USTC-ICTS/PCFT-22-03

Geometric Algebra and Algebraic Geometry
of Loop and Potts Models

Janko Böhm1, Jesper Lykke Jacobsen2,3,4, Yunfeng Jiang5,6***Corresponding
author, Yang Zhang7,8

1Department of Mathematics, Technische Universität Kaiserslautern, 67663
Kaiserslautern, Germany
2Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL,
CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
3Sorbonne Université, École Normale Supérieure, CNRS,
Laboratoire de Physique (LPENS), F-75005 Paris, France
4Institut de Physique Théorique, Paris Saclay, CEA, CNRS, 91191 Gif-sur-
Yvette, France
5School of Physics, Southeast University, Nanjing 211189, China
6Shing-Tung Yau Center, Southeast University, Nanjing 210096, China
7Interdisciplinary Center for Theoretical Study, University of Science and
Technology of China, Hefei, Anhui 230026, China
8Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China

Abstract

We uncover a connection between two seemingly separate subjects in integrable
models: the representation theory of the affine Temperley-Lieb algebra, and
the algebraic structure of solutions to the Bethe equations of the XXZ spin
chain. We study the solution of Bethe equations analytically by computational
algebraic geometry, and find that the solution space encodes rich information
about the representation theory of Temperley-Lieb algebra. Using these
connections, we compute the partition function of the completely-packed loop
model and of the closely related random-cluster Potts model, on medium-size
lattices with toroidal boundary conditions, by two quite different methods. We
consider the partial thermodynamic limit of infinitely long tori and analyze
the corresponding condensation curves of the zeros of the partition functions.
Two components of these curves are obtained analytically in the full
thermodynamic limit.

###### Contents

  1. 1 Introduction
  2. 2 The affine Temperley-Lieb algebra
    1. 2.1 Diagrammatic representation
    2. 2.2 Standard modules
    3. 2.3 Resonances and quotient modules
  3. 3 The XXZ spin chain
  4. 4 The loop model
    1. 4.1 Markov trace
    2. 4.2 Torus partition function
    3. 4.3 Explicit results
    4. 4.4 Further comments
  5. 5 The Potts model
    1. 5.1 Potts model as a loop model
    2. 5.2 Torus partition function
    3. 5.3 Explicit results
  6. 6 Rational $Q$-system of twisted spin chains
    1. 6.1 Twisted XXZ spin chain and Bethe ansatz
    2. 6.2 Twisted $Q$-system of the XXX spin chain
    3. 6.3 Twisted $Q$-system of the XXZ spin chain
    4. 6.4 An explicit example
  7. 7 Resonance condition from algebraic geometry
    1. 7.1 Example 1: $L=4$, $M=1$
    2. 7.2 Example 2: $L=4$, $M=2$
    3. 7.3 Example 3: $L=6$, $M=3$
    4. 7.4 General comments
  8. 8 Algebraic-geometry approach to the torus partition function
    1. 8.1 General strategy
    2. 8.2 $TQ$-relation of the twisted spin chain
    3. 8.3 The full partition function
  9. 9 Analytical results in closed form
    1. 9.1 Analytic result for the loop model
    2. 9.2 Analytic result for the Potts model
  10. 10 Zeros of partition functions
    1. 10.1 Condensation curves
    2. 10.2 Results
  11. 11 Conclusions and discussion
  12. A More on computational algebraic geometry
    1. A.1 Basic notions of computational algebraic geometry
    2. A.2 Implementation notes
  13. B Partition function results
    1. B.1 Loop model
    2. B.2 Potts model

##  1 Introduction

Integrable lattice model and computational algebraic geometry seem to be two
distant subjects _a priori_. However, a bit of thought shows that this should
not be the case. Computational algebraic geometry stems from studying
solutions of algebraic equations. At the same time, such algebraic equations
are ubiquitous in integrable lattice models, where they arise as the famous
Bethe ansatz equations. Therefore, it is natural to expect that computational
algebraic geometry should play a useful role in the study of integrable
models.

Surprisingly, this possibility has not been pursued until very recently
[1].111See [2, 3] for the application of algebraic geometry to the
inhomogeneous XXZ spin chain. Here we are making a connection to
_computational_ algebraic geometry, whose focus and the techniques involved
(such as Gröbner basis) are different from [2, 3]. It has been shown, as
expected, that standard methods in computational algebraic geometry, such as
Gröbner bases and companion matrices, provide powerful tools to study lattice
integrable models. Important applications involve testing the completeness of
the Bethe ansatz, and computing exact partition functions of the six-vertex
model at its isotropic point, both for torus [4] and cylinder [5] geometries.
In this paper, we emphasize the relevance of these exact results for
intermediate sizes of the lattice, which are not small enough to be tackled by
hand or by brute force computational approaches, and yet not large enough to
be approximated by the thermodynamic limit. More recently, these techniques
have also been applied to a much wider class of observables, such as
quantities arising in quench dynamics [6].

So far the development of the algebro-geometric method has focussed on the
Bethe equations of XXX type, which can be written as polynomial equations of
the Bethe roots. The current work is devoted to the generalization to Bethe
equations of XXZ type. Written in terms of Bethe roots, the Bethe equations
involve hyperbolic functions. By a change of variable, we can work with the
exponential of Bethe roots, in terms of which the Bethe equations still take a
polynomial form.

The XXZ-type Bethe equations show up in many integrable models. In this work,
we are interested in two such models that have connections with the affine
Temperley-Lieb (aTL) algebra [7, 8, 9], an extension of the well-known
Temperley-Lieb algebra [10] to periodic boundary conditions. These models are
the completely-packed O($\mathfrak{m}$) loop model [11] and the $Q$-state
Potts model in the random-cluster formulation due to Fortuin and Kasteley (FK)
[12], henceforth referred to simply as the loop model and the Potts model.
Both of these models are fundamental models of statistical mechanics, and many
specific values of $\mathfrak{m}$ and $Q$ harbor cases of particular physical
interest. We shall consider both models on a square lattice with toroidal
boundary conditions. A configuration in the loop model is a set of loops
living on the lattice edges, subject to the constraints that each edge is
covered by exactly one loop, and that loops do not cross at the vertices. The
loop model partition function is then defined by attributing a weight
$\mathfrak{m}$ to each loop and summing over the configurations. The
configurations in the Potts model are exactly the same, but the weights are
slightly different. Color each face of the graph defined by the loops in black
or white, subject to the constraints that a fixed reference point (the origin)
lives on a black face, and that adjacent faces have different colors. In this
formulation, each black face coincides with an FK cluster. The Potts model
partition function is then defined by attributing a weight $Q$ to each black
face and summing over the configurations. Setting $Q=\mathfrak{m}^{2}$, the
two models would be strictly equivalent if they were defined on a planar
lattice (use the Euler identity), but on the torus there are subtle
differences (see below).

The loop model and the Potts model can be studied by means of representation
theory of the affine Temperley-Lieb algebra [7, 8, 9], and more precisely a
quotient thereof, known as the Jones-Temperley-Lieb algebra [13, 14], that we
shall define precisely below. Our goal is to study the torus partition
function for these models. It is computed by taking the Markov traces [15] of
the transfer matrix in the so-called standard modules of the aTL algebra,
denoted $\mathcal{W}_{j,\rho^{2}}$, which can be constructed in the basis of
link patterns. The parameters $j$ and $\rho^{2}$ are defined by choosing a
quantization scheme, in which one principal direction of the torus is taken as
‘space’ and the other as (imaginary) ‘time’. Then $2j$ is the number of non-
contractible loop strands, henceforth called through-lines, running along the
time direction, while $\rho^{2}$ is the pseudo-momentum corresponding to the
movement of through-lines across the periodic space direction. The transfer
matrices in the aTL standard modules are related to the transfer matrices of
the XXZ spin chain with a diagonal twist, which can be solved by Bethe ansatz.
The correct twist is related to $\rho^{2}$, so this parameter enters the Bethe
equations. In the framework of the Bethe ansatz, the eigenvalues of the
transfer matrix are functions of Bethe roots which are physical solutions of
the Bethe equations. Therefore one can expect physical solutions of the Bethe
equations to contain information about the representation theory of the aTL
algebra.

Confirming such a connection and offering a method to see it explicitly are
part of the main results of the current work. To this end, we shall rely on
analytical tools from computational algebraic geometry (CAG). It is customary
to parametrize $\mathfrak{m}=-q-q^{-1}$, with $q=-e^{i\gamma}$. We focus in
this paper on the so-called generic situation where $q$ is not a root of unity
(_i.e._ , $\gamma\in\mathbb{R}\setminus\pi\mathbb{Q}$); the root of unity case
will be discussed elsewhere. In this context we have identified the following
links between aTL algebra, AG and the Bethe ansatz:

  1. 1.

The dimension of a standard module equals the number of physical solutions of
the corresponding XXZ Bethe equations.

  2. 2.

When the twist variable $\rho^{2}$ takes specific values that satisfy a
certain resonance condition [9, 7], the standard module
$\mathcal{W}_{j,\rho^{2}}$ becomes reducible. This resonance condition also
shows up in the Bethe equation in an interesting way. When $\rho^{2}$
satisfies the resonance condition, the corresponding _Gröbner basis_ of the
Bethe equations has vanishing leading coefficients.

  3. 3.

The transfer matrices constructed in the standard module can be identified
with the _companion matrix_ of the transfer matrix of the twisted XXZ spin
chain. In general they take different forms, but have the same eigenvalues.

Based on these identifications, the torus partition function, or more
precisely the transfer matrices that generate it, can be computed in two
rather different ways. One way is based on the link-pattern basis construction
of the standard module. Since it gives nice diagrammatical representations of
the aTL algebra, we call this approach the _geometric algebra_ approach; the
other approach is by a combination of Bethe ansatz and computational algebraic
geometry, which we call the _algebraic geometry_ approach. We consider the
torus partition function for both models on an $L\times N$ lattice with
periodic boundary conditions in both directions (space and time). We give
closed-form analytical results for $L=4,6$ and generic $N$. For $L=8,10$ we
compute the partition function analytically for large $N\sim 1000$.

The partition functions that we obtain are polynomials with integer
coefficients. All information about such polynomials are contained in their
zeros over $\mathbb{C}$. We therefore investigate the distribution of the
zeros of the partition functions. We find that for fixed $L$ and in the limit
$N\to\infty$, the zeros condense on certain curves. We analyze these
condensation curves for $L=6,8,10,12,14$ and compare them with the finite-$N$
results for the zeros for $L=6,8,10$.

The rest of the paper is structured as follows. We introduce the aTL algebra
in section 2. Its connection with the algebraic Bethe Ansatz for the twisted
XXZ chain is discussed in section 3. The loop model and the Potts model will
be introduced and discussed in sections 4 and 5, respectively. We provide
their connection with the XXZ chain, and also show how to produce their torus
partition functions from aTL representation theory. To study the physical
solutions of the Bethe equations, by which we mean solutions that also solve
the original eigenvalue problem of the transfer matrix, it is more convenient
to apply the rational $Q$-system approach. The generalization of this approach
to the twisted case is a byproduct of our analysis and is given in section 6.
We then apply algebraic geometry to analyze the rational $Q$-system and give
the resonance condition from a different perspective in section 7. The closed
form results for $L=4,6$ and generic $N$ are presented in section 9. The
distribution of zeros and the partition functions for $L=6,8,10$ are studied
in section 10. For large $N$ they are in accord with the condensation curves,
which we provide as well for the larger sizes $L=12,14$. Furthermore we obtain
two components of these curves analytically in the full thermodynamic limit.
We conclude and discuss future directions in section 11. The appendices
contain more background and technical details for the main text.

##  2 The affine Temperley-Lieb algebra

In this section, we introduce the affine Temperley-Lieb algebra. We first give
the definition and then study its finite-dimensional representations called
standard modules.

##### Definitions.

We first define the usual Temperley-Lieb (TL) algebra, which is generated by
$L-1$ generators $e_{j}$ $(j=1,\ldots,L-1)$, together with the identity
$\mathbf{1}$, subject to the following relations [10]

$\displaystyle e_{j}^{2}$ $\displaystyle=$
$\displaystyle\mathfrak{m}\,e_{j}\,,$ (2.1) $\displaystyle e_{j}e_{j\pm
1}e_{j}$ $\displaystyle=$ $\displaystyle e_{j}\,,$ $\displaystyle e_{j}e_{k}$
$\displaystyle=$ $\displaystyle e_{k}e_{j}\qquad(\mbox{for }j\neq k,~{}k\pm
1)\,.$

We shall here consider the case of periodic boundary conditions, for which the
indices $j$ are interpreted modulo $L$ (so in particular we also have an
element $e_{0}$), and moreover there exist additional elements $U$ and
$U^{-1}$ which generate cyclic translations by one site to the right and to
the left, respectively. They satisfy the following relations

$\displaystyle Ue_{j}U^{-1}$ $\displaystyle=$ $\displaystyle e_{j+1}\,,$ (2.2)
$\displaystyle U^{2}e_{L-1}$ $\displaystyle=$ $\displaystyle e_{1}\cdots
e_{L-1}\,.$

It is clear that $U^{\pm L}$ is a central element. The algebra generated by
$e_{1}$ and $U^{\pm 1}$ together with the relations (2.1) and (2.2) is called
the affine Temperley–Lieb algebra [7, 8, 9] and will be denoted by
$\mathsf{T}^{\rm a}_{L}$. In the usual quantization scheme of two-dimensional
Euclidean models, the site index $j$ is a discretization of the space
direction, while the algebra action is a discretization of the imaginary time
evolution.

The remainder of this section reviews some well-known facts about the aTL
algebra, along the lines of [16]. We focus on the material that is essential
for our purposes, in order to keep the presentation self-contained.

###  2.1 Diagrammatic representation

The aTL algebra can be represented by particular diagrams on an annulus. This
is called the _loop representation_. In the geometry where time propagates
upwards (and which is natural from the point of view of the algebra) the
identity operator $\mathbf{1}$ and the TL generator $e_{j}$ are represented as

$\mathbf{1}=\raisebox{-14.22636pt}{ \leavevmode\hbox to17.87pt{\vbox
to34.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-17.47182pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-17.07182pt}\pgfsys@lineto{0.0pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{17.07182pt}{-17.07182pt}\pgfsys@lineto{17.07182pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,,\qquad\qquad
e_{j}=\raisebox{-14.22636pt}{ \leavevmode\hbox to17.87pt{\vbox
to34.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-17.47182pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{{{{}{}{}{}}}{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-17.07182pt}\pgfsys@lineto{0.0pt}{-8.5359pt}\pgfsys@curveto{0.0pt}{-4.76447pt}{3.82161pt}{-1.70718pt}{8.5359pt}{-1.70718pt}\pgfsys@curveto{13.2502pt}{-1.70718pt}{17.07181pt}{-4.76447pt}{17.07181pt}{-8.5359pt}\pgfsys@lineto{17.07182pt}{-17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{{}{}{}{}}}
{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{17.07182pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@curveto{0.0pt}{4.76447pt}{3.82161pt}{1.70718pt}{8.5359pt}{1.70718pt}\pgfsys@curveto{13.2502pt}{1.70718pt}{17.07181pt}{4.76447pt}{17.07181pt}{8.5359pt}\pgfsys@lineto{17.07182pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,,$ (2.3)

where we have only depicted the sites of index $j$ and $j+1$. The remaining
sites evolve trivially as in $\mathbf{1}$. One of the advantages of the loop
representation is that we can attribute a non-local weight

$\mathfrak{m}=2\cos\gamma$ (2.4)

to each loop in the diagrammatic formulation. This is very useful for
describing geometrical problems, such as percolation hulls or dense polymers.
It is also closely related to the cluster representation of the $Q$-state
Potts model with $Q=\mathfrak{m}^{2}$ [17], as we will discuss below.

A general basis element in the algebra of diagrams corresponds to a diagram of
$L$ sites on the inner and $L$ on the outer boundary of the annulus. We will
always restrict to even $L$ for simplicity. The sites are connected in pairs,
and only configurations that can be represented using simple curves inside the
annulus that do not cross are allowed. Such diagrams are called affine.
Examples of affine diagrams are shown in Fig. 2.1, where we draw them in a
slightly different geometry: we cut the annulus and transform it to a
rectangle, which we call framing, with the sites labeled from left to right.

An important parameter is the number of through-lines, which we denote by
$2j$; each through-line is a simple curve connecting a site on the inner and a
site on the outer boundary of the annulus; the $2j$ sites on the inner
boundary attached to a through-line we call free or non-contractible. The
inner (resp. outer) boundary of the annulus corresponds to the bottom (resp.
top) side of the framing rectangle.

Multiplication of two affine diagrams, $a$ and $b$, is defined in a natural
way, by joining the inner boundary of the annulus containing $a$ to the outer
boundary of the annulus containing $b$, and removing the interior sites.
Accordingly, $ab$ is obtained by joining the bottom side of $a$’s framing
rectangle to the top side of $b$’s framing rectangle, and removing the
corresponding joined sites. Whenever a closed contractible loop is produced in
the process of multiplying diagrams, this loop must be replaced by a numerical
factor $\mathfrak{m}$.

,  ,

Figure 2.1: Examples of affine diagrams for $L=4$, with the left and right
sides of the framing rectangle identified, so as to form an annulus. The first
diagram represents the generator $e_{4}$, the second is $e_{2}e_{4}$, and the
third is $e_{4}e_{3}e_{1}$.

###  2.2 Standard modules

With the defining relations (2.1)–(2.2), the algebra $\mathsf{T}^{\rm a}_{L}$
is infinite-dimensional. There are two mechanisms at play [13, 14] that
prevent the algebra from being finite-dimensional:

  1. 1.

any loop which is non-homotopic to a point (with respect to the periodic space
direction) cannot be reduced to a number (such as $\mathfrak{m}$) by the mere
application of the relations (2.1)–(2.2) and such loops may hence accumulate
in arbitrary numbers, and

  2. 2.

through-lines may wind around the periodic space direction, in one direction
or the other, an arbitrary number of times.

