#  $U_{q}[OSp(3|2)]$ quantum chains with quantum group invariant boundaries

Holger Frahm  Institut für Theoretische Physik, Leibniz Universität Hannover,
Appelstraße 2, 30167 Hannover, Germany  Márcio J. Martins  Departamento de
Física, Universidade Federal de São Carlos, C.P. 676, 13565-905 São Carlos
(SP), Brazil

###### Abstract

Based on the finite-size analysis of the spectrum of the quantum group
invariant deformation of the $OSp(3|2)$ superspin chain we identify the
operator content of the conformal field theory describing the model in its
scaling limit. We find that the macroscopic degeneracy of the conformal
weights observed in the thermodynamic limit of the isotropic superspin chain
is lifted by the deformation.

##  I Introduction

During the recent past years there has been an increasing activity on the
study of families of massless one-dimensional integrable models with intricate
critical behaviour similar to that expected for conformal field theories
(CFTs) based on non-compact symmetries. A common property among of these
lattice models is the presence of towers of lower energy states leading to the
same scaling dimension in the thermodynamic limit whose degeneracies is
typically lifted by logarithmic corrections on the finite interval. It has
been argued that examples of such systems are not restricted to specific
models but instead range from staggered two-dimensional vertex models to
families of spin chains based on twisted Lie algebras and as well as on
supergroup symmetries Essler  _et al._ (2005); Ikhlef  _et al._ (2008); Frahm
and Martins (2011, 2012); Vernier  _et al._ (2014, 2016); Frahm  _et al._
(2019). Probably best understood among these models is the staggered six-
vertex model Ikhlef  _et al._ (2008, 2012); Candu and Ikhlef (2013); Frahm and
Seel (2014); Bazhanov  _et al._ (2019, 2021a): here the spectrum of conformal
weights has been shown to have continuous components arising from the non-
compact target space of the underlying CFT. Based on finite size studies
probing properties of the algebra of extended conformal symmetry the low
energy effective theory of this model has been argued to be a Lorentzian black
hole nonlinear sigma model Bazhanov  _et al._ (2021a, b).

For conventional conformal field theories it is well known that boundary
conditions are able to select some of the conformal dimensions predicted for
the operator content of the underlying conformal field theory from symmetry
considerations. Therefore, a natural question to ask is whether or not the
presence of towers of low energy states and the presence of continuous
components in the finite-size spectrum is robust against changing the system
boundary conditions. Indeed, this kind of question has recently been
investigated in the context of the staggered six-vertex model Robertson  _et
al._ (2019, 2021); Frahm and Gehrmann (2022, 2023): in this case fine-tuning
of the boundary conditions is necessary for the low energy degrees of freedom
to be described by a non-compact boundary CFT. Moreover, based on numerical
work a boundary RG flow to a fixed point with discrete conformal spectrum has
been identified.

The purpose of this work is to begin an investigation about the effect of
boundary condition on the operator content of the ${q}$-deformed $OSp(3|2)$
superspin chain. In the isotropic ${q}\rightarrow 1$ limit this model is a
lattice realization of the Goldstone phase of intersecting loops and it has
been pointed out that, for both periodic and free boundary conditions, there
exist several towers of states with integer conformal dimensions subject to
strong subleading logarithmic corrections to scaling in the finite system
Martins  _et al._ (1998); Jacobsen  _et al._ (2003); Frahm and Martins (2015,
2022). On the other hand, it has been observed that some of the towers are
eliminated in the integrable anisotropic deformation of the periodic model
Frahm  _et al._ (2019). At the same time the corresponding corrections to
scaling vanish as power laws depending on the deformation parameter ${q}$
rather than logarithmic in the thermodynamic limit. This behaviour has been
linked to the observation that part of the energies of the ${q}$-deformed
$OSp(3|2)$ superspin chain coincides with the eigenvalues of the integrable
$XXZ$ spin $S=1$ chain for a special value of the deformation parameter ${q}$.

Here we will study the critical properties of such ${q}$-deformed superspin
chain with boundary conditions restoring the full $U_{q}[OSp(3|2)]$ quantum
group invariance. This is the simplest boundary condition retaining part of
the spectral degeneracies of the isotropic model and at the same time recovers
the free boundary conditions in the isotropic limit ${q}\to 1$. In addition,
the $U_{q}[OSp(3|2)]$ spin chain is solvable by the Bethe ansatz which allows
allows to study its spectral properties for very large systems. For our
analysis of the critical properties we exploit the relationship between the
eigenenergies $E_{n}(L)$ of the finite system and the central charge $c$ and
the conformal weights $h_{n}$ (appearing as surface exponents describing the
asymptotic behaviour of boundary correlation functions) of the corresponding
operators in the boundary CFT describing the model in the scaling limit Blöte
_et al._ (1986); Alcaraz  _et al._ (1987),

$E_{n}(L)=L\varepsilon_{\infty}+f_{\infty}-\frac{\pi v_{F}}{24L}\,c+\frac{\pi
v_{F}}{L}\,h_{n}+o\left(\frac{1}{L}\right)\,,$ (1)

where $\varepsilon_{\infty}$ is the energy per site of the ground state,
$f_{\infty}$ is the surface energy resulting from the open boundary
conditions, and $v_{F}$ is the Fermi velocity of the massless low-lying
excitations. A peculiar feature of $U_{q}[OSp(3|2)]$-invariant model is that
the ground state energy has no finite-size corrections beyond the surface
energy $f_{\infty}$ implying a vanishing central charge $c$ for generic values
of the deformation parameter ${q}$ in the massless regime. This has to be
contrasted with other integrable quantum group invariant spin chains such as
those based on the spin-$s$ representation of the $U_{q}[SU(2)]$ quantum
algebra in which the ground state finite-size corrections only vanish for a
specific choice of ${q}$ Alcaraz  _et al._ (1987); Martins (1990).

Given the quantum group symmetry the eigenenergies of the superspin chain
appear in the multiplets of $OSp(3|2)$. Following Ref. Van der Jeugt (1984)
they may be labelled by two indices $(\mathtt{p};\mathtt{q})$. Except for the
trivial representation $(0;0)$ the index $\mathtt{p}$ takes nonnegative
integer values while $\mathtt{q}\geq\frac{1}{2}$ is an integer or half integer
– as in the isotropic case. The presence of these levels in the spectrum of
the superspin chain of length $L$ requires the selection rule
$L-\mathtt{p}-2\mathtt{q}\in 2\mathbb{N}$ to be satisfied. An exception are
the ’atypical’ representations such as $(1;1)$ which appears in the Hilbert
space of chains of any length as part of reducible but indecomposable
representations.

Combining exact diagonalization for small systems with the numerical solution
of the Bethe equations we have investigated a significant number of low energy
states in the sectors with different values of the two $U(1)$ charges of the
$U_{q}[OSp(3|2)]$ superspin chain. Using (1) with data for sufficiently large
systems we provide strong evidence that the conformal weights in terms of the
quantum numbers $(\mathtt{p};\mathtt{q})$ are given as

$\displaystyle h_{(\mathtt{p};\mathtt{q})}$
$\displaystyle=\frac{\pi-\gamma}{2\pi}\,\mathtt{p}(\mathtt{p}+1)+\frac{\gamma}{2\pi}\,2\mathtt{q}(2\mathtt{q}-1)\,$
(2)

in the massless region $0\leq\gamma\leq\frac{\pi}{2}$. The anisotropic
parameter $\gamma$ is related to the quantum group deformation by
${q}=e^{i\gamma/2}$. By way of contrast with the periodic case studied in
Frahm  _et al._ (2019) we find no trace of the presence of towers of states in
the eigenspectrum for the quantum group boundary condition. We anticipate
however that some of the low lying states have strong power law corrections to
finite-size scaling which required the study of large system sizes. In any
case this result further elucidates our earlier findings for the isotropic
model Frahm and Martins (2015, 2022): (i) as $\gamma\rightarrow 0$ the second
term of Eq.(2) vanishes and we are left with the conformal weights identified
in the finite-size analysis of the periodic chain. (ii) For the isotropic
model with free boundary conditions (2) results in a tower of states with
integer conformal dimensions labelled on the finite lattice by the index
$\mathtt{p}$. Since the number of available values for $\mathtt{q}$ grows with
the lattice size this leads to a macroscopic degeneracy of the low energy
states which is lifted by corrections to scaling vanishing as $1/\log L$ in
the thermodynamic limit.

This paper is organized as follows: in the following section we formulate the
integrable $U_{q}[OSp(3|2)]$-invariant superspin chain based on particular
solutions of the boundary Yang-Baxter algebra Cherednik (1984); Sklyanin
(1988). Then we present the solution to this spectral problem of the model in
terms of Bethe equations. Being based on a superalgebra there exist two such
solutions corresponding to different orderings of the fermionic and bosonic
states in the local basis. In Section IV we uncover an exact correspondence
between the spectra of the ${q}$-deformed $OSp(3|2)$ superspin chain and the
$XXZ$ spin-$1$ spin chain, both with quantum-group invariant boundary
conditions, for a particular value of the deformation parameter ${q}$.
Together with our finite-size analysis of the $U_{q}[OSp(3|2)]$-symmetric
super spin chain this correspondence supports our proposal (2) for the
operator content of the boundary CFT describing the critical point of the
model. We close with a discussion putting the present results in the context
of earlier work on the $OSp(3|2)$ superspin chain and its deformations as well
as on the influence of boundary conditions in other integrable spin chains
with a continuous component to the conformal spectrum. The
$U_{q}[OSp(3|2)]$-invariant Hamiltonian in terms of the $OSp(3|2)$
superalgebra is presented in Appendix A.

##  II Formulation of the model

In this section we describe the integrable ${q}$-deformed $OSp(3|2)$ spin
chain with quantum group invariance. We recall that the main tools for dealing
with open boundary conditions within the the quantum inverse scattering method
have been introduced by Cherednik and Sklyanin Cherednik (1984); Sklyanin
(1988). This pioneering work was further elaborated and in particular it has
been shown how integrable quantum group invariant models can be constructed
for a variety of distinct affine Lie algebras in Mezincescu and Nepomechie
(1991); Kulish and Sklyanin (1991). Later on similar constructions have been
pursued to include integrable models with underlying graded affine
superalgebras Links and Gould (1996); Bracken  _et al._ (1998). Specifically,
the right and left boundary conditions are encoded in reflection matrices
$K^{\pm}(\lambda)$ satisfying the so-called boundary Yang-Baxter equations,
e.g.

