Scaling limit of the staggered six-vertex model with
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant boundary conditions

Holger Frahm, Sascha Gehrmann and Gleb A. Kotousov

Institut f$\ddot{{\rm u}}$r Theoretische Physik, Leibniz Universit$\ddot{{\rm
a}}$t Hannover

Appelstraße 2, 30167 Hannover, Germany

Abstract

We study the scaling limit of a statistical system, which is a special case of
the integrable inhomogeneous six-vertex model. It possesses
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariance due to the choice of open
boundary conditions imposed. An interesting feature of the lattice theory is
that the spectrum of scaling dimensions contains a continuous component. By
applying the ODE/IQFT correspondence and the method of the Baxter $Q$ operator
the corresponding density of states is obtained. In addition, the partition
function appearing in the scaling limit of the lattice model is computed,
which may be of interest for the study of nonrational CFTs in the presence of
boundaries. As a side result of the research, a simple formula for the matrix
elements of the $Q$ operator for the general, integrable, inhomogeneous six-
vertex model was discovered, that has not yet appeared in the literature. It
is valid for a certain one parameter family of diagonal open boundary
conditions in the sector with the $z$ -projection of the total spin operator
being equal to zero.

##  1 Introduction

Onsager’s solution of the square lattice Ising model [1] opened an era in the
application of integrable systems to the study of phase transitions. Exactly
solvable lattice models in 2D have been found to exhibit an interesting array
of effects and powerful analytic/numerical techniques are available to explore
them. Among such phenomena are, for instance, exactly marginal deformations
[2, 3, 4], strong/weak coupling dualities [5], and the appearance of extended
conformal symmetry in the scaling limit [6], which have collectively inspired
and refined our understanding of (conformal) QFT. Of special mention is the
Pott’s model, a generalization of the 2D Ising model. In the antiferromagnetic
case it possesses a phase, where the ground state degeneracy is macroscopic
with its logarithm being proportional to the system size; similar to fractons
that have recently attracted attention (see, e.g., [7] for a review). This
observation was originally made in the 80’s by Berker and Kadanoff [8]. The
universal properties were further studied in [9] via the mapping of the
critical Potts to the six-vertex model [10] (see also [11]) and the
application of the methods of Yang-Baxter integrability and 2D CFT.

In the work [8] the Potts model is taken to be homogeneous and isotropic, so
that the coupling between any two nearest neighbour pairs of ‘spins’ is the
same. One can consider an anisotropic version defined on the square lattice,
which has two coupling constants — one associated to the vertical and the
other to the horizontal edges of the lattice that join the neighbouring spins
together. As explained in ref. [12], focusing on the curve in the parameter
space where the model is critical leads one to the so-called staggered six-
vertex model, which is a special case of the integrable, inhomogeneous six-
vertex model introduced by Baxter in [13]. It gets its name from the fact that
the inhomogeneities are distributed along the square lattice in a checkerboard
(staggered) pattern. In addition, in the case of the antiferromagnetic Potts
model, they are fixed to a special value for which the system is ‘self-dual’,
i.e., possesses an extra ${\cal Z}_{2}$ symmetry.

The critical behaviour of the staggered six-vertex model at the self-dual
point was considered in the works [14, 12]. Valuable results about the
spectrum of scaling dimensions were obtained by studying the low energy
spectrum of the Hamiltonian, which is expressed in terms of a logarithmic
derivative of the two row transfer-matrix. The Hamiltonian, unlike the
transfer-matrix, is given by a sum of operators, which act locally on
$\mathscr{V}_{2L}=\mathbb{C}^{2}_{1}\otimes\mathbb{C}^{2}_{2}\otimes\ldots\otimes\mathbb{C}^{2}_{2L}$,
where $2L$ is the number of lattice columns. The precise formula reads as
$\mathbb{H}=-\frac{{\rm i}}{q^{2}-q^{-2}}\sum_{J=1}^{2L}{\cal O}_{J}$
with111The formula for the Hamiltonian (7.6) in the work [15] is identical to
the one given above except that the overall sign in front of the term
$\propto(q-q^{-1})$ in the second line of eq. (1), containing the product of
three Pauli matrices, is flipped. This comes about because that paper uses the
different convention for the quantum space:
$\mathscr{V}_{2L}=\mathbb{C}^{2}_{2L}\otimes\mathbb{C}^{2}_{2L-1}\otimes\ldots\otimes\mathbb{C}^{2}_{1}$,
see eq.(2.1) therein. The two Hamiltonians are related via the similarity
transformation ${\cal U}\,:\ {\cal U}^{2}=1$, which acts on the local spin
operators as ${\cal U}\,\sigma^{A}_{J}\,{\cal U}=\sigma_{2L-J+1}^{A}$.

$\displaystyle{\cal O}_{J}$ $\displaystyle=$
$\displaystyle\big{(}q-q^{-1}\big{)}^{2}\,\sigma^{z}_{J}\,\sigma^{z}_{J+1}+2\,\big{(}\sigma^{x}_{J}\,\sigma^{x}_{J+2}+\sigma^{y}_{J}\,\sigma^{y}_{J+2}+\sigma^{z}_{J}\,\sigma^{z}_{J+2}\big{)}$
$\displaystyle-$
$\displaystyle(q-q^{-1})\,\Big{(}\sigma^{z}_{J}\,\big{(}\sigma_{J+1}^{x}\sigma_{J+2}^{x}+\sigma_{J+1}^{y}\sigma_{J+2}^{y}\big{)}-\big{(}\sigma_{J}^{x}\sigma_{J+1}^{x}+\sigma_{J}^{y}\sigma_{J+1}^{y}\big{)}\,\sigma^{z}_{J+2}\Big{)}-\big{(}q^{2}+q^{-2}\big{)}\,\hat{{\bf
1}}\,.$

Here $\sigma^{A}_{J}$ with $A=x,y,z$ are the Pauli matrices that act on site
$J$ subject to periodic boundary conditions
$\sigma_{J+2L}^{A}=\sigma^{A}_{J}$, while the parameter $q$ is known as the
anisotropy. The system turns out to be critical when $q$ is a unimodular
number and different universal behaviour occurs depending on whether
$\arg(q)\in(0,\frac{\pi}{2})$ or $\arg(q)\in(\frac{\pi}{2},\pi)$. The regime

$|q|=1\qquad{\rm and}\qquad\arg(q)\in(0,\tfrac{\pi}{2})$ (1.2)

has attracted the most amount of attention. The reason for this is that the
corresponding spectrum of scaling dimensions was found to possess a continuous
component [14].

The subsequent study of the regime (1.2) saw a remarkable interchange of ideas
between statistical mechanics and formal high energy theory. On the one hand,
the conjecture from ref.[16], that the scaling limit of the lattice system is
governed by the 2D Euclidean black hole sigma model introduced in refs. [17,
18, 19], uses the results of [20] and its development [21] which come from the
string theory literature. On the other, the detailed study of the vertex model
performed in [15] led to the solution of the spectral problem for the 2D
Euclidean black hole CFT, including the computation of the density of states
of the continuous spectrum. Perhaps the most surprising output of the research
is the following. While it has been confirmed that one half of the partition
function arising in the scaling limit of the vertex model with
(quasi-)periodic boundary conditions coincides with the partition function of
the 2D Euclidean black hole sigma model on the torus, the original conjecture
of [16] has been refined. It was proposed in [15] that a part of the Hilbert
space of the lattice model in the scaling limit should coincide with the
pseudo-Hilbert space of the black hole sigma model with Lorentzian signature.

The above mentioned works all focus on the case when the lattice is
(quasi)-periodic in the horizontal direction. In the recent papers [22, 23],
motivated by the possibility of making precise contact with $D$-brane
constructions of non-compact boundary CFTs [24, 25], the statistical system
has been considered with certain integrable, open boundary conditions imposed.
In this case the Hamiltonian is given by

$\displaystyle\mathbb{H}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{{\rm
i}}{q^{2}-q^{-2}}\bigg{(}\sum_{J=1}^{2L-2}{\cal
O}_{J}-(q+q^{-1})\,\big{(}\sigma_{1}^{x}\sigma_{2}^{x}+\sigma_{1}^{y}\sigma_{2}^{y}+\sigma_{2L-1}^{x}\sigma_{2L}^{x}+\sigma_{2L-1}^{y}\sigma_{2L}^{y}\big{)}-(q^{2}-q^{-2})\,(\sigma^{z}_{2L}-\sigma^{z}_{1})$
(1.3) $\displaystyle-$
$\displaystyle\\!\\!2\,(\sigma^{z}_{1}\sigma^{z}_{2}-\hat{{\bf
1}})+(q^{2}+q^{-2})\ (\sigma^{z}_{2L-1}\sigma^{z}_{2L}-\hat{{\bf 1}})\bigg{)}\
,$

where ${\cal O}_{J}$ is defined in eq. (1). A special feature of such a choice
of boundary terms is that the model possesses
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ symmetry. To explain, notice that
$\mathbb{H}$ commutes with the $z$ -projection of the total spin operator:

$\mathbb{S}^{z}=\frac{1}{2}\,\sum_{J=1}^{2L}\sigma^{z}_{J}\,.$ (1.4)

One may check that it also commutes with

$\mathbb{S}_{q}^{\pm}=\ (\mp{\rm
i})\,\sum_{J=1}^{2L}\bigg{(}\,\prod_{\ell=J+1}^{2L}q^{-\frac{\sigma^{z}_{\ell}}{2}}\,\bigg{)}\,(-1)^{J}\
\sigma_{J}^{\pm}\
\bigg{(}\,\prod_{\ell=1}^{J-1}q^{+\frac{\sigma^{z}_{\ell}}{2}}\,\bigg{)}\ ,$
(1.5)

which, together with $\mathbb{S}^{z}$, satisfy the defining relations of the
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ algebra:222The factor $(\mp{\rm i})$ in
eq. (1.5) together with the term $(-1)^{J}$ appearing in the summand may be
removed via a similarity transformation by a diagonal matrix.

$\big{[}\mathbb{S}^{z}\,,\,\mathbb{S}_{q}^{\pm}\big{]}=\pm\mathbb{S}_{q}^{\pm}\,,\qquad\qquad\big{[}\mathbb{S}_{q}^{+},\mathbb{S}_{q}^{-}\big{]}=\frac{q^{2\mathbb{S}^{z}}-q^{-2\mathbb{S}^{z}}}{q-q^{-1}}\,.$
(1.6)

As a result, the eigenstates of the Hamiltonian form irreps of this algebra.
These are labelled by the eigenvalues of the Casimir,

$2\mathds{C}=(q+q^{-1})\,[\mathbb{S}^{z}]_{q}^{2}+\mathbb{S}_{q}^{+}\,\mathbb{S}_{q}^{-}+\mathbb{S}_{q}^{-}\,\mathbb{S}_{q}^{+}\
,$ (1.7)

which are given by $[{\cal S}]_{q}\,[{\cal S}+1]_{q}$ with integer ${\cal
S}=0,1,2,\ldots,L$ (here we use the standard notation
$[m]_{q}=(q^{m}-q^{-m})/(q-q^{-1})$). The presence of
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ symmetry simplifies the diagonalization
problem for $\mathbb{H}$. In particular, one can restrict to the
$(2L)!/(L!)^{2}$ dimensional subspace spanned by the eigenstates of the
operator $\mathbb{S}^{z}$, whose eigenvalues are zero.

It was observed in refs. [22, 23] that the spectrum of scaling dimensions of
the staggered six-vertex model with $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$
invariant open boundary conditions possesses a continuous component in the
regime (1.2). However, the corresponding density of states had not been
obtained. This was among the open problems that inspired our research.

In this paper we perform a systematic study of the low energy spectrum of the
Hamiltonian (1.3) at large system size $L\gg 1$. It is carried out via a
mixture of methods, including a numerical analysis of the Bethe Ansatz
equations as well as the powerful analytical technique of the ODE/IQFT
correspondence. It turns out that the ODEs describing the scaling limit fall
within the class of differential equations considered in refs.[15, 26], where
the universal behaviour of the vertex model with quasi-periodic boundary
conditions imposed was studied. They are described in detail in sec. 3.2
below. As for the Bethe Ansatz equations, they read as

$\bigg{(}\frac{1+\zeta_{m}^{2}q^{+2}}{1+\zeta_{m}^{2}q^{-2}}\bigg{)}^{2L}=q^{4+4{\cal
S}}\prod_{j=1\atop j\neq m}^{L-{\cal
S}}\frac{\big{(}\zeta_{j}-q^{+2}\zeta_{m}\big{)}\,\big{(}1-q^{+2}\zeta_{m}\zeta_{j}\big{)}}{\big{(}\zeta_{j}-q^{-2}\zeta_{m}\big{)}\,\big{(}1-q^{-2}\zeta_{m}\zeta_{j}\big{)}}\
,$ (1.8)

where ${\cal S}$ stands for the $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ total
spin of the state. Having at hand a solution set
$\\{\zeta_{m}\\}^{L-\mathcal{S}}_{m=1}$ to the above algebraic system, the
energy of the corresponding state is computed via the formula:

${\cal E}=\sum_{m=1}^{L-{\cal S}}\frac{4{\rm
i}\,(q^{2}-q^{-2})}{\zeta^{2}_{m}+\zeta^{-2}_{m}+q^{2}+q^{-2}}\ .$ (1.9)

Notice that eqs. (1.8) are invariant upon making the transformation
$\zeta_{m}\mapsto\zeta_{m}^{-1}$ of any one of the Bethe roots. This allows
one, without loss of generality, to assume that

$|\zeta_{m}|\leq 1\qquad\qquad(m=1,2,\ldots,L-{\cal S})\,.$ (1.10)

The paper is organized as follows. In section 2 we present a formula for the
matrix elements of the Baxter $Q$ operator that was used in our numerical
work. It is valid in the sector $S^{z}=0$ and for a one parameter family of
open boundary conditions. The outcomes of our study of the low energy spectrum
of the Hamiltonian (1.3) in the regime (1.2) is given in section 3. The first
subsection thereof, for the most part, is a review of the results contained in
refs.[22, 23]. In the next two, the ODE/IQFT correspondence is described and,
on the basis of this, the so-called ‘quantization condition’ is obtained. The
latter is what allows us to perform a full characterization of the low energy
space of states of the lattice model, which is detailed in section 4. In
section 5, the formula for the partition function appearing in the scaling
limit of the lattice model is given, which may be of interest to those
studying non-rational boundary CFT. The last section is devoted to a
discussion and includes a summary of the main results.

##  2 The $Q$ operator

In analysing the low energy spectrum of the Hamiltonian of a critical lattice
system in the scaling limit one meets with immediate issues. An important one
concerns the construction of RG trajectories, where low energy stationary
states at different lattice sizes $L$ are grouped into families
$\\{|\Psi_{L}\rangle\\}$. It is clear how to define the RG flow for the ground
state or, for that matter, the lowest energy states in the disjoint sectors of
the Hilbert space, say, the sector with given eigenvalue of $\mathbb{S}^{z}$
(1.4). However, assigning an $L$ dependence to a low energy excited state and
then continuing $|\Psi_{L}\rangle$ to $L\gg 1$ seems to be a non-trivial task.

In the case at hand the model is integrable and the construction of RG
trajectories is facilitated by the Bethe Ansatz solution. A formulation of the
procedure is provided in ref. [26], see also the work [15]. The low energy
stationary states can be chosen to be the Bethe states, which are simultaneous
eigenvectors of the full family of operators commuting with the Hamiltonian.
They are labeled by the Bethe roots $\\{\zeta_{m}\\}$, which solve the Bethe
Ansatz equations (1.8). The eigenvalues of the commuting operators for the
Bethe state are given in terms of the corresponding $\\{\zeta_{m}\\}$, as in
formula (1.9) for the energy, and their computation does not require any
explicit diagonalization. Suppose one has at hand the Bethe roots for a low
energy Bethe state for a lattice of $L=L_{\rm in}$ sites. The state
$|\Psi_{L_{\rm in}+2}\rangle$ is specified such that the pattern of Bethe
roots qualitatively remains the same. These can be obtained by numerically
solving the Bethe Ansatz equations (1.8), where the initial approximation for
the iterative solution is constructed from the Bethe roots corresponding to
$|\Psi_{L_{\rm in}}\rangle$. By iterating this procedure an RG trajectory
$\\{|\Psi_{L}\rangle\\}$ for increasing $L$ is obtained.

A method is required in order to extract the Bethe roots corresponding to the
state $|\Psi_{L_{\rm in}}\rangle$ which seeds the RG trajectory. Typically for
$|\Psi_{L_{\rm in}}\rangle$ generic, $\\{\zeta_{m}\\}$ are complex numbers
that do not resemble a simple pattern in the complex plane, see Fig. 1 for an
example. Finding all of the possible solution sets of the Bethe Ansatz
equations (1.8) even for $L_{\rm in}\lesssim 10$ is impossible to carry out on
a modern laptop because the system is to complicated. Moreover, searching for
the set $\\{\zeta_{m}\\}$ by applying the Newton method to (1.8), trying out
various initial approximations, is time consuming and not guaranteed to work.

$u$$\Im m(u)=\frac{\pi}{2}$

$u$$\Im m(u)=\frac{\pi}{2}$

Figure 1: Displayed is the pattern of Bethe roots in the complex
$u=-\frac{1}{2}\log(\zeta)$ plane for a low energy Bethe state that was
generically chosen. For the left panel $L=10$ and the set $\\{\zeta_{m}\\}$
was obtained from the eigenvalue of the $Q$ operator computed on the state.
The latter was used to seed an RG trajectory $\\{|\Psi_{L}\rangle\\}$ and, on
the right panel, shown are the Bethe roots for the corresponding state with
$L=40$. The trajectory is labelled by $\mathcal{S}=1$, while the anisotropy
parameter was taken to be $q=\mbox{e}^{\frac{10\pi{\rm i}}{49}}$.

