# Finite size spectrum of the staggered six-vertex model with
$U_{q}(\mathfrak{sl}(2))$-invariant boundary conditions

Holger Frahm  Sascha Gehrmann  Institut für Theoretische Physik, Leibniz
Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

###### Abstract

The finite size spectrum of the critical $\mathbb{Z}_{2}$-staggered spin-$1/2$
XXZ model with quantum group invariant boundary conditions is studied. For a
particular (self-dual) choice of the staggering the spectrum of conformal
weights of this model has been recently been shown to have a continuous
component, similar as in the model with periodic boundary conditions whose
continuum limit has been found to be described in terms of the non-compact
$SU(2,\mathbb{R})/U(1)$ Euclidean black hole conformal field theory (CFT).
Here we show that the same is true for a range of the staggering parameter. In
addition we find that levels from the discrete part of the spectrum of this
CFT emerge as the anisotropy is varied. The finite size amplitudes of both the
continuous and the discrete levels are related to the corresponding
eigenvalues of a quasi-momentum operator which commutes with the Hamiltonian
and the transfer matrix of the model.

##  I Introduction

The analysis of the finite size spectra of two-dimensional vertex models and
the corresponding $(1+1)$-dimensional quantum spin chains has long been used
e.g. to identify the effective field theories describing the low energy
behaviour of correlated many-body systems in the presence of strong quantum
fluctuations. Recent studies of (super) spin chains related to network models
for quantum Hall transitions, the anti-ferromagnetic Potts model, intersecting
loops or two-dimensional polymers have shown that the continuum descriptions
may involve conformal field theories (CFTs) with a non-compact target space
leading to a continuous component to the spectrum of conformal weights in the
thermodynamic limit Essler  _et al._ (2005a); Ikhlef  _et al._ (2008); Frahm
and Martins (2011); Ikhlef  _et al._ (2012); Vernier  _et al._ (2014); Frahm
and Martins (2015, 2018); Frahm  _et al._ (2019).

In this paper we study the $\mathbb{Z}_{2}$-staggered six-vertex model with
anisotropy $0<\gamma<\pi/2$ and staggering parameter
$\gamma<\alpha<\pi-\gamma$. At the ’self-dual’ point, $\alpha=\pi/2$, this
model is equivalent to the critical anti-ferromagnetic Potts model. At low
energies it can be described effectively in terms of the
$SL(2,\mathbb{R})_{k}/U(1)$ sigma model, a CFT on the two-dimensional
Euclidean black hole background Witten (1991); Hanany  _et al._ (2002);
Ribault and Schomerus (2004), at level $k=\pi/\gamma$ Ikhlef  _et al._ (2012);
Candu and Ikhlef (2013); Frahm and Seel (2014); Bazhanov  _et al._ (2021a, b):
for periodic boundary conditions the observed finite size spectrum of the
lattice model and the density of states in the continua emerging in the
thermodynamic limit have been found to agree with what is known for this CFT.
Moreover, the quantum number describing the states in the continuum has been
related to a conserved quasi-momentum operator in the lattice model. The
construction of this operator relies on the existence of the staggering of the
vertex model in the vertical direction: the two-row transfer matrix of the
periodic model generating conserved quantities such as the Hamiltonian
factorizes into a product of two commuting single-row transfer matrices taken
at spectral parameters shifted by the staggering parameter. The quasi-momentum
operator, on the other hand, is obtained in an expansion of the ratio of these
single-row operators.

More recently, the effect of boundary conditions on the spectrum of this model
has been studied: at the self-dual point the staggered six-vertex model has
been shown to be related to the $R$-matrix of the $D_{2}^{(2)}$ affine Lie
algebra Robertson  _et al._ (2020). This has motivated the construction of a
$D_{2}^{(2)}$ spin chain with a particular choice of integrable open boundary
conditions which also possesses a continuous spectrum of conformal weights
related to the Euclidean black hole CFT Robertson  _et al._ (2021). Similar to
the periodic case, the transfer matrix of this $D_{2}^{(2)}$ model can be
factorized into products of transfer matrices of the six-vertex model
Nepomechie and Retore (2021). This procedure maps the boundary terms of the
$D_{2}^{(2)}$ chain to $U_{q}(\mathfrak{sl}(2))$ quantum group invariant open
boundary conditions of the six-vertex model Pasquier and Saleur (1990); Kulish
and Sklyanin (1991). In the latter formulation an integrable model with these
boundary conditions can be extended to generic values of the staggering
parameter $\alpha$. Furthermore, it allows for the definition of an analog of
the quasi-momentum operator for the open boundary model. This turns out to be
particularly useful for the identification of states from the discrete part of
the CFT spectrum which are not present in the periodic model (although these
states do appear under a twist, see e.g. Vernier  _et al._ (2014); Frahm and
Hobuß (2017); Bazhanov  _et al._ (2021b)).

Below we recall the construction of the double-row transfer matrix of the
inhomogeneous six-vertex model with quantum group invariant boundary
conditions and its Bethe ansatz solution for $\mathbf{Z}_{2}$-staggered
inhomogeneities $\pm i\alpha/2$. Introducing the same staggering in the
auxiliary direction, we obtain commuting four-row transfer matrices, pairs of
which can be related by a duality transformation changing the staggering
parameter as $\alpha\to\pi-\alpha$. As for the periodic staggered six-vertex
model another family of commuting integrals of motion is generated by a
quotient of the double-row transfer matrices. Representative members of these
families are the Hamiltonian and the so-called quasi-momentum operator of the
staggered XXZ spin chain constructed in Section III. The Temperley-Lieb
representation of the Hamiltonian and its relation to other models in certain
limiting cases is shown. Based on our numerical diagonalization of the
Hamiltonian and the quasi-momentum operator for small lattice sizes, we
identify the solutions of the Bethe equations relevant for the low energy part
of the spectrum. Using the root density formalism Yang and Yang (1969) the
ground state of the system in the thermodynamic limit is characterized. For
the analysis of the finite size spectrum, we solve the Bethe equations
numerically for large system sizes. This uncovers the role of the quasi-
momentum in the characterization of the continuous part of the conformal
spectrum and the emergence of discrete states as the anisotropy $\gamma$ is
varied. The paper ends with a summary of our findings.

##  II Definition of the model

We use the following convention for the $R$-matrix for the XXZ-model,

$\displaystyle R(u)=\begin{pmatrix}\sinh{(u+i\gamma)}&0&0&0\\\
0&\sinh{(u)}&\sinh{(i\gamma)}&0\\\ 0&\sinh{(i\gamma)}&\sinh{(u)}&0\\\
0&0&0&\sinh{(u+i\gamma)}\\\ \end{pmatrix}$ (1)

This four by four matrix can be interpreted as an operator on $V_{1}\otimes
V_{2}$, $V_{j}\sim\mathbb{C}^{2}$. The parameter $\gamma$ measures the
anisotropy of the model. The $R$-matrix has the following properties

$\displaystyle R(0)$ $\displaystyle=\sinh(i\gamma)P\,,$ (2a) $\displaystyle
PR(u)P$ $\displaystyle=R(u)\,,$ (2b) $\displaystyle R^{T}(u)$
$\displaystyle=R(u)\,,$ (2c) $\displaystyle R(u)R(-u)$
$\displaystyle=\frac{1}{2}(\cos(2\gamma)-\cosh(2u))\,\mathbbm{1}=:\rho(u)\,\mathbbm{1}\,,$
(2d) $\displaystyle R^{t_{1}}(u)R^{t_{1}}(-u-2i\gamma)$
$\displaystyle=\frac{1}{2}(\cos(2\gamma)-\cosh(2u+2i\gamma))\,\mathbbm{1}=\rho(u+i\gamma)\,\mathbbm{1}\,,$
(2e)

where $P$ denotes the permutation operator. In addition, the $R$-matrix
satisfies the Yang-Baxter equation (YBE)

$\displaystyle
R_{2,3}(v)R_{1,3}(u)R_{1,2}(u-v)=R_{1,2}(u-v)R_{1,3}(u)R_{2,3}(v),$ (3)

where the subscripts indicate the factors of $V_{1}\otimes V_{2}\otimes V_{3}$
where the $R$-matrices act non-trivially. Based on the $R$-matrix, we can
construct a monodromy matrix $T\left(u,\\{\delta_{j}\\}\right)$ depending on
the spectral parameter $u$ and a set of parameters $\\{\delta_{j}\\}$, which
are called the inhomogeneities:

$\displaystyle T_{0}(u,\\{\delta_{j}\\})=$ $\displaystyle
R_{0,2L}(u-\delta_{2L})R_{0,2L-1}(u-\delta_{2L-1})...R_{0,1}(u-\delta_{1})\,.$
(4)

Each $R$-matrix acts on the auxiliary space $V_{0}\sim\mathbb{C}^{2}$ and one
of the quantum spaces $V_{j}\sim\mathbb{C}^{2}$ represented by the second
index $j=1,\dots,2L$. The YBE (3) and the fact that it just depends on the
difference of its parameters ensure that the monodromy matrix $T$ is a
representation of the Yang-Baxter algebra with commutation relations defined
by the $RTT$-relation

$\displaystyle R_{1,2}(u-v)T_{1}(u)T_{2}(v)=T_{2}(v)T_{1}(u)R_{1,2}(u-v)$ (5)

for arbitrary values of the inhomogeneities $\delta_{j}$.

While the above is enough to define an integrable model with periodic boundary
conditions, to study open boundary conditions, one needs two $K$-matrices
related to the $R$-matrix of the periodic model via the so-called reflection
algebras Sklyanin (1988)

$\displaystyle R_{1,2}(u-v)K_{1,-}(u)R_{1,2}(u+v)K_{2,-}(v)$
$\displaystyle=K_{2,-}(v)R_{1,2}(u+v)K_{1,-}(u)R_{1,2}(u-v)\,,$ (6)
$\displaystyle
R_{1,2}(-u+v)K^{t_{1}}_{1,+}(u)R_{1,2}(-u-v-2i\gamma)K^{t_{2}}_{2,+}(v)$
$\displaystyle=K^{t_{2}}_{2,+}(v)R_{1,2}(-u-v-2i\gamma)K^{t_{1}}_{1,+}(u)R_{1,2}(-u+v)\,.$

Here we take the reflection matrix $K_{-}$ to be

$\displaystyle K_{-}(u)=\begin{pmatrix}e^{u}&0\\\ 0&e^{-u}\end{pmatrix},$ (7)

and use the $X$-isomorphism Sklyanin (1988) of the reflection algebras to
construct the second one as $K_{-}(-u-i\gamma)$. For notational reasons, we
define the matrix

$\displaystyle K_{+}(u)=\begin{pmatrix}e^{-u}&0\\\ 0&e^{u}\end{pmatrix}.$ (8)

The $R$-matrix and the $K$-matrices allow the construction of a family of
commuting operators for arbitrary inhomogeneities $\delta_{j}$Sklyanin (1988)

