# Symmetry indicators in commensurate magnetic flux

Yuan Fang  Department of Physics and Astronomy, Stony Brook University, Stony
Brook, New York 11974, USA  Jennifer Cano  Department of Physics and
Astronomy, Stony Brook University, Stony Brook, New York 11974, USA  Center
for Computational Quantum Physics, Flatiron Institute, New York, New York
10010, USA

###### Abstract

We derive a framework to apply topological quantum chemistry in systems
subject to magnetic flux. We start by deriving the action of spatial symmetry
operators in a uniform magnetic field, which extends Zak’s magnetic
translation groups to all crystal symmetry groups. Ultimately, the magnetic
symmetries form a projective representation of the crystal symmetry group. As
a consequence, band representations acquire an extra gauge invariant phase
compared to the non-magnetic theory. Thus, the theory of symmetry indicators
is distinct from the non-magnetic case. We give examples of new symmetry
indicators that appear at $\pi$ flux. Finally, we apply our results to an
obstructed atomic insulator with corner states in a magnetic field. The
symmetry indicators reveal a topological-to-trivial phase transition at finite
flux, which is confirmed by a Hofstadter butterfly calculation. The bulk phase
transition provides a new probe of higher order topology in certain obstructed
atomic insulators.

##  I  Introduction

Threading magnetic flux through a two-dimensional crystal changes the single
particle band spectrum into a Hofstadter butterfly spectrum that exhibits a
fractal structure with an infinitude of mini gaps [1]. The Hofstadter
butterfly is the lattice counterpart of Landau levels in the continuum. While
the Landau levels of a continuum model are often easier to compute than the
Hofstadter butterfly of the corresponding lattice model, diagnosing band
topology in the presence of magnetic flux requires the lattice because
topological invariants are defined over the entire Brillouin zone. The
topology of Hofstadter bands has been a subject of intense recent study [2, 3,
4, 5].

In the absence of magnetic flux, the topology of a band structure can be
classified by the theory of topological quantum chemistry (TQC) [6, 7, 8, 9,
10, 11, 12, 13]. A practical diagnosis comes from studying the space group
representations of bands at high symmetry momenta, which are known as symmetry
indicators [14]. However, in its present form, TQC cannot be directly applied
to systems in a magnetic field because it does not account for the Aharonov-
Bohm phase.

Figure 1: Framework for TQC in the presence of magnetic flux. A representation
of the site-symmetry group induces a band representation of the entire space
group, which is subduced to a representation of the little co-group at each
high-symmetry momentum, i.e., the symmetry indicator. The new element
introduced by the magnetic field is that the symmetry operators form a
projective representation of the space group. The red font indicates
differences between TQC with and without magnetic flux.

In the present manuscript we derive a framework to generalize TQC and the
classification of symmetry indicators to band structures in the presence of a
rational magnetic flux per unit cell. The workflow is shown in Fig. 1. We find
that the two key ingredients in the theory of TQC – the irreducible
representations of bands at high symmetry points in momentum space and the
induced representations of localized orbitals in real space – are modified
from their non-magnetic counterparts due to the presence of magnetic flux. The
essential reason for this modification is that the commutation relations
between crystal symmetries change in the presence of magnetic flux due to the
Aharonov-Bohm phase. As a result, the symmetry operators form non-trivial
projective representations of the space group. The earliest example of this is
Zak’s magnetic translation group [15, 16]. Our theory builds on Zak’s theory
by including crystalline symmetries.

Our theory of TQC in commensurate magnetic flux is distinct from “magnetic
TQC” [13]. While magnetic TQC classifies topological band structures according
to the representations of magnetic space groups, which describe the symmetry
of magnetically ordered crystals, magnetic TQC does not yet accommodate
magnetic flux through each unit cell, as it deals with zero flux
configurations of orbitals.

Our manuscript proceeds as follows. In Sec. II, we derive the space group
symmetry operators at rational flux in both real and momentum space. The
results are at the crux of the theory of TQC in a magnetic field that we
derive in Sec. IV. We then use the theory to compute symmetry indicators for
three magnetic layer groups, $p2$, $p4$ and $p4/m^{\prime}$, at $\pi$ flux per
unit cell. The strong indicator in $p2$ recovers an earlier formula for the
Chern number in Ref. 17, which is a stronger version of the formula in Ref.
18. In group $p4/m^{\prime}$, our theory gives rise to a new strong
$\mathbb{Z}_{2}$ indicator, which is simply the filling per unit cell mod $2$.
This $\mathbb{Z}_{2}$ non-trivial phase is protected by translation symmetry:
the non-trivial phase does not permit exponentially localized and symmetric
Wannier functions, but such Wannier functions exist when translation
symmetries within the magnetic unit cell are broken. We also find a new
indicator for $p4$.

In Sec. V, we study a tight binding model introduced in Ref. 19 that realizes
an obstructed atomic limit (OAL) on the square lattice at zero flux. The
Hofstadter butterfly spectrum shows that the system undergoes a gap-closing
phase transition at finite flux after which the corner states that were
present in the OAL phase disappear. By applying our theory of TQC in a
magnetic field to this model, we show that the gap closing corresponds to a
phase transition from an OAL to a trivial phase that can be diagnosed by
symmetry indicators.

##  II  Magnetic symmetries

In quantum mechanics the coupling of a magnetic field to a charged particle is
described by replacing the momentum $\mathbf{P}$ of the particle with the
canonical momentum $\mathbf{p}=\mathbf{P}+\mathbf{A}$ in the Hamiltonian
(without loss of generality, we have used natural units and assumed positive
unit charge). To account for the Aharonov-Bohm phase, terms in the single-
particle tight binding Hamiltonian are modified by the usual Peierls
substitution:

$c^{\dagger}_{\mathbf{r}_{2}}c_{\mathbf{r}_{1}}\mapsto
e^{i\int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{A}(\mathbf{r})\cdot
d\mathbf{r}}c^{\dagger}_{\mathbf{r}_{2}}c_{\mathbf{r}_{1}},$ (1)

where the path of the integral is the straight line connecting
$\mathbf{r}_{1}$ and $\mathbf{r}_{2}$.

However, if the zero-field Hamiltonian is invariant under a crystal symmetry
$\hat{g}:c_{\mathbf{r}}\mapsto c_{\hat{g}\mathbf{r}}$, the Hamiltonian
modified by the Peierls substitution in Eq. (1) is not necessarily invariant
under $\hat{g}$, even if the physical system is unchanged by the symmetry.
Consequently, the operator $\hat{g}$ must be modified from its zero-field form
by a gauge transformation that accounts for the Aharonov-Bohm phase.
Specifically, the magnetic field requires $\hat{g}$ be replaced by
$g\equiv\tilde{G}_{g}\hat{g}$, where
$\tilde{G}_{g}=e^{i\sum_{\mathbf{x}}\lambda_{g}(\mathbf{x})c^{\dagger}_{\mathbf{x}}c_{\mathbf{x}}}$
is a gauge transformation that acts on the electron annihilation operators by
[2]:

$\displaystyle\tilde{G}_{g}c_{\mathbf{r}}\tilde{G}_{g}^{-1}$
$\displaystyle=e^{-i\lambda_{g}(\mathbf{r})}c_{\mathbf{r}},$ (2)
$\displaystyle\tilde{G}_{g}c^{\dagger}_{\mathbf{r}}\tilde{G}_{g}^{-1}$
$\displaystyle=e^{i\lambda_{g}(\mathbf{r})}c^{\dagger}_{\mathbf{r}},$ (3)

where $\lambda_{g}$ is a scalar function defined for each symmetry $\hat{g}$
that we will derive momentarily. Acting on terms in the Hamiltonian in the
form of Eq. (1), $\tilde{G}_{g}$ has the effect of mapping
$\mathbf{A}(\mathbf{r})\mapsto\mathbf{A}(\mathbf{r})+\nabla\lambda_{g}(\mathbf{r})$.

Similar gauge transformations were introduced by Zak for the magnetic
translation operators in Refs. [15, 16]. More recently, the magnetic operators
for rotations about the origin and for time-reversal symmetry were considered
in Refs. [20, 17, 2]. Here, we develop a general theory for any symmetry group
in the presence of a magnetic field, thereby extending previous works to
include more general rotations and glide reflection symmetries. Doing so
allows us to apply the theory of symmetry indicators to diagnose topological
phases in the presence of a magnetic field.

We now derive the gauge transformation $\lambda_{g}$ in Eq. (2): we require
that if a single-particle Hamiltonian in zero field is invariant under a
symmetry $\hat{g}$, then in the presence of a magnetic field that preserves
$\hat{g}$, the Hamiltonian modified by the Peierls substitution in Eq. (1) is
invariant under the combined symmetry operation $g\equiv\tilde{G}_{g}\hat{g}$,
i.e., we require

$g:e^{i\int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{A}(\mathbf{r})\cdot
d\mathbf{r}}c^{\dagger}_{\mathbf{r}_{2}}c_{\mathbf{r}_{1}}\mapsto
e^{i\int_{g\mathbf{r}_{1}}^{g\mathbf{r}_{2}}\mathbf{A}(\mathbf{r}^{\prime})\cdot
d\mathbf{r}^{\prime}}c^{\dagger}_{g\mathbf{r}_{2}}c_{g\mathbf{r}_{1}}$ (4)

Acting on the left-hand-side by $g=\tilde{G}_{g}\hat{g}$, using the definition
of $\tilde{G}_{g}$ in Eqs. (2) and (3), and equating with the right-hand-side
yields

$e^{i\int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{A}(\mathbf{r})\cdot
d\mathbf{r}+i\int_{g\mathbf{r}_{1}}^{g\mathbf{r}_{2}}\nabla\lambda(\mathbf{r}^{\prime})\cdot
d\mathbf{r}^{\prime}}=e^{i\int_{g\mathbf{r}_{1}}^{g\mathbf{r}_{2}}\mathbf{A}(\mathbf{r}^{\prime})\cdot
d\mathbf{r}^{\prime}}$ (5)

A few lines of algebra (detailed in Appendix A) show that Eq. (5) is satisfied
when $\lambda_{g}(\mathbf{r})$ satisfies

$\nabla\lambda_{g}(\mathbf{r})=\mathbf{A}(\mathbf{r})-R_{g}\mathbf{A}(g^{-1}\mathbf{r}),$
(6)

where $R_{g}$ is the point group part of $g$. Eq. (6) determines each
$\lambda_{g}$ up to a constant. We choose the constant such that for
translation [2]

$\lambda_{T(\mathbf{a})}(\mathbf{r})=\int_{\mathbf{r}-\mathbf{a}}^{\mathbf{r}}\mathbf{A}(\mathbf{r}^{\prime})\cdot
d\mathbf{r}^{\prime}+\mathbf{B}\cdot\mathbf{a}\times\mathbf{r}$ (7)

and that a $2\pi$ rotation is implemented by the identity matrix. This choice
of constants ensures that the commutation relations between translation and
rotations about the origin are the same as at zero field as we show in the
Appendix B. This choice of gauge is fixed throughout this paper; later when we
refer to a gauge choice, we are referring to the gauge of the vector
potential. So far, we have only considered lattice degrees of freedom; orbital
and spin degrees of freedom can be included by an extra unitary transformation
in the action of $\hat{g}$, which does not change $\lambda_{g}$. We will
include these degrees of freedom in later sections.

Eq. (6), which serves as the definition of $\lambda_{g}$, is the first key
result of this manuscript. Combining it with the spatial action of the
symmetry yields the explicit form of the magnetic symmetry operator:

$\displaystyle g$
$\displaystyle=e^{i\sum_{\mathbf{x}^{\prime}}\lambda_{g}(\mathbf{x}^{\prime})c^{\dagger}_{\mathbf{x}^{\prime}}c_{\mathbf{x}^{\prime}}}\sum_{\mathbf{x}}c^{\dagger}_{\hat{g}\mathbf{x}}c_{\mathbf{x}}$
(8)
$\displaystyle=\sum_{\mathbf{x}}e^{i\lambda_{g}(\hat{g}\mathbf{x})}c^{\dagger}_{\hat{g}\mathbf{x}}c_{\mathbf{x}},$
(9)

where $\lambda_{g}$ is determined by Eq. (6). The second equality holds when
$g$ acts on the single-particle Hilbert space.

We now explain why changing $\lambda_{g}$ up to a constant does not change the
representation of the magnetic symmetry operators defined in Eq. (9). These
operators furnish projective representations of the space group. A projective
representation $\rho$ of a group satisfies the following multiplication rule

$\rho(h_{1})\rho(h_{2})=\omega(h_{1},h_{2})\rho(h_{1}h_{2}),$ (10)

where $h_{1}$, $h_{2}$ are group elements and $\omega(h_{1},h_{2})$ is called
the 2-cocycle. If $\omega(h_{1},h_{2})\equiv 1$, then $\rho$ is an ordinary
linear representation. In general, the magnetic symmetry operators in Eq. (9)
will have non-trivial 2-cocycles, as we show in the next sections. The $U(1)$
gauge freedom in Eq. (6) corresponding to the gauge transformation
$\lambda_{g}\mapsto\lambda_{g}+C_{g}$, $g\mapsto e^{iC_{g}}g$ leaves the
representation in the same group cohomology class, i.e. the transformed
projective representation is equivalent to the previous one. Essential
properties of projective representations are presented in Appendix. C.

In the next two subsections, we apply this formalism to two examples, first
rederiving Zak’s magnetic translation group and then reviewing the symmetries
of the square lattice in a magnetic field.

###  II.1  Zak’s magnetic translation group

In Refs. 15 and 16 Zak introduced the continuous magnetic translation
symmetries. We reproduce Zak’s result by taking the continuum limit of Eq.
(9).