$\displaystyle
R_{12}(\lambda-\mu)K_{1}^{-}(\lambda)R_{21}(\lambda+\mu)K_{2}^{-}(\mu)=K_{2}^{-}(\mu)R_{12}(\lambda+\mu)K_{1}^{-}(\lambda)R_{21}(\lambda-\mu)\,$
(3)

for the left boundary matrix $K^{-}(\lambda)$. Here $K^{-}_{j}(\lambda)$ is a
copy of the reflection matrix acting non-trivially on the space $V_{j}$ and
$R_{ij}(\lambda)\in\text{End}\left(V_{i}\otimes V_{j}\right)$ is the
$R$-matrix of the ${q}$-deformed $OSp(3|2)$ vertex model in the fundamental
representation for which $V_{j}$ is a five-dimensional $\mathbb{Z}_{2}$-graded
vector space Galleas and Martins (2004). The $R$-matrix satisfies the
commutation property $[\check{R}_{12}(\lambda),\check{R}_{12}(\mu)]=0$ where
$\check{R}_{12}(\lambda)=P_{12}\,R_{12}(\lambda)$ with the graded permutation
operator $P_{12}\,a\otimes b=(-1)^{p_{a}p_{b}}b\otimes a$ on $V\otimes
V$.111This is because $\check{R}$ belongs to the algebra generated by a braid
operator and can therefore be obtained by means of the Baxterization method
Jones (1990). Recall here, that $p_{a}$ stands for the grading (or parity),
i.e. $p_{a}=0$ if $a$ is an even (bosonic) state and $p_{a}=1$ if $a$ is odd
(fermionic). As a consequence it is easy to see that (3) is satisfied by the
trivial solution $K^{-}(\lambda)\equiv\mathbf{1}$. Given this solution of (3)
$K^{+}(\lambda)$ is easily obtained using the crossing properties of the
$R$-matrix Sklyanin (1988). Since the explicit form of the $R$-matrix depends
on the choice of the $\mathbb{Z}_{2}$-grading the same is true for the
corresponding $K^{+}(\lambda)$. For instance, ordering the elements of the
five dimensional basis in the $fbbbf$ grading we obtain,

$K^{+}(\lambda)=\left(\begin{array}[]{ccccc}{q}^{2}&0&0&0&0\\\
0&{q}^{2}&0&0&0\\\ 0&0&{q}&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&1\\\
\end{array}\right)\,.$ (4)

With these objects the double row transfer matrix generating commuting
integrals of motion acting on the Hilbert space $\otimes_{j=1}^{L}V_{j}$ of a
chain of length $L$ then reads

$T(\lambda)=Str_{0}[K^{+}_{0}(\lambda)R_{0L}(\lambda)\cdots
R_{01}(\lambda)K^{-}_{0}(\lambda)R_{01}^{-1}(-\lambda)\cdots
R_{0L}^{-1}(-\lambda)]$ (5)

where the symbol $Str_{0}$ denotes the supertrace taken over an auxiliary
five-dimensional $OSp(3|2)$ superspace. The respective Hamiltonian with open
boundary is obtained by expanding the double-row transfer matrix (5) up to the
first order in the spectral parameter $\lambda$. For the choice of reflection
matrices introduced above the Hamiltonian can be written as

$H=\sum_{j=1}^{L-1}\left.{\frac{\partial}{\partial\lambda}{\check{R}}_{j,j+1}(\lambda)}\right|_{\lambda=0}$
(6)

up to an additive constant. Recall that the $R$-matrix is constructed by means
of Baxterization from its braid limit and the $\check{R}$-matrix can be
written as a sum over projectors appearing in the tensor product of two copies
of the five-dimensional vector representations. As has been argued before this
property ensures that the Hamiltonian (6) commutes with the generators of
$U_{q}[OSp(3|2)]$ (see also our discussion of the isotropic model below) Links
and Gould (1996).

In principle it is also possible to express the Hamiltonian (6) in terms of
the generators of the $OSp(3|2)$ superalgebra. To this end we recall that
representations of this superalgebra have been previously investigated by Van
der Jeugt Van der Jeugt (1984). The even part of the $OSp(3|2)$ superalgebra
is isomorphic to $SO(3)\oplus Sp(2)$ and we shall denote the corresponding
generators by the operators $\\{\tau^{z},\tau^{\pm}\\}$ and
$\\{\sigma^{z},\sigma^{\pm}\\}$, respectively. The odd subspace of the
$OSp(3|2)$ is constituted by another six fermionic generators which here are
going to be represented by the operators $\\{c^{\pm},d^{\pm},f^{\pm}\\}$. In
terms of the $5\times 5$ Weyl matrices $e_{ij}$, whose entries are 1 on the
$i$-th row and the $j$-th column and zero elsewhere, the twelve generators of
the $OSp(3|2)$ superalgebra can be given explicitely in the $fbbbf$ grading.
For the reader’s convenience we have listed them together with the originial
notation used in Ref. Van der Jeugt (1984) in Table 1.

Generator |  Weyl Matrix |  $\mathbb{Z}_{2}$ parity
---|---|---
our notation | notation from Van der Jeugt (1984) |  |
$\tau^{z}$ | $s^{z}$ | $e_{22}-e_{44}$ | $\mathrm{even}$
$\tau^{+}$ | $s^{+}$ | $\sqrt{2}(e_{23}-e_{34})$ | $\mathrm{even}$
$\tau^{-}$ | $s^{-}$ | $\sqrt{2}(e_{32}-e_{43})$ | $\mathrm{even}$
$\sigma^{z}$ | $t^{z}$ | $(e_{11}-e_{55})/2$ | $\mathrm{even}$
$\sigma^{+}$ | $t^{+}$ | $-e_{15}$ | $\mathrm{even}$
$\sigma^{-}$ | $t^{-}$ | $-e_{51}$ | $\mathrm{even}$
$c^{+}$ | $R_{1,\frac{1}{2}}$ | $e_{25}-e_{14}$ | $\mathrm{odd}$
$c^{-}$ | $R_{-1,-\frac{1}{2}}$ | $-e_{52}-e_{41}$ | $\mathrm{odd}$
$d^{+}$ | $R_{0,-\frac{1}{2}}$ | $e_{31}+e_{53}$ | $\mathrm{odd}$
$d^{-}$ | $R_{0,\frac{1}{2}}$ | $e_{35}-e_{13}$ | $\mathrm{odd}$
$f^{+}$ | $R_{-1,\frac{1}{2}}$ | $e_{12}-e_{45}$ | $\mathrm{odd}$
$f^{-}$ | $R_{1,-\frac{1}{2}}$ | $e_{21}+e_{54}$ | $\mathrm{odd}$
Table 1: The five-dimensional representation of the six bosonic and six
fermionic generators of the $OSp(3|2)$ superalgebra in terms of the Weyl
matrices ordered in the $fbbbf$ grading.

The Hamiltonian of the open spin chain (6) commutes with $U(1)$ charges of the
Cartan subalgebra of $OSp(3|2)$ which are directly related to the azimuthal
bosonic generators of the $SO(3)$ and $Sp(2)$ subalgebras:

$[H,\sum_{j=1}^{L}\tau_{j}^{z}]=[H,\sum_{j=1}^{L}\sigma_{j}^{z}]=0\,.$ (7)

We further remark that the quadratic Casimir $C_{j,j+1}$ of $OSp(3|2)$ acting
on pair of sites $(j,j+1)$ in terms of these generators is

$\displaystyle C_{j,j+1}$ $\displaystyle=$
$\displaystyle\frac{\left(\tau_{j}^{+}\tau_{j+1}^{-}+\tau_{j}^{-}\tau_{j+1}^{+}\right)}{2}+\tau_{j}^{z}\tau_{j+1}^{z}-2\left(\sigma_{j}^{+}\sigma_{j+1}^{-}+\sigma_{j}^{+}\sigma_{j+1}^{-}\right)-4\sigma_{j}^{z}\sigma_{j+1}^{z}$
(8) $\displaystyle+$
$\displaystyle\left(c_{j}^{+}c_{j+1}^{-}-c_{j}^{-}c_{j+1}^{+}\right)+\left(d_{j}^{+}d_{j+1}^{-}-d_{j}^{-}d_{j+1}^{+}\right)+\left(f_{j}^{+}f_{j+1}^{-}-f_{j}^{-}f_{j+1}^{+}\right)$

where the tensor products among the fermionic degrees of freedom have to be
understood in the graded sense. The degeneracies of the eigenvalues of this
Casimir operator are compatible with expected Clebsch-Gordon decomposition
$(0;\frac{1}{2})\otimes(0;\frac{1}{2})=(0;0)\oplus(0;1)\oplus(1;\frac{1}{2})$,
corresponding to the identity, the $12$-dimensional adjoint and a non-
fundamental representation Van der Jeugt (1984). At this point we have the
basic ingredients to represent the quantum group Hamiltonian (6) in terms of
the generators of the superalgebra $OSp(3|2)$. The final expression for the
Hamiltonian is quite cumbersome and is given in Appendix A.

In the limit $\gamma\rightarrow 0$ the Hamiltonian (6) can be written in terms
of the nearest-neighbour Casimir operator,

$H=-\sum_{j=1}^{L-1}\left(C_{j,j+1}-3C_{j,j+1}^{2}\right)-4(L-1)\,,$ (9)

making explicit the the $OSp(3|2)$ invariance at the isotropic point.

##  III The Bethe ansatz solution

The Hamiltonian of the ${q}$-deformed $OSp(3|2)$ superspin chain with
_periodic_ boundary condition has been diagonalized using the algebraic Bethe
ansatz in Ref. Galleas and Martins (2004). Within the framework of the
analytical Bethe ansatz it has been found that the Bethe equations for the
corresponding _open_ spin chain with quantum algebra invariance can be
obtained from those for the periodic model based on the so-called ’doubling
postulate’ Mezincescu and Nepomechie (1992); Artz  _et al._ (1995a, b); Yung
and Batchelor (1995). We have checked that the postulate applies for the
$U_{q}[OSp(3|2)]$ model by comparing the eigenenergies obtained by exact
diagonalization of the Hamiltonian (6) with those obtained solving the doubled
Bethe equations given below for several low-lying states up to $L=8$.

With these Bethe equations the spectrum of the model is built starting from
suitable highest weight reference states. For models based on superalgebras it
is well known that the explicit form of the Bethe equations depend on the
choice of the grading, see. e.g. Essler and Korepin (1992). For the
$U_{q}[OSp(3|2)]$ superspin chain this amounts to two different formulations
of the Bethe ansatz, i.e. in the grading $fbbbf$ starting from a reference
state in the $(L-1;\frac{1}{2})$-multiplet and in the grading $bfbfb$ starting
from the $(0;L/2)$-multiplet Frahm and Martins (2015). In what follows we
shall discuss the form of the Bethe equations for these two possible gradings.

###  III.1 The fbbbf grading

Applying the doubling procedure to the $fbbbf$ Bethe equations of the periodic
$U_{q}[OSp(3|2)]$ model Galleas and Martins (2004); Frahm  _et al._ (2019) we
find that the eigenstates of the open $U_{q}[OSp(3|2)]$ invariant superspin
chain (6) are parametrized by solutions to the following set of Bethe
equations,

$\displaystyle\left[f_{1/2}\left(\lambda_{j}^{(1)}\right)\right]^{{2L}}=\prod_{k=1}^{{L-n_{1}-n_{2}}}f_{1/2}\left({\lambda_{j}^{(1)}-\lambda_{k}^{(2)}}\right)\,f_{1/2}\left(\lambda_{j}^{(1)}+\lambda_{k}^{(2)}\right)\,,\quad
j=1,\cdots,{L-n_{1}},$ (10)
$\displaystyle\prod_{k=1}^{{L-n_{1}}}f_{1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(1)}\right)\,f_{1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(1)}\right)=$
$\displaystyle\qquad=\prod_{\stackrel{{\scriptstyle k=1}}{{k\neq
j}}}^{{L-n_{1}-n_{2}}}f_{1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(2)}\right)\,f_{1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(2)}\right)\,,\quad
j=1,\cdots,{L-n_{1}-n_{2}}\,,$

where the function $f_{s}(\lambda)$ is defined as

$f_{s}(\lambda)=\frac{\sinh(\lambda+is\gamma)}{\sinh(\lambda-is\gamma)}\,.$
(11)

In (10) $n_{1}$, $n_{2}$ are the eigenvalues of the $U(1)$ charges (7) for a
highest weight state in the multiplet
$(\mathtt{p};\mathtt{q})=(n_{1}-1;\frac{1}{2}(n_{2}+1))$. Its energy is given
in terms of the Bethe roots from the first level as,

$E(\\{\lambda_{j}^{(a)}\\},L)=\sum^{L-n_{1}}_{j=1}\frac{2\sin\gamma}{\cos\gamma-\cosh(2\lambda_{j}^{(1)})}\,.$
(12)