To describe lattice models with a finite number of degrees of freedom per site
we shall need some finite-dimensional representations of $\mathsf{T}^{\rm
a}_{L}$. These are the so-called standard modules $\mathcal{W}_{j,e^{i\phi}}$,
which depend on two parameters. In terms of diagrams, the first parameter
defines the number of through-lines $2j$, with $j=0,1,\ldots,\frac{L}{2}$. We
also stipulate that if an action contracts two or more free sites, a situation
which reduces the number of through-lines, the result is zero. This implies
that the action of $\mathsf{T}^{\rm a}_{L}$ does not connect standard modules
with different $j$. Notice that for a given non-zero value of $j$, it is
possible, using the action of the algebra, to cyclically permute the free
sites. This motivates the introduction of a pseudomomentum, which we
parametrize by the second parameter $\phi$: whenever a through-line winds
around the annulus once, we attribute to it a phase $e^{\pm i\phi/2}$, where
the sign plus (resp. minus) is for a clockwise (resp. counterclockwise)
winding.

A more convenient formulation of $\mathcal{W}_{j,e^{i\phi}}$ can be obtained
as follows. As the free sites are not allowed to be contracted, the pairwise
connections among non-free sites on the inner boundary cannot be changed by
the action of the algebra. This part of the diagrammatic information is thus
inessential and can be omitted. Even so, the representation remains infinite-
dimensional, in particular because the $2j$ through-lines connecting the inner
and outer boundaries may still spiral around the periodic boundary condition.
As a first step to obtain the desired finite-dimensional representation we
therefore decree that a full turn of the inner boundary with respect to the
outer one is irrelevant. Such a turn is equivalent to a full turn of all the
through-lines, so by definition of the pseudomomentum we should set

$(e^{\pm i\phi/2})^{2j}=1\,.$ (2.5)

Having made this choice, it is now enough to concentrate on the upper halves
of the affine diagrams, obtained by cutting the affine diagrams across its
$2j$ through-lines. Each upper half is then called a link state, and for
simplicity the “half” through-lines attached to the free sites on the outer
boundary of the annulus (or top side of the framing rectangle) are still
called through-lines. The phase

$\rho\equiv e^{i\phi/2}$ (2.6)

(resp. $\rho^{-1}=e^{-i\phi/2}$) is now attributed each time one of these
through-lines moves through the periodic boundary condition of the framing
rectangle in the rightward (resp. leftward) direction. The quantisation
condition on the pseudomomentum then reads

$(\rho^{2})^{j}=1\,.$ (2.7)

With these conventions, it is readily seen that the Temperley-Lieb algebra
action obtained by stacking the affine diagrams on top of the link states
gives rise to exactly the same representations $\mathcal{W}_{j,\rho^{2}}$ as
defined above.

The representations $\mathcal{W}_{j,e^{i\phi}}$ over $\mathsf{T}^{\rm
a}_{L}(\mathfrak{m})$ having through-lines (_i.e._ , with $j>0$) are now
finite-dimensional. Their dimensions are easily found by counting the link
states, giving

$\hat{d}_{j}=\binom{L}{\frac{L}{2}+j}\,,\qquad\mbox{for }j>0\,.$ (2.8)

Note that these dimensions do not depend on $\phi$ (but representations with
different $e^{i\phi}$ are not isomorphic). These standard modules
$\mathcal{W}_{j,e^{i\phi}}$ are also called cell modules in the seminal work
[7].

The situation without through-lines (_i.e._ , when $j=0$) is a bit different.
There is no pseudomomentum, but representations are still characterized by a
parameter other than $j$, which now specifies the weight given to loops which
are non-contractible, _i.e._ , which wrap the periodic space direction of the
lattice. Notice that since the curves cannot cross, non-contractible loops
cannot coexist with through-lines, and hence are not possible for $j>0$.
Parametrizing the weight of a non-contractible loop as
$\mathfrak{n}=z+z^{-1}$, the corresponding standard module of $\mathsf{T}^{\rm
a}_{N}(\mathfrak{m})$ is denoted $\mathcal{W}_{0,z^{2}}$. This module is
isomorphic to $\mathcal{W}_{0,z^{-2}}$.

It is natural to require that $\mathfrak{n}=\mathfrak{m}$, _i.e._ , to set
$z=q$, so that contractible and non-contractible loops get the same weight.
Imposing this is however not without consequences for the nature of the
corresponding aTL representation, as is most easily seen by focussing on the
dimension. If we do not distinguish between contractible and non-contractible
loops, the corresponding link state needs not distinguish whether two sites
are connected strictly inside the framing rectangle, or by going through the
periodic boundary condition.222To be precise, the distinction is whether the
link connecting the two sites crosses the periodic boundary condition an even
or an odd number of times. By deforming the curves, these crossing numbers can
always be reduced to 0 or 1. Taking therefore all connections strictly inside
the framing rectangle, we arrive at the same dimension as in the usual (non-
periodic) TL algebra, which is

$\bar{d}_{0}=\binom{L}{\frac{L}{2}}-\binom{L}{\frac{L}{2}+1}\,,\qquad\mbox{for
}j=0\,.$ (2.9)

We denote the corresponding representation by
$\overline{\mathcal{W}}_{\\!0,q^{2}}$, and below we shall make clear how it is
related to $\mathcal{W}_{0,z^{2}}$ from an algebraic point of view.

To avoid too much drawing, it is convenient to adopt a no-frill notation for
specifying the link states. We denote the (half) through-lines by a bar and an
arc connecting two sites by a pair of matching parentheses. An arc represented
by a matching $)($ pair then straddles the periodic boundary condition,
whereas a $()$ pair does not. For instance, the link states describing the
upper halves of the diagrams in Figure 2.1 are written as $)||($, $)()($ and
$))(($ respectively.

###  2.3 Resonances and quotient modules

We now parametrize

$\mathfrak{m}=-q-q^{-1}=2\cos\gamma\,.$ (2.10)

The standard modules $\mathcal{W}_{j,e^{i\phi}}$ are irreducible for generic
values of $q$ and $\phi$. However, degeneracies appear whenever the following
resonance criterion is satisfied [9, 7]:333In [7] this criterion appears with
some extra liberty in the form of certain $\pm$ signs, but we shall not need
these signs here.

$\displaystyle\rho^{2}\equiv e^{i\phi}$ $\displaystyle=$ $\displaystyle
q^{2j+2k},\qquad\mbox{for }k\in\mathbb{Z}_{+}\,.$ (2.11)

The representation $\mathcal{W}_{j,q^{2j+2k}}$ then becomes reducible, and
contains a submodule isomorphic to $\mathcal{W}_{j+k,q^{2j}}$. The quotient
$\mathcal{W}_{j,q^{2j+2k}}/\mathcal{W}_{j+k,q^{2j}}$ is generically
irreducible, with dimension

$\bar{d}_{j}:=\hat{d}_{j}-\hat{d}_{j+k}\,,\qquad\mbox{for }j>0\,.$ (2.12)

When $q$ is a root of unity, there are infinitely many solutions to (2.11),
leading to a complex pattern of degeneracies.

We now return to the case $j=0$ without through-lines and with non-
contractible loops having the weight $\mathfrak{n}=z+z^{-1}$. If we make the
identification

$z=e^{i\phi/2}$ (2.13)

the resonance criterion (2.11) still applies. Imposing this therefore leads to
the module $\mathcal{W}_{0,q^{2}}$, which is reducible even for generic $q$.
Indeed, (2.11) is satisfied with $j=0$, $k=1$, and hence
$\mathcal{W}_{0,q^{2}}$ contains a submodule isomorphic to $\mathcal{W}_{1,1}$
(and more precisely $\rho=-1$, so that $e^{i\phi}=\rho^{2}=1$). Taking the
quotient $\mathcal{W}_{0,q^{2}}/\mathcal{W}_{1,1}$ leads to a simple module
for generic $q$ which we denote by

$\overline{\mathcal{W}}_{\\!0,q^{2}}=\mathcal{W}_{0,q^{2}}/\mathcal{W}_{1,1}\,.$
(2.14)

This module is isomorphic to $\overline{\mathcal{W}}_{\\!0,q^{-2}}$. As
already anticipated in (2.9), it has the dimension

$\bar{d}_{0}=\binom{L}{\frac{L}{2}}-\binom{L}{\frac{L}{2}+1}\,,$ (2.15)

in agreement with the general formula (2.12) for $k=1$.

We now discuss in some more detail the geometrical meaning of the difference
between $\mathcal{W}_{0,q^{2}}$ and $\overline{\mathcal{W}}_{0,q^{2}}$, which
was already sketched above. In the latter case, one only keeps track of which
sites are connected to which in the diagrams, while in the former, one also
keeps information of how the connectivities wind around the periodic direction
of the annulus (the ambiguity does not arise when there are through-lines
propagating). Formally, this corresponds to the existence of a surjection
$\psi$ between different quotients of the $\mathsf{T}^{\rm a}_{L}$ algebra:

$\textstyle{\mbox{}\quad\xrightarrow{{\mbox{}\quad\psi\quad}}\quad}$ (2.16)

The definition of link states as the upper halves of the affine diagrams also
makes sense for $j=0$. The representation $\mathcal{W}_{0,q^{2}}$ requires
keeping track of whether each pairwise connection between the sites on the
outer boundary goes through the periodic boundary condition, whereas in the
quotient module $\overline{\mathcal{W}}_{0,q^{2}}$ this information is
omitted. In both cases, it is easy to see that the number of link states
coincides with the dimension $d_{0}$ or $\bar{d}_{0}$, respectively.

The quotient $\overline{\mathcal{W}}_{0,q^{2}}$ is just one example of
representations that appear more generally in $\mathsf{T}^{\rm a}_{L}$ when
$q$ is still generic, but $z$ takes particular values [7, 18]. Indeed, the
standard module $\mathcal{W}_{j^{\prime},z^{\prime}}$ has a non-zero
homomorphism to $\mathcal{W}_{j,z}$,

$\mathcal{W}_{j^{\prime},z^{\prime}}\hookrightarrow\mathcal{W}_{j,z}\,,$
(2.17)

if and only if $j^{\prime}-j\in\mathbb{N}_{0}$ and the pairs
$(j^{\prime},z^{\prime})$ and $(j,z)$ satisfy

$(z^{\prime})^{2}=(-q)^{2\epsilon j}\qquad\text{and}\qquad
z^{2}=(-q)^{2\epsilon j^{\prime}},\qquad\text{for}\quad\epsilon=\pm 1.$ (2.18)

When $q$ is not a root of unity, there is at most one solution to (2.17). When
there is one, the module $\mathcal{W}_{j,z}$ is not irreducible, but has a
unique proper irreducible submodule isomorphic to
$\mathcal{W}_{j^{\prime},z^{\prime}}$. One can then obtain a simple module by
taking the quotient

$\overline{\mathcal{W}}_{\\!j,z}\equiv\mathcal{W}_{j,z}/\mathcal{W}_{j^{\prime},z^{\prime}}$
(2.19)

of dimension

$\overline{d}_{j}=\dim\overline{\mathcal{W}}_{\\!j,z}=\binom{L}{\frac{L}{2}+j}-\binom{L}{\frac{L}{2}+j^{\prime}}\,.$
(2.20)

The quotient $\overline{\mathcal{W}}_{\\!0,q^{2}}$ appearing above is the
simplest example (with $j=0$, $z=q^{2}$ and $j^{\prime}=1$, $z^{\prime}=1$) of
this situation, and it is the only such quotient that will turn out to be
relevant for the Potts model at generic $q$.

##  3 The XXZ spin chain

There exists another representation of the aTL algebra which is called the XXZ
representation. It makes manifest a connection to an integrable model, the
twisted XXZ spin chain. This connection is of paramount importance to us,
since the application of the AG approach goes via the Bethe equations of an
integrable model.

To make the link, we first recall the definition of the XXZ spin chain in the
framework of algebraic Bethe ansatz. The transfer matrix of the XXZ spin chain
can be defined by

$t(u)={\rm tr}_{a}\,R_{{a},1}(u)R_{{a},2}(u)\cdots R_{{a},L}(u)\,.$ (3.1)

where $R_{a,n}(u)$ is the $R$-matrix acting between the auxiliary space and
the $n$-th site of the chain. The corresponding $\check{R}$-matrix, defined by
$\check{R}_{a,n}(u)=P_{a,n}R_{a,n}(u)$ where $P$ denotes the permutation
operator, can be written in terms of the Temperley-Lieb generators as

$\check{R}_{{a},j}(u)=\sin(\gamma-u)\mathbf{1}+\sin(u)e_{j}\,,$ (3.2)

where $u$ denotes the spectral parameter, and the crossing parameter is
$\gamma\in[0,\pi]$. The auxiliary space represents a horizontal row of lattice
edges, and $t(u)$ is the transfer matrix going from one row of vertical edges
to the next.

This presence of the auxiliary space implies a change of geometry with respect
to the algebraic consideration made in section 2, namely that imaginary time
does no longer flow upwards but rather towards the North-East. For instance,
in the present setup the diagrammatic rendering of the aTL generators,
formerly drawn as (2.3), now looks like

$\mathbf{1}=\raisebox{-14.22636pt}{ \leavevmode\hbox to34.94pt{\vbox
to34.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-17.47182pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{}{{{}{}{}{}}}{} {}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{17.07182pt}{-17.07182pt}\pgfsys@lineto{17.07182pt}{-3.4143pt}\pgfsys@curveto{17.07182pt}{-1.5286pt}{18.60046pt}{0.00005pt}{20.48618pt}{0.00005pt}\pgfsys@lineto{34.14365pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,,\qquad\qquad
e_{j}=\raisebox{-14.22636pt}{ \leavevmode\hbox to34.94pt{\vbox
to34.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-17.47182pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{}{{{}{}{}{}}}{} {}
{}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.6575pt}{0.0pt}\pgfsys@curveto{15.54321pt}{0.0pt}{17.07185pt}{-1.52864pt}{17.07185pt}{-3.41435pt}\pgfsys@lineto{17.07182pt}{-17.07182pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}{{{}{}{}{}}}{} {} {}{} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{17.07182pt}{17.07182pt}\pgfsys@lineto{17.07182pt}{3.4143pt}\pgfsys@curveto{17.07182pt}{1.5286pt}{18.60046pt}{-0.00005pt}{20.48618pt}{-0.00005pt}\pgfsys@lineto{34.14365pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,.$ (3.3)

If we take the isotropic value $u=\gamma/2$ of the spectral parameter in
(3.2), we have $\check{R}_{{a},j}(u)\propto\mathbf{1}+e_{j}$ and the transfer
matrix covers the lattice by the locally equal-weighted linear combination of
those two operators. This isotropic value is particularly important for the
applications to the loop and Potts model, to be presented in the following
sections. Recall that in the loop representation of section 2, the commutation
relations satisfied by the $e_{j}$ gave rise to a non-local weight
$\mathfrak{m}$ per loop. The amount of non-locality is controlled by $\gamma$
and goes away at the special value $\gamma=\frac{\pi}{3}$, since then
$\mathfrak{m}=1$ and the non-local loops are simply not counted.

The corresponding XXZ spin chain ensues by expanding $t(u)$ around another
special value, the fully anisotropic value $u=0$. To first order in this
expansion, the logarithmic derivative of $t(u)$ becomes the XXZ Hamiltonian

$\mathcal{H}=-\frac{\gamma}{\pi\sin\gamma}\sum^{N}_{j=1}(e_{j}-e_{\infty})\,.$
(3.4)

Here the prefactor is chosen to ensure relativistic invariance at low energy,
and $e_{\infty}$ is a constant energy density that cancels out extensive
contributions to the ground state. Its value is

$e_{\infty}=\sin\gamma\>I_{0}\,,$ (3.5)

with $I_{0}$ given by the integral [19]

$I_{0}=\int^{\infty}_{-\infty}\frac{\sinh(\pi-\gamma)t}{\sinh(\pi
t)\cosh(\gamma t)}\,\mathrm{d}t\,.$ (3.6)

In (3.1) and (3.4), the generators $e_{j}$ can be taken to act in different
representations of the aTL algebra $\mathsf{T}^{\rm a}_{L}(\mathfrak{m})$. The
loop representation has already been shown in (3.3). But from the usual
formulation of the XXZ chain in terms of spins $|\\!\uparrow\rangle$ and
$|\\!\downarrow\rangle$, it is more natural to consider the XXZ representation
in which the $e_{j}$ act on $(\mathbb{C}^{2})^{\otimes L}$ with

$e_{j}=-\sigma_{j}^{-}\sigma_{j+1}^{+}-\sigma_{j}^{+}\sigma_{j+1}^{-}-\frac{\cos\gamma}{2}\sigma_{j}^{z}\sigma_{j+1}^{z}-\frac{i\sin\gamma}{2}(\sigma_{j}^{z}-\sigma_{j+1}^{z})+\frac{\cos\gamma}{2}\,,$
(3.7)

where the $\sigma_{j}$ are the Pauli matrices, so the Hamiltonian is the
familiar XXZ spin chain

$\mathcal{H}=\frac{\gamma}{2\pi\sin\gamma}\sum^{N}_{j=1}\left[\sigma^{x}_{j}\sigma^{x}_{j+1}+\sigma^{y}_{j}\sigma^{y}_{j+1}+\Delta(\sigma^{z}_{j}\sigma^{z}_{j+1}-1)+2e_{\infty}\right]$
(3.8)

with anisotropy parameter

$\Delta=\cos(\gamma)\,.$ (3.9)

In the spin basis, the generator $e_{j}$ then acts on sites $j,j+1$ (with
periodic boundary conditions) as

$e_{j}=\cdots\otimes\mathbf{1}\otimes\left(\begin{array}[]{cccc}0&0&0&0\\\
0&-q^{-1}&-1&0\\\ 0&-1&-q&0\\\
0&0&0&0\end{array}\right)\otimes\mathbf{1}\otimes\cdots\,,$ (3.10)

where the quantum-group related parameter $q$ is defined as

$q=-e^{i\gamma}\,.$ (3.11)

One may introduce a twist in the spin chain without changing the expression
(3.4), by modifying the expression of the Temperley-Lieb generator acting
between the first and last spin with a twist parametrized by $\phi$. In terms
of the Pauli matrices, this twist imposes the boundary conditions
$\sigma^{z}_{L+1}=\sigma^{z}_{1}$ and $\sigma^{\pm}_{L+1}=e^{\mp
i\phi}\sigma^{\pm}_{1}$. The value of the energy density $e_{\infty}$ is
independent of $\phi$ and is still given by (3.5).

##  4 The loop model

We are now ready to define the loop model precisely and place it in the
context of lattice algebras (see section 2) and integrable models (see section
3).

