The $D^{(2)}_{3}$ spin chain and its finite-size spectrum

Holger Frahm1, Sascha Gehrmann1,
Rafael I. Nepomechie2, Ana L. Retore3

1Institut f$\ddot{{\rm u}}$r Theoretische Physik, Leibniz Universit$\ddot{{\rm
a}}$t Hannover

Appelstraße 2, 30167 Hannover, Germany

2Physics Department, PO Box 248046

University of Miami, Coral Gables, FL 33124 USA

3Department of Mathematical Sciences, Durham University

Durham, DH1 3LE, UK

Abstract

Using the analytic Bethe ansatz, we initiate a study of the scaling limit of
the quasi-periodic $D^{(2)}_{3}$ spin chain. Supported by a detailed symmetry
analysis, we determine the effective scaling dimensions of a large class of
states in the parameter regime $\gamma\in(0,\frac{\pi}{4})$. Besides two
compact degrees of freedom, we identify two independent continuous components
in the finite-size spectrum. The influence of large twist angles on the latter
reveals also the presence of discrete states. This allows for a conjecture on
the central charge of the conformal field theory describing the scaling limit
of the lattice model.

##  1 Introduction

Low-dimensional critical lattice models whose low-energy behavior is captured
by field theories with a non-compact target space have attracted increasing
attention in recent years. Such theories may support continuous spectra of
critical exponents, which have been argued to describe the multifractal
scaling of the critical wave functions at the integer quantum Hall plateau
transition and other disorder-driven quantum phase transitions [1, 2]. At the
same time, finite lattice realizations of such models with a compact
configuration space may facilitate numerical studies of questions arising in
the context of AdS/CFT dualities [3]. This is particularly true if these
lattice models are integrable and allow for a solution of their spectral
problem by means of Bethe ansatz methods.

One approach to study two-dimensional disorder problems such as quantum Hall
systems or dirty $d$-wave superconductors is based on supersymmetric
reformulations in terms of spin chains based on supergroup symmetries [4, 5,
6, 7]. A characteristic feature of these spin chains is that their Hilbert
space has a $Z_{2}$ staggering, i.e. the local degrees of freedom in these
spin chains are different (conjugate) representations of the underlying
algebra on even and odd sites respectively. Motivated by these findings,
integrable deformations of such staggered superspin chains have been
constructed [8, 9, 10, 11, 12, 13]. Beginning with Ref. [11] on an integrable
$sl(2|1)$ superspin chain with alternating three-dimensional quark and
antiquark representations as the local degrees of freedom, finite-size studies
of these models have provided evidence for the presence of continuous
components in their spectrum of critical exponents. The latter manifest
themselves through towers with a macroscopic number of energy levels that
extrapolate to the same scaling dimension in the thermodynamic limit, but
exhibit strong (logarithmic) corrections to scaling that lift their
degeneracies in the finite system. By now the most studied and best understood
of the models with such properties is the staggered six-vertex model related
to the antiferromagnetic Potts model [14, 15, 16, 17, 18, 19], whose low
energy effective theory has been identified to be the $SL(2,{R})/U(1)$ black
hole conformal field theory featuring one compact and one non-compact bosonic
degree of freedom [20].

A common property of the models mentioned above is that they are staggered
(either by choosing alternating representations of the underlying algebra, or
by considering inhomogeneous shifts of the spectral parameter in the vertex
model). This staggering allows for the construction of a conserved ‘quasi
momentum’ operator, which has been crucial in the identification of the
scaling limit. Note however, there also exist translation-invariant models,
e.g. based on twisted Lie algebras including several spin chains from the
$A^{(2)}_{n}$ series in their regime III [21, 22, 23], for which signatures
for the existence of continuous components of the conformal spectrum have been
observed. Based on finite-size estimates of their central charge and the
spectra of elementary excitations, it has been argued that the $A^{(2)}_{n}$
models allow for two (or more) non-compact degrees of freedom (in addition to
the compact ones) in the scaling limit for the higher rank cases with $n>3$.
Due to numerical and analytical difficulties, a reliable description of the
conformal spectra in these models has not been possible, though.

Interestingly, the staggered six-vertex model has recently been shown to allow
for a mapping to a homogeneous integrable spin chain constructed from the
twisted affine $D_{2}^{(2)}$ Lie algebra [24]; as a consequence, a quasi
momentum can be constructed in one formulation but not in the other. Such a
factorization is not known for the higher-rank models based on $D^{(2)}_{n}$.
Nevertheless, obvious questions to ask are whether these, too, give rise to a
series of non-compact CFTs; what is the counting of compact and non-compact
degrees of freedom; and finally, what is the operator content of the CFTs
describing the low energy spectrum in the scaling limit.

With this paper, we begin the investigation of these questions for the
simplest case beyond the staggered six-vertex model, i.e. the $D^{(2)}_{3}$
spin chain. Our paper is organized as follows: in Sec. 2, after recalling the
underlying integrable structures, we construct the transfer matrix of the
model subject to generic diagonal twisted boundary conditions, and we identify
some of its symmetries. Generalizing the analytical Bethe ansatz for the
periodic case [25], we obtain the eigenvalues of the transfer matrix and the
resulting Hamiltonian with local, i.e. nearest-neighbor interactions. In Sec.
3, combining results from the exact diagonalization of small systems with the
numerical solution of the Bethe equations, we identify the root configurations
of the low-lying states. This is then used in Sec. 4 to compute the ground
state energy density in the thermodynamic limit, and in Sec. 5 to construct
the renormalization group trajectories for the ground state and excitations
used in the finite-size scaling analysis of the spectrum. The latter is done
separately for the compact and the non-compact parts of the spectrum, both
with and without the twist. The flow of the compact modes under the twist
resembles that of two compact bosons with compactification radii depending on
the anisotropy. In addition, we observe the emergence of discrete states from
the continuous parts of the spectrum of conformal weights for sufficiently
large twists, similar as in other lattice realizations of the black hole CFT.
Based on the analytical dependence of these discrete weights on the twist, we
conjecture a result for the central charge of the model in its scaling limit
which is consistent with that of two copies of the $SL(2,R)/U(1)$ sigma model.

##  2 The $D^{(2)}_{3}$ spin chain

In this section, we define the $D^{(2)}_{3}$ spin chain, describe its
symmetries, and review its Bethe ansatz solution.

###  2.1 The R-matrix

The main building block of the $D^{(2)}_{3}$ integrable quantum spin chain is
the $36\times 36$ $D^{(2)}_{3}$ R-matrix obtained in [26]. We follow here the
conventions in Eqs. (A.8-10) of [27]; i.e. our R-matrix $\mathbb{R}(u)$ is
related to the one of Jimbo $R_{J}(x)$ [26] by the following identifications
and rescaling:

$\displaystyle\mathbb{R}(u)={\rm e}^{-2u-6\eta}R_{J}(x)\,,\qquad x={\rm
e}^{u}\,,\qquad k={\rm e}^{2\eta}\,.$ (2.1)

Moreover, we henceforth set $\eta={\rm i}\gamma$, so that $\gamma$ is our
anisotropy parameter. This R-matrix satisfies the Yang-Baxter equation

$\mathbb{R}_{12}(u-v)\,\mathbb{R}_{13}(u)\,\mathbb{R}_{23}(v)=\mathbb{R}_{23}(v)\,\mathbb{R}_{13}(u)\,\mathbb{R}_{12}(u-v)\,,$
(2.2)

and has the two $U(1)$ symmetries

$\left[\mathbb{R}(u)\,,(\mathbbm{h}_{j}\otimes\mathbb{I}+\mathbb{I}\otimes\mathbbm{h}_{j})\right]=0\,,\qquad
j=1,2\,,$ (2.3)

where

$\mathbbm{h}_{1}=e^{(1,1)}-e^{(6,6)}\,,\qquad\mathbbm{h}_{2}=e^{(2,2)}-e^{(5,5)}\,,$
(2.4)

and $e^{(k,l)}$ denotes the elementary $6\times 6$ matrix with 0 everywhere
except for one 1 at position $(k,l)$; and $\mathbb{I}$ is the identity matrix.

Further important properties of this R-matrix include PT symmetry:

$\mathbb{R}_{21}(u):=\mathcal{P}_{12}\,\mathbb{R}_{12}(u)\,\mathcal{P}_{12}=\mathbb{R}_{12}^{t_{1}t_{2}}(u)\,,$
(2.5)

where $\mathcal{P}$ is the permutation matrix, and $t$ denotes transposition;
braiding unitarity:

$\mathbb{R}_{12}(u)\,\mathbb{R}_{21}(-u)=\xi(u)\,\xi(-u)\,\mathbb{I}^{\otimes
2}\,,\qquad\xi(u)=4\sinh(u+2{\rm i}\gamma)\sinh(u+4{\rm i}\gamma)\,;$ (2.6)

regularity:

$\mathbb{R}_{12}(0)=\xi(0)\,\mathcal{P}_{12}\,;$ (2.7)

crossing symmetry:

$\mathbb{R}_{12}(u)=V_{1}\,\mathbb{R}_{12}^{t_{2}}(4{\rm
i}\gamma-u)\,V_{1}=V_{2}^{t_{2}}\,\mathbb{R}_{12}^{t_{1}}(4{\rm
i}\gamma-u)\,V_{2}^{t_{2}}\,,\qquad V=\left(\begin{smallmatrix}&&&&&{\rm
e}^{-3{\rm i}\gamma}\\\ &&&&{\rm e}^{-{\rm i}\gamma}&\\\ &&&1&&\\\ &&1&&&\\\
&{\rm e}^{{\rm i}\gamma}&&&&\\\ {\rm e}^{3{\rm
i}\gamma}&&&&&\end{smallmatrix}\right)\,,$ (2.8)

with $V^{2}=\mathbb{I}$ and

$V_{1}\,\mathbb{R}_{12}(u)\,V_{1}=V_{2}\,\mathbb{R}_{21}(u)\,V_{2}\,;$ (2.9)

quasi-periodicity:

$\mathbb{R}_{12}(u+i\pi)=U_{1}\,\mathbb{R}_{12}(u)\,U_{1}=U_{2}\,\mathbb{R}_{12}(u)\,U_{2}\,,\qquad
U=\left(\begin{smallmatrix}1&&&&&\\\ &1&&&&\\\ &&0&1&&\\\ &&1&0&&\\\
&&&&&1&\\\ &&&&&&1\end{smallmatrix}\right)\,,$ (2.10)

with $U^{2}=\mathbb{I}$; and the two $Z_{2}$ symmetries:

$\displaystyle\mathbb{R}_{12}(u)$
$\displaystyle=U_{1}\,U_{2}\,\mathbb{R}_{12}(u)\,U_{1}U_{2}\,,$ (2.11)
$\displaystyle\ \mathbb{R}_{12}(u)$
$\displaystyle=W_{1}(u)\,W_{2}(0)\,\mathbb{R}_{12}(u)\,W_{1}(u)\,W_{2}(0)\,,\qquad
W(u)=\left(\begin{smallmatrix}&&&&{\rm e}^{-u}&\\\ &&&&&-{\rm e}^{-u}\\\
&&1&&&\\\ &&&-1&&\\\ {\rm e}^{u}&&&&&\\\ &-{\rm
e}^{u}&&&&\end{smallmatrix}\right)\,.$ (2.12)

with $W(u)^{2}=\mathbb{I}$.

