jumping by $\ell_{2}$ each time, until we reach the eigenstate beyond the
equator at the bottom. For example, the descendant tower obtained in this way
from the pseudovacuum that we considered above is illustrated in Fig. 4. This
structure is easily verified by explicitly constructing the Q operator, and
one sees that the eigenvalues of the transfer matrices $\mathbf{T}_{s}$ are
degenerate (up to a sign) for all descendant states.

In particular it follows that the number of descendant states within the
magnetisation sector with number of down spins fixed to $M=M_{0}+n\,\ell_{2}$
is equal to $\begin{pmatrix}n_{\textsc{fm}}\\\ n\end{pmatrix}$, i.e. grows
binomially in the number $n_{\textsc{fm}}$ of FM strings for the state beyond
the equator. By adding up all descendant states at the occurring values of $M$
we find that the total number within the descendant tower is

$n_{\mathrm{total}}=\sum_{n=0}^{n_{\textsc{fm}}}\begin{pmatrix}n_{\textsc{fm}}\\\
n\end{pmatrix}=2^{n_{\textsc{fm}}},$ (8.6)

in agreement with the loop-algebra prediction, cf. Eq. (1.19) of Ref. [32].
The the descendant tower is exponentially large in the number
$n_{\textsc{fm}}$ of FM strings present in the state beyond the equator
corresponding to the primitive state.

${M{=}0}$${\lvert\\{\varnothing\\}\rangle}$${M{=}3}$${\lvert\\{\alpha_{1}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{2}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{3}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{4}^{\textsc{fm}}\\}\rangle}$${M{=}6}$${\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{3}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{4}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{2}^{\textsc{fm}},\alpha_{3}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{2}^{\textsc{fm}},\alpha_{4}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{3}^{\textsc{fm}},\alpha_{4}^{\textsc{fm}}\\}\rangle}$${M{=}9}$${\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}},\alpha_{3}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}},\alpha_{4}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{3}^{\textsc{fm}},\alpha_{4}^{\textsc{fm}}\\}\rangle}$${\lvert\\{\alpha_{2}^{\textsc{fm}},\alpha_{3}^{\textsc{fm}},\alpha_{4}^{\textsc{fm}}\\}\rangle}$${\
\,M{=}12}$${\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}},\alpha_{3}^{\textsc{fm}},\alpha_{4}^{\textsc{fm}}\\}\rangle}$

Figure 4: Illustration of a descendant tower from the pseudovacuum for $N=12$,
$\Delta=1/2$, $\phi=0$. Arrows of different colours correspond to the addition
of different FM strings.

####  8.1.2 Descendant towers with pairs of roots at infinity

Next we turn to an example of a descendant tower that is generated by adding
FM strings as well as pairs $+\infty,-\infty$ to the Bethe roots of a
primitive state. Recall that Bethe roots at infinity correspond to
applications of the lowering operators of $U_{q}(\mathfrak{sl}_{2})$ (see
Appendix A.1). At root of unity $\ell_{2}$ applications of either of
$\mathbf{S}^{\pm}$ or $\bar{\mathbf{S}}^{\pm}$ already gives zero, so the
total numbers of Bethe roots at $+\infty$ and $-\infty$ must be smaller than
$\ell_{2}$: we need $n_{\pm\infty}<\ell_{2}$. As we will illustrate, pairs of
Bethe roots at $+\infty,-\infty$ play a similar role as FM strings.

We continue with the example $N=12$, $\Delta=\frac{1}{2}$
($\eta=\frac{\mathrm{i}\pi}{3}$), $\phi=0$. Let us now start from a primitive
state with one down spin ($M_{0}=1$), $\lvert\\{u_{1}\\}\rangle$, say for the
solution to Bethe equations (2.7) given by

$u_{1}=\frac{1}{2}\log\frac{\sqrt{3}+1}{\sqrt{3}-1}.$ (8.7)

Consider again the corresponding state beyond the equator,
$\lvert\\{v_{m^{\prime}}\\}_{m^{\prime}=1}^{11}\rangle\propto\prod_{j=1}^{12}\sigma^{x}_{j}\
\lvert\\{u_{1}\\}\rangle$. We obtain its eigenvalue for the Q operator by
solving (7.15), yielding the Q function

$\begin{split}Q(t)&\propto\prod_{m^{\prime}=1}^{11}\bigl{(}t-e^{2\mspace{2.0mu}v_{m^{\prime}}}\,t^{-1}\bigr{)}\\\
&=\biggl{(}t-\frac{\sqrt{3}+1}{\sqrt{3}-1}\,t^{-1}\biggr{)}\biggl{(}t^{6}-\frac{59}{74}(-8+3\sqrt{3})+\frac{91-48\sqrt{3}}{37}\,t^{-6}\biggr{)}.\end{split}$
(8.8)

The zeroes of this Q function are $u_{1}$, two FM strings, along with (cf.
Section 5.5) two pairs of roots $+\infty,-\infty$:

$\begin{split}v_{1}&=u_{1},\\\
v_{3(n-1)+k+1}&=\alpha^{\textsc{fm}}_{n}+\frac{\mathrm{i}(k-2)\pi}{3},\qquad
1\leq n\leq n_{\textsc{fm}}=2\,,\quad 1\leq k\leq\ell_{2}=3\,,\\\
v_{8}&=v_{10}=+\infty,\quad v_{9}=v_{11}=-\infty.\end{split}$ (8.9)

Here the FM strings are centred at

$\alpha_{1}^{\textsc{fm}}=\alpha_{1}+\mathrm{i}\frac{\pi}{6},\quad\alpha_{2}^{\textsc{fm}}=\alpha_{2}+\mathrm{i}\frac{\pi}{6},$
(8.10)

with real parts that have numerical values

$\alpha_{1}=-0.38464681,\quad\alpha_{2}=0.12649136,$ (8.11)

satisfying the quantisation condition (7.15). Note that
$n_{\pm\infty}=2<\ell_{2}=3$.

Unlike for FM strings, Bethe roots at infinity cannot added one by one:
$\\{u_{1},\pm\infty\\}$ does not satisfy the Bethe equations, as it violates
the condition (2.12). On the other hand, using (A.7) we can easily check that

$\lvert\\{u_{1},2\times\pm\infty\\}\rangle\coloneqq\lvert\\{u_{1},+\infty,-\infty,+\infty,-\infty\\}\rangle\propto(\mathbf{S}^{-})^{2}\,(\bar{\mathbf{S}}^{-})^{2}\,\lvert\\{u_{1}\\}\rangle$
(8.12)

is an eigenstate with the same eigenvalues for $\mathbf{T}_{s}$ as
$\lvert\\{u_{1}\\}\rangle$. That is, a pair of Bethe roots $+\infty,-\infty$
can be viewed as a ‘bound state’: these two roots always appear together in
order to satisfy condition (2.15).101010 Another viewpoint is that, unlike in
the isotropic case, the coproduct of $U_{q}(\mathfrak{sl}_{2})$ is ‘chiral’:
the operators $\mathbf{S}^{\pm}$ and $\bar{\mathbf{S}}^{\pm}$ from (2.5) break
the left-right (parity) symmetry of the Hamiltonian. The combination
$\mathbf{S}^{-}\,\bar{\mathbf{S}}^{-}$, however, is compatible with parity
invariance. Like FM strings, pairs $+\infty,-\infty$ do not scatter with other
magnons. Therefore, when the corresponding descendant state beyond the equator
contains one pair of infinite Bethe roots as in our example, the total number
of states in the descendant tower is

$n_{\mathrm{total}}=\sum_{m=0}^{n_{\textsc{fm}}+1}\begin{pmatrix}n_{\textsc{fm}}+1\\\
m\end{pmatrix}=2^{n_{\textsc{fm}}+1},$ (8.13)

as illustrated in Fig. 5.

${M=1}$${\lvert\\{u_{1}\\}\rangle}$${M=4}$${\lvert\\{u_{1},\alpha_{1}^{\textsc{fm}}\\}\rangle}$${\lvert\\{u_{1},\alpha_{2}^{\textsc{fm}}\\}\rangle}$${M=5}$${\lvert\\{u_{1},2\times\pm\infty\\}\rangle}$${M=7}$${\lvert\\{u_{1},\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}}\\}\rangle}$${M=8}$${\lvert\\{u_{1},\alpha_{1}^{\textsc{fm}},2\times\pm\infty\\}\rangle}$${\lvert\\{u_{1},\alpha_{2}^{\textsc{fm}},2\times\pm\infty\\}\rangle}$${\
\,M=11}$${\lvert\\{u_{1},\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}},2\times\pm\infty\\}\rangle}$
Figure 5: The descendant tower of $\lvert\\{u_{1}\\}\rangle$ with finite
$u_{1}$ for $N=12$, $\Delta=1/2$, $\phi=0$.

Almost all (primitive) states at $M=1$ give rise to a descendant tower of this
form, only differing in the locations of the FM string centres. There are two
exceptions, to which we turn next.

####  8.1.3 Mirroring descendant towers

There is one more interesting feature present in the spectrum for $N=12$,
$\Delta=\frac{1}{2}$ ($\eta=\frac{\mathrm{i}\pi}{3}$), $\phi=0$. At $M=1$ down
spin we find from (2.15) that there are two eigenstates, namely
$\lvert\\{+\infty\\}\rangle$ and $\lvert\\{-\infty\\}\rangle$, which have an
infinite Bethe root. These are primitive: their $\mathbf{T}_{s}$-eigenvalues
are different from those of the pseudovacuum, reflecting that we are not at
the isotropic point. We will concentrate on $\lvert\\{+\infty\\}\rangle$ in
this section; the situation for $\lvert\\{-\infty\\}\rangle$ is analogous.

In general, in the (anti)periodic case $\phi\in\\{0,\pi\\}$, the existence of
an eigenstate of the form
$\lvert\\{u_{m}\\}_{m=1}^{M}\cup\\{n\times+\infty\\}\rangle$ for some $n>0$
implies the existence of another eigenstate
$\lvert\\{u_{m}\\}_{m=1}^{M}\cup\\{(\ell_{2}-n)\times-\infty\\}\rangle$ with
the same eigenvalues of $\mathbf{T}_{s}$ (up to a sign), whose Bethe roots can
be found from the Bethe equations (2.7). (Likewise, the presence of a state
$\lvert\\{u_{m}\\}_{m=1}^{M}\cup\\{n\times-\infty\\}\rangle$ implies that of
$\lvert\\{u_{m}\\}_{m=1}^{M}\cup\\{(\ell_{2}-n)\times+\infty\\}\rangle$.)

One can verify that the state beyond the equator corresponding to
$\lvert\\{+\infty\\}\rangle$ does _not_ belong to the same descendant tower as
$\lvert\\{+\infty\\}\rangle$, unlike for the examples presented in the
previous sections. The eigenvalue for the Q operator on the corresponding
state beyond the equator,
$\lvert\\{v_{m^{\prime}}\\}_{m^{\prime}=1}^{11}\\}\rangle\propto\prod_{j=1}^{12}\sigma^{x}_{j}\
\lvert\\{+\infty\\}\rangle$, is obtained through (7.15), yielding

$Q(t)\propto\prod_{m^{\prime}=1}^{11}\\!\left(t-e^{2\mspace{2.0mu}v_{m^{\prime}}}\,t^{-1}\right)=t^{11}+\frac{165}{4}\,t^{5}+66\,t^{-1}+\frac{11}{2}\,t^{-7}.$
(8.14)

Its zeroes contain three FM strings and two Bethe roots at $-\infty$,

$\displaystyle v_{1}$ $\displaystyle=v_{2}=-\infty,$ (8.15) $\displaystyle
v_{3(n-1)+k+2}$
$\displaystyle=\alpha_{n}^{\textsc{fm}}+\frac{\mathrm{i}(k-2)\pi}{3},\qquad
1\leq n\leq n_{\textsc{fm}}=3,\quad 1\leq k\leq\ell_{2}=3,$

where the FM strings have centres

$\alpha_{n}^{\textsc{fm}}=\alpha_{n}+\mathrm{i}\frac{\pi}{6},\quad\alpha_{1}=-0.404723313,\quad\alpha_{2}=0.075767627,\quad\alpha_{3}=0.613080368,$
(8.16)

which satisfy the quantisation condition (7.15).

The resulting descendant-tower structure is as follows. The primitive state
$\lvert\\{+\infty\\}\rangle$ gives rise to descendants via the addition of the
above FM strings. This yields a descendant tower that accounts for half of the
states with the same eigenvalues (possibly up to a sign) of the transfer
matrices $\mathbf{T}_{s}$. The other half of the states form a ‘mirroring
tower’, obtained from the first tower by spin reversal. At the top of the
mirroring tower we find the primitive eigenstate
$\lvert\\{2\times-\infty\\}\rangle$, whose eigenvalues for $\mathbf{T}_{s}$
are the same as those of $\lvert\\{+\infty\\}\rangle$ except for a sign for
$s=1/2$. Spin reversal relates states on opposite sides of the equator as

$\prod_{j=1}^{12}\\!\sigma_{j}^{x}\
\lvert\\{u_{m}\\}_{m=1}^{M}\rangle\,\propto\,\lvert\\{v_{m^{\prime}}\\}_{m^{\prime}=1}^{12-M}\rangle=\lvert\\{+\infty,2\times-\infty,\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}},\alpha_{3}^{\textsc{fm}}\\}\setminus\\{u_{m}\\}_{m=1}^{M}\rangle.$
(8.17)

See Fig. 6 for an illustration.

Figure 6: Illustration of descendant towers from degenerate primitive states
at $M_{0}=1,2$ with Bethe roots at infinity for $N=12$, $\Delta=1/2$,
$\phi=0$. Global spin reversal acts by reflection through the middle of the
figure: the states with the same colour are pairs corresponding to each other
beyond the equator.

###  8.2 Descendant towers for nonzero commensurate twist

Now we turn to commensurate twist (6.15) with $\phi\notin\\{0,\pi\\}$ away
from the (anti)periodic case. It will be instructive to consider the two
commensurate twists $\pm\phi$ simultaneously. Like with the mirror-pair of
descendant towers in Section 8.1.3 we need to consider two copies of
descendant towers. This time, however, the two copies include states at twist
$\phi$ as well as states at _opposite_ twist $-\phi$. Indeed, in Appendix C.3
we show that the transfer matrices $\mathbf{T}_{s}(u,\phi)$ with twist $\phi$
are related to those with opposite twist by flipping all spins,

$\prod_{j=1}^{N}\\!\sigma^{x}_{j}\
\mathbf{T}_{s}(u,\phi)\prod_{j=1}^{N}\\!\sigma^{x}_{j}=e^{2s\mathrm{i}\phi}\,\mathbf{T}_{s}(u,-\phi).$
(8.18)

Thus $\prod_{j=1}^{N}\sigma^{x}_{j}\ \lvert\\{u_{m}\\}_{m=1}^{M}\rangle$ is an
eigenstate of (the Hamiltonian, and more generally) the transfer matrices
$\mathbf{T}_{s}(u,-\phi)$ whenever $\lvert\\{u_{m}\\}_{m=1}^{M}\rangle$ is so
for $\mathbf{T}_{s}(u,\phi)$, cf. (C.17). (Observe that $\phi=0,\pi$ are the
only two values for which $\phi=-\phi\,\mathrm{mod}\,2\pi$.)

We study the example $N=6$, $\Delta=\frac{1}{2}$ (so $\eta=\mathrm{i}\pi/3$)
and commensurate twist $\phi=\pm\frac{2\pi}{3}$. We wish to understand the
descendant tower for primitive states with $M_{0}=1$ down spin. These ‘twisted
magnons’ are discussed in detail in Appendix B.2. Consider
$\lvert\\{u_{1}\\}\rangle$ with $u_{1}=\arctan\frac{\tan(\pi/18)}{\sqrt{3}}$,
which is an eigenvector for the Hamiltonian with twist $\phi=\frac{2\pi}{3}$.
Its quasimomentum, see Eq. (B.9), is

$\widetilde{p}=\mathrm{i}\log\frac{\sinh(u_{1}-\eta/2)}{\sinh(u_{1}+\eta/2)}=\frac{6\pi-\phi}{6}=\frac{8\pi}{9}.$
(8.19)

Next consider the primitive state $\lvert\\{u_{2}\\}\rangle^{\prime}$ with
$u_{2}=-\\!\arctan\frac{\tan(\pi/18)}{\sqrt{3}}$, where we decorate the ket by
a prime to indicate that it is an eigenstate for the Hamiltonian with twist
$\phi^{\prime}=-\frac{2\pi}{3}$.111111 With our conventions (see Section 3)
the off-shell Bethe vectors are independent of the twist, and the dependence
on $\phi$ only enters via the Bethe equations. The notation
$\lvert\\{\dots\\}\rangle^{\prime}$ therefore is not really necessary, as the
values of the Bethe roots implicitly determine the sign of the twist, yet we
believe that it helps clarifying the discussion. This second state has
opposite quasimomentum

$\widetilde{p}\mspace{2.0mu}^{\prime}=\mathrm{i}\log\frac{\sinh(u_{2}-\eta/2)}{\sinh(u_{2}+\eta/2)}=\frac{-6\pi-\phi^{\prime}}{6}=-\frac{8\pi}{9}=\frac{10\pi}{9}\,\mathrm{mod}\,2\pi.$
(8.20)

These two primitive states have the same eigenvalues for the energy (cf.
Appendix B.2)

$\begin{aligned}
H(\phi)\,\lvert\\{u_{1}\\}\rangle\hphantom{{}^{\prime}}&=E\hphantom{{}^{\prime}}\,\lvert\\{u_{1}\\}\rangle,\\\
H(-\phi)\,\lvert\\{u_{2}\\}\rangle^{\prime}&=E^{\prime}\,\lvert\\{u_{2}\\}\rangle^{\prime},\end{aligned}\qquad
E=E^{\prime}=-\\!\cos\frac{\pi}{9}-\frac{1}{2},$ (8.21)

and, as $\varepsilon^{n_{\textsc{fm}}}=(-1)^{0}=1$ in (7.8), the same
‘twisted’ momentum
$p=\widetilde{p}+\phi/N=\widetilde{p}\mspace{2.0mu}^{\prime}-\phi/N=\pi\,\mathrm{mod}\,2\pi$.
We note, however, that these two states are _not_ degenerate for the transfer
matrices. For example, the fundamental transfer matrix $\mathbf{T}_{1/2}$ acts
by

$\mathbf{T}_{1/2}(u,\phi)\,\lvert\\{u_{1}\\}\rangle=\frac{e^{\mathrm{i}\phi/2}}{64}\,\bigl{(}a_{1}\,t^{6}+a_{2}\,t^{4}+a_{3}\,t^{2}+a_{4}+a_{5}\,t^{-2}+a_{6}\,t^{-4}+a_{7}\,t^{-6}\bigr{)}\,\lvert\\{u_{1}\\}\rangle,$
(8.22)

in terms of $t=e^{u}$ as usual, while

$\mathbf{T}_{1/2}(u,\phi^{\prime})\,\lvert\\{u_{2}\\}\rangle^{\prime}=\frac{e^{\mathrm{i}\phi^{\prime}\\!/2}}{64}\,\bigl{(}a_{7}\,t^{6}+a_{6}\,t^{4}+a_{5}\,t^{2}+a_{4}+a_{3}\,t^{-2}+a_{2}\,t^{-4}+a_{1}\,t^{-6}\bigr{)}\,\lvert\\{u_{2}\\}\rangle^{\prime}$
(8.23)

has the opposite order of the coefficients. The coefficients read

$\begin{split}&a_{1}=-2,\qquad
a_{2}=a_{5}=9-6\sin\frac{\pi}{18}-6\cos\frac{\pi}{9},\\\
&a_{3}=-6\,\Bigl{(}1-3\cos\frac{\pi}{9}+\cos\frac{2\pi}{9}-4\sin\frac{\pi}{18}\Bigr{)},\\\
&a_{4}=-1+18\cos\frac{2\pi}{9}-18\sin\frac{\pi}{18},\\\
&a_{6}=-6\,\Bigl{(}1+\cos\frac{2\pi}{9}-\sin\frac{\pi}{18}\Bigr{)},\qquad
a_{7}=1.\end{split}$ (8.24)

This reproduces the above values of $\widetilde{p},\widetilde{p}\,^{\prime}$,
and thus $p$, through (B.9) and $E,E^{\prime}$ via (B.6).