Consider a square lattice of width $L$ and height $N$, with the usual
(‘axial’) orientation, corresponding to auxiliary spaces along the periodic
horizontal (or ‘space’) direction. On this lattice, draw a set of loops by
splitting each vertex in one of the two ways shown in (3.3). We wish to give
the same weight (viz., one) to each splitting, and up to an unimportant
proportionality constant this is accomplished by the $\check{R}$-matrix (3.2)
at the isotropic point $u=\frac{\gamma}{2}$. The corresponding row-to-row
transfer matrix (3.1) then adds one row to the lattice and, at the same time,
generates all possible configurations of splittings.

We recall from the introduction that the loop model is defined by giving a
weight $\mathfrak{m}^{\ell}$ to each configuration, where $\ell$ is the number
of loops. We further wish to impose toroidal boundary conditions on the
$L\times N$ lattice. The periodic boundary conditions in the horizontal
direction is accounted for by the trace in (3.1). The aTL algebra can handle
this boundary condition, and at the same time it ensures that each closed loop
gets the correct weight $\mathfrak{m}$.

It follows that $t\left(\gamma/2\right)^{N}$ generates the partition function
of the loop model, up to one last subtlety. Namely, we need also to impose
periodic boundary conditions in the vertical (or ‘time’) direction, giving
still a weight $\mathfrak{m}$ per loop. In fact, $t\left(\gamma/2\right)^{N}$
is a weighted sum of words in the algebra, such as those shown in figure 2.1,
so what remains is to “glue” the top and the bottom of the framing rectangle
of each diagram generated by the algebra, and replace each loop in the glued
diagram by the weight $\mathfrak{m}$. For instance, the gluing of the three
diagrams shown in Figure 2.1 will result in the respective weights
$\mathfrak{m}^{3}$, $\mathfrak{m}^{2}$ and $\mathfrak{m}$. This may sound like
a difficult task, but fortunately there is an algebraic construction that does
just what we need.

###  4.1 Markov trace

The algebraic construction enabling this gluing is called the Markov trace
[15], denoted ${\rm Tr}$ to distinguish it from the usual matrix trace ${\rm
tr}$. For the partition function $Z(L,N)$ on the $L\times N$ torus we then
have

$Z(L,N)={\rm Tr}\,\left[t\left(\frac{\gamma}{2}\right)^{N}\right]\,.$ (4.1)

Below we shall always assume that both $L$ and $N$ are even.

We shall need the following fundamental result, which is well-known for the
case of open transverse boundary conditions (cylinder geometry), but which
carries over to periodic boundary conditions (toroidal geometry) as well: The
Markov trace ${\rm Tr}$ can be decomposed as a linear combination of usual
matrix traces ${\rm tr}$ over the standard modules. The coefficients in this
decomposition, which we shall call eigenvalue multiplicities, are denoted
$\Lambda_{j,m}$.

For $j\in\mathbb{N}$ and $m$ a positive divisor of $j$, the multiplicity
$\Lambda_{j,m}$ turns out to be

$\Lambda_{j,m}=2\sum_{d>0\,:\,d|j}\frac{\mu\left(\frac{m}{m\wedge
d}\right)\varphi\left(\frac{j}{d}\right)}{j\,\varphi\left(\frac{m}{m\wedge
d}\right)}\>T_{2d}\left(\frac{\mathfrak{m}}{2}\right)\,.$ (4.2)

Here, the sum is over all positive divisors of $j$. Also, $m\wedge d$ denotes
the greatest common divisor of $m$ and $d$, and $\mu$ and $\varphi$ are
respectively the Möbius function and Euler’s totient function [20]. The Möbius
function $\mu$ is defined by $\mu(n)=(-1)^{r}$, if $n$ is an integer that is a
product $n=\prod_{i=1}^{r}p_{i}$ of $r$ distinct primes, $\mu(1)=1$, and
$\mu(x)=0$ otherwise or if $x$ is not an integer. Euler’s totient function
$\varphi$ is defined for positive integers $n$ as the number of integers
$n^{\prime}$ such that $1\leq n^{\prime}\leq n$ and $n\wedge n^{\prime}=1$.
Finally, $T_{n}(x)$ is the $n$’th order Chebyshev polynomial of the first
kind. If we parameterise the loop weight $\mathfrak{m}$ as in (2.10) then

$T_{2d}\left(\frac{\mathfrak{m}}{2}\right)=\cos(2d\gamma)\,.$ (4.3)

The crucial formula (4.2) originated in sketchy form in [21], was first
written as above in the appendix of [22] and justified by a field-theoretic
reasoning, and finally a rigorous combinatorial derivation was given in [23].

###  4.2 Torus partition function

We can now state the general result for the decomposition of the torus
partition function over the standard modules. Recall first that the twist
variable $\rho^{2}$ has to be a $j$’th root of unity by the quantisation
condition (2.7). We can write the possible solutions as

$\rho^{2}=\omega_{k}\equiv\exp\left(\frac{2\pi ik}{j}\right)\,,\quad\mbox{for
}k=1,2,\ldots,j\,.$ (4.4)

Let $m$ be a positive divisor of $j$. Define $\kappa(m)$ to be the subset of
$k\in\\{1,2,\ldots,j\\}$, such that $k\in\kappa(m)$ if and only if $m$ is the
smallest integer satisfying $km\in\mathbb{N}j$. For example, with $j=12$ we
have

$\displaystyle\kappa(1)$ $\displaystyle=$ $\displaystyle\\{12\\}\,,$
$\displaystyle\kappa(2)$ $\displaystyle=$ $\displaystyle\\{6\\}\,,$
$\displaystyle\kappa(3)$ $\displaystyle=$ $\displaystyle\\{4,8\\}\,,$
$\displaystyle\kappa(4)$ $\displaystyle=$ $\displaystyle\\{3,9\\}\,,$
$\displaystyle\kappa(6)$ $\displaystyle=$ $\displaystyle\\{2,10\\}\,,$
$\displaystyle\kappa(12)$ $\displaystyle=$ $\displaystyle\\{1,5,7,11\\}\,.$

Note that $\\{1,2,\ldots,j\\}$ is the disjoint union $\cup_{m|j}\kappa(m)$.

We then claim that within $\mathsf{T}^{\rm a}_{L}$ we can express the torus
partition function as follows:

$Z(L,N)={\rm
tr}_{\mathcal{W}_{0,q^{2}}}\left[t\left(\frac{\gamma}{2}\right)^{N}\right]+\sum_{j=1}^{L/2}\sum_{m|j\atop
m\geq 1}\Lambda_{j,m}\sum_{k\in\kappa(m)}{\rm
tr}_{\mathcal{W}_{j,\rho^{2}=\omega_{k}}}\left[t\left(\frac{\gamma}{2}\right)^{N}\right]\,.$
(4.5)

Note that the first term has multiplicity one, while the multiplicities of the
remaining terms are the $\Lambda_{j,m}$ defined in (4.2). We also notice that
the total multiplicity for a fixed $j$ is simply

$\Lambda_{j}\equiv\sum_{m|j\atop m\geq
1}|\kappa(m)|\;\Lambda_{j,m}=2\;T_{2j}\left(\frac{\mathfrak{m}}{2}\right)\,,$
(4.6)

with the Chebyshev polynomial being defined in (4.3). Thus, in the classical
(_i.e._ , not ${q}$-deformed) limit $\mathfrak{m}=-2$ we have $\Lambda_{j}=2$
for any $j\geq 1$. Since $T_{2j}\left(\frac{\mathfrak{m}}{2}\right)$ is a
polynomial in $\mathfrak{m}^{2}$, the same is true for $\mathfrak{m}=2$.

###  4.3 Explicit results

To illustrate our general discussions on the torus partition function of the
loop model, we give some explicit examples in this subsection. These include
the closed-form expressions of the torus partition functions
$Z_{\text{loop}}(2,N)$ and $Z_{\text{loop}}(4,N)$.

When assembling the ingredients corresponding to (4.5), it is convenient to
let $t_{0}$ denote the matrix representation of the transfer matrix $t$ acting
in the representation $\mathcal{W}_{0,q^{2}}$, while $t_{j,\rho}$ denotes $t$
acting in $\mathcal{W}_{j,\rho^{2}}$.

####  4.3.1 Results for $L=2$

For $L=2$, we have $j=0,1$. Therefore, we can define the standard modules
$\mathcal{W}_{0,q^{2}}$ and $\mathcal{W}_{1,\rho^{2}}$.

Within $\mathcal{W}_{0,q^{2}}$ we pick the link-state basis $\\{(),)(\\}$. The
(isotropic) transfer matrix reads

$t_{0}=\left(\begin{array}[]{cc}\mathfrak{m}&2+\mathfrak{m}\\\
2+\mathfrak{m}&\mathfrak{m}\end{array}\right)$ (4.7)

with eigenvalues

$\left\\{2+2\mathfrak{m},-2\right\\}\,.$ (4.8)

To obtain this result (and those given below) it is important to realise that
the construction (3.1) with an auxiliary space is compatible with the aTL
formalism. The crucial part is to perform the trace over the auxiliary space
in the loop representation. This is done by decomposing $t$ as a product of
$\check{R}$-matrices. Before acting with the first $\check{R}$-matrix of a
row, we need to insert a pair of mutually connected extra sites, _i.e._ , to
append $()$ to the left of each link state. The second of those sites plays
the role of the auxiliary space $a$ on which the first factor
$\check{R}_{a,1}$ acts. The space corresponding to $a$ then moves to the right
after each $\check{R}$ multiplication, and after acting with the product of
all $\check{R}$-matrices, the ‘open’ (non-traced) auxiliary space is
represented by the first and last sites. Identifying those two sites by acting
with the corresponding aTL operator $e_{0}$ attributes the correct weight,
corresponding to the trace operation ${\rm tr}_{a}$ in (3.1). We may then
finally delete the two extra sites, and relabel the remaining $L$ sites, so as
to obtain a completed row.

Within $\mathcal{W}_{1,\rho^{2}}$ the only link state is $||$, and the
transfer matrix is

$t_{1,\rho}=\rho+\rho^{-1}\,.$ (4.9)

The partition function (4.5) is then

$\displaystyle Z_{\text{loop}}(2,N)$ $\displaystyle=$ $\displaystyle{\rm
tr}\,\big{(}t_{0}\big{)}^{N}+\Lambda_{1,1}\,{\rm
tr}\,\big{(}t_{1,1}\big{)}^{N}$ (4.10) $\displaystyle=$
$\displaystyle(2+2\mathfrak{m})^{N}+(-2)^{N}+(\mathfrak{m}^{2}-2)\times
2^{N}\,,$

where we have set $\rho=1$ in $t_{1,\rho}$.

####  4.3.2 Results for $L=4$

For $L=4$, we $j=0,1,2$. The corresponding standard modules are
$\mathcal{W}_{0,q^{2}}$, $\mathcal{W}_{1,\rho^{2}}$ and
$\mathcal{W}_{2,\rho^{2}}$.

Within $\mathcal{W}_{0,q^{2}}$ we pick the link-state basis

$\\{()(),(()),())(,)((),)()(,))((\\}$ (4.11)

and the transfer matrix reads

$t_{0}=\left(\begin{array}[]{cccccc}\mathfrak{m}&2\mathfrak{m}+\mathfrak{m}^{2}&4+2\mathfrak{m}&4+2\mathfrak{m}&4+4\mathfrak{m}+\mathfrak{m}^{2}&2\mathfrak{m}+\mathfrak{m}^{2}\\\
0&\mathfrak{m}&2+\mathfrak{m}&2+\mathfrak{m}&2+\mathfrak{m}&0\\\
2+\mathfrak{m}&2+\mathfrak{m}&\mathfrak{m}&0&0&2+\mathfrak{m}\\\
2+\mathfrak{m}&2+\mathfrak{m}&0&\mathfrak{m}&0&2+\mathfrak{m}\\\
4+4\mathfrak{m}+\mathfrak{m}^{2}&4+2\mathfrak{m}&2\mathfrak{m}+\mathfrak{m}^{2}&2\mathfrak{m}+\mathfrak{m}^{2}&\mathfrak{m}&4+2\mathfrak{m}\\\
0&0&2+\mathfrak{m}&2+\mathfrak{m}&2+\mathfrak{m}&\mathfrak{m}\end{array}\right).$
(4.12)

The eigenvalues can be found analytically, and they read

$\\{8+7\mathfrak{m}+\mathfrak{m}^{2},-\mathfrak{m}-\mathfrak{m}^{2}\\}\cup\\{-8-3\mathfrak{m},\mathfrak{m},\mathfrak{m},\mathfrak{m}\\}\,.$
(4.13)

Within $\mathcal{W}_{1,\rho^{2}}$ we pick the link-state basis

$\\{()||,|()|,||(),)||(\\}$ (4.14)

and the transfer matrix reads

$t_{1,\rho}=\left(\begin{array}[]{cccc}\rho+\rho^{-1}&2\rho^{-1}+\mathfrak{m}\rho^{-1}&2\rho^{-1}+\mathfrak{m}\rho^{-1}&2+\mathfrak{m}\\\
2\rho+\mathfrak{m}\rho&\rho+\rho^{-1}&2\rho^{-1}+\mathfrak{m}\rho^{-1}&2+\mathfrak{m}\\\
2\rho+\mathfrak{m}\rho&2\rho+\mathfrak{m}\rho&\rho+\rho^{-1}&2+\mathfrak{m}\\\
2+\mathfrak{m}&2+\mathfrak{m}&2+\mathfrak{m}&\rho+\rho^{-1}\end{array}\right)\,.$
(4.15)

The eigenvalues for $\rho=1$ read

$\\{8+3\mathfrak{m},-\mathfrak{m},-\mathfrak{m},-\mathfrak{m}\\}\,,$ (4.16)

while those for $\rho=-1$ all have the opposite sign. Note that the latter
eigenvalues correspond to the second of the sets in (4.13). This is consistent
with the general result (2.17) that $\mathcal{W}_{0,q^{2}}$ contains a
submodule isomorphic to $\mathcal{W}_{1,\rho^{2}}$ with $\rho=-1$ [set $j=0$,
$j^{\prime}=1$, $z=q^{2}$ and $z^{\prime}=\rho^{2}$ in (2.18)]. Although this
permits us to define a quotient module $\overline{\mathcal{W}}_{0,q^{2}}$, of
dimension $6-4=2$ in this case, it is the full module $\mathcal{W}_{0,q^{2}}$
that we need to construct the partition function.

Finally, within $\mathcal{W}_{2,\rho^{2}}$ the only link state is $||||$, and
the transfer matrix is

$t_{2,\rho}=\rho+\rho^{-1}\,.$ (4.17)

The partition function (4.5) is then

$\displaystyle Z_{\text{loop}}(4,N)$ $\displaystyle=$ $\displaystyle{\rm
tr}\,\big{(}t_{0}\big{)}^{N}+\Lambda_{1,1}\,{\rm
tr}\,\big{(}t_{1,1}\big{)}^{N}+\Lambda_{2,1}\,{\rm
tr}\,\big{(}t_{2,1}\big{)}^{N}+\Lambda_{2,2}\,{\rm
tr}\,\big{(}t_{2,i}\big{)}^{N}$ (4.18) $\displaystyle=$
$\displaystyle(8+7\mathfrak{m}+\mathfrak{m}^{2})^{N}+3\mathfrak{m}^{N}+(-\mathfrak{m}-\mathfrak{m}^{2})^{N}+(-8-3\mathfrak{m})^{N}$
$\displaystyle+$
$\displaystyle(\mathfrak{m}^{2}-2)\times\left[(8+3\mathfrak{m})^{N}+3(-\mathfrak{m})^{N}\right]+\frac{1}{2}(\mathfrak{m}^{4}-3\mathfrak{m}^{2})\times
2^{N}\,.$

Note that $t_{2,i}$ is zero in this case; this will obviously not be so for
higher values of $L$.

From (4.10) and (4.18) we can make the crucial check of modular invariance

$Z_{\text{loop}}(2,4)=Z_{\text{loop}}(4,2)=16\mathfrak{m}^{4}+64\mathfrak{m}^{3}+112\mathfrak{m}^{2}+64\mathfrak{m}\,.$
(4.19)

####  4.3.3 Results for $L\geq 6$

For $L=6$, the analysis is similar, which leads to the following partition
function

$\displaystyle Z_{\text{loop}}(6,N)$ $\displaystyle=$ $\displaystyle{\rm
tr}\,\big{(}t_{0}\big{)}^{N}+\Lambda_{1,1}\,{\rm
tr}\,\big{(}t_{1,1}\big{)}^{N}+\Lambda_{2,1}\,{\rm
tr}\,\big{(}t_{2,1}\big{)}^{N}+\Lambda_{2,2}\,{\rm
tr}\,\big{(}t_{2,i}\big{)}^{N}$ (4.20)
$\displaystyle+\Lambda_{3,1}\big{(}t_{3,1}\big{)}^{N}+\Lambda_{3,3}\left[\big{(}t_{3,e^{i\pi/3}}\big{)}^{N}+\big{(}t_{3,e^{2i\pi/3}}\big{)}^{N}\right]\,.$

We see here the first manifestation of the regrouping of several traces with a
common multiplicity, mirroring the fact that $\kappa(3)=\\{1,2\\}$.

Using the computer algebra system Mathematica and Singular [24], we have
generated the matrix representations of the transfer matrix within each
standard module for sizes $L=6,8,10$. From this we have constructed the
corresponding $Z(L,N)$ for $L\leq 10$. To give one example, we find that

$\displaystyle Z_{\text{loop}}(6,4)$ $\displaystyle=$ $\displaystyle
786432\mathfrak{m}+2916256\mathfrak{m}^{2}+4685824\mathfrak{m}^{3}+4346912\mathfrak{m}^{4}+2603712\mathfrak{m}^{5}+1063398\mathfrak{m}^{6}$
(4.21)
$\displaystyle+304032\mathfrak{m}^{7}+61242\mathfrak{m}^{8}+8560\mathfrak{m}^{9}+798\mathfrak{m}^{10}+48\mathfrak{m}^{11}+2\mathfrak{m}^{12}\,,$

and we can check that $Z_{\text{loop}}(6,4)=Z_{\text{loop}}(4,6)$. More
generally, we have checked the modular invariance,
$Z_{\text{loop}}(L,N)=Z_{\text{loop}}(N,L)$, for all values $L,N\leq 10$. Some
of the results for $Z_{\text{loop}}(L,L)$ are given in Appendix B.1.