Consider a two-dimensional infinite plane without a lattice and denote
operators that translate by
${\mathbf{\Delta}}=\Delta_{x}\hat{\mathbf{x}}+\Delta_{y}\hat{\mathbf{y}}$ by
$\hat{T}(\mathbf{\Delta})\equiv\hat{T}(\Delta_{x},\Delta_{y})$, where
$\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ denote the unit vectors.

We first work in the symmetric gauge:
$\mathbf{A}(\mathbf{r})=\frac{B}{2}(-r_{y},r_{x})$. Then from Eq. (6):

$\displaystyle\lambda_{T({\mathbf{\Delta}})}(\mathbf{r})=\frac{B}{2}(\Delta_{x}r_{y}-\Delta_{y}r_{x})$
(11)

For continuous translations, we replace the sum
$\sum_{\mathbf{x}}c^{\dagger}_{\hat{T}(\mathbf{\Delta})\mathbf{x}}c_{\mathbf{x}}$
in Eq. (8) with $e^{-ip_{x}\Delta_{x}-ip_{y}\Delta_{y}}$. Then the magnetic
translation by vector $\mathbf{\Delta}$ is

$\displaystyle T(\mathbf{\Delta})$
$\displaystyle=e^{i(\frac{1}{2}B\Delta_{x}(r_{y}+\Delta_{y})-\frac{1}{2}B\Delta_{y}(r_{x}+\Delta_{x}))}e^{-i(p_{x}\Delta_{x}+p_{y}\Delta_{y})}$
$\displaystyle=e^{-i((p_{x}-\frac{1}{2}Br_{y})\Delta_{x}+(p_{y}+\frac{1}{2}Br_{x})\Delta_{y})},$
(12)

where the Baker–Campbell–Hausdorff formula is considered. Therefore, the
generators of the magnetic translations in $\hat{\mathbf{x}}$ and
$\hat{\mathbf{y}}$ directions are

$\displaystyle K_{x}$ $\displaystyle=p_{x}-\frac{1}{2}Br_{y}$ $\displaystyle
K_{y}$ $\displaystyle=p_{y}+\frac{1}{2}Br_{x},$ (13)

which is exactly Zak’s definition from his 1964 paper [15].

In the remainder of the manuscript it will be easier to use the Landau gauge
$\mathbf{A}(\mathbf{r})=(-Br_{y},0)$. Repeating the calculation of
$\lambda_{g}$ in the Landau gauge yields

$\displaystyle\lambda_{T(\mathbf{\Delta})}(\mathbf{r})=-B\Delta_{y}r_{x}$ (14)

One important property of the magnetic translation operators is the gauge-
invariant noncommutativity:

$T(\Delta_{x}\hat{\mathbf{x}})T(\Delta_{y}\hat{\mathbf{y}})=T(\Delta_{y}\hat{\mathbf{y}})T(\Delta_{x}\hat{\mathbf{x}})e^{iB\Delta_{x}\Delta_{y}},$
(15)

which reproduces the Aharonov-Bohm phase. More generally, for two translations
$\mathbf{a}_{1}$ and $\mathbf{a}_{2}$, the gauge invariant multiplication
equation is [15]

$T(\mathbf{a}_{1})T(\mathbf{a}_{2})=T(\mathbf{a}_{1}+\mathbf{a}_{2})e^{\frac{i}{2}\mathbf{B}\cdot(\mathbf{a}_{1}\times\mathbf{a}_{2})}$
(16)

The gauge invariant phase term
$e^{\frac{i}{2}\mathbf{B}\cdot(\mathbf{a}_{1}\times\mathbf{a}_{2})}$ is the
2-cocycle of magnetic translations, which shows the magnetic translation
operators form a non-trivial projective representation of the translation
group.

###  II.2  Magnetic symmetries of the square lattice

$g$ | $T(\Delta_{x}\hat{\mathbf{x}})$ | $T(\Delta_{y}\hat{\mathbf{y}})$ | $C_{2}(\bar{x},\bar{y})$ | $C_{4}(\bar{x},\bar{y})$ | $I(\bar{x},\bar{y})$ | $Um_{x}(\bar{x})$ | $Um_{y}(\bar{y})$
---|---|---|---|---|---|---|---
$\hat{g}=\\{R_{g}|\tau_{g}\\}$ | $\\{0|(\Delta_{x},0)\\}$ | $\\{0|(0,\Delta_{y})\\}$ | $\\{\hat{C}_{2}|(2\bar{x},2\bar{y})\\}$ | $\\{\hat{C}_{4}|(\bar{x}+\bar{y},\bar{y}-\bar{x})\\}$ | $\\{\hat{I}|(2\bar{x},2\bar{y})\\}$ | $U\\{\hat{m}_{x}|(2\bar{x},0)\\}$ | $U\\{\hat{m}_{y}|(0,2\bar{y})\\}$
$\lambda_{g}(x,y)$ | 0 | $-B\Delta_{y}x$ | $-2B\bar{y}(x-\bar{x})$ | $-B(x-\bar{x})(y-\bar{y})$ | $-2B\bar{y}(x-\bar{x})$ | $0$ | $-2B\bar{y}x$
|  |  |  | $+B\bar{y}((y-{\bar{y}})-(x-\bar{x}))$ |  |  |
Table 1: The gauge transformation $\lambda_{g}(x,y)$ for symmetries of the
square lattice in Landau gauge. For each symmetry $g$ in the first row, the
second row lists the symmetry in the notation $\\{R_{g}|\tau_{g}\\}$, where
$\hat{g}:\mathbf{r}\mapsto R_{g}\mathbf{r}+\tau_{g}$. The third row provides
$\lambda_{g}$ from Eq. (6).

As a second example, we consider discrete symmetries of the two-dimensional
square lattice using the Landau gauge $\mathbf{A}(\mathbf{r})=(-Br_{y},0)$.
When $B=0$, the square lattice is invariant under the layer group $p4/mmm$,
which is generated by a four-fold rotation symmetr and the mirrors $m_{x}$ and
$m_{z}$. Without a magnetic field, the system is also invariant under time-
reversal symmetry, $\cal T$.

When $B\neq 0$, only the symmetries that leave the magnetic field invariant
(four-fold rotations and $m_{z}$) remain; the resulting layer group is $p4/m$.
To determine how these symmetries act on the electron creation/annihilation
operators, one must compute the gauge transformation $\lambda_{g}$ from Eq.
(6). We summarize the results in Table 1. Notice that $\lambda_{g}$ depends on
the rotation or inversion center; thus, it is necessary to introduce the
notation

$C_{n}(\bar{x},\bar{y})\equiv T(\bar{x},\bar{y})C_{n}T(-\bar{x},-\bar{y})$
(17)

to denote an $n$-fold rotation about the point $(\bar{x},\bar{y})$; we use
$C_{n}\equiv C_{n}(\bar{x}=0,\bar{y}=0)$ to denote a rotation about the
origin. We adopt analogous notation for inversions and reflections about
different points and planes.

The symmetries $m_{x}$, $m_{(110)}$ and $\mathcal{T}$ flip the magnetic field
and thus are not symmetries at finite $B$. However, the product of these
symmetries with a magnetic flux shifting operator can leave the system
invariant at special values of flux, as we now describe.

A lattice Hamiltonian coupled to a magnetic field is periodic in $B$: the
period corresponds to the minimal magnetic field such that every possible
closed hopping path encloses an integer multiple of $2\pi$ flux. Let $\phi$
denote the magnetic flux per unit cell and $\Phi=2\pi n$ its periodicity,
where $n$ is an integer. Following Ref. [2], we define the unitary matrix $U$
that shifts $\phi\mapsto\phi+\Phi$ by

$\displaystyle{U}$
$\displaystyle=e^{i\sum_{\mathbf{x}^{\prime}}\lambda_{U}(\mathbf{x}^{\prime})c^{\dagger}_{\mathbf{x}^{\prime}}c_{\mathbf{x}^{\prime}}}$
(18) $\displaystyle\lambda_{U}(\mathbf{r})$
$\displaystyle=\int^{\mathbf{r}}_{\mathbf{r}_{0}}\widetilde{\mathbf{A}}(\mathbf{r})\cdot
d\mathbf{r},$ (19)

where $\mathbf{r}_{0}$ is a reference lattice point and $\tilde{\mathbf{A}}$
is the magnetic vector potential corresponding to $\Phi$ flux, i.e.,
$\nabla\times\widetilde{\mathbf{A}}=\Phi$.

Notice that for any symmetry $g$ that flips $\phi\mapsto-\phi$, the product
$Ug$ is a symmetry in the special case where $\phi=\Phi/2$. In the case of the
square lattice, the products $Um_{x}$, $Um_{y}$ and $U{\cal T}$ are recovered
as symmetries of the system at the special value of $\phi=\Phi/2$. We list the
gauge transformations for $Um_{x}$ and $Um_{y}$ at $\phi=\Phi/2$ in Table 1.

In the special case of a square lattice and Landau gauge, $\lambda_{U}=-\Phi
yx,$ where $x=(\mathbf{r}-\mathbf{r}_{0})\cdot\hat{\mathbf{x}}$. Since $\Phi$
is a multiple of $2\pi$ and $x,y$ are integers, this phase is also a multiple
of $2\pi$. The flux translation matrix is given by $U=\mathbb{I}$, where
$\mathbb{I}$ is the identity matrix.

In summary, we have explicitly extended Zak’s translation operators in a
magnetic field to the discrete symmetries of the square lattice. In Appendix D
we generalize the results to the symmetries of the triangular lattice.

###  II.3 BBH model

We apply the results of the previous section to derive the symmetry operators
in the Benalcazar-Bernevig-Hughes (BBH) model [21, 22]. The model describes
spinless electrons on a square lattice. The Hamiltonian consists of nearest-
neighbor hopping terms, whose amplitudes $\lambda_{x/y}$ and $\gamma_{x/y}$
are depicted in Fig. 2. Since $\lambda_{x/y}\neq\gamma_{x/y}$, each unit cell
contains four atoms. Further, each square plaquette has $\pi$ flux, for a
total flux $\phi=4\pi$ per unit cell. The flux periodicity is $\Phi=8\pi$,
corresponding to $2\pi$ flux per square plaquette.

Figure 2: Lattice and hopping terms of the BBH model. The black dashed square
indicates the unit cell. Blue dots indicate atoms, each with one orbital. The
origin is at the left-bottom atom in the unit cell indicated by $0$. The
hopping amplitudes $\gamma_{x/y}$ and $\lambda_{x/y}$ are real; the minus
signs result from the magnetic flux $\phi=4\pi$, i.e., applying Eq. (1) in
Landau gauge.

We now derive the symmetry operations in the presence of the magnetic field;
these commutation relations were stated in Refs. 21, 22, but here we derive
them as an application of our formalism.

We start with the mirror symmetries: in zero field, the Hamiltonian is
invariant under $m_{x}(\bar{x})$ and $m_{y}(\bar{y})$ where $\bar{x},\bar{y}$
are half-integers. (The Hamiltonian is not invariant under reflections about
lines containing the origin because $\gamma_{x/y}\neq\lambda_{x/y}$.) These
mirror reflections flip the sign of the magnetic field and thus generically
are not symmetries of the Hamiltonian at finite flux. However, since
$\phi=4\pi=\Phi/2$, the combined operations $Um_{x}(\bar{x})$ and
$Um_{y}(\bar{y})$ are symmetries. We showed in the previous section that in
the Landau gauge, the flux shifting operator $U=\mathbb{I}$ for this model.
Therefore, at $\phi=4\pi$, $m_{x}(\bar{x})$ and $m_{y}(\bar{y})$ are in fact
symmetries of the Hamiltonian. The effect of the magnetic field is to change
their commutation relations: using Table 1 with $U=\mathbb{I}$ yields
$m_{x}(\bar{x})m_{y}(\bar{y})=m_{y}(\bar{y})m_{x}(\bar{x})e^{4iB\bar{x}\bar{y}}$.
Since $B=\pi$ and $\bar{x},\bar{y}$ are half-integers

$\\{m_{x}(\bar{x}),m_{y}(\bar{y})\\}=0,$ (20)

i.e., mirror symmetries in the BBH model anti-commute.

We now consider a four-fold rotation. When $\gamma_{x}=\gamma_{y}$,
$\lambda_{x}=\lambda_{y}$, the BBH model has a four-fold rotation symmetry
$C_{4}(\frac{1}{2},\frac{1}{2})$, as well as other four-fold rotation axes
related by translation. Since $\phi=\Phi/2$, the system also has an effective
time-reversal symmetry, $U\cal T$; since we established $U=\mathbb{I}$ for
this model, $\mathcal{T}$ is a symmetry even at this finite field and acts by
complex conjugation. In the absence of a magnetic field, time-reversal pairs
eigenstates with $\pm i$ rotation eigenvalues. We now show that in the
presence of a magnetic field, time-reversal pairs eigenstates of
$C_{4}(\bar{x},\bar{y})$ in a more complicated way.

Since our origin is chosen such that all lattice sites have integer
coordinates, $x,y\in\mathbb{Z}$, the phase $e^{i\lambda_{C_{4}}}$ in Table 1
takes values of $\pm e^{-i\pi/4}$, so that $e^{-2i\lambda_{C_{4}}}=i$.
Therefore, given an eigenstate $|\xi\rangle$ of $C_{4}(\bar{x},\bar{y})$ with
eigenvalue $\xi$, $\cal T|\xi\rangle$ is also an eigenstate of
$C_{4}(\bar{x},\bar{y})$:

$\displaystyle C_{4}(\bar{x},\bar{y}){\cal T}|\xi\rangle$
$\displaystyle=e^{i\lambda_{C_{4}}}\hat{C_{4}}(\bar{x},\bar{y}){\cal
T}|\xi\rangle$ $\displaystyle={\cal
T}e^{-i\lambda_{C_{4}}}\hat{C_{4}}(\bar{x},\bar{y})|\xi\rangle$
$\displaystyle={\cal
T}e^{-2i\lambda_{C_{4}}}C_{4}(\bar{x},\bar{y})|\xi\rangle$
$\displaystyle={\cal T}i\xi|\xi\rangle=-i\xi^{*}{\cal T}|\xi\rangle.$ (21)

Thus, $\cal T$ pairs $C_{4}(\bar{x},\bar{y})$ eigenstates with eigenvalues
$\xi$ and $-i\xi^{*}$. This is an example of symmetry operators acting in
unusual ways at finite field.

