# Interacting Topological Quantum Chemistry in 2D: Many-body Real Space
Invariants

Jonah Herzog-Arbeitman1 B. Andrei Bernevig1,2,3 Zhi-Da Song1,4 1Department of
Physics, Princeton University, Princeton, NJ 08544  2Donostia International
Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain
3IKERBASQUE, Basque Foundation for Science, Bilbao, Spain  4International
Center for Quantum Materials, School of Physics, Peking University, Beijing
100871, China

###### Abstract

The topological phases of non-interacting fermions have been classified by
their symmetries, culminating in a modern electronic band theory where
wavefunction topology can be obtained (in part) from momentum space. Recently,
Real Space Invariants (RSIs) have provided a spatially local description of
the global momentum space indices. The present work generalizes this real
space classification to interacting 2D states. We construct many-body local
RSIs as the quantum numbers of a set of symmetry operators on open boundaries,
but which are independent of the choice of boundary. Using the $U(1)$ particle
number, they yield many-body fragile topological indices, which we use to
identify which single-particle fragile states are many-body topological or
trivial at weak coupling. To this end, we construct an exactly solvable
Hamiltonian with single-particle fragile topology that is adiabatically
connected to a trivial state through strong coupling. We then define global
many-body RSIs on periodic boundary conditions. They reduce to Chern numbers
in the band theory limit, but also identify strongly correlated stable
topological phases with no single-particle counterpart. Finally, we show that
the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in
the topological quantum field theory describing the phase.

Introduction. The symmetries of a Hamiltonian are essential to the
classification of topological phases in crystals. For instance, the Ten-Fold
Way Ryu et al. (2010); Kitaev (2009), Topological Quantum Chemistry
(TQC)Bradlyn et al. (2017); Elcoro et al. (2020), and symmetry indicators
Aroyo et al. (2006a); Song et al. (2019a); Po et al. (2017); Kruthoff et al.
(2017) have redefined our understanding of non-interacting electronic states
of matter with the symmetry group of the Hamiltonian taking center stage. The
success of this program motivates us to extend its reach to interacting
Hamiltonians where many-body effects accompany band topology in the
groundstate Else et al. (2019); Thorngren and Else (2018); Lessnich et al.
(2021); Iraola et al. (2021); Misawa and Yamaji (2022); Barkeshli et al.
(2022); Herzog-Arbeitman et al. (2022a); Chen et al. (2021); Herzog-Arbeitman
et al. (2022b); Huang et al. (2022a); Lu and Vishwanath (2012); Peotta and
Törmä (2015); Bernevig et al. (2021); Herzog-Arbeitman et al. (2022c); Xie et
al. (2020); Chiu et al. (2016); Han et al. (2017); Klein Kvorning et al.
(2020); Shiozaki et al. (2018); Wang et al. (2021a); Manjunath and Barkeshli
(2021); Wang et al. (2021b); May-Mann and Hughes (2021); Sen et al. (2020);
Fan et al. (2022); Rachel (2018); Kashihara et al. (2022). This work focuses
on 2D systems with space group $G$ and $U(1)$ charge conservation at an
integer filling per unit cell.

The classifications of single-particle topology originally relied on momentum
space calculations such as the Wilson loop Bouhon et al. (2018); Ahn et al.
(2019); Song et al. (2019b); Alexandradinata et al. (2014); Yu et al. (2011);
Yang et al. (2018); Benalcazar et al. (2017); Bernevig and Hughes (2013) and
band structure irreps Po et al. (2017); Cano and Bradlyn (2020); Watanabe et
al. (2018); Kruthoff et al. (2017); Khalaf et al. (2018); Song et al. (2018).
Physically, however, nontrivial topology is completely diagnosed in real space
where a topological index serves as an obstruction to the atomic limit,
defined by a representation of the groundstate with localized, symmetric
Wannier states Marzari et al. (2012); Soluyanov and Vanderbilt (2011); Brouder
et al. (2007). Recently, Ref. Song et al. (2020) extended this idea to
symmetry-protected phases by developing Real Space Invariants (RSIs) — local
indices which can be considered as Noether charges associated to discrete
symmetries — which may be calculated from the irreps formed by the Wannier
states in the unit cell, even if they are not exponentially localized. These
RSIs are gauge-invariant and classify all 2D and 3D symmetry eigenvalue-
indicated single-particle topology Po (2020); Aroyo et al. (2006a); Song et
al. (2019a); Xu et al. (2021); Herzog-Arbeitman et al. (2022d), but may not
detect some topological states which are classified by cohomology or not
protected by symmetry Gurarie (2011); Essin and Gurarie (2011); Zhang et al.
(2022); Tran et al. (2022); Fidkowski and Kitaev (2010). Our paper extends
this technique by defining _many-body_ RSIs (henceforth referred to as RSIs
for brevity) of two types.

First we construct _local_ RSIs on open boundary conditions (OBCs) in all 2D
point groups. These RSIs classify adiabatically distinct many-body atomic
statesElse et al. (2019); Song et al. (2017); Schindler et al. (2020);
Benalcazar et al. (2020, 2017, 2017). We then define many-body fragile
topological states Else et al. (2019); Po et al. (2018); Song et al. (2019a);
Peri et al. (2020, 2019); Song et al. (2019b); Wieder and Bernevig (2018) by
an obstruction to all many-body atomic limit states and derive their
topological invariants in terms of inequalities between the RSIs and the
$U(1)$ particle number. We also present an exactly solvable model verifying
that interactions trivialize certain fragile non-interacting states identified
by our classification. To study many-body stable topological states Haldane
(1988); Bernevig et al. (2006); Kane and Mele (2005); Bradlyn et al. (2017),
we introduce _global_ RSIs defined on periodic boundary conditions and show
that they are many-body stable topological invariants. Finally, we show that
the local RSIs appear as quantized coefficients in the topological response
theory generalizing the Chern-Simons action of a Chern insulator.

_Many-Body Local RSIs._ Axiomatically, we define a many-body topological state
by an obstruction to adiabatically deforming it into a many-body atomic
(trivial) state while respecting the symmetries of the space group $G$. The
results of this paper rely on the following definition. A many-body atomic
state is any state which is adiabatically connected to a trivial many-body
atomic _limit_ which is $\mathbf{(1)}$ non-degenerate, $\mathbf{(2)}$
spatially decoupled, and $\mathbf{(3)}$ endowed with a many-body gap. This
limit allows for arbitrarily strong interactions in the Hamiltonian as long as
they are strictly local. A many-body atomic limit is the groundstate of the
tight-binding Hamiltonian

$\displaystyle H_{AL}$
$\displaystyle=\sum_{\mathbf{R},\mathbf{r}}H_{\mathbf{R},\mathbf{r}},\qquad
H_{\mathbf{R},\mathbf{r}}=T_{\mathbf{R}}H_{\mathbf{r}}T_{\mathbf{R}}^{\dagger}$
(1)

where $\mathbf{R}$ are the lattice vectors, $T_{\mathbf{R}}\in G$ are the
translation operators, $\mathbf{r}$ are the locations of orbitals in the unit
cell, and $H_{\mathbf{R},\mathbf{r}}$ is supported only on the orbitals at
$\mathbf{R}+\mathbf{r}$ (there are no hoppings). This ensures
$[H_{\mathbf{R},\mathbf{r}},H_{\mathbf{R}^{\prime},\mathbf{r}^{\prime}}]=0$
and thus the groundstate of $H$ can be written as
$\ket{GS}=\prod_{\mathbf{R},\mathbf{r}}\mathcal{O}_{\mathbf{R},\mathbf{r}}\ket{0}$
and $\mathcal{O}_{\mathbf{R},\mathbf{r}}$ creates the groundstate of
$H_{\mathbf{R},\mathbf{r}}$. In the thermodynamic limit, the filling is
$\nu=N_{occ}/N_{orb}$ where $N_{orb}$ is the number of orbitals per unit cell
and $N_{occ}$ is the filling, e.g.
$\prod_{\mathbf{r}}\mathcal{O}_{\mathbf{R},\mathbf{r}}$ creates $N_{occ}$
electrons. Eq. 1 ensures $\mathbf{(2)}$ holds, and we _require_ that
$\mathbf{(1)}$ and $\mathbf{(3)}$ are satisfied, as is natural in an insulator
(see App. B).

We now define local RSIs in many-body atomic limits at a Wyckoff position
$\mathbf{x}$ protected only by the symmetries of the point group
$G_{\mathbf{x}}\in G$. To do so, we identify a set of discrete symmetry
operators whose eigenvalues are the local RSIs. This ensures the local RSIs
are adiabatic invariants since they are discrete quantum numbers. To ensure
locality, we define the local RSI on OBCs by imposing a spatial cutoff around
$\mathbf{x}$ _and_ requiring invariance under the particular choice of cutoff.

Let $\ket{GS,R}$ be the groundstate of $H_{AL}$ but restricted to OBCs by
including only sites $|\mathbf{R}+\mathbf{r}-\mathbf{x}|\leq R$ in Eq. 1. The
cutoff breaks translational symmetry but preserves the point group symmetries
$g\in G_{\mathbf{x}}\subset G$. The quantum numbers of $\ket{GS,R}$ are

$\displaystyle\hat{N}\ket{GS,R}=N\ket{GS,R},\
g\ket{GS,R}=e^{i\lambda[g]}\ket{GS,R}$ (2)

where $\hat{N}$ is the number operator and $e^{i\lambda[g]}$ is a 1D irrep of
$G_{\mathbf{x}}$. Note that $\ket{GS,R}$ is a non-degenerate trivial many-body
atomic limit and must transform in a 1D irrep $(\mathbf{1})$. However, the
quantum numbers $N$ and $\lambda[g]$ are not of the cutoff, as we now show, so
they cannot be local RSIs. Consider the spinless rotation groups
$G_{\mathbf{x}}=n$. Because all terms in $H_{AL}$ are strictly local
$\mathbf{(2)}$, we can write the groundstate at a larger cutoff $R^{\prime}>R$
as

$\displaystyle\ket{GS,R^{\prime}}=\prod_{i=0}^{n-1}\mathcal{O}_{i}\,\ket{GS,R},\quad\mathcal{O}_{i}=\prod_{\mathbf{R}+\mathbf{r}\in\mathcal{D}}C^{i}_{n}\mathcal{O}_{\mathbf{R},\mathbf{r}}C^{{\dagger}i}_{n}$
(3)

where $\mathcal{D}$ is the annulus sector between $R$ and $R^{\prime}$ of
angle $2\pi/n$ and $C_{n}\in G_{\mathbf{x}},C_{n}^{n}=+1$ is an $n$-fold
rotation. Define the total charge $N_{\mathcal{O}}$ by
$[\hat{N},\mathcal{O}_{i}]=N_{\mathcal{O}}\mathcal{O}_{i}$ so that
$\mathcal{O}_{i}\mathcal{O}_{j}=(-1)^{N_{\mathcal{O}}}\mathcal{O}_{j}\mathcal{O}_{i}$.
Since the operators $\mathcal{O}_{i}$ commute/anti-commute if
$N_{\mathcal{O}}$ is even/odd, Eq. 3 gives

$\displaystyle
C_{n}\ket{GS,R^{\prime}}=e^{i\lambda[C_{n}]}(-1)^{N_{\mathcal{O}}}\ket{GS,R^{\prime}},\quad(n\text{
even})\ .$ (4)

Thus the $C_{n}$ eigenvalue $\lambda[C_{n}]$ is _not_ invariant (for even $n$)
under expanding the cutoff because $N_{\mathcal{O}}$ is arbitrary. Similarly,
$\hat{N}\ket{GS,R^{\prime}}=(N+nN_{\mathcal{O}})\ket{GS,R^{\prime}}$ so $N$ is
clearly not invariant. However, we can easily produce symmetry operators which
are invariant under an arbitrary expansion of the cutoff. Indeed,
$e^{i\frac{\pi}{n}\hat{N}}C_{n}$ is invariant because

$\displaystyle\braket{GS,R^{\prime}}{e^{i\frac{\pi}{n}\hat{N}}C_{n}}{GS,R^{\prime}}=e^{i\frac{\pi}{n}(N+nN_{\mathcal{O}})}(-1)^{N_{\mathcal{O}}}e^{i\lambda[C_{n}]}$
$\displaystyle\qquad=e^{i\frac{\pi}{n}N}e^{i\lambda[C_{n}]}=\braket{GS,R}{e^{i\frac{\pi}{n}\hat{N}}C_{n}}{GS,R}\
,$ (5)

and from Eq. 4, we see immediately that $C_{n}^{2}$ is also invariant (for $n$
even). Hence we have found symmetry operators whose eigenvalues only depend on
the local properties of the groundstate near $\mathbf{x}$ but are invariant
under the imposed cutoff. Their eigenvalues are the local RSIs defined by

$\displaystyle e^{i\frac{\pi}{n}\hat{N}}C_{n}\ket{GS}$
$\displaystyle=e^{i\frac{\pi}{n}\Delta_{1}}\ket{GS},\quad\Delta_{1}\in\mathds{Z}_{2n}$
(6) $\displaystyle C^{2}_{n}\ket{GS}$
$\displaystyle=e^{i\frac{2\pi}{n/2}\Delta_{2}}\ket{GS},\quad\Delta_{2}\in\mathds{Z}_{n/2}$

using $(e^{i\frac{\pi}{n}\hat{N}}C_{n})^{2n}=(C_{n}^{2})^{n/2}=+1$ which
quantizes $\Delta_{1},\Delta_{2}$. Although we derived Eq. 6 in many-body
atomic _limits_ , we now prove the local RSIs remain well-defined in general
many-body atomic states. First, observe that the operators in Eq. 6 remain
symmetries as hoppings and off-site interactions are added to $H_{AL}$ and
their eigenvalues (the local RSIs) remain well-defined. Then because we assume
a many-body gap $\mathbf{(3)}$ and the local RSIs are quantized, they cannot
change as $H_{AL}$ is adiabatically deformed out of the strict atomic limit.

We have explicitly constructed local RSIs at a single Wyckoff position
$\mathbf{x}$ on OBCs. But on infinite boundary conditions, the unit cell
contains multiple Wyckoff positions and it should be possible to define local
RSIs $\Delta_{\mathbf{x},1},\Delta_{\mathbf{x},2}$ at each $\mathbf{x}$. For
instance in $G=p2$ with $C_{2}$ and translations, there are four Wyckoff
positions 1a $=(0,0)$, 1b $=(1/2,0)$, 1c $=(0,1/2)$, and 1d $=(1/2,1/2)$ whose
point groups are generated by $C_{2},T_{1}C_{2},T_{2}C_{2}$ and
$T_{1}T_{2}C_{2}$ respectively. Even though imposing OBCs at one Wyckoff
position breaks the symmetries of the others, we argue that the local RSIs are
still well-defined on infinite boundary conditions. This is because the local
RSIs are independent of the cutoff, so sending the cutoff to infinity recovers
the full space group by restoring translations.

In App. B.3, we extend our results to construct local RSIs in all 2D spinless
and spinful point groups (see Tables 3 and 4). In the spinless groups, we find
that mirrors and time-reversal restrict the $C_{n}$ eigenvalue on the
groundstate to be real, reducing the $\mathds{Z}_{2n}\times\mathds{Z}_{n/2}$
classification of Eq. 6 to $\mathds{Z}_{2n}$ for even $n$. For odd $n$, the
reduction is $\mathds{Z}_{n}\times\mathds{Z}_{n}\to\mathds{Z}_{n}$. For even
$n$ in the spinful groups, mirrors and time-reversal also force $C_{n}=\pm 1$
to be real on any non-degenerate state, which yields a $\mathds{Z}_{2}$ factor
in the local RSI classification. We check that $C_{n}=+1$ holds on all product
states (Slater determinants). The $-1$ eigenvalue is only possible with strong
interactions, and can be obtained in “trivial atomic Mott insulators” Soldini
et al. (2022). In all cases, the classifying groups are abelian, so the local
RSIs are additive: the local RSIs of a tensor product of states is the sum of
their individual local RSIs. This is crucial for defining many-body fragile
topology.

_Many-body Fragile Topology_. We now consider a state on infinite boundary
conditions with charge density $\nu=N_{occ}/N_{orb}$. Recall that single-
particle fragile topology is characterized by an obstruction to adiabatic
deformation into an atomic state, but this obstruction is removed if
additional (trivial) orbitals in a specific representation are added Song et
al. (2019a). The topological indices for the single-particle fragile states
are inequalities and mod equations relating $N_{occ}$ and the RSIs in the unit
cell Song et al. (2020).