$\displaystyle\tau(u)$
$\displaystyle=\text{tr}_{0}\left(K_{0,+}\left(u+\frac{i\gamma}{2}\right)T_{0}\left(u-\frac{i\gamma}{2},\\{\delta_{j}\\}\right)K_{0,-}\left(u-\frac{i\gamma}{2}\right)T^{-1}_{0}\left(-\left(u-\frac{i\gamma}{2}\right),\\{\delta_{j}\\}\right)\right)\,,$
(9) $\displaystyle 0$
$\displaystyle=\left[\tau(u),\tau(v)\right]\,,\qquad\forall u,v\in\mathbb{C}$

which act on the Hilbert space $\mathcal{H}=\otimes_{j=1}^{2L}V_{j}$. We will
refer to $\tau(u)$ as the double-row transfer matrix depending on the spectral
parameter $u$. Due to the boundary matrices’ particular choice, the double-row
transfer matrix commutes with the generators of the algebra
$U_{q}(\mathfrak{sl}(2))$ on $\mathcal{H}$ Kulish and Sklyanin (1991):

$\displaystyle
S_{z}=\frac{1}{2}\Big{(}\sigma^{z}_{1}+...+\sigma^{z}_{2L}\Big{)}\,,\qquad
X^{\pm}=\sum^{2L}_{j=1}e^{\pm\delta_{j}}e^{\frac{i\gamma}{2}\sum^{j-1}_{i=1}\sigma^{z}_{i}}\sigma^{\pm}_{j}e^{-\frac{i\gamma}{2}\sum^{2L}_{i=j+1}\sigma^{z}_{i}}$
(10)

In this study, we will restrict to a $\mathbb{Z}_{2}$-staggering, meaning that

$\displaystyle\delta_{2j-1}=\frac{i\alpha}{2}=-\delta_{2j}\,,\qquad
j=1,...,L\,,$ (11)

where we call $\alpha$ the staggering parameter. For the
$\mathbb{Z}_{2}$-staggering, the double-row transfer matrix $\tau(u)$ becomes
in terms of the $R$-matrices

$\displaystyle\tau(u)=c_{\tau}\,\text{tr}_{0}$
$\displaystyle\left(K_{0,+}(u+\frac{i\gamma}{2})R_{0,2L}(u+\frac{i\alpha}{2}-\frac{i\gamma}{2})\dots
R_{0,1}(u-\frac{i\alpha}{2}-\frac{i\gamma}{2})\right.$ (12)
$\displaystyle\quad\times\left.K_{0,-}(u-\frac{i\gamma}{2})R_{1,0}(u+\frac{i\alpha}{2}-\frac{i\gamma}{2})\dots
R_{2L,0}(u-\frac{i\alpha}{2}-\frac{i\gamma}{2})\right),$

where we picked up the factor

$\displaystyle
c_{\tau}=\left({\rho\left(-u+\frac{i\gamma}{2}+\frac{i\alpha}{2}\right)\rho\left(-u+\frac{i\gamma}{2}-\frac{i\alpha}{2}\right)}\right)^{-L},$
(13)

due to the evaluation of the inverse monodromy matrix using the unitarity
relation (2d). Introducing the pictorial representations of the $K$-matrices
and the $R$-matrix displayed in Figure 1,

$\alpha$$\beta$$u$$(K_{+})^{\beta}_{\alpha}(u)=$$\beta$$\alpha$$u$,
$(K_{-})^{\beta}_{\alpha}(u)=$,
$R^{\gamma\delta}_{\alpha\beta}(u)=$$u$$\alpha$$\gamma$$\beta$$\delta$ Figure
1: $R$-matrix and $K$-matrices in graphical notation

we can represent the double-row transfer matrix graphically in Figure 2.

$u-\frac{i\gamma}{2}-\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}+\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}-\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}+\frac{i\alpha}{2}$$u+\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}-\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}+\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}-\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}-\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}+\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}+\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}-\frac{i\alpha}{2}$$u-\frac{i\gamma}{2}$$2L$$2L-1$$2L-2$$2L-3$$2$$1$(a)

$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\alpha}{2}+\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\alpha}{2}-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u+\frac{i\alpha}{2}+\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}$$u+\frac{i\alpha}{2}-\frac{i\gamma}{2}$$2L$$2L-1$$2L-2$$2L-3$$2$$1$(b)

Figure 2:  (a) The double row transfer matrix $\tau(u)$. The horizontal lines
correspond to the auxiliary space $V_{0}$, while the vertical lines represent
the quantum spaces $V_{j}$ as labeled underneath. (b) Graphical representation
of the four-row transfer matrix $\mathcal{T}(u)$ with two auxiliary spaces.
Observe the staggering in the horizontal and vertical direction.

The double-row transfer matrix (12) can be diagonalized by means of the
algebraic Bethe ansatz. The Bethe equations and the eigenvalues of the
transfer matrix have been derived in Kulish and Sklyanin (1991). Note that our
normalization of the transfer matrix (12) differs from the one used in Kulish
and Sklyanin (1991), where it is multiplied by the quantum determinant of the
monodromy matrix

$\displaystyle q\det(T(-u))=$
$\displaystyle\sinh^{L}\left(\frac{i\alpha}{2}-\frac{i\gamma}{2}-u\right)\sinh^{L}\left(\frac{i\alpha}{2}+\frac{3i\gamma}{2}-u\right)$
(14)
$\displaystyle\times\sinh^{L}\left(-\frac{i\alpha}{2}-\frac{i\gamma}{2}-u\right)\sinh^{L}\left(-\frac{i\alpha}{2}+\frac{3i\gamma}{2}-u\right).$

Taking this normalization difference into account, the eigenvalues of (12) in
the sector with $S_{z}=L-M$ read

$\displaystyle\Lambda(u)=$
$\displaystyle\frac{\sinh(2u+i\gamma)}{\sinh(2u)}\sinh^{2L}\left(u+\frac{i(\alpha+\gamma)}{2}\right)\sinh^{2L}\left(u-\frac{i(\alpha-\gamma)}{2}\right)$
(15)
$\displaystyle\quad\times\frac{1}{q\det(T(-u))}\prod^{M}_{m=1}\frac{\sinh(u-v_{m}-i\gamma)\sinh(u+v_{m}-i\gamma)}{\sinh(u-v_{m})\sinh(u+v_{m})}$
$\displaystyle+\frac{\sinh(2u-i\gamma)}{\sinh(2u)}\sinh^{2L}\left(u+\frac{i(\alpha-\gamma)}{2}\right)\sinh^{2L}\left(u-\frac{i(\alpha+\gamma)}{2}\right)$
$\displaystyle\quad\times\frac{1}{q\det(T(-u))}\prod^{M}_{m=1}\frac{\sinh(u-v_{m}+i\gamma)\sinh(u+v_{m}+i\gamma)}{\sinh(u-v_{m})\sinh(u+v_{m})}$

and the Bethe roots $v_{m}$, $m=1,\dots,M$, satisfy the Bethe equations

$\displaystyle\left(\frac{\sinh{(v_{m}+\frac{i\gamma-i\alpha}{2})}\sinh{(v_{m}+\frac{i\gamma+i\alpha}{2})}}{\sinh{(v_{m}-\frac{i\gamma-i\alpha}{2})}\sinh{(v_{m}-\frac{i\gamma+i\alpha}{2})}}\right)^{2L}=\prod_{\begin{subarray}{c}k=1{}\\\
k\neq
m\end{subarray}}^{M}\frac{\sinh(v_{m}-v_{k}+i\gamma)\sinh(v_{m}+v_{k}+i\gamma)}{\sinh(v_{m}-v_{k}-i\gamma)\sinh(v_{m}+v_{k}-i\gamma)}\,.$
(16)

Similar to the periodic case Ikhlef  _et al._ (2008), we introduce a vertical
staggering by multiplying two transfer matrices with arguments differing by
$i\alpha$. This procedure leads to the four-row transfer matrix

$\displaystyle\mathcal{T}(u)=\tau\left(u+\frac{i\alpha}{2}\right)\tau\left(u-\frac{i\alpha}{2}\right),$
(17)

shown in Figure 2. By construction $\tau(u)$ and $\mathcal{T}(u)$ commute for
different arguments. The four-row transfer matrix $\mathcal{T}(u)$ allows a
duality transformation regarding the staggering parameter $\alpha$. The
spectrum of $\mathcal{T}(u)$ is invariant under the transformation
$\mathcal{D}$ which sends $\alpha\to\pi-\alpha$. Concretely, the
$\mathcal{D}$-transformed transfer matrix
$\mathcal{D}\left(\mathcal{T}(u)\right)$ is similar to $\mathcal{T}(u)$:

$\displaystyle\mathcal{D}\left(\mathcal{T}(u)\right)=\left(\prod^{L}_{i=1}\sigma^{z}_{2j}\right)\mathcal{C}(\alpha)\mathcal{T}(u)\mathcal{C}^{-1}(\alpha)\left(\prod^{L}_{i=1}\sigma^{z}_{2j}\right)\,,$
(18)

where $\mathcal{C}(\alpha)$ is given as a product of local operators

$\displaystyle\mathcal{C}(\alpha)=\prod^{L}_{i=1}c_{2i-1,2i}(\alpha)\quad\text{with}\quad
c_{i,j}(\alpha)=P_{i,j}R_{i,j}(i\alpha)\,.$ (19)

For the case $\alpha=\pi/2$ the transformation $\mathcal{D}$ becomes the
identity and so the transfer matrix $\mathcal{T}(u)$ is invariant under the
action of $\left(\prod^{L}_{i=1}\sigma^{z}_{2j}\right)\mathcal{C}(\alpha)$.
Hence, an additional symmetry arises for this choice of $\alpha$ and we will
refer to the parameter subspace
$\\{(\alpha=\pi/2,\gamma)|\gamma\in(0,\pi/2)\\}$ as the self dual line or as
the self-dual point regarding the parameter interval of $\alpha$. The general
duality (18) can also be seen on the level of the Bethe equations (16). If we
perform the duality transformation and the following redefinition of the Bethe
roots

$\displaystyle\alpha$ $\displaystyle\to\pi-\alpha\,,$ (20) $\displaystyle
v_{k}$ $\displaystyle\to v_{k}+\frac{i\pi}{2}\,,$

solutions of (16) are mapped to solutions of (16). This can be easily seen by
taking into account another symmetry transformation of the Bethe equations:
Given a solution $\\{v_{k}\\}$, the transformed set

$v_{k}\to v_{k}+i\pi\qquad v_{k}\in\Omega$

for an arbitrary subset $\Omega$ of the Bethe roots $\\{v_{k}\\}$ also solves
(16). Furthermore, changing the sign of arbitrary many Bethe roots $v_{k}\to-
v_{k}$ still yields a solution of (16). Hence, one can restrict the solutions
of (16) to the strip region $\\{v_{k}\in\mathbb{C}|\text{Re}(v_{k})\geq
0,-\frac{\pi}{2}<\text{Im}(v_{k})\leq\frac{\pi}{2}\\}$ in the complex plane.

Using the unitarity relation (2d), it is straightforward to show that the
four-row transfer matrix (17) is proportional to the identity operator for the
spectral parameter $u=u_{0}:=i\gamma/2$, see Figure 3.

