Rational $Q$-Systems at Root of Unity
I. Closed Chains

Jue Hou1, Yunfeng Jiang1, Yuan Miao2

1School of physics & Shing-Tung Yau Center, Southeast University,
Nanjing 211189, P. R. China
2 Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan

Abstract

The solution of Bethe ansatz equations for XXZ spin chain with the parameter
$q$ being a root of unity is infamously subtle. In this work, we develop the
rational $Q$-system for this case, which offers a systematic way to find all
physical solutions of the Bethe ansatz equations at root of unity. The
construction contains two parts. In the first part, we impose additional
constraints to the rational $Q$-system. These constraints eliminate the so-
called Fabricius–McCoy (FM) string solutions, yielding all primitive
solutions. In the second part, we give a simple procedure to construct the
descendant tower of any given primitive state. The primitive solutions
together with their descendant towers constitute the complete Hilbert space.
We test our proposal by extensive numerical checks and apply it to compute the
torus partition function of the 6-vertex model at root of unity.

###### Contents

  1. 1 Introduction
  2. 2 XXZ spin chain at root of unity
    1. 2.1 Integrability of XXZ spin chain
    2. 2.2 Transfer matrices
  3. 3 States of XXZ model at root of unity
    1. 3.1 Exact Fabricius–McCoy strings
    2. 3.2 Bethe roots at infinity
  4. 4 Primitive states and constrained $Q$-system
    1. 4.1 Rational $Q$-system for XXZ spin chain
    2. 4.2 Constraints on $Q$-functions
    3. 4.3 Primitive solutions with roots at infinity
  5. 5 Descendant states
    1. 5.1 General discussions
    2. 5.2 Examples
  6. 6 Torus Partition Function of 6-vertex model
  7. 7 Conclusions and discussions
  8. A Reality of $\\{c_{j}\\}$
  9. B Location of FM-strings
  10. C Combinatorics for the number of primitive states
  11. D Coefficients for the torus partition function

##  1 Introduction

Rational $Q$-system [1] provides a powerful way to obtain physical solutions
of Bethe equations. The key feature of this method is that it leads to all and
only physical solutions. Therefore rational $Q$-system is intimately related
to the completeness of Bethe ansatz. Up to now, rational $Q$-system has been
constructed for various spin chain models including the XXX [1, 2] and XXZ
spin chains [3] and their higher rank generalizations [4, 5] with periodic,
twisted [6] and some open [3, 7] boundary conditions. Generalization to
spin-$s$ case was achieved very recently [8].

The rational $Q$-systems of XXX and XXZ models have significant differences.
For XXX spin chain, solutions of $Q$-system give the Bethe roots of the
highest weight states of SU(2) symmetry. Descendant states are taken into
account at a later stage by adding Bethe roots at infinity, related to the
spin lowering operator of the global SU(2) symmetry. Therefore for fixed
length $L$ and magnon number $M$, the number of solutions for XXX spin chain
is ${L\choose M}-{L\choose M-1}$, which is less than the dimension of the
Hilbert space in the $M$-magnon sector ${L\choose M}$. For XXZ spin chain with
generic $q$ parameter, _i.e._ not at root of unity, the SU(2) symmetry is
broken down to U(1). The number of solutions simply equals ${L\choose M}$. Due
to this reason, it is generally harder to solve the rational $Q$-system of the
XXZ spin chain.

There is a situation which sits in between of XXX spin chain and XXZ at
generic $q$ and is (in)famously subtle. This is the XXZ spin chain with $q$ at
root of unity. The reason is that although SU(2) symmetry is still broken,
there is an enhancement of symmetry. As a result, descendant states with
respect to a different type of symmetry emerges, leading to exponential
degeneracies of the transfer matrices. So far the exact structure of this
symmetry is not fully understood. Proposals include an $\mathfrak{sl}_{2}$
loop algebra [9, 10, 11] and Onsager algebra [12, 13]. The existence of such
descendant states is reflected in the solution of Bethe ansatz equations.
There exists two special kinds of solutions, which are roots at infinity [14]
and the so-called Fabricius–McCoy (FM) strings [15, 16]. The latter are bound
states (Bethe strings) without string deviation that are also related to the
root of unity value. Adding such Bethe roots to any _primitive states_ , a
notion which will be introduced below, does not change the value of the energy
and all higher (quasi)-local charges. Therefore these states can be regarded
as the descendant states of the primitive state, sharing the same eigenvalue
of the transfer matrices up to a possible minus sign.

The existence of FM-strings poses new challenge for solving Bethe ansatz
equations, or its reformulations such as $TQ$-relation and rational
$Q$-system. In fact, if we naively apply the rational $Q$-system constructed
for generic $q$ and simply take $q$ at root of unity, we find that the
$Q$-system has infinitely many solutions. The reason is that the center of FM-
strings cannot be fixed by rational $Q$-system and any value of the center
gives a solution. Constructing the rational $Q$-system for XXZ Bethe ansatz
equations at root of unity is the goal of the current work. To this end, we
take the following strategy. We first impose additional constraints to
eliminate the FM-strings. We will call the resulting $Q$-system the
_constrained_ rational $Q$-system. We tested extensively that the constrained
$Q$-system gives all the physical _primitive_ solutions required by
completeness of Bethe ansatz. There are three types of primitive solutions: 1)
solutions without roots at infinity; 2) Solutions with finite roots and roots
at $+\infty$; 3) Solutions with finite roots and roots at $-\infty$. Notice
that primitive solutions cannot have both positive and negative infinities.
The three types of constrained $Q$-systems are constructed separately.

Once all the primitive solutions are obtained, the rest of solutions can be
constructed by adding FM-strings and infinity pairs, _i.e._ roots of the form
$\\{+\infty^{n_{\infty}},-\infty^{n_{\infty}}\\}$. However, due to the
intricacy of the underlying symmetry algebra, constructing all the descendant
states require some extra non-trivial work. To start with, the possible number
of roots at infinity $n_{\infty}$ are highly constrained and we need to
distinguish for which quantum numbers we can add such solutions. Second, for
given length $L$ and magnon number $M$, we need to determine the center of the
FM-strings. The location of the center of the FM-strings can be fixed via the
fusion relations of transfer matrices [12, 17]. Despite that such fusion
relations [18, 19] are well-studied, to the best of our knowledge, the fact
that the locations of the FM-strings are hidden in the zeros of the higher-
spin transfer matrix has not been reported in the literature. Finally, we need
to determine how many different ways we can add the FM-strings and infinity
pairs, which can be summarized nicely in terms of a Hesse diagram. The
primitive state with all its descendants form a descendant tower. The Hilbert
space consists of all such descendant towers.

The rest of the paper is structured as follows. In section 2, we introduce the
XXZ spin chain at root of unity and discuss its integrability. In section 3,
we discuss the structure of Hilbert space with emphasis on FM-strings and
roots at infinities. In section 4, we construct the constrained rational
$Q$-system which gives all primitive states. The construction of the
descendant towers are discussed in section 5. As an application of our method,
we compute the torus partition function of the 6-vertex model at root of unity
in section 6. We conclude in section 7 and discuss future directions.
Technical details and necessary backgrounds are summarized in the appendices.
Of special interest is that in appendix C we give a simple algorithm for
computing the number of primitive states for given quantum numbers.

##  2 XXZ spin chain at root of unity

The model that we will consider is the quantum XXZ spin-1/2 chain with the
anisotropy parameter $\Delta$. The Hamiltonian of a length $L$ spin chain is
given by

$\mathbf{H}_{\rm
XXZ}=\sum_{j=1}^{L-1}\sigma^{+}_{j}\sigma^{-}_{j+1}+\sigma^{-}_{j}\sigma^{+}_{j+1}+\frac{\Delta}{2}(\sigma^{z}_{j}\sigma^{z}_{j+1}-1)+\kappa\sigma^{+}_{L}\sigma^{-}_{1}+\kappa^{-1}\sigma^{-}_{L}\sigma^{+}_{1}+\frac{\Delta}{2}(\sigma^{z}_{L}\sigma^{z}_{1}-1),$
(2.1)

where $\sigma^{\alpha}_{j}$ are Pauli matrices acting on the $j$-th site. Here
we consider the twisted boundary condition with the twist
$\kappa=e^{\mathrm{i}\phi}$. For $\phi=0$, we recover the periodic boundary
condition.

We parametrize the anisotropy parameter $\Delta$ as

$\Delta=\frac{q+q^{-1}}{2}=\cosh\eta,\qquad q=e^{\eta}.$ (2.2)

The parameter $q$ is the q-deformation parameter from the isotropic XXX model.
When the parameter $q$ satisfies the following relations,

$\eta=\mathrm{i}\pi\,\frac{\ell_{1}}{\ell_{2}},\qquad
q^{\ell_{2}}=\varepsilon=\pm 1,$ (2.3)

where $\ell_{1}$ and $\ell_{2}$ are two co-prime integers, we say that the
q-deformation parameter $q$ (hence the anisotropy parameter $\Delta$) is at
root of unity value. We are particularly interested in understanding the
physical properties of the XXZ spin chain at root of unity. The intricacy of
the XXZ spin chain at root of unity is due to the special properties of the
representation theory of the quantum enveloping algebra
$\mathcal{U}_{q}(\mathfrak{sl}_{2})$, which plays a crucial role in the
integrability of the model [20].

###  2.1 Integrability of XXZ spin chain

The XXZ spin chain is integrable and can be solved by algebraic Bethe ansatz
(ABA). In the ABA framework, the Lax operator of the XXZ spin chain reads

$\displaystyle\mathbf{L}_{aj}(u)={}$
$\displaystyle\sinh(u)\,\frac{\mathbf{K}_{a}+\mathbf{K}^{-1}_{a}}{2}+\cosh(u)\,\frac{\mathbf{K}_{a}-\mathbf{K}^{-1}_{a}}{2}\,\sigma^{z}_{j}+\sinh(\eta)\bigl{(}\mathbf{S}^{+}_{a}\,\sigma^{-}_{j}+\mathbf{S}^{-}_{a}\,\sigma^{+}_{j}\bigr{)}$
(2.4) $\displaystyle={}$
$\displaystyle\frac{1}{2}\begin{pmatrix}e^{u}\,\mathbf{K}_{a}-e^{-u}\,\mathbf{K}^{-1}_{a}&2\sinh(\eta)\,\mathbf{S}^{-}_{a}\\\
2\sinh(\eta)\,\mathbf{S}^{+}_{a}&e^{u}\,\mathbf{K}^{-1}_{a}-e^{-u}\,\mathbf{K}_{a}\end{pmatrix}_{\\!j},$

where the operators acting on the auxiliary space $a$ are in the highest-
weight irreducible representation of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$,

$\mathbf{K}_{a}\,\mathbf{S}^{\pm}_{a}\,\mathbf{K}^{-1}_{a}=q^{\pm
1}\,\mathbf{S}^{\pm}_{a},\quad\left[\mathbf{S}^{+}_{a},\mathbf{S}^{-}_{a}\right]=\frac{\mathbf{K}^{2}_{a}-\mathbf{K}^{-2}_{a}}{q-q^{-1}}.$
(2.5)

One remark is that in (2.4) when writing the Lax operator in the matrix form,
the matrix elements are operators in the _auxiliary space_. This is slightly
different from the usual way of writing the Lax operator where matrix elements
are operators acting on site-$j$ of the spin chain. At the moment we do not
specify the representation of the auxiliary space. The Lax operator (2.5)
satisfies the $RLL$-relation

$\mathbf{R}_{ab}(u-v)\mathbf{L}_{aj}(u)\mathbf{L}_{bj}(v)=\mathbf{L}_{bj}(v)\mathbf{L}_{aj}(v)\mathbf{R}_{ab}(u-v),\quad\forall
u,v\in\mathbb{C}\,,$ (2.6)

where $\mathbf{R}_{ab}$ is the quantum $R$-matrix, which satisfies the Yang-
Baxter equation. An explicit form of $\mathbf{R}_{ab}$ can be found in [12].
We define the monodromy matrix as follows,

$\mathbf{M}_{a}(u,\phi,\\{\xi_{j}\\})=\left(\prod_{j=L}^{1}\mathbf{L}_{aj}(u,\xi_{j})\right)\mathbf{E}_{a}(\phi),$
(2.7)

with the inhomogeneities $\\{\xi_{j}\\}$ and twist operator
$\mathbf{E}_{a}(\phi)=\mathrm{diag}(\cdots,\exp{(2\mathrm{i}\phi)},\exp{(\mathrm{i}\phi)},1)=\mathrm{diag}(\cdots,\kappa^{2},\kappa,1)$.
In this paper, we only consider the homogeneous case, _i.e._ all $\xi_{j}=0$.