This structure extends to the many-body case. We define a topological state
with $N_{occ}$ particles per unit cell as many-body fragile iff it can be
adiabatically connected to a many-body trivial atomic state with
$N_{occ}+\tilde{N}$ particles per unit cell by the addition of $\tilde{N}>0$
many-body atomic statesElse et al. (2019); Latimer and Wang (2021). Let
$\Delta_{N_{occ}+\tilde{N}}$ (resp. $\Delta_{\tilde{N}}$) denote the set of
RSIs of the trivial state with $N_{occ}+\tilde{N}$ particles (resp. the
trivial state of the additional $\tilde{N}$ orbitals) at all Wyckoff positions
in the unit cell. The local RSIs of the $N_{occ}$-particle many-body fragile
state are defined by
$\Delta_{frag}=\Delta_{N_{occ}+\tilde{N}}-\Delta_{\tilde{N}}$ (see App. C.2
for details). Crucially, $\Delta_{N_{occ}+\tilde{N}}$ is well-defined because
the $N_{occ}+\tilde{N}$-particle state is trivial atomic. This underlies the
essentially difference between fragile and stable topological many-body states
(to be defined shortly), where the latter _cannot_ be connected to any many-
body atomic state via the addition of any many-body atomic statesSong et al.
(2020).

To assess whether a state is topological given a set of local RSIs, we can
enumerate all possible atomic limits in $G$ formed from $N_{occ}$ orbitals —
note that only a finite number are possible. If the local RSIs of the
groundstate do _not_ appear in this set, then the state must be fragile
topological by definition. In practice, we find a simpler method by deriving
inequality constraints that relate the local RSIs and the $U(1)$ electron
density. To illustrate this, we again consider $G=p2$ with Wyckoff positions
$\mathbf{x}=$1a,1b,1c,1d. There are four RSIs given by the eigenvalues
$e^{i\frac{\pi}{2}\Delta_{1,\mathbf{x}}}$ of
$e^{i\frac{\pi}{2}\hat{N}}C_{2,\mathbf{x}}$ where $C_{2,\mathbf{x}}$ is a
rotation centered at $\mathbf{x}$. It is convenient to define
$\Delta_{1,\mathbf{x}}\in\\{-1,0,1,2\\}$. For many-body atomic states on
arbitrary OBCs respecting $G_{\mathbf{x}}$, it is easy to prove (see App. C)
that $N_{\mathbf{x}}\geq|\Delta_{1,\mathbf{x}}|$ where $N_{\mathbf{x}}$ is the
total number of particles. Note that $\Delta_{1\mathbf{x}}$ is does not depend
on the OBC cutoff, whereas $N_{\mathbf{x}}$ obviously does. In a many-body
atomic _limit_ where we can take $N_{\mathbf{x}}=N_{\mathcal{O}}$ (see Eq. 3)
by choosing a cutoff surrounding $\mathbf{x}$ only, we can bound the total
density by summing over the number of states at the high-symmetry Wyckoff
positions in a single unit cell:

$\displaystyle N_{occ}$
$\displaystyle\geq\sum_{\mathbf{x}=1a,1b,1c,1d}N_{\mathbf{x}}\geq\sum_{\mathbf{x}=1a,1b,1c,1d}|\Delta_{\mathbf{x}}|$
(7)

which is a lower bound because only the high-symmetry Wyckoff positions are
counted. The bound in Eq. 7 is ultimately written in terms of RSIs and the
charge density which are well-defined quantum numbers in any many-body atomic
state. Eq. 7 holds in all many-body atomic states. Hence if Eq. 7 is
_violated_ , then the RSIs impose an obstruction to deformation into a many-
body atomic state, proving many-body fragile topology.

We can now prove that certain single-particle fragile states remain fragile
topological as interactions are added. As a first step, we determine a formula
for the local RSI when acting on product states (which are groundstates
without interactions). With $C_{2}$, the irreps are $A$ and $B$ which are even
and odd under $C_{2}$ respectively. Noting that
$e^{i\frac{\pi}{2}\hat{N}}C_{2}\ket{GS}=e^{i\frac{\pi}{2}(m(A)+m(B))+i\pi
m(B)}\ket{GS}$, Eq. 6 yields $\Delta_{1}=m(A)-m(B)\mod 4$ where $m(\rho)$ is
the multiplicity of the $\rho$ irrep in the product state. In fact, this
expression can be understood perturbatively. The single-particle RSI with
$C_{2}$ is $\delta_{1}=m(B)-m(A)\in\mathds{Z}$ Song et al. (2020). With
interactions, a state with $m(A)=2$ can be scattered into a state with
$m(B)=2$ since both have even parity. Thus states with $\delta_{1}=\pm 2$ are
identified with interactions Liu et al. (2019), and only $\delta_{1}\mod
4=\Delta_{1}$ is invariant. Let us now consider the single-particle
$N_{occ}=2$ fragile state $(A_{1a}\oplus A_{1b}\oplus A_{1c}\ominus
A_{1d})\uparrow G=2\Gamma_{1}\oplus 2X_{1}\oplus 2Y_{1}\oplus 2M_{2}$ which
can be thought as stacking a Chern $+1$ state with a Chern $-1$ state.
Although their total Chern number vanishes, there is still fragile topology
protected by $C_{2}$. We compute the RSIs to be
$\Delta_{1a}=\Delta_{1b}=\Delta_{1c}=1,\Delta_{1d}=-1$. Evaluating the
topological obstruction $\sum_{\mathbf{x}}|\Delta_{\mathbf{x}}|=4$ in Eq. 7,
we find that this single-particle state violates the inequality since
$N_{o}cc=2$ and is _many-body_ fragile. Adiabatically adding interactions
cannot trivialize the state.

We generalize the inequality criterion of Eq. 7 to all wallpaper groups in
App. C, obtaining a classification of many-body fragile phases. In Tables
LABEL:tab:rsiredint and LABEL:tab:mpgrsi, we give expressions for the local
RSIs on product states in terms of irrep multiplicities and also single-
particle RSIs which are readily computed from momentum space irreps Song et
al. (2020). Our results determine the stability of any single-particle fragile
phase when interactions are added.

_Trivializing Fragile Topology._

\begin{overpic}[height=78.04842pt,tics=10]{fragmodellattice}
\end{overpic}\begin{overpic}[height=78.04842pt,tics=10]{fragmodelorb}
\end{overpic}

\begin{overpic}[height=73.7146pt,tics=10]{phasediagrammain_flat} \end{overpic}

Figure 1: Trivializing fragile topology. $(a)$ Orbitals and Wyckoff positions
with $w^{\dagger}_{A_{1a}}$ shown in orange. $(b)$ Phase diagram of
$H_{0}+H_{I}$ where shading denotes the many-body gap. The only gapless point
(blue) occurs at $M=1/2,U=0$ separating the single-particle fragile and
trivial phases at $U=0$. Both phases have the same (trivial) _many-body_ RSIs.
Along the dashed line, the many-body gap is equal to 1 (App. E.3).

If the local RSIs are compatible with a many-body atomic state, our method
indicates there is no obstruction to trivialization — even if the single-
particle RSIs are nontrivial. To show this, we present an exactly solvable
model where we adiabatically deform a single-particle fragile state into a
single-particle trivial state by going through a strong coupling region
Fidkowski and Kitaev (2010).

Our strategy is to build a non-interacting Hamiltonian with fragile valence
bands and obstructed atomic conduction bands. Importantly, we choose the
conduction bands to have Wannier functions which are nonzero on a _finite_
number of orbitals Schindler and Bernevig (2021). We choose $G=p3$ which has
three Wyckoff positions shown in Fig. 1a. Each has spinless PG 3, whose irreps
we denote $A,{}^{1}\\!E,{}^{2}\\!E$ carrying $C_{3}$ eigenvalues
$1,e^{i2\pi/3},e^{-i2\pi/3}$ respectively. We create the atomic orbitals
$A_{1b},{}^{2}\\!E_{1b},{}^{2}\\!E_{1c}$ and form the state

$\displaystyle
w^{\dagger}_{0,A_{1a}}\ket{0}=\frac{1}{3}\sum_{j=0}^{2}C_{3}^{j}(c^{\dagger}_{0,A_{1b}}+c^{\dagger}_{0,{}^{2}\\!E_{1b}}+c^{\dagger}_{0,{}^{2}\\!E_{1c}})\ket{0}$
(8)

in the $\mathbf{R}=0$ unit cell and
$w^{\dagger}_{\mathbf{R},A_{1a}}=T_{\mathbf{R}}w^{\dagger}_{\mathbf{R},A_{1a}}T_{\mathbf{R}}^{\dagger}$.
$w^{\dagger}_{0,A_{1a}}$ creates an $A$ irrep at 1a (see Fig. 1a). The
complementary two bands on the $A_{1b},{}^{2}\\!E_{1b},{}^{2}\\!E_{1c}$
orbitals are fragile and cannot be induced from local orbitals (see App. E).
We also add the atomic orbitals ${}^{1}E_{1a}$ and ${}^{2}E_{1a}$ to the
valence band (which do not _not_ trivialize the fragile topology) and a
$A_{1a}$ orbital to the conduction band. Note that $A_{1a}$ would trivialize
the valence band. To construct the non-interacting Hamiltonian $H_{0}$, we
choose all bands to be perfectly flat so $H_{0}$ is local in the Wannier
basis. As discussed in App. E.2, we set

$\displaystyle H_{0}$
$\displaystyle=\sum_{\mathbf{R}}(1-M)w^{\dagger}_{\mathbf{R},A_{1a}}w_{\mathbf{R},A_{1a}}$
(9)
$\displaystyle\qquad+M(n_{\mathbf{R},{}^{1}\\!E_{1a}}+n_{\mathbf{R},{}^{2}\\!E_{1a}})+(1-M)n_{\mathbf{R},A_{1a}}$

where $n_{\mathbf{R},\rho}=c^{\dagger}_{\mathbf{R},\rho}c_{\mathbf{R},\rho}$.
All terms in $H_{0}$ are strictly local because $w_{\mathbf{R},A_{1a}}$ is
finitely supported. At filling $\nu=N_{occ}/N_{orb}=4/6$, $M$ tunes between a
fragile phase:

$\displaystyle{}^{1}\\!E_{1a}\oplus{}^{2}\\!E_{1a}\oplus[A_{1b}\oplus{}^{2}\\!E_{1b}\oplus{}^{2}\\!E_{1c}\ominus
A_{1a}]\ \uparrow G$ (10)

for $M\in(0,1/2)$ and a trivial phase $A_{1a}\oplus
A_{1b}\oplus{}^{2}\\!E_{1b}\oplus{}^{2}\\!E_{1c}$ for $M\in(1/2,1)$. The two
fragile bands in Eq. 10 are in brackets. A gap closing at $M=1/2$ separates
the two phases. However Table LABEL:tab:mpgrsi shows the RSIs are the same in
both phases: $\Delta_{1a}=(1,0),\ \Delta_{1b}=(2,1),\ \Delta_{1c}=(1,1)$ and
indicate a many-body atomic limit (as can be checked using the topological
indices in Table LABEL:tab:fragilenoSOC). Accordingly, the single-particle
fragile phase can be connected to the trivial phase without a gap closing by
adding interactions. We now add the symmetry-preserving term

$\displaystyle H_{I}$
$\displaystyle=U\sum_{\mathbf{R}}w^{\dagger}_{\mathbf{R},A_{1a}}c^{\dagger}_{\mathbf{R},A_{1a}}c_{\mathbf{R},{}^{1}\\!E_{1a}}c_{\mathbf{R},{}^{2}\\!E_{1a}}+h.c.$
(11)

which is strictly local. Physically, $H_{I}$ implements the interaction-
allowed conversionElse et al. (2019) ${}^{1}\\!E_{1a}\oplus{}^{2}\\!E_{1a}\to
A_{1a}\oplus A_{1a}$ and removes the fragile obstruction symbolized as
$\ominus A_{1a}$ in Eq. 10 (but note that $H_{I}$ annihilates the fragile
bands). Because $H_{0}+H_{I}$ acts independently on the Wannier states in each
unit cell, the Hamiltonian is entirely decoupled and is trivial to solve
Mizoguchi et al. (2021). We show the phase diagram in Fig. 1b and see that the
gap closing at $U=0$ separating the single-particle phases can be opened with
interactions, adiabatically connecting the phases.

_Many-body Stable Topology._ Non-interacting stable topological states (such
as Chern and quantum spin Hall insulators where many-body invariants are known
Thouless et al. (1982); Fu and Kane (2006); Dehghani et al. (2021); Wang and
Zhang (2012); Lee and Ryu (2008)) cannot be trivialized by coupling to any
local orbitals – unlike fragile topology. This is reflected in the single-
particle RSIs, which take on fractional values in stable states Song et al.
(2020). For instance with $C_{2}$, the real-space derivation of the single-
particle RSI $\delta_{1}=m(A)-m(B)\in\mathds{Z}$ relies on the existence of
Wannier functions such that $m(A),m(B)$ are well-defined integers. It is only
by generalizing the definition of $\delta_{1}$ to momentum space (on periodic
boundary conditions) that the possibility of fractional values emerges. In
analogy to the non-interacting case, we define a many-body stable topological
phase to be robust against coupling to all many-body atomic states. It is
impossible to compute local (many-body) RSIs on OBCs in this case because the
edge states, a signature of stable topology, prevent our assumption
$\mathbf{(1)}$ of non-degeneracy and cannot be removed by coupling to any
atomic states.

We now propose a definition of _global_ RSIs $\Delta^{G}_{\mathbf{x},i}$ in
many-body stable topological phases at Wyckoff position $\mathbf{x}$ in the
unit cell. Their definition is identical to Eq. 6 but evaluated on a spatial
torus, i.e. periodic boundary conditions (PBCs). Explicitly, with $C_{n}\in
G_{\mathbf{x}}$ for $n$ even,

$\displaystyle e^{i\frac{\pi}{n}\hat{N}}C_{n}\ket{GS,PBC}$
$\displaystyle=e^{i\frac{\pi}{n}\Delta^{G}_{\mathbf{x},1}}\ket{GS,PBC},$
$\displaystyle C_{n}^{2}\ket{GS,PBC}$
$\displaystyle=e^{i\frac{2\pi}{n/2}\Delta^{G}_{\mathbf{x},2}}\ket{GS,PBC}$
(12)

defines the global RSIs. The PBCs are essential for $\ket{GS,PBC}$ to be non-
degenerate so $\Delta^{G}_{\mathbf{x},i}$ are quantum numbers. We claim that
if $\Delta^{G}_{\mathbf{x},i}\neq 0$, $\ket{GS,PBC}$ is many-body stable. In
other words, the global RSIs are topological invariants. To support this
claim, we prove two properties of $\Delta^{G}_{\mathbf{x},i}$. First, all 2D
many-body atomic phases have $\Delta^{G}_{\mathbf{x},i}=0$, from which it
follows that all many-body fragile topological phases also have
$\Delta^{G}_{\mathbf{x},i}=0$. Second, we evaluate $\Delta^{G}_{\mathbf{x},i}$
on Slater determinants and find that they are determined by the Chern number
$C$. As a corollary, this demonstrates the well-known fact that Chern
insulators are robust to weak interactions.

Consider a many-body atomic limit
$\prod_{\mathbf{R},\mathbf{r}}\mathcal{O}_{\mathbf{R},\mathbf{r}}\ket{0}$ with
$G=p2$ generated by $C_{2},T_{\mathbf{R}}$ where $C_{2}\in G_{\mathbf{x}}$ is
a rotation around the point $\mathbf{x}=(0,0)$. Now consider placing the state
on $L_{1}\times L_{2}$ PBCs with $L_{1},L_{2}$ even. Observe that there are
four points invariant under $C_{2}$ denoted
$\mathbf{x}^{G}=\\{(0,0),(L_{1}/2,0),(0,L_{2}/2),(L_{1}/2,L_{2}/2)\\}$. Since
the $C_{2}$ operator is a symmetry of each point, it protects a local RSI
$\Delta_{\mathbf{x},1}$ given by
$e^{i\frac{\pi}{2}\Delta_{1,\mathbf{x}}}\mathcal{O}_{\mathbf{x}}=(e^{i\frac{\pi}{2}\hat{N}}C_{2})\mathcal{O}_{\mathbf{x}}(e^{i\frac{\pi}{2}\hat{N}}C_{2})^{\dagger}$
for each $\mathbf{x}\in\mathbf{x}^{G}$. In fact, the $\Delta_{1,\mathbf{x}}$
is the same at each $\mathbf{x}\in\mathbf{x}^{G}$ because of translations:
$\mathcal{O}_{\mathbf{x}}=T_{\mathbf{x}}\mathcal{O}_{(0,0)}T_{\mathbf{x}}^{\dagger}$
and $C_{2}T_{\mathbf{x}}C_{2}^{\dagger}=T_{\mathbf{x}}$ since
$\mathbf{x}=-\mathbf{x}$ for $\mathbf{x}\in\mathbf{x}^{G}$ on PBCs.