$-i\alpha$$-i\alpha$$-\frac{i\alpha}{2}+i\gamma$$-i\alpha$$-i\alpha$$-i\alpha$$-i\alpha$$-\frac{i\alpha}{2}$$i\alpha$$i\alpha$$\frac{i\alpha}{2}+i\gamma$$i\alpha$$i\alpha$$i\alpha$$i\alpha$$\frac{i\alpha}{2}$$2L$$2L-1$$2L-2$$2L-3$$2$$1$(a)

$2L$$2L-1$$2L-2$$2L-3$$2L-4$$3$$2$$1$(b)$-i\alpha$$-i\alpha$$-i\alpha$$-i\alpha$$-i\alpha$$-i\alpha$$-i\alpha$$-i\alpha$$i\alpha$

Figure 3: (a) The transfer matrix $\mathcal{T}$ (17) evaluated at
$u_{0}=i\gamma/2$, where many $R$-matrices become proportional to the
permutation operator. Using the unitarity condition (2d) for the vertices in
the second and third row, then between the first and fourth row one obtains
the identity operator in the bulk. For the right boundary, one notes that the
weights of the $K_{-}$-matrices differ by a minus sign, leading to the
identity operator if one takes the explicit form of $K_{-}$ in (7) into
account. The left boundary will simplify to a loop-diagram, giving also the
identity when evaluated using the explicit form of $K_{+}$ (8). (b) Graphical
representation of
${\tau\left(u-\frac{i\alpha}{2}\right)}/{\tau\left(u+\frac{i\alpha}{2}\right)}$
evaluated at $u_{0}$, Eq. (21). The tiny crosses stand for the operator
insertion of $\exp\left[{\pm i\alpha\sigma^{z}}/{2}\right]$.

Note that the expansion of $\mathcal{T}(u)$ around $u_{0}$ yields a family of
commuting operators including the Hamiltonian (23) constructed below which are
even under the duality transformation (18). This family can be complemented by
considering the quotient of two double-row transfer matrices,
$\tau(u-i\alpha/2)/\tau(u+i\alpha/2)$. For $u=u_{0}$ and using the explicit
form of $\tau(u)$, we can express this operator quotient as a product of the
operators $c_{i,j}(-\alpha)$ defined in (19):

$\displaystyle\left.\frac{\tau\left(u-\frac{i\alpha}{2}\right)}{\tau\left(u+\frac{i\alpha}{2}\right)}\right|_{u=\frac{i\gamma}{2}}=$
$\displaystyle\Bigg{[}\left(\prod^{L}_{i=1}c_{2i-1,2i}(-\alpha)\right)K_{-,1}\left(-\frac{i\alpha}{2}\right)\left(\prod^{L-1}_{i=1}c_{2i,2i+1}(-\alpha)\right)$
(21)
$\displaystyle\times\text{tr}_{0}\left(c_{0,2L}(-\alpha)K_{+,0}\left(i\gamma-\frac{i\alpha}{2}\right)\right)\frac{1}{\rho(i\alpha)^{L}}\Bigg{]}^{2}\frac{\cos(2\gamma)-\cos(2\alpha)}{\cos(4\gamma)-\cos(2\alpha)}$
$\displaystyle=$
$\displaystyle\left[\left(\prod^{L}_{i=1}c_{2i-1,2i}(-\alpha)\right)\exp\left\\{-\frac{i\alpha}{2}\sigma^{z}_{1}\right\\}\left(\prod^{L-1}_{i=1}c_{2i,2i+1}(-\alpha)\right)\exp\left\\{\frac{i\alpha}{2}\sigma^{z}_{2L}\right\\}\right]^{2}$
$\displaystyle\times\frac{i^{2}\sin^{2}(\alpha-2\gamma)}{\rho(i\alpha)^{2L}}\frac{\cos(2\gamma)-\cos(2\alpha)}{\cos(4\gamma)-\cos(2\alpha)}\,.$

Compared to corresponding operator in the periodic model Ikhlef  _et al._
(2012) we see that the quantum group invariant boundaries are adding phase
shifts on the quantum spaces related to the boundary, similar as in the
presence of a twist in the periodic case Candu and Ikhlef (2013). The operator
(21) can be represented graphically in Figure 3.

##  III The Hamiltonian limit

By expanding $\mathcal{T}$ around $u_{0}$, we can define a local Hamiltonian
Sklyanin (1988):

$\displaystyle\frac{\text{d}}{\text{d}u}\mathcal{T}(u)|_{u=u_{0}}$
$\displaystyle=aH+b,$ (22)

where $a$ and $b$ are constants given by

$\displaystyle a=$
$\displaystyle-\frac{2i(\cos(4\gamma)-\cos(2\alpha))}{\cos(2\alpha)-\cos(2\gamma)}\,,$
$\displaystyle b=$
$\displaystyle-\frac{2i\cot(\gamma)(\cos(2\alpha)+4\cos(2\gamma)-3\cos(4\gamma)-2)}{\cos(2\alpha)-\cos(2\gamma)}$
$\displaystyle-\frac{2i\cot(\gamma)(\cos(4\gamma)-\cos(2\alpha))((1-2L)\cos(2\alpha)+(4L-3)\cos(2\gamma)-2L+2)}{(\cos(2\alpha)-\cos(2\gamma))^{2}}\,.$

The defined Hamiltonian reads in terms of the Pauli-matrices
$\sigma_{j}^{\alpha}$:

$\displaystyle H=-\frac{1}{2\sin(\gamma)\rho(i\alpha))}$
$\displaystyle\bigg{\\{}\quad\;\,\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-2\sin^{2}(\gamma)\sum^{2L-1}_{j=1}\cos(\gamma)\sigma^{z}_{j}\sigma^{z}_{j+1}+2\cos(\alpha)(\sigma^{+}_{j}\sigma^{-}_{j+1}+\sigma^{-}_{j}\sigma^{+}_{j+1})$
(23)
$\displaystyle+\cos(\gamma)\sin^{2}(\alpha)\sum^{2L-2}_{j=1}\sigma^{z}_{j}\sigma^{z}_{j+2}+2(\sigma^{+}_{j}\sigma^{-}_{j+2}+\sigma^{-}_{j}\sigma^{+}_{j+2})$
$\displaystyle+\sin(\alpha)\sin(2\gamma)\sum^{2L-2}_{j=1}(-1)^{j+1}\sigma^{z}_{j}\sigma^{+}_{j+1}\sigma^{-}_{j+2}+(-1)^{j}\sigma^{z}_{j}\sigma^{-}_{j+1}\sigma^{+}_{j+2}$
$\displaystyle+\sin(\alpha)\sin(2\gamma)\sum^{2L-2}_{j=1}(-1)^{j+1}\sigma^{+}_{j}\sigma^{-}_{j+1}\sigma^{z}_{j+2}+(-1)^{j}\sigma^{-}_{j}\sigma^{+}_{j+1}\sigma^{z}_{j+2}$
$\displaystyle+\sin(\gamma)\sin(2\alpha)\sum^{2L-2}_{j=1}(-1)^{j+1}\sigma^{-}_{j}\sigma^{z}_{j+1}\sigma^{+}_{j+2}+(-1)^{j}\sigma^{+}_{j}\sigma^{z}_{j+1}\sigma^{-}_{j+2}$
$\displaystyle+\cos(\gamma)\sin^{2}(\alpha)(\sigma^{z}_{1}\sigma^{z}_{2}+\sigma^{z}_{2L-1}\sigma^{z}_{2L})$
$\displaystyle+i(\sin(\alpha)\cos(2\gamma)-\sin(\alpha)e^{2i\alpha})(\sigma^{+}_{1}\sigma^{-}_{2}+\sigma^{+}_{2L-1}\sigma^{-}_{2L})$
$\displaystyle-i(\sin(\alpha)\cos(2\gamma)-\sin(\alpha)e^{-2i\alpha})(\sigma^{-}_{1}\sigma^{+}_{2}+\sigma^{-}_{2L-1}\sigma^{+}_{2L})$
$\displaystyle+2\rho(i\alpha)\sinh(i\gamma)(\sigma^{z}_{1}-\sigma^{z}_{2L})$
$\displaystyle+\cos(\gamma)(L\cos(2\alpha)+(1-2L)\cos(2\gamma)+L-1)\bigg{\\}}\,.$

We see that the Hamiltonian is indeed local since its maximal interaction
range amounts to three neighboring lattice sites. Due to the particular choice
of the constants $a,b$ the energy takes a simply form in terms of the Bethe
roots $\\{v_{j}\\}$

$\displaystyle E=\sum^{M}_{j=1}\epsilon_{0}(v_{j}),$ (24)

where the bare energies $\epsilon_{0}$ are given as in the periodic case Frahm
and Seel (2014):

$\displaystyle\epsilon_{0}(x)$
$\displaystyle=-\frac{2\sin(\alpha-\gamma)}{\cosh(2x)-\cos(\alpha-\gamma)}+\frac{2\sin(\alpha+\gamma)}{\cosh(2x)-\cos(\alpha+\gamma)}\,.$
(25)

The Hamiltonian (23) takes a particularly simple form when written in terms of
the generators $e_{i,i+1}$ of the Temperley-Lieb (TL) algebra

$\displaystyle e_{i,i+1}^{2}$ $\displaystyle=-2\cos(\gamma)e_{i,i+1}\,,$ (26)
$\displaystyle e_{i,i+1}e_{i+1,i+2}e_{i,i+1}$ $\displaystyle=e_{i,i+1}\,,$
$\displaystyle e_{i+1,i+2}e_{i,i+1}e_{i+1,i+2}$ $\displaystyle=e_{i+1,i+2}\,,$
$\displaystyle e_{i,i+1}e_{j,j+1}$
$\displaystyle=e_{j,j+1}e_{i,i+1}\,,\qquad|i-j|>1\,.$

Employing the vertex representation of the Temperley-Lieb generators

$\displaystyle e_{j,j+1}=\left(\mathbbm{1}_{\mathbb{C}^{2}}\right)^{\otimes
j-1}\otimes\begin{pmatrix}0&0&0&0\\\ 0&-e^{-i\gamma}&1&0\\\
0&1&-e^{i\gamma}&0\\\
0&0&0&0\end{pmatrix}\otimes\left(\mathbbm{1}_{\mathbb{C}^{2}}\right)^{\otimes
2L-j-1}\,,$ (27)

one can express the Hamiltonian (23) as

$\displaystyle
U^{odd}_{\mathcal{H}}(i\alpha)HU^{odd}_{\mathcal{H}}(-i\alpha)=-\frac{1}{\sin(\gamma)\rho(i\alpha)}\Bigg{(}$
$\displaystyle\sum^{2L-1}_{j=1}2\rho(i\alpha)e_{j,j+1}$ (28)
$\displaystyle+\sin(\alpha)$
$\displaystyle\sum^{2L-1}_{j=2}\sin(\alpha+(-1)^{j+1}\gamma)e_{j,j+1}e_{j-1,j}$
$\displaystyle+\sin(\alpha)$
$\displaystyle\sum^{2L-1}_{j=2}\sin(\alpha+(-1)^{j}\gamma)e_{j-1,j}e_{j,j+1}\Bigg{)}\,,$

where $U^{odd}_{\mathcal{H}}(i\alpha)$ is the rotation of the spin variables
on the odd lattice sites by $-i\alpha$ in the $x,y$-spin plane:

$\displaystyle
U^{odd}_{\mathcal{H}}(i\alpha)\sigma^{\pm}_{2j-1}U^{odd}_{\mathcal{H}}(-i\alpha)=e^{\mp
i\alpha}\sigma^{\pm}_{2j-1}\,,\qquad j=1,...,L.$ (29)

