# Integrable boundary conditions for staggered vertex models

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

###### Abstract

Yang-Baxter integrable vertex models with a generic
$\mathbb{Z}_{2}$-staggering can be expressed in terms of composite
$\mathbb{R}$-matrices given in terms of the elementary $R$-matrices.
Similarly, integrable open boundary conditions can be constructed through
generalized reflection algebras based on these objects and their
representations in terms of composite boundary matrices $\mathbb{K}^{\pm}$. We
show that only two types of staggering yield a local Hamiltonian with
integrable open boundary conditions in this approach. The staggering in the
underlying model allows for a second hierarchy of commuting integrals of
motion (in addition to the one including the Hamiltonian obtained from the
usual transfer matrix), starting with the so-called quasi momentum operator.
In this paper, we show that this quasi momentum operator can be obtained
together with the Hamiltonian for both periodic and open models in a unified
way from enlarged Yang-Baxter or reflection algebras in the composite picture.
For the special case of the staggered six-vertex model, this allows
constructing an integrable spectral flow between the two local cases.

##  I Introduction

Integrable lattice models based on representations of the Yang-Baxter algebra
have proven to be extremely useful in the understanding of non-perturbative
phenomena in one-dimensional many-body systems. Given a particular $R$-matrix,
it is possible to consider variations in the spectral parameter leading to
local inhomogeneities preserving the integrability of such models. This has
first been used by Baxter in the context of the six-vertex model Baxter
(1971). In particular, models with periodically repeating inhomogeneities (or
staggered models) have proven to be applicable to a wide range of problems.
Apart from the construction of integrable spin chains with larger unit cells
Popkov and Zvyagin (1993); Frahm and Rödenbeck (1996) inhomogeneous vertex
models have been used, e.g., to formulate the Potts model as a
$\mathbb{Z}_{2}$-staggered six-vertex model Baxter (1973), for the lattice
regularization of field theories such as the principal chiral model Faddeev
and Reshetikhin (1986); Destri and de Vega (1989), in the quantum transfer
matrix approach to the thermodynamics of integrable models Klümper (1992), and
to study integrable perturbations of conformal field theories Reshetikhin and
Saleur (1994). Alternatively, the staggering can be realized by choosing
alternating local representations of the underlying symmetry algebra. Such a
staggering appears quite naturally in the superspin formulation of network
models describing the disorder induced plateau transition in integer quantum
Hall systems Chalker and Coddington (1988); Zirnbauer (1997) which can be made
integrable by fine-tuning of the coupling constants Gade (1999); Essler  _et
al._ (2005). Extensions of such staggered models to open boundary conditions
with their integrability encoded in representations of the corresponding
reflection algebra Cherednik (1984); Sklyanin (1988) allow for the
construction of spin chains with soliton non-preserving boundary conditions
Doikou (2000). Similar algebraic structures emerge in the study of certain
AdS/CFT-type integrability theories Prinsloo  _et al._ (2015); Bai  _et al._
(2019).

Interestingly, finite-size studies of certain staggered models based both on
variations of the spectral parameter or the local representations have
revealed that their continuum limit – in spite of the compact formulation as a
spin chain – is described by conformal field theories with a non-compact
target space Essler  _et al._ (2005); Ikhlef  _et al._ (2008); Frahm and
Martins (2011, 2012). Among these the most studied example is the periodically
staggered six-vertex model whose low energy effective theory has been
identified to be the $SL(2,\mathbb{R})/U(1)$ black hole CFT Ikhlef  _et al._
(2008, 2012); Candu and Ikhlef (2013); Frahm and Seel (2014); Bazhanov  _et
al._ (2021a, b). More recently, the influence of open $U_{q}(sl(2))$-invariant
boundary conditions in this model has been studied, both for the self-dual
staggering related to the Potts model Robertson  _et al._ (2020a, 2021) and
also away from the self-dual line Frahm and Gehrmann (2022). These studies
have shown that boundary conditions have a profound effect: depending on their
choice the symmetry of the ground state may be spontaneously broken or the
continuous component of the conformal spectrum disappears completely.

A conserved quantity existing in these models which has been particularly
useful for the identification of the conformal field theory is the so-called
quasi momentum operator. Its role in staggered models without a non-compact
continuum limit, however, has not been studied yet. The definition of the
quasi momentum relies on the possibility to introduce a staggering in the
vertical direction of the vertex model which is compatible with the horizontal
one Ikhlef  _et al._ (2012); Frahm and Seel (2014); Frahm and Hobuß (2017);
Frahm and Gehrmann (2022). That such an operator cannot be defined in the
homogeneous case has impeded progress in the analysis of the spectrum other
models where indications for a continuous spectrum of conformal weights have
been observed, namely the $a_{N-1}^{(2)}$ models and a family of
orthosymplectic superspin chains Vernier  _et al._ (2014, 2016); Martins  _et
al._ (1998); Frahm and Martins (2015, 2018); Frahm  _et al._ (2019); Frahm and
Martins (2022).

In this work, we will use a different perspective on the staggered models to
address the question of whether the quasi momentum can be constructed in an
alternative approach which may be applicable for homogeneous models, too.
After a brief review of the construction of integrable models with periodic
and open boundary conditions based on an ’elementary’ $R$-matrix solving the
Yang-Baxter equation and corresponding boundary matrices, we construct
’composite’ $\mathbb{R}$-matrices using the co-multiplication property of the
Yang-Baxter algebra. These $\mathbb{R}$-matrices satisfy a generalized Yang-
Baxter equation (18) and depend on the staggering parameters through
additional arguments. For periodic boundary conditions this allows to define a
homogeneous transfer matrix generating both the local integrals of motions
such as the Hamiltonian and the quasi momentum operator.

In Section III we generalize this procedure to the open case, where we express
composite boundary matrices in terms of the elementary ones. Depending on the
properties of the elementary $R$-matrices (and unlike in the periodic case) we
identify two different choices of the staggering parameters leading to a
transfer matrix constructed from the composite $\mathbb{R}$\- and boundary
matrices which generates a Hamiltonian with local interactions in the bulk
(similar as in Refs. Nepomechie and Retore (2021a); Li  _et al._ (2022)). For
one of these choices a second homogeneous transfer matrix with boundary
matrices satisfying a different reflection equation generates the quasi
momentum. The commutativity of these objects is guaranteed by a set of
intertwining relations between the two sets of boundary matrices.

Finally we apply our findings to the self-dual staggered six-vertex model.
Based on this construction the spectral flow between the models with compact
and non-compact continuum limits can be studied in a family of integrable
models. Although one has to give up locality at the intermediate steps one
finds that the two endpoints of this scheme are separated by two first-order
transitions where massive degeneracies lead to a reordering of levels. Based
on the numerical solution of the Bethe equations we provide some insights into
the role of the quasi momentum in the model with compact continuum limit.

##  II Basic Ingredients

###  II.1 Yang-Baxter integrable models

Let
$\mathbb{V}=\mathcal{V}_{0}\otimes\mathcal{V}_{\bar{0}}\otimes\bigotimes^{L^{\prime}}_{j=1}\mathcal{V}_{j}$
be the tensor product of $2+L^{\prime}$ copies of a vector space
$\mathcal{V}$. Given an operator $A$ acting on the space $\mathcal{V}^{\otimes
n}$ we define $A_{j_{1},\dots,j_{n}}$ to be the operator on $\mathbb{V}$ which
acts as $A$ on
$\bigotimes^{n}_{m=1}\mathcal{V}_{j_{m}}\cong\mathcal{V}^{\otimes n}$ and as
the identity on all the other factors (assuming implicitly that all $j_{k}$
are different). We will use this notation throughout the study. Denote by
$R(u)$ a linear operator depending meromorphically on $u\in\mathbb{C}$ and
acting on the twofold tensor product $\mathcal{V}\otimes\mathcal{V}$ which
satisfies the Yang-Baxter equation (YBE)

$\displaystyle
R_{i,j}(u-v)R_{i,k}(u)R_{j,k}(v)=R_{j,k}(v)R_{i,k}(u)R_{i,j}(u-v)\,.$ (1)

In the following we will call $R$ the $R$-matrix. We assume it to satisfy the
initial condition

$\displaystyle R_{i,j}(0)$ $\displaystyle=P_{i,j}\,,$ (2a) with the
permutation operator $P_{i,j}$ on $\mathcal{V}_{i}\otimes\mathcal{V}_{j}$.
Note that initial condition guarantees that $R_{i,j}(u)$ is differentiable
near $u=0$. In addition to this regularity condition we require several
properties of the $R$-matrix throughout this paper, namely unitarity,
$PT$-symmetry, crossing symmetry and crossing unitarity111The stated
properties are not independent from each other. The initial condition paired
with the Yang-Baxter equation gives unitarity. In turn, the combination of
(2b),(2c) and (2d) imply (2e). $\displaystyle R_{i,j}(u)R_{j,i}(-u)$
$\displaystyle=\xi(u)\mathbf{1}\,,$ (2b) $\displaystyle
R^{t_{i}t_{j}}_{i,j}(u)$ $\displaystyle=R_{j,i}(u)\,,$ (2c) $\displaystyle
R_{i,j}(u)$ $\displaystyle=V_{i}R^{t_{j}}_{i,j}(-u-\eta)V^{-1}_{i}$ (2d)
$\displaystyle R^{t_{i}}_{i,j}(u)M_{i}R^{t_{j}}_{i,j}(-u-2\eta)M^{-1}_{i}$
$\displaystyle=\xi(u+\eta)\,\mathbf{1}\,,$ (2e) with the crossing parameter
$\eta\in\mathbb{C}$ , a scalar function $\xi(u)$ and some invertible matrix
$V\in\mathrm{End}(\mathcal{V})$ and $M=V^{t}V=M^{t}$ being a symmetry
transformation of the $R$-matrix $\displaystyle M^{-1}_{i}R_{i,j}(u)M_{i}$
$\displaystyle=M_{j}R_{i,j}(u)M^{-1}_{j}\,.$ (2f)

The first study of such type of $R$-matrix in the context of open spin chains
was carried out in Mezincescu and Nepomechie (1991).

In a later section we will assume that the $R$-matrix is quasi periodic i.e.

$\displaystyle R_{i,j}(u+p)=f_{p}G_{i}R_{i,j}(u)G^{-1}_{i},$ (3)

where $p\in\mathbb{C}$ is non-zero constant. Using the properties (2) of the
$R$-matrix, we find that $f^{2}_{p}=1$ and $G$ is an invertible matrix, in
fact $G^{-1}\propto V^{-1}GV$. Further, one concludes that $G$ must be a
symmetric or anti-symmetric matrix. Using its (anti-)symmetry one deduces by
using PT-symmetry in the transposed version of (3) with $i$ and $j$
interchanged that $R$ is $G$-invariant, i.e.

$\displaystyle G_{i}G_{j}R_{i,j}(u)G^{-1}_{i}G^{-1}_{j}=R_{i,j}(u).$ (4)

Trigonometric and elliptic $R$-matrices which obey the quasi periodicity
condition (3) have been constructed for example in Belavin (1981); Bazhanov
(1987). We will state explicitly whenever we use the quasi periodicity
assumption in addition to Eqs. (2).

Based on the Yang-Baxter equation (1) an algebraic structure equipped with a
coproduct can be introduced. This allows for the construction of a monodromy
matrix

$\displaystyle T_{0}(u,\\{u_{\ell}\\})=$ $\displaystyle
R_{0,L^{\prime}}(u+u_{L^{\prime}})R_{0,L^{\prime}-1}(u+u_{L^{\prime}-1})\cdots
R_{0,1}(u+u_{1})\,,$ (5)

which obeys the following equivalent ($RTT$-)relations

$\displaystyle R_{i,j}(u-v)T_{i}(u,\\{u_{\ell}\\})T_{j}(v,\\{u_{\ell}\\})$
$\displaystyle=T_{j}(v,\\{u_{\ell}\\})T_{i}(u,\\{u_{\ell}\\})R_{i,j}(u-v)\,,$
(6a) $\displaystyle
T_{i}(u,\\{u_{\ell}\\})R_{i,j}(u+v)T^{-1}_{j}(-v,\\{u_{\ell}\\})$
$\displaystyle=T^{-1}_{j}(-v,\\{u_{\ell}\\})R_{i,j}(u+v)T_{i}(u,\\{u_{\ell}\\})\,,$
(6b) $\displaystyle
R_{i,j}(u-v)T^{-1}_{i}(u,\\{u_{\ell}\\})T^{-1}_{j}(v,\\{u_{\ell}\\})$
$\displaystyle=T^{-1}_{j}(v,\\{u_{\ell}\\})T^{-1}_{i}(u,\\{u_{\ell}\\})R_{i,j}(u-v)\,.$
(6c)

Each $R$-matrix in (5) acts on the auxiliary space $\mathcal{V}_{0}$ and one
of the factors in the quantum space
$\mathcal{H}=\otimes_{j=1}^{L^{\prime}}\mathcal{V}_{j}$. The parameters
$\\{u_{\ell}\\}$ are called inhomogeneities. Taking the trace of (5) one
obtains the transfer matrix

$\displaystyle\tau^{\mathrm{pbc}}(u,\\{u_{\ell}\\})$
$\displaystyle=\text{tr}_{0}\bigg{(}T_{0}(u,\\{u_{\ell}\\})\bigg{)}\,,$ (7)

which, as a consequence of (6a), commutes for different values of the spectral
parameter $u$. Therefore, it generates integrals of motion for a model defined
on the Hilbert space $\mathcal{H}$ corresponding to an $L^{\prime}$ site
lattice subject to periodic boundary conditions.

For models with integrable open boundary condition one needs, in addition,
representations $K^{\pm}(u)\in\text{End}(\mathcal{V})$ of the reflection
algebras Cherednik (1984)

$\displaystyle R_{i,j}(u-v)K^{-}_{i}(u)R_{j,i}(u+v)K^{-}_{j}(v)$
$\displaystyle=K^{-}_{j}(v)R_{i,j}(u+v)K^{-}_{i}(u)R_{j,i}(u-v)\,,$ (8a) and
$\displaystyle R_{i,j}(-u+v)$
$\displaystyle\left(K^{+}_{i}(u)\right)^{t_{i}}M^{-1}_{i}R_{j,i}(-(u+v)-2\eta)M_{i}\left(K^{+}_{j}(v)\right)^{t_{j}}$
(8b)
$\displaystyle=\left(K^{+}_{j}(v)\right)^{t_{j}}M_{i}R_{i,j}(-(u+v)-2\eta)M^{-1}_{i}\left(K^{+}_{i}(u)\right)^{t_{i}}R_{j,i}(-u+v)\,.$

For the six- and eight-vertex models (or spin-$1/2$ chains) the most general
$c$-number solutions to these equation have been constructed in Refs. Sklyanin
(1988); Mezincescu and Nepomechie (1991); de Vega and González-Ruiz (1994).
$K$-matrices for more general cases of quantum affine algebras have been
constructed Delius and George (2002); Delius and MacKay (2003); Appel and
Vlaar (2020). In the following we will assume the unitarity property

$\displaystyle K^{-}(u)K^{-}(-u)\propto\mathbf{1}\,,$ (9)

which can be shown to hold for a large number of $K$-matrices, see e.g. Appel
and Vlaar (2022) and references therein. Given the form of the reflection
equation (8a) with a quasi periodic $R$-matrix (3) we further assume
that222Note that both conditions (10) and (11) are satisfied by the general
$K$-matrices for the anisotropic spin-$1/2$ chains de Vega and González-Ruiz
(1994).

$\displaystyle K^{-}(u+p)\propto GK^{-}(u)G\,.$ (10)

Finally, it is natural, especially with regard to _local_ Hamiltonian’s
discussed in section III.5, to assume that $K^{-}(u)$ is meromorphic in $u$,
too, and obeys

$\displaystyle K^{-}(0)\propto\mathbf{1}\,,\quad K^{-}(\frac{p}{2})\propto
G\,.$ (11)

The corresponding properties for $K^{+}$ follow from the isomorphisms of the
algebras given in Sklyanin (1988). Given representations of (8) one obtains
the transfer matrix

$\displaystyle\tau(u,\\{u_{\ell}\\})=\text{tr}_{0}\big{(}K^{+}_{0}(u)T_{0}(u,\\{u_{\ell}\\})K^{-}_{0}(u)T^{-1}_{0}(-u,\\{u_{\ell}\\})\big{)}\,,$
(12)

which can be shown Sklyanin (1988); Mezincescu and Nepomechie (1991) to
commute for different values of the spectral parameter i.e.

$\displaystyle\left[\tau(u,\\{u_{\ell}\\}),\tau(v,\\{u_{\ell}\\})\right]=0\,.$
(13)

Hence, it generates commuting integrals of motion of a lattice model on
$\mathcal{H}$ with boundary conditions defined by $K^{\pm}(u)$.