###  4.4 Further comments

Before ending the section, let us make several comments.

##### Special values of $\mathfrak{m}$.

Although a closed-form expression for $Z(L,N)$ as a function of $\mathfrak{m}$
for arbitrary $L$ and $N$ is beyond reach, at special values of $\mathfrak{m}$
the partition function can be written down. Two important cases are
$\mathfrak{m}=0,1$.

For $\mathfrak{m}=0$, we have obviously $Z_{\text{loop}}(L,N)=0$ for any
values of $L,M$, because each configuration consists of at least one loop. A
non-trivial quantity is however provided by

$E(L,N)=\left.\frac{{\rm d}}{{\rm
d}\mathfrak{m}}Z(L,N)\right|_{\mathfrak{m}=0}\,,$ (4.22)

which gives the number of configurations with precisely one loop. These are
known in physics as dense polymers and in graph theory as Eulerian circuits,
here on the $L\times N$ torus. Unlike in the planar case, we do not have the
bijection between dense polymers and spanning trees—note that the latter can
be counted by the matrix-tree theorem and exploiting properties of the
discrete Laplacian [25] (see also [26] and references therein). We shall
however have more to say about the numbers $E(L,N)$ in section 5.3 below. For
$L=N$ they can be read off from the first coefficient of the results in
Appendix B.1.

For $\mathfrak{m}=1$, we have the trivial result $Z(L,N)=2^{LN}$, since in
this case the number of loops is not counted and we have independently one of
the two configurations (3.3) at each vertex.

Apart from this we have made two non-trivial observations, based on all
$Z(L,N)$ with $L,N\leq 10$. They read

$\displaystyle Z(L,N)$ $\displaystyle=$ $\displaystyle
0\,,\quad\quad\quad\mbox{for }\mathfrak{m}=-1\,,$ (4.23a) $\displaystyle
Z(L,N)$ $\displaystyle=$ $\displaystyle 2^{L+N}\,,\quad\ \mbox{for
}\mathfrak{m}=-2\,.$ (4.23b)

We shall be able to prove (4.23a) in section 5 by a reasoning that involves
the Potts model. However (4.23b) remains elusive, and it is particularly
intriguing that the power of two is reminiscent of a surface effect, despite
the torus having of course no boundary.

##### Momentum sectors.

We have further block-diagonalized the three $L=4$ transfer matrices with
respect to lattice momentum. This leads to the same results, but using at most
matrices of dimension 2. Note that there are two momenta in this problem: the
pseudomomentum of the through-lines (controlled by the parameter $\rho$) and
the lattice momentum. In the corresponding XXZ spin chain the pseudomomentum
is controlled by the twist of the chain, while the lattice momentum has its
usual significance.

##### Quotients and further resonances.

We wish to illustrate as well the more general construction of quotient
modules in (2.19). The reason is that the corresponding singular values of $z$
in $\mathcal{W}_{j,z^{2}}$ also appear from the point of view of the
$QQ$-relations, to be discussed in section 6. As we will see, such special
values lead to vanishing leading coefficients in the Gröbner basis.

To provide this illustration, we first generalize the construction of the
module $\mathcal{W}_{0,q^{2}}$ so that contractible loops still have a weight
$\mathfrak{m}=q+q^{-1}$, while non-contractible loops now have a different
weight $\mathfrak{n}=z+z^{-1}$. In this way (4.12) is replaced by

$t_{0}=\left(\begin{array}[]{cccccc}\mathfrak{n}&2\mathfrak{n}+\mathfrak{m}\mathfrak{n}&4+2\mathfrak{m}&4+2\mathfrak{m}&4+4\mathfrak{m}+\mathfrak{m}^{2}&2\mathfrak{n}+\mathfrak{m}\mathfrak{n}\\\
0&\mathfrak{n}&2+\mathfrak{m}&2+\mathfrak{m}&2+\mathfrak{m}&0\\\
2+\mathfrak{m}&2+\mathfrak{m}&\mathfrak{n}&0&0&2+\mathfrak{m}\\\
2+\mathfrak{m}&2+\mathfrak{m}&0&\mathfrak{n}&0&2+\mathfrak{m}\\\
4+4\mathfrak{m}+\mathfrak{m}^{2}&4+2\mathfrak{m}&2\mathfrak{n}+\mathfrak{m}\mathfrak{n}&2\mathfrak{n}+\mathfrak{m}\mathfrak{n}&\mathfrak{n}&4+2\mathfrak{m}\\\
0&0&2+\mathfrak{m}&2+\mathfrak{m}&2+\mathfrak{m}&\mathfrak{n}\end{array}\right).$
(4.24)

We have already seen the resonance criterion (2.18) at work in the case
$(j,j^{\prime})=(0,1)$, corresponding to $z^{\prime}=1$ and $z=-q$. We saw
then that 4 of the eigenvalues in $t_{0}$ coincided with those of $t_{1}$, so
that the corresponding quotient module
$\overline{\mathcal{W}}_{\\!\\!j,(-q)^{2}}$ of dimension 2 could be defined.

Another case allowed by (2.18) is $(j,j^{\prime})=(0,2)$, corresponding to
$z^{\prime}=1$ and $z=q^{2}$. We have then

$\mathfrak{n}=q^{2}+q^{-2}=\mathfrak{m}^{2}-2\,.$ (4.25)

Inserting this into (4.24) we find the eigenvalues

$\left\\{2,\mathfrak{m}^{2}-2,\mathfrak{m}^{2}-2,2+6\mathfrak{m}+3\mathfrak{m}^{2},\frac{1}{2}\left(-12-6\mathfrak{m}+\mathfrak{m}^{2}\pm(\mathfrak{m}+2)\sqrt{32-7\mathfrak{m}^{2}}\right)\right\\}$
(4.26)

The first of these eigenvalues (namely $2$) is also the (unique) eigenvalue of
$t_{2,\rho=1}$, and another quotient module (2.19) of dimension 5 can be
defined.

Finally, we consider (2.18) with $(j,j^{\prime})=(1,2)$, corresponding to
$z^{\prime}=-q$ and $z=q^{2}$. The eigenvalues of $t_{1,q^{2}}$ can be found
from (4.15) as

$\left\\{-\mathfrak{m},\mathfrak{m}(2\mathfrak{m}+3),q^{-2}\pm
iq^{-2}(q-1)(q+1)^{3}+q^{-1}(-1+q^{3}-q^{2}-2q)\right\\}\,.$ (4.27)

The first of these eigenvalues (namely $-\mathfrak{m}$) coincides with the
(unique) eigenvalue of $t_{2,\rho=-q}$, and so we can define a quotient module
(2.19) of dimension 3.

##  5 The Potts model

Having discussed the torus partition function of the loop model, we turn to
the Potts model in this section. It will turn out possible to construct the
torus partition function of the Potts model from the same ingredients that
have been employed to study the loop model [21].

###  5.1 Potts model as a loop model

The $Q$-state Potts model on the graph $G=(V,E)$ with vertex set $V$ and edge
set $E$ is originally defined for $Q\in\mathbb{N}^{*}$ by [27]

$Z_{\rm Potts}=\sum_{\\{\sigma\\}}\exp\left(\sum_{(ij)\in
E}K\delta_{\sigma_{i},\sigma_{j}}\right)\,,$ (5.1)

where the $Q$-component spins $\sigma_{i}=1,\ldots,Q$ interact along edges
with coupling constant $K$ through the Kronecker symbol ($\delta_{x,y}=1$ if
$x=y$, and $\delta_{x,y}=0$ otherwise). Using the identity ${\rm
e}^{K\delta_{\sigma_{i},\sigma_{j}}}=1+v\delta_{\sigma_{i},\sigma_{j}}$ with
$v={\rm e}^{K}-1$ (valid because $\delta_{\sigma_{i},\sigma_{j}}=0$ or $1$),
expanding the product over edges of this two-term expression, and performing
the sum over spins $\\{\sigma\\}$, this can be rewritten in the random-cluster
form due to Fortuin and Kasteleyn (FK) [12]

$Z_{\rm Potts}=\sum_{A\subseteq E}v^{|A|}Q^{k(A)}\,,$ (5.2)

where $|A|$ and $k(A)$ denote respectively the number of elements in the edge
subset $A$ and the corresponding number of connected components (often called
FK clusters) in the induced graph $G_{A}=(V,A)$. It is important to notice
that the FK form makes sense for any $Q\in\mathbb{R}$ (or even
$Q\in\mathbb{C}$).

This can be further transformed into a loop model on the medial lattice
$G_{\rm m}=(V_{\rm m},E_{\rm m})$ with vertices $V_{\rm m}$ located on the
edges of $G$, and edges $E_{\rm m}$ between vertex pairs in $V_{\rm m}$ whose
corresponding edges in $E$ are indicident on a common vertex in $V$ [11]. Take
now $G_{\rm m}$ to be the same $L\times N$ square lattice on which we defined
the loop model in section 4, which, as the reader will recall, was axially
oriented (_i.e._ , with horizontal and vertical edges). Supposing again $L$
and $N$ to be even, it is easy to see that there exists a graph $G$ of which
$G_{\rm m}$ is the medial: indeed, $G$ is another square lattice, but
diagonally oriented (_i.e._ , rotated through $\frac{\pi}{4}$ with respect to
$G_{\rm m}$ and scaled up by a factor $\sqrt{2}$). There is a bijection
between FK clusters on $G$ and loops on $G_{\rm m}$. Indeed, split the
vertices $V_{\rm m}$ in one of two ways, as in (3.3), so that the loops
intersect none of the edges in $A$ and all of the edges in $E\setminus A$.
Stated less precisely, but more intuitively, the loops wrap tightly around the
FK clusters.

Now, if $G$ were a planar graph, application of the Euler relation would give
[11]

$Z_{\rm Potts}=\mathfrak{m}^{|V|}\sum_{A\subseteq
E}\left(\frac{v}{\mathfrak{m}}\right)^{|A|}\mathfrak{m}^{\ell(A)}\,,$ (5.3)

where $\ell(A)$ is the number of loops, and $Q=\mathfrak{m}^{2}$. Remarkably,
at the critical point on the square lattice, which is $v_{\rm c}=\mathfrak{m}$
[28], the weight conjugate to $|A|$ becomes one, so, up to the unimportant
overall factor, $Z_{\rm Potts}$ is just the partition function of a loop model
with weight $\mathfrak{m}$ per loop. This reasoning however breaks down (as
does the Euler relation) on the torus, so although the Potts model can still
be formulated in terms of loops the precise weighting is slightly different.

###  5.2 Torus partition function

On the torus, the decomposition of the Potts-model partition function with
respect to the aTL algebra is subject to a number of subtleties, which have
their root in the non-planarity. This can however be solved by a subtle
reweighting of the FK configurations in which there exists a cluster
(necessarily unique) that wraps around both periodic directions. In the
original paper [21] such a doubly-wrapping FK cluster was called a ‘cluster
with cross topology’.

The end result is very similar to (4.5) and reads [21]

$\displaystyle Z_{\rm Potts}(L,N)$ $\displaystyle=$
$\displaystyle\mathfrak{m}^{LN/2}\left\\{{\rm
tr}_{\mathcal{W}_{0,q^{2}}}\left[t\left(\frac{\gamma}{2}\right)^{N}\right]+\frac{Q-1}{2}{\rm
tr}_{\mathcal{W}_{0,-1}}\left[t\left(\frac{\gamma}{2}\right)^{N}\right]\right.$
(5.4) $\displaystyle+$ $\displaystyle\left.\sum_{j=1}^{L/2}\sum_{m|j\atop
m\geq 1}\widetilde{\Lambda}_{j,m}\sum_{k\in\kappa(m)}{\rm
tr}_{\mathcal{W}_{j,\rho^{2}=\omega_{k}}}\left[t\left(\frac{\gamma}{2}\right)^{N}\right]\right\\}\,.$

The overall factor $\mathfrak{m}^{LN/2}$ obviously has the same origin as the
overall factor in (5.3). Much more interestingly, the second term in the
bracket is new with respect to (4.5) and is related to the doubly-wrapping FK
clusters. Let us sketch the derivation of the form (5.4), giving an argument a
bit different from that presented in the original paper [21]. The Euler
relation used to derive (5.3) on a planar graph almost holds on the torus, the
only difference being that two loops are ‘lost’ in configurations with a
doubly-wrapping FK cluster, as is readily seen by a bit of drawing. These
wrapping clusters correspond to a situation in which non-contractible loops
are disallowed, whence the particular role of $\mathcal{W}_{0,-1}$ (in which
$z^{2}=-1$ corresponds to a vanishing weight $\mathfrak{n}=z+z^{-1}=0$ of the
non-contractible loops). This sector must acquire an extra weight of
$Q=\mathfrak{m}^{2}$ to make up for the two ‘lost’ loops in the Euler-type
argument, explaining the factor $Q-1$. The further factor of $\frac{1}{2}$ is
explaining by the fact (well known from aTL representation theory, but which
can also be derived by a duality argument [29]) that exactly when
$\mathfrak{n}=0$ the representation $\mathcal{W}_{0,z^{2}}$ breaks up as the
direct sum of two isomorphic representations, of which we need only one copy.
The same kind of argument shows that the eigenvalue multiplicities are
affected by the reweighting and become [21, 22, 23]

$\widetilde{\Lambda}_{j,m}=\Lambda_{j,m}(\mathfrak{m})+\frac{Q-1}{2}\Lambda_{j,m}(0)\,,$
(5.5)

where $\Lambda_{j,m}(\mathfrak{m})$ now denotes the same quantity as (4.2).

It is interesting to notice that $\widetilde{\Lambda}_{1,1}=-1$. This implies
that in the expression (5.4) the $j=1$ term in the sum (for which $m=1$ is the
only divisor) can be subtracted from the first, $\mathcal{W}_{0,q^{2}}$ term
[16]. This subtraction is precisely what defines the quotient module (2.14).
An expression equivalent to (5.4) is then obtained by replacing
$\mathcal{W}_{0,q^{2}}$ by the quotient $\overline{\mathcal{W}}_{0,q^{2}}$ and
starting the sum at $j=2$. This implies, somewhat surprisingly, that
$\mathcal{W}_{1,1}$ does not contribute to the partition function of the Potts
model, although it contributes to the loop model. Conversely, as we have seen,
$\mathcal{W}_{0,-1}$ contributes to the Potts model but not to the loop model.

###  5.3 Explicit results

As for the loop model, we have used the computer-generated matrix
representations of the transfer matrix within each standard module to compute
the corresponding $Z_{\rm Potts}(L,N)$ for $L\leq 10$. As an example, we have

$\displaystyle Z_{\rm Potts}(6,4)$ $\displaystyle=$ $\displaystyle
393216\mathfrak{m}^{13}+1810336\mathfrak{m}^{14}+3679232\mathfrak{m}^{15}+4388576\mathfrak{m}^{16}$
(5.6) $\displaystyle+$ $\displaystyle
3453776\mathfrak{m}^{17}+1920258\mathfrak{m}^{18}+794064\mathfrak{m}^{19}+255294\mathfrak{m}^{20}+65984\mathfrak{m}^{21}$
$\displaystyle+$ $\displaystyle
13854\mathfrak{m}^{22}+2312\mathfrak{m}^{23}+289\mathfrak{m}^{24}+24\mathfrak{m}^{25}+\mathfrak{m}^{26}\,,$

and we can check that $Z_{\rm Potts}(6,4)=Z_{\rm Potts}(4,6)$. More generally,
we have checked the modular invariance, $Z_{\rm Potts}(L,N)=Z_{\rm
Potts}(N,L)$, for all values $L,N\leq 10$. Some of the results for $Z(L,L)$
are given in Appendix B.2.

Recall that the graph $G$ on which the original Potts model (5.2) is defined
is a tilted square lattice with $|V|=\frac{LN}{2}$ spins and $|E|=LN$ edges.
Since we have $v=\mathfrak{m}$ and $Q=\mathfrak{m}^{2}$, the lowest power of
$\mathfrak{m}$ is obtained from configurations with a single cluster and the
least possible number of edges, of weight
$v^{|V|-1}Q=\mathfrak{m}^{\frac{LN}{2}+1}$, corresponding to spanning trees.
It is not hard to see that no other types of configurations can produce such a
small power of $\mathfrak{m}$, so the coefficient of
$\mathfrak{m}^{\frac{LN}{2}+1}$ counts the number of spanning trees on $G$.
The highest power of $\mathfrak{m}$ is meanwhile obtained from configurations
with the largest possible number of edges, of weight
$v^{|E|}Q=\mathfrak{m}^{LN+2}$. Since there is only one such configuration,
the coefficient of $\mathfrak{m}^{LN+2}$ is one. All configurations with
strictly more than one cluster correspond to terms with non-extremal powers of
$\mathfrak{m}$.

##### Special values of $\mathfrak{m}$.

For $\mathfrak{m}=0$ the number of spanning trees can hence be found from the
derivative

$S(L,N)=s!\left.\frac{{\rm d}^{s}}{{\rm
d}\mathfrak{m}^{s}}Z(L,N)\right|_{\mathfrak{m}=0}\,,\qquad\mbox{with
}s=\frac{LN}{2}+1\,.$ (5.7)

Let ${\cal L}_{G}$ denote the discrete Laplacian of the corresponding graph
$G$, and let ${\cal L}^{\prime}_{G}$ be any one of its minors, obtained by
deleting some row (_e.g._ , the last one) and its corresponding column. We
have verified that $S(L,N)=\det{\cal L}^{\prime}_{G}$, in agreement with the
matrix-tree theorem [25, 26]. By comparing (4.21) with (5.6)—or the results of
Appendix B.1 with those of Appendix B.2—we see that, in fact, the number of
Eulerian circuits is precisely twice the number of spanning trees:

$E(L,N)=2S(L,N)\,.$ (5.8)

This is of course no coincidence. Indeed, an Eulerian circuit either wraps
tightly around a spanning tree on the graph $G$, or around a spanning tree of
the dual $G^{*}$. On a planar graph the dual of a spanning tree is another
spanning tree, so the same Eulerian circuit sees one tree on its inside and
another on its outside. However, on a torus a spanning tree is homotopic to a
point, so its dual contains a double-wrapping FK cluster, hence cannot be a
spanning tree (and vice versa). So we have just proven that, in general,
$E(L,N)$ is the number of spanning trees on $G$ plus the number of spanning
trees on the dual $G^{*}$. For the case of a selfdual lattice ($G=G^{*}$),
such as the square lattice considered here, we infer (5.8).