To construct the descendant towers we turn to the corresponding states beyond
the equator. For $\lvert\\{u_{1}\\}\rangle$ this is
$\prod_{j=1}^{6}\sigma^{x}_{j}\
\lvert\\{u_{1}\\}\rangle\propto\lvert\\{u_{1},-\infty,\alpha_{1}^{\textsc{fm}}\rangle^{\prime}$,
whose eigenvalue for the Q operator $\mathbf{\tilde{Q}}(t,\phi^{\prime})$ is

$\displaystyle Q^{\prime}(t)$ $\displaystyle\propto
t\,\bigl{(}t-e^{2\mspace{1.0mu}u_{1}}t^{-1}\bigr{)}\bigl{(}t^{3}-e^{6\mspace{2.0mu}\alpha_{1}^{\textsc{fm}}}t^{-3}\bigr{)}$
(8.25) $\displaystyle\approx
t^{5}-1.2266816\,t^{3}+3.4456224\,t^{-1}-4.2266816\,t^{-3}.$

The state beyond the equator corresponding to
$\lvert\\{u_{2}\\}\rangle^{\prime}$ is
$\lvert\\{u_{2},+\infty,\alpha_{2}^{\textsc{fm}}\rangle$, with eigenvalue for
the Q operator $\mathbf{\tilde{Q}}(t,\phi)$ given by

$\displaystyle Q(t)$ $\displaystyle\propto
t^{-1}\,\bigl{(}t-e^{2\mspace{1.0mu}u_{2}}t^{-1}\bigr{)}\bigl{(}t^{3}-e^{6\mspace{2.0mu}\alpha_{2}^{\textsc{fm}}}t^{-3}\bigr{)}$
(8.26) $\displaystyle\approx
t^{3}-0.8152075\,t+0.2902233\,t^{-3}-0.2365922\,t^{-5}.$

Here the string centres of the two FM strings are

$\alpha_{1}^{\textsc{fm}}=0.20618409+\frac{\mathrm{i}\pi}{6},\quad\alpha_{2}^{\textsc{fm}}=-0.20618409+\frac{\mathrm{i}\pi}{6}.$
(8.27)

The descendant towers can now readily be constructed using the ‘free
fermion’-like property as in Section 8.1.1. The resulting tower structure is
depicted in Fig. 7, where in general states come in pairs that have the same
(possibly up to a sign, as always) eigenvalues for all transfer matrices
$e^{\mp\mathrm{i}s\phi}\,\mathbf{T}_{s}(u,\pm\phi)$, where the sign in
$\pm\phi$ is determined by the Hamiltonian for which the state is an
eigenvector.

Figure 7: Illustration of descendant towers from the primitive states
$\lvert\\{u_{1}\\}\rangle$ and $\lvert\\{u_{2}\\}\rangle^{\prime}$ for $N=6$,
$\Delta=1/2$ and commensurate twist $\phi=\frac{2\pi}{3}$ and
$\phi^{\prime}=-\phi$. Like in Fig. 6 states of the same colour correspond to
each other via global spin reversal.

A key difference with the (anti)periodic case is that, in order to construct
the descendant tower by considering the state beyond the equator corresponding
to the primitive state, we are led to consider the systems with twists $\phi$
and $\phi^{\prime}=-\phi$ simultaneously. The states _within_ either tower in
Fig. 7, e.g. $\lvert\\{u_{1}\\}\rangle$ and
$\lvert\\{u_{1},-\infty\\}\rangle^{\prime}$, have degenerate (possibly up to a
sign) eigenvalues for all $e^{\mp\mathrm{i}s\phi}\,\mathbf{T}_{s}(u,\pm\phi)$,
with sign of the twist determined by the states. On the other hand, _between_
the two different towers, the eigenstates are only degenerate for the momentum
(possibly up to a shift by $\pi$) and energy: as we saw for the two primitive
states this does _not_ extend to the higher charges generated by the transfer
matrices. This is the second important difference with the (anti)periodic case
from Section 8.1.3, where all eigenstates in Fig. 6 have degenerate (up to a
possible sign) eigenvalues for $\mathbf{T}_{s}$. To conclude this discussion
we remark that the descendant-tower structure for twists
$\pm\phi\notin\\{0,\pi\\}$ can get more complicated for roots of unity with
$\ell_{2}>3$.

###  8.3 Full spectrum at root of unity: an example

Although we have not been able to find an algorithmic description of the
structure of the spectrum for given values of the spin-chain parameters
$N,\Delta,\phi$, our numerical investigations of numerous examples suggest
that the full spectrum can be understood case by case in terms of the
descendant towers that we have described in this section. Let us demonstrate
this for a system with $N=10$, $\Delta=\frac{1}{2}$
($\eta=\frac{\mathrm{i}\pi}{3}$) and $\phi=0$. The resulting description of
the full spectrum is given in Table 1.

We stress that, due to the vanishing Wronskian relation (6.14) and the ‘free
fermion’-like behaviour of FM strings, each primitive state is at the top of a
descendant tower, with descendants that are obtained by adding FM strings or
pairs of roots $+\infty,-\infty$. By (2.18) the states come in three
categories, classified by the number of Bethe roots located at infinity:

  1. i)

$n_{+\infty}=n_{-\infty}$ ($=0$ for primitive states),

  2. ii)

$n_{+\infty}>n_{-\infty}=0$,

  3. iii)

$n_{-\infty}>n_{+\infty}=0$.

More precisely, the conditions (2.15) and (2.17) allow for
$n_{\pm\infty}=M+1\,\mathrm{mod}\,3$. Since of course
$n_{+\infty}+n_{-\infty}\leq M$ infinite roots can first occur at $M=3$
($n_{\pm\infty}=1$), then at $M=4$ ($n_{\pm\infty}=2$), not at $M=5$
($n_{\pm\infty}=0$), after which the pattern repeats.

Start with
$\lvert\uparrow\\!\cdots\\!\uparrow\rangle=\lvert\\{\varnothing\\}\rangle$.
From the zeroes of the Q function (7.23) we find (cf. Footnote 9 on p. 9) that
the corresponding state beyond the equator can be written as
$\lvert\downarrow\\!\cdots\\!\downarrow\rangle\propto\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}},2\times\pm\infty\\}\rangle$,
with FM strings of length $\ell_{2}=3$. We proceed as in Section 8.1.2. Draw
the candidate descendant tower using the ‘free-fermion’ property. Note that
(2.15) and (2.17) exclude all potential descendant states in this tower with
one pair $+\infty,-\infty$ of infinite roots: all four infinite roots must be
added at the same time. The primitive state at $M=0$ thus gives rise to a
descendant tower with two descendants at $M=3$ and at $M=7$, and one
descendant at each of $M=4,6,10$.

All ten states $\lvert\\{u_{1}\\}\rangle$ at $M=1$ must be primitive. Their
Bethe root are finite in view of (2.15) and (2.17). The corresponding states
beyond the equator have room for two FM strings and one pair of infinite roots
($n_{+\infty}=n_{-\infty}=1$, allowed at $M=9$). The descendant tower is as in
Fig. 5, except that there is just a single pair of infinite roots. All
intermediate descendants occur: the pair of infinite roots at $M=3,6,9$ is
allowed by (2.15) and (2.17). Each $M=1$ vector thus sits at the top of a
descendant tower containing one descendant at each of $M=3,7$ and two
descendants at $M=4,6$.

The 45 states at $M=2$ are also primitive, and must have finite Bethe roots.
The corresponding states beyond the equator allow for two FM strings. By the
free-fermion property each descendant tower contains four states.

Next we turn to $M=3$, where $120-12=108$ primitive states remain. Here it
starts to become a little more complicated to predict the structure by hand,
as infinite Bethe roots ($n_{\pm\infty}=1$) may occur. Due to parity
invariance there must be equally many states in the classes (ii) and (iii). It
is possible to find out how many such states occur. Indeed, suppose that
$n_{+\infty}=1$, $n_{-\infty}=0$ (the opposite case is treated analogously).
Then two Bethe roots remain to be determined. Their Bethe equations
effectively acquire a twist due to the presence of the infinite root, cf. the
discussion preceding (2.12). One can solve these Bethe equations, and delete
all solutions that themselves contain infinite roots. Alternatively one can
directly use the eigenvalues of the truncated two-parameter transfer matrix
$\tilde{\mathbb{T}}(x,y,\phi)$ to compute the eigenvalues of the Q operator
$\tilde{\mathbf{Q}}$ numerically. In either case, the corresponding states
beyond the equator need four more Bethe roots. For the primitive states with
$n_{+\infty}=n_{-\infty}=0$ these extra roots come from
$n_{+\infty}=n_{-\infty}=2$, while for the primitive states with
$n_{\pm\infty}=1$ and $n_{\mp\infty}=0$ these descendants have
$n_{\pm\infty}=2$, $n_{\mp\infty}=0$ plus one FM string.

One proceeds analogously for the $210-12=198$ primitive states left at $M=4$.
The corresponding states beyond the equator need two more Bethe roots, which
come from adding a pair $+\infty,-\infty$ for the primitive states in class
(i). The situation is a bit more tricky for the primitive states in the other
classes. For class (ii) they are of the form
$\lvert\\{u_{1},u_{2},+\infty,+\infty\\}\rangle$, but the corresponding states
beyond the equator are only allowed to have one infinite root. In this case
one of the infinite roots is removed to make place for an FM string: the
descendants are $\lvert\\{u_{1},u_{2},\alpha^{\textsc{fm}},+\infty\\}\rangle$.

The $252-90=162$ primitive states left at the equator must fall in class (i).

The resulting spectrum is summarised in Table 1, where ‘$\times k$’ counts the
different FM string configurations (choices of $\alpha^{\textsc{fm}}_{n}$) for
the descendant states at a given magnetisation. For example, the entry
$1\times\\!2+10$ comes from $\lvert\\{\alpha^{\textsc{fm}}_{n}\\}\rangle$ for
$n=1,2$ (descendants from $M=0$) along with
$\lvert\\{u_{1},\pm\infty\\}\rangle$ for the ten different Bethe roots $u_{1}$
at $M=1$. The entry $1+10\times\\!2$ similarly represents
$\lvert\\{2\times\pm\infty\\}\rangle$ and
$\lvert\\{u_{1},\alpha^{\textsc{fm}}_{n}\\}\rangle$ for $n=1,2$, respectively.

| total $\\#$ |  $\\#$ primimitive states |  $\\#$ descendant states
---|---|---|---
$M$ |  |  $n_{\pm\infty}=0$ |  $n_{+\infty}>0$ |  $n_{-\infty}>0$ |  $n_{+\infty}=n_{-\infty}$ |  $n_{+\infty}>0$ |  $n_{-\infty}>0$
0 | 1 |  1 |  0 |  0 |  0 |  0 |  0
1 | 10 |  10 |  0 |  0 |  0 |  0 |  0
2 | 45 |  45 |  0 |  0 |  0 |  0 |  0
3 | 120 |  40 |  34 |  34 |  $1\times\\!2+10$ |  0 |  0
4 | 210 |  121 |  34 |  34 |  $1+10\times\\!2$ |  0 |  0
5 | 252 |  162 |  0 |  0 |  $45\times\\!2$ |  0 |  0
6 | 210 |  0 |  0 |  0 |  $1+10\times\\!2+121$ |  $34$ |  $34$
7 | 120 |  0 |  0 |  0 |  $1\times\\!2+10+40$ |  $34$ |  $34$
8 | 45 |  0 |  0 |  0 |  $45$ |  0 |  0
9 | 10 |  0 |  0 |  0 |  $10$ |  0 |  0
10 | 1 |  0 |  0 |  0 |  $1$ |  0 |  0
Table 1: The full spectrum for $N=10$, $\Delta=\frac{1}{2}$
($\eta=\frac{\mathrm{i}\pi}{3}$) and $\phi=0$, organised in terms of primitive
and descendant states with $n_{+\infty}=n_{-\infty}$,
$n_{+\infty}>n_{-\infty}=0$, $n_{-\infty}>n_{+\infty}=0$.

###  8.4 Deformations of FM strings

At the start of Section 7.3 we showed that FM strings occur precisely when the
system is at root of unity with commensurate twist (6.15). Let us further
substantiate this by studying what happens to the Bethe roots constituting an
FM string under small changes to the twist $\phi$ or the anisotropy parameter
$\eta$. (Observe that such deformations immediately break the relation (6.15),
as $\ell_{2}$ fluctuates wildly when the value of $\eta$ varies through
$\mathrm{i}\,\mathbb{Q}\subset\mathrm{i}\,\mathbb{R}$.)

Turning on a small twist $\phi$ (Appendix E.1) leads to smooth changes of the
Bethe roots associated to FM strings, and the same is true for the eigenvalues
of the Q operator. This is closely related to the string hypothesis in the
thermodynamic limit, as explained in Section 10.2: under small variations of
the twist an FM string decomposes into (string deviations of) two ordinary
Bethe strings with slightly different string centres. See also our conjecture
in Section 10.2.

Instead, even a tiny change of $\eta$ away from the root-of-unity values
completely changes the Bethe roots associated to the FM strings (Appendix
E.2): unlike before, FM strings cannot be continuously reconstructed when
$\eta$ is deformed, and the eigenvalues of the Q operator change drastically.
This is expected from the representation theory of $U_{q}(\mathfrak{sl}_{2})$,
which is very sensitive to the precise root of unity. However, as one would
expect on physical grounds, physical quantities such as the eigenvalues of the
transfer matrices $\mathbf{T}_{s}$ _do_ change continuously as $\eta$ is
varied. Further investigations are required in order to fully understand this
behaviour.

###  8.5 Connection to the work of Fabricius and McCoy

In [31] Fabricius and McCoy derived an equation for the string centre of FM
strings at zero twist, see (1.11) therein, by linearising the Bethe equations
(2.7) around an arbitrary root of unity. In other words, these string centres
are obtained by continuity from their values in an infinitesimal neighbourhood
of the root of unity. Let us explain why this is not in contradiction with the
results from Section 8.4 and Appendix E.2.

By ‘Bethe roots’ we mean the zeroes of the eigenvalue of Q operator
constructed from the two-parameter transfer matrix. We thus determine the FM
string centres at commensurate twist (6.15) from the zeroes of the Q operator.
Such zeroes associated with FM strings are among the zeroes of the Drinfeld
polynomial (parameters of the evaluation representation). Note that the
Drinfeld polynomial $Y\mspace{-1.0mu}(v)$ in (1.42) of Ref. [32], cf. (4.2)
therein and [39], is proportional to $Q_{\mathrm{s}}\,P_{\mathrm{s}}$ in our
notation, as can be seen by comparing (1.42) in [32] with our (7.15).

Next, our roots associated with FM strings do _not_ satisfy (1.11) in [31].
This can most easily been seen when $N$ is a multiple of $\ell_{2}$ and
$\ell_{1}$ is even (i.e. $L$ a multiple of $N$ and $r$ even in the notation of
[31]), so that (1.11) from [31] reduces to (1.10) therein. In that case the
denominator of $Y\mspace{-1.0mu}(v)$ is one, and the zeroes of
$Y\mspace{-1.0mu}(v)$ coincide (up to a simple change of notation) with the
zeroes of (7.23), which we identify with the roots constituting the FM string.
This would coincide with the solutions of [31] if one would ignore the second
factor in parenthesis in (1.10).

