{\theta_{\geq 2}})$ such
that $x_{j}\in\\{1_{b},\mathbf{b}^{+}\\}$ and
$y_{j}\in\\{1_{r},\mathbf{b}^{-}\\}$. However, since $\langle
0|\mathbf{b}^{+}=0$ and $\mathbf{b}^{-}|0\rangle=0$, terms that can survive
should satisfy $x_{j}=1_{b}$ and $y_{j}=1_{r}$ for all $j$. By the same
reason, the $\mathcal{F}_{(n-2,2)}$ components of terms in $\mathfrak{X}$ that
can survive should be of the form $1_{b}^{\theta_{\leq
1}}\mathbf{t}^{\theta_{2}}1_{r}^{\theta_{\geq 3}}$. Therefore, from Lemma 3.1,
we find that the $\mathcal{F}_{(n-2,1)}$ components of the survived terms are
of the form $(x_{1}\dots
x_{\theta_{0}})\mathbf{t}^{\theta_{1}+\theta_{2}}(y_{1}\dots
y_{\theta_{>2}})$. For the same reason we previously noted, terms that can
survive should be of the form
$1_{b}^{\theta_{0}}\mathbf{t}^{\theta_{1}+\theta_{2}}1_{r}^{\theta_{>2}}$. We
can repeat this procedure and show that the $\mathcal{F}_{kl}$ components of
terms that can survive should be of the form
$1_{b}^{\theta_{<l}}\mathbf{t}^{\theta_{l}+\cdots+\theta_{n-k}}1_{r}^{\theta_{>n-k}}$.
Therefore, all the possible configurations at each intersection must consist
of (i) two blue lines, (ii) two red lines, or (iii) one vertical red line and
one horizontal blue line. Figure 5 presents the only possible configuration on
a slice corresponding to $X_{i}(z)$, the contribution of which is $z^{i}$.
Multiplying these factors gives (3.2). ∎

Figure 5: The unique configuration for the operator $X_{i}(z)$ in
$\langle\Omega|X_{i_{1}}(z_{1})X_{i_{2}}(z_{2})\cdots
X_{i_{m-1}}(z_{m-1})X_{i_{m}}(z_{m})|\Omega\rangle$ when $n\geq i_{m}\geq
i_{m-1}\geq\dots\geq i_{2}\geq i_{1}\geq 0$. One notes from this configuration
that from $X_{i}(z)$ the factor $z^{i}$ contributes.

The operators $X_{i}(z)$ are shown to satisfy the following relations.

######  Theorem 3.3.

[17, 19] The operators $X_{i}(z)$ satisfy the Zamolodchikov-Faddeev algebra
relations

$\displaystyle
X_{i}(x)X_{j}(y)=\left\\{\begin{array}[]{ll}X_{i}(y)X_{j}(x)+(1-x/y)X_{j}(y)X_{i}(x)&i<j,\\\
X_{i}(y)X_{i}(x)&i=j,\\\ x/yX_{i}(y)X_{j}(x)&i>j,\end{array}\right..$ (3.6)

Theorem 3.3 is proved in [17, 19] as a consequence of the tetrahedron equation
(see Appendix A for the tetrahedron equation).

Recall the Schur polynomials. For a sequence of integers
$\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{n})$ where
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0$, the Schur
polynomials $s_{\lambda}(z_{1},\dots,z_{n})$ is

$\displaystyle s_{\lambda}(z_{1},\dots,z_{n})=\frac{\det_{1\leq i,j\leq
n}(z_{i}^{\lambda_{j}+n-j})}{\prod_{1\leq i<j\leq n}(z_{i}-z_{j})}.$ (3.7)

Let us introduce several shorthand notations. For a set of variables ${\bf
z}=\\{z_{1},\dots,z_{|{\bf z}|}\\}$ denote $X_{i}({\bf z}):=\prod_{j=1}^{|{\bf
z}|}X_{i}(z_{j})$. Note that due to commutativity
$X_{i}(z)X_{i}(w)=X_{i}(w)X_{i}(z)$, the ordering of variables in the product
does not matter. For an arbitrary integer $p$, define ${\bf
z}^{p}:=\prod_{j=1}^{|{\bf z}|}z_{j}^{p}$. We also introduce the following
notations ${\bf z}-{\bf w}:=\prod_{j=1}^{|{\bf z}|}\prod_{k=1}^{|{\bf
w}|}(z_{j}-w_{k})$, $1-{\bf z}/{\bf w}:=\prod_{j=1}^{|{\bf
z}|}\prod_{k=1}^{|{\bf w}|}(1-z_{j}/w_{k})$ for two sets of variables ${\bf
z},{\bf w}$.

We use the following formula for Schur functions ($t=0$ case of [27],
Proposition 4).

######  Proposition 3.4.

[27] For ${\bf z}=({\bf z}_{1},{\bf z}_{2},\dots,{\bf z}_{m})$, we have

$\displaystyle s_{(\lambda_{1}^{|{\bf z}_{1}|},\lambda_{2}^{|{\bf
z}_{2}|},\dots,\lambda_{m}^{|{\bf z}_{m}|})}({\bf z})=\sum_{{\bf w}=({\bf
w}_{1},{\bf w}_{2},\dots,{\bf w}_{m})}\frac{1}{\prod_{1\leq j<k\leq m}({\bf
w}_{j}-{\bf w}_{k})}{\bf w}_{1}^{\lambda_{1}+m-1}{\bf
w}_{2}^{\lambda_{2}+m-2}\cdots{\bf w}_{m}^{\lambda_{m}}.$ (3.8)

Here, we take the sum over all ${\bf w}=({\bf w}_{1},{\bf w}_{2},\dots,{\bf
w}_{m})$ such that ${\bf w}_{i}\ (i=1,\dots,m)$ are unordered sets of
variables satisfying $|{\bf w}_{i}|=|{\bf z}_{i}|$ and ${\bf w}_{1}\cup{\bf
w}_{2}\cup\cdots\cup{\bf w}_{m}={\bf z}_{1}\cup{\bf z}_{2}\cup\cdots\cup{\bf
z}_{m}$ .

We prepare the following multiple commutation relations between the operators
$X_{i}(z)$.

######  Proposition 3.5.

For $n\geq i_{1}>i_{2}>\cdots>i_{m}\geq 0$, we have the following commutation
relations:

$\displaystyle\frac{X_{i_{1}}({\bf z}_{1})X_{{i_{2}}}({\bf z}_{2})\cdots
X_{i_{m-1}}({\bf z}_{m-1})X_{i_{m}}({\bf z}_{m})}{\prod_{k=1}^{m}{\bf
z}_{k}^{m-k}}$ $\displaystyle=$ $\displaystyle\sum_{{\bf w}=({\bf w}_{1},{\bf
w}_{2},\dots,{\bf w}_{m})}\frac{1}{\prod_{1\leq j<k\leq m}({\bf w}_{j}-{\bf
w}_{k})}X_{i_{m}}({\bf w}_{m})X_{i_{m-1}}({\bf w}_{m-1})\cdots X_{i_{2}}({\bf
w}_{2})X_{i_{1}}({\bf w}_{1}).$ (3.9)

Here, we take the sum over all ${\bf w}=({\bf w}_{1},{\bf w}_{2},\dots,{\bf
w}_{m})$ such that ${\bf w}_{i}\ (i=1,\dots,m)$ are unordered sets of
variables satisfying $|{\bf w}_{i}|=|{\bf z}_{i}|$ and ${\bf w}_{1}\cup{\bf
w}_{2}\cup\cdots\cup{\bf w}_{m}={\bf z}_{1}\cup{\bf z}_{2}\cup\cdots\cup{\bf
z}_{m}$ .

###### Proof.

We can give a proof using the argument given in [25, 28]. We rewrite the
Zamolodchikov-Faddeev algebra relations (3.3) in the following forms

$\displaystyle X_{i}(x)X_{j}(y)$
$\displaystyle=(1-y/x)^{-1}X_{j}(y)X_{i}(x)-(1-y/x)^{-1}X_{j}(x)X_{i}(y),\
i>j,$ (3.10) $\displaystyle X_{i}(x)X_{i}(y)$
$\displaystyle=X_{i}(y)X_{i}(x),$ (3.11) $\displaystyle X_{i}(x)X_{j}(y)$
$\displaystyle=x/yX_{i}(y)X_{j}(x),\ i>j.$ (3.12)

From these commutation relations, one notes there exist commutation relations
of the following form

$\displaystyle\frac{X_{i_{1}}({\bf z}_{1})X_{{i_{2}}}({\bf z}_{2})\cdots
X_{i_{m-1}}({\bf z}_{m-1})X_{i_{m}}({\bf z}_{m})}{\prod_{k=1}^{m}{\bf
z}_{k}^{m-k}}$ $\displaystyle=$ $\displaystyle\sum_{{\bf w}=({\bf w}_{1},{\bf
w}_{2},\dots,{\bf w}_{m})}A({\bf w}_{1},{\bf w}_{2},\dots,{\bf
w}_{m})X_{i_{m}}({\bf w}_{m})X_{i_{m-1}}({\bf w}_{m-1})\cdots X_{i_{2}}({\bf
w}_{2})X_{i_{1}}({\bf w}_{1}),$ (3.13)

and the problem is to determine $A({\bf w}_{1},{\bf w}_{2},\dots,{\bf w}_{m})$
explicitly. Fix ${\bf w}=({\bf w}_{1},{\bf w}_{2},\dots,{\bf w}_{m})$ and
first note that the left hand side of (3.13) can be rewritten using (3.11),
(3.12) as

$\displaystyle\frac{X_{i_{1}}({\bf z}_{1})X_{{i_{2}}}({\bf z}_{2})\cdots
X_{i_{m-1}}({\bf z}_{m-1})X_{i_{m}}({\bf z}_{m})}{\prod_{k=1}^{m}{\bf
z}_{k}^{m-k}}$ $\displaystyle=$ $\displaystyle\frac{X_{i_{1}}({\bf
w}_{1})X_{{i_{2}}}({\bf w}_{2})\cdots X_{i_{m-1}}({\bf w}_{m-1})X_{i_{m}}({\bf
w}_{m})}{\prod_{k=1}^{m}{\bf w}_{k}^{m-k}}.$ (3.14)

Next, we reverse the order of the operators in the right hand side of (3.14)
using (3.10). We note that to get the term $X_{i_{m}}({\bf
w}_{m})X_{i_{m-1}}({\bf w}_{m-1})\cdots X_{i_{2}}({\bf w}_{2})X_{i_{1}}({\bf
w}_{1})$ in this way, we only need to keep track of the first term of the
right hand side of (3.10), and we get the factor $\frac{1}{\prod_{1\leq
j<k\leq m}(1-{\bf w_{k}}/{\bf w_{j}})}$. Hence we find $A({\bf w}_{1},{\bf
w}_{2},\dots,{\bf w}_{m})$ is

$\displaystyle A({\bf w}_{1},{\bf w}_{2},\dots,{\bf
w}_{m})=\frac{1}{\prod_{k=1}^{m}{\bf w}_{k}^{m-k}}\frac{1}{\prod_{1\leq
j<k\leq m}(1-{\bf w_{k}}/{\bf w_{j}})}=\frac{1}{\prod_{1\leq j<k\leq m}({\bf
w}_{j}-{\bf w}_{k})}.$ (3.15)

These coefficients are uniquley determined by Lemma 3.2. ∎

Using Lemma 3.2, Proposition 3.4, Proposition 3.5, we determine the explicit
forms of the partition functions.

######  Theorem 3.6.

For $n\geq i_{1}>i_{2}>\cdots>i_{m}\geq 0$, we have

$\displaystyle\langle\Omega|X_{i_{1}}({\bf z}_{1})X_{{i_{2}}}({\bf
z}_{2})\cdots X_{i_{m-1}}({\bf z}_{m-1})X_{i_{m}}({\bf z}_{m})|\Omega\rangle$
$\displaystyle=$ $\displaystyle\prod_{k=1}^{m}{\bf
z}_{k}^{m-k}s_{((i_{1}-m+1)^{|{\bf z}_{1}|},(i_{2}-m+2)^{|{\bf
z}_{2}|},\dots,i_{m}^{|{\bf z}_{m}|})}({\bf z}).$ (3.16)

In particular, the simplest case $|{\bf z}_{1}|=\cdots=|{\bf z}_{m}|=1$ gives
the following.