For $\mathfrak{m}=1$ we again have the trivial result $Z(L,N)=2^{LN}$ for the
same reason as in the loop model case.

Furthermore, based on all $Z_{\rm Potts}(L,N)$ with $L,N\leq 10$ we have made
the observations

$\displaystyle Z_{\rm Potts}(L,N)$ $\displaystyle=$ $\displaystyle
0\,,\quad\quad\quad\quad\quad\quad\ \ \mbox{for }\mathfrak{m}=-1\,,$ (5.9a)
$\displaystyle Z_{\rm Potts}(L,N)$ $\displaystyle=$ $\displaystyle
2^{LN/2+L+N-2}\,,\quad\ \ \mbox{for }\mathfrak{m}=-2\,.$ (5.9b)

We can in fact derive (5.9a) by using results of section 5.1. Since we have

$v={\rm e}^{K}-1=\mathfrak{m}=-1\,,$ (5.10)

and $Q=\mathfrak{m}^{2}=1$, it follows that we have a one-state Potts model in
which a pair of equal neighbouring spins have the Boltzmann weight ${\rm
e}^{K}=0$. Therefore, $Z_{\rm Potts}(L,N)=0$ at $\mathfrak{m}=-1$. Moreover,
since the difference with the loop-model partition function $Z(L,N)$ is always
proportional to $Q-1$, as we have seen, this argument also proves (4.23a).

On the other hand, the result (5.9b) remais elusive.

##  6 Rational $Q$-system of twisted spin chains

In section 3, we have shown that the XXZ spin chain (3.8) can be written in
terms of aTL generators. Since both the loop model and the Potts model have
been related to the aTL algebra, this implies that both models are intimately
related to the XXZ spin chain. The XXZ spin chain is integrable and can be
solved by the Bethe ansatz. This opens the interesting possibility of studying
the standard modules, relevant for producing the torus partition function of
the loop and Potts models, from the Bethe ansatz perspective.

In this section, we first review the twisted XXZ spin chain. Then we turn to
the discussion of the physical solutions of the Bethe equations. It is known
that the Bethe equations contain unphysical solutions that do not solve the
eigenvalue problem of the transfer matrix. A better formulation that eschews
unphysical solutions is the rational $Q$-system. This approach was first
introduced as an efficient method of solving the Bethe equations of the XXX
spin chain in [30]. The properties of the $Q$-system and its relation to the
physicality of solutions were further clarified in [31], and subsequently
generalized to the XXZ case [32]. We here go one step further and generalize
the rational $Q$-system to both the XXX and XXZ spin chains with diagonal
twists.

###  6.1 Twisted XXZ spin chain and Bethe ansatz

Up to an irrelevant global factor and a constant shift, we can define the XXZ
spin chain (3.8) by the following Hamiltonian

$\displaystyle\mathcal{H}_{\text{XXZ}}=\sum_{m=1}^{L}\left(\sigma_{m}^{+}\sigma_{m+1}^{-}+\sigma_{m}^{-}\sigma_{m+1}^{y}+\Delta\,\sigma_{m}^{z}\sigma_{m+1}^{z}\right)\,,$
(6.1)

where $L$ is the length of the spin chain, which is identified with the number
of non-trivial generators of $\mathsf{T}^{\rm a}_{L}(\mathfrak{m})$. We
consider the algebraic Bethe ansatz. The $R$-matrix is given by

$\displaystyle
R_{an}(u)\propto\left(\begin{array}[]{cccc}\sinh(u+\tfrac{i\gamma}{2})&0&0&0\\\
0&\sinh(u-\tfrac{\eta}{2})&\sinh(i\gamma)&0\\\
0&\sinh(i\gamma)&\sinh(u-\tfrac{i\gamma}{2})&0\\\
0&0&0&\sinh(u+\tfrac{i\gamma}{2})\\\ \end{array}\right).$ (6.6)

The precise normalization will be fixed by comparing results with the loop
model given in section 4.3. We consider diagonal twisted boundary condition,
which can be implemented in the algebraic Bethe ansatz by a constant matrix

$\displaystyle K_{a}=\left(\begin{array}[]{cc}\kappa^{+}&0\\\ 0&\kappa^{-}\\\
\end{array}\right)$ (6.9)

in the auxiliary space. The monodromy matrix and the transfer matrix are then
defined as

$\displaystyle\widetilde{\mathbb{M}}_{a}(u)=K_{a}\,R_{a1}(u)R_{a2}(u)\ldots
R_{aL}(u)\,,\qquad\widetilde{\mathbb{T}}(u)=\text{tr}_{a}\widetilde{\mathbb{M}}_{a}(u)\,,$
(6.10)

and seeing the former as a $2\times 2$ matrix in auxiliary space of operators
acting on the $L$ quantum spaces, we may write as usual

$\displaystyle\widetilde{\mathbb{M}}_{a}(u)=\left[\begin{array}[]{cc}\tilde{A}(u)&\tilde{B}(u)\\\
\tilde{C}(u)&\tilde{D}(u)\end{array}\right]\,,\qquad\widetilde{\mathbb{T}}(u)=\tilde{A}(u)+\tilde{D}(u)\,.$
(6.13)

The transfer matrix can be diagonalized by the Bethe ansatz. We start with the
pseudovacuum $|\Omega\rangle=|\\!\uparrow^{L}\rangle$, which is an eigenvector
of $\tilde{A}$ and $\tilde{D}$, and is annihilated by $\tilde{C}$. We have

$\displaystyle\tilde{A}(u)|\Omega\rangle=a(u)|\Omega\rangle\,,\qquad\tilde{D}(u)|\Omega\rangle=d(u)|\Omega\rangle\,,$
(6.14)

where

$\displaystyle
a(u)=\kappa^{+}\,\left(\sinh(u+\tfrac{i\gamma}{2})\right)^{L},\qquad
d(u)=\kappa^{-}\,\left(\sinh(u-\tfrac{i\gamma}{2})\right)^{L}.$ (6.15)

Bethe states $|\mathbf{u}\rangle$ are constructed by acting on
$|\Omega\rangle$ by the $\tilde{B}(u)$ operators,

$\displaystyle|\mathbf{u}\rangle=\tilde{B}(u_{1})\ldots\tilde{B}(u_{M})|\Omega\rangle\,.$
(6.16)

If the Bethe roots $\mathbf{u}=\\{u_{1},\ldots,u_{M}\\}$ satisfy the Bethe
ansatz equations

$\displaystyle\left(\frac{\sinh(u_{j}+\tfrac{i\gamma}{2})}{\sinh(u_{j}-\tfrac{i\gamma}{2})}\right)^{L}=-\frac{\kappa^{-}}{\kappa^{+}}\prod_{k=1}^{M}\frac{\sinh(u_{j}-u_{k}+i\gamma)}{\sinh(u_{j}-u_{k}-i\gamma)}\,,$
(6.17)

the state $|\mathbf{u}\rangle$ diagonalizes the transfer matrix

$\displaystyle\widetilde{\mathbb{T}}(u)|\mathbf{u}\rangle=\tilde{\tau}(u)|\mathbf{u}\rangle\,.$
(6.18)

The corresponding eigenvalue,

$\displaystyle\tilde{\tau}(u)\propto
a(u)\frac{\tilde{Q}(u-i\gamma)}{\tilde{Q}(u)}+d(u)\frac{\tilde{Q}(u+i\gamma)}{\tilde{Q}(u)}\,,$
(6.19)

can be expressed in terms of the $Q$-function defined by

$\displaystyle\tilde{Q}(u)=\prod_{j=1}^{M}\sinh(u-u_{j})\,.$ (6.20)

To precisely relate the results to the loop model, we fix the normalization
and take the eigenvalue of the transfer matrix to be

$\displaystyle\tilde{\tau}(u)=\kappa^{+}\left(\frac{\sinh(u+\tfrac{i\gamma}{2})}{\sinh(u-\tfrac{i\gamma}{2})}\right)^{L}\frac{\tilde{Q}(u-i\gamma)}{\tilde{Q}(u)}+\kappa^{-}\frac{\tilde{Q}(u+i\gamma)}{\tilde{Q}(u)}\,.$
(6.21)

To apply the computational algebraic-geometry method, it is more convenient to
work with the multiplicative variables

$\displaystyle q=e^{i\gamma}\,,\qquad t=e^{u}\,,\qquad t_{k}=e^{u_{k}}\,,$
(6.22)

in terms of which the Bethe equations (6.17) become

$\displaystyle\left(\frac{t_{j}^{2}\,q-1}{t_{j}^{2}-q}\right)^{L}=-\frac{\kappa^{-}}{\kappa^{+}}\prod_{k=1}^{M}\frac{t_{j}^{2}\,q^{2}-t_{k}^{2}}{t_{j}^{2}-t_{k}^{2}\,q^{2}}\,.$
(6.23)

The eigenvalue of the transfer matrix then takes the form

$\displaystyle\tau(t)=\kappa^{+}\left(\frac{t^{2}q-1}{t^{2}-q}\right)^{L}\frac{Q(t\,q^{-1})}{Q(t)}+\kappa^{-}\,\frac{Q(t\,q)}{Q(t)}\,,$
(6.24)

where we have defined

$\displaystyle Q(t)=\prod_{k=1}^{M}(t-t^{-1}t_{k}^{2})\,.$ (6.25)

$Q(t)$ is a Laurent polynomial of degree $M$ which can be expanded as

$\displaystyle
Q(t)=t^{M}+\sum_{k=1}^{M-1}c_{k}\,t^{2k-M}+\frac{1}{t^{M}c_{0}}\,.$ (6.26)

To reproduce the result in the loop model, we take the twist parameters
$\kappa^{\pm}=\rho^{\pm 1}$, with $\rho=e^{i\phi/2}$.

###  6.2 Twisted $Q$-system of the XXX spin chain

As mentioned before, to find the physical solutions of the Bethe quation, we
apply the rational $Q$-system. To this end, we need to generalize the method
of [31, 32] to include twisted boundary conditions. We first consider the XXX
model in this subsection. The XXZ case will be discussed in the next
subsection.

Let us recall briefly the twist-less rational $Q$-system of the XXX spin
chain. Each $Q$-system is related to a Young tableaux as is shown in figure
6.2.

Figure 6.2: The Young tableaux associated to the rational $Q$-system. At each
node of the Young diagram we have one $Q$-function denoted by $Q_{a,s}$. The
number in each circle denotes the power of the polynomial.

For the $\mathfrak{su}(2)$ spin chain with length $L$ and magnon number $M$,
the Young tableaux has two rows with the number of boxes given by $(M,L-M)$,
where $L-M\geq M$. At each node of the Young tableaux, we define a
$Q$-function. The $Q$-functions satisfy the following $QQ$-relation

$\displaystyle
Q_{a+1,s}(u)Q_{a,s+1}(u)=Q_{a+1,s+1}^{-}(u)Q_{a,s}^{+}(u)-Q_{a+1,s+1}^{+}(u)Q_{a,s}^{-}(u)\,,$
(6.27)

where we have introduced the notation $f^{\pm}(u)\equiv
f(u\pm\frac{i}{2})$.444This notation only applies to $Q$-functions and has no
relation to the superscripts used for the twist parameters $\kappa^{\pm}$
above. We impose the boundary conditions $Q_{a,s}=1$ at the upper-right
boundary, together with $Q_{0,0}(u)=u^{L}$. One can then parameterize the
$Q_{1,0}$ by

$\displaystyle
Q_{1,0}(u)=\prod_{j=1}^{M}(u-u_{j})=u^{M}+\sum_{k=0}^{M-1}c_{k}u^{k}.$ (6.28)

Requiring all the $Q_{a,s}$ functions on the Young tableaux to be
_polynomials_ , we obtain a set of equations for the coefficients
$\\{c_{k}\\}$ in (6.28), which can be solved analytically or numerically. The
zeros of $Q_{1,0}(u)$ then give the corresponding Bethe roots.

A comment is in order here. The $QQ$-relations (6.27) are in fact defined by
proportionality. This means the $QQ$-relation

$\displaystyle
Q_{a+1,s}(u)Q_{a,s+1}(u)={\kappa_{a}}\left(Q_{a+1,s+1}^{+}(u)Q_{a,s}^{-}(u)-Q_{a+1,s+1}^{-}(u)Q_{a,s}^{+}(u)\right)\,,$
(6.29)

is equivalent to (6.27) in the sense that they lead to the same Bethe roots.

Now we turn to the twisted spin chain. To incorporate the twist in the
rational $Q$-system, we modify the $QQ$-relation [33] as follows

$\displaystyle
Q_{a+1,s}(u)Q_{a,s+1}(u)={\kappa_{a}}\,Q_{a+1,s+1}^{-}(u)Q_{a,s}^{+}(u)-Q_{a+1,s+1}^{+}(u)Q_{a,s}^{-}(u)\,,$
(6.30)

where $\kappa_{a}$ are $u$ independent constants. Notice that we multiply
different factors onto the two terms on the right-hand side. We will call this
deformed $QQ$-relation (6.30) the $\kappa QQ$-relation.

The $\kappa QQ$-relation can be defined for the $\mathfrak{su}(N)$ spin chain
and provides a method to solve the corresponding twisted Bethe equations. For
our purpose, we consider only the $\mathfrak{su}(2)$ case where the Young
tableaux have two rows. We impose the same boundary conditions $Q_{2,s}=1$ at
the upper-right boundary, and $Q_{0,0}(u)=u^{L}$ in the lower-left corner. For
this simple situation, the $\kappa QQ$ relations can be solved explicitly as
follows.

First, consider the $\kappa QQ$ relation for $a=1$. By taking into account the
boundary condition we find

$\displaystyle Q_{1,s+1}(u)=\kappa_{1}\,Q_{1,s}^{+}(u)-Q_{1,s}^{-}(u)\,.$
(6.31)

This is a finite-difference equation whose solution is given by [31, 32]

$\displaystyle
Q_{1,n}(u)=Q^{(n)}_{\kappa_{1}}(u)\equiv\widetilde{D}^{n}_{\kappa_{1}}Q(u)\,,$
(6.32)

where the twisted discrete derivative is defined by

$\displaystyle\widetilde{D}_{\kappa}f(u)={\kappa}f(u+\tfrac{i}{2})-f(u-\tfrac{i}{2})\,.$
(6.33)

Taking then $a=0$ in the $\kappa QQ$-relation, we obtain

$\displaystyle
Q_{0,n}=\frac{{\kappa_{0}}\,Q^{(n)-}_{\kappa_{1}}Q_{0,s}^{+}-Q^{(n)+}_{\kappa_{1}}Q_{0,s}^{-}}{Q^{(n-1)}_{\kappa_{1}}}\,.$
(6.34)

##### Bethe equations.

Let us derive the Bethe equations from the $\kappa QQ$-relations. Consider the
following $\kappa QQ$-relations

$\displaystyle Q_{1,1}(u)=$
$\displaystyle\,\kappa_{1}\,Q_{1,0}^{+}(u)-Q_{1,0}^{-}(u)\,,$ (6.35a)
$\displaystyle Q_{1,0}(u)Q_{0,1}(u)=$
$\displaystyle\,\kappa_{0}\,Q_{1,1}^{-}(u)Q_{0,0}^{+}(u)-Q_{1,1}^{+}(u)Q_{0,0}^{-}(u)\,.$
(6.35b)

The Bethe equations can be derived by requiring that all the $Q_{a,s}$ be
polynomials in $u$. As for the non-twisted case, the zeros of $Q_{1,0}(u)$ are
the Bethe roots, which we will denote by $\\{u_{k}\\}$. We evaluate the
equations (6.35) at one of the Bethe roots $u_{k}$. From (6.35a) we obtain

$\displaystyle Q_{1,1}^{+}(u_{k})=\kappa_{1}\,Q_{1,0}^{++}(u_{k})\,,\qquad
Q_{1,1}^{-}(u_{k})=-Q_{1,0}^{--}(u_{k})\,,$ (6.36)

where the double signs mean double shifts, $f^{\pm\pm}(u)\equiv f(u\pm i)$.
From (6.35b) we have

$\displaystyle\kappa_{0}\,Q_{1,1}^{-}(u_{k})Q_{0,0}^{+}(u_{k})-Q_{1,1}^{+}(u_{k})Q_{0,0}^{-}(u_{k})=0\,.$
(6.37)

Combining these two equations, we find that

$\displaystyle\kappa_{0}Q_{1,0}^{--}(u_{k})Q_{0,0}^{+}(u_{k})+\kappa_{1}Q_{1,0}^{++}(u_{k})Q_{0,0}^{-}(u_{k})=0\,.$
(6.38)

Using the fact that

$\displaystyle Q_{0,0}(u)=u^{L},\qquad Q_{1,0}(u)=\prod_{j=1}^{M}(u-u_{j})\,,$
(6.39)

we see that (6.38) is precisely the twisted Bethe equation

$\displaystyle-\frac{\kappa_{1}}{\kappa_{0}}=\left(\frac{u_{k}+\tfrac{i}{2}}{u_{k}-\tfrac{i}{2}}\right)^{L}\prod_{j=1}^{M}\frac{u_{k}-u_{j}-i}{u_{k}-u_{j}+i}\,.$
(6.40)

Similar to the untwisted case, the requirement that all $Q_{a,s}$ be
polynomials in $u$ leads to a set of algebraic equations—the _zero-remainder
conditions_ (ZRC)—for the coefficients of the $Q$-functions. In our case, the
ZRC now depend on the twists. The ZRC for fixed values of $\kappa_{0}$ and
$\kappa_{1}$ can be solved numerically. We find that for the system with
length $L$ and magnon number $M$, the number of solutions for non-trivial
twist is $L\choose M$. We have checked numerically that the solutions of the
$\kappa QQ$-relation indeed solve the twisted Bethe equation.