The solutions to (1.11) in [31] can be used to construct the eigenvectors of
$\mathbf{H}$. Indeed, Fabricius and McCoy proposed a creation operator for FM
strings, see (1.38) and (1.41)–(1.42) in [32]. In examples we find that,
taking the spectral parameter therein to by a solution to (1.11) in [31], the
result is an eigenvector of $\mathbf{H}$ and $\mathbf{T}_{1/2}$, but _not_ of
the two-parameter transfer matrix or Q operator. (Note that we cannot take the
spectral parameter to be the FM string centres as found by our methods, cf.
Section 8.5, since these are zeroes of $Y\mspace{-1.0mu}(v)$ which appears as
a denominator in (1.38) from [32].) Instead, the FM string creation operator
from [32] yields a linear combination of the eigenvectors of the Q operator.
In Appendix E.2 we illustrate this with an example.

In the next section we propose creation and annihilation operators for FM
strings that do yield eigenvectors of the Q operator.

##  9 Conjectures for FM creation and annihilation operators

Vernier et al. [36] proposed to use semicyclic representations in order to
construct degenerate states with different magnetisation. These
representations have also been used to construct quasilocal charges [77].
Building on these ideas we present conjectures for explicit constructions of
the FM string creation and annihilation operators that commute with the XXZ
transfer matrix while changing the magnetisation in steps of $\ell_{2}$.

###  9.1 Case $q^{\ell_{2}}=+1$

We start with roots of unity obeying $\varepsilon=q^{\ell_{2}}=+1$. Pick the
semicyclic representation for the auxiliary space: $V_{a}^{\text{sc}}$ is
$\ell_{2}$ dimensional, has spin $s\in\mathbb{C}$ and furthermore depends on a
parameter $\beta\in\mathbb{C}$, see Appendix A.2. Replace $\textbf{X}_{a}$
from (4.1), (6.3) by
$\textbf{X}^{\text{sc}}_{a}=\textbf{X}_{a}+\beta\,x\,y/(x-y)\,|0\rangle\langle
l-1|_{a}$, which still obeys the commutation relation
$\textbf{W}_{\\!a}\,\textbf{X}^{\text{sc}}_{a}=q\,\textbf{X}_{a}^{\text{sc}}\,\textbf{W}_{\\!a}$.
Substituting this in (4.7) modifies the matrix entry
$\textbf{S}^{+}_{a}=\textbf{L}^{21}_{a}\mspace{-1.0mu}/\mspace{-1.0mu}\sinh\eta$
to

$\textbf{S}^{+,\,\text{sc}}_{a}=\textbf{S}^{+}_{a}+\beta\,|0\rangle\langle\ell_{2}-1|_{a},$
(9.1)

while keeping the other three matrix elements in (4.7) unchanged. In this way
we obtain a Lax operator $\mathbf{L}^{\text{sc}}_{aj}$, and the usual
construction (3.5)–(3.6) gives the transfer matrix

$\mathbf{T}^{\rm sc}_{s}(u,\phi,\beta)=\mathrm{tr}_{a}\bigl{[}\mathbf{L}^{\rm
sc}_{aN}(u,\beta)\cdots\mathbf{L}^{\rm
sc}_{a1}(u,\beta)\,\mathbf{E}_{a}(\phi)\bigr{]}$ (9.2)

depending on $s\in\mathbb{C}$ and $\beta\in\mathbb{C}$. Here the twist
$\mathbf{E}_{a}(\phi)$ acts on $V_{a}^{\text{sc}}$ by the usual expression
(B.1). The matrix elements of (9.2) change the magnetisation $(N-2M)/2$ by
$-n\,\ell_{2}$ and are proportional to $\beta^{n}$ for positive
$n\in\mathbb{Z}_{>0}$. For example, in a chain of length $N=\ell_{2}$, the
expansion of the transfer matrix contains a term
$\mathrm{tr}_{a}[(\mathbf{S}^{+,\,\text{sc}}_{a})^{N}]\,\prod_{j=1}^{N}\sigma^{-}_{j}$
that changes the magnetisation by $-\ell_{2}$, creating $\ell_{2}$ magnon
excitations.

The matrices $\textbf{T}_{s}^{\text{sc}}(u,\phi)$ do _not_ commute with each
other at different values of $u$, but _do_ commute with
$\textbf{T}_{{1/2}}(u,\phi)$ when the twist is commensurate. To see this, note
that the construction of the semicyclic Lax operator guarantees that the
following version of the RLL relation (3.3) with (six-vertex) R-matrix (3.4)
holds true for any $\beta$:

$\textbf{R}_{jk}(u-v)\,\textbf{L}^{\text{sc}}_{aj}(u,\beta)\,\textbf{L}^{\text{sc}}_{ak}(v,\beta)=\textbf{L}^{\text{sc}}_{ak}(v,\beta)\,\textbf{L}^{\text{sc}}_{aj}(u,\beta)\,\textbf{R}_{jk}(u-v).$
(9.3)

This is an identity of operators on $V_{j}\otimes V_{k}\otimes
V_{a}^{\text{sc}}$. Let us reinterpret it by thinking of the spin-1/2 space
labelled by $k$ as an auxiliary space, $V_{k}\rightsquigarrow V_{b}$. The
symmetry property $\textbf{R}_{jk}=\textbf{R}_{kj}$ of (3.4) allows us to view
the R-matrix as a Lax matrix $\textbf{L}_{bj}$, whereas
$\textbf{L}^{\text{sc}}_{ak}$ takes the role of an R-matrix
$\textbf{R}^{\text{sc}}_{ab}$. Reversing the two sides of (9.3) and changing
$v\mapsto u-v$ we arrive at an RLL relation on $V_{a}^{\text{sc}}\otimes
V_{b}\otimes V_{j}$:

$\mathbf{R}^{\text{sc}}_{ab}(u-v,\beta)\,\textbf{L}^{\text{sc}}_{aj}(u,\beta)\,\textbf{L}_{bj}(v)=\textbf{L}_{bj}(v)\,\textbf{L}^{\text{sc}}_{aj}(u,\beta)\,\textbf{R}^{\text{sc}}_{ab}(u-v,\beta).$
(9.4)

Here $a$ corresponds to the semicyclic auxiliary space, $b$ to a spin-$1/2$
auxiliary space, and $k$ to a spin-$1/2$ physical space. The train argument
implies that the twisted semicyclic transfer matrix (9.2) commutes with the
twisted fundamental transfer matrix (3.5)–(3.6), _provided_
$\textbf{R}^{\text{sc}}_{ab}$ commutes with the tensor product of the twists
matrices. This requires the twist to be commensurate,
$e^{\mathrm{i}\ell_{2}\phi}=1$, cf. (6.15).

By fusion (9.2) further commutes with $\textbf{T}_{s^{\prime}}(u,\phi)$ for
any $2\,s^{\prime}\in\mathbb{Z}_{>0}$:

$\left[\mathbf{T}^{\rm
sc}_{s}(u,\phi,\beta),\textbf{T}_{s^{\prime}}(v,\phi)\right]=0,\qquad
s\in\mathbb{C},\quad\beta\in\mathbb{C},\quad 2s^{\prime}\in\mathbb{Z}_{>0}.$
(9.5)

Since the twisted semicyclic transfer matrix $\mathbf{T}^{\rm
sc}_{s}(u,\phi,\beta)$ changes the magnetisation of an eigenstate of the Q
operator in steps of $\ell_{2}$, it mixes states that are degenerate for
$\mathbf{T}_{s^{\prime}}(u,\phi)$ within each descendant tower.

One can use $\mathbf{T}^{\rm sc}_{s}(u,\phi,\beta)$ to construct eigenstates
of the Q operator itself. Because the part of $\mathbf{T}^{\rm
sc}_{s}(u,\phi,\beta)$ of first order in $\beta$ changes the magnetisation of
eigenstates of the Q operator by $-\ell_{2}$, we make

Conjecture 1. When $\varepsilon=q^{\ell_{2}}=+1$ the linearisation in $\beta$
of (9.2) at $s=(\ell_{2}-1)/2$,

$\mathbf{B}^{\textsc{fm}}(u)\,\coloneqq\,\partial_{\beta}\,\mathbf{T}^{\rm
sc}_{s}(u,\phi,\beta)\big{|}_{\beta=0,\,2s=\ell_{2}-1},$ (9.6)

is the creation operator for the FM string
$\\{\alpha^{\textsc{fm}}\\}=\\{u,u+\frac{\mathrm{i}\pi}{\ell_{2}},\cdots\mspace{-1.0mu},u+\mathrm{i}\pi\frac{\ell_{2}-1}{\ell_{2}}\\}$.
The spectral parameter can be taken to be any Bethe root from the FM string,
e.g. $u$ as in (9.6).

Note that at $\beta=0$ and $2s=\ell_{2}-1$ the semicyclic Lax operator
coincides with the Lax operator whose auxiliary space is the
$\ell_{2}$-dimensional highest-weight representation. By (9.2) the operator
(9.6) can thus be expressed more explicitly as

$\mathbf{B}^{\textsc{fm}}(u)=\sum_{j=1}^{N}\mathrm{tr}_{a}\bigl{[}\mathbf{L}_{aN}(u)\cdots\mathbf{L}_{a,j+1}(u)\;\mathbf{e}^{0,\ell_{2}-1}_{a}\mspace{1.0mu}\sigma^{-}_{j}\;\mathbf{L}_{a,j-1}(u)\cdots\mathbf{L}_{a1}(u)\,\mathbf{E}_{a}(\phi)\bigr{]},$
(9.7)

where we write $\mathbf{e}^{n,n^{\prime}}_{a}=|n\rangle\langle
n^{\prime}|_{a}$ for the matrix units on $V_{a}$.

The construction of the FM-string creation operator (9.6) is just like that of
the generating function of the quasilocal Y charges proposed in Ref. [77],
except that the transfer matrices are evaluated at different values of $s$.
Therefore $\mathbf{B}^{\textsc{fm}}(u)$, like those Y charges, commutes with
$\mathbf{T}_{s}(u)$ with $2s\in\mathbb{Z}_{>0}$ but not with
$\tilde{\mathbf{T}}_{s^{\prime}}(u)$ when
$2s^{\prime}\in\mathbb{C}\setminus\mathbb{Z}_{>0}$.

Example. We have verified our conjecture in many examples. To illustrate this
consider $N=6$, $\Delta=-\frac{1}{2}$ ($\eta=\frac{2\mathrm{i}\pi}{3}$) and
$\phi=0$. The descendant tower of the pseudovacuum
$\lvert\\{\varnothing\\}\rangle=\lvert\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\rangle$
contains three descendants:

$\lvert\\{\alpha^{\textsc{fm}}_{1}\\}\rangle,\quad\lvert\\{\alpha^{\textsc{fm}}_{2}\\}\rangle,\quad\lvert\\{\alpha^{\textsc{fm}}_{1},\alpha^{\textsc{fm}}_{2}\\}\rangle,$
(9.8)

with FM strings centred at

$\alpha^{\textsc{fm}}_{1}=-\frac{\log(10+3\sqrt{11})}{6},\quad\alpha^{\textsc{fm}}_{2}=+\frac{\log(10+3\sqrt{11})}{6}.$
(9.9)

One can verify that (9.6) does indeed create these FM strings:

$\lvert\\{\alpha^{\textsc{fm}}_{n}\\}\rangle\propto\mathbf{B}^{\textsc{fm}}(\alpha^{\textsc{fm}}_{n})\,\lvert\\{\varnothing\\}\rangle,\quad
n=1,2,$ (9.10)

and the consistency condition
$\lvert\\{\alpha^{\textsc{fm}}_{1},\alpha^{\textsc{fm}}_{2}\\}\rangle\propto\mathbf{B}^{\textsc{fm}}(\alpha^{\textsc{fm}}_{1})\,\lvert\\{\alpha^{\textsc{fm}}_{2}\\}\rangle=\mathbf{B}^{\textsc{fm}}(\alpha^{\textsc{fm}}_{2})\,\lvert\\{\alpha^{\textsc{fm}}_{1}\\}\rangle$
holds, even though the $\mathbf{B}^{\textsc{fm}}$ do not commute in general.
More generally, whenever two FM strings with centres
$\alpha^{\textsc{fm}}_{n},\alpha^{\textsc{fm}}_{n^{\prime}}$ occur among the
descendants of some primitive state $\lvert\\{u_{m}\\}_{m}\rangle$ we find
that
$\mathbf{B}^{\textsc{fm}}(\alpha^{\textsc{fm}}_{n})\,\lvert\\{u_{m}\\}_{m}\cup\\{\alpha^{\textsc{fm}}_{n^{\prime}}\\}\rangle=\mathbf{B}^{\textsc{fm}}(\alpha^{\textsc{fm}}_{n^{\prime}})\,\lvert\\{u_{m}\\}_{m}\cup\\{\alpha^{\textsc{fm}}_{n}\\}\rangle\propto\lvert\\{u_{m}\\}_{m}\cup\\{\alpha^{\textsc{fm}}_{n},\alpha^{\textsc{fm}}_{n^{\prime}}\\}\rangle$.
That is, our creation operator (9.6) can be used to construct the whole
descendant tower described in Section 8.1.1.

Next we turn to the annihilation operator. Let us denote the parameter of the
cyclic representation by $\gamma$. If we repeat the same construction with the
transposed Lax matrix $\mathbf{L}_{aj}(\mathbf{v}_{y},\mathbf{u}_{x})$ from
(4.9), this time replacing the entry
$\hat{\textbf{S}}{}^{-}_{a}\coloneqq\textbf{L}^{12}_{a}(\textbf{v}_{y},\textbf{u}_{x})/\sinh\eta$
by

$\hat{\textbf{S}}{}^{-,\,\text{sc}}_{a}=\hat{\textbf{S}}{}^{-}_{a}+\gamma\,|\ell_{2}-1\rangle\langle
0|_{a},$ (9.11)

we obtain a semicyclic transfer matrix

$\hat{\mathbf{T}}^{\rm
sc}_{s}(u,\phi,\gamma)=\mathrm{tr}_{a}\bigl{[}\\!\hat{\,\mathbf{L}}^{\rm
sc}_{aN}(u,\gamma)\cdots\\!\hat{\,\mathbf{L}}^{\rm
sc}_{a1}(u,\gamma)\,\mathbf{E}_{a}(\phi)\bigr{]}$ (9.12)

that changes the magnetisation by positive multiples of $\ell_{2}$. Like
before we arrive at

Conjecture 2. When $\varepsilon=q^{\ell_{2}}=+1$ the linearisation in $\gamma$
at $s=(\ell_{2}-1)/2$,

$\begin{split}\mathbf{C}^{\textsc{fm}}(u)\coloneqq{}&\partial_{\gamma}\,\hat{\mathbf{T}}^{\rm
sc}_{s}(u,\phi,\gamma)\big{|}_{\gamma=0,\,2s=\ell_{2}-1}\,\\\
={}&\sum_{j=1}^{N}\mathrm{tr}_{a}\bigl{[}\mathbf{L}_{aN}(u)\cdots\mathbf{L}_{a,j+1}(u)\;\mathbf{e}^{\ell_{2}-1,0}_{a}\mspace{1.0mu}\sigma^{+}_{j}\;\mathbf{L}_{a,j-1}(u)\cdots\mathbf{L}_{a1}(u)\,\mathbf{E}_{a}(\phi)\bigr{]},\end{split}$
(9.13)

annihilates the FM string with Bethe roots
$\\{u,u+\frac{\mathrm{i}\pi}{\ell_{2}},\cdots\mspace{-1.0mu},u+\mathrm{i}\pi\frac{\ell_{1}-1}{\ell_{2}}\\}$.

We have again checked this in many examples including the one above. We find
that the consistency condition
$\mathbf{C}^{\textsc{fm}}(\alpha_{n}^{\textsc{fm}})\,\mathbf{B}^{\textsc{fm}}(\alpha_{n^{\prime}})\,\lvert\\{u_{m}\\}_{m}\cup\\{\alpha^{\textsc{fm}}_{n}\\}\rangle=\lvert\\{u_{m}\\}_{m}\cup\\{\alpha_{n^{\prime}}^{\textsc{fm}}\\}\rangle=\mathbf{B}^{\textsc{fm}}(\alpha_{n^{\prime}})\,\mathbf{C}^{\textsc{fm}}(\alpha_{n}^{\textsc{fm}})\,\lvert\\{u_{m}\\}_{m}\cup\\{\alpha^{\textsc{fm}}_{n}\\}\rangle$
holds so long as
$\lvert\\{u_{m}\\}_{m}\cup\\{\alpha_{n}^{\textsc{fm}}\\}\rangle$ and
$\lvert\\{u_{m}\\}_{m}\cup\\{\alpha_{n^{\prime}}^{\textsc{fm}}\\}\rangle$ are
eigenstates of the Q operator, even though we do not know the commutation
relations between $\mathbf{B}^{\textsc{fm}}(u)$ and
$\mathbf{C}^{\textsc{fm}}(v)$ in general. We postpone such ‘off-shell’
relations to future investigations.