Using (15) the eigenvalues of the other conserved quantities of the model can
be expressed in terms of the Bethe roots. One particular important quantity is
the so-called quasi-momentum, which is related to the quotient of transfer
matrices (21) by:

$\displaystyle K=$
$\displaystyle\log\left(\left.\frac{\tau\left(u-\frac{i\alpha}{2}\right)}{\tau\left(u+\frac{i\alpha}{2}\right)}\right)\right|_{u=\frac{i\gamma}{2}}$
(30) $\displaystyle=$ $\displaystyle
2\log\left[\frac{1}{\rho(i\alpha)^{L}}\left(\prod^{L}_{i=1}c_{2i-1,2i}(-\alpha)\right)e^{-i\alpha/2\sigma^{z}_{1}}\left(\prod^{L-1}_{i=1}c_{2i,2i+1}(-\alpha)\right)e^{i\alpha/2\sigma^{z}_{2L}}\right]$
$\displaystyle+2\log\big{[}i\sin(\alpha-2\gamma)\big{]}+\log\left[\frac{\cos(2\gamma)-\cos(2\alpha)}{\cos(4\gamma)-\cos(2\alpha)}\right]$

The eigenvalues $\mathcal{K}$ of the quasi-momentum operator $K$ can be
expressed by the Bethe roots

$\displaystyle\mathcal{K}=$
$\displaystyle\left.\log\left(\frac{\Lambda(u-\frac{i\alpha}{2})}{\Lambda(u+\frac{i\alpha}{2})}\right)\right|_{u=\frac{i\gamma}{2}}$
(31) $\displaystyle=$
$\displaystyle\sum^{M}_{i=1}k_{0}(v_{i})+\log\left[\frac{\sin(2\gamma-\alpha)}{\sin(2\gamma+\alpha)}\right]+\left(2L-1\right)\log\left[\frac{\sin(\gamma-\alpha)}{\sin(\gamma+\alpha)}\right],$

where the bare quasi-momentum $k_{0}(u)$ takes the form:

$\displaystyle
k_{0}(u)=2\log\left[\frac{\cosh(2u)-\cos(\alpha+\gamma)}{\cosh(2u)-\cos(\alpha-\gamma)}\right]$
(32)

Note that the energy $E$ (24) is invariant under the duality transformation
(20) while the quasi-momentum changes the sign $\mathcal{K}\to-\mathcal{K}$.

We end this section by discussing some limiting cases of the Hamiltonian. It
has recently been shown by Robertson _et al._ Robertson  _et al._ (2020) that
the transfer matrices of the staggered six-vertex model at the self-dual point
$\alpha=\pi/2$ can be mapped to that of a spin chain constructed from the
twisted quantum algebra $D^{(2)}_{2}$ with suitably chosen integrable boundary
conditions.111Such a relation has been observed in Ref. Frahm and Martins
(2012) before within the analysis of the spectra of these models with periodic
boundary conditions In fact the transfer matrices of the $D_{2}^{(2)}$ model
with both closed and open boundary conditions have been shown to factorize
into the product of transfer matrices based on the $R$-matrix (1) considered
in the present paper Nepomechie and Retore (2021). For $\alpha=\pi/2$ the
Hamiltonian (28)

$U^{odd}_{\mathcal{H}}(\frac{i\pi}{2})HU^{odd}_{\mathcal{H}}(-\frac{i\pi}{2})=-\frac{2}{\sin(2\gamma)}\left(\sum^{2L-1}_{j=1}2\cos(\gamma)e_{j,j+1}+\sum^{2L-1}_{j=2}e_{j,j+1}e_{j-1,j}+e_{j-1,j}e_{j,j+1}\right),$
(33)

(or (23) in the spin-formulation) reduces via a suitable parameter
identification to the open integrable $D^{(2)}_{2}$ spin chain with boundary
matrices from Refs. Martins and Guan (2000); Nepomechie and Pimenta (2018).
This model has recently been studied in Robertson  _et al._ (2021); Nepomechie
and Retore (2021). Inserting a crossing relation into the four-row transfer
matrix (17) and using the YBE to rearrange $R$-matrices as in Figure 4, the
$D^{(2)}_{2}$ $K$-matrices in the vertex representation can be expressed in
terms of those of the staggered six-vertex model for the self-dual case
$\alpha=\pi/2$ by a suitable shift in the spectral parameter, i.e.:

$\displaystyle K^{-}_{D^{(2)}_{2}}(u)=$ $\displaystyle
G_{\bar{0}}\Big{(}\frac{i\pi}{2}\Big{)}B(u)K_{-,\bar{0}}(-u+\frac{i\pi}{4})R_{\bar{0}0}(-2u)K_{-,0}(-u-\frac{i\pi}{4})P_{\bar{0}0}B(u)G_{\bar{0}}\Big{(}-\frac{i\pi}{2}\Big{)}$
(34) $\displaystyle\times\text{csch}(2u+i\gamma),$

where we used the transformations

$\displaystyle G(u)=\text{diag}(1,e^{u})\,,\qquad
B(u)=\text{diag}(e^{u},1,1,e^{-u})\,.$ (35)

Note that the transformation $B(u)$ has recently been used to prove the
$U_{q}(B_{n-p})\times U_{q}(B_{p})$ invariance of models based on the
$D^{(2)}_{n+1}$ $K$-matrices Nepomechie and Pimenta (2018).

$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\alpha}{2}+\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}-i\alpha$$u-\frac{i\alpha}{2}-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u+\frac{i\alpha}{2}+\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}+i\alpha$$u-\frac{i\gamma}{2}$$u+\frac{i\alpha}{2}-\frac{i\gamma}{2}$$2u-i\gamma$$-2u-i\gamma$$2L$$2L-1$$2L-2$$2L-3$$2$$1$
Figure 4: Graphical representation of the swapped four-row transfer matrix for
the relation of the $D^{(2)}_{2}$-model. The resulting new $K^{New}$-matrices
(red boxes) acting on the enlarged unit cells can be related to the ones of
the $D^{(2)}_{2}$-model via the spectral shift $u\to-u+\frac{i\gamma}{2}$ and
some gauge transformations see (34).

Finally, we note that the staggered model can be related to another well-known
model in the limit $\gamma\to 0$. In this case, the Hamiltonian becomes, up to
an overall scale factor unitary equivalent to the one of the ferromagnetic XXX
Heisenberg chain with periodic boundary conditions for all staggering
parameters $\alpha$ (see Robertson  _et al._ (2021) for the self-dual case):

$\displaystyle\lim_{\gamma\to
0}\sin(\gamma)\,U^{odd}_{\mathcal{H}}(i\alpha)HU^{odd}_{\mathcal{H}}(-i\alpha)=$
$\displaystyle-\frac{1}{2}\sum^{2L-2}_{j=1}\sigma^{z}_{j}\sigma^{z}_{j+2}+2(\sigma^{+}_{j}\sigma^{-}_{j+2}+\sigma^{-}_{j}\sigma^{+}_{j+2})$
(36)
$\displaystyle-\frac{1}{2}\left(\sigma^{z}_{1}\sigma^{z}_{2}+2\left(\sigma^{+}_{1}\sigma^{-}_{2}+\sigma^{-}_{1}\sigma^{+}_{2}\right)\right)$
$\displaystyle-\frac{1}{2}\left(\sigma^{z}_{2L-1}\sigma^{z}_{2L}+2\left(\sigma^{+}_{2L-1}\sigma^{-}_{2L}+\sigma^{-}_{2L-1}\sigma^{+}_{2L}\right)\right)$
$\displaystyle+L\,.$

##  IV Numerical Study of small number of lattice sites

As a basis for our study of the finite size spectrum of the staggered six-
vertex model using its Bethe ansatz solution we have numerically diagonalized
the Hamiltonian (23) and the quasi-momentum $K$ (30) for small lattices with
$L$ unit cells. The quantum group symmetry of the system with our choice of
boundary conditions allows to decompose its Hilbert space
$\left(\mathbb{C}^{2}\right)^{\otimes 2L}$ into a direct sum of sectors
spanned by $U_{q}(\mathfrak{sl}(2))$ states with total spin $S=0,1,\dots,L$.

From our numerics we find that majority of the spectrum has real energies
although we also find complex ones. Since the latter come with large real
parts, we expect that they do not play a role at low energies. In Figures 5-5
we present the real parts of the eigenenergies as a function of the anisotropy
$\gamma$ for $L=3$ and $4$ and different values of the staggering parameter.

Figure 5:  (a) Spectrum of the Hamiltonian for $L=3$ and $\alpha=\pi/2$, which
is completely real for this choice of parameters. The solid lines indicate
levels with real quasi-momentum, while the dashed ones represent a purely
imaginary quasi-momentum. The colours indicate the quantum number $S$, i.e.
the highest $S_{z}$ of the multiplet leading to the displayed energy level.
(b) Real part of the spectrum of the model with staggering $\alpha=9\pi/4$ and
$L=3$. Notation as used in Figure 5. (c) Real part of the spectrum of the
self-dual model $\alpha=\pi/2$ for $L=4$. Same Notation as in 5. We see a new
ground state crossing appearing at $\gamma=\pi/8$ in comparison to the $L=3$
case.

Surprisingly, we find the spin of the model’s ground state (GS) depends on the
anisotropy parameter: ground state crossings are observed at certain rational
fractions of $\gamma/\pi$. For the small lattices which are accessible to the
numerical diagonalization we find as an approximate rule that the ground state
has $U_{q}(\mathfrak{sl}(2))$ spin $S^{GS}=1,2,\dots,L-1$ for anisotropies

$\frac{\pi}{2(S^{GS}+1)}\lesssim\gamma\lesssim\frac{\pi}{2S^{GS}}\,,$ (37)

(we will confirm that these inequalities become exact for larger systems
through our Bethe ansatz analysis below). For $0\leq\gamma\lesssim\pi/{2L}$
the ground state is in the sector with maximum $S^{GS}=L$, matching our
observation that the Hamiltonian (23) becomes that of the _ferromagnetic_
Heisenberg chain in the limit $\gamma\to 0$. Hence the
$U_{q}(\mathfrak{sl}(2))$ is maximal spontaneously broken in this range of
$\gamma$. This is different from the periodic model, where the ground state is
a unique state with total $S^{z}=0$ for all anisotropies Ikhlef  _et al._
(2008).

Furthermore, we find that the eigenvalues of the quasi-momentum operator may
transmute from real into purely imaginary ones when the anisotropy $\gamma$ is
lowered. The lower $\gamma$, the more energy levels acquire a purely imaginary
quasi-momentum. Such a transmutation has been found to be related to the
appearance of discrete states in the spectrum of conformal weights of a
staggered superspin chain based on a deformation of the algebra $sl(2|1)$
Frahm and Hobuß (2017).

The above analysis has been carried out for small lattice sizes only. However,
we will see how the same results appear when using the root density formalism
valid for all lattice sizes.