Now we compute the global RSI. Using the $C_{n}$ symmetry, the groundstate can
generically be written as

$\displaystyle\ket{GS,PBC}=\prod_{\mathbf{x}\in\mathbf{x}^{G}}\\!\mathcal{O}_{\mathbf{x}}\
\prod_{i=1}^{2}C^{i}_{2}\mathcal{O}C_{2}^{i{\dagger}}\ket{0}$ (13)

for some $\mathcal{O}$ (App. D.1). Following Eq. 4, we compute

$\displaystyle
e^{i\frac{\pi}{2}\hat{N}}C_{2}\ket{GS,PBC}=\prod_{\mathbf{x}\in\mathbf{x}^{G}}e^{i\frac{\pi}{2}\Delta_{\mathbf{x},1}}\
\ket{GS,PBC}$ (14)

which is intuitive because operators $\mathcal{O}$ off the $C_{2}$ centers
contribute trivially. Using Eq. Interacting Topological Quantum Chemistry in
2D: Many-body Real Space Invariants, the global RSI is

$\displaystyle\Delta_{\mathbf{x},1}^{G}=\sum_{\mathbf{x}\in\mathbf{x}^{G}}\Delta_{\mathbf{x},1}=4\Delta_{\mathbf{x},1}=0\mod
4$ (15)

since we proved that $\Delta_{\mathbf{x},1}$ are all equal. We extend this
proof to all point groups in App. D.1.

Next, we relate $\Delta^{G}_{\mathbf{x},1}$ to the Chern number $C$ in product
states. We need two existing results. With $C_{2}$, $(-1)^{C}$ is equal to the
product of inversion eigenvalues at the high-symmetry points in the Brillouin
zone (BZ)Hughes et al. (2011); Po (2020); Po et al. (2017); Cano and Bradlyn
(2020), and secondly $C_{2}\ket{GS,PBC}=(-1)^{C}(-1)^{N/2}\ket{GS,PBC}$ where
$N$ is the number of states in the BZ which must be even since $L_{1},L_{2}$
are even Fang et al. (2013). Evaluating the global RSI with Eq. Interacting
Topological Quantum Chemistry in 2D: Many-body Real Space Invariants yields
$\Delta_{1a,1}^{G}=2C\mod 4$. We can also compute $\Delta^{G}_{\mathbf{x},1}$
at other Wyckoff positions taking e.g. $C_{2}\to T_{1}C_{2}$. Then because
$\ket{GS,PBC}$ has zero total many-body momentum in 2D, we find

$\displaystyle\Delta^{G}_{1a,1}=\Delta^{G}_{1b,1}=\Delta^{G}_{1c,1}=\Delta^{G}_{1d,1}=2C\mod
4\ .$ (16)

This result is reminiscent of the half-integer valued single-particle RSIs in
Chern insulators: Eq. 15 gives
$\Delta^{G}_{\mathbf{x},1}=4\Delta_{\mathbf{x},1}$ when local RSIs are well-
defined, so Eq. 16 is suggestive of a half-integer local RSI in odd Chern
states.

We compute the global RSIs of all Slater determinants in App. D.2. Our
classification reveals the possibility of many stable topological phases which
cannot exist in band theory but are enabled by strong interactions, agreeing
with an extending earlier results Liu et al. (2019); Shapourian et al. (2017);
Shiozaki et al. (2017). We give a three illustrative examples. In $p2$, a
state $\ket{\psi}$ with $\Delta_{1a,1}^{G}=\Delta_{1d,1}^{G}=2$,
$\Delta_{1b,1}^{G}=\Delta_{1c,1}^{G}=0$ obeys
$e^{i\frac{\pi}{2}\hat{N}}C_{n}\ket{\psi}=-\ket{\psi}$ like in a Chern state,
but with nonzero total momentum $T_{1}\ket{\psi}=T_{2}\ket{\psi}=-\ket{\psi}$.
In $p2mm$, mirrors ensure $C=-C=0$ without interactions. But the global RSI
$\Delta^{G}_{1}$ retains a $\mathds{Z}_{4}$ classification after adding
mirrors, allowing an “interaction-enabled” state like Eq. 16 following the
heuristic $2C=-2C\mod 4$ Lapa et al. (2014); Isobe and Fu (2015). Finally, Eq.
16 also shows that $\Delta^{G}_{\mathbf{x},1}$ must be even in Slater
determinants, so $\Delta^{G}_{\mathbf{x},1}$ odd is interaction-enabled. By
Eq. Interacting Topological Quantum Chemistry in 2D: Many-body Real Space
Invariants, odd $\Delta^{G}_{\mathbf{x},1}$ requires an odd number of
particles in the groundstate and indicates a gapless state (a correlated
topological semimetal) since $L_{1},L_{2}$ are evenSong et al. (2018); Huang
et al. (2022b).

_Topological Response Theory._ RSIs are quantized, symmetry-protected
invariants beyond the Chern number. Then, since a nonzero Chern number is
encoded in the continuum topological response theory as a Chern-Simons termQi
et al. (2008), it might be expected that RSIs appear as well. In the presence
of crystalline symmetries, the response theory includes Wen-Zee-type termsWen
and Zee (1992); Hung and Wen (2013); Manjunath and Barkeshli (2021); Barkeshli
et al. (2022); Wen (2013); Huang et al. (2022a); Abanov and Gromov (2014);
Zhang et al. (2022a); Mera (2017); Manjunath and Barkeshli (2021); Barkeshli
et al. (2022):

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{C}{4\pi}A\,dA+\frac{s}{2\pi}A\,d\omega+\frac{\ell}{4\pi}\omega\,d\omega$
(17)

where $A=A_{\mu}dx^{\mu}$ is the $U(1)$ gauge field and
$\omega=\omega_{\mu}dx^{\mu}$ is the rotational gauge field, or spin
connection Abanov and Gromov (2014); Thorngren and Else (2018); Manjunath and
Barkeshli (2021); Else et al. (2021). Physically, $\int\\!d\\!A$ and
$\int\\!d\omega$ are the total flux and total disclination angle. Eq. 17
neglects translational gauge fieldsHung and Wen (2013); Zhang et al. (2022b)
and hence ignores the unit cell structure, so $\mathcal{L}$ describes an
expansion around a fixed Wyckoff position $\mathbf{x}$. We will show that $s$
and $\ell$ are the local RSIs at $\mathbf{x}$.

The coefficients $s$ and $\ell$ can be understood from the equation of motion
for the charge density $\rho$ and angular momentum density $L$:

$\displaystyle\rho$ $\displaystyle=\frac{\delta\mathcal{L}}{\delta
A_{0}}=\frac{C}{2\pi}dA+\frac{s}{2\pi}d\omega,\
L=\frac{\delta\mathcal{L}}{\delta\omega_{0}}=\frac{s}{2\pi}dA+\frac{\ell}{2\pi}d\omega\
.$ (18)

Let us first consider $\rho$. If $d\omega=0$, Eq. 18 reduces to the Streda
formulaStreda (1982). If $dA=0$, $s$ describes the charge bound to a
disclination center. A partial disclination with $s\neq 0$ reveals the
fractional charge at $\mathbf{x}$Li et al. (2020); May-Mann and Hughes (2021);
Zhang et al. (2022a). As such, taking $\int d\omega=2\pi$ to be a complete
disclination (in analogy to inserting a full flux), Eq. 18 shows $s=\int\rho$
is the total charge at $\mathbf{x}$. Since the total charge is related to the
local RSIs, Eq. 6 gives (for $n$ even)

$\displaystyle
e^{i\frac{2\pi}{n}s}=e^{i\frac{2\pi}{n}\hat{N}}=(e^{i\frac{\pi}{n}\hat{N}}C_{n})^{2}(C_{n}^{2})^{\dagger}=e^{i\frac{2\pi}{n}(\Delta_{1,\mathbf{x}}-2\Delta_{2,\mathbf{x}})}$
(19)

acting on $\ket{GS}$ with OBCs. Hence $s$ is the local charge

$\displaystyle s=\Delta_{1,\mathbf{x}}-2\Delta_{2,\mathbf{x}}\mod
n\quad\text{($n$ even)}\ .$ (20)

In many-body fragile or atomic states where $\Delta_{i,\mathbf{x}}$ are
integers, $s\in\mathds{Z}_{n}$. Eq. 20 suggests a straightforward
generalization to Chern states. Without interactions, the single-particle
local RSIs are well-defined but fractional, and a formula for the charge $s$
at $\mathbf{x}$ is known Song et al. (2020); Herzog-Arbeitman et al. (2022d).
Proving the many-body extension of this result will be the subject of
forthcoming work. For now in the $C_{2}$ case, note that Eq. 15 (proved only
at $C=0$) shows $4\Delta_{\mathbf{x},1}=\Delta^{G}_{\mathbf{x},1}$, while Eq.
16 shows $\Delta^{G}_{\mathbf{x},1}=2C\mod 4$. Then at least heuristically,
the local RSI $\Delta_{\mathbf{x},1}=C/2\mod 4$ can be half-integer in
agreement with Ref. Zhang et al. (2022a).

We now consider $L$ in Eq. 18. Setting $d\omega=0$ shows that $s$ shifts the
angular momentum after inserting a full flux. Indeed, single-particle RSIs can
enforce irrep flow due to angular momentum pumping in fluxHerzog-Arbeitman et
al. (2022d). Setting $dA=0$ shows that $\ell$ describes the angular momentum
bound to a disclination center and should be identified with the eigenvalue of
the rotation operator $e^{i\frac{\pi}{n}\hat{N}}C_{n}$. Hence we propose
$\ell=\frac{1}{2}\Delta_{1,\mathbf{x}}\in\frac{1}{2}\mathds{Z}_{2n}$ for $n$
even, matching the classification of Ref. Zhang et al. (2022a).

_Discussion._ TQC is a unifying theory of non-interacting materials. Given
atomic orbitals, their symmetries, and the number of electrons, the
topological invariants of TQC classify possible gapped, degenerate, and
gapless phases. The present work achieves the first case in interacting
Hamiltonians by defining local and global RSIs, many-body topological indices,
and the effective field theory that governs them. In so doing, we revealed
which single-particle fragile phases survive interactions and identified
undiscovered stable topological states with no single-particle counterpart. We
leave the study of global and magnetic symmetries, spontaneous symmetry
breaking, the numerical investigation of interaction-enabled many-body stable
topology, and the study of boundary signatures to future work.

_Acknowledgements._ We thank Adrian Po for influential comments in the early
stages of this work, Frank Schindler for useful insight, Abhinav Prem, Jiabin
Yu, and Biao Lian for enlightening consultations, and Titus Neupert, Glenn
Wagner, and Martina Soldini for sharing their unpublished paper on many-body
atomic limits in the final states of this work. JHA thanks the Donostia
International Physics Center for their hospitality during the completion of
this manuscript. B.A.B. and Z-D.S. were supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 101020833), the ONR Grant No. N00014-20-1-2303,
the Schmidt Fund for Innovative Research, Simons Investigator Grant No.
404513, the Packard Foundation, the Gordon and Betty Moore Foundation through
the EPiQS Initiative, Grant GBMF11070 and Grant No. GBMF8685 towards the
Princeton theory program. Further support was provided by the NSF-MRSEC Grant
No. DMR-2011750, BSF Israel US foundation Grant No. 2018226, and the Princeton
Global Network Funds. JHA is supported by a Hertz Fellowship and ONR Grant No.
N00014-20-1-2303, as well as a Marshall Scholarship during the early stages of
this project.

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###### Contents

  1. A Symmetries in Interacting Hamiltonians
  2. B Many-Body Atomic Limits and Construction of RSIs
    1. B.1 1D Many-Body RSI example
    2. B.2 Defining Many-Body RSIs in the Atomic Limit
    3. B.3 Extension to General Point Groups
    4. B.4 Weakly Interacting Limit
  3. C Fragile Topology and Constraints on Particle Number
    1. C.1 Orbital Number Constraints
    2. C.2 Defining Many-Body Local RSIs in Fragile States
    3. C.3 Fragile Inequalities
  4. D Stable Topology and Global RSIs
    1. D.1 Global Many-body RSIs in Many-body Atomic and Fragile States
    2. D.2 Many-Body Global RSIs in the Band Theory Limit
  5. E Details of the Exactly Solvable Hamiltonian
    1. E.1 Construction of the Fragile Flat Band Hamiltonian
    2. E.2 Trivialization by Interactions
    3. E.3 Exact Solution of $H$

##  Appendix A Symmetries in Interacting Hamiltonians

In this Appendix, we lay out our notations for the Hamiltonians, symmetries,
and irreducible representations (irreps) defined in this work. Tables 1 and 2
list the character tables in all point groups (PGs) with and without time-
reversal symmetry.

Let $\mathbf{a}_{1},\mathbf{a}_{2}$ be the 2D Bravais lattice vectors, which
we normalize for convenience by requiring
$\mathbf{a}_{1}\times\mathbf{a}_{2}=+1$, fixing the area of the unit cell to
1. Greek indices $\alpha=1,\dots,N_{orb}$ denote the orbitals of the unit cell
(i.e. the local Hilbert space dimension). We find it convenient to work in an
index convention where $c^{\dagger}_{\mathbf{R},\alpha}$ is the creation
operator in the
$\mathbf{R}=r_{1}\mathbf{a}_{1}+r_{2}\mathbf{a}_{2},r_{i}\in\mathds{Z}$ unit
cell of orbital $\alpha$, which is located at
$\mathbf{R}+\boldsymbol{\delta}_{\alpha}$. We raise and lower indices on the
electron operators to save space in some expressions.

We now discuss crystalline symmetries which form the space group (also called
a wallpaper group in two spatial dimensions) $G$. Comprehensive details of the
space group may be found in Ref. [44, 3]. In brief, $G$ contains an infinite
subgroup composed of the translations $T_{i}$ and a finite set of generators
of rotations and in-plane mirrors which yield the point groups (PGs). $G$ may
also contain time-reversal $\mathcal{T}$, but we do not consider magnetic
space groups explicitly here. However, one can extend our classification to
this case.

A space group symmetry $g\in G$ is defined by its action on the election
operators which reads

$\displaystyle g^{\dagger}c_{\mathbf{R},\alpha}g$
$\displaystyle=\sum_{\beta}D[g]^{\alpha}_{\beta}c_{g(\mathbf{R}+\boldsymbol{\delta}_{\alpha})-\boldsymbol{\delta}_{\beta},\beta}\
.$ (21)

where $D[g]^{\alpha}_{\beta}$, the $N_{orb}\times N_{orb}$ representation
matrix of $g$ on the orbitals, is nonzero only when
$g\boldsymbol{\delta}_{\alpha}=\boldsymbol{\delta}_{\beta}\mod\mathbf{a}_{i}$.
Hence $g(\mathbf{R}+\boldsymbol{\delta}_{\alpha})-\boldsymbol{\delta}_{\beta}$
is a lattice vector and Eq. 21 is well-defined. Spinless representations
describe systems without spin-orbit coupling (SOC) and have
$D[C_{n}]^{n}=D[M]^{2}=+1$ and spinful systems with SOC satisfy
$D[C_{n}]^{n}=D[M]^{2}=-1$, where $C_{n}$ is an $n$-fold rotation and $M$ is a
mirror. In groups with time-reversal $\mathcal{T}$, spinless particles have
$\mathcal{T}^{2}=+1$ and spinful particles have $\mathcal{T}^{2}=-1$. The
representation of $\mathcal{T}$ on the orbitals is denoted $D[\mathcal{T}]K$
where $K$ is the complex conjugation operator, and hence
$D[\mathcal{T}]D^{*}[\mathcal{T}]=\pm 1$ for spinless/spinful particles. The
irreps in all 2D PGs with and without SOC, with and without time-reversal are
shown in Tables 1 and 2. They can also be found, with full group theory data,
on the Bilbao Crystallographic Server [100, 5] as the $k_{z}=0$ slice of the
3D layer groups.