###  9.2 Case $q^{\ell_{2}}=-1$

When $\varepsilon=q^{\ell_{2}}=-1$ the RLL relation (9.4) has to be modified
by including some signs:

$\mathbf{R}^{\rm sc}_{ab}(u-v,\beta)\,\textbf{L}^{\rm
sc}_{aj}(u,-\beta)\,\textbf{L}_{bj}(v)=\textbf{L}_{bj}(v)\,\textbf{L}^{\rm
sc}_{aj}(u,\beta)\,\textbf{R}^{\rm sc}_{aj}(u-v,-\beta).$ (9.14)

This RLL relation can be shown by direct calculation on $V_{b}\otimes V_{j}$.
The resulting monodromy matrix is ‘staggered’, with alternating sign of
$\beta$. For general $\varepsilon=q^{\ell_{2}}=\pm 1$ the semicyclic monodromy
matrix can thus be defined as

$\mathbf{M}^{\rm
sc}_{a}(u,s,\phi,\beta)\coloneqq\mathbf{L}_{aN}^{\text{sc}}(u,s,\varepsilon^{N}\beta)\cdots\mathbf{L}_{aj}^{\text{sc}}(u,s,\varepsilon^{j}\beta)\cdots\mathbf{L}_{a1}^{\text{sc}}(u,s,\varepsilon\beta)\,\mathbf{E}_{a}(\phi).$
(9.15)

When $\varepsilon=+1$ all $\beta$s have the same sign, while for
$\varepsilon=-1$ the sign alternates. Taking the trace we obtain the general
expression for the semicyclic transfer matrix

$\mathbf{T}^{\rm
sc}_{s}(u,\phi,\beta)=\mathrm{tr}_{a}\bigl{[}\mathbf{L}_{aN}^{\text{sc}}(u,s,\varepsilon^{N}\beta)\cdots\mathbf{L}_{aj}^{\text{sc}}(u,s,\varepsilon^{j}\beta)\cdots\mathbf{L}_{a1}^{\text{sc}}(u,s,\varepsilon\beta)\,\mathbf{E}_{a}(\phi)\bigr{]}.$
(9.16)

For $\varepsilon=+1$ this reduces to (9.16); in particular, the use of the
same notation as in (9.16) should not cause any confusion.

Figure 8: Graphical proof of Eq. (9.18) for an even number $N$ of sites.
Figure 9: Graphical proof of Eq. (9.18) for an odd number $N$ of sites.

Recall that for $\varepsilon=-1$ the commensurate twist $\phi$ depends on the
system size $N$ through the condition $e^{\mathrm{i}\ell_{2}\phi}=(-1)^{N}$
from (6.15). Therefore on $V_{a}^{\text{sc}}\otimes V_{b}$ we have

$\mathbf{E}_{a}(\phi)\,\mathbf{E}_{b}(\phi)\,\mathbf{R}^{\rm
sc}_{ab}(u,\beta)=\mathbf{R}^{\rm
sc}_{ab}\bigl{(}u,(-1)^{N}\beta\bigr{)}\,\mathbf{E}_{a}(\phi)\,\mathbf{E}_{b}(\phi).$
(9.17)

Combining (9.14) and (9.17) the train argument, illustrated in Figs. 8 and 9,
yields

$\mathbf{R}^{\rm sc}_{ab}(u-v,\beta)\,\mathbf{M}^{\rm
sc}_{a}(u,s,\beta,\phi)\,\mathbf{M}_{b}(v,\phi)=\mathbf{M}_{b}(v,\phi)\,\mathbf{M}^{\rm
sc}_{a}(u,s,-\beta,\phi)\,\mathbf{R}^{\rm sc}_{ab}(u-v,\beta).$ (9.18)

Multiplying by $\mathbf{R}^{\rm sc}_{ab}(u-v,\beta)^{-1}=\mathbf{R}^{\rm
sc}_{ab}(v-u+\eta,-\beta)/\sinh^{2}(u-v+\eta)$, provided it exists, and taking
the trace over the $\ell_{2}$-dimensional auxiliary space we see that the
semicyclic transfer matrix (9.16) commutes with the fundamental transfer
matrix in the sense that

$\mathbf{T}^{\rm
sc}_{s}(u,\beta,\phi)\,\mathbf{T}_{1/2}(v,\phi)=\mathbf{T}_{1/2}(v,\phi)\,\mathbf{T}^{\rm
sc}_{s}(u,-\beta,\phi),\qquad u,v,s\in\mathbb{C}.$ (9.19)

With the aid of the fusion relation we obtain the commutation relations

$\mathbf{T}^{\rm
sc}_{s}(u,\beta,\phi)\,\mathbf{T}_{s^{\prime}}(v,\phi)=\mathbf{T}_{s^{\prime}}(v,\phi)\,\mathbf{T}^{\rm
sc}_{s}\bigl{(}u,(-1)^{2s^{\prime}}\beta,\phi\bigr{)},\qquad
u,v,s\in\mathbb{C},\quad 2s^{\prime}\in\mathbb{Z}_{>0}.$ (9.20)

In particular, the semicyclic transfer matrix commutes with
$\mathbf{T}_{s^{\prime}}$ for integer $s^{\prime}$.

Remark. Another way to get this result is to evaluate the monodromy matrix
$\mathbf{M}^{\rm sc}_{a^{\prime}}$ with an $2\ell_{2}$-dimensional semicyclic
auxiliary space $V_{a^{\prime}}$ when $\varepsilon=-1$. One can verify that
$\mathrm{tr}_{a^{\prime}}\mspace{1.0mu}\mathbf{M}^{\rm
sc}_{a^{\prime}}=\mathbf{T}^{\rm sc}_{s}(u,\beta,\phi)+\mathbf{T}^{\rm
sc}_{s}(u,-\beta,\phi)$, while
$(-1)^{N}\,\mathrm{tr}_{a^{\prime}}\bigl{[}\mathbf{M}^{\rm
sc}_{a^{\prime}}\,\mathbf{E}^{\prime}\bigr{]}=\mathbf{T}^{\rm
sc}_{s}(u,\beta,\phi)-\mathbf{T}^{\rm sc}_{s}(u,-\beta,\phi)$ if an extra
twist $\mathbf{E}^{\prime}=\sum_{n=0}^{2\ell_{1}-1}\,|n+\ell_{2}\
\mathrm{mod}\,2\mspace{1.0mu}\ell_{2}\rangle\langle n|_{a^{\prime}}$ is
included.

In either way we are led to

Conjecture 3. For any $\varepsilon=q^{\ell_{2}}=\pm 1$ the linearisation in
$\beta$ of (9.16) at $s=(\ell_{2}-1)/2$,

$\begin{split}\mathbf{B}^{\textsc{fm}}(u)\,\coloneqq\,{}&\partial_{\beta}\,\mathbf{T}^{\rm
sc}_{s}(u,\phi,\beta)\big{|}_{\beta=0,\,2s=\ell_{2}-1}\\\
=\,{}&\sum_{j=1}^{N}\varepsilon^{j}\mathrm{tr}_{a}\bigl{[}\mathbf{L}_{aN}(u)\cdots\mathbf{L}_{a,j+1}(u)\;\mathbf{e}^{0,\ell_{2}-1}_{a}\mspace{1.0mu}\sigma^{-}_{j}\;\mathbf{L}_{a,j-1}(u)\cdots\mathbf{L}_{a1}(u)\,\mathbf{E}_{a}(\phi)\bigr{]},\end{split}$
(9.21)

creates an FM string with Bethe roots
$\\{u^{\prime},u^{\prime}+\frac{\mathrm{i}\pi}{\ell_{2}},\cdots\mspace{-1.0mu},u^{\prime}+\mathrm{i}\pi\frac{\ell_{2}-1}{\ell_{2}}\\}$.
The spectral parameter in (9.21) is related to the FM string by $u=u^{\prime}$
if $\varepsilon=+1$ and $u=u^{\prime}-\frac{\mathrm{i}\pi}{2\ell_{2}}$ when
$\varepsilon=-1$.

This conjecture has been successfully tested on all examples in Section 8, as
well as for various cases when $\ell_{2}$ is even.

Let us focus on $\varepsilon=-1$ again. Taking the derivative of (9.20) with
respect to the parameter $\beta$ we obtain the (anti)commutation relations

$\displaystyle\bigl{\\{}\mathbf{B}^{\textsc{fm}}(u),\mathbf{T}_{s^{\prime}}(v)\bigr{\\}}$
$\displaystyle=0,\qquad 2s^{\prime}+1\in\mathbf{Z}_{>0},$ (9.22)
$\displaystyle\bigl{[}\mathbf{B}^{\textsc{fm}}(u),\mathbf{T}_{s^{\prime}}(v)\bigr{]}$
$\displaystyle=0,\qquad s^{\prime}\in\mathbf{Z}_{>0}.$

The _anti_ commutation of $\mathbf{B}^{\textsc{fm}}(u)$ with
$\mathbf{T}_{1/2}(v)$ for $\varepsilon=-1$ might be surprising, but can be
understood as follows. When $\varepsilon=-1$ an FM string adds $\pi$ to the
momentum without affecting the energy or other charges. This requires
$\mathbf{B}^{\textsc{fm}}(u)$ to anticommute with the translation operator,
and to commute with all higher conserved charges (logarithmic derivatives of
$\mathbf{T}_{1/2}$). This is guaranteed by the anticommutation.

The annihilation operator for FM strings can be defined like in (9.13).
Generalise the semicyclic transfer matrix (9.12) to arbitrary
$\varepsilon\in\\{+1,-1\\}$ by including staggered parameters $\pm\gamma$ like
in (9.16). Then we propose

Conjecture 4. When $\varepsilon=q^{\ell_{2}}=\pm 1$ the operator

$\begin{split}\mathbf{C}^{\textsc{fm}}(u)\,\coloneqq\,{}&\partial_{\gamma}\,\hat{\mathbf{T}}^{\rm
sc}_{s}(u,\phi,\gamma)\big{|}_{\gamma=0,\,2s=\ell_{2}-1}\,\\\
=\,{}&\sum_{j=1}^{N}\varepsilon^{j}\,\mathrm{tr}_{a}\bigl{[}\mathbf{L}_{aN}(u)\cdots\mathbf{L}_{a,j+1}(u)\;\mathbf{e}^{\ell_{2}-1,0}_{a}\mspace{1.0mu}\sigma^{+}_{j}\;\mathbf{L}_{a,j-1}\cdots\mathbf{L}_{a1}(u)\,\mathbf{E}_{a}(\phi)\bigr{]}\end{split}$
(9.23)

annihilates an FM string with Bethe roots
$\\{u^{\prime},u^{\prime}+\frac{\mathrm{i}\pi}{\ell_{2}},\cdots\mspace{-1.0mu},u^{\prime}+\mathrm{i}\pi\frac{\ell_{2}-1}{\ell_{2}}\\}$.
The spectral parameter is again related to the FM string by $u=u^{\prime}$ if
$\varepsilon=1$ and $u=u^{\prime}-\frac{\mathrm{i}\pi}{2\ell_{2}}$ for
$\varepsilon=-1$.

##  10 Thermodynamic limit

###  10.1 FM strings and Z charges

In Section 7.2 we have seen that all states within a descendant tower have the
same eigenvalues (up to a possible minus sign) for $\mathbf{T}_{s}$. This may
lead one to wonder if those states can be distinguished at all using charges
that are (quasi)local in the thermodynamic limit. The answer to this question
is positive: the (exponentially many) degeneracies can be lifted by taking
into account the quasilocal _Z charges_ [45, 42, 43, 44] that can be
constructed at root of unity as logarithmic derivatives of the truncated
transfer matrix $\mathbf{\tilde{T}}_{s}$ from Section 6. The quasilocality of
the Z charges in the thermodynamic limit was demonstrated in Refs. [45, 42,
43, 44]. The Z charges were used to study out-of-equilibrium phenomena such as
quantum quenches in Ref. [41]. Here we focus on their role in distinguishing
states that are degenerate for $\mathbf{T}_{s}$.

In Section 6 we already studied the key ingredient for the construction of the
Z charges, viz. the truncated two-parameter transfer matrix
$\tilde{\mathbb{T}}(x,y,\phi)=\mathbf{\tilde{T}}_{s}(u,\phi)$ at root of
unity. Similar to Ref. [41] we define the generating function of the Z charges
as

$\mathbf{Z}(u,\phi)\coloneqq-\mathrm{i}\,\partial_{s}\log\mathbf{\tilde{T}}_{s}(u,\phi)\big{|}_{2s=\ell_{2}-1}=-\mathrm{i}\,\mathbf{\tilde{T}}_{(\ell_{2}-1)/2}^{-1}(u,\phi)\,\partial_{s}\mathbf{\tilde{T}}_{s}(u,\phi)\big{|}_{2s=\ell_{2}-1}.$
(10.1)

Like in (3.8) the Z charges are the coefficients of the expansion around
$u=\frac{\eta}{2}$,

$\textbf{Z}^{(j)}\coloneqq-\mathrm{i}\,\frac{\mathrm{d}^{j-1}}{\mathrm{d}u^{j-1}}\,\mathbf{Z}(u,\phi)\Big{|}_{u=\eta/2}.$
(10.2)

The Z charges are able to tell apart each member of the descendant tower. Let
us illustrate this for the example $N=6$, $\Delta=\frac{1}{2}$
($\eta=\frac{\mathrm{i}\pi}{3}$) and $\phi=0$ from Section 2.3.Consider the
descendant tower formed by

$\lvert\\{\varnothing\\}\rangle=\lvert\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\rangle,\quad\lvert\\{\alpha_{1}^{\textsc{fm}}\\}\rangle,\quad\lvert\\{\alpha_{2}^{\textsc{fm}}\\}\rangle,\quad\lvert\\{\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}}\\}\rangle\propto\lvert\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\rangle,$
(10.3)

where the string centres for the two FM strings are

$\alpha_{1}^{\textsc{fm}}=-\frac{\log(10+3\sqrt{11})}{6}+\frac{\mathrm{i}\pi}{6},\quad\alpha_{2}^{\textsc{fm}}=\frac{\log(10+3\sqrt{11})}{6}+\frac{\mathrm{i}\pi}{6}.$
(10.4)

The generating function (10.1) acts on the descendant tower by

$\begin{split}\mathbf{Z}(u,0)\,|\\{\varnothing\\}\rangle&=\hphantom{-}\frac{12\pi\cosh(6u)}{10-\cosh(6u)}\,|\\{\varnothing\\}\rangle,\\\
\mathbf{Z}(u,0)\,|\\{\alpha_{1}^{\textsc{fm}}\\}\rangle&=\hphantom{-}\frac{6\sqrt{11}\pi}{10-\cosh(6u)}\,|\\{\alpha_{1}^{\textsc{fm}}\\}\rangle,\\\
\mathbf{Z}(u,0)\,|\\{\alpha_{2}^{\textsc{fm}}\\}\rangle&=-\frac{6\sqrt{11}\pi}{10-\cosh(6u)}\,|\\{\alpha_{2}^{\textsc{fm}}\\}\rangle,\\\
\mathbf{Z}(u,0)\,|\\{\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}}\\}\rangle&=-\frac{12\pi\cosh(6u)}{10-\cosh(6u)}\,|\\{\alpha_{1}^{\textsc{fm}},\alpha_{2}^{\textsc{fm}}\\}\rangle.\end{split}$
(10.5)

The eigenvalues of $\mathbf{Z}(u)$ are different for each eigenstate.

Now consider the thermodynamic limit $N\to\infty$. There is a subtlety in the
presence of FM strings. For instance, when $\phi=0$ and
$\eta=\mathrm{i}\pi\ell_{1}/\ell_{2}$ with $\ell_{1}$ odd, (6.15) implies that
for odd $N$ there are no FM strings in the spectrum, while for even $N$ there
are (exponentially many) states associated with FM strings. The spectrum of
systems at finite size is therefore sensitive to the parity of $N$, and it is
not a priori clear whether the thermodynamic limit is well defined. However,
numerics suggests that at odd $N$ there are states in the spectrum that differ
by terms that vanish as $N\to\infty$ to give rise to the same asymptotic
degeneracies as obtained when taking the limit via even $N$. Thus the result
in the thermodynamic limit should be independent of the way in which the limit
$N\to\infty$ is taken.

The role of the Z charges in separating the degeneracies in the presence of FM
strings has important implications for the thermodynamic limit: Z charges have
to be included when constructing the generalised Gibbs ensemble (GGE) for the
XXZ spin chain at root of unity. Z charges are also important to obtain non-
vanishing spin Drude weight with XXZ model at root of unity [42, 43]. This is
further supported by the discussion in the remainder of this section.

###  10.2 TBA, string-charge duality, and a conjecture for string centres of
FM strings

One of the cornerstones in the study of the thermodynamic properties of
quantum integrable models is the thermodynamic Bethe ansatz (TBA), which has
been used extensively to study out-of-equilibrium problems such as transport,
quantum quenches and generalised hydrodynamics.

In order to solve the TBA equations for the XXZ spin chain at root of unity
one relies on the _string hypothesis_ [78], which stipulates the types of
bound states that the model permits. This is a powerful tool that has yielded
numerous results for thermodynamic properties of the XXZ model. A general
review can be found in [79]. In the following we will use the notation of Ref.
[46].

In [46], Ilievski et al. made the remarkable observation that the root
densities of the Bethe strings (bound states) are in one-to-one correspondence
with the generating functions of (quasi)local charges (3.8) of the model. This
_string-charge duality_ is of great use and convenient for the study of
expectation values of (quasi)local charges, especially for quantum quenches
[12]. The string-charge duality furthermore provides the root densities for
the long-time non-equilibrium steady states [46, 80].

When we are at root of unity we have to take into account the generating
function of the Z charges to obtain the root densities of the ‘last two string
types’ in the list of Takahashi [78, 79]. This result has been derived in [46,
41].121212 In Ref. [41], the authors considered only principal root of unity
($\ell_{1}=1$). Similar results for arbitrary root of unity can be obtained
analogously, cf. the supplementary material of [81]. Moreover, the total root
densities of the ‘last two string types’ are the same from the TBA calculation
(see Eq. (9.66) of [79]), revealing a deep connection between the two.