Note that the unitarity relation (2b) allows to rewrite the inverse monodromy
matrix $T_{0}^{-1}$ in (12) as

$\displaystyle
T_{0}^{-1}(-u,\\{u_{\ell}\\})=R_{1,0}(u-u_{1})R_{2,0}(u-u_{2})\cdots
R_{L^{\prime},0}(u-u_{L^{\prime}})\,\left(\prod^{L^{\prime}}_{j=1}\xi(-u+u_{j})\right)^{-1}\,.$
(14)

In the following of this study we will consider lattices of even length
$L^{\prime}=2L$ and restrict to a $\mathbb{Z}_{2}$-staggering of the
inhomogeneities, i.e.

$\displaystyle u_{2j}=\delta_{1},\quad u_{2j-1}=\delta_{2}\,,\qquad
j=1,\dots,L\,,$ (15)

where $\delta_{1},\delta_{2}\in\mathbb{C}$. The tuple of inhomogeneities
$\\{\delta_{2},\delta_{1},.....\delta_{2},\delta_{1}\\}$ will be abbreviated
by $\\{\delta_{1},\delta_{2}\\}$ below. Further, we define the function

$\displaystyle
c_{\tau}(u,\\{\delta_{1},\delta_{2}\\})=\left[\xi(-u+\delta_{1})\xi(-u+\delta_{2})\right]^{-L}\,.$
(16)

.

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

###  II.2 The composite $\mathbb{R}$-matrix

Using the coproduct of the Yang-Baxter algebra a solution of the YBE (1) can
be extended to act as an endomorphism on $\mathcal{W}\otimes\mathcal{W}$ with
$\mathcal{W}=\mathcal{V}\otimes\mathcal{V}$ (or, more generally, $n$-fold
tensor products of the vector space $\mathcal{V}$). Specifically, we define

$\displaystyle\mathbb{R}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})=R_{i,\ell}(u+\Delta_{ij})R_{i,k}(u+\Delta_{ij}-\Delta_{k\ell})R_{j,\ell}(u)R_{j,k}(u-\Delta_{k\ell})\,,$
(17)

where $\Delta_{ij},\Delta_{k\ell}$ are arbitrary parameters. The explicit form
is motivated by a general choice of inhomogeneities in both the horizontal and
vertical direction as displayed in Fig. 2 using the graphical notation
introduced in Fig. 1.

$u+\delta_{i}-\delta_{\ell}$$u+\delta_{j}-\delta_{\ell}$$u+\delta_{i}-\delta_{k}$$u+\delta_{j}-\delta_{k}$$\delta_{i}$$\delta_{j}$$\delta_{k}$$\delta_{\ell}$$u\to
u-\delta_{j}+\delta_{\ell}$$u+\Delta_{ij}$$u$$-\Delta_{kl}$$u-\Delta_{kl}$$u+\Delta_{ij}$$\delta_{i}$$\delta_{j}$$\delta_{k}$$\delta_{\ell}$
Figure 2: The composite $\mathbb{R}$-matrix (17), where
$\Delta_{ij}=\delta_{i}-\delta_{j}$

The index notation implies that $\mathbb{R}_{i,j|k,\ell}$ acts on the tensor
product of the two copies $(\mathcal{V}_{i}\otimes\mathcal{V}_{j})$ and
$(\mathcal{V}_{k}\otimes\mathcal{V}_{\ell})$ of $\mathcal{W}$. We will use
this notation throughout this paper. By construction this $R$-matrix satisfies
the generalized Yang-Baxter equation

$\displaystyle\mathbb{R}_{i,j|k,\ell}(u-v,\Delta_{ij},\Delta_{k\ell})$
$\displaystyle\mathbb{R}_{i,j|m,n}(u,\Delta_{ij},\Delta_{mn})\mathbb{R}_{k,\ell|m,n}(v,\Delta_{k\ell},\Delta_{mn})=$
(18)
$\displaystyle\mathbb{R}_{k,\ell|m,n}(v,\Delta_{k\ell},\Delta_{mn})\mathbb{R}_{i,j|m,n}(u,\Delta_{ij},\Delta_{mn})\mathbb{R}_{i,j|k,\ell}(u-v,\Delta_{ij},\Delta_{k\ell})\,,$

and therefore allows to introduce commuting transfer matrices as in Section
II.1. Since this construction relies on the properties (2) of $R(u)$ it is
natural to ask which of these are inherited to the $\mathbb{R}$-matrix. It
turns out that the $\mathbb{R}$-matrix obeys the following properties

$\displaystyle\mathbb{R}_{i,j|k,\ell}(0,\Delta,\Delta)$
$\displaystyle=\xi(\Delta)\,\mathbb{P}_{i,j|k,\ell},$ (19a)
$\displaystyle\mathbb{R}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})\mathbb{R}_{k,\ell|i,j}(-u,\Delta_{k\ell},\Delta_{ij})$
$\displaystyle=\Xi(u,\Delta_{ij},\Delta_{k\ell})\,\mathbf{1}\,,$ (19b)
$\displaystyle\mathbb{R}^{t_{i}t_{j}t_{k}t_{\ell}}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})$
$\displaystyle=\mathbb{R}_{k,\ell|i,j}(u,-\Delta_{k\ell},-\Delta_{ij})\,,$
(19c)
$\displaystyle\mathbb{R}^{t_{i}t_{j}}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})\mathbb{M}_{i,j}\mathbb{R}_{i,j|k,\ell}^{t_{k}t_{\ell}}(-u-2\eta,-\Delta_{ij},-\Delta_{k\ell})\mathbb{M}^{-1}_{i,j}$
$\displaystyle=\Xi(u+\eta,\Delta_{ij},\Delta_{k\ell})\,\mathbf{1}\,,$ (19d)
$\displaystyle\mathbb{M}^{-1}_{i,j}\mathbb{R}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})\mathbb{M}_{i,j}$
$\displaystyle=\mathbb{M}_{k,\ell}\mathbb{R}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})\mathbb{M}^{-1}_{k,\ell}\,,$
(19e)
$\displaystyle\mathbb{G}^{-1}_{i,j}\mathbb{R}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})\mathbb{G}_{i,j}$
$\displaystyle=\mathbb{G}_{k,\ell}\mathbb{R}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})\mathbb{G}^{-1}_{k,\ell}\,,$
(19f) $\displaystyle\mathbb{R}_{i,j|k,\ell}(u+p,\Delta_{ij},\Delta_{k\ell})$
$\displaystyle=G_{i}G_{j}\mathbb{R}_{i,j|k\ell}(u,\Delta_{ij},\Delta_{k\ell})G^{-1}_{i}G^{-1}_{j}\,,$
(19g)

which we discuss in the following order by order. Eq. (19a) is a regularity
property (with a different normalization) where $\mathbb{P}_{i,j|k,\ell}$ is
the enlarged permutation operator acting on the state $(a\otimes
b)\otimes(c\otimes
d)\in\mathcal{W}\otimes\mathcal{W}=(\mathcal{V}_{i}\otimes\mathcal{V}_{j})\otimes(\mathcal{V}_{k}\otimes\mathcal{V}_{\ell})$
as

$\displaystyle\mathbb{P}_{i,j|k,\ell}(a\otimes b)\otimes(c\otimes d)=(c\otimes
d)\otimes(a\otimes b)\,.$

Eq. (19b) is the unitarity condition where the proportionality constant is
given by

$\displaystyle\Xi(u,\Delta_{ij},\Delta_{k\ell})=\xi(u+\Delta_{ij})\xi(u+\Delta_{ij}-\Delta_{k\ell})\xi(u)\xi(u-\Delta_{k\ell})\,.$

The third property, Eq. (19c), is a _generalized_ $PT$-symmetry of
$\mathbb{R}$, where the parameters in $\mathbb{R}$ change sign due to the
reordering caused by the transposition:

$\displaystyle\mathbb{R}^{t_{i}t_{j}t_{k}t_{\ell}}_{i,j|k,\ell}(u,\Delta_{ij},\Delta_{k\ell})$
$\displaystyle=\left(R_{i,\ell}(u+\Delta_{ij})R_{i,k}(u+\Delta_{ij}-\Delta_{k\ell})R_{j,\ell}(u)R_{j,k}(u-\Delta_{k\ell})\right)^{t_{i}t_{j}t_{k}t_{\ell}}$
$\displaystyle=R^{t_{j}t_{k}}_{j,k}(u-\Delta_{k\ell})R^{t_{i}t_{k}}_{i,k}(u+\Delta_{ij}-\Delta_{k\ell})R^{t_{j}t_{\ell}}_{j,\ell}(u)R^{t_{i}t_{\ell}}_{i,\ell}(u+\Delta_{ij})$
$\displaystyle=R_{k,j}(u-\Delta_{k\ell})R_{k,i}(u+\Delta_{ij}-\Delta_{k\ell})R_{\ell,j}(u)R_{\ell,i}(u+\Delta_{ij})$
$\displaystyle=\mathbb{R}_{k,\ell|i,j}(u,-\Delta_{k\ell},-\Delta_{ij})\,.$

As the $PT$-symmetry is related to the crossing unitary, the
$\mathbb{R}$-matrix satisfies a _generalized_ crossing unitarity relation
given in the fourth equation where $\mathbb{M}_{i,j}=M_{i}M_{j}$.
Finally, also the symmetry relation (2f) and the quasi periodicity (3) and its
implication (4) can be directly transferred to the $\mathbb{R}$-matrix,
yielding the last three equations of (19) where
$\mathbb{G}_{i,j}=G_{i}G_{j}$.333Note that the $\mathbb{R}$-matrix, depending
on the choice of the parameters in (17), may have an extended symmetry. This
has been discussed recently in the context of the antiferromagnetic Potts
model where this construction leads to an integrable model based on the affine
$D_{2}^{(2)}$ Lie algebra starting from the $U_{q}[\mathfrak{sl}(2)]$ (or
$A_{1}^{(1)}$) invariant $R$-matrix of the six-vertex model Robertson  _et
al._ (2020b).

###  II.3 Staggered vertex models with periodic boundary conditions

These properties suffice to construct $\mathbb{Z}_{2}$-staggered models with
periodic boundary conditions. Here we show how their formulations in terms of
the elementary and the composite $R$-matrices are related. Consider the
product of two transfer matrices (7) with different spectral parameters
corresponding to a staggering in the auxiliary direction ($0$ and
$\overline{0}$ label different auxiliary spaces)

$\displaystyle\mathcal{T}^{\mathrm{pbc}}(u,\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\})$
$\displaystyle=\tau^{\mathrm{pbc}}(u+\delta_{0},\\{\delta_{1},\delta_{2}\\})\,\tau^{\mathrm{pbc}}(u+\delta_{\overline{0}},\\{\delta_{1},\delta_{2}\\})$
(20)
$\displaystyle=\text{tr}_{0}\bigg{(}R_{0,2L}(u+\delta_{0}+\delta_{1})R_{0,2L-1}(u+\delta_{0}+\delta_{2})\cdots
R_{0,1}(u+\delta_{0}+\delta_{2})\bigg{)}$
$\displaystyle\quad\times\text{tr}_{\overline{0}}\bigg{(}R_{\overline{0},2L}(u+\delta_{\overline{0}}+\delta_{1})R_{\overline{0},2L-1}(u+\delta_{\overline{0}}+\delta_{2})\cdots
R_{\overline{0},1}(u+\delta_{\overline{0}}+\delta_{2})\bigg{)}\,.$

By reordering the $R$-matrices we obtain

$\displaystyle\mathcal{T}^{\mathrm{pbc}}(u,$
$\displaystyle\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\})=$
$\displaystyle\text{tr}_{0\overline{0}}\bigg{(}$ $\displaystyle
R_{0,2L}(u+\delta_{0}+\delta_{1})R_{\overline{0},2L}(u+\delta_{\overline{0}}+\delta_{1})R_{0,2L-1}(u+\delta_{0}+\delta_{2})R_{\overline{0},2L-1}(u+\delta_{\overline{0}}+\delta_{2})\cdots$
$\displaystyle\cdots
R_{0,2}(u+\delta_{0}+\delta_{1})R_{\overline{0},1}(u+\delta_{\overline{0}}+\delta_{2})R_{0,1}(u+\delta_{0}+\delta_{2})R_{\overline{0},1}(u+\delta_{\overline{0}}+\delta_{2})\bigg{)}\,.$

The products of four $R$-matrices appearing in each row can be expressed in
terms of the composite $\mathbb{R}$-matrix (17). Shifting the spectral
parameter as $u\to u-\delta_{\overline{0}}-\delta_{1}$ we obtain a
_homogeneous_ transfer matrix

$\displaystyle\mathcal{T}^{\mathrm{pbc}}(u,\\{\Delta_{0\overline{0}},\Delta_{12}\\})=\text{tr}_{0\overline{0}}\bigg{(}$
$\displaystyle\mathbb{R}_{0,\overline{0}|2L-1,2L}(u,\Delta_{0\overline{0}},\Delta_{12})\cdots\mathbb{R}_{0,\overline{0}|1,2}(u,\Delta_{0\overline{0}},\Delta_{12})\bigg{)}\,.$
(21)

For the physical interpretation as a lattice model with local (i.e. finite
range) interactions additional conditions have to be satisfied. Typically,
locality can be derived from the regularity property of the $R$-matrix, i.e.
that it becomes a permutation operator at a shift point $u=u_{0}$. In the
present case, (19a), we have $u_{0}=0$ and need to tune the staggering
parameters such that

$\displaystyle\Delta_{0\overline{0}}=\Delta_{12}\equiv\Delta\,.$ (22)

With this constraint a Hamiltonian coupling the degrees of freedom from
nearest neighbour quantum spaces $\mathcal{W}$ is obtained from

$\displaystyle H^{\mathrm{pbc}}\propto\left.\frac{\partial}{\partial
u}\log(\mathcal{T}^{\mathrm{pbc}}(u,\Delta,\Delta))\right|_{u=0}\,,$ (23)

where we have assumed, in addition, that $R$ is differentiable at $\pm\Delta$.
The staggering in the auxiliary direction allows to construct another
operator, generating a family of commuting integrals deriving from (7):
instead of (20) we can consider the so-called quasi shift operator, given by
the quotient of single row transfer matrices

$\displaystyle\frac{\tau^{\mathrm{pbc}}(u+\delta_{\overline{0}},\\{\delta_{1},\delta_{2}\\})}{\tau^{\mathrm{pbc}}(u+\delta_{0},\\{\delta_{1},\delta_{2}\\})}\,.$

As done for the product of transfer matrices, we shift the spectral parameter
$u\to u-\delta_{\overline{0}}-\delta_{1}$, leading to

$\displaystyle\widetilde{\mathcal{Q}}^{\mathrm{pbc}}(u,\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\})=\frac{\tau^{\mathrm{pbc}}(u-\delta_{1},\\{\delta_{1},\delta_{2}\\})}{\tau^{\mathrm{pbc}}(u+\Delta_{0\overline{0}}-\delta_{1},\\{\delta_{1},\delta_{2}\\})}\,.$
(24)

Restricting the staggering parameters $\delta_{1},\delta_{2},\delta_{0}$ and
$\delta_{\overline{0}}$ to be compatible with (22) and taking the logarithm of
this operator at the shift point, $u_{0}=0$, we obtain the ’quasi momentum’

$\displaystyle\mathbb{Q}^{\mathrm{pbc}}=\log\left[\widetilde{\mathcal{Q}}^{\mathrm{pbc}}(0,\\{\delta_{1},\delta_{2}\\})\right]=\log\left[\frac{\tau^{\mathrm{pbc}}(-\delta_{1},\\{\delta_{1},\delta_{2}\\})}{\tau^{\mathrm{pbc}}(-\delta_{2},\\{\delta_{1},\delta_{2}\\})}\right]\,.$
(25)

This operator has proven to be particularly useful for the characterization of
low energy effective behavior of several staggered vertex models, see e.g.
Ikhlef  _et al._ (2012); Candu and Ikhlef (2013); Frahm and Seel (2014); Frahm
and Hobuß (2017); Frahm and Gehrmann (2022) and has found recently application
as a Floquet Hamiltonian Miao  _et al._ (2022).