We can now write down a Hamiltonian $H$ with arbitrary interactions that
preserve the space group symmetries. We decompose $H=\sum_{q}H_{q}$ where
$H_{q}$ is a “$q$-body” term. We call $H$ interacting if it contains $2$-body
or higher terms. Explicitly, the $q$-body term is written

$\displaystyle H_{q}$
$\displaystyle=\sum_{\mathbf{R}^{\prime}_{1}\dots\mathbf{R}^{\prime}_{q},\mathbf{R}_{1}\dots\mathbf{R}_{1}}t^{\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\\{\mathbf{R}^{\prime},\mathbf{R}\\})c^{\dagger}_{\mathbf{R}^{\prime}_{1},\alpha_{1}}\dots
c^{\dagger}_{\mathbf{R}_{q}^{\prime},\alpha_{q}}c_{\mathbf{R}_{1},\beta_{1}}\dots
c_{\mathbf{R}_{q},\beta_{q}}$ (22)

where we have summed over the repeated greek indices. The tensor $t$ obeys

$\displaystyle\text{fermion anti-symmetry}:$ $\displaystyle
t^{\alpha_{2}\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\mathbf{R}^{\prime}_{2}\mathbf{R}_{1}^{\prime},\dots)=-t^{\alpha_{1}\alpha_{2}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\mathbf{R}_{1}^{\prime}\mathbf{R}^{\prime}_{2}\dots),$
(23) $\displaystyle\text{Hermiticity}:$
$\displaystyle\left(t^{\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\\{\mathbf{R}^{\prime},\mathbf{R}\\})\right)^{*}=t_{\alpha_{1}\dots\alpha_{q}}^{\beta_{1}\dots\beta_{q}}(\\{\mathbf{R},\mathbf{R}^{\prime}\\})\
.$

By imposing translation invariance, we find

$\displaystyle
t^{\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\\{\mathbf{R}^{\prime},\mathbf{R}\\})$
$\displaystyle=t^{\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\\{\mathbf{R}^{\prime}+\mathbf{a}_{i},\mathbf{R}+\mathbf{a}_{i}\\})\
.$ (24)

Continuing to impose that $g\in G$ is a symmetry of $H_{q}$ for general
elements of the space group, we require

$\displaystyle g^{\dagger}H_{q}g$
$\displaystyle=\sum_{\\{\mathbf{R},\mathbf{R}^{\prime}\\}}t^{\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\\{\mathbf{R}^{\prime},\mathbf{R}\\})g^{\dagger}c^{\dagger}_{\mathbf{R}^{\prime}_{1},\alpha_{1}}g\dots
g^{\dagger}c^{\dagger}_{\mathbf{R}_{q}^{\prime},\alpha_{q}}gg^{\dagger}c_{\mathbf{R}_{1}}^{\beta_{1}}g\dots
g^{\dagger}c_{\mathbf{R}_{q}}^{\beta_{q}}g$ (25)
$\displaystyle=\sum_{\\{\mathbf{R},\mathbf{R}^{\prime}\\}}(D^{\dagger}[g]_{\alpha_{1}}^{\alpha^{\prime}_{1}}\dots
D^{\dagger}[g]_{\alpha_{q}}^{\alpha^{\prime}_{q}})t^{\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\\{g^{-1}(\mathbf{R}^{\prime}+\boldsymbol{\delta}_{\alpha^{\prime}})-\boldsymbol{\delta}_{\alpha},g^{-1}(\mathbf{R}+\boldsymbol{\delta}_{\beta^{\prime}})-\boldsymbol{\delta}_{\beta}\\})(D[g]^{\beta_{1}}_{\beta^{\prime}_{1}}\dots)c^{\dagger}_{\mathbf{R}_{1}^{\prime},\alpha_{1}^{\prime}}\dots
c^{\dagger}_{\mathbf{R}_{q}^{\prime},\alpha^{\prime}_{q}}c_{\mathbf{R}_{1}}^{\beta^{\prime}_{1}}\dots
c_{\mathbf{R}_{q}}^{\beta^{\prime}_{q}}$

The $\alpha_{1},\dots,\alpha_{q}$ and $\beta_{1},\dots,\beta_{q}$ indices are
fully anti-symmetric because they contract with the fully anti-symmetric
$t_{\beta_{1},\dots,\beta_{q}}^{\alpha_{1},\dots,\alpha_{q}}$ tensor, and the
$\alpha^{\prime}_{1},\dots,\alpha^{\prime}_{q}$ and
$\beta^{\prime}_{1},\dots,\beta^{\prime}_{q}$ indices are fully anti-symmetric
because they contract with the electron operators. Now using
$gH_{q}g^{\dagger}=H_{q}$, we obtain

$\displaystyle(D^{\dagger}[g]_{\alpha_{1}}^{\alpha^{\prime}_{1}}\dots
D^{\dagger}[g]_{\alpha_{q}}^{\alpha^{\prime}_{q}})t^{\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\\{\mathbf{R}^{\prime}+\boldsymbol{\delta}_{\alpha^{\prime}})-\boldsymbol{\delta}_{\alpha},g^{-1}(\mathbf{R}+\boldsymbol{\delta}_{\beta^{\prime}})-\boldsymbol{\delta}_{\beta}\\}\\}(D[g]^{\beta_{1}}_{\beta^{\prime}_{1}}\dots
D[g]^{\beta_{q}}_{\beta^{\prime}_{q}})$
$\displaystyle=t^{\alpha^{\prime}_{1}\dots\alpha^{\prime}_{q}}_{\beta^{\prime}_{1}\dots\beta^{\prime}_{q}}(\\{\mathbf{R}^{\prime},\mathbf{R}\\})$
(26)

so $t$ transforms in the representation
$\bigwedge_{i=1}^{q}D[g]=D[g]\wedge\dots\wedge D[g]$. Here $\wedge$ denotes
the exterior (anti-symmetric) product which anti-symmetrizes
$D^{\dagger}[g]_{\alpha_{1}}^{\alpha^{\prime}_{1}}\dots
D^{\dagger}[g]_{\alpha_{q}}^{\alpha^{\prime}_{q}}$ in its top and bottom
indices. Time reversal behaves similarly. Repeating the calculation, we find

$\displaystyle(D^{\dagger}[\mathcal{T}]_{\alpha_{1}}^{\alpha^{\prime}_{1}}\dots
D^{\dagger}[\mathcal{T}]_{\alpha_{q}}^{\alpha^{\prime}_{q}}){t^{*}}^{\alpha_{1}\dots\alpha_{q}}_{\beta_{1}\dots\beta_{q}}(\\{\mathbf{R}^{\prime},\mathbf{R}\\})(D[\mathcal{T}]^{\beta_{1}}_{\beta^{\prime}_{1}}\dots
D[\mathcal{T}]^{\beta_{q}}_{\beta^{\prime}_{q}})$
$\displaystyle=t^{\alpha^{\prime}_{1}\dots\alpha^{\prime}_{q}}_{\beta^{\prime}_{1}\dots\beta^{\prime}_{q}}(\\{\mathbf{R}^{\prime},\mathbf{R}\\})\
.$ (27)

Finally, we remark that our expression for $H$ in Eq. 22 is particle-number
preserving. By construction, $H$ commutes with the total $U(1)$ charge
operator

$\displaystyle U(\theta)$
$\displaystyle=e^{i\theta\hat{N}},\quad\hat{N}=\sum_{\mathbf{R}}c^{\dagger}_{\mathbf{R},\alpha}c_{\mathbf{R},\alpha}\
$ (28)

for all $\theta$, and we have set the electron charge to 1. $U(\theta)$
generates the $U(1)$ global symmetry and allows us to fix the total $U(1)$
particle number to $N$. The filling in the thermodynamic limit is
$\nu=N/A=N_{occ}/N_{orb}$ where $A$ is the number of unit cells, $N_{orb}$ is
the number of orbitals per unit cell, and $N_{occ}$ is the number of electrons
per unit cell.

PG $2$ | 1 | $2$
---|---|---
$A$ | 1 | 1
$B$ | 1 | $-1$
${}^{2}\overline{E}$ | 1 | $-i$
${}^{1}\overline{E}$ | 1 | $i$

PG $m$ | 1 | $m$
---|---|---
$A^{\prime}$ | $A^{\prime}$ | 1 | 1
$A^{\prime\prime}$ | $A^{\prime\prime}$ | 1 | $-1$
${}^{2}E_{1/2}$ | ${}^{2}\overline{E}$ | 1 | $-i$
${}^{1}E_{1/2}$ | ${}^{1}\overline{E}$ | 1 | $i$

PG $2mm$ | 1 | $2$ | $m_{100}$ | $m_{010}$
---|---|---|---|---
$A_{1}$ | $A_{1}$ | 1 | 1 | 1 | 1
$A_{2}$ | $A_{2}$ | 1 | 1 | $-1$ | $-1$
$B_{1}$ | $B_{1}$ | 1 | $-1$ | $-1$ | 1
$B_{2}$ | $B_{2}$ | 1 | $-1$ | 1 | $-1$
$E_{1/2}$ | $\overline{E}$ | 2 | 0 | 0 | 0

PG $4$ | 1 | $4^{+}$ | $2$ | $4^{-}$
---|---|---|---|---
$A$ | $A$ | 1 | 1 | 1 | 1
$B$ | $B$ | 1 | $-1$ | 1 | $-1$
${}^{1}E$ | ${}^{1}E$ | 1 | $-i$ | $-1$ | $i$
${}^{2}E$ | ${}^{2}E$ | 1 | $i$ | $-1$ | $-i$
${}^{1}E_{1/2}$ | ${}^{2}\overline{E}_{1}$ | 1 | $e^{-i\frac{\pi}{4}}$ | $-i$ | $e^{i\frac{\pi}{4}}$
${}^{2}E_{1/2}$ | ${}^{1}\overline{E}_{1}$ | 1 | $e^{i\frac{\pi}{4}}$ | $i$ | $e^{-i\frac{\pi}{4}}$
${}^{1}E_{3/2}$ | ${}^{2}\overline{E}_{2}$ | 1 | $e^{i\frac{3\pi}{4}}$ | $-i$ | $e^{-i\frac{3\pi}{4}}$
${}^{2}E_{3/2}$ | ${}^{1}\overline{E}_{2}$ | 1 | $e^{-i\frac{3\pi}{4}}$ | $i$ | $e^{i\frac{3\pi}{4}}$

PG $4mm$ | 1 | $\\{4^{+},4^{-}\\}$ | $2$ | $\\{m_{010},m_{100}\\}$ | $\\{m_{110},m_{1-10}\\}$
---|---|---|---|---|---
$A_{1}$ | $A_{1}$ | 1 | 1 | 1 | 1 | 1
$A_{2}$ | $A_{2}$ | 1 | 1 | 1 | $-1$ | $-1$
$B_{1}$ | $B_{1}$ | 1 | $-1$ | 1 | 1 | $-1$
$B_{2}$ | $B_{2}$ | 1 | $-1$ | 1 | $-1$ | 1
$E$ | $E$ | 2 | 0 | $-2$ | 0 | 0
$E_{1/2}$ | $\overline{E}_{1}$ | 2 | $\sqrt{2}$ | 0 | 0 | 0
$E_{3/2}$ | $\overline{E}_{2}$ | 2 | $-\sqrt{2}$ | 0 | 0 | 0

PG 3 | 1 | $3^{+}$ | $3^{-}$
---|---|---|---
$A$ | $A$ | 1 | 1 | 1
${}^{2}E$ | ${}^{2}E$ | 1 | $e^{i\frac{2\pi}{3}}$ | $e^{-i\frac{2\pi}{3}}$
${}^{1}E$ | ${}^{1}E$ | 1 | $e^{-i\frac{2\pi}{3}}$ | $e^{i\frac{2\pi}{3}}$
$A_{3/2}$ | $\overline{E}$ | 1 | $-1$ | $-1$
${}^{2}E_{1/2}$ | ${}^{1}\overline{E}$ | 1 | $e^{-i\frac{\pi}{3}}$ | $e^{i\frac{\pi}{3}}$
${}^{1}E_{1/2}$ | ${}^{2}\overline{E}$ | 1 | $e^{i\frac{\pi}{3}}$ | $e^{-i\frac{\pi}{3}}$

PG $3m$ | 1 | $\\{3^{+},3^{-}\\}$ | $\\{m_{120},m_{210},m_{1-10}\\}$
---|---|---|---
$A_{1}$ | $A_{1}$ | 1 | 1 | 1
$A_{2}$ | $A_{2}$ | 1 | 1 | $-1$
$E$ | $E$ | 2 | $-1$ | 0
$E_{1/2}$ | $\overline{E}_{1}$ | 2 | 1 | 0
${}^{1}E_{3/2}$ | ${}^{1}\overline{E}$ | 1 | $-1$ | $i$
${}^{2}E_{3/2}$ | ${}^{2}\overline{E}$ | 1 | $-1$ | $-i$

PG 6 | 1 | $6^{+}$ | $3^{+}$ | $2$ | $3^{-}$ | $6^{-}$
---|---|---|---|---|---|---
$A$ | $A$ | 1 | 1 | 1 | 1 | 1 | 1
$B$ | $B$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$
${}^{1}E_{1}$ | ${}^{1}E_{2}$ | 1 | $e^{-i\frac{\pi}{3}}$ | $e^{-i\frac{2\pi}{3}}$ | $-1$ | $e^{i\frac{2\pi}{3}}$ | $e^{i\frac{\pi}{3}}$
${}^{2}E_{1}$ | ${}^{2}E_{2}$ | 1 | $e^{i\frac{\pi}{3}}$ | $e^{i\frac{2\pi}{3}}$ | $-1$ | $e^{-i\frac{2\pi}{3}}$ | $e^{-i\frac{\pi}{3}}$
${}^{1}E_{2}$ | ${}^{1}E_{1}$ | 1 | $e^{i\frac{2\pi}{3}}$ | $e^{i\frac{4\pi}{3}}$ | 1 | $e^{i\frac{2\pi}{3}}$ | $e^{i\frac{4\pi}{3}}$
${}^{2}E_{2}$ | ${}^{2}E_{1}$ | 1 | $e^{-i\frac{2\pi}{3}}$ | $e^{-i\frac{4\pi}{3}}$ | 1 | $e^{-i\frac{2\pi}{3}}$ | $e^{-i\frac{4\pi}{3}}$
${}^{1}E_{1/2}$ | ${}^{2}\overline{E}_{3}$ | 1 | $e^{i\frac{\pi}{6}}$ | $e^{i\frac{\pi}{3}}$ | $i$ | $e^{-i\frac{\pi}{3}}$ | $e^{-i\frac{\pi}{6}}$
${}^{2}E_{1/2}$ | ${}^{1}\overline{E}_{3}$ | 1 | $e^{-i\frac{\pi}{6}}$ | $e^{-i\frac{\pi}{3}}$ | $-i$ | $e^{i\frac{\pi}{3}}$ | $e^{i\frac{\pi}{6}}$
${}^{1}E_{3/2}$ | ${}^{2}\overline{E}_{1}$ | 1 | $-i$ | $-1$ | $i$ | $-1$ | $i$
${}^{2}E_{3/2}$ | ${}^{1}\overline{E}_{1}$ | 1 | $i$ | $-1$ | $-i$ | $-1$ | $-i$
${}^{1}E_{5/2}$ | ${}^{2}\overline{E}_{2}$ | 1 | -$e^{i\frac{\pi}{6}}$ | $e^{i\frac{\pi}{3}}$ | $-i$ | $e^{-i\frac{\pi}{3}}$ | -$e^{-i\frac{\pi}{6}}$
${}^{2}E_{5/2}$ | ${}^{1}\overline{E}_{2}$ | 1 | -$e^{-i\frac{\pi}{6}}$ | $e^{-i\frac{\pi}{3}}$ | $i$ | $e^{i\frac{\pi}{3}}$ | -$e^{i\frac{\pi}{6}}$

PG $6mm$ | 1 | $\\{6^{+},6^{-}\\}$ | $\\{3^{+},3^{-}\\}$ | $2$ | $\\{m_{100},m_{010},m_{110}\\}$ | $\\{m_{120},m_{210},m_{1-10}\\}$
---|---|---|---|---|---|---
$A_{1}$ | $A_{1}$ | 1 | 1 | 1 | 1 | 1 | 1
$A_{2}$ | $A_{2}$ | 1 | 1 | 1 | 1 | $-1$ | $-1$
$B_{1}$ | $B_{1}$ | 1 | $-1$ | 1 | $-1$ | $-1$ | 1
$B_{2}$ | $B_{2}$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$
$E_{1}$ | $E_{1}$ | 2 | 1 | $-1$ | $-2$ | 0 | 0
$E_{2}$ | $E_{2}$ | 2 | $-1$ | $-1$ | 2 | 0 | 0
$E_{1/2}$ | $\overline{E}_{1}$ | 2 | $\sqrt{3}$ | 1 | 0 | 0 | 0
$E_{3/2}$ | $\overline{E}_{3}$ | 2 | 0 | $-2$ | 0 | 0 | 0
$E_{5/2}$ | $\overline{E}_{2}$ | 2 | $-\sqrt{3}$ | 1 | 0 | 0 | 0

Table 1: The character tables of 2D magnetic PGs. The irrep names are shown in
the first column using the Altmann-Herzig notation and second columns using
the Bilbao Crystallographic notation. However, in $p6$, the notation differs
from the BCS but is consistent with Ref. [51]. In each table, the linear
irreps above the second horizontal line are the single-valued irreps (no-SOC),
and the projective irreps below the second horizontal line are the double-
valued irreps (SOC).