The ‘last two string types’ are related to FM strings too. In fact, two Bethe
strings, one of each of the last two string types, that have coinciding real
parts of their string centres (cf. Footnote 9 on p. 9) together form an FM
string. For example, consider $\Delta=\frac{1}{2}$
($\eta=\frac{\mathrm{i}\pi}{3}$), for which the string hypothesis says that
the last two string types are $(2,+)$ and $(1,-)$. Call a bound state with $n$
magnons whose Bethe roots have the same real part an ‘$n$-string’. Then
‘$(2,+)$’ denotes a 2-string with even parity (centred at the real axis, i.e.
a complex conjugate pair), and ‘$(1,-)$’ a 1-string with odd parity (centred
at $\mathrm{Im}\,u_{m}=\frac{\pi}{2}\equiv-\frac{\pi}{2}$). For
$\eta=\frac{\mathrm{i}\pi}{3}$ the imaginary parts of the 2-string with even
parity are $\pm\frac{\pi}{6}$ while the imaginary part of the 1-string with
odd parity is $\frac{\pi}{2}$. On the other hand, according to the examples in
Section 8, FM strings for $\eta=\frac{\mathrm{i}\pi}{3}$ can be expressed as

$u_{1}=\alpha-\frac{\mathrm{i}\pi}{6},\quad
u_{2}=\alpha+\frac{\mathrm{i}\pi}{6},\quad
u_{3}=\alpha+\frac{\mathrm{i}\pi}{2},\qquad\alpha\in\mathbb{R}.$ (10.6)

But this can be viewed as two strings, one of type $(2,+)$ and one of type
$(1,-)$, with coinciding real part of the string centre.

We expect that for any system size with commensurate twist FM strings can be
decomposed in terms of the last two string types of the string hypothesis in
this way, even at finite system size. Indeed, for any principal root of unity
$\eta=\frac{\mathrm{i}\pi}{\ell_{2}}$

the last two string types are $(\ell_{2}-1,+)$ and $(1,-)$ [78]. We have
investigated FM strings for all $\ell_{2}\leq 6$ at various $N$ and found that
they can all be viewed as consisting of strings of the last two string types
with equal real parts of the string centres. It even appears to hold for non-
principal root of unity, see Appendix F for examples.

By studying the string centres of FM strings as the value of $\phi$ approaches
the commensurate twist (6.15) as in Appendix E.1 we are led to

Conjecture 5. For any finite system size $N$ and root of unity
$\eta=\mathrm{i}\pi\frac{\ell_{1}}{\ell_{2}}$ with commensurate twist, any
Fabricius–McCoy string

$\alpha_{k}=\alpha^{\textsc{fm}}+\frac{\mathrm{i}\pi}{\ell_{2}}\,\biggl{(}\frac{\ell_{2}+1}{2}-k\biggr{)},\quad
1\leq k\leq\ell_{2},$ (10.7)

has string centre with imaginary part

$\text{Im}\,\alpha^{\textsc{fm}}=\begin{cases}\
\frac{\pi}{2\ell_{2}}&\text{if}\ \ell_{1}\ \text{is odd},\\\ \ 0&\text{if}\
\ell_{1}\ \text{is even}.\end{cases}$ (10.8)

This conjecture has been confirmed for all examples in Section 8.

In the thermodynamic limit, the conjecture (10.8) is compatible with the known
TBA results [79]. Due to the exponentially large degeneracies of the
descendant tower, FM strings in the thermodynamic limit will contribute to the
thermodynamic quantities. Through conjecture (10.8) the density of FM strings
has been included already in the root densities of the ‘last two string
types’. It implies that we do not need to include any new functional relation
to the TBA formalism in [79].

Combining the conjecture with the knowledge that the generating function of
the Z charges is crucial to obtain the root densities of ‘last two string
types’, we conclude that the Z charges are directly related to the FM strings.

Remark. There is a key difference between the thermodynamics of the
‘descendant states’ in our article and the $\mathfrak{sl}_{2}$-descendants in
spin-$1/2$ Heisenberg XXX spin chain. For the latter the
$\mathfrak{sl}_{2}$-descendant states are associated with Bethe roots at
infinity (vanishing quasimomenta), and do not enter in the TBA calculations.
However, for the XXZ spin chain at root of unity the ‘descendant states’ in
our terminology (see Section 7.1) do enter the TBA calculation through the
density of FM strings, which is in principle determined by the density of the
last two string types.

###  10.3 FM strings and spin Drude weight

One of the most important physical consequences of the quasilocal Z charges is
the non-vanishing high-temperature spin Drude weight131313 The full frequency-
dependent conductivity $\sigma(\omega)$, which would be of greater
experimental relevance than the Drude weight, remains challenging to compute
[82]. of the XXZ model at root of unity, due to the non-commutativity between
the spin flip operator $\prod_{j}\sigma^{x}_{j}$ and the Z charges [45, 42,
43]. It can be considered as a manifestation of the exponentially many
degeneracies in the thermodynamic limit.

In Appendix E.2 we demonstrate that perturbing the anisotropy parameter $\eta$
away from root of unity can change the structure of the Bethe roots
dramatically, especially for states with FM strings. On the other hand, the
spin Drude weight [45] is also known to change significantly under such
perturbations. This hints at a connection between the existence of FM strings
and the fractal structure of the spin Drude weight.

Another hint for the intimate relation between FM strings and non-vanishing
spin Drude weight at root of unity comes from the domain-wall quench, i.e. the
time evolution of an initial state
$\lvert\uparrow\\!\cdots\\!\uparrow\uparrow\downarrow\downarrow\\!\cdots\\!\downarrow\rangle$,
for the XXZ model at root of unity. Here ballistic spin transport (non-
vanishing spin Drude weight) can be treated analytically using generalised
hydrodynamics [83]. The right half of the system, i.e. the fully polarised
state $\lvert\downarrow\downarrow\\!\cdots\\!\downarrow\rangle$, has Q
function given in Section 7.5 for finite size. In the thermodynamic limit,
according to the TBA, this fully polarised state consists of a filled ‘Fermi
sea’ with Bethe roots of the last two string types [79, 83], cf. Section 10.2.
In this case, each pair of Bethe strings with one string from each of the last
two string types has the same real parts of the string centres. From the
definition of the density of Bethe strings it follows that the densities of
the last two string types must coincide. The conjecture (10.8) implies that
the Bethe roots of $\lvert\downarrow\downarrow\\!\cdots\\!\downarrow\rangle$
consists solely of FM strings. For the domain-wall quench the ballistic spin
transport from the right half of the system is solely carried out by the
quasilocal Z charges [83]. Notice that, even though the FM strings do not
contribute to the dynamics, the ‘FM strings’ of the right half of the system
are not true excitations from the perspective of the whole system. Thus they
_do_ contribute to the dynamics, as combinations of Bethe strings of the last
two string types, resulting in the domain-wall melting phenomenon.

##  11 Conclusion

We have studied the full spectrum of the transfer matrices associated to the
quantum spin-$\frac{1}{2}$ Heisenberg XXZ chain, focussing on root of unity
with arbitrary twist. To this end we constructed the Baxter’s Q operator, and
the P operator, from the factorisation of a two-parameter transfer matrix
(4.25). The eigenvalues of the Q operator, i.e. Q functions, are polynomials
whose zeroes encode the physical solutions of the Bethe equations (2.7).

As a by-product of our construction we rederived the matrix TQ relation (5.10)
and transfer-matrix fusion relations (5.12) from a decomposition of the two-
parameter transfer matrix, providing a simplification of the conventional
approach. At root of unity we further derived truncated transfer-matrix fusion
and Wronskian relations from the two-parameter transfer matrix with auxiliary
space truncated to a finite-dimensional space. We also proved an
interpolation-type formula conjectured in Refs. [32, 34, 41].

Equipped with these algebraic tools we obtained analytic results about the
spectrum at root of unity (Section 7). These results enabled us to demonstrate
the presence of descendant towers for the XXZ model at root of unity with
commensurate twist, for with explicit examples for the various scenarios that
occur (Section 8). We elucidated the exponential growth of the degeneracies at
root of unity. Since we can construct the Q operator explicitly at root of
unity, via a trace over a finite-dimensional auxiliary space, we obtained
analytic results for rather large system size compared to previous works. From
the quantisation condition (7.15) we found that FM strings associated with the
descendant states behave like free fermions within each descendant tower
(Section 8.1.1). We have found new semicyclic transfer matrices that satisfy
unconventional RLL relations, see e.g. (9.18), from which we conjectured an
explicit expression for the creation and annihilation operators of FM strings
(Section 9).

Even though our main results and discussions concentrate on systems with
finitely many sites, we moreover compared our results with recent works on the
thermodynamic limit (Section 10). We explained the relation between the
truncated two-parameter transfer matrices and the quasilocal Z charges, which
are of crucial importance in many applications such as quantum quenches and
spin transport at root of unity. Inspired by the string-charge duality we
found a connection between the last two string types à la Takahashi, the Z
charge and the FM strings in the XXZ model at root of unity. This led us to a
conjecture about the imaginary part of the string centres of FM strings based
on the string hypothesis.

##### Outlook.

Several interesting aspects remain to be explored. First of all we hope to
apply the construction of the two-parameter transfer matrix to other quantum
integrable models, such as the XXZ spin chain with other boundary conditions
as well as their higher-spin generalisations. It is not known how the
construction should be generalised to integrable models with higher-rank
symmetry, where there exist several Q operators.

We would like to understand more properties of the FM strings in the
thermodynamic limit, e.g. whether quasilocality of the Z charges has a deep
relation to FM strings (cf. Section 10). In addition, the semiclassical limit
of the domain-wall quench in the gapless regime yields a similar ballistic
spin transport behaviour [84, 85] to the quantum counterpart [83]. However, it
is not clear what the semiclassical limit of the Z charges would be since the
classical Landau–Lifshitz field theory does not have an underlying quantum-
group structure. It would be desirable to clarify the relation between the
mechanisms in both quantum and classical regime when a qualitative classical-
quantum correspondence of spin transport [84] applies.

The structure of descendant towers from Section 8 implies the existence of a
hidden Onsager algebra at least for a part of the physical Hilbert space [86].
At the free-fermion point $\Delta=0$, and at $\eta=\frac{\mathrm{i}\pi}{3}$
for the integrable spin-1 XXZ chain, the Onsager algebra plays a crucial role.
The properties of the Onsager algebra may allow one to obtain results like in
Ref. [36]. The relation between the Onsager generators and the semicyclic
transfer matrices from Section 9 needs to be elucidated. In particular we
would like to discover algebraic structures like those in [36] in the
extensively studied, yet not completely understood, XXZ spin chain.

The full spectrum of the XXZ model at root of unity has potential physical
applications. For instance, equipped with our results, it would be interesting
to calculate the partition function of the six-vertex model at root of unity
analytically using algebraic geometry, following recent works on the isotropic
case [87, 88]. Furthermore, other physically relevant quantities such as the
density matrix and overlaps with a generic state can be extracted from the
full spectrum at finite system size, which could shed light on the studies of
quantum entanglement and non-equilibrium dynamics in exact solvable models.

## Acknowledgements

We are indebted to Jean-Sébastien Caux for collaboration in the early stages
of this work and for discussions throughout the course of this work. Y.M.
thanks Paul Fendley, Etienne Granet, Enej Ilievski, Tomaž Prosen, Eyzo Stouten
and Eric Vernier for useful discussions and correspondences. V.P. thanks
Olivier Babelon and Alexandre Lazarescu for illuminating discussions. We thank
the referees for their comments.

Y.M. acknowledges the hospitality of Institut Henri Poincaré during the
workshop “Systems out of equilibrium”, where part of the work has been
conducted. Y.M. acknowledges support from the European Research Council under
ERC Advanced grant 743032 dynamint. J.L. acknowledges the Australian Research
Council Centre of Excellence for Mathematical and Statistical Frontiers
(acems) for financial support.

##  Appendix A Quantum $\mathfrak{sl}_{2}$

The quantum group $U_{q}(\mathfrak{sl}_{2})$ is the unital associative algebra
with generators $\mathbf{S}^{+},\mathbf{S}^{-}$ along with $\mathbf{K}$ (which
is invertible) subject to the commutation relations

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

We take the coproduct to be
$\mathbf{S}^{\pm}\mapsto\mathbf{S}^{\pm}\otimes\mathbf{K}^{-1}+\mathbf{K}\otimes\mathbf{S}^{\pm}$
and $\mathbf{K}\mapsto\mathbf{K}\otimes\mathbf{K}$. There is a counit and
antipode, see e.g. Eqs. (1.2)–(1.4) in Ref. [47]. A good (but hard to find)
introduction is [89].

In this appendix we summarise the representations of
$U_{q}(\mathfrak{sl}_{2})$ that we will use. For each case it is easy to check
that the commutation relations (A.1) hold and that
$\mathbf{K}=\exp(\eta\,\mathbf{S}^{z})$. The spin of an irrep is defined by
the eigenvalue $[s]_{q}\,[s+1]_{q}$ of the quantum Casimir operator

$\frac{1}{2}\bigl{(}\mathbf{S}^{+}\,\mathbf{S}^{-}+\mathbf{S}^{-}\,\mathbf{S}^{+})+\frac{[2]_{q}}{2}\biggl{(}\frac{\mathbf{K}-\mathbf{K}^{-1}}{q-q^{-1}}\biggr{)}^{\\!2},$
(A.2)

which generates the centre of $U_{q}(\mathfrak{sl}_{2})$.

###  A.1 Global representation

The physical Hilbert space $(\mathbb{C}^{2})^{\otimes N}$ of the spin chain
comes with two ‘global’ representations. When $N=1$ the representation is
given by $\mathbf{S}^{\pm}=\sigma^{\pm}$ and $\mathbf{K}=q^{\sigma^{z}\\!/2}$.
For $N>2$ repeated application of the coproduct gives the (reducible)
representation

$\begin{split}&\mathbf{S}^{\pm}=\sum_{j=1}^{N}q^{\sigma^{z}_{1}/2}\otimes\cdots\otimes
q^{\sigma^{z}_{j-1}/2}\otimes\sigma^{\pm}_{j}\otimes
q^{-\sigma^{z}_{j+1}/2}\otimes\cdots\otimes q^{-\sigma^{z}_{N}/2},\\\
&\mathbf{K}=q^{\mathbf{S}^{z}}=q^{\sigma^{z}_{1}/2}\otimes
q^{\sigma^{z}_{2}/2}\otimes\cdots\otimes q^{\sigma^{z}_{N}/2},\end{split}$
(A.3)

from (2.5). By reversing the factors (taking the opposite coproduct) we obtain
another (reducible) representation:

$\begin{split}&\bar{\mathbf{S}}^{\pm}=\sum_{j=1}^{N}q^{-\sigma^{z}_{1}/2}\otimes\cdots\otimes
q^{-\sigma^{z}_{j-1}/2}\otimes\sigma^{\pm}_{j}\otimes
q^{\sigma^{z}_{j+1}/2}\otimes\cdots\otimes q^{\sigma^{z}_{N}/2},\\\
&\bar{\mathbf{K}}=q^{\bar{\mathbf{S}}^{z}}=q^{\sigma^{z}_{1}/2}\otimes
q^{\sigma^{z}_{2}/2}\otimes\cdots\otimes
q^{\sigma^{z}_{N}/2}=\mathbf{K}.\end{split}$ (A.4)

All of these operators can be obtained from the entries (3.9) of the monodromy
matrix is the (‘braid’) limits $u\to\pm\infty$. In particular, the
$\mathbf{B}$ operator from the QISM is closely related to the above spin-
lowering generators. To see this we write the Lax operator (3.2) with
spin-$\frac{1}{2}$ auxiliary space (see Footnote 4 on p. 4) in the form

$\mathbf{L}_{aj}(u)=\begin{pmatrix}\mathbf{a}_{j}(u)&\mathbf{b}_{j}(u)\\\
\mathbf{c}_{j}(u)&\mathbf{d}_{j}(u)\end{pmatrix}_{\\!a},\quad\mathbf{b}_{j}(u)=\sinh(\eta)\,\sigma^{-}_{j},\quad\mathbf{c}_{j}(u)=\sinh(\eta)\,\sigma^{+}_{j}.$
(A.5)

In the (‘braid’) limits $u\to\pm\infty$ the diagonal entries behave as

$\mathbf{a}_{j}(u)\sim\pm\frac{e^{\pm
u}}{2}\,q^{\sigma^{z}_{j}/2},\qquad\mathbf{d}_{j}(u)\sim\pm\frac{e^{\pm
u}}{2}\,q^{-\sigma^{z}_{j}/2},\quad\qquad u\to\pm\infty.$ (A.6)

Therefore, using the definition of monodromy matrix (3.9), we have

$\displaystyle\lim_{u\to-\infty}(-2\,e^{u})^{N-1}\,\mathbf{B}(u)$
$\displaystyle=\sinh(\eta)\,\mathbf{S}^{-},$ (A.7)
$\displaystyle\lim_{u\to+\infty}\,(2\,e^{-u})^{N-1}\,\mathbf{B}(u)$
$\displaystyle=\sinh(\eta)\,\bar{\mathbf{S}}^{-}.$

One similarly recovers the spin-raising operators from $\mathbf{C}$, and
$\mathbf{K},\mathbf{K}^{-1}$ from either of $\mathbf{A},\mathbf{D}$.

###  A.2 Auxiliary representations

In this article we use various choices for the auxiliary space, which is
always a representation of $U_{q}(\mathfrak{sl}_{2})$. We summarise the key
ingredient in this appendix.