##  V The root density approach for the ground state

To proceed with our studies, we have to identify the Bethe root configurations
describing the low energy states. We have solved the Bethe equations (16) for
small lattice sizes and compared the resulting energies (24) and quasi-momenta
with the results from the direct diagonalization of the last chapter. We find
that the Bethe states parameterized by $M=L-S$ Bethe roots always realize the
highest-weight state in a given spin-$S$ multiplet.
Moreover, this allows to identify the configurations of the Bethe roots, which
correspond to the low-energy regime of the spin-chain. For this regime,
renormalization trajectories can be defined by fixing the deviation of the
pattern of Bethe roots from that for the ground state.
Regarding the ground states, we find in the entire regime of anisotropies
$0<\gamma<\pi/2$ that the root configurations consist of two types of Bethe
roots, either completely real or with an imaginary part $\pi/2$:

$\displaystyle v^{0}_{m}$ $\displaystyle=x_{m}\,,\qquad\qquad
m=1,2,3,\dots,M^{0}\,,$ (38) $\displaystyle v^{\frac{\pi}{2}}_{n}$
$\displaystyle=y_{n}+\frac{i\pi}{2}\,,\qquad
n=1,2,3,\dots,M^{\frac{\pi}{2}}\,.$

This observation enables us to study the model further in the root density
formalism Yang and Yang (1969): plugging (38) into the Bethe equations (16) we
obtain equations for their yet undetermined real parts, $x_{m}$ and $y_{n}$.
By taking the logarithm, we obtain the following coupled equations:

$\displaystyle 2\pi I^{x}_{m}$
$\displaystyle=-2L\phi\left(x_{m},\frac{\gamma-\alpha}{2}\right)-2L\phi\left(x_{m},\frac{\alpha+\gamma}{2}\right)+\sum_{k=1,\neq
m}^{M^{0}}\phi\left(x_{m}-x_{k},\gamma\right)$ (39)
$\displaystyle+\sum_{k=1,\neq
m}^{M^{0}}\phi\left(x_{m}+x_{k},\gamma\right)-\sum_{k=1}^{M^{\frac{\pi}{2}}}\psi\left(x_{m}-y_{k},\gamma\right)-\sum_{k=1}^{M^{\frac{\pi}{2}}}\psi\left(x_{m}+y_{k},\gamma\right)\,,\quad
m=1,\dots,M^{0}\,,$ $\displaystyle 2\pi I^{y}_{n}$
$\displaystyle=2L\psi\left(y_{n},\frac{\gamma-\alpha}{2}\right)+2L\psi\left(y_{n},\frac{\alpha+\gamma}{2}\right)-\sum_{k=1}^{M^{0}}\psi\left(y_{n}-x_{k},\gamma\right)$
$\displaystyle-\sum_{k=1}^{M^{0}}\psi\left(y_{n}+x_{k},\gamma\right)+\sum_{k=1,\neq
m}^{M^{\frac{\pi}{2}}}\phi\left(y_{n}-y_{k},\gamma\right)+\sum_{k=1,\neq
m}^{M^{\frac{\pi}{2}}}\phi\left(y_{n}+y_{k},\gamma\right)\,,\quad
n=1,\dots,M^{\frac{\pi}{2}}\,.$

Here we have introduced the quantum numbers $I^{x,y}_{m}\in\mathbb{N}$
characterizing the different branches of the logarithm and further have
defined

$\displaystyle\phi(x,y)$ $\displaystyle=2\arctan\left(\tanh(x)\cot(y)\right)$
$\displaystyle\psi(x,y)$ $\displaystyle=2\arctan\left(\tanh(x)\tan(y)\right)$

The solutions of these equations become dense on the whole real lines in the
thermodynamic limit $L\to\infty$ with $M^{0,\frac{\pi}{2}}/L$ fixed. This
allows to describe the distributions of the Bethe roots for the ground state
by two densities. The coupled linear integral equations fixing these densities
can be derived by the doubling procedure of the Bethe roots (see e.g. Asakawa
and Suzuki (1996)) and are given by:

$\displaystyle\rho^{x}(x)$
$\displaystyle=\sigma^{x}_{0}(x)+\frac{\tau^{x}_{0}(x)}{L}+\int_{-\infty}^{\infty}\text{d}x^{\prime}K_{0}(x-x^{\prime})\rho^{x}(x^{\prime})+\int_{-\infty}^{\infty}\text{d}x^{\prime}K_{1}(x-x^{\prime})\rho^{y}(x^{\prime})+\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}\,,$
(40) $\displaystyle\rho^{y}(x)$
$\displaystyle=\sigma^{y}_{0}(x)+\frac{\tau^{y}_{0}(x)}{L}+\int_{-\infty}^{\infty}\text{d}x^{\prime}K_{1}(x-x^{\prime})\rho^{x}(x^{\prime})+\int_{-\infty}^{\infty}\text{d}x^{\prime}K_{0}(x-x^{\prime})\rho^{y}(x^{\prime})+\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}\,.$

The driving terms and the integral kernels are given by the following
expressions, where the prime denotes the derivative in respect to the first
argument:

$\displaystyle\sigma^{x}_{0}(x)=$
$\displaystyle-\frac{1}{\pi}\phi^{\prime}\left(x,\frac{\gamma-\alpha}{2}\right)-\frac{1}{\pi}\phi^{\prime}\left(x,\frac{\alpha+\gamma}{2}\right)\,,$
(41) $\displaystyle\sigma^{y}_{0}(x)=$
$\displaystyle\frac{1}{\pi}\psi^{\prime}\left(y,\frac{\gamma-\alpha}{2}\right)+\frac{1}{\pi}\psi^{\prime}\left(y,\frac{\alpha+\gamma}{2}\right)\,,$
$\displaystyle\tau^{x}_{0}(x)=$
$\displaystyle-\frac{1}{\pi}\phi^{\prime}(2x,\gamma)-\frac{1}{2\pi}\phi^{\prime}(x,\gamma)+\frac{1}{2\pi}\psi^{\prime}(x,\gamma)\,,$
$\displaystyle\tau^{y}_{0}(x)=$
$\displaystyle-\frac{1}{\pi}\phi^{\prime}(2x,\gamma)-\frac{1}{2\pi}\phi^{\prime}(x,\gamma)+\frac{1}{2\pi}\psi^{\prime}(x,\gamma)\,,$
$\displaystyle K_{0}(x)=$
$\displaystyle\frac{1}{2\pi}\phi^{\prime}(x,\gamma)\,,\quad
K_{1}(x)=-\frac{1}{2\pi}\psi^{\prime}(x,\gamma)$

Note that for $\alpha=\pi/2$, the driving terms coincide, reflecting the self-
duality of the model for this value of the staggering parameter. The integral
equations can be solved order by order in $1/L$ by Fourier transformation. We
obtain the results for the first two orders:

$\displaystyle\sigma^{x}(x)$
$\displaystyle=\frac{2\sin\left(\frac{\pi(\alpha-\gamma)}{\pi-2\gamma}\right)}{\pi-2\gamma}\frac{1}{\cosh\left(\frac{2\pi
x}{\pi-2\gamma}\right)-\cos\left(\frac{\pi(\alpha-\gamma)}{\pi-2\gamma}\right)}\,,$
(42) $\displaystyle\sigma^{y}(x)$
$\displaystyle=\frac{2\sin\left(\frac{\pi(\alpha-\gamma)}{\pi-2\gamma}\right)}{\pi-2\gamma}\frac{1}{\cosh\left(\frac{2\pi
x}{\pi-2\gamma}\right)+\cos\left(\frac{\pi(\alpha-\gamma)}{\pi-2\gamma}\right)}\,,$
$\displaystyle\tau^{x}(x)$
$\displaystyle=\tau^{y}(x)=\frac{1}{4\pi}\int^{\infty}_{-\infty}\text{d}\omega
e^{i\omega
x}\frac{\sinh\left(\frac{3\gamma-\pi}{4}\omega\right)}{\sinh\left(\frac{\gamma\omega}{4}\right)\cosh\left(\frac{2\gamma-\pi}{4}\omega\right)}\,.$

Note that $\sigma^{x}(x)\leftrightarrow\sigma^{y}(x)$ under the duality
transformation $\alpha\to\pi-\alpha$, cf. (20). Furthermore, the staggering
has to be restricted to values $\gamma<\alpha<\pi-\gamma$ for the bulk parts
of the root densities to be positive Frahm and Martins (2012). From these
densities we compute the number of Bethe roots describing the ground state and
obtain:

$\displaystyle\frac{2M^{0}_{GS}+1}{L}$
$\displaystyle=2\cdot\frac{\pi-\alpha-\gamma}{\pi-2\gamma}+\frac{1}{L}\left(\frac{3}{2}-\frac{\pi}{2\gamma}\right)+\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}\,,$
(43) $\displaystyle\frac{2M^{\frac{\pi}{2}}_{GS}+1}{L}$
$\displaystyle=2\cdot\frac{\alpha-\gamma}{\pi-2\gamma}+\frac{1}{L}\left(\frac{3}{2}-\frac{\pi}{2\gamma}\right)+\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}\,.$

The individual numbers of Bethe roots are $\alpha$ and $\gamma$ dependent.
With the above expression and the fact that all Bethe states are highest
weight states, we can compute the sector $S$ in which the ground state is
realized. The surface contribution would imply a non-zero spin
$-\frac{1}{2}+\frac{\pi}{2\gamma}$ of the ground state which is independent of
the staggering $\alpha$ but is non-integer due to the explicit $\gamma$
dependence. This can be resolved by rounding the number of Bethe roots (43)
and the resulting ground state spin $S^{GS}$ to the nearest integer number,
giving

$\displaystyle S^{GS}=\left[-\frac{1}{2}+\frac{\pi}{2\gamma}\right],$ (44)

where the brackets indicate the rounding. Inverting this relation we obtain a
range of anisotropies $\gamma$ for which the ground state is realized in the
sector with spin $S^{GS}$:

$\displaystyle\frac{\pi}{2S^{GS}+2}<\gamma<\frac{\pi}{2S^{GS}}.$ (45)

This formula refines the approximate rule (37), which we have conjectured
based on our numerical investigations of small systems above. The minor
differences between (45) and the numerically observations for small lattices
can be interpreted as a result of the excessive influence of the boundary
terms for small $L$. Note that (44) tends to infinity as $\gamma\to 0$
reflecting the relation to the ferromagnetic XXX Heisenberg chain in this
limit. This generalizes our findings regarding the spontaneously broken
$U_{q}(\mathfrak{sl}(2))$ symmetry from small system sizes to general $L$.