###  2.2 The transfer matrix and its symmetries

We consider a closed homogeneous spin chain of length $L$ with diagonally-
twisted boundary conditions. The corresponding transfer matrix
$\mathbbm{t}(u)$ is therefore given by [28, 29]

$\displaystyle\mathbbm{t}(u):=\mathrm{tr}_{0}\,\mathbb{K}_{0}\,\mathbb{T}_{0}(u)\,,$
(2.13)

where $\mathbb{T}_{0}(u)$ is the monodromy matrix

$\mathbb{T}_{0}(u):=\mathbb{R}_{0L}(u)\dots\mathbb{R}_{01}(u)\,.$ (2.14)

Moreover, the diagonal twist matrix $\mathbb{K}$ is given by

$\mathbb{K}={\rm e}^{\sum_{j=1}^{2}{\rm
i}\phi_{j}\,\mathbbm{h}_{j}}=\text{diag}\left({\rm e}^{{\rm i}\phi_{1}}\,,{\rm
e}^{{\rm i}\phi_{2}}\,,1\,,1\,,{\rm e}^{-{\rm i}\phi_{2}}\,,{\rm e}^{-{\rm
i}\phi_{1}}\right)\,,$ (2.15)

where $\phi_{1}$ and $\phi_{2}$ are twist angles, which we restrict to real
$\phi_{1,2}$. Note that the local Hilbert space at each site has dimension 6.
Furthermore,

$\left[\mathbb{R}(u)\,,\mathbb{K}\otimes\mathbb{K}\right]=\left[\mathbb{R}(u)\,,{\rm
e}^{\sum_{j=1}^{2}{\rm
i}\phi_{j}\,(\mathbbm{h}_{j}\otimes\mathbb{I}+\mathbb{I}\otimes\mathbbm{h}_{j})}\right]=0\,,$
(2.16)

where the last equality follows from (2.3). The transfer matrix (2.13) has the
commutativity property

$\left[\mathbbm{t}(u)\,,\mathbbm{t}(v)\right]=0\,,$ (2.17)

which is the hallmark of quantum integrability, as a consequence of (2.2),
(2.16).

The $U(1)$ symmetries of the R-matrix (2.3) are inherited by the transfer
matrix

$\left[\mathbbm{t}(u)\,,\mathbbm{h}_{j}^{(L)}\right]=0\,,\qquad j=1,2\,,$
(2.18)

where111We will henceforth abbreviate $\mathbbm{h}_{j}^{(L)}$ as
$\mathbbm{h}_{j}$ when the meaning is clear from the context.

$\mathbbm{h}_{j}^{(L)}=\sum_{i=1}^{L}\left(\mathbbm{h}_{j}\right)_{i}\,,$
(2.19)

and $\left(\mathbbm{h}_{j}\right)_{i}$ denotes the generator $\mathbbm{h}_{j}$
(2.4) at site $i$, that is

$\left(\mathbbm{h}_{j}\right)_{i}=\mathbb{I}\otimes\cdots\otimes\mathbb{I}\otimes\underbrace{\mathbbm{h}_{j}}_{i}\otimes\mathbb{I}\otimes\cdots\otimes\mathbb{I}\,.$
(2.20)

Further properties inherited from the R-matrix by the transfer matrix include
crossing symmetry:

$\mathbbm{t}^{t}(u;\\{\phi_{j}\\})=\mathbbm{t}(4{\rm
i}\gamma-u;\\{-\phi_{j}\\})\,,$ (2.21)

periodicity:

$\mathbbm{t}(u+i\pi)=\mathbbm{t}(u)\,,$ (2.22)

and $Z_{2}$ symmetry:

$\mathbbm{t}(u)=U^{\otimes L}\,\mathbbm{t}(u)\,U^{\otimes L}\,,$ (2.23)

see Eqs. (2.8), (2.10), (2.11), respectively. The $Z_{2}$ symmetry (2.23) is a
generalization of the $Z_{2}$ symmetry found for $D^{(2)}_{2}$ [14, 24,
30].222In [30], the $Z_{2}$ symmetry of the $D^{(2)}_{2}$ closed-chain
transfer matrix is expressed (3.32) in terms of an operator constructed from a
matrix $C$ (2.10), in a gauge that is specified by a matrix $B$ (2.9), with
$B\,C\,B=\left(\begin{smallmatrix}1&&&\\\ &0&1&\\\ &1&0&\\\
&&&&1\end{smallmatrix}\right)\,,$ which is evidently a reduction of the matrix
$U$ in (2.10). For generic values of the twists $\phi_{1},\phi_{2}$, we also
have

$\displaystyle W(0)^{\otimes
L}\,\mathbbm{t}(u;\phi_{1},\phi_{2})\,W(0)^{\otimes
L}=\mathbbm{t}(u;-\phi_{2},-\phi_{1})\,.$ (2.24)

Hence, if the twist angles satisfy $\phi_{1}+\phi_{2}=0$ (but not for generic
values), then the transfer matrix (2.13) also has the $Z_{2}$ symmetry

$\displaystyle W(0)^{\otimes L}\,\mathbbm{t}(u;\phi,-\phi)\,W(0)^{\otimes
L}=\mathbbm{t}(u;\phi,-\phi)\,.$ (2.25)

Finally, we note that the transfer matrix has the CPT-like symmetry

$V^{\otimes L}\,\Pi\,\mathbbm{t}^{t}(u;\\{-\phi_{j}\\})\,\Pi\,V^{\otimes
L}=\mathbbm{t}(u;\\{\phi_{j}\\})\,,$ (2.26)

where $\Pi$ is the parity operator

$\Pi=\prod_{i=1}^{\lfloor\frac{L}{2}\rfloor}\mathcal{P}_{i,L+1-i}\,,$ (2.27)

which on any local operator $X_{i}$ at site $i$ acts as
$\Pi\,X_{i}\,\Pi=X_{L+1-i}$, see e.g. [31]; in Eq. (2.27) $\lfloor x\rfloor$
denotes the floor of $x$. Proofs of the symmetries (2.21), (2.24) and (2.26)
are sketched in Appendix A.

###  2.3 Bethe ansatz

Let $|\Lambda\rangle$ denote a simultaneous (normalized) eigenstate of the
transfer matrix $\mathbbm{t}(u)$ (2.13) and of the $U(1)$ generators
$\mathbbm{h}_{j}$

$\displaystyle\mathbbm{t}(u)\,|\Lambda\rangle$
$\displaystyle=\Lambda(u)\,|\Lambda\rangle\,,$ (2.28)
$\displaystyle\mathbbm{h}_{j}\,|\Lambda\rangle$
$\displaystyle=h_{j}\,|\Lambda\rangle\,,\qquad j=1,2\,.$ (2.29)

The eigenvalues $\Lambda(u)$ are given by

$\displaystyle\Lambda(u)=$ $\displaystyle\left(4\sinh(u-2{\rm
i}\gamma)\sinh(u-4{\rm i}\gamma)\right)^{L}{\rm e}^{{\rm i}\phi_{1}}\,A(u)$
$\displaystyle+\left(4\sinh(u-4{\rm i}\gamma)\sinh u\right)^{L}\left[{\rm
e}^{{\rm i}\phi_{2}}\,B_{1}(u)+B_{2}(u)+B_{3}(u)+{\rm e}^{-{\rm
i}\phi_{2}}\,B_{4}(u)\right]$ $\displaystyle+\left(4\sinh(u-2{\rm
i}\gamma)\sinh u\right)^{L}{\rm e}^{-{\rm i}\phi_{1}}\,C(u)\,,$ (2.30)

with

$\displaystyle A(u)$
$\displaystyle=\prod_{j=1}^{m_{1}}\frac{\sinh(u-u^{[1]}_{j}+{\rm
i}\gamma)}{\sinh(u-u^{[1]}_{j}-{\rm i}\gamma)}\,,$ (2.31) $\displaystyle
B_{1}(u)$ $\displaystyle=\prod_{j=1}^{m_{1}}\frac{\sinh(u-u^{[1]}_{j}-3{\rm
i}\gamma)}{\sinh(u-u^{[1]}_{j}-{\rm
i}\gamma)}\,\prod_{j=1}^{m_{2}}\frac{\sinh(u-u^{[2]}_{j})}{\sinh(u-u^{[2]}_{j}-2{\rm
i}\gamma)}\,,$ $\displaystyle B_{2}(u)$
$\displaystyle=\prod_{j=1}^{m_{2}}\frac{2\cosh\left(\frac{1}{2}\left(u-u^{[2]}_{j}\right)\right)\,\sinh\left(\frac{1}{2}\left(u-u^{[2]}_{j}-4{\rm
i}\gamma\right)\right)}{\sinh(u-u^{[2]}_{j}-2{\rm i}\gamma)}\,,$

and

$C(u)=\bar{A}(4{\rm i}\gamma-u)\,,\qquad B_{3}(u)=\bar{B_{2}}(4{\rm
i}\gamma-u)\,,\qquad B_{4}(u)=\bar{B_{1}}(4{\rm i}\gamma-u)\,,$ (2.32)

where the barred quantities are obtained by negating all the Bethe-roots (i.e.
$u^{[l]}_{j}\mapsto-u^{[l]}_{j}$), that is,

$\displaystyle\bar{A}(u)$
$\displaystyle=\prod_{j=1}^{m_{1}}\frac{\sinh(u+u^{[1]}_{j}+{\rm
i}\gamma)}{\sinh(u+u^{[1]}_{j}-{\rm i}\gamma)}\,,$ (2.33)
$\displaystyle\bar{B}_{1}(u)$
$\displaystyle=\prod_{j=1}^{m_{1}}\frac{\sinh(u+u^{[1]}_{j}-3{\rm
i}\gamma)}{\sinh(u+u^{[1]}_{j}-{\rm
i}\gamma)}\,\prod_{j=1}^{m_{2}}\frac{\sinh(u+u^{[2]}_{j})}{\sinh(u+u^{[2]}_{j}-2{\rm
i}\gamma)}\,,$ $\displaystyle\bar{B}_{2}(u)$
$\displaystyle=\prod_{j=1}^{m_{2}}\frac{2\cosh\left(\frac{1}{2}\left(u+u^{[2]}_{j}\right)\right)\,\sinh\left(\frac{1}{2}\left(u+u^{[2]}_{j}-4{\rm
i}\gamma\right)\right)}{\sinh(u+u^{[2]}_{j}-2{\rm i}\gamma)}\,.$

If $u$ and $\eta={\rm i}\gamma$ are real, then the bar has the interpretation
of complex conjugation, and the periodic transfer-matrix eigenvalue has the
crossing symmetry

$\bar{\Lambda}(u)=\Lambda(4{\rm i}\gamma-u)\,,$ (2.34)

similarly to the XXZ model [32]. The result for the periodic case
($\phi_{1}=\phi_{2}=0$) was obtained by Reshetikhin via the analytical Bethe
ansatz in [25]. The generalization to the twisted case is presented in
Appendix B. Note that we have redefined the Bethe-roots from [25] as

$\displaystyle u^{[1]}_{j}=2{\rm i}x_{j}\,,\qquad u^{[2]}_{j}=2{\rm
i}y_{j}\,.$ (2.35)

We also note that the eigenvalues (2.30) have the periodicity
$\Lambda(u+i\pi)=\Lambda(u)$, consistent with (2.22).