######  Theorem 3.7.

For $n\geq i_{1}>i_{2}>\cdots>i_{m}\geq 0$, we have

$\displaystyle\langle\Omega|X_{i_{1}}(z_{1})X_{i_{2}}(z_{2})\cdots
X_{i_{m}}(z_{m})|\Omega\rangle$ $\displaystyle=$
$\displaystyle\prod_{k=1}^{m}z_{k}^{m-k}s_{(i_{1}-m+1,i_{2}-m+2,\dots,i_{m})}(z_{1},\cdots,z_{m}).$
(3.17)

###### Proof.

From (3.9), we have the following relation between partition functions

$\displaystyle\langle\Omega|X_{i_{1}}({\bf z}_{1})X_{{i_{2}}}({\bf
z}_{2})\cdots X_{i_{m-1}}({\bf z}_{m-1})X_{i_{m}}({\bf z}_{m})|\Omega\rangle$
$\displaystyle=$ $\displaystyle\prod_{k=1}^{m}{\bf z}_{k}^{m-k}\sum_{{\bf
w}=({\bf w}_{1},{\bf w}_{2},\dots,{\bf w}_{m})}\frac{1}{\prod_{1\leq j<k\leq
m}({\bf w}_{j}-{\bf w}_{k})}$ $\displaystyle\times\langle\Omega|X_{i_{m}}({\bf
w}_{m})X_{i_{m-1}}({\bf w}_{m-1})\cdots X_{i_{2}}({\bf w}_{2})X_{i_{1}}({\bf
w}_{1})|\Omega\rangle.$ (3.18)

For $n\geq i_{1}>i_{2}>\cdots>i_{m}\geq 0$, we have

$\displaystyle\langle\Omega|X_{i_{m}}({\bf w}_{m})X_{i_{m-1}}({\bf
w}_{m-1})\cdots X_{i_{2}}({\bf w}_{2})X_{i_{1}}({\bf
w}_{1})|\Omega\rangle={\bf w}_{1}^{i_{1}}{\bf w}_{2}^{i_{2}}\cdots{\bf
w}_{m-1}^{i_{m-1}}{\bf w}_{m}^{i_{m}}.$ (3.19)

by Lemma 3.2. Combining (3.18) and (3.19) we have

$\displaystyle\langle\Omega|X_{i_{1}}({\bf z}_{1})X_{{i_{2}}}({\bf
z}_{2})\cdots X_{i_{m-1}}({\bf z}_{m-1})X_{i_{m}}({\bf z}_{m})|\Omega\rangle$
$\displaystyle=$ $\displaystyle\prod_{k=1}^{m}{\bf z}_{k}^{m-k}\sum_{{\bf
w}=({\bf w}_{1},{\bf w}_{2},\dots,{\bf w}_{m})}\frac{1}{\prod_{1\leq j<k\leq
m}({\bf w}_{j}-{\bf w}_{k})}{\bf w}_{1}^{i_{1}}{\bf w}_{2}^{i_{2}}\cdots{\bf
w}_{m-1}^{i_{m-1}}{\bf w}_{m}^{i_{m}}.$ (3.20)

Using (3.8), we note the right hand side of (3.20) can be rewritten as

$\displaystyle\prod_{k=1}^{m}{\bf z}_{k}^{m-k}s_{((i_{1}-m+1)^{|{\bf
z}_{1}|},(i_{2}-m+2)^{|{\bf z}_{2}|},\dots,i_{m}^{|{\bf z}_{m}|})}({\bf z}).$
(3.21)

∎

Let us introduce hat operators [19] and natural generalizations

$\displaystyle\hat{X}_{i}^{k}(z):=\frac{d^{k}}{dz^{k}}X_{i}(z),\ \ \
k=0,1,\dots.$ (3.22)

From (3.17), we have the following.

######  Corollary 3.8.

For $n\geq i_{1}>i_{2}>\cdots>i_{m}\geq 0$, we have

$\displaystyle\langle\Omega|\hat{X}_{i_{1}}^{j_{1}}(z_{1})\hat{X}_{i_{2}}^{j_{2}}(z_{2})\cdots\hat{X}_{i_{m}}^{j_{m}}(z_{m})|\Omega\rangle$
$\displaystyle=$ $\displaystyle\frac{\partial^{j_{1}}}{\partial
z_{1}^{j_{1}}}\frac{\partial^{j_{2}}}{\partial
z_{2}^{j_{2}}}\cdots\frac{\partial^{j_{m}}}{\partial
z_{m}^{j_{m}}}\Bigg{(}\prod_{k=1}^{m}z_{k}^{m-k}s_{(i_{1}-m+1,i_{2}-m+2,\dots,i_{m})}(z_{1},\cdots,z_{m})\Bigg{)}.$
(3.23)

Let us see some examples. Consider the case $m=n$, $i_{j}=n-j+1$
$(j=1,\dots,\ell)$, $i_{j}=n-j$ $(j=\ell+1,\dots,n)$. From (3.17), we have

$\displaystyle\langle\Omega|X_{n}(z_{1})\cdots
X_{n-\ell+1}(z_{\ell})X_{n-\ell-1}(z_{\ell+1})\cdots
X_{0}(z_{n})|\Omega\rangle$ $\displaystyle=$
$\displaystyle\prod_{k=1}^{n}z_{k}^{n-k}s_{(1^{\ell},0^{n-\ell})}(z_{1},\dots,z_{n})=\prod_{k=1}^{n}z_{k}^{n-k}e_{\ell}(z_{1},\dots,z_{n}),$
(3.24)

where $e_{\ell}(z_{1},\dots,z_{n})$ is the elementary symmetric functions
$e_{\ell}(z_{1},\dots,z_{n})=\sum_{1\leq k_{1}<k_{2}<\cdots<k_{\ell}\leq
n}z_{k_{1}}z_{k_{2}}\cdots z_{k_{\ell}}$. Using

$\displaystyle\frac{\partial}{\partial
z_{k}}e_{\ell}(z_{1},\dots,z_{n})=e_{\ell-1}(z_{1},\dots,z_{k-1},z_{k+1},\dots,z_{n}),$
(3.25)

we have

$\displaystyle\langle\Omega|\hat{X}_{n}(z_{1})X_{n-1}(z_{2})\cdots
X_{n-\ell+1}(z_{\ell})X_{n-\ell-1}(z_{\ell+1})\cdots
X_{0}(z_{n})|\Omega\rangle$ $\displaystyle=$
$\displaystyle(n-1)z_{1}^{n-2}\prod_{k=2}^{n}z_{k}^{n-k}e_{\ell}(z_{1},\dots,z_{n})+\prod_{k=1}^{n}z_{k}^{n-k}e_{\ell-1}(z_{2},\dots,z_{n}).$
(3.26)

More generally, first recall the Jacobi-Trudi formula for the Schur functions

$\displaystyle s_{\lambda}(z_{1},\dots,z_{n})=\det_{1\leq
i,j\leq\ell(\lambda^{\prime})}(e_{\lambda_{i}^{\prime}-i+j}(z_{1},\dots,z_{n})),$
(3.27)

where
$\lambda_{i}^{\prime}=(\lambda_{1}^{\prime},\lambda_{2}^{\prime},\dots,\lambda_{\ell(\lambda^{\prime})}^{\prime})$
is the conjugate partition of $\lambda$, and $\ell(\lambda^{\prime})$ is the
length of $\lambda^{\prime}$. Using (3.17), (3.25), (3.27) and the property of
differential operators acting on determinants, we have

$\displaystyle\frac{\partial}{\partial z_{k}}s_{\lambda}(z_{1},\dots,z_{n})$
$\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\ell(\lambda^{\prime})}\det_{1\leq
i,j\leq\ell(\lambda^{\prime})}\Bigg{(}(1-\delta_{j,\ell}+\delta_{j,\ell}\frac{\partial}{\partial
z_{k}})e_{\lambda_{i}^{\prime}-i+j}(z_{1},\dots,z_{n})\Bigg{)}$
$\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\ell(\lambda^{\prime})}\det_{1\leq
i,j\leq\ell(\lambda^{\prime})}\Bigg{(}(1-\delta_{j,\ell})e_{\lambda_{i}^{\prime}-i+j}(z_{1},\dots,z_{n})+\delta_{j,\ell}e_{\lambda_{i}^{\prime}-i+j-1}(z_{1},\dots,z_{k-1},z_{k+1},\dots,z_{n})\Bigg{)}.$
(3.28)

Hence we get

$\displaystyle\langle\Omega|\hat{X}_{\lambda_{1}+m-1}(z_{1})X_{\lambda_{2}+m-2}(z_{2})\cdots
X_{\lambda_{m}}(z_{m})|\Omega\rangle$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
z_{1}}\Bigg{(}\prod_{k=1}^{m}z_{k}^{m-k}s_{(\lambda_{1},\lambda_{2},\dots,\lambda_{m})}(z_{1},\cdots,z_{m})\Bigg{)}$
$\displaystyle=$
$\displaystyle(m-1)z_{1}^{m-2}\prod_{k=2}^{m}z_{k}^{m-k}s_{(\lambda_{1},\lambda_{2},\dots,\lambda_{m})}(z_{1},\cdots,z_{m})+\prod_{k=1}^{m}z_{k}^{m-k}$
$\displaystyle\times\sum_{\ell=1}^{\ell(\lambda^{\prime})}\det_{1\leq
i,j\leq\ell(\lambda^{\prime})}\Bigg{(}(1-\delta_{j,\ell})e_{\lambda_{i}^{\prime}-i+j}(z_{1},\dots,z_{m})+\delta_{j,\ell}e_{\lambda_{i}^{\prime}-i+j-1}(z_{1},\dots,z_{k-1},z_{k+1},\dots,z_{m})\Bigg{)}.$
(3.29)

Let us also see the case when the variables are all specialized to 1. Recall
the factorization of Schur polynomials when specializing the variables
$z_{1}=z_{2}=\dots=z_{n}=1$

$\displaystyle s_{\lambda}(1^{n})=\prod_{1\leq k<\ell\leq
n}\frac{\lambda_{k}-\lambda_{\ell}+\ell-k}{\ell-k},$ (3.30)

which is well-known, which follows from the determinant form for example.
Applying this factorization to (3.17), we have the following.

######  Corollary 3.9.

For $n\geq i_{1}>i_{2}>\cdots>i_{m}\geq 0$, we have

$\displaystyle\langle\Omega|X_{i_{1}}(1)X_{i_{2}}(1)\cdots
X_{i_{m}}(1)|\Omega\rangle=\prod_{1\leq k<\ell\leq
m}\frac{i_{k}-i_{\ell}}{\ell-k}.$ (3.31)

This corollary corresponds to counting the number of configurations
contributing to the partition functions.

From (3.29) and (3.31) we have the following.

######  Corollary 3.10.

We have

$\displaystyle\frac{\langle\Omega|\hat{X}_{\lambda_{1}+m-1}(1)X_{\lambda_{2}+m-2}(1)\cdots
X_{\lambda_{m}}(1)|\Omega\rangle}{\langle\Omega|X_{\lambda_{1}+m-1}(1)X_{\lambda_{2}+m-2}(1)\cdots
X_{\lambda_{m}}(1)|\Omega\rangle}=m-1$ $\displaystyle+\prod_{1\leq k<\ell\leq
m}\frac{\ell-k}{\lambda_{k}-\lambda_{\ell}+\ell-k}\sum_{\ell=1}^{\ell(\lambda^{\prime})}\det_{1\leq
i,j\leq\ell(\lambda^{\prime})}\Bigg{(}(1-\delta_{j,\ell})\binom{m}{\lambda_{i}^{\prime}-i+j}+\delta_{j,\ell}\binom{m}{\lambda_{i}^{\prime}-i+j-1}\Bigg{)}.$
(3.32)

Here, $\binom{n}{k}:=n!/k!(n-k)!$, $k!:=\prod_{j=1}^{k}j$.

The simplest cases are

$\displaystyle\frac{\langle\Omega|\hat{X}_{n}(1)X_{n-1}(1)\cdots
X_{n-\ell+1}(1)X_{n-\ell-1}(1)\cdots
X_{0}(1)|\Omega\rangle}{\langle\Omega|X_{n}(1)X_{n-1}(1)\cdots
X_{n-\ell+1}(1)X_{n-\ell-1}(1)\cdots
X_{0}(1)|\Omega\rangle}=n-1+\frac{1}{\ell}.$ (3.33)

which can also be derived using (3.26).

Let us finally discuss an application/interpretation of this result to
probability theory/statistical physics. The shape for the partition functions
which we consider can be viewed as a prism with two bases which are triangles
and three lateral faces which are rectangles. To each of the lateral faces,
there are $mn$ points associated, each point colored with either red or blue.
To each edge of the triangles corresponding to bases, there are $n$ points
associated, each of which is colored with red or blue.

The following quantity

$\displaystyle\frac{\langle\Omega|\hat{X}_{i_{1}}(1)X_{i_{2}}(1)\cdots
X_{i_{m}}(1)|\Omega\rangle}{\langle\Omega|X_{i_{1}}(1)X_{i_{2}}(1)\cdots
X_{i_{m}}(1)|\Omega\rangle},$ (3.34)

corresponds to the average number of points lying on one edge of the base is
colored with red, under the boundary condition such that coloring on $mn$
points on one of the lateral faces is fixed which can be labelled by
$(i_{1},i_{2},\dots,i_{m})$, and the boundaries corresponding to the other two
lateral faces are free. (3.33) means that for the case when the fixed boundary
is labelled by
$(i_{1},i_{2},\dots,i_{n})=(n,n-1,\dots,n-\ell+1,n-\ell-1,\dots,0)$, the
average number of points on one edge of the base colored with red is
$\displaystyle n-1+\frac{1}{\ell}$.

##  4 Inhomogeneous three-dimensional partition functions and loop elementary
symmetric functions

Figure 6: The operator-valued matrix elements of $\mathcal{L}(z)$.
$\mathbf{b}^{+}$ and $\mathbf{b}^{-}$ in $\mathcal{R}$ is replaced by
$z\mathbf{b}^{+}$ and $z^{-1}\mathbf{b}^{-}$.