In our study of the spin chain, to find the Bethe roots corresponding to a
generic low energy state $|\Psi_{L_{\rm in}}\rangle$ systematically, we used
the technique based on the notion of the Baxter $Q$ operator
$\mathbb{Q}(\zeta)$ [27]. The latter commutes with itself for different values
of the spectral parameter as well as the transfer-matrix,

$\mathbb{Q}(\zeta)\,:\
\qquad[\mathbb{Q}(\zeta),\mathbb{Q}(\zeta^{\prime})]=[\mathbb{Q}(\zeta),\mathbb{T}(\zeta^{\prime})]=0\,.$
(2.1)

In addition, its eigenvalues for a Bethe state is a polynomial in $\zeta$,
whose zeroes coincide with the corresponding solution set to the Bethe Ansatz
equations (see, e.g., formula (2.29) below). For a small number of lattice
sites it is possible to obtain the first few hundred low lying stationary
states of the spin chain via an explicit diagonalization of the Hamiltonian.
For each of them, the corresponding set $\\{\zeta_{m}\\}$ is extracted by
computing the eigenvalue of $\mathbb{Q}(\zeta)$ (some further details can be
found in section 18 of ref.[15]). In turn, the RG trajectory
$\\{|\Psi_{L}\rangle\\}$ is continued from $L=L_{\rm in}$ to large $L$ via the
procedure outlined above.

Below we present the explicit formula for the $Q$-operator that was used in
our analysis. It allowed us to go beyond the results of the previous papers
[22, 23]. Our formula was obtained based on the results of ref.[28], which is
part of an interesting group of recent papers [28, 29, 30, 31] appearing in
the mathematics literature. Among other things, they give $\mathbb{Q}(\zeta)$
as a trace of a monodromy matrix over a $q$-oscillator representation for
models associated with the rational [29] and trigonometric [30, 31] $R$-matrix
for $\mathfrak{sl}(2)$ with a two parameter family of open Boundary Conditions
(BCs). The matrix elements of $\mathbb{Q}(\zeta)$ from [28] take the form of
an infinite sum, which converges only in a certain parameteric domain that
excludes the model with $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant BCs we
are considering. As such, some analysis was required in order to bring the
expression to a form, which is literally applicable to the case at hand and is
efficient for numerical purposes.

Let’s consider the more general case of a lattice system with an arbitrary set
of inhomogeneities $\\{\eta_{J}\\}_{J=1}^{2L}$. Moreover, for technical
reasons, we’ll take a one parameter family of open BCs depending on
$\epsilon$, such that the $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant case
is recovered at

$\epsilon=0\qquad\qquad\qquad\qquad\Big{(}U_{q}\big{(}\mathfrak{sl}(2)\big{)}\
{\rm invariant}\ {\rm BCs}\Big{)}\ .$ (2.2)

Rather than following the conventions of ref.[23], we’ll use the
multiplicative spectral parameter $\zeta$ and arrange the definitions such
that the matrix elements of both $\mathbb{T}(\zeta)$ and $\mathbb{Q}(\zeta)$
are mainifestly polynomials in $\zeta$. The $R$-matrix reads as

$R(\zeta)=\left(\begin{array}[]{cccc}q-q^{-1}\,\zeta&0&0&0\\\\[0.0pt]
0&1-\zeta&q-q^{-1}&0\\\\[0.0pt] 0&(q-q^{-1})\,\zeta&1-\zeta&0\\\\[0.0pt]
0&0&0&q-q^{-1}\,\zeta\end{array}\right)\,.$ (2.3)

It can be interpreted as an operator $R(\zeta)=R_{I,J}(\zeta)$ acting on the
tensor product $\mathbb{C}_{I}^{2}\otimes\mathbb{C}^{2}_{J}$, where the
convention is used such that $\mathbb{C}_{J}^{2}$ with $J=0$ stands for the
auxiliary space, while for $J=1,\dots,2L$ it is the $J$-th factor of the
quantum space

$\mathscr{V}_{2L}=\mathbb{C}_{1}^{2}\otimes\mathbb{C}_{2}^{2}\otimes\ldots\otimes\mathbb{C}_{2L}^{2}\
.$ (2.4)

$\mathbb{T}(\zeta
q^{-1})=$$K^{+}$$K^{-}$$\eta_{4}$$\frac{1}{2}$$\eta_{3}$$\frac{1}{2}$$\eta_{2}$$\frac{1}{2}$$\eta_{1}$$\frac{1}{2}$$\eta_{2L-1}$$\frac{1}{2}$$\eta_{2L}$$\frac{1}{2}$

Figure 2: A graphical representation of the transfer-matrix for the
inhomogeneous six-vertex model. Open BCs are imposed, as indicated by the
presence of the reflection matrices $K^{\pm}$ given by (2.7) acting in the
two-dimensional auxiliary space.

The transfer-matrix, which is graphically depicted in Fig. 2, is given
by333The formula for the transfer matrix (2.6) may be equivalently written in
index notation:  $\displaystyle\big{[}\mathbb{T}(\zeta
q^{-1})\big{]}^{b_{1}\dots b_{2L}}_{a_{1}\dots a_{2L}}=q^{-2L}\sum_{c_{1}\dots
c_{2L}=\pm}\sum_{\alpha_{1}\dots\alpha_{2L+1}=\pm\atop\beta_{1}\dots\beta_{2L+1}=\pm}$
$\displaystyle\big{[}K^{+}(\zeta)\big{]}^{\beta_{2L+1}}_{\alpha_{2L+1}}\,\big{[}R(\zeta\eta^{-1}_{2L})\big{]}^{\alpha_{2L+1}b_{2L}}_{\alpha_{2L}c_{2L}}\dots\big{[}R(\zeta\eta^{-1}_{1})\big{]}^{\alpha_{2}b_{1}}_{\alpha_{1}c_{1}}\,$
(2.5)
$\displaystyle\big{[}K^{-}(\zeta)\big{]}^{\alpha_{1}}_{\beta_{1}}\,\big{[}R(\zeta\eta_{1})\big{]}^{c_{1}\beta_{1}}_{a_{1}\beta_{2}}\dots\big{[}R(\zeta\eta_{2L})\big{]}^{c_{2L}\beta_{2L}}_{a_{2L}\beta_{2L+1}}\,.$
Here $\big{[}K^{\pm}(\zeta)\big{]}^{\beta}_{\alpha}$ with $\alpha,\beta=\pm$
stand for the entries of the diagonal matrices (2.7) and similar for
$\big{[}R(\zeta)\big{]}^{\beta b}_{\alpha a}$, e.g.,
$\big{[}R(\zeta)\big{]}^{+-}_{-+}=q-q^{-1}$ and
$\big{[}R(\zeta)\big{]}^{-+}_{+-}=(q-q^{-1})\,\zeta$.

$\mathbb{T}(\zeta
q^{-1})=q^{-2L}\,\mathrm{tr}_{0}\Big{(}K^{+}_{0}(\zeta)\,R_{0,2L}(\zeta\,\eta^{-1}_{2L})\dots
R_{0,1}(\zeta\,\eta^{-1}_{1})\,K^{-}_{0}(\zeta)\,R_{1,0}(\zeta\,\eta_{1})\dots
R_{2L,0}(\zeta\,\eta_{2L})\Big{)}\,.$ (2.6)

Here the trace is taken over the auxiliary space, while $K^{\pm}$ stand for
the diagonal matrices

$K^{-}(\zeta)=\begin{pmatrix}1+\zeta\epsilon&0\\\
0&\zeta^{2}+\zeta\epsilon\end{pmatrix},\quad\qquad
K^{+}(\zeta)=\begin{pmatrix}q^{-2}\zeta+\epsilon&0\\\
0&\zeta^{-1}+q^{-2}\epsilon\end{pmatrix}\,.$ (2.7)

It is straightforward to check that $\mathbb{T}(\zeta)$ is a polynomial of
order $4L+2$ in the spectral parameter and satisfies the conditions

$\mathbb{T}(0)=\epsilon\,\big{(}q^{2\mathbb{S}^{z}}+q^{-2\mathbb{S}^{z}}\big{)}\
,\qquad\qquad\qquad\mathbb{T}(\zeta^{-1})=\zeta^{-4L-2}\ \mathbb{T}(\zeta)\ .$
(2.8)

To get back the case of the staggered six-vertex model with
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant BCs, at the self-dual point,
one should fix the inhomogeneities $\eta_{J}$ and the extra parameter
$\epsilon$ to be

$\eta_{J}={\rm i}\,(-1)^{J}\qquad\qquad(J=1,2,\ldots
2L)\,,\qquad\qquad\epsilon=0\,.$ (2.9)

We will use the notation444The transfer-matrix as defined in formula (2.6) for
arbitrary inhomogenieties $\eta_{J}$ and $\epsilon=0$ is related to $\tau(u)$
given by (2.9) of ref. [23] via a similarity transformation by a diagonal
matrix and an overall multiplicative factor. Namely,
$\mathbb{T}^{(0)}(\zeta)=2^{4L}\mbox{e}^{-4Lu-2u}\bigg{(}\prod^{2L}_{J=1}\rho(-u+\tfrac{{\rm
i}\gamma}{2}-\delta_{J})\bigg{)}\ \mathcal{U}\,\tau(u)\,\mathcal{U}^{-1}.$
(2.10) Here $\mathcal{U}=G_{1}(\delta_{1})\otimes\dots\otimes
G_{2L}(\delta_{2L})$ with $G(u)={\rm{diag}}(1,\mbox{e}^{-u})$, while the
parameters $u,\gamma,\delta_{J}$ need to be identified with $\zeta,q,\eta_{J}$
as  $\zeta=\mbox{e}^{-2u},\qquad q=\mbox{e}^{{\rm
i}\gamma},\qquad\eta_{J}=\mbox{e}^{-2\delta_{J}}\,.$ (2.11) Also, the function
$\rho(u)$ is given by
$\rho(u)=\frac{1}{2}\big{(}\cos(2\gamma)-\cosh(2u)\big{)}$.

$\mathbb{T}^{(0)}(\zeta)\equiv\mathbb{T}(\zeta)\big{|}_{\epsilon=0}\,.$ (2.12)

For our purposes it is sufficient to focus on the sector where the eigenvalue
of the $z$ -projection of the total spin operator is zero, i.e., $S^{z}=0$.
This is because for the model possessing $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$
invariance, the states come in multiplets $\mathcal{M}_{\mathcal{S}}$ each of
which has a representative in that sector. Let the tuples $(a_{1}a_{2}\dots
a_{2L})$ and $(b_{1}b_{2}\dots b_{2L})$ with $a_{J},b_{J}=\pm$ be the
input/output indices for the space $\mathscr{V}_{2L}$ (2.4). To present the
formula for the $Q$-operator, introduce

$\displaystyle\left(\begin{array}[]{cc}\big{[}{A}(\zeta;m)\big{]}^{+}_{+}&\big{[}{A}(\zeta;m)\big{]}^{-}_{+}\\\\[5.69046pt]
\big{[}{A}(\zeta;m)\big{]}^{+}_{-}&\big{[}{A}(\zeta;m)\big{]}^{-}_{-}\end{array}\right)=\left(\begin{array}[]{cc}q^{m}&q^{m}\\\\[5.69046pt]
\zeta q^{-m+1}&q^{-m}\end{array}\right)\,,$ (2.17)
$\displaystyle\left(\begin{array}[]{cc}\big{[}{\widetilde{A}}(\zeta;m)\big{]}^{+}_{+}&\big{[}{\widetilde{A}}(\zeta;m)\big{]}^{-}_{+}\\\\[5.69046pt]
\big{[}{\widetilde{A}}(\zeta;m)\big{]}^{+}_{-}&\big{[}{\widetilde{A}}(\zeta;m)\big{]}^{-}_{-}\end{array}\right)=\left(\begin{array}[]{cc}q^{m}&\zeta
q^{m+2}\\\\[5.69046pt] q^{-m-1}&q^{-m}\end{array}\right)\,.$ (2.22)

The matrix elements of $\mathbb{Q}(\zeta)$, valid in the sector $S^{z}=0$, are
given by

$\big{[}\mathbb{Q}(\zeta)\big{]}^{b_{1}\dots b_{2L}}_{a_{1}\dots
a_{2L}}=\sum_{c_{1}\dots c_{2L}=\pm}q^{(S^{z}_{c})^{2}}(\epsilon
q^{2}\zeta)^{S^{z}_{c}}\prod^{2L}_{J=1}\big{[}{A(\zeta\eta^{-1}_{J},m_{b,J})}\big{]}^{b_{J}}_{c_{J}}\,\big{[}{\widetilde{A}}(\zeta\eta_{J},m_{a,J+1})\big{]}^{c_{J}}_{a_{J}}\,.$
(2.23)

Here the symbol $S_{\rm c}^{z}$, which should not be confused with the
eigenvalue of the $z$ -projection of the total spin operator, stands for

$S_{\rm c}^{z}=\frac{1}{2}\sum_{J=1}^{2L}c_{J}$ (2.24)

and provides a natural grading for the sum over $c_{J}$. The internal indices
$\\{m_{a,J},m_{b,J}\\}$ come from the product over the auxiliary space. They
are fixed completely by the ice-rule to be

$m_{x,J}=\frac{1}{2}\sum_{\ell=J}^{2L}(x_{\ell}-c_{\ell})\qquad\qquad(J=1,\ldots,2L+1;\
x=a,b)\,.$ (2.25)

Despite the presence of the factor $\zeta^{S^{z}_{c}}$ in formula (2.23),
where the exponent can be negative, the matrix elements of $\mathbb{Q}(\zeta)$
turn out to be polynomials in $\zeta$ of degree $2L$. Moreover, it is
straightforward to show that they satisfy:

$\mathbb{Q}(\zeta^{-1})=\zeta^{-2L}\
\mathbb{Q}(\zeta)\,,\qquad\qquad\mathbb{Q}(0)=\mathds{1}\,.$ (2.26)

For arbitrary values of the inhomogeneities $\eta_{J}$ and parameter
$\epsilon$, the $Q$-operator obeys the commutativity conditions (2.1) as well
as the $TQ$ relation with the transfer-matrix

$\displaystyle\big{(}1-\zeta^{2}\big{)}\,\mathbb{Q}(\zeta)\,\mathbb{T}(\zeta)$
$\displaystyle=$
$\displaystyle\,\big{(}\epsilon+q^{+1}\,\zeta\big{)}\,\big{(}1+\zeta\,q^{+1}\,\epsilon\big{)}\,f(q^{-1}\,\zeta)\,\mathbb{Q}\big{(}\zeta\,q^{+2}\big{)}$
$\displaystyle+$
$\displaystyle\big{(}\epsilon+q^{-1}\,\zeta\big{)}\,\big{(}1+\zeta\,q^{-1}\,\epsilon\big{)}\,f(q^{+1}\zeta)\,\mathbb{Q}\big{(}\zeta\,q^{-2}\big{)}\qquad\qquad(S^{z}=0)\,,$

where

$f(\zeta)=(1-\zeta^{2})\prod_{J=1}^{2L}\big{(}\zeta-\eta_{J}^{-1}\big{)}\big{(}\zeta-\eta_{J}\big{)}\
.$ (2.28)

The above implies the Bethe Ansatz equations for the integrable model in the
usual way: One considers both sides of eq. (2) evaluated on a common
eigenvector. In view of (2.26), the eigenvalues of $\mathbb{Q}(\zeta)$ take
the form

$Q(\zeta)=\prod_{j=1}^{L}\,(1-\zeta/\zeta_{j})\,(1-\zeta\zeta_{j})\,.$ (2.29)

Combining the above with (2) and setting $\zeta=\zeta_{m}$ into that formula,
one arrives at the coupled system of algebraic equations for the Bethe roots:

$\displaystyle\prod_{J=1}^{2L}\frac{\big{(}\zeta_{m}q^{+1}-\eta_{J}^{-1}\big{)}\,\big{(}\zeta_{m}q^{+1}-\eta_{J}\big{)}}{\big{(}\zeta_{m}q^{-1}-\eta_{J}^{-1}\big{)}\,\big{(}\zeta_{m}q^{-1}-\eta_{J}\big{)}}$
$\displaystyle=$
$\displaystyle\frac{\big{(}\epsilon+q^{+1}\,\zeta_{m}\big{)}\,\big{(}q^{+1}\,\epsilon+\zeta^{-1}_{m}\big{)}}{\big{(}\epsilon+q^{-1}\,\zeta_{m}\big{)}\,\big{(}q^{-1}\,\epsilon+\zeta^{-1}_{m}\big{)}}$
$\displaystyle\times$ $\displaystyle q^{2}\prod_{j=1\atop j\neq
m}^{L}\frac{\big{(}\zeta_{j}-q^{+2}\zeta_{m}\big{)}\,\big{(}1-q^{+2}\zeta_{m}\zeta_{j}\big{)}}{\big{(}\zeta_{j}-q^{-2}\zeta_{m}\big{)}\,\big{(}1-q^{-2}\zeta_{m}\zeta_{j}\big{)}}\qquad\qquad(S^{z}=0)\,.$

These are equivalent to the Bethe Ansatz equations obtained in the original
work [32], specialized to the sector $S^{z}=0$ and with $\xi^{+}=-\xi^{-}$.
The parameters should be identified as
$(q,\eta_{J},\epsilon)\mapsto(\mbox{e}^{\eta},\mbox{e}^{-2u_{J}},\mbox{e}^{2\xi^{\pm}})$,
while $\zeta_{m}\mapsto\mbox{e}^{-2v_{m}}$.