We now want to generate the quasi momentum from an operator constructed from
the composite $\mathbb{R}$-matrix (17). It is straightforward to invert the
single row transfer matrix (7) in the denominator of (25) by using regularity
and unitarity of the $R$-matrix:

$\displaystyle\left(\tau^{\mathrm{pbc}}(-\delta_{2},\\{\delta_{1},\delta_{2}\\})\right)^{-1}$
$\displaystyle\propto\text{tr}_{0}\left(R_{1,0}(0)R_{2,0}(-\Delta)\cdots
R_{2L-1,0}(0)R_{2L,0}(-\Delta)\right)\,.$ (26)

With the definition of

$\displaystyle Q^{\mathrm{pbc}}(u)$
$\displaystyle=\text{tr}_{0}\left(R_{1,0}(-u)R_{2,0}(-u-\Delta)\cdots
R_{2L-1,0}(-u)R_{2L,0}(-u-\Delta)\right)$
$\displaystyle\qquad\times\text{tr}_{\overline{0}}\left(R_{\overline{0},2L}(u)R_{\overline{0},2L-1}(u-\Delta)\cdots
R_{\overline{0},2}(u)R_{\overline{0},1}(u-\Delta)\right)\,,$

we obtain a _product_ of $R$-matrices depending on a spectral parameter which
after taking the logarithm becomes proportional to (25) at the shift point.
Using crossing symmetry (2d) and expressing the result in terms of the
composite $\mathbb{R}$-matrices we find:

$\displaystyle Q^{\mathrm{pbc}}(u)$
$\displaystyle=\text{tr}_{0\overline{0}}\big{(}R_{0,2L}(u+\Delta-\eta)R_{0,2L-1}(u-\eta)\cdots
R_{0,2}(u+\Delta-\eta)R_{0,1}(u-\eta)$ (27) $\displaystyle\qquad\qquad\times
R_{\overline{0},2L}(u)R_{\overline{0},2L-1}(u-\Delta)\cdots
R_{\overline{0},2}(u)R_{\overline{0},1}(u-\Delta)\big{)}$
$\displaystyle=\text{tr}_{0\overline{0}}\Big{(}\mathbb{R}_{0,\overline{0}|2L-1,2L}(u,\Delta-\eta,\Delta)\cdots\mathbb{R}_{0,\overline{0}|1,2}(u,\Delta-\eta,\Delta)\Big{)}$
$\displaystyle=\mathcal{T}^{\mathrm{pbc}}\left(u,\\{\Delta-\eta,\Delta\\}\right)\,.$

Note that this becomes the product of single row transfer matrices with
arguments differing by the crossing parameter $\eta$ in the homogeneous limit,
$\Delta\to 0$,

$\displaystyle\lim_{\Delta\to
0}Q^{\mathrm{pbc}}(u)=\text{tr}_{0\overline{0}}\big{(}R_{0,2L}(u-\eta)R_{\overline{0},2L}(u)\cdots
R_{0,1}(u-\eta)R_{\overline{0},1}(u)\big{)}=\tau^{\mathrm{pbc}}(u-\eta)\,\tau^{\mathrm{pbc}}(u)\,.$
(28)

This product can be related to the higher-spin transfer matrices through the
$T$-system bilinear functional relations Kuniba  _et al._ (2011).

In summary the transfer matrices

$\displaystyle\mathcal{T}^{\mathrm{pbc}}(u,\\{\theta,\Delta\\})$
$\displaystyle=\text{tr}_{0\overline{0}}\bigg{(}\mathbb{R}_{0,\overline{0}|2L-1,2L}(u,\theta,\Delta)\cdots\mathbb{R}_{0,\overline{0}|1,2}(u,\theta,\Delta)\bigg{)}$
(29)

provide a unified framework generating both local integrals of motion such as
the Hamiltonian under the locality condition (22), i.e. $\theta=\Delta$, and
the quasi momentum (25) for $\theta=\Delta-\eta$. Note that the third
arguments of all $\mathbb{R}$-matrices in (29) coincide. Therefore,
commutativity of
$\mathcal{T}^{\mathrm{pbc}}\left(u,\\{\theta,\Delta\\}\right)$ for different
$u$ and $\theta$ (which includes $Q^{\mathrm{pbc}}(u)$) follows directly from
the generalized YBE (18). Moreover, let us note that
$\mathbb{Q}^{\mathrm{pbc}}$, unlike the Hamiltonian, is a _non-local_
operator.

##  III Integrable open boundary conditions for staggered models

###  III.1 Composite picture for open models

We now want to address the question to which extent this procedure can be
applied to construct staggered models with open boundary conditions. The
strategy is the same: we begin by considering the product of two transfer
matrices (12) built out of generic $R$ and $K$-matrices satisfying the Yang-
Baxter and reflection equations, respectively, i.e.

$\displaystyle\mathcal{T}(u,\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\})$
$\displaystyle=\text{tr}_{0}\bigg{(}X_{0}(u+\delta_{0})\bigg{)}\text{tr}_{\overline{0}}\bigg{(}Y_{\overline{0}}(u+\delta_{\overline{0}})\bigg{)}=\text{tr}_{0\overline{0}}\bigg{(}X_{0}(u+\delta_{0})Y^{t_{\overline{0}}}_{\overline{0}}(u+\delta_{\overline{0}})\bigg{)}\,,$
(30)

where we have defined

$\displaystyle X_{0}(u)\equiv$ $\displaystyle
K^{+}_{0}(u)T_{0}(u,\\{\delta_{1},\delta_{2}\\})K^{-}_{0}(u)T^{-1}_{0}(-u,\\{\delta_{1},\delta_{2}\\})\,,$
$\displaystyle Y_{\overline{0}}(u)\equiv$ $\displaystyle
T_{\overline{0}}(u,\\{\delta_{1},\delta_{2}\\})K^{-}_{\overline{0}}(u)T^{-1}_{\overline{0}}(-u,\\{\delta_{1},\delta_{2}\\})K^{+}_{\overline{0}}(u)\,.$

(see Figure 3 for a graphical representation of $\mathcal{T}(u)$).

$u+\delta_{\overline{0}}-\delta_{1}$$u+\delta_{\overline{0}}-\delta_{2}$$u+\delta_{\overline{0}}-\delta_{1}$$u+\delta_{\overline{0}}-\delta_{2}$$u+\delta_{\overline{0}}$$u+\delta_{\overline{0}}+\delta_{1}$$u+\delta_{\overline{0}}+\delta_{2}$$u+\delta_{\overline{0}}+\delta_{1}$$u+\delta_{\overline{0}}+\delta_{2}$$u+\delta_{\overline{0}}-\delta_{1}$$u+\delta_{\overline{0}}-\delta_{2}$$u+\delta_{\overline{0}}+\delta_{1}$$u+\delta_{\overline{0}}+\delta_{2}$$u+\delta_{\overline{0}}$$u+\delta_{0}-\delta_{1}$$u+\delta_{0}-\delta_{2}$$u+\delta_{0}-\delta_{1}$$u+\delta_{0}-\delta_{2}$$u+\delta_{0}$$u+\delta_{0}+\delta_{1}$$u+\delta_{0}+\delta_{2}$$u+\delta_{0}+\delta_{1}$$u+\delta_{0}+\delta_{2}$$u+\delta_{0}-\delta_{1}$$u+\delta_{0}-\delta_{2}$$u+\delta_{0}+\delta_{1}$$u+\delta_{0}+\delta_{2}$$u+\delta_{0}$$2L$$2L-1$$2L-2$$2L-3$$2$$1$
Figure 3: Graphical representation of the product (30) of two transfer
matrices with arbitrary $\mathbb{Z}_{2}$ staggering by using the conventions
defined in Fig. 1.

Inserting a crossing unitarity (2e) and using cyclicity of the trace and
$PT$-symmetry we obtain

$\displaystyle\mathcal{T}(u,\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\})\xi(2u+\delta_{0}+\delta_{\overline{0}}+\eta)$
$\displaystyle\qquad=\text{tr}_{0\overline{0}}\bigg{(}M_{0}R_{0,\overline{0}}(-2u-\delta_{0}-\delta_{\overline{0}}-2\eta)M^{-1}_{0}X_{0}(u+\delta_{0}))R_{\overline{0},0}(2u+\delta_{0}+\delta_{\overline{0}})Y_{\overline{0}}(u+\delta_{\overline{0}})\bigg{)}$
$\displaystyle\qquad=\text{tr}_{0\overline{0}}\bigg{(}M_{0}R_{0,\overline{0}}(-2u-\delta_{0}-\delta_{\overline{0}}-2\eta)M^{-1}_{0}K^{+}_{0}(u+\delta_{0})$
$\displaystyle\qquad\qquad\qquad\times
T_{0}(u+\delta_{0},\\{\delta_{1},\delta_{2}\\})K^{-}_{0}(u+\delta_{0})T^{-1}_{0}(-u-\delta_{0},\\{\delta_{1},\delta_{2}\\})R_{\overline{0},0}(2u+\delta_{0}+\delta_{\overline{0}})$
$\displaystyle\qquad\qquad\qquad\times
T_{\overline{0}}(u+\delta_{\overline{0}},\\{\delta_{1},\delta_{2}\\})K^{-}_{\overline{0}}(u+\delta_{\overline{0}})T^{-1}_{\overline{0}}(-u-\delta_{\overline{0}},\\{\delta_{1},\delta_{2}\\})K^{+}_{\overline{0}}(u+\delta_{\overline{0}})\bigg{)}\,.$

Now we use Eq. (6b) and rearrange the $K$-matrices to get:

$\displaystyle\mathcal{T}(u,\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\})=$
$\displaystyle\text{tr}_{0\overline{0}}\bigg{(}K^{+}_{\overline{0}}(u+\delta_{\overline{0}})M_{0}R_{0,\overline{0}}(-2u-\delta_{0}-\delta_{\overline{0}}-2\eta)M^{-1}_{0}K^{+}_{0}(u+\delta_{0})$
$\displaystyle\qquad\times
T_{0}(u+\delta_{0},\\{\delta_{1},\delta_{2}\\})T_{\overline{0}}(u+\delta_{\overline{0}},\\{\delta_{1},\delta_{2}\\})$
$\displaystyle\qquad\times
K^{-}_{0}(u+\delta_{0})R_{\overline{0},0}(2u+\delta_{0}+\delta_{\overline{0}})K^{-}_{\overline{0}}(u+\delta_{\overline{0}})$
$\displaystyle\qquad\times
T^{-1}_{0}(-u-\delta_{0},\\{\delta_{1},\delta_{2}\\})T^{-1}_{\overline{0}}(-u-\delta_{\overline{0}},\\{\delta_{1},\delta_{2}\\})\bigg{)}$
$\displaystyle\qquad\times\xi^{-1}(2u+\delta_{0}+\delta_{\overline{0}}+\eta)\,.$

Finally, using the expression for the monodromy matrices in terms of the
elementary $R$-matrices this transfer matrix can be represented graphically as
shown in Fig. 4 (up to a scalar factor).

$u+\delta_{\overline{0}}-\delta_{1}$$u+\delta_{\overline{0}}-\delta_{2}$$u+\delta_{\overline{0}}-\delta_{1}$$u+\delta_{\overline{0}}-\delta_{2}$$u+\delta_{\overline{0}}$$u+\delta_{\overline{0}}+\delta_{1}$$u+\delta_{\overline{0}}+\delta_{2}$$u+\delta_{\overline{0}}+\delta_{1}$$u+\delta_{\overline{0}}+\delta_{2}$$u+\delta_{\overline{0}}-\delta_{1}$$u+\delta_{\overline{0}}-\delta_{2}$$u+\delta_{\overline{0}}+\delta_{1}$$u+\delta_{\overline{0}}+\delta_{2}$$u+\delta_{\overline{0}}$$u+\delta_{0}-\delta_{1}$$u+\delta_{0}-\delta_{2}$$u+\delta_{0}-\delta_{1}$$u+\delta_{0}-\delta_{2}$$u+\delta_{0}$$u+\delta_{0}+\delta_{1}$$u+\delta_{0}+\delta_{2}$$u+\delta_{0}+\delta_{1}$$u+\delta_{0}+\delta_{2}$$u+\delta_{0}-\delta_{1}$$u+\delta_{0}-\delta_{2}$$u+\delta_{0}+\delta_{1}$$u+\delta_{0}+\delta_{2}$$u+\delta_{0}$$2u+\delta_{0}+\delta_{\overline{0}}$$-2u-\delta_{0}-\delta_{\overline{0}}-2\eta$$2L$$2L-1$$2L-2$$2L-3$$2$$1$
Figure 4: Graphical representation of the product of transfer matrix (31)
after the merging procedure. The merged $\mathbb{R}$-matrix here is given by
four vertices as in Fig. 2 as indicated as an example by the red box. Further
we see we obtain some enlarged boundary matrices as emphasized by the blue
box.

Clearly, this can be expressed in terms of the composite $\mathbb{R}$-matrices
(17) giving

$\displaystyle\mathcal{T}(u,\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\})=c_{\tau}(u+\delta_{0},\\{\delta_{1},\delta_{2}\\})c_{\tau}(u+\delta_{\overline{0}},\\{\delta_{1},\delta_{2}\\})\xi^{-1}(2u+\delta_{0}+\delta_{\overline{0}}+\eta)$
(31)
$\displaystyle\qquad\times\text{tr}_{0\overline{0}}\bigg{(}K^{+}_{\overline{0}}(u+\delta_{\overline{0}})M_{0}R_{0,\overline{0}}(-2u-\delta_{0}-\delta_{\overline{0}}-2\eta)M^{-1}_{0}K^{+}_{0}(u+\delta_{0})$
$\displaystyle\qquad\qquad\qquad\times\mathbb{R}_{0,\overline{0}|2L-12L}(u+\delta_{\overline{0}}+\delta_{1},\Delta_{0\overline{0}},\Delta_{12})\cdots\mathbb{R}_{0,\overline{0}|1,2}(u+\delta_{\overline{0}}+\delta_{1},\Delta_{0\overline{0}},\Delta_{12})$
$\displaystyle\qquad\qquad\qquad\times
K^{-}_{0}(u+\delta_{0})R_{\overline{0},0}(2u+\delta_{0}+\delta_{\overline{0}})K^{-}_{\overline{0}}(u+\delta_{\overline{0}})$
$\displaystyle\qquad\qquad\qquad\times\mathbb{R}_{1,2|\overline{0},0}(u+\delta_{0}-\delta_{1},\Delta_{12},\Delta_{0\overline{0}})\cdots\mathbb{R}_{2L-1,2L|\overline{0},0}(u+\delta_{0}-\delta_{1},\Delta_{12},\Delta_{0\overline{0}})\bigg{)}\,.$

###  III.2 Local interactions I: alternating staggering

As for periodic boundary conditions the staggering parameters
$\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\}$ have to satisfy
constraints to generate local interactions from this open boundary transfer
matrix. Nearest neighbour interactions between the composite degrees of
freedom of the staggered model are obtained by taking the derivative (assuming
all quantities to be differentiable at the corresponding points) of
$\mathcal{T}(u)$ with respect to the spectral parameter Sklyanin (1988)

$\displaystyle H\propto\left.\frac{\partial}{\partial
u}\mathcal{T}\left(u,\\{\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}\\}\right)\right|_{u=u_{0}}\,.$
(32)

Again, locality derives from the regularity of $\mathbb{R}$. To make use of
(19a) three conditions are needed to be met:

  1. i)

The $\mathbb{R}$-matrices in (31) need to act on the same auxiliary space e.g.
$\mathcal{W}_{0\overline{0}}$.

  2. ii)

As in the periodic case the staggering parameters have to satisfy the
constraint (22).

  3. iii)

The staggering parameters have to be chosen such that _all_ $\mathbb{R}$ in
(31) can simultaneously be evaluated at the shift point $u_{0}=0$.

For (i) we use the identity
$\mathbb{R}_{i,j|\overline{0},0}=P_{0,\overline{0}}\mathbb{R}_{i,j|0,\overline{0}}P_{0,\overline{0}}$.
Conditions (ii) and (iii) are achieved by choosing the staggering parameters
to be opposite and equal in both the horizontal and the vertical direction,
i.e.

$\displaystyle\delta_{0}=-\delta_{\overline{0}}=\delta_{1}=-\delta_{2}\,$ (33)

(depicted in Fig. 5c) or the equivalent choice of parameters obtained by
changing $\delta_{0}\to-\delta_{0}$ (see Fig. 5b).

$2\delta_{0}$$-2\delta_{0}$$-2\delta_{0}$$2\delta_{0}$$2j-1$$2j$(b)$2\delta_{0}$$-2\delta_{0}$$-2\delta_{0}$$2\delta_{0}$$-\frac{p}{2}$$-\frac{p}{2}$$\frac{p}{2}$$\frac{p}{2}$$2j-1$$2j$(c)$2j-1$$2j$(d)$u+\delta_{\overline{0}}-\delta_{1}$$u+\delta_{0}-\delta_{1}$$u+\delta_{\overline{0}}+\delta_{1}$$u+\delta_{0}+\delta_{1}$$u+\delta_{\overline{0}}-\delta_{2}$$u+\delta_{0}-\delta_{2}$$u+\delta_{\overline{0}}+\delta_{2}$$u+\delta_{0}+\delta_{2}$$2j$$2j-1$(a)$2j-1$$2j$(e)
Figure 5: (a) To establish bulk locality at $u=0$ the parameters
$\delta_{0},\delta_{\overline{0}},\delta_{1},\delta_{2}$ have to be fine tuned
such that the conditions (i)-(iii) hold. The diagrammatic schemes of the bulk
elementary cells at the shift point $u=0$ for all possible non-trivial choices
of staggering are displayed in (b) & (c) (alternating case (33)) and (d)
(quasi periodic case (37)). Using regularity (2a), unitarity (2b), and in (d)
quasi periodicity (3) of the elementary $R$-matrix gives the identity (e) in
the bulk.