First of all we use the finite-dimensional unitary spin-$s$ representation of
$U_{q}(\mathfrak{sl}_{2})$. Denote the (orthonormal) basis of
$\mathbb{C}^{2s+1}$ by $|n\rangle$ for $n=0,1,\cdots\mspace{-1.0mu},2s$. Then
the generators are given by

$\begin{split}\mathbf{S}^{+}&=\sum_{n=0}^{2s-1}\sqrt{[2s-n]_{q}[n+1]_{q}}\
|n+1\rangle\langle n|,\\\
\mathbf{S}^{-}&=\sum_{n=0}^{2s-1}\sqrt{[2s-n]_{q}[n+1]_{q}}\ |n\rangle\langle
n+1|,\\\ \mathbf{K}\ &=\sum_{n=0}^{2s}q^{-s+n}\,|n\rangle\langle
n|,\qquad\mathbf{S}^{z}=\sum_{n=0}^{2s}(-s+n)\,|n\rangle\langle n|,\\\
\end{split}\qquad 2s\in\mathbb{Z}_{\geq 0}.$ (A.8)

When $s=1/2$ this gives the case $N=1$ of (A.3) when we identify $\lvert
1\rangle=\lvert\uparrow\rangle$ and $\lvert 0\rangle=\lvert\downarrow\rangle$.
Note that we are thus led to a _de_ creasing ordering of the basis: we prefer
conventions such that $\mathbf{S}^{\pm}\,\lvert n\rangle\propto\lvert n\pm
1\rangle$ and the basis is labelled by non-negative integers; then the
matrices match the standard choice provided we order the basis as $\lvert
2\,s\rangle,\lvert 2\,s-1\rangle,\cdots\mspace{-1.0mu},\lvert 0\rangle$.

Next we use the complex spin-$s$ highest weight representation of
$U_{q}(\mathfrak{sl}_{2})$. It is defined on an infinite-dimensional Hilbert
space with orthonormal basis $\langle n\rvert$ indexed by
$n\in\mathbb{Z}_{\geq 0}$. The generators are given by

$\begin{split}\mathbf{S}^{+}&=\sum_{n=0}^{\infty}[2s-n]_{q}\,|n+1\rangle\langle
n|,\\\ \mathbf{S}^{-}&=\sum_{n=0}^{\infty}[n+1]_{q}\,|n\rangle\langle n+1|,\\\
\mathbf{K}\ &=\sum_{n=0}^{\infty}q^{-s+n}\,|n\rangle\langle
n|,\qquad\mathbf{S}^{z}=\sum_{n=0}^{\infty}(-s+n)\,|n\rangle\langle
n|,\end{split}\qquad 2s\in\mathbb{C}.$ (A.9)

This is related to the transfer matrices $\mathbf{T}^{\mathrm{hw}}_{s}$
(4.19). In Figs. 2 and 3 we write down explicit matrices; let us stress that —
albeit perhaps somewhat unusual in an infinite-dimensional context — as above
we take the basis to be ordered _de_ creasingly: $\cdots\mspace{-1.0mu},\lvert
2\rangle,\lvert 1\rangle,\lvert 0\rangle$. When
$s\in\frac{1}{2}\mathbb{Z}_{\geq 0}$ the infinite-dimensional highest-weight
representation contains a finite-dimensional submodule. Indeed, the entry of
$\mathbf{S}^{+}$ for $n=2s$ vanishes (cf. Fig. 2), so the subspace labelled by
$0\leq n\leq 2s$ is preserved by all of (A.9). Thus we can truncate to a
representation of dimension $2s+1$ with generators

$\begin{aligned}
\mathbf{S}^{+}&=\sum_{n=0}^{2s-1}[2s-n]_{q}\,|n+1\rangle\langle n|,\\\
\mathbf{S}^{-}&=\sum_{n=0}^{2s-1}[n+1]_{q}\,|n\rangle\langle n+1|,\\\
\mathbf{K}\ &=\sum_{n=0}^{2s}q^{-s+n}\,|n\rangle\langle
n|,\qquad\mathbf{S}^{z}=\sum_{n=0}^{2s}(-s+n)\,|n\rangle\langle
n|,\end{aligned}\qquad 2s\in\mathbb{Z}_{\geq 0}.$ (A.10)

This representation is equivalent to (A.8) by a gauge transformation
(conjugation).

More importantly, when $\eta=\mathrm{i}\pi\ell_{1}/\ell_{2}$ there exists
another truncation yielding an $\ell_{2}$-dimensional representation,
illustrated in Fig. 3. This is sometimes referred to as a _nilpotent_
representation, due to the fact that $(\mathbf{S}^{\pm})^{\ell_{2}}=0$ in this
case. The generators act on the subspace with basis $|n\rangle$ for $0\leq
n\leq\ell_{2}-1$ by

$\begin{aligned}
\mathbf{S}^{+}&=\sum_{n=0}^{\ell_{2}-2}[2s-n]_{q}\,|n+1\rangle\langle n|,\\\
\mathbf{S}^{-}&=\sum_{n=0}^{\ell_{2}-2}[n+1]_{q}\,|n\rangle\langle n+1|,\\\
\mathbf{K}\ &=\sum_{n=0}^{\ell_{2}-1}q^{-s+n}\,|n\rangle\langle
n|,\qquad\mathbf{S}^{z}=\sum_{n=0}^{\ell_{2}-1}(-s+n)\,|n\rangle\langle
n|,\end{aligned}\qquad 2s\in\mathbb{C}.$ (A.11)

There is one more truncated $\ell_{2}$-dimensional representation that we will
use at root of unity: the _semicyclic_ representation. It is similar to the
truncated highest-weight representation (A.11) with an additional entry in
$\mathbf{S}^{+}$:

$\begin{aligned}
\mathbf{S}^{+}&=\beta\,|0\rangle\langle\ell_{2}-1|+\sum_{n=0}^{\ell_{2}-2}[2s-n]_{q}\,|n+1\rangle\langle
n|,\\\ \mathbf{S}^{-}&=\sum_{n=0}^{\ell_{2}-2}[n+1]_{q}\,|n\rangle\langle
n+1|,\\\ \mathbf{K}\ &=\sum_{n=0}^{\ell_{2}-1}q^{-s+n}\,|n\rangle\langle
n|,\qquad\mathbf{S}^{z}=\sum_{n=0}^{\ell_{2}-1}(-s+n)\,|n\rangle\langle
n|,\end{aligned}\qquad\begin{aligned} &2s\in\mathbb{C},\\\ &\
\beta\in\mathbb{C}.\end{aligned}$ (A.12)

##  Appendix B Quasiperiodicity

###  B.1 Twist operator

We define the twist operator $\mathbf{E}_{a}(\phi)$ for the auxiliary space.
Each of the $U_{q}(\mathfrak{sl}_{2})$ representations on the auxiliary space
from Appendix A.2 is expressed in terms of an orthonormal basis $\\{\lvert
d-1\rangle,\cdots,\lvert 1\rangle,\lvert 0\rangle\\}$ with $d$ the dimension
of the representation. Here $d=2\,s+1$ for the unitary spin-$s$ representation
with $s\in\frac{1}{2}\,\mathbb{Z}$, $d=\infty$ for the highest-weight
representation with $s\in\mathbb{C}$, and $d=\ell_{2}$ for the truncation at
root of unity. We consider diagonal twist operator $\mathbf{E}_{a}(\phi)$
given by

$\mathbf{E}_{a}(\phi)=\sum_{n=0}^{d-1}e^{\mathrm{i}\mspace{1.0mu}\phi\mspace{1.0mu}n}\,|n\rangle\langle
n|_{a}.$ (B.1)

In view of our ordering of the basis this yields the twist from (3.9) for
$s=1/2$ ($d=2$).

In particular, the complex spin-$s$ representation yields monodromy matrix
$\mathbf{M}_{s}^{\mathrm{hw}}$

$\mathbf{M}_{s}^{\mathrm{hw}}(u,\phi)=\mathbf{L}_{sN}(u)\cdots\mathbf{L}_{s2}(u)\,\mathbf{L}_{s1}(u)\,\mathbf{E}_{s}^{\mathrm{hw}}(\phi),\qquad\mathbf{E}_{s}^{\mathrm{hw}}(\phi)=\sum_{n=0}^{\infty}e^{\mathrm{i}\mspace{1.0mu}\phi\mspace{1.0mu}n}\,|n\rangle\langle
n|_{s},$ (B.2)

resulting in the transfer matrix
$\mathbf{T}^{\mathrm{hw}}_{s}(u,\phi)=\mathrm{tr}_{s}\,\mathbf{M}_{s}^{\mathrm{hw}}(u,\phi)$.
When $|q|\leq 1$ the diagonal matrix elements of
$\mathbf{M}_{s}^{\mathrm{hw}}$ can be bounded by
$A_{N}\,|q^{N}\,e^{\mathrm{i}\phi}|^{n}$ for some constant $A_{N}$, and so the
trace is convergent if $|e^{\mathrm{i}\phi}|<|q|^{N}$. At this point we do not
know if $\mathbf{T}^{\mathrm{hw}}_{s}$ can be analytically continued outside
this disc of convergence.

The truncation at root of unity $\eta=\mathrm{i}\pi\ell_{1}/\ell_{2}$ likewise
has

$\mathbf{\tilde{M}}_{s}(u,\phi)=\mathbf{L}_{sN}(u)\,\cdots\mathbf{L}_{s2}(u)\,\mathbf{L}_{s1}(u)\,\mathbf{\tilde{E}}_{s}(\phi),\qquad\mathbf{\tilde{E}}_{s}(\phi)=\sum_{n=0}^{\ell_{2}-1}e^{\mathrm{i}\mspace{1.0mu}\phi\mspace{1.0mu}n}\,|n\rangle\langle
n|_{s},$ (B.3)

and transfer matrix
$\mathbf{\tilde{T}}_{s}(u,\phi)=\mathrm{tr}_{s}\,\mathbf{\tilde{M}}_{s}(u,\phi)$.
In this case the trace is well defined for any value of of the twist $\phi$.

###  B.2 Twisted momentum and magnons

Let us compute the first few charges (3.8) generated by the (six-vertex)
transfer matrix with auxiliary space $V_{a}\cong\mathbb{C}^{2}$ of spin
$\frac{1}{2}$ and twist
$\mathbf{E}(\phi)=\mathrm{diag}(e^{\mathrm{i}\mspace{1.0mu}\phi},1)$ as in
(3.9)–(3.10). In the periodic case ($\phi=0$) we get the cyclic (right)
translation operator $\mathbf{G}(0)$,

$\mathbf{T}(\eta/2,0)=\sinh^{N}\\!\eta\
\mathbf{G}(0),\qquad\mathbf{G}(0)=\mathbf{P}_{12\cdots
N}=\mathbf{P}_{12}\,\mathbf{P}_{23}\cdots\mathbf{P}_{N-1,N},$ (B.4)

so the charge $\mathbf{I}^{(1)}=-\mathrm{i}\log\mathbf{G}$ is the usual
momentum operator. In the quasiperiodic case the twist deforms the translation
operator to

$\mathbf{T}(\eta/2,\phi)=\sinh^{N}\\!\eta\
\mathbf{G}(\phi),\qquad\mathbf{G}(\phi)=e^{\mathrm{i}\mspace{1.0mu}\phi\mspace{1.0mu}(\sigma^{z}_{1}+1)/2}\,\mathbf{G}(0)=\mathbf{G}(0)\,e^{\mathrm{i}\mspace{1.0mu}\phi\mspace{1.0mu}(\sigma^{z}_{N}+1)/2}.$
(B.5)

In analogy with the periodic case write its eigenvalue as
$e^{\mathrm{i}\mspace{1.0mu}p}$; we will get back to the ‘twisted momentum’
$p$ soon. The next charge is the XXZ Hamiltonian (2.1) up to some constants:

$\mathbf{I}^{(2)}=-\mathrm{i}\mathbf{T}(\eta/2,\phi)^{-1}\,\mathbf{T}^{\prime}(\eta/2,\phi)=-\frac{2\mathrm{i}}{\sinh\eta}\,\Bigl{(}\mathbf{H}+\frac{N\,\Delta}{2}\Bigr{)},$
(B.6)

where the prime denotes the derivative with respect to the first argument. The
twist spoils homogeneity in the traditional sense: for $\phi\neq 0$ (B.6) does
not commute with (B.4). However, (3.7) guarantees that (B.6) is ‘twisted
homogeneous’ in that it commutes with (B.5). For fun let us show that this
immediately gives the eigenvectors for $M=1$.

The twisted translation (B.5) shows the quasiperiodic boundary conditions by
$\mathbf{G}(\phi)^{N}=\exp[\mathrm{i}\mspace{1.0mu}\phi\mspace{1.0mu}(\mathbf{S}^{z}+N/2)]$.
Since the transfer matrix commutes with $\mathbf{S}^{z}$ the ‘improved’
translation operator
$\mathbf{G}_{\star}(\phi)\coloneqq\exp[-\mathrm{i}\mspace{1.0mu}\phi\mspace{1.0mu}(\mathbf{S}^{z}+N/2)/N]\,\mathbf{G}(\phi)=\mathbf{G}(\phi)\exp[-\mathrm{i}\mspace{1.0mu}\phi\mspace{1.0mu}(\mathbf{S}^{z}+N/2)/N]$
still commutes with the Hamiltonian. The ‘improved momentum’ is
$p_{\star}=p-(1-M/N)\,\phi$. Formally this setting provides a fully
translationally-invariant, and in particular periodic, structure:
$\mathbf{G}_{\star}(\phi)\,\mathbf{H}\,\mathbf{G}_{\star}(\phi)^{-1}=\mathbf{H}$
and $\mathbf{G}_{\star}(\phi)^{N}=\mathbbm{1}$. For $M=1$ it is easy to build
the eigenvectors of $\mathbf{G}_{\star}$ explicitly (see Lemma 1.14 in [90]
for a variant of this construction): the cyclicity of
$\mathbf{G}_{\star}(\phi)$ ensures that with
$\lvert\Omega\rangle=\lvert\uparrow\uparrow\cdots\uparrow\rangle$ the vector

$\sum_{j=1}^{N}e^{\mathrm{i}\mspace{1.0mu}p_{\star}\,j}\,\mathbf{G}_{\star}(\phi)^{1-j}\,\sigma^{-}_{N}\,\lvert\Omega\rangle=e^{\mathrm{i}\mspace{1.0mu}(p_{\star}-\phi)}\sum_{j=1}^{N}e^{-\mathrm{i}\mspace{1.0mu}(p_{\star}-\phi/N)\mspace{1.0mu}j}\,\sigma^{-}_{j}\,\lvert\Omega\rangle$
(B.7)

has $\mathbf{G}_{\star}(\phi)$-eigenvalue $e^{\mathrm{i}\,p_{\star}}$ where
$p_{\star}=2\pi k/N$, $0\leq k\leq N-1$, is quantised. When $\phi$ is real
$\mathbf{G}_{\star}(\phi)$ is unitary so the twisted magnons (B.7) are
linearly independent and form a basis for the $M=1$ sector, so they are also
eigenvectors of (B.5) and (B.6). In terms of $\widetilde{p}\coloneqq(2\pi
k-\phi)/N$ the (twisted) momentum is $p=\widetilde{p}+\phi/N$, the dispersion
is $\varepsilon=\cos(\widetilde{p}\mspace{2.0mu})-\Delta$, and the right-hand
side of (B.7) looks just like an ordinary magnon,
$\sum_{j}e^{-\mathrm{i}\mspace{1.0mu}\widetilde{p}\mspace{1.0mu}j}\,\sigma^{-}_{j}\,\lvert\Omega\rangle$.
(The sign in the exponential is because we work with the _right_ translation
operator.)

Let us finally make contact with the algebraic Bethe ansatz: for $M=1$ (3.11)
gives

$\lvert\\{v\\}\rangle=\mathbf{B}(v)\,\lvert\Omega\rangle=\frac{\sinh\eta}{\sinh(v-\eta/2)}\,\sinh^{N}(v+\eta/2)\,\sum_{j=1}^{N}e^{-\mathrm{i}\mspace{1.0mu}\widetilde{p}j}\,\sigma^{-}_{j}\,\lvert\Omega\rangle,$
(B.8)

where the quasimomentum

$\widetilde{p}=\mathrm{i}\log\frac{\sinh(v-\eta/2)}{\sinh(v+\eta/2)}=\frac{2\pi
k-\phi}{N}$ (B.9)

solves the Bethe equation
$e^{\mathrm{i}\mspace{1.0mu}N\mspace{1.0mu}\widetilde{p}}=e^{-\mathrm{i}\mspace{1.0mu}\phi}$.
The (twisted) momentum and energy given above match the result (2.8)–(2.9)
obtained from (3.12) using (B.5) and (B.6). The difference between
quasimomentum and (twisted) momentum for $M=1$ can be avoided by taking
$\widetilde{\mathbf{G}}\coloneqq
e^{-\mathrm{i}\mspace{1.0mu}\phi}\,\mathbf{G}$ to be the twisted translation.

##  Appendix C Bethe roots

###  C.1 Numerical recipe for finding Bethe roots

Here we review a numerical recipe, called McCoy’s method, for solving the
functional TQ relation. It appears to have been published first in [91]. We
follow the description of Haldane [92]. A similar method was also described by
Baxter [20].

The idea is that rather than solve the (coupled, nonlinear) Bethe equations
one can obtain the Bethe roots by solving a few sets of linear equations. This
is done by exploiting the known form of the eigenvalues of the (fundamental)
transfer matrix and the Q operator, whose zeroes are the Bethe roots (see
Section 5.5). The recipe goes as follows:

  * i.

Construct the transfer matrix $\mathbf{T}_{1/2}(u)$ at some $u\in\mathbb{C}$
(almost any value will do), and numerically diagonalise it; this is much more
efficient than diagonalisation when $u$ is kept free. One obtains $2^{N}$
eigenstates that span the physical Hilbert space. From the Bethe ansatz we
know that the eigenvectors are independent of the spectral parameter, so these
will be eigenvectors for $\mathbf{T}_{1/2}(u)$ for any $u$.

  * ii.

The eigenvalues, however, depend on $u$. Pick one of the eigenvectors. Its
eigenvalue is found by acting with $\mathbf{T}_{1/2}(u)$. This may again be
done numerically by writing the eigenvalue as a Laurent polynomial in
$t=e^{u}$ of order $N$,

$T_{1/2}(t)=\text{cst}_{T}\prod_{n=1}^{N}(\tau_{n}^{-1}\,t-\tau_{n}\,t^{-1})$
(C.1)

with zeroes $\tau_{n}$ that can be fixed by acting with
$\mathbf{T}_{1/2}(u_{n})$ for $N$ distinct values $u_{n}\in\mathbb{C}$.