Transfer matrix is defined by taking trace of the monodromy matrix in the
auxiliary space111One should not be confused by the index $a$ on the transfer
matrix. After taking the trace in auxiliary space, the transfer matrix is an
operator acting on the quantum space. However, for different choices of
auxiliary space, the explicit form of the transfer matrix can be different.
Here the label $a$ is simply used to highlight this fact.

$\mathbf{T}_{a}(u,\phi)={\rm tr}_{a}\mathbf{M}_{a}(u,\phi).$ (2.8)

From the $RLL$-relation (2.6), we can prove that the transfer matrices are in
involution,

$\left[\mathbf{T}_{a}(u,\phi),\mathbf{T}_{b}(v,\phi)\right]=0,\quad\forall
u,v\in\mathbb{C}.$ (2.9)

This establishes integrability of the model.

###  2.2 Transfer matrices

The above discussions are valid for general auxiliary spaces. In this
subsection, we introduce a number of transfer matrices which are important for
later discussions. For more details of these representations, we refer to
appendices of [12].

#### Spin-$s$ representations

The first important class of transfer matrices correspond to taking the
auxiliary space to be the spin-$s$ ($2s\in\mathbb{Z}_{+}$) highest weight
representation of $\mathcal{U}_{q}(\mathfrak{sl}_{2})$. We will denote these
transfer matrices by $\mathbf{T}_{s}(u,\phi)$. The special case $s=1/2$
corresponds to the transfer matrix of the 6-vertex model. The transfer
matrices $\mathbf{T}_{s}(u,\phi)$ will play a crucial role in the fusion
relations that will be introduced shortly.

####  $\ell_{2}$-dimensional representations

At root of unity $q=\exp\left(\mathrm{i}\ell_{1}\pi/\ell_{2}\right)$, another
important class of transfer matrices correspond to taking the auxliary space
to be the $\ell_{2}$-dimensional highest weight representations. These
transfer matrices, which will be denoted by $\mathbf{T}^{\text{hw}}(u,s)$
where $s$ is the complex spin, has been crucial to derive the quasi-local
charges of the XXZ spin chain at root of unity [21, 22, 23, 24] and to
construct the Baxter’s $Q$-operator [12].

A remarkable property of $\mathbf{T}^{\text{hw}}(u,s)$ is that it can be
factorized into two operators [12]

$\begin{split}&\mathbf{T}^{\rm hw}(u,s)=\mathbf{Q}(x)\mathbf{P}(y),\quad
u,s\in\mathbf{C},\\\ &x=u+\frac{2s+1}{2}\eta,\quad
y=u-\frac{2s+1}{2}\eta,\end{split}$ (2.10)

where $\mathbf{Q}(x)$ and $\mathbf{P}(y)$ can be diagonalized simultaneously
by on-shell Bethe states.

The $\mathbf{Q}$-operator satisfies the following $TQ$-relation [25]

$\mathbf{T}_{1/2}(u)\mathbf{Q}(u)=\kappa^{1/2}T_{0}(u+\eta/2)\mathbf{Q}(u-\eta)+\kappa^{-1/2}T_{0}(u-\eta/2)\mathbf{Q}(u+\eta)\,,$
(2.11)

where

$\displaystyle T_{0}(u)=\sinh(u)^{L}\,.$ (2.12)

Therefore $\mathbf{Q}(u)$ is identified with Baxter’s $Q$-operator. For an
$M$-magnon state, the eigenvalue of $\mathbf{Q}(u)$ is a Laurent polynomial of
$t=e^{u}$ of order $M$

$Q(u)\propto\prod_{m=1}^{M}\sinh(u-u_{m})\propto\prod_{m=1}^{M}\left(t-t^{-1}t_{m}^{2}\right).$
(2.13)

Taking the limit $u\to u_{m}$ in (2.11), we obtain the Bethe ansatz equations

$\left(\frac{\sinh(u_{j}+\eta/2)}{\sinh(u_{j}-\eta/2)}\right)^{L}=\kappa^{-1}\prod_{k\neq
j}^{M}\frac{\sinh(u_{j}-u_{k}+\eta)}{\sinh(u_{j}-u_{k}-\eta)}.$ (2.14)

We label the eigenstates of the transfer matrices by the zeros of the
eigenvalues of the $\mathbf{Q}$-operator, _i.e._ Bethe roots

$|\\{u_{m}\\}_{m=1}^{M}\rangle.$ (2.15)

These eigenstates can be constructed by the algebraic Bethe ansatz (ABA),

$|\\{u_{m}\\}_{m=1}^{M}\rangle\propto\prod_{m=1}^{M}\mathbf{B}(u_{m})|\Uparrow\rangle,$
(2.16)

where $\mathbf{B}(u)$ is the B-operator in ABA [26], and the vacuum
$|\Uparrow\rangle$ is the ferromagnetic state with all spin-ups.

Now we turn to the operator $\mathbf{P}(y)$ whose eigenvalue will be denoted
by $P(u)$. For an $M$-magnon state, $P(u)$ is a Laurent polynomial in $t$ of
order $(L-M)$. The zeros of $P(u)$ corresponds to the Bethe roots of the dual
state obtained by flipping all spins simultaneously

$\displaystyle|\\{v_{m}^{\prime}\\}_{m=1}^{L-M}\rangle\propto\prod_{j=1}^{L}\sigma_{j}^{x}|\\{u_{m}\\}\rangle_{m=1}^{M}\,.$
(2.17)

Similarly, for an $M$-magnon eigenstate, the eigenvalue of $\mathbf{P}(u)$ is
a Laurent polynomial of $t=e^{u}$ of order $(L-M)$

$\begin{split}&\mathbf{P}(y)|\\{u_{m}\\}\rangle_{m=1}^{M}=P(u)|\\{u_{m}\\}\rangle_{m=1}^{M},\\\
&P(u)\propto\prod_{m^{\prime}=1}^{L-M}\sinh(u-v_{m^{\prime}})\propto\prod_{m=1}^{L-M}\left[t-t^{-1}(t^{\prime}_{m})^{2}\right].\end{split}$
(2.18)

In terms of ABA, analogously we have

$|\\{v_{m}^{\prime}\\}_{m=1}^{L-M}\rangle\propto\prod_{m=1}^{L-M}\mathbf{B}(v_{m}^{\prime})|\Uparrow\rangle,\quad|\\{u_{m}\\}_{m=1}^{M}\rangle\propto\prod_{m=1}^{L-M}\mathbf{C}(v_{m}^{\prime})|\Downarrow\rangle,$
(2.19)

where $\mathbf{C}(u)$ is the C-operator in ABA [26], and the vacuum
$|\Downarrow\rangle$ is the ferromagnetic state with all spin-downs.

It is thus tempting to identify $P(u)$ with the dual solution of $Q(u)$ of the
$TQ$-relation. Indeed these two quantities are intimately related, but with
subtle differences depending on the choice of twists. When the twist $\phi$ is
not commensurate [12], $P(u)$ is indeed identified with the dual solution of
$Q(u)$. However, when the twist $\phi$ is commensurate, the dual solution of
$Q(u)$ becomes a quasi-polynomial [3], while $P(u)$ remains a Laurent
polynomial of order $(L-M)$. We will explain the relations between the dual
solution of $TQ$-relation and the eigenvalues of $\mathbf{P}(u)$ in more
details in [27].

##  3 States of XXZ model at root of unity

In this section, we discuss the structure of the Hilbert space of the XXZ spin
chain at root of unity. Before doing so, let us first briefly recall the
structure for the XXX spin chain and XXZ spin chain with generic $q$. For XXX
spin chain with periodic boundary condition, states in the Hilbert space are
organized in terms of SU(2) multiplets. Each multiplet consists of a highest
weight state, corresponding to an on-shell Bethe state with _finite_ Bethe
roots, together with its descendants obtained by adding roots at infinity. An
$M$-magnon Bethe state and its descendants form a spin-$(\tfrac{L}{2}-M)$
representation of SU(2) algebra with dimension $(L-2M+1)$. All the states in
the same multiplet share the same eigenvalue of the transfer matrix
$\mathbf{T}_{1/2}(u)$.

For XXZ spin chain with generic $q$, SU(2) symmetry is broken into a U(1)
symmetry. For each $M$-magnon Bethe state with finite Bethe roots, in general
there is only one state that is degenerate and share the same eigenvalue of
$\mathbf{T}_{1/2}(u)$. This is the state with all spins flipped
simultaneously.

At root of unity, the situation is more complicated. The degeneracy of
$\mathbf{T}_{1/2}(u)$ is generally larger than 2, which signifies an enhanced
symmetry. However, since SU(2) invariance is broken, this symmetry cannot be
SU(2), but has to be something else. The precise nature of this new symmetry
is not fully understood. Conjectures include hidden Onsager algebraic symmetry
of the model [12, 13] and the $\mathfrak{sl}_{2}$-loop algebra [9, 10, 11]. In
the XXX case, degenerate states are obtained by adding roots at infinity.
Similarly, at root of unity, degenerate states can also be obtained by adding
special types of roots to a given solution. The difference is that, now there
are two types of special roots we can add : 1) Fabricius–McCoy strings; 2)
Infinity pairs. Here by ‘infinity pairs’, we mean the roots with equal number
of positive and negative infinities, _i.e._
$\\{+\infty^{n_{\infty}},-\infty^{n_{\infty}}\\}$. One remark is that
different from the XXX spin chain where the sign of infinities are not
relevant, for the XXZ case it is necessary to distinguish the sign of the
infinities because they lead to different analytic behaviors of the
$Q$-function.

It is straightforward to show that by adding these two types of special roots
to a given solution, the eigenvalue of the transfer matrix
$\mathbf{T}_{1/2}(u)$ can at most change a sign. This leads us to the notion
of a _descendant tower_. A descendant tower is defined as all the states that
are degenerate with respect to the eigenvalues of the transfer matrix
$\mathbf{T}_{1/2}(u)$ up to a possible minus sign. The primitive state is the
one with smallest number of Bethe roots. The existence of the descendant
towers is closely related to the twist $\kappa$, as explained in [12]. For
special situations where neither the FM-string nor the infinity pairs are
allowed, the Hilbert space has the same structure as the generic $q$ case.