Some care is needed in taking the limit $\epsilon\to 0$ of the $Q$ operator.
It is clear from the explicit formula (2.23) that the matrix elements of
$\mathbb{Q}(\zeta)$ generically diverge due to the presence of the factor
$\epsilon^{S^{z}_{\rm c}}$, where the exponent may be negative. This is a
manifestation of the $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariance
possessed by the model at the point $\epsilon=0$, so that states in different
sectors of $S^{z}$ form multiplets of the symmetry group that have the same
eigenvalue of $\mathbb{Q}(\zeta)$. In ref.[33] a similar phenomenon was
studied in the context of the XXX spin chain with twisted boundary conditions
controlled by the parameter $\phi$. At $\phi=0$ the model possesses global
$\mathfrak{su}(2)$ symmetry and the matrix elements of the $Q$ operator become
infinite. It was explained in that paper how to take the limit $\phi\to 0$ of
$\mathbb{Q}(\zeta)$ so that one obtains a well defined result. The discussion
is readily adapted to the case at hand.

Recall that the quadratic Casimir for the
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ algebra is given by:

$2\mathds{C}=(q+q^{-1})\,[\mathbb{S}^{z}]_{q}^{2}+\mathbb{S}_{q}^{+}\,\mathbb{S}_{q}^{-}+\mathbb{S}_{q}^{-}\,\mathbb{S}_{q}^{+}\
,$ (2.31)

where $\mathbb{S}^{z}$ is defined in formula (1.4) in the introduction, while
for arbitrary values of the inhomogeneities,

$\mathbb{S}_{q}^{\pm}=\sum_{J=1}^{2L}\bigg{(}\,\prod_{\ell=J+1}^{2L}q^{-\frac{\sigma^{z}_{\ell}}{2}}\,\bigg{)}\
\eta_{J}^{\mp 1}\sigma_{J}^{\pm}\
\bigg{(}\,\prod_{\ell=1}^{J-1}q^{+\frac{\sigma^{z}_{\ell}}{2}}\,\bigg{)}\ .$
(2.32)

The eigenvalues of $\mathds{C}$ are given by

$[{\cal S}]_{q}\,[{\cal S}+1]_{q}$ (2.33)

with ${\cal S}=0,1,2,\ldots,L\ $. One can consider ${\cal S}$ as an operator,
which for an eigenstate of the quadratic Casimir with eigenvalue $[{\cal
S}]_{q}\,[{\cal S}+1]_{q}$ gives back the non-negative integer ${\cal S}\geq
0$. Then, following the work [33], it turns out that the limit

$\mathbb{Q}^{(0)}(\zeta)=\lim_{\epsilon\to 0}\,\epsilon^{\frac{{\cal
S}}{2}}\,\mathbb{Q}(\zeta;\epsilon)\,\epsilon^{\frac{{\cal S}}{2}}$ (2.34)

exists and yields the $Q$ operator for the inhomogeneous six-vertex model with
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant BCs in the sector $S^{z}=0$.
The commutativity condition (2.1) and the $TQ$ relation (2) with the
substitutions
$\big{(}\mathbb{Q},\mathbb{T}\big{)}\mapsto\big{(}\mathbb{Q}^{(0)},\mathbb{T}^{(0)}\big{)}$
are satisfied provided, for the latter formula, one sets $\epsilon=0$. We
note, however, that the normalisation (2.26) no longer holds true. Instead,

$\mathbb{Q}^{(0)}(\zeta)|\Psi_{L}\rangle=C\,\zeta^{\mathcal{S}}\prod_{j=1}^{L-\mathcal{S}}\,(1-\zeta/\zeta_{j})\,(1-\zeta\zeta_{j})\,|\Psi_{L}\rangle\,,$
(2.35)

where $C$ is a constant depending only on $q$ and $\mathcal{S}$.

The $TQ$ relation specialized to $\epsilon=0$, combined with (2.35), leads to
the Bethe Ansatz equations

$\prod_{J=1}^{2L}\frac{\big{(}\zeta_{m}q^{+1}-\eta_{J}^{-1}\big{)}\,\big{(}\zeta_{m}q^{+1}-\eta_{J}\big{)}}{\big{(}\zeta_{m}q^{-1}-\eta_{J}^{-1}\big{)}\,\big{(}\zeta_{m}q^{-1}-\eta_{J}\big{)}}=q^{4+4\mathcal{S}}\prod_{j=1\atop
j\neq
m}^{L-\mathcal{S}}\frac{\big{(}\zeta_{j}-q^{+2}\zeta_{m}\big{)}\,\big{(}1-q^{+2}\zeta_{m}\zeta_{j}\big{)}}{\big{(}\zeta_{j}-q^{-2}\zeta_{m}\big{)}\,\big{(}1-q^{-2}\zeta_{m}\zeta_{j}\big{)}}\
.$ (2.36)

Upon taking the inhomogeneities to be as in (2.9) one gets back eq. (1.8) that
appeared in the introduction. Notice that (2.36) also follows from (2) by
taking $\epsilon\to 0$ and assuming that ${\cal S}$ of the roots $\zeta_{j}$
with $j\neq m$ vanish in this limit.

##  3 Low energy spectrum in the scaling limit

###  3.1 Preliminaries

The low energy spectrum of the Hamiltonian of a 1D critical quantum spin chain
at large system size contains important information about the universal
behaviour of the model [36, 37, 38, 35]. As such, upon the construction of the
RG trajectories $\\{|\Psi_{L}\rangle\\}$, the corresponding eigenvalue of the
Hamiltonian, $E(L)$, for $L\gg 1$ is one of the first quantities that may be
studied. For the staggered six-vertex model with
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant boundary conditions in the
critical regime $q=\mbox{e}^{{\rm i}\gamma}$ with $\gamma\in(0,\frac{\pi}{2})$
the energy of a certain class of low energy states was analyzed in the works
[22, 23]. The leading and sub-leading behaviour is described as $E\asymp
Le_{\infty}+f_{\infty}+o(1)$, where the specific bulk energy $e_{\infty}$ and
surface contribution to the energy $f_{\infty}$ were obtained in ref.[23]
within the root density approach:

$\displaystyle e_{\infty}$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{+\infty}\mbox{d}\omega\,\frac{\sinh(\frac{\pi\omega}{2n+4})}{\sinh(\frac{\pi\omega}{4})\,\cosh(\frac{n\pi\omega}{4n+8})}$
(3.1) $\displaystyle f_{\infty}$ $\displaystyle=$ $\displaystyle
4\tan(\tfrac{\pi}{n+2})+\int^{\infty}_{-\infty}\mbox{d}\omega\,\frac{\cosh(\frac{\pi\omega}{4n+8})\,\sinh(\frac{\pi\omega(n-1)}{4n+8})}{\sinh(\frac{\pi\omega}{4})\,\cosh(\frac{n\pi\omega}{4n+8})}\,.$

Here and below the anisotropy $q$ is swapped for the parameter $n>0$ according
to the relation

$q=\mbox{e}^{{\rm i}\gamma}\,:\ \ \ \ \ \ \ \ \
\gamma=\frac{\pi}{n+2}\qquad\qquad\qquad(n>0)\,.$ (3.2)

The next order term in the asymptotic expansion of the energy goes as $1/L$.
Unlike $e_{\infty}$ and $f_{\infty}$, which are the same for all the low
energy states, the coefficient for the $1/L$ correction depends on the
particular RG trajectory $\\{|\Psi_{L}\rangle\\}$ one is considering. Among
other things, it contains the conformal dimension characterising the scaling
limit of that state. In the works [22, 23] the following asymptotic formula
was proposed:

$E\asymp L\,e_{\infty}+f_{\infty}+\frac{\pi v_{\rm
F}}{L}\bigg{(}\frac{p^{2}}{n+2}+\frac{b^{2}}{n}-\frac{1}{12}+{\tt
d}\bigg{)}+o(L^{-1-\varepsilon})\ .$ (3.3)

Here the Fermi velocity $v_{\rm F}$ is non-universal and depends on the
overall multiplicative normalization of the Hamiltonian. In our conventions
for $\mathbb{H}$, one has

$v_{\rm F}=2+\frac{4}{n}\ .$ (3.4)

The term $p$ is shorthand for the expression

$p=\frac{1}{2}\,\big{(}2{\cal S}+1+{\tt w}\,(n+2)\big{)}\,\qquad{\rm
with}\qquad{\tt w}=-1\ ,$ (3.5)

where ${\cal S}$ is the $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ spin of
$|\Psi_{L}\rangle$, while the notation ${\tt d}$ stands for ‘descendent’ and
is a non-negative integer ${\tt d}=0,1,2,\ldots\ $. Also, the correction term
$o(L^{-1-\varepsilon})$ contains an infinitesimally small $\varepsilon>0$.

The quantity $b$ in the asymptotic formula (3.3) requires special comment.
First of all, it turns out to depend on the system size $b=b(L)$. In ref.[23]
it was found that $b(L)$ is related to the eigenvalue $B$ of the so-called
‘quasi-shift’ operator $\mathbb{B}$ computed on $|\Psi_{L}\rangle$. This
operator commutes with the transfer-matrix and coincides, up to an overall
factor,555The factor is given by $-\frac{q^{4L+2}}{1+q^{2}}\,f^{2}({\rm
i}q^{-2})$ with $f$ being the function defined in (2.28). with
$\left(\mathbb{T}^{(0)}({\rm i}q^{-1})\right)^{2}$, see also ref. [39]. The
precise relation reads as

$b(L)=\frac{n}{2\pi}\,\log\big{(}\sqrt{B}\,\big{)}\ .$ (3.6)

Here $\sqrt{B}$ is given in terms of the Bethe roots corresponding to a Bethe
state as

$\sqrt{B}=\prod_{m=1}^{L-{\cal S}}\,\frac{\big{(}\zeta_{m}+{\rm
i}q^{-1}\big{)}\,\big{(}\zeta_{m}-{\rm i}q\,\big{)}}{\big{(}\zeta_{m}-{\rm
i}q^{-1}\big{)}\,\big{(}\zeta_{m}+{\rm i}q\,\big{)}}\ .$ (3.7)

The asymptotic formula (3.3) for the low energy spectrum should be understood
with $b$, therein, substituted for $b(L)$ obtained via (3.6).

Some comment is required on the choice of the branch for the logarithm in the
expression for $b(L)$ (3.6). For all the trajectories we constructed, it
turned out that consistency with the asymptotic formula for the energy (3.3)
requires that:

$-\frac{n}{2}<\Im m\big{(}b(L)\big{)}\leq\frac{n}{2}\ .$ (3.8)

The question of which of the boundaries $\Im
m\big{(}b(L)\big{)}=\pm\frac{n}{2}$ to include in the domain of $b(L)$ does
not matter for the following reason. The only RG trajectories of the spin
chain which were observed such that $\Im m\big{(}b(L)\big{)}\to\pm\frac{{\rm
i}n}{2}$ as $L\to\infty$ had vanishing real part in the scaling limit. The
typical pattern of Bethe roots for one of these is depicted in Fig. 3. In this
case, one notes that the asymptotic formula for the energy (3.3) yields the
same result for $\lim_{L\to\infty}L\,(E-Le_{\infty}-f_{\infty})$ regardless of
whether $\lim_{L\to\infty}b(L)$ coincides with $-\frac{{\rm i}n}{2}$ or
$+\frac{{\rm i}n}{2}$.

$u$$\Im m(u)=\frac{\pi}{2}$

$L$$\Im m(b(L))$

Figure 3: Displayed is numerical data for an RG trajectory where $b(L)$ tends
to $-\frac{{\rm i}n}{2}$. The left panel depicts the pattern of Bethe roots in
the complex $u=-\frac{1}{2}\log(\zeta)$ plane for the state $|\Psi_{L}\rangle$
with $L=10$. The right panel is a plot of $b(L)$ as a function of $L$, where
the dashed line represents its limiting value. Here $\mathcal{S}=1$ and $n$,
which parameterizes $q$ as in (3.2), is given by $n=2.9$.

The appearance of a term like $b=b(L)$ in the $1/L$ corrections to $E(L)$,
that depends on the low energy state, was originally observed in the context
of 1D quantum spin chains with periodic boundary conditions imposed [40, 14].
The problem of extracting the spectrum of conformal dimensions for these
models was studied in detail in the later works [16, 15]. The analysis therein
carries over to the staggered six-vertex model with open,
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant boundary conditions we are
considering. As an illustration, let’s discuss it first for the class of
states that were considered in the papers [23, 22].

There exist solutions to the Bethe Ansatz equations (1.8) such that all the
Bethe roots are real. Different solution sets are distinguished by the integer
$\mathfrak{m}$, which stands for the difference between the number of roots
lying on the positive and negative real axes (see Fig.4):

$\\{\zeta_{m}\\}_{m=1}^{M}=\\{\zeta_{m}^{(+)}\\}_{m=1}^{M+\frac{\mathfrak{m}}{2}}\,\cup\,\\{\zeta_{m}^{(-)}\\}_{m=1}^{M-\frac{\mathfrak{m}}{2}}\qquad\qquad{\rm
with}\qquad\qquad\zeta_{m}^{(\pm)}\gtrless 0\qquad\qquad(M=L-{\cal S})\,.$
(3.9)

Comparison of the energy computed from the solution set $\\{\zeta_{m}\\}$ via
eq. (1.9) with that coming from the direct diagonalization of the Hamiltonian,
for small lattice sizes, shows that the corresponding Bethe states are low
energy states of the spin chain provided $\mathfrak{m}\ll L$. The large $L$
asymptotic of the energy turns out to obey (3.3) with ${\tt d}=0$ so long as
there are no significant gaps in the distribution of Bethe roots along the
positive and negative rays $\zeta\lessgtr 0$. Computing the value of $b(L)$
from the definition (3.7), (3.6) one finds it to be a real number, which turns
out to satisfy the large $L$ asymptotic behaviour [23]

$b_{\mathfrak{m}}(L)=\frac{\pi\mathfrak{m}}{2\log(L)}+O\big{(}1/(\log
L)^{2}\big{)}\qquad\qquad\qquad(\mathfrak{m}\,-\,{\rm fixed})\,.$ (3.10)

This way as $L$ becomes large, $\Delta
b_{\mathfrak{m}}(L)=b_{\mathfrak{m}+2}(L)-b_{\mathfrak{m}}(L)\propto 1/\log
L$, so that the values of $b_{\mathfrak{m}}(L)$ are densely distributed in
some segment of the real line. The latter is given by $(-b_{\mathfrak{m}_{\rm
max}},+b_{\mathfrak{m}_{\rm max}})$, where $\mathfrak{m}_{\rm max}(L)\ll L$ is
the maximum value of the integer $\mathfrak{m}$ such that the state with Bethe
roots (3.9) is still of low energy. Assuming that $\mathfrak{m}_{\rm max}$
grows faster than $\log(L)$ as $L\to\infty$, this segment becomes the entire
real line in the scaling limit.

When assigning an $L$ dependence $|\Psi_{L}\rangle$ to the class of states
discussed above, it is tempting to keep the integer $\mathfrak{m}$ fixed.
Then, in view of formula (3.10), the value of $b(L)$ would go to zero as
$L\to\infty$. However, there is another way of organizing the RG flow of the
states. One may increase $\mathfrak{m}$ as $\sim\log(L)$ so that the value of
$b(L)$ tends to a finite, non zero limit as $L\to\infty$. Such an RG
trajectory would be characterized by

$s=\operatorname*{slim}_{L\to\infty}b(L)\,,$ (3.11)

which can be arranged to be an arbitrary real number. Let’s compare the
asymptotic formula for the energy (3.3) for the RG trajectory with the general
CFT prediction for a lattice system with open boundary conditions imposed [36,
35]:

$E\asymp L\,e_{\infty}+f_{\infty}+\frac{\pi v_{\rm
F}}{L}\bigg{(}\Delta-\frac{c}{24}\bigg{)}+o(L^{-1-\varepsilon})\ .$ (3.12)

Here $\Delta$ is the conformal dimension, while $c$ stands for the central
charge. One finds that

$\Delta-\frac{c}{24}=\frac{p^{2}}{n+2}+\frac{s^{2}}{n}-\frac{1}{12}+{\tt d}$
(3.13)

with ${\tt d}=0$. Thus we conclude that the spectrum of scaling dimensions
develops a continuous component labeled by the parameter $s\in\mathbb{R}$.

It should be mentioned that in the work [23] RG trajectories were also
constructed for which $b(L)$ tends to a pure imaginary number in the scaling
limit. Among them is the low energy state mentioned in the paragraph
containining formula (3.8), whose typical pattern of Bethe roots in the
complex $u=-\frac{1}{2}\log(\zeta)$ plane is depicted in Fig. 3. Moreover, the
authors of ref.[23] propose that the imaginary values of $s$ that appear in
the scaling limit of the lattice model must satisfy the condition

$s=\pm{\rm i}\,\Big{(}-p-\frac{1}{2}-a\,\Big{)}\qquad\qquad{\rm
with}\qquad\qquad a=0,1,2,\ldots<-p-\tfrac{1}{2}\ \qquad\qquad({\tt d}=0)\,.$
(3.14)

$u$$\Im m(u)=\frac{\pi}{2}$

$u$$\Im m(u)=\frac{\pi}{2}$

Figure 4: On the left and right panels displayed are the Bethe roots in the
complex $u=-\frac{1}{2}\log(\zeta)$ plane for the ground state and an excited
state, respectively, of the spin chain with $L=10$. The excitation is built by
disbalancing the number of roots on the two lines with respect to the ground
state pattern. The states have total $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$
spin $\mathcal{S}=2$ while $n=2.9$.