PG $21^{\prime}$ | 1 | $2$
---|---|---
$A$ | 1 | 1
$B$ | 1 | $-1$
${}^{1}\overline{E}{}^{2}\overline{E}$ | 2 | $0$

PG $m1^{\prime}$ | 1 | $m$
---|---|---
$A^{\prime}$ | $A^{\prime}$ | 1 | 1
$A^{\prime\prime}$ | $A^{\prime\prime}$ | 1 | $-1$
${}^{2}E_{1/2}{}^{1}E_{1/2}$ | ${}^{2}\overline{E}{}^{1}\overline{E}$ | 2 | $0$

PG $2mm1^{\prime}$ | 1 | $2$ | $m_{100}$ | $m_{010}$
---|---|---|---|---
$A_{1}$ | $A_{1}$ | 1 | 1 | 1 | 1
$A_{2}$ | $A_{2}$ | 1 | 1 | $-1$ | $-1$
$B_{1}$ | $B_{1}$ | 1 | $-1$ | $-1$ | 1
$B_{2}$ | $B_{2}$ | 1 | $-1$ | 1 | $-1$
$E_{1/2}$ | $\overline{E}$ | 2 | 0 | 0 | 0

PG $41^{\prime}$ | 1 | $4^{+}$ | $2$ | $4^{-}$
---|---|---|---|---
$A$ | $A$ | 1 | 1 | 1 | 1
$B$ | $B$ | 1 | $-1$ | 1 | $-1$
${}^{1}E{}^{2}E$ | ${}^{1}E{}^{2}E$ | 2 | $0$ | $-2$ | $0$
${}^{1}E_{1/2}{}^{2}E_{1/2}$ | ${}^{1}\overline{E}_{1}{}^{2}\overline{E}_{1}$ | 2 | $\sqrt{2}$ | $0$ | $\sqrt{2}$
${}^{1}E_{3/2}{}^{2}E_{3/2}$ | ${}^{1}\overline{E}_{2}{}^{2}\overline{E}_{2}$ | 2 | $-\sqrt{2}$ | $0$ | $-\sqrt{2}$

PG $4mm1^{\prime}$ | 1 | $\\{4^{+},4^{-}\\}$ | $2$ | $\\{m_{010},m_{100}\\}$ | $\\{m_{110},m_{1-10}\\}$
---|---|---|---|---|---
$A_{1}$ | $A_{1}$ | 1 | 1 | 1 | 1 | 1
$A_{2}$ | $A_{2}$ | 1 | 1 | 1 | $-1$ | $-1$
$B_{1}$ | $B_{1}$ | 1 | $-1$ | 1 | 1 | $-1$
$B_{2}$ | $B_{2}$ | 1 | $-1$ | 1 | $-1$ | 1
$E$ | $E$ | 2 | 0 | $-2$ | 0 | 0
$E_{1/2}$ | $\overline{E}_{1}$ | 2 | $\sqrt{2}$ | 0 | 0 | 0
$E_{3/2}$ | $\overline{E}_{2}$ | 2 | $-\sqrt{2}$ | 0 | 0 | 0

PG $31^{\prime}$ | 1 | $3^{+}$ | $3^{-}$
---|---|---|---
$A$ | $A$ | 1 | 1 | 1
${}^{1}E{}^{2}E$ | ${}^{1}E{}^{2}E$ | 2 | $-1$ | $-1$
$A_{3/2}A_{3/2}$ | $\overline{EE}$ | 2 | $-2$ | $-2$
${}^{2}E_{1/2}{}^{1}E_{1/2}$ | ${}^{1}\overline{E}{}^{2}\overline{E}$ | 2 | $1$ | $1$

PG $3m1^{\prime}$ | 1 | $\\{3^{+},3^{-}\\}$ | $\\{m_{120},m_{210},m_{1-10}\\}$
---|---|---|---
$A_{1}$ | $A_{1}$ | 1 | 1 | 1
$A_{2}$ | $A_{2}$ | 1 | 1 | $-1$
$E$ | $E$ | 2 | $-1$ | 0
$E_{1/2}$ | $\overline{E}_{1}$ | 2 | 1 | 0
${}^{1}E_{3/2}{}^{2}E_{3/2}$ | ${}^{1}\overline{E}{}^{2}\overline{E}$ | 2 | -2 | $0$

PG $61^{\prime}$ | 1 | $6^{+}$ | $3^{+}$ | $2$ | $3^{-}$ | $6^{-}$
---|---|---|---|---|---|---
$A$ | $A$ | 1 | 1 | 1 | 1 | 1 | 1
$B$ | $B$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$
${}^{1}E_{1}{}^{2}E_{1}$ | ${}^{1}E_{2}{}^{2}E_{2}$ | 2 | $1$ | $-1$ | $-2$ | $-1$ | $1$
${}^{1}E_{2}{}^{2}E_{2}$ | ${}^{1}E_{1}{}^{2}E_{1}$ | 2 | $-1$ | $-1$ | 2 | $-1$ | $-1$
${}^{1}E_{1/2}{}^{2}E_{1/2}$ | ${}^{1}\overline{E}_{3}{}^{2}\overline{E}_{3}$ | 2 | $\sqrt{3}$ | $1$ | $0$ | $1$ | $\sqrt{3}$
${}^{1}E_{3/2}{}^{2}E_{3/2}$ | ${}^{1}\overline{E}_{1}{}^{2}\overline{E}_{1}$ | 2 | $0$ | $-2$ | $0$ | $-2$ | $0$
${}^{1}E_{5/2}{}^{2}E_{5/2}$ | ${}^{1}\overline{E}_{2}{}^{2}\overline{E}_{2}$ | 2 | $-\sqrt{3}$ | $1$ | $0$ | $1$ | $-\sqrt{3}$

PG $6mm1^{\prime}$ | 1 | $\\{6^{+},6^{-}\\}$ | $\\{3^{+},3^{-}\\}$ | $2$ | $\\{m_{100},m_{010},m_{110}\\}$ | $\\{m_{120},m_{210},m_{1-10}\\}$
---|---|---|---|---|---|---
$A_{1}$ | $A_{1}$ | 1 | 1 | 1 | 1 | 1 | 1
$A_{2}$ | $A_{2}$ | 1 | 1 | 1 | 1 | $-1$ | $-1$
$B_{1}$ | $B_{1}$ | 1 | $-1$ | 1 | $-1$ | $-1$ | 1
$B_{2}$ | $B_{2}$ | 1 | $-1$ | 1 | $-1$ | 1 | $-1$
$E_{1}$ | $E_{1}$ | 2 | 1 | $-1$ | $-2$ | 0 | 0
$E_{2}$ | $E_{2}$ | 2 | $-1$ | $-1$ | 2 | 0 | 0
$E_{1/2}$ | $\overline{E}_{1}$ | 2 | $\sqrt{3}$ | 1 | 0 | 0 | 0
$E_{3/2}$ | $\overline{E}_{3}$ | 2 | 0 | $-2$ | 0 | 0 | 0
$E_{5/2}$ | $\overline{E}_{2}$ | 2 | $-\sqrt{3}$ | 1 | 0 | 0 | 0

Table 2: The character tables of 2D PGs with time-reversal (sometimes called
the gray groups). Here and throughout $1^{\prime}$ denotes $\mathcal{T}$ as a
group element. The irrep names are shown in the first column using the
Altmann-Herzig notation and second columns using the Bilbao Crystallographic
notation. In each table, the irreps above the second horizontal line are the
single-valued irreps (no-SOC), and the irreps below the second horizontal line
are the double-valued irreps (SOC). The traces of the anti-unitary operators
are not shown because they are not invariant under unitary transforms.

##  Appendix B Many-Body Atomic Limits and Construction of RSIs

In this Appendix, we construct many-body local Real Space Invariants (RSIs).
First App. B.1 gives a simple example of how single-particle RSIs breaks down
with interactions. App. B.2 then defines the many-body atomic limit states and
gives explicit expressions for the many-body local RSIs of rotation groups.
App. B.3 generalizes these results to all 2D point groups with mirrors, time-
reversal, and spin-orbit coupling. Lastly, App. B.4 evaluates the many-body
local RSIs on product states to give expressions in terms of irrep
multiplicities and single-particle RSIs.

###  B.1 1D Many-Body RSI example

Ref. [51] obtained the single-particle Real Space Invariants (RSIs) as
adiabatic invariants in non-interacting Hamiltonians. States with different
single-particle RSIs cannot be connected without a gap closing in non-
interacting Hamiltonians. We give a minimal example of the breakdown of these
single-particle RSIs when interactions are added by showing that two states
with different single-particle RSIs can be brought into superposition by
interaction terms.

Let us consider a 1D system with inversion ($C_{2}$) symmetry. We propose a
three site system with open boundaries. The site $R=0$ has two $s$ orbitals
denoted $s,s^{\prime}$, and the sites at $R=1,-1$ have two $p$ orbitals each
denoted $p,p^{\prime}$. Inversion $\mathcal{I}$ takes $R\to-R$. We set the
non-interacting Hamiltonian to be

$\displaystyle
H_{0}=-\frac{t}{2}(c^{\dagger}_{0,s}c_{0,s}+c^{\dagger}_{0,s^{\prime}}c_{0,s^{\prime}})+\frac{t}{2}(c^{\dagger}_{-1,p}c_{-1,p}+c^{\dagger}_{-1,p^{\prime}}c_{-1,p^{\prime}}+c^{\dagger}_{1,p}c_{1,p}+c^{\dagger}_{1,p^{\prime}}c_{1,p^{\prime}}),\quad
t>0$ (29)

so the $s$ orbitals are at lower energy than the $p$ orbitals. The groundstate
of $H_{0}$ at filling $2$ is given by
$\ket{GS}=c^{\dagger}_{0,s}c^{\dagger}_{0,s^{\prime}}\ket{0}$. We add a pair-
hopping interaction term in the form

$\displaystyle H_{int}$
$\displaystyle=\frac{1}{\sqrt{2}}U(c^{\dagger}_{1,p}c^{\dagger}_{1,p^{\prime}}c_{0,s}c_{0,s^{\prime}}+c^{\dagger}_{-1,p}c^{\dagger}_{-1,p^{\prime}}c_{0,s}c_{0,s^{\prime}})+h.c.$
(30)

which respects inversion symmetry, using
$\mathcal{I}c^{\dagger}_{r,\alpha}\mathcal{I}^{\dagger}=\pm
c^{\dagger}_{-r,\alpha}$ for $\alpha=s/p$. Intuitively, $H_{int}$ connects the
states $c^{\dagger}_{0,s}c^{\dagger}_{0,s^{\prime}}\ket{0}$ and
$\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p}c^{\dagger}_{1,p^{\prime}}+c^{\dagger}_{-1,p}c^{\dagger}_{-1,p^{\prime}})\ket{0}$.
This would be impossible at the single-particle level. Alternatively, we can
diagonalize $H_{0}+H_{int}$ to obtain explicit energies and states. There are
6 orbitals in the model, so filling 2 gives a $15$-dimensional Hilbert space.
Inversion symmetry splits the Hilbert space into $7$ even parity states and 8
odd parity states. A basis for the even parity states is

$\displaystyle\ket{i}$
$\displaystyle=c^{\dagger}_{0,s}c^{\dagger}_{0,s^{\prime}}\ket{0},\
w^{\dagger}_{0,s}w^{\dagger}_{0,s^{\prime}}\ket{0},\
w^{\dagger}_{0,p}w^{\dagger}_{0,p^{\prime}}\ket{0},\
c^{\dagger}_{0,s}w^{\dagger}_{0,s}\ket{0},\
c^{\dagger}_{0,s^{\prime}}w^{\dagger}_{0,s}\ket{0},\
c^{\dagger}_{0,s}w^{\dagger}_{0,s^{\prime}}\ket{0},\
c^{\dagger}_{0,s^{\prime}}w^{\dagger}_{0,s^{\prime}}\ket{0}$ (31)

where we introduced the Wannier functions

$\displaystyle w^{\dagger}_{0,p}$
$\displaystyle=\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p}+c^{\dagger}_{-1,p}),\qquad
w^{\dagger}_{0,s}=\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p}-c^{\dagger}_{-1,p})$
(32) $\displaystyle w^{\dagger}_{0,p^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p^{\prime}}+c^{\dagger}_{-1,p^{\prime}}),\quad\
\
w^{\dagger}_{0,s^{\prime}}=\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p^{\prime}}-c^{\dagger}_{-1,p^{\prime}})$

which obey, for instance,
$\mathcal{I}w^{\dagger}_{0,s}\mathcal{I}=w^{\dagger}_{0,s}$ and
$\mathcal{I}w^{\dagger}_{0,p}\mathcal{I}=-w^{\dagger}_{0,p}$. The Wannier
states are centered at $R=0$, and hence behave as $s/p$ orbitals at the $R=0$
site under inversion. The states in Eq. 32 are simply the induced
representations of $G_{0}=\\{1,\mathcal{I}\\}$ formed by the $p$ orbitals at
$R=\pm 1$. One can also check that this Wannier basis is orthogonal, e.g.
$\\{w^{\dagger}_{0,p},w^{\dagger}_{0,s}\\}=0$.

In the order of Eq. 31, the Hamiltonian can be written (in the even parity
sector) as

$\displaystyle\braket{i}{H_{0}+H_{int}}{j}$
$\displaystyle=\begin{pmatrix}-t&U/\sqrt{2}&U/\sqrt{2}&&&&\\\
U/\sqrt{2}&t&0&&&&\\\ U/\sqrt{2}&0&t&&&&\\\ &&&0&&&\\\ &&&&0&&\\\ &&&&&0&\\\
&&&&&&0\\\ \end{pmatrix}_{ij}$ (33)

whose spectrum is $\pm\sqrt{U^{2}+t^{2}},t,0,0,0,0,$. The groundstate is at
energy $-\sqrt{U^{2}+t^{2}}$ and remains gapped for all $U$. At $t=0$, the
groundstate is $c^{\dagger}_{0,s}c^{\dagger}_{0,s^{\prime}}\ket{0}$, but
interactions yield the entangled groundstate wavefunction

$\displaystyle\ket{GS(U)}\propto\sqrt{2}(\sqrt{U^{2}+t^{2}}-t)c^{\dagger}_{0,s}c^{\dagger}_{0,s^{\prime}}\ket{0}+U\left(c^{\dagger}_{1,p}c^{\dagger}_{1,p^{\prime}}+c^{\dagger}_{-1,p}c^{\dagger}_{-1,p^{\prime}}\right)\ket{0}$
(34)

which we left normalized, and used the simple identity

$\displaystyle
w^{\dagger}_{0,s}w^{\dagger}_{0,s^{\prime}}+w^{\dagger}_{0,p}w^{\dagger}_{0,p^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p}-c^{\dagger}_{-1,p})\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p^{\prime}}-c^{\dagger}_{-1,p^{\prime}})+\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p}+c^{\dagger}_{-1,p})\frac{1}{\sqrt{2}}(c^{\dagger}_{1,p^{\prime}}+c^{\dagger}_{-1,p^{\prime}})$
(35)
$\displaystyle=c^{\dagger}_{1,p}c^{\dagger}_{1,p^{\prime}}+c^{\dagger}_{-1,p}c^{\dagger}_{-1,p^{\prime}}\
.$

We now show that $\ket{GS(U)}$ has a well-defined many-body RSI. The two-
particle product states $c^{\dagger}_{0,s}c^{\dagger}_{0,s^{\prime}}$,
$w^{\dagger}_{0,s}w^{\dagger}_{0,s^{\prime}}$, and
$w^{\dagger}_{0,p}w^{\dagger}_{0,p^{\prime}}$ each have a definite single-
particle RSI (they are non-interacting Wannier product states), but their
single-particle RSIs are different. Because
$w^{\dagger}_{0,s}w^{\dagger}_{0,s^{\prime}}$ and
$w^{\dagger}_{0,p}w^{\dagger}_{0,p^{\prime}}$ have the same _many-body_ RSI,
they can be brought into superposition, necessarily creating entanglement. To
be explicit, the single-particle RSI $\delta=m(s)-m(p)$ and many-body RSI
$\Delta=\delta\mod 4$ of the states are

$\displaystyle
c^{\dagger}_{0,s}c^{\dagger}_{0,s^{\prime}},w^{\dagger}_{0,s}w^{\dagger}_{0,s^{\prime}}:$
$\displaystyle\qquad\delta=2,\quad\Delta=2\mod 4,$ (36) $\displaystyle
w^{\dagger}_{0,p}w^{\dagger}_{0,p^{\prime}}:$
$\displaystyle\qquad\delta=-2,\quad\Delta=2\mod 4\ .$

The fact that the two states have different single-particle RSIs means they
are adiabatically distinct without interactions. But, because they have the
same many-body RSIs, these two states can be adiabatically connected by
interactions. As such, the correlated state $\ket{GS(U)}$ has a well-defined
many-body RSI given by $\Delta=2\mod 4$. In the following section App. B.2, we
give a rigorous argument for the definition of many-body RSIs in 2D PGs.