In order to study the thermodynamic limit properties we first diagonalized the
Hamiltonian (6) for lattice sizes $L\leq 8$. We next solve numerically the
Bethe equations (10) for some $U(1)$-sectors $(n_{1},n_{2})$ and compare the
eigenenergies (12) with the spectrum obtained by exact diagonalization of
Hamiltonian (6). For low-lying energy states, similar as in the periodic case
Frahm  _et al._ (2019), we find that the Bethe root configurations on both
levels are dominated by two-strings with
$\mathrm{Re}\left(\lambda_{j}^{(1,2)}\right)\geq 0$ and
$\mathrm{Im}\left(\lambda_{j}^{(1,2)}\right)\simeq\pm\gamma/4$. As the system
size grows the difference among the root configurations of the two levels
becomes exponentially close. Therefore, for $L\rightarrow\infty$ the
respective string hypothesis may be formulated as

$\lambda^{(1)}_{j}=\xi_{j}\pm
i\frac{\gamma}{4}\,,\quad\lambda^{(2)}_{j}=\xi_{j}\pm
i\frac{\gamma}{4}\,,\quad\xi_{j}\in\mathbb{R}^{+}\,.$ (13)

In the thermodynamic limit these strings fill the positive part of the real
axis and the root configuration can be described in terms of their density
$\sigma_{L}(\xi)$ within the root density approach Yang and Yang (1969).
Symmetrizing the density around the origin we obtain the following linear
integral equation for $\sigma_{L}(\xi)$:

$\displaystyle 2\pi\sigma_{L}(\xi)$
$\displaystyle+\int_{-\infty}^{+\infty}\mathrm{d}\xi^{{}^{\prime}}\left[2\Psi\left(\xi-\xi^{{}^{\prime}},\frac{\gamma}{2}\right)+\Psi\left(\xi-\xi^{{}^{\prime}},\gamma\right)\right]\sigma_{L}(\xi^{{}^{\prime}})$
(14)
$\displaystyle=2\left[\Psi\left(\xi,\frac{3\gamma}{4}\right)+\Psi\left(\xi,\frac{\gamma}{4}\right)\right]+\frac{1}{L}\left[2\Psi\left(\xi,\frac{\gamma}{2}\right)+\Psi\left(\xi,\gamma\right)-2\Psi\left(2\xi,\gamma\right)\right]\,,$

where $\Psi(\xi,\gamma)=\frac{2\sin(2\gamma)}{\cosh(2x)-\cos(2\gamma)}$. This
equation can be solved by Fourier transformation order by order in $L$. To
leading order one finds

$\sigma_{\infty}(x)=\frac{2}{\gamma\cosh(2\pi x/\gamma)}\,,$ (15)

from which we reproduce the ground state energy density

$\varepsilon_{\infty}=-2\cot{\frac{\gamma}{2}}\,.$ (16)

The low energy excitations above the ground state are gapless with a linear
dispersion relation $\epsilon(p)\simeq v_{F}|p|$ where the Fermi velocity is
$v_{F}=2\pi/\gamma$. These quantities are already known from the
${q}$-deformed model with periodic boundary conditions Frahm  _et al._ (2019).
Similarly, one obtains the surface energy $f_{\infty}$ from the
$\mathcal{O}(L^{-1})$ contribution to the solution of (14). After some
simplifications we find

$f_{\infty}^{(OSp)}=2\cot\frac{\gamma}{2}\,,$ (17)

which is just the value for the ground state energy per site with the opposite
sign.

###  III.2 The grading bfbfb

Alternatively we can use the doubling procedure in the Bethe ansatz solution
of the model with periodic boundaries for the grading $bfbfb$ Frahm  _et al._
(2019). In this case the spectrum of the open $U_{q}[OSp(3|2)]$ invariant
superspin chain is parametrized by solutions of a different set of Bethe
equations,

$\displaystyle\left[f_{1/2}\left(\lambda_{j}^{(1)}\right)\right]^{{2L}}=\prod_{k=1}^{{L-m_{1}-m_{2}}}f_{1/2}\left({\lambda_{j}^{(1)}-\lambda_{k}^{(2)}}\right)\,f_{1/2}\left(\lambda_{j}^{(1)}+\lambda_{k}^{(2)}\right)\,,\quad
j=1,\cdots,{L-m_{2}},$ (18)
$\displaystyle\prod_{k=1}^{{L-m_{2}}}f_{1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(1)}\right)\,f_{1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(1)}\right)=$
$\displaystyle\qquad=\prod_{\stackrel{{\scriptstyle k=1}}{{k\neq
j}}}^{{L-m_{1}-m_{2}}}f_{-1/2}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(2)}\right)\,f_{-1/2}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(2)}\right)\
f_{1}\left(\lambda_{j}^{(2)}-\lambda_{k}^{(2)}\right)\,f_{1}\left(\lambda_{j}^{(2)}+\lambda_{k}^{(2)}\right)\,,$
$\displaystyle\qquad\qquad j=1,\cdots,{L-m_{1}-m_{2}}\,,$

and the energy of the Hamiltonian (6) corresponding to a particular root
configuration is

$E\left(\\{\lambda_{j}^{(a)}\\},L\right)=-\sum^{L-m_{2}}_{j=1}\frac{2\sin\gamma}{\cos\gamma-\cosh(2\lambda_{j}^{(1)})}-2(L-1)\cot\gamma\,.$
(19)

As mentioned above, the Bethe states for $fbbbf$ and $bfbfb$ are constructed
starting from different reference states. Solutions to (18) parametrize the
state with $U(1)$-charges $(n_{1},n_{2})=(m_{1}+1,m_{2}-1)$ in a
$(\mathtt{p};\mathtt{q})=(m_{1};{m_{2}}/2)$ multiplet.

It turns out that the thermodynamic limit in the $bfbfb$ grading simplifies
studies in the subsector with quantum number $m_{1}=0$ because the number of
Bethe roots in both levels are the same. Numerical solution of the Bethe
equations (18) for small systems tell us that the Bethe root configurations
are dominated by pairs of complex rapidities with positive real parts and
$\mathrm{Im}\left(\lambda_{j}^{(1)}\right)\simeq\pm 3\gamma/4$,
$\mathrm{Im}\left(\lambda_{j}^{(2)}\right)\simeq\pm\gamma/4$. Once again the
difference of their real parts become exponentially small for large $L$.
Similar to what has been found for the periodic case Frahm  _et al._ (2019),
the string hypothesis for the $bfbfb$ grading is given by,

$\lambda^{(1)}_{j,\pm}=\xi_{j}\pm
i\frac{3\gamma}{4}\,,\quad\lambda^{(2)}_{j,\pm}=\xi_{j}\pm
i\frac{\gamma}{4}\,,\quad\xi_{j}\in\mathbb{R}^{+}\,.$ (20)

The root density approach based on this string assumption can be used to
reproduce the bulk energy density, Fermi velocity of low energy excitations
and surface energy obtained in the grading $fbbbf$.

##  IV The $U_{q}[SU(2)]$ spin $S=1$ $XXZ$ model

The Hamiltonian of the integrable $S=1$ $XXZ$ model with quantum algebra
symmetry is Pasquier and Saleur (1990); Batchelor  _et al._ (1990)

$H=\sum_{j=1}^{L-1}H_{j,j+1}+i(S_{L}^{z}-S_{1}^{z})$ (21)

where the nearest-neighbour bulk terms $H_{j,j+1}$ are those of the integrable
spin-1 model introduced by Zamolodchikov and Fateev Zamolodchikov and Fateev
(1980),

$\displaystyle H_{j,j+1}=$ $\displaystyle\frac{1}{\sin(\gamma)}\left[{\bf
S}_{j}\cdot{\bf S}_{j+1}-({\bf S}_{j}\cdot{\bf
S}_{j+1})^{2}\right]-\tan(\frac{\gamma}{2})\left[S_{j}^{z}S_{j+1}^{z}+(S_{j}^{z})^{2}+(S_{j+1}^{z})^{2}-(S_{j}^{z}S_{j+1}^{z})^{2}\right]$
(22)
$\displaystyle+\frac{4\sin^{2}(\frac{\gamma}{4})}{\sin(\gamma)}\left[(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})S_{j}^{z}S_{j+1}^{z}+S_{j}^{z}S_{j+1}^{z}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})\right]$
$\displaystyle+\left(1+2\tan(\frac{\gamma}{2})\right)\,\mathbf{1}_{j}\mathbf{1}_{j+1}\,,$

where the last constant term is a convenient normalization for the spectral
relationship to the $U_{q}[OSp(3|2)]$ superspin chain. The ${\bf
S}=(S^{x},S^{y},S^{z})$ are the spin-1 generators of $SU(2)$

$S^{x}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&1\\\ 0&1&0\\\
\end{array}\right)\,,\quad
S^{y}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&-i\\\ 0&i&0\\\
\end{array}\right)\,,\quad S^{z}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\\\ \end{array}\right)$ (23)

and $\mathbf{1}$ is the $3\times 3$ identity. The Hamiltonian (21) has been
diagonalized using the algebraic Bethe ansatz Mezincescu  _et al._ (1990). The
eigenstates of this model are parameterized by the $L-n$ complex roots
$\lambda_{k}$ of the Bethe equations,

$\left[f_{1/2}\left(\lambda_{k}\right)\right]^{{2L}}=\prod_{\stackrel{{\scriptstyle\ell\neq
k}}{{\ell=1}}}^{{L-n}}f_{1/2}\left({\lambda_{k}-\lambda_{\ell}}\right)\,f_{1/2}\left(\lambda_{k}+\lambda_{\ell}\right)\,,\quad
k=1,\cdots,{L-n}\,.$ (24)

where $n$ is the $U(1)$-charge $\sum_{j=1}^{L}S_{j}^{z}$ for the highest
weight state in a $U_{q}[SU(2)]$ spin-$j$ multiplet, i.e. $n=j$. In terms of
these roots the corresponding energies of the Hamiltonian (21) are

$E(\\{\lambda_{k}\\},L)=\sum^{L-n}_{k=1}\frac{2\sin\gamma}{\cos\gamma-\cosh(2\lambda_{k})}\,.$
(25)

For large $L$, the Bethe roots configurations for the low-lying energies are
essentially dominated by pairs of complex complex conjugate roots

$\lambda_{k,\pm}\simeq\xi_{k}\pm
i\frac{\gamma}{4}\,,\quad\xi_{k}\in\mathbb{R}^{+}\,.$ (26)

In the thermodynamic limit the density of these $2$-strings is given by the
integral equation

$\displaystyle 2\pi\sigma_{L}(\xi)$
$\displaystyle+\int_{-\infty}^{+\infty}d\xi^{{}^{\prime}}\left[2\Psi\left(\xi-\xi^{{}^{\prime}},\frac{\gamma}{2}\right)+\Psi\left(\xi-\xi^{{}^{\prime}},\gamma\right)\right]\sigma_{L}(\xi^{{}^{\prime}})$
(27)
$\displaystyle=2\left[\Psi\left(\xi,\frac{3\gamma}{4}\right)+\Psi\left(\xi,\frac{\gamma}{4}\right)\right]+\frac{1}{L}\left[2\Psi\left(\xi,\frac{\gamma}{2}\right)+\Psi\left(\xi,\gamma\right)+2\Psi\left(2\xi,\gamma\right)\right]$

obtained within the root density approach.