We would like to emphasize that, unlike the XXX spin chain, primitive
solutions can contain roots at infinity. However, in this case it can contain
only positive or negative infinities, but not both. In other words, we have
the following three types of primitive solutions

  1. 1.

Finite Bethe roots

$\displaystyle\\{u_{1},\ldots,u_{N}\\},\qquad-\infty<u_{k}<\infty$ (3.1)

  2. 2.

Solutions with positive infinities

$\displaystyle\\{u_{1},\ldots,u_{N},\underbrace{\infty,\ldots,\infty}_{n_{+}}\\},\qquad-\infty<u_{k}<\infty$
(3.2)

  3. 3.

Solutions with negative infinities

$\displaystyle\\{u_{1},\ldots,u_{N},\underbrace{-\infty,\ldots,-\infty}_{n_{-}}\\},\qquad-\infty<u_{k}<\infty$
(3.3)

In the rest of this section, we give more detailed discussions of the FM-
string and roots at infinities.

The $Q$-function of a generic solution of Bethe ansatz equations has the
following decomposition

$Q(t)=Q_{\rm reg}(t)Q_{+\infty}(t)Q_{-\infty}(t)Q_{\rm FM}(t).$ (3.4)

The zeros of the regular part

$Q_{\rm reg}(t)=\prod_{m=1}^{n_{\rm reg}}\left(t-t^{-1}t_{m}^{2}\right)$ (3.5)

consists of finite Bethe roots $u_{m}=\log t_{m}$ that are not part of FM-
strings.

The other parts are

$Q_{+\infty}(t)=t^{-n_{+}},\quad Q_{-\infty}(t)=t^{n_{-}},\quad Q_{\rm
FM}(t)=\prod_{m=1}^{n_{\rm
FM}}\left(t^{\ell_{2}}-t^{-\ell_{2}}t_{\mathrm{FM},m}^{2\ell_{2}}\right)\,,$
(3.6)

where $n_{\pm}$ are the number of Bethe roots at $\pm\infty$ respectively.
Notice that for the Bethe roots at infinities, we need to take proper limit of
the $Q$-functions. Specifically, we need the following two limits

$-\lim_{t_{m}\rightarrow+\infty}\left(t_{m}^{-2}t-t^{-1}\right)=t^{-1},$ (3.7)

$\lim_{t_{m}\rightarrow-\infty}\left(t-t^{-1}t_{m}^{2}\right)=t,$ (3.8)

which justify the form of $Q_{\pm\infty}$ in (3.6).

###  3.1 Exact Fabricius–McCoy strings

Fabricius–McCoy strings are a special type of bound states in the XXZ model at
root of unity. They are a set of $\ell_{2}$ Bethe roots of the following form

$u_{n}=u_{0}+\frac{\mathrm{i}\pi(n-1)}{\ell_{2}},\quad
u_{0}\in\mathbb{C},\qquad n\in\\{1,2,\cdots\ell_{2}\\}.$ (3.9)

Such bound states have interesting properties. For example, the scattering
phase between an FM-string and any other Bethe roots is trivial, _i.e._

$\prod_{n=1}^{\ell_{2}}S(u,u_{n})=1,\qquad
S(u,v)=\frac{\sinh(u-v-\eta)}{\sinh(u-v+\eta)}\,.$ (3.10)

In addition, they do not carry any conserved charges except for a $\pi$
momentum when $\varepsilon=q^{\ell_{2}}=-1$. This implies that the states that
have the same regular parts but different FM-strings are degenerate with
respect to the transfer matrices $\mathbf{T}_{s}$ up to a possible sign
$\varepsilon$ [12]. The FM-strings are also related to the phantom states in
the XXZ model at root of unity [28, 29].

Finally, and most relevant to our current discussions, the presence of FM-
strings makes it difficult to solve Bethe ansatz equations or $TQ$-relation.
Consider the eigenvalue equations of the $TQ$-relation (2.11)

$\displaystyle
T_{1/2}(u)Q(u)=\kappa^{1/2}T_{0}(u+\eta/2)Q(u-\eta)+\kappa^{-1/2}T_{0}(u-\eta/2)Q(u+\eta)\,.$
(3.11)

Suppose $Q(u)$ is a solution to this equation. We can construct another
solution by

$\displaystyle\widetilde{Q}(u)=Q(u)Q_{\text{FM}}(u)$ (3.12)

where $Q_{\text{FM}}(u)$ is the Baxter polynomial of an FM-string which has
the following property

$\displaystyle
Q_{\text{FM}}(u\pm\eta)=\varepsilon\,Q_{\text{FM}}(u),\qquad\varepsilon=q^{\ell_{2}}=\pm
1\,.$ (3.13)

It is then clear that $\widetilde{Q}(u)$ satisfy the following $TQ$-relation

$\displaystyle\varepsilon\,T_{1/2}(u)\widetilde{Q}(u)=\kappa^{1/2}T_{0}(u+\eta/2)\widetilde{Q}(u-\eta)+\kappa^{-1/2}T_{0}(u-\eta/2)\widetilde{Q}(u+\eta)\,.$
(3.14)

When $\varepsilon=1$, this is the same $TQ$-relation as (3.11). However, when
constructing $\widetilde{Q}(u)$ we did not specify the center $u_{0}$ of the
FM-string. This means for any choice of $u_{0}$, $\widetilde{Q}(u)$ is a
solution of the $TQ$-relation. So there are infinitely many solutions to the
$TQ$-relation. This fact makes it impossible to find solutions with FM-
strings. Similar analysis for Bethe ansatz equations using the property (3.10)
leads to the same problem. This seems to be a contradiction, as the dimension
of the Hilbert space is finite. The resolution is that the center of the FM-
string $u_{0}$ is not arbitrary and need to be determined by other means. One
way is by diagonalizing the $\mathbf{Q}$-operator constructed in [12]. We will
give another way to fix the value of $u_{0}$ in section 5 by using the fusion
relations of the transfer matrices $\mathbf{T}_{s}$.

###  3.2 Bethe roots at infinity

Now we discuss Bethe roots at infinity. When we say a Bethe root is at
infinity, we always mean the Bethe root given in the additive variable $u_{k}$
instead of the multiplicative variable $t_{k}=e^{u_{k}}$. Whether roots at
infinities are allowed depends on quantum numbers $L,N$ and the parameters of
the model including twist $\kappa$ and anisotropy $q$. Let us denote the
number of positive and negative infinities by $n_{+}$ and $n_{-}$
respectively.

To begin with, we notice that the solutions with $n_{+}=n_{-}=0$ are always
possible. For $n_{\pm}\neq 0$, they have to satisfy several constraints. The
first constraints come from Bethe ansatz equations. By taking the limit
$u_{j}\to\pm\infty$, (2.14) become

$q^{\pm L}=\kappa^{-1}q^{\pm 2(N-n_{\pm})},$ (3.15)

Here we assume that the Bethe roots at either $+\infty$ or $-\infty$ do not
scatter with themselves. In addition, we have the constraint that

$\displaystyle 0\leq n_{\pm}\leq\ell_{2}-1.$ (3.16)

For more explanations of this constraint, see [12].

Equation (3.15) and the constraint (3.16) is our starting point for analyzing
roots at infinities. When $\kappa$ is not an integer power of $q$, the Bethe
equation (3.15) does not permit any solution for integer $n_{\pm}$. This
implies that for a generic twist, there is no solution with roots at infinity.

However, when $\kappa$ is an integer power of $q$, there will be solutions
with the integers $n_{\pm}\leq\ell_{2}-1$. Suppose that $\kappa=q^{k}$ with
$k\in\mathbb{Z}$, (3.15) becomes

$\left(\pm 2n_{\pm}\pm L+k\mp 2N\right)\,\mathrm{mod}\,(x\,\ell_{2})=0,$
(3.17)

where $x=1$ for even $\ell_{1}$ and $x=2$ for odd $\ell_{1}$.

When $\kappa\neq\pm 1$, the only permitted solutions to (3.15) are

$n_{+}\neq 0,n_{-}=0\qquad\text{or}\qquad n_{+}=0,n_{-}\neq 0$ (3.18)

These solutions correspond to primitive states in the absence of FM-strings.

Let us focus on the special case when $\kappa=1$ (_i.e._ periodic boundary
condition) with no FM-string. There are three possibilities for the solution
to (3.15) with $n_{\pm}\neq 0$,

$n_{+}\neq 0,n_{-}=0\quad\mathrm{or}\quad n_{-}\neq
0,n_{+}=0\quad\mathrm{or}\quad n_{+}=n_{-}\neq 0.$ (3.19)

The first two cases correspond to primitive states, while the third case
corresponds to descendant states. The reason is as follows:

For the first two cases with $n_{\pm}\neq 0$, using the decomposition (3.4)
and the $TQ$-relation, we see that the regular part of the $Q$-polynomial is a
solution to a _twisted_ $TQ$-relation with effective twist $q^{\pm 2n_{\pm}}$
that originates from roots at infinities,

$T_{1/2}(u)Q_{\rm reg}(u)=q^{\pm n_{\pm}}T_{0}(u+\eta/2)Q_{\rm
reg}(u-\eta)+q^{\mp n_{\pm}}T_{0}(u-\eta/2)Q_{\rm reg}(u+\eta).$ (3.20)

$Q_{\rm reg}(u)$ in this case is not a solution to the original $TQ$-relation,
due to the presence of the effective twist. Hence the first two cases are
primitive and they have descendants with additional FM-strings.

For the third case, it is easy to observe that since
$Q_{+\infty}(t)Q_{-\infty}(t)=1$, the regular part of the $Q$-polynomial
corresponds to a primitive state satisfying the _original_ $TQ$-relation,

$T_{1/2}(u)Q_{\rm reg}(u)=T_{0}(u+\eta/2)Q_{\rm
reg}(u-\eta)+T_{0}(u-\eta/2)Q_{\rm reg}(u+\eta).$ (3.21)

Therefore, the states with infinity pairs
$\\{(+\infty)^{n_{\infty}},(-\infty)^{n_{\infty}}\\}$
$(n_{\infty}=n_{+}=n_{-})$ are descendants of the state corresponding to
$Q_{\text{reg}}$.

##  4 Primitive states and constrained $Q$-system

In this section, we construct the constrained $Q$-system which gives all
primitive solutions of the XXZ Bethe ansatz equations at root of unity. Our
strategy is imposing extra constraints to eliminate solutions containing FM-
strings.

As discussed in the previous section, we need to consider two situations : 1)
Solutions with finite Bethe roots; and 2) Solutions containing roots at
infinities. In what follows, we will first review the rational $Q$-system for
XXZ spin chain at generic $q$ and explain why this construction cannot be
applied directly to the case when $q$ is a root of unity. We will then discuss
the constrained $Q$-system for the primitive states with finite Bethe roots
and roots at infinity.

###  4.1 Rational $Q$-system for XXZ spin chain

Let us recall the rational $Q$-system for the XXZ spin chain with twisted
boundary condition for generic $q$. Each $Q$-system is associated with a Young
diagram as in figure 4.1. For XXZ spin chain with length $L$ and magnon number
$M$, we consider a two-row Young diagram with number of boxes $(M,L-M)$ where
$L-M\geq M$. We define a $Q$-function at each node of the Young diagram
labelled by $Q_{a,s}(t)$ where $a$ and $s$ are the lattice coordinates of the
node. Here we work with multiplicative variable $t=e^{u}$.