###  B.2 Defining Many-Body RSIs in the Atomic Limit

We will now construct many-body local RSIs at a fixed Wyckoff position
$\mathbf{x}$ in 2D many-body atomic states protected by the symmetries of a
point group $G_{\mathbf{x}}$. Our construction proceeds as follows. First we
define a class of many-body atomic _limit_ states with zero correlation
length. We then show that, after imposing a cutoff to create open boundary
conditions, there is a natural set of quantum numbers protected by
$G_{\mathbf{x}}$ which is invariant as the cutoff is changed in a symmetry-
preserving fashion. We identity these invariants as many-body local RSIs and
we show that they remain adiabatically well-defined beyond the zero
correlation length limit. Since the cutoff can be taken to infinity, the many-
body local RSIs remain well-defined as the open boundary conditions approach
infinite boundary conditions, where we can simultaneously define many-body
local RSIs at each Wyckoff position.

Recall that the non-interacting atomic limit is defined by taking the lattice
constant to infinity so that all hoppings in the Hamiltonian go to zero (but
orbitals on the same site may still be connected by single-particle terms or
interactions). The groundstate of such a system is a product state of occupied
local orbitals. Our generalization of atomic limits to interacting
Hamiltonians is analogous. We consider a general lattice with $N_{orb}$
electron orbitals per unit cell located at positions
$\mathbf{r}_{\alpha}\in\\{\mathbf{r}\\},\ \alpha=1,\dots,N_{orb}$. Denote the
number of orbitals at position $\mathbf{r}$ to be $n_{\mathbf{r}}$. An
interacting Hamiltonian in the atomic limit only contains terms that are
totally local in $\mathbf{r}$, e.g. onsite non-interacting potentials
$H_{\mathbf{R},\mathbf{r}}\sim
c^{\dagger}_{\mathbf{R},\alpha}c_{\mathbf{R},\beta}$ or onsite interactions
$H_{\mathbf{R},\mathbf{r}}\sim
c^{\dagger}_{\mathbf{R},\alpha}c^{\dagger}_{\mathbf{R},\alpha^{\prime}}c_{\mathbf{R},\beta}c_{\mathbf{R},\beta^{\prime}}$
for
$\mathbf{r}_{\alpha}=\mathbf{r}_{\alpha^{\prime}}=\mathbf{r}_{\beta}=\mathbf{r}_{\beta^{\prime}}$.
Such a Hamiltonian can be written

$\displaystyle H_{AL}$
$\displaystyle=\bigoplus_{\mathbf{R}}\bigoplus_{\mathbf{r}\in\\{\mathbf{r}\\}}H_{\mathbf{R},\mathbf{r}}=\bigoplus_{\mathbf{R}}\bigoplus_{\mathbf{r}\in\\{\mathbf{r}\\}}T^{\dagger}_{\mathbf{R}}H_{\mathbf{r}}T_{\mathbf{R}}$
(37)

where we imposed translation invariance in the second equality. Each
$H_{\mathbf{r}}$ acts on a strictly local Hilbert space whose Fock space is
$2^{n_{\mathbf{r}}}$ dimensional. Because $H_{\mathbf{r}}$ is strictly local,
$[H_{\mathbf{r}},T_{\mathbf{R}}^{\dagger}H_{\mathbf{r}^{\prime}}T_{\mathbf{R}}]=0$
for all $\mathbf{r},\mathbf{r}^{\prime},\mathbf{R}$. Thus $H_{AL}$ is composed
of commuting terms (different sites are decoupled) and its ground state at
fixed density $\nu=N_{occ}/N_{orb},\ N_{occ}\in\mathbb{N}$ can be written

$\displaystyle\ket{GS}=\prod_{\mathbf{R},\mathbf{r}\in\\{\mathbf{r}\\}}T^{\dagger}_{\mathbf{R}}\mathcal{O}^{\dagger}_{\mathbf{r}}T_{\mathbf{R}}\
\ket{0},\qquad[\hat{N},\prod_{\mathbf{r}\in\\{\mathbf{r}\\}}\mathcal{O}^{\dagger}_{\mathbf{r}}]=N_{occ}\prod_{\mathbf{r}\in\\{\mathbf{r}\\}}\mathcal{O}^{\dagger}_{\mathbf{r}}$
(38)

where each $\mathcal{O}^{\dagger}_{\mathbf{r}}$ creates a groundstate of
$H_{\mathbf{r}}$ and $\hat{N}$ is the number operator. One can think of
$H_{\mathbf{r}}$ as being a quantum dot Hamiltonian, and
$\mathcal{O}^{\dagger}_{\mathbf{r}}\ket{0}$ as the groundstate of the quantum
dot.

To move away from this strict atomic limit, hopping terms and/or interactions
coupling different sites can be added to $H_{AL}$. As long as the many-body
gap does not close, we say that the ground state is a many-body atomic phase
since, by construction, it is adiabatically connected to an atomic limit. Note
that an obstructed atomic limit[51] is not a trivial atomic limit as defined
here: there is an obstruction to deforming it into a zero correlation length
state. For example, the dimerized limit of the SSH chain requires strong
inter-site hoppings, and cannot be connected to the trivial atomic limit.
Indeed, a trivial atomic limit is spatially decoupled and so has no corner or
edge states on open boundary conditions.

We now define many-body local RSIs on the many-body atomic limit states Eq.
38. Recall that non-interacting local RSIs[51] are adiabatic invariants (they
do not change value unless a single-particle gap closes) protected by point
group symmetries, and are defined in single-particle states at fixed filling.
We now explicitly show the existence of symmetry-protected adiabatic
invariants which are the quantum numbers of certain symmetry operators in
_many-body_ atomic limit groundstates (Eq. 38). We call these quantum numbers
many-body local RSIs.

A key part of our construction is to define the many-body local RSIs on open
boundary conditions which break the space group $G$ to $G_{\mathbf{x}}$. This
serves to show that they are local (since they only depend on the groundstate
within an arbitrary range of $\mathbf{x}$ set by a cutoff) and are protected
only by the point group symmetries of $G_{\mathbf{x}}$. However, it is then
crucial to show that the many-body local RSIs do not depend on the choice of
cutoff. This ensures that the many-body local RSIs are well-defined invariants
of the thermodynamic groundstate Eq. 38 since the spatial cutoff can be sent
to infinity. They then serve to define invariants on infinite boundary
conditions such that the gap is the thermodynamic gap. Define the open
boundary conditions by

$\displaystyle H_{AL,R,\mathbf{x}}$
$\displaystyle=\bigoplus_{|\mathbf{R}+\mathbf{r}-\mathbf{x}|<R}H_{\mathbf{R},\mathbf{r}}$
(39)

for a Wyckoff position $\mathbf{x}$ and cutoff $R$ (see Fig. 2a). The choice
of a circular cutoff is just for convenience: the following argument hold for
any cutoff that preserves $G_{\mathbf{x}}$. At cutoff $R$, the total electron
number $N_{R}$ is given by $[N,\mathcal{O}_{GS,R}]=N_{R}\mathcal{O}_{GS,R}$
where
$\mathcal{O}_{GS,R}=\prod_{|\mathbf{r}-\mathbf{x}|<R}\mathcal{O}_{\mathbf{r}}$.
As the cutoff $R$ is taken to infinity, the filling approaches the
thermodynamic filling $\nu=N_{occ}/N_{orb}$.

\begin{overpic}[height=173.44534pt,tics=10]{GSR}
\end{overpic}\begin{overpic}[height=173.44534pt,tics=10]{GSR2} \end{overpic}

Figure 2: (a) We depict the groundstate $\ket{GS,R}$ on OBCs and the
additional symmetry-related operators which are included upon expanding the
cutoff to $R^{\prime}$ (see Eq. 41). The many-body local RSIs are invariant
under the expansion. (b) Given a fixed cutoff $R$, all operators inside of $R$
but not at the $C_{n}$-invariant point $\mathbf{x}$ are symmetry-related and
so do not contribute to the many-body local RSI (see Eq. 46). Only the
operator $O_{\mathbf{x}}$ at $\mathbf{x}$ transforms locally under
$\mathbf{x}$. Its quantum numbers determine the many-body local RSI.

For each cutoff $R$, denote the ground state of $H_{AL,R,\mathbf{x}}$ by
$\ket{GS,R}$ (we omit $\mathbf{x}$ henceforth because it is fixed throughout
the calculation). It is important that $\ket{GS,R}$ is non-degenerate and
gapped for all $R$ at the exact filling. It is natural to require these two
properties because a many-body atomic phase has no edge states so the ground
state is unique and is an insulator, so there is a many-body gap. Note that
non-degeneracy excludes single-particle obstructed atomic or fragile
topological phase (to be addressed in App. C.2), and spontaneously broken
symmetries like a charge density wave. We now construct many-body local RSIs
of $H_{AL,R,\mathbf{x}}$ at a Wyckoff position $\mathbf{x}$ protected by the
point group symmetries of $G_{\mathbf{x}}$. We will then use an adiabaticity
argument to show that the many-body local RSIs are still well-defined in a
many-body atomic phase once weak hoppings and off-site interactions are
introduced. Because $\ket{GS,R}$ is non-degenerate (it is a strict atomic
limit state), it transforms in a 1D irrep of $G_{\mathbf{x}}$. Thus
$\ket{GS,R}$ has the quantum numbers

$\displaystyle\hat{N}\ket{GS,R}$ $\displaystyle=N\ket{GS,R},$ (40)
$\displaystyle g\ket{GS,R}$ $\displaystyle=e^{i\lambda[g]}\ket{GS,R},\qquad
g\in G_{\mathbf{x}}\ .$

However, the quantum numbers $N,e^{i\lambda[g]}$ depend on the cutoff — as we
now show. To begin, we consider the spinless rotation groups
$G_{\mathbf{x}}=\\{1,C_{n},\dots,C_{n}^{n-1}\\}$ with $C_{n}^{n}=+1$.
Extending the cutoff $R\to R^{\prime}$ yields a new ground state which can be
written

$\displaystyle\ket{GS,R^{\prime}}=\prod_{R<|\mathbf{r}|<R^{\prime}}\mathcal{O}_{\mathbf{r}}\
\ket{GS,R}=\prod_{i=0}^{n-1}C_{n}^{i}\mathcal{O}C_{n}^{{\dagger}i}\
\ket{GS,R}$ (41)

where, up to an irrelevant overall phase,
$\mathcal{O}=\prod_{\mathbf{r}\in\mathcal{D}}\mathcal{O}_{\mathbf{r}}$ is a
product of $\mathcal{O}_{\mathbf{r}}$ in the region $\mathcal{D}$ which obeys
$\\{R<|\mathbf{r}|<R^{\prime}\\}=\\{\mathcal{D},C_{n}\mathcal{D},\dots,C_{n}^{-1}\mathcal{D}\\}$
and $\mathcal{D}\cap C_{n}^{i}\mathcal{D}=\emptyset$. To be concrete, a
possible choice is
$\mathcal{D}=\\{\mathbf{r}|R<r<R^{\prime},0\leq\theta<2\pi/n\\}$. Eq. 41 holds
because in the strict atomic limit, $C_{n}\mathcal{O}C_{n}^{\dagger}$ has zero
overlap with $\mathcal{O}$. We call $N_{\mathcal{O}}$ the charge (number of
particles) of $\mathcal{O}$. The quantum numbers of the state with cutoff
$R^{\prime}$ are

$\displaystyle\hat{N}\ket{GS,R^{\prime}}$
$\displaystyle=(N+nN_{\mathcal{O}})\ket{GS,R^{\prime}}\ .$ (42) $\displaystyle
C_{n}\ket{GS,R^{\prime}}$
$\displaystyle=\left(\prod_{i=1}^{n-1}C_{n}^{i}\mathcal{O}C_{n}^{{\dagger}i}\right)C_{n}^{n}\mathcal{O}C_{n}^{n{\dagger}}\
C_{n}\ket{GS,R}$
$\displaystyle=e^{i\lambda[C_{n}]}\prod_{i=1}^{n-1}C_{n}^{i}\mathcal{O}C_{n}^{{\dagger}i}\
\mathcal{O}\ \ket{GS,R}$
$\displaystyle=e^{i\lambda[C_{n}]}(-1)^{(n-1)N_{\mathcal{O}}^{2}}\
\ket{GS,R^{\prime}}$
$\displaystyle=e^{i\lambda[C_{n}]}\begin{cases}(-1)^{N_{\mathcal{O}}},&n\text{
even}\\\ 1,&n\text{ odd}\\\ \end{cases}\ \ket{GS,R^{\prime}}$

where in the last line we used that fact that
$\mathcal{O}\,C_{n}^{i}\mathcal{O}C_{n}^{{\dagger}i}=(-1)^{N_{\mathcal{O}}^{2}}C_{n}^{i}\mathcal{O}C_{n}^{{\dagger}i}\,\mathcal{O}$
since $C_{n}\mathcal{O}C_{n}^{\dagger}$ and $\mathcal{O}$ are supported on
non-overlapping regions and each contain $N_{\mathcal{O}}$ fermionic
operators. We see that both the $\hat{N}$ and $C_{n}$ eigenvalue (if $n$ is
even) change under the expansion of the cutoff. The $(-1)^{N_{\mathcal{O}}}$
factor is simply from the fermionic parity of the operator. In a system of
bosons, no $(-1)^{N_{\mathcal{O}}}$ factor appears, and the RSI groups (not
considered here) are altered.

However, there are two quantities invariant under expansion of the cutoff for
arbitrary $N_{\mathcal{O}}$:

$\displaystyle e^{i\frac{2\pi}{n}\hat{N}}\ket{GS,R}$
$\displaystyle=e^{i\frac{2\pi}{n}\hat{N}}\ket{GS,R^{\prime}}$ (43)
$\displaystyle e^{i\frac{\pi}{n}\hat{N}}C_{n}\ket{GS,R}$
$\displaystyle=e^{i\frac{\pi}{n}\hat{N}}C_{n}\ket{GS,R^{\prime}}\quad\text{($n$
even)}$ $\displaystyle C_{n}\ket{GS,R}$
$\displaystyle=C_{n}\ket{GS,R^{\prime}}\quad\quad\quad\,\text{($n$ odd)}$

corresponding to the total charge mod $n$ and the many-body angular momentum
of the $e^{i\frac{\pi}{n}\hat{N}}C_{n}$ [101, 65]. (Note that
$e^{i\frac{\pi}{n}\hat{N}}C_{n}$ is effectively a “spinful” rotation since
$(e^{i\frac{\pi}{n}\hat{N}}C_{n})^{n}=(-1)^{\hat{N}}$ and $C_{n}$ is a spin-
less operator since $C_{n}^{n}=+1$.) Because the eigenvalues of
$e^{i\frac{2\pi}{n}\hat{N}},e^{i\frac{\pi}{n}\hat{N}}C_{n}$ (or $C_{n}$ is $n$
is odd) are independent of the cutoff, they are good quantum numbers for all
choices of the cutoff. Then we can take $R\to\infty$ at constant density,
which is the thermodynamic limit on infinite boundary conditions. Secondly,
the eigenvalues of $e^{i\frac{2\pi}{n}\hat{N}}$ and $C_{n}$ are quantized, so
they cannot change if the ground state is perturbed away from the strict
atomic limit without closing the gap. Thus small hoppings and off-site
interactions can be added without changing the quantum numbers in Eq. 43.
Hence we have shown that the eigenvalues of the operators in Eq. 43 are local,
symmetry-protected quantum numbers invariant under the adiabatic expansion of
the cutoff: they are many-body local RSIs. We remark that the many-body local
RSIs are not typical invariants of $\ket{GS}$ in the thermodynamic limit.
Usually, one computes the quantum numbers of a state with the Hilbert space
fixed, whereas we demonstrated the existence of quantum numbers which are
invariants upon enlarging the Hilbert space (while preserving density and
symmetry). It would be desirable to have equivalent expressions for the many-
body local RSIs which can be computed on a fixed Hilbert space without
requiring OBCs. This is left for future work.