###  3.2 ODE/IQFT correspondence

In order to characterize the scaling limit of a lattice system, where the
spectrum of scaling dimensions possesses a continuous component, mathematical
techniques are required that go beyond those used in the works [22, 23]. One
of these is an approach to the study of the scaling limit that is based on the
so-called ODE/IQFT correspondence [41, 42, 43, 44]. We found that the ODEs
which describe the scaling limit of the staggered six-vertex model with
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant BCs lie within the class of
differential equations considered in refs.[26, 15] in the context of the
lattice system with quasi-periodic BCs. This made the analysis performed in
our work possible. In addition, it allowed us to transfer over many previous
results concerning the differential equations. Here we briefly discuss the
application of the ODE/IQFT correspondence to the study of the scaling limit
of the spin chain, while referring to ref.[15] for technical details.

The primary Bethe states, i.e., the RG trajectories where the energy obeys the
asymptotic formula (3.3) with ${\tt d}=0$, are labeled by the
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ spin ${\cal S}$ and the RG invariant $s$
defined in eq. (3.11). The ODE/IQFT correspondence implies a relation between
the scaling limit of the eigenvalue of the $Q$-operator for $|\Psi_{L}\rangle$
and the spectral determinant of the ODE:

$\displaystyle\Bigg{[}\,-\frac{\mbox{d}^{2}}{{\mbox{d}z}^{2}}+\frac{p^{2}-\frac{1}{4}}{z^{2}}+\frac{2{\rm
i}s}{z}+1+\mu^{-2-n}\ z^{n}\,\Bigg{]}\,\psi=0\ .$ (3.15)

Here $p$ is given in terms of ${\cal S}$ as in eq. (3.5), $n>0$ is related to
the anisotropy parameter of the spin chain, see (3.2), while $\mu$ arises in
the scaling limit of $\zeta$. The spectral determinant $D(\mu\,|\,p,s)$ is
defined in the following way. One specifies a solution to the differential
equation by its behaviour in the vicinity of the singular point $z=0$:

$\psi_{p}(z)\to z^{\frac{1}{2}+p}\ \ \ \ \ \ \ {\rm as}\ \ \ \ \ \ \ z\to 0\
.$ (3.16)

For large $z$ the term $\mu^{-2-n}\,z^{n}$ in (3.15) becomes dominant and one
can define another solution through the $z\to+\infty$ asymptotic:

$\Xi(z)\asymp\Big{(}\frac{z}{\mu}\Big{)}^{-\frac{n}{4}}\,\exp\bigg{[}-\frac{2}{n+2}\
\Big{(}\frac{z}{\mu}\Big{)}^{\frac{n}{2}+1}\,{}_{2}F_{1}\Big{(}-\tfrac{1}{2},-\tfrac{n+2}{2n},\tfrac{n-2}{2n}\,\big{|}-\mu^{n+2}\,z^{-n}\Big{)}\
+\ o(1)\,\bigg{]}$ (3.17)

with ${}_{2}F_{1}(a,b,c|z)$ being the usual Gauss hypergeometric function and
we make the technical assumption that $\mu>0$ and $n\neq\frac{2}{2k-1}$ with
$k=1,2,\ldots\ $. The spectral determinant $D(\mu)=D(\mu\,|\,p,s)$ is given by

$D(\mu)=\sqrt{\pi}\ (n+2)^{-\frac{2p}{n+2}-\frac{1}{2}}\ \mu^{-p+\frac{1}{2}}\
\frac{W[\,\psi_{p},\Xi\,]}{\Gamma(1+\frac{2p}{n+2})}\,,$ (3.18)

where
$W[\,\psi_{p},\Xi\,]=\Xi\,\partial_{z}\psi_{p}-\psi_{p}\,\partial_{z}\Xi$ is
the Wronskian. The overall factor has been chosen to ensure the normalization

$D(0)=1\ .$ (3.19)

It should be mentioned that the procedure for computing the spectral
determinant based on formula (3.18) with the solutions $\psi_{p}$ and $\Xi$
obtained via a numerical integration of the ODE (3.15) works literally only
for $\Re e(p)\geq 0$. Nevertheless $D(\mu\,|\,p,s)$ turns out to be a
meromorphic function of $p$ and can be defined for generic complex values of
this variable via analytic continuation.

Rather than considering the eigenvalue of $\mathbb{Q}^{(0)}(\zeta)$ (2.35) for
a primary Bethe state, we instead discuss the scaling limit of

$A(\zeta)=\prod_{j=1}^{L-{\cal
S}}\,(1-\zeta/\zeta_{j})\,(1-\zeta\zeta_{j})\,.$ (3.20)

It does not involve the overall factor $\zeta^{\cal S}$ and the normalization
has been imposed such that $A(0)=1$. Then, the scaling relation between
$A(\zeta)$ and the spectral determinant reads as

$\displaystyle\operatorname*{slim}_{L\to\infty\atop b(L)\to
s}G^{(L)}\Big{(}-\mu^{2}\Big{|}\tfrac{2}{n+2}\Big{)}\
A\Big{(}\big{(}L/L_{0}\big{)}^{-\frac{n}{n+2}}\ {\rm
i}\mu\Big{)}={D}(\mu)\,.\qquad$ (3.21)

Here the function $G$ has been chosen to ensure the convergence of the limit
and is given by

$\displaystyle
G^{(L)}(E\,|g)=\exp\left(\sum^{\big{[}\frac{1}{2(1-g)}\big{]}}_{m=1}\frac{(-1)^{m}\,L}{m\cos(\pi
mg)}\left(\frac{L}{L_{0}}\right)^{2m(g-1)}E^{m}\right)\,,$ (3.22)

(this should be compared with formula (5.48) in the work [15]), where the
brackets $[\ldots]$ stand for the integer part, while

$L_{0}=\frac{\sqrt{\pi}\,\Gamma\big{(}1+\frac{1}{n}\big{)}}{4\Gamma\big{(}\tfrac{3}{2}+\frac{1}{n}\big{)}}\,.$
(3.23)

As above, the technical assumption

$n\neq\frac{2}{2k-1}$ (3.24)

with $k=1,2,\ldots$ is being made, see ref.[15] for details concerning the
case $n=\frac{2}{2k-1}$.

For the RG trajectories, where ${\tt d}$ entering into the asymptotic formula
for the energy (3.3) is greater than zero, the scaling relation (3.21) is
modified as follows. The l.h.s. remains the same, while for the r.h.s. one
takes $D(\mu)$ to be the spectral determinant for the differential equation:

$\displaystyle\Bigg{[}\,-\frac{\mbox{d}^{2}}{{\mbox{d}z}^{2}}+\frac{p^{2}-\frac{1}{4}}{z^{2}}+\frac{2{\rm
i}s}{z}+1+\sum_{a=1}^{\tt
d}\bigg{(}\frac{2}{(z-w_{a})^{2}}+\frac{n}{z(z-w_{a})}\bigg{)}+\mu^{-2-n}\
z^{n}\,\Bigg{]}\,\psi=0\ .$ (3.25)

Here

$\bm{w}=(w_{1},\ldots,w_{\tt d})$ (3.26)

are not arbitrary parameters. They are restricted by the condition that any
solution $\psi(z)$ of the differential equation must be single valued in the
vicinity of $z=w_{a}$. This leads to the coupled algebraic system:

$\displaystyle 4n\,w_{a}^{2}\\!\\!$ $\displaystyle+$ $\displaystyle\\!\\!8{\rm
i}s\,(n+1)\,w_{a}-(n+2)\ \big{(}(n+1)^{2}-4p^{2}\,\big{)}$ $\displaystyle+$
$\displaystyle\\!\\!4\ \sum_{b\not=a}^{{\tt
d}}\frac{w_{a}\,\big{(}\,(n+2)^{2}\,w_{a}^{2}-n(2n+5)\,w_{a}w_{b}+n(n+1)\,w_{b}^{2}\,\big{)}}{(w_{a}-w_{b})^{3}}=0\
\ \ \ \ \ \ \ \ (a=1,\ldots,{\tt d})\,.$

For generic $n$, $s$ and $p$ the number of solutions
$\bm{w}=\\{w_{a}\\}_{a=1}^{\tt d}$, up to permutations of the $w_{a}$’s, is
given by ${\rm par}_{2}({\tt d})$ – the number of bipartitions of ${\tt d}$.
The generating function for this combinatorial quantity reads as:

$\sum_{{\tt d}=0}^{\infty}{\rm par}_{2}({\tt d})\,{\tt q}^{\tt
d}=\prod_{j=1}^{\infty}\frac{1}{(1-{\tt q}^{j})^{2}}\ .$ (3.28)

For applications to the staggered six-vertex model with
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant BCs $p$ is not generic, but
should be taken as in (3.5), i.e., $2p=2{\cal S}+1-(n+2)$. Then, it turns out
that the number of solutions of the coupled equations (3.2) is typically less
than ${\rm par}_{2}({\tt d})$. To explain this phenomenon, let’s replace $p$
with $p_{\varepsilon}=p+\varepsilon^{2{\cal S}+1}$ where $0<\varepsilon\ll 1$.
Of the ${\rm par}_{2}({\tt d})$ solution sets of (3.2) with $p\mapsto
p_{\varepsilon}$ there exists those where

$w_{a}=O(\varepsilon)\,\qquad\qquad{\rm for}\qquad\qquad a=1,2,\ldots,2{\cal
S}+1\ .$ (3.29)

The other variables $\\{w_{a}\\}_{a=2{\cal S}+2}^{\tt d}$ tend to a finite,
nonvanishing limit as $\varepsilon\to 0$. Their limiting values obey (3.2)
with the replacements $p\mapsto{\cal S}+\frac{1}{2}+\frac{1}{2}\,(n+2)$ and
${\tt d}\mapsto{\tt d}-2{\cal S}-1$. In counting the solution sets of the
algebraic system on $w_{a}$ with $2p=2{\cal S}+1-(n+2)$ we only consider those
to be admissible where none of the $w_{a}$ are zero. It is easy to see that

$N({\tt d}\,|\,\mathcal{S}):=\\#\ {\rm of\ solution\ sets\ of\
\eqref{sksksk1}\ with}\ p\ {\rm as\ in\ \eqref{asmn12bass}}={\rm par}_{2}({\tt
d})-{\rm par}_{2}({\tt d}-2{\cal S}-1)$ (3.30)

(we take by definition ${\rm par}_{2}({\tt d})=0$ when its argument is a
negative integer).

We suppose that for a given trajectory $\\{|\Psi_{L}\rangle\\}$ with RG
invariants ${\cal S}$, $s$ and ${\tt d}$ there exists a solution set $\bm{w}$
of (3.2) such that the scaling relation (3.21) holds true with
$D(\mu)=D(\mu\,|\,\bm{w},p,s)$ being the spectral determinant for the
differential equation (3.25). Note that eqs. (3.16)-(3.18) for the definition
of $D(\mu)$ still remain valid since the inclusion of the extra sum in the ODE
has no impact on the leading asymptotics of $\psi_{p}$ and $\Xi$.

Unfortunately, we do not know of a way of rigorously proving the above
statement. We checked it numerically for a variety of cases using the so-
called sum rules. The analysis is not included here, as it is essentially the
same as that presented in section 11 of the work [15] concerning the staggered
six-vertex model with quasi-periodic BCs. We wish, however, to mention a
scaling relation involving the products over the Bethe roots,

$\Pi_{\pm}=\prod_{m=1}^{L-{\cal S}}q\,\big{(}\zeta_{m}\pm{\rm
i}q^{-1}\big{)}\,\big{(}\zeta_{m}^{-1}\pm{\rm i}q^{-1}\big{)}\,,$ (3.31)

that will become important later. It involves the coefficients
$\mathfrak{C}^{(\pm)}_{p,s}=\mathfrak{C}^{(\pm)}_{p,s}(\bm{w})$, which occur
in the large $\mu$ asymptotic expansion of $D(\mu)$:

$D(\mu\,|\,\bm{w},p,s)\,\asymp\,{\mathfrak{C}}_{p,s}^{(\pm)}(\bm{w})\
\big{(}\pm\mu\big{)}^{\pm\frac{{\rm
i}(n+2)s}{n}-p}\,\exp\bigg{(}\,\frac{2L_{0}}{\cos(\frac{\pi}{n})}\
\big{(}\pm\mu\big{)}^{\frac{n+2}{n}}+o(1)\,\bigg{)}\ \ \ \,{\rm for}\ \ \ \
\Re e(\pm\mu)>0$ (3.32)

(again, we assume that $n\neq\frac{2}{2k-1}$ with $k=1,2,\ldots\ $). For the
case when ${\tt d}=0$, the coefficients are given by

$\mathfrak{C}_{p,s}^{(0,\pm)}=\sqrt{\frac{2\pi}{n+2}}\ \ \ 2^{-p\pm\frac{{\rm
i}(n+2)s}{n}}\ (n+2)^{-\frac{2p}{n+2}}\
\frac{\Gamma(1+2p)}{\Gamma(1+\frac{2p}{n+2})\,\Gamma(\frac{1}{2}+p\pm{\rm
i}s)}\ \,.$ (3.33)

In general,

$\mathfrak{C}^{(\pm)}_{p,s}(\bm{w})=\mathfrak{C}_{p,s}^{(0,\pm)}\,\check{\mathfrak{C}}^{(\pm)}_{p,s}(\bm{w})\,,$
(3.34)

where $\check{\mathfrak{C}}^{(\pm)}$ are normalized to be one for ${\tt d}=0$.
A closed form expression for $\check{\mathfrak{C}}^{(\pm)}$ for general ${\tt
d}=0,1,2,\ldots$ was obtained in ref.[45] and, for the reader’s convenience,
is reproduced in this paper in Appendix A. The scaling relation reads as

$\Pi_{\pm}\asymp\frac{C}{2\cos(\frac{\pi}{n+2})}\,\mbox{e}^{\pm\frac{\pi}{n}s}\,{\mathfrak{C}}_{p,s}^{(\pm)}(\bm{w})\,\left(\frac{L}{L_{0}}\right)^{-\frac{np}{n+2}\pm{\rm
i}s}\left(\frac{4n}{n+2}\right)^{L}\,\big{(}1+O(L^{-\epsilon})\big{)}\ .$
(3.35)

It involves a non-universal constant $C$, which is expressed in terms of

$\hat{\tau}(\omega)=-\frac{1}{4\pi}\
\frac{\sinh(\frac{\pi(n-1)}{4(n+2)}\,\omega)}{\sinh(\frac{\pi\omega}{4(n+2)})\cosh(\frac{n\pi\omega}{4(n+2)})}$
(3.36)

and the Lerch transcendent

$\Phi(z,s,a)=\sum_{m=0}^{\infty}\frac{z^{m}}{(m+a)^{s}}\,.$ (3.37)

Namely,

$C=\exp\Bigg{(}2\int_{-\infty}^{\infty}\mbox{d}\omega\,\bigg{(}\frac{\hat{\tau}(\omega)}{\omega}\,\Big{(}\Im
m\Big{[}\mbox{e}^{\frac{2{\rm i}\pi}{n+2}}\,\Phi\big{(}\mbox{e}^{-\frac{{\rm
i}n\pi}{n+2}}\,,\,1\,,\,1-\tfrac{{\rm
i}\omega}{4}\big{)}\Big{]}-\frac{\pi}{n+2}-\frac{2}{\omega}\,\Big{)}-\frac{n-1}{2\pi\omega^{2}}\bigg{)}\Bigg{)}\
.$ (3.38)

Also, the notation $O(L^{-\epsilon})$ with some $\epsilon>0$ means that the
correction terms fall off faster than any power of the logarithm of $L$. The
asymptotic formula (3.35) is the analogue for the lattice system with open
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant BCs of a product rule
presented in the work [15], see (11.19) therein.

###  3.3 Quantization condition

Consider the problem of computing the spectrum of conformal dimensions of the
critical lattice system and the classification of the space of states ${\cal
H}$ appearing in the scaling limit. The asymptotic relation for the energy
(3.3), together with the numerical analysis from refs.[22, 23] concerning the
large $L$ behaviour of $b(L)$ for a class of states, shows that ${\cal H}$
contains two components. There is a ‘continuous component’, where
$s=\operatorname*{slim}_{L\to\infty}b(L)$ parameterising the conformal
dimensions as in eq. (3.13) may take any real value, as well as a ‘discrete’
one for which $s$ belongs to a finite set of pure imaginary numbers. In the
next section a full description of these linear spaces will be given. Among
other things, this includes the admissible values of pure imaginary $s$ as
well as the density of states characterizing the continuous spectrum. The
results are based on an analysis of the so-called ‘quantization condition’ for
$b(L)$, which we shall obtain below.