  * iii.

The corresponding Bethe roots are the zeroes of the Q operator, found by
solving the functional TQ relation (5.19), i.e.

$T_{1/2}(u,\phi)\,Q(u,\phi)=T_{0}(u-\eta/2)\,Q(u+\eta,\phi)+e^{\mathrm{i}\phi}\,T_{0}(u+\eta/2)\,Q(u-\eta,\phi).$
(C.2)

Here $T_{0}(u)=\sinh^{N}(u)$ and the eigenvalues are of the form

$Q(t)=\text{cst}_{Q}\prod_{m=1}^{M}(t_{m}^{-1}\,t-t_{m}\,t^{-1}),$ (C.3)

where $M$ is the number of down spins of the eigenvector under consideration.
The zeroes $t_{m}$ can once more be found numerically by taking $t=e^{u}$
equal to the zeroes $\tau_{n}$ of $T_{1/2}$ and solving the linear problem.

The zeroes give the Bethe roots $u_{m}=\log t_{m}$. One needs to be careful to
interpret the result correctly in the presence of Bethe roots at infinity:
$u_{m}=\pm\infty$ corresponds to $t_{m}\in\\{0,\infty\\}$ so the corresponding
factor in (C.3) collapses to $t^{\pm 1}$, yielding (5.21):

$Q(t)=\text{cst}_{Q}\times
t^{n_{-\infty}-n_{+\infty}}\prod_{n=1}^{M-n_{+\infty}-n_{-\infty}}(t_{n}^{-1}\,t-t_{n}\,t^{-1}).$
(C.4)

The numerical recipe works very well for the XXZ model away from root of
unity, as well as for the XXX model ($\Delta=\pm 1$). However, one cannot find
all the Bethe roots for the XXZ spin chain at root of unity, precisely due to
the existence of degenerate eigenstates of the transfer matrix
$\mathbf{T}_{1/2}$. In that case, our construction for the Q operator still
works and gives the correct results, sometimes even analytically.

###  C.2 Relation between Bethe roots for anisotropies $\Delta$ and $-\Delta$

In the gapless regime ($-1<\Delta<1$) there is a simple relation between Bethe
roots of all the physical solutions at anisotropy $\Delta$ and those at
anisotropy $-\Delta$, even though the corresponding eigenstates are different
since the $\mathbf{B}$-operators in the algebraic Bethe ansatz differ. We will
denote the parameters of the second spin chain by primes:
$\Delta^{\prime}=-\Delta$ and

$\eta=\mathrm{arccosh}(\Delta)\in\mathrm{i}\,\mathbb{R},\qquad\eta^{\prime}=\mathrm{arccosh}(\Delta^{\prime})=\mathrm{i}\,\pi-\eta.$
(C.5)

Consider any solution to Bethe equation (2.7) with $\eta$, system size $N$ and
twist $\phi$: assume that the Bethe roots $\\{u_{m}\\}_{m=1}^{M}$ obey

$\biggl{(}\frac{\sinh(u_{m}+\eta/2)}{\sinh(u_{m}-\eta/2)}\biggr{)}^{\\!N}\prod_{n(\neq
m)}^{M}\frac{\sinh(u_{m}-u_{n}-\eta)}{\sinh(u_{m}-u_{n}+\eta)}=e^{-\mathrm{i}\phi}.$
(C.6)

Then define $\\{u^{\prime}_{m}\\}_{m=1}^{M}$ by

$u^{\prime}_{m}=-u_{m}-\frac{\mathrm{i}\pi}{2},\qquad 1\leq m\leq M.$ (C.7)

In terms of these parameters (C.6) reads

$\biggl{(}\frac{\sinh(-u^{\prime}_{m}+\mathrm{i}\pi/2+\eta/2)}{\sinh(-u^{\prime}_{m}+\mathrm{i}\pi/2-\eta/2)}\biggr{)}^{\\!N}\prod_{n(\neq
m)}^{M}\frac{\sinh(-u^{\prime}_{m}+u^{\prime}_{n}-\eta)}{\sinh(-u^{\prime}_{m}+u^{\prime}_{n}+\eta)}=e^{-\mathrm{i}\phi}$
(C.8)

This precisely of the form (C.6) with $\eta^{\prime}=\mathrm{i}\pi-\eta$ and
twist $\phi^{\prime}$ chosen such that
$e^{-\mathrm{i}\phi^{\prime}}=(-1)^{N}\,e^{-\mathrm{i}\phi}$. This shows that
for each solution $\\{u_{m}\\}_{m=1}^{M}$ at anisotropy $\Delta$ there is a
corresponding solution $\\{u^{\prime}_{m}\\}_{m=1}^{M}$ at
$\Delta^{\prime}=-\Delta$ provided the twist is modified to

$\phi^{\prime}=\begin{cases}\phi&N\ \text{even},\\\ \phi+\pi\qquad&N\
\text{odd}.\end{cases}$ (C.9)

The two eigenstates are related by the unitary gauge transformation

$\mathbf{U}=\exp\biggl{(}\mathrm{i}\pi\sum_{j=1}^{N}\frac{j}{2}\,\sigma^{z}_{j}\biggr{)}=e^{\mathrm{i}\pi\,L(L+1)/4}\prod_{j=1}^{\lceil
N/2\rceil}\\!\\!\sigma^{z}_{2j-1}.$ (C.10)

(Removing the prefactor in the expression on the right yields a simpler
transformation that also does the job and is its own inverse.) It is easy to
check that this transformation changes the sign of $\Delta$ in the Hamiltonian
(2.1):

$\mathbf{U}\,\mathbf{H}(\Delta,\phi)\,\mathbf{U}^{-1}=-\mathbf{H}(-\Delta,\phi^{\prime}).$
(C.11)

Moreover, the eigenstates are related by

$\lvert\\{u^{\prime}_{m}\\}\rangle\propto\mathbf{U}\,\lvert\\{u_{m}\\}\rangle.$
(C.12)

###  C.3 Relation between eigenstates with opposite twist

Recall from Section 5.5 that an $M$-particle Bethe state
$\lvert\\{u_{m}\\}_{m=1}^{M}\rangle$ for the XXZ model obeys

$\displaystyle\mathbf{Q}(u,\phi)\,\lvert\\{u_{m}\\}_{m=1}^{M}\rangle$
$\displaystyle=Q(u)\,\lvert\\{u_{m}\\}_{m=1}^{M}\rangle,$ (C.13)
$\displaystyle\mathbf{P}(u,\phi)\,\lvert\\{u_{m}\\}_{m=1}^{M}\rangle$
$\displaystyle=P(u)\,\lvert\\{u_{m}\\}_{m=1}^{M}\rangle$

with eigenvalue $Q(u)$ and $P(u)$ of the form

$\begin{aligned}
&Q(u)=\mathrm{cst}_{Q}\times\prod_{m=1}^{M}\left(t_{m}^{-1}\,t-t_{m}\,t^{-1}\right),\qquad&t_{m}&=e^{u_{m}},\\\
&P(u)=\mathrm{cst}_{P}\times\prod_{n=1}^{N-M}\left(\tau_{n}^{-1}\,t-\tau_{n}\,t^{-1}\right),\quad&\tau_{n}&=e^{v_{n}},\end{aligned}\mspace{-64.0mu}t=e^{u},$
(C.14)

and where the zeroes $u_{m}=\log t_{m}$ of $Q$ are the Bethe roots. Let us
show that the $v_{n}$ can similarly be interpreted as the Bethe roots of the
spin-flipped counterpart of $\lvert\\{u_{m}\\}_{M=1}^{M}\rangle$ ‘beyond the
equator’ with opposite twist.

Consider the transfer matrices $\mathbf{T}_{s}(u)$ with
$s\in\frac{1}{2}\mathbb{Z}_{\geq 0}$. Under global spin inversion these
operators simply behave as

$\prod_{j=1}^{N}\\!\sigma^{x}_{j}\
\mathbf{T}_{s}(u,\phi)\prod_{j=1}^{N}\\!\sigma^{x}_{j}=e^{2s\mspace{1.0mu}\mathrm{i}\phi}\,\mathbf{T}_{s}(u,-\phi).$
(C.15)

To see this note that for the unitary spin-$s$ representation (A.8) the Lax
operator (3.2) is invariant under total spin reversal, which acts by
conjugation by $\sigma^{x}_{j}$ in the physical space and by the antidiagonal
matrix $\mathbf{U}\coloneqq\sum_{n=0}^{2s}\,|2s-n\rangle\langle n|$ in the
auxiliary space. Spin reversal in the physical space is thus equivalent to
spin reversal in the auxiliary space. This property is inherited by the
monodromy matrix. In the periodic case it follows that spin flip in the
physical space does not affect the transfer matrix, as the trace is invariant
under reordering the basis. In the twisted case we only have to correct for
the twist (B.1), which is reversed:
$\mathbf{U}\,\mathbf{E}(\phi)\,\mathbf{U}^{-1}=e^{2s\mathrm{i}\phi}\,\mathbf{E}(-\phi)$.
This proves (C.15).

Now consider the TQ equation (5.10) with $\phi$ inverted to $-\phi$. By
conjugating both sides with the global spin-flip operator
$\prod_{j=1}^{N}\\!\sigma^{x}_{j}$, using (C.15) and multiplying both sides by
$e^{\mathrm{i}\phi}$ we see that
$\bar{\mathbf{Q}}(u,\phi)\coloneqq\prod_{j=1}^{N}\\!\sigma^{x}_{j}\
\mathbf{Q}(u,-\phi)\prod_{j=1}^{N}\\!\sigma^{x}_{j}$ precisely obeys the TP
equation (5.11). Moreover, comparing the eigenvalues in (C.14) shows that the
eigenvalues of $\bar{\mathbf{Q}}(u,\phi)$ on $M$-particle Bethe vectors are
trigonometric polynomials of degree $N-M$, just as for the P operator. It
follows that the eigenvalues of $\bar{\mathbf{Q}}(u,\phi)$ are proportional to
those of the P operator; in particular they have the same zeroes:

$\bar{Q}(u,\phi)\propto
P(u,\phi)\propto\prod_{n=1}^{N-M}\left(\tau_{n}^{-1}\,t-\tau_{n}\,t^{-1}\right).$
(C.16)

Since $\bar{\mathbf{Q}}(u,\phi)$ and $\mathbf{Q}(u,-\phi)$ have the same
characteristic polynomial this shows that the $M$-particle eigenvalues of the
P operator are the same as the $(N-M)$-particle eigenvalues of
$\mathbf{Q}(u,-\phi)$. But we know that the latter can be interpreted as the
Bethe roots. Therefore the zeroes of the P operator can be interpreted as the
Bethe roots of the spin-reversed Bethe vector beyond the equator with opposite
twist.

Finally notice that the Bethe vectors (3.11) are constructed using the
B-operator, which is independent of the twist, see (3.9). This implies that
the result of reversing all spins on an off-shell Bethe vector (for the
Hamiltonian with original twist $\phi$) is

$\prod_{l=1}^{N}\\!\sigma^{x}_{l}\,\lvert\\{u_{m}\\}_{m=1}^{M}\rangle=\lvert\\{v_{n}\\}_{n=1}^{N-M}\rangle,$
(C.17)

where the Bethe roots $v_{n}$ beyond the equator are related to the zeroes of
eigenvalues of the P operator on $\lvert\\{u_{m}\\}_{m=1}^{M}\rangle$. (The
dependence of the on-shell Bethe vectors on the twist enters through the Bethe
equations.)

##  Appendix D Alternative proof of Eq. (6.13)

In this appendix we give another proof of (6.13), i.e.

$\mathbf{\tilde{T}}_{s}(u,\phi)-e^{\mathrm{i}(2s+1)\phi}\,\mathbf{\tilde{T}}_{-s-1}(u,\phi)=\bigl{(}1-\varepsilon^{N}e^{\mathrm{i}\ell_{2}\phi}\bigr{)}\mathbf{T}_{s}(u,\phi).$
(D.1)

Consider $2s\in\mathbb{Z}_{\geq 0}$ with $s<\frac{\ell_{2}}{2}-1$. Proceeding
as in Section 5.1 we obtain a decomposition like in (5.5):

$\mathbf{\tilde{T}}_{s}(u,\phi)=\mathbf{T}_{s}(u,\phi)+e^{\mathrm{i}(2s+1)\phi}\,\mathbf{T}^{*}_{s}(u,\phi).$
(D.2)

Here $\mathbf{T}^{\ast}_{s}$ denotes the result of restricting the trace over
the auxiliary space to the subspace $V_{a}^{*}$ spanned by $\lvert n\rangle$
with $2s+1\leq n\leq\ell_{2}-1$.

The restricted Lax matrix $\mathbf{L}^{\ast}_{sj}(u)$ coincides with the
transpose (in both auxiliary and physical space)
$\mathbf{L}_{-s-1,j}(u)^{\mathrm{T}}$ of the Lax operator
$\mathbf{L}_{-s-1,j}(u)$, whose auxiliary space has dimension $\ell_{2}-2s-1$.
Therefore $\mathbf{T}^{*}_{s}(u,\phi)=\mathbf{T}_{-s-1}(u,\phi)$, and we get

$\mathbf{\tilde{T}}_{s}(u,\phi)=\mathbf{T}_{s}(u,\phi)+e^{\mathrm{i}(2s+1)\phi}\,\mathbf{T}_{-s-1}(u,\phi).$
(D.3)

Now consider the transfer matrix $\mathbf{\tilde{T}}^{*}_{s}$ obtained like
$\mathbf{\tilde{T}}_{s}$ truncating the trace to the basis of $\tilde{V}_{a}$
translated (raised) by $2s+1$, i.e. to $\lvert n+2s+1\rangle$ with $0\leq
n\leq\ell_{2}$. Again, since $2s+1<\ell_{2}$ and
$\mathbf{S}^{+}\,\lvert\ell_{2}\rangle=0$, the trace can be split into two
transfer matrices. The first one coincides with $\mathbf{T}^{\ast}_{s}$, while
the second one is obtained as $\mathbf{T}_{s}$ in the basis shifted by
$\ell_{2}$, i.e. $\lvert n+\ell_{2}\rangle$ with $0\leq n\leq 2s+1$, and
therefore picks up a factor $\varepsilon^{N}$ compared to $\,\mathbf{T}_{s}$
times the twist contribution. Proceeding as before, one can verify that the
second transfer matrix in the decomposition of $\mathbf{\tilde{T}}^{*}_{s}$
and $\mathbf{\tilde{T}}_{-s-1}$ derive from similar Lax matrices and are both
equal to $e^{\mathrm{i}(\ell_{2}-2s-1)\phi}\,\varepsilon^{N}\,\mathbf{T}_{s}$.
Thus one has

$\mathbf{\tilde{T}}^{*}_{s}(u,\phi)=\mathbf{\tilde{T}}_{-s-1}(u,\phi)=\mathbf{T}_{-s-1}(u,\phi)+\varepsilon^{N}e^{\mathrm{i}(\ell_{2}-2s-1)\phi}\,\mathbf{T}_{s}\left(u,\phi\right).$
(D.4)

Subtracting $e^{\mathrm{i}(\ell_{2}-2s-1)\phi}$ times (D.3) from (D.4) we
obtain (6.13).

##  Appendix E Deforming FM strings

###  E.1 Tuning a small twist

We study the Bethe roots for states that include FM strings at $\phi=0$
numerically in the presence of a small twist. The results illustrate that FM
strings are formed by combining the last two string types from the string
hypothesis.

Consider the case $N=6$ and $\Delta=\frac{1}{2}$, like the example in Section
2.3. At zero twist $\phi=0$ the primitive state
$\lvert\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\rangle$ has, by the
results of Section 7.5, corresponding state beyond the equator with Bethe
roots (2.21). The descendant tower further includes the states

$\lvert\\{\alpha_{1}^{\textsc{fm}}\\}\rangle=\lvert\\{u_{m}\\}_{m=1}^{3}\rangle,\qquad\lvert\\{\alpha_{2}^{\textsc{fm}}\\}\rangle=\lvert\\{v_{m}\\}_{m=1}^{3}\rangle,$
(E.1)

with energy eigenvalues $E=0$.

Now we turn on a very small twist and study what happens to the corresponding
states by computing the eigenstates with $M=3$ and eigenvalues for the
transfer matrices $\mathbf{T}_{s}$ that are very close to those of (E.1). The
resulting Bethe roots are collected in Tables 2–3, where we have expressed the
analytic initial values (2.21) numerically to facilitate the comparison. We
observe that the Bethe roots are string deviations of two $(2,+)$ strings,
namely $u_{1}$, $u_{2}$ and $v_{1}$, $v_{2}$, with string deviations that
decrease as $\phi$ approaches zero. In the limit $\phi\to 0$ an FM string
forms when two strings, here of type $(2,+)$ and $(1,-)$ respectively, have
coinciding real part of the string centres. This motivates our conjecture in
Section 10.2 about the relation between the last two string types of the
string hypothesis and the string centres of FM strings, even for systems with
finite system sizes.