##  III Momentum space representations

We now define how the magnetic symmetry operators act in momentum space. This
requires first defining how the symmetries act on Bloch wave functions and
then labelling the Bloch wave functions by irreducible representations
(irreps) of the symmetry group at each momentum point. However, in the
presence of magnetic flux, we cannot immediately define the Bloch wave
functions because Bloch’s theorem does not apply when the translation
operators do not commute.

To apply Bloch’s theorem, we define an enlarged “magnetic unit cell,” chosen
to contain an integer multiple of $2\pi$ flux. The translation vectors that
span the magnetic unit cell are referred to as magnetic translation vectors.
From Eq. (15), the magnetic translation operators commute and thus can be
simultaneously diagonalized, forming an abelian subgroup $\mathbb{T}_{M}$ of
the full translation group $\mathbb{T}$. Consequently, Bloch’s theorem applies
to the magnetic unit cell and eigenstates of the Hamiltonian can be labelled
by wave vectors in the magnetic Brillouin zone.

In Sec. III.1, we define the Fourier transformed electron creation and
annihilation operators in the magnetic unit cell. The operators necessarily
have a “sublattice” index because the magnetic unit cell contains more than
one non-magnetic unit cell.

In Sec. III.2, we address how to label the Bloch wave functions by irreps of
the little co-group at each momentum. Here we encounter another subtle point:
since the little co-group is defined as a quotient group obtained from the
space group mod magnetic translations, the little co-group only has a group
structure if the magnetic translation group is a normal subgroup of the space
group. Thus, not all magnetic unit cells are equal: to label Bloch wave
functions by irreps of the little co-group, we must choose a magnetic unit
cell such that $\mathbb{T}_{M}$ is a normal subgroup. After addressing this
issue, we explain how to find the irreducible projective representations of
the little co-group.

###  III.1  Symmetries in the magnetic Brillouin zone

Figure 3:  Examples of (a) a $q$-by-$1$ unit cell and (b) a $q$-by-$q$ unit
cell, taking $q=2$.

We first consider the minimal magnetic unit cell in Landau gauge, which is a
$q$-by-$1$ unit cell (see Fig. 3(a)). For this choice of unit cell, the
magnetic translation group $\mathbb{T}_{M}$ is generated by
$T(\hat{\mathbf{x}})$ and $T(q\hat{\mathbf{y}})$.

Now consider the layer group $p2$, generated by $C_{2}$ and lattice
translations, for which $\mathbb{T}_{M}$ is a normal subgroup. $C_{2}$ acts
identically to the non-magnetic case, mapping a Bloch wave function at
$\mathbf{k}$ to one at $-\mathbf{k}$. However, $T(\hat{\mathbf{y}})$ acts in
an unusual way, by mapping $k_{x}$ to $k_{x}+\phi$. This can be understood as
follows: let $|\mathbf{k}\rangle$ be an eigenstate of $T(\hat{\mathbf{x}})$
such that
$T(\hat{\mathbf{x}})|\mathbf{k}\rangle=e^{ik_{x}}|\mathbf{k}\rangle$. Then
$T(\hat{\mathbf{y}})|\mathbf{k}\rangle$ is also an eigenstate of
$T(\hat{\mathbf{x}})$, with eigenvalue $k_{x}+\phi$, i.e.,

$T(\hat{\mathbf{x}})\left[T(\hat{\mathbf{y}})|\mathbf{k}\rangle\right]=e^{i(k_{x}+\phi)}T(\hat{\mathbf{y}})|\mathbf{k}\rangle$
(22)

Thus, $T(\hat{\mathbf{y}})$ shifts the eigenvalue of $T(\hat{\mathbf{x}})$ by
$e^{i\phi}$. Nonetheless, both $C_{2}$ and $T(\hat{\mathbf{y}})$ have the
usual property that a Bloch state at $\mathbf{k}$ is mapped to another Bloch
state at $\mathbf{k}^{\prime}$.

This is not the case for the layer group $p4$, with respect to which
$\mathbb{T}_{M}$ is not a normal subgroup. As we will show below, the symmetry
operator $C_{4}$ mixes a Bloch state at $\mathbf{k}$ into a linear combination
of Bloch states at other momenta, forming a $q^{2}$-dimensional
representation. Thus, we are motivated to consider a $q$-by-$q$ unit cell (see
Fig. 3(b)), where, although the magnetic unit cell is larger, the symmetry
matrices are the same size as in the $q$-by-$1$ case. In Appendix E, we show
that the representations obtained from these two choices of magnetic unit cell
are the same up to a unitary transformation. However, the $q$-by-$q$ unit cell
is a more suitable to apply topological quantum chemistry because the
corresponding magnetic translation group is a normal Abelian subgroup of the
layer group $p4$. We now consider the $q$-by-$1$ and $q$-by-$q$ unit cells in
detail for the group $p4$ to illustrate these points.

####  III.1.1 $q$-by-$1$ unit cell for $p4$

We first consider the $q$-by-$1$ unit cell shown in Fig. 3(a). The coordinates
of lattice sites are labeled by $(x,y)=(R_{x},qR_{y}+j)$ where
$R_{x},R_{y}\in\mathbb{Z}$, $j=0,\dots,q-1$. The Fourier transformed electron
creation and annihilation operators are defined by

$\displaystyle c^{\dagger}_{\mathbf{R},j,\alpha}$
$\displaystyle=\frac{q}{(2\pi)^{2}}\int
d\mathbf{k}e^{i(k_{x}R_{x}+k_{y}qR_{y})}c^{\dagger}_{\mathbf{k},j,\alpha}$
(23) $\displaystyle c_{\mathbf{R},j,\alpha}$
$\displaystyle=\frac{q}{(2\pi)^{2}}\int
d\mathbf{k}e^{-i(k_{x}R_{x}+k_{y}qR_{y})}c_{\mathbf{k},j,\alpha},$ (24)

where $\alpha$ labels orbital degrees of freedom on each site. For now, we
ignore the $\alpha$ degree of freedom, but will add it later when necessary.
The magnetic Brillouin zone is a torus with $k_{x}\in[0,2\pi)$,
$k_{y}\in[0,2\pi/q)$.

Using the Fourier transforms in Eqs. (23) and (24), we find the action of the
symmetry operators in momentum space. A translation by one (non-magnetic)
lattice vector in the $\hat{\mathbf{y}}$ direction is implemented by

$\displaystyle T(\hat{\mathbf{y}})$
$\displaystyle=\frac{q}{(2\pi)^{2}}\sum_{j}\int
d\mathbf{k}~{}e^{ik_{y}}c^{\dagger}_{\mathbf{k}+(\phi,0),j}c_{\mathbf{k},j-1}$
(25)

Unlike the non-magnetic case, $T(\hat{\mathbf{y}})$ does not leave each
$\mathbf{k}$ point invariant: it maps $(k_{x},k_{y})$ to $(k_{x}+\phi,k_{y})$.
Translations by the magnetic lattice vectors do leave $\mathbf{k}$ invariant.

We now consider the four-fold rotation operator. Using the function
$\lambda_{C_{4}}$ in Table 1,

$\displaystyle C_{4}$ $\displaystyle=\frac{q}{(2\pi)^{2}}\int
d\mathbf{k}\sum_{j,j^{\prime}}\sum_{n=0}^{q-1}\frac{1}{q}e^{i(\phi
jj^{\prime}-(k_{y}+2\pi n/q)j-k_{x}j^{\prime})}$ $\displaystyle
c^{\dagger}_{(k_{x},k_{y}),j}c_{(k_{y}+2\pi
n/q,~{}-k_{x}~{}\text{mod}~{}2\pi/q),j^{\prime}}$ (26)

Thus, the situation for $C_{4}$ is much worse than for $T(\hat{\mathbf{y}})$:
$C_{4}$ does not rotate one $\mathbf{k}$ point to another, but instead mixes a
state at $(k_{x},k_{y})$ into a linear combination of states at the different
points $(k_{y}+2\pi n/q,-k_{x})$ $n=0,1,\dots,q-1$.

####  III.1.2 $q$-by-$q$ unit cell for $p4$

We now consider the $q$-by-$q$ unit cell shown in Fig. 3(b). The coordinates
of lattice sites are labeled by $(x,y)=(qR_{x}+j_{x},qR_{y}+j_{y})$ where
$R_{x},R_{y}\in\mathbb{Z}$ label a magnetic unit cell and
$j_{x},j_{y}=0,\dots,q-1$ label the coordinates of atoms within. The Fourier
transformed electron creation and annihilation operators are defined by

$\displaystyle c^{\dagger}_{\mathbf{R},\mathbf{j},\alpha}$
$\displaystyle=(\frac{q}{2\pi})^{2}\int
d\mathbf{k}e^{i(k_{x}qR_{x}+k_{y}qR_{y})}c^{\dagger}_{\mathbf{k},\mathbf{j},\alpha}$
(27) $\displaystyle c_{\mathbf{R},\mathbf{j},\alpha}$
$\displaystyle=(\frac{q}{2\pi})^{2}\int
d\mathbf{k}e^{-i(k_{x}qR_{x}+k_{y}qR_{y})}c_{\mathbf{k},\mathbf{j},\alpha}$
(28)

Again we omit the orbital degrees of freedom $\alpha$ in this section. The
magnetic Brillouin zone is a torus with $k_{x},k_{y}\in[0,2\pi/q)$.

Using the Fourier transforms in Eqs. (27) and (28) and plugging $\lambda_{g}$
from Table 1 into Eq. (9), the magnetic $T(\hat{\mathbf{y}})$ and $C_{4}$
symmetries are [2]

$\displaystyle T(\hat{\mathbf{y}})$ $\displaystyle=(\frac{q}{2\pi})^{2}\int
d\mathbf{k}~{}e^{ik_{y}}\sum_{j_{x},j_{y}}e^{-i\phi
j_{x}}c^{\dagger}_{\mathbf{k},j_{x},j_{y}}c_{\mathbf{k},j_{x},j_{y}-1}$ (29)

and

$\displaystyle C_{4}$ $\displaystyle=(\frac{q}{2\pi})^{2}\int
d\mathbf{k}~{}\sum_{j_{x},j_{y}}e^{-i\phi
j_{x}j_{y}}e^{-i(C_{4}\mathbf{k}\cdot\mathbf{j}-\mathbf{k}\cdot\mathbf{j}^{\prime})}$
$\displaystyle\qquad
c^{\dagger}_{(-k_{y},k_{x}),j_{x},j_{y}}c_{(k_{x},k_{y}),j_{x}^{\prime},j_{y}^{\prime}}$
(30)

where $\mathbf{j}^{\prime}=(j_{x}^{\prime},j_{y}^{\prime})$ is a function of
$j_{x},j_{y}$ that satisfies $j_{x}^{\prime}=j_{y}\mod q$ and
$j_{y}^{\prime}=-j_{x}\mod q$. In Eqs. (29) and (30), the action of the
symmetry operator on $\mathbf{k}$ is identical to its action in the absence of
a magnetic field, i.e., translation leaves $\mathbf{k}$ invariant and a
rotation in space rotates $\mathbf{k}$. This is an improvement over the
$q$-by-$1$ magnetic unit cell (Eqs. (25) and (26)), for which a rotation mixed
a Bloch state into a linear combination of several Bloch states.

###  III.2 Irreps at high symmetry points

We now address how to determine irreps of the symmetry group at each momentum.
A Bloch wave function at a particular momentum $\mathbf{k}$ transforms as a
representation of the little group at $\mathbf{k}$, denoted $G_{\mathbf{k}}$,
which consists of all the space group operations that leave $\mathbf{k}$
invariant up to a reciprocal lattice vector:

$G_{\mathbf{k}}=\\{g\in G|g\mathbf{k}\equiv\mathbf{k}\\},$ (31)

where $\equiv$ is defined by equality up to a reciprocal lattice vector. Since
the lattice translations are always represented by Bloch phases in the
representations, it is useful to label the wave functions by irreps of the
little co-group, defined as

$\widetilde{G}_{\mathbf{k}}=G_{\mathbf{k}}/{\mathbb{T}}_{M}$ (32)

As mentioned above, for the little co-group to satisfy the definition of a
group, $\mathbb{T}_{M}$ must be a normal subgroup of $G_{\mathbf{k}}$, i.e.,
for all $g\in G_{\mathbf{k}}$, $t\in\mathbb{T}_{M}$,
$g^{-1}tg\in\mathbb{T}_{M}$. One can check that for the $q$-by-$1$ unit cell,
the magnetic translation group is a normal subgroup of the layer group $p2$,
but it is not normal for the layer groups containing three- or four-fold
rotations (because, for example,
$C_{4}^{-1}T(\hat{\mathbf{x}})C_{4}=T(\hat{\mathbf{y}})^{-1}$, which is not in
the magnetic translation group for the $q$-by-$1$ unit cell.) Thus, we use the
$q$-by-$q$ unit cell for layer groups with three- or four-fold rotations.

Thus, under magnetic flux, the little co-groups and their irreps differ from
their zero-flux analogues in two important ways: first, in the presence of
magnetic flux, the little co-groups include sublattice translation symmetries;
and second, the irreps of little co-groups in the presence of magnetic flux
are projective representations corresponding to the 2-cocyle defined by the
flux.