###  6.3 Twisted $Q$-system of the XXZ spin chain

We are now ready to generalize the $\kappa QQ$-relation to the $q$-deformed
case, for which the XXX chain just considered is the special case $q=1$. We
consider the multiplicative variable $t=e^{u}$ as in (6.22). The $\kappa
QQ$-relation takes the same form

$\displaystyle
Q_{a+1,s}(t)Q_{a,s+1}(t)={\kappa_{a}}\,Q_{a+1,s+1}^{-}(t)Q_{a,s}^{+}(t)-Q_{a+1,s+1}^{+}(u)Q_{a,s}^{-}(t)\,,$
(6.41)

where now $f^{\pm}(t)\equiv f(tq^{\pm 1/2})$, and we have introduced twist
parameters $\kappa_{0}$ and $\kappa_{1}$. We focus on the $U_{q}(sl(2))$ case
where the corresponding Young tableaux has two rows. The boundary conditions
are555One might notice that the normalization of $Q_{1,0}(t)$ is different
from the normalization of $Q(t)$ in (6.26). Since the $\kappa QQ$-relation is
defined by proportionality, this difference is harmless.

$\displaystyle Q_{0,0}(t)=(t-t^{-1})^{L}\,,\qquad
Q_{1,0}(t)=Q(t)=\prod_{j=1}^{M}(tt_{j}^{-1}-t^{-1}t_{j})\,.$ (6.42)

Both $Q$-functions are Laurent polynomials in $t$. The twisted discrete
derivative in the $q$-deformed case is defined by

$\displaystyle\widetilde{D}_{q,\kappa}f(t)={\kappa}\,f(tq^{+1/2})-f(tq^{-1/2})\,.$
(6.43)

Again we can solve the $\kappa QQ$-relation explicitly, and find that
$Q_{1,n}(t)=Q^{(n)}(t)=\widetilde{D}^{n}_{q,\kappa_{1}}Q(t)$. Similarly,
$Q_{0,n}(t)$ is given by

$\displaystyle
Q_{0,n}(t)=\frac{{\kappa_{0}}\,Q^{(n)+}Q_{0,n-1}^{-}-Q^{(n)-}Q_{0,n-1}^{+}}{Q^{(n-1)}}\,.$
(6.44)

By requiring all $Q_{0,n}(t)$ to be polynomials in $t$ and $t^{-1}$, we obtain
a set of zero-remainder conditions. Going through the same steps as the XXX
case, we find

$\displaystyle\left(\frac{t_{j}^{2}\,q-1}{t_{j}^{2}-q}\right)^{L}=-\frac{\kappa_{1}}{\kappa_{0}}\prod_{k=1}^{M}\frac{t_{j}^{2}\,q^{2}-t_{k}^{2}}{t_{j}^{2}-t_{k}^{2}\,q^{2}}\,.$
(6.45)

By identifying $\kappa_{1}=\kappa^{-}$ and $\kappa_{0}=\kappa^{+}$, we
reproduce the Bethe equation of the twisted XXZ spin chain (6.23). For generic
$q$ and $\kappa^{\pm}$, the ZRCs can be solved numerically. We have checked
that for length $L$ and magnon number $M$, the number of solutions of the ZRC
is ${L\choose M}$ and they give the correct Bethe roots. The situation where
$q$ is a root of unity is also interesting, but more subtle, and it will be
analyzed elsewhere.

###  6.4 An explicit example

In order to illustrate that we can reproduce the result for the loop model
from the twisted XXZ spin chain, we consider the example with $L=2$ in this
subsection. In aTL language, there are two standard modules with $j=0,1$,
which correspond to the two sectors of the XXZ spin chain with magnon numbers
$M=1,0$ respectively. From now on, we take the twists to be
$\kappa^{\pm}=\rho^{\pm 1}$.

  * •

$j=1$. This corresponds to the sector $M=0$. We have $Q(t)=1$ and the transfer
matrix (6.24) reads

$\displaystyle
t_{1,\rho}(t)=\rho\,\left(\frac{t^{2}q-1}{t^{2}-q}\right)^{2}+\rho^{-1}\,.$
(6.46)

The transfer matrix of the loop model is given by fixing the spectral
parameter to a special value $t^{2}=-1$. Plugging this into (6.46), we obtain

$\displaystyle t_{1,\rho}\equiv t_{1,\rho}(i)=\rho+\rho^{-1}\,,$ (6.47)

in agreement with (4.9).

  * •

$j=0$. This corresponds to the sector $M=1$. We evaluate the transfer matrix
at $t=i$,

$\displaystyle
t_{0}(i)=\rho\frac{Q(iq^{-1})}{Q(i)}+\rho^{-1}\frac{Q(iq)}{Q(i)}\,.$ (6.48)

The $Q$-function is given by

$\displaystyle Q(t)=t+\frac{1}{c_{0}t}\,,$ (6.49)

where $c_{0}$ satisfies the ZRC

$\displaystyle(q^{2}-\rho^{2})c_{0}^{2}+2q(1-\rho^{2})c_{0}+(1-q^{2}\rho^{2})=0\,.$
(6.50)

Assuming for the moment $q^{2}-\rho^{2}\neq 0$, we have two solutions for
$c_{0}$:

$\displaystyle c_{0}^{(1)}=-\frac{q\rho+1}{\rho+q}\,,\qquad
c_{0}^{(2)}=-\frac{q\rho-1}{\rho-q}\,.$ (6.51)

Plugging into the transfer matrix (6.48), the corresponding two eigenvalues
are

$\displaystyle t_{0}^{(1)}=-2+(q+q^{-1})+(\rho+\rho^{-1})\,,\qquad
t_{0}^{(2)}=2-(q+q^{-1})+(\rho+\rho^{-1})\,.$ (6.52)

Now taking $\rho=-q$, we obtain

$\displaystyle t_{0}^{(1)}=-2\,,\qquad t_{0}^{(2)}=2+2\mathfrak{m}\,,$ (6.53)

in accord with (4.8), where $\mathfrak{m}$ is defined in (2.10). We see that
(6.53) gives precisely the two eigenvalues of the transfer matrix of the aTL
loop model.

Before ending the section, let us make a comment here. We find that in the
ZRCs (6.50), the case $\rho=\pm q$ is special. In this case, the quadratic
equation becomes a linear equation and the number of solutions is reduced from
2 to 1. As we will explain in more detail in the next section, this is a
general phenomenon. When $\rho$ and $q$ satisfy certain relations, the number
of solutions of the ZRC changes. These relations, as we shall see, are
precisely the resonance conditions of the standard module.

##  7 Resonance condition from algebraic geometry

As we have seen in section 2, for generic values of $q$ and $\rho$, the
standard module $\mathcal{W}_{j,\rho^{2}}$ is irreducible with dimension
$\hat{d}_{j}$. However, when $\rho$ satisfies the resonance condition, the
representation becomes reducible and we can define the quotient module. From
the Bethe ansatz perspective, the dimension of the standard module corresponds
to the number of physical solutions of the Bethe equations. We therefore
expect the structure of solutions to change at these special values of $\rho$.
One simple example has been given in the previous section.

How can we see such a change in the number of solutions? One approach is by
using the Gröbner basis. For a given set of algebraic equations, which are the
ZRCs in our context, we can compute the Gröbner basis by following the
standard algorithm. The number of solutions is given by the dimension of the
quotient ring, whose basis is characterized by the leading terms of the
Gröbner basis. Therefore, we expect that at special values, some coefficients
of the leading terms of the Gröbner basis vanish.

To test this expectation, we consider three explicit examples,
$(L,M)=(4,1),(4,2)$ and $(L,M)=(6,3)$, in what follows. We will take a careful
look at the leading terms of the Gröbner basis. At special value of the
twists, some leading terms of the Gröbner basis vanish and the dimension of
the quotient ring changes.

###  7.1 Example 1: $L=4$, $M=1$

For $L=4$, $M=1$, the $Q_{1,0}(t)$ is given by

$\displaystyle Q_{1,0}(t)=t+\frac{1}{c_{0}t}\,.$ (7.1)

The rational $Q$-system leads to the following ZRC

$\displaystyle 0=$
$\displaystyle\,(\rho^{2}-q^{4})(q^{2}-1){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{4}}+4q(\rho^{2}-q^{2})(q^{2}-1){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{3}}+6q^{2}(q^{2}-1)(\rho^{2}-1){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{2}}$
(7.2)
$\displaystyle\,+4q(\rho^{2}q^{2}-1)(q^{2}-1){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}}+(\rho^{2}q^{4}-1)(q^{2}-1)\,.$

This is a single-variable polynomial equation. Note that for $q=\pm 1$, the
right-hand side is zero identically, which implies that there are infinitely
many solutions to the ZRC in this case. This is nothing but a manifestation of
the subtleties at $q$ being a root of unity. Here we focus on generic $q$,
therefore we can strip off the global factor $q^{2}-1$. The Gröbner basis can
be computed straightforwardly and is given by

$\displaystyle
g_{1}=(q^{4}-\rho^{2}){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{4}}+4q(q^{2}-\rho^{2}){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{3}}-6q^{2}(\rho^{2}-1){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{2}}+4q(1-q^{2}\rho^{2}){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}}+(1-q^{4}\rho^{2})\,.$
(7.3)

The leading term of the Gröbner basis is

$\displaystyle\text{LT}[g_{1}]=(q^{4}-\rho^{2}){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{4}}\,,$
(7.4)

from which we can read off the quotient-ring basis

$\displaystyle\\{c_{0}^{3},c_{0}^{2},c_{0},1\\}.$ (7.5)

This implies that there are 4 physical solutions to the twisted Bethe equation
with $L=4$, $M=1$.

##### Special twist.

It is easy to see that the leading term’s coefficient (7.4) vanishes at the
specific values of the twist $\rho^{2}=q^{4}$. This hints that if we set
$\rho=\pm q^{2}$, the number of solutions may be different from the generic
case. Indeed, taking $\rho=q^{2}$, we find the following ZRC

$\displaystyle 0=$
$\displaystyle\,4q^{3}(q^{2}-1)^{2}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{3}}+6q^{2}(q^{2}+1)(q^{2}-1)^{2}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{2}}$
(7.6)
$\displaystyle\,+4q(1-q^{2}-q^{6}+q^{8})\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}}+(1-q^{2}-q^{8}+q^{10})\,.$

The corresponding Gröbner basis is given by

$\displaystyle
g_{1}=4q^{3}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{3}}+6q^{2}(1+q^{2})\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{2}}+4q(1+q^{2}+q^{4})\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}}+(1+q^{2})(1+q^{4})\,.$
(7.7)

The leading term is

$\displaystyle\text{LT}[g_{1}]=4q^{3}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{3}}\,,$
(7.8)

which leads to the quotient ring basis

$\displaystyle\\{c_{0}^{2},c_{0},1\\}\,.$ (7.9)

Therefore, we see that at $\rho=q^{2}$, the number of solution is 3.

##### Resonance condition.

Let us now compare what we have found with the resonance condition of the aTL
representation discussed in section 7. At the special value $\rho=q^{2}$, the
corresponding standard module $\mathcal{W}_{1,q^{2}}$ satisfies the resonance
condition (2.18) with $(j,z)=(1,q^{2})$ and $(j^{\prime},z^{\prime})=(2,q)$.
The quotient module is

$\displaystyle\overline{\mathcal{W}}_{1,q^{2}}=\mathcal{W}_{1,q^{2}}/\mathcal{W}_{2,q}\,,$
(7.10)

with the corresponding dimension

$\displaystyle\bar{d}_{1}=\text{dim}\,\overline{\mathcal{W}}_{1,q^{2}}={4\choose
2+1}-{4\choose 2+2}=3\,.$ (7.11)

This matches nicely with the dimension of the quotient ring (7.9).

###  7.2 Example 2: $L=4$, $M=2$

Now we consider a more complicated case with $L=4$, $M=2$. In this case, the
$Q_{1,0}$ function is given by

$\displaystyle Q_{1,0}(t)=t^{2}+c_{1}+\frac{1}{c_{0}\,t^{2}}\,.$ (7.12)

The ZRC gives a set of algebraic equations for $\\{c_{0},c_{1}\\}$. These
equations are too long to be presented here. The Gröbner basis of the ZRC in
this case, with respect to degree-reverse lexicographic ordering, consists of
four polynomials $\\{g_{1},g_{2},g_{3},g_{4}\\}$. The leading terms are given
by

$\displaystyle\text{LT}[g_{1}]=q^{2}(q^{2}\rho^{2}-1)^{3}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{1}^{3}}\,,$
(7.13)
$\displaystyle\text{LT}[g_{2}]=q^{2}(\rho^{2}-1)(q^{2}\rho^{2}-1)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}c_{1}^{2}}\,,$
$\displaystyle\text{LT}[g_{3}]=(q^{2}-\rho^{2})^{2}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{2}c_{1}}\,,$
$\displaystyle\text{LT}[g_{4}]=(q^{4}-\rho^{2})(q^{2}-\rho^{2})^{3}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{3}}\,,$

where the blue-colored parts are the monomials in $c_{0}$ and $c_{1}$, and the
rest are coefficients. At generic values of $\rho$ and $q$, the basis of the
quotient ring is given by

$\displaystyle\\{c_{0}^{2},\,c_{1}^{2},\,c_{0}c_{1},\,c_{0},\,c_{1},\,1\\}\,,$
(7.14)

which implies that the corresponding twisted Bethe equations have 6 physical
solutions.

##### Special twists and resonance conditions.

From (7.13), we see that one or more of the leading terms’ coefficients vanish
when

$\displaystyle\rho^{2}=1\,,\qquad\rho^{2}=q^{\pm 2}\,,\qquad\rho^{2}=q^{4}\,.$
(7.15)

The case $\rho^{2}=1$ corresponds to the twist-less case. The other two cases,
$\rho=q^{\pm 1},q^{\pm 2}$, correspond exactly to the two resonance conditions
$(j,j^{\prime})=(0,1),(0,2)$ given in (2.18).

Since the coefficient vanish at the special points, they need to be treated
separately. We can set $\rho$ to these special values in the $\kappa
QQ$-relation, and then calculate the Gröbner basis. For $\rho=q$, the leading
terms for the Gröbner basis now become

$\displaystyle\text{LT}[g_{1}]=(q^{2}+1){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{1}}\,,$
(7.16)
$\displaystyle\text{LT}[g_{2}]=q^{2}(q^{2}+4q+1)(q^{2}-4q+1){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{2}}\,,$

from which it is easy to see that the dimension of the quotient ring is 2.
This matches the dimension of the quotient module given in (2.20) with $L=4$
and $(j,j^{\prime})=(0,1)$.

Taking instead $\rho=q^{2}$, the leading terms of the Gröbner basis become

$\displaystyle\text{LT}[g_{1}]=6q^{6}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{2}}\,,$
(7.17)
$\displaystyle\text{LT}[g_{2}]=(q^{2}-q+1)^{3}(q^{2}+q+1)^{3}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{1}^{3}}\,,$
$\displaystyle\text{LT}[g_{3}]=(q^{2}-q+1)(q^{2}+q+1)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}c_{1}^{2}}\,,$

from which we find that the dimension of the quotient ring is now 5. This
matches the dimension of the quotient module given in (2.20) with $L=4$ and
$(j,j^{\prime})=(0,2)$.

###  7.3 Example 3: $L=6$, $M=3$

In this case, the $Q_{1,0}$ function reads

$\displaystyle
Q_{1,0}(t)=t^{3}+c_{2}\,t+\frac{c_{1}}{t}+\frac{1}{c_{0}t^{3}}\,.$ (7.18)

The ZRC leads to a set of algebraic equations in $\\{c_{0},c_{1},c_{2}\\}$.
The corresponding Gröbner basis consists of 12 polynomials
$\\{g_{1},\ldots,g_{12}\\}$. The leading terms are given by

$\displaystyle\text{LT}[g_{1}]=(q^{2}-\rho^{2})(q^{4}-\rho^{2})\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{2}c_{1}}\,,$
(7.19)
$\displaystyle\text{LT}[g_{2}]=q^{4}(q^{2}-\rho^{2})(q^{2}\rho^{2}-1)^{2}(q^{4}\rho^{2}-1)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}^{4}}\,,$
$\displaystyle\text{LT}[g_{3}]=q^{4}(q^{2}\rho^{2}-1)^{3}G_{1}(q,\rho)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{1}c_{2}^{3}}\,,$
$\displaystyle\text{LT}[g_{4}]=q^{4}(q^{2}-\rho^{2})(\rho^{2}-1)(q^{2}\rho^{2}-1)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}c_{2}^{3}}\,,$
$\displaystyle\text{LT}[g_{5}]=q^{6}(\rho^{2}-1)^{2}(q^{2}\rho^{2}-1)^{3}G_{2}(q,\rho)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{1}^{2}c_{2}^{2}}\,,$
$\displaystyle\text{LT}[g_{6}]=q^{4}(\rho^{2}-1)^{2}(q^{2}\rho^{2}-1)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}c_{1}c_{2}^{2}}\,,$
$\displaystyle\text{LT}[g_{7}]=q^{6}(\rho^{2}-1)^{2}(q^{4}-\rho^{2})(q^{2}\rho^{2}-1)^{4}(q^{4}\rho^{2}-1)^{2}G_{1}(q,\rho)G_{2}(q,\rho)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{1}^{3}c_{2}}\,,$
$\displaystyle\text{LT}[g_{8}]=q^{2}(\rho^{2}-1)(q^{2}-\rho^{2})(q^{4}-\rho^{2})(q^{2}\rho^{2}-1)^{2}\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}c_{1}^{2}c_{2}}\,,$
$\displaystyle\text{LT}[g_{9}]=(q^{2}-\rho^{2})^{4}(q^{4}-\rho^{2})^{3}(q^{4}\rho^{2}-1)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{3}c_{2}}\,,$
$\displaystyle\text{LT}[g_{10}]=q^{4}(\rho^{2}-1)(q^{2}-\rho^{2})(q^{4}-\rho^{2})(q^{2}\rho^{2}-1)^{4}(q^{4}\rho^{2}-1)^{4}G_{1}(q,\rho)G_{2}(q,\rho)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{1}^{4}}\,,$
$\displaystyle\text{LT}[g_{11}]=q^{4}(\rho^{2}-1)(q^{2}-\rho^{2})^{2}(q^{4}-\rho^{2})(q^{2}\rho^{2}-1)^{2}(q^{4}\rho^{2}-1)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}c_{1}^{3}}\,,$
$\displaystyle\text{LT}[g_{12}]=(q^{2}-\rho^{2})^{4}(q^{4}-\rho^{2})^{4}(q^{6}-\rho^{2})(q^{4}\rho^{2}-1)G_{1}(q,\rho)G_{2}(q,\rho)\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{0}^{4}}\,,$

where the polynomials $G_{1}(q,\rho)$ and $G_{2}(q,\rho)$ are given by

$\displaystyle G_{1}(q,\rho)=$
$\displaystyle\,-2q^{6}\,\rho^{4}+(q^{8}+2q^{6}-2q^{4}+2q^{2}+1)\rho^{2}-2q^{2}\,,$
(7.20) $\displaystyle G_{2}(q,\rho)=$
$\displaystyle\,q^{6}\rho^{6}+(q^{10}-3q^{8}-2q^{6}+3q^{4}-q^{2}-1)\rho^{4}$
$\displaystyle+(q^{10}+q^{8}-3q^{6}+2q^{4}+3q^{2}-1)-q^{4}\,.$

##### Special twists and resonance conditions.

We observe that there are several cases where the leading terms’ coefficient
vanish. Firstly, $\rho^{2}=1$ corresponds to the twist-less case. Secondly we
have the following conditions

$\displaystyle\rho^{2}=q^{\pm 2}\,,\qquad\rho^{2}=q^{\pm
4}\,,\qquad\rho^{2}=q^{\pm 6}\,,$ (7.21)

which correspond to the resonance conditions
$(j,j^{\prime})=(0,1),(0,2),(0,3)$ in (2.18).