Figure 4.1: Young diagram with two rows.

####  $QQ$-relation

$Q$-functions at different nodes are related by the $QQ$-relation

$\displaystyle
Q_{a+1,s}(t)Q_{a,s+1}(t)=\kappa_{a}\,Q_{a+1,s+1}^{-}(t)Q_{a,s}^{+}(t)-Q_{a+1,s+1}^{+}(t)Q_{a,s}^{-}(t)\,,$
(4.1)

where $f^{\pm}(t)\equiv f(tq^{\pm 1/2})$ and $\kappa_{a}$ are related to
twists. The $QQ$-relation is defined up to proportionality. Therefore what is
important is the ratio between $\kappa_{0}$ and $\kappa_{1}$, which is
identified with the twist $\kappa$ in (2.14) as
$\kappa=\kappa_{0}/\kappa_{1}$.

#### Boundary condition

Certain $Q$-functions at the boundaries of the Young diagram are fixed
completely or partially. These are called boundary conditions. More precisely,
the upper boundary is completely fixed $Q_{2,s}=1$ and the $Q$-functions at
the left boundary are

$\displaystyle Q_{0,0}(t)=(t-t^{-1})^{L},\qquad
Q_{1,0}(t)\propto\prod_{j=1}^{M}(t\,t_{j}^{-1}-t^{-1}t_{j})$ (4.2)

where the zeros of $Q_{1,0}(t)$ correspond to Bethe roots, which we want to
find by solving the $Q$-system. Since $Q_{1,0}(t)$ is a Laurent polynomial of
degree $M$, we can write the following ansatz

$\displaystyle
Q_{1,0}(t)=Q(t)=t^{M}+\sum_{k=1}^{M-1}c_{k}\,t^{2k-M}+\frac{t^{-M}}{c_{0}}\,.$
(4.3)

We will derive a set of algebraic equations for the coefficients
$\\{c_{j}\\}$, which can be solved and give $Q(t)$. Finding the zeros of
$Q(t)$, we obtain the Bethe roots. The algebraic equations come from the zero
remainder conditions, which we now turn to.

#### Zero remainder condition

We require that all the $Q$ functions on the Young diagram to be Laurent
polynomials in $t$, or polynomials in $t$ and $t^{-1}$. From the $QQ$-relation
and the boundary condition, we can easy solve $Q_{1,n}(t)$

$\displaystyle Q_{1,n}(t)=D_{\kappa_{1}}^{n}Q(t)$ (4.4)

where $D_{\kappa}$ is the finite difference operator defined by

$\displaystyle D_{\kappa}f(t)=\kappa\,f(tq^{1/2})-f(tq^{-1/2})\,.$ (4.5)

From (4.4), it is clear that all $Q_{1,n}(t)$ are Laurent polynomials in $t$.
Now we turn to $Q_{0,n}$. From $QQ$-relation, we have

$\displaystyle
Q_{0,n}(t)=\frac{\kappa_{0}\,(D_{\kappa_{1}}^{n}Q)^{+}Q_{0,n-1}^{-}-(D_{\kappa_{1}}^{n}Q)^{-}Q^{+}_{0,n-1}}{D_{\kappa_{1}}^{n}Q}\,.$
(4.6)

We can use this relation to find $Q_{0,n}$ recursively. Now $Q_{0,n}(t)$ is no
longer a Laurent polynomial for generic $Q(t)$. In order for $Q_{0,n}$ to be
polynomials, we require that the remainders of the ratio of two polynomials in
(4.6) to vanish. Such conditions are called zero remainder conditions, which
can be written as a set of algebraic equations for $\\{c_{j}\\}$. We then
solve this system of algebraic equations and find the solutions.

#### Bethe ansatz equations

The Bethe ansatz equations and related $TQ$-relations can be derived from the
$QQ$-relation. From the $QQ$-relation of the first row, we find

$\displaystyle Q_{1,1}^{+}(t_{k})=\kappa_{1}\,Q^{++}_{1,0}(t_{k}),\qquad
Q_{1,1}^{-}=-Q_{1,0}^{--}(t_{k})$ (4.7)

while from the second row we have

$\displaystyle\kappa_{0}\,Q_{1,1}^{-}(t_{k})Q_{0,0}^{+}(t_{k})-Q_{1,1}^{+}(t_{k})Q_{0,0}^{-}(t_{k})=0\,.$
(4.8)

Combining these two equations and eliminating $Q_{1,1}(t)$, we find

$\displaystyle\kappa_{0}\,Q_{1,0}^{--}(t_{k})Q_{0,0}^{+}(t_{k})+\kappa_{1}\,Q_{1,0}^{++}(t_{k})Q_{0,0}^{-}(t_{k})=0\,.$
(4.9)

Plugging in the explicit form of $Q_{1,0}(t)$ and $Q_{0,0}(t)$, we obtain the
Bethe ansatz equation

$\displaystyle\left(\frac{t_{j}^{2}q-1}{t_{j}^{2}-q}\right)^{L}=-\frac{\kappa_{1}}{\kappa_{0}}\prod_{k=1}^{M}\frac{t_{j}^{2}q^{2}-t_{k}^{2}}{t_{j}^{2}-t_{k}^{2}q^{2}}=\kappa^{-1}\prod_{k=1}^{M}\frac{t_{j}^{2}q^{2}-t_{k}^{2}}{t_{j}^{2}-t_{k}^{2}q^{2}}\,,$
(4.10)

which is identical to the BAE mentioned previously (2.14) after a change of
variables.

#### Root of unity

Let us now discuss why the above construction is problematic at root of unity.
The main reason is due to the appearance of FM-strings.

Similar to the $TQ$-relation, suppose we have found a solution of the zero
remainder condition $Q(t)$, which means all $Q_{0,n}$ calculated from

$\displaystyle
Q_{0,n}=\frac{\kappa_{0}\,(D_{\kappa_{1}}^{n}Q)^{+}Q_{0,n-1}^{-}-(D_{\kappa_{1}}^{n}Q)^{-}Q^{+}_{0,n-1}}{D_{\kappa_{1}}^{n}Q}$
(4.11)

are Laurent polynomials of $t$. Now consider the following $Q(t)$ function by
multiplying the $Q$-function of an FM-string

$\displaystyle\widetilde{Q}(t)=Q(t)Q_{\text{FM}}(t)\,.$ (4.12)

Let us first consider the case where $\ell_{1}$ is even. In this case, we have
$Q_{\text{FM}}^{+}(t)=Q_{\text{FM}}^{-}(t)$ and

$\displaystyle
D_{\kappa_{1}}\widetilde{Q}=\kappa_{1}\,Q^{+}Q^{+}_{\text{FM}}-Q^{-}Q^{-}_{\text{FM}}=Q_{\text{FM}}^{+}\left(\kappa_{1}Q^{+}-Q^{-}\right)=Q_{\text{FM}}^{+}\,D_{\kappa_{1}}Q$
(4.13)

It is then easy to prove that

$\displaystyle
D_{\kappa_{1}}^{n}\widetilde{Q}=(Q_{\text{FM}}^{+})^{n}\,D_{\kappa_{1}}^{n}Q\,.$
(4.14)

Therefore

$\displaystyle{Q}_{0,n}=$
$\displaystyle\,\frac{\kappa_{0}\,(D_{\kappa_{1}}^{n}\widetilde{Q})^{+}Q_{0,n-1}^{-}-(D_{\kappa_{1}}^{n}\widetilde{Q})^{-}Q^{+}_{0,n-1}}{D_{\kappa_{1}}^{n}\widetilde{Q}}$
(4.15) $\displaystyle=$
$\displaystyle\,\left(\frac{q_{\text{FM}}}{q_{\text{FM}}^{+}}\right)^{n}\frac{\kappa_{0}\,(D_{\kappa_{1}}^{n}q)^{+}Q_{0,n-1}^{-}-(D_{\kappa_{1}}^{n}Q)^{-}Q^{+}_{0,n-1}}{D_{\kappa_{1}}^{n}q}\
$

For even $\ell_{1}$, we have $Q_{\text{FM}}^{+}(t)=\pm Q_{\text{FM}}(t)$,
therefore the factor $(Q_{\text{FM}}(t)/Q_{\text{FM}}^{+}(t))^{n}=(\pm 1)^{n}$
is a simple global factor. As a result, $Q_{0,n}$ in (4.15) are polynomials.
This implies that $\widetilde{Q}(t)$ is another solution of the zero remainder
condition. Since we have not specify the center of the FM-string, it can be a
free parameter. This means the zero remainder condition in this case has
infinitely many solutions.

The analysis for odd $\ell_{1}$ is similar. If we consider an even number of
FM strings, the situation is the same as the $\ell_{1}$ even case. In the case
for odd number of FM-strings solutions
$Q_{\text{FM}}^{+}(t)=-Q_{\text{FM}}^{-}(t)$. The zero-remainder condition
will be modified slightly, which effectively corresponds to adding a $\pi$
twist to the system (or a $\pi$ boost for the momentum).

To conclude, due to the presence of FM-strings, the rational $Q$-system will
yield infinitely many solutions at root of unity. This is easily confirmed by
numerical calculations. Clearly there is a discrepancy between the infinitely
many solutions of the rational $Q$-system and the finitely many physical
solutions bound by the dimension of the entire Hilbert space. This can be
resolved by adding constraints on the $Q$-functions, resulting in the correct
number of physical primitive solutions, as shown in the next section.

###  4.2 Constraints on $Q$-functions

Since the existence of FM-strings makes it hard to solve $Q$-system, we shall
forbid such solutions in the first step and add them back in a later stage.
This can be achieved by imposing constraints for the $Q$-functions.

We start with $Q$-functions with only finite Bethe roots. We first notice that
if the Bethe roots contain FM-strings, the coefficients of the $Q$-function
are not independent, but satisfy specific relations. These are precisely the
relations we want to forbid. The $Q$-function is parameterized as

$\displaystyle
Q(t)=t^{M}+\sum_{k=1}^{M-1}c_{k}t^{2k-M}+\frac{t^{-M}}{c_{0}}=\prod_{j=1}^{M}\left(t-\frac{t_{j}^{2}}{t}\right)\,.$
(4.16)

where $t_{j}$ are Bethe roots. Note that the coefficient for $t^{-M}$ is
$1/c_{0}$. The reason for this unusual choice (instead of the seemingly more
natural choice $c_{0}$) is to ensure the coefficient $1/c_{0}$ to be non-zero,
which is true for $Q$-functions corresponding to finite Bethe roots.