While the spin $S=L-(M^{0}_{GS}+M_{GS}^{\frac{\pi}{2}})$ of the ground state
is independent of $\alpha$, the difference or the ratio of the numbers
$M^{0,\frac{\pi}{2}}_{GS}$ of the corresponding Bethe roots does depend on the
staggering parameter: from the bulk contributions to (43) we obtain

$\displaystyle dN_{GS}=$ $\displaystyle
M^{0}_{GS}-M^{\frac{\pi}{2}}_{GS}=L\frac{\pi-2\alpha}{\pi-2\gamma}\,,$ (46)
$\displaystyle\frac{2M^{0}_{GS}+1}{2M^{\frac{\pi}{2}}_{GS}+1}=$
$\displaystyle\frac{\pi-\alpha-\gamma}{\alpha-\gamma}+\mathcal{O}\Big{(}\frac{1}{L}\Big{)}\,.$
(47)

Hence, by varying $\alpha$, crossings between different spin-$S$ states may be
induced. To realize the corresponding root configurations on a given lattice
the numbers of roots should be commensurate with $L$, i.e. have a rational
ratio and a simple scaling of the difference of the number of Bethe roots for
the bulk contribution. This is achieved by fixing $\alpha$ as

$\displaystyle\alpha=\frac{p\gamma+q(\pi-\gamma)}{p+q},$ (48)

where $p$ and $q$ are positive integers and relatively prime to each other
Frahm and Seel (2014). With that condition, one would obtain the following
expression for the ratio and difference between the two types of roots

$\displaystyle\frac{2M^{0}_{GS}+1}{2M^{\frac{\pi}{2}}_{GS}+1}$
$\displaystyle=\frac{p}{q}+\mathcal{O}(\frac{1}{L})$ (49) $\displaystyle
M^{0}_{GS}-M^{\frac{\pi}{2}}_{GS}$ $\displaystyle=L\frac{p-q}{p+q}.$

Note that by setting $p=q=1$, which corresponds to the self-dual case
$\alpha=\pi/2$, the numbers of the two types of Bethe roots become the same
for all $\gamma$, corresponding to the additional degeneracy of the spectrum.

Knowing the ground state densities (42) the bulk and boundary contributions to
the expectation values of the conserved quantities in the thermodynamic limit
can be calculated: being bulk quantities, the energy density and the Fermi
velocity agree with those of the periodic model Frahm and Seel (2014)

$\displaystyle e_{\infty}$
$\displaystyle=-2\int^{\infty}_{-\infty}\text{d}\omega\frac{\sinh(\frac{\gamma\omega}{2})\left(\sinh\left(\frac{\pi\omega}{2}-\frac{\omega\gamma}{2}\right)\cosh(\frac{\omega\pi}{2}-\alpha\omega)-\sinh(\frac{\gamma\omega}{2})\right)}{\sinh(\frac{\omega\pi}{2})\sinh((\frac{\pi-2\gamma}{2})\omega)}\,,\qquad
v_{F}=\frac{2\pi}{\pi-2\gamma}\,,$ (50)

while the surface contribution to the energy reads

$\displaystyle
f_{\infty}=-\int^{\infty}_{-\infty}\text{d}\omega\frac{\cosh\left(\frac{1}{4}(\pi-2\alpha)\omega\right)\sinh\left(\frac{1}{4}(3\gamma-\pi)\omega\right)\cosh\left(\frac{\gamma\omega}{4}\right)}{\cosh\left(\frac{1}{4}(\pi-2\gamma)\omega\right)\sinh\left(\frac{\pi\omega}{4}\right)}-\frac{4\sin(2\gamma)}{\cos(2\alpha)-\cos(2\gamma)}.$
(51)

Similarly, we obtain an expression for the value of quasi-momentum of the
ground state in the thermodynamic limit

$\displaystyle\mathcal{K}_{thermo}=Lk_{\infty}+k_{s}+\log\Big{[}\frac{\sin(2\gamma-\alpha)}{\sin(2\gamma+\alpha)}\Big{]}+\left(2L-1\right)\log\Big{[}\frac{\sin(\gamma-\alpha)}{\sin(\gamma+\alpha)}\Big{]}+\mathcal{O}\big{(}\frac{1}{L}\big{)},$
(52)

where the bulk $k_{\infty}$ and surface $k_{s}$ contributions read

$\displaystyle k_{\infty}=$ $\displaystyle
4\int^{\infty}_{-\infty}\text{d}\omega\frac{\sinh(\frac{\omega\gamma}{2})\sinh(\frac{\pi\omega}{2}-\alpha\omega)\sinh(\frac{\pi-\gamma}{2}\omega)}{\omega\sinh(\frac{\omega\pi}{2})\sinh(\frac{\pi-2\gamma}{2}\omega)}\,,$
(53) $\displaystyle k_{s}=$ $\displaystyle
2\int^{\infty}_{-\infty}\text{d}\omega\frac{\sinh(\frac{3\gamma-\pi}{4}\omega)\cosh(\frac{\gamma\omega}{4})\sinh(\frac{\pi-2\alpha}{4}\omega)}{\omega\sinh(\frac{\omega\pi}{4})\cosh(\frac{2\gamma-\pi}{4}\omega)}$
$\displaystyle-\log\left(\frac{\cos(\alpha+\gamma)-1}{\cos(\alpha-\gamma)-1}\right)-\log\left(\frac{\cos(\alpha+\gamma)+1}{\cos(\alpha-\gamma)+1}\right)\,.$
(54)

Having these explicit expressions, we can identify the effective field theory
describing the low-energy regime of the spin-chain for large system sizes by
studying the finite size spectrum. Due to the criticality of the model and the
open boundary condition, we expect that the field theory is a boundary
conformal field theory (BCFT). The scaling dimensions of the BCFT in the
finite strip geometry formulation of the BCFT can be accessed by calculating
the energy $E_{(n,d)}(L)$ of a state in the finite lattice. The explicit
relationship is given by Blöte  _et al._ (1986); Cardy (1984)

$\displaystyle E_{(n,d)}(L)=Le_{\infty}+f_{\infty}+\frac{\pi
v_{F}}{L}\left(-\frac{c}{24}+h_{n}+d\right),$ (55)

where $c$ is the central charge of the CFT, $h_{n}$ the conformal weight of
the corresponding primary field and $d$ is the level of the descendants. Note
here, since the true central charge and the true scaling dimensions appear in
(55) as a sum, the spectrum of the lattice model provides only the effective
central charge and effective conformal weights by the formulae

$\displaystyle c_{\text{eff}}$ $\displaystyle=-\frac{24L}{\pi v_{F}}\Delta
E_{0}\,,$ (56) $\displaystyle h^{n}_{\text{eff}}$ $\displaystyle=\frac{L}{\pi
v_{F}}\Delta E_{n}\,,$

where $\Delta E_{n}$ is the energy gap of a state with energy $E_{n}$ in
respect to thermodynamic ground state:

$\displaystyle\Delta E_{n}=E_{(n,0)}(L)-Le_{\infty}-f_{\infty}\,.$ (57)

##  VI Analysis of the finite size spectrum

###  VI.1 Continuous part

To determine the (effective) scaling dimensions, one needs to take low lying
excitations above the ground state into account. We find that most root
configurations describing the low energy regime are still given by (38) but
with different quantum numbers $S$ and $dN$ as compared to the ground state
(45), (46). Hence, the low-lying excited states can be described in the
framework of the root density formalism analogously as already done for the
ground state. However, the integral boundaries of the linear integral equation
for the excited states will differ from those of the ground state representing
the different values of quantum numbers $S$, $dN$.

For the model considered in this paper with two branches of excitations and
the same Fermi velocity the resulting finite size energies can be expressed in
terms of these quantum numbers as de Vega and Woynarowich (1985); Bogoliubov
_et al._ (1986); Suzuki (1988); Frahm and Yu (1990) (for the particular case
of open boundary conditions see Asakawa and Suzuki (1996); Essler  _et al._
(2005b))

$\displaystyle E(L)=Le_{\infty}+f_{\infty}+\frac{\pi
v_{F}}{L}\left\\{\frac{1}{2}\Delta\vec{M}^{T}\left(ZZ^{\top}\right)^{-1}\Delta\vec{M}\right\\}+o\left(L^{-1}\right)\,,$
(58)

where

$\displaystyle\Delta\vec{M}=\begin{pmatrix}M^{0}-M^{0}_{GS}\\\
M^{\frac{\pi}{2}}-M^{\frac{\pi}{2}}_{GS}\end{pmatrix}\,,\qquad
Z=\lim_{x\to\infty}\begin{pmatrix}\xi_{11}(x)&&\xi_{12}(x)\\\
\xi_{21}(x)&&\xi_{22}(x)\end{pmatrix}\,,$

We recall that $S=L-M^{0}-M^{\frac{\pi}{2}}$ is the $S_{z}$-quantum number of
the Bethe state, i.e. highest weight state in the corresponding spin-$S$
multiplet, and $dN=M^{0}-M^{\frac{\pi}{2}}$ is the difference in the number of
Bethe roots of the two different types (38). $\mathbf{\xi}$ is the so-called
dressed charge matrix defined by linear integral equations similar to (40):

$\displaystyle\xi_{11}(x)$ $\displaystyle=1+$
$\displaystyle\int^{\infty}_{-\infty}\text{d}x^{\prime}\,K_{0}(x-x^{\prime})\xi_{11}(x^{\prime})+\int^{\infty}_{-\infty}\text{d}x^{\prime}\,K_{1}(x-x^{\prime})\xi_{21}(x^{\prime})\,,$
(59) $\displaystyle\xi_{21}(x)$ $\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}\text{d}x^{\prime}\,K_{1}(x-x^{\prime})\xi_{11}(x^{\prime})+\int^{\infty}_{-\infty}\text{d}x^{\prime}\,K_{0}(x-x^{\prime})\xi_{21}(x^{\prime})\,,$
$\displaystyle\xi_{12}(x)$ $\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}\text{d}x^{\prime}\,K_{0}(x-x^{\prime})\xi_{12}(x^{\prime})+\int^{\infty}_{-\infty}\text{d}x^{\prime}\,K_{1}(x-x^{\prime})\xi_{22}(x^{\prime})\,,$
$\displaystyle\xi_{22}(x)$ $\displaystyle=1+$
$\displaystyle\int^{\infty}_{-\infty}\text{d}x^{\prime}\,K_{1}(x-x^{\prime})\xi_{12}(x^{\prime})+\int^{\infty}_{-\infty}\text{d}x^{\prime}\,K_{0}(x-x^{\prime})\xi_{22}(x^{\prime})\,.$

By means of the Wiener-Hopf method one finds that Frahm and Yu (1990)

$\displaystyle\left(ZZ^{\top}\right)=\begin{pmatrix}1-\int_{-\infty}^{\infty}\text{d}x\,K_{0}(x)&&-\int_{-\infty}^{\infty}\text{d}x\,K_{1}(x)\\\
-\int_{-\infty}^{\infty}\text{d}x\,K_{1}(x)&&1-\int_{-\infty}^{\infty}\text{d}x\,K_{0}(x)\end{pmatrix}^{-1}$

giving

$\displaystyle E(L)=Le_{\infty}+f_{\infty}+\frac{\pi
v_{F}}{L}\left(-\frac{1}{12}+\frac{\gamma}{4\pi}\left(2S+1-\frac{\pi}{\gamma}\right)^{2}+\frac{1}{4}\frac{(dN-
dN_{GS})^{2}}{\tilde{Z}_{D}^{2}}+n_{ph}\right),$ (60)

where $dN_{GS}$ has been defined in (46), $n_{ph}$ is the number of particle-
hole excitations in the vicinity of the Fermi points and

$\displaystyle\tilde{Z}_{D}=\lim_{\omega\to
0}\left(1-\int_{-\infty}^{\infty}\text{d}x\,\text{e}^{i\omega
x}\,(K_{0}(x)-K_{1}(x))\right)^{-1}\,.$