We now study some examples: in Tables 2, 3 and 4 we summarize the projective
irreps at high symmetry points for the layer groups $p2$, $p4$,
$p4/m^{\prime}$ at flux $\phi=\pi$. For later convenience we have assumed
there is spin-orbit coupling, i.e., $C_{n}^{n}=-1$. Notice that the character
tables are not square, which is a general feature of projective
representations. The projective irreps corresponding to a particular 2-cocycle
can be considered as a subset of non-projective representations of a larger
group; the character table of that larger group will be square.

To ensure that we have found all the projective irreps, we use the theorem by
Schur [23] stating that for irreducible projective representations with a
particular $2$-cocyle,

$\sum_{\rho}\left(\text{dim}(\rho)\right)^{2}=|\widetilde{G}_{\mathbf{k}}|,$
(33)

where the sum runs over all projective irreps $\rho$ of
$\widetilde{G}_{\mathbf{k}}$ with the specified $2$-cocyle and
$\widetilde{G}_{\mathbf{k}}$ is the little co-group defined above. (Notice
this formula does not apply to anti-unitary groups.)

The calculation of the irreps of little co-groups are shown in Appendix F with
the (anti)-commutation relations for the magnetic symmetries shown in Appendix
B. In the remainder of this section, we sketch the calculation for the
simplest non-trivial case, layer group $p2$ at $\pi$ flux.

For the $2$-by-$1$ unit cell, the group of magnetic lattice translations is
${\mathbb{T}}_{M}=\\{T(n_{1}{\hat{\mathbf{x}}}+2n_{2}{\hat{\mathbf{y}}})|n_{1},n_{2}\in\mathbb{Z}\\}$
and the Brillouin zone is $[-\pi,\pi)\times[-\pi/2,\pi/2)$. We now determine
the high-symmetry points. Since $C_{2}$ symmetry maps $(k_{x},k_{y})$ to
$(-k_{x},-k_{y})$, there are four momenta that are symmetric under $C_{2}$ up
to a magnetic reciprocal lattice vector:
$(0,0),(0,\pi/2),(\pi,0),(\pi,\pi/2)$. Since $T(\hat{\mathbf{y}})$ maps
$(k_{x},k_{y})$ to $(k_{x}+\pi,k_{y})$ (Eq. (22)), $T(\hat{\mathbf{y}})C_{2}$
maps $(k_{x},k_{y})$ to $(-k_{x}+\pi,-k_{y})$. Therefore, there are four
$T(\hat{\mathbf{y}})C_{2}$ symmetric momenta, $(\pm\pi/2,0)$ and
$(\pm\pi/2,\pi/2)$.

We derive in Appendix F that the $C_{2}$ eigenvalues at $(\pi,0)$ are the same
as $(0,0)$, while the $C_{2}$ eigenvalues at $(0,\pi/2)$ are opposite of
$(\pi,\pi/2)$. The same relations hold for the $T(\hat{\mathbf{y}})C_{2}$
symmetric points. In conclusion, there are two independent $C_{2}$ symmetric
points, $\Gamma=(0,0)$ and $Y=(0,\pi/2)$, and we find that each has two one-
dimensional irreps labeled by $C_{2}$ eigenvalue $+i$, $-i$. There are also
two independent $T(\hat{\mathbf{y}})C_{2}$ symmetric points, $X=(\pi/2,0)$ and
$M=(\pi/2,\pi/2)$, and each has two one-dimensional irreps labeled by
$T(\hat{\mathbf{y}})C_{2}$ eigenvalue $+i$, $-i$. Since each little co-group
contains the identity element and either $C_{2}$ or
$T(\hat{\mathbf{y}})C_{2}$, $|\widetilde{G}_{\mathbf{k}}|=2$ for these points.
Thu, Eq. (33) is satisfied, which means we have found all the projective
irreps.

| $X(\pi/2,0)$ | $Y(0,\pi/2)$ | $\Gamma(0,0)$ | $M(\pi/2,\pi/2)$
---|---|---|---|---
Irrep | $X_{1}^{(p2)}$ | $X_{2}^{(p2)}$ | $Y_{1}^{(p2)}$ | $Y_{2}^{(p2)}$ | $\Gamma_{1}^{(p2)}$ | $\Gamma_{2}^{(p2)}$ | $M_{1}^{(p2)}$ | $M_{2}^{(p2)}$
$C_{2}$ |  |  | $+i$ | $-i$ | $+i$ | $-i$ |  |
$T(\hat{\mathbf{y}})C_{2}$ | $+i$ | $-i$ |  |  |  |  | $+i$ | $-i$
Table 2: High symmetry momenta (first row) and the irreps (second row) of
their little co-group for the group $p2$. The third and fourth rows list the
eigenvalue of the indicated symmetry; the row is blank if the symmetry is not
in the little co-group.

| $X(\pi/2,0)$ | $Y(0,\pi/2)$
---|---|---
Irrep | $X_{1}$ | $X_{2}$ | $Y_{1}$ | $Y_{2}$
$C_{2}$ | $i\sigma_{z}$ | $-i\sigma_{z}$ | $i\sigma_{z}$ | $-i\sigma_{z}$
$T(\hat{\mathbf{x}})$ | $\sigma_{x}$ | $\sigma_{x}$ | $\sigma_{z}$ | $\sigma_{z}$
$T(\hat{\mathbf{y}})$ | $\sigma_{z}$ | $\sigma_{z}$ | $\sigma_{y}$ | $\sigma_{y}$
$T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})$ | $-i\sigma_{y}$ | $-i\sigma_{y}$ | $-i\sigma_{x}$ | $-i\sigma_{x}$

| $\Gamma(0,0)$
---|---
Irrep | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$
$C_{4}T(\hat{\mathbf{x}})$ | $\begin{pmatrix}1&\\\ &i\end{pmatrix}$ | $\begin{pmatrix}i&\\\ &-1\end{pmatrix}$ | $\begin{pmatrix}-1&\\\ &-i\end{pmatrix}$ | $\begin{pmatrix}-i&\\\ &1\end{pmatrix}$
$T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})$ | $i\sigma_{z}$ | $i\sigma_{z}$ | $i\sigma_{z}$ | $i\sigma_{z}$

| $M(\pi/2,\pi/2)$
---|---
Irrep | $M_{1}$ | $M_{2}$ | $M_{3}$ | $M_{4}$
$C_{4}$ | $\begin{pmatrix}\epsilon&\\\ &\epsilon^{*}\end{pmatrix}$ | $\begin{pmatrix}-\epsilon^{*}&\\\ &\epsilon\end{pmatrix}$ | $\begin{pmatrix}-\epsilon&\\\ &-\epsilon^{*}\end{pmatrix}$ | $\begin{pmatrix}\epsilon^{*}&\\\ &-\epsilon\end{pmatrix}$
$T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})$ | $i\sigma_{z}$ | $i\sigma_{z}$ | $i\sigma_{z}$ | $i\sigma_{z}$

Table 3: High symmetry momenta (first row) and the irreps (second row) of their little co-group for the group $p4$. Subsequent rows list the eigenvalue of the indicated symmetry with $\epsilon=e^{i\pi/4}$; the row is blank if the symmetry is not in the little co-group. | $X(\pi/2,0)$ | $Y(0,\pi/2)$ | $\Gamma(0,0)$ | $M(\pi/2,\pi/2)$
---|---|---|---|---
Irrep | $X_{1}X_{2}$ | $Y_{1}Y_{2}$ | $\Gamma_{1}\Gamma_{4}$ | $\Gamma_{2}\Gamma_{3}$ | $M_{1}M_{1}$ | $M_{3}M_{3}$ | $M_{2}M_{4}$
Table 4: High symmetry momenta (first row) and the irreps (second row) of
their little co-group for the group $p4/m^{\prime}$. The notation
$\Pi_{i}\Pi_{j}$ indicates that $\Pi_{i}$ and $\Pi_{j}$ are paired by $\cal
TI$ symmetry, where $\Pi_{i}$ is an irrep of the corresponding little co-group
with respect to layer group $p4$, shown in Table 3.

##  IV Topological quantum chemistry in a magnetic field

Finally we turn to the theory of TQC. TQC classifies topological crystalline
insulators (TCIs) by enumerating all trivial phases in each space group, where
a trivial phase is defined as one where exponentially localized Wannier
functions exist and transform locally under all symmetries. A group of bands
can be identified as a TCI if it is not in the space of trivial phases.

Together, the Wannier functions corresponding to a single band (or group of
bands) transform as a representation of the full space group, called a band
representation [24, 25, 26, 27, 28, 6, 9]. TQC labels each band representation
by how its Bloch wave functions transform under symmetry at high symmetry
momenta, i.e., by a set of irreps of the little co-group at each high symmetry
momentum; this label is known as a symmetry indicator [14]. Symmetry
indicators provide a practical way to identify many TCIs: specifically, a
group of bands whose irreps at high symmetry momenta are not consistent with
any of the trivial phases must be topological.

In Sec. IV.1 we describe how to construct a basis of symmetric magnetic
Wannier functions. We use this basis in Sec. IV.2 to derive how the space
group symmetries act on the Wannier functions; the symmetry matrices comprise
the band representation. Fourier transforming the band representation yields
its symmetry indicator.

###  IV.1 Magnetic Wannier functions

We now describe how to construct a basis of symmetric Wannier functions for a
magnetic unit cell. Given a site $\mathbf{q}$, which will serve as a Wannier
center, the site-symmetry group $G_{\mathbf{q}}$ is defined as the set of
symmetries that leave $\mathbf{q}$ invariant, i.e., $G_{\mathbf{q}}=\\{g\in
G|g\mathbf{q}=\mathbf{q}\\}$. The site-symmetry group defines a coset
decomposition of the space group,

$G=\bigcup_{\alpha}g_{\alpha}G_{\mathbf{q}}\ltimes{\mathbb{T}}_{M},$ (34)

where $G$ is the space group, ${\mathbb{T}}_{M}$ is the magnetic lattice
translation group, and $\alpha=1,\dots,n$, where
$n=|G/{\mathbb{T}}_{M}|/|G_{\mathbf{q}}|$ is the multiplicity of the Wyckoff
position containing $\mathbf{q}$. The symmetries $g_{\alpha}$ are coset
representatives. The choice of coset representatives is not unique; a
different choice will yield a band representation related to the original by a
unitary transformation, while the symmetry indicator is unchanged.

The coset representatives define positions
${\mathbf{q}}_{\alpha}=g_{\alpha}{\mathbf{q}}$ that form the orbit of
$\mathbf{q}$ within the magnetic unit cell. Together, these points are part of
the same Wyckoff position, whose multiplicity $n$ is equal to the number of
points in the orbit of $\mathbf{q}$ in the magnetic unit cell. Unlike the case
of zero magnetic field, the set of coset representatives $g_{\alpha}$ includes
pure translations within the magnetic unit cell. Fig. 4 shows the Wyckoff
positions for the groups $p2$, and $p4$, $p4/m^{\prime}$.

Figure 4: Wyckoff positions in a magnetic flux $\pi$ for (a) the $2$-by-$1$
unit cell for group $p2$ and (b) the $2$-by-$2$ unit cell for the groups $p4$
and $p4/m^{\prime}$. Each Wyckoff position is labelled by its multiplicity and
an alphanumeric character. The general Wyckoff position, whose site-symmetry
group consists of only the identity, is not shown.

Suppose there are $n_{\mathbf{q}}$ orbitals centered at $\mathbf{q}$. These
orbitals are described by $n_{\mathbf{q}}$ Wannier functions $|W_{i1}\rangle$,
where $i=1\dots n_{\mathbf{q}}$. The Wannier functions transform under
symmetries $g\in G_{\mathbf{q}}$ as a projective representation $\rho$ of
$G_{\mathbf{q}}$,

$g|W_{i1}\rangle=\sum_{j=1}^{n_{\mathbf{q}}}[\rho(g)]_{ji}|W_{j1}\rangle.$
(35)

Applying the representatives $g_{\alpha}$ in the coset decomposition of the
space group $G$ in Eq. (34) to $|W_{i1}\rangle$ gives another Wannier function

$|W_{i\alpha}\rangle=g_{\alpha}|W_{i1}\rangle,$ (36)

localized at ${\mathbf{q}}_{\alpha}$. All these Wannier functions
$|W_{i\alpha}\rangle$, where $i=1\dots n_{\mathbf{q}}$ and $\alpha=1\dots n$,
form an induced representation of $G$, as we now explain.

###  IV.2 Induced representation

In this section, we derive how the space group symmetries act on the Wannier
functions. This provides an explicit construction of a band representation
with Wannier functions as a basis. Fourier transforming the band
representation gives the irreps of the little co-group at each high-symmetry
point, i.e., the symmetry indicator.

Consider a group element $hg_{\alpha}\in G$. The coset decomposition in Eq.
(34) implies that $hg_{\alpha}$ can be written in the form

$hg_{\alpha}=e^{if_{\alpha\beta}(h)}\\{E|{\mathbf{t}}_{\alpha\beta}(h)\\}g_{\beta}g$
(37)

where
$\mathbf{t}_{\alpha\beta}(h)=h{\mathbf{q}}_{\alpha}-{\mathbf{q}}_{\beta}$ and
$\\{E|\mathbf{t}_{\alpha\beta}\\}\in\mathbb{T}_{M}$, $g\in G_{\mathbf{q}}$,
and the coset representative $g_{\beta}$ are uniquely determined by the coset
decomposition. The remaining phase factor $f_{\alpha\beta}(h)$ is due to the
non-trivial 2-cocycles. For the two-dimensional systems without magnetic
field, $f_{\alpha\beta}(h)\equiv 0$. For the case with magnetic field, in
general $f_{\alpha\beta}(h)$ is nonzero.