An eigenvalue $\Lambda(u)$ of the transfer matrix is a Fourier polynomial and
hence an analytic function. By requiring that the residues of (2.30) at the
apparent poles

$u=u^{[1]}_{j}+{\rm i}\gamma\,,\qquad u=u^{[2]}_{j}+2{\rm i}\gamma$ (2.36)

vanish, one obtains the Bethe equations (BE)

$\displaystyle\left(\frac{\sinh\left(u^{[1]}_{j}-{\rm
i}\gamma\right)}{\sinh\left(u^{[1]}_{j}+{\rm i}\gamma\right)}\right)^{L}={\rm
e}^{{\rm i}(\phi_{2}-\phi_{1})}\prod_{k\neq
j}^{m_{1}}\frac{\sinh\left(u^{[1]}_{j}-u^{[1]}_{k}-2{\rm
i}\gamma\right)}{\sinh\left(u^{[1]}_{j}-u^{[1]}_{k}+2{\rm
i}\gamma\right)}\,\prod_{k=1}^{m_{2}}\frac{\sinh\left(u^{[1]}_{j}-u^{[2]}_{k}+{\rm
i}\gamma\right)}{\sinh\left(u^{[1]}_{j}-u^{[2]}_{k}-{\rm i}\gamma\right)}\,,$
(2.37)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
j=1,\dots,m_{1}\,,$
$\displaystyle\prod_{k=1}^{m_{1}}\frac{\sinh\left(u^{[2]}_{j}-u^{[1]}_{k}-{\rm
i}\gamma\right)}{\sinh\left(u^{[2]}_{j}-u^{[1]}_{k}+{\rm i}\gamma\right)}={\rm
e}^{-{\rm i}\phi_{2}}\prod_{k\neq
j}^{m_{2}}\frac{\sinh\frac{1}{2}\left(u^{[2]}_{j}-u^{[2]}_{k}-2{\rm
i}\gamma\right)}{\sinh\frac{1}{2}\left(u^{[2]}_{j}-u^{[2]}_{k}+2{\rm
i}\gamma\right)}\,,\quad j=1,\dots,m_{2}\,,$

see also [25, 33].

Note that the BE are invariant under the transformations $u^{[1]}_{j}\to
u^{[1]}_{j}+{\rm i}\pi$ and $u^{[2]}_{j}\to u^{[2]}_{j}+2{\rm i}\pi$. Hence,
one can restrict

$\displaystyle-\frac{\pi}{2}<\Im m(u^{[1]}_{j})\leq\frac{\pi}{2}\,,$ (2.38)
$\displaystyle-\pi<\Im m(u^{[2]}_{j})\leq\pi\,.$ (2.39)

Notice that we are also allowed to do $u^{[2]}_{j}\to u^{[2]}_{j}+{\rm i}\pi$,
but only if we shift all the Bethe-roots (i.e. for every $j$) at the same
time.

The eigenvalues $h_{j}$ of the $U(1)$ generators $\mathbbm{h}_{j}$ (see Eq.
(2.29)) are given by [25]

$\displaystyle h_{1}$ $\displaystyle=L-m_{1}\,,$ (2.40) $\displaystyle h_{2}$
$\displaystyle=m_{1}-m_{2}\,.$

As usual, the Bethe ansatz provides solutions for

$\displaystyle L\geq m_{1}\geq m_{2}\geq 0\,.$ (2.41)

Although an algebraic Bethe ansatz construction of the eigenstates
$|u^{[1]}_{1},\ldots,u^{[1]}_{m_{1}}\,;u^{[2]}_{1},\ldots,u^{[2]}_{m_{2}}\rangle$
of the $D^{(2)}_{3}$ transfer-matrix (2.13) has not yet been worked out in
detail, we expect that these states can be constructed as follows: the first
level of nesting (introducing type-1 Bethe roots
$u^{[1]}_{1},\ldots,u^{[1]}_{m_{1}}$) can be accomplished [34] using certain
elements of the monodromy matrix (2.14), reducing the problem to
$D^{(2)}_{2}$. The transfer matrix for the latter can be expressed as a
product of $A^{(1)}_{1}$ transfer matrices [30], which can then be
diagonalized by the usual algebraic Bethe ansatz (introducing the type-2 Bethe
roots $u^{[2]}_{1},\ldots,u^{[2]}_{m_{2}}$). We therefore expect that the
$Z_{2}$ symmetry (2.23) shifts all type-2 Bethe roots by ${\rm i}\pi$

$U^{\otimes
L}\,|u^{[1]}_{1},\ldots,u^{[1]}_{m_{1}}\,;u^{[2]}_{1},\ldots,u^{[2]}_{m_{2}}\rangle\propto|u^{[1]}_{1},\ldots,u^{[1]}_{m_{1}}\,;u^{[2]}_{1}+{\rm
i}\pi,\ldots,u^{[2]}_{m_{2}}+{\rm i}\pi\rangle\,.$ (2.42)

Indeed, the $Z_{2}$ symmetry in the $D^{(2)}_{2}$ case shifts all Bethe roots
by ${\rm i}\pi$ (see (3.34) in [30]); and, in the $D^{(2)}_{3}$ case, these
Bethe roots correspond to type-2 Bethe roots.

###  2.4 Hamiltonian

Commuting integrals of motion for the $D^{(2)}_{3}$ spin chain are obtained by
expanding the transfer matrix (2.13) about the regular point $u=0$ (2.7). The
leading term is

$\displaystyle\mathbbm{t}(0)=\left(4\sinh(2{\rm i}\gamma)\sinh(4{\rm
i}\gamma)\right)^{L}\,{\rm e}^{{\rm i}\,\mathbb{P}}\,,$ (2.43)

where $e^{{\rm i}\,\mathbb{P}}$ is the one-site translation operator of the
model with quasi-periodic boundary conditions, whose matrix elements are given
by

$\displaystyle\left[{\rm e}^{{\rm
i}\,\mathbb{P}}\right]^{b_{1},\dots,b_{L}}_{a_{1},\dots,a_{L}}=\exp{\left\\{{\rm
i}\phi_{1}\left(\delta^{b_{1}}_{1}-\delta^{b_{1}}_{6}\right)+{\rm
i}\phi_{2}\left(\delta^{b_{1}}_{2}-\delta^{b_{1}}_{5}\right)\right\\}}\delta^{b_{2}}_{a_{1}}\delta^{b_{3}}_{a_{2}}\dots\delta^{b_{L}}_{a_{L-1}}\delta^{b_{1}}_{a_{L}}\,.$
(2.44)

From (2.30) we find that its eigenvalues ${\rm e}^{{\rm i}P}$ are
parameterized by the Bethe roots as

$\displaystyle{\rm e}^{{\rm i}\,P}={\rm e}^{{\rm
i}\phi_{1}}\,\prod^{m_{1}}_{k=1}\frac{\sinh(u^{[1]}_{k}-{\rm
i}\gamma)}{\sinh(u^{[1]}_{k}+{\rm i}\gamma)}\,.$ (2.45)

Similarly, we define the local Hamiltonian of the $D^{(2)}_{3}$ spin chain
as333With this sign the Hamiltonian generalizes the $D^{(2)}_{2}$ spin chain
which has been related to the antiferromagnetic Potts model [24] or
respectively the corresponding staggered-six vertex model [30, 35] .

$\displaystyle\mathbb{H}=\sinh(2{\rm
i}\gamma)\frac{\mathrm{d}}{\mathrm{d}u}\log\left(\mathbbm{t}(u)\right)\Big{|}_{u=0}+L\sinh(2{\rm
i}\gamma)\left[\coth(2{\rm i}\gamma)+\coth(4{\rm
i}\gamma)\right]\mathbb{I}^{\otimes L}\,.$ (2.46)

the eigenenergies are

$\displaystyle
E=\sum^{m_{1}}_{k=1}\epsilon_{0}(u^{[1]}_{k})=-\sum^{m_{1}}_{k=1}\frac{2\sinh^{2}(2{\rm
i}\gamma)}{\cosh\left(2u^{[1]}_{k}\right)-\cosh\left(2{\rm
i}\gamma\right)}\,.$ (2.47)

The Hamiltonian of course inherits the $U(1)$ and $Z_{2}$ symmetries of the
transfer matrix

$\displaystyle\left[\mathbb{H}\,,\mathbbm{h}_{j}\right]$
$\displaystyle=0\,,\qquad j=1\,,2\,,$ (2.48)
$\displaystyle\left[\mathbb{H}\,,U^{\otimes L}\right]$ $\displaystyle=0\,,$
(2.49) $\displaystyle\left[\mathbb{H}(\phi,-\phi)\,,W(0)^{\otimes L}\right]$
$\displaystyle=0\,,$ (2.50)

see Eqs. (2.18), (2.23), (2.25), respectively. The Hamiltonian also has the
additional CP symmetry,

$\left[\mathbb{H}\,,V^{\otimes L}\,\Pi\right]=0\,,$ (2.51)

see Appendix A. We emphasize that this symmetry does not extend to the full
transfer matrix, which has only the CPT symmetry (2.26).

The $U(1)$ generators commute with the $Z_{2}$ symmetry (2.23)

$\left[\mathbbm{h}_{j}\,,U^{\otimes L}\right]=0\,,\qquad j=1\,,2\,,$ (2.52)

and they transform into each other under $W(0)$ (2.12)

$\displaystyle W(0)^{\otimes L}\,\mathbbm{h}_{1}\,W(0)^{\otimes L}$
$\displaystyle=-\mathbbm{h}_{2}\,,$ $\displaystyle W(0)^{\otimes
L}\,\mathbbm{h}_{2}\,W(0)^{\otimes L}$ $\displaystyle=-\mathbbm{h}_{1}\,.$
(2.53)

Under the CP symmetry, the $U(1)$ generators transform as

$V^{\otimes L}\,\Pi\,\mathbbm{h}_{j}\,\Pi\,V^{\otimes
L}=-\mathbbm{h}_{j}\,,\qquad j=1\,,2\,.$ (2.54)

The symmetry transformations (2.50) together with (2.51) induce degeneracies
in the energy spectrum between different sectors of the $U(1)$-charges. For
the analysis of the finite-size spectrum, it is sufficient to focus on one
representative of a given energy level, keeping these degeneracies implicit.
The symmetries (2.50), (2.51) allow one to restrict to the case
$h_{1}\geq|h_{2}|$ for suitably chosen twist angles. In addition, we found by
exact diagonalization of the Hamiltonian for small system sizes that one can
further restrict to

$\displaystyle 0\leq h_{2}\leq h_{1}\,.$ (2.55)

Note that all the sectors specified by (2.55) can be accessed by the above
Bethe ansatz, see (2.41) and (2.40). Further, we should stress that the
defined Hamiltonian is non-Hermitian. This leads to complex eigenvalues. On
numerical grounds, we find, however, that the energies of the ground state and
lowest excitations are real. In the rest of this work, we study states
parameterized by the classes of Bethe root configurations listed in the
following section. These states, too, turn out to have real energies.

##  3 Methodology of studying the scaling limit

To study the scaling limit of a lattice model, one should define an
$L$-dependence to the low-lying energy states. Such an assignment
$\ket{\Psi_{L}}$ will be called an RG-trajectory. For the ground state or for
the lowest energy states in the disjoint sectors $(h_{1},h_{2})$ of the
Hilbert space, such an assignment is clear. On the other hand, constructing
individual RG trajectories $\ket{\Psi_{L}}$ for generic low-energy states is
not a trivial task. The fact that the considered model is integrable allows
for the following strategy: For small initial lattice sizes
$L=L_{\rm{in}}\lesssim 6$, we diagonalize the Hamiltonian in the subspace
spanned by eigenvectors of the $U(1)$ generators with given eigenvalues
$h_{1}$, $h_{2}$.444Note that the full Hilbert space has dimension $6^{L}$.
The eigenvalues are then matched via (2.47) with a solution of the Bethe
ansatz equations $\\{u^{[1]},u^{[2]}\\}^{m_{1}}_{m_{2}}$ where $m_{1}$ and
$m_{2}$ are determined by (2.40). The state $\ket{\Psi_{L}}$ at higher
$L=L_{\rm{in}}+2$ is obtained by solving the Bethe ansatz equations for a
pattern of Bethe roots that qualitatively resembles the one of the state
$\ket{\Psi_{L_{\rm{in}}}}$. Via this procedure, we construct the RG trajectory
$\ket{\Psi_{L}}$ up to $L\sim 2000$ without relying on a direct
diagonalization of the Hamiltonian, which is an impossible task for $L\gg 1$
since the size of the Hilbert space grows exponentially.