These constraints on the staggering parameters imply
$\Delta_{0\overline{0}}=\Delta_{12}=2\delta_{0}$ with $\delta_{0}$ remaining
as a _free_ parameter. The resulting transfer matrix is

$\displaystyle\mathcal{T}(u,\\{\delta_{0},-\delta_{0},\delta_{0},-\delta_{0}\\})=c_{\tau}(u+\delta_{0},\\{\delta_{0},-\delta_{0}\\})c_{\tau}(u-\delta_{0},\\{\delta_{0},-\delta_{0}\\})$
$\displaystyle\qquad\text{tr}_{0\overline{0}}\bigg{(}\mathbb{K}^{+}_{0,\overline{0}}(u,2\delta_{0})\mathbb{R}_{0,\overline{0}|2L-1,2L}(u,2\delta_{0},2\delta_{0})\cdots\mathbb{R}_{0,\overline{0}|1,2}(u,2\delta_{0},2\delta_{0})$
$\displaystyle\qquad\qquad\times\mathbb{K}^{-}_{0,\overline{0}}(u,2\delta_{0})\mathbb{R}_{1,2|0,\overline{0}}(u,2\delta_{0},2\delta_{0})\cdots\mathbb{R}_{2L-1,2L|0,\overline{0}}(u,2\delta_{0},2\delta_{0})\bigg{)},$

where we have introduced

$\displaystyle\mathbb{K}^{-}_{i,j}(u,2\delta_{0})$
$\displaystyle=P_{i,j}K^{-}_{j}(u+\delta_{0})R_{i,j}(2u)K^{-}_{i}(u-\delta_{0})\,,$
(34a) $\displaystyle\mathbb{K}^{+}_{i,j}(u,2\delta_{0})$
$\displaystyle=\frac{1}{\xi(2u+\eta)}P_{i,j}K^{+}_{j}(u-\delta_{0})M_{i}R_{i,j}(-2u-2\eta)M^{-1}_{i}K^{+}_{i}(u+\delta_{0})\,.$
(34b)

In terms of the monodromy matrix built from the composite
$\mathbb{R}$-matrices,

$\displaystyle\mathbb{T}_{0,\overline{0}}(u,\Delta_{0\overline{0}},\Delta_{12})=\mathbb{R}_{0,\overline{0}|2L-1,2L}(u,\Delta_{0\overline{0}},\Delta_{12})\cdots\mathbb{R}_{0,\overline{0}|1,2}(u,\Delta_{0\overline{0}},\Delta_{12})\,,$
(35)

the transfer matrix for alternating staggering (33) is brought into standard
form (12)444If one drops the constraint 33, a shift in the of the spectral
parameter in (31) leads to a transfer matrix with a moving boundary Nepomechie
and Retore (2021a). This transfer matrix does not lead in general, however, to
a local Hamiltonian.

$\displaystyle\mathcal{T}^{\text{alt}}(u,2\delta_{0})$
$\displaystyle\equiv\mathcal{T}(u,\\{\delta_{0},-\delta_{0},\delta_{0},-\delta_{0})$
(36)
$\displaystyle=\text{tr}_{0\overline{0}}\bigg{(}\mathbb{K}^{+}_{0,\overline{0}}(u,2\delta_{0})\mathbb{T}_{0,\overline{0}}(u,2\delta_{0},2\delta_{0})\mathbb{K}^{-}_{0,\overline{0}}(u,2\delta_{0})\mathbb{T}^{-1}_{0,\overline{0}}(-u,2\delta_{0},2\delta_{0})\bigg{)}\,.$

###  III.3 Local interactions II: quasi periodic staggering

Interestingly, there exists a second choice of the staggering parameters
leading to a local Hamiltonian (32) when the elementary $R$-matrix is quasi
periodic (3), namely

$\displaystyle\delta_{0}=\frac{p}{2}\,,\quad\delta_{\overline{0}}=0\,,\quad\delta_{1}=0\,,\quad\delta_{2}=\frac{p}{2}\,,$
(37)

which is displayed in Fig. 5d. Again, we have to implement three steps to
bring the transfer matrix into a form generating a local Hamiltonian: for step
(i), i.e. switching the auxiliary space $\mathcal{W}_{\overline{0},0}$ to
$\mathcal{W}_{0,\overline{0}}$, we use the Yang-Baxter equation (1) for
$v=-p/2$ giving

$\mathbb{R}_{i,j|\overline{0},0}\left(u+\frac{p}{2},-\frac{p}{2},\frac{p}{2}\right)\,R_{0,\overline{0}}\left(-\frac{p}{2}\right)=R_{0,\overline{0}}\left(-\frac{p}{2}\right)\,\mathbb{R}_{i,j|0,\overline{0}}\left(u,-\frac{p}{2},-\frac{p}{2}\right)\,.$
(38)

For step (ii), i.e. preparing $\mathbb{R}$ such that the regularity (19a) can
be exploited, we use the quasi periodicity of $R$ which implies

$\mathbb{R}_{0,\overline{0}|i,j}\left(u,\frac{p}{2},-\frac{p}{2}\right)=G_{0}\,\mathbb{R}_{0,\overline{0}|i,j}\left(u,-\frac{p}{2},-\frac{p}{2}\right)\,G_{0}^{-1}\,.$
(39)

Together with the unitarity condition
$R_{0,\overline{0}}(-\frac{p}{2})R_{\overline{0},0}(\frac{p}{2})=\xi(\frac{p}{2})\mathbf{1}$
these identities allow to rewrite the transfer matrix (31) such that also
condition (iii) is satisfied, i.e.

$\displaystyle\mathcal{T}(u,\\{\frac{p}{2},0,0,\frac{p}{2}\\})=c_{\tau}(u+\frac{p}{2},\\{0,\frac{p}{2}\\})\,c_{\tau}(u,\\{0,\frac{p}{2}\\})\xi^{-1}\left(\frac{p}{2}\right)$
(40)
$\displaystyle\qquad\text{tr}_{0\overline{0}}\bigg{(}\overline{\mathbb{K}}^{+}_{0,\overline{0}}(u,-\frac{p}{2})\mathbb{R}_{0,\overline{0}|2L-1,2L}(u,-\frac{p}{2},-\frac{p}{2})\cdots\mathbb{R}_{0,\overline{0}|1,2}(u,-\frac{p}{2},-\frac{p}{2})$
$\displaystyle\qquad\qquad\times\overline{\mathbb{K}}^{-}_{0,\overline{0}}(u,-\frac{p}{2})\mathbb{R}_{1,2|0,\overline{0}}(u,-\frac{p}{2},-\frac{p}{2})\cdots\mathbb{R}_{2L-1,2L|0,\overline{0}}(u,-\frac{p}{2},-\frac{p}{2})\bigg{)},$

with

$\displaystyle\overline{\mathbb{K}}^{-}_{i,j}\left(u,-\frac{p}{2}\right)$
$\displaystyle=G^{-1}_{i}K^{-}_{i}\left(u+\frac{p}{2}\right)R_{j,i}\left(2u+\frac{p}{2}\right)K^{-}_{j}(u)R_{i,j}\left(-\frac{p}{2}\right)\,,$
(41a)
$\displaystyle\overline{\mathbb{K}}^{+}_{i,j}\left(u,-\frac{p}{2}\right)$
$\displaystyle=\frac{1}{\xi\left(2u+\frac{p}{2}+\eta\right)}R_{j,i}\left(\frac{p}{2}\right)K^{+}_{j}(u)M_{i}R_{i,j}\left(-2u-\frac{p}{2}-2\eta\right)M^{-1}_{i}K^{+}_{i}\left(u+\frac{p}{2}\right)\,G_{i}\,.$
(41b)

Using the monodromy matrix (35) for the composite $\mathbb{R}$-matrices the
transfer matrix for the quasi periodic staggering (37) can be written as

$\displaystyle\mathcal{T}^{\text{qp}}(u,-\frac{p}{2})\equiv$
$\displaystyle\mathcal{T}(u,\\{-\frac{p}{2},0,0,-\frac{p}{2}\\})$ (42)
$\displaystyle=$
$\displaystyle\text{tr}_{0\overline{0}}\bigg{(}\overline{\mathbb{K}}^{+}_{0,\overline{0}}(u,-\frac{p}{2})\mathbb{T}_{0,\overline{0}}(u,-\frac{p}{2},-\frac{p}{2})\overline{\mathbb{K}}^{-}_{0,\overline{0}}(u,-\frac{p}{2})\mathbb{T}^{-1}_{0,\overline{0}}(u,-\frac{p}{2},-\frac{p}{2})\bigg{)}$
$\displaystyle\times\xi^{-1}\left(\frac{p}{2}\right)\,.$

Note that the composite monodromy matrix $\mathbb{T}$ (35) enters in the
transfer matrices (36) and (42) with identical arguments for the particular
choice of $2\delta_{0}=-\frac{p}{2}$. Hence, the bulk of these models
coincides while the reflection matrices $\mathbb{K}$ (34) and
$\overline{\mathbb{K}}$ (41) correspond to different boundary
conditions.555This does not lead to different models in the case of periodic
boundary conditions where different choices of the horizontal staggering can
be related by a shift in the spectral parameter (see Figure 6). This has been
discussed recently in the context of a staggered six-vertex (or $A_{1}^{(1)}$)
model Robertson  _et al._ (2020b); Nepomechie and Retore (2021a); Li  _et al._
(2022): for $2\delta_{0}=-\frac{p}{2}$ the resulting composite model is a
vertex model based on the twisted affine Lie algebra $D_{2}^{(2)}$. The
$D_{2}^{(2)}$ boundary matrices corresponding to $\mathbb{K}$ and
$\overline{\mathbb{K}}$ were known previously Nepomechie and Pimenta (2018)
and can be factorized into objects of the six-vertex model subject to
$U_{q}[\mathfrak{sl}_{2}]$ boundary conditions.

$u+\delta_{1}$$u+\delta_{2}$$u\to
u-\delta_{1}$$u$$u+\delta_{2}-\delta_{1}$$u-\delta_{1}$$u-\delta_{2}$$u+\delta_{2}$$u+\delta_{1}$$u\to
u-\delta_{1}$$u-2\delta_{1}$$u-\delta_{2}-\delta_{1}$$u+\delta_{2}-\delta_{1}$$u$(a)(b)
Figure 6: In the periodic case (a) a shift in the spectral parameter allows to
adjust one inhomogeneity to zero e.g. $\delta_{1}=0$. This is not possible in
the open case (b) where each inhomogeneity appears twice with different signs.

###  III.4 Associated reflection algebras and composite $\mathbb{K}$-matrices

We are left to prove that the reflection matrices (34) and (41) are indeed
representations of a reflection algebra associated with the
$\mathbb{R}$-matrix. We define the following _generalized_ reflection
algebras:

$\displaystyle\mathbb{R}_{i,j|k,\ell}(u-v,\theta,\theta)$
$\displaystyle\mathbb{K}^{-}_{i,j}(u,\theta)\mathbb{R}_{k,\ell|i,j}(u+v,\theta,\theta)\mathbb{K}^{-}_{k,\ell}(v,\theta)$
(43a)
$\displaystyle=\mathbb{K}^{-}_{k,\ell}(v,\theta)\mathbb{R}_{i,j|k,\ell}(u+v,\theta,\theta)\mathbb{K}^{-}_{i,j}(u,\theta)\mathbb{R}_{k,\ell|i,j}(u-v,\theta,\theta)$
and
$\displaystyle\mathbb{R}_{i,j|k,\ell}(-u+v,-\theta,-\theta)\left(\mathbb{K}^{+}_{i,j}(u,\theta)\right)^{t_{i}t_{j}}\mathbb{M}^{-1}_{i,j}\mathbb{R}_{k,\ell|i,j}(-u-v-2\eta,-\theta,-\theta)\mathbb{M}_{i,j}\left(\mathbb{K}^{+}_{k,\ell}(v,\theta)\right)^{t_{k}t_{\ell}}=$
$\displaystyle\,\,\left(\mathbb{K}^{+}_{k,\ell}(v,\theta)\right)^{t_{k}t_{\ell}}\mathbb{M}_{i,j}\mathbb{R}_{i,j|k,\ell}(-u-v-2\eta,-\theta,-\theta)\mathbb{M}^{-1}_{i,j}\left(\mathbb{K}^{+}_{i,j}(u,\theta)\right)^{t_{i}t_{j}}\mathbb{R}_{k,\ell|i,j}(-u+v,-\theta,-\theta)\,.$
(43b)

Note that the sign of the free parameter $\theta$ in the arguments of the
composite $\mathbb{R}$-matrices differs between (43a) and (43b). One of the
main results is that for given $K^{\pm}$ satisfying Eqs. (8) the matrices
$\mathbb{K}$ (34) and $\overline{\mathbb{K}}$ (41) obey the equations (43)
with the composite $\mathbb{R}$-matrix $\mathbb{R}(u,\theta,\theta)$ for
$\theta=2\delta_{0}$, $\delta_{0}$ arbitrary, and $\theta=-\frac{p}{2}$,
respectively. The proof for (34b) is given in Appendix A. The one for (34a)
works along the same line, for (41) one needs to use multiple times the quasi
periodicity in addition.

Based on the reflection algebra it is straightforward to show the
commutativity of the transfer matrices for both alternating and quasi periodic
boundary conditions. i.e.

$\displaystyle\left[\mathcal{T}(u,\epsilon),\mathcal{T}(v,\epsilon)\right]=0\,,\qquad\epsilon=0,1\,.$

###  III.5 Boundary terms in the Hamiltonian

Above we have identified two types of staggering, (33) and (37), allowing for
the construction of a local bulk Hamiltonian from the corresponding transfer
matrix of the composite model. For a compact presentation we define

$\displaystyle\mathfrak{R}_{i,j|k,\ell}(u,\epsilon)=\begin{cases}\mathbb{R}_{i,j|k,\ell}(u,-\frac{p}{2},-\frac{p}{2})&\epsilon=0\\\
\mathbb{R}_{i,j|k,\ell}(u,2\delta_{0},2\delta_{0})&\epsilon=1\end{cases},\qquad\mathfrak{K}^{\pm}_{i,j}(u,\epsilon)=\begin{cases}\overline{\mathbb{K}}^{\pm}_{i,j}(u,-\frac{p}{2})&\epsilon=0\\\
\mathbb{K}^{\pm}_{i,j}(u,2\delta_{0})&\epsilon=1\end{cases}$ (44)

where $\epsilon=1$ ($0$) corresponds to the alternating (36) and the quasi
periodic staggering (42), respectively. Note that
$\mathfrak{K}^{-}(0,\epsilon)\propto\mathbf{1}$ by (9) for $\epsilon=1$ and by
(11) for $\epsilon=0$. Hence, we obtain a local 666It is noteworthy, that for
$K^{-}(0)\not\propto\mathbf{1}$ , inducing an alternating staggering is
sufficient to define a local Hamiltonian. See also Doikou (2000) for a similar
approach. Hamiltonian Sklyanin (1988) via (32) whose bulk contribution is
found to be

$\displaystyle
H^{\epsilon}_{bulk}=\frac{2}{\xi(\Delta)}\sum^{L-1}_{j=1}\mathbb{P}_{2j,2j-1|2j+2,2j+1}\mathfrak{R}^{\prime}_{2j,2j-1|2j+2,2j+1}(0,\epsilon)\,.$
(45)

Here and in the following the prime indicates the derivative with respect to
the first argument where we assume that all quantities are differentiable at
the corresponding points. The boundary contributions read

$\displaystyle H^{\epsilon}_{left}$
$\displaystyle=\frac{\text{tr}_{0\overline{0}}\bigg{(}\mathfrak{K}^{{}^{\prime}+}_{0,\overline{0}}(0,\epsilon)\bigg{)}}{\text{tr}_{0\overline{0}}\bigg{(}\mathfrak{K}^{+}_{0,\overline{0}}(0,\epsilon)\bigg{)}}+\frac{2\text{tr}_{0\overline{0}}\bigg{(}\mathfrak{K}^{+}_{0,\overline{0}}(0,\epsilon)\mathbb{P}_{0,\overline{0}|2L-1,2L}\mathfrak{R}^{\prime}_{0,\overline{0}|2L-1,2L}(0,\epsilon)\bigg{)}}{\text{tr}_{0\overline{0}}\bigg{(}\mathfrak{K}^{+}_{0,\overline{0}}(0,\epsilon)\bigg{)}\xi(\Delta)}\,,$
(46) $\displaystyle H^{\epsilon}_{right}$
$\displaystyle=\frac{\mathfrak{K}^{-^{\prime}}_{1,2}(0,\epsilon)}{\mathfrak{K}^{-}_{1,2}(0,\epsilon)}\,.$

Note that, to obtain the spectrum of the above Hamiltonians it is sufficient
to use the Bethe Ansatz for the _single_ double row transfer matrix $\tau(u)$
(12). Knowing the eigenvalue $\Lambda(u)$ of $\tau(u)$ the energies can be
calculated via equations (30) and (32) with operators replaced by their
eigenvalues.

##  IV Quasi momentum for Open Systems

As pointed out for the periodic case above, there exist two families of
conserved quantities for the staggered models considered in this paper: in
addition to the ones generated to the product of elementary transfer matrices
(30) (or Eqs. (36) and (42) for the two cases discussed above) one can
consider operators such as the quasi momentum generated from the _quotient_ of
elementary transfer matrices. For the staggered model with open boundary
conditions built from arbitrary elementary $R$\- and $K$-matrices we replace
(25) by

$\displaystyle\mathbb{Q}=\log\left[\frac{\tau(-\delta_{1},\\{\delta_{1},\delta_{2}\\})}{\tau(-\delta_{2},\\{\delta_{1},\delta_{2}\\})}\right]\,.$
(47)

To express this operator in the composite picture, we adopt the idea from the
periodic case: we look for a generating function built out of a _product_ of
transfer matrices giving (47) as the leading term.