If $Q(t)$ contains an FM-string, it must take the form

$\displaystyle Q(t)=Q_{\text{FM}}(t)\tilde{Q}(t)$ (4.17)

where

$\displaystyle
Q_{\text{FM}}(t)=\prod_{j=1}^{\ell_{2}}\left(t-\frac{(t_{j}^{\text{FM}})^{2}}{t}\right)$
(4.18)

and $\tilde{Q}(t)$ is another Laurent polynomial of degree $(M-\ell_{2})$ .
From the ansatz (4.16), we have

$\displaystyle
Q(t)=t^{M}+\sum_{k=1}^{M-1}c_{k}t^{2k-M}+\frac{t^{-M}}{c_{0}}=\prod_{j=1}^{\ell_{2}}\left(t-\frac{(t_{j}^{\text{FM}})^{2}}{t}\right)\prod_{k=\ell_{2}+1}^{M}\left(t-\frac{t_{k}^{2}}{t}\right)$
(4.19)

Expanding right hand side of (4.19), we can solve $\\{c_{j}\\}$ in terms of
$\\{t_{j}^{\text{FM}}\\}$ and $\\{t_{k}\\}$ ($k=\ell_{2}+1,\ldots,M$) and
write

$\displaystyle c_{l}=F_{l}(\\{t_{j}^{\text{FM}}\\},\\{t_{k}\\}),\qquad
l=0,1,\ldots,M-1\,.$ (4.20)

We can then eliminate $\\{t_{j}^{\text{FM}}\\}$ and $\\{t_{k}\\}$ from (4.20),
which leads to a set of equations involving only $\\{c_{j}\\}$

$\displaystyle R_{k}(c_{0},\ldots,c_{M-1})=0,\qquad k=1,\ldots,N_{c}\,.$
(4.21)

If the solution contains an FM-string, the coefficients of $Q(t)$ satisfy
additional relations (4.21) . Therefore, to eliminate FM-strings, we can add
the following constraint to the rational $Q$-system

$\displaystyle 1+w\left(|R_{1}|^{2}+\ldots+|R_{N_{c}}|^{2}\right)=0\,$ (4.22)

where $|R_{k}|^{2}=R_{k}R_{k}^{*}$ and $w$ is an extra variable which we
introduce. The equation (4.22) implies that we cannot have the situation
$R_{1}=R_{2}=\cdots R_{N_{c}}=0$. One might worry that the constraints (4.22)
might involve the complex conjugate $\\{c_{j}^{*}\\}$ and make it complicated.
However, we can in fact prove that all $\\{c_{j}\\}$ are real. The details of
the proof is given in Appendix A.

#### Example 3.1

Let us give an example to illustrate how the above procedure works. To this
end, we consider the following quantum numbers

$\displaystyle L=8,\quad M=4,\quad\eta=\frac{\mathrm{i}\pi}{3}\,.$ (4.23)

For this case, the FM-string is a length-3 bound state consisting of the
following 3 Bethe roots

$\displaystyle
u_{1}^{\text{FM}}=\alpha_{\text{FM}}-\frac{2\pi\mathrm{i}}{3},\qquad
u_{2}^{\text{FM}}=\alpha_{\text{FM}},\qquad
u_{3}^{\text{FM}}=\alpha_{\text{FM}}+\frac{2\pi\mathrm{i}}{3}$ (4.24)

Note that the imaginary parts of the Bethe roots are defined
$\mathrm{mod}\,\pi$. Plugging (4.24) into (4.19), we find that the solution

$\displaystyle c_{0}=\frac{e^{-6\alpha_{\text{FM}}}}{t_{4}^{2}},\qquad
c_{1}=-e^{6\alpha_{\text{FM}}},\qquad c_{2}=0,\qquad c_{3}=-t_{4}^{2}\,.$
(4.25)

Eliminating $\alpha_{\text{FM}}$ and $t_{4}$ from (4.25), we find the
following relations among $\\{c_{j}\\}$222Such computations can be performed
straightforwardly by computer algebraic systems, such as the function
Eliminate of Wolfram Mathematica.

$\displaystyle c_{2}=0,\qquad c_{0}c_{1}c_{3}-1=0\,.$ (4.26)

Therefore to obtain the constrained $Q$-system, we add the following equation

$\displaystyle 1+w\left[c_{2}^{2}+(c_{0}c_{1}c_{3}-1)^{2}\right]=0\,,$ (4.27)

to the zero remainder conditions.

###  4.3 Primitive solutions with roots at infinity

Now we consider primitive states with roots at infinity. The strategy is the
same, but the ansatz is slightly different.

#### Positive infinity

For $n_{+}\neq 0$, $n_{-}=0$, the $Q$-function takes the form

$\displaystyle
Q(t)=\frac{t^{-M}}{c_{0}}+\sum_{k=1}^{M-n_{+}-1}c_{k}t^{2k-M}+t^{M-2n_{+}}\,.$
(4.28)

To find the constraints for the FM-string, we solve $\\{c_{k}\\}$ from

$\displaystyle
Q(t)=\frac{t^{-M}}{c_{0}}+\sum_{k=1}^{M-n_{+}-1}c_{k}t^{2k-M}+t^{M-2n_{+}}=\frac{1}{t^{n_{+}}}\prod_{j=1}^{\ell_{2}}\left(t-\frac{(t_{j}^{\text{FM}})^{2}}{t}\right)\prod_{k=\ell_{2}+1}^{M-n_{+}}\left(t-\frac{t_{k}^{2}}{t}\right)$
(4.29)

and eliminate $\\{t_{j}^{\text{FM}}\\}$ and $\\{t_{k}\\}$ from the solution.

#### Negative infinity

For $n_{+}=0,n_{-}\neq 0$, the $Q$-function takes the form

$\displaystyle Q(t)=t^{M}+\sum_{k=n_{-}}^{M-1}c_{k}t^{2k-M}$ (4.30)

To find the constraints for the FM-string, we solve $\\{c_{k}\\}$ from

$\displaystyle
Q(t)=t^{M}+\sum_{k=n_{-}}^{M-1}c_{k}t^{2k-M}=t^{n_{-}}\prod_{j=1}^{\ell_{2}}\left(t-\frac{(t_{j}^{\text{FM}})^{2}}{t}\right)\prod_{k=\ell_{2}+1}^{M-n_{-}}\left(t-\frac{t_{k}^{2}}{t}\right)$
(4.31)

and eliminate $\\{t_{j}^{\text{FM}}\\}$ and $\\{t_{k}\\}$ from the solution.

#### Example 3.2

Here we present an example with root at infinity. We consider the following
quantum numbers

$\displaystyle L=10,\qquad M=5,\qquad\eta=\frac{\mathrm{i}\pi}{3},\qquad
n_{+}=1\,.$ (4.32)

The Bethe roots of the FM-string are still (4.24). Plugging into (4.28), we
find that the solution is given by

$\displaystyle c_{0}=\frac{e^{-6\alpha_{\text{FM}}}}{t_{4}^{2}},\qquad
c_{1}=-e^{6{\alpha_{\text{FM}}}},\qquad c_{2}=0,\qquad c_{3}=-t_{4}^{2}.$
(4.33)

Eliminating $\alpha_{\text{FM}}$ and $t_{4}$ from (4.25), we find the
following relations among $\\{c_{j}\\}$

$\displaystyle c_{2}=0,\qquad c_{0}c_{1}c_{3}=1\,.$ (4.34)

Therefore to obtain the constrained $Q$-system, we add the following equation

$\displaystyle
1+w\left[c_{2}^{2}+\left(c_{0}c_{1}c_{3}-1\right){}^{2}\right]=0\,,$ (4.35)

to the zero remainder conditions.

##  5 Descendant states

In this section, we discuss the construction of descendant states of a given
primitive state. Combined together, they form a multiplet which we shall call
a _descendant tower_. States within each descendant tower share the same
regular part of the $Q$-functions, by definition.

###  5.1 General discussions

#### Descendant tower

Let us denote the Bethe roots of the primitive state by $\\{u_{m}\\}$,
$m=1,\ldots,M$ with $M\leq L/2$. The Bethe roots of the descendant states are
given by adding _infinity pairs_ 333Recall that these are the Bethe roots
which consists of $n_{\infty}$ pairs of $+\infty$ and $-\infty$ roots. or FM-
strings to $\\{u_{m}\\}$, until one reaches the unique ‘bottom’ state. The
bottom state is nothing but the dual state of $|\\{u_{m}\\}\rangle$, defined
by flipping all spins simultaneously

$\displaystyle|\\{v_{m^{\prime}}\\}\rangle\equiv\prod_{j=1}^{L}\sigma_{j}^{x}|\\{u_{m}\\}\rangle\,,$
(5.1)

where $\\{v_{m^{\prime}}\\}$, $m^{\prime}=1,2,\ldots,L-2M$ are the Bethe roots
of the dual state. All the descendant states of a primitive state form a
diamond shaped Hesse diagram whose end points are the primitive state and its
dual state. Examples of such diagrams can be found in figure 5.2,5.3,5.4 of
the next subsection.

#### FM-strings

Now we discuss how to find all the FM-strings of a given primitive state. We
consider a state with length $L$, magnon number $M$, twist
$\kappa=e^{\mathrm{i}\phi}$ and anisotropy
$\eta=\mathrm{i}\pi\ell_{1}/\ell_{2}$. Let us define the $Q$-function of the
primitive solution

$\displaystyle Q(u)=\prod_{j=1}^{M}\sinh(u-u_{j})\,.$ (5.2)

Our claim is that the FM-strings are given by the zeros of the following
quantity

$\displaystyle
F(u)=\sum_{k=0}^{\ell_{2}-1}e^{\mathrm{i}k\phi}\frac{\left[\sinh\left(u+(k+\tfrac{1}{2})\eta\right)\right]^{L}}{Q(u+k\eta)Q(u+(k+1)\eta)}\,.$
(5.3)

In other words, we have

$\displaystyle
F(u)\propto\prod_{j=1}^{n_{\text{FM}}}\left(\prod_{m=1}^{\ell_{2}}\sinh\left(u-\alpha_{j}^{\text{FM}}-\frac{2m-1-\ell_{2}}{2\ell_{2}}\mathrm{i}\pi\right)\right)$
(5.4)

where each set of Bethe roots

$\displaystyle
u_{j}^{(m)}=\alpha_{j}^{\text{FM}}+\frac{2m-1-\ell_{2}}{2\ell_{2}}\mathrm{i}\pi,\qquad
m=1,\ldots,\ell_{2}$ (5.5)

form an FM-string with center $\alpha_{j}^{\text{FM}}$. The total number of
the FM-strings is denoted by $n_{\text{FM}}$. For a proof of this claim, see
appendix B .

#### Infinity pairs

From the definition (5.3), one might expect that $F(u)$ is a trigonometric
polynomial in $u$ (or Laurent polynomial in the multiplicative variable) of
degree $L-2M$. However, there are cases when this is not the case, which
signifies the appearance of infinity pairs
$\\{(-\infty)^{n_{\infty}},(+\infty)^{n_{\infty}}\\}$. The number of infinity
pairs $n_{\infty}$ is given by

$\displaystyle n_{\infty}=\frac{1}{2}(L-2M-\deg F)\,.$ (5.6)

#### Hesse diagram

The structure of the descendant tower can be seen clearly from a Hesse
diagram. It consists of all possible paths from the primitive state to the
dual state, at each step one adds an FM-string or infinity pairs.

#### The algorithm

To summarize, given a primitive solution $\\{u_{m}\\}$, the descendant tower
can be constructed by the following procedure

  1. 1.

Construct the $Q$-polynomial (5.2) and compute the corresponding $F(u)$ in
(5.3);

  2. 2.

Compute the number of infinity pairs by (5.6);

  3. 3.

Determine the center of FM-strings $\\{\alpha_{j}^{\text{FM}}\\}$ by finding
the zeros of $F(u)$;

  4. 4.

Construct the Hesse diagram of the descendant tower by adding either infinity
pairs or an FM-string once at a time, until one reaches the dual state.

By using the solutions to the constraint $Q$-systems and the algorithm to
construct the descendant towers, we are able to find all the physical
solutions to the XXZ spin chain at root of unity. Those structures are
essential to compute the torus partition functions of the 6-vertex model, cf.
Section 6, and they reveal certain intriguing combinatorics as shown in
Appendix C.