###  3.1 Considered class of states

In the above procedure, it is essential to understand the structure of the
low-energy spectrum in terms of the Bethe roots. However, the particular
structure depends essentially on the domain of the parameters $\gamma$,
$\phi_{1}$ and $\phi_{2}$. In the regime

$\displaystyle\gamma\in(0,\tfrac{\pi}{4})$ (3.1)

and for small $\phi_{1,2}$, we found that the bulk of the Bethe root
configurations corresponding to low energy states consists of 4-strings, each
containing a pair of conjugate roots on both levels centered at real $x_{j}$:

$\displaystyle u^{[1]}\longrightarrow\quad
v^{[1]}_{j}=x_{j}+\delta^{[1]}_{j}+\frac{{\rm i}\pi}{2}-{\rm i}\gamma-{\rm
i}\epsilon^{[1]}_{j}\,,\quad\bar{v}^{[1]}_{j}=x_{j}+\delta^{[1]}_{j}-\frac{{\rm
i}\pi}{2}+{\rm i}\gamma+{\rm i}\epsilon^{[1]}_{j}\,,$ (3.2) $\displaystyle
u^{[2]}\longrightarrow\quad v^{[2]}_{j}=x_{j}+\delta^{[2]}_{j}+\frac{{\rm
i}\pi}{2}+{\rm
i}\epsilon^{[2]}_{j}\,,\qquad\quad\,\bar{v}^{[2]}_{j}=x_{j}+\delta^{[2]}_{j}-\frac{{\rm
i}\pi}{2}-{\rm i}\epsilon^{[2]}_{j}\,,$

where $j\leq\frac{L}{2}$ and $\delta^{[k]}_{j},\epsilon^{[k]}_{j}$ are small
real deviations. For even system sizes, the ground state of the system is
realized in the sectors $h_{1}=h_{2}=0$ with $\epsilon^{[2]}_{j}\equiv 0$. See
Fig. 1 for the ground state of the $L=18$ chain. The low-energy spectrum is
described by various root configurations. In this work, we focus on a
particular class of states described by the following additional roots outside
these 4-string configurations:

  1. i)

Level-$1$ roots on the line $\frac{{\rm i}\pi}{2}$

  2. ii)

Level-$2$ roots placed on the line ${\rm i}\pi$.

  3. iii)

Level-$2$ roots placed on the real line.

subject to the constraint (2.41).

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$-\gamma+\frac{\pi}{2}$$-\frac{\pi}{2}$$\gamma-\frac{\pi}{2}$$-\pi$

Figure 1: Bethe root configuration of the ground state for $L=18$ and
$\gamma=0.4$ plotted in the complex $u$-plane. Blue (red) symbols denote level
1 (2) roots. One can clearly see the pattern (3.2).

##  4 Root density approach for the ground state

For even $L$ the ground state is parameterized by roots arranged in the
configuration (3.2) where $j$ runs from one to $\frac{L}{2}$ and
$\epsilon^{[2]}_{j}$ is set to zero. Further, we find numerically that the
remaining deviations $\delta^{[k]}_{j},\epsilon^{[1]}_{j}$ in (3.2) tend to
zero as $L\to\infty$. Hence, we can study the ground state in the root density
approach [36]. By inserting (3.2) into the Bethe equations and taking the
logarithm one obtains the following counting function for the real centers:

$\displaystyle z^{x}(x)$
$\displaystyle=\frac{1}{2\pi}\psi(x,2\gamma)+\frac{1}{2\pi
L}\sum^{\frac{L}{2}}_{k=1}\chi(x-x_{k},4\gamma)\,,$ (4.1)

where

$\displaystyle\chi(x,y)$
$\displaystyle=2\arctan\left(\tanh(x)\cot(y)\right)\,,\quad\psi(x,y)=2\arctan\left(\tanh(x)\tan(y)\right)\,.$
(4.2)

Upon differentiation we obtain the following linear integral equation for the
root density defined by $\rho^{x}(x)=\partial_{x}z^{x}(x)$:

$\displaystyle\rho^{x}(x)$
$\displaystyle=\frac{1}{2\pi}\psi^{\prime}(x,2\gamma)+\frac{1}{2\pi}\int^{\infty}_{-\infty}\mathrm{d}x^{\prime}\,\chi^{\prime}(x-x^{\prime},4\gamma)\rho^{x}(x^{\prime})\,,$
(4.3)

which is solved by Fourier transform giving:

$\displaystyle\rho^{x}(x)$
$\displaystyle=\frac{1}{2(\pi-4\gamma)}\frac{1}{\cosh(\frac{\pi
x}{\pi-4\gamma})}\,.$ (4.4)

The density is positive and becomes singular at $\gamma=\tfrac{\pi}{4}$,
giving additional support to our choice of the parameter domain (3.1) for
$\gamma$. Similarly, the _dressed_ energy $\epsilon^{x}(x)$ of excitations
corresponding to the removal of a four-string is obtained from the same linear
integral equation as (4.3) but with the driving term $\psi$ replaced by
$\epsilon^{x}_{0}(x)=\epsilon_{0}(x+\frac{{\rm i}\pi}{2}-{\rm
i}\gamma)+\epsilon_{0}(x-\frac{{\rm i}\pi}{2}+{\rm i}\gamma)$, where
$\epsilon_{0}$ has been defined in (2.47) above. For $\gamma$ in the domain
(3.1), these excitations turn out to be gapless with a linear dispersion. The
corresponding Fermi velocity is

$\displaystyle v_{F}$
$\displaystyle:=\frac{1}{2\pi}\lim_{\Lambda\to\infty}\frac{1}{\rho^{x}(\Lambda)}\frac{\mathrm{d}}{\mathrm{d}\Lambda}\epsilon^{x}(\Lambda)=\frac{\pi\sin(2\gamma)}{\pi-4\gamma}\,.$
(4.5)

Finally, using (4.4), we obtain the energy density $e_{\infty}$ in the
thermodynamic limit

$\displaystyle e_{\infty}$
$\displaystyle=-\frac{\sin(2\gamma)}{2}\int^{\infty}_{-\infty}\mathrm{d}\omega\,\frac{\sinh(2\gamma\omega)}{\sinh\left(\frac{\pi\omega}{2}\right)}\frac{1}{\cosh(\frac{1}{2}(\pi-4\gamma)\omega)}\,.$
(4.6)

##  5 Analysis of the finite-size spectrum

As the model is critical, the spectrum of low-energy excitations can be
described within the framework of a conformal field theory. In this paper, we
extract the central charge and investigate the first features of the
underlying CFT, such as the spectrum of scaling dimensions. The following
prediction from conformal field theory is expected to hold [37, 38, 39],

$\displaystyle\frac{L}{2\pi v_{F}}\left(E-Le_{\infty}\right)$
$\displaystyle\simeq-\frac{c}{12}+h+\bar{h}\,,$ (5.1)

where $e_{\infty}$ and $v_{F}$ are given by (4.6) and (4.5), respectively.
Hence, by studying the asymptotic behavior of the energies on the lattice, one
can access the central charge $c$ and the conformal weights $h,\,\bar{h}$ of
primaries in the underlying CFT. Note that on the right-hand side of (5.1),
only the sum of the central charge and the scaling dimension $X=h+\bar{h}$
appears, such that one cannot determine $c$ or $h,\,\bar{h}$ on their own.
Suitable measures of the scaling dimensions and the central charge are given
by the _effective_ scaling dimensions $X_{\mathrm{eff}}$ and the _effective_
central charge $c_{\mathrm{eff}}$ defined by

$\displaystyle X_{\mathrm{eff}}$ $\displaystyle=\frac{L}{2\pi
v_{F}}\left(E-Le_{\infty}\right)\,,$ (5.2) $\displaystyle c_{\mathrm{eff}}$
$\displaystyle=-\frac{6L}{\pi v_{F}}\left(E_{GS}-Le_{\infty}\right)\,.$ (5.3)

The quantity $E_{GS}$ in (5.3) is the ground state energy.

Besides (5.1) we have also the following relation between the scaling
dimensions and the eigenvalue of one-site translation operator (2.44)

$\displaystyle{\rm e}^{{\rm i}\,P}$ $\displaystyle={\rm e}^{\frac{2{\rm
i}\pi}{L}\left(h-\bar{h}\right)}\,.$ (5.4)

One can easily show that

$\displaystyle{\rm e}^{{\rm i}\,\mathbb{P}\,L}=\exp{\left\\{2{\rm
i}\pi\left(\mathbbm{h}_{1}\,\frac{\phi_{1}}{2\pi}+\mathbbm{h}_{2}\,\frac{\phi_{2}}{2\pi}\right)\right\\}}\,.$
(5.5)

Comparing with (5.4), one analytically obtains the result for the difference
$h-\bar{h}$

$\displaystyle
h-\bar{h}=h_{1}\,\frac{\phi_{1}}{2\pi}+h_{2}\,\frac{\phi_{2}}{2\pi}\mod 1\,.$
(5.6)

To proceed further in our analysis, we relied on numerical methods whose
results we present in the following sections.

###  5.1 Compact part

In this section, we investigate two classes of fundamental excitation
patterns. In terms of the Bethe roots, the first class is simply built from
configurations following the structure (3.2) but with a non-zero $h_{1}$ in
contrast to the ground state. Here, the eigenvalue $h_{2}$ of the
$U(1)$-charge $\mathbbm{h}_{2}$ is kept the same as for the ground state, i.e.
$h_{2}=0$. See Fig. 2 for an illustration.

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

Figure 2: Left (right) plot displays the Bethe-root configuration in the
complex $u$-plane of an excited state for $L=18$, $\gamma=0.4,\phi_{1,2}=0$ in
the sector $h_{1}=4$ ($8$). Blue (red) symbols denote level 1 (2) roots. This
excitation corresponds to removing $2$ ($4$) four-strings from the
configuration of the ground state, see Fig. 1.

The second class of excitations gives $h_{2}$ a non-vanishing value. This is
accomplished on the level of the Bethe roots by mechanism (i), i.e. by placing
additional level-1 roots on the line $\frac{{\rm i}\pi}{2}$. We have
illustrated this type of excitations in Fig. 3.

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

Figure 3: Left (right) plot displays the Bethe-root configuration in the
complex plane of an excited state for $L=18,(19)$, $\gamma=0.4,\phi_{1,2}=0$
in the sector $h_{1}=4$ and $h_{2}=2,(3)$. Blue (red) symbols denote level 1
(2) roots. This excitation is built by placing 2(3) level-$1$ roots ($\times$)
on the line $\frac{{\rm i}\pi}{2}$ in addition to the bulk roots ($\bullet$).

We start our numerical analysis by investigating the scaling behavior of the
ground state. We obtain

$\displaystyle c_{\rm{eff}}=4\,.$ (5.7)

Further, for periodic boundary condition $\phi_{1,2}=0$, we have constructed
the RG trajectories for various excited states based on the mechanisms
discussed above. Exemplary plots of the numerical data for finite $L$
calculated by the Bethe ansatz and their extrapolations to $L\to\infty$ are
given in Figs. 5-5. Here, the extrapolation procedure is based on the
assumption that the effective scaling dimensions are rational functions of
$\frac{1}{\log(L)}$. We conclude that they flow to the following effective
scaling dimensions:

$\displaystyle
X^{\rm{Com}}_{\mathrm{eff}}=-\frac{4}{12}+\frac{(h_{1})^{2}}{2\,{\tt
k}}+\frac{(h_{2})^{2}}{2\,{\tt k}}\,.$ (5.8)

In (5.8) the first term accounts for the effective central charge and agrees
with our findings (5.7). Note the exchange symmetry of $h_{1}$ and $h_{2}$
observed in the spectrum. The parameter ${\tt k}$ specifying the amplitudes is
related to the anisotropy by

$\displaystyle{\tt k}=\frac{\pi}{\gamma}\,.$ (5.9)

Figure 4: Finite-size scaling up to $L\sim 2000$ of the ground state (black)
and states with pattern (see Fig. 2) similar to the ground state with
$h_{2}=0$ but with different $U(1)$-charge $\mathbbm{h}_{1}$, i.e.,
$h_{1}=1,2,3,4,6$ in increasing order from below. The crosses are the
numerically obtained effective scaling dimensions. The dashed lines are given
by formula (5.8) with the associated $h_{1,2}$. Solid lines are given by a
rational extrapolation. Here $\gamma=0.4$ and $\phi_{1,2}=0$.