$\phi$ | $u_{1}$ | $u_{2}$ | $u_{3}$
---|---|---|---
$0$ | $-0.49887047-0.52359877\,\mathrm{i}$ | $-0.49887047+0.52359877\,\mathrm{i}$ | $-0.49887047+1.5707963\,\mathrm{i}$
$10^{-5}\pi$ | $-0.49886307-0.52359861\,\mathrm{i}$ | $-0.49886307+0.52359861\,\mathrm{i}$ | $-0.49888095+1.5707963\,\mathrm{i}$
$10^{-4}\pi$ | $-0.49879646-0.52359713\,\mathrm{i}$ | $-0.49879646+0.52359713\,\mathrm{i}$ | $-0.49897526+1.5707963\,\mathrm{i}$
$10^{-3}\pi$ | $-0.49813029-0.52358228\,\mathrm{i}$ | $-0.49813029+0.52358228\,\mathrm{i}$ | $-0.49991897+1.5707963\,\mathrm{i}$
$10^{-2}\pi$ | $-0.49147893-0.52344350\,\mathrm{i}$ | $-0.49147893+0.52344350\,\mathrm{i}$ | $-0.50942065+1.5707963\,\mathrm{i}$

Table 2: Numerical results for the Bethe roots of the Bethe vector $\lvert\\{u_{m}\\}_{m=1}^{3}\rangle$ deforming $\lvert\\{\alpha_{1}^{\textsc{fm}}\\}\rangle$ as a small twist $\phi$ is turned on. $\phi$ | $v_{1}$ | $v_{2}$ | $v_{3}$
---|---|---|---
$0$ | $0.49887047-0.52359877\,\mathrm{i}$ | $0.49887047+0.52359877\,\mathrm{i}$ | $0.49887047+1.5707963\,\mathrm{i}$
$10^{-5}\pi$ | $0.49887788-0.52359894\,\mathrm{i}$ | $0.49887788+0.52359894\,\mathrm{i}$ | $0.49886000+1.5707963\,\mathrm{i}$
$10^{-4}\pi$ | $0.49894452-0.52360045\,\mathrm{i}$ | $0.49894452+0.52360045\,\mathrm{i}$ | $0.49876570+1.5707963\,\mathrm{i}$
$10^{-3}\pi$ | $0.49961090-0.52361550\,\mathrm{i}$ | $0.49961090+0.52361550\,\mathrm{i}$ | $0.49782342+1.5707963\,\mathrm{i}$
$10^{-2}\pi$ | $0.50628580-0.52377622\,\mathrm{i}$ | $0.50628580+0.52377622\,\mathrm{i}$ | $0.48846413+1.5707963\,\mathrm{i}$
Table 3: Numerical results for the Bethe roots of the Bethe vector
$\lvert\\{v_{m}\\}_{m=1}^{3}\rangle$ deforming
$\lvert\\{\alpha_{2}^{\textsc{fm}}\\}\rangle$ when introducing a small twist
$\phi$.

###  E.2 Tuning the anisotropy

Next we study the behaviour of Bethe roots for states including FM strings at
$\phi=0$ when $\eta$ is slightly deformed. We give two examples, both with
$\Delta\approx 1/2$.

First we revisit the example with $N=6$ from Section 2.3 and Appendix E.1.
When $\Delta=1/2$ the states (E.1) with Bethe roots (2.21), i.e. numerical
values given by the first rows of Tables 2–3, are degenerate for the
fundamental transfer matrix, with eigenvalue (recall that $t=e^{u}$)

$\displaystyle T_{1/2}(t)={}$
$\displaystyle\frac{1}{32}\,t^{6}-\frac{3}{32}\,t^{4}-\frac{15}{64}\,t^{2}+\frac{5}{8}-\frac{15}{64}\,t^{-2}-\frac{3}{32}\,t^{-4}+\frac{1}{32}\,t^{-6}$
(E.2) $\displaystyle={}$ $\displaystyle
0.03125\,t^{6}-0.093750000\,t^{4}-0.23437500\,t^{2}+0.62500000$
$\displaystyle-0.23437500\,t^{-2}-0.093750000\,t^{-4}+0.03125\,t^{-6}\,,$

where we give the numerical values for later convenience. In particular their
energy is $E=0$. Now we vary the anisotropy a little, taking
$\eta=\mathrm{i}\,(\frac{\pi}{3}\pm\epsilon)$ for small $\epsilon>0$, to move
away from root of unity. The numerically obtained values of the Bethe roots of
the two states at $M=3$ with energy $E\approx 0$ are given in Table 4,
including the values that one would obtain as limits at $\epsilon=0$. (These
values obey Eq. (1.11) from [31].) Note that the limiting values at root of
unity are rather different from the Bethe roots (2.21) obtained from the
zeroes of the Q operator. Here we use the notation
$\\{u^{\prime}_{m}\\}_{m},\\{u^{\prime\prime}_{m}\\}_{m}$ rather than
$\\{u_{m}\\}_{m},\\{v_{m}\\}_{m}$ because we cannot distinguish between the
two FM strings arising as the zeroes of the Q function at $\epsilon=0$, i.e.
we cannot say which of
$\\{u^{\prime}_{m}\\}_{m},\\{u^{\prime\prime}_{m}\\}_{m}$ jump to which of
$\\{u_{m}\\}_{m},\\{v_{m}\\}_{m}$ in the limit.

$\eta-\mathrm{i}\pi/3$ |  | $u^{\prime}_{1}$ | $u^{\prime}_{2}$ | $u^{\prime}_{3}$ |  | $u^{\prime\prime}_{1}$ | $u^{\prime\prime}_{2}$ | $u^{\prime\prime}_{3}$
---|---|---|---|---|---|---|---|---
$-\mathrm{i}\,10^{-2}$ |  | $0$ | $-1.03763873\,\mathrm{i}$ | $1.03763873\,\mathrm{i}$ |  | $1.5707963\,\mathrm{i}$ | $-0.51859878\,\mathrm{i}$ | $0.51859878\,\mathrm{i}$
$-\mathrm{i}\,10^{-4}$ |  | $0$ | $-1.04710210\,\mathrm{i}$ | $1.04710210\,\mathrm{i}$ |  | $1.5707963\,\mathrm{i}$ | $-0.52354878\,\mathrm{i}$ | $0.52354878\,\mathrm{i}$
$0$ |  | $0\mathrlap{\,?}$ | $-1.04719755\,\mathrm{i}\mathrlap{\,?}$ | $1.04719755\,\mathrm{i}\mathrlap{\,?}$ |  | $1.5707963\,\mathrm{i}\mathrlap{\,?}$ | $-0.52359878\,\mathrm{i}\mathrlap{\,?}$ | $0.52359878\,\mathrm{i}\mathrlap{\,?}$
$\hphantom{+}\mathrm{i}\,10^{-4}$ |  | $0$ | $-1.04729300\,\mathrm{i}$ | $1.04729300\,\mathrm{i}$ |  | $1.5707963\,\mathrm{i}$ | $-0.52364878\,\mathrm{i}$ | $0.52364878\,\mathrm{i}$
$\hphantom{+}\mathrm{i}\,10^{-2}$ |  | $0$ | $-1.05672931\,\mathrm{i}$ | $1.05672931\,\mathrm{i}$ |  | $1.5707963\,\mathrm{i}$ | $-0.52859878\,\mathrm{i}$ | $0.52859878\,\mathrm{i}$

Table 4: The numerical Bethe roots of the two states at $M=3$ with energy
close to zero as $\eta$ changes away from $\mathrm{i}\pi/3$, including the
values $0,\mp\mathrm{i}\pi/3$ and $\mathrm{i}\pi/2,\mp\mathrm{i}\pi/6$
(indicated with a question mark) that one would extrapolate as $\phi\to 0$.

The corresponding states

$\lim_{\epsilon\to
0}\,\lvert\\{u^{\prime}_{m}\\}_{m=1}^{3}\rangle,\qquad\lim_{\epsilon\to
0}\,\lvert\\{u^{\prime\prime}_{m}\\}_{m=1}^{3}\rangle,$ (E.3)

can be constructed via the FM string creation operators of [32] (see Section
(8.5)) to produce eigenvectors of $\mathbf{H}$ with $E=0$. However, these
vectors are not eigenvectors of the two-parameter transfer matrix or the Q
operator at $\Delta=1/2$. We find that nontrivial linear combinations of them
(namely: a multiple of their sum and difference) equal
$\lvert\\{u_{m}\\}_{m=1}^{3}\rangle$ and $\lvert\\{v_{m}\\}_{m=1}^{3}\rangle$,
which are eigenvectors of Q operator, respectively.

Despite the jump in the Bethe roots $\\{u_{m}\\}_{m}$ and $\\{v_{m}\\}_{m}$ as
$\epsilon\to 0$ we stress that physical quantities are continuous. For
instance, fundamental transfer matrix has eigenvalue

$\displaystyle\begin{aligned}
T_{1/2}(t)={}&0.03125\,t^{6}-0.093758120\,t^{4}-0.23428568\,t^{2}+0.62482949\\\
&-0.23428568\,t^{-2}-0.093758120\,t^{-4}+0.03125\,t^{-6}\end{aligned}\quad$
$\displaystyle\epsilon=-10^{-4},$ (E.4) $\displaystyle\begin{aligned}
T_{1/2}(t)={}&0.03125\,t^{6}-0.093758120\,t^{4}-0.23428568\,t^{2}+0.62482949\\\
&-0.23428568\,t^{-2}-0.093758120\,t^{-4}+0.03125\,t^{-6}\end{aligned}\quad$
$\displaystyle\epsilon=+10^{-4},$

for $\lvert\\{u^{\prime}_{m}\\}_{m=1}^{3}\rangle$ and

$\displaystyle\begin{aligned}
T_{1/2}(t)={}&0.03125\,t^{6}-0.093750738\,t^{4}-0.23430930\,t^{2}+0.62486049\\\
&-0.23430930\,t^{-2}-0.093750738\,t^{-4}+0.03125\,t^{-6}\end{aligned}\quad$
$\displaystyle\epsilon=-10^{-4},$ (E.5) $\displaystyle\begin{aligned}
T_{1/2}(t)={}&0.03125\,t^{6}-0.093766977\,t^{4}-0.23437426\,t^{2}+0.62505535\\\
&-0.23437426\,t^{-2}-0.093766977\,t^{-4}+0.03125\,t^{-6}\end{aligned}\quad$
$\displaystyle\epsilon=+10^{-4},$

for $\lvert\\{u^{\prime\prime}_{m}\\}_{m=1}^{3}\rangle$. These polynomials are
very close to (E.2).

The second example is $N=10$. There is an eigenstate
$\lvert\\{u_{m}\\}_{m=1}^{4}\rangle$ with energy eigenvalue $E=1/2$ when
$\Delta=\frac{1}{2}$. Its Bethe roots are

$\begin{split}&u_{1}=\frac{\mathrm{i}\pi}{2},\,\\\
&u_{2}=-0.56101744-\frac{\mathrm{i}\pi}{6},\quad
u_{3}=-0.56101744+\frac{\mathrm{i}\pi}{6},\quad
u_{4}=-0.56101744+\frac{\mathrm{i}\pi}{2}.\end{split}$ (E.6)

Here $u_{2}$, $u_{3}$, $u_{4}$ form an FM string. The numerically obtained
eigenvalue for $\mathbf{T}_{1/2}$ is

$\begin{split}\mathbf{T}_{1/2}(u,0)\,\lvert\\{u_{m}\\}_{m=1}^{4}\rangle=\big{(}\\!&\,0.0009765625\,t^{10}-0.022460937\,t^{8}+0.076171875\,t^{6}\\\
&+0.084960938\,t^{4}-0.72949219\,t^{2}+1.1806641\\\
&-0.72949219\,t^{-2}+0.084960938\,t^{-4}+0.076171875\,t^{-6}\\\
&-0.022460937\,t^{-8}+0.0009765625\,t^{-10}\big{)}\,\lvert\\{u_{m}\\}_{m=1}^{4}\rangle,\end{split}$
(E.7)

where $t=e^{u}$.

The behaviour of the Bethe roots under a small change of $\eta$ is given in
Table 5. We see that the Bethe roots change drastically, jumping to form a
2-string and two 1-strings with odd parity. The same is true for the
eigenvalues of the Q operator. On the other hand, one can check that the
eigenvalues of $\mathbf{T}_{s}$ change smoothly. For instance, the eigenvalue
of $\mathbf{T}_{1/2}$ for the corresponding state at
$\eta=\mathrm{i}(\pi/3+10^{-5})$ is very close to (E.7):

$\begin{split}\mathbf{T}_{1/2}(u,0)\,\lvert\\{u_{m}\\}_{m=1}^{4}\rangle=\big{(}\\!&\,0.00097654559\,t^{10}-0.022460999\,t^{8}+0.076172376\,t^{6}\\\
&+0.084963582\,t^{4}-0.72950736\,t^{2}+1.18068832\\\
&-0.72950736\,t^{-2}+0.084963582\,t^{-4}+0.076172376\,t^{-6}\\\
&-0.022460999\,t^{-8}+0.00097654559\,t^{-10}\big{)}\,\lvert\\{u_{m}\\}_{m=1}^{4}\rangle.\end{split}$
(E.8)

$\eta-\mathrm{i}\pi/3$ | $u_{1}$ | $u_{2}$ | $u_{3}$ | $u_{4}$
---|---|---|---|---
$0$ | $1.5708\,\mathrm{i}$ | $-0.56102+1.5708\,\mathrm{i}$ | $-0.56102-0.52360\,\mathrm{i}$ | $-0.56102+0.52360\,\mathrm{i}$
$\mathrm{i}\,10^{-6}$ | $-4.9936\times 10^{-4}+1.5708\,\mathrm{i}$ | $4.9936\times 10^{-4}+1.5708\,\mathrm{i}$ | $2.6516\times 10^{-5}-0.52360\,\mathrm{i}$ | $2.6516\times 10^{-5}+0.52360\,\mathrm{i}$
$\mathrm{i}\,10^{-5}$ | $-1.5382\times 10^{-3}+1.5708\,\mathrm{i}$ | $1.5382\times 10^{-3}+1.5708\,\mathrm{i}$ | $2.9364\times 10^{-6}-0.52360\,\mathrm{i}$ | $2.9364\times 10^{-6}+0.52360\,\mathrm{i}$
$\mathrm{i}\,10^{-4}$ | $-4.8592\times 10^{-3}+1.5708\,\mathrm{i}$ | $4.8592\times 10^{-3}+1.5708\,\mathrm{i}$ | $2.5441\times 10^{-7}-0.52360\,\mathrm{i}$ | $2.5441\times 10^{-7}+0.52360\,\mathrm{i}$
$\mathrm{i}\,10^{-3}$ | $-0.015342+1.5708\,\mathrm{i}$ | $0.015342+1.5708\,\mathrm{i}$ | $3.2550\times 10^{-8}-0.52360\,\mathrm{i}$ | $3.2550\times 10^{-8}+0.52360\,\mathrm{i}$

Table 5: The numerical Bethe roots of $\lvert\\{u_{m}\\}_{m=1}^{4}\rangle$ as
$\eta$ changes away from $\mathrm{i}\pi/3$.

##  Appendix F Examples of last two string types of TBA at non-principal root
of unity

We present several examples on the last two string types of the TBA at root of
unity using Takahashi’s notation [78, 79] at non-principal root of unity to
illustrate our conjecture (10.8) for the string centre of FM strings.

We start with $\eta=\frac{2\mathrm{i}\pi}{3}$, which is a non-principal root
of unity. The allowed string types are $(1,-)$, $\underline{(2,+)}$,
$\underline{(1,+)}$. Here we have underlined the last two string types, which
are of the form

$\displaystyle\alpha_{1}$
$\displaystyle=\alpha-\frac{\mathrm{i}\pi}{3},\quad\alpha_{2}=\alpha+\frac{\mathrm{i}\pi}{3},\qquad$
$\displaystyle\alpha\in\mathbb{R},$ (F.1) $\displaystyle\alpha_{3}$
$\displaystyle=\alpha^{\prime},$ $\displaystyle\alpha^{\prime}\in\mathbb{R}.$

According to the conjecture (10.8) the FM string with
$\eta=\frac{2\mathrm{i}\pi}{3}$ should be expressed as

$\alpha^{\prime}_{1}=\alpha^{\textsc{fm}}+\frac{\mathrm{i}\pi}{3},\quad\alpha^{\prime}_{2}=\alpha^{\textsc{fm}},\quad\alpha^{\prime}_{3}=\alpha^{\textsc{fm}}-\frac{\mathrm{i}\pi}{3},\qquad\alpha^{\textsc{fm}}\in\mathbb{R}.$
(F.2)

Clearly, (F.1) and (F.2) describe the same FM string when the real parts of
the string centres coincide. This results are confirmed by all the examples in
Section 8 with finite-size calculations.

For another example with non-principal root of unity we consider
$\eta=\frac{2\mathrm{i}\pi}{5}$. Using the method in Chapter 9.2 of [79], we
obtain the allowed string types which are $(1,+)$, $(1,-)$,
$\underline{(3,+)}$, $\underline{(2,+)}$, where we again underline the last
two string types, given by

$\displaystyle\alpha_{1}=\alpha-\frac{2\mathrm{i}\pi}{5},\quad\alpha_{2}=\alpha,\quad\alpha_{3}=\alpha+\frac{2\mathrm{i}\pi}{5},\qquad$
$\displaystyle\alpha\in\mathbb{R},$ (F.3)
$\displaystyle\alpha_{4}=\alpha^{\prime}-\frac{\mathrm{i}\pi}{5},\quad\alpha_{5}=\alpha^{\prime}+\frac{\mathrm{i}\pi}{5},,\qquad$
$\displaystyle\alpha^{\prime}\in\mathbb{R}.$

From the conjecture (10.8) the FM string should be expressed a

$\begin{split}&\alpha^{\prime}_{1}=\alpha^{\textsc{fm}}+\frac{2\mathrm{i}\pi}{5},\quad\alpha^{\prime}_{2}=\alpha^{\textsc{fm}}+\frac{\mathrm{i}\pi}{5},\quad\alpha^{\prime}_{3}=\alpha^{\textsc{fm}},\\\
&\alpha^{\prime}_{4}=\alpha^{\textsc{fm}}-\frac{\mathrm{i}\pi}{5},\quad\alpha^{\prime}_{5}=\alpha^{\textsc{fm}}-\frac{2\mathrm{i}\pi}{5},\end{split}\qquad\alpha^{\textsc{fm}}\in\mathbb{R}.$
(F.4)

Again, when the real parts of the string centres in (F.3) and (F.4) coincide
they describe the same FM string.

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