Comparing Eq. (27) with the corresponding result for the $U_{q}[OSp(3|2)]$
superspin chain given by Eq. (14) we note that they only differ in the sign of
the driving term $2\Psi(2\xi,\gamma)/L$. Therefore both the bulk energy
density $\varepsilon_{\infty}$ and the Fermi velocity $v_{F}$ of the models
(6) and (21) coincide in the critical region and only the corresponding
surface energies $f_{\infty}$ differ. Considering the thermodynamic limit of
the relation (27) we find

$f_{\infty}^{(XXZ)}=2\int_{-\infty}^{+\infty}\mathrm{d}x\,\frac{\cosh[(\pi-\gamma)x]\tanh(\gamma
x)}{\sinh(\pi x)}$ (28)

for the integrable quantum group invariant $U_{q}[SU(2)]$ spin-$1$ chain.

Note, however, that $\Psi(2\xi,\gamma)$ vanishes for $\gamma=\pi/2$ and
therefore both the bulk and surface energies of the two models coincide. It
has already been observed that the $XXZ$ spin-$1$ chain and the ${q}$-deformed
$OSp(3|2)$ model with periodic boundary conditions have some common
eigenvalues for this value of the anisotropyFrahm  _et al._ (2019). Motivated
by that we have compared the eigenspectra of the quantum group invariant
Hamiltonians (6) and (21) for small lattice sizes using exact
diagonalizations. Remarkably, we find perfect matching of the spectra of these
Hamiltonians for $\gamma=\frac{\pi}{2}$ apart from degeneracies due to the
different sizes of the Hilbert space, see Figure 1.

Figure 1: Spectra of the $U_{q}[OSp(3|2)]$ superspin chain and the quantum
group invariant $XXZ$ spin-$1$ model as a function of the anisotropy $\gamma$
for $L=3$ (left panel) and $L=4$ (right panel). Note that for $\gamma=\pi/2$
the spectra coincide up to degeneracies.

Within our numerical precision this feature has also been checked for the
complete spectrum of both models up to the size $L=8$. In addition we have
observed this correspondence of the spectra continues to hold from the point
of view of solutions of the corresponding Bethe ansatz equations: for small
$L$ we find that the level-$1$ Bethe roots for the state with
$(\mathtt{p};\mathtt{q})=(0;1)$ in the $fbbbf$ grading approach those for the
$n=1$ state in the $U_{q}[SU(2)]$ chain as $\gamma\to\pi/2$. As a consequence
of (12) and (25) their energies coincide in this limit.

#### The critical behaviour.

The low energy spectrum of the integrable spin-$1$ $XXZ$ spin chain in the
critical regime $0\leq\gamma\leq{\pi}$ subject to various boundary conditions
has been studied extensively di Francesco  _et al._ (1988); Alcaraz and
Martins (1989, 1990); Frahm  _et al._ (1990); Martins (1990); Koo and Saleur
(1993). The finite-size spectrum of the quantum group invariant model has
first been investigated for the lowest states with $U_{q}[SU(2)]$-spin $j=0$
and $1$ in Ref. Martins (1990) for even length chains giving the conformal
anomaly and the conformal weight $h_{j=1}$ as a function of the anisotropy

$c_{\text{qg}}=\frac{3}{2}\left(1-\frac{2\gamma^{2}}{\pi(\pi-\gamma)}\right)\,,\quad
h_{j=1}=1-\frac{\gamma}{\pi}\,.$ (29)

Later on it has been proposed that the full operator content of the
$U_{q}[SU(2)]$-invariant spin chain can be obtained by combining a twisted
free boson field with compactification radius depending on the anisotropy and
an Ising field with free boundaries. More precisely the conformal weights, in
our notation, are Koo and Saleur (1993)

$h_{j}=\frac{j}{2}\left(j-\frac{\gamma}{\pi}\left(j+1\right)\right)+h_{I}\,,$
(30)

where $h_{I}=0$ or $\frac{1}{2}$ depending on whether $j$ is even or odd is
related to the Ising degree of freedom von Gehlen and Rittenberg (1986). Note
that $c_{\text{qg}}$ vanishes for $\gamma=\frac{\pi}{2}$ together with the
subleading corrections to scaling to the ground state. This is accordance with
the mentioned correspondence with the $U_{q}[OSp(3|2)]$ symmetric superspin
chain.

As a consequence of the antiferromagnetic character of the quantum group
invariant spin chain the ground state is typically frustrated which may lead
to an excited state with distinct exponent when the thermodynamic limit is
taken for $L$ odd. Based on our numerical results shown in Fig. 2

Figure 2:  Finite-size estimates $h_{j}(L)=(L/\pi
v_{F})\left(E_{j}(L)-E_{0}(L)\right)$ of the conformal weights corresponding
to the ground states of the quantum group invariant spin-$1$ chain (21) in the
sectors with total spin $j=0,1,2,3$ for $\gamma=2\pi/9$ and odd length $L$.
Red diamonds indicate the values proposed in (30) with (31).

we are led to complement the proposal of Ref. Koo and Saleur (1993) for the
operator content of the conformal field theory describing the continuum limit
of the $U_{q}[SU(2)]$ spin-$1$ chain to be given by (30 but with

$h_{I}=\begin{cases}0&\text{for~{}}L+j\text{~{}even}\\\
\frac{1}{2}&\text{for~{}}L+j\text{~{}odd}\end{cases}\,.$ (31)

In addition we have identified the Bethe root configuration for the first
excitation in the $j=0$ sector for chains with even $L$. This is a descendent
of the identity with conformal weight $h=2$ independent of the anisotropy
which we identify with the stress tensor. The results are exhibited Figure 3.

Figure 3:  Finite-size estimates $h(L)=(L/\pi
v_{F})\left(E_{0,x}(L)-E_{0}(L)\right)$ of the conformal weight corresponding
to the first excitation in the total spin $j=0$ sector of the quantum group
invariant spin-$1$ chain (21) for various $\gamma$. The dotted lines
connecting data for even lengths $L=2^{k}$, $k=2,\dots,7$ are a guide to the
eye only. The data converge to $h\equiv 2$ independent of $\gamma$ as
$L\to\infty$ (red dashed line).

We emphasize that this spectrum of conformal weights is discrete for all
values of $\gamma$. At rational values of $\gamma/\pi$ the conformal weights
can be rearranged in terms of an extended algebra with a finite number of
primaries. Specifically, for $\gamma=\pi/2$ (30) with (31) take only integer
and half-odd integer values.

Given the spectral correspondence with the $U_{q}[OSp(3|2)]$ symmetric
superspin chain the critical properties of the quantum group invariant
spin-$1$ chain will provide additional input to elaborate on the conformal
content of the $U_{q}[OSp(3|2)]$ symmetric superspin chain below.

##  V Finite-size spectrum of the $U_{q}[OSp(3|2)]$ chain

Based on the exact diagonalization of small systems we find that the ground
state of the $U_{q}[OSp(3|2)]$ superspin chain with quantum-group invariant
boundary conditions is a $(0;0)$-singlet for $L$ even ($(0;1/2)$-quintet for
$L$ odd) with energy

$E_{0}\equiv
L\epsilon_{\infty}+f_{\infty}^{(OSp)}=-2(L-1)\cot\left(\gamma/2\right)\,,$
(32)

i.e. without any finite-size corrections implying that the effective central
charge of the field theory is $c_{\text{eff}}=0$.222This has also been found
for the isotropic $OSp(3|2)$ superspin chain with both periodic and free
boundary conditions Frahm and Martins (2015, 2022). Note that (32) is _not_
the lowest energy in the ${q}$-deformed $OSp(3|2)$ model subject to periodic
boundary conditions Frahm  _et al._ (2019). For odd $L$ Eq. (32) can be
verified by solving the $bfbfb$ Bethe equations (18) for the quintet where we
find root configurations containing $(L-1)/2$ complex conjugate pairs of roots
on each level which are arranged in groups similar to (20). The root
configurations for even $L$ contain degenerate roots.

Next we have studied the lowest states in the sector
$(\mathtt{p};\mathtt{q})=(0;1)$, see Figure 4: in the thermodynamic limit the
numerical estimates for the conformal weights

$h_{\text{eff}}(L)=\frac{L}{\pi
v_{F}}\,\left(E(L)-L\,\epsilon_{\infty}-f_{\infty}^{(OSp)}\right)\,,$ (33)

converge to the $h_{(0;1)}=\gamma/\pi$ (plus positive integers for the
descendent fields) in accordance with our proposal Eq. (2). For all states we
observe subleading corrections to scaling in the conformal weights vanishing
as $h_{\text{eff}}(L)-h_{(0;1)}\propto L^{-\alpha}$ with an exponent
$\alpha=\gamma/(\pi-\gamma)$. Such power law corrections have also been
observed in the periodic $U_{q}[OSp(3|2)]$ model Frahm  _et al._ (2019).
Typically they originate from the presence of irrelevant operators with
scaling dimensions larger than two Cardy (1986). Note that this perturbation
becomes marginal ($\alpha\to 0$) in the isotropic limit where it becomes the
source of the logarithmic fine structure observed in the conformal spectrum of
the $OSp(3|2)$ chain with free boundaries Frahm and Martins (2022). We note
that the finite size estimates of the conformal weights for some of the
descendent fields are complex. We find, however, that the imaginary parts are
again subleading corrections to scaling which vanish as $L^{-\alpha}$ in the
thermodynamic limit, see Fig. 4.

Figure 4:  Effective conformal weights $h_{\mathrm{eff}}(L)$ of the lowest
states in the sector $(\mathtt{p};\mathtt{q})=(0;1)$: in the top panels the
finite size data are displayed for $\gamma=2\pi/9$ (left panel) and
$\gamma=2\pi/7$ (right panel). Red diamonds indicate the conformal weights
$h_{(0;1)}=\gamma/\pi+k$, $k=0,1,2,3$. Note that two of the descendent fields
correspond to complex energies in the finite size spectrum. For these the real
part of the conformal weight is shown (marked by $\triangle$ and
$\triangledown$). The lower left panel shows finite size data of the lowest
states which extrapolate to $h_{(0;1)}=\gamma/\pi$ in the thermodynamic limit
(indicated by red $\diamond$). The dashed lines indicate the conjectured
corrections to scaling $\propto L^{-\alpha}$ with
$\alpha=\gamma/(\pi-\gamma)$. Similarly, in the lower right panel the
vanishing of the imaginary parts of the complex energies for $\gamma=2\pi/9$
with the same subleading power law $\propto L^{-\alpha}$ is shown.

Proceeding in the same way for the lowest states in the sectors
$(0;\mathtt{q})$ with $\mathtt{q}=\frac{3}{2}$ and $2$ we find that the finite
size data converge to the proposed values $h_{(0;\frac{3}{2})}=3\gamma/\pi$
and $h_{(0;2)}=6\gamma/\pi$ again with subleading power law corrections
$\propto L^{-\alpha}$, $\alpha={\gamma/(\pi-\gamma)}$, see Figure 5. Again,
the energies of the level-2 and -3 descendents of the $(0;\frac{3}{2})$
primary are complex for finite chains with their imaginary parts vanishing as
$L^{-\alpha}$ the thermodynamic limit.

Figure 5: Finite size data for the primary and some descendents in the
$(\mathtt{p};\mathtt{q})=(0;\frac{3}{2})$\- (left panel) and $(0;2)$-sector
(right panel) for $\gamma=2\pi/9$. Red diamonds indicate the conformal weights
$h_{(0;\frac{3}{2})}=3\gamma/\pi+k$, $k=0,1,2,3$ and
$h_{(0;2)}=6\gamma/\pi+k$, $k=0,1$. Note that the level-2 and -3 descendents
of the $(0;\frac{3}{2})$ appear as complex energies in the finite size
spectrum (triangular symbols). Similar as for the complex levels in the
$(0;1)$-sector shown in Fig. 4 the imaginary part of the effective conformal
weights vanishes with the subleading power law in the thermodynamic limit.