The key observation is that the square root of the eigenvalue of the quasi-
shift operator (3.7), used in the computation of $b(L)$ (3.6), may be
expressed as

$\sqrt{B}=(-1)^{L-{\cal S}}\ \frac{\Pi_{+}}{\Pi_{-}}\ .$ (3.39)

Here $\Pi_{\pm}$ stand for the products over the Bethe roots defined in
(3.31). Let us substitute these products for their asymptotics (3.35) with $s$
replaced by the ‘running coupling’ $b(L)$. Upon rearranging and making use of
eq. (3.6), one finds

$\bigg{(}\frac{L}{L_{0}}\bigg{)}^{2{\rm i}s}\mbox{e}^{\frac{{\rm
i}}{2}\delta(\bm{w},p,s)}\Big{|}_{s=b(L)}=\sigma+O(L^{-\epsilon})$ (3.40)

with

$\mbox{e}^{\frac{{\rm
i}}{2}\delta(\bm{w},p,s)}=\frac{\mathfrak{C}^{(+)}_{p,s}(\bm{w})}{\mathfrak{C}^{(-)}_{p,s}(\bm{w})}$
(3.41)

and

$\sigma=(-1)^{L-{\cal S}}\ .$ (3.42)

The above relation, which will be henceforth referred to as the quantization
condition, is interpreted in the following way. Given an RG trajectory
$\\{|\Psi_{L}\rangle\\}$ one computes $p$ from the value of the
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ spin ${\cal S}$ via the definition (3.5)
as well as the sign factor $\sigma=(-1)^{L-{\cal S}}$. The latter is kept
fixed along $\\{|\Psi_{L}\rangle\\}$ since, in the construction of RG
trajectories, $L$ is always increased by two (see section 2). Then $b(L)$
computed from the Bethe roots according to eqs. (3.7) and (3.6) obeys (3.40)
for some solution set $\bm{w}=\\{w_{a}\\}_{a=1}^{\tt d}$ of the algebraic
system (3.2) with $s$ replaced by $b(L)$. The ‘phase shift’ $\delta$ is given
in terms of the coefficients $\mathfrak{C}^{(\pm)}_{p,s}(\bm{w})$ (3.34),
which were introduced in the previous subsection. For the primary Bethe states
with ${\tt d}=0$, one has

$\mbox{e}^{\frac{{\rm i}}{2}\delta(\emptyset,p,s)}=2^{\frac{2{\rm
i}(n+2)s}{n}}\ \frac{\Gamma(\frac{1}{2}+p-{\rm i}s)}{\Gamma(\frac{1}{2}+p+{\rm
i}s)}\qquad\qquad\qquad({\tt d}=0)\,.$ (3.43)

For ${\tt d}=1,2,3,\ldots$ one must make use of eqs. (3.33), (3.34) together
with the explicit formula for $\check{\mathfrak{C}}^{(\pm)}_{p,s}(\bm{w})$ as
a function of $p$, $s$ and $\bm{w}=\\{w_{a}\\}_{a=1}^{\tt d}$ contained in
Appendix A.

Let’s take a moment to discuss the quantization condition (3.40) for the
primary Bethe states in the context of the results of the previous works [23,
22]. We start with the asymptotic (3.10) for $b(L)$ that was observed for a
class of RG trajectories labelled by the integer $\mathfrak{m}$. In this case,
it is useful to take the logarithm of both sides of formula (3.40) with the
phase shift as in (3.43) and write it in the form:

$2b_{\mathfrak{m}}\log\bigg{(}\frac{L}{L_{0}}\bigg{)}-{\rm
i}\log\bigg{[}2^{\frac{2{\rm i}(n+2)b_{\mathfrak{m}}}{n}}\
\frac{\Gamma(\frac{1}{2}+p-{\rm i}b_{\mathfrak{m}})}{\Gamma(\frac{1}{2}+p+{\rm
i}b_{\mathfrak{m}})}\bigg{]}=\pi\mathfrak{m}+O(L^{-\epsilon})\ .$ (3.44)

For the class of states we are considering $b_{\mathfrak{m}}(L)$ goes to zero
as $L\to\infty$. As a result, the second term in the l.h.s. of the above
relation containing the $\Gamma$-functions also tends to zero and, to a first
approximation, can be ignored. This way one obtains (3.10). Formula (3.44)
provides a refinement to the large $L$ asymptotic behaviour of
$b_{\mathfrak{m}}(L)$ which takes into account all power law corrections in
$1/\log(L)$. To demonstrate its accuracy, some numerical data obtained from
the Bethe roots for a primary Bethe state $|\Psi_{L}\rangle$ is compared with
the predictions coming from the quantization condition in Fig. 5.

$b(L)$$L$ Figure 5: The numerical data comes from an RG trajectory
$\\{|\Psi_{L}\rangle\\}$, where the representative state for $L=10$ is the one
whose pattern of Bethe roots is displayed on the right panel of fig. 4. In
particular, it has ${\tt d}=0$; total $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$
spin ${\cal S}=2$; while the integer $\mathfrak{m}$ — the difference between
the number of roots lying on the real line and the line $\Im
m(u)=\frac{\pi}{2}$ in the complex $u$ plane — is held fixed along the flow to
be $\mathfrak{m}=4$. The crosses depict the numerical values of
$b(L)=b_{\mathfrak{m}}(L)$ for different $L$ which were computed from the
Bethe roots corresponding to $|\Psi_{L}\rangle$ via formulae (3.6) and (3.7).
The dashed line gives the predictions coming from the quantization condition
(3.44) with $\mathfrak{m}=4$ and the correction terms ignored. Note that the
branch of the logarithm was fixed by requiring that the l.h.s. of (3.44) is a
continuous function for real $b_{\mathfrak{m}}$ which vanishes at
$b_{\mathfrak{m}}=0$.

Another possibility of how (3.40) could be satisfied for $L\gg 1$ is if $b(L)$
approaches a singularity of the phase shift. The explicit formula (3.43),
valid for ${\tt d}=0$, shows that these occur for pure imaginary $s$ when
$\frac{1}{2}+p\pm{\rm i}s$ is a positive integer. If the imaginary part of
$b(L)$ is positive, then the vanishing of the first term in the l.h.s. of
(3.40) may be compensated if $b(L)$ tends to a pole of $\mbox{e}^{\frac{{\rm
i}}{2}\delta}$, i.e.,

$\operatorname*{slim}_{L\to\infty}b(L)=s\qquad\qquad{\rm with}\qquad\qquad
s={\rm i}\big{(}-p-\tfrac{1}{2}-\ell\big{)}\qquad\qquad\big{(}\Im
m(b(L))>0\big{)}$ (3.45)

and $\ell=0,1,2,\ldots\ $. This is the same as eq. (3.14) with the sign factor
chosen to be ‘$+$’. The upper bound on $\ell$ in that equation ensures the
condition $\Im m(b(L))>0$. The minus version of the relation is deduced from
(3.40) by means of similar arguments.

A verification of the quantization condition (3.40) was carried out using
numerical data obtained from the lattice model with $L=10$. The spin chain
Hamiltonian was constructed and the first few hundred lowest energy Bethe
states were found via a direct diagonalization procedure. Note that, because
of the $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ symmetry, it was sufficient to
focus on the sector with $S^{z}=0$ as there is always one state
$|\Psi_{L}\rangle$ from the $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ multiplet
${\cal M}_{\cal S}$ lying in this sector. For each Bethe state, apart from the
energy, the eigenvalue of the quasi-shift operator was computed from which we
extracted $b$. The numerical data for $b(L)$ was compared with $b_{*}(L)$ —
the predictions coming from the quantization condition. The latter was
obtained by considering (3.40) with $L=10$ and the correction term
$O(L^{-\epsilon})$ ignored as an equation from which $b_{*}(L)$ could be
determined numerically. Note that the phase shift $\mbox{e}^{\frac{{\rm
i}}{2}\delta}$ therein depends on $b$ transcendentally via the Gamma functions
as in (3.43) and algebraically through the set $\bm{w}$, which solves the
coupled system (3.2) with $s\mapsto b$. For given ${\tt d}\leq 3$ we took the
${\rm par}_{2}({\tt d})-{\rm par}_{2}({\tt d}-2{\cal S}-1)$ equations which
are obtained from the quantization condition by specializing the phase shift
$\delta=\delta(\bm{w},p,s)$ to different solution sets
$\bm{w}=\\{w_{a}\\}^{\tt d}_{a=1}$ of (3.2). For each of them we found all
possible solutions $b_{*}(L)$ that lie in a suitably chosen finite portion of
the strip $\big{|}\Im m(b_{*})\big{|}<\frac{n}{2}+\varepsilon$ with
$0<\varepsilon\ll 1$ of the complex $b$ plane. Some of the results for the
comparison of $b(L)$ and $b_{*}(L)$ for $L=10$ are presented in Fig. 6. They
motivated us to make the following conjecture.

$b$$\Im m(b)=\frac{n}{2}$${\tt d}=2$$\Im m(b)=-\frac{n}{2}$

$b$$\Im m(b)=\frac{n}{2}$$\Im m(b)=-\frac{n}{2}$${\tt d}=1$

Figure 6: Numerical data for $b(L)$ and $b_{*}(L)$ is plotted in the complex
$b$ plane for the lattice system with $L=10$. Out of the few hundred states
that were considered, only those with ${\tt d}=2$ (top panel), ${\tt d}=1$
(bottom panel) and $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ spin ${\cal S}=2$
were used to produce the figure. The open circles correspond to $b(L)$ that
was extracted from the Bethe roots by means of eqs. (3.7), (3.6). The filled
shapes represent $b_{*}(L)$ obtained from an analysis of the quantization
condition (3.40). The green squares and blue circles are used to distinguish
whether in the scaling limit $b_{*}(L)$ becomes a pure imaginary number or a
real number, respectively. The two green squares in the top panel for which
$\operatorname*{slim}_{L\to\infty}b_{*}(L)=\pm\frac{{\rm i}n}{2}$ correspond
to the same state. It seems interesting to note that the agreement between
$b(L)$ and $b_{*}(L)$ is better than in the case of the lattice model with
quasi-periodic boundary conditions imposed, compare the above figures with the
ones contained in Appendix C of ref.[15]. The parameter $n$ was taken to be
$n=2.9$.

Conjecture: For any RG trajectory $\\{|\Psi_{L}\rangle\\}$ labeled by ${\cal
S}$, ${\tt d}$ and a solution set $\bm{w}=\\{w_{a}\\}_{a=1}^{\tt d}$ of eq.
(3.2) the corresponding value of
$b(L)=\frac{n}{2\pi}\log\big{(}\sqrt{B}\,\big{)}$, with $\sqrt{B}$ computed
according to formula (3.7), obeys the quantization condition (3.40).
Conversely, let $b_{*}(L)\,:\ -\frac{n}{2}<\Im
m\big{(}b_{*}(L)\big{)}\leq\frac{n}{2}$ be a solution of the relation (3.40)
with the correction terms ignored. Then, there exists a unique
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ multiplet ${\cal M}_{\cal S}$ for which
$\sqrt{B}$ obtained from $|\Psi_{L}\rangle\in{\cal M}_{\cal S}$ is such that
$\sqrt{B}-\exp\big{(}\frac{2\pi}{n}\,b_{*}(L)\big{)}$ goes to zero faster than
any power of the logarithm of $L$.

##  4 Space of states in the scaling limit

###  4.1 Continuous and discrete spectrum

A key result of this paper is the conjecture, above, which was motivated by
our numerical work. It describes a certain one to one relation between $b(L)$
and $b_{*}(L)$. The former is computed via the Bethe roots corresponding to a
state $|\Psi_{L}\rangle$ in a multiplet ${\cal M}_{\cal S}$, labeled by
$2p=2{\cal S}+1-(n+2)$, the non-negative integer ${\tt d}$ and one of the
$N({\tt d}\,|\,{\cal S})={\rm par}_{2}({\tt d})-{\rm par}_{2}({\tt d}-2{\cal
S}-1)$ solution sets $\bm{w}=\\{w_{a}\\}_{a=1}^{\tt d}$ of the algebraic
system (3.2). The notation $b_{*}(L)$ stands for a solution of the
quantization condition (3.40) treated as an equation for $b(L)$ with the
correction terms ignored. Accepting the conjecture to be true, one can
determine the spectrum of $b(L)$ for the low energy states at large $L$ via a
study of the quantization condition. In turn, the results allow one to
characterize the spectrum of conformal dimensions as well as the space of low
energy states in the scaling limit.

Consider the quantization condition (3.40) and suppose that the phase shift
$\delta(\bm{w},p,s)|_{s\mapsto b(L)}\ll\log(L)$. Then the first term dominates
and one can develop an asymptotic expansion for $b(L)$ in $1/\log(L)$. The
leading and subleading asymptotic behaviour reads as

$b_{\mathfrak{m}}(L)=\frac{2\pi\mathfrak{m}-\delta_{0}}{4\log\big{(}\mbox{e}^{\frac{1}{4}\delta^{\prime}_{0}}L/L_{0}\big{)}}+O\big{(}(\log
L)^{-3}\big{)}\qquad\qquad\qquad\big{(}L\gg 1,\ \mathfrak{m}\ {\rm-\
fixed}\big{)}\ .$ (4.1)

Here we use the notation

$\delta_{0}=\delta|_{s=0}\,,\qquad\qquad\qquad\delta^{\prime}_{0}=\partial_{s}\delta|_{s=0}\,,$
(4.2)

while $\mathfrak{m}$, which labels the different $b(L)$ obeying the
quantization condition, comes about as a result of taking the logarithm of
(3.40) and is an even or odd integer for $\sigma=+1$ or $\sigma=-1$,
respectively. Formula (4.1) shows that, in general, $b_{\mathfrak{m}}(L)$ is a
complex number since $\delta_{0}$ and $\delta^{\prime}_{0}$ are generically
complex. However, while $\Im m\big{(}b_{\mathfrak{m}}(L)\big{)}\sim
1/\log(L)\to 0$ as $L\to\infty$ the magnitude of the real part is controlled
by the integer $\mathfrak{m}$ which, for the low energy states, may take any
values up to some $\mathfrak{m}_{\rm max}\ll L$. Numerical work leads us to
suppose that $\lim_{L\to\infty}b_{\mathfrak{m}_{\rm max}}=\infty$.

Let ${\cal H}_{L|{\cal S}}^{({\rm cont})}$ denote the set of low energy states
$|\Psi_{L}\rangle$ with fixed value of the
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ spin ${\cal S}=0,1,2,\ldots$ such that
$\Im m\big{(}b(L)\big{)}\to 0$ as $L\to\infty$. Recall that the states come in
multiplets ${\cal M}_{\cal S}$ and one can choose a basis for that multiplet
in which the $z$ -projection of the total spin,
$\mathbb{S}^{z}=\frac{1}{2}\sum_{J=1}^{2L}\sigma_{J}^{z}$, is diagonal. This
yields the refinement

${\cal H}_{L|{\cal S}}^{({\rm cont})}=\bigcup_{S^{z}=-{\cal S}}^{{\cal
S}}{\cal H}_{L|{\cal S},S^{z}}^{({\rm cont})}\ .$ (4.3)

Each low energy Bethe state in ${\cal H}_{L|{\cal S},S^{z}}^{({\rm cont})}$ is
uniquely specified by the non-negative integer ${\tt d}$, a solution set
$\bm{w}=\\{w_{a}\\}_{a=1}^{\tt d}$ of the algebraic system (3.2) with
$s\mapsto b(L)$, as well as the even or odd integer $\mathfrak{m}$ that enters
into the asymptotics (4.1). For $L\gg 1$ the value of $b_{\mathfrak{m}}(L)$
becomes densely distributed in the segment $(-b_{\mathfrak{m}_{\rm
max}}(L),+b_{\mathfrak{m}_{\rm max}}(L))$. The density of states is obtained
from the quantization condition (3.40) written in logarithmic form:

$\big{[}4s\log(L/L_{0})+\delta(\bm{w},p,s)\big{]}|_{s\mapsto
b_{\mathfrak{m}}(L)}=2\pi\mathfrak{m}+O(L^{-\epsilon})\ .$ (4.4)

Here the branch of the logarithm needed to define $\delta$ from the relation
(3.41) is taken such that the phase shift is a continuous function of $s$ in
the strip $|\Im m(s)|<\varepsilon$ for some $\varepsilon>0$ (it is being
assumed that $\mbox{e}^{\frac{{\rm i}}{2}\delta}$ contains no zeroes or poles
for real $s$). The term in the square brackets in the left hand side of (4.4)
is a monotonic function of $s$ for $L$ sufficiently large. This way one
concludes that the number of states in ${\cal H}_{L|{\cal S},S^{z}}^{({\rm
cont})}$ with fixed ${\tt d}$ such that $\Re e\big{(}b(L)\big{)}$ lies in the
interval $(s,s+\Delta s)\in(-b_{\mathfrak{m}_{\rm
max}}(L),+b_{\mathfrak{m}_{\rm max}}(L))$ is given by $\rho^{({\tt
d})}_{p}(s)\,\Delta s$ with666This line of arguments is analogous to the
standard derivation in the root density approach. One introduces a monotonic
increasing counting function which evaluates to (half-)integers at the Bethe
roots similar as the l.h.s of (4.4) evalutes to odd/even integers
$\mathfrak{m}$ multiplied by $\pi$ when $s$ is swapped for
$b_{\mathfrak{m}}(L)$ and $L\gg 1$. Differentiating the counting function in
the root density approach yields the root density, while we obtain the density
of states (4.5) by differentiating (4.4) and dividing by $4\pi$.