###  IV.1 Alternating staggering

For the alternating staggering case the quasi momentum operator can be
directly related (up to an additive constant) to a single double row-transfer
matrix

$\displaystyle\mathbb{Q}^{\mathrm{alt}}=\log\bigg{(}\tau^{2}(-\delta_{0},\\{\delta_{0},-\delta_{0}\\})\bigg{)}.$
(48)

In this case the quasi momentum can be represented in the rotated geometry as
displayed in Figure 7.

$-\delta_{0}$$-2\delta_{0}$$-2\delta_{0}$$-2\delta_{0}$$-2\delta_{0}$$-2\delta_{0}$$-2\delta_{0}$$-2\delta_{0}$$-\delta_{0}$$2L$$2L-1$$2L-2$$2L-3$$2L-4$$3$$2$$1$
Figure 7: Graphical representation of the quasi momentum operator for
alternating staggering in the rotated geometry. One can see that the quasi
momentum operator is acting non-locally. The loops at the right and left
ending are due to the influences of the boundary matrices.

Starting from $\tau(u-\delta_{0},\\{\delta_{0},-\delta_{0}\\})^{2}$ and
repeating the steps in Section III.1 to reach (31) and then the manipulations
(i)-(ii) in Section III.2 we obtain another generating functional for the
quasi momentum operator:

$\displaystyle
Q(u)=\text{tr}_{0\overline{0}}\bigg{(}\mathcal{K}^{+}_{0,\overline{0}}(u,2\delta_{0})\mathbb{T}_{0,\overline{0}}(u,0,2\delta_{0})\mathcal{K}^{-}_{0,\overline{0}}(u,2\delta_{0})\mathbb{T}^{-1}_{0,\overline{0}}(-(u-2\delta_{0}),0,2\delta_{0})\bigg{)}\,,$
(49)

where the $\mathcal{K}$-matrices

$\displaystyle\mathcal{K}^{-}_{i,j}(u,2\delta_{0})$
$\displaystyle=P_{i,j}K^{-}_{j}(u-\delta_{0})R_{i,j}(2u-2\delta_{0})K^{-}_{i}(u-\delta_{0})\,,$
(50a) $\displaystyle\mathcal{K}^{+}_{i,j}(u,2\delta_{0})$
$\displaystyle=\frac{1}{\xi(2u-2\delta_{0}+\eta)}P_{i,j}K^{+}_{j}(u-\delta_{0})M_{i}R_{i,j}(-2u+2\delta_{0}-2\eta)M^{-1}_{i}K^{+}_{i}(u-\delta_{0})\,,$
(50b)

obey the reflection algebras

$\displaystyle\mathbb{R}_{i,j|k,\ell}$
$\displaystyle(u-v,0,0)\mathcal{K}^{-}_{i,j}(u,2\delta_{0})\mathbb{R}_{k,\ell|i,j}(u+v-2\delta_{0},0,0)\mathcal{K}^{-}_{k,\ell}(v,2\delta_{0})=$
(51a)
$\displaystyle\mathcal{K}^{-}_{k,\ell}(v,2\delta_{0})\mathbb{R}_{i,j|k,\ell}(u+v-2\delta_{0},0,0)\mathcal{K}^{-}_{i,j}(u,2\delta_{0})\mathbb{R}_{k,\ell|i,j}(u-v,0,0)$
and
$\displaystyle\mathbb{R}_{i,j|k,\ell}(-u+v,0,0)\left(\mathcal{K}^{+}_{i,j}(u,2\delta_{0})\right)^{t_{i}t_{j}}\mathbb{M}^{-1}_{i,j}\mathbb{R}_{k,\ell|i,j}(-u-v+2\delta_{0}-2\eta,0,0)\mathbb{M}_{i,j}\left(\mathcal{K}^{+}_{k,\ell}(v,2\delta_{0})\right)^{t_{k}t_{\ell}}=$
(51b)
$\displaystyle\,\,\,\left(\mathcal{K}^{+}_{k,\ell}(v,2\delta_{0})\right)^{t_{k}t_{\ell}}\mathbb{M}_{i,j}\mathbb{R}_{i,j|k,\ell}(-u-v+2\delta_{0}-2\eta,0,0)\mathbb{M}^{-1}_{i,j}\left(\mathcal{K}^{+}_{i,j}(u,2\delta_{0})\right)^{t_{i}t_{j}}\mathbb{R}_{k,\ell|i,j}(-u+v,0,0)\,,$

respectively. Again, the proof is analogous to the one shown in Appendix A for
$\mathbb{K}^{+}$. Reflection algebras are of this type were introduced by
Nepomechie and Retore Nepomechie and Retore (2021b, a): they describe a moving
boundary where reflection of a particle at the boundary not only changes the
sign of its rapidity but also leads to the shift by $2\delta_{0}$ appearing in
the argument of $\mathbb{R}$-matrix containing the sum of $u+v$.

These reflection algebras together with the generalized YBE (18) ensure the
commutativity of the $Q$ with itself for different arguments. Finally, we need
to prove the commutativity with the transfer matrix (36) in the composite
picture. For the open chain with alternating staggering this is not obvious
because the boundary matrices $\mathbb{K}^{\pm}$ and $\mathcal{K}^{\pm}$ are
representations of _different_ reflection algebras. Remarkably it turns out
that they are intertwined by the following relations

$\displaystyle\mathbb{R}_{i,j|k,\ell}$
$\displaystyle(u-v,0,-2\delta_{0})\mathcal{K}^{-}_{i,j}(u,2\delta_{0})\mathbb{R}_{k,\ell|i,j}(u+v-2\delta_{0},2\delta_{0},0)\mathbb{K}^{-}_{k,\ell}(v,2\delta_{0})=$
(52a)
$\displaystyle\mathbb{K}^{-}_{k,\ell}(v,2\delta_{0})\mathbb{R}_{i,j|k,\ell}(u+v,0,-2\delta_{0})\mathcal{K}^{-}_{i,j}(u,2\delta_{0})\mathbb{R}_{k,\ell|i,j}(u-v-2\delta_{0},2\delta_{0},0)$
and
$\displaystyle\mathbb{R}_{i,j|k,\ell}(-u+v,0,2\delta_{0})\left(\mathcal{K}^{+}_{i,j}(u,2\delta_{0})\right)^{t_{i}t_{j}}\mathbb{M}^{-1}_{i,j}\mathbb{R}_{k,\ell|i,j}(-u-v+2\delta_{0}-2\eta,-2\delta_{0},0)$
(52b)
$\displaystyle\times\mathbb{M}_{i,j}\left(\mathbb{K}_{k,\ell}^{+}(v,2\delta_{0})\right)^{t_{k}t_{\ell}}=\left(\mathbb{K}^{+}_{k,\ell}(v,2\delta_{0})\right)^{t_{k}t_{\ell}}\mathbb{M}_{i,j}\mathbb{R}_{i,j|k,\ell}(-u-v-2\eta,0,2\delta_{0})$
$\displaystyle\times\mathbb{M}^{-1}_{i,j}\left(\mathcal{K}^{+}_{i,j}(u,2\delta_{0})\right)^{t_{i}t_{j}}\mathbb{R}_{k,\ell|i,j}(-u+v+2\delta_{0},-2\delta_{0},0)\,.$

Again this can be proven as in Appendix A. Using these algebras one can show
on the composite level that

$\displaystyle
Q(u)\mathcal{T}^{\text{alt}}(v,2\delta_{0})=\mathcal{T}^{\text{alt}}(v,2\delta_{0})Q(u)\,.$
(53)

We want to stress that the intertwining relations (52) ensure the
commutativity of transfer matrices with _different_ boundary matrices. It
would interesting to address whether similar relations between already other
known boundary matrices exists.

###  IV.2 Quasi periodic staggering

For the quasi periodic staggering the single ingredients
$\tau(-\frac{p}{2},\\{0,\frac{p}{2}\\})$ and $\tau(0,\\{0,\frac{p}{2}\\})$
become trivial in the bulk, see Figure 8.

$\frac{p}{2}$$\frac{p}{2}$$\frac{p}{2}$$-\frac{p}{2}$$-\frac{p}{2}$$\frac{p}{2}$$-\frac{p}{2}$$\frac{p}{2}$$2L$$2L-1$$2L-2$$2L-3$$2$$1$$-\frac{p}{2}$$-\frac{p}{2}$$0$$\frac{p}{2}$$\frac{p}{2}$$-\frac{p}{2}$$\frac{p}{2}$$0$$2L$$2L-1$$2L-2$$2L-3$$2$$1$
Figure 8: The two double row transfer matrices defining the quasi momentum as
in (47) for quasi periodic staggering evaluated at the shift point. We see
that both at their own become trivial in the bulk and if one assumes that
$K^{-}(0),K^{-}\left(\frac{p}{2}\right)\propto 1$ then the whole transfer
matrices become essentially the identity leading to a trivial quasi momentum.

Under the assumptions (11) the quasi momentum operator is trivial

$\displaystyle\mathbb{Q}^{\mathrm{qp}}\propto\mathbf{1}.$ (54)

Instead of constructing a generating functional in the composite picture for
this trivial quasi momentum we consider the next to leading term in the
expansion of the corresponding quasi shift operator, i.e.

$\displaystyle\overline{\mathbb{Q}}^{\mathrm{qp}}\propto\left.\frac{\text{d}}{\text{d}u}\frac{\tau(u,\\{0,\frac{p}{2}\\})}{\tau(u-\frac{p}{2},\\{0,\frac{p}{2}\\})}\right|_{u=0}\,,$
(55)

where we assume that $\tau$ is differentiable at $u=0,-p/2$. Below we study
the properties of $\overline{\mathbb{Q}}^{\mathrm{qp}}$ in the special case of
the staggered six-vertex model with quasi periodic staggering in the following
chapter.

##  V Example: The staggered $A^{(1)}_{1}$ (or $D^{(2)}_{2}$) model

In this last section we want to apply our findings to the staggered six-vertex
model with elementary $R$-matrix

$\displaystyle
R(u)=\begin{pmatrix}\sinh{(u+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}&0&0&0\\\
0&\sinh{(u)}&\sinh{(\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}&0\\\
0&\sinh{(\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}&\sinh{(u)}&0\\\
0&0&0&\sinh{(u+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}\\\
\end{pmatrix}\,.$ (56)

Up to different normalizations of the regularity and crossing symmetry this
$R$-matrix obeys the characteristic Eqs. (1), (2) and (3) with

$\displaystyle M$
$\displaystyle=\mathbf{1},\qquad\eta=\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma\,,\qquad
f_{p}=-1\,,\qquad V=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\,,$ (57)
$\displaystyle p$
$\displaystyle=\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\,,\qquad
G=\sigma^{z}\,,\qquad\xi(u)=\frac{1}{2}(\cos(2\gamma)-\cosh(2u))\,.$

Moreover, we restrict ourselves to the case of
$U_{q}(\mathfrak{sl}(2))$-invariant boundary conditions

$\displaystyle K^{-}(u)=\begin{pmatrix}e^{u}&0\\\ 0&e^{-u}\end{pmatrix},\qquad
K^{+}(u)=\left(K^{-}(-u-\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\right)^{t}.$
(58)

The double row transfer matrix (12) can be diagonalized by means of the
algebraic Bethe Ansatz Kulish and Sklyanin (1991). For the
$\mathbb{Z}_{2}$-staggering (15) its eigenvalues are given as

$\displaystyle\Lambda(u)=$
$\displaystyle\frac{\sinh(2u+2\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}{\sinh(2u+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}\bigg{(}\sinh(u-\delta_{1}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\sinh(u-\delta_{2}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\sinh(u+\delta_{1}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\sinh(u+\delta_{2}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\bigg{)}^{L}$
(59)
$\displaystyle\quad\times\frac{1}{q\det(T(-u-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2}))}\prod^{M}_{m=1}\frac{\sinh(u-v_{m}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})\sinh(u+v_{m}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}{\sinh(u-v_{m}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})\sinh(u+v_{m}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}$
$\displaystyle+\frac{\sinh(2u)}{\sinh(2u+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}\bigg{(}\sinh(u+\delta_{1})\sinh(u+\delta_{2})\sinh(u-\delta_{1})\sinh(u-\delta_{2})\bigg{)}^{L}$
$\displaystyle\quad\times\frac{1}{q\det(T(-u-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2}))}\prod^{M}_{m=1}\frac{\sinh(u-v_{m}+\frac{3\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})\sinh(u+v_{m}+\frac{3\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}{\sinh(u-v_{m}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})\sinh(u+v_{m}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}\,$

in terms of the parameters $v_{m}$, $m=1\dots M$, solving the Bethe equations

$\displaystyle\bigg{(}\frac{\sinh(v_{m}-\delta_{1}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}{\sinh(v_{m}+\delta_{1}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}$
$\displaystyle\frac{\sinh(v_{m}-\delta_{2}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}{\sinh(v_{m}+\delta_{2}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}\frac{\sinh(v_{m}+\delta_{1}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}{\sinh(v_{m}-\delta_{1}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}\frac{\sinh(v_{m}+\delta_{2}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}{\sinh(v_{m}-\delta_{2}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2})}\bigg{)}^{L}$
(60) $\displaystyle=\prod^{M}_{k=1\neq m}$
$\displaystyle\frac{\sinh(v_{m}-v_{k}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\sinh(v_{m}+v_{k}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}{\sinh(v_{m}-v_{k}-\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\sinh(v_{m}+v_{k}-\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}\,.$

In (59) the quantum determinant reads

$\displaystyle q\det(T(u))=$
$\displaystyle\sinh^{L}\left(u+\delta_{1}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2}\right)\sinh^{L}\left(u+\delta_{1}+\frac{3\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2}\right)$
(61)
$\displaystyle\times\sinh^{L}\left(u+\delta_{2}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2}\right)\sinh^{L}\left(u+\delta_{2}+\frac{3\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2}\right)\,.$

From our discussion above we know that only the alternating or quasi periodic
staggering leads to a local Hamiltonian. Both cases have been studied
extensively in Robertson  _et al._ (2021); Frahm and Gehrmann (2022) and
Robertson  _et al._ (2020a) respectively. Using the vertex representation of
the Temperley-Lieb generators $e_{i,i+1}$:

$\displaystyle e_{j,j+1}=\left(\mathds{1}_{\mathbb{C}^{2}}\right)^{\otimes
j-1}\otimes\begin{pmatrix}0&0&0&0\\\
0&-e^{-\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}&1&0\\\
0&1&-e^{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}&0\\\
0&0&0&0\end{pmatrix}\otimes\left(\mathds{1}_{\mathbb{C}^{2}}\right)^{\otimes
2L-j-1}\,,$ (62)

obeying

$\displaystyle e_{j,j+1}^{2}$ $\displaystyle=-2\cos(\gamma)e_{j,j+1}\,,$ (63)
$\displaystyle e_{j,j+1}e_{j+1,j+2}e_{j,j+1}$ $\displaystyle=e_{j,j+1}\,,$
$\displaystyle e_{j+1,j+2}e_{j,j+1}e_{j+1,j+2}$ $\displaystyle=e_{j+1,j+2}\,,$
$\displaystyle e_{k,k+1}e_{j,j+1}$
$\displaystyle=e_{j,j+1}e_{k,k+1}\,,\qquad|k-j|>1\,.$

the Hamiltonian with alternating staggering (33) can be written as

$\displaystyle H^{alt}(2\delta_{0})=$
$\displaystyle-\frac{1}{\sin(\gamma)\xi(2\delta_{0})}\Bigg{(}\sum^{2L-1}_{j=1}2\xi(2\delta_{0})e_{j,j+1}-\sinh(2\delta_{0})\sum^{2L-1}_{j=2}\sinh(2\delta_{0}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(-1)^{j+1}\gamma)e_{j,j+1}e_{j-1,j}$
$\displaystyle\qquad\qquad\qquad\qquad-\sinh(2\delta_{0})\sum^{2L-1}_{j=2}\sinh(2\delta_{0}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(-1)^{j}\gamma)e_{j-1,j}e_{j,j+1}\Bigg{)}\,.$

As discussed in Section III.3 the model where the free staggering parameter is
related to the quasi period as
$2\delta_{0}=\frac{p}{2}=\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{2}$
deserves special attention. In this case the Hamiltonian of the staggered
model becomes

$\displaystyle
H^{alt}\left(\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{2}\right)=$
$\displaystyle-\frac{2}{\sin(2\gamma)}\left(2\cos(\gamma)\sum^{2L-1}_{j=1}e_{j,j+1}+\sum^{2L-1}_{j=2}e_{j,j+1}e_{j-1,j}+e_{j-1,j}e_{j,j+1}\right)\,.$
(64)

This expression coincides with the Hamiltonian of the model with quasi
periodic staggering (37) up to boundary terms

$\displaystyle H^{qper}=$ $\displaystyle
H^{alt}\left(\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{2}\right)+\frac{2}{\sin(2\gamma)\cos(\gamma)}\left(e_{1,2}+e_{2L-1,2L}\right)\,.$
(65)

As mentioned earlier, the staggered model for
$2\delta_{0}=\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{2}$
has an extended underlying $D_{2}^{(2)}$ symmetry. In this context the
boundary matrices (34), (41) are different representations Martins and Guan
(2000); Nepomechie  _et al._ (2017); Nepomechie and Pimenta (2018) of the
corresponding $D^{(2)}_{2}$ reflection algebra.