Note that the conformal weights of the $(0;\mathtt{q})$-primaries for
$\mathtt{q}=1,\frac{3}{2},2,\dots$ degenerate with the $(0;0)$\- and
$(0;\frac{1}{2})$-vacua of the even and odd length chain with $h=0$ in the
thermodynamic limit. The apparent degeneracy is lifted by the subleading
corrections which become logarithmic in the limit $\gamma\to 0$. This is
consistent with what has been found for the periodic model Martins  _et al._
(1998); Jacobsen  _et al._ (2003); Frahm and Martins (2015) and the
observation in our previous study of the $OSp(3|2)$-symmetric superspin chain
subject to free boundary conditions Frahm and Martins (2022) where we found

$\left.h_{(0;\mathtt{q})}\right|_{\gamma=0}\simeq\frac{2\mathtt{q}(2\mathtt{q}-1)}{\log
L}\,.$ (34)

A similar behaviour can be observed for the conformal weights of the
$(1;\mathtt{q})$-primaries: for $\mathtt{q}=\frac{1}{2}$ and $\frac{3}{2}$
these are realized in the finite size spectrum of even $L$ chains. The
corresponding finite size data for the primaries and some descendents are
shown in Figure 6.

Figure 6: Finite size data for the primary and some descendents in the
$(\mathtt{p};\mathtt{q})=(1;\frac{1}{2})$\- (left panel) and
$(1;\frac{3}{2})$-sector (right panel) for $\gamma=2\pi/9$. Red diamonds
indicate the conformal weights $h_{(1;\frac{1}{2})}=1-\gamma/\pi+k$,
$k=0,1,2,3$ and $h_{(1;\frac{3}{2})}=1+2\gamma/\pi+k$, $k=0,1,2$.

In the thermodynamic limit the effective conformal weights of the primaries in
these sectors converge to $h_{(1,\frac{1}{2})}=1-\gamma/\pi$ and
$h_{(1,\frac{3}{2})}=1+2\gamma/\pi$, respectively, in agreement with our
proposal (2). Again one observes power law corrections to scaling vanishing as
$L^{-\alpha}$.

As mentioned in the introduction the $(1;1)$ representation is atypical and
appears chains of length with either parity as part of indecomposables.
According to (2) the conformal weights of operators for this representation
are integers, independent of the anisotropy $\gamma$. This is supported by the
finite size estimates of the lowest two conformal weights for both odd and
even length presented in Fig. 7.

Figure 7: Finite size data for the primary and some descendents in the
$(\mathtt{p};\mathtt{q})=(1;1)$-sector for $\gamma=2\pi/9$ (left panel) and
$\gamma=2\pi/7$ (right panel). Red diamonds indicate the conformal weights
$h_{(1;1)}=1+k$, $k=0,1,2$. Open (filled) symbols denote data for odd (even)
lattice length.

For $L$ odd the ground state in this sector extrapolates to $h_{(1;1)}=1$,
independent of the anisotropy. The lowest $(1;1)$-level observed for even
length corresponds to a field with conformal weight $h=2$ which we may
identify with the stress tensor.

As in the case of the $\mathtt{p}=0$ the
$(\mathtt{p};\mathtt{q})=(1,\mathtt{q})$-primaries degenerate to give the
integer conformal weight $\left.h_{(1;\mathtt{q})}\right|_{\gamma=0}=1$ in the
thermodynamic limit. The subleading corrections become logarithmic consistent
with the critical behaviour observed for the isotropic $OSp(3|2)$ model Frahm
and Martins (2015).

Finally, we have studied the finite size spectrum of levels in the
$(\mathtt{p};\mathtt{q})=(2;\mathtt{q})$-representations, see Figure 8.

Figure 8: Finite size data for the primary and first descendents in the
$(\mathtt{p};\mathtt{q})=(2;\frac{1}{2})$\- (left panel) and $(2;1)$-sector
(right panel) for $\gamma=2\pi/9$. Red diamonds indicate the conformal weights
$h_{(2;\frac{1}{2})}=3-3\gamma/\pi+k$ and $h_{(2;1)}=3-2\gamma/\pi+k$, both
for $k=0,1$.

For $\mathtt{q}=\frac{1}{2},1$ the effective conformal weights of primaries
and first descendents extrapolate to the proposal (2) for $h_{(2;\mathtt{q})}$
and $h_{(2;\mathtt{q})}+1$. As for the sectors with $\mathtt{p}=0,1$ they
degenerate in the isotropic limit $\gamma\to 0$, consistent with what has been
found previously Jacobsen  _et al._ (2003); Frahm and Martins (2015).

##  VI Summary and Conclusion

Based on our analysis of the finite-size spectrum of the
$U_{q}[OSp(3|2)]$-invariant superspin chain in the previous section together
with the spectral correspondence to the quantum group invariant $XXZ$ spin-$1$
spin chain we arrive at the proposal (2) for the operator content of the
former. We emphasize that this proposal is consistent with previous results
for the isotropic superspin chain: for $\gamma\to 0$ the conformal weights
$h_{(p;q)}=\mathtt{p}(\mathtt{p}+1)/2$ coincide with those identified in the
periodic model Frahm and Martins (2015). Moreover, the amplitudes of the
subleading logarithmic corrections in the $(0;\mathtt{q})$-multiplets of the
isotropic model with free boundaries Frahm and Martins (2022) have the same
$\mathtt{q}$-dependence as (2).

For the models with $\gamma>0$ a similar link to the $q$-deformed superspin
chain with periodic boundary conditions can not be established. In view of
what is known for the periodic model the most striking property of the quantum
group invariant one studied in the present paper is its purely discrete
spectrum of conformal weights, see Fig. 9,

Figure 9: Conformal weights (2) of primaries $(\mathtt{p};\mathtt{q})$ with
$\mathtt{p}=0,1,2,3$, $\mathtt{q}\leq\frac{5}{2}$ in the
$U_{q}[OSp(3|2)]$-invariant superspin as function of the anisotropy: the
degeneracies of weights with different $\mathtt{q}$ in the Goldstone phase of
the isotropic model are lifted for any $\gamma>0$. For $\gamma=\pi/2$ the
possible weights coincide with those of the quantum group invariant $XXZ$
spin-$1$ chain (30) and (31) (red triangles, filled (open) for even (odd)
chain lengths).

while there have been indications for the presence of contiuous components in
the periodic model Frahm  _et al._ (2019). This resembles the behaviour
observed in the open staggered six-vertex model: depending on the choice of
boundary conditions this system flows to different fixed points with the non-
compact one being unstable, see Refs. Robertson  _et al._ (2019, 2021); Frahm
and Gehrmann (2022, 2023).

At this point we remark that solutions of the boundary Yang-Baxter equation
for the $q$-deformed $OSp(3|2)$ spin chain beyond the quantum group invariant
one have been investigated in the literature: in Lima-Santos (2009) the author
claims that there exist two additional diagonal solutions (the most amenable
ones for a Bethe ansatz solution). Just as the constant boundary matrices
$K^{\pm}$ considered here, none of these does contain any free parameter
though. It may well be that non-compact degrees of freedom are present in the
scaling limit of the superspin chain with one of these additional boundary
conditions (or a combination of two of the three known ones). In the absence
of a free parameter such integrable models would however be isolated points in
the space of boundary parameters and the system sizes needed for tests of
whether or not the corresponding critical properties are robust against
perturbations appear to be out of reach for numerical approaches.

Another obstacle is the lack of knowledge on the conformal spectrum of the
periodic superspin chain where the connection to the spectrum of the periodic
Zamolodchikov-Fateev spin-$1$ chain for $\gamma=\pi/2$ is limited to certain
charge-sectors. While this has been taken as a hint that the fields appearing
in the effective low energy theory for the periodic $U_{q}[OSp(3|2)]$ model
may be composites of two Gaussian fields and an Ising operator the preliminary
proposal for the scaling dimensions of the periodic superspin chain is far
from a complete description of the full operator content of the critical
model. In particular, the dependence of the scaling dimensions on the two
possible twist angles related to the conserved $U(1)$ charges has not been
addressed.

## Data availability

The Bethe roots and finite size spectral data used in Figures 2–8 are
available from the Research Data Repository at Leibniz Universität Hannover
Frahm and Martins (2023).

###### 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). MJM was partially supported
by the Brazilian agency CNPq under grant no. 305617/2021-4.

##  Appendix A The quantum group invariant Hamiltonian

Here we write the the quantum group invariant Hamiltonian (6) in the following
form,

$H=\sum_{j=1}^{L-1}H^{b}_{j,j+1}+H^{s}_{L,1}$ (35)

where $H^{b}_{j,j+1}$ represents the two-body bulk term and $H^{s}_{L,1}$
encodes the surface interactions.

The surface term can be represented in the terms of azimuthal bosonic
operators as follows

$H^{s}_{L,1}=-i(\tau^{z}_{L}-\tau^{z}_{1})-2i(\sigma^{z}_{L}-\sigma^{z}_{1})$
(36)

The bulk term encodes both the bosonic and the fermionic generators and it is
given by,

$\displaystyle H^{b}_{j,j+1}$ $\displaystyle=$
$\displaystyle-\frac{\left(C_{j,j+1}-3C_{j,j+1}^{2}\right)}{\sin(\gamma)}+\delta_{1}\tau^{z}_{j}\tau^{z}_{j+1}+\delta_{2}\sigma^{z}_{j}\sigma^{z}_{j+1}+\sin(\gamma)\left(\tau^{z}_{j}\tau^{z}_{j+1}\right)^{2}+\delta_{3}\left(\sigma_{j}^{z}\sigma_{j+1}^{z}\right)^{2}$
(37) $\displaystyle+$
$\displaystyle\frac{\tan(\frac{\gamma}{2})}{2}\left[(\tau_{j}^{+}\tau_{j+1}^{-}+\tau_{j}^{-}\tau_{j+1}^{+})^{2}+2(\tau_{j}^{z})^{2}+2(\tau_{j+1}^{z})^{2}+24(\sigma_{j}^{z})^{2}+24(\sigma_{j+1}^{z})^{2}\right]$
$\displaystyle-$ $\displaystyle
i\left[2+\cos(\gamma)\right]\tau_{j}^{z}\tau_{j+1}^{z}(\tau_{j+1}^{z}-\tau_{j}^{z})+8i\cos(\gamma)\sigma_{j}^{z}\sigma_{j+1}^{z}(\sigma_{j+1}^{z}-\sigma_{j}^{z})$
$\displaystyle-$ $\displaystyle
t_{1}^{-}\left(c_{j}^{+}c_{j+1}^{-}c_{j}^{-}c_{j+1}^{+}+f_{j}^{-}f_{j+1}^{+}f_{j}^{+}f_{j+1}^{-}+\tau_{j}^{z}\tau_{j+1}^{z}c_{j}^{+}c_{j+1}^{-}-4\sigma_{j}^{z}\sigma_{j+1}^{z}c_{j}^{-}c^{+}_{j+1}\right)$
$\displaystyle-$ $\displaystyle
t_{1}^{+}\left(c_{j}^{-}c_{j+1}^{+}c_{j}^{+}c_{j+1}^{-}+f_{j}^{+}f_{j+1}^{-}f_{j}^{-}f_{j+1}^{+}-\tau_{j}^{z}\tau_{j+1}^{z}c_{j}^{-}c_{j+1}^{+}+4\sigma_{j}^{z}\sigma_{j+1}^{z}c_{j}^{+}c^{-}_{j+1}\right)$
$\displaystyle-$
$\displaystyle\frac{t_{2}^{+}}{2}\left[(\tau_{j}^{+}\tau_{j+1}^{-}+\tau_{j}^{-}\tau_{j+1}^{+})\tau_{j}^{z}\tau_{j+1}^{z}-(\tau_{j}^{+}\tau_{j+1}^{-}+\tau_{j}^{-}\tau_{j+1}^{+})f_{j}^{-}f_{j+1}^{+}-8\sigma_{j}^{z}\sigma_{j+1}^{z}d_{j}^{-}d_{j+1}^{+}\right]$
$\displaystyle-$
$\displaystyle\frac{t_{2}^{-}}{2}\left[\tau_{j}^{z}\tau_{j+1}^{z}(\tau_{j}^{+}\tau_{j+1}^{-}+\tau_{j}^{-}\tau_{j+1}^{+})+(\tau_{j}^{+}\tau_{j+1}^{-}+\tau_{j}^{-}\tau_{j+1}^{+})f_{j}^{+}f_{j+1}^{-}+8\sigma_{j}^{z}\sigma_{j+1}^{z}d_{j}^{+}d_{j+1}^{-}\right]$
$\displaystyle+$ $\displaystyle
t_{3}^{+}\left(\tau_{j}^{z}\tau_{j+1}^{z}f_{j}^{-}f_{j+1}^{+}-4\sigma_{j}^{z}\sigma_{j+1}^{z}f_{j}^{+}f_{j+1}^{-}\right)-t_{3}^{-}\left(\tau_{j}^{z}\tau_{j+1}^{z}f_{j}^{+}f_{j+1}^{-}-4\sigma_{j}^{z}\sigma_{j+1}^{z}f_{j}^{-}f_{j+1}^{+}\right)$
$\displaystyle+$ $\displaystyle
4i\cos(\frac{\gamma}{2})\left(f_{j}^{-}f_{j+1}^{+}d_{j}^{-}d_{j+1}^{+}-d_{j}^{-}d_{j+1}^{+}f_{j}^{-}f_{j+1}^{+}\right)-2\left(\tan(\frac{\gamma}{2})+\frac{2}{\sin(\gamma)}\right)\mathrm{I_{j}}\mathrm{I_{j+1}}$