###  5.2 Examples

To see how the outlined procedure works in practice, let us consider a few
examples in this subsection. It is more convenient to work with multiplicative
variables. With slight abuse of notation, we denote the $Q$-polynomial in
multiplicative variables by

$\displaystyle Q(t)=\prod_{j=1}^{M}\left(t\,t_{j}^{-1}-t^{-1}t_{j}\right)$
(5.7)

The quantity $F(u)$ in (5.3) written in terms of multiplicative variables
(omitting an unimportant factors $2^{-L+2M}$) reads

$\displaystyle
F(t)=\sum_{k=0}^{\ell_{2}-1}e^{\mathrm{i}k\phi}\frac{\left[tq^{k+1/2}-1/(tq^{k+1/2})\right]^{L}}{Q(tq^{k})Q(tq^{k+1})}\,.$
(5.8)

#### Example 5.1

Consider a spin chain with $L=6$, $\phi=0$, $\eta=2\mathrm{i}\pi/3$
$(\ell_{1}=2,\ell_{2}=3)$. We consider the $M=0$ sector, which corresponds to
the pseudovacuum state
$|\\!\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\rangle$. Its dual state
is the one with all spins down
$|\\!\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\rangle$. The
$Q$-polynomial in this case is a constant

$\displaystyle Q(t)=1\,.$ (5.9)

Plugging in (5.8), we obtain

$\displaystyle F(t)=3(t^{6}-20+t^{-6})$ (5.10)

We find that $\deg F=6=L$, therefore we do not have infinity pairs. Since each
FM-string is of length-3 in this case, we conclude we have 2 FM-strings.
Denoting their centers by $t_{1}^{\text{FM}}$ and $t_{2}^{\text{FM}}$, we have

$\displaystyle
F(t)\propto(t^{6}-20+t^{-6})=\prod_{j=1}^{2}(t/t_{j}^{\text{FM}}-t_{j}^{\text{FM}}/t)(qt/t_{j}^{\text{FM}}-t_{j}^{\text{FM}}/qt)(t/qt_{j}^{\text{FM}}-qt_{j}^{\text{FM}}/t),$
(5.11)

from which we find that

$\displaystyle t_{1}^{\text{FM}}=\left(10-3\sqrt{11}\right)^{1/6},\qquad
t_{2}^{\text{FM}}=\left(10+3\sqrt{11}\right)^{1/6}\,.$ (5.12)

The Hesse diagram for the descendant tower is given by figure 5.2. This
descendant tower contains 4 states.

Figure 5.2: Hesse diagram for the descendant tower of Example 5.1.

#### Example 5.2

Consider a spin chain with $L=8$, $\phi=0$, $\eta=2\mathrm{i}\pi/3$
$(\ell_{1}=2,\ell_{2}=3)$.

We consider the one magnon sector $M=1$ with $t_{1}=\mathrm{i}$. The
$Q$-polynomial is given by

$\displaystyle Q(t)=t+t^{-1}\,.$ (5.13)

The corresponding $F(t)$ reads

$\displaystyle F(t)=3(t^{6}-29+t^{-6})$ (5.14)

Since $\deg F=6=L-2M$, we do not have infinity pairs. Since $\ell_{2}=3$, each
FM-string is of length 3. Therefore we have 2 FM-strings. Similar to Example
5.1, we can find the center of the FM-strings

$\displaystyle
t_{1}^{\text{FM}}=\left(\frac{1}{2}(29+3\sqrt{93})\right)^{1/6},\qquad
t_{2}^{\text{FM}}=\left(\frac{1}{2}(29-3\sqrt{93})\right)^{1/6}\,.$ (5.15)

The Hesse diagram for the descendant tower is given in figure 5.3. This
descendant tower contains 4 states.

Figure 5.3: Hesse diagram for the descendant tower of Example 5.2.

#### Example 5.3

Let us consider an example with infinity pairs. We take $L=12$, $\phi=0$,
$\eta=\mathrm{i}\pi/3$ $(\ell_{1}=1,\ell_{2}=3)$. We consider the one-magnon
state with

$\displaystyle t_{1}=\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)^{1/2},\qquad
Q(t)=\sqrt{2-\sqrt{3}}t-\frac{\sqrt{2+\sqrt{3}}}{t}\,.$ (5.16)

The corresponding $F(t)$ is given by

$\displaystyle
F(t)=-9\left(2(8+3\sqrt{3})t^{6}+59+2(8-3\sqrt{3})t^{-6}\right)$ (5.17)

We see that in this case $\deg F=6<L-2M=10$. Therefore the number of infinity
pairs is $n_{\infty}=2$. The Hesse diagram for the descendant tower is given
in figure 5.4. This descendant tower contains 8 states.

Figure 5.4: Hesse diagram for the descendant tower of Example 5.3. $\pm\infty$
should be understood as the value of the Bethe roots. In terms of the variable
$t=e^{u}$, we have $\lim_{u\rightarrow+\infty}t=+\infty$ and
$\lim_{u\rightarrow-\infty}t=0$.

#### Example 5.4

$L=8$, $\eta=\mathrm{i}\pi/3$, $\phi=0$. A primitive state with one $+\infty$
root and one regular root ($M=2$ and $n_{+}=1$):

$Q_{1}(t)=t^{-2}-\frac{1}{2}(\sqrt{3}+1)\propto t^{-1}(t-t^{-1}t_{1}^{2}).$
(5.18)

The regular root $t_{1}$ is

$t_{1}=\sqrt{\sqrt{3}-1}.$ (5.19)

In the meantime, we need to consider another primitive state with
$(\ell_{2}-1)=2$ roots at $-\infty$ and the same regular root (5.19)

$Q_{2}(t)=t-\frac{1}{2}(\sqrt{3}+1)t^{3}\propto t^{2}(t-t^{-1}t_{1}^{2}).$
(5.20)

We can calculate the corresponding $F(t)$, which is the same for both
primitive states up to a normalization factor,

$\begin{split}&F_{1}(t)\propto
t^{3}\left[\sqrt{\frac{31-15\sqrt{3}}{22}}t^{3}-\sqrt{\frac{22}{31-15\sqrt{3}}}t^{-3}\right],\\\
&F_{2}(t)\propto
t^{-3}\left[\sqrt{\frac{31-15\sqrt{3}}{22}}t^{3}-\sqrt{\frac{22}{31-15\sqrt{3}}}t^{-3}\right],\end{split}$
(5.21)

where the FM string centre is located as

$t^{\rm FM}_{1}=\mathrm{i}\left(\frac{31+15\sqrt{3}}{13}\right)^{1/6}.$ (5.22)

There is a subtlety on understanding the function $F(t)$, which is related to
the state beyond the equator (the dual state) with respect to the primitive
state. For instance,

$F_{1}(t)=\frac{Q^{\rm dual}_{1}(t)}{Q_{1}(t)}=t^{3}Q_{\rm FM}(t),$ (5.23)

where the $Q$-function associated with the dual state is

$Q^{\rm dual}_{1}(t)=t^{2}(t-t^{-1}t_{1}^{2})Q_{\rm FM}(t).$ (5.24)

This is reflected by the fact that the dual state is the descendant state of
the mirrored tower in figure 5.5

Figure 5.5: Hesse diagram for the descendant tower of Example 5.4. $\pm\infty$
should be understood as the value of the Bethe roots, the same as previously
explained. It is clear that the dual states of the two primitive states are in
the mirrored descendant towers, respectively.

The descendant tower in this case has a mirroring structure: It splits into
two parts with the same regular root(s) and FM-strings, while possessing with
different number of Bethe roots at $+\infty$ and $-\infty$, respectively. The
total number in the mirroring descendant tower is hence $2^{n_{\rm FM}+1}$.

##  6 Torus Partition Function of 6-vertex model

With the rational $Q$-system we are able to correctly find all the primitive
solutions as well as their descendants. This is very useful in many
applications. In this section, we apply our method to compute the torus
partition functions of 6-vertex model at root of unity on a $N\times M$
lattice, as is shown in figure 6.6.

Figure 6.6: Torus partition function of 6-vertex model.

Using the transfer matrix approach, the torus partition function is defined as

$W_{N,M}(t)=\mathrm{Tr}\big{(}\mathbf{T}_{1/2}^{M}(t)\big{)},$ (6.1)

where $\mathbf{T}_{1/2}(t)$ is the transfer matrix with $N$ sites. The torus
partition function $W_{N,M}(t)$ is a Laurent polynomial in $t$ of order $(NM)$
Alternatively, one can rotate the lattice by $\pi/2$ and obtain the same
partition function from the other channel

$W_{M,N}(t)=W_{N,M}(t)=\mathrm{Tr}\big{(}\tilde{\mathbf{T}}_{1/2}^{N}(t)\big{)},$
(6.2)

where $\tilde{\mathbf{T}}_{1/2}(t)$ is the transfer matrix with $M$ sites.
This symmetry can be used as a consistency check of our results.

We first solve the $Q$-system to obtain all the primitive $Q$-functions, as
well as the descendants (the degeneracies are crucial here). The eigenvalues
of the transfer matrix $\mathbf{T}_{1/2}(t)$ can be found by $TQ$-relation

$T_{1/2}(t)=\frac{1}{Q}\big{(}T_{0}^{+}Q^{--}+\kappa T_{0}^{-}Q^{++}\big{)}.$
(6.3)

For the descendant states, we observe that

$T_{1/2}^{\rm des}(t)=\varepsilon^{n_{\rm FM}}T_{1/2}^{\rm pri}(t),$ (6.4)

where $T_{1/2}^{\rm pri}$ and $T_{1/2}^{\rm des}$ stand for the eigenvalues of
the transfer matrix for the primitive and its descendant states respectively.

The advantage of using the $Q$-systems to calculate the torus partition
functions is that we would not have to diagonalize a $2^{L}\times 2^{L}$
transfer matrix. Instead, we solve the $Q$-system, which are polynomial
equations. The number of primitive solutions of $M$ magnons are typically much
smaller than the dimension of the Hilbert space for the $M$-magnon sector.
This is clearly a significant improvement when the system size $L$ becomes
bigger.

We exemplify the calculation for the 6-vertex model with $N=8$ and $M=6$,
$\Delta=-1/2$, $\phi=0$. In this case, we have 144 primitive states (28 out of
144 are primitive states with root(s) at infinities) and 112 descendant
states, as described in details in table 1.