Figure 5: Finite-size scaling up to $L\sim 2000$ for states with similar root
configuration, as depicted in Fig. 3. Crosses are the numerically obtained
effective scaling dimensions. The $U(1)$-charges $(h_{1},h_{2})$ take the
values $(4,0),(4,1),(4,2),(4,3),(4,4)$ labelled from below. Dashed lines are
given by equation (5.8) and the solid lines are given by a rational
extrapolation. Here $\gamma=0.4$ and $\phi_{1,2}=0$.

####  5.1.1 Spectrum flow of the compact modes under twists

We now turn to the extension of the formula (5.8) to small non-vanishing twist
angles. The analytic expression (5.6) and symmetry arguments suggest the
following generalization

$\displaystyle X^{\rm{Com}}_{\rm{eff}}(h_{1},h_{2},\phi_{1},\phi_{2})=$
$\displaystyle-\frac{4}{12}+\frac{(h_{1}+{\tt
k}\,\frac{\phi_{1}}{2\pi})^{2}}{4\,{\tt k}}+\frac{(h_{2}+{\tt
k}\,\frac{\phi_{2}}{2\pi})^{2}}{4\,{\tt k}}+\frac{(h_{1}-{\tt
k}\,\frac{\phi_{1}}{2\pi})^{2}}{4\,{\tt k}}+\frac{(h_{2}-{\tt
k}\,\frac{\phi_{2}}{2\pi})^{2}}{4\,{\tt k}}\,.$ (5.10)

We have numerically verified the above expression by using the data of the
periodic model by applying the following iterative method: We start with a
solution $\\{u^{[1]},u^{[2]}\\}^{\phi^{\rm{in}}_{1}}_{\phi^{\rm{in}}_{2}}$ of
the BAE (LABEL:Twisted_BAE) in logarithmic form with a particular initial set
$(\phi^{\rm{in}}_{1},\phi^{\rm{in}}_{2})$ of twist values e.g.,
$(\phi^{\rm{in}}_{1},\phi^{\rm{in}}_{2})=(0,0)$. We see that the maximal error
using $\\{u^{[1]},u^{[2]}\\}^{\phi^{\rm{in}}_{1}}_{\phi^{\rm{in}}_{2}}$ as an
initial approximation for the BAE for new values
$(\phi^{\rm{in}}_{1}+\Delta\phi_{1},\phi^{\rm{in}}_{2}+\Delta\phi_{2})$ behave
as $\max\left\\{|\Delta\phi_{1}|,|\Delta\phi_{2}-\Delta\phi_{1}|\right\\}$.
Hence, by taking the steps sizes $\Delta\phi_{1},\Delta\phi_{2}$ small enough,
we can iteratively obtain the state at some
$(\phi^{\rm{end}}_{1},\phi^{\rm{end}}_{2})$.

Note that the above form of the effective scaling dimensions is compatible
with the symmetries (2.25), (2.51). To interpret these results further,
consider the conformal weights of a twisted free boson given by [40]

$\displaystyle
h_{n,\omega}=\frac{1}{2}\left(\frac{n}{2R}+R(\omega+\varphi)\right)^{2}\qquad\bar{h}_{n,\omega}=\frac{1}{2}\left(\frac{n}{2R}-R(\omega+\varphi)\right)^{2}$
(5.11)

where the integers $n,\omega$ label charge and winding while $\varphi$
parameterizes the twisted boundary condition. By comparing (5.11) and (5.10)
we see that the excitations (5.10) mimic two independent twisted compact
bosonic modes with the same compactification radii $R_{1,2}=\sqrt{\frac{{\tt
k}}{2}}$ with charges $n_{1,2}=h_{1,2}$ and zero windings. Despite extensive
study of root patterns of the low-lying excitations, we have not been able to
identify any state with non-zero winding. Note that the functional dependence
on the compactification radii induced by non-zero twists $\phi_{1,2}$ is
exactly the same as for two compact bosons as expected from the symmetries of
the model.

We want to end this section by the following important remark. The above
expressions for the scaling dimensions capture the leading finite-size
behavior only. Corrections to (5.1) can arise, e.g., due to perturbations of
the fixed-point Hamiltonian by terms involving irrelevant operators present in
the lattice model (2.46) [39]. In the presence of a marginally irrelevant
operator, one expects these subleading corrections to contain logarithms [41].
In the present case we observe such corrections, see e.g., Figs. 5 and 5. As
we will argue below, however, these are also a signature of non-compact
degrees of freedom in the effective theory describing the critical behavior.

###  5.2 Continuous part of the spectrum

Interestingly, there also exist excitations whose scaling dimensions coincide
with compact ones (5.10) up to logarithmic corrections. These excitations can
be characterized by the presence of roots of type (ii) and (iii). Consider
first the ones of type (ii). Examples of their root configurations are
displayed in Fig. 6.

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

Figure 6: Left (right) plot displays the Bethe-root configuration in the
complex plane of an excited state for $L=19,(18)$, $\gamma=0.4$ in the sector
$h_{1}=1$ and $h_{2}=0$. Blue (red) symbols denote level 1 (2) roots. This
excitation is built by placing 2(3) level $1$ roots ($\times$) on the line
$\frac{{\rm i}\pi}{2}$ and 2,(3) level-2 roots ($\square$) on the line ${\rm
i}\pi$ in addition to the bulk roots ($\bullet$). Further, one (and a half
four-string) has been removed with respect to the lowest energy state
configuration in this sector.

By replacing more and more 4-strings by roots of type (i) and (ii), one can
generate an infinite tower of excitations labelled by the number $M_{\pi}$ of
type (ii) roots. All of them flow to the same scaling dimension (5.10), see
Fig. 7.

Figure 7:  Left (right) plot displays the finite-Size Scaling up to $L\sim
2000$ in the sectors $(h_{1},h_{2})$: $(0,0)$ $\times$ , $(2,0)$ $\diamond$
($(1,0)$ $\triangle$) for states with $M_{\pi}=0,1,2,3$ in increasing order
from below (blue, red, green, cyan). Solid lines are rational extrapolation.
One can see clearly the logarithmic dependence of the scaling dimensions. The
parameters are set to $\gamma=0.4$ and $\phi_{1,2}=0$.

Using the symmetry $u^{[2]}_{j}\to u^{[2]}_{j}+{\rm i}\pi$ that exchanges the
(ii) and (iii) types of roots, one deduces that the RG trajectories of
excitations with root configuration built by mechanism (iii) instead of (ii),
see e.g. Fig. 8, also flow to the same scaling dimensions. Let’s label them by
the number $M_{0}$ of type (iii) roots.

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

Figure 8: Left (right) plot displays the Bethe-root configuration in the
complex plane of an excited state for $L=19,(18)$, $\gamma=0.4$ in the sector
$h_{1}=1$ and $h_{2}=0$. Blue (red) symbols denote level 1 (2) roots. This
excitation is built by placing 2(3) level-$1$ roots ($\times$) on the line
$\frac{{\rm i}\pi}{2}$ and 2,(3) level-2 roots ($\triangle$) on the real line
in addition to the bulk roots ($\bullet$). Further, one (and a half four-
string) has been removed with respect to the lowest energy state configuration
in this sector. It is the state displayed in Fig. 6 transformed by
$u^{[2]}+{\rm i}\pi$, and so it has the same energy.

Further, it turns out that combinations of the two above excitation patterns
are possible, see for example Fig. 9 for the Bethe root configuration of a
mixed state of both fundamental excitations (ii) and (iii). Further, see Fig.
10 to see how such states fit in within the scaling behavior of other states.

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

Figure 9: Left (right) plot displays the Bethe-root configuration in the
complex plane of an excited state for $L=18$, $\gamma=0.4$ in the sector
$h_{1}=0$ and $h_{2}=0$. Blue (red) symbols denote level 1 (2) roots. This
excitation is built by placing 2(4) level-$1$ roots ($\times$) on the line
$\frac{{\rm i}\pi}{2}$ and 1,(2) level-2 roots ($\triangle$) on the real line
and 1,(2) level-2 roots ($\square$) on the line ${\rm i}\pi$ in addition to
the bulk roots ($\bullet$). This is an excitation of both non-compact modes.

Figure 10:  Left (right) plot displays the finite-size scaling up to $L\sim
300$ in the sectors $(h_{1},h_{2})$: $(1,0)$ $\circ$, $(2,1)$ $\triangle$
($(1,1)$ $\times$, $(2,2)$ $\square$) for states with
$(M_{0},M_{\pi})=(0,0),(0,1),(1,1),(0,2),(1,2),(2,2)$ in increasing order from
below (black, blue, cyan, green, red, orange). The solid lines are obtained by
a rational extrapolation. The dashed lines depict the limiting value given by
(5.10). One can see clearly the logarithmic dependence of the scaling
dimensions. The parameters are set to $\gamma=0.4$, $\phi_{1,2}=0$.

The obtained numerical data for various RG trajectories with different
$M_{0},M_{\pi}$ can be used to extract the form of the logarithmic
corrections: The rational extrapolation in Fig. 10 suggest a general quadratic
decay as $\propto\frac{1}{\log(L)^{2}}$. Further, the existence of two
excitation mechanism refines this ansatz to
$\propto\frac{C_{1}}{\log(L)^{2}}+\frac{C_{2}}{\log(L)^{2}}$ with state
dependent constants $C_{1,2}$. As the $Z_{2}$ symmetry interchanges these two
contributions, we conclude that we must have $C_{1}=C_{2}$. Multiplying the
numerical data with $\log(L)^{2}$ extrapolating $L\to\infty$, we can access,
by considering ratios, the dominant state dependence for each
$\ket{\Psi_{L}}$. This numerical work reveals the following behavior:

$\displaystyle
X_{\rm{eff}}=X^{\rm{Com}}_{\rm{eff}}(h_{1},h_{2},0,0)+\frac{\mathcal{A}(\gamma)(M_{0}+\frac{2}{3})^{2}}{\log(L/L_{0})^{2}}+\frac{\mathcal{A}(\gamma)(M_{\pi}+\frac{2}{3})^{2}}{\log(L/L_{0})^{2}}\,.$
(5.12)

Here $L_{0}$ is a non-universal, state dependent constant, which we do not
attempt to calculate here. We are left to extract the amplitude
$\mathcal{A}(\gamma)$. As it is the same for all states, we determine it by
considering the effective scaling dimensions of the ground state where we
expect the subleading logarithmic corrections to be the smallest:

$\displaystyle
X^{GS}_{\rm{eff}}(L)=-\frac{4}{12}+\frac{8}{9}\frac{\mathcal{A}(\gamma)}{\log(L/L_{0})^{2}}\,.$
(5.13)

Following [14], we eliminate $L_{0}$ by using data points for two system size
$L^{1}$ and $L^{2}$:

$\displaystyle\mathcal{A}(\gamma)=\frac{9}{8}\Bigg{[}\frac{\log(\frac{L^{1}}{L^{2}})}{(X^{GS}_{\rm{eff}}(L^{1})+\frac{4}{12})^{-\frac{1}{2}}-(X^{GS}_{\rm{eff}}(L^{2})+\frac{4}{12})^{-\frac{1}{2}}}\Bigg{]}^{2}\,.$
(5.14)

The numerical results are displayed in Fig. 11. Based on these, we conjecture
that

$\displaystyle\mathcal{A}(\gamma)=\frac{5\pi-4\gamma}{4\gamma}\,.$ (5.15)

Figure 11: Amplitude $\mathcal{A}(\gamma)$ calculated via (5.14) for
$L^{1}=2000$, $L^{2}=1000$ for various $\gamma$-values. The dashed line is the
conjecture (5.15). One see a fairly good matching. At the boundaries
$\gamma\approx 0,(\frac{\pi}{4})$ one sees deviations which are assumed to be
due to increasing finite-size corrections, see also [14].