This section discusses an inhomogeneous generalization of the partition
function. We derive explicit representations for one of the simplest cases by
brute force computation. We use the following three-dimensional $L$-operator
$\mathcal{L}(z)$, which is a degeneration of the one in [5] (see [17], (18.23)
for example)

$\displaystyle\mathcal{L}(z)(v_{i}\otimes
v_{j})=\sum_{a=0}^{1}\sum_{b=0}^{1}v_{a}\otimes
v_{b}[\mathcal{L}(z)]_{ij}^{ab},$ (4.1)

where

$\displaystyle[\mathcal{L}(z)]_{00}^{00}$ $\displaystyle=1,\ \ \
[\mathcal{L}(z)]_{11}^{11}=1,\ \ \
[\mathcal{L}(z)]_{10}^{01}=z\mathbf{b}^{+},$ (4.2)
$\displaystyle[\mathcal{L}(z)]_{01}^{10}$
$\displaystyle=z^{-1}\mathbf{b}^{-},\ \ \
[\mathcal{L}(z)]_{01}^{01}=\mathbf{t},$ (4.3)

and $[\mathcal{L}(z)]_{ij}^{ab}=0$ otherwise. Note
$\mathcal{L}(1)=\mathcal{R}$. See Figure 6.

Let $z=(z^{(k,\ell)})_{(k,\ell)\in D_{n}}$ be a set of independent variables.
Each $z^{(k,\ell)}$ is associated with the line $L_{k\ell}$. We define the
linear operator $\overline{X}_{i}(z)$, which acts on $\bigotimes_{(k,\ell)\in
D_{n}}\mathcal{F}_{k\ell}\otimes\mathbb{C}[z^{(k,\ell)}]$, as the sum of
matrix elements of $\bigotimes_{(k,\ell)\in D_{n}}\mathcal{L}(z^{(k,\ell)})$
over all possible configurations that we mentioned in Section 2. See Figure 7.
Although $\overline{X}_{i}(z)$ depends on several variables $z^{(k,\ell)}$
$((k,\ell)\in D_{n})$, we keep to use $z$. Hereinafter, we refer to such $z$
as a multiple variable. When $z^{(k,\ell)}=z$ for all $(k,\ell)$,
$\overline{X}_{i}(z)$ reduces to $z^{-i}X_{i}(z)$.

Figure 7: The operator $\overline{X}_{i}(z)$. We use
$\mathcal{L}(z^{(k,\ell)})$ in the $k$-th row and $\ell$-th column $(1\leq
k,\ell\leq n-1,k+\ell\leq n)$.

For a string ${\bf z}=(z_{1},z_{2}\dots,z_{m})$ of $m$ multiple variables, we
define the operator $\displaystyle\overline{X}_{i}({\bf
z}):=\overline{X}_{i}(z_{1})\overline{X}_{i}(z_{2})\cdots\overline{X}_{i}(z_{m})$.

For $n$ strings of multiple variables
$\mathbf{z}_{i}:=(z_{i1},z_{i2},\dots,z_{i,m_{i}})$, $(i=1,2,\dots,n,\
z_{ij}=(z_{ij}^{(k,\ell)})_{(k,\ell)\in D_{n}})$, we consider the partition
function

$\langle\Omega|\overline{X}_{n}({\bf z}_{1})\overline{X}_{n-1}({\bf
z}_{2})\cdots\overline{X}_{2}({\bf z}_{n-1})\overline{X}_{0}({\bf
z}_{n})|\Omega\rangle.$ (4.4)

Let $m_{i}=|\mathbf{z}_{i}|$ denote the number of elements of
$\mathbf{z}_{i}$. Note the operator
$\overline{X}_{n}({\bf z}_{1})\overline{X}_{n-1}({\bf
z}_{2})\cdots\overline{X}_{2}({\bf z}_{n-1})\overline{X}_{0}({\bf z}_{n})$
depends on $\frac{(m_{1}+\dots+m_{n})n(n-1)}{2}$ variables $z_{ij}^{(k,\ell)}$
in principle. However, as shown in the theorem below, the partition functions
(4.4) which is sandwiched by the vacuum state and its dual actually depend on
fewer variables.

######  Theorem 4.1.

Assume ${\bf z}_{i}\neq\phi$ for all $i$. Let $M_{0}:=0$,
$M_{i}:=m_{1}+m_{2}+\dots+m_{i}$, $V:=M_{n-1}$, and $W:=M_{n}$. Then, we have

$\displaystyle\langle\Omega|\overline{X}_{n}({\bf
z}_{1})\overline{X}_{n-1}({\bf z}_{2})\cdots\overline{X}_{2}({\bf
z}_{n-1})\overline{X}_{0}({\bf z}_{n})|\Omega\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\prod_{i=1}^{n-1}\prod_{j=1}^{m_{i}}z_{ij}^{(i,1)}}\sum_{1\leq
k_{1}<k_{2}<\cdots<k_{V}\leq
W}\prod_{i=1}^{n-1}\prod_{j=1}^{m_{i}}w_{k_{M_{i-1}+j}}^{(i,1)},$ (4.5)

where $(w_{1},\dots,w_{W})$ is the concatenation of the strings:
$(w_{1},\dots,w_{W})=\mathbf{z}_{1}\mathbf{z}_{2}\dots\mathbf{z}_{n}$.

In particular, when $m_{1}=m_{2}=\cdots=m_{n-1}=1$, we have $V=n-1$,
$W=n-1+m_{n}$ and

$\sum_{1\leq k_{1}<k_{2}<\cdots<k_{V}\leq
W}\prod_{i=1}^{n-1}\prod_{j=1}^{m_{i}}w_{k_{M_{i-1}+j}}^{(i,1)}=\sum_{1\leq
k_{1}<k_{2}<\cdots<k_{n-1}\leq n-1+m_{n}}\prod_{i=1}^{n-1}z_{k_{i}}^{(i,1)}.$

This polynomial is a special case of loop elementary symmetric functions [22,
23, 30].

Since the notations become complicated and can be proved in the same way for
the general case, we show the following case corresponding to
$m_{1}=m_{2}=\cdots=m_{n-1}=1$.

######  Theorem 4.2.

We have

$\displaystyle\langle\Omega|\overline{X}_{n}(z_{1})\overline{X}_{n-1}(z_{2})\cdots\overline{X}_{2}(z_{n-1})\overline{X}_{0}({\bf
z}_{n})|\Omega\rangle=\frac{1}{\prod_{i=1}^{n-1}z_{i}^{(i,1)}}\sum_{1\leq
k_{1}<k_{2}<\cdots<k_{n-1}\leq n-1+m_{n}}\prod_{i=1}^{n-1}z_{k_{i}}^{(i,1)}.$
(4.6)

Figure 8:  One-column operators
$T^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})_{{\boldsymbol{i}},j}^{{\boldsymbol{a}},b}$
(left), $Y_{\ell}^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})\
(\ell=0,\dots,m-1)$ (middle) and
$Y_{m}^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})$ (right).

Before proving Theorem 4.2, let us introduce several notations. Let
$z=(z^{(k,l)})_{(k,l)\in D_{n}}$ be a multiple variable. For brevity, we write
$z^{(i)}=z^{(i,1)}$. For two sets of indices
${\boldsymbol{a}}=(a_{1},a_{2},\dots,a_{m})\in\\{0,1\\}^{m}$ and
${\boldsymbol{i}}=(i_{1},i_{2},\dots,i_{m})\in\\{0,1\\}^{m}$, we introduce the
linear operator

$\displaystyle
T^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})_{{\boldsymbol{i}},j}^{{\boldsymbol{a}},b}$
$\displaystyle:=$
$\displaystyle\sum_{k_{1},\dots,k_{m-1}}\mathcal{L}(z^{(n)})_{i_{1}k_{1}}^{a_{1}b}\otimes\mathcal{L}(z^{(n+1)})_{i_{2}k_{2}}^{a_{2}k_{1}}\otimes\cdots\otimes\mathcal{L}(z^{(n+m-2)})_{i_{m-1}k_{m-1}}^{a_{m-1}k_{m-2}}\otimes\mathcal{L}(z^{(n+m-1)})_{i_{m}j}^{a_{m}k_{m-1}},$
(4.7)

which acts on
$(\mathcal{F}_{n1}\otimes\mathbb{C}[z^{(n)}])\otimes(\mathcal{F}_{n+1,1}\otimes\mathbb{C}[z^{(n+1)}])\otimes\dots\otimes(\mathcal{F}_{n+m-1,1}\otimes\mathbb{C}[z^{(n+m-1)}])$.

We also introduce the following operators which are certain sums of the
operators which we have just introduced

$\displaystyle Y_{\ell}^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})$
$\displaystyle=\sum_{b,i_{1},\dots,i_{\ell+1}}T^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})_{(i_{1},\dots,i_{\ell+1},1^{m-\ell-1}),1}^{(0^{\ell},1^{m-\ell}),b},$
(4.8)

for $\ell=0,\dots,m-1$ and

$\displaystyle Y_{m}^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})=$
$\displaystyle\sum_{b,{\boldsymbol{i}}}T^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})_{{\boldsymbol{i}},0}^{(0^{m}),b}.$
(4.9)

Graphical descriptions of these operators are given in Figure 8.

If there is no confusion, we use the abbreviations

$\begin{gathered}{T^{[n,n+m-1]}}(z)_{{\boldsymbol{i}},j}^{{\boldsymbol{a}},b}:=T^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)})_{{\boldsymbol{i}},j}^{{\boldsymbol{a}},b},\\\
Y_{\ell}^{[n,n+m-1]}(z):=Y_{\ell}^{[n,n+m-1]}(z^{(n)},z^{(n+1)},\dots,z^{(n+m-1)}).\end{gathered}$

Let

${T^{(m)}}(z)_{{\boldsymbol{i}},j}^{{\boldsymbol{a}},b}:={T^{[1,m]}}(z)_{{\boldsymbol{i}},j}^{{\boldsymbol{a}},b},\qquad
Y_{\ell}^{(m)}(z):=Y_{\ell}^{[1,m]}(z).$

We also introduce the following notations for (dual) vectors

$\displaystyle|i_{k},\dots,i_{k+m}\rangle_{[k,k+m]}$
$\displaystyle=|i_{k}\rangle_{k1}\otimes\cdots\otimes|i_{k+m}\rangle_{k+m,1}\in\mathcal{F}_{k1}\otimes\cdots\otimes\mathcal{F}_{k+m,1},$
(4.10) $\displaystyle{}_{[k,k+m]}\langle i_{k},\dots,i_{k+m}|$
$\displaystyle={}_{k1}\langle i_{k}|\otimes\cdots\otimes{}_{k+m,1}\langle
i_{k+m}|\in\mathcal{F}^{*}_{k1}\otimes\cdots\otimes\mathcal{F}^{*}_{k+m,1}.$
(4.11)

Figure 9: The operator $\overline{X}_{i}(z_{n+1-i})$ $(2\leq i\leq n)$ in the
inhomogeneous partition functions
$\langle\Omega|\overline{X}_{n}(z_{1})\overline{X}_{n-1}(z_{2})\cdots\overline{X}_{2}(z_{n-1})\overline{X}_{0}(z_{n})\cdots\overline{X}_{0}(z_{n+k})|\Omega\rangle$.
We find the $L$-operators from the second to the $(n-1)$-th columns are all
fixed, and the $L$-operators in the first column form the operator
$Y_{n-i}^{(n-1)}(z_{n+1-i})$.

######  Lemma 4.3.

For $k$ a non-negative integer, we have

$\displaystyle\langle\Omega|\overline{X}_{n}(z_{1})\overline{X}_{n-1}(z_{2})\cdots\overline{X}_{2}(z_{n-1})\overline{X}_{0}(z_{n})\cdots\overline{X}_{0}(z_{n+k})|\Omega\rangle$
$\displaystyle=$ $\displaystyle{}_{[1,n-1]}\langle
0^{n-1}|Y_{0}^{(n-1)}(z_{1})Y_{1}^{(n-1)}(z_{2})\cdots
Y_{n-2}^{(n-1)}(z_{n-1})Y_{n-1}^{(n-1)}(z_{n})\cdots
Y_{n-1}^{(n-1)}(z_{n+k})|0^{n-1}\rangle_{[1,n-1]}.$ (4.12)

###### Proof.

This can be shown in a similar manner to the proof of Lemma 3.2.