We note that $\tilde{Z}_{D}$ diverges in the limit $\omega\to 0$ as a
consequence of the degeneracy of the integral kernel in (59). This is a
characteristic feature in several lattice models with a continuous spectrum of
scaling dimensions emerging in the continuum limit, see e.g. Essler  _et al._
(2005a); Ikhlef  _et al._ (2008); Frahm and Martins (2011): as a consequence
of this singularity the penultimate term in (60) does not contribute to the
finite size scaling for any finite $dN-dN_{GS}$ in the limit $L\to\infty$. For
large but finite $L$ one finds that the energy gaps between states with
different $dN-dN_{GS}$ vanish as $1/L(\log L)^{2}$, forming a continuum in the
thermodynamic limit. As in the periodic model we find that these logarithmic
corrections are determined in terms of the eigenvalues of the quasi-momentum
operator: to bring the logarithmic corrections in the scaling dimensions under
control we introduce the continuous quantum number $s$ based on the difference
between the quasi-momentum of the renormalization trajectory and the one of
the thermodynamic ground state

$\displaystyle s$
$\displaystyle=\frac{\pi-2\gamma}{4\pi\gamma}\Big{(}\mathcal{K}-\mathcal{K}_{thermo}\Big{)}$
$\displaystyle=\frac{\pi-2\gamma}{4\pi\gamma}\Big{(}\sum^{M}_{i=1}k_{0}(v_{i})-Lk_{\infty}-k_{s}\Big{)}\,.$
(61)

As in the periodic model the variable $s$ can be related to the deviation of
the quantum number $dN$ from the one of the ground state $dN_{GS}$ (note that
$s$ is real for root configurations (38)):

$\displaystyle dN-
dN_{GS}=\frac{2s}{\pi}\left(\log\left(\frac{L}{L_{0}}\right)+B(s)\right),$
(62)

where $L_{0}$ is a non-universal length, which depends on the anisotropy
$\gamma$, while $B(s)$ is the density of states in the continuum. For the
quantum group invariant open boundary case we have checked this identification
numerically for levels in the spin sectors with several $S=S^{GS}$ and
corresponding anisotropies $\gamma$ from Eq. (45) on the self-dual line. For
$\gamma\lesssim\pi/2S$, i.e. close to the _right_ boundary of these intervals,
we find that the quotient $(dN-dN_{GS})/s$ for different $dN$ (and $L$)
collapse to a single line $\propto\log L$ with slope $2/\pi$ independent of
$\gamma$, as predicted by (62), see Figure 6.

Figure 6:  $(dN-dN_{GS})/s$ vs. $\log(L)$ on the self-dual line
$\alpha=\pi/2$. Different symbols indicate different combinations of $S$ and
$\gamma$ as labeled in the legend. (a) Collapse of data for spin-$S$ states
and anisotropies $\gamma\lesssim\pi/2S$ corresponding to the continuous part
of the conformal spectrum with different $dN$ (encoded by coloring: black
$dN=1$, red $dN=2$, green $dN=3$, blue $dN=4$ and magenta $dN=5$). (b) Lifting
of this degeneracy w.r.t. $dN$ for the states in the $S=1$-continuum for
anisotropy $\gamma=23\pi/80\gtrsim\pi/4$, where the ground state crosses into
the $S=2$ sector.

.

On the other hand, for values of $\gamma\gtrsim\pi/(2S+2)$, i.e. close to the
transition $S^{GS}\to S^{GS}+1$, we observe a splitting into lines for
different values of $dN$, although still with slope $2/\pi$ for sufficient
large system sizes $L$, see Figure 6. In Section VI.2 below we will see that
the resulting modification of the density of states $B(s)$ in (62) can be
attributed to the transmutation of levels near the bottom of the continuous
part of the spectrum into discrete states. We emphasize, however, that this
does not affect the quality of the parameterization of the logarithmic
corrections to scaling in the continuum part of the spectrum, see Figures 8,9
and 10 below.

The subleading corrections in (62), in particular the density of states
$B(s)$, provide additional information for the description of the low energy
behavior of the system by means of an effective field theory. A reliable
determination of $B(s)$, however, requires the use of methods allowing the
analysis of systems which are significantly larger than what is possible in
the approach used here, e.g. based on the formulation of the spectral problem
in terms of non-linear integral equations as in Candu and Ikhlef (2013); Frahm
and Seel (2014). While we do not address this problem here let us note that
the identification (61) of the continuous quantum number $s$ with the
eigenvalues of the quasi-momentum operator is an essential condition for
further progress in this direction.

Comparing (60), written in terms of quantum number $s$, with the prediction
(55) we find the effective conformal weights corresponding to levels in the
sector with spin $S$ to be

$\displaystyle h_{\text{eff}}$
$\displaystyle=-\frac{1}{12}+\frac{\gamma}{4\pi}\left(2S+1-\frac{\pi}{\gamma}\right)^{2}+\frac{\gamma
s^{2}}{\pi-2\gamma}+n_{ph},\qquad s\in\mathbb{R}\,.$ (63)

The ground state of the lattice model is realized in the sector with spin
$S=S^{GS}(\gamma)$, Eq. (44), and $s=0$. This leads to the effective central
charge Robertson  _et al._ (2021)

$\displaystyle c_{\text{eff}}$
$\displaystyle=2-\frac{6\gamma}{\pi}\left(2S^{GS}(\gamma)+1-\frac{\pi}{\gamma}\right)^{2}=2-\frac{24\gamma}{\pi}\,\left(\mathrm{frac}\left(\frac{\pi}{2\gamma}\right)-\frac{1}{2}\right)^{2}\,,$
(64)

where $\mathrm{frac}\left(\frac{\pi}{2\gamma}\right)$ denotes the fractional
part of $\frac{\pi}{2\gamma}$. The cusps due to the fractional part
$(\mathrm{frac}(\frac{\pi}{2\gamma})-\frac{1}{2})$ of the effective central
charge are a consequence of the ground state crossings occurring at integer
values of $\pi/2\gamma$ in the staggered six-vertex model, see Figure 7.

Figure 7: The effective central charge (64) of the staggered spin chain as a
function of the anisotropy is represented as the solid black line. Dashed
colored lines are plots of (64) in sectors with given spin $S$, crosses
represent the effective central charges $c_{\text{eff}}$ obtained from the
Bethe ansatz solutions for large $L$. Vertical lines represent the ground
state crossings.

The emergence of a continuous spectrum, parameterized by the quantum number
$s$, in the thermodynamic limit is shown in Figures 8, 9 and 10 where we have
computed the effective conformal weights $h_{\mathrm{eff}}$ from the finite
size energies using (56) and, using the Bethe ansatz results for $s$, from
(63) for various anisotropies $\gamma$ and staggering $\alpha$ in the spin
sector containing the ground state. Extrapolation of the finite size data to
$L\to\infty$ by means of a rational function of $1/\log L$ shows that various
levels with $dN\neq dN_{GS}$ converge to the bottom of the corresponding
continuum (given by the Bethe state with $dN=dN_{GS}$).

Figure 8: Spectrum of conformal weights for the self-dual model (i.e.
$dN_{GS}=0$): for each $S=1,2,3$ we have chosen two values of the anisotropy
in the intervals (45) $S$ is the spin of the ground state. Black symbols are
the effective scaling dimensions obtained from Eq. (56) using the finite size
energies obtained from the solution of the Bethe equations. Red symbols
represent the scaling dimensions obtained from (63) using the Bethe ansatz
results for the quasi-momentum $s$. Note that the latter provide an excellent
parameterization of the logarithmic corrections via the continuous variable
$s$ for all $\gamma$ values considered (e.g. in the top-right figure for the
parameters used in Figure 6 showing the lifting of the degeneracy of (62)
w.r.t. $dN$).

Figure 9: Similar as Fig. 8, but for anisotropies where the ground state is
found in the sectors $S=7,8,10$.

Figure 10: Similar as Fig. 8 but for values of $\alpha$ away from the self-
dual point ($\Delta N=dN-dN_{GS}$).

Motivated by previous studies we interpret these results in the context of the
$SL(2,\mathbb{R})/U(1)$ black hole CFT at level $k$: this model has been found
to describe the continuum limit of the staggered six-vertex model for periodic
boundary conditions Ikhlef  _et al._ (2012); Candu and Ikhlef (2013); Frahm
and Seel (2014) and, at the self-dual point, for open boundary conditions in
the $D^{(2)}_{2}$-formulation Robertson  _et al._ (2021). The central charge
$c_{BH}$ and and conformal weights $h_{BH}$ of the primary fields in the black
hole CFT are given by Hanany  _et al._ (2002); Ribault and Schomerus (2004)

$\displaystyle c_{BH}=2+\frac{6}{(k-2)}\,,\quad
h_{BH}=\frac{(n+wk)^{2}}{4k}-\frac{J(J-1)}{k-2}\quad\text{with}\quad
J=\frac{1}{2}+i\tilde{s},\quad\tilde{s}\in\mathbb{R}^{+}_{0}\,,$ (65)

where the integers $n$ and $w$ label the momentum and winding in the compact
direction of the semi-infinite cigar-shaped target space of the CFT while
$i\tilde{s}=J-1/2$ is the momentum along the uncompactified direction. The
resulting effective conformal weights are

$\displaystyle
h^{BH}_{\text{eff}}=-\frac{1}{12}+\frac{(n+wk)^{2}}{4k}-\frac{1}{k-2}\,\left(J-\frac{1}{2}\right)^{2}+d\,,$
(66)

where $d$ describes the level of descendant. The identification of the studied
spin chain as a lattice regularization of the black hole CFT can be made by
comparing (66) with (63) leading to the following correspondence222Note $s$ as
defined in (61) can have either sign and its relation to $J-1/2$ can be fixed
up to this sign only. Similarly, we could replace $(n,w)\to(-n,-w)$. Here we
follow the convention used in Ref. Robertson  _et al._ (2021).

$\displaystyle k=\frac{\pi}{\gamma},\qquad n=-2S-1,\qquad
w=1,\qquad\left(J-\frac{1}{2}\right)^{2}=(is)^{2},\qquad d=n_{ph}.$ (67)

The appearance of a non-vanishing winding number $w$ in the open boundary
model is a consequence of the boundary conditions (7), (8) which lead to an
effective twist for the propagating modes in the spin chain. Note that, as
expected from the presence of the continuous quantum number $s$, this is a
non-rational CFT. The Virasoro vacuum is not normalizable which is reflected
by the fact that the identity field with $h_{BH}=0$ appears neither in the
spectrum of conformal weights (65) nor in the discrete part of the spectrum
discussed in the following section.