We want to briefly comment how the above leads to two continuous components in
the spectrum of scaling dimensions. So far, we have defined the RG
trajectories $\ket{\Psi_{L}}$ by keeping the numbers $M_{0,\pi}$ fixed,
leading at first view to infinite degeneracies in the scaling limit. However,
we can also organize the RG trajectories differently. Instead of keeping
$M_{0,\pi}$ fixed, we can also let them run under the RG flow. In particular,
we group states into trajectories $\ket{\Psi_{L}}$ such that

$\displaystyle M_{0}\sim\log(L)\,,\quad M_{\pi}\sim\log(L)\,.$ (5.16)

Here (5.16) is subject to the constraint $M_{0,\pi}\ll L$ such that
$\ket{\Psi_{L}}$ is still a low energy state for any finite $L$, i.e. its
energy obeys (5.1). This restriction is essential as RG trajectories leave the
low energy spectrum once $M_{0,\pi}\sim L$. In fact, it is straightforward to
show within the root density approach that the state with $M_{0,\pi}=L$ is
highly excited: in comparison to the energy $Le_{\infty}$ of the ground state
(4.6), its energy is of the order $\sim L$ (similar as e.g. in the staggered
$sl(2|1)$ superspin chain [11]). This supports the interpretation of the
findings above as evidence for the existence of continuous components in the
conformal spectrum labelled by two continuous quantum numbers $M_{0,\pi}/\log
L$.

The redefinition (5.16) of RG trajectories enables that (5.12) can tend as
$L\to\infty$ to a different scaling dimension than
$X^{\rm{Com}}_{\rm{eff}}(h_{1},h_{2},0,0)$. In fact, by suitably arranging the
concrete behavior (5.16), the scaling dimensions can take any value larger or
equal to $X^{\rm{Com}}_{\rm{eff}}(h_{1},h_{2},0,0)$. Note that $X_{\rm{eff}}$
for trajectories with similar $M_{0,\pi}(L)$ for fixed $L$ become densely
distributed ($\sim\frac{1}{\log(L)}$), leading to continuous spectrum of
scaling dimensions. As the two excitation mechanisms labelled by $M_{0}$ and
$M_{\pi}$ are independent of each other, we must have two continuous
variables, call them
$s_{0}\sim\frac{M_{0}}{\log(L)},s_{\pi}\sim\frac{M_{\pi}}{\log(L)}$, whose
limits label the state in the scaling limit. The existence of two continua is
also further supported by our finding for finite twist angles discussed below.

We want to stress that for the identification of the underlying CFT with two
continuous components, a more rigorous definition of the scaling limit than in
(5.16) is needed. A proper scaling limit can be defined in inhomogeneous
models, where the logarithmic corrections can be parameterized by a conserved
operator of the lattice model, the so-called quasi-momentum operator. However,
the definition of this operator in these models relies on their inhomogeneity,
and is therefore not applicable to our model. For details, we refer to the
extensive study of the staggered-six vertex model [19, 42] and the pioneering
work [15].

####  5.2.1 Spectral flow for continuum states

Having identified the finite-size spectrum for vanishing twist angles, we now
turn to the question of what happens when these angles are tuned on. We follow
the procedure described below Eq. (5.10) starting from
$(\phi^{\rm{in}}_{1},\phi^{\rm{in}}_{2})=(0,0)$ and iterating to the higher
twists. This procedure can be technically involved, as certain roots of a
given configuration can tend to infinity as the twists approach certain
values. If these specific twist values are exceeded, then the infinite roots
come back to a finite value. To avoid numerical problems caused by these
infinities, it is suitable to transform to a different coordinate set. By
using $\zeta={\rm e}^{-u}$ infinitely large roots are mapped to zero in this
coordinate frame.

We start by discussing the lowest states ($M_{0}=M_{\pi}=0$) in the two
continua first. Some of our results are represented in Fig. 12. One can see
that for small twist angles, the scaling dimensions follow (5.12) but with the
first term replaced by (5.10) with non-vanishing twist. However, after
critical values of the twists given by

$\displaystyle\phi^{c}_{1,2}=2\gamma(h_{1,2}+1)=\frac{2\pi(h_{1,2}+1)}{{\tt
k}}\,,$ (5.17)

the behavior changes drastically to

$\displaystyle
X_{\rm{eff}}=X^{\rm{Com}}_{\rm{eff}}(h_{1},h_{2},\phi_{1},\phi_{2})-\frac{(h_{1,2}+1-{\tt
k}\frac{\phi_{1,2}}{2\pi})^{2}}{2({\tt
k}-2)}\,,\qquad\phi^{c}_{1,2}<\phi_{1,2}<\widetilde{\phi}^{c}_{1,2}\,,$ (5.18)

with the absence of logarithmic corrections. Numerical work suggests that the
choice between $1$ and $2$ in the above formulae seems to taken in a way such
that the second term in (5.18) always incorporates the bigger twist angle and,
if both twists are equal, minimizing the critical twist angle (see Fig. 12).
It turns out that (5.18) is valid just until the twist exceeds another
critical twist angle $\widetilde{\phi}^{c}_{1,2}$. For example, for twisting
only with $\phi_{1}$ or $\phi_{2}$ in the lowest sector $h_{1}=h_{2}=0$, we
find that

$\displaystyle\left.\widetilde{\phi}^{c}_{1,2}\right|_{h_{1}=h_{2}=0}=2\pi-\left.\phi^{c}_{1,2}\right|_{h_{1}=h_{2}=0}\,.$
(5.19)

Figure 12: The left (right) plot displays the effective scaling dimensions for
the lowest state in the continuum in the sectors $(h_{1},h_{2})=(0,0)$,
$((1,0))$ for $L=2000$, $(1999)$, under various twists
$(\phi_{1},\phi_{2})=(\phi,0),(0,\phi),(\phi,\phi),(\phi,\frac{1}{2}\phi)$
(Black, Blue, Red, Green). The solid lines display the expected behavior
(5.10) for small twist angles excluding the strong logarithmic corrections.
The crosses or circles display the numerical data obtain from the Bethe ansatz
for twist angles as far as possible in the numerical procedure. The vertical
lines designate the critical twist values, where the agreement with (5.10)
breaks down. The dashed lines indicate the conjectured formula (5.18) for the
scaling dimensions valid beyond the critical points. Note that the matching
with the conjecture is extremely accurate. We interpret this as the emergence
of discrete states having less logarithmic corrections.

So far, we have considered only the lowest states in the continua. If we twist
excited states, their scaling dimensions follow (5.18) but again spoiled by
decreasing logarithmic corrections. To further investigate this phenomenon, we
have searched for twists for which the Bethe root configurations are again
regular enough to define RG trajectories. We find that a suitable point is
$(\phi_{1},\phi_{2})=(\pi,0)$. Here, the Bethe roots parameterizing the low-
energy states consists mainly of

$\displaystyle u^{[1]}\longrightarrow$ $\displaystyle\quad x_{j}+\frac{{\rm
i}\pi}{2}-{\rm i}\gamma-{\rm i}\epsilon^{[1]}_{j}\,,\quad x_{j}-\frac{{\rm
i}\pi}{2}+{\rm i}\gamma+{\rm i}\epsilon^{[1]}_{j}\,$ (5.20) $\displaystyle
u^{[2]}\longrightarrow$ $\displaystyle\quad z_{l}+\frac{{\rm
i}\pi}{2}\,\,\,,\quad l=1,\dots,M_{z},$ $\displaystyle\quad w_{k}-\frac{{\rm
i}\pi}{2},\quad k=1,\dots,M_{w}\,.$

In addition there are level-1 roots sitting exponentially close to the
following values depending on the parity of $dN:=M_{z}-M_{w}$:

$\displaystyle{\rm i}\gamma,-{\rm i}\gamma$ $\displaystyle\text{if }dN\text{
even}$ (5.21) $\displaystyle 2{\rm i}\gamma,0,-2{\rm i}\gamma$
$\displaystyle\text{if }dN\text{ odd}$

Plots of typical configurations are shown in Fig. 13. It turns out that states
with different distributions $dN$ of level-2 roots on the two lines
$\pm\frac{{\rm i}\pi}{2}$ flow to the same conformal dimensions (5.18) with
$\phi_{1}=\pi$ and $\phi_{2}=0$, see e.g. Fig. 14. We find that the scaling of
dimensions of excited states for $\phi_{1}=\pi,\phi_{2}=0$ are given by

$\displaystyle
X_{\rm{eff}}=\left.X^{\rm{Com}}_{\rm{eff}}(h_{1},h_{2},\phi_{1},\phi_{2})\right|_{\phi_{1}=\pi,\phi_{2}=0}-\left.\frac{(h_{1}+1-{\tt
k}\frac{\phi_{1}}{2\pi})^{2}}{2({\tt
k}-2)}\right|_{\phi_{1}=\pi}+\tilde{\mathcal{A}}(\gamma)\frac{dN^{2}}{\log(L/\tilde{L}_{0})^{2}}\,.$
(5.22)

where again $\tilde{L}_{0}$ is a non-universal constant which we do not
attempt to calculate here. Further, the above formula also holds true for
small deviations around the twist angles i.e. $\phi_{1}\approx\pi$,
$\phi_{2}\approx 0$ with the obvious modifications. We conjecture that the
amplitude $\tilde{\mathcal{A}}(\gamma)$ is given by

$\displaystyle\tilde{\mathcal{A}}(\gamma)=\frac{2(2-5\gamma)\gamma}{3(1-4\gamma)^{2}}\,.$
(5.23)

In order to interpret these results, let us recall that the scaling limit of
models such as the staggered-six vertex model or the $A^{(2)}_{2}$-model
possess one continuous parameter, call it $s_{*}$. Besides this class of
states, there also exist states where $s_{*}$ takes values in a discrete set.
The lattice regularization of those states does not possess logarithmic
corrections. Further, a state belonging to the family with continuous $s_{*}$
can become a discrete state under a twist [19, 21]. We interpret the above
finding in an analogous way for our model. Consider a state whose scaling
limit is described by the two continuous variables $s_{0},s_{\pi}$. Under a
twist, one of the continuous variables changes its class to the discrete one,
while the other remains in the continuous family. The latter still induces
logarithmic corrections on the level of the lattice regularization as seen in
(5.22).

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

$\Im m(u)$$\Re
e(u)$$\pi$$\frac{\pi}{2}$$\frac{\pi}{2}-\gamma$$-\frac{\pi}{2}$$-\frac{\pi}{2}+\gamma$$-\pi$

Figure 13: Left (right) plot displays the Bethe-root configuration in the
complex $u$-plane of an excited state for $L=18$, $\gamma=0.4$ in the sector
$h_{1}=0$ and $h_{2}=0$. Blue (red) symbols denote level 1 (2) roots. The left
figure shows the ground state configuration, while the right excitation is
built by unbalancing $dN=4$ the number of level-2 on the lines $\pm\frac{{\rm
i}\pi}{2}$. The two level-1 roots with vanishing real part have imaginary
close to $\pm\gamma$.  Figure 14: Effective scaling dimensions up to $L\sim
2000$ for $\phi_{1}=\pi$, $\phi_{2}=0$, $\gamma=0.4$ in the sector
$h_{1}=h_{2}=0$ for $dN=0,1,2,3,4$ in increasing order from below (black,
blue, cyan, green, red). The crosses are the numerical data obtain from the
Bethe ansatz. The solid lines are rational extrapolation. Further, the dashed
pink line is given by the constant limit value (5.18). In order to obtain the
numerical data, we have assumed that the roots which are exponentially close
to (5.21) actually sit on these values. This leads to a small offset (see the
green and blue crosses on the far right) for small system sizes, as here the
approximation is inducing an error.