Let
$\mathfrak{Y}:=\overline{X}_{n}(z_{1})\overline{X}_{n-1}(z_{2})\cdots\overline{X}_{2}(z_{n-1})\overline{X}_{0}(z_{n})\cdots\overline{X}_{0}(z_{n+k})$.
By investigating the bosonic Fock spaces $\mathcal{F}_{k\ell}$,
$(k,\ell)=(n-2,2)\to(n-3,3)\to(n-3,2)\to(n-4,4)\to(n-4,3)\to(n-4,2)\to\cdots\to(1,2)$,
we find that configurations that survive in
$\langle\Omega|\mathfrak{Y}|\Omega\rangle$ should satisfy the following
property: On the slice that corresponds to $\overline{X}_{i}(z_{n+1-i})$
$(2\leq i\leq n)$, the matrix element assigned to $(k,l)$ is (i) $1_{r}$ if
$k>n-i$, and (ii) $\mathbf{t}$ if $k\leq n-i$ and $l>1$. (Figure 9 presents a
such configuration.) This fact implies that
$\langle\Omega|\mathfrak{Y}|\Omega\rangle$ can be factorized as a product of
the partition function constructed only from the first column and that
constructed from the other columns. Since the latter becomes $\prod_{(k,l)\in
D_{n},k\neq 1}{}_{k\ell}\langle 0|0\rangle_{k\ell}=1$,
$\langle\Omega|\mathfrak{Y}|\Omega\rangle$ is equal to the one constructed
from the first column, i.e. we note that the three-dimensional partition
functions are reduced to partition functions

$\displaystyle{}_{[1,n-1]}\langle
0^{n-1}|Y_{0}^{(n-1)}(z_{1})Y_{1}^{(n-1)}(z_{2})\cdots
Y_{n-2}^{(n-1)}(z_{n-1})Y_{n-1}^{(n-1)}(z_{n})\cdots
Y_{n-1}^{(n-1)}(z_{n+k})|0^{n-1}\rangle_{[1,n-1]},$ (4.13)

consisting only of $L$-operators in the first column, i.e., operators
$Y_{n-i}^{(n-1)}(z)$, sandwiched by reduced (dual) vacuum vectors
${}_{[1,n-1]}\langle 0^{n-1}|$, $|0^{n-1}\rangle_{[1,n-1]}$ which are
reductions of the original vacuum vectors $\langle\Omega|,|\Omega\rangle$. ∎

Figure 10:  Another class of partition functions constructed from one-row
operators ${}_{[1,k+\ell]}\langle
1^{k},0^{\ell}|Y_{k}^{(k+\ell)}(z_{1})Y_{k+1}^{(k+\ell)}(z_{2})\cdots
Y_{k+\ell-1}^{(k+\ell)}(z_{\ell})Y_{k+\ell}^{(k+\ell)}(z_{\ell+1})\cdots
Y_{k+\ell}^{(k+\ell)}(z_{n})|0^{k+\ell}\rangle_{[1,k+\ell]}$.

From Lemma 4.3, the proof of Theorem 4.2 reduces to showing that the right-
hand side of (4.12) is given by the loop elementary symmetric function (4.6).
For this, we prepare the following Lemma 4.4 about some special type of
partition functions constructed from one-column operators (Figure 10) and loop
elementary symmetric functions.

Figure 11: Configurations which contribute to the partition functions.
Investigating the edges connecting the last two dashed lines (with which
$z^{(k+\ell-1)}$ and $z^{(k+\ell)}$ are associated) which were not colored in
Figure 10, we find only one of the edges is colored with red, and the others
are colored with blue. We also note that starting from that red edge, there is
a straight red line passing through that column, and the columns right to that
column are all colored with blue (top panel). Repeating this argument, we find
every nontrivial configuration can be graphically represented as the bottom
panel.

######  Lemma 4.4.

For $k\geq 1$ and $k+\ell\geq n$, we have

$\displaystyle{}_{[1,k+\ell]}\langle
1^{k},0^{\ell}|Y_{k}^{(k+\ell)}(z_{1})Y_{k+1}^{(k+\ell)}(z_{2})\cdots
Y_{k+\ell-1}^{(k+\ell)}(z_{\ell})Y_{k+\ell}^{(k+\ell)}(z_{\ell+1})\cdots
Y_{k+\ell}^{(k+\ell)}(z_{n})|0^{k+\ell}\rangle_{[1,k+\ell]}$ $\displaystyle=$
$\displaystyle(z_{1}^{(k+1)}\cdots z_{\ell}^{(k+\ell)})^{-1}\sum_{1\leq
j_{1}<j_{2}<\cdots<j_{k+\ell}\leq n}z_{j_{1}}^{(1)}\cdots
z_{j_{k+\ell}}^{(k+\ell)}.$ (4.14)

###### Proof.

We present a proof based on a graphical representation. See Figure 10. First,
we observe that from the $\ell$-th to the $n$-th columns, only one of the
edges with which $z^{(k+\ell-1)}$ and $z^{(k+\ell)}$ are associated connecting
the last two dashed lines is colored with red, and the rest are colored with
blue. One can observe this as follows. In the bottom layer, one can see from
Figure 10 that there can be at most one annihilation operator coming from the
$\ell$-th column. If there is no annihilation operator, then the edges in the
$\ell$-th column are colored with red, and the edges in the $(\ell+1)\sim
n$-th columns are colored with blue since the operators are sandwiched by
$|0\rangle$ and $\langle 0|$ and neither creation nor annihilation operators
can appear. When there is one annihilation operator arising from the $\ell$-th
column, there must be exactly one annihilation operator arsing from one of the
$(\ell+1)\sim n$-th columns which the uncolored edges must be colored with
red, and the other columns corresponding to identity operators must be colored
with blue.

Next, we note that colors of edges in the column layer which contains the red-
colored connecting edge are uniquely determined, and then note all colors in
the column layers right to that layer are determined to be blue (Figure 11,
top). Let us see this in some more detail. When looking the $(k+\ell-1)$-th
row of the column layer which contains the red-colored connecting edge, we
note the operator-valued $L$-operator assigned has the form
$\mathcal{L}_{i1}^{0j}$. In this case, for this to be nonzero, $i$ and $j$
must be $i=0$, $j=1$. One can continue this observation, and we find there is
a red line passing through the column, and zero-number projection operators
arise from the first to the $(k+\ell-1)$-th row. From this, we note the edges
in the column layers right to that layer are all colored with blue. This can
be understood in the same way. When looking the $(k+\ell-1)$-th row of those
column layers, we note every operator-valued $L$-operator assigned has the
form $\mathcal{L}_{i0}^{0j}$. For this to be nonzero, either $i=j=0$ which
corresponds to the identity operator or $i=j=1$ which corresponds to the
creation operator. However, since we have to multiply by the zero-number
projection operators after all, using $i=j=1$ case does not contribute to the
partition function, and one gets $i=j=0$. One can repeat this argument and
find all edges in those layers are colored with blue.

Repeating the argument above, we find that all admissible configurations can
be represented as Figure 11, bottom. There are $k+\ell$ red lines partially
passing through the column, which can be labelled by $k+\ell$ integers $1\leq
j_{1}<j_{2}<\dots<j_{k+\ell}\leq n$. Let us denote the contribution to the
partition function coming from this configuration as
$Z_{j_{1},j_{2},\dots,j_{k+\ell}}$.

Computing the products of weights for this configuration, we find

$\displaystyle Z_{j_{1},j_{2},\dots,j_{k+\ell}}=(z_{1}^{(k+1)}\cdots
z_{\ell}^{(k+\ell)})^{-1}z_{j_{1}}^{(1)}\cdots z_{j_{k+\ell}}^{(k+\ell)}.$
(4.15)

Taking all configurations into account, i.e. taking sum $\sum_{1\leq
j_{1}<j_{2}<\dots<j_{k+\ell}\leq n}Z_{j_{1},j_{2},\dots,j_{k+\ell}}$ gives the
explicit expression of the partition function (4.14). ∎

Finally, we show the partition functions given in Figure 12 give loop
elementary functions.

Figure 12:  Partition functions ${}_{[1,k]}\langle
0^{k}|Y_{0}^{(k)}(z_{1})Y_{1}^{(k)}(z_{2})\cdots
Y_{k-1}^{(k)}(z_{k})Y_{k}^{(k)}(z_{k+1})\cdots
Y_{k}^{(k)}(z_{n})|0^{k}\rangle_{[1,k]}$ constructed from one-column
operators.

######  Proposition 4.5.

We have

$\displaystyle{}_{[1,k]}\langle
0^{k}|Y_{0}^{(k)}(z_{1})Y_{1}^{(k)}(z_{2})\cdots
Y_{k-1}^{(k)}(z_{k})Y_{k}^{(k)}(z_{k+1})\cdots
Y_{k}^{(k)}(z_{n})|0^{k}\rangle_{[1,k]}$ $\displaystyle=$
$\displaystyle(z_{1}^{(1)}\cdots z_{k}^{(k)})^{-1}\sum_{1\leq
j_{1}<j_{2}<\cdots<j_{k}\leq n}z_{j_{1}}^{(1)}\cdots z_{j_{k}}^{(k)}.$ (4.16)

###### Proof.

We prove by induction on $k$ with the help of Lemma 4.4. Suppose (4.16) with
$k$ replaced by $k-1$ holds. For $m\geq 0$, consider the projection operator

$|m\rangle_{1}{}_{1}\langle
m|:\mathcal{F}_{11}\otimes\mathcal{F}_{21}\otimes\dots\otimes\mathcal{F}_{k1}\to\mathcal{F}_{11}\otimes\mathcal{F}_{21}\otimes\dots\otimes\mathcal{F}_{k1},$

which sends $|i_{1},i_{2},\dots,i_{k}\rangle_{[1,k]}$ to
$|m,i_{2},\dots,i_{k}\rangle_{[1,k]}\cdot\langle m|i_{1}\rangle$. Then the
identity operator $\mathrm{id}$ decomposes into

$\displaystyle\mathrm{id}=\sum_{m\geq 0}|m\rangle_{1}{}_{1}\langle m|.$ (4.17)

By using (4.17), we decompose the left-hand side of (4.16) as

$\displaystyle{}_{[1,k]}\langle
0^{k}|Y_{0}^{(k)}(z_{1})|0\rangle_{1}{}_{1}\langle 0|Y_{1}^{(k)}(z_{2})\cdots
Y_{k-1}^{(k)}(z_{k})Y_{k}^{(k)}(z_{k+1})\cdots
Y_{k}^{(k)}(z_{n})|0^{k}\rangle_{[1,k]}$ $\displaystyle+$
$\displaystyle\sum_{i=2}^{k-1}{}_{[1,k]}\langle
0^{k}|Y_{0}^{(k)}(z_{1})|1\rangle_{1}{}_{1}\langle
1|Y_{1}^{(k)}(z_{2})|1\rangle_{1}\cdots{}_{1}\langle
1|Y_{i-2}^{(k)}(z_{i-1})|1\rangle_{1}{}_{1}\langle
1|Y_{i-1}^{(k)}(z_{i})|0\rangle_{1}$ $\displaystyle\times{}_{1}\langle
0|Y_{i}^{(k)}(z_{i+1})\cdots Y_{k-1}^{(k)}(z_{k})Y_{k}^{(k)}(z_{k+1})\cdots
Y_{k}^{(k)}(z_{n})|0^{k}\rangle_{[1,k]}$ $\displaystyle+$
$\displaystyle\sum_{i=k}^{n}{}_{[1,k]}\langle
0^{k}|Y_{0}^{(k)}(z_{1})|1\rangle_{1}{}_{1}\langle
1|Y_{1}^{(k)}(z_{2})|1\rangle_{1}\cdots{}_{1}\langle
1|Y_{k-1}^{(k)}(z_{k})|1\rangle_{1}$ $\displaystyle\times{}_{1}\langle
1|Y_{k}^{(k)}(z_{k+1})|1\rangle_{1}\cdots{}_{1}\langle
1|Y_{k}^{(k)}(z_{i})|0\rangle_{1}{}_{1}\langle 0|Y_{k}^{(k)}(z_{i+1})\cdots
Y_{k}^{(k)}(z_{n})|0^{k}\rangle_{[1,k]}.$ (4.18)

Note that here we use the fact that one can further restrict the summands of
the identity operator (4.17) only to $|0\rangle\langle 0|$ and
$|1\rangle\langle 1|$. This is because in the top slice, only the first column
can produce an annihilation operator (see Figure 12), and all the operators
coming from the first row are finally sandwiched by $|0\rangle$ and $\langle
0|$, which means $|m\rangle$ $(m\geq 2)$ do not appear since we cannot change
to $|0\rangle$ by acting just one annihilation operator, and we can neglect
$|m\rangle\langle m|$ $(m\geq 2)$.

We first examine the first term in (4.18). One can see the following relations
hold

$\displaystyle{}_{[1,k]}\langle
0^{k}|Y_{0}^{(k)}(z_{1})|0\rangle_{1}={}_{[2,k]}\langle 0^{k-1}|,$ (4.19)

and

$\displaystyle{}_{[2,k]}\langle 0^{k-1}|{}_{1}\langle
0|Y_{1}^{(k)}(z_{2})\cdots Y_{k-1}^{(k)}(z_{k})Y_{k}^{(k)}(z_{k+1})\cdots
Y_{k}^{(k)}(z_{n})|0^{k}\rangle_{[1,k]}$ $\displaystyle={}_{[2,k]}\langle
0^{k-1}|Y_{1}^{(k-1)}(z_{2}^{\prime})\cdots
Y_{k-1}^{(k-1)}(z_{k}^{\prime})Y_{k}^{(k-1)}(z_{k+1}^{\prime})\cdots
Y_{k}^{(k-1)}(z_{n}^{\prime})|0^{k-1}\rangle_{[2,k]}.$ (4.20)

Here, $z_{j}^{\prime}=(z_{j}^{(2)},z_{j}^{(3)},\dots,z_{j}^{(k)})$.