$\rho^{({\tt d})}_{p}(s)=\frac{1}{\pi}\,N({\tt d}\,|\,{\cal
S})\,\log\big{(}2^{\frac{n+2}{n}}\,L/L_{0}\big{)}+\frac{1}{2\pi{\rm i}}\
\partial_{s}\log\Bigg{[}\bigg{(}\frac{\Gamma(\frac{1}{2}+p-{\rm
i}s)}{\Gamma(\frac{1}{2}+p+{\rm i}s)}\bigg{)}^{N({\tt d}\,|\,{\cal
S})}\,\prod_{\bm{w}\atop{\tt
d}{\rm\,-\,fixed}}\frac{\check{\mathfrak{C}}_{p,s}^{(+)}(\bm{w})}{\check{\mathfrak{C}}^{(-)}_{p,s}(\bm{w})}\,\Bigg{]}$
(4.5)

up to corrections which vanish as $L\to\infty$. The product over $\bm{w}$
appearing in the r.h.s. goes over all the $N({\tt d}\,|\,\mathcal{S})$ (3.30)
solution sets of the algebraic system (3.2) with ${\tt d}$ fixed. Also, recall
that $2p=2{\cal S}+1-(n+2)$.

In the work [45] a formula is presented for a product over $\bm{w}$ similar to
the one appearing in the r.h.s. of (4.5) (see also Appendix B of [15]). It is
valid for the case of generic $p$ and $n$ when the number of solution sets
$\bm{w}$ of (3.2) is ${\rm par}_{2}({\tt d})$. Based on this, one can derive
the result:

$\displaystyle\prod_{\bm{w}\atop{\tt
d}{\rm\,-\,fixed}}\frac{\check{\mathfrak{C}}_{p,s}^{(+)}(\bm{w})}{\check{\mathfrak{C}}^{(-)}_{p,s}(\bm{w})}$
$\displaystyle=$ $\displaystyle(-1)^{{\rm par}_{2}({\tt d}-2{\cal
S}-1)}\prod_{a=0}^{{\tt d}-1}\bigg{(}\frac{\frac{1}{2}+a+p-{\rm
i}s}{\frac{1}{2}+a+p+{\rm i}s}\bigg{)}^{N({\tt d}\,|\,{\cal S})-N_{a}^{+}({\tt
d}\,|\,{\cal S})}$ (4.6) $\displaystyle\times$ $\displaystyle\prod_{a=0}^{{\tt
d}-1}\bigg{(}\frac{\frac{1}{2}+a-p-{\rm i}s}{\frac{1}{2}+a-p+{\rm
i}s}\bigg{)}^{N({\tt d}\,|\,{\cal S})-N_{a}^{-}({\tt d}\,|\,{\cal S})}$

with the integers $N^{\pm}_{a}$ being defined through their generating
function as

$\sum_{{\tt d}=0}^{\infty}N_{a}^{\pm}({\tt d}\,|\,{\cal S})\,{\tt q}^{\tt
d}=\bigg{(}\prod_{j=1}^{\infty}(1-{\tt
q}^{j})^{-2}\bigg{)}\,\sum_{m=0}^{\infty}(-1)^{m}\,\big{(}1-{\tt q}^{(1\pm
m)(2{\cal S}+1)}\big{)}\,{\tt q}^{ma+\frac{m(m+1)}{2}}\ .$ (4.7)

Notice that

$N_{a}^{+}({\tt d}\,|\,{\cal S})=N({\tt d}\,|\,{\cal S})-N_{-a-1}^{-}({\tt
d}\,|\,{\cal S})\,.$ (4.8)

The scaling limit of the RG trajectory $L\mapsto|\Psi_{L}\rangle\in{\cal
H}_{L|{\cal S}}^{({\rm cont})}$ labeled by real $s$, the integers ${\cal S}$,
$S^{z}$, ${\tt d}$ and the solution set $\bm{w}$ yields

$\operatorname*{slim}_{L\to\infty}|\Psi_{L}\rangle=|\psi_{p,s}^{(S^{z})}(\bm{w})\rangle\
.$ (4.9)

One can define the linear span ${\cal H}_{{\cal S}}^{({\rm cont})}$ of all
such possible states with fixed ${\cal S}$. The above discussion implies that
this linear space admits the decomposition

${\cal H}_{\cal S}^{({\rm cont})}=\bigoplus_{S^{z}=-{\cal S}}^{{\cal S}}{\cal
H}_{{\cal S},S^{z}}^{({\rm cont})}\,,$ (4.10)

where each of the spaces ${\cal H}_{{\cal S},S^{z}}^{({\rm cont})}$ is
isomorphic to ${\cal H}_{{\cal S},{\cal S}}^{({\rm cont})}$ and

${\cal H}_{{\cal S},{\cal S}}^{({\rm
cont})}=\int^{\oplus}_{\mathbb{R}}\mbox{d}s\ \bigoplus_{{\tt
d}=0}^{\infty}{\cal H}^{({\rm cont},{\tt
d})}_{p,s}\qquad\qquad\qquad\qquad\qquad\big{(}2p=2{\cal S}+1-n-2\big{)}\,.$
(4.11)

The components appearing inside the direct sum are finite dimensional such
that

$\dim\big{(}{\cal H}_{p,s}^{({\rm cont},{\tt d})}\big{)}=N({\tt
d}\,|\,\mathcal{S})\ .$ (4.12)

For the low energy states where the value of $\Im m\big{(}b(L)\big{)}$ is
nonvanishing in the limit $L\to\infty$ so that they do not belong to ${\cal
H}_{L|{\cal S}}^{({\rm cont})}$, one may repeat the similar analysis that was
performed in ref.[15] in the context of the staggered six-vertex model with
quasi-periodic BCs (see Appendix B therein). Let’s denote by ${\cal
H}_{L|{\cal S},S^{z}}^{({\rm disc})}$ the set of such states
$|\Psi_{L}\rangle$ with given quantum numbers ${\cal S}$ and $S^{z}$. The
quantization condition (3.40) implies that the set $\bm{w}$ and
$s=\operatorname*{slim}_{L\to\infty}b(L)$ labelling the RG trajectory
$\\{|\Psi_{L}\rangle\\}$ must be such that

$\mbox{e}^{-\frac{{\rm i}}{2}\delta(\bm{w},p,s)}=0\qquad{\rm if}\qquad\Im
m(s)>0\,,\qquad\qquad\qquad\mbox{e}^{+\frac{{\rm
i}}{2}\delta(\bm{w},p,s)}=0\qquad{\rm if}\qquad\Im m(s)<0\,.$ (4.13)

We supplement this with the additional constraint on the imaginary part on
$s$:

$-\frac{n}{2}<\Im m(s)\leq\frac{n}{2}\,.$ (4.14)

It comes from the inequality (3.8), while the line $\Im m(s)=-\frac{{\rm
i}n}{2}$ was excluded from the interval in order to avoid double counting
states with $\operatorname*{slim}_{L\to\infty}b(L)=\pm\frac{{\rm i}n}{2}$. It
turns out that the phase shift satisfies:

$\mbox{e}^{\frac{{\rm i}}{2}\delta(\bm{w},p,s)}=\mbox{e}^{-\frac{{\rm
i}}{2}\delta(-\bm{w},p,-s)}\,.$ (4.15)

Here $-\bm{w}$ denotes the set $\\{-w_{a}\\}_{a=1}^{\tt d}$, where if $\bm{w}$
obeys the algebraic system (3.2) then $-\bm{w}$ obeys the same set of
equations with $s\mapsto-s$. This allows one to focus on the case with $0<\Im
m(s)\leq\frac{n}{2}$, while results for $-\frac{n}{2}<\Im m(s)<0$ follow by
simply flipping the sign $s\mapsto-s$.

The analysis of (4.13) is greatly facilitated by the relation

$\prod_{\bm{w}\atop{\tt d}{\rm\,-\,fixed}}\mbox{e}^{\frac{{\rm
i}}{2}\delta(\bm{w},p,s)}=\bigg{(}2^{(2n+4)\frac{{\rm i}s}{n}}\
\frac{\Gamma(\frac{1}{2}+p-{\rm i}s)}{\Gamma(\frac{1}{2}+p+{\rm
i}s)}\bigg{)}^{N({\tt d}\,|\,{\cal S})}\,\prod_{\bm{w}\atop{\tt
d}{\rm\,-\,fixed}}\frac{\check{\mathfrak{C}}_{p,s}^{(+)}(\bm{w})}{\check{\mathfrak{C}}^{(-)}_{p,s}(\bm{w})}$
(4.16)

with the last term in the r.h.s. being given by the product (4.6). It follows
from the definitions (3.41), (3.33) and (3.34). Also, we’ll need the following
assumptions on the positions of the poles and zeroes of the function
$\mbox{e}^{\frac{{\rm i}}{2}\delta(\bm{w},p,s)}$, which were verified
numerically for small ${\tt d}\leq 3$:

  * (i)

The points where $\mbox{e}^{\frac{{\rm i}}{2}\delta(\bm{w},p,s)}$ is singular
do not coincide with the location of any zero of $\mbox{e}^{\frac{{\rm
i}}{2}\delta(\bm{w}^{\prime},p,s)}$ with $\bm{w}^{\prime}$ being some other
solution set of (3.2).

  * (ii)

All singularities of $\mbox{e}^{\frac{{\rm i}}{2}\delta(\bm{w},p,s)}$ in the
complex $s$ plane are simple poles. Notice that, in view of eq. (4.15), this
implies that all of its zeroes are simple as well.

From assumption (i), any pole or zero of $\mbox{e}^{\frac{{\rm
i}}{2}\delta(\bm{w},p,s)}$ must appear as a pole/zero in the r.h.s. of (4.16).
This way, one finds that the values of $s$ for which the first condition in
(4.13) is obeyed are $s=\pm s_{a}$, with

$s_{a}={\rm i}\big{(}-p-\tfrac{1}{2}-a\big{)}={\rm i}\big{(}\tfrac{n}{2}-{\cal
S}-a\big{)}\qquad\qquad{\rm and}\qquad\qquad 0\leq a+{\cal S}<\tfrac{n}{2}\,,\
\ \ a\in\mathbb{Z}\,,$ (4.17)

where the bound on $a+{\cal S}$ comes from (4.14). Moreover, due to (ii) one
can determine the number of solution sets $\bm{w}$ with (4.13) being satisfied
at $s=s_{a}$ by reading off the multiplicity of that pole/zero from eqs.
(4.16) and (4.6). This would coincide with the dimension of the linear space
${\cal H}_{p,s}^{({\rm disc},{\tt d})}$, which is the span of all states of
the form $|\psi_{p,s}^{(S^{z})}(\bm{w})\rangle$ having fixed ${\cal S}$,
$S^{z}$, ${\tt d}$ and $s$ with $\Im m(s)\neq 0$. One finds the number of such
$\bm{w}$ to be $N_{a}^{+}({\tt d})$.

Define the space ${\cal H}_{{\cal S},S^{z}}^{({\rm disc})}$ as the linear span
of all the states that appear in the scaling limit of ${\cal H}_{L|{\cal
S},S^{z}}^{({\rm disc})}$. These are isomorphic to ${\cal H}_{{\cal S},{\cal
S}}^{({\rm disc})}$and the analysis of the quantization condition performed
above implies that:

${\cal H}_{{\cal S},{\cal S}}^{({\rm
disc})}=\bigoplus_{s\in\Sigma^{+}\cup\Sigma^{-}}\ \bigoplus_{{\tt
d}=0}^{\infty}{\cal H}_{p,s}^{({\rm disc},{\tt d})}\ .$ (4.18)

Here $\Sigma(\pm)$ denote the finite sets of pure imaginary numbers:

$\Sigma^{+}=\big{\\{}s\,:\ \tfrac{n}{2}+{\rm i}s\in\mathbb{Z}\,,\ \Im
m(s)\in(0,\tfrac{n}{2}]\big{\\}}\,,\quad\qquad\Sigma^{-}=\big{\\{}s\,:\
\tfrac{n}{2}-{\rm i}s\in\mathbb{Z}\,,\ \Im
m(s)\in(-\tfrac{n}{2},0)\big{\\}}\,,$ (4.19)

which incorporate the bound on the imaginary part of $s$ (4.14). Each
component ${\cal H}_{p,s}^{({\tt d})}$ is finite dimensional and

$\dim\big{(}{\cal H}_{p,s}^{({\rm disc},{\tt d})}\big{)}=N^{+}_{a}({\tt
d}\,|\,{\cal S})\qquad\qquad{\rm with}\qquad\qquad a=\tfrac{n}{2}-{\cal
S}\pm{\rm i}s\in\mathbb{Z}$ (4.20)

(here and below, when a condition involving $\pm{\rm i}s$ appears we mean it
is to be satisfied for some choice of the sign $+$ or $-$).

The following comment is in order here. For the case $a<0$, the integers
$N_{a}^{+}({\tt d}\,|\,{\cal S})$ (4.7) are all zero for ${\tt d}=0$:

$N_{a}^{+}(0\,|\,{\cal S})=0\qquad{\rm for}\qquad a=-1,-2,-3,\ldots\ .$ (4.21)

As a result, for the primary Bethe states the limiting values of $\Im
m\big{(}b(L)\big{)}$ are given by $s=\pm s_{a}$ (4.17) with the extra
condition imposed that $a\geq 0$. Thus one recovers the results of the work
[23], see also formula (3.14). Of course, RG trajectories exist for the spin
chain which are not primary Bethe states that are labeled by $s=\pm s_{a}$
with $a<0$.

We conjecture that any RG trajectory of the lattice model with given
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ spin ${\cal S}$ and eigenvalue of the
$z$ -projection of the total spin operator $S^{z}$ belongs either to ${\cal
H}_{L|{\cal S},S^{z}}^{({\rm cont})}$ or ${\cal H}_{L|{\cal S},S^{z}}^{({\rm
disc})}$. Thus, the full space of low energy states of the lattice system in
the scaling limit becomes the linear space

${\cal H}={\cal H}^{({\rm cont})}\oplus{\cal H}^{({\rm disc})}$ (4.22)

with

${\cal H}^{({\rm cont})}=\bigoplus_{{\cal
S}=0}^{\infty}\,\bigoplus_{S^{z}=-{\cal S}}^{{\cal S}}{\cal H}_{{\cal
S},S^{z}}^{({\rm cont})}\,,\qquad\qquad{\cal H}^{({\rm
disc})}=\bigoplus_{{\cal S}=0}^{\infty}\,\bigoplus_{S^{z}=-{\cal S}}^{{\cal
S}}{\cal H}_{{\cal S},S^{z}}^{({\rm disc})}\ .$ (4.23)

We call ${\cal H}^{({\rm cont})}$ the ‘continuous spectrum’ due to the
presence of a direct integral over $s$ in its decomposition, see (4.11). The
space ${\cal H}^{({\rm disc})}$ will be referred to as the ‘discrete
spectrum’.

###  4.2 ${\cal W}_{\infty}$ algebra

In the scaling limit the critical lattice system possesses extended conformal
symmetry. The corresponding algebra is expected to be the ${\cal W}_{\infty}$
algebra from ref.[46] with central charge $c<2$. This is the same one that
appears in the scaling limit of the staggered six-vertex model with
(quasi-)periodic BCs [26, 15]. Among other things, such a statement implies
that the graded linear spaces

$\bigoplus_{{\tt d}=0}^{\infty}{\cal H}_{p,s}^{({\tt d})}\,,\qquad\qquad{\cal
H}_{p,s}^{({\tt d})}=\begin{cases}{\cal H}_{p,s}^{({\rm cont},{\tt d})}&\ \ \
{\rm for}\qquad s\in\mathbb{R}\\\\[5.69046pt] {\cal H}_{p,s}^{({\rm disc},{\tt
d})}&\ \ \ {\rm for}\qquad p+\tfrac{1}{2}\pm{\rm i}s\in\mathbb{Z}\end{cases}$
(4.24)

are isomorphic to a (irreducible) representation of ${\cal W}_{\infty}$. Then
formulae (4.23), (4.11) and (4.18) would provide a classification of the space
of states ${\cal H}$ occuring in the scaling limit of the lattice model in
terms of irreps of the algebra of extended conformal symmetry. In order to
demonstrate this we briefly mention some details concerning the ${\cal
W}_{\infty}$ algebra and its representations, while referring the reader to
section 16 of ref.[15] for a deeper discussion.

The ${\cal W}_{\infty}$ algebra is generated by a set of currents $W_{j}(u)$
of Lorentz spin $j=2,3,\ldots\ $. These satisfy an infinite system of Operator
Product Expansions (OPEs). Its first few members can be chosen to be

$\displaystyle W_{2}(u)\,W_{2}(0)$ $\displaystyle=$
$\displaystyle\frac{c}{2u^{4}}-\frac{2}{u^{2}}\ W_{2}(0)-\frac{1}{u}\ \partial
W_{2}(0)+O(1)$ $\displaystyle W_{2}(u)\,W_{3}(0)$ $\displaystyle=$
$\displaystyle-\frac{3}{u^{2}}\ W_{3}(0)-\frac{1}{u}\ \partial W_{3}(0)+O(1)$
(4.25) $\displaystyle W_{3}(u)\,W_{3}(0)$ $\displaystyle=$
$\displaystyle-\frac{c(c+7)(2c-1)}{9(c-2)u^{6}}+\frac{(c+7)(2c-1)}{3(c-2)u^{4}}\
\big{(}W_{2}(u)+W_{2}(0)\big{)}-\frac{1}{u^{2}}\ \Big{(}W_{4}(u)+W_{4}(0)$
$\displaystyle+$ $\displaystyle
W^{2}_{2}(u)+W^{2}_{2}(0)+\frac{2c^{2}+22c-25}{30(c-2)}\,\big{(}\partial^{2}W_{2}(u)+\partial^{2}W_{2}(0)\big{)}\Big{)}+O(1)\
,$

where in the last line $W_{2}^{2}$ is a composite field which coincides with
the first regular term in the OPE $W_{2}(u)W_{2}(0)$. Notice that there is
some ambiguity in the definition of $W_{j}$ for $j\geq 3$. Apart from the
freedom in the overall multiplicative normalization, $W_{j}\mapsto CW_{j}$, it
is possible to add to $W_{j}$ any differential polynomial of Lorentz spin $j$
involving the lower spin currents $W_{k}$ with $k<j$. Here, the $W_{3}$
current was fixed by the requirement that it be a primary field of spin three,
so that its OPE with $W_{2}$ takes the form of the second line in formula
(4.2). As for $W_{4}$, one can not arrange for it to be a primary field by
adding linear combinations of $W_{2}^{2}$, $\partial W_{3}$ and
$\partial^{2}W_{2}$. Defined such that it appears in the OPE of
$W_{3}(u)W_{3}(0)$ as above, it turns out that $W_{2}(u)W_{4}(0)$ takes a
simpler form,

$W_{2}(u)\,W_{4}(0)=\frac{(c+10)(17c+2)}{15(c-2)\,u^{4}}\
W_{2}(0)-\frac{4}{u^{2}}\ W_{4}(0)-\frac{1}{u}\ \partial W_{4}(0)+O(1)\ ,$
(4.26)

where the singular terms $\propto u^{-6}$ and $\propto u^{-3}$ are absent.