where $I_{j}$ denotes the $5\times 5$ identity acting on the $jth$ lattice
site. The dependence of the coupling parameters on the anisotropy $\gamma$ is,

$\displaystyle\delta_{1}=2\sin^{2}(\frac{\gamma}{2}){\tan(\frac{\gamma}{2})},~{}~{}\delta_{2}=4\left[2+\cos(\gamma)\right]\tan(\frac{\gamma}{2}),~{}~{}\delta_{3}=-16\left[6+\cos(\gamma)\right]\tan(\frac{\gamma}{2})$
$\displaystyle t_{1}^{\pm}=\pm i\frac{\exp(\pm
i\frac{\gamma}{2})}{\cos(\frac{\gamma}{2})},~{}~{}t_{2}^{\pm}=\pm
i\frac{\exp(\pm i\frac{\gamma}{2})}{\cos(\frac{\gamma}{2})}\left(1+\exp(\pm
i\frac{\gamma}{4})\frac{\cos(\frac{\gamma}{2})}{\cos(\frac{\gamma}{4})}\right)$
$\displaystyle t_{3}^{\pm}=\pm i\frac{\exp(\pm
i\frac{\gamma}{2})}{\cos(\frac{\gamma}{2})}\left(1+2\exp(\pm
i\frac{\gamma}{2})\cos(\frac{\gamma}{2})\right)$ (38)

We finally remark that one may use the commutation relations among the bosonic
and fermionic generators of the $OSp(3|2)$ superalgebra to write alternative
expressions to the bulk Hamiltonian.

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# Nodal Line Topological Superfluid and Multiply Protected Majorana Fermi Arc
in a Three-Dimensional Time-Reversal-Invariant Superfluid Model

Beibing Huang<EMAIL_ADDRESS>Department of Physics, Yancheng Institute
of Technology, Yancheng, 224051, China  Yujie Bai  Department of Physics,
Yancheng Institute of Technology, Yancheng, 224051, China  Ning Xu
<EMAIL_ADDRESS>Department of Physics, Yancheng Institute of Technology, Yancheng,
224051, China

###### Abstract

We theoretically study a time-reversal-invariant three-dimensional superfluid
model by stacking in $z$ direction identical bilayer models with intralayer
spin-orbit coupling and contrary Zeeman energy splitting for different layer,
which has been suggested recently to realize two-dimensional time-reversal-
invariant topological superfluid. We find that this model shows two kinds of
topologically nontrivial phases: gapless phases with nodal lines in pairs
protected by chiral symmetry and a gapped phase, both of which support time-
reversal-invariant Majorana Fermi arc (MFA) on the $yz$ and $xz$ side surface.
These MFA abide by time-reversal and particle-hole symmetries and are
topologically protected by the winding numbers in mirror subspaces and $Z_{2}$
numbers of two-dimensional DIII class topological superfluid, different from
MFA in the time-reversal broken Weyl superfluid protected by nonzero Chern
number. This important observation means that MFA in our model represents a
new type of topological state not explored previously. The Zeeman field
configuration in our model is relevant to antiferromagnetic topological
insulator MnBi2Te4, thus our work stimulates the further studies on
superconducting effects in the realistic antiferromagnetic topological
insulator.

##  I Introduction

Accompanying the hottest discussion on gapped topological phase hasan ; Qi ;
Schnyder08 ; Kitaev , gapless topological phase with Fermi surface has
gradually become another studying focus horava ; volovik . Considering all ten
Altland-Zirnbauer symmetry classes, topologically stable Fermi surfaces with
all kinds of codimension $p$ have been classified according to their
topological charges from the K-theory wangzd1 ; wangzd2 . A typical
representative explored in various cold atom systems Gong ; zhangchuanwei1 ;
zhangchuanwei2 ; vincent ; huhui ; liuxiaji ; ketterler ; jianghua  and solid
state materials wanxiangang ; hongming ; shiming ; suyang1 ; bqlv ; suyang2 ;
suyang3 ; nxu ; zhenwang ; balents ; meng ; sau ; fangzhong1  is three-
dimensional (3D) Weyl semimetal/superfluid with codimension $p=3$, which
characterize Weyl point (WP) Fermi surfaces. The WP can be regarded as the
magnetic monopole in momentum space fangzhong2 ; hosur ; cheny  and
topologically protected by nonzero Chern number defined for any two-
dimensional (2D) surface enclosing the WP.

The bulk-boundary correspondence signifies the nontrivial surface states in
Weyl semimetal/superfluid. For any surfaces not perpendicular to lines
connecting WPs, there exists open Fermi arc, which connects projections of the
WPs with opposite Chern number on the surface Brillouin zone wanxiangang . In
terms of Weyl superfluids, Fermi arc possesses Majorana character and is also
denoted by Majorana Fermi arc (MFA) due to inherent particle-hole symmetry
(PHS). To our knowledge in all existing models realizing the Weyl superfluid
Gong ; zhangchuanwei1 ; zhangchuanwei2 ; vincent ; huhui ; liuxiaji ; meng ;
sau , time-reversal symmetry (TRS) is explicitly broken, and MFA is protected
by nonzero Chern number.

In this work, we suggest a 3D theoretical model to study MFA not protected by
nonzero Chern number. Our main results are as followings. The suggested model
realizes two kinds of topologically nontrivial phases, both of which support
MFA. The first kind of phase is gapless. Different from Weyl superfluids, this
gapless phase has nodal lines instead of WPs. The topological stabilities of
nodal lines are protected by chiral symmetry wangzd1 ; beri ; sato . By tuning
the parameters of the model, these nodal lines disappear in pairs, driving the
system into a fully gapped phase, which shows the MFA in the whole Brillouin
zone in the $z$ direction and is different from the conventional gapped
topological phases. Moreover MFA in two kinds of phases are topologically
protected by the winding numbers in mirror subspaces and $Z_{2}$ numbers of 2D
DIII class topological superfluid, but not nonzero Chern number. This
important observation means that MFA in our model represents a new type of
topological state not explored previously.

##  II Theoretical model

The model we consider can be regarded as 3D extension of a bilayer model with
intralayer spin-orbit coupling and contrary Zeeman energy splitting for
different layer, which has been suggested recently to realize 2D time-
reversal-invariant topological superfluid bbhuang . Let $H_{i}$ and
$\mathcal{V}_{i}$ denote single particle Hamiltonian and pairings for ith
bilayer, $H_{i,i+1}$ the coupling between ith and (i+1)th bilayers, the
Hamiltonian of 3D system is

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i}\left[H_{i}+\mathcal{V}_{i}+H_{i,i+1}\right],$ (1)
$\displaystyle H_{i}$ $\displaystyle=$ $\displaystyle\sum_{{\bf
k_{\bot}}\alpha\beta s}\psi_{{\bf k_{\bot}}i\alpha
s}^{\dagger}\left\\{\left[\epsilon_{{\bf
k_{\bot}}}+\Gamma_{\alpha}\sigma_{z}+\lambda_{y}k_{y}\sigma_{x}\right.\right.$
$\displaystyle\left.\left.-\lambda_{x}k_{x}\sigma_{y}\right]_{ss^{\prime}}\delta_{\alpha\beta}-t(1-\delta_{\alpha\beta})\delta_{ss^{\prime}}\right\\}\psi_{{\bf
k_{\bot}}i\beta s^{\prime}},$ $\displaystyle\mathcal{V}_{i}$ $\displaystyle=$
$\displaystyle-\Delta\sum_{{\bf k_{\bot}}\alpha}\psi_{{\bf
k_{\bot}}i\alpha\uparrow}^{{\dagger}}\psi^{{\dagger}}_{-{\bf
k_{\bot}}i\alpha\downarrow}+h.c.,$ $\displaystyle H_{i,i+1}$ $\displaystyle=$
$\displaystyle-t\sum^{\prime}_{{\bf k_{\bot}}s}\psi_{{\bf k_{\bot}}i\alpha
s}^{\dagger}\psi_{{\bf k_{\bot}}i+1\beta s}+h.c.$

where $\psi_{{\bf k_{\bot}}i\alpha s}^{\dagger}$ is the creation operator for
the fermion particle with momentum ${\bf k_{\bot}}$, spin
$s=\uparrow,\downarrow$ and sublayer index $\alpha=1,2$ in a bilayer.
$\epsilon_{{\bf k_{\bot}}}={\bf k^{2}_{\bot}}/2m-\mu$ is in-plane kinetic
energy measured from chemical potential $\mu$. The prime over the summation in
$H_{i,i+1}$ means that the sublayer indexes are limited to $\alpha=2$ and
$\beta=1$, i.e. we only consider the nearest neighborhood tunneling between
the bilayers. $t$ is the tunneling between different sublayers. We assume the
distance between any two nearest layer to be $a/2$, so the length for unit
cell in $z$ direction is $a$. $\lambda_{x}$ and $\lambda_{y}$ are anisotropic
intralayer spin-orbit coupling. Our main investigations are on the isotropic
case $\lambda_{x}=\lambda_{y}=\lambda$. We will briefly discuss the effects of
the anisotropy. $\Gamma_{\alpha}$ are sublayer-dependent effective Zeeman
energy splittings $\Gamma_{1}=-\Gamma_{2}=\Gamma$. Such Zeeman field
configuration can be relevant to Van der Waals layered material MnBi2Te4
zhongdongqin ; lijiaheng ; mikhail , in which the intralayer exchange coupling
is ferromagnetic, giving 2D ferromagnetism in their septuple layer; while the
interlayer exchange coupling is antiferromagnetic, forming 3D A-type
antiferromagnetism in their bulk. On the other hand this required Zeeman field
may also be realized in ultracold atoms using spin-dependent optical lattice
bbhuang ; detush . Under the Nambu basis $\Phi_{{\bf k}}=(\psi_{{\bf
k}},\psi_{-{\bf k}}^{{\dagger}})^{T}$ with $\psi_{{\bf k}}=(\psi_{{\bf
k}1\uparrow},\psi_{{\bf k}1\downarrow},\psi_{{\bf k}2\uparrow},\psi_{{\bf
k}2\downarrow})$ and ${\bf k}=({\bf k_{\bot}},k_{z})$, the Hamiltonian $H$ can
be arranged into the following Bogoliubov-de Gennes (BdG) equation $H={1\over
2}\sum_{{\bf k}}\Phi_{{\bf k}}^{\dagger}H({\bf k})\Phi_{{\bf k}}$ with