One can apply the inductive assumption to the right hand side of (4.20), which
is

$\displaystyle{}_{[2,k]}\langle 0^{k-1}|Y_{1}^{(k-1)}(z_{2}^{\prime})\cdots
Y_{k-1}^{(k-1)}(z_{k}^{\prime})Y_{k}^{(k-1)}(z_{k+1}^{\prime})\cdots
Y_{k}^{(k-1)}(z_{n}^{\prime})|0^{k-1}\rangle_{[2,k]}$
$\displaystyle=(z_{2}^{(2)}\cdots z_{k}^{(k)})^{-1}\sum_{2\leq
j_{2}<\cdots<j_{k}\leq n}z_{j_{2}}^{(2)}\cdots z_{j_{k}}^{(k)},$ (4.21)

hence combining with (4.19) and (4.20), we find the first term of the left
hand side of (4.18) is (4.21).

Figure 13:  The first term in (4.18) is the case when 1 appears in the
leftmost inner part in the top dashed line. After coloring parts in the solid
lines which are already fixed to be red or blue, we get the top panel. We can
remove the leftmost column layer as everything is fixed. We can also remove
the top row layer as the matrix elements of the $L$-operators from the third
to the last columns are all identity operators. Removing the leftmost column
and the top row layers, we get the bottom panel, which is nothing but (4.21).

Figure 13 gives a graphical understanding of the computations above to get
(4.21).

Next we deal with the summands in (4.18). Let us see the summands in the first
sum. One can show the first factor is

$\displaystyle{}_{[1,k]}\langle
0^{k}|Y_{0}^{(k)}(z_{1})|1\rangle_{1}=(z_{1}^{(1)})^{-1}{}_{[2,k]}\langle
0^{k-1}|.$ (4.22)

Acting the second factor on ${}_{[2,k]}\langle 0^{k-1}|$, we get

$\displaystyle{}_{[2,k]}\langle 0^{k-1}|{}_{1}\langle
1|Y_{1}^{(k)}(z_{2})|1\rangle_{1}=(z_{2}^{(2)})^{-1}{}_{[2,k]}\langle
1,0^{k-2}|.$ (4.23)

One can repeat this computation. Using

$\displaystyle{}_{[2,k]}\langle 1^{j},0^{k-1-j}|{}_{1}\langle
1|Y_{1}^{(k)}(z_{2+j})|1\rangle_{1}$
$\displaystyle=(z_{2+j}^{(2+j)})^{-1}{}_{[2,k]}\langle 1^{1+j},0^{k-2-j}|,\ \
\ j=0,\dots,i-3,$ (4.24) $\displaystyle{}_{[2,k]}\langle
1^{i-2},0^{k+1-i}|{}_{1}\langle 1|Y_{1}^{(k)}(z_{i})|1\rangle_{1}$
$\displaystyle=z_{i}^{(1)}(z_{i}^{(i)})^{-1}{}_{[2,k]}\langle
1^{i-1},0^{k-i}|,$ (4.25)

we get

$\displaystyle{}_{[1,k]}\langle
0^{k}|Y_{0}^{(k)}(z_{1})|1\rangle_{1}{}_{1}\langle
1|Y_{1}^{(k)}(z_{2})|1\rangle_{1}\cdots{}_{1}\langle
1|Y_{i-2}^{(k)}(z_{i-1})|1\rangle_{1}{}_{1}\langle
1|Y_{i-1}^{(k)}(z_{i})|0\rangle_{1}$ $\displaystyle=$ $\displaystyle
z_{i}^{(1)}(z_{1}^{(1)}\cdots z_{i}^{(i)})^{-1}{}_{[2,k]}\langle
1^{i-1},0^{k-i}|.$ (4.26)

Next we note that

$\displaystyle{}_{[2,k]}\langle 1^{i-1},0^{k-i}|{}_{1}\langle
0|Y_{i}^{(k)}(z_{i+1})\cdots Y_{k-1}^{(k)}(z_{k})Y_{k}^{(k)}(z_{k+1})\cdots
Y_{k}^{(k)}(z_{n})|0^{k}\rangle_{[1,k]}$ $\displaystyle=$
$\displaystyle{}_{[2,k]}\langle
1^{i-1},0^{k-i}|Y_{i}^{(k-1)}(z_{i+1}^{\prime})\cdots
Y_{k-1}^{(k-1)}(z_{k}^{\prime})Y_{k}^{(k-1)}(z_{k+1}^{\prime})\cdots
Y_{k}^{(k-1)}(z_{n}^{\prime})|0^{k-1}\rangle_{[2,k]}.$ (4.27)

Applying (4.14) in Lemma 4.4 to the right hand side of (4.27), we get

$\displaystyle{}_{[2,k]}\langle
1^{i-1},0^{k-i}|Y_{i}^{(k-1)}(z_{i+1}^{\prime})\cdots
Y_{k-1}^{(k-1)}(z_{k}^{\prime})Y_{k}^{(k-1)}(z_{k+1}^{\prime})\cdots
Y_{k}^{(k-1)}(z_{n}^{\prime})|0^{k-1}\rangle_{[2,k]}$ $\displaystyle=$
$\displaystyle(z_{i+1}^{(i+1)}\cdots z_{k}^{(k)})^{-1}\sum_{i+1\leq
j_{2}<\cdots<j_{k}\leq n}z_{j_{2}}^{(2)}\cdots z_{j_{k}}^{(k)}.$ (4.28)

Together with (4.26) and (4.27), the first sum of the left hand side of (4.18)
is given by

$\displaystyle\sum_{i=2}^{k-1}(z_{1}^{(1)}\cdots
z_{k}^{(k)})^{-1}z_{i}^{(1)}\sum_{i+1\leq j_{2}<\cdots<j_{k}\leq
n}z_{j_{2}}^{(2)}\cdots z_{j_{k}}^{(k)}.$ (4.29)

Figure 14:  A summand of the first sum in (4.18) is graphically represented as
the top panel after coloring parts in the solid lines which are already fixed
to be red or blue. Removing the leftmost column and the top row layers, we get
the bottom panel, which corresponds to (4.28). The removed parts contribute
the factor $z_{i}^{(1)}(z_{1}^{(1)}\cdots z_{i}^{(i)})^{-1}$. Multiplying with
(4.28) gives (4.29).

See Figure 14 for a graphical derivation of (4.29).

In the same way, we can prove that the second sum of the left hand side of
(4.18) is given by

$\displaystyle\sum_{i=k}^{n-k+1}(z_{1}^{(1)}\cdots
z_{k}^{(k)})^{-1}z_{i}^{(1)}\sum_{i+1\leq j_{2}<\cdots<j_{k}\leq
n}z_{j_{2}}^{(2)}\cdots z_{j_{k}}^{(k)}.$ (4.30)

Finally, we note that the sum of (4.21), (4.29) and (4.30) can be written as

$\displaystyle(z_{1}^{(1)}\cdots z_{k}^{(k)})^{-1}\sum_{1\leq
j_{1}<j_{2}<\cdots<j_{k}\leq n}z_{j_{1}}^{(1)}\cdots z_{j_{k}}^{(k)},$ (4.31)

which completes the proof of (4.16).

∎

Combining Lemma 4.3 and Proposition 4.5, we get Theorem 4.2.

## Acknowledgement

S.I. is supported by Grant-in-Aid for Scientific Research 19K03605, 22K03239,
23K03056, JSPS. K.M. is supported by Grant-in-Aid for Scientific Research
21K03176, 20K03793, JSPS. R.O. is supported by Grant-in-Aid for Scientific
Research 21K03180, JSPS.

## Appendix A: Tetrahedron equation

Figure 15:  Tetrahedron equation. The left and right figure represents
$\mathcal{M}_{126}(z_{12})\mathcal{M}_{346}(z_{34})\mathcal{L}_{135}(z_{13})\mathcal{L}_{245}(z_{24})$
and
$\mathcal{L}_{245}(z_{24})\mathcal{L}_{135}(z_{13})\mathcal{M}_{346}(z_{34})\mathcal{M}_{126}(z_{12}),$
respectively.

We record the tetrahedron equation and the original three-dimensional
$R$-matrix for completeness. See [5, 17, 19] for more details. (4.1) is the
$q=0$ degeneration of the following version

$\displaystyle\mathcal{L}(z)(v_{i}\otimes
v_{j})=\sum_{a=0}^{1}\sum_{b=0}^{1}v_{a}\otimes
v_{b}[\mathcal{L}(z)]_{ij}^{ab},$

where

$\displaystyle[\mathcal{L}(z)]_{00}^{00}$ $\displaystyle=1,\ \ \
[\mathcal{L}(z)]_{11}^{11}=1,\ \ \
[\mathcal{L}(z)]_{10}^{01}=z\mathbf{a}^{+},$
$\displaystyle[\mathcal{L}(z)]_{01}^{10}$
$\displaystyle=z^{-1}\mathbf{a}^{-},\ \ \
[\mathcal{L}(z)]_{01}^{01}=\mathbf{k},\ \ \
[\mathcal{L}(z)]_{10}^{10}=-q\mathbf{k},$

and $[\mathcal{L}(z)]_{ij}^{ab}=0$ otherwise. Here, $\mathbf{a}^{+}$,
$\mathbf{a}^{-}$, $\mathbf{k}$, together with $\mathbf{k}^{-1}$ form the
$q$-Oscillator algebra whose defining relations are

$\displaystyle\mathbf{k}\mathbf{a}^{+}=q\mathbf{a}^{+}\mathbf{k},\ \ \
\mathbf{k}\mathbf{a}^{-}=q^{-1}\mathbf{a}^{-}\mathbf{k},\ \ \
\mathbf{a}^{-}\mathbf{a}^{+}=1-q^{2}\mathbf{k}^{2},$
$\displaystyle\mathbf{a}^{+}\mathbf{a}^{-}=1-\mathbf{k}^{2},\ \ \
\mathbf{k}\mathbf{k}^{-1}=\mathbf{k}^{-1}\mathbf{k}=1,$

The operators act on the basis of $q$-bosonic Fock space
$\mathcal{F}_{q}=\displaystyle\oplus_{m=0}^{\infty}\mathbb{C}(q)|m\rangle$ as

$\displaystyle\mathbf{k}|m\rangle=q^{m}|m\rangle,\ \ \
\mathbf{a}^{+}|m\rangle=|m+1\rangle,\ \ \
\mathbf{a}^{-}|m\rangle=(1-q^{2m})|m-1\rangle.$

Define also $\mathcal{M}(z):=\mathcal{L}(z)|_{q\to-q}$. $\mathcal{L}(z)$ and
$\mathcal{M}(z)$ can be viewed as operators acting on $V\otimes
V\otimes\mathcal{F}_{q}$. We can naturally extend this to operators acting on
the tensor product of many more two-dimensional and Fock spaces by acting on
three spaces nontrivially and the other spaces as identity. It is customary to
indicate the three spaces as subscripts. $\mathcal{L}_{ijk}(z)$ and
$\mathcal{M}_{ijk}(z)$ imply they act on $V_{i}$, $V_{j}$ and
$(\mathcal{F}_{q})_{k}$ nontrivially and the other spaces as identity. Using
these notations, the operators satisfy the following tetrahedron equation
(Figure 15) [5, 17, 19]

$\displaystyle\mathcal{M}_{126}(z_{12})\mathcal{M}_{346}(z_{34})\mathcal{L}_{135}(z_{13})\mathcal{L}_{245}(z_{24})$
$\displaystyle=$
$\displaystyle\mathcal{L}_{245}(z_{24})\mathcal{L}_{135}(z_{13})\mathcal{M}_{346}(z_{34})\mathcal{M}_{126}(z_{12}),$

where $z_{ij}=z_{i}/z_{j}$. Both hand sides act on $V_{1}\otimes V_{2}\otimes
V_{3}\otimes V_{4}\otimes(\mathcal{F}_{q})_{5}\otimes(\mathcal{F}_{q})_{6}$.
The Zamolodchikov-Faddeev algebra commutation relations (3.3) are derived by
using the tetrahedron equation [17, 19]. The $q=0$ degeneration of
$\mathcal{L}(z)$ in this Appendix gives the one used in section 4. Note
$\lim_{q\to 0}\mathbf{a}^{+}=\mathbf{b}^{+},\lim_{q\to
0}\mathbf{a}^{-}=\mathbf{b}^{-},\lim_{q\to 0}\mathbf{k}=\mathbf{t}$.

## Appendix B: Figures of configurations

We give several figures in this appendix. Figure 16 is an example of Lemma 3.2
which corresponds to the case
$\langle\Omega|X_{1}^{(4)}(z_{1})X_{2}^{(4)}(z_{2})X_{3}^{(4)}(z_{3})X_{3}^{(4)}(z_{4})X_{4}^{(4)}(z_{5})|\Omega\rangle$.
The figure is the unique configuration. The layer corresponding to
$X_{1}^{(4)}(z_{1})$, $X_{2}^{(4)}(z_{2})$, $X_{3}^{(4)}(z_{3})$,
$X_{3}^{(4)}(z_{4})$ and $X_{4}^{(4)}(z_{5})$ contributes the weight $z_{1}$,
$z_{2}^{2}$, $z_{3}^{3}$, $z_{4}^{3}$ and $z_{5}^{4}$ respectively, and
multiplying them gives $z_{1}z_{2}^{2}z_{3}^{3}z_{4}^{3}z_{5}^{4}$.