Ultimately, this conjecture should imply the existence of purely discrete
states (apart from the $dN=0$ state) without any logarithmic corrections. We
have checked that this is indeed the case. Starting from the twist
$(\phi_{1},\phi_{2})=(\pi,0)$, we turn on the second twist significantly. We
find that the first excited states $dN=1,2$ in the sector $h_{1}=0=h_{2}$
become purely discrete states when the second twist angle exceeds the critical
value $\phi_{2}=2\gamma$. It has effective scaling dimensions

$\displaystyle
X_{\rm{eff}}=X^{\rm{Com}}_{\rm{eff}}(h_{1},h_{2},\pi,\phi_{2})-\frac{(h_{1}+1-{\tt
k}\frac{\pi}{2\pi})^{2}}{2({\tt k}-2)}-\frac{(h_{2}+1-{\tt
k}\frac{\phi_{2}}{2\pi})^{2}}{2({\tt
k}-2)}\quad\text{with}\quad\phi_{2}>2\gamma\,.$ (5.24)

Note that this check can be done on the level of small $L$, as we expect that
this state does not possess any logarithmic corrections, see Fig. 15. The
purely discrete scaling dimensions (5.24) are valid for large twist angle
only; however, one can analytically continue back the scaling dimensions
(5.24) to zero twist. For the lowest state with $h_{1}=h_{2}=0$, we obtain in
this way:

$\displaystyle X_{\rm{eff}}=-\frac{4}{12}+\frac{1}{{\tt k}-2}\,.$ (5.25)

Assuming that the conformal weights $h,\bar{h}$ vanish in this procedure as it
is for example in the staggered six-vertex model or the $A^{(2)}_{2}$ model
[21], we obtain on speculative grounds that the central charge is given by

$\displaystyle c$ $\displaystyle=4-\frac{12}{{\tt k}-2}=2\left(2-\frac{6}{{\tt
k}-2}\right)\,,$ (5.26)

which formally coincides with two black hole CFTs [43, 44, 3].

Figure 15: Effective scaling dimensions for small $L$ and
$\phi_{1}=\frac{9\pi}{10}$, $\gamma=0.4+{\rm i}\epsilon$ in the sector
$h_{1}=h_{2}=0$ for $dN=0,1,2,3$ (black, blue, green, red) under variation of
the second twist $\phi_{2}$. The sightly complex value of the anisotropy and
offset of the first twist angle from $\pi$ is due to numerical purposes. The
black solid line displays
$X^{\rm{Com}}_{\rm{eff}}(h_{1},h_{2},\frac{9\pi}{10},\phi_{2})$ while the blue
line is given by (5.24). The black vertical line hallmarks the appearance of a
purely discrete state.

##  6 Conclusion

Starting from the periodic $D^{(2)}_{3}$ spin chain [25], we have generalized
it to the quasi-periodic case. The boundary conditions are found to be
parameterized by two twist angles $\phi_{1}$, $\phi_{2}$. The appearance of
the twists $\phi_{1}$, $\phi_{2}$ can be accounted for in the analytic Bethe
ansatz such that the model can be exactly solved in the Bethe ansatz sense
(2.30), (LABEL:Twisted_BAE). As the rank of $D^{(2)}_{3}$ is two, the Bethe
ansatz is two-level nested. As usual, the model possesses an infinite family
of commuting operators (2.17). Among these, a local Hamiltonian can be defined
in the standard way by the first logarithmic derivative of the transfer matrix
(2.46).

The model has a rich symmetry structure, see (2.23), (2.25), (2.26). The most
interesting result is that a generalization (2.23) of the $Z_{2}$-symmetry of
the lower-rank case $D^{(2)}_{2}$ is identified. It is related to the quasi-
periodicity of the R-matrix (2.10); and on the level of the Bethe ansatz, this
symmetry maps states among each other whose level-$2$ Bethe roots differ by
${\rm i}\pi$. Further, we have found that the transfer matrix is CPT-invariant
(2.26), while on the Hamiltonian level this symmetry reduces to CP-symmetry
(2.51).

Turning to the analysis of the scaling limit, we have concentrated on the
regime of the anisotropy $\gamma\in(0,\frac{\pi}{4})$. The spin chain is found
to be critical, as it possesses gapless excitations with a linear dispersion
relation. Hence, the effective theory of its low-lying excitations arising in
the thermodynamic limit $L\to\infty$ should be governed by a conformal field
theory. We have identified certain classes of the low-lying energy states
which are parameterized by the $U(1)$-charges. We have found that their
effective scaling dimensions give rise to two compact modes in the scaling
limit. More precisely, these modes mimic two compact bosons with zero winding
(5.10). Indeed, despite considerable numerical effort, we did not find any
non-zero winding states. Whether non-trivial winding states exist is left open
for future investigation.

In addition to the two compact modes, we found two types of decreasing
logarithmic corrections. The corrections are generated by the number of
level-2 Bethe roots on the real line and on the line with imaginary part
$\pi$. We provide evidence that these logarithmic corrections give rise to two
non-compact degrees of freedom in the scaling limit. The two non-compact modes
are interchanged by the $Z_{2}$ symmetry (2.23). Furthermore, we have
considered the influence of large twists. We found that, beyond certain
critical twist angles, some of the logarithmic corrections disappear. We
interpret this phenomenon as the emergence of discrete states under twists.
For the case of $\phi_{1}=\pi$ and small $\phi_{2}$, we find that one of the
continua becomes totally discrete, while the other persists. For the extreme
case of $\phi_{1}=\pi$ and large $\phi_{2}$, we observe the existence of
purely discrete states (5.24). By analytical continuation of its effective
scaling dimension to zero twists, and under the assumption that the conformal
dimensions vanish there [21, 45], we access the true central charge (5.26).
Formally, it agrees with the sum of two Black Hole CFT central charges. One
copy of this CFT describes the scaling limit of the lower-rank $D^{(2)}_{2}$
model [30, 15, 19].

For a rigorous identification of the underlying CFT, we would need a conserved
operator that parameterizes the non-compact degrees of freedom on the lattice.
Such an operator, the so-called quasi-momentum, has so far been defined only
in staggered models in which either the representation of the R-matrix [45] or
the associated spectral parameter [15, 46] of the quantum spaces varies
periodically along the chain. Its definition relies on the type of
inhomogeneity; hence, this construction is not applicable for the case of the
homogeneous $D^{(2)}_{3}$ model we are considering here. The search for such
an operator might be an interesting research direction, supporting the
analysis of the scaling limit. It could help with the identification of the
space of states in the scaling limit, and especially the calculation of the
density of states of the continua.

It should be possible to study the influence of open boundary conditions for
selecting certain sectors of the underlying CFT, as has been done for the
lower rank case $D^{(2)}_{2}$ [47, 48, 24, 30, 49, 35, 50]. It might also be
interesting to investigate the different parameter regimes
$\gamma\in(\frac{\pi}{4},\frac{\pi}{2})$. Another natural but challenging
topic might be the generalization to $D^{(2)}_{n}$ with $n>3$, as has been
done for the $A^{(2)}_{n}$ series [23].

## Acknowledgement

The authors thank Yacine Ikhlef, Gleb A. Kotousov and Marcio J. Martins for
valuable discussions. HF and SG acknowledge funding provided by the Deutsche
Forschungsgemeinschaft (DFG) under grant No. Fr 737/9-2 as part of the
research unit Correlations in Integrable Quantum Many-Body Systems (FOR2316).
RN was supported in part by the National Science Foundation under Grant No.
NSF 2310594 and by a Cooper fellowship. ALR was supported by a UKRI Future
Leaders Fellowship (grant number MR/T018909/1). Part of the numerical work has
been performed on the LUH compute cluster, which is funded by the Leibniz
Universität Hannover, the Lower Saxony Ministry of Science and Culture and the
DFG.

##  Appendix A Proofs of symmetries

We sketch here proofs of some of the symmetries noted in the main text.

###  A.1 Crossing symmetry of the transfer matrix (2.21)

To prove that the transfer matrix (2.13) has the crossing symmetry (2.21), we
begin by noting that the _transposed_ transfer matrix can be expressed as

$\displaystyle\mathbbm{t}^{t}(u;\\{\phi_{j}\\})$
$\displaystyle=\mathrm{tr}_{0}\,\left(\mathbb{K}_{0}(\\{\phi_{j}\\})\,\mathbb{R}_{0L}(u)\dots\mathbb{R}_{01}(u)\right)^{t_{0}t_{1}\cdots
t_{L}}$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{R}_{01}^{t_{0}t_{1}}(u)\cdots\mathbb{R}_{0L}^{t_{0}t_{L}}(u)\,\mathbb{K}^{t_{0}}_{0}(\\{\phi_{j}\\})$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(\\{\phi_{j}\\})\,\mathbb{R}_{10}(u)\cdots\mathbb{R}_{L0}(u)\,,$
(A.1)

where we have passed to the final line using the PT symmetry (2.5) and the
fact that the twist matrix (2.15) is symmetric $\mathbb{K}^{t}=\mathbb{K}$. We
then observe that the transfer matrix itself can be expressed as

$\displaystyle\mathbbm{t}(u;\\{\phi_{j}\\})$
$\displaystyle=\mathrm{tr}_{0}\,\left(\mathbb{K}_{0}(\\{\phi_{j}\\})\,\mathbb{R}_{0L}(u)\dots\mathbb{R}_{01}(u)\right)^{t_{0}}$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{R}_{01}^{t_{0}}(u)\cdots\mathbb{R}_{0L}^{t_{0}}(u)\,\mathbb{K}^{t_{0}}_{0}(\\{\phi_{j}\\})$
$\displaystyle=\mathrm{tr}_{0}\,V_{0}^{t_{0}}\,\mathbb{R}_{10}(4{\rm
i}\gamma-u)\,V_{0}^{t_{0}}\cdots V_{0}^{t_{0}}\,\mathbb{R}_{L0}(4{\rm
i}\gamma-u)\,V_{0}^{t_{0}}\,\mathbb{K}_{0}(\\{\phi_{j}\\})$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(\\{-\phi_{j}\\})\,\mathbb{R}_{10}(4{\rm
i}\gamma-u)\,\cdots\mathbb{R}_{L0}(4{\rm i}\gamma-u)\,$
$\displaystyle=\mathbbm{t}^{t}(4{\rm i}\gamma-u;\\{-\phi_{j}\\})\,.$ (A.2)

In passing to the third equality, we have used (2.8) and (2.5); and in passing
to the fourth equality, we have used the fact

$V\,\mathbb{K}(\\{\phi_{j}\\})=\mathbb{K}(\\{-\phi_{j}\\})\,V\,.$ (A.3)

Finally, to pass to the last line of (A.2), we have used the result (A.1).