$\displaystyle H({\bf k})=\begin{pmatrix}\mathcal{H}_{0}({\bf
k})&\hat{\Delta}\\\ \hat{\Delta}^{\dagger}&-\mathcal{H}_{0}^{\ast}(-{\bf
k})\end{pmatrix},$ (2)

where $\mathcal{H}_{0}({\bf k})=\epsilon_{{\bf
k_{\bot}}}s_{0}\otimes\sigma_{0}+\Gamma
s_{z}\otimes\sigma_{z}-2t\cos{(k_{z}a/2)}s_{x}\otimes\sigma_{x}+\lambda_{y}k_{y}s_{0}\otimes\sigma_{x}-\lambda_{x}k_{x}s_{0}\otimes\sigma_{y}$,
$\hat{\Delta}=-i\Delta s_{0}\otimes\sigma_{y}$. $s_{0,x,y,z}$,
$\sigma_{0,x,y,z}$ are two sets of Pauli matrices acting on the layer and spin
spaces. We will also introduce $\tau_{0,x,y,z}$ as Pauli matrix acting on the
particle-hole space. A feature of this Hamiltonian is that it is free in $xy$
plane and constrained by a lattice in $z$ direction.

The Hamiltonian (2) has PHS $\Sigma H^{\ast}({\bf k})\Sigma^{-1}=-H(-{\bf k})$
with $\Sigma=\tau_{x}\otimes s_{0}\otimes\sigma_{0}$ and TRS
$\mathcal{T}H^{\ast}({\bf k})\mathcal{T}^{-1}=H(-{\bf k})$ with
$\mathcal{T}=i\tau_{0}\otimes s_{x}\otimes\sigma_{y}$. Thus our model belongs
to 3D DIII class. The combination of TRS and PHS gives rise to a chiral
symmetry $C=i\mathcal{T}\Sigma$ with $CH({\bf k})C^{-1}=-H({\bf k})$. In
addition, the model (2) also has spatial inversion symmetry $IH({\bf
k})I^{-1}=H(-{\bf k})$ with $I=\tau_{z}\otimes s_{0}\otimes\sigma_{z}$ and
mirror symmetries
$\mathcal{M}_{x}H(k_{x},k_{y},k_{z})\mathcal{M}^{-1}_{x}=H(-k_{x},k_{y},k_{z})$,
$\mathcal{M}_{y}H(k_{x},k_{y},k_{z})\mathcal{M}^{-1}_{y}=H(k_{x},-k_{y},k_{z})$,
$\mathcal{M}_{z}H(k_{x},k_{y},k_{z})\mathcal{M}^{-1}_{z}=H(k_{x},k_{y},-k_{z})$
with $\mathcal{M}_{x}=I\mathcal{T}$, $\mathcal{M}_{y}=-i\mathcal{T}$ and
$\mathcal{M}_{z}=\tau_{0}\otimes s_{0}\otimes\sigma_{0}$. These mirror
symmetries are very vital to topological stabilities of MFA in the model (2),
as demonstrated below.

Figure 1: Possible phases in the model (2). The schematic plots for two
equations in (4) at $k_{z}=0$ plane. The radius of black circle (dotted) is
$2t/\lambda$, while the radii of red (solid) and blue (dashed) circles are
$\sqrt{W_{c}^{+}}$, $\sqrt{W_{c}^{-}}$ if they can be defined. In 3D the red
(blue) circle corresponds to an open cylindrical surface along the $k_{z}$
axis, while the black circle gradually decreases to zero when $k_{z}$ reaches
the boundary of Brillouin zone to form an irregular spherical surface. Thus as
long as one of blue and red circles is into the black one, the cylindrical
surface(s) will cross irregular spherical surface to form gapless nodal lines
of the model (2). From this criterion (a), (e), (f) are gapped phases, (b) is
gapless phase with four nodal lines and (c), (d) are gapless phases with two
nodal lines. In the maintext, (a) [(e) and (f)] is (are) denoted by gapped
phase I (III), while (c) and (d) [(b)] are (is) denoted by gapless phase II
(IV).

##  III Possible phases and bulk topology

The structure of phase diagram can be obtained from analyzing the determinant
of $H({\bf k})$, which can be calculated with the help of the chiral symmetry.
By finding a unitary matrix $U=(\tau_{x}\otimes
s_{x}\otimes\sigma_{y}+\tau_{z}\otimes s_{0}\otimes\sigma_{0})/\sqrt{2}$ to
diagonalize $C$ that $UCU^{\dagger}=\tau_{z}\otimes s_{0}\otimes\sigma_{0}$,
we can transform the Hamiltonian into an off-diagonal form

$UH({\bf k})U^{\dagger}=\begin{pmatrix}0&Q\\\ Q^{\dagger}&0\end{pmatrix},$ (3)

with $Q=\mathcal{H}_{0}({\bf k})\cdot(s_{x}\otimes\sigma_{y})-\hat{\Delta}$.
Thus $\text{Det}(H({\bf k}))=\text{Det}(Q)\cdot\text{Det}(Q^{\dagger})\geq 0$
and the gap closing condition is $\text{Det}(Q)=0$. Following this method, we
find for our model gap closing signifies that the equations

$\displaystyle\epsilon_{{\bf
k_{\bot}}}^{2}+\Delta^{2}=\Gamma^{2},\lambda^{2}{\bf
k_{\bot}}^{2}=4t^{2}\cos^{2}{\frac{k_{z}a}{2}}$ (4)

have solutions for ${\bf k_{\bot}}$ and $k_{z}$. With the second equation in
(4) defining an irregular spherical surface, the first equation defines
possible cylindrical surfaces depending on the values
$W_{c}^{\pm}=2m(\mu\pm\sqrt{\Gamma^{2}-\Delta^{2}})$. Thus the solutions of
(4) correspond to the intersections of spherical surface and cylindrical
surfaces and define some gapless nodal lines. Since the number of cylindrical
surfaces is at most equal to 2, the maximal number of nodal lines is 4. The
relation between all possible phases and parameters $2t/\lambda$,
$W_{c}^{\pm}$ can be found in Fig.1. Here we notice that the sign of $\mu$
affects the phases available. For $\mu<0$ the maximal number of phases is 3,
but for $\mu>0$ the maximal number of phases is 4.

Below we discuss the topological properties of all phases from the 3D bulk
viewpoint. For the gapless phases, there are pairwise nodal lines. The
featured structures of nodal lines can be visualized from the circles at
$k_{z}=0$ plane in Fig.1. To characterize their stabilities, we calculate the
winding number wangzd1 ; beri ; sato  $N_{1}=\frac{1}{2\pi}\oint_{S^{1}}d{\bf
k}\cdot\partial_{{\bf k}}\text{Im}[\ln\text{Det}(Q)]$, where the contour
$S^{1}$ encircles the one of nodal lines from its transverse direction. A
nonzero $N_{1}$ signifies topological stability. In our model $N_{1}=\pm 1$.
The topological superfluid with nodal lines can be regarded as the supefluid
counterpart of nodal line semimetal widely studied in the solid state
materials Ca3P2 nl1 , SrIrO3 nl2 , PbTaSe2 nl3  and stable 3D carbon allotrope
nl4 ; nl5 . While for gapped phase, we could consider a strong topological
invariant Schnyder08  $N_{2}=\int\frac{d^{3}{\bf
k}}{24\pi^{2}}\epsilon^{ijk}\text{Tr}[(Q^{-1}\partial_{i}Q)(Q^{-1}\partial_{j}Q)(Q^{-1}\partial_{k}Q)]$
with $i,j,k=k_{x},k_{y},k_{z}$ defined in the whole momentum space, but
$N_{2}=0$ due to trivial mirror symmetry $\mathcal{M}_{z}$ in our model, which
leads to $Q(-k_{z})=Q(k_{z})$ zhangfan . $N_{2}=0$ means our model cannot
realize a strong 3D DIII class topological superfluid, but as illustrated in
the next section, the gapped phase III owns anomalous surface states in $xz$
and $yz$ side surfaces, thus it remains topological.

##  IV Majorana Fermi arc and topological protection

In this section surface states on the $yz$ side surface are discussed. We
consider a slab with periodic boundary condition in $y$, $z$ directions and a
finite thickness $L$ along $x$ direction. Expanding the wavefunction
$\psi(x)=\sum_{n=1}^{\infty}c_{n}\sin{(n\pi x/L)}$ dong ; huang , the
calculated results for $k_{y}=0$ are presented in Fig.2. There exists MFA with
four fold degeneracy in two gapless phases and gapped phase III.

Figure 2: Surface states on $yz$ side surfaces at $k_{y}=0$ for four phases in
our model (2). (a)-(d) correspond to phase I-IV, respectively. In (a)
$\Gamma=0.8E_{F}$, $\Delta=1.23E_{F}$, $\mu=-0.07E_{F}$, (b)
$\Gamma=1.3E_{F}$, $\Delta=0.63E_{F}$, $\mu=-0.11E_{F}$, (c)
$\Gamma=1.8E_{F}$, $\Delta=0.22E_{F}$, $\mu=-0.43E_{F}$, (d)
$\Gamma=0.41E_{F}$, $\Delta=0.23E_{F}$, $\mu=0.42E_{F}$, with $t=0.55E_{F}$,
$\lambda k_{F}=1.0E_{F}$. For $E_{F}$ and $k_{F}$, see Fig.3.

To correctly illustrate the observed MFA, we introduce two different
topological invariants. Firstly we regard 3D system as a set of 2D subsystems,
parameterized by the momentum component $k_{z}$. Generally the symmetries
satisfied by a 3D system must not remain symmetries of 2D subsystem,
especially for TRS and PHS, since they correlate the momentum k with
$-\textbf{k}$ non-locally. However in our model due to trivial MS
$\mathcal{M}_{z}$ furusaki , TRS $\mathcal{T}H({\bf
k_{\bot}},k_{z})\mathcal{T}^{-1}=H(-{\bf k_{\bot}},k_{z})$ and PHS $\Sigma
H({\bf k_{\bot}},k_{z})\Sigma^{-1}=-H(-{\bf k_{\bot}},k_{z})$ always exist for
2D subsystems. Thus every $k_{z}$ fixed plane belongs to 2D DIII class and its
topological properties can be distinguished by the $Z_{2}$ number qixiaoliang

$\displaystyle\nu_{2}(k_{z})=\prod_{j}[\text{sgn}(\delta_{j})]^{p_{j}},$ (5)

where $p_{j}$ is the number of time-reversal-invariant points enclosed by the
$j$-th Fermi surface, $\delta_{j,\mathbb{k}}=\langle
j\mathbb{k}|(is_{x}\otimes\sigma_{y})\cdot\hat{\Delta}^{{\dagger}}|j\mathbb{k}\rangle$
and sgn$(\delta_{j})$ is the sign of pairing gap on the Fermi surface with
$|j\mathbb{k}\rangle$ being the eigenstates of $\mathcal{H}_{0}(\mathbb{k})$.