Figure 16:  The unique nontrivial configuration for
$\langle\Omega|X_{1}^{(4)}(z_{1})X_{2}^{(4)}(z_{2})X_{3}^{(4)}(z_{3})X_{3}^{(4)}(z_{4})X_{4}^{(4)}(z_{5})|\Omega\rangle$.
The layer corresponding to $X_{1}^{(4)}(z_{1})$, $X_{2}^{(4)}(z_{2})$,
$X_{3}^{(4)}(z_{3})$, $X_{3}^{(4)}(z_{4})$ and $X_{4}^{(4)}(z_{5})$
contributes the weight $z_{1}$, $z_{2}^{2}$, $z_{3}^{3}$, $z_{4}^{3}$ and
$z_{5}^{4}$ respectively, and multiplying them gives
$z_{1}z_{2}^{2}z_{3}^{3}z_{4}^{3}z_{5}^{4}$.

Figure 17 gives an example of checking Theorem 3.6 which corresponds to the
case
$\langle\Omega|X_{3}^{(4)}(z_{1})X_{3}^{(4)}(z_{2})X_{1}^{(4)}(z_{3})|\Omega\rangle$.
There are three nontrivial configurations, and each of them contributes the
weight $z_{1}^{3}z_{2}^{2}z_{3}^{2}$, $z_{1}^{3}z_{2}^{3}z_{3}$ and
$z_{1}^{2}z_{2}^{3}z_{3}^{2}$ respectively. Taking sum of the weights gives
$z_{1}^{3}z_{2}^{2}z_{3}^{2}+z_{1}^{3}z_{2}^{3}z_{3}+z_{1}^{2}z_{2}^{3}z_{3}^{2}=z_{1}z_{2}s_{(2,2,1)}(z_{1},z_{2},z_{3})$.

Figure 17:  Three nontrivial configurations for
$\langle\Omega|X_{3}^{(4)}(z_{1})X_{3}^{(4)}(z_{2})X_{1}^{(4)}(z_{3})|\Omega\rangle$.
Configuration in top left, top right and bottom panel contributes the weight
$z_{1}^{3}z_{2}^{2}z_{3}^{2}$, $z_{1}^{3}z_{2}^{3}z_{3}$ and
$z_{1}^{2}z_{2}^{3}z_{3}^{2}$ respectively, and taking sum gives
$z_{1}z_{2}s_{(2,2,1)}(z_{1},z_{2},z_{3})$.

## References

  * [1] R. J. Baxter, Partition function for the eight-vertex lattice model, Ann. Phys. 70, 193–-228 (1972).
  * [2] R. J. Baxter, On Zamolodchikov’s Solution of the Tetrahedron Equations, Commun. Math. Phys. 88, 185–205 (1983).
  * [3] R. J. Baxter, The Yang-Baxter equations and the Zamolodchikov model, Physica D 18, 321–-347 (1986).
  * [4] V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, Quantum geometry of three-dimensional lattices, J. Stat. Mech. P07004 (2008).
  * [5] V. V. Bazhanov and S. M. Sergeev, Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups J. Phys. A: Math. General 39, 3295 (2006).
  * [6] V. V. Bazhanov and Y. G. Stroganov, Free fermions on a three-dimensional lattice and tetrahedron equations, Nucl. Phys. B 230, 435–454 (1984).
  * [7] B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a ID asymmetric exclusion model using a matrix formulation, J. Phys. A : Math. Gen. 26, 1493–1617 (1993).
  * [8] V. G. Drinfeld, Quantum Groups, Vol. 1 of Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, pp.198–-820 (1987).
  * [9] L. D. Faddeev, N. Y. Reshetikhin, and L. A. Takhtajan, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1, 193–-225 (1990).
  * [10] L. D. Faddeev, E. K. Sklyanin, and L. A. Takhtajan, The quantum inverse problem method. 1, Theor. Math. Phys. 40, 688–-706 (1979).
  * [11] P. A. Ferrari and J. B. Martin, Stationary distributions of multi type totally asymmetric exclusion processes, Ann. Probab. 35, 807–832 (2007).
  * [12] M. Jimbo. A $q$-difference analogue of $U$($\mathfrak{g}$) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–-69 (1985).
  * [13] R. M. Kashaev, On discrete three-dimensional equations associated with the local Yang-Baxter relation. Lett. Math. Phys. 38, 389–-397 (1996).
  * [14] R. M. Kashaev, I. G. Korepanov, and S. M. Sergeev, Functional tetrahedron equation, Theor. Math. Phys. 117, 1402–1413 (1998).
  * [15] M. Kashiwara, Crystalizing the $q$-analogue of universal enveloping algebras, Commun. Math. Phys. 133 249–-260 (1990).
  * [16] I. G. Korepanov, Tetrahedral Zamolodchikov algebras corresponding to Baxter’s $L$-operators, Commun. Math. Phys. 154, 85–97 (1993).
  * [17] A. Kuniba, Quantum Groups in Three-Dimensional Integrability, Springer, Singapore (2022).
  * [18] A. Kuniba, S. Maruyama, and M. Okado, Multispecies TASEP and combinatorial $R$, J. Phys. A: Math. Theor. 48, 34FT02 (2015).
  * [19] A. Kuniba, S. Maruyama, and M. Okado, Multispecies TASEP and tetrahedron equation, J. Phys. A: Math. Theor. 49, 114001 (2016).
  * [20] A. Kuniba, S. Matsuike, and A. Yoneyama New Solutions to the Tetrahedron Equation Associated with Quantized Six-Vertex Models, Commun. Math. Phys. 401, 3247–-3276 (2023).
  * [21] A. Kuniba, M. Okado, and S. Sergeev, Tetrahedron equation and generalized quantum groups, J. Phys. A: Math. Theor. 48, 304001 (2015).
  * [22] T. Lam, Loop symmetric functions and factorizing matrix polynomials, Fifth International Congress of Chinese Mathematicians, AMS/IP Studies in Advanced Mathematics, 51, (2012), 609-628.
  * [23] T. Lam and P. Pylyavskyy, Total positivity in loop groups, I: Whirls and curls, Adv. Math. 230, 1222–1271, (2012).
  * [24] V. V. Mangazeev, V. V. Bazhanov, and S. M. Sergeev, An integrable 3D lattice model with positive Boltzmann weights J. Phys. A: Math. Theor. 46, 465206 (2013).
  * [25] K. Motegi, Yang-Baxter algebra, higher rank partition functions and K-theoretic Gysin map for partial flag bundles, arXiv:2210.03965.
  * [26] A. Nakayashiki and Y. Yamada, Kostka polynomials and energy functions in solvable lattice models, Sel. Math, New Ser. 3, 547–599 (1997).
  * [27] P. Pragacz, A Gysin formula for Hall-Littlewood polynomials, Proceedings of the American Mathematical Society, 143, 4705–-4711 (2015).
  * [28] K. Shigechi and M. Uchiyama, Boxed Skew Plane Partition and Integrable Phase Model, J. Phys. A: Math. Gen. 38, 10287 (2005).
  * [29] F. Spitzer, Interaction of Markov processes, Adv. Math. 5, 246–290 (1970).
  * [30] Y. Yamada, A birational representation of Weyl group, combinatorial $R$ matrix and discrete Toda equation, in Physics and Combinatorics 2000 (Nagoya), ed. by A. N. Kirillov, N. Liskova (World Scientific, Singapore, 2001), pp. 305–-319.
  * [31] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19, 1312–-1315 (1967).
  * [32] A. B. Zamolodchikov, Tetrahedra equations and integrable systems in three-dimensional space, Soviet Phys. JETP 79, 641–664 (1980).
  * [33] A. B. Zamolodchikov, Tetrahedron equations and the relativistic $S$-matrix of straight strings in 2+1 dimensions, Commun. Math. Phys. 79, 489–505 (1981).


# Three-boson stability for boosted interactions towards the zero-range limit

K. Mohseni<EMAIL_ADDRESS>A. J. Chaves  D. R. da Costa  T. Frederico  M. R.
Hadizadeh  Instituto Tecnológico de Aeronáutica, DCTA, 12228-900 São José dos
Campos, Brazil  Departamento de Física, Universidade Federal do Ceará, Caixa
Postal 6030, Campus do Pici, 60455-900 Fortaleza, Ceará, Brazil  College of
Engineering, Science, Technology and Agriculture, Central State University,
Wilberforce, OH 45384, USA  Department of Physics and Astronomy, Ohio
University, Athens, OH 45701, USA

###### Abstract

We study the three-boson bound-state mass and wave functions for ground and
excited states within the three-body relativistic framework with Kamada and
Glöcke boosted potentials in the limit of a zero-range interaction. We adopt a
nonrelativistic short-range separable potential, with Yamaguchi and Gaussian
form factors, and drive them towards the zero-range limit by letting the form
factors’ momentum scales go to large values while keeping the two-body binding
fixed. We show that the three-boson relativistic masses and wave functions are
model-independent towards the zero-range limit, and the Thomas collapse is
avoided, while the nonrelativistic limit kept the Efimov effect. Furthermore,
the stability in the zero-range limit is a result of the reduction of boosted
potential with the increase of the virtual pair center of mass momentum within
the three-boson system. Finally, we compare the present results with Light-
Front and Euclidean calculations.

###### keywords:

Faddeev equations, relativistic interaction, zero-range limit

††journal: Physics Letters B

The interpretation of novel results for the spectrum of hadron and multi-
hadron states from Lattice Quantum Chromodynamics (LQCD) simulations serves as
a new motivation to address relativistic few-body systems.

In this context, it is known that two-body (2B) phase shifts can be extracted
from the volume dependence of the spectrum by using the Lüscher formula with
two-particle quantization condition [1, 2]. Beyond 2B, the spectra of multi-
pion states in maximum isospin levels were computed by the NPLQCD
collaboration [3, 4] in a finite volume more than a decade ago, and more
recently, LQCD calculations were performed for the two- and three-pion finite-
volume spectra for isospin $I=2$ and $3$, respectively [5]. The $I=3$ spectrum
was calculated within a unitary three-dimensional framework for the two- and
three-pion scattering amplitudes in a finite-volume discretized with periodic
boundary conditions [6] (see also [7]). Furthermore, the finite-volume energy
spectrum of the $K^{-}K^{-}K^{-}$ system was also obtained from LQCD
calculations [8].

A field theory on a four-dimensional (4D) lattice is defined with periodic
(anti-periodic) boundary conditions in the case of bosonic (fermionic) fields,
which includes the time direction as well (see e.g. [9]). In correspondence,
the 4D finite-volume formulation of the Faddeev-like equations in Euclidean
space was recently implemented [10] in the 4D three-boson Faddeev-Bethe-
Salpeter (FBS) equation for the contact interaction [11].

An important issue that permeates these continuum treatments of the
relativistic three-boson system is its stability in the limit of a zero-range
interaction, as it is known that the Thomas collapse [12, 13] is present in
the nonrelativistic three-boson system with contact potentials. The stability
was shown by solving the Light-Front (LF) reduction of the FBS equation
truncated at the valence state [14, 15]. The stability of the three-boson
system was also shown by solving the 4D FBS equation for the contact
interaction in Euclidean space [16]. It was then confirmed the relative
importance of the implicit three-body interactions of relativistic origin
[17], which is missing in the formulation of the valence state integral
equation.

This context motivates us to study the stability of the three-boson system in
the limit of contact interactions within other frameworks to formulate the
relativistic Faddeev equations, for example, using boosted potentials [18].
The boost concept comes from the moving 2B subsystem in the rest frame of the
three-particle system. One approach for calculating 2B boosted $T-$matrices is
solving the relativistic Lippmann-Schwinger equation for boosted 2B
potentials. At the 2B level, the relativistic 2B potentials are designed to
preserve the 2B observables for bound and scattering states. Kamada and
Glöckle have shown that relativistic and boosted potentials can be obtained
directly from nonrelativistic potentials by solving a quadratic equation using
an iterative scheme [18]. Once the boosted potential is obtained, the 2B
$T-$matrix in the rest frame of the three-body system can be computed as
required for the kernel of the relativistic Faddeev equations [19, 20].

The above formulation will be explored to study the three-boson stability
separable interactions (Yamaguchi-type and Gaussian-type) driven to the zero-
range limit. Our goal is to solve the relativistic Faddeev equations, and
obtain both the binding energies and associated wave functions, and study
their properties when the interaction range is driven towards zero.