###  VI.2 Discrete part

In the black hole CFT, besides the continuous scaling dimensions (65) with
$J=\frac{1}{2}+i\tilde{s}$ for real $\tilde{s}$ considered above, there exists
also a set of conformal weights with discrete values $J$, related to bound
states localized near the tip of the cigar-shaped target space of the model
Maldacena and Ooguri (2001); Hanany  _et al._ (2002); Ribault and Schomerus
(2004). For the corresponding states to be normalizable and to guarantee non-
negative conformal weights the possible values of $J$ are restricted to obey
the following two conditions Ribault and Schomerus (2004); Hanany  _et al._
(2002):

$\displaystyle\frac{1}{2}<J<\frac{(k-1)}{2}\,,\quad
J=\frac{|kw|-|n|}{2}-\ell,\qquad\ell=0,1,2\dots\,.$ (68)

From the first of these conditions we expect that a discrete state with fixed
$J$ is realized in the spin-chain spectrum for anisotropies $\gamma$ ranging
from $0$ to some root of unity. Specifically, a discrete state with given $J$
may be realized for anisotropies

$\displaystyle 0<\gamma<\frac{\pi}{2J+1}\,.$ (69)

The discrete CFT states and their realization conditions can also be related
to the quasi-momentum of the Bethe state. In our study of the continuous part
of the spectrum above we found that the Bethe states are parameterized by root
configurations (38) resulting in real eigenvalues for the quasi-momentum.
However, as seen in the study of small lattice sizes, the quasi-momentum can
also change from real values to purely imaginary ones when the anisotropy
$\gamma$ is lowered. On the level of Bethe roots, this translates into a root
pattern changing from (38) to more complicated complex configurations: the
real parts of one or more of the roots diverge as the anisotropy approaches
from above certain rational multiples of $\pi$. Reducing the anisotropy
further these roots reappear in the finite domain with different imaginary
parts. Depending on the state considered this process may be repeated several
times until the root configuration acquires the following remarkable pattern:
for states parameterized by an even number $M$ of Bethe roots they come in
pairs $v_{j}$, $\bar{v}_{j}$ mirrored at the line $i\pi/4$

$\displaystyle
v_{j}=x_{j}+iy_{j}\,,\qquad\bar{v}_{j}=x_{j}+i\left(\frac{\pi}{2}-y_{j}\right)\,$
(70) $\displaystyle\text{with}\quad x_{j},y_{j}\geq 0\,\qquad
j=1,2\dots\frac{M}{2}\,.$

If the number $M$ of Bethe roots is odd there appears an additional root with
imaginary part $\pi/4$, i.e.

$\displaystyle
v_{j}=x_{j}+iy_{j}\,,\quad\bar{v}_{j}=x_{j}+i\left(\frac{\pi}{2}-y_{j}\right)\,,\quad
v_{M}=x+\frac{i\pi}{4}$ (71) $\displaystyle\text{with}\quad x_{j},y_{j},x\geq
0\,,\qquad j=1,2\dots\frac{M-1}{2}\,.$

Examples of such root configurations for spin-$1$ states on lattices with
$L=40$ ($41$) sites, i.e. $M=39$, $M=40$, evolving from the (38) with
$dN=M^{0}-M^{\frac{\pi}{2}}=1$ and $2$, respectively, as $\gamma$ is lowered
are shown in Figure 11.

Figure 11: Pattern of Bethe roots leading to an imaginary quasi-momentum. The
configurations are obtained by following configurations given as (38) with
$dN$ as indicated by lowering the anisotropy. One the left, we have an odd
number of Bethe roots and so one root lies on the line $i\pi/4$, while the
rest is paired. On the right, we have an even number of roots, hence, the
pairing works.

Following the transmutation of configurations (38) to (70), (71) under the
variation of $\gamma$ for small values of $dN$ we are able to observe what
happens to the scaling dimensions when the quasi-momentum changes from from
real to imaginary: as discussed above the scaling dimensions corresponding to
spin-$1$ states are in the continuous part of the spectrum (63) for
anisotropies $\pi/4<\gamma<\pi/2$ where they are separated by finite size gaps
$\sim(dN/\log L)^{2}$.333In the limit $\gamma\to\pi/2$ one observes a
crossover to a linear dependence on $1/\log L$. This can be understood from
the fact that the staggered six-vertex model at $\gamma=\pi/2$ coincides with
the integrable $OSp(2|2)$ model which is in a different universality class
Martins  _et al._ (1998); Jacobsen  _et al._ (2003). As $\gamma$ is reduced
further the lowest levels approach the lower bound of the $S=1$ continuum,
leaving it when the quasi-momentum becomes purely imaginary, see Figure 12 for
the lowest states in this spin sector. Specifically we find that the finite
size energies of the states with $dN=1$, $2$ ($dN=3$, $4$), realized for even
and odd lattice sizes $L$ respectively, lead to the conformal weights (66) of
the black hole CFT with $J=(k-3)/2$ ($(k-5)/2$) in the regime where $s\in
i\mathbb{R}$.

Figure 12: Effective scaling dimensions vs. anisotropy $\gamma$ for the self-
dual model, $\alpha=\pi/2$, of size $L=40,41$ derived for the lowest states in
the $S=1$ sector, showing the transmutations of continuous states into
discrete ones. The black shaded area represents the continuum of levels
starting at the spin-$S^{GS}=1$ ground state. Grey shading indicates the
continua in the spin $S^{GS}>1$ sectors, which overlap with each other and the
spin-$S^{GS}=1$ one. The lower edge of the continua corresponds to $c_{eff}$
(64) up to the factor of $-24$. Blue symbols denote the effective scaling
dimensions obtained from finite size data (56) corresponding to weights from
the continuous part with $S=1$ of the CFT spectrum given in terms of root
configurations (38) with different $dN$. Red and green symbols depict the
continuation of the corresponding states (same symbol shape) to anisotropies
where the Bethe root patterns change to (70) or (71) and the quasi-momentum
$s$ becomes imaginary. Red (green) solid lines are the effective scaling
dimensions (66) of the primary fields from the discrete part of the CFT
spectrum with $n=-3$, $w=1$ and $J=(k-3)/2$ ($(k-5)/2$) for $k=\pi/\gamma$.
Dashed lines are their continuation to anisotropies $\gamma>\pi/(2J+1)$ where
the corresponding operators in the CFT become non-normalizable.

It turns out that the finite size formula (63) continues to hold for purely
imaginary $s$: in this case $J=\frac{1}{2}+is$ has to be an element of the
discrete set (68). In the CFT sector with quantum numbers $(n,w)=(-2S-1,1)$
for the compact degree of freedom this leads to the following condition on the
allowed imaginary values of $s$ in the lattice model (as mentioned above the
sign of $s$ is not fixed by (67)):

$\displaystyle s_{\ell}=\pm
i\left(\frac{\pi}{2\gamma}-S-1-\ell\right)\,,\qquad\ell=0,1,2,\dots<\frac{\pi}{2\gamma}-S-1\,.$
(72)

Note that this implies that the thresholds for the appearance of discrete
levels of the black hole CFT resulting from the unitarity condition (68)
coincide with the anisotropies (45) where the spin of the ground state in the
lattice model changes. This prediction can be compared with our numerically
findings for the $S=1$ states with $J=(k-3)/2-\ell$, $\ell=0,1$, considered in
Figure 12: their quasi-momentum, once it becomes imaginary, is expected to
match the condition above for $\ell=0,1$, i.e.

$\displaystyle s_{\ell=0}=\pm
i\left(\frac{\pi}{2\gamma}-2\right)\quad\text{for~{}}\gamma<\frac{\pi}{4}\,,$
(73) $\displaystyle s_{\ell=1}=\pm
i\left(\frac{\pi}{2\gamma}-3\right)\quad\text{for~{}}\gamma<\frac{\pi}{6}\,.$

Our numerical data show that the change from real to imaginary quasi-momentum
takes place at or slightly below these values of $\gamma$. Overall, i.e. up to
finite size corrections near these thresholds, Eqs. (73) match our results for
the quantum number $s$ obtained for system sizes $L=40$ and $41$, see Figure
13. We presume that this transmutation of low-lying levels from the continuous
part of the spectrum into discrete ones is the origin of the modification of
the density of states $B(s)$ for small $s$ close to these anisotropies
observed in the previous section.

Figure 13: Real (blue symbols) and imaginary (red and green symbols) part of
the quasi-momentum $s$ vs. anisotropy $\gamma$ for the spin $S=1$ states
considered in Figure 12. The solid red and green lines depict the quasi-
momentum given by the CFT predictions via (73). Dashed lines are obtained by
continuation of the CFT data into the region where the unitarity condition
(68) is violated. Note that $s\equiv 0$ for the $dN=0$ state.

Therefore, the expression for the effective scaling dimensions (63) as
obtained from the finite size analysis of the staggered vertex model correctly
describes both the continuous and the discrete part of the spectrum of the
$SL(2,\mathbb{R})/U(1)$ sigma model at level $k$ via the identifications (67).

##  VII Summary

Starting from the six-vertex model with quantum group invariant boundary
conditions we have constructed an integrable anisotropic
$\mathbb{Z}_{2}$-staggered spin-$1/2$ chain. The
$U_{q}(\mathfrak{sl}(2))$-symmetry of the model is spontaneously broken: in
contrast to the periodic chain where the lowest state is always in the sector
with $S_{z}=0$, the ground state of the open chain has non-zero spin depending
on the anisotropy $\gamma$, becoming completely polarized for sufficiently
small $\gamma$ ($\gamma\to 0$ in the thermodynamic limit).

For the self-dual choice of the staggering, i.e. $\alpha=\pi/2$, this model is
equivalent to the $D_{2}^{(2)}$-spin chain with integrable boundary conditions
studied in Refs. Robertson  _et al._ (2021); Nepomechie and Retore (2021).
This model has been identified as a lattice regularization of the
$SL(2,\mathbb{R})_{k}/U(1)$ sigma model whose non-compact degrees of freedom
lead to a continuous spectrum of conformal weights in the thermodynamic limit.
We find that, similar as in the staggered six-vertex model with periodic
boundary conditions Frahm and Seel (2014), this identification can be extended
to the entire range of staggering parameters $\gamma<\alpha<\pi-\gamma$.

As in previous studies of the periodic model Ikhlef  _et al._ (2012); Candu
and Ikhlef (2013); Frahm and Seel (2014) the formulation as a staggered model
facilitates the definition of a conserved quasi-momentum operator $K$ (30).
Unlike the Hamiltonian (23) and the other conserved quantities generated by
the four-row transfer matrix (17) the quasi-momentum is odd under the duality
transformation. The eigenvalues of this operator allow to distinguish states
corresponding to the levels in the continuous (discrete) part of the spectrum
of the sigma model (see also Frahm and Hobuß (2017)): already for the very
small systems that have been diagonalized numerically the corresponding quasi-
momenta are real (imaginary), respectively. Moreover, the eigenvalues of the
quasi-momentum can be related to the momentum along the non-compact direction
of the semi-infinite cigar-shaped target space of the black hole CFT. As a
consequence they determine the amplitudes of the strong logarithmic
corrections to scaling of the states leading to the continua and reflect the
appearance of the quantized discrete states in the spectrum of the lattice
model. An open problem is the calculation of the density of states in the
continuous part of the CFT spectrum from the finite size data for the vertex
model and in particular the effect of the ground state crossings between
sectors with different $U_{q}(\mathfrak{sl}(2))$ spin. We expect that this
calculation is facilitated by the relation (61) between the continuous quantum
number $s$ in the CFT and the quasi-momentum $\mathcal{K}$ for the Bethe
states, see Refs. Candu and Ikhlef (2013); Frahm and Seel (2014) for the
periodic chain.

###### Acknowledgements.

Funding for this work has been provided by the Deutsche Forschungsgemeinschaft
under grant No. Fr 737/9-2 as part of the research unit _Correlations in
Integrable Quantum Many-Body Systems_ (FOR2316).

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