For the study of the ${\cal W}_{\infty}$ algebra it is useful to know that it
admits a realization in terms of two independent chiral Bose fields. We
normalize them as

$\partial\vartheta(u)\,\partial\vartheta(0)=-\frac{1}{2u^{2}}+O(1)\,,\qquad\qquad\partial\varphi(u)\,\partial\varphi(0)=-\frac{1}{2u^{2}}+O(1)\,,$
(4.27)

while $\partial\varphi(u)\partial\vartheta(0)=O(1)$. One may check that as a
consequence of the free field OPEs, the currents

$\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle(\partial\vartheta)^{2}+(\partial\varphi)^{2}+\frac{{\rm
i}}{\sqrt{n+2}}\ \partial^{2}\varphi$ (4.28) $\displaystyle W_{3}$
$\displaystyle=$
$\displaystyle\frac{6n+8}{3n+6}\,(\partial\vartheta)^{3}+2\,(\partial\varphi)^{2}\partial\vartheta+{\rm
i}\sqrt{n+2}\ \partial^{2}\varphi\,\partial\vartheta-\frac{{\rm
i}n}{\sqrt{n+2}}\ \partial\varphi\,\partial^{2}\vartheta+\frac{n}{6(n+2)}\
\partial^{3}\vartheta$

obey the algebra (4.2). The parameter $n$ entering above is related to the
central charge $c$ as

$c=\frac{2(n-1)}{n+2}$ (4.29)

so that if $n$ is real and positive, the central charge $c$ is less than two.
Notice that, while an expression for $W_{4}$ in terms of $\partial\vartheta$
and $\partial\varphi$ has not been provided, it can be deduced from the OPEs
(4.2) and the formula (4.28) for $W_{2}$ and $W_{3}$. One simply computes
$W_{3}(u)W_{3}(0)$ with $W_{3}$ written in terms of free fields and compares
the coefficient $\propto u^{-2}$ with the same coefficient appearing in the
last two lines of eq. (4.2). It turns out that the higher spin currents always
appear in the OPEs involving the lower spin ones. This way, starting from
(4.28) and recursively computing OPEs, one can determine the realization of
$W_{j}$ in terms of the free fields $\partial\varphi$ and $\partial\vartheta$
for any $j=4,5,6,\ldots\ $.

A stepping stone for the construction of highest weight irreducible
representations of the ${\cal W}_{\infty}$ algebra is the Verma module. It is
defined using the Fourier modes of $W_{j}(u)$, which we assume to be periodic
functions of the variable $u\sim u+2\pi$:

$W_{j}=-\frac{c}{24}\,\delta_{j,2}+\sum_{m=-\infty}^{\infty}{\widetilde{W}}_{j}(m)\
\mbox{e}^{-{\rm i}mu}\ .$ (4.30)

Introduce the highest state, which is specified by the conditions:

$\widetilde{W}_{j}(m)\,|\bm{\omega}\rangle=0\qquad\qquad(\forall
m>0)\,,\qquad\qquad\widetilde{W}_{j}(0)\,|\bm{\omega}\rangle=\omega_{j}\,|\bm{\omega}\rangle$
(4.31)

with $j=2,3$. The highest weight is given by
$\bm{\omega}=(\omega_{2},\omega_{3})$, where the component $\omega_{2}$ is
equal to the conformal dimension of the highest state, while $\omega_{3}$ is
the eigenvalue of $\widetilde{W}_{3}(0)$, which commutes with
$\widetilde{W}_{2}(0)$. The Verma module is spanned by the states that are
obtained by acting with the ‘creation modes’ of the spin 2 and spin 3 currents
on the highest state:

$\widetilde{W}_{2}(-\ell_{1})\ldots\widetilde{W}_{2}(-\ell_{m})\,\widetilde{W}_{3}(-\ell_{1}^{\prime})\ldots\widetilde{W}_{3}(-\ell_{m^{\prime}}^{\prime})|\bm{\omega}\rangle$
(4.32)

with $\ell_{j},\,\ell^{\prime}_{j^{\prime}}\geq 1$. It possesses a natural
grading given by

${\tt d}=\sum_{j=1}^{m}\ell_{j}+\sum_{j=1}^{m^{\prime}}\ell_{j}^{\prime}$
(4.33)

and the dimensions of its level subspace with fixed ${\tt d}$ is the number of
bi-partitions of ${\tt d}$, i.e., ${\rm par}_{2}({\tt d})$ (3.28). In what
follows we will parameterize the highest weight for the Verma module ${\cal
V}_{\rho,\nu}$ as

$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\frac{\rho^{2}-\frac{1}{4}}{n+2}+\frac{\nu^{2}}{n}$ (4.34)
$\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle\frac{2\nu}{\sqrt{n}}\,\Big{(}\,\frac{\rho^{2}}{n+2}+\frac{(3n+4)\,\nu^{2}}{3n\,(n+2)}-\frac{2n+3}{12\,(n+2)}\,\Big{)}\
.$

This is motivated by the free field realization (4.28). Supposing that the
highest state is an eigenvector of the operators
$\int\mbox{d}u\,\partial\vartheta(u)$ and $\int\mbox{d}u\,\partial\varphi(u)$
with eigenvalues $\frac{\nu}{\sqrt{n}}$ and $\frac{\rho}{\sqrt{n+2}}$,
respectively, formula (4.34) follows from (4.28). The highest weight is an
even function of $\rho$. As a result the spaces ${\cal V}_{\rho,\nu}$ and
${\cal V}_{-\rho,\nu}$ should be identified. In the parameterization (4.34),
the conformal dimensions of a state in the Verma module at level ${\tt d}$ is
such that

$\widetilde{W}_{2}(0)-\frac{c}{24}=\frac{\rho^{2}}{n+2}+\frac{\nu^{2}}{n}-\frac{1}{12}+{\tt
d}\,,$ (4.35)

which should be compared with eq. (3.13).

For generic complex values of $\rho$ and $\nu$ the Verma module ${\cal
V}_{\rho,\nu}$ is an irreducible representation of the ${\cal W}_{\infty}$
algebra. However, with $\rho$, $\nu$ obeying certain constraints, the Verma
module contains null vectors – highest states occuring at non-zero levels.
Then the highest weight irrep ${\cal W}_{\rho,\nu}$ can be obtained from
${\cal V}_{\rho,\nu}$ by factoring out all of the invariant subspace(s)
generated by the null vector(s). In view of applications to the scaling limit
of the lattice model of particular interest is when
$\rho=\pm\frac{1}{2}\big{(}r-m(n+2)\big{)}$ with $r,m=1,2,\ldots\ $. In this
case a null vector occurs at level ${\tt d}=mr$ and the Verma module splits
into the direct sum of two representations, which are irreducible for generic
$n$ and $\nu$:

${\cal V}_{\rho,\nu}={\cal W}_{\rho,\nu}\oplus{\cal
W}_{\rho^{\prime},\nu}\qquad\qquad{\rm
with}\qquad\qquad\begin{array}[]{ll}\rho\phantom{{}^{\prime}}=\frac{1}{2}\,(r-m\,(n+2))\qquad\qquad(n,\,\nu\,{\rm-\,generic})\\\\[5.69046pt]
\rho^{\prime}=\frac{1}{2}\,(r+m\,(n+2))\qquad\qquad(r,m=1,2,\ldots)\end{array}\,.$
(4.36)

The space ${\cal W}_{\rho^{\prime},\nu}$ is isomorphic to the Verma module and
the dimensions of its level subspaces is ${\rm par}_{2}({\tt d})$, while for
${\cal W}_{\rho,\nu}$, the level subspaces are ${\rm par}_{2}({\tt d})-{\rm
par}_{2}({\tt d}-mr)$ dimensional. Consider again the components ${\cal
H}^{({\rm cont},{\tt d})}_{p,s}$, which appear in the decomposition of the
continuous spectrum of the space of states occuring in the scaling limit of
the spin chain. Taking into account formulae (4.12) and (3.30) it is clear
that

${\cal W}_{p,s}\cong\bigoplus_{{\tt d}=0}^{\infty}{\cal H}^{({\rm cont},{\tt
d})}_{p,s}\qquad\qquad\qquad\qquad\big{(}2p=2{\cal S}+1-(n+2)\,,\ s\
{\rm-\,real}\big{)}\,.$ (4.37)

To describe the discrete spectrum in terms of irreps of the ${\cal
W}_{\infty}$ algebra, it is necessary to analyze the case when $\nu$ is such
that $\rho+\frac{1}{2}\pm{\rm i}\nu$ is an integer for some choice of the sign
$\pm$. As explained in, e.g., section 16.2 of the work [15] the Verma module
with $\rho+\frac{1}{2}+{\rm i}\nu=-a_{+}=0,\pm 1,\pm 2,\ldots$ contains a null
vector at level $|a_{+}+\frac{1}{2}|+\frac{1}{2}$, while for
$-\rho+\frac{1}{2}+{\rm i}\nu=-a_{-}=0,\pm 1,\pm 2,\ldots$ there is a null
vector at level $|a_{-}+\frac{1}{2}|+\frac{1}{2}$. Assuming $\rho$ is generic
for now the character of the irreducible representation,

${\rm ch}_{\rho,\nu}({\tt q})\equiv{\rm Tr}_{{\cal W}_{\rho,\nu}}\Big{[}\,{\tt
q}^{\widetilde{W}_{2}(0)-\frac{c}{24}}\,\Big{]}\,,$ (4.38)

is given by [48] (see also [47, 49])

${\rm ch}_{\rho,\nu}({\tt q})={\tt
q}^{-\frac{1}{12}+\frac{\nu^{2}}{n}+\frac{\rho^{2}}{n+2}}\
\bigg{(}\prod_{m=1}^{\infty}\frac{1}{(1-{\tt
q}^{m})^{2}}\bigg{)}\sum_{m=0}^{\infty}(-1)^{m}\ {\tt
q}^{m|a+\frac{1}{2}|+\frac{m^{2}}{2}}\qquad\quad\begin{array}[]{l}\rho+\tfrac{1}{2}\pm{\rm
i}\nu=-a\in\mathbb{Z}\\\\[5.69046pt] n,\,\rho\ \ {\rm generic}\end{array}\,.$
(4.39)

If, in addition to $\nu$ being constrained as above,
$\rho\to\frac{1}{2}\,(2{\cal S}-n-1)$ then the irrep ${\cal W}_{\rho,\nu}$
further breaks up into two irreducible representations. One of them is
generated by the null vector which appears at level $2{\cal S}+1$ and has
highest weights $(\rho^{\prime},\nu)$ with $\rho^{\prime}=\frac{1}{2}\,(2{\cal
S}+n+3)$. Its character is given by (4.39) with $\rho$ replaced by
$\rho^{\prime}$ and $a\mapsto a^{\prime}=-2({\cal S}+1)-a$. Taking the
difference ${\rm ch}_{\rho,\nu}({\tt q})-{\rm ch}_{\rho^{\prime},\nu}({\tt
q})$ with $\rho\to\frac{1}{2}\,(2{\cal S}-n-1)$ and
$\rho^{\prime}\to\frac{1}{2}\,(2{\cal S}+n+3)$ yields for the character of the
irreducible representation ${\cal W}_{\rho,\nu}$ with

$\rho+\tfrac{1}{2}\pm{\rm i}\nu=-a\in\mathbb{Z}\qquad\qquad{\rm
and}\qquad\qquad 2\rho=2{\cal S}-n-1$ (4.40)

that

${\rm ch}_{\rho,\nu}={\tt
q}^{-\frac{1}{12}+\frac{\nu^{2}}{n}+\frac{\rho^{2}}{n+2}}\
\bigg{(}\prod_{j=1}^{\infty}\frac{1}{(1-{\tt
q}^{j})^{2}}\bigg{)}\sum_{m=0}^{\infty}(-1)^{m}\,q^{\frac{m^{2}}{2}}\,\Big{(}q^{m|a+\frac{1}{2}|}-q^{2{\cal
S}+1+m|2{\cal S}+a+\frac{3}{2}|}\Big{)}\ .$ (4.41)

For the case $a\geq 0$ the above expression, apart from the overall factor
${\tt q}^{-\frac{1}{12}+\frac{\nu^{2}}{n}+\frac{\rho^{2}}{n+2}}$, coincides
with the generating function (4.7) for the integers $N_{a}^{+}({\tt
d}\,|\,{\cal S})$. This way, one concludes

${\cal W}_{p,s}\cong\bigoplus_{{\tt d}=0}^{\infty}{\cal H}^{({\rm disc},{\tt
d})}_{p,s}\qquad\qquad\qquad\big{(}p={\cal
S}+\tfrac{1}{2}-\tfrac{1}{2}(n+2)\,,\ \tfrac{n}{2}-{\cal S}\pm{\rm
i}s=a\in\mathbb{Z}_{\geq 0}\big{)}\,.$ (4.42)

The remaining case to be considered is when $-\mathcal{S}\leq a<0$. The lower
bound comes from the condition $s\in(-\frac{n}{2},\frac{n}{2}]$ which implies
that $\pm{\rm i}s=\tfrac{n}{2}-{\mathcal{S}}-a\leq\frac{n}{2}$. From the
definition of the integers $N^{+}_{a}({\tt d}\,|\,{\cal S})$ (4.7), which give
the dimensions of the level subspaces ${\cal H}_{p,s}^{({\rm disc},{\tt
d})}\subset{\cal H}^{({\rm disc})}$, one finds

$\dim\big{(}{\cal H}_{p,s}^{({\tt d})}\big{)}=0\qquad{\rm for}\qquad{\tt
d}=0,1,\ldots,|a|-1\qquad\qquad\big{(}-p-\tfrac{1}{2}\pm{\rm
i}s=a\in\mathbb{Z}\,,\ -{\cal S}\leq a<0\big{)}$ (4.43)

Thus the corresponding irrep (4.24) has highest state whose conformal
dimension is given by:

$\Delta=\frac{p^{2}}{n+2}+\frac{s^{2}}{n}+|a|\qquad\qquad\big{(}-\mathcal{S}\leq
a<0\big{)}\,.$ (4.44)

This turns out to be an irreducible representation of the ${\cal W}_{\infty}$
algebra,

${\cal W}_{\rho,\nu}=\bigoplus_{{\tt d}=0}^{\infty}{\cal H}_{p,s}^{({\tt
d})}\,,$ (4.45)

with highest weight parameterized as in (4.34), where

$\rho=\mathcal{S}+\frac{1}{2}\,,\qquad\qquad\nu=\begin{cases}s-\frac{{\rm
i}n}{2}&{\rm for}\ \ \ (-{\rm i}s)>0\\\\[5.69046pt] s+\frac{{\rm i}n}{2}&{\rm
for}\ \ \ (-{\rm i}s)<0\end{cases}\qquad\qquad\big{(}\tfrac{n}{2}-{\cal
S}\pm{\rm i}s=a\in\mathbb{Z}_{<0}\,,\ -{\cal S}\leq a<0\big{)}\,.$

Assuming $n$ is irrational, the character of such a representation is given by

${\rm ch}_{\rho,\nu}({\tt q})={\tt
q}^{-\frac{1}{12}+\frac{\nu^{2}}{n}+\frac{\rho^{2}}{n+2}}\
\bigg{(}\prod_{m=1}^{\infty}\frac{1}{(1-{\tt q}^{m})^{2}}\bigg{)}\
\sum_{m=0}^{\infty}(-1)^{m}\,{\tt q}^{\frac{m^{2}}{2}}\big{(}\,{\tt
q}^{m|\,|\rho|-|\nu|\,|}-{\tt
q}^{(m+1)(|\rho|+|\nu|+1)-\frac{1}{2}}\,\big{)}\,.$ (4.46)

One can check that the dimensions of the level subspaces, obtained by
expanding ${\rm ch}_{\rho,\nu}({\tt q})$ in a series in ${\tt q}$, coincides
with the integers $N^{+}_{a}({\tt d}|\,{\cal S})$ with $-{\cal S}\leq a<0$ and
${\tt d}=|a|,\,|a|+1,\,|a|+2,\ldots$ .