###  V.1 Spectral flow between the integrable points

Remarkably, the choice of boundary conditions has a profound influence on the
low energy properties of the staggered models: the effective theory of (64)
with anisotropy $\gamma<2\delta_{0}<\pi-\gamma$ has been identified to be the
$SL(2,\mathbb{R})/U(1)$ sigma model at level $k=\pi/\gamma$ with a non-compact
spectrum of conformal weights Robertson  _et al._ (2021); Frahm and Gehrmann
(2022). On the contrary, Robertson _et al._ found that the continuum limit of
the model (65) is compact Robertson  _et al._ (2020b). The boundary RG flow
between these two critical fixed points has been studied numerically: based on
finite size estimates of the gap between the ground state and the lowest
excitation when the amplitude of the boundary term in (65) is varied between
the two integrable points it has been concluded that the fixed points
corresponding to (64) and (65) are unstable and stable, respectively Robertson
_et al._ (2021).

In the setting established in this paper both (64) and (65) originate from a
staggered vertex model with the $A_{1}^{(1)}$ $R$-matrix (56). This allows to
study the spectral flow between (64) and (65) in an integrable setting with a
fixed choice of the boundary matrices under the variation of the bulk
inhomogeneities. The price to pay for integrability is giving up locality of
the Hamiltonian at the intermediate points. Using the same staggering in the
vertical and horizontal directions of the vertex model, i.e.
$\\{\delta_{0},\delta_{\overline{0}}\\}=\\{\delta_{1},\delta_{2}\\}$, we tune
$\delta_{1}$ and $\delta_{2}$ to interpolate between the integrable models
with local interactions. We choose the following normalization of the non-
local ’Hamiltonian’

$\displaystyle H=$
$\displaystyle\frac{q\det\left(T\left(-\delta_{1}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2}\right)\right)q\det\left(T\left(-\delta_{2}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}{2}\right)\right)}{2\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}f(\delta_{1})f(\delta_{2})}$
(66)
$\displaystyle\times\left(\left.\frac{\text{d}}{\text{d}u}\right|_{u=0}\mathcal{T}(u,\\{\delta_{1},\delta_{2},\delta_{1},\delta_{2}\\})-\left.\frac{\text{d}}{\text{d}u}\right|_{u=0}f(u+\delta_{1})f(u+\delta_{2})\right)$

with

$\displaystyle f(u)=$
$\displaystyle\frac{\sinh(2u+2\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}{\sinh(2u+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)}\sinh^{L}(u-\delta_{1}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\sinh^{L}(u-\delta_{2}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)$
(67)
$\displaystyle\times\sinh^{L}(u+\delta_{1}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\sinh^{L}(u+\delta_{2}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma)\,.$

Specifically, we choose for the remaining inhomogeneities the parameterization

$\displaystyle\delta_{1}=\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\vartheta}{2}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}\,,\qquad\delta_{2}=\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\vartheta}{2}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}\,,\qquad-\frac{\pi}{2}\leq\vartheta\leq
0\,,$ (68)

resulting in alternating (quasi periodic) staggering for $\vartheta=0$ and
$-\pi/2$, respectively.

The eigenvalues of the ’Hamiltonian’ (66) in this parameterization are given
in terms of the Bethe roots $\\{v_{m}\\}$ solving (60) as

$\displaystyle E=$
$\displaystyle\left(-4L\cot(2\gamma)+L\frac{2\sin(2\vartheta)}{\sin(2\gamma)\sin(2(\gamma+\vartheta))}-\frac{2\sin(\vartheta)}{\cos(\gamma)\cos(\gamma+\vartheta)}+\frac{2\sin(\vartheta)}{\cos(2\gamma)\cos(2\gamma+\vartheta)}\right.$
(69)
$\displaystyle\qquad\qquad\left.+\frac{2\tan(\gamma)}{\cos(2\gamma)}\right)\times\left(\prod^{M}_{m=1}\frac{\cos(2(\gamma-\vartheta))+\cosh(4v_{m})}{\cos(2(\gamma+\vartheta))+\cosh(4v_{m})}-1\right)$
$\displaystyle-4\sin(2\gamma)\sum^{M}_{k=1}\left\\{\frac{\cos(2\gamma)+\cos(\vartheta)\cosh(4v_{k})}{(\cos(2(\gamma+\vartheta))+\cosh(4v_{k}))^{2}}\right\\}\times\prod^{M}_{\begin{subarray}{c}m=1{}\\\
m\neq
k\end{subarray}}\frac{\cos(2(\gamma-\vartheta))+\cosh(4v_{m})}{\cos(2(\gamma+\vartheta))+\cosh(4v_{m})}\,.$

As expected, this expression reduces to a sum of bare quasi-particle energies
$\epsilon_{0}(v_{m})$ for $\vartheta=0,\pm\pi/2$ where the Hamiltonian becomes
local. Away from these points, the normalization of (69) leads to
singularities at particular values of the flow parameter: the one at
$\vartheta=\frac{\pi}{2}-2\gamma$ can be removed by multiplying the
Hamiltonian by the $\vartheta$-dependent factor $\cos(2\gamma+\vartheta)$
while $\vartheta=\frac{\pi}{2}-\gamma>0$ does not lie on the spectral flow
(68). The remaining singularity at $\vartheta_{c1}=-\gamma$ depends on the
state considered. In terms of the corresponding Bethe root configuration this
can be related to the low energy root configurations of the quasi periodic
model (65). These consists of pairs of complex conjugate roots with imaginary
part $\pm\frac{\pi}{4}$ and an additional root at
$\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}$
for $M^{\mathrm{qper}}$ odd Robertson  _et al._ (2020b):

$\displaystyle
v^{\mathrm{qper}}\in\left\\{x_{m}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4},x_{m}-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}\bigg{|}x_{m}\in\mathbb{R}_{>0}\,,m=1,...,\left\lfloor\frac{M^{\mathrm{qper}}}{2}\right\rfloor\right\\}\cup\left\\{\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}\right\\}\,.$
(70)

Here $\lfloor\dots\rfloor$ denotes the Gaussian bracket. Exact diagonalization
of the Hamiltonian for small systems together with the determination of the
corresponding Bethe roots shows that root patterns of this type persist
throughout the interval $\vartheta=-\pi/2\dots\vartheta_{c1}$. At
$\vartheta_{c1}$, however, several roots become purely imaginary,
$v_{m}=\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi/4$,
changing the order of the pole in (69). This singularity can be removed by
renormalization of the spectrum by a factor $\sin^{\nu}(\vartheta+\gamma)$
with an appropriate choice of an integer $\nu$.

The spectral flow starting from the alternating model (64), $\vartheta=0$, can
be studied in a similar way: here a class of low energy states (including the
ground state) is known Frahm and Gehrmann (2022); Robertson  _et al._ (2021)
to be described by configurations consisting of real roots and ones having an
imaginary part of $\frac{\pi}{2}$:

$\displaystyle
v^{\mathrm{alt}}=\left\\{x_{m},y_{n}+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{2}\bigg{|}x_{m},y_{n}\in\mathbb{R}_{>0},\,m=1,...,M_{0},\,n=1,...,M_{\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{2}}\right\\}\,.$
(71)

Configurations of this type exist in the interval
$\vartheta=\vartheta_{c2}\dots 0$ where roots with vanishing real parts appear
for $\vartheta_{c2}=\gamma-\pi/2$. These do not, however, lead to
singularities in the eigenvalues of (69).

Both at $\vartheta_{c1}$ and $\vartheta_{c2}$ the appearance of purely
imaginary Bethe roots leads to degeneracies involving many states, as shown in
results from exact diagonalization for a spin chain with $2L=8$ sites,
anisotropy $\gamma=0.9$ for the charge sector $S_{z}=2$ shown in Figure 9.
Under the spectral flow low energy states of the local Hamiltonian (64) are
mapped to high energy ones for (65) and vice versa. The crossing of a large
number of levels indicates the presence of first-order transitions when the
flow parameter is $\vartheta_{c1}$ or $\vartheta_{c2}$.

Figure 9: Rescaled real parts $\tilde{E}$ of the eigenenergies of the
staggered six-vertex Hamiltonian with $2L=8$ sites, anisotropy $\gamma=0.9$
for the charge sector $S_{z}=2$ for the spectral flow (68). Energies have been
multiplied with $-\cos(\vartheta+2\gamma)\sin^{5}(\vartheta+\gamma)$ to
regularize the singularities as described in the main text. The mapping
between low and high energy states in the local models is clearly seen. In the
lower plots the level crossings at $\vartheta=\vartheta_{c1},\vartheta_{c2}$
of low lying states evolving from the respective ground states are resolved
within the $\vartheta$-intervals indicated in the upper image details.

Support for this interpretation is obtained by studying the spectral flow
within the root density formalism Yang and Yang (1969): in the thermodynamic
limit the densities of roots in the configuration (71) of the alternating
model are found to be

$\displaystyle\rho^{a}(v)=\sigma^{a}(v)+\frac{1}{L}\tau^{a}(v)\,,\quad
a=x,y\,,$ (72)

with bulk and surface contributions

$\displaystyle\sigma^{x}(v)=\sigma^{y}(v)=$
$\displaystyle\frac{\cos\left(\frac{\pi\,\vartheta}{\pi-2\gamma}\right)}{\pi-2\gamma}\left[\frac{1}{\cosh\left(\frac{2\pi
v}{\pi-2\gamma}\right)+\sin\left(\frac{\pi\,\vartheta}{\pi-2\gamma}\right)}+\frac{1}{\cosh\left(\frac{2\pi
v}{\pi-2\gamma}\right)-\sin\left(\frac{\pi\,\vartheta}{\pi-2\gamma}\right)}\right]\,,$
$\displaystyle\tau^{x}(v)=\tau^{y}(v)=$
$\displaystyle\frac{1}{4\pi}\int^{\infty}_{-\infty}\text{d}\omega
e^{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\omega
v}\frac{\sinh\left(\frac{3\gamma-\pi}{4}\omega\right)}{\sinh\left(\frac{\gamma\omega}{4}\right)\cosh\left(\frac{2\gamma-\pi}{4}\omega\right)}\,.$

Similarly, the density $\bar{\rho}(x)$ of root configurations (70) of the
quasi periodic model is found to be

$\displaystyle\bar{\sigma}(x)=$
$\displaystyle\frac{4}{\pi-2\gamma}\frac{\cos\left(\frac{\pi}{2}\frac{\pi+2\vartheta}{\pi-2\gamma}\right)\cosh\left(\frac{2\pi
x}{\pi-2\gamma}\right)}{\cosh\left(\frac{4\pi
x}{\pi-2\gamma}\right)+\cos\left(\pi\frac{\pi+2\vartheta}{\pi-2\gamma}\right)}\,,\quad\bar{\tau}(x)=\frac{1}{\pi-2\gamma}\frac{1}{\cosh\left(\frac{2\pi
x}{\pi-2\gamma}\right)}\,.$ (73)

That the bulk densities $\sigma^{x,y}(v)$ ($\bar{\sigma}(x)$) vanish at
$\vartheta_{c2}$ ($\vartheta_{c1}$) indicates a transition into a different
state in accordance with our results for small system sizes.

###  V.2 Role of the quasi momentum in the quasi periodic model

As mentioned above the continuum limit of the alternating model (64) is
described by a non-compact conformal field theory. In the lattice model this
is manifest in the finite size gaps closing as

$\Delta
E^{\mathrm{alt}}_{n}=E^{\mathrm{alt}}_{n}-Le^{\mathrm{alt}}_{\infty}-f^{\mathrm{alt}}_{\infty}\sim\frac{\pi
v^{\mathrm{alt}}_{F}}{L}\left(\text{const.}+\text{const.}\frac{(dN^{\mathrm{alt}})^{2}}{\log(L)^{2}}\right)\,,$
(74)

where $Le^{\mathrm{alt}}_{\infty}$ and $f^{\mathrm{alt}}_{\infty}$ are the
bulk and surface contributions to the energy and $v^{\mathrm{alt}}_{F}$ is the
Fermi velocity. On the level of the Bethe configurations (71) logarithmic
corrections arise when the numbers $M_{0}$
($M_{\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{2}}$)
of roots with $\text{Im}(v_{m}^{alt})=0$ ($\frac{\pi}{2}$) are different, i.e.
$dN^{\mathrm{alt}}=M_{0}-M_{\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{2}}\neq
0$. For large $L$ the eigenvalues of the quasi momentum operator are
proportional to $dN^{\mathrm{alt}}/\log(L)$, which allows a direct
identification of the underlying CFT Ikhlef  _et al._ (2012); Frahm and Seel
(2014); Frahm and Gehrmann (2022).

In this section, we reconsider the finite-size analysis of the quasi periodic
chain using the definition (55). Motivated by the insights gained in the
alternating model we consider Bethe configurations (70) with different numbers
$M_{\pm\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}}$
of roots on the lines $\text{Im}(v_{m}^{alt})=\pm\frac{\pi}{4}$. To
investigate the scaling behavior of those states, we consider the rescaled
energy gaps

$\displaystyle h^{n}_{\text{eff}}$ $\displaystyle=\frac{L}{\pi
v^{\mathrm{qp}}_{F}}\left(E^{\mathrm{qp}}_{n}-Le^{\mathrm{qp}}_{\infty}-f^{\mathrm{qp}}_{\infty}\right)\,.$
(75)

The energies $e^{\mathrm{qp}}_{\infty}$, $f^{\mathrm{qp}}_{\infty}$ and Fermi
velocity of the quasi periodic model are obtained in the root density
formalism using (73) with $\vartheta=-\frac{\pi}{2}$

$\displaystyle e^{\mathrm{qp}}_{\infty}=2f^{\mathrm{qp}}_{\infty}=$
$\displaystyle-\frac{1}{2}\int^{\infty}_{-\infty}\text{d}\omega\frac{\sinh\left(\frac{\gamma\omega}{2}\right)}{\sinh\left(\frac{\pi\omega}{4}\right)\cosh\left(\frac{1}{4}(\pi-2\gamma)\omega\right)}\,,\qquad
v^{\mathrm{qp}}_{F}=\frac{2\pi}{\pi-2\gamma}\,.$ (76)

Using the Bethe-Ansatz we have calculated $h^{n}_{\text{eff}}$ for states with
various
$dN^{\mathrm{qp}}=M_{-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}}-M_{\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}}$
as displayed in Fig. 10.

Figure 10: Scaling of energy gaps (75) of the quasi periodic chain in the
sector $S_{z}=7$ weighted by $1/\left(dN^{\mathrm{qp}}\right)^{2}$ obtained by
solving the Bethe Ansatz equations for various $dN^{\mathrm{qp}}$ and $L$ and
fixed $\gamma=0.9$. We see that the scaling dimensions of those state depend
on $\left(dN^{\mathrm{qp}}\right)^{2}$ and display a clear logarithmic
divergence as $L\to\infty$. While solving the Bethe equations numerically, we
found that the $dN^{\mathrm{qp}}$ needs to be smaller than $S_{z}$ for
numerical convergence.

Differing from the alternating model, we see that the $h^{n}_{\text{eff}}$
_diverge_ logarithmically with an amplitude proportional to
$\left(dN^{\mathrm{qp}}\right)^{2}$. We interpret these diverging scaling
dimensions in the quasi periodic case in the manner that states with
$dN^{\mathrm{qp}}\neq 0$ disappear from the low energy sector in the
thermodynamic limit. Only states with $dN^{\mathrm{qp}}=0$ stay in the low
energy regime as $L$ tends to infinity which have been extensively studied in
Robertson  _et al._ (2020b).