In this more complicated example, we also have the possibility that at certain
values of $\rho$, the rather complicated polynomials $G_{1}(q,\rho)$ and
$G_{2}(q,\rho)$ vanish. It is important to notice that the nature of these two
types of special values of $\rho$ are different. By tuning $\rho$ to the
special values $q^{\pm 1},q^{\pm 2},q^{\pm 3}$ which satisfy the resonance
conditions (7.21) in the $\kappa QQ$-relation, we can verify that the
dimensions of the resulting quotient rings are $5,14,19$ respectively. These
are indeed the dimensions of the quotient modules.

On the other hand, we find numerically that if we take $\rho$ to be one of the
values where $G_{1}(q,\rho)$ or $G_{2}(q,\rho)$ vanish, the dimension of the
quotient ring does not change, and remains equal to 20. Let us take a more
careful look at this point.

For generic value of $\rho$ and $q$, the corresponding quotient-ring basis is
given by

$\displaystyle\left\\{c_{0}^{2}c_{2}^{2},c_{0}^{3},c_{0}c_{1}^{2},c_{1}^{3},c_{0}^{2}c_{2},c_{0}c_{1}c_{2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}^{2}c_{2}},c_{0}c_{2}^{2},c_{1}c_{2}^{2},c_{2}^{3},c_{0}^{2},c_{0}c_{1},c_{1}^{2},c_{0}c_{2},c_{1}c_{2},c_{2}^{2},c_{0},c_{1},c_{2},1\right\\}\,.$
(7.22)

Now we consider the special value of $\rho$ such that $G_{1}(q,\rho)$ is
vanishing. The leading-term monomials are then given by

$\displaystyle\left\\{c_{1}^{2}c_{2},c_{0}^{2}c_{1},c_{2}^{4},c_{0}c_{2}^{3},c_{0}c_{1}c_{2}^{2},c_{0}^{3}c_{2},c_{1}^{4},c_{0}c_{1}^{3},c_{0}^{4}\right\\}\,.$
(7.23)

We see that there are now only 9 polynomials in the Gröbner basis.
Nevertheless, the quotient ring is still of dimension 20, with the following
basis

$\displaystyle\left\\{c_{0}^{2}c_{2}^{2},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}c_{2}^{3}},c_{0}^{3},c_{0}c_{1}^{2},c_{1}^{3},c_{0}^{2}c_{2},c_{0}c_{1}c_{2},c_{0}c_{2}^{2},{c_{1}c_{2}^{2}},c_{2}^{3},c_{0}^{2},c_{0}c_{1},c_{1}^{2},c_{0}c_{2},c_{1}c_{2},c_{2}^{2},c_{0},c_{1},c_{2},1\right\\}\,.$
(7.24)

We find that the two quotient-ring bases are not exactly the same in (7.22)
and (7.24). The difference stems from the red-colored monomials.

An alternative way to see the difference between the two types of special
values of $\rho$ is by considering different orderings. For a different
ordering, the Gröbner basis in general takes a different form. Nevertheless,
at the special values which satisfy resonance conditions, the coefficients of
the leading terms still vanish, such that the dimension of the quotient ring
basis is different. On the other hand, for a differential ordering, the zeros
of $G_{1}(q,\rho)$ and $G_{2}(q,\rho)$ in (7.20) in general no longer lead to
vanishing leading terms of the new Gröbner basis. Therefore, these values are
simply artifacts of the specific ordering that we choose.

###  7.4 General comments

From the above three examples, and by exploring more examples, we can make the
following observations:

  1. 1.

The resonance conditions of the aTL algebra can be obtained by requiring the
coefficients of the leading terms of the Gröbner basis to vanish.

  2. 2.

There are additional values of $\rho$ at which some of the leading terms’
coefficients vanish. Such additional values are artifacts of the specific
basis we are working with. By choosing a different monominal ordering, they do
not lead to vanishing coefficients anymore. Thus these values do not reflect
the intrinsic properties of the equations themselves.

  3. 3.

We can set the values of $\rho$ to satisfy the resonance condition in the
$\kappa QQ$-relation. The number of solutions for such $\kappa QQ$-relation is
precisely the dimension of the quotient module. We can compute the transfer
matrices of the quotient module using these $\kappa QQ$-relations. There is
then agreement between the dimension of the quotient ring and the dimension of
the quotient module (2.19).

We emphasize that these are observations based on explicit examples and guided
by physical intuitions. It would be very interesting to provide a more
rigorous and general proof for these observations.

##  8 Algebraic-geometry approach to the torus partition function

In this section, we present the algebraic-geometry approach to computing the
torus partition function of the loop model and the Potts model. Using this
approach, we obtain closed-form expressions for the partition function
$Z(L,N)$ for $L=4,6$ and arbitrary integer $N$, which will be presented in
section 9. For larger $L$, we can compute the partition function for fixed $L$
up to 10 and for large $N$. The latter results are too long to be presented in
the paper. Instead, for such cases we give the zeros of the partition
functions. The zeros condense on certain curves in the $N\to\infty$ limit.
Such condensation curves can be generated by numerical methods and perfectly
match our results. This will be discussed in section 10.

###  8.1 General strategy

The main quantities to be computed are the companion matrices of the
eigenvalues of the transfer matrices. For the chain on $L$ sites, the possible
values of $j$ are $j=0,1,\ldots,\tfrac{L}{2}$. For the loop model, the value
$j$ is the number of through-lines. It is related to the XXZ spin chain magnon
numbers by $j=\tfrac{L}{2}-M$.

For each fixed $L$ and $j\neq 0$, the values of the twists are given by
$\rho^{2}=\omega_{k}$ (4.4).666Notice that the quantization condition is for
$\rho^{2}$. For the value of $\rho$, we can choose two possible signs,
according to (2.18). This raises the question which branch we should choose.
For the partition functions with even $L$ and $M$, the choice of the branch
does not matter and we can choose either sign. We denote the companion matrix
of the sector $j$ with the twist $\rho^{2}=\omega_{k}$ by
$\mathbf{T}_{j,\omega_{k}}$.

The case $j=0$ is special. For $j=0$, the twist is given by $\rho^{2}=q^{2}$,
which satisfies the resonance condition. This implies that the corresponding
standard module $\mathcal{W}_{0,q^{2}}$ is reducible and we have
$\overline{\mathcal{W}}_{0,q^{2}}=\mathcal{W}_{0,q^{2}}/\mathcal{W}_{1,1}$.
Therefore, to compute the companion matrices of the transfer matrix, we can
perform the computation over $\overline{\mathcal{W}}_{0,q^{2}}$ and
$\mathcal{W}_{1,1}$ separately and then take the direct sum. More precisely,
we denote the companion matrix of the transfer matrix over
$\overline{\mathcal{W}}_{0,q^{2}}$ by $\overline{\mathbf{T}}_{0,q^{2}}$. We
then have

$\displaystyle\mathbf{T}_{0,q^{2}}=\overline{\mathbf{T}}_{0,q^{2}}\oplus\mathbf{T}_{1,1}.$
(8.1)

As discussed in the previous section, the quotient ring of the ZRC which
corresponds to $\overline{\mathcal{W}}_{0,q^{2}}$ is constructed by taking
$\rho^{2}=q^{2}$ in the ZRC and then performing the standard algebraic-
geometry computation.

##### Momentum decomposition.

The computation of the companion matrix can be simplified further by
decomposing the quotient ring. In the current context, one natural way
proceeds by performing a decomposition by the _total lattice momentum_ of the
spin chain. This method has been applied previously to the XXX spin chain [4].
The momentum decomposition for the twisted spin chain can be obtained as
follows. The Bethe equations (6.17) can be written as

$\displaystyle
e^{ip(u_{l})L}=-\rho^{-2}\prod_{k=1}^{M}\frac{\sinh(u_{l}-u_{k}+\eta)}{\sinh(u_{l}-u_{k}-\eta)}\,,\qquad
l=1,2,\ldots,M\,,$ (8.2)

where $p(u_{l})$ is the momentum of each magnon. Taking the product of (8.2)
for all $l$, we obtain

$\displaystyle e^{iP_{M}L}=\rho^{-2M}\,,$ (8.3)

where $P_{M}$ is the total momentum

$\displaystyle P_{M}=p(u_{1})+p(u_{2})+\cdots+p(u_{M})\,.$ (8.4)

Therefore we find that

$\displaystyle e^{iP_{M}}=e^{\frac{2\pi in}{L}}\rho^{-\frac{2M}{L}}\,,\qquad
n=1,2,\ldots,L\,.$ (8.5)

On the other hand, we have

$\displaystyle
e^{iP_{M}}=\prod_{j=1}^{M}\frac{\sinh(u_{l}+\tfrac{\eta}{2})}{\sinh(u_{l}-\tfrac{\eta}{2})}=\prod_{j=1}^{M}\frac{t_{l}^{2}q-1}{t_{l}^{2}-q}=\frac{Q(q^{-1/2})}{Q(q^{+1/2})}\,.$
(8.6)

Therefore, the momentum-decomposition condition is given by

$\displaystyle\frac{Q(q^{-1/2})}{Q(q^{+1/2})}=e^{\frac{2\pi
in}{L}}\rho^{-\frac{2M}{L}}\,,\qquad n=1,2,\ldots\,.$ (8.7)

Thus, for a given through-line number $j\neq 0$, we have two independent
decompositions. Each sector in the double decomposition is labeled by two
integers $(n,k)$ where $n=1,\ldots,L$ and $k=1,\ldots,j$.

###  8.2 $TQ$-relation of the twisted spin chain

If we apply (6.24) to compute the companion matrix of the transfer matrix, we
need to take the inverse of the companion matrix of $Q(i)$; see for example
(6.48). There are two drawbacks for this strategy: (1) It is time consuming to
take the inverse of the companion matrix analytically, especially when the
dimension of the matrix is large; (2) Sometimes the determinant of the
companion matrix of $Q(i)$ vanishes, so the matrix does not have an inverse,
even though the eigenvalues of the transfer matrix are perfectly well-defined.

To avoid these problems, we can use the $TQ$-relation to express the
eigenvalues of the transfer matrix directly in terms $\\{c_{k}\\}$, which are
the coefficients of $Q(t)$. The procedure is as follows. Consider the transfer
matrix (with a different normalization) for generic spectral parameter $t$. We
have the following $TQ$-relation

$\displaystyle\tilde{\tau}(t)Q(t)=\rho(t^{2}q-1)^{L}\,Q(tq^{-1})+\rho^{-1}(t^{2}-q)^{L}Q(tq)\,.$
(8.8)

The eigenvalue of the transfer matrix $\tilde{\tau}(t)$ is a polynomial in
$t$,

$\displaystyle\tilde{\tau}(t)=\sum_{k=0}^{L}a_{k}\,t^{2k}\,.$ (8.9)

Plugging this and (6.25) into the $TQ$-relation (8.8), and comparing the
coefficients of different powers in $t$, we obtain a set of algebraic
equations in $\\{a_{k}\\}$ and $\\{c_{k}\\}$. This set of equations are linear
both in $\\{a_{k}\\}$ and $\\{c_{k}\\}$ and can be solved straightforwardly
for $\\{a_{k}\\}$ in terms of $\\{c_{k}\\}$. Plugging back into (8.9), we can
write $\tilde{\tau}(t)$ in terms of $\\{c_{k}\\}$. Once this is done, the
transfer matrices that we need for the computation of the torus partition
functions are given by

$\displaystyle\tau_{M}=\frac{\tilde{\tau}(i)}{(-q-1)^{L}}\,.$ (8.10)

The algebraic-geometry computation gives the companion matrix of
$\\{c_{k}\\}$. Using the $TQ$-relation as described above, we obtain the
companion matrix of $\tau_{M}$, which is the $\mathbf{T}_{j,\omega_{k}}$ we
are after.

Compared to the transfer matrices appearing in (4.5), which are constructed
using the link-pattern basis for the standard module, our companion matrix
$\mathbf{T}_{j,\omega_{k}}$ takes a different form, but it share the same
eigenvalues, viz., the matrices are related by a similarity transformation.
Therefore we have

$\displaystyle\text{tr}_{\mathcal{W}_{j,\rho^{2}=\omega_{k}}}\left[t\left(\frac{\gamma}{2}\right)^{N}\right]=\text{tr}\,\mathbf{T}_{j,\omega_{k}}^{N}$
(8.11)

for any $N$.

###  8.3 The full partition function

In order to compute the torus partition function, we need to take the Markov
trace of the transfer matrices. This is done just as before. Therefore the
torus partition function can be computed as in (4.5),

$\displaystyle
Z_{\text{loop}}(L,N)=\text{tr}\,\mathbf{T}_{0,q^{2}}^{N}+\sum_{j=1}^{L/2}\sum_{m|j}\Lambda_{j,m}\sum_{k\in\kappa_{j}(m)}\text{tr}\,\mathbf{T}_{j,\omega_{k}}^{N}\,.$
(8.12)

The same ingredients can be used to compute the torus partition function of
Potts model,

$\displaystyle
Z_{\text{Potts}}(L,N)=\mathfrak{m}^{LN/2}\left\\{\text{tr}\,\mathbf{T}_{0,q^{2}}^{N}+\frac{Q-1}{2}\text{tr}\,{\mathbf{T}_{0,-1}^{N}}+\sum_{j=1}^{L/2}\sum_{m|j}\widetilde{\Lambda}_{j,m}\sum_{k\in\kappa_{j}(m)}\text{tr}\,\mathbf{T}_{j,\omega_{k}}^{N}\right\\}\,,$
(8.13)

where $\widetilde{\Lambda}_{j,m}$ has been defined in (5.5).

Before presenting the explicit results, let us make a comment on the
computational procedure. In principle, we can follow the standard algorithm
and compute the Gröbner basis and companion matrices with generic parameters
$\rho$ and $q$. In practice, however, keeping $\rho$ and $q$ generic slows
down the computation considerably because bulky analytic expressions are
generated in the intermediate steps. Therefore, we employ a slightly different
strategy in the actual computation. The basic idea is to compute the results
for different sets of values of $(\rho,q)$ and then interpolate the final
result using this data. We refer to Appendix A for details.

##  9 Analytical results in closed form

In this section, we present the exact closed-form results for $L=4,6$ and
general $N$.

###  9.1 Analytic result for the loop model

The analytic result for $L=4$ has been given in (4.18). This case is rather
simple and we do not need to perform the decomposition to find the eigenvalues
of the transfer matrix, although it makes the computation slightly more
efficient. Using the algebraic-geometry approach, we reproduced the closed-
form partition function (4.18), which we will not repeat here.

Let us now consider the $L=6$ case. The partition function is given by

$\displaystyle Z_{\text{Loop}}(6,N)=$
$\displaystyle\,\text{tr}\,\mathbf{T}_{0,q^{2}}^{N}+\Lambda_{1,1}\,\text{tr}\,\mathbf{T}_{1,1}^{N}+\Lambda_{2,1}\,\text{tr}\,\mathbf{T}_{2,1}^{N}+\Lambda_{2,2}\,\text{tr}\,\mathbf{T}_{2,-1}^{N}$
(9.1)
$\displaystyle\,+\Lambda_{3,1}\,\text{tr}\,\mathbf{T}_{3,1}^{N}+\Lambda_{3,3}\left[\mathbf{T}_{3,e^{2i\pi/3}}^{N}+\mathbf{T}_{3,e^{4i\pi/3}}^{N}\right]\,.$

The multiplicities $\Lambda_{i,j}$ are given by

$\displaystyle\Lambda_{1,1}=$ $\displaystyle\,\mathfrak{m}^{2}-2,$ (9.2)
$\displaystyle\Lambda_{2,1}=$
$\displaystyle\,\frac{1}{2}\mathfrak{m}^{2}(\mathfrak{m}^{2}-3)\,,$
$\displaystyle\Lambda_{2,2}=$
$\displaystyle\,\frac{1}{2}(\mathfrak{m}^{4}-5\mathfrak{m}^{2}+4)\,,$
$\displaystyle\Lambda_{3,1}=$
$\displaystyle\,\frac{1}{3}(\mathfrak{m}^{6}-6\mathfrak{m}^{4}+11\mathfrak{m}^{2}-6)\,,$
$\displaystyle\Lambda_{3,3}=$
$\displaystyle\,\frac{1}{3}\mathfrak{m}^{2}(\mathfrak{m}^{4}-6\mathfrak{m}^{2}+8)\,.$

To obtain an analytic result, we need to find the eigenvalues of the companion
matrices in the four sectors with $j=0,1,2,3$. The dimension of these
companion matrices are $20,15,6,1$ respectively.

##### The $j=0$ case.

For $j=0$, there are no through-lines. We take the value of the twist to be
$\rho=-q$. Such choice of the twist satisfies the resonance condition and we
can compute the eigenvalues for the quotient module
$\overline{\mathcal{W}}_{0,q^{2}}$ and the standard module
$\mathcal{W}_{1,1}$.