Additionally we consider the mirror symmetry $\mathcal{M}_{y}$. For $k_{y}=0$,
the Hamiltonian commutes with $\mathcal{M}_{y}$ and can be partitioned into
two independent mirror subspaces
$M_{\pm}(k_{x},k_{z})=-\epsilon_{k_{x}}\sigma_{z}\otimes\sigma_{0}-\Gamma\sigma_{z}\otimes\sigma_{z}-(\lambda
k_{x}\pm
2t\cos{\frac{k_{z}a}{2}})\sigma_{z}\otimes\sigma_{y}+\Delta\sigma_{y}\otimes\sigma_{y}$
under the basis of diagonalizing $\mathcal{M}_{y}$ ms1 ; ms2 ; ms3 ; ms4 ; ms5
. Since $\mathcal{M}_{y}$ anticommutes with TRS and PHS, every mirror subspace
belongs to AIII class with the common chiral operator
$S=\sigma_{x}\otimes\sigma_{0}$. Thus every mirror subspace can be further
written into the nondiagonal form

$\displaystyle M_{\pm}(k_{x},k_{z})\sim\begin{pmatrix}0&Q_{\pm}\\\
Q_{\pm}^{\dagger}&0\end{pmatrix}$ (6)

with $Q_{\pm}=\epsilon_{k_{x}}\sigma_{0}-\Gamma\sigma_{z}-(i\Delta+\lambda
k_{x}\pm 2t\cos{\frac{k_{z}a}{2}})\sigma_{y}$. In the mirror subspaces, every
1D subsystem can be classified by the winding number

$\displaystyle
N_{\pm}(k_{z})=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk_{x}\partial_{k_{x}}\text{Im}[\ln\text{Det}Q_{\pm}].$
(7)

Since two mirror subspaces are correlated with each other by TRS and PHS, the
winding numbers $N_{\pm}(k_{z})$ are not independent and we can prove exactly
$N_{+}(k_{z})=N_{-}(k_{z})$.

We have numerically calculated $\nu_{2}(k_{z})$, $N_{\pm}(k_{z})$ and find
that two topological invariants are nonzero as long as
$\Gamma^{2}>\left[2t^{2}\cos^{2}{\frac{k_{z}a}{2}}/(m\lambda^{2})-\mu\right]^{2}+\Delta^{2}$.
In gapped phase I (III), two invariants are trivial (nontrivial) for all
$k_{z}$ fixed subsystems; In gapless phase II, only subsystems with
$k_{z}\in(-\pi/a,-k_{z}^{+})\cup(k_{z}^{+},\pi/a)$ or
$k_{z}\in(-k_{z}^{-},k_{z}^{-})$ are nontrivial, where
$\cos{(k_{z}^{\pm}a/2)}=\sqrt{\lambda^{2}W_{c}^{\pm}/(4t^{2})}$, depending on
that the gapless phase II corresponds to Fig.1(b) or Fig.1(d); While in
gapless IV only subsystems with
$k_{z}\in(-k_{z}^{-},-k_{z}^{+})\cup(k_{z}^{+},k_{z}^{-})$ are nontrivial. We
also find that the range in which two topological invariants are nonzero is
exactly the same as that MFA exist, in other words MFA in the model (2) is
protected by $\nu_{2}(k_{z})$, $N_{\pm}(k_{z})$. No matter which topological
invariant protect the MFA in Fig.2, it is different from MFA in the time-
reversal broken Weyl superfluid protected by nonzero Chern number. This
important observation means that MFA in our model represents a new type of
topological state not explored previously. The same MFA also exists in $xz$
surface and can be analyzed similarly if we consider the mirror symmetry
$\mathcal{M}_{x}$.

##  V phase diagram with self-consistent pairing gap

Within BCS mean-field theory, let $-U$ ($U>0$) denote the effective attraction
strength in each layer. Then the pairing gaps $\Delta_{\alpha}=U\sum_{{\bf
k}}\langle\psi_{-{\bf k}\alpha\downarrow}\psi_{{\bf k}\alpha\uparrow}\rangle$
and the free energy at zero temperature $F={1\over 2}\sum_{{\bf
k}}\left[\text{Tr}\,\mathcal{H}_{0}({\bf k})-\sum_{\eta=1}^{4}E_{\eta{\bf
k}}\right]+(|\Delta_{1}|^{2}+|\Delta_{2}|^{2})/U+\mu N$. Here $E_{\eta{\bf
k}}$ are four positive eigenvalues of BdG Hamiltonian $H({\bf k})$ and $N$ is
the particle number. The integral for ${\bf k}_{\bot}$ is divergent and we
choose to use the relation $1/U=\sum_{{\bf k_{\bot}}}({\bf
k^{2}_{\bot}}/m+\varepsilon_{b})^{-1}$ with $\varepsilon_{b}$ being the
binding energy in 2D free space randeria  to regularize $F$. The physical
properties of the ground state can be obtained by minimizing $F$. Notice that
the conclusions of nodal line topological superfluid and multiply protected
MFA in model (2) is highly depend on whether the order parameters
$\Delta_{\alpha}$ of two layers are equivalent. The same magnitudes for
$\Delta_{\alpha}$ can be directly anticipated from the bilayer’s symmetry. In
order to check whether $\Delta_{\alpha}$ have the same phases, we consider the
ansatz $\Delta_{1}=\Delta$, $\Delta_{2}=\Delta e^{i\phi}$ to investigate the
behavior of free energy $F$ as the function of the relative phase $\phi$ by
self-consistently solving order parameter $\Delta$ and chemical potential
$\mu$. We find that $F$ always obtains minimum at $\phi=0$. Fig.3(a) shows the
landscapes of the free energy for two chosen parameter sets. When
$\Delta_{1}=\Delta_{2}$, TRS $\mathcal{T}$ and chiral symmetry $C$ are
recovered and our conclusions about bulk topology and MFA are valid. The phase
diagram from the self-consistent calculation is shown in Fig.3(b).

Figure 3: The landscapes of free energy $F$ as the function of the relative
phase $\phi$ of two order parameters $\Delta_{\alpha}$ (a) and phase diagram
from the self-consistent calculation (b). In (a) $\varepsilon_{b}=1.0E_{F}$,
$\Gamma=0.8E_{F}$ for the black (upper) line and $\Gamma=1.3E_{F}$ for the red
(lower) line. The step structure in the middle of red (lower) line corresponds
to the normal state with $\Delta_{\alpha}=0$. The other parameters are
$t=0.55E_{F}$, $\lambda k_{F}=1.0E_{F}$. We define the Fermi momentum
$k_{\text{F}}=\sqrt{\pi n}$ from particle number density $n$ for every bilayer
and Fermi energy $E_{F}=k^{2}_{F}/2m$ as the unit of inverse length and
energy. For the definitions of different phases, see Fig.1.

##  VI Discussions

We want to emphasize the following comments. Let us note that the model (2)
also have the chiral symmetry $C$. Thus at $k_{y}=0$ we can define similar
topological invariant (7) by substituting $Q$ for $Q_{\pm}$. The numerical
results suggest that this winding number is often zero and cannot protect the
MFA observed in Fig.2. (This winding number in essence is the same as $N_{1}$,
but its integral path cannot embrace the nodal lines.) However this chiral
symmetry can protect some other interesting surface states. For example when
the projections of nodal lines on certain surfaces do not overlap completely,
there are zero-energy Andreev bound states on this surface located within the
projected nodal rings ryu . Here we are not interested in them.

In our Hamiltonian, Rashba spin-orbit coupling has been assumed, which ensures
flat nodal lines and the same MFA in $yz$ and $xz$ surfaces. However the
existences of gapless nodal lines and MFA are also robust for the anisotropic
spin-orbit coupling. When we change $\lambda_{y}$ ($\lambda_{x}$) slightly,
every flat nodal line will develop into a spatial curve and have a finite
extension along $k_{z}$ axis, while the MFA on $yz$ ($xz$) surfaces will not
change when other parameters are fixed. But for a large anisotropy, the number
of nodal lines and MFA can change once the red or blue circle intersects with
the black ellipse at $k_{z}=0$ in Fig.1. As an example, let us decrease
$\lambda_{y}$ from Fig.1(e) to induce the crossings. As a result gapped phase
III transits into gapless phase II and the MFA in $xz$ surface has the similar
structure with Fig.2(b), with the MFA in $yz$ surface invariant. Since a
nonzero 2D topological invariant $\nu_{2}(k_{z})$ means the simultaneous
presence of edge states in $yz$ and $xz$ surfaces, thus generally for
$\lambda_{x}\neq\lambda_{y}$ the MFA is only protected by $N_{\pm}(k_{z})$.
Moreover the above example also demonstrates that for
$\lambda_{x}\neq\lambda_{y}$ the different phases can own the same MFA in $yz$
or $xz$ surface.

One of the motivations considering sublayer-dependent effective Zeeman energy
splittings $\Gamma_{1}=-\Gamma_{2}=\Gamma$ comes from the recent discovery of
layer antiferromagnetic topological insulator MnBi2Te4 zhongdongqin ;
lijiaheng ; mikhail . In the low energy effective model of MnBi2Te4, there are
$4$ bands for every septuple layer, thus for the antiferromagnetic bilayer
there are $8$ bands totally; While we only consider a two-band case for the
single layer, which is usually used to describe semiconductors or Fermi gases
with spin-orbit coupling and is also the starting point for studying D class
topological superfluid in 2D dc1 ; dc2 ; dc3 . Additionally from the 3D
perspective the single-particle model of MnBi2Te4 has been topological, but it
is not true for our model. Although the model (2) is more simple, it has shown
lots of exotic quantum phenomena, as demonstrated before. Thus to investigate
superconducting effects in the realistic antiferromagnetic topological
insulator should be deserved. The superconductivities in time-reversal-
invariant topological insulator, Dirac and Weyl semimetal have been widely
studied and bring interesting results ti1 ; ti2 ; ti3 ; dw .

Experimentally, four kinds of phases in our model can be solely identified by
radio-frequency spectroscopy in the cold atom gases and angle-resolved
photoemission spectroscopy in condensed matter materials zhengkun ; zhangjing
; jiangkaijun ; arpes1 ; arpes2 ; arpes3 . On one hand gapless and gapped
phases can be distinguished by nodal lines in the gapless phases, meantime
different number of nodal lines can also discriminate different gapless
phases. On the other hand in contrast to the gapped phase I, the existence of
MFA in gapped phase III enhances the low-energy spectral function and becomes
a smoking gun for identifying the different gapped phases.

##  VII Conclusion

To conclude we theoretically study a time-reversal-invariant three-dimensional
superfluid model by stacking a series of bilayers with intralayer spin-orbit
coupling and contrary Zeeman energy splitting for different layer. This model
shows two kinds of topologically nontrivial phases: gapless phases with nodal
lines in pairs protected by chiral symmetry and a gapped time-reversal-
invariant weak topological superfluid phase, both of which support time-
reversal-invariant Majorana Fermi arc on the $yz$ and $xz$ side surface, which
is topologically protected by the winding numbers in mirror subspaces and
$Z_{2}$ numbers of two-dimensional DIII class topological superfluid,
different from Majorana Fermi arc in the time-reversal broken Weyl superfluid
protected by nonzero Chern number. The exotic Majorana Fermi arc predicted
here represents a new type of topological state and provides fertile grounds
for exploring exotic quantum matters.

Acknowledgements. We would like to thank Ming Gong for discussions. This work
is supported by NSFC under Grant No. 11547047 and No. 11704324.

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