Before presenting our study in detail, we should remark that the relativistic
Faddeev approach with boosted interactions is not based on field theory, but
it uses a relativistic version of the phenomenological 2B potential. It
belongs to one of the three forms of relativistic dynamics proposed by Dirac
in 1949 [21]: the instant form, the LF form, and the point form. The different
forms of dynamics are characterized by the number of kinematical and dynamical
boosts, in correspondence with the generators of the Poincaré group. The most
commonly used forms of dynamics are the instant and LF forms, and we restrict
ourselves to those ones. The kinematical boosts keep the initial state hyper-
surface invariant and do not contain the interaction, while the dynamical ones
depend on the interaction. In the instant form, six out of ten generators of
the Poincaré group are kinematical, while four are dynamical and contain the
interaction. In the LF form, seven generators are kinematical, and three are
dynamical. This form corresponds to the one with a maximal number of
kinematical generators (for a thorough discussion of this form of dynamics
applied to nuclear few-body systems, see e.g. Ref. [22]). The relativistic
framework developed by Kamada and Glöckle is within the class of the instant
form dynamics. Their development applied to few-nucleon systems was designed
to keep the relativistic deuteron binding energy and nucleon-nucleon phase
shifts unaltered from the results obtained with the nonrelativistic
calculations. Furthermore, it provides energy states with good angular
momentum quantum numbers, as usual in the nonrelativistic frameworks. In this
way, the relativistic Faddeev approach with boosted potential can be viewed as
one possible and practical implementation of the instant form of dynamics. We
will also quantitatively illustrate the difference between the Kamada and
Glöckle instant form framework outcomes and the results from LF and field
theoretical models in the limit of zero-range interactions.

Relativistic Faddeev approach with boosted interactions. The relativistic
Faddeev equations for the bound state of three identical particles were
recently derived in momentum space as a function of relativistic Jacobi
momentum vectors [19, 20]. A partial wave projection of relativistic Faddeev
equation in an $s-$wave channel is given by

$\psi(p,k)=4\pi\,G^{r}_{0}(p,k)\,\int_{0}^{\infty}\hbox{d}k^{\prime}{k^{\prime}_{2}}^{2}\int_{-1}^{1}\hbox{d}x^{\prime}\,N(k,k^{\prime},x^{\prime})\
T_{k}(p,\tilde{\pi};\epsilon)\ \psi(\pi,k^{\prime}),$ (1)

where $p$ and $k$ are the relativistic Jacobi momenta,
$G^{r}_{0}(p,k)=[M_{t}-\omega_{k}(p)-\Omega(k)]^{-1}$ is the relativistic free
propagator, where $M_{t}=E_{t}+3m$ is the 3B mass eigenvalue, $m$ is the mass
of each particle, $\omega_{k}(p)=\sqrt{\omega^{2}(p)+k^{2}}$,
$\omega(p)=2\sqrt{p^{2}+m^{2}}$, and $\Omega(k)=\sqrt{m^{2}+k^{2}}$. The
definitions of the shifted momentum $\pi$ and $\tilde{\pi}$, and the remaining
quantities are given in A. In the nonrelativistic limit where the momenta are
much smaller than the masses, the Jacobian function $N$, given in A, reduces
to one. In addition, the relativistic Jacobi momenta $p$ and $k$ reduce to the
corresponding nonrelativistic Jacobi momenta, and similarly, the shifted
momentum arguments $\tilde{\pi}$ and $\pi$ reduce to the corresponding
nonrelativistic ones of Ref. [23]. The boosted 2B transition matrix
$T_{k}(p,p^{\prime};\epsilon)$ for 2B subsystem energies
$\epsilon=M_{t}-\Omega(k)$ is obtained from relativistic Lippmann-Schwinger
equation as

$T_{k}(p,p^{\prime};\epsilon)=V_{k}(p,p^{\prime})+4\pi\int
dp^{\prime\prime}p^{\prime\prime
2}\,\frac{V_{k}(p,p^{\prime\prime})}{\epsilon-\omega_{k}(p^{\prime\prime})}\,T_{k}(p^{\prime\prime},p^{\prime};\epsilon).$
(2)

The matrix elements of boosted potential $V_{k}$ can be obtained directly from
the nonrelativistic potential $V_{nr}$, by solving a quadratic integral
equation [18]

$V_{k}(p,p^{\prime})+\frac{4\pi}{\omega_{k}(p)+\omega_{k}(p^{\prime})}\int
dp^{\prime\prime}p^{\prime\prime
2}\,V_{k}(p,p^{\prime\prime})\,V_{k}(p^{\prime\prime},p^{\prime})=\frac{4m\,V_{nr}(p,p^{\prime})}{\omega_{k}(p)+\omega_{k}(p^{\prime})}.$
(3)

An important physical property of the boosted potential, clear in Eq. (3), is
the damping with the increase of the momentum of the spectator particle $k$.
This behavior corresponds in practice to an effective three-body repulsive
effect, working in the ultraviolet region (UV), which balances the attraction
at the short range. We will illustrate these properties when studying the
three-boson bound state when driving the potential range to zero.

Results. In the following, we present our numerical results for the solution
of relativistic Faddeev integral equation (1) for boosted potentials obtained
from one-term separable nonrelativistic potentials, which are generally
defined as $V_{nr}(p,p^{\prime})=\lambda_{nr}\,g(p)\,g(p^{\prime})$, where
$\lambda_{nr}$ is the potential strength and $g(p)$ is the form factor in the
momentum basis. In this work we use two models of separable potentials,
Yamaguchi-type form factor $g(p)=1/(p^{2}+\beta^{2})$ [24] and the Gaussian
form factor $g(p)=\exp{(-p^{2}/\Lambda^{2})}$ [25].

Figure 1: The diagonal matrix elements of nonrelativistic and boosted
potentials calculated for a wide range of form factor parameters $\beta$ for
Yamaguchi-type (upper panel) and $\Lambda$ for the Gaussian potentials (lower
panel) obtained with different boost momentum $k$.

The inputs for the solution of relativistic Lippmann–Schwinger equation (2)
are the matrix elements of boosted potentials $V_{k}(p,p^{\prime})$ which can
be obtained directly from nonrelativistic interaction $V_{nr}(p,p^{\prime})$
by solving the integral Eq. (3) using an iterative scheme proposed by Kamada
and Glöckle [18] and successfully implemented in a three-dimensional scheme
[26, 27]. The iteration starts with the initial guess

$V_{k}^{(0)}(p,p^{\prime})=\frac{4m\,V_{nr}(p,p^{\prime})}{\omega_{k}(p)+\omega_{k}(p^{\prime})}\,,$
(4)

and continues to reach convergence in the matrix elements of the boosted
potential with a relative error of $10^{-16}$ MeV fm3 at each set points
$(p,p^{\prime})$. In Fig. 1, we show the diagonal matrix elements of boosted
potentials with different values of boost momentum $k$ calculated for a wide
range of form factor parameters $\beta$ for Yamaguchi-type and $\Lambda$ for
the Gaussian potentials. As one can see, the boosted potentials are getting
smaller by increasing the boost momentum $k$.

Figure 2: Left panel: the ground and first excited state binding energies as a
function of the form factor parameters $\beta$ (solid line for Yamaguchi-type
potential) and $\Lambda$ (full circles for the Gaussian potential) obtained
for a fixed 2B binding energy of $-2.225$ MeV. The first excited state binding
energies are multiplied by a factor of $50$. Right panel: the value of
$M_{3}/m$ as a function of $M_{2}/m$ calculated from the three-body ground and
first excited states obtained with form factor parameters $\beta=\Lambda=2000$
fm-1 for Yamaguchi-type (solid lines) and Gaussian potentials (full circles).
For comparison, our results with the Bethe-Salpeter (dashed line) and Light-
Front (dash-dotted line) zero-range calculations of Ref. [16] are added to the
plot.

By having the matrix elements of boosted potentials $V_{k}(p,p^{\prime})$, we
solve the relativistic Lippmann-Schwinger integral equation (2) to calculate
fully off-shell boosted $T-$matrices $T_{k}(p,p^{\prime};\epsilon)$ for 2B
subsystem energies $\epsilon=M_{t}-\Omega(k)$ dictated by the boost momentum
$k$. Then, by solving the integral equation (1) with the Lanczos technique
(see Appendix C2 of Ref. [28]), we obtain relativistic 3B binding energies
$E_{t}$ and Faddeev components $\psi(p,k)$ for ground and excited states. We
use the Gauss-Legendre quadratures with hyperbolic plus linear mapping for
Jacobi momenta and linear mapping for angle variables to discretize continuous
momentum and angle variables [26]. The cutoffs of Jacobi momenta and the
distribution of their mesh points strongly depend on the potential form factor
parameters $\beta$ and $\Lambda$.

In the left panel of Fig. 2, we show three-boson ground and first excited
state binding energies as a function of the potential range parameters $\beta$
and $\Lambda$, for the Yamaguchi and Gaussian form factors, respectively, with
a nonrelativistic 2B binding energy kept at -2.225 MeV, namely, the deuteron
binding energy. For $\beta$ and $\Lambda$ increasing, the three-boson system
follows the Thomas collapse, with the binding energy $\propto-\beta^{2}$ and
$\propto-\Lambda^{2}$, happening in our examples up to values around 200 fm-1
or corresponding to a momentum of 0.4 GeV/c comparable to the nucleon mass.
Then, the boost effects take place and stabilize the system through an induced
repulsion that tends to counterbalance the singular behavior of the collapse
with the binding energy reaching a plateau, regardless of the short-range
potential model, suggesting that a well-defined zero-range limit exists for
the Glöckle-Kamada boosted potentials within the relativistic three-body
framework.

In the right panel of Fig. 2, we present three-body and single-particle mass
ratio $M_{3}/m$ as a function of the 2B and single-particle mass ratio
$M_{2}/m$, obtained for three-body ground and first excited states using large
form factor parameters $\beta=\Lambda=2000$ fm-1. The plot presents the mass
in units of the particle mass, which can be compared with previous
calculations using the Bethe-Salpeter (BS) and Light-Front equations [16]. As
one can see in the right panel of Fig. 2, our results with large form factor
parameters for both Yamaguchi-type and Gaussian potentials reveal a universal
behavior. Our numerical results for three-boson first excited bound state mass
obtained from the boosted potentials with large form factor parameters, i.e.
$\beta=2000$ fm-1 and $\Lambda=2000$ fm-1, show a weaker attraction when
compared to the results obtained with the LF and BS frameworks [16]. First, we
observe that, in the 2B bound state region, both the LF and BS approaches have
an unphysical three-body ground state with $M_{3}^{2}<0$, which are possible
as the homogeneous integral equations, only depend on $M_{3}^{2}$ [16]. The
physical “ground” state with $0<M^{2}_{3}<(m+M_{2})^{2}$ from the solution of
the LF and BS equations are indeed an excited state, and in this way, they are
denoted in the right panel of the figure.

Figure 3: Relativistic Faddeev components for three-body ground
$\psi^{(0)}(p,k)$ and first excited $\psi^{(1)}(p,k)$ states calculated for
Yamaguchi-type potential with form factor parameters (lines)
$\beta=200,~{}2000$ fm-1 and Gaussian potential with form factor parameters
(symbols) $\Lambda=200,~{}2000$ fm-1.

The difference between the LF and BS approaches is the attractive three-body
effective interactions appearing in the BS approach due to the implicit
inclusion of an infinite set of LF Fock-components, which is missing in the LF
equation where the truncation is made at the valence level [16]. What is
noticeable is that the boosted potential brings less attraction to the three-
body system, and furthermore in the UV region is much less attractive due to
the softening of the relativistic potential owing the boost coming with the
solution of Eq. (3). This effect can be appreciated by noticing the slowest
decrease of $M_{3}$ by decreasing $M_{2}$ for the boosted potential
calculation with respect to BS and LF results. The softening in the decrease
of the three-body mass with the increase of the 2B binding in the adopted
relativistic framework, as seen in the right panel of Fig. 2, suggests that
the increase of the binding turns the three-body bound state more compact,
which forces the system to explore the UV region, where the boosted potential
becomes weak. On the other hand, the 2B amplitude considered in the LF and BS
equations is much less damped in the UV region, presenting a $\sim 1/\log(k)$
behavior (cf. Eq. (2) in Ref. [16]), quite soft compared to the decrease of
the boosted potential matrix elements, with $\sim 1/k$, as the non-linear term
is not relevant for $k>>m$ attaining very large values, as the driving term of
the iterative solution for the boosted potential, Eq. (4), follows
$V^{(0)}_{k}(p,p^{\prime})|_{k>>m}\propto 1/k$. Thus, this discussion is
indeed indicating a weaker kernel of the Faddeev equation provided by the
boosted potential with respect to the LF and BS ones.

Finally, in Fig. 3, we show the Faddeev component of the wave functions
corresponding to the ground state, $\psi^{(0)}(p,k)$, and first excited one,
$\psi^{(1)}(p,k)$, as a function of Jacobi momenta $p$ and $k$, for two sets
of small and large potential form factor parameters $\beta$ and $\Lambda$. The
corresponding binding energies are listed in Table 1. The first striking
observation is the universality, i.e., the model independence, of
$\psi^{(0)}(p,k)$ and $\psi^{(1)}(p,k)$ up to momenta about $\beta$ or
$\Lambda$. For larger momenta, both potential models clearly show different
decay behaviors, and the Gaussian model shows, as expected, the faster damping
of the Faddeev component, while the Yamaguchi one still seems a power-law
behavior.

The interesting aspect seen in Fig. 3 is the model independence of the power-
law damping of the wave function happening both for the third particle
spectator momentum $m<k<\beta(\Lambda)$, and the two-particle relative one
$m<p<\beta(\Lambda)$. Furthermore, the power-law behavior in the region of
momenta between $\sim m$ and $\sim\beta(\Lambda)$ is independent of the
excitation state of the three-boson system for both momenta. The fit of the
power-law functions in the region $m<k<\beta(\Lambda)$, scales with
$\psi^{(0)}(p,k=0)\sim p^{-1.9}$ and $\psi^{(0)}(p=0,k)\sim k^{-3.7}$.