The phase factor $f_{\alpha\beta}(h)$ is the new ingredient that appears in a
magnetic field and is a key result of the present work; it does not appear in
the non-magnetic theory (for example, it does not appear in Eq. (6) in Ref.
9). This phase factor is gauge invariant because it results from the
commutations between rotations and translations (see Appendix B). We briefly
give two examples to show how this phase factor appears.

As a first example, consider the layer group $p1$ with a $2$-by-$2$ unit cell,
corresponding to $\pi$ flux. Starting from a Wannier function centered at a
general position $\mathbf{q}=(x,y)$, the coset representatives in Eq. (34) can
be chosen as $g_{1}=\\{E|0\\}$, $g_{2}=T(\hat{\mathbf{x}})$,
$g_{3}=T(\hat{\mathbf{y}})$, $g_{4}=T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})$.
Now consider the left-hand-side of Eq. (37) with $h=T(\hat{\mathbf{y}})$,
$g_{\alpha}=T(\hat{\mathbf{x}})$. Then on the right-hand-side of Eq. (37),
$g_{\beta}=T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})$, $g=E$, and
${\mathbf{t}}_{\alpha\beta}=0$. Since
$T(\hat{\mathbf{y}})T(\hat{\mathbf{x}})=e^{i\pi}T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})$,
$f_{\alpha\beta}(h)=\pi$.

As a second example, consider layer group $p4$ with a $2$-by-$2$ unit cell,
corresponding again to $\pi$ flux. Starting from a Wannier function centered
at $\mathbf{q}=(\frac{1}{2},\frac{1}{2})$, the coset representatives in Eq.
(34) can be chosen as $g_{1}=\\{E|0\\}$, $g_{2}=T(\hat{\mathbf{x}})$,
$g_{3}=T(\hat{\mathbf{y}})$, $g_{4}=T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})$.
Now consider the left-hand-side of Eq. (37) with $h=C_{4}$,
$g_{\alpha}=T(\hat{\mathbf{x}})$. The coset decomposition uniquely determines
$g_{\beta}=T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})$,
$g=C_{4}(\frac{1}{2},\frac{1}{2})$, and $\mathbf{t}_{\alpha\beta}=(-2,0)$ on
the right-hand-side of Eq. (37). Since
$C_{4}T(\hat{\mathbf{x}})=e^{i3\pi/4}\\{E|(-2,0)\\}T(\hat{\mathbf{x}})T(\hat{\mathbf{y}})C_{4}(1/2,1/2)$,
the extra phase term $f_{\alpha\beta}(h)=3\pi/4$.

As discussed at the start of Sec. IV, the set of Wannier functions centered at
all $\mathbf{q}_{\alpha}$ form a basis for a band representation, which we
denote $\rho_{G}$. Given a space group symmetry $h\in G$, Eq. (37) determines
the matrix elements of $\rho_{G}(h)$ in the basis of Wannier functions defiend
in Eq. (36) by:

$\displaystyle\rho_{G}(h)|W_{i\alpha}(\mathbf{r}-\mathbf{t})\rangle$
$\displaystyle=e^{if_{\alpha\beta}(h)}[\rho(g)]_{ji}|W_{j\beta}(\mathbf{r}-R\mathbf{t}-{\mathbf{t}}_{\alpha\beta})\rangle$
(38)

where $R$ is the rotational part of $h$, $\rho(g)$ is the given representation
defined in Eq. (35),
$\mathbf{t}_{\alpha\beta}(h)=h{\mathbf{q}}_{\alpha}-{\mathbf{q}}_{\beta}$ and
sum over $j=1\dots n_{\mathbf{q}}$ is implied.

Substituting the Fourier transformed Wannier functions,

$\displaystyle|W_{j\beta}(\mathbf{r}-\mathbf{t})\rangle=\int
d\mathbf{k}e^{i{\mathbf{k}}\cdot{\mathbf{t}}}|a_{j\beta}(\mathbf{k},\mathbf{r})\rangle,$
(39)
$\displaystyle|a_{j\beta}(\mathbf{k},\mathbf{r})\rangle=\sum_{\mathbf{t}}e^{-i{\mathbf{k}}\cdot{\mathbf{t}}}|W_{j\beta}(\mathbf{r}-\mathbf{t})\rangle,$
(40)

into Eq. (38) yields

$\displaystyle\rho_{G}(h)|a_{i\alpha}(\mathbf{k},\mathbf{r})\rangle$
$\displaystyle=e^{if_{\alpha\beta}(h)-iR{\mathbf{k}}\cdot{\mathbf{t}}_{\alpha\beta}}[\rho(g)]_{ji}|a_{j\beta}(R\mathbf{k},\mathbf{r})\rangle$
(41)

From Eq. (41), a representation of the little co-group (defined in Sec. III.2)
is determined from $\rho_{G}$ by restricting each matrix
$\rho_{G}(h\in\widetilde{G}_{\mathbf{k}})$ to only the rows and column
corresponding to Fourier-transformed Wannier functions at $\mathbf{k}$. The
set of irreps obtained at all $\mathbf{k}$ determines the symmetry indicator
following the procedure we introduced in Ref. 29, which is summarized in
Appendix G.

We now derive the symmetry indicator classification for a few examples.

###  IV.3 Examples

We apply our formalism of TQC in a magnetic flux to three magnetic layer
groups with $\pi$ flux: $p2$, $p4$, and $p4/m^{\prime}$. In each case, we
discuss the stable symmetry indicator classification; the derivations are in
Appendix G.

####  IV.3.1 p2

For layer group $p2$ with $\pi$ flux, we choose a $2$-by-$1$ magnetic unit
cell, following the discussion in Sec. III.1. The symmetry indicator has a
$\mathbb{Z}_{4}$ classification. The indicator for a particular group of bands
is

$\text{index}=\\#\Gamma_{1}-\\#\Gamma_{2}+\\#X_{1}-\\#X_{2}+\\#Y_{1}-\\#Y_{2}\\\
+\\#M_{1}-\\#M_{2}-N\mod 4$ (42)

where $\\#\Pi_{i}$ indicates the number of times the irrep $\Pi_{i}$ at the
high symmetry point $\Pi$ appears in the bands and
$N=\\#\Gamma_{1}+\\#\Gamma_{2}=\\#X_{1}+\\#X_{2}=\\#Y_{1}+\\#Y_{2}=\\#M_{1}+\\#M_{2}$
is the filling per magnetic unit cell.

This indicator Eq. (42) is exactly the same as the Chern number indicator Eq.
(3) in Ref. 17 in the special case of $\pi$-flux and spinful electrons, i.e.,

$e^{i\pi(C/q-\bar{\rho})}=(-)^{2SN}w_{C_{2}}^{\Gamma}w_{C_{2}}^{Y}w_{T(\hat{\mathbf{y}})C_{2}}^{X}w_{T(\hat{\mathbf{y}})C_{2}}^{M}$
(43)

where at flux $\pi$, $q=2$; $\bar{\rho}=N/2$ is the filling per non-magnetic
primitive unit cell; $S=1/2$ is the spin (angular momentum) quantum number;
and $w_{g}^{\Pi}$ is the product of eigenvalues of the symmetry $g$ for all
filled bands at momentum $\Pi$.

####  IV.3.2 p4

For layer group $p4$ with $\pi$ flux, we choose $2$-by-$2$ unit cell,
following the discussion in Sec. III.1. The symmetry indicator has a
$\mathbb{Z}_{8}$ classification. The indicators for a particular group of
bands are determined by:

$\text{index}=2\\#\Gamma_{1}+4\\#\Gamma_{2}-2\\#\Gamma_{3}+\\#M_{1}+3\\#M_{2}\\\
-3\\#M_{3}-\\#M_{4}+4\\#X_{1}\mod 8$ (44)

To understand this indicator, we compare the new index to the symmetry
indicator formula for Chern number in Ref. 18:

$e^{i\frac{\pi}{2}C}=(-)^{2SN}w_{C_{4}}^{\Gamma}w_{C_{2}}^{X}w_{C_{4}}^{M},$
(45)

which, in terms of irreps, is given by (derivation in Appendix H)

$\displaystyle C=2N$
$\displaystyle+\\#\Gamma_{1}+\\#\Gamma_{3}-\\#\Gamma_{2}-\\#\Gamma_{4}$
$\displaystyle+2(\\#M_{2}+\\#M_{4})\mod 4$ (46)

$N$ is always an even integer due to the two-dimensional irreps (shown in
Table 3). We conclude

$C=\text{index}\mod 4$ (47)

The topological phase with index equal to $4\mod 8$ goes beyond the earlier
symmetry indicator given by Eq. (45). We leave it to future work to determine
whether this index is a stronger indicator for the Chern number or has a
different physical meaning.

####  IV.3.3 p4/m’

For layer group $p4/m^{\prime}$ with $\pi$ flux, we choose $2$-by-$2$ unit
cell, following the discussion in Sec. III.1. The symmetry indicator has a
$\mathbb{Z}_{2}$ classification. The indicator for a particular group of bands
is determined by:

index $\displaystyle=N/4\mod 2,$ (48)

where $N/4\equiv\bar{\rho}$ is the filling per original unit cell. Notice each
band in this group is four-fold degenerate (see Table 4 in Sec. III.2), and
hence $\bar{\rho}\in\mathbb{Z}$.

The group $p4/m^{\prime}$ is generated by a four-fold rotation and the product
of time-reversal and inversion symmetry $\cal TI$. As is well known, $\cal TI$
prevents a non-vanishing Chern number [30] and the absence of $\cal T$
prevents the existence of strong topological insulator [31]. Since $\cal T$
and $\cal I$ are not separately symmetries, there is no mirror symmetry and
hence no mirror Chern number. Thus, our stable index is a new phase that only
exists in systems with magnetic flux.

This phase is realized in the model we present in Sec. V. However, it does not
realize an anomalous gapless boundary state because when the boundary is
opened, the sublattice translation symmetries that protect the phase are
broken.

##  V  Application to a quadrupole insulator

In this section, we apply our results to a model on the square lattice. At
zero flux, this model is a quadrupole insulator that exhibits corner states.
Since the symmetries that protect the corner states are preserved in the
presence of a perpendicular magnetic field, the corner states must survive
when magnetic flux is introduced. We use the formalism developed in the
previous sections to verify the presence of corner states using symmetry
indicators. Finally, we show that at a critical magnetic flux, the bulk gap
closes and the corner states disappear, as shown in the Hofstadter butterfly
spectrum in Fig. 5. We use the symmetry indicators to verify that when the
corner states disappear, the symmetry indicator vanishes.

Our results provide a new probe of the higher order topology in the model,
i.e., the presence of a gap closing phase transition in the presence of a
magnetic field, which may be easier to observe than probing the corner states
directly.

###  V.1  Model

Figure 5: Hofstadter spectrum of the OAL model. The grey states are calculated
with periodic boundary conditions and show the bulk gap closing at a critical
flux. The red states are calculated with open boundary conditions and show the
disappearance of the corner states upon bulk gap closure. The spectrum is
computed for dimensions $L_{x}=200$, $L_{y}=10$ and parameters $\lambda=1$,
$\gamma=0.5$.

We study a model proposed by Wieder et al in Ref. 19 at zero flux that has the
same momentum space Hamiltonian as the $C_{4}$ symmetric Bernevig-Benalcazar-
Hughes (BBH) model which is at $4\pi$ flux per unit cell. [22, 21] This model
was given as an example in Fig. 3 and Appendix A of Ref. 19. Yet the two
models have some fundamental differences: while the BBH model has four atoms
per unit cell and one orbital per atom, Wieder’s model has one atom sitting at
the origin of the unit cell and four orbitals per atom. Since the position of
atoms in the unit cell will be important when we include magnetic flux, the
two models have different Hofstadter spectra. Further, the BBH model describes
spinless fermions, while Wieder’s model describes fermions with spin-orbit
coupling. As a result, the symmetry representations for the two models are
different.

In momentum space the Hamiltonian is

$\displaystyle H(\mathbf{k})$
$\displaystyle=(v_{m}+t_{1}(\cos(k_{x})+\cos(k_{y})))\Gamma_{3}$
$\displaystyle+t_{2}(\cos(k_{x})-\cos(k_{y}))\Gamma_{4}$
$\displaystyle+u\sin(k_{x})\Gamma_{1}+u\sin(k_{y})\Gamma_{2},$ (49)

where $\Gamma_{1}=\tau_{y}\sigma_{y}$, $\Gamma_{2}=\tau_{y}\sigma_{x}$,
$\Gamma_{3}=\tau_{z}$, and $\Gamma_{4}=\tau_{x}$. The Pauli matrices $\tau$
and $\sigma$ together span the orbital space of each atom. In the limit
$t_{1}=t_{2}=u/\sqrt{2}=\lambda/\sqrt{2}$ and $v_{m}=\sqrt{2}\gamma$, this
Hamiltonian is equivalent to the BBH Hamiltonian after a basis change. In this
section, we adopt these parameters and set $\lambda=1$ and $\gamma=0.5$ so
that the system is a quadrupole insulator at zero flux.