To parameterize this behavior in terms of the quasi momentum we consider (55)
(recall that the lowest order term in the expansion of the quasi shift
operator (54) is trivial for the quasi periodic model). For the staggered six-
vertex model this operator can be expressed in terms of Pauli matrices as

$\displaystyle\overline{\mathbb{Q}}^{\mathrm{qp}}=\bigg{\\{}$ $\displaystyle
2\cos(\gamma)\left(\sum_{j=1}^{2L-2}(\sigma^{-}_{j}\sigma^{+}_{j+1}\sigma^{z}_{j+2}-\sigma^{+}_{j}\sigma^{-}_{j+1}\sigma^{z}_{j+2}+\sigma^{z}_{j}\sigma^{-}_{j+1}\sigma^{+}_{j+2}-\sigma^{z}_{j}\sigma^{+}_{j+1}\sigma^{-}_{j+2})\right)$
(77)
$\displaystyle+\frac{\cos(\gamma)}{\sin(\gamma)}\left(\sum_{j=1}^{2L-2}(-1)^{j}(2(\sigma^{-}_{j+2}\sigma^{+}_{j}+\sigma^{+}_{j+2}\sigma^{-}_{j})+\sigma^{z}_{j}\sigma^{z}_{j+2})\right)$
$\displaystyle-2\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sin(\gamma)(\sigma^{+}_{1}\sigma^{-}_{2}-\sigma^{-}_{1}\sigma^{+}_{2})+2\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sin(\gamma)(\sigma^{-}_{2L}\sigma^{+}_{2L-1}-\sigma^{+}_{2L}\sigma^{-}_{2L-1})$
$\displaystyle-\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma^{z}_{1}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma^{z}_{2}+\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma^{z}_{2L-1}-\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sigma^{z}_{2L}\bigg{\\}}\frac{1}{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\cos^{2}(\gamma)}\,.$

Note this this a sum of local operators in contrast to the alternating case,
see Fig. 7. The normalization of in (77) is chosen such that its eigenvalue
$\overline{\mathcal{Q}}^{\mathrm{qp}}$ can be expressed in a simple form in
terms of the Bethe roots:

$\displaystyle\overline{\mathcal{Q}}^{\mathrm{qp}}=\sum_{j}q_{0}(v_{j})\quad\text{with}\quad
q_{0}(u)=\frac{16\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sin(\gamma)\cosh(2v_{j})}{\cosh(4u)-\cos(2\gamma)}\,.$
(78)

Note that $\overline{\mathcal{Q}}^{\mathrm{qp}}$ measures the difference of
the number of Bethe roots on the lines
$\pm\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4}$:

$\displaystyle
q_{0}(x+\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4})=-q_{0}(x-\frac{\mathrm{i}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi}{4})\,.$
(79)

This is similar to the role of (48) in the alternating case. In the present
case, however, the $\overline{\mathcal{Q}}^{\mathrm{qp}}\propto
dN^{\mathrm{qp}}$ as $L\to\infty$, see Fig. 11.

Figure 11: The quasi momentum of Bethe states having a non vanishing quantum
number $dN^{\mathrm{qp}}$.

Hence, $\overline{\mathcal{Q}}^{\mathrm{qp}}$ does not capture the
$L$-dependence observed in Fig. 10.

##  VI Conclusion

Starting from an elementary solution $R$ of the YBE (1) we have constructed
the composite $\mathbb{R}$-matrix (17). In addition to the spectral parameter
the composite matrix depends on two free parameters related to the staggering
of the elementary vertices (see Figure 2). $PT$-symmetry, unitarity,
regularity and crossing unitarity are inherited from the elementary
$R$-matrix. Most importantly, the composite $\mathbb{R}$-matrix obeys a
generalized YBE (18). In this picture the commuting transfer matrices of
arbitrary $\mathbb{Z}_{2}$-staggered models can be rewritten as homogeneous
ones where the staggering parameters enter through the additional arguments of
the composite $\mathbb{R}$-matrices (and reflection matrices in the case of
open boundary conditions). Integrals of motion (including the quasi momentum
operator) whose natural definition relies on the staggering of the model have
been described in the _homogeneous_ picture based on the composite
$\mathbb{R}$-matrix. In the case of open boundary conditions the Hamiltonian
and the quasi momentum are defined in terms of _different_ representations of
the reflection algebra intertwined by (52). This construction may provide
insights into a definition of this operator in homogeneous models (lacking a
known factorization of the transfer matrix). This is of particular interest
for such models featuring a continuous component of the conformal spectrum at
criticality Vernier  _et al._ (2014, 2016); Frahm and Martins (2018); Frahm
_et al._ (2019). Knowing the quasi momentum operator in these models is
expected to foster the identification of the CFT describing the low energy
regime.

Demanding locality in the Hamiltonian limit leads to constraints on the
staggering parameters: in the case of periodic boundary conditions they have
to be tuned to satisfy (22). For open boundary conditions the staggering has
to satisfy Eq. (33) for ’alternating staggering’. Moreover, for quasi periodic
$R$\- (and $\mathbb{R}$-) matrices the inequivalent choice of ’quasi periodic
staggering’ (37) leads to a different Hamiltonian with local interactions. For
both cases we have identified the corresponding composite boundary matrices
which generalizes the findings of the factorization Li  _et al._ (2022) for
the $D^{(2)}_{2}$ boundary matrices Martins and Guan (2000); Nepomechie  _et
al._ (2017); Nepomechie and Pimenta (2018) to arbitrary algebras.

Applying our construction to this model we have studied the spectral flow
between the alternating and the quasi periodic model. Following this flow
along a line of integrable models we find that the endpoints are separated by
two first-order transitions which is consistent with the different properties
of the corresponding spectra observed previously. The different role of the
quasi momentum operator in the alternating and quasi periodic model are
briefly discussed.

###### 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).

##  Appendix A Proof of the reflection algebra (43b) for $\mathbb{K}^{+}$

Using the definitions of the composite quantities in the LHS of the reflection
algebra (43b) we obtain (we omit the prefactor of $\mathbb{K}^{+}$ and set
$ijk\ell=1234$ for notational clarity):

$\displaystyle\mathbb{R}_{1,2|3,4}$
$\displaystyle(-u+v,-\theta,-\theta)\left(\mathbb{K}^{+}_{1,2}(u,\theta)\right)^{t_{1}t_{2}}\mathbb{M}^{-1}_{1,2}\mathbb{R}_{3,4|1,2}(-u-v-2\eta,-\theta,-\theta)\mathbb{M}_{1,2}\left(\mathbb{K}^{+}_{3,4}(v,\theta)\right)^{t_{3}t_{4}}$
$\displaystyle=R_{1,4}(-u+v-\theta)R_{1,3}(-u+v)R_{2,4}(-u+v)R_{2,3}(-u+v+\theta)$
$\displaystyle\times\left(P_{1,2}K^{+}_{2}(u-\frac{\theta}{2})M_{1}R_{1,2}(-2u-2\eta)M^{-1}_{1}K^{+}_{1}(u+\frac{\theta}{2})\right)^{t_{1}t_{2}}M^{-1}_{1}M^{-1}_{2}$
$\displaystyle\times
R_{3,2}(-u-v-2\eta-\theta)R_{3,1}(-u-v-2\eta)R_{4,2}(-u-v-2\eta)R_{4,1}(-u-v-2\eta+\theta)$
$\displaystyle\times
M_{1}M_{2}\left(P_{3,4}K^{+}_{4}(v-\frac{\theta}{2})M_{3}R_{3,4}(-2v-2\eta)M^{-1}_{3}K^{+}_{3}(v+\frac{\theta}{2})\right)^{t_{3}t_{4}}\,.$

Resolving the transpositions and reordering the permutation operators gives:

$\displaystyle\mathbb{R}_{1,2|3,4}(-u+v,-\theta,-\theta)\left(\mathbb{K}^{+}_{1,2}(u,\theta)\right)^{t_{1}t_{2}}\mathbb{M}^{-1}_{1,2}\mathbb{R}_{3,4|1,2}(-u-v-2\eta,-\theta,-\theta)\mathbb{M}_{1,2}\left(\mathbb{K}^{+}_{3,4}(v,\theta)\right)^{t_{3}t_{4}}$
$\displaystyle=P_{3,4}R_{1,3}(-u+v-\theta)R_{1,4}(-u+v)R_{2,3}(-u+v)R_{2,4}(-u+v+\theta)$
$\displaystyle\times\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}M^{-1}_{1}R_{2,1}(-2u-2\eta)M_{1}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}M^{-1}_{1}M^{-1}_{2}$
$\displaystyle\times
R_{4,1}(-u-v-2\eta-\theta)R_{4,2}(-u-v-2\eta)R_{3,1}(-u-v-2\eta)R_{3,2}(-u-v-2\eta+\theta)$
$\displaystyle\times
M_{1}M_{2}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}M^{-1}_{4}R_{3,4}(-2v-2\eta)M_{4}\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}P_{1,2}\,.$

We present from now on also the graphical proof for maximal clarity:
$2$$1$$4$$3$$-u+v-\theta$$-u+v$$-u+v$$-u+v+\theta$$-u-v-2\eta$$-u-v-2\eta+\theta$$-u-v-2\eta$$-u-v-2\eta-\theta$$-2u-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$$-2v-2\eta$
$\displaystyle=P_{3,4}R_{1,3}(-u+v-\theta)R_{1,4}(-u+v)R_{2,3}(-u+v)$
$\displaystyle\times
R_{2,4}(-u+v+\theta)\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}M^{-1}_{1}$
$\displaystyle\times
R_{2,1}(-2u-2\eta)M_{1}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}$
$\displaystyle\times M^{-1}_{1}M^{-1}_{2}R_{4,1}(-u-v-2\eta-\theta)$
$\displaystyle\times R_{4,2}(-u-v-2\eta)R_{3,1}(-u-v-2\eta)$
$\displaystyle\times
R_{3,2}(-u-v-2\eta+\theta)M_{1}M_{2}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}$
$\displaystyle\times
M^{-1}_{4}R_{3,4}(-2v-2\eta)M_{4}\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}P_{1,2}\,.$
Canceling the operator insertions gives:
$2$$1$$4$$3$$-u+v-\theta$$-u+v$$-u+v$$-u+v+\theta$$-u-v-2\eta$$-u-v-2\eta+\theta$$-u-v-2\eta$$-u-v-2\eta-\theta$$-2u-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$$-2v-2\eta$
$\displaystyle=P_{3,4}R_{1,3}(-u+v-\theta)R_{1,4}(-u+v)R_{2,3}(-u+v)$
$\displaystyle\times
R_{2,4}(-u+v+\theta)\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}M^{-1}_{1}$
$\displaystyle\times
R_{2,1}(-2u-2\eta)\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}$
$\displaystyle\times M^{-1}_{2}R_{4,1}(-u-v-2\eta-\theta)$
$\displaystyle\times R_{4,2}(-u-v-2\eta)R_{3,1}(-u-v-2\eta)$
$\displaystyle\times
R_{3,2}(-u-v-2\eta+\theta)M_{1}M_{2}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}$
$\displaystyle\times
M^{-1}_{4}R_{3,4}(-2v-2\eta)M_{4}\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}P_{1,2}\,.$
Using the YBE to pass the weight $-2u-2\eta$ to the right side gives:
$2$$1$$4$$3$$-u+v-\theta$$-u-v-2\eta$$-u+v$$-u-v-2\eta-\theta$$-u+v$$-u-v-2\eta+\theta$$-u-v-2\eta$$-u+v+\theta$$-2u-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$$-2v-2\eta$
$\displaystyle=P_{3,4}R_{1,3}(-u+v-\theta)R_{1,4}(-u+v)\left(K^{+}_{1}(u+\frac{\theta}{2})\right)^{t_{1}}$
$\displaystyle\times M^{-1}_{1}R_{4,1}(-u-v-2\eta-\theta)R_{3,1}(-u-v-2\eta)$
$\displaystyle\times R_{2,1}(-2u-2\eta)R_{2,3}(-u+v)R_{2,4}(-u+v+\theta)$
$\displaystyle\times\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}M^{-1}_{2}R_{4,2}(-u-v-2\eta)$
$\displaystyle\times
R_{3,2}(-u-v-2\eta+\theta)M_{1}M_{2}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}$
$\displaystyle\times
M^{-1}_{4}R_{3,4}(-2v-2\eta)M_{4}\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}P_{1,2}\,.$
Using the reflection algebra (8b) move the $K^{+}$-matrix with weight
$v+\theta/2$ upwards we obtain:
$2$$1$$4$$3$$-u+v-\theta$$-u-v-2\eta$$-u+v$$-u-v-2\eta-\theta$$-u+v$$-u-v-2\eta+\theta$$-u-v-2\eta$$-u+v+\theta$$-2u-2\eta$$-2v-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$
$\displaystyle=P_{3,4}R_{1,3}(-u+v-\theta)R_{1,4}(-u+v)\left(K^{+}_{1}(u+\frac{\theta}{2})\right)^{t_{1}}$
$\displaystyle\times M^{-1}_{1}R_{4,1}(-u-v-2\eta-\theta)R_{3,1}(-u-v-2\eta)$
$\displaystyle\times
R_{2,1}(-2u-2\eta)R_{2,3}(-u+v)\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}$
$\displaystyle\times
M_{2}R_{2,4}(-u-v-2\eta)M^{-1}_{2}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}$
$\displaystyle\times
R_{4,2}(-u+v+\theta)M^{-1}_{2}R_{3,2}(-u-v-2\eta+\theta)M_{1}$
$\displaystyle\times
M_{2}M^{-1}_{4}R_{3,4}(-2v-2\eta)M_{4}\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}P_{1,2}\,.$
Using (8b) again this yields:
$2$$1$$4$$3$$-u+v-\theta$$-u-v-2\eta$$-u-v-2\eta-\theta$$-u+v$$-u+v$$-u-v-2\eta+\theta$$-u-v-2\eta$$-u+v+\theta$$-2u-2\eta$$-2v-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$
$\displaystyle=P_{3,4}R_{1,3}(-u+v-\theta)\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}M_{1}$
$\displaystyle\times
R_{1,4}(-u-v-2\eta-\theta)M^{-1}_{1}\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}$
$\displaystyle\times R_{4,1}(-u+v)M^{-1}_{1}R_{3,1}(-u-v-2\eta)$
$\displaystyle\times R_{2,1}(-2u-2\eta)R_{2,3}(-u+v)$ $\displaystyle\times
M_{2}R_{2,4}(-u-v-2\eta)M^{-1}_{2}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}$
$\displaystyle\times
R_{4,2}(-u+v+\theta)M^{-1}_{2}R_{3,2}(-u-v-2\eta+\theta)M_{1}$
$\displaystyle\times
M_{2}M^{-1}_{4}R_{3,4}(-2v-2\eta)M_{4}\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}P_{1,2}\,.$
Reshuffling the operator insertions via (2f) we get
$2$$1$$4$$3$$-u+v-\theta$$-u-v-2\eta$$-u-v-2\eta-\theta$$-u+v$$-u+v$$-u-v-2\eta+\theta$$-u-v-2\eta$$-u+v+\theta$$-2u-2\eta$$-2v-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$
$\displaystyle=P_{3,4}R_{1,3}(-u+v-\theta)\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}M^{-1}_{4}$
$\displaystyle\times
R_{1,4}(-u-v-2\eta-\theta)\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}M^{-1}_{1}$
$\displaystyle\times R_{4,1}(-u+v)R_{3,1}(-u-v-2\eta)R_{2,1}(-2u-2\eta)$
$\displaystyle\times
R_{2,3}(-u+v)R_{2,4}(-u-v-2\eta)\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}$
$\displaystyle\times M^{-1}_{2}R_{4,2}(-u+v+\theta)R_{3,2}(-u-v-2\eta+\theta)$
$\displaystyle\times
M_{1}M_{2}R_{3,4}(-2v-2\eta)M_{4}\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}P_{1,2}\,.$
By using the YBE to bring the weight $-2v-2\eta$ to the top we obtain:
$2$$1$$3$$4$$-u-v-2\eta-\theta$$-u+v$$-u+v-\theta$$-u-v-2\eta$$-u-v-2\eta$$-u+v+\theta$$-u+v$$-u-v-2\eta+\theta$$-2u-2\eta$$-2v-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$
$\displaystyle=P_{3,4}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}M^{-1}_{4}R_{3,4}(-2v-2\eta)$
$\displaystyle\times R_{1,4}(-u-v-2\eta-\theta)R_{1,3}(-u+v-\theta)$
$\displaystyle\times\left(K^{+}_{1}(u+\frac{\theta}{2})\right)^{t_{1}}M^{-1}_{1}R_{3,1}(-u-v-2\eta)R_{4,1}(-u+v)$
$\displaystyle\times R_{2,1}(-2u-2\eta)R_{2,4}(-u-v-2\eta)R_{2,3}(-u+v)$
$\displaystyle\times\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}M^{-1}_{2}R_{3,2}(-u-v-2\eta+\theta)$
$\displaystyle\times
R_{4,2}(-u+v+\theta)M_{1}M_{2}M_{4}\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}P_{1,2}\,.$
Similar as above we use the reflection algebra (8b) twice to get
$2$$1$$3$$4$$-u-v-2\eta-\theta$$-u+v$$-u-v-2\eta$$-u+v-\theta$$-u-v-2\eta$$-u+v+\theta$$-u+v$$-u-v-2\eta+\theta$$-2u-2\eta$$-2v-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$
$\displaystyle=P_{3,4}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}M^{-1}_{4}R_{3,4}(-2v-2\eta)$
$\displaystyle\times\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}R_{1,4}(-u-v-2\eta-\theta)M_{1}$
$\displaystyle\times
R_{1,3}(-u-v-2\eta)M_{1}^{-1}\Big{(}K^{+}_{1}(u+\frac{\theta}{2})\Big{)}^{t_{1}}R_{3,1}(-u+v-\theta)$
$\displaystyle\times
M^{-1}_{1}R_{4,1}(-u+v)R_{2,1}(-2u-2\eta)R_{2,4}(-u-v-2\eta)$
$\displaystyle\times
M_{2}R_{2,3}(-u-v-2\eta+\theta)M^{-1}_{2}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}$
$\displaystyle\times
R_{3,2}(-u+v)M_{2}^{-1}R_{4,2}(-u+v+\theta)M_{1}M_{2}M_{4}P_{1,2}\,.$
We reshuffle the operator insertions again to obtain:
$2$$1$$3$$4$$-u-v-2\eta-\theta$$-u+v$$-u-v-2\eta$$-u+v-\theta$$-u-v-2\eta$$-u+v+\theta$$-u+v$$-u-v-2\eta+\theta$$-2u-2\eta$$-2v-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$
$\displaystyle=P_{3,4}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}M^{-1}_{4}R_{3,4}(-2v-2\eta)$
$\displaystyle\times\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}R_{1,4}(-u-v-2\eta-\theta)M_{1}$
$\displaystyle\times
R_{1,3}(-u-v-2\eta)M_{1}^{-1}\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}M^{-1}_{1}$
$\displaystyle\times
M^{-1}_{3}R_{3,1}(-u+v-\theta)R_{4,1}(-u+v)R_{2,1}(-2u-2\eta)$
$\displaystyle\times R_{2,4}(-u-v-2\eta)R_{2,3}(-u-v-2\eta+\theta)$
$\displaystyle\times
M_{3}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}R_{3,2}(-u+v)M_{2}^{-1}$
$\displaystyle\times R_{4,2}(-u+v+\theta)M_{1}M_{2}M_{4}P_{1,2}\,.$
Now we can use the YBE to bring the weight $-2u-2\eta$ back to left.
$2$$1$$3$$4$$-u-v-2\eta-\theta$$-u-v-2\eta$$-u-v-2\eta$$-u-v-2\eta+\theta$$-u+v$$-u+v+\theta$$-u+v-\theta$$-u+v$$-2u-2\eta$$-2v-2\eta$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$
$\displaystyle=P_{3,4}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}M^{-1}_{4}R_{3,4}(-2v-2\eta)$
$\displaystyle\times\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}R_{1,4}(-u-v-2\eta-\theta)M_{1}$
$\displaystyle\times
R_{1,3}(-u-v-2\eta)M_{1}^{-1}\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}M^{-1}_{1}$
$\displaystyle\times M^{-1}_{3}R_{2,4}(-u-v-2\eta)R_{2,3}(-u-v-2\eta+\theta)$
$\displaystyle\times R_{2,1}(-2u-2\eta)R_{3,1}(-u+v-\theta)R_{4,1}(-u+v)$
$\displaystyle\times
M_{3}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}R_{3,2}(-u+v)M_{2}^{-1}$
$\displaystyle\times R_{4,2}(-u+v+\theta)M_{1}M_{2}M_{4}P_{1,2}\,.$
Now we reshuffle the operator insertion a last time (2f) to obtain finally:
$2$$1$$4$$3$$-u-v-2\eta-\theta$$-u-v-2\eta$$-u-v-2\eta$$-u-v-2\eta+\theta$$-u+v$$-u+v+\theta$$-u+v$$-u+v-\theta$$-2v-2\eta$$-2u-2\eta$$v+\frac{\theta}{2}$$v-\frac{\theta}{2}$$u+\frac{\theta}{2}$$u-\frac{\theta}{2}$
$\displaystyle=P_{3,4}\left(K^{+}_{4}\left(v+\frac{\theta}{2}\right)\right)^{t_{4}}M^{-1}_{4}R_{3,4}(-2v-2\eta)M_{4}$
$\displaystyle\times\left(K^{+}_{3}\left(v-\frac{\theta}{2}\right)\right)^{t_{3}}M_{1}M_{2}R_{1,4}(-u-v-2\eta-\theta)$
$\displaystyle\times R_{1,3}(-u-v-2\eta)R_{2,4}(-u-v-2\eta)$
$\displaystyle\times
R_{2,3}(-u-v-2\eta+\theta)M^{-1}_{1}M^{-1}_{2}\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}$
$\displaystyle\times
M_{1}^{-1}R_{2,1}(-2u-2\eta)M_{1}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}$
$\displaystyle\times R_{3,1}(-u+v-\theta)R_{4,1}(-u+v)M_{3}R_{3,2}(-u+v)$
$\displaystyle\times R_{4,2}(-u+v+\theta)M_{1}M_{2}M_{4}P_{1,2}\,.$
Now we proceed algebraically. We reorder the permutation operators