Table 1: Relativistic three-body ground and first excited state binding energies for Faddeev components shown in Fig. 3. The 2B binding energy is $-2.225$ MeV. Potential parameter (fm-1) |  $E_{t}^{(0)}$ (MeV) |  $E_{t}^{(1)}$ (MeV)
---|---|---
Yamaguchi-type potential
$\beta=200$ | $-454.1$ | $-6.076$
$\beta=2000$ | $-353.7$ | $-4.891$
Gaussian potential
$\Lambda=200$ | $-465.2$ | $-6.176$
$\Lambda=2000$ | $-357.7$ | $-4.914$

We should observe that the power-law property of the wave function component
in the relative momentum $p$ is somewhat expected, as the relativistic
propagator behavior dominates it at large $p$ values. On the other hand, the
power-law in $k$ is in contrast with the log-periodic behavior characteristic
of the Thomas collapsed states in the limit of the zero-range interaction for
the nonrelativistic three-boson system [29, 30]. On the other hand, the Efimov
effect [31], when the 2B binding is let to zero, is kept by the Faddeev
equations with boosted potentials.

Summary. In this work, we solved the relativistic three-boson bound state
problem within the Kamada and Glöcke framework of building the boosted
potential in the limit of a zero-range interaction. The starting point is a
nonrelativistic short-range separable potential, with Yamaguchi and Gaussian
form factors, which are driven to the contact interaction by letting their
momentum scales to large values in comparison with the boson mass scale, while
the 2B binding energy was kept fixed.

The solutions of the relativistic three-boson Faddeev equation with the
boosted potentials are stable, and the three-boson relativistic masses and
wave functions are finite and model-independent towards the limit of the zero-
interaction. The Thomas collapse is avoided, while the Efimov physics for
large scattering lengths are kept. The boosted potential provides an effective
repulsive short-range effect that decreases the interaction intensity and
guarantees the stability of the relativistic three-boson system towards the
zero-range limit.

We found that the Faddeev equation with the boosted potential provides a much
weaker attraction to bind the three-boson system than to what was found by
solving the LF and BS ones [16]. That effective weaker attraction can be
partially traced back to the more strong damping of the boosted potential, and
equally well the 2B T-matrix with the spectator momentum with a characteristic
behavior $\sim 1/k$, while within the LF and BS formulations, the 2B amplitude
is damped as $\sim 1/\log k$, quite soft compared to the decrease of the
boosted potential matrix elements. Furthermore, we have confirmed that the
stability of the three-boson system with boosted potentials in the zero-range
limit is accompanied by a universal power-law behavior of the wave function
component when the momenta are large. This is in contrast with the log-
periodic behavior characteristic of the Thomas collapsed states in the limit
of the zero-range interaction.

We want to add that the adopted framework may find applications beyond hadron
physics due to the renewed interest in few-body complexes in condensed matter,
motivated by the recently synthesized two dimensional materials (see, e.g.,
[32]). For example, in the case of monolayers of gapped honeycomb materials
like Transition Metal Dichalcogenides [33], anisotropic 2D semiconductors [34]
and hexagon Boron Nitride [35], the dispersion relation is hyperbolic [36,
37], thus allowing the use of relativistic frameworks in 2D, but requiring the
consideration of the spinor property of the quasi-particles [38]. These
possible applications are left for future studies.

Acknowledgments. T.F. thanks E. Ydrefors for the discussions. This work is a
part of the project INCT-FNA proc. No. 464898/2014-5. K.M. acknowledges a PhD
scholarship from the Brazilian agency CNPq (Conselho Nacional de
Desenvolvimento Científico e Tecnológico). T.F. acknowledges the CNPq Grant
No. 308486/2015-3 and Fundação de Amparo à Pesquisa do Estado de São Paulo
(FAPESP) under Thematic Project 2017/05660-0. The work of M.R.H. was supported
by the National Science Foundation under Grant No. NSF-PHY-2000029 with
Central State University. This work was supported by CNPq Grant No.
400789/2019-0.

##  Appendix A Definitions

The definitions and notation shown in this appendix follow Refs. [19, 20]. The
shifted momentum arguments appearing in Eq. (1) are given by

$\displaystyle\tilde{\pi}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{4}C^{2}(k,k^{\prime},x^{\prime})k^{2}+k^{\prime
2}+C(k,k^{\prime},x^{\prime})kk^{\prime}x^{\prime}},$ (5) $\displaystyle\pi$
$\displaystyle=$
$\displaystyle\sqrt{k^{2}+\frac{1}{4}C^{2}(k^{\prime},k,x^{\prime})k^{\prime
2}+C(k^{\prime},k,x^{\prime})kk^{\prime}x^{\prime}},$ (7)

where the permutation coefficients $C(k,k^{\prime},x^{\prime})$ are defined as

$C(k,k^{\prime},x^{\prime})=1+\frac{\Omega(k^{\prime})-\Omega(\mathscr{K})}{\Omega(k^{\prime})+\Omega(\mathscr{K})+\sqrt{\bigl{(}\Omega(k^{\prime})+\Omega(\mathscr{K})\bigr{)}^{2}-k^{2}}}.$
(8)

As a note, permutation coefficients $C$ reduces to one in the nonrelativistic
limit. The Jacobian function $N(k,k^{\prime},x^{\prime})$ is defined as

$\displaystyle N(k,k^{\prime},x^{\prime})$ $\displaystyle=$
$\displaystyle\left(\frac{4\Omega(k^{\prime})\Omega(\mathscr{K})}{\sqrt{\biggl{(}\Omega(k^{\prime})+\Omega(\mathscr{K})\biggr{)}^{2}-k^{2}}\biggl{(}\Omega(k^{\prime})+\Omega(\mathscr{K})\biggr{)}}\
\right)^{-\frac{1}{2}}$ (9) $\displaystyle\times$
$\displaystyle\left(\frac{4\Omega(k)\Omega(\mathscr{K})}{\sqrt{\biggl{(}\Omega(k)+\Omega(\mathscr{K})\biggr{)}^{2}-k^{\prime
2}}\ \biggl{(}\Omega(k)+\Omega(\mathscr{K})\biggr{)}}\right)^{-\frac{1}{2}},$
(10)

where $\mathscr{K}=\sqrt{k^{2}+k^{\prime 2}+2kk^{\prime}x^{\prime}}$.

## References

  * [1] M. Lüscher, Commun. Math. Phys. 105 (1986) 153. https://doi.org/10.1007/BF01211097
  * [2] M. Lüscher, Nucl. Phys. B 354 (1991) 531. https://doi.org/10.1016/0550-3213(91)90366-6
  * [3] S. R. Beane, W. Detmold, T. C. Luu, K. Orginos, M. J. Savage, and A. Torok (NPLQCD Collaboration), Phys. Rev. Lett. 100 (2008) 082004. https://doi.org/10.1103/physrevlett.100.082004
  * [4] W. Detmold, M. J. Savage, A. Torok, S. R. Beane, T. C. Luu, K. Orginos, and A. Parreño (NPLQCD Collaboration), Phys. Rev. D 78 (2008) 014507. https://doi.org/10.1103/PhysRevD.78.014507
  * [5] B. Hörz and A. Hanlon, Phys. Rev. Lett. 123 (2019) 142002. https://doi.org/10.1103/PhysRevLett.123.142002
  * [6] T. D. Blanton, F. Romero-López, and S. R. Sharpe, Phys. Rev. Lett. 124 (2020) 032001. https://doi.org/10.1103/PhysRevLett.124.032001
  * [7] M. Mai and M. Döring, Phys. Rev. Lett. 122 (2019) 062503. https://doi.org/10.1103/PhysRevLett.122.062503
  * [8] A. Alexandru, R. Brett, C. Culver, M. Döring, D. Guo, F. X. Lee, and M. Mai, Phys. Rev. D 102 (2020) 114523. https://doi.org/10.1103/PhysRevD.102.114523
  * [9] C. Gattringer and C. B. Lang, Quantum chromodynamics on the lattice. Vol. 788. Berlin: Springer, 2010 https://doi.org/10.1007/978-3-642-01850-3
  * [10] T. Frederico and E. Ydrefors, Few-Body Syst. 62 (2021) 8. https://doi.org/10.1007/s00601-021-01594-4
  * [11] T. Frederico, Phys. Lett. B 282 (1992) 409. https://doi.org/10.1016/0370-2693(92)90661-M
  * [12] L. H. Thomas, Phys. Rev. 47 (1935) 903. https://doi.org/10.1103/PhysRev.47.903
  * [13] F. A. B. Coutinho, J. F. Perez, and W. F. Wreszinski, Math. Phys. 36 (1995) 1625. https://doi.org/10.1063/1.531074
  * [14] J. Carbonell and V. A. Karmanov, Phys. Rev. C 67 (2003) 037001. https://doi.org/10.1103/PhysRevC.67.037001
  * [15] V. A. Karmanov and J. Carbonell, Few-Body Syst. 34 (2004) 85. https://doi.org/10.1007/s00601-004-0027-5
  * [16] E. Ydrefors, J. H. Alvarenga Nogueira, V. Gigante, T. Frederico, and V. A. Karmanov, Phys. Lett. B 770 (2017) 131. https://doi.org/10.1016/j.physletb.2017.04.035
  * [17] V. A. Karmanov and P. Maris, Few-Body Syst. 46 (2009) 95. https://doi.org/10.1007/s00601-009-0054-3
  * [18] H. Kamada and W. Glöckle, Phys. Lett. B 655 (2007) 119. https://doi.org/10.1016/j.physletb.2007.07.071
  * [19] M. R. Hadizadeh, Ch. Elster, and W. N. Polyzou, Phys. Rev. C 90 (2014) 054002. https://doi.org/10.1103/PhysRevC.90.054002
  * [20] M. R. Hadizadeh, M. Radin, and K. Mohseni, Sci. Rep. 10 (2020) 1949. https://doi.org/10.1038/s41598-020-58577-4
  * [21] P. A. M. Dirac, Rev. Mod. Phys. 21 (1949) 392. https://doi.org/10.1103/RevModPhys.21.392
  * [22] J. Carbonell, B. Desplanques, V. A. Karmanov, J.-F.Mathiot, Phys. Rep. 300 (1998) 215. https://doi.org/10.1016/S0370-1573(97)00090-2
  * [23] Ch. Elster, W. Schadow, A. Nogga, and W. Glöckle, Few-Body Syst. 27 (1999) 83. https://doi.org/10.1007/s006010050124
  * [24] Y. Yamaguchi, Phys. Rev. 95 (1954) 1628. https://doi.org/10.1103/PhysRev.95.1628
  * [25] A. Deltuva, R. Lazauskas, and L. Platter, Few-Body Syst. 51 (2011) 235. https://doi.org/10.1007/s00601-011-0227-8
  * [26] M. R. Hadizadeh and M. Radin, Eur. Phys. Jour. A 53 (2017) 18. https://doi.org/10.1140/epja/i2017-12209-6
  * [27] M. R. Hadizadeh, M. Radin, and M. Nazari, Sci. Rep. 11 (2021) 17550. https://doi.org/10.1038/s41598-021-96924-1
  * [28] M. R. Hadizadeh, M. T. Yamashita, L. Tomio, A. Delfino, and T. Frederico, Phys. Rev. A 85 (2012) 023610. https://doi.org/10.1103/PhysRevA.85.023610
  * [29] G. V. Skornyakov and K. A. Ter-Martirosyan, Sov. Phys. JETP 4 (1956) 648.
  * [30] G. S. Danilov. Sov. Phys. JETP 13 (1961) 349.
  * [31] V. Efimov, Phys. Lett. B 33 (1970) 563. https://doi.org/10.1016/0370-2693(70)90349-7
  * [32] T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, Phys. Rev. B 88 (2013) 045318. https://doi.org/10.1103/PhysRevB.88.045318
  * [33] G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Rev. Mod. Phys. 90 (2018) 021001. https://doi.org/10.1103/RevModPhys.90.021001
  * [34] L. S. R. Cavalcante, D. R. da Costa, G. A. Farias, D. R. Reichman, and A. Chaves, Phys. Rev. B 98 (2018) 245309. https://doi.org/10.1103/PhysRevB.98.245309
  * [35] K. Zhang, Y. Feng, F. Wang, Z. Yang, and J. Wang, Jour. Mat. Chem. C 46 (2017) 11992. https://doi.org/10.1039/C7TC04300G
  * [36] A. Kormányos, G. Burkard, M. Gmitra, J. Fabian, V. Zólyomi, N. D. Drummond, and V. Fal’ko, 2D Materials 2 (2015) 022001. https://doi.org/10.1088/2053-1583/2/4/049501
  * [37] F. Ferreira, A. J. Chaves, N. M. R. Peres, and R. M. Ribeiro, JOSA B 36 (2019) 674. https://doi.org/10.1364/JOSAB.36.000674
  * [38] A. J. Chaves, R. M. Ribeiro, T. Frederico, and N. M. R. Peres, 2D Materials 4 (2017) 025086. https://doi.org/10.1088/2053-1583/aa6b72