The generators of the symmetries of this Hamiltonian take the matrix forms:

$\displaystyle C_{4z}$
$\displaystyle=\tau_{z}\left(\frac{{\mathbb{I}}_{\sigma}-\sigma_{z}}{2}\right),$
(50) $\displaystyle M_{x}$ $\displaystyle=\sigma_{x},$ (51)
$\displaystyle{\cal TI}$ $\displaystyle=\sigma_{y}{\cal K},$ (52)

where $\cal K$ is the complex conjugation. There is also a chiral symmetry
that anti-commutes with the Hamiltonian

$\Gamma_{5}=\tau_{y}\sigma_{z}$ (53)

To incorporate the effect of a magnetic field, we need the real space
Hamiltonian, given by:

$\displaystyle H$
$\displaystyle=\sum_{i,j\in\mathbb{Z}}\mathbf{t}_{x}c^{\dagger}_{(i+1,j)}c_{(i,j)}+\mathbf{t}_{y}c^{\dagger}_{(i,j+1)}c_{(i,j)}+h.c.$
$\displaystyle\quad+v_{m}\Gamma_{3}c^{\dagger}_{(i,j)}c_{(i,j)}$ (54)

where $\mathbf{t}_{x,y}$ are hopping matrices given by

$\displaystyle\mathbf{t}_{x}=(t_{1}\Gamma_{3}+t_{2}\Gamma_{4})/2-iu\Gamma_{1}/2$
(55)
$\displaystyle\mathbf{t}_{y}=(t_{1}\Gamma_{3}-t_{2}\Gamma_{4})/2-iu\Gamma_{2}/2$
(56)

When a magnetic field in the $z$-direction is turned on, the Hamiltonian in
Eq. (54) requires the Peierls substitution [1]. Working in Landau gauge,
$\mathbf{A}(x,y)=(-\phi y,0)$ where $\phi=B$ is the flux per unit cell and the
substitution is given by

$\displaystyle c^{\dagger}_{(i+1,j)}c_{(i,j)}$ $\displaystyle\mapsto e^{-i\phi
j}c^{\dagger}_{(i+1,j)}c_{(i,j)}$ (57) $\displaystyle
c^{\dagger}_{(i,j+1)}c_{(i,j)}$ $\displaystyle\mapsto
c^{\dagger}_{(i,j+1)}c_{(i,j)}$ (58)

The momentum space Hamiltonian at finite flux can be obtained by Fourier
transforming Eq. (54) using the convention in Eqs. (23) and (24) when the flux
is rational $\phi=2\pi p/q$.

In Fig. 5, we numerically compute the Hofstadter spectrum for this model.

###  V.2  Symmetry analysis

The model has a $2\pi$ periodicity in $\phi$, the flux per unit cell. At zero
flux and $\pi$-flux the system is invariant under the symmetry group
$p4/m^{\prime}mm$, while at other fluxes the symmetry group is $p4$. Using the
formalism developed in this manuscript, we apply TQC in a magnetic field to
compute the symmetry indicators at $\pi$ flux. Indicators at other fluxes are
discussed in Appendix I. Ultimately, we will show that the symmetry indicator
at $\pi$ flux corresponds to an absence of corner states, from which we deduce
there must be a gap closing phase transition at a critical flux between zero
and $\pi$.

At $\pi$-flux, the magnetic unit cell is $2$-by-$2$ and the Brillouin zone is
$[-\pi/2,\pi/2]\times[-\pi/2,\pi/2]$. According to Sec. III.1, the four-fold
rotation symmetry operators at $\Gamma=(0,0)$ and $M=(\pi/2,\pi/2)$ are

$\displaystyle D(C_{4z},\Gamma)$ $\displaystyle=\begin{pmatrix}1&&&\\\ &&1&\\\
&1&&\\\ &&&-1\end{pmatrix}\otimes C_{4z},$ (59) $\displaystyle D(C_{4},M))$
$\displaystyle=\begin{pmatrix}1&&&\\\ &&-1&\\\ &1&&\\\
&&&1\end{pmatrix}\otimes C_{4z},$ (60)

where the first matrix acts on the sublattice basis, and the $C_{4z}$ matrix
acts on the orbital basis as defined in Eq. (50). The magnetic translation
symmetries at $\mathbf{k}$ are implemented by

$\displaystyle D(T(\hat{\mathbf{x}}),\mathbf{k})$
$\displaystyle=e^{ik_{x}}\begin{pmatrix}&1&&\\\ 1&&&\\\ &&&1\\\
&&1&\end{pmatrix}\otimes\tau_{0}\sigma_{0}$ (61) $\displaystyle
D(T(\hat{\mathbf{y}}),\mathbf{k})$
$\displaystyle=e^{ik_{y}}\begin{pmatrix}&&1&\\\ &&&-1\\\ 1&&&\\\
&-1&&\end{pmatrix}\otimes\tau_{0}\sigma_{0},$ (62)

where $\tau_{0}$ and $\sigma_{0}$ are identity matrices.

The irreps of the occupied bands are listed in Table 5. Each band is four-fold
degenerate because ${\cal TI}^{2}=-1$ and
$\\{T(\hat{\mathbf{x}}),T(\hat{\mathbf{y}})\\}=0$, as explained in Appendix F.
Using the computed irreps in Table 5, the symmetry indicators are listed in
Table 6.

band index | $1$ | $2$ | $3$ | $4$
---|---|---|---|---
irrep at $\Gamma$ | $\Gamma_{2}\Gamma_{3}$ | $\Gamma_{1}\Gamma_{4}$ | $\Gamma_{2}\Gamma_{3}$ | $\Gamma_{1}\Gamma_{4}$
irrep at $X$ | $X_{1}X_{2}$ | $X_{1}X_{2}$ | $X_{1}X_{2}$ | $X_{1}X_{2}$
irrep at $M$ | $M_{2}M_{4}$ | $M_{3}M_{3}$ | $M_{1}M_{1}$ | $M_{2}M_{4}$
Table 5: Band representation of the four four-fold degenerate bands at $\pi$-flux. The ordering of band index is from lowest energy to highest energy, i.e., half-filling corresponds to filling bands 1 and 2. Each irrep $\Pi_{i}\Pi_{j}$ is four-dimensional and defined in Table 4.  band index | $n=1,4$ | $n=2,3$ | $1\oplus 2$ | $3\oplus 4$
---|---|---|---|---
$\mathbb{Z}_{2}$ phase (Eq. (48)) | $1$ | $1$ | $0$ | $0$
$e_{4a}\mod 8$ |  |  | $2$ | $2$
$e_{4b}\mod 8$ |  |  | $0$ | $0$
$e_{8c}\mod 4$ |  |  | $0$ | $0$
$e_{1a^{\prime}}\mod 4$ | $0$ | $2$ | $2$ | $2$
$e_{1b^{\prime}}\mod 4$ | $0$ | $2$ | $2$ | $2$
$e_{2c^{\prime}}\mod 2$ | $2$ | $0$ | $2$ | $2$
Table 6: Symmetry indicators at $\pi$-flux. The second column corresponds to
each four-fold degenerate band individually, while the last two columns
correspond to sums of bands. The second row shows the strong topological index
in Eq. (48) is $1\mod 2$ for each band, while for two occupied/empty bands the
index is $0\mod 2$. Since symmetric and exponentially localized Wannier
functions exist for the two occupied or two empty bands, in the next three
rows, $e_{\mathbf{q}}$ indicates the number of Wannier functions centered at
the Wyckoff position $\mathbf{q}$, computed using Eqs. (133) – (135) in
Appendix G. If the sublattice translation symmetry within each magnetic unit
cell is broken, the number of Wannier functions centered at the indicated
Wyckoff positions in the lower symmetry group are shown in Fig. 6.  Figure 6:
(a) Wyckoff position $4a$ in $2$-by-$2$ unit cell with space group $G$ splits
into (b) Wyckoff positions $1a^{\prime}$, $1b^{\prime}$, $2c^{\prime}$ in the
same unit cell with symmetry group $G/(\mathbb{T}/\mathbb{T}_{M})$, i.e. no
sublattice translations.

Below the gap at half filling, the two occupied bands together ($1\oplus 2$ in
Table 6) are topologically trivial. They admit symmetric exponentially
localized Wannier functions located at the $4a$ Wyckoff position. Since the
atoms are also at $4a$ Wyckoff position, there is no corner charge. This
analysis agrees with the Hofstadter spectrum shown in Fig. 5.

Open boundary conditions break the lattice translation symmetries and, in
particular, break the sublattice translation symmetries within the magnetic
unit cell. Once the sublattice translation symmetries are broken, the little
co-group (Eq. (32)) is identical to the non-magnetic case. Thus, the
crystalline symmetry protected phases with open boundary condition should be
labelled by the usual symmetry indicators in zero flux but with respect to the
enlarged magnetic unit cell; these indicators are computed in Ref. 29. The
results are shown in the lower half of Table 6. In this reduced symmetry
group, the magnetic $4a$ Wyckoff position splits into positions:
$1a^{\prime}=(0,0)$, $1b^{\prime}=(1,1)$, $2c^{\prime}=(1,0),(0,1)$, as shown
in Fig. 6.

###  V.3 Corner states

The spectrum with periodic boundary condition has a gap at half filling at
$\phi=0$ and $\phi=\pi$. This gap closes at some $\phi^{*}$ between $0$ and
$\pi$ as shown in Fig. 5. For the spectrum with open boundary condition, there
are higher-order topological states when $-\phi^{*}<\phi<\phi^{*}$ that are
corner localized. Due to the chiral symmetry (53), they are at zero energy in
this model. The corner states can be understood by the non-zero quadrupole
moment [21] or the non-zero filling anomaly [32, 33, 19, 29].

The corner states have four-fold degeneracy, consistent with the non-magnetic
symmetry analysis in Refs. 29, 34. The corner states with open boundary
condition always come in a group of $d$ states. This degeneracy $d$ is
determined by the point group of the finite system. Let $w$ be the general
Wyckoff position of the point group, with multiplicity $n_{w}$. Denote the
site symmetry group $G_{w}$. It has only one irrep, $\rho(G_{w})$. The
degeneracy of corner states is [34]

$d=\text{dim}(\rho(G_{w}))\times n_{w}$ (63)

where dim$(\rho(G_{w}))=2$ for spinful systems with time-reversal symmetry
that squares to $-1$, otherwise dim$(\rho(G_{w}))=1$.

In the present case, at zero flux the system is invariant under the symmetry
group $p4/m^{\prime}mm$, while at any small flux the symmetry group reduces to
$p4$. For $p4/m^{\prime}mm$ and $p4$, the point groups of the finite size
system are $4/m^{\prime}mm$ and $4$ respectively. Each has a general Wyckoff
position $w$ with $n_{w}=4$ and $\text{dim}(\rho(G_{w}))=1$; thus, Eq. (63)
yields a degeneracy of $4$ [34]. Since the degeneracy of corner states is the
same for zero flux and finite flux, the corner states do not split when the
magnetic flux is introduced (The chiral symmetry in this model pins the corner
states to zero energy, but even in the absence of chiral symmetry, the nonzero
filling anomaly will remain the same for $0\leq\phi<\phi^{*}$.)

We have also shown from the symmetry indicators that at half filling and $\pi$
flux, the system is in the trivial phase, without corner states. Thus, the
corner states must terminate at $\phi=\phi^{*}$ by either a bulk or edge gap
closing. There is indeed a bulk gap closing at flux $\phi^{*}$ as Fig. 5
shows. This is consistent with the Wannier centers of the occupied bands,
which can be deduced from the symmetry indicators: the Wannier centers are at
the $4b$ Wyckoff position at zero flux and the $4a$ Wyckoff position at $\pi$
flux (see Table 6 and Appendix I). Symmetries prevent four Wannier functions
from moving continuously between the $4a$ and $4b$ positions [29, 35]. A
discontinuous jump of the Wannier centers implies the bulk gap closes between
zero and $\pi$ flux.

In Appendix I we compute the symmetry indicator at intermediate flux
$\phi=2\pi/5<\phi^{*}$ and $\phi=4\pi/5>\phi^{*}$ to verify that symmetry
indicators are consistent with the presence and absence of corner states
between zero and $\pi$. In Appendix J we show that the presence and absence of
corner states also agrees with the nested Wilson loops [21].

##  VI  Conclusion

In conclusion, we derived a general framework to apply TQC and the theory of
symmetry indicators to crystalline systems at rational flux per unit cell.
Applying our results to some simple examples at $\pi$ flux revealed new
symmetry indicators that did not appear at zero flux. Finally, the symmetry
indicators enable us to study a quadrupole insulator at finite field, which
reveals a gap closing topological-to-trivial phase transition as a function of
magnetic field. Observing this phase transition could be particularly
promising in moiré systems where higher flux is attainable for reasonable
magnetic fields.

While preparing our work, we became aware of a related study [36], which gives
criteria for when such a bulk gap closing at finite flux can be predicted from
the band structure at zero flux. The two bulk gap closings between zero and
$\Phi=2\pi$ flux of our model in Sec. V are indicated by the real space
invariant in Ref. 36.

We note that the Zeeman term is neglected in this manuscript. When the Zeeman
term is present, the periodicity in the flux direction is broken. Thus there
is no magnetic time-reversal symmetry or other mirror symmetries that flip
magnetic flux. At large magnetic field where Zeeman term dominates, the two
dimensional system must be in the trivial atomic limit where Wannier centers
locate at the atom positions.

Our work is also restricted to a spatially constant magnetic field. It would
be interesting to extend our results to a spatially varying periodic magnetic
field that maintains a commensurate flux per unit cell. This more general
theory might be relevant to magnetically ordered crystals.

As a final note, we draw a connection between our results and the theory of
phase space quantization, where one seeks a symmetric and exponentially
localized Wannier basis that can continuously reduce to points in the
classical phase space by setting the Planck constant $h\rightarrow 0$ [37].
However, such a basis can never be found due to the Balian-Low theorem, which
forbids the existence of exponentially localized translational symmetric basis
for any single particle [38]. The magnetic Wannier functions in two dimensions
share a similar translation group structure to the one-dimensional quantum
phase space and the non-vanishing Chern number for any single magnetic band
also forbids Wannierization [39], as we explain in Appendix K. Thus, there is
an interesting open question: since in two dimensional magnetic systems, it is
possible to find a Wannier basis for a group of bands, we conjecture that the
continuous quantization of phase space may be realized by constructing Wannier
functions for groups of particles.

##  VII Acknowledgements

We acknowledge useful conversations with Andrei Bernevig, Aaron Dunbrack,
Sayed Ali Akbar Ghorashi, Jonah Herzog-Arbeitman, and Oskar Vafek. We thank
Jonah, Andrei, and their collaborators for sharing their unpublished
manuscript.