$\displaystyle\mathbb{R}_{1,2|3,4}$
$\displaystyle(-u+v,-\theta,-\theta)\left(\mathbb{K}^{+}_{1,2}(u,\theta)\right)^{t_{1}t_{2}}\mathbb{M}^{-1}_{1,2}\mathbb{R}_{3,4|1,2}(-u-v-2\eta,-\theta,-\theta)\mathbb{M}_{1,2}\left(\mathbb{K}^{+}_{3,4}(v,\theta)\right)^{t_{3}t_{4}}$
$\displaystyle=$
$\displaystyle\left(K^{+}_{3}(v+\frac{\theta}{2})\right)^{t_{3}}M^{-1}_{3}R_{4,3}(-2v-2\eta)M_{3}\left(K^{+}_{4}(v-\frac{\theta}{2})\right)^{t_{4}}P_{3,4}M_{1}M_{2}$
$\displaystyle\times
R_{1,4}(-u-v-2\eta-\theta)R_{1,3}(-u-v-2\eta)R_{2,4}(-u-v-2\eta)R_{2,3}(-u-v-2\eta+\theta)$
$\displaystyle\times
M^{-1}_{1}M^{-1}_{2}\left(K^{+}_{1}\left(u+\frac{\theta}{2}\right)\right)^{t_{1}}M_{1}^{-1}R_{2,1}(-2u-2\eta)M_{1}\left(K^{+}_{2}\left(u-\frac{\theta}{2}\right)\right)^{t_{2}}P_{1,2}$
$\displaystyle\times
R_{3,2}(-u+v-\theta)R_{4,2}(-u+v)M_{3}R_{3,1}(-u+v)R_{4,1}(-u+v+\theta)M_{1}M_{2}M_{4}\,,$

and reintroduce the transposition, while using (2c) to obtain

$\displaystyle\mathbb{R}_{1,2|3,4}$
$\displaystyle(-u+v,-\theta,-\theta)\left(\mathbb{K}^{+}_{1,2}(u,\theta)\right)^{t_{1}t_{2}}\mathbb{M}^{-1}_{1,2}\mathbb{R}_{3,4|1,2}(-u-v-2\eta,-\theta,-\theta)\mathbb{M}_{1,2}\left(\mathbb{K}^{+}_{3,4}(v,\theta)\right)^{t_{3}t_{4}}$
$\displaystyle=$
$\displaystyle\left(P_{3,4}K^{+}_{4}(v-\frac{\theta}{2})M_{3}R_{3,4}(-2v-2\eta)M^{-1}_{3}K^{+}_{3}(v+\frac{\theta}{2})\right)^{t_{3}t_{4}}M_{1}M_{2}$
$\displaystyle\times
R_{1,4}(-u-v-2\eta-\theta)R_{1,3}(-u-v-2\eta)R_{2,4}(-u-v-2\eta)R_{2,3}(-u-v-2\eta+\theta)$
$\displaystyle\times
M^{-1}_{1}M^{-1}_{2}\left(P_{1,2}K^{+}_{2}(u-\frac{\theta}{2})M_{1}R_{1,2}(-2u-2\eta)M^{-1}_{1}K^{+}_{1}(u+\frac{\theta}{2})\right)^{t_{1},t_{2}}$
$\displaystyle\times
R_{3,2}(-u+v-\theta)R_{4,2}(-u+v)M_{3}R_{3,1}(-u+v)R_{4,1}(-u+v+\theta)M_{1}M_{2}M_{4}$
$\displaystyle=$
$\displaystyle\left(\mathbb{K}^{+}_{3,4}(v,\theta)\right)^{t_{3}t_{4}}\mathbb{M}_{1,2}\mathbb{R}_{3,4|1,2}(-u-v-2\eta,-\theta,-\theta)\mathbb{M}^{-1}_{1}\left(\mathbb{K}^{+}_{1,2}(u,\theta)\right)^{t_{1}t_{2}}\mathbb{R}_{1,2|3,4}(-u+v,-\theta,-\theta)\,,$

which completes the proof.

## References

  * Baxter (1971) R. J. Baxter, Stud. Appl. Math. L, 51 (1971).
  * Popkov and Zvyagin (1993) V. Yu. Popkov and A. A. Zvyagin, Phys. Lett. A 175, 295 (1993).
  * Frahm and Rödenbeck (1996) H. Frahm and C. Rödenbeck, Europhys. Lett. 33, 47 (1996), cond-mat/9502090 .
  * Baxter (1973) R. J. Baxter, J. Phys. C 6, L445 (1973).
  * Faddeev and Reshetikhin (1986) L. D. Faddeev and N. Yu. Reshetikhin, Ann. Phys. (NY) 167, 227 (1986).
  * Destri and de Vega (1989) C. Destri and H. J. de Vega, J. Phys. A: Math. Gen. 22, 1329 (1989).
  * Klümper (1992) A. Klümper, Ann. Phys. 1, 540 (1992).
  * Reshetikhin and Saleur (1994) N. Yu. Reshetikhin and H. Saleur, Nucl. Phys. B 419, 507 (1994), hep-th/9309135 .
  * Chalker and Coddington (1988) J. T. Chalker and P. D. Coddington, J. Phys. C 21, 2665 (1988).
  * Zirnbauer (1997) M. R. Zirnbauer, J. Math. Phys. 38, 2007 (1997), erratum _ibid._ 40, 2197 (1999), cond-mat/9701024 .
  * Gade (1999) R. M. Gade, J. Phys. A: Math. Gen. 32, 7071 (1999).
  * Essler  _et al._ (2005) F. H. L. Essler, H. Frahm, and H. Saleur, Nucl. Phys. B 712 [FS], 513 (2005), cond-mat/0501197 .
  * Cherednik (1984) I. V. Cherednik, Theor. Math. Phys. 61, 977 (1984).
  * Sklyanin (1988) E. K. Sklyanin, J. Phys. A: Math. Gen. 21, 2375 (1988).
  * Doikou (2000) A. Doikou, J. Phys. A: Math. Gen. 33, 8797 (2000).
  * Prinsloo  _et al._ (2015) A. Prinsloo, V. Regelskis, and A. Torrielli, Phys. Rev. D 92, 106006 (2015), arXiv:1505.06767 .
  * Bai  _et al._ (2019) N. Bai, H.-H. Chen, H. Ouyang, and J.-B. Wu, J. High. Energ. Phys. , 03, 193 (2019), arXiv:1901.03949 .
  * Ikhlef  _et al._ (2008) Y. Ikhlef, J. L. Jacobsen, and H. Saleur, Nucl. Phys. B 789, 483 (2008), cond-mat/0612037 .
  * Frahm and Martins (2011) H. Frahm and M. J. Martins, Nucl. Phys. B 847, 220 (2011), arXiv:1012.1753 .
  * Frahm and Martins (2012) H. Frahm and M. J. Martins, Nucl. Phys. B 862, 504 (2012), arXiv:1202.4676 .
  * Ikhlef  _et al._ (2012) Y. Ikhlef, J. L. Jacobsen, and H. Saleur, Phys. Rev. Lett. 108, 081601 (2012), arXiv:1109.1119 .
  * Candu and Ikhlef (2013) C. Candu and Y. Ikhlef, J. Phys. A: Math. Theor. 46, 415401 (2013), arXiv:1306.2646 .
  * Frahm and Seel (2014) H. Frahm and A. Seel, Nucl. Phys. B 879, 382 (2014), arXiv:1311.6911 .
  * Bazhanov  _et al._ (2021a) V. V. Bazhanov, G. A. Kotousov, S. M. Koval, and S. L. Lukyanov, Nucl. Phys. B 965, 115337 (2021a), arXiv:2010.10613 .
  * Bazhanov  _et al._ (2021b) V. V. Bazhanov, G. A. Kotousov, S. M. Koval, and S. L. Lukyanov, SIGMA 17, 025 (2021b), arXiv:2010.10615 .
  * Robertson  _et al._ (2020a) N. F. Robertson, M. Pawelkiewicz, J. L. Jacobsen, and H. Saleur, J. High. Energ. Phys. , 05, 144 (2020a), arXiv:2003.03261 .
  * Robertson  _et al._ (2021) N. F. Robertson, J. L. Jacobsen, and H. Saleur, J. High. Energ. Phys. , 02, 180 (2021), arXiv:2012.07757 .
  * Frahm and Gehrmann (2022) H. Frahm and S. Gehrmann, J. High. Energ. Phys. , 01, 070 (2022), arXiv:2111.00850 .
  * Frahm and Hobuß (2017) H. Frahm and K. Hobuß, J. Phys. A: Math. Theor. 50, 294002 (2017), arXiv:1703.08054 .
  * Vernier  _et al._ (2014) É. Vernier, J. L. Jacobsen, and H. Saleur, J. Phys. A: Math. Theor. 47, 285202 (2014), arXiv:1404.4497 .
  * Vernier  _et al._ (2016) E. Vernier, J. L. Jacobsen, and H. Saleur, Nucl. Phys. B 911, 52 (2016), arXiv:1601.01559 .
  * Martins  _et al._ (1998) M. J. Martins, B. Nienhuis, and R. Rietman, Phys. Rev. Lett. 81, 504 (1998), cond-mat/9709051 .
  * Frahm and Martins (2015) H. Frahm and M. J. Martins, Nucl. Phys. B 894, 665 (2015), arXiv:1502.05305 .
  * Frahm and Martins (2018) H. Frahm and M. J. Martins, Nucl. Phys. B 930, 545 (2018), arXiv:1802.05191 .
  * Frahm  _et al._ (2019) H. Frahm, K. Hobuß, and M. J. Martins, Nucl. Phys. B 946, 114697 (2019), arXiv:1906.00655 .
  * Frahm and Martins (2022) H. Frahm and M. J. Martins, Nucl. Phys. B 980, 115799 (2022), arXiv:2202.13405 .
  * Nepomechie and Retore (2021a) R. I. Nepomechie and A. L. Retore, J. High. Energ. Phys. , 03, 089 (2021a), arXiv:2012.08367 .
  * Li  _et al._ (2022) G.-L. Li, J. Cao, W.-L. Yang, K. Shi, and Y. Wang, J. High. Energ. Phys. , 04, 101 (2022), arXiv:2202.06531 [math-ph] .
  * Mezincescu and Nepomechie (1991) L. Mezincescu and R. I. Nepomechie, J. Phys. A: Math. Gen. 24, L17 (1991).
  * Belavin (1981) A. A. Belavin, Nucl. Phys. B 180, 189 (1981).
  * Bazhanov (1987) V. V. Bazhanov, Comm. Math. Phys. 113, 471 (1987).
  * de Vega and González-Ruiz (1994) H. J. de Vega and A. González-Ruiz, J. Phys. A: Math. Gen. 27, 6129 (1994), hep-th/9306089 .
  * Delius and George (2002) G. W. Delius and A. George, Lett. Math. Phys. 62, 211 (2002), math/0208043 .
  * Delius and MacKay (2003) G. W. Delius and N. J. MacKay, Comm. Math. Phys. 233, 173 (2003), hep-th/0112023 .
  * Appel and Vlaar (2020) A. Appel and B. Vlaar, Represent. Theory 26, 764 (2020), arXiv:2007.09218 .
  * Appel and Vlaar (2022) A. Appel and B. Vlaar, preprint (2022), arXiv:2203.16503 .
  * Robertson  _et al._ (2020b) N. F. Robertson, M. Pawelkiewicz, J. L. Jacobsen, and H. Saleur, J. High. Energ. Phys. , 05, 144 (2020b), arXiv:2003.03261 .
  * Miao  _et al._ (2022) Y. Miao, V. Gritsev, and D. V. Kurlov, “The Floquet Baxterisation,” (2022), arXiv:2206.15142 .
  * Kuniba  _et al._ (2011) A. Kuniba, T. Nakanishi, and J. Suzuki, J. Phys. A: Math. Theor. 44, 103001 (2011), arXiv:1010.1344 .
  * Nepomechie and Pimenta (2018) R. I. Nepomechie and R. A. Pimenta, J. Phys. A: Math. Theor. 51, 39LT02 (2018), arXiv:1805.10144 .
  * Nepomechie and Retore (2021b) R. I. Nepomechie and A. L. Retore, J. High. Energ. Phys. , 08, 053 (2021b), arXiv:2105.04982 .
  * Kulish and Sklyanin (1991) P. P. Kulish and E. K. Sklyanin, J. Phys. A: Math. Gen. 24, L435 (1991).
  * Martins and Guan (2000) M. J. Martins and X. W. Guan, Nucl. Phys. B 583, 721 (2000), nlin/0002050 .
  * Nepomechie  _et al._ (2017) R. I. Nepomechie, R. A. Pimenta, and A. L. Retore, Nucl. Phys. B 924, 86 (2017), arXiv:1707.09260 .
  * Yang and Yang (1969) C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1115 (1969).