More explicitly, the torus partition function with $N=8$ and $M=6$ is

$W_{8,6}=\frac{1}{140737488355328}\sum_{m=1}^{24}c_{m}\big{(}t^{2m}+t^{-2m}\big{)}+c_{0},$
(6.5)

where the exact coefficients are curated in Appendix D.

| total $\\#$ |  $\\#$ primitive states |  $\\#$ descendant states
---|---|---|---
$M$ |  |  $n_{\pm}=0$ |  $n_{+}>0$ |  $n_{-}>0$ |  $n_{+}=n_{-}$ |  $n_{+}>0$ |  $n_{-}>0$
0 | 1 |  1 |  0 |  0 |  0 |  0 |  0
1 | 8 |  8 |  0 |  0 |  0 |  0 |  0
2 | 28 |  13 |  7 |  7 |  1 |  0 |  0
3 | 56 |  40 |  7 |  7 |  $1\times\\!2$ |  0 |  0
4 | 70 |  54 |  0 |  0 |  $8\times\\!2$ |  0 |  0
5 | 56 |  0 |  0 |  0 |  40 |  7 |  7
6 | 28 |  0 |  0 |  0 |  13 |  7 |  7
7 | 8 |  0 |  0 |  0 |  8 |  0 |  0
8 | 1 |  0 |  0 |  0 |  1 |  0 |  0
Table 1: The number of primitive states and their descendants for $L=8$, $\Delta=-1/2$, $\phi=0$, organized by the number of magnons of each state $M$. The number of primitive states is $\mathcal{N}^{\rm pri}=144$, while the number of descendant states is $\mathcal{N}^{\rm des}=112$. | total $\\#$ |  $\\#$ primitive states |  $\\#$ descendant states
---|---|---|---
$M$ |  |  $n_{\pm}=0$ |  $n_{+}>0$ |  $n_{-}>0$ |  $n_{+}=n_{-}$ |  $n_{+}>0$ |  $n_{-}>0$
0 | 1 |  1 |  0 |  0 |  0 |  0 |  0
1 | 6 |  4 |  1 |  1 |  0 |  0 |  0
2 | 15 |  13 |  1 |  1 |  0 |  0 |  0
3 | 20 |  18 |  0 |  0 |  $1\times\\!2$ |  0 |  0
4 | 15 |  0 |  0 |  0 |  13 |  1 |  1
5 | 6 |  0 |  0 |  0 |  4 |  1 |  1
6 | 1 |  0 |  0 |  0 |  1 |  0 |  0
Table 2: The number of primitive states and their descendants for $L=6$,
$\Delta=-1/2$, $\phi=0$, organized by the number of magnons of each state
$M$.The number of primitive states is $\mathcal{N}^{\rm pri}=34$, while the
number of descendant states is $\mathcal{N}^{\rm des}=30$.

For the other channel with $L=6$, $\Delta=-1/2$, $\phi=0$. We have 40
primitive states (4 out of 40 are primitive states with root(s) at infinity)
and 24 descendant states, as shown explicitly in table 2.

Without much surprise, we arrive at

$W_{6,8}=\frac{1}{140737488355328}\sum_{m=1}^{24}c_{m}\big{(}t^{2m}+t^{-2m}\big{)}+c_{0}=W_{8,6}.$
(6.6)

This simply implies the torus partition function remains the same after a
$\pi/2$ rotation.

##  7 Conclusions and discussions

In this paper, we address the question of finding all physical solutions of
the XXZ spin chain with parameter $q$ being a root of unity. Our method is a
refined version of the rational $Q$-system, which has been successfully
constructed for various spin chain models [1, 3, 7, 4, 8]. A naive application
of the rational $Q$-system for XXZ spin chain with generic $q$ fails at root
of unity, due to the appearance of FM-strings. The main new results of this
work are: 1) A method which eliminate solutions with FM-strings, rendering
only primitive solutions; 2) A systematic method to construct the descendant
tower of a given primitive state. With these new developments, we now have an
efficient method of obtaining all physical solutions of XXZ spin chain at root
of unity.

Even though our method is straightforward to implement, the structure of the
solutions is rather intricate. For a given length $L$ and magnon number $M$,
whether there are primitive solutions with root(s) at infinities is sensitive
to the parameters $\kappa$ and $q$ and need to be considered separately. In
addition, for a given primitive solution, the number of FM-strings and
infinite pairs are not immediately clear and need to be worked out with some
effort. These of course reflect the intricacy of the enhanced symmetry at root
of unity. Therefore it would be interesting to further clarify the structure
of Bethe solutions with respect to the representation theory of the enhanced
symmetry, such as the conjectured Onsager symmetry [13]. In this regard, it is
useful to investigate the open chain with $U_{q}(\mathfrak{sl}_{2})$ quantum
group symmetry [20, 30, 31], where the representation theory of the
$U_{q}(\mathfrak{sl}_{2})$ algebra will be of vital importance. Our $Q$-system
can be generalized to this case, following the lines of [32]. The advantage
for the open chain case is that, the representation theory is fully understood
and it is a clean starting point for clarifying the representation theoretical
meaning of the Bethe solutions. A more detailed analysis will be presented
elsewhere [33].

The rational $Q$-system constructed in this work has a number of potential
applications. Combining with computational algebraic geometry method, the
rational $Q$-system can be used to compute the lattice partition function of
various lattice models of intermediate size. Through the renowned vertex-face
correspondence [34, 35, 36, 37, 38], the partition functions of the 8-vertex
(and 6-vertex) models can be mapped into the partition functions of the Solid-
on-Solid (SOS) model, an archetypical example of the interaction-round-a-face
(IRF) model [37, 39]. It would be useful if one can apply the rational
$Q$-system method to the 8-vertex and SOS models too, extending the current
paradigm to the $Q$-systems without the U(1) symmetry.

Another intriguing direction, which was initiated in the work [20] is further
exploring the relation with the representation theory of affine Temperley-Lieb
(aTL) algebra and the physical solution space of Bethe ansatz equations. It is
particularly interesting at root of unity, where rich structure emerges. The
representation theory of the aTL algebra has been studied in [20, 40, 41] and
it would be very interesting to discover such structures from the solutions of
rational $Q$-system.

Finally, it is also interesting to generalize the constrained $Q$-system in
this work to higher rank [5] and higher spin models [8] at root of unity,
where more $Q$-functions are involved and the nested Bethe ansatz plays a
crucial role.

## Acknowledgement

The work of J.H. is supported by Jiangsu Funding Program for Excellent
Postdoctoral Talent. The work of Y.J. is partly supported by Startup Funding
no. 3207022217A1 of Southeast University. Y.M.’s work was supported by World
Premier International Research Center Initiative (WPI), MEXT, Japan. Y.M. is
grateful to Jules Lamers and Vincent Pasquier for collaborations on related
topics.

##  Appendix A Reality of $\\{c_{j}\\}$

In this appendix, we prove that $\\{c_{j}\\}$ are real for $|q|=1$. We start
from the $TQ$-relation

$\displaystyle\tau(t)Q(t)=Q_{0,0}(t\,q^{-1/2})Q(t\,q)+Q_{0,0}(uq^{1/2})Q(t\,q^{-1})\,.$
(A.1)

In the regime $|\Delta|\leq 1$ ($|q|=1$), we have $q^{*}=1/q$. 444The
following is obviously true when $q$ is at root of unity.

Let us define

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

Taking complex conjugate of both sides of (A.1)555Here $t$ is regarded as a
formal variable and complex conjugate does not act on it.

$\displaystyle\tau(t)\overline{Q}(t)=Q_{0,0}(tq^{1/2})\overline{Q}(tq^{-1})+Q_{0,0}(tq^{-1/2})\overline{Q}(tq)$
(A.3)

where we have used the fact that $Q_{0,0}^{*}(tq^{\pm})=Q_{0,0}(tq^{\mp})$ and
$\tau^{*}(t)=\tau(t)$, which will be proven shortly. Equations (A.1) and (A.3)
indicate that both $Q(t)$ and $\overline{Q}(t)$ are solutions of the
$TQ$-relation. Since $TQ$-relation is a second order finite difference
equation, it has two independent solutions. However, the other solution is
‘beyond the equator’ and has higher degree. It cannot be identified with
$\overline{Q}(t)$. Therefore we must have $Q(t)\propto\overline{Q}(t)$. Since
$Q(t)$ and $\overline{Q}(t)$ are both monic, we then conclude that
$Q(t)=\overline{Q}(t)$, which implies that $c_{j}^{*}=c_{j}$.

Now we show that $\tau^{*}(t)=\tau(t)$, namely the eigenvalues of the transfer
matrix are real for real $t$. This amounts to prove that the transfer matrix
$\mathbf{T}_{1/2}(t)$ is Hermitian. For XXX spin chain, using the standard
representation for the Lax operator

$\displaystyle
L_{an}(u)=\left(\begin{array}[]{cc}u+\mathrm{i}\,S_{n}^{z}&\mathrm{i}\,S_{n}^{-}\\\
\mathrm{i}\,S_{n}^{+}&u-\mathrm{i}\,S_{n}^{z}\end{array}\right)_{a}$ (A.6)

we have (assuming $u$ is real)

$\displaystyle
L_{an}^{\dagger}(u)=\left(\begin{array}[]{cc}u-\mathrm{i}\,S_{n}^{z}&-\mathrm{i}\,S_{n}^{+}\\\
-\mathrm{i}\,S_{n}^{-}&u+\mathrm{i}\,S_{n}^{z}\end{array}\right)_{a}=\sigma_{a}^{y}L_{an}(u)\sigma_{a}^{y}\,,$
(A.9)

where we use the simple fact that

$\displaystyle\sigma^{y}\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\sigma^{y}=\left(\begin{array}[]{cc}D&-C\\\
-B&A\end{array}\right)\,.$ (A.14)

The transfer matrix for a length $L$ spin chain is defined as

$\displaystyle\mathbf{T}_{1/2}(u)={\rm tr}_{a}\left[L_{a1}(u)\ldots
L_{aL}(u)\right]$ (A.15)

We therefore have

$\displaystyle\mathbf{T}_{1/2}^{\dagger}(u)=$ $\displaystyle\,{\rm
tr}_{a}\left[L_{aL}^{\dagger}(u)\ldots L_{a1}^{\dagger}(u)\right]$ (A.16)
$\displaystyle=$ $\displaystyle\,{\rm
tr}_{a}\left[\sigma_{a}^{y}L_{aL}(u)\sigma_{y}^{a}\ldots\sigma_{y}^{a}\,L_{a1}(u)\,\sigma_{y}^{a}\right]$
$\displaystyle=$ $\displaystyle\,{\rm tr}_{a}\left[L_{a1}(u)L_{a2}(u)\ldots
L_{aL}(u)\right]=\mathbf{T}_{1/2}(u)$

For XXZ spin chain, the Lax operator reads

$\displaystyle
L_{an}(t)=\left(\begin{array}[]{cc}t\,q^{S_{n}^{z}}-t^{-1}q^{-S_{n}^{z}}&(q-q^{-1})S_{n}^{-}\\\
(q-q^{-1})S_{n}^{+}&t\,q^{-S_{n}^{z}}-t^{-1}q^{S_{n}^{z}}\end{array}\right)$
(A.19)

Once again we have
$L_{an}^{\dagger}(t)=\sigma_{a}^{y}\,L_{an}(t)\,\sigma_{a}^{y}$ for real $t$.
It then follows that $\mathbf{T}_{1/2}^{\dagger}(t)=\mathbf{T}_{1/2}(t)$.

##  Appendix B Location of FM-strings

The center of the FM-strings can be determined by the fusion of transfer
matrices. We first review some results from [12].