Our manuscript is based upon work supported by the National Science Foundation
under Grant No. DMR-1942447. The work was performed in part at the Aspen
Center for Physics, which is supported by National Science Foundation grant
PHY-1607611. J.C. also acknowledges the support of the Flatiron Institute, a
division of the Simons Foundation.

## References

  * Hofstadter [1976] D. R. Hofstadter, Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields, Physical Review B 14, 2239 (1976).
  * Herzog-Arbeitman  _et al._ [2020] J. Herzog-Arbeitman, Z.-D. Song, N. Regnault, and B. A. Bernevig, Hofstadter topology: noncrystalline topological materials at high flux, Physical review letters 125, 236804 (2020).
  * Zuo  _et al._ [2021] Z.-W. Zuo, W. A. Benalcazar, Y. Liu, and C.-X. Liu, Topological phases of the dimerized hofstadter butterfly, Journal of Physics D: Applied Physics 54, 414004 (2021).
  * Asaga and Fukui [2020] K. Asaga and T. Fukui, Boundary-obstructed topological phases of a massive Dirac fermion in a magnetic field, Physical Review B 102, 155102 (2020).
  * Otaki and Fukui [2019] Y. Otaki and T. Fukui, Higher-order topological insulators in a magnetic field, Physical Review B 100, 245108 (2019).
  * Bradlyn  _et al._ [2017] B. Bradlyn, L. Elcoro, J. Cano, M. Vergniory, Z. Wang, C. Felser, M. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547, 298 (2017).
  * Vergniory  _et al._ [2017] M. Vergniory, L. Elcoro, Z. Wang, J. Cano, C. Felser, M. Aroyo, B. A. Bernevig, and B. Bradlyn, Graph theory data for topological quantum chemistry, Physical Review E 96, 023310 (2017).
  * Elcoro  _et al._ [2017] L. Elcoro, B. Bradlyn, Z. Wang, M. G. Vergniory, J. Cano, C. Felser, B. A. Bernevig, D. Orobengoa, G. Flor, and M. I. Aroyo, Double crystallographic groups and their representations on the Bilbao Crystallographic Server, Journal of Applied Crystallography 50, 1457 (2017).
  * Cano  _et al._ [2018] J. Cano, B. Bradlyn, Z. Wang, L. Elcoro, M. Vergniory, C. Felser, M. I. Aroyo, and B. A. Bernevig, Building blocks of topological quantum chemistry: Elementary band representations, Physical Review B 97, 035139 (2018).
  * Bradlyn  _et al._ [2018] B. Bradlyn, L. Elcoro, M. Vergniory, J. Cano, Z. Wang, C. Felser, M. Aroyo, and B. A. Bernevig, Band connectivity for topological quantum chemistry: Band structures as a graph theory problem, Physical Review B 97, 035138 (2018).
  * Vergniory  _et al._ [2019] M. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, A complete catalogue of high-quality topological materials, Nature 566, 480 (2019).
  * Cano and Bradlyn [2021] J. Cano and B. Bradlyn, Band representations and topological quantum chemistry, Annual Review of Condensed Matter Physics 12, 225 (2021).
  * Elcoro  _et al._ [2021] L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, Magnetic topological quantum chemistry, Nature communications 12, 1 (2021).
  * Po  _et al._ [2017] H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nature communications 8, 1 (2017).
  * Zak [1964a] J. Zak, Magnetic translation group, Physical Review 134, A1602 (1964a).
  * Zak [1964b] J. Zak, Magnetic translation group. II. irreducible representations, Physical Review 134, A1607 (1964b).
  * Matsugatani  _et al._ [2018] A. Matsugatani, Y. Ishiguro, K. Shiozaki, and H. Watanabe, Universal relation among the many-body chern number, rotation symmetry, and filling, Physical Review Letters 120, 096601 (2018).
  * Fang  _et al._ [2012] C. Fang, M. J. Gilbert, and B. A. Bernevig, Bulk topological invariants in noninteracting point group symmetric insulators, Physical Review B 86, 115112 (2012).
  * Wieder  _et al._ [2020] B. J. Wieder, Z. Wang, J. Cano, X. Dai, L. M. Schoop, B. Bradlyn, and B. A. Bernevig, Strong and fragile topological Dirac semimetals with higher-order fermi arcs, Nature communications 11, 1 (2020).
  * De Nittis and Lein [2011] G. De Nittis and M. Lein, Exponentially localized wannier functions in periodic zero flux magnetic fields, Journal of mathematical physics 52, 112103 (2011).
  * Benalcazar  _et al._ [2017a] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators, Physical Review B 96, 245115 (2017a).
  * Benalcazar  _et al._ [2017b] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357, 61 (2017b).
  * Bradley and Cracknell [2010] C. Bradley and A. Cracknell,  _The mathematical theory of symmetry in solids: representation theory for point groups and space groups_ (Oxford University Press, 2010).
  * Zak [1980] J. Zak, Symmetry specification of bands in solids, Phys. Rev. Lett. 45, 1025 (1980).
  * Zak [1981] J. Zak, Band representations and symmetry types of bands in solids, Phys. Rev. B 23, 2824 (1981).
  * Michel and Zak [1999] L. Michel and J. Zak, Connectivity of energy bands in crystals, Physical Review B 59, 5998 (1999).
  * Michel and Zak [2000] L. Michel and J. Zak, Elementary energy bands in crystalline solids, EPL (Europhysics Letters) 50, 519 (2000).
  * Michel and Zak [2001] L. Michel and J. Zak, Elementary energy bands in crystals are connected, Physics Reports 341, 377 (2001).
  * Fang and Cano [2021a] Y. Fang and J. Cano, Filling anomaly for general two- and three-dimensional ${C}_{4}$ symmetric lattices, Phys. Rev. B 103, 165109 (2021a).
  * Bernevig [2013] B. A. Bernevig, Topological insulators and topological superconductors, in  _Topological Insulators and Topological Superconductors_ (Princeton university press, 2013).
  * Fu and Kane [2006] L. Fu and C. L. Kane, Time reversal polarization and a z 2 adiabatic spin pump, Physical Review B 74, 195312 (2006).
  * Benalcazar  _et al._ [2019] W. A. Benalcazar, T. Li, and T. L. Hughes, Quantization of fractional corner charge in $c_{n}$-symmetric higher-order topological crystalline insulators, Physical Review B 99, 245151 (2019).
  * Schindler  _et al._ [2019] F. Schindler, M. Brzeziíska, W. A. Benalcazar, M. Iraola, A. Bouhon, S. S. Tsirkin, M. G. Vergniory, and T. Neupert, Fractional corner charges in spin-orbit coupled crystals, Phys. Rev. Research 1, 033074 (2019).
  * Fang and Cano [2021b] Y. Fang and J. Cano, Classification of Dirac points with higher-order fermi arcs, Physical Review B 104, 245101 (2021b).
  * Song  _et al._ [2017] Z. Song, Z. Fang, and C. Fang, (d- 2)-dimensional edge states of rotation symmetry protected topological states, Physical review letters 119, 246402 (2017).
  * Herzog-Arbeitman  _et al._ [2022a] J. Herzog-Arbeitman, Z.-D. Song, L. Elcoro, and B. A. Bernevig, Hofstadter topology and reentrant projective symmetries, In prep. (2022a).
  * Von Neumann [2018] J. Von Neumann, Mathematical foundations of quantum mechanics, in  _Mathematical Foundations of Quantum Mechanics_ (Princeton university press, 2018).
  * Benedetto  _et al._ [1994] J. J. Benedetto, C. Heil, and D. F. Walnut, Differentiation and the balian-low theorem, Journal of Fourier Analysis and Applications 1, 355 (1994).
  * Zak [1997] J. Zak, Balian-low theorem for Landau levels, Physical review letters 79, 533 (1997).
  * Herzog-Arbeitman  _et al._ [2022b] J. Herzog-Arbeitman, A. Chew, and B. A. Bernevig, Magnetic bloch theorem and reentrant flat bands in twisted bilayer graphene at $2\pi$ flux, arXiv preprint arXiv:2206.07717 (2022b).
  * Gorenstein [2007] D. Gorenstein,  _Finite groups_ , Vol. 301 (American Mathematical Soc., 2007).
  * Song  _et al._ [2020a] Z.-D. Song, L. Elcoro, Y.-F. Xu, N. Regnault, and B. A. Bernevig, Fragile phases as affine monoids: classification and material examples, Physical Review X 10, 031001 (2020a).
  * Song  _et al._ [2020b] Z.-D. Song, L. Elcoro, and B. A. Bernevig, Twisted bulk-boundary correspondence of fragile topology, Science 367, 794 (2020b).
  * Song  _et al._ [2018] Z. Song, T. Zhang, Z. Fang, and C. Fang, Quantitative mappings between symmetry and topology in solids, Nature communications 9, 1 (2018).
  * Watanabe and Ono [2020] H. Watanabe and S. Ono, Corner charge and bulk multipole moment in periodic systems, Physical Review B 102, 165120 (2020).
  * Alexandradinata  _et al._ [2014] A. Alexandradinata, X. Dai, and B. A. Bernevig, Wilson-loop characterization of inversion-symmetric topological insulators, Physical Review B 89, 155114 (2014).
  * Bradlyn  _et al._ [2019] B. Bradlyn, Z. Wang, J. Cano, and B. A. Bernevig, Disconnected elementary band representations, fragile topology, and wilson loops as topological indices: An example on the triangular lattice, Physical Review B 99, 045140 (2019).
  * Resta and Vanderbilt [2007] R. Resta and D. Vanderbilt, Theory of polarization: a modern approach, in  _Physics of Ferroelectrics_ (Springer, 2007) pp. 31–68.
  * Wheeler  _et al._ [2019] W. A. Wheeler, L. K. Wagner, and T. L. Hughes, Many-body electric multipole operators in extended systems, Physical Review B 100, 245135 (2019).
  * Battle [1988] G. Battle, Heisenberg proof of the balian-low theorem, Letters in Mathematical Physics 15, 175 (1988).
  * Moscolari and Panati [2019] M. Moscolari and G. Panati, Symmetry and localization for magnetic Schroedinger operators: Landau levels, Gabor frames and all that, Acta Applicandae Mathematicae 162, 105 (2019).
  * Streda [1982] P. Streda, Theory of quantised hall conductivity in two dimensions, Journal of Physics C: Solid State Physics 15, L717 (1982).
  * Thouless [1984] D. Thouless, Wannier functions for magnetic sub-bands, Journal of Physics C: Solid State Physics 17, L325 (1984).

##  Appendix A Derivation of $\lambda_{g}$ in Eq. (6)

In this appendix, we derive Eq. (6) starting from Eq. (5), which we rewrite
here for convenience:

$\int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{A}(\mathbf{r})\cdot
d\mathbf{r}+\int_{\hat{g}\mathbf{r}_{1}}^{\hat{g}\mathbf{r}_{2}}\nabla\lambda(\mathbf{r}^{\prime})\cdot
d\mathbf{r}^{\prime}=\int_{\hat{g}\mathbf{r}_{1}}^{\hat{g}\mathbf{r}_{2}}\mathbf{A}(\mathbf{r}^{\prime})\cdot
d\mathbf{r}^{\prime}$ (64)

Substituting $\mathbf{r}=\hat{g}^{-1}\mathbf{r}^{\prime}$ in the first term on
the left side of Eq. (64) yields

$\displaystyle\int_{\mathbf{r}_{1}}^{\mathbf{r}_{2}}\mathbf{A}(\mathbf{r})\cdot
d\mathbf{r}$
$\displaystyle=\int_{\hat{g}\mathbf{r}_{1}}^{\hat{g}\mathbf{r}_{2}}\mathbf{A}(\hat{g}^{-1}\mathbf{r}^{\prime})\cdot
d(\hat{g}^{-1}\mathbf{r}^{\prime})$
$\displaystyle=\int_{\hat{g}\mathbf{r}_{1}}^{\hat{g}\mathbf{r}_{2}}\mathbf{A}(\hat{g}^{-1}\mathbf{r}^{\prime})\cdot
R_{g}^{-1}d\mathbf{r}^{\prime}$
$\displaystyle=\int_{\hat{g}\mathbf{r}_{1}}^{\hat{g}\mathbf{r}_{2}}R_{g}\mathbf{A}(\hat{g}^{-1}\mathbf{r}^{\prime})\cdot
d\mathbf{r}^{\prime}$ (65)

In the second equality we have defined the rotation $R_{g}$ by the action
$\hat{g}:\mathbf{r}\mapsto R_{g}\mathbf{r}+\tau_{g}$, from which it follows
that $\hat{g}^{-1}:\mathbf{r}\mapsto R_{g}^{-1}\mathbf{r}-R_{g}^{-1}\tau_{g}$,
and, consequently,
$d(\hat{g}^{-1}\mathbf{r}^{\prime})=R_{g}^{-1}d\mathbf{r}^{\prime}$, since
$d(R_{g}^{-1}\tau_{g})=0$. Eqs. (64) and (65) together are exactly the
integral form of Eq. (6).

##  Appendix B Gauge invariant commutation relations between translation and
rotation operators

In this Appendix, we derive the commutation relations for magnetic rotation
and translation symmetry operators in real space. We show these relations are
gauge invariant. The results are used in Secs. II and IV and Appendix F.