Finally, we mention that the states
$|\psi_{p,s}^{(S^{z})}(\bm{w})\rangle\in{\cal W}_{\rho,\nu}$ appearing in the
scaling limit of the Bethe states, see eq.(4.9), have an important
interpretation. They are the simultaneous eigenstates of the so-called quantum
AKNS integrable structure [50, 51]. The function $\mbox{e}^{\frac{{\rm
i}}{2}\delta}$ (3.41) entering into the quantization condition coincides with
the eigenvalue of a certain so-called reflection operator [52] computed on
$|\psi_{p,s}^{(S^{z})}(\bm{w})\rangle$, see ref.[45] for details.

##  5 Partition function in the scaling limit

In the case of the lattice model with (quasi-)periodic Boundary Conditions
(BCs) imposed, it was proposed in the work [16] and then verified numerically
in ref.[15] that the partition function appearing in the scaling limit of the
lattice system, $Z^{\rm(scl)}$, coincides with twice the partition function
for the 2D Euclidean black hole CFT. The latter was constructed in refs. [20,
21] by computing a functional integral with the worldsheet being taken to be a
torus. The results presented in the previous section allow one to easily
compute $Z^{\rm(scl)}$ for the staggered six-vertex model subject to
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant open BCs. One may expect
$\frac{1}{2}\,Z^{({\rm scl})}$ to coincide with the partition function for the
2D Euclidean black hole CFT on the open segment $x\in(0,R)$, with certain
conditions imposed on the fields at $x=0,R$.

Consider the lattice partition function

$Z_{L}^{({\rm lattice})}={\rm
Tr}_{\mathscr{V}_{2L}}\big{[}\mbox{e}^{-M\,\mathbb{H}}\big{]}\,,$ (5.1)

where the Hamiltonian $\mathbb{H}$ is given by (1.3) with
$q=\mbox{e}^{\frac{{\rm i}\pi}{n+2}}$ and $n\geq 0$, while the trace is taken
over the $2^{2L}$ dimensional space of states:
$\mathscr{V}_{2L}=\mathbb{C}_{1}\otimes\mathbb{C}_{2}\otimes\ldots\otimes\mathbb{C}_{2L}$.
Keeping the ratio

$\tau=\frac{v_{\rm F}M}{2L}$ (5.2)

fixed as $L\to\infty$, one finds that the large $L$ behaviour of the lattice
partition function is given by

$Z_{L}^{({\rm lattice})}\asymp\mbox{e}^{-MLe_{\infty}-Mf_{\infty}}\,Z^{({\rm
scl})}\,.$ (5.3)

Here $Z^{({\rm scl})}$ takes the form of a trace over the space of states
${\cal H}$ appearing in the scaling limit of the lattice model:

$Z^{({\rm scl})}={\rm Tr}_{\cal H}\Big{(}{\tt q}^{\hat{H}_{\rm
CFT}}\Big{)}\qquad{\rm with}\qquad{\tt q}=\mbox{e}^{-2\pi\tau}\,.$ (5.4)

It involves the ‘CFT Hamiltonian’ $\hat{H}_{\rm CFT}$ which when restricted to
the finite dimensional spaces ${\cal H}^{(\rm{cont},{\tt d})}_{p,s}$ or ${\cal
H}^{(\rm{disc},{\tt d})}_{p,s}$ appearing in the decomposition of ${\cal H}$
coincides with the identity operator multiplied by the factor

$E_{\rm CFT}=\frac{p^{2}}{n+2}+\frac{s^{2}}{n}-\frac{1}{12}+{\tt d}\ .$ (5.5)

Notice that the asymptotic formula for the energy (3.3) can be re-written as
the formal relation

$\hat{H}_{\rm CFT}=\operatorname*{slim}_{L\to\infty}\ \frac{L}{\pi v_{\rm
F}}\,\Big{(}\mathbb{H}-L\,e_{\infty}-f_{\infty}\Big{)}\ .$ (5.6)

In subsection 4.1 the space of states ${\cal H}$ was expressed as a direct sum
of the continuous spectrum ${\cal H}^{({\rm cont})}$ and the discrete one
${\cal H}^{({\rm disc})}$, see formula (4.22). The contribution of the states
to the trace in eq. (5.4) for each of these spaces will be denoted as
$Z^{({\rm cont})}$ and $Z^{({\rm disc})}$, respectively, so that

$Z^{({\rm scl})}=Z^{({\rm cont})}+Z^{({\rm disc})}\,,$ (5.7)

where

$Z^{({\rm disc})}={\rm Tr}_{{\cal H}^{({\rm disc})}}\,\Big{(}{\tt
q}^{\hat{H}_{\rm CFT}}\Big{)}\ ,\qquad\qquad Z^{({\rm cont})}={\rm Tr}_{{\cal
H}^{({\rm cont})}}\,\Big{(}{\tt q}^{\hat{H}_{\rm CFT}}\Big{)}\ .$ (5.8)

Let’s first focus on the computation of $Z^{({\rm disc})}$. The space ${\cal
H}^{({\rm disc})}$ is made up of the components ${\cal H}_{{\cal
S},S^{z}}^{({\rm disc})}\cong{\cal H}_{{\cal S},{\cal S}}^{({\rm disc})}$,
which admit the decomposition (4.18) into finite dimensional spaces. Introduce
the notation:

$\chi_{a,{\cal S}}({\tt q})={\tt q}^{-\frac{(\frac{n}{2}-{\cal
S}-a)^{2}}{n}+\frac{p^{2}}{n+2}-\frac{1}{12}}\
\bigg{(}\prod_{j=1}^{\infty}(1-{\tt
q}^{j})^{-2}\bigg{)}\,\sum_{m=0}^{\infty}(-1)^{m}\,\big{(}1-{\tt
q}^{(1+m)(2{\cal S}+1)}\big{)}\,{\tt q}^{ma+\frac{m(m+1)}{2}}\ ,$ (5.9)

where, aside from the prefactor, the function $\chi_{a,{\cal S}}({\tt q})$
coincides with the generating function for the dimensions of the level
subspaces ${\cal H}_{{\cal S},{\cal S}}^{({\rm disc,{\tt d}})}$, see eqs.
(4.20) and (4.7). Then, the contribution of the discrete spectrum to the
partition function reads as:

$Z^{({\rm disc})}=\sum_{{\cal S}\geq 0}(2{\cal S}+1)\bigg{(}\chi_{-{\cal
S},{\cal S}}({\tt q})+2\\!\\!\\!\\!\\!\sum_{a\in\mathbb{Z}\atop 0<a+{\cal
S}<\frac{n}{2}}\\!\\!\\!\\!\\!\chi_{a,{\cal S}}({\tt q})\bigg{)}\ .$ (5.10)

Each term in the sum over ${\cal S}$ has multiplicity $(2{\cal S}+1)$ as a
result of the $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ symmetry of the lattice
model. Also, for every state with given $s=s_{a}$ (4.17) there exists another
one with $s=-s_{a}$ which yields the same contribution to the partition
function, except for the case when $s=\pm\frac{{\rm i}n}{2}$, where they are
identified as the same state. This explains why the functions $\chi_{a,{\cal
S}}({\tt q})$ come with a factor of two except the one with $a=-{\cal S}$
(recall that $s_{a}={\rm i}(\tfrac{n}{2}-{\cal S}-a)$ and hence
$s_{a}=\frac{{\rm i}n}{2}$ for $a=-{\cal S}$).

The contribution of the continuous spectrum to the partition function is given
by

$Z^{({\rm cont})}=\sum_{{\cal S}\geq 0}(2{\cal
S}+1)\,\int_{-\infty}^{\infty}\mbox{d}s\ \sum_{{\tt d}\geq 0}\rho^{({\tt
d})}_{p}(s)\,{\tt q}^{\frac{s^{2}}{n}+\frac{p^{2}}{n+2}-\frac{1}{12}+{\tt
d}}\,.$ (5.11)

Here $\rho^{({\tt d})}_{p}(s)$ is the density of states defined in formulae
(4.5) and (4.6), while recall that $2p=2{\cal S}+1-(n+2)$. Notice that
$Z^{({\rm cont})}$ becomes singular as $L\to\infty$:

$Z^{({\rm cont})}=Z^{({\rm sing})}+O(1)\,,$ (5.12)

where the singular part goes as $\log(L)$ and reads as

$Z^{({\rm sing})}=\sqrt{\frac{n}{2\tau}}\
\frac{\log\big{(}2^{\frac{n+2}{n}}\,L/L_{0}\big{)}}{\pi\,{\tt
q}^{\frac{1}{24}}\prod_{m=1}^{\infty}(1-{\tt q}^{m})}\
\sum^{\infty}_{\mathcal{S}=0}\,\,(2\mathcal{S}+1)\ {\tt
q}^{-\frac{1}{24}+\frac{p^{2}}{n+2}}\ \frac{1-{\tt
q}^{2\mathcal{S}+1}}{\prod_{m=1}^{\infty}(1-{\tt q}^{m})}\,.$ (5.13)

The factor out the front of the sum is easily recognized to be the partition
function of a boson taking values in the segment $\sim\log(L)$ with Neumann
BCs imposed at the endpoints of the field at $x=0,R$. As for the remaining
term,

$\sum^{\infty}_{\mathcal{S}=0}\,\,(2\mathcal{S}+1)\ {\tt
q}^{-\frac{1}{24}+\frac{p^{2}}{n+2}}\ \frac{1-{\tt
q}^{2\mathcal{S}+1}}{\prod_{m=1}^{\infty}(1-{\tt q}^{m})}\,,$ (5.14)

in all likelihood it corresponds to a boundary state which is a superposition
of Ishibashi states associated with a degenerate representation of the
Virasoro algebra with generic central charge $c$ (see ref. [53] for the $c=1$
case). Note that (5.14) also appears in the scaling limit of the XXZ spin
$\frac{1}{2}$ chain with open $U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant
BCs imposed [54].

Formulae (5.10) and (5.11) do not seem to correspond to the published results
in the literature concerning branes in the 2D Euclidean black hole CFT, in
particular, ref. [24]. As such, a separate investigation is required in order
to establish a possible relation between the partition function $Z^{({\rm
scl})}=Z^{({\rm cont})}+Z^{({\rm disc})}$ and that of the black hole CFTs in
the presence of boundaries.

##  6 Discussion

In this work the universal behaviour of the staggered six-vertex model with
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant boundary conditions imposed
was considered. We focused on the so-called self-dual case in the critical
regime (1.2). The problem was reformulated in the Hamiltonian picture, where a
central rôle belongs to $\mathbb{H}$ (1.3), which commutes with the transfer-
matrix of the vertex model. The study of the $1/L$ corrections to the
$L\to\infty$ behaviour of the energy of $\mathbb{H}$ for the low energy states
allowed us to extract the spectrum of scaling dimensions of the statistical
system. Our treatment, involving the use of novel numerical and analytic
techniques, represents an advance on the type of analysis of the scaling limit
of integrable lattice systems with open boundary conditions that typically
exists in the literature.

The numerical construction of the RG trajectories was achieved via the method
of the $Q$ operator. A key rôle was played by the formula (2.23) for the
matrix elements of $\mathbb{Q}(\zeta)$, valid for a one parameter family of
open boundary conditions in the sector $S^{z}=0$. This is an original result
of our work. The advantage of (2.23), as opposed to the expressions for
$\mathbb{Q}(\zeta)$ appearing in the literature [28, 29, 30, 31], is that it
contains no infinite sums; works literally for any (generic) complex values of
$q$ and the boundary parameter $\epsilon$; and can be programmed efficiently
on the computer. We believe that in view of the important applications of the
$Q$ operator, it would be worthwhile to extend (2.23) to the other sectors of
the Hilbert space with $S^{z}\neq 0$ and the case of more general open
boundary conditions.

The powerful analytic technique, which was crucial to our investigation, is
the ODE/IQFT approach to the study of the scaling limit of integrable,
critical lattice systems. For the staggered six-vertex model in the regime
(1.2) with quasi-periodic boundary conditions imposed, it was developed in the
works [26, 15]. We found that it was applicable to the case of
$U_{q}\big{(}\mathfrak{sl}(2)\big{)}$ invariant open boundary conditions as
well. This points to the versatility of the ODE/IQFT approach, where the
analysis for one set of boundary conditions can be readily carried over to
another.

As was already observed in refs. [22, 23], the set of scaling dimensions of
the statistical system possesses a continuous component, labeled by the
quantum number $s=\operatorname*{slim}_{L\to\infty}b(L)\in\mathbb{R}$ with
$b(L)$ from eq. (3.6) [23]. One of the results of this work is the explicit
formula for the density of states $\rho^{({\tt d})}(s)$ (4.5), (4.6), which
characterizes the continuous spectrum. In addition, we studied the RG
trajectories $\\{|\Psi_{L}\rangle\\}$, where $s$ becomes a pure imaginary
number in the scaling limit. Building on the analysis of [23], the discrete
set $\Sigma\equiv\Sigma^{+}\cup\Sigma^{-}$ (4.19) of all admissible values of
pure imaginary $s$ was found. We also determined the dimension of the linear
span of states occuring in the scaling limit of $|\Psi_{L}\rangle$ with given
$s\in\Sigma$ and conformal dimensions $\Delta$ (3.13).

Our work includes a full characterization of the linear space ${\cal H}$
appearing in the scaling limit of the space of low energy states of the
lattice system. To describe it, ${\cal H}$ was decomposed into a direct sum of
the ‘continuous spectrum’ ${\cal H}^{({\rm cont})}$ and the ‘discrete
spectrum’ ${\cal H}^{({\rm disc})}$. The former, when expressed in terms of
finite dimensional spaces involves a direct integral over $s$, see eqs.
(4.23), (4.10), (4.11), while the latter contains a direct sum (4.18). We
explained how the graded linear spaces $\oplus_{{\tt d}=0}^{\infty}{\cal
H}_{p,s}^{({\rm cont},{\tt d})}$ and $\oplus_{{\tt d}=0}^{\infty}{\cal
H}_{p,s}^{({\rm disc},{\tt d})}$ are irreps. of the ${\cal W}_{\infty}$
algebra – the algebra of extended conformal symmetry of the model.

Perhaps the most interesting question is the relation between the scaling
limit of the lattice system and the 2D Euclidean/Lorentzian black hole CFTs
[17, 18, 19]. We believe that the formula for the partition function $Z^{({\rm
scl})}$ provided in sec. 5 may be of help. Unfortunately, it does not seem to
correspond to known results in the literature on branes in the 2D black hole
CFTs. It is likely that progress in this direction would require a separate
and detailed study.

## Acknowledgments

The authors are grateful to M. Flohr and V. Schomerus for their interest in
the work and for valuable discussions. GK would also like to thank V. V.
Bazhanov, S. L. Lukyanov and J. Teschner for useful and stimulating
interactions.

The research of the authors is supported by the Deutsche
Forschungsgemeinschaft (DFG) under grant No. Fr 737/9-2. Part of the numerical
work has been performed on the LUH computer cluster, which is funded by the
Leibniz Universität Hannover, the Lower Saxony Ministry of Science and Culture
and the DFG.

##  Appendix A The asymptotic coefficients $\mathfrak{C}_{p,s}^{(\pm)}$

Here we provide a closed form expression for the coefficients
$\mathfrak{C}_{p,s}^{(\pm)}(\bm{w})$ that were obtained in the work [45].
Among other things, they enter into the quantization condition (3.40), (3.41).

One has

$\mathfrak{C}^{(\pm)}_{p,s}(\bm{w})=\mathfrak{C}_{p,s}^{(0,\pm)}\,\check{\mathfrak{C}}^{(\pm)}_{p,s}(\bm{w})\,,$
(A.1)

where

$\mathfrak{C}_{p,s}^{(0,\pm)}=\sqrt{\frac{2\pi}{n+2}}\ \ \ 2^{-p\pm\frac{{\rm
i}(n+2)s}{n}}\ (n+2)^{-\frac{2p}{n+2}}\
\frac{\Gamma(1+2p)}{\Gamma(1+\frac{2p}{n+2})\,\Gamma(\frac{1}{2}+p\pm{\rm
i}s)}\ ,$ (A.2)

while $\check{\mathfrak{C}}^{(\pm)}$ are one for ${\tt d}=0$. In the general
case, they are given by the determinant of a ${\tt d}\times{\tt d}$ matrix:

$\displaystyle\check{\mathfrak{C}}^{(\pm)}_{p,s}(\bm{w})=\frac{(\mp 1)^{\tt
d}\ \det\big{(}w_{a}^{b-1}\,{U}^{(\pm)}_{a}(b)\big{)}}{\prod_{a=1}^{\tt
d}w_{a}\ \prod_{b>a}(w_{b}-w_{a})\ \prod_{a=1}^{\tt d}\big{(}2p+2a-1\pm 2{\rm
i}s\big{)}}$ (A.3)

with

$\displaystyle{U}^{(\pm)}_{a}(D)$ $\displaystyle=$
$\displaystyle(D-1)^{2}-\bigg{(}2p+2+n\mp 2w_{a}+\sum_{b\not=a}^{\tt
d}\frac{4w_{a}}{w_{a}-w_{b}}\bigg{)}\ (D-1)$ $\displaystyle+$
$\displaystyle\tfrac{1}{2}\,n^{2}+\big{(}p+\tfrac{3}{2}\big{)}\
n\mp(n+1+2p+2{\rm i}s)\ w_{a}+2p+1$ $\displaystyle+$
$\displaystyle\bigg{(}\sum_{b\not=a}^{\tt
d}\frac{2w_{a}}{w_{a}-w_{b}}\,\bigg{)}^{2}+\,\big{(}2\,(2p+1+n\mp
2w_{a})-n\,\big{)}\,\sum_{b\not=a}^{\tt d}\frac{w_{a}}{w_{a}-w_{b}}\ .$

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