#### Factorization of transfer matrix

At root of unity, we have

$\displaystyle q=\exp\left(\mathrm{i}\pi\frac{\ell_{1}}{\ell_{2}}\right)$
(B.1)

It was proven in [12] that the transfer matrix $\mathbf{T}^{\rm hw}(t,s)$
factorizes into two parts

$\displaystyle\mathbf{T}^{\rm hw}(t,s)=\mathbf{Q}(x)\mathbf{P}(y)\,,$ (B.2)

where

$\displaystyle x=t\,q^{\frac{2s+1}{2}},\qquad y=t\,q^{-\frac{2s+1}{2}}\,,$
(B.3)

and $s$ is a complex number. By construction, the eigenvalue of
$\mathbf{T}^{\rm hw}(t,s)$ is a Laurent polynomial of $t$ of order $L$. When
$s=(\ell_{2}-1)/2$ (namely $2s+1=\ell_{2}$), we have

$\displaystyle\mathbf{T}^{\rm
hw}\left(t,\frac{\ell_{2}-1}{2}\right)=\mathbf{T}_{\frac{\ell_{2}-1}{2}}(t),$
(B.4)

where $\mathbf{T}_{s}(t)$ is the transfer matrix where the auxiliary space is
in the unitary spin-$s$ representation. In this case,

$\displaystyle\mathbf{T}_{\frac{\ell_{2}-1}{2}}(t)=\varepsilon^{L}Q\big{(}t\,\varepsilon^{1/2}\big{)}P\big{(}t\,\varepsilon^{1/2}\big{)}\,.$
(B.5)

Let us consider corresponding equation for the eigenvalue

$\displaystyle
T_{{(\ell_{2}-1)}/{2}}(t)=\varepsilon^{L}Q\big{(}t\,\varepsilon^{1/2}\big{)}P\big{(}t\,\varepsilon^{1/2}\big{)}\,.$
(B.6)

This equation implies that the zeros of $Q(t)$ and $P(t)$ are contained in the
zeros of $T_{(\ell_{2}-1)/2}(t)$. An important fact is for us is that the
polynomials $Q(t)$ is the solution of the usual $TQ$-relation. The zeros of
$Q(t)$ are the Bethe roots, while the zeros of $P(t)$ corresponds to the
solution ‘beyond the equator’.

For a given primitive solution $Q(t)$, if we know $T_{(\ell_{2}-1)/2}(t)$, we
can find the dual solution $P(t)$. More specifically, for a primitive
$Q$-function $Q(t)$, the corresponding eigenvalue of the transfer matrix can
be thus written as

$T_{(\ell_{2}-1)/2}(t)=\varepsilon^{L}Q^{2}(t\varepsilon^{1/2})F(t\varepsilon^{1/2}),$
(B.7)

where the polynomial $F(t)$ of (5.8) from the eigenvalue
$T_{(\ell_{2}-1)/2}(t)$ reveals the locations of the FM-strings associated
with the dual state of the primitive state. From the eigenvalues of
$\mathbf{T}_{(\ell_{2}-1)/2}$ and the primitive $Q$-functions, we are able to
derive and calculate the polynomial $F(t)$ of (5.8) via (B.7), justifying
(5.3) and (5.8).

In what follows, we want to construct all descendant states of a given
primitive state. Therefore we assume that $Q(t)$ corresponds to a primitive
solution. As is illustrated in [12], its descendant states form a descendant
tower. The top and bottom of the descendant tower correspond to $Q(t)$ and
$P(t)$. Once we know $P(t)$, the descendant states can be worked out
straightforwardly.

#### Fusion of transfer matrix

It is proven in [18] that the transfer matrices satisfy the following fusion
relations

$\displaystyle\mathbf{T}_{1/2}(tq^{(2s+1)/2})\mathbf{T}_{s}(t)=\mathbf{T}_{0}(tq^{(s+1)})\mathbf{T}_{s-1/2}(tq^{-1/2})+\mathbf{T}_{0}(tq^{s})\mathbf{T}_{s+1/2}(tq^{1/2})\,.$
(B.8)

The eigenvalue of $\mathbf{T}_{0}(t)$ is known, and $\mathbf{T}_{1/2}(t)$ can
be worked out by the usual $TQ$-relation. Using these two initial conditions,
we can work out the eigenvalue of $\mathbf{T}_{(\ell_{2}-1)/2}$ recursively.
This allows us to obtain the eigenvalues of the higher-spin transfer matrices
from the eigenvalues of the spin-$1/2$ transfer matrix, and equivalently the
locations of the FM-strings at root of unity.

##  Appendix C Combinatorics for the number of primitive states

In some applications, it is useful to know the number of primitive solutions
for given quantum numbers. In this case, since we do not need to know the
explicit solutions, there are more efficient methods than solving the
constrained $Q$-system. In this appendix, we present such an algorithm, based
on a conjecture which we have tested numerically. We focus on the case with no
twist, _i.e._ $\kappa=1$ and $\eta=\frac{\mathrm{i}\pi}{3}$ for simplicity.
Generalizations to other parameters can be achieved similarly.

In order to determine the number of primitive states with a fixed system size
$L$ and number of down spins $M$, we need to carry out the calculation
recursively by subtracting the number of descendant states with number of down
spins $m<M$.

Another subtlety is number of primitive states with Bethe roots at infinity.
Essentially it amounts to counting the number of primitive solutions with only
finite Bethe roots of a twisted system with effective twist $\kappa=q^{\pm
n_{\pm}}$ and effective magnon number $M^{\prime}=M-n_{\pm}$.

We shall consider the case with even $L$, so that the descendant states are
always present.

For $M>\frac{L}{2}$, there is no primitive states, and

$\mathcal{N}^{\rm des}(L,M)=\begin{pmatrix}L\\\ M\end{pmatrix},\quad
M>\frac{L}{2}.$ (C.1)

So we can focus on finding the number of primitive states for
$M\leq\frac{L}{2}$.

Before introducing the algorithm, we make the following conjecture for the
number of primitive states with infinite Bethe root(s) by observing the
numerical results:

$\mathcal{N}_{\pm\infty}^{\rm pri}(L,M,n_{\pm})=\begin{pmatrix}L\\\
M-n_{\pm}\end{pmatrix}-\sum_{x=0}^{M-n_{\pm}-1}\begin{pmatrix}L\\\
x\end{pmatrix},$ (C.2)

when $n_{\pm}=n_{+}=n_{-}$ is a solution to (3.15). When there is no solution
to (3.15), $\mathcal{N}_{\pm\infty}^{\rm pri}(L,M,n_{\pm})=0$. It will be
useful in the algorithm to obtain the number of primitive states.

Moreover, for each $(L,M)$, the number of primitive states are

$\mathcal{N}^{\rm pri}(L,M)=\begin{pmatrix}L\\\
M\end{pmatrix}-\mathcal{N}^{\rm des}(L,M),$ (C.3)

which are further decomposed into three parts,

$\mathcal{N}^{\rm pri}(L,M)=\mathcal{N}^{\rm pri}_{\rm
reg}(L,M)+\mathcal{N}_{+\infty}^{\rm
pri}(L,M,n_{+})+\mathcal{N}_{-\infty}^{\rm pri}(L,M,n_{-}).$ (C.4)

The number of descendant states for a given $M$ is determined by the number of
primitive states with $\tilde{M}<M$ as well as the structure of the descendant
tower, which is explained in Section 5.

#### The algorithm

The algorithm to determine the number of primitive states is achieved
recursively.

  1. 1.

We start with $M=0$, i.e. the ferromagnetic vacuum. In this case
$\mathcal{N}^{\rm pri}(L,0)=1$. We also know its descendants by constructing
the descendant tower.

  2. 2.

Then we move to $M=1$. $\mathcal{N}^{\rm des}(L,1)=0$ and $\mathcal{N}^{\rm
pri}(L,1)=L$. We need to distinguish $\mathcal{N}^{\rm pri}_{\rm reg}(L,1)$
and $\mathcal{N}^{\rm pri}_{\rm\pm\infty}(L,1)$ because they have different
descendant tower structures, cf. Section 5. From (C.2), we obtain
$\mathcal{N}^{\rm pri}_{\rm\pm\infty}(L,1)$ and $\mathcal{N}^{\rm pri}_{\rm
reg}(L,1)=L-\mathcal{N}^{\rm pri}_{+\infty}(L,1)-\mathcal{N}^{\rm
pri}_{\rm-\infty}(L,1)$. Their descendants are obtained accordingly.

  3. 3.

For $M=2$, $\mathcal{N}^{\rm des}(L,2)$ is determined by the descendant towers
of states with $M<2$. Therefore, we have $\mathcal{N}^{\rm
pri}(L,2)=\begin{pmatrix}L\\\ 2\end{pmatrix}-\mathcal{N}^{\rm des}(L,2)$.
$\mathcal{N}^{\rm pri}_{\rm\pm\infty}(L,2)$ are obtained from (C.2) and
$\mathcal{N}^{\rm pri}_{\rm reg}(L,2)=\mathcal{N}^{\rm
pri}(L,2)-\mathcal{N}^{\rm pri}_{+\infty}(L,2)-\mathcal{N}^{\rm
pri}_{\rm-\infty}(L,2)$. We again construct their descendant towers.

  4. 4.

For $M\geq 3$, we repeat the same procedure as $M=2$, increasing $M$ one by
one. We terminate the algorithm when $M>\frac{L}{2}$.

From the algorithm above, we computed the number of primitive and descendant
states for $L=6$ and $8$, which are in accord with the results of the
$Q$-system calculations as shown in tables 1 and 2.

Another curious observation is that

$\mathcal{N}^{\rm pri}_{\rm reg}(L,L/2)=\mathcal{N}^{\rm pri}(L,L/2)=2\cdot
3^{L/2-1},$ (C.5)

as shown in table 3.Even though the number of primitive states grows
exponentially at the equator, it is still of a vanishing factor when going to
the thermodynamic limit $L\to\infty$,

$\lim_{L\to\infty}\frac{\mathcal{N}^{\rm pri}(L,L/2)}{\mathcal{N}(L,L/2)}=0,$
(C.6)

with $\mathcal{N}(L,L/2)=\begin{pmatrix}L\\\ L/2\end{pmatrix}$. This implies
that we cannot simply ignore the contribution of the descendant states to
physical quantities such as correlation functions in the thermodynamic limit,
e.g. thermodynamic Bethe ansatz (TBA). A similar discussion on the relation
between TBA and the descendant states can be found in [12].

A full-fledged understanding of the combinatorics of primitive states would be
achieved through studying carefully of the conjectured Onsager symmetry of the
XXZ spin chain with periodic boundary condition, which will be reserved for
later investigations.

$L$ | 6 | 8 | 10 | 12 | 14
---|---|---|---|---|---
$\mathcal{N}^{\rm pri}(L,L/2)$ | 18 | 54 | 162 | 1458 | 4374
Table 3: The number of primitive states at the equator $(L,L/2)$ for $L$ even,
$\Delta=1/2$ and $\kappa=1$.

##  Appendix D Coefficients for the torus partition function

$\begin{split}&c_{0}=57267357244328424573,\,\,c_{1}=54076521230629032240,\\\
&c_{2}=45526563458718976224,\,\,c_{3}=34160205646639767616,\\\
&c_{4}=22830887730248006100,\,\,c_{5}=13580296134202058160,\\\
&c_{6}=7181465323611005168,\,\,c_{7}=3371637072931630992,\\\
&c_{8}=1403048821385237139,\,\,c_{9}=516445615459374816,\\\
&c_{10}=167746535765622792,\,\,c_{11}=47940464791005984,\\\
&c_{12}=12015558187203100,\,\,c_{13}=2630770706010480,\\\
&c_{14}=501082343957136,\,\,c_{15}=82600192303376,\\\
&c_{16}=11723781340854,\,\,c_{17}=1420831897920,\\\
&c_{18}=145966283304,\,\,c_{19}=12459660480,\\\
&c_{20}=881811720,\,\,c_{21}=48704080,\\\
&c_{22}=2323824,\,\,c_{23}=68160,\,\,c_{24}=2837.\end{split}$ (D.1)

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