We now determine the group structure formed by the many-body local RSIs (Eq.
43). For even $n$, we have $(e^{i\frac{\pi}{n}\hat{N}}C_{n})^{2n}=+1$, so the
eigenvalues of $e^{i\frac{\pi}{n}\hat{N}}C_{n}$ are
$\mathds{Z}_{2n}$-classified. The eigenvalues of $e^{i\frac{2\pi}{n}\hat{N}}$
are $\mathds{Z}_{n}$-classified, but they are not all independent from the
eigenvalues of $e^{i\frac{\pi}{n}\hat{N}}C_{n}$ because
$(e^{i\frac{\pi}{n}\hat{N}}C_{n})^{n}=e^{i\pi\hat{N}}=(e^{i\frac{2\pi}{n}\hat{N}})^{n/2}$.
Hence $e^{i\frac{2\pi}{n}\hat{N}}$ only provides $n/2$ more independent
quantum numbers, and we obtain a $\mathds{Z}_{2n}\times\mathds{Z}_{n/2}$
classification for $n$ even. To get a set of
$\mathds{Z}_{2n}\times\mathds{Z}_{n/2}$ independent quantum numbers, we define
the eigenvalues of the symmetry operators in Eq. 43 as

$\displaystyle e^{i\frac{\pi}{n}\hat{N}}C_{n}\ket{GS}$
$\displaystyle=e^{i\frac{\pi}{n}\Delta_{1}}\ket{GS},\qquad\Delta_{1}\in\mathds{Z}_{2n},\qquad\text{$n$
even}$ (44) $\displaystyle
e^{-i\frac{2\pi}{n}\hat{N}}(e^{i\frac{\pi}{n}\hat{N}}C_{n})^{2}\ket{GS}$
$\displaystyle=C_{n}^{2}\ket{GS}=e^{i\frac{2\pi}{n/2}\Delta_{2}}\ket{GS},\qquad\Delta_{2}\in\mathds{Z}_{n/2}$

since $(C_{n}^{2})^{n/2}=+1$. We will refer to $\Delta_{1}$ and $\Delta_{2}$
as the many-body local RSIs. Note that Eq. 42 can be used to show directly
that $\Delta_{2}$ is invariant under the cutoff, since
$C_{n}^{2}\ket{GS,R^{\prime}}=e^{i2\lambda[C_{n}]}(-1)^{2N_{\mathcal{O}}}\ket{GS,R^{\prime}}=e^{i2\lambda[C_{n}]}\ket{GS,R^{\prime}}$
with the $N_{\mathcal{O}}$-dependent phase canceling.

For $n$ odd (only $n=3$ is relevant for the crystalline point groups), the
$\mathds{Z}_{n}\times\mathds{Z}_{n}$ group structure is obvious from Eq. 43
with $e^{i\frac{2\pi}{n}\hat{N}}$ and $C_{n}$ providing independent
$\mathds{Z}_{n}$ quantum numbers. We define the eigenvalues as

$\displaystyle e^{i\frac{2\pi}{n}\hat{N}}\ket{GS}$
$\displaystyle=e^{i\frac{2\pi}{n}\Delta_{1}}\ket{GS},\qquad\Delta_{1}\in\mathds{Z}_{n},\qquad\text{$n$
odd}$ (45) $\displaystyle C_{n}\ket{GS}$
$\displaystyle=e^{i\frac{2\pi}{n}\Delta_{2}}\ket{GS},\qquad\Delta_{2}\in\mathds{Z}_{n}$

We will refer to $\Delta_{1}$ and $\Delta_{2}$ as the many-body local RSIs.

Having obtained our key results, we offer a brief alternative perspective from
the strict atomic limit. At any cutoff $R$, we can write the wavefunction of
the groundstate of the OBC Hamiltonian Eq. 39 respecting the rotation group
$G_{\mathbf{x}}$ as

$\displaystyle\ket{GS,R}$
$\displaystyle=\mathcal{O}_{\mathbf{x}}\prod_{i=1}^{n}\mathcal{O}_{i}\ket{0},\qquad\mathcal{O}_{i}=C^{i}_{n}\mathcal{O}C_{n}^{{\dagger}i}$
(46)

in analogy to Eq. 41, where
$\mathcal{O}=\prod_{\mathbf{r}\in\mathcal{W}}\mathcal{O}_{\mathbf{r}}$ and
$\mathcal{W}$ is the wedge
$\\{\mathbf{r}|0<|\mathbf{r}-\mathbf{x}|<R,\text{arg}(\mathbf{r}-\mathbf{x})\in(0,2\pi/n]\\}$
(see Fig. 2b). Note that it is important $\mathcal{W}$ not contain
$\mathbf{x}$, since $C_{n}\mathcal{W}$ must not overlap $\mathcal{W}$. For
this reason, the operator $\mathcal{O}_{\mathbf{x}}$ is separated out in Eq.
46. Since the many-body local RSIs are constructed so that the operator
$\prod_{i=1}^{n}\mathcal{O}_{i}$, which depends on the cutoff, does not
contribute, we observe that the many-body local RSIs of $\ket{GS,R}$ are
simply the many-body local RSIs of $\mathcal{O}_{\mathbf{x}}\ket{0}$. This
statement relies on the strict atomic limit where $\mathcal{O}_{\mathbf{x}}$
can be defined since all operators are onsite, but Eq. 46 provides a useful
intuitive picture.

As we have shown, the many-body local RSIs in Eqs. 44 and 45 remain invariant
as the cutoff is taken to infinity. In this infinite limit, it is natural to
consider the many-body local RSIs of each Wyckoff position in the unit cell.
As an example in the space group $p2$, there are four Wyckoff positions
$\mathbf{x}=(0,0),(1/2,0),(0,1/2),(1/2,1/2)$ with $G_{\mathbf{x}}=2$ (in each
unit cell). The point groups are generated by the symmetries
$C_{2},T_{1}C_{2},T_{2}C_{2},T_{1}T_{2}C_{2}$ respectively where $T_{i}$ is
the translation operator along the $i$th lattice vector. Choosing finite
boundary conditions respecting the symmetries of a particular point group will
necessarily break the symmetries of the other point groups, and thus our
construction of the many-body local RSIs does not immediately imply they can
be simultaneously defined.

While we do not provide a rigorous general argument, we claim that the many-
body local RSIs are well-defined (for interacting atomic phases) in the
infinite limit at each Wyckoff position in the unit cell. Consider the
following heuristic argument. On infinite boundary conditions, the many-body
local RSIs at each Wyckoff positions are well-defined in the strict atomic
limit because the correlation length is zero and each Wyckoff position is
decoupled. Hence their many-body local RSIs can be defined at each Wyckoff
position. Adding weak hoppings and off-site interactions will make the
correlation length nonzero, but the quantization of the RSI eigenvalues means
they remain invariant throughout this process. Of course, the formal
difficulty is that the eigenvalue of $\hat{N}$, the number of particles, is
not well-defined in the infinite limit. We extend all the results of this
section to general 2D point groups in App. B.3. We will give explicit symmetry
operators whose quantum numbers (eigenvalues) are many-body local RSIs.

Lastly, we prove that the many-body local RSIs at two symmetry-related
positions are identical. This statement holds on infinite boundary conditions
(where the full wallpaper group is intact) according to the argument in the
preceding paragraph, such that the many-body local RSIs on infinite boundary
conditions are understood as the infinite-cutoff limit on OBCs. For instance,
1a sites in different unit cells of a crystal are related by translations and
have the same many-body local RSIs. (In concrete terms, we can compute the
many-body local RSI at the same Wyckoff position in two different unit cells
by choosing two different OBCs which respect the symmetries of each site
individually, though it is impossible to respect both at once. The resulting
many-body RSIs will be the same since they are related by translations on
infinite boundary conditions.) Alternatively, the two sites in the 2c
$=\\{(1/2,0),(0,1/2)\\}$ position in wallpaper group $p4$ are related by
$C_{4}$ and have the same many-body local RSIs protected by $T_{1}C_{2}$ and
$T_{2}C_{2}$ respectively. As such, we can define
$\Delta_{2c,1}=\Delta_{(1/2,0),1}=\Delta_{(0,1/2),1}\in\mathds{Z}_{4}$. In
other words, we can refer to the Wyckoff position 2c rather than either of its
individual sites.

We now prove the general case. Consider a wallpaper group with two sites
$\mathbf{x},\mathbf{x}^{\prime}$ related by a symmetry:
$\mathbf{x}^{\prime}=g^{\prime}\mathbf{x},g^{\prime}\in G$. The site symmetry
groups $G_{\mathbf{x}},G_{\mathbf{x}^{\prime}}$ are isomorphic and related by
conjugation: $G_{\mathbf{x}}\cong G_{\mathbf{x}}^{\prime}$ with
$G_{\mathbf{x}^{\prime}}=\\{h^{\prime}=g^{\prime}hg^{\prime{\dagger}}|h\in
G_{\mathbf{x}},g^{\prime}\notin G_{\mathbf{x}}\\}$. For instance,
$\mathbf{x}=(1/2,0),\mathbf{x}^{\prime}=(0,1/2)$ in $G=p4$ are the two sites
of the 2c position with $g^{\prime}=C_{4}$ discussed above, or in $G=p2$, we
could consider $\mathbf{x}=(0,0),\mathbf{x}^{\prime}=(1,0)$, which are 1a
sites related by a translation $g^{\prime}=T_{1}$. In the atomic limit where
$\mathcal{O}_{\mathbf{r}}$ is the creation operator of the strictly local
Hamiltonian $H_{\mathbf{r}}$, we now prove that $\mathcal{O}_{\mathbf{x}}$ and
$\mathcal{O}_{\mathbf{x}}^{\prime}$ have the same quantum numbers. To do so,
define the transformation of $\mathcal{O}_{\mathbf{x}}$ by
$h\mathcal{O}_{\mathbf{x}}h^{\dagger}=e^{iO[h]}\mathcal{O}_{\mathbf{x}}$. Then
note that

$\displaystyle h^{\prime}\mathcal{O}_{\mathbf{x}^{\prime}}h^{\prime{\dagger}}$
$\displaystyle=g^{\prime}hg^{\prime{\dagger}}\mathcal{O}_{\mathbf{x}^{\prime}}g^{\prime}h^{\dagger}g^{\prime{\dagger}}=e^{iO[g^{\prime}]}g^{\prime}h\mathcal{O}_{\mathbf{x}}h^{\dagger}g^{\prime{\dagger}}=e^{iO[h]}e^{iO[g^{\prime}]}g^{\prime}\mathcal{O}_{\mathbf{x}}g^{\prime{\dagger}}=e^{iO[h]}\mathcal{O}_{\mathbf{x}^{\prime}}$
(47)

using
$g^{\prime{\dagger}}\mathcal{O}_{\mathbf{x}^{\prime}}g^{\prime}=e^{iO[g^{\prime}]}\mathcal{O}_{\mathbf{x}}$,
with $e^{iO[g^{\prime}]}$ being an irrelevant phase factor. We saw earlier in
Eq. 46 that the quantum numbers of $\mathcal{O}_{\mathbf{x}}$ completely
determined the many-body local RSIs of the groundstate
$\ket{GS,R,\mathbf{x}}=\mathcal{O}_{\mathbf{x}}\prod_{j=1}^{n}\mathcal{O}_{j}\ket{0}$.
Because the quantum numbers of $\mathcal{O}_{\mathbf{x}}$ and
$\mathcal{O}_{\mathbf{x}^{\prime}}$ are the same, the RSIs of
$\ket{GS,R,\mathbf{x}}$ and $\ket{GS,R,\mathbf{x}^{\prime}}$ must be the same.
This holds for all adiabatically connected states, completing the proof.

###  B.3 Extension to General Point Groups

In App. B.2, we gave a detailed derivation of the many-body local RSIs in the
spinless point groups generated by $C_{n}$. To complete our RSI classification
for other point groups with mirrors, time-reversal, and spin-orbit coupling
(SOC), we develop the general theory of many-body RSIs. As before, we begin by
working in zero correlation length atomic limit states, and then extend to
adiabatically connected atomic states. Physically, expanding the cutoff and
including more orbitals at a position $\mathbf{x}^{\prime}\neq\mathbf{x}$ adds
$C_{n}$-related copies of the orbitals to the Hilbert space. Like in the non-
interacting case [51], these orbitals transforms as an irrep of the subgroup
$G_{\mathbf{x}^{\prime}}\in G_{\mathbf{x}}$. For instance with $C_{n}$ and a
mirror $M_{x}$, the addition of orbitals in the strict atomic limit described
by
$\mathcal{O}=\prod_{\mathbf{r}\in\mathcal{D}^{\prime}}\mathcal{O}_{\mathbf{r}}$
where
$\mathcal{D}^{\prime}=\\{R<|\mathbf{r}|<R^{\prime},-\pi/n<\theta<\pi/n\\}$ is
mapped to a distinct region by $C_{n}$ but is preserved under the mirror $M$
taking $y\to-y$. Thus $\mathcal{O}$ transforms under an irrep of the
reflection group $M$. In the rotation group cases we have considered so far
with $G_{\mathbf{x}}=n$, the subgroup is trivial
($G_{\mathbf{x}^{\prime}}=1$), but in general there can be nontrivial
subgroups of $G_{\mathbf{x}}$. There is a further requirement on
$\mathcal{O}$: because we are in the strict atomic limit with a non-degenerate
ground state, $\mathcal{O}$ must transform in a 1D irrep. Explicitly, for
$G_{\mathbf{x}}=\\{g_{1}H,\dots,g_{d}H\\}$ in a coset construction with
$G_{\mathbf{x}^{\prime}}=H$ and $d=|H|/|G|$, we have

$\displaystyle h\mathcal{O}h^{\dagger}=e^{iO[h]}\mathcal{O},\qquad\forall h\in
H\subset G_{\mathbf{x}}$ (48)

and the operators $g_{i}\mathcal{O}g_{i}^{\dagger}$ are supported on different
orbitals for all $i=1,\dots,d$ in the strict atomic limit. The representation
$e^{iO[h]},h\in H$ is a 1D irrep of $H$. For instance, if $\mathcal{O}$ is a
product of orbital creation operators with definite parity under $M$, then
$O[h=M]$ is simply the total many-body parity. Under rotations,
$C_{n}\mathcal{O}C_{n}^{\dagger}$ is a new operator supported on different
orbitals in the region $C_{n}\mathcal{D}$.