The dimension of the quotient module $\overline{\mathcal{W}}_{0,q^{2}}$ is 5.
The 5 eigenvalues of $\mathbf{T}_{0,q^{2}}$ are given by

$\displaystyle\lambda_{0,-q}^{\text{quo}}(1)=$
$\displaystyle\,-\mathfrak{m}^{2}+2\,,$ (9.3)
$\displaystyle\lambda_{0,-q}^{\text{quo}}(2)=$
$\displaystyle\,-\mathfrak{m}^{2}+2\,,$
$\displaystyle\lambda_{0,-q}^{\text{quo}}(3)=$
$\displaystyle\,-\mathfrak{m}^{3}-6\mathfrak{m}^{2}-8\mathfrak{m}-2\,,$
$\displaystyle\lambda_{0,-q}^{\text{quo}}(4)=$
$\displaystyle\,\frac{1}{2}(\mathfrak{m}^{3}+8\mathfrak{m}^{2}+28\mathfrak{m}+28)-\frac{1}{2}(\mathfrak{m}+2)^{2}\sqrt{\mathfrak{m}^{2}+48}\,,$
$\displaystyle\lambda_{0,-q}^{\text{quo}}(5)=$
$\displaystyle\,\frac{1}{2}(\mathfrak{m}^{3}+8\mathfrak{m}^{2}+28\mathfrak{m}+28)+\frac{1}{2}(\mathfrak{m}+2)^{2}\sqrt{\mathfrak{m}^{2}+48}\,.$

The dimension of the standard module $\mathcal{W}_{1,1}$ is 15. To find the
eigenvalues $\mathbf{T}_{1,1}$ analytically, we perform the momentum
decomposition. The value of the twist is taken to be $\rho=-1$. We can take
the momentum quantum number to be $p=1,2,\ldots,6$ for non-singular solutions
$\mathrm{Q}_{M}(q^{1/2})\neq 0$, together with the singular case
$\mathrm{Q}_{M}(q^{1/2})=\mathrm{Q}_{M}(q^{-1/2})=0$. We denote the
eigenvalues in each sector by $\lambda_{0,-q}^{\text{std},p}(j)$ and
$\lambda_{0,-q}^{\text{std},\text{s}}(j)$. We list the eigenvalues in each
sector in what follows.

  * •

$p=1$. There are 2 eigenvalues:

$\displaystyle\lambda_{0,-q}^{\text{std},1}(1)=$
$\displaystyle\,-\frac{1}{2}(\mathfrak{m}^{2}+4\mathfrak{m}+8)+\frac{1}{2}(\mathfrak{m}+2)\sqrt{\mathfrak{m}^{2}+12}\,,$
(9.4) $\displaystyle\lambda_{0,-q}^{\text{std},1}(2)=$
$\displaystyle\,-\frac{1}{2}(\mathfrak{m}^{2}+4\mathfrak{m}+8)-\frac{1}{2}(\mathfrak{m}+2)\sqrt{\mathfrak{m}^{2}+12}\,.$

  * •

$p=2$. There are 3 eigenvalues. The explicit forms of the eigenvalues are
somewhat bulky and we first give the characteristic equation

$\displaystyle x^{3}+(4\mathfrak{m}^{2}+20\mathfrak{m}+30)x^{2}$ (9.5)
$\displaystyle+2(\mathfrak{m}^{4}+10\mathfrak{m}^{3}+30\mathfrak{m}^{2}+40\mathfrak{m}+30)x+4(\mathfrak{m}^{5}+5\mathfrak{m}^{4}+10\mathfrak{m}^{3}+10\mathfrak{m}^{2}+4\mathfrak{m}+2)=0\,.$

We can use the standard formula to find the solution of the cubic equation.
The solutions are given by

$\displaystyle\lambda_{0,-q}^{\text{std},2}(k)=-\frac{1}{3}\left(b+\xi^{k}C^{\tfrac{1}{3}}+\frac{\Delta}{\xi^{k}C^{\tfrac{1}{3}}}\right),\qquad
k=1,2,3$ (9.6)

where $\xi=(-1+i\sqrt{3})/2$, and

$\displaystyle b=$ $\displaystyle\,4\mathfrak{m}^{2}+20\mathfrak{m}+30\,,$
(9.7) $\displaystyle\Delta=$
$\displaystyle\,10(\mathfrak{m}+2)^{2}(\mathfrak{m}^{2}+6\mathfrak{m}+18)\,,$
$\displaystyle C=$
$\displaystyle\,2(\mathfrak{m}+2)^{3}(14\mathfrak{m}^{3}+153\mathfrak{m}^{2}+594\mathfrak{m}+1188)$
$\displaystyle+6i\sqrt{3}(\mathfrak{m}+2)^{4}\sqrt{2\mathfrak{m}^{4}+9\mathfrak{m}^{2}+432}\,.$

  * •

$p=3$. There are 2 eigenvalues, which are the same as in the $p=1$ case.

  * •

$p=4$. There are 3 eigenvalues. The characteristic equation is

$\displaystyle x^{3}-2(\mathfrak{m}^{2}+5\mathfrak{m}+3)x^{2}$ (9.8)
$\displaystyle+(-\mathfrak{m}^{4}-\mathfrak{m}^{3}+6\mathfrak{m}^{2}-4\mathfrak{m}-12)x+\mathfrak{m}^{5}+2\mathfrak{m}^{4}-2\mathfrak{m}^{3}+4\mathfrak{m}^{2}+16\mathfrak{m}+8=0\,.$

The solutions are again given by

$\displaystyle\lambda_{0,-q}^{\text{std},4}(k)=-\frac{1}{3}\left(b+\xi^{k}C^{\tfrac{1}{3}}+\frac{\Delta}{\xi^{k}C^{\tfrac{1}{3}}}\right),\qquad
k=1,2,3$ (9.9)

where now

$\displaystyle b=$ $\displaystyle\,-2(\mathfrak{m}^{2}+5\mathfrak{m}+3)\,,$
(9.10) $\displaystyle\Delta=$
$\displaystyle\,(\mathfrak{m}+2)^{2}(7\mathfrak{m}^{2}+15\mathfrak{m}+18)\,,$
$\displaystyle C=$
$\displaystyle\,-\frac{1}{2}(\mathfrak{m}+2)^{3}(34\mathfrak{m}^{3}+117\mathfrak{m}^{2}+216\mathfrak{m}+108)$
$\displaystyle+\frac{3i\sqrt{3}}{2}(\mathfrak{m}+2)^{4}\sqrt{8\mathfrak{m}^{4}+9\mathfrak{m}^{2}+108}\,.$

  * •

$p=5$. There is 1 eigenvalue,

$\displaystyle\lambda_{0,-q}^{\text{std},5}(1)=2\mathfrak{m}^{2}+6\mathfrak{m}+2\,.$
(9.11)

  * •

$p=6$. There are 3 eigenvalues, which are the same as in the $p=4$ case.

  * •

Singular solution. There is 1 eigenvalue which is given by

$\displaystyle\lambda_{0,-q}^{\text{std},\text{s}}=2\mathfrak{m}+2\,.$ (9.12)

##### The $j=1$ case.

For $j=1$, we have 2 through-lines and we take the value of the twist to be
$\rho=-1$. This gives precisely the module $\mathcal{W}_{1,1}$. The
eigenvalues of $\mathbf{T}_{1,1}$ has already been discussed in the $j=0$
case.

##### The $j=2$ case.

For $j=2$, we have 4 through-lines and we need to consider two values of the
twist: $\rho=1$ and $\rho=i$. The companion matrices $\mathbf{T}_{2,1}$ and
$\mathbf{T}_{2,-1}$ are 6-dimensional. The corresponding eigenvalues are given
by:

  * •

Eigenvalues of $\mathbf{T}_{2,1}$:

$\displaystyle\\{-\mathfrak{m},-\mathfrak{m},-\mathfrak{m},-\mathfrak{m},-\mathfrak{m},12+5\mathfrak{m}\\}\,.$
(9.13)

  * •

Eigenvalues of $\mathbf{T}_{2,-1}$:

$\displaystyle\\{\pm(\mathfrak{m}+2),\pm(2+\sqrt{3})(\mathfrak{m}+2),\pm(2-\sqrt{3})(\mathfrak{m}+2)\\}\,.$
(9.14)

##### The $j=3$ case.

For $j=3$, we have 6 through-lines. This case is trivial and for generic value
of twist $\rho$, we have

$\displaystyle\mathbf{T}_{3,\rho^{2}}=\rho+\rho^{-1}\,.$ (9.15)

We need to take $\rho=e^{i\pi/3},e^{2i\pi/3},1$ and thus

$\displaystyle\mathbf{T}_{3,1}=2\,,\qquad\mathbf{T}_{3,e^{2i\pi/3}}=1\,,\qquad\mathbf{T}_{3,e^{4i\pi/3}}=-1\,.$
(9.16)

Plugging all the eigenvalues of the transfer matrices into (9.1) gives the
analytic expression of the partition function $Z_{\text{loop}}(6,N)$ of the
loop model for any $N$.

###  9.2 Analytic result for the Potts model

To construct the partition function for the Potts model we need an additional
piece, which is $\mathbf{T}_{0,-1}$ in the $j=0$ sector. For $L=4,6$, the
companion matrix $\mathbf{T}_{0,-1}$ has dimension 6 and 20, respectively. We
can find the analytic eigenvalues of the transfer matrix for $L=4$, but not
for $L=6$. Therefore, we will write down the analytic expression for $L=4$

$\displaystyle\frac{Z_{\text{Potts}}(4,N)}{\mathfrak{m}^{2N}}=\text{tr}\,\mathbf{T}_{0,q^{2}}^{N}+\frac{\mathfrak{m}^{2}-1}{2}\text{tr}\,\mathbf{T}_{0,-1}^{N}+\tilde{\Lambda}_{1,1}\,\text{tr}\,\mathbf{T}_{1,1}^{N}+\tilde{\Lambda}_{2,1}\,\text{tr}\,\mathbf{T}_{2,1}^{N}+\tilde{\Lambda}_{2,2}\,\text{tr}\,\mathbf{T}_{2,-1}^{N}\,,$
(9.17)

where we have

$\displaystyle\tilde{\Lambda}_{1,1}=-1\,,\qquad\tilde{\Lambda}_{2,1}=\frac{1}{2}\mathfrak{m}^{2}(\mathfrak{m}^{2}-3)\,,\qquad\tilde{\Lambda}_{2,2}=\frac{1}{2}(\mathfrak{m}^{2}-2)(\mathfrak{m}^{2}-1)\,.$
(9.18)

There are 6 eigenvalues of $\mathbf{T}_{0,-1}$, given by

$\displaystyle\left\\{0,0,\frac{1}{2}(\mathfrak{m}+2)\big{(}\mathfrak{m}+4\pm\sqrt{\mathfrak{m}^{2}+16}\big{)},-\frac{1}{2}(\mathfrak{m}+2)\big{(}\mathfrak{m}+4\pm\sqrt{\mathfrak{m}^{2}+16}\big{)}\right\\}.$
(9.19)

Assuming $N$ is even, we find that the contribution of $\mathbf{T}_{1,1}^{N}$
cancels the same contribution in $\mathbf{T}_{0,q^{2}}^{N}$ and we find
finally

$\displaystyle Z_{\text{Potts}}(4,N)=$
$\displaystyle\,\mathfrak{m}^{2N}\left\\{(\mathfrak{m}^{2}+7\mathfrak{m}+8)^{N}+(\mathfrak{m}^{2}+\mathfrak{m})^{N}+2^{N-1}\mathfrak{m}^{2}(\mathfrak{m}^{2}-3)\right.$
(9.20)
$\displaystyle\,\left.+(\mathfrak{m}^{2}-1)(\mathfrak{m}+2)^{N}\left[(\mathfrak{m}+4+\sqrt{\mathfrak{m}^{2}+16})^{N}+(\mathfrak{m}+4-\sqrt{\mathfrak{m}^{2}+16})^{N}\right]\right\\}.$

##  10 Zeros of partition functions

In this section we investigate the zeros of the partition function $Z(L,N)$ in
the plane of complex $\mathfrak{m}$. For simplicity, we shall focus on the
loop model, where we recall that $\mathfrak{m}$ is the weight given to each
loop; we leave the Potts model aside for future work. We are particularly
interested in the partial thermodynamic limit, where $L$ is fixed and finite,
but $N\gg L$.

###  10.1 Condensation curves

A convenient tool for describing the partial thermodynamic limit is provided
by the Beraha-Kahane-Weiss (BKW) theorem [34]. It states that in the limit
$N/L\to\infty$ the partition function zeros condense on certain curves in the
complex $\mathfrak{m}$-plane, henceforth called condensation curves.

Let $\\{\Lambda_{i}(\mathfrak{m})\\}$ with $i=1,2,\ldots$ denote the union of
eigenvalues of all the transfer matrices that build up the torus partition
function (4.5). As we have seen, they can be obtained by diagonalizing
$t(\gamma/2)$ in any one of the standard modules entering (4.5). For a given
$\mathfrak{m}$, we order these eigenvalues by norm,
$|\Lambda_{1}(\mathfrak{m})|\geq|\Lambda_{2}(\mathfrak{m})|\geq\cdots$, and we
say that $\Lambda_{i}(\mathfrak{m})$ is dominant at $\mathfrak{m}$ if no other
eigenvalue has a strictly greater norm. The BKW theorem [34] then states that
the condensation curves are given by the loci where there are (at least) two
dominant eigenvalues, $|\Lambda_{1}(u)|=|\Lambda_{2}(u)|$.

We shall use a numerical technique for tracing out the condensation curves
that has already been detailed in our previous paper [4]. It combines an
efficient method for the numerically exact diagonalization of the sparse
transfer matrices with a direct-search method that allows us to trace out the
condensation curves. The sparse matrix decomposition follows directly from the
writing (3.1) of $t(u)$ as a product of $R$-matrices. The geometrical action
can then be inferred from (3.3), where we take a link-pattern basis for each
standard module. To take into account the possibility that the dominant
eigenvalues may come from sectors with different through-line numbers we
actually diagonalize the direct sum of $t(u)$ over all standard modules
entering (4.5).

###  10.2 Results

The results for the condensation curves with $L=6,8,10,12,14$ are shown in
Figure 10.3. Notice that the curves are invariant under changing the sign of
${\rm Im}\,\mathfrak{m}$. The first three cases, $L=6,8,10$, are compared to
the partition function zeros for a system of large aspect ratio ($N=1024$),
obtained from the algebraic-geometry computations. The agreement is seen to be
perfect, but it should be noticed that the density of zeros on the curves
exhibits considerable variation.

Apart from this, the condensation curves display several noteworthy features.
For any $L$ they go through the points $\mathfrak{m}=0$ and $\mathfrak{m}=-2$
exactly. There is a central bubble ${\cal B}_{1}$ containing those two points,
a number of branches extending out towards infinity, and a “necklace” region
with many bifurcations and small enclosed regions that appears to produce, as
$L$ increases, a larger bubble ${\cal B}_{2}$ that surrounds ${\cal B}_{1}$.

Figure 10.3: Condensation curves in the complex $\mathfrak{m}$-plane for
partition-function zeros of the loop model on $L\times N$ tori, in the limit
$N\to\infty$. The panels show, in reading direction, the cases
$L=6,8,10,12,14$. The first three panels also superpose the partition-function
zeros for a finite system with $N=1024$ (in red color). The last panel
compares the $L=14$ curves with an exact prediction for the thermodynamic
limit ($L\to\infty$).

We can provide the exact expressions of ${\cal B}_{1}$ and ${\cal B}_{2}$ in
the thermodynamic limit ($L\to\infty$) by the following argument. The loop
model is known to be critical along a segment of the real line, ${\cal
S}=\\{\mathfrak{m}|-2<\mathfrak{m}\leq 2\\}$, and its continuum limit is
described by the Coulomb Gas (CG) approach to conformal field theory [35]. It
is natural to assume [36] that the model remains critical in a larger region
${\cal R}$ of the complex plane, with ${\cal S}\subset{\cal R}$, and that
within ${\cal R}$ all the CG results remain valid by analytic continuation.

It is customary to parametrize the loop weight as

$\mathfrak{m}=-2\cos(\pi g)\,,$ (10.1)

where $g$ is called the CG coupling constant. Inside ${\cal S}$ we have
$0<g\leq 1$. The critical exponents of the loop model are conveniently written
in the Kac-table parametrization of conformal weights

$h_{r,s}=\frac{(rg-s)^{2}-(g-1)^{2}}{4g}\,.$ (10.2)

The model contains [35] in particular the identity operator $I$ of weight
$h_{1,1}=0$, the energy operator $\varepsilon$ of weight $h_{3,1}$, and for
any $j\in\mathbb{N}^{*}$ the $2j$-leg operators ${\cal O}_{j}$ of weight
$h_{j,0}$. Notice that ${\cal O}_{j}$ is the dominant operator inside the
standard module with $2j$ through-lines.

For $-2<\mathfrak{m}<0$, or $0<g<\frac{1}{2}$, we have that $h_{1,0}<0$, so
the 2-leg operator is more relevant than the identity. Moreover, it is the
operator of most negative conformal weight among the ${\cal O}_{j}$. The
power-law behavior of correlation functions is governed by the real part of
the conformal weights, so we can expect a phase transition to take place when
${\rm Re}\,h_{1,0}=0$. We hypothesize that the locus of this transition is
precisely ${\cal B}_{1}$, in the thermodynamic limit. Since
$h_{1,0}=\frac{1}{2}-\frac{1}{4g}$ by (10.2), we arrive at the prediction

${\cal B}_{1}=-2\cos\left(\frac{\pi}{2+i\mathbb{R}}\right)\,.$ (10.3)

This is in excellent agreement with the condensation curves, as witnessed by
the comparison with the $L=14$ curve in the last panel of Figure 10.3.

The termination of critical behavior in the CG is related [35] to the fact
that the energy operator $\varepsilon$, of weight $h_{3,1}=2g-1$, becomes
marginal when $g\uparrow 1$ (_i.e._ , $\mathfrak{m}\uparrow 2$). We therefore
expect termination of critical behavior when ${\rm Re}\,h_{3,1}=1$.
Hypothesizing that the locus of this transition is precisely ${\cal B}_{2}$ we
arrive at our second prediction,

${\cal B}_{2}=-2\cos\left(\frac{\pi}{1+i\mathbb{R}}\right)\,.$ (10.4)

As seen from the last panel of Figure 10.3, this cuts right through the
“necklace” region, and although finite-$L$ effects are definitely larger than
in the case of ${\cal B}_{1}$, we expect this part of the condensation curves