###  A.2 $W(0)$ symmetry of the transfer matrix (2.24)

The transfer matrix (2.13) transforms under $W(0)^{\otimes L}$ as

$\displaystyle W(0)^{\otimes
L}\,\mathbbm{t}(u;\phi_{1},\phi_{2})\,W(0)^{\otimes L}$
$\displaystyle=W(0)^{\otimes
L}\,\mathrm{tr}_{0}\,\left(\mathbb{K}_{0}(\phi_{1},\phi_{2})\,\mathbb{R}_{0L}(u)\dots\mathbb{R}_{01}(u)\,\right)W(0)^{\otimes
L}$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(\phi_{1},\phi_{2})\,W_{L}(0)\,\mathbb{R}_{0L}(u)\,W_{L}(0)\,\dots
W_{1}(0)\,\mathbb{R}_{01}(u)\,W_{1}(0)$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(\phi_{1},\phi_{2})\,W_{0}(u)\,\mathbb{R}_{0L}(u)\,W_{0}(u)\,\dots
W_{0}(u)\,\mathbb{R}_{01}(u)\,W_{0}(u)$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(-\phi_{2},-\phi_{1})\,\mathbb{R}_{0L}(u)\,\dots\,\mathbb{R}_{01}(u)$
$\displaystyle=\mathbbm{t}(u;-\phi_{2},-\phi_{1})\,,$ (A.4)

where we have passed to the third equality using (2.12), and to the fourth
equality using the fact

$W(u)\,\mathbb{K}(\phi_{1},\phi_{2})\,W(u)=\mathbb{K}(-\phi_{2},-\phi_{1})\,.$
(A.5)

###  A.3 CPT symmetry of the transfer matrix (2.26)

In order to prove that the transfer matrix (2.13) has the CPT symmetry (2.26),
we begin by observing that the parity operator (2.27) acts as

$\displaystyle\Pi\,\mathbbm{t}(u;\\{\phi_{j}\\})\,\Pi$
$\displaystyle=\mathrm{tr}_{0}\,\Pi\left(\mathbb{K}_{0}(\\{\phi_{j}\\})\,\mathbb{R}_{0L}(u)\dots\mathbb{R}_{01}(u)\right)\Pi$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(\\{\phi_{j}\\})\,\mathbb{R}_{01}(u)\dots\mathbb{R}_{0L}(u)\,.$
(A.6)

It then follows that

$\displaystyle V^{\otimes
L}\,\Pi\,\mathbbm{t}(u;\\{\phi_{j}\\})\,\Pi\,V^{\otimes L}$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(\\{\phi_{j}\\})\,V_{1}\,\mathbb{R}_{01}(u)\,V_{1}\dots
V_{L}\,\mathbb{R}_{0L}(u)\,V_{L}$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(\\{\phi_{j}\\})\,V_{0}\,\mathbb{R}_{10}(u)\,V_{0}\,\dots
V_{0}\,\mathbb{R}_{L0}(u)\,V_{0}$
$\displaystyle=\mathrm{tr}_{0}\,V_{0}\,\mathbb{K}_{0}(\\{\phi_{j}\\})\,V_{0}\,\mathbb{R}_{10}(u)\,\dots\mathbb{R}_{L0}(u)$
$\displaystyle=\mathrm{tr}_{0}\,\mathbb{K}_{0}(\\{-\phi_{j}\\})\,\mathbb{R}_{10}(u)\,\dots\mathbb{R}_{L0}(u)$
$\displaystyle=\mathbbm{t}^{t}(u;\\{-\phi_{j}\\})\,,$ (A.7)

where we have passed to the second equality using (2.9), to the fourth
equality using (A.3), and to the last equality using (A.1).

###  A.4 CP symmetry of the Hamiltonian (2.51)

In order to prove that the Hamiltonian has the CP symmetry (2.51), we note the
more explicit expression that follows from its definition (2.46)

$\mathbb{H}\sim\mathbbm{t}^{-1}(0)\,\mathbbm{t}^{\prime}(0)=\sum_{i=1}^{L-1}H_{i,i+1}+\mathbb{K}^{-1}_{L}\,H_{L,1}\,\mathbb{K}_{L}\,,\qquad
H_{i,i+1}=\mathcal{P}_{i,i+1}\,\mathbb{R}^{\prime}_{i,i+1}(0)\,,$ (A.8)

and proceed to show that the “bulk” and “boundary” terms are separately
invariant.

For the “bulk” terms in (A.8), we observe that parity acts as

$\Pi\,H_{i,i+1}\Pi=H_{L+1-i,L-i}=\mathcal{P}_{L+1-i,L-i}\,\mathbb{R}^{\prime}_{L+1-i,L-i}(0)\,.$
(A.9)

It follows that CP acts as

$\displaystyle V^{\otimes L}\,\Pi\,H_{i,i+1}\Pi\,V^{\otimes L}$
$\displaystyle=\mathcal{P}_{L+1-i,L-i}\,V_{L-i}\,V_{L+1-i}\,\mathbb{R}^{\prime}_{L+1-i,L-i}(0)\,V_{L-i}\,V_{L+1-i}$
$\displaystyle=\mathcal{P}_{L+1-i,L-i}\,\mathbb{R}^{\prime}_{L-i,L+1-i}(0)\,$
$\displaystyle=H_{L-i,L+1-i}\,,$ (A.10)

where we have passed to the second equality using (2.9). Summing over $i$, we
obtain

$V^{\otimes L}\,\Pi\,\left(\sum_{i=1}^{L-1}H_{i,i+1}\right)\,\Pi\,V^{\otimes
L}=\sum_{i=1}^{L-1}H_{L-i,L+1-i}=\sum_{j=1}^{L-1}H_{j,j+1}\,,$ (A.11)

where we have performed the change of variables $j=L-i$ to pass to the final
equality. In short, the “bulk” terms in (A.8) are CP invariant.

For the “boundary” term in (A.8), we observe that parity acts as

$\Pi\,\mathbb{K}^{-1}_{L}\,H_{L,1}\,\mathbb{K}_{L}\Pi=\mathbb{K}^{-1}_{1}\,H_{1,L}\,\mathbb{K}_{1}\,,$
(A.12)

and therefor CP acts as

$\displaystyle V^{\otimes
L}\,\Pi\,\mathbb{K}^{-1}_{L}\,H_{L,1}\,\mathbb{K}_{L}\Pi\,V^{\otimes L}$
$\displaystyle=V_{1}\,V_{L}\,\mathbb{K}^{-1}_{1}\,H_{1,L}\,\mathbb{K}_{1}\,V_{1}\,V_{L}$
$\displaystyle=\mathbb{K}_{1}\,V_{1}\,V_{L}\,H_{1,L}\,V_{1}\,V_{L}\,\mathbb{K}^{-1}_{1}$
$\displaystyle=\mathbb{K}_{1}\,H_{L,1}\,\mathbb{K}^{-1}_{1}$
$\displaystyle=\mathbb{K}^{-1}_{L}\,H_{L,1}\,\mathbb{K}_{L}\,,$ (A.13)

where we have passed to the second equality using (A.3), to the third equality
using (2.9), and to the final equality using the fact

$\left[H_{L,1}\,,\mathbb{K}_{L}\,\mathbb{K}_{1}\right]=0\,.$ (A.14)

We conclude that the “boundary” term is also CP invariant, and therefore, so
is the full Hamiltonian (A.8).

##  Appendix B Introducing a diagonal twist in the Bethe ansatz

We present here a derivation of the result (2.30) for the eigenvalue
$\Lambda(u)$ of the (twisted) transfer matrix, starting from the untwisted
result [25]

$\displaystyle\Lambda(u)\Big{|}_{\phi_{1}=\phi_{2}=0}=$
$\displaystyle\left(4\sinh(u-2{\rm i}\gamma)\sinh(u-4{\rm
i}\gamma)\right)^{L}A(u)+\left(4\sinh(u-4{\rm i}\gamma)\sinh
u\right)^{L}\sum_{\ell=1}^{4}B_{\ell}(u)\,$ (B.1)
$\displaystyle+\left(4\sinh(u-2{\rm i}\gamma)\sinh u\right)^{L}C(u)\,,$

where $A(u),B_{\ell}(u),C(u)$ are given in Eqs. (2.31),(2.32).

We proceed, in the same spirit as [25], using a kind of analytical Bethe
ansatz approach. The asymptotic behavior of the monodromy matrix (2.14) is
given by

$\mathbb{T}(u)\underset{u\rightarrow\infty}{\sim}({\rm e}^{2u-4{\rm
i}\gamma})^{L}\left\\{\text{diag}\left({\rm e}^{-2{\rm
i}\gamma\mathbbm{h}_{1}}\,,{\rm e}^{-2{\rm
i}\gamma\mathbbm{h}_{2}}\,,\mathbb{I}\,,\mathbb{I}\,,{\rm e}^{2{\rm
i}\gamma\mathbbm{h}_{2}}\,,{\rm e}^{2{\rm
i}\gamma\mathbbm{h}_{1}}\right)+\ldots\right\\}\,,$ (B.2)

where the ellipsis denotes off-diagonal terms that will not contribute to the
final result. It follows that the twisted transfer matrix (2.13) has the
asymptotic behavior

$\mathbbm{t}(u)\underset{u\rightarrow\infty}{\sim}({\rm e}^{2u-4{\rm
i}\gamma})^{L}\left\\{2\,\mathbb{I}+\sum_{j=1}^{2}\left({\rm e}^{{\rm
i}\phi_{j}}{\rm e}^{-2{\rm i}\gamma\mathbbm{h}_{j}}+{\rm e}^{-{\rm
i}\phi_{j}}{\rm e}^{2{\rm i}\gamma\mathbbm{h}_{j}}\right)+\ldots\right\\}\,.$
(B.3)

We assume that the eigenvalue $\Lambda(u)$ is given by the following “dressed”
version of the periodic result (B.1)

$\displaystyle\Lambda(u)=$ $\displaystyle\left(4\sinh(u-2{\rm
i}\gamma)\sinh(u-4{\rm i}\gamma)\right)^{L}a\,A(u)+\left(4\sinh(u-4{\rm
i}\gamma)\sinh u\right)^{L}\sum_{\ell=1}^{4}b_{\ell}\,B_{\ell}(u)\,$
$\displaystyle+\left(4\sinh(u-2{\rm i}\gamma)\sinh u\right)^{L}c\,C(u)\,,$
(B.4)

where the parameters $a\,,b_{\ell}\,,c$ are still to be determined. Evidently,
we have from (2.28)

$\langle\Lambda|\mathbbm{t}(u)|\Lambda\rangle=\Lambda(u)\,.$ (B.5)

For $u\rightarrow\infty$, we evaluate the LHS of (B.5) using (B.3) and (2.29),
obtaining

$\displaystyle\langle\Lambda|\mathbbm{t}(u)|\Lambda\rangle\underset{u\rightarrow\infty}{\sim}$
$\displaystyle({\rm e}^{2u-4{\rm i}\gamma})^{L}\Big{\\{}{\rm e}^{{\rm
i}\phi_{1}}{\rm e}^{-2{\rm i}\gamma(L-m_{1})}+{\rm e}^{{\rm i}\phi_{2}}{\rm
e}^{-2{\rm i}\gamma(m_{1}-m_{2})}$ $\displaystyle+2+{\rm e}^{-{\rm
i}\phi_{1}}{\rm e}^{2{\rm i}\gamma(L-m_{1})}+{\rm e}^{-{\rm i}\phi_{2}}{\rm
e}^{2{\rm i}\gamma(m_{1}-m_{2})}\Big{\\}}\,;$ (B.6)

and we evaluate the RHS of (B.5) using (B.4) to obtain

$\displaystyle\Lambda(u)\underset{u\rightarrow\infty}{\sim}$
$\displaystyle({\rm e}^{2u-4{\rm i}\gamma})^{L}\Big{\\{}a\,{\rm e}^{-2{\rm
i}\gamma(L-m_{1})}+b_{1}\,{\rm e}^{-2{\rm i}\gamma(m_{1}-m_{2})}$
$\displaystyle+b_{2}+b_{3}+b_{4}\,{\rm e}^{2{\rm
i}\gamma(m_{1}-m_{2})}+c\,{\rm e}^{2{\rm i}\gamma(L-m_{1})}\Big{\\}}\,.$ (B.7)

Comparing (B.6) and (B.7), we conclude that the parameters are given by

$a={\rm e}^{{\rm i}\phi_{1}}\,,\quad b_{1}={\rm e}^{{\rm i}\phi_{2}}\,,\quad
b_{2}=b_{3}=1\,,\quad b_{4}={\rm e}^{-{\rm i}\phi_{2}}\,,\quad c={\rm
e}^{-{\rm i}\phi_{1}}\,,$ (B.8)

which is the result in (2.30).

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