Now expanding the cutoff to introduce new operators at
$g_{1}\mathbf{x}^{\prime},\dots,g_{d}\mathbf{x}^{\prime}$ (we always take
$g_{1}=\mathbb{1}$) yields the ground state

$\displaystyle\ket{GS,R^{\prime}}$
$\displaystyle=\prod_{i=1}^{d}g_{i}\mathcal{O}g_{i}^{\dagger}\ \ket{GS,R}\ .$
(49)

Importantly, we must find quantum numbers which are invariant under _all_
possible expansions of the cutoff, which can include different orbitals
transforming under any of the possible subgroups. To do so systematically, we
recall some basic facts from the theory of induced representations. Because
$\mathcal{O}$ transforms in a representation of $H$ (in this case, a 1D irrep
as required by non-degeneracy), it forms an induced representation of
$G_{\mathbf{x}}$. To be explicit, this representation is the following. Define
$\mathcal{O}_{i}=g_{i}\mathcal{O}g_{i}^{\dagger},i=1,\dots,d$ and the induced
representation $R_{ij}[g]$ by

$\displaystyle
g\mathcal{O}_{i}g^{\dagger}=\sum_{j=1}^{d}R_{ij}[g]\mathcal{O}_{j},\qquad\forall
g\in G_{\mathbf{x}}\ .$ (50)

For the spinless symmetries considered here where $C_{n}^{n}=+1$, $R[g]$ is a
spinless representation of $G_{\mathbf{x}}$. Since $\mathcal{O}_{i}$ are
distinct operators for each $i$ in the strict atomic limit
($\mathcal{O}_{i},\mathcal{O}_{j}$ are supported on non-overlapping sites for
$i\neq j$), $R_{ij}[g]$ is a complex permutation matrix [51] in this basis:
there is only one nonzero entry per row. As an example, consider
$G_{\mathbf{x}}=4mm$ which has $G_{\mathbf{x}^{\prime}}=m$ as a nontrivial
subgroup. Expand the cutoff to include the operator $\mathcal{O}$ supported in
a finite region $\mathcal{D}^{\prime}$ centered around the point
$(x^{\prime},0)\neq\mathbf{q}$. Note that $M\in G_{\mathbf{x}}$ is also a
symmetry of $G_{(x,0)}$, where $M$ is a reflection acting as
$M\hat{y}=-\hat{y}$, and denote the irrep of $\mathcal{O}$ by
$M\mathcal{O}M^{\dagger}=\pm\mathcal{O}$. Since $(x,0)\neq\mathbf{x}$, $C_{4}$
ensures the groundstate also contains the operators
$C_{4}^{i}\mathcal{O}C_{4}^{i{\dagger}},i=1,2,3$ at position
$(0,x),(-x,0),(0,-x)$ respectively. From Eq. 50, the induced representation is

$\displaystyle R[C_{4}]=\begin{pmatrix}0&&&1\\\ 1&0&&\\\ &1&0&\\\ &&1&0\\\
\end{pmatrix},\quad R[M]=\pm\begin{pmatrix}1&&&\\\ &0&&1\\\ &&1&\\\ &1&&0\\\
\end{pmatrix}\ .$ (51)

We now need to compute the representation of $g$ on the groundstate Eq. 49. To
do so, we recall that $g\ket{GS,R}=e^{i\lambda[g]}\ket{GS,R}$ and
$\mathcal{O}_{i}\mathcal{O}_{j}=(-1)^{N_{\mathcal{O}}^{2}}\mathcal{O}_{j}\mathcal{O}_{i}=(-1)^{N_{\mathcal{O}}}\mathcal{O}_{j}\mathcal{O}_{i}$
in which case, upon enlarging the cutoff from $R^{\prime}$ to $R$ gives

$\displaystyle g\ket{GS,R^{\prime}}$
$\displaystyle=g\prod_{i=1}^{d}\mathcal{O}_{i}\ g^{\dagger}g\ket{GS,R}$ (52)
$\displaystyle=e^{i\lambda[g]}\prod_{i=1}^{d}\left(\sum_{j}R_{ij}[g]\mathcal{O}_{j}\right)\
\ket{GS,R}$
$\displaystyle=e^{i\lambda[g]}\left(\det\\!{}_{(-1)^{N_{\mathcal{O}}}}R[g]\right)\,\prod_{i=1}^{d}\mathcal{O}_{i}\
\ket{GS,R}$
$\displaystyle=e^{i\lambda[g]}\left(\det\\!{}_{(-1)^{N_{\mathcal{O}}}}R[g]\right)\,\ket{GS,R^{\prime}}$

where $\det{}_{\pm}$ is the permanent for $+$ and determinant for $-$.
Explicitly for a $d\times d$ matrix $A$,

$\displaystyle\det\\!{}_{\pm}\,A$ $\displaystyle=\sum_{\sigma}(\pm
1)^{\\#_{\sigma}}A_{1,\sigma_{1}}\dots A_{d,\sigma_{d}}$ (53)

where $\sigma$ are the $d!$ permutations of the numbers $1,\dots,d$ and
$\\#_{\sigma}$ is the number of interchanges in the permutation. Because
$D[g]$ is a complex permutation matrix with only one nonzero element per row,
only one permutation $\sigma^{\prime}$ in the sum Eq. 53 over $\sigma$ is
nonzero. Thus

$\displaystyle\det\\!{}_{\pm}\,R[g]$ $\displaystyle=(\pm
1)^{\\#_{\sigma^{\prime}}}R[g]_{1,\sigma_{1}^{\prime}}\dots
R[g]_{d,\sigma_{d}^{\prime}}$ (54)

so that only the overall sign in Eq. 52 depends on the parity of
$N_{\mathcal{O}}$ (through the signature $(-1)^{\\#_{\sigma^{\prime}}}$ of
$g$).

We now give an example of this procedure in the point groups generated by
$C_{4}$ and $M$. Consider expanding the cutoff to include the new operators
$\mathcal{O}_{i}=C_{4}^{i}\mathcal{O}C_{4}^{{\dagger}i}$ where
$\mathcal{O}=\prod_{\mathbf{r}\in\mathcal{D}^{\prime}}\mathcal{O}_{\mathbf{r}}$.
There are two possible nontrivial subgroups that $\mathcal{O}$ could transform
under: $G_{\mathbf{x}^{\prime}}=\\{1,M\\}$ if $\mathcal{D}^{\prime}$ is
centered around e.g. $(x,0)$ or $G_{\mathbf{x}^{\prime}}=\\{1,C_{4}M\\}$ if
$\mathcal{D}^{\prime}$ is centered around e.g. $(x,x)$. Here $x\neq 0$ so
$\mathcal{D}^{\prime}$ cannot be invariant under $C_{4}$, and thus we do not
consider the subgroups with $C_{4}$, as in the single-particle case [51]. For
both cases, the number of particles increases as $N\to N+4N_{\mathcal{O}}$
where $N_{\mathcal{O}}$ is the number of particles in $\mathcal{O}$. Note that
$N_{\mathcal{O}}$ can be even or odd, for instance if $\mathcal{D}$ encloses a
single site with an even or odd number of orbitals. We start by considering
the subgroup $\\{1,M\\}$. With Eq. 52, we find that

$\displaystyle C_{4}\ket{GS,R^{\prime}}$
$\displaystyle=e^{i\lambda[C_{4}]}(-1)^{N_{\mathcal{O}}}\,\ket{GS,R^{\prime}}$
(55)

since the signature $(-1)^{\\#_{\sigma^{\prime}}}=-1$ of $C_{4}$ is negative.
This obviously matches Eq. 42, recalling that
$e^{i\lambda[C_{4}]},e^{i\lambda[M]}$ are the eigenvalues of $C_{4}$ and $M$
on $\ket{GS,R}$ respectively. We now need the representation of $M$. We find

$\displaystyle M\ket{GS,R^{\prime}}$
$\displaystyle=e^{i\lambda[M]}(-1)^{N_{\mathcal{O}}}\,\ket{GS,R^{\prime}},\qquad
H=\\{1,M\\}$ (56)

since $R[M]$ in Eq. 51 also has odd signature. Now we consider the subgroup
$\\{1,C_{4}M\\}$. The representation matrices are now

$\displaystyle R[C_{4}]=\begin{pmatrix}0&&&1\\\ 1&0&&\\\ &1&0&\\\ &&1&0\\\
\end{pmatrix},\quad R[M]=\pm\begin{pmatrix}&&&1\\\ &&1&\\\ &1&&\\\ 1&&&\\\
\end{pmatrix}$ (57)

and notably $R[M]$ now has even signature. So in this case

$\displaystyle M\ket{GS,R^{\prime}}$
$\displaystyle=e^{i\lambda[M]}\,\ket{GS,R^{\prime}},\qquad H=\\{1,C_{4}M\\}\
.$ (58)

Let us scrutinize Eqs. 55 and 58. On the original groundstate, the mirror
eigenvalue is $M\ket{GS,R}=e^{i\lambda[M]}\ket{GS,R}$. Expanding the cutoff to
obtain $\ket{GS,R^{\prime}}$ can include operators transforming in one of two
possible nontrivial subgroups. In both cases, the mirror eigenvalue is well-
defined and is given by Eqs. 55 and 58, but crucially the mirror eigenvalue
_differs_ between the two cases by a factor of $(-1)^{N_{\mathcal{O}}}$. In
order to obtain a many-body local RSI, it is essential to have a symmetry
operator whose eigenvalues are invariant under any (symmetric) choice of
cutoff, such as $e^{i\frac{\pi}{4}\hat{N}}C_{n}$. We see from Eqs. 55 and 58
that, because $N_{\mathcal{O}}$ is unrestricted, we do not obtain another
cutoff-independent quantum number from $M$.

In fact, this is the same as in the non-interacting RSIs. Ref. [51] showed
that the non-interacting RSIs of $4mm$ can all be obtained by reducing the
non-interacting RSIs of $4$ in the presence of mirrors, but that mirrors did
not introduce any new non-interacting RSIs. We check that for all even $n$,
inducing from the $\\{1,M\\}$ and $\\{1,C_{n}M\\}$ subgroups gives factors of
$e^{i\lambda[M]}$ and $e^{i\lambda[M]}(-1)^{N_{\mathcal{O}}}$ as in the $4mm$
example, and thus there are no RSIs protected by $M$. For $n$ odd, we find
that inducing the odd and even mirror irreps from $\\{1,M\\}$ brings factors
of $\pm e^{i\lambda[M]}(-1)^{N_{\mathcal{O}}}$ which also prevents $M$ from
protecting a new RSI.

Lastly, we must consider how $M$ restricts the possible eigenvalues of
$C_{n}$, thereby reducing the $C_{n}$-protected many-body local RSIs. First we
consider the case of $n$ even. Since $MC_{n}M^{\dagger}=C_{n}^{\dagger}$ and
$M\ket{GS}=e^{i\lambda[M]}\ket{GS}$ on a given groundstate, the only allowed
$C_{n}$ eigenvalues are real: $\pm 1$. Then Eq. 44 shows that $\Delta_{2}=0$,
reducing the local RSI group from
$\mathds{Z}_{2n}\times\mathds{Z}_{n/2}\to\mathds{Z}_{2n}$. If $n$ is odd, then
$MC_{n}M^{\dagger}=C_{n}^{\dagger}$ enforces the $C_{n}$ eigenvalue to be
real, and thus $+1$ is the only possibility since $-1$ cannot be a $C_{n}$
eigenvalue for $n$ odd. Eq. 45 then shows $\Delta_{2}=0$, and the many-body
local RSI group is reduced from
$\mathds{Z}_{n}\times\mathds{Z}_{n}\to\mathds{Z}_{n}$.

We now consider spinless time-reversal. This case is simpler since the only
subgroup is $1^{\prime}$ which with $\mathcal{T}^{2}=+1$ has only the trivial
irrep, so the induction is identical to the $C_{n}$ case alone. Thus to derive
the many-body local RSI, we only have to consider the constraints
$\mathcal{T}$ imposes on the $C_{n}$ eigenvalues. This is also very simple
since, because $\mathcal{T}^{2}=+1$, we can choose the overall phase of the
groundstate such that $\mathcal{T}\ket{GS,R}=\ket{GS,R}$. Then
$e^{i\lambda[C_{n}]}$ must be real because $[C_{n},\mathcal{T}]=0$. This is
the same condition enforced by $M$, so following the same argument we find
$\Delta_{2}=0$.

We remark that in all cases, the many-body local RSIs of all 2D point groups
can obtained by reduction from their rotation subgroups.

Table 3 summarizes the full classification obtained from adding mirrors and
time-reversal to the rotation groups. We also must remark on the case of a
point group $G_{\mathbf{x}}=m$ with only mirrors. This case is somewhat
degenerate because $M$ is a quasi-1D symmetry and its Wyckoff positions are
extended lines, not points. When expanding the cutoff in 2D, it is possible to
include an arbitrary number of each irrep on the mirror axis, and thus our
prescription does not yield a many-body local RSI protected by $M$ or by
$e^{i\pi\hat{N}}$. In 1D there is no such issue because expanding the cutoff
always induces orbitals from off the mirror plane.

PG | RSI Operators on $\ket{GS}$ | Classification
---|---|---
$m$ |  | $\Delta_{1}\in\mathds{Z}_{1}$
2 | $e^{i\frac{\pi}{2}\hat{N}}C_{2}=e^{i\frac{\pi}{2}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{4}$
$2mm$ | $e^{i\frac{\pi}{2}\hat{N}}C_{2}=e^{i\frac{\pi}{2}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{4}$
3 | $e^{i\frac{2\pi}{3}\hat{N}}=e^{i\frac{2\pi}{3}\Delta_{1}},\ C_{3}=e^{i\frac{2\pi}{3}\Delta_{2}}$ | $\Delta_{1}\in\mathds{Z}_{3},\Delta_{2}\in\mathds{Z}_{3}$
$3m$ | $e^{i\frac{2\pi}{3}\hat{N}}=e^{i\frac{2\pi}{3}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{3}$
4 | $e^{i\frac{\pi}{4}\hat{N}}C_{4}=e^{i\frac{\pi}{4}\Delta_{1}},\ C_{4}^{2}=e^{i\pi\Delta_{2}}$ | $\Delta_{1}\in\mathds{Z}_{8},\Delta_{2}\in\mathds{Z}_{2}$
$4mm$ | $e^{i\frac{\pi}{4}\hat{N}}C_{4}=e^{i\frac{\pi}{4}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{8}$
6 | $e^{i\frac{\pi}{6}\hat{N}}C_{6}=e^{i\frac{\pi}{6}\Delta_{1}},\ C_{6}^{2}=e^{i\frac{2\pi}{3}\Delta_{2}}$ | $\Delta_{1}\in\mathds{Z}_{12},\Delta_{2}\in\mathds{Z}_{3}$
$6mm$ | $e^{i\frac{\pi}{6}\hat{N}}C_{6}=e^{i\frac{\pi}{6}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{12}$

PG | RSI Operators on $\ket{GS}$ | Classification
---|---|---
$m$ |  | $\Delta_{1}\in\mathds{Z}_{1}$
$21^{\prime}$ | $e^{i\frac{\pi}{2}\hat{N}}C_{2}=e^{i\frac{\pi}{2}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{4}$
$2mm1^{\prime}$ | $e^{i\frac{\pi}{2}\hat{N}}C_{2}=e^{i\frac{\pi}{2}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{4}$
$31^{\prime}$ | $e^{i\frac{2\pi}{3}\hat{N}}=e^{i\frac{2\pi}{3}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{3}$
$3m1^{\prime}$ | $e^{i\frac{2\pi}{3}\hat{N}}=e^{i\frac{2\pi}{3}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{3}$
$41^{\prime}$ | $e^{i\frac{\pi}{4}\hat{N}}C_{4}=e^{i\frac{\pi}{4}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{8}$
$4mm1^{\prime}$ | $e^{i\frac{\pi}{4}\hat{N}}C_{4}=e^{i\frac{\pi}{4}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{8}$
$61^{\prime}$ | $e^{i\frac{\pi}{6}\hat{N}}C_{6}=e^{i\frac{\pi}{6}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{12}$
$6mm1^{\prime}$ | $e^{i\frac{\pi}{6}\hat{N}}C_{6}=e^{i\frac{\pi}{6}\Delta_{1}}$ | $\Delta_{1}\in\mathds{Z}_{12}$

Table 3: Many-Body Local RSIs without SOC.

We now consider spin-orbit coupling (SOC) where
$C_{n}^{n}=M^{2}=\mathcal{T}^{2}=(-1)^{\hat{N}}$. Without $M$ and
$\mathcal{T}$, the SOC case can be mapped immediately to the spinless (no SOC)
case via $C_{n}=e^{i\frac{\pi}{n}\hat{N}}\tilde{C}_{n}$ where
$\tilde{C}_{n}^{n}=+1$ is a spinless operator. Then Eq. 43 yields the same
classification for Eqs. 44 and 45. This mapping would take
$\tilde{C}_{n}^{2}\to e^{-i\frac{2\pi}{n}\hat{N}}C_{n}^{2}$ for even $n$, but
we find it more convenient to use the operator
$e^{+i\frac{2\pi}{n}\hat{N}}C_{n}^{2}$ in order to obtain simple expressions
matching the conventions of Ref. [51]. This is clearly equivalent because the
difference is a multiple of $e^{i\frac{2\pi}{n}\hat{N}}$ which is also a
symmetry operator invariant under expansions of the cutoff, and
$(e^{+i\frac{2\pi}{n}\hat{N}}C_{n}^{2})^{n/2}=e^{i2\pi\hat{N}}=+1$.
Explicitly, for $n$ even we have (dropping the cutoff dependence $R$ for
brevity)

$\displaystyle C_{n}\ket{GS}$
$\displaystyle=e^{i\frac{\pi}{n}\Delta_{1}}\ket{GS},\qquad\Delta_{1}\in\mathds{Z}_{2n}$
(59) $\displaystyle e^{i\frac{2\pi}{n}\hat{N}}C_{n}^{2}\ket{GS}$
$\displaystyle=e^{i\frac{4\pi}{n}\Delta_{2}}\ket{GS},\qquad\Delta_{2}\in\mathds{Z}_{n/2}$

which is to be compared with Eq. 44, and for $n$ odd we have

$\displaystyle e^{i\frac{2\pi}{n}\hat{N}}\ket{GS}$
$\displaystyle=e^{i\frac{2\pi}{n}\Delta_{1}}\ket{GS},\qquad\Delta_{1}\in\mathds{Z}_{n}$
(60) $\displaystyle e^{-i\frac{\pi}{n}\hat{N}}C_{n}\ket{GS}$
$\displaystyle=e^{i\frac{2\pi}{n}\Delta_{2}}\ket{GS},\qquad\Delta_{2}\in\mathds{Z}_{n}$