We denote an $n$-fold magnetic rotation operator centered about
$(\bar{x}_{1},\bar{x}_{2})$ as $C_{n}(\bar{x}_{1},\bar{x}_{2})$. For rotations
about the origin, we simplify the notation by dropping the argument
$(\bar{x}_{1},\bar{x}_{2})$, i.e., $C_{n}\equiv
C_{n}(\bar{x}_{1}=0,\bar{x}_{2}=0)$. We define $C_{n}(\bar{x},\bar{y})$ by
conjugating magnetic translation operators with $C_{n}$

$C_{n}(\bar{x}_{1},\bar{x}_{2})\equiv
T(\bar{x}_{1}\hat{\mathbf{x}}_{1}+\bar{x}_{2}\hat{\mathbf{x}}_{2})C_{n}T(-\bar{x}_{1}\hat{\mathbf{x}}_{1}-\bar{x}_{2}\hat{\mathbf{x}}_{2})$
(66)

where $\hat{\mathbf{x}}_{1}$ and $\hat{\mathbf{x}}_{2}$ are unit vectors in
the $x_{1}$ and $x_{2}$ directions. This definition is a general form for any
gauge choice of vector potential. In this definition
$(C_{n}(\bar{x}_{1},\bar{x}_{2}))^{n}=C_{n}^{n}$ is the identity. The
definition is independent of the ordering of translations. For example,

$\displaystyle C_{n}(\bar{x}_{1},\bar{x}_{2})$
$\displaystyle=T(\bar{x}_{2}\hat{\mathbf{x}}_{2})T(\bar{x}_{1}\hat{\mathbf{x}}_{1})C_{n}T(-\bar{x}_{1}\hat{\mathbf{x}}_{1})T(-\bar{x}_{2}\hat{\mathbf{x}}_{2})$
$\displaystyle=T(\bar{x}_{1}\hat{\mathbf{x}}_{1})T(\bar{x}_{2}\hat{\mathbf{x}}_{2})C_{n}T(-\bar{x}_{2}\hat{\mathbf{x}}_{2})T(-\bar{x}_{1}\hat{\mathbf{x}}_{1})$

We now derive commutation relations between the magnetic rotations and
translations. First, we derive a gauge invariant commutation relation between
a translation $T(\mathbf{a})$ and $C_{n}$,

$C_{n}T(\mathbf{a})=T(R_{n}\mathbf{a})C_{n},$ (67)

which is the same as the commutation relation in zero field. We now prove Eq.
(67); Ref. 40 gives a different proof of Eq. (67).

Eq. (9) provides explicit operator forms for $C_{n}$ and $T(\mathbf{a})$:

$\displaystyle C_{n}$
$\displaystyle=\sum_{\mathbf{x}}e^{i\lambda_{C_{n}}(R_{n}\mathbf{x})}c^{\dagger}_{R_{n}\mathbf{x}}c_{\mathbf{x}}$
(68) $\displaystyle T(\mathbf{a})$
$\displaystyle=\sum_{\mathbf{x}}e^{i\lambda_{T(\mathbf{a})}(\mathbf{x}+\mathbf{a})}c^{\dagger}_{\mathbf{x}+\mathbf{a}}c_{\mathbf{x}}$
(69)

The gauge dependence of the operators are encoded in the definition of
$\lambda_{g}$. The explicit expression of $\lambda_{C_{n}}$ and
$\lambda_{T(\mathbf{a})}$ from Eqs. (6) and (7) in our definition are

$\displaystyle\lambda_{C_{n}}(\mathbf{x})$
$\displaystyle=\int^{\mathbf{x}}_{\mathbf{r}_{0}}\mathbf{A}(\mathbf{x}^{\prime})-R_{n}\mathbf{A}(R_{n}^{-1}\mathbf{x}^{\prime})\cdot
d\mathbf{x}^{\prime}$
$\displaystyle=\left[\int^{\mathbf{x}}_{\mathbf{r}_{0}}-\int^{R_{n}^{-1}\mathbf{x}}_{R_{n}^{-1}\mathbf{r}_{0}}\right]\mathbf{A}(\mathbf{x}^{\prime})\cdot
d\mathbf{x}^{\prime}$ (70) $\displaystyle\lambda_{T(\mathbf{a})}(\mathbf{x})$
$\displaystyle=\int_{\mathbf{x}-\mathbf{a}}^{\mathbf{x}}\mathbf{A}(\mathbf{x}^{\prime})\cdot
d\mathbf{x}^{\prime}+\mathbf{B}\cdot\mathbf{a}\times\mathbf{x}$ (71)

where $\mathbf{r}_{0}$ is a reference point that will cancel below.

Plugging into Eq. (67) and considering the action on single particle states,
the equation holds when the phase terms on both sides are equal, i.e.,

$\lambda_{C_{n}}(R_{n}(\mathbf{x}+\mathbf{a}))+\lambda_{T(\mathbf{a})}({\mathbf{x}+\mathbf{a}})\\\
=\lambda_{T(R_{n}\mathbf{a})}({R_{n}(\mathbf{x}+\mathbf{a})})+\lambda_{C_{n}}(R_{n}\mathbf{x}).$
(72)

Applying Eq. (B) yields

$\displaystyle\lambda_{C_{n}}(R_{n}(\mathbf{x}+\mathbf{a}))-\lambda_{C_{n}}(R_{n}\mathbf{x})$
$\displaystyle=\left[\int^{R_{n}(\mathbf{x}+\mathbf{a})}_{\mathbf{r}_{0}}-\int^{\mathbf{x}+\mathbf{a}}_{R_{n}^{-1}\mathbf{r}_{0}}-\int^{R_{n}\mathbf{x}}_{\mathbf{r}_{0}}+\int^{\mathbf{x}}_{R_{n}^{-1}\mathbf{r}_{0}}\right]\mathbf{A}(\mathbf{x}^{\prime})\cdot
d\mathbf{x}^{\prime}$
$\displaystyle=\left[\int^{R_{n}(\mathbf{x}+\mathbf{a})}_{R_{n}\mathbf{x}}+\oint_{R_{n}\cal
C}-\int^{\mathbf{x}+\mathbf{a}}_{\mathbf{x}}-\oint_{\cal
C}\right]\mathbf{A}(\mathbf{x})\cdot d\mathbf{r}$
$\displaystyle=\left[\int^{R_{n}(\mathbf{x}+\mathbf{a})}_{R_{n}\mathbf{x}}-\int^{\mathbf{x}+\mathbf{a}}_{\mathbf{x}}\right]\mathbf{A}(\mathbf{x})\cdot
d\mathbf{r},$ (73)

where the path $\cal C$ is a triangle whose vertices are $R_{n}^{-1}r_{0}$,
$\mathbf{x}$, $\mathbf{x}+\mathbf{a}$. The last equality follows because the
two closed loops enclose the same amount of flux. Then applying Eq. (71)
yields

$\displaystyle\lambda_{T(R_{n}\mathbf{a})}$
$\displaystyle({R_{n}(\mathbf{x}+\mathbf{a})})-\lambda_{T(\mathbf{a})}({\mathbf{x}+\mathbf{a}})$
$\displaystyle=\left[\int^{R_{n}(\mathbf{x}+\mathbf{a})}_{R_{n}\mathbf{x}}-\int^{\mathbf{x}+\mathbf{a}}_{\mathbf{x}}\right]\mathbf{A}(\mathbf{x})\cdot
d\mathbf{r}.$ (74)

The two expressions are equal, proving Eq. (67).

Combined with the gauge invariant relation for magnetic translations (Eq. (16)
in the main text),

$T(\mathbf{a}_{1})T(\mathbf{a}_{2})=T(\mathbf{a}_{1}+\mathbf{a}_{2})e^{\frac{i}{2}\mathbf{B}\cdot(\mathbf{a}_{1}\times\mathbf{a}_{2})},$
(75)

Eq. (67) implies that the phases that appear in the commutation relations
between $C_{n}(\bar{x}_{1},\bar{x}_{2})$ and $T(\mathbf{a})$ are all gauge
invariant. For example, for two-fold and four-fold magnetic rotations on a
square lattice

$\displaystyle C_{2}(\bar{x},\bar{y})T(\Delta\hat{\mathbf{x}})$
$\displaystyle=T(-\Delta\hat{\mathbf{x}})C_{2}(\bar{x},\bar{y})e^{-2iB\Delta\bar{y}}$
$\displaystyle C_{2}(\bar{x},\bar{y})T(\Delta\hat{\mathbf{y}})$
$\displaystyle=T(-\Delta\hat{\mathbf{y}})C_{2}(\bar{x},\bar{y})e^{2iB\Delta\bar{x}}$
$\displaystyle C_{4}(\bar{x},\bar{y})T(\Delta\hat{\mathbf{x}})$
$\displaystyle=T(\Delta\hat{\mathbf{y}})C_{4}(\bar{x},\bar{y})e^{-iB\Delta(\bar{x}+\bar{y})}$
$\displaystyle C_{4}(\bar{x},\bar{y})T(\Delta\hat{\mathbf{y}})$
$\displaystyle=T(-\Delta\hat{\mathbf{x}})C_{4}(\bar{x},\bar{y})e^{iB\Delta(\bar{x}-\bar{y})}$
(76)

and

$\displaystyle C_{2}(\bar{x}_{1},\bar{y}_{1})C_{2}(\bar{x}_{2},\bar{y}_{2})$
$\displaystyle=C_{2}(-\bar{x}_{2},-\bar{y}_{2})C_{2}(-\bar{x}_{1},-\bar{y}_{1})$
$\displaystyle\qquad
e^{4iB(\bar{x}_{1}\bar{y}_{1}-\bar{x}_{2}\bar{y}_{2}+\bar{x}_{2}\bar{y}_{1}-\bar{x}_{1}\bar{y}_{2})}$
$\displaystyle C_{4}(\bar{x}_{1},\bar{y}_{1})C_{4}(\bar{x}_{2},\bar{y}_{2})$
$\displaystyle=C_{4}(-\bar{y}_{2},\bar{x}_{2})C_{4}(\bar{y}_{1},-\bar{x}_{1})$
$\displaystyle\qquad
e^{iB((\bar{x}_{1}+\bar{x}_{2})^{2}-(\bar{y}_{1}-\bar{y}_{2})^{2})}$ (77)

These relations are used when we study irreducible representations in Sec.
III.2 and topological quantum chemistry in Sec. IV.

##  Appendix C  Projective representation

In general, a projective representation $\rho$ of a group $G$ satisfies

$\rho(g_{1})\rho(g_{2})=\omega(g_{1},g_{2})\rho(g_{1}g_{2})$ (78)

for all $g_{1},g_{2}\in G$, where $\omega(g_{1},g_{2})$ is called the
2-cocycle (or Schur multiplier, or factor system). When
$\omega(g_{1},g_{2})\equiv 1$, the representation is an ordinary linear
representation.

Under a gauge transformation of $\rho$ by $\rho^{\prime}(g)=f(g)\rho(g)$,
2-cocyles satisfy the cocycle condition
$\omega^{\prime}(g_{1},g_{2})=f(g_{1}g_{2})f(g_{1})^{-1}f(g_{2})^{-1}\omega(g_{1},g_{2})$,
which defines an equivalence class $[\omega(g_{1},g_{2})]$ of cocycles. The
classification of the equivalence classes is determined by the group
cohomology.

In the remaining part of this section, we explain the standard way to derive
irreducible projective representations through the group extension. The
magnetic symmetries can be viewed as projective representations of the non-
magnetic symmetries. These projective representations are projected from the
ordinary representations of a larger group that can be described as a group
extension of the non-magnetic group. For example, the continuous magnetic
translations form the Heisenberg group $Heis$. It is extended from the non-
magnetic translations $\mathbb{R}^{2}$ by the abelian group $U(1)$. There are
many irreducible representations of $Heis$, but only some of them are
compatible with the 2-cocycles, which are the phases due to gauge
transformations of the magnetic symmetries. For example, the trivial
irreducible representation is not compatible with the particular 2-cocycles.

We now describe the group extension for the translation symmetries on a square
lattice as a concrete example. Consider a model with only nearest neighbor
hopping. Then the phase introduced by the magnetic translations in Eq. (15)
with commensurate magnetic flux $\phi=2\pi p/q$ belongs to $\mathbb{Z}_{q}$.
We first consider the extension of full translation group
$\mathbb{T}=\mathbb{Z}\times\mathbb{Z}$ by $\mathbb{Z}_{q}$. This real space
Heisenberg group is defined by the central group extension, which is a short
exact sequence

${1}$${\mathbb{Z}_{q}}$${Heis}$${\mathbb{Z}\times\mathbb{Z}}$${1}$$\scriptstyle{i}$$\scriptstyle{\pi}$
(79)

where each arrow is a group homomorphism and $\text{im}(i)=\text{ker}(\pi)$.
Next we consider the momentum space Heisenberg group $Heis_{k}$ (for the
$q$-by-$q$ unit cell), which is the sublattice translation group
$\mathbb{T}/\mathbb{T}_{M}$ extended by $\mathbb{Z}_{q}$. It’s defining
central group extension is

${1}$${\mathbb{Z}_{q}}$${Heis_{k}}$${\mathbb{Z}_{q}\times\mathbb{Z}_{q}}$${1}$$\scriptstyle{i}$$\scriptstyle{\pi}$
(80)

It can be shown that this $Heis_{k}$ is an extra special group of order
$q^{3}$. It has $q^{2}$ $1$-dimensional irreducible representations and $q-1$
$q$-dimensional irreducible representations [41]. The abelian group
$\mathbb{Z}_{q}$ acts trivially on the $1$-dimensional irreps. Thus, the
projective representations are projected from the $q$-dimensional
representations. Therefore, the momentum space states satisfying magnetic
translation symmetries are $q$-fold degenerate. For example, at $\pi$ flux,
$Heis_{k}$ is isomorphic to the group $D_{4}$ and the irreducible projective
representation of translations $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ is
projected from the only two-dimensional irreducible representation $E$ of
$D_{4}$.

For more complicated cases with magnetic rotation symmetries, in principle one
can use the same formalism to get the corresponding extended group and the
irreducible projective representations of the unextended symmetry group.
However, practically it is easier to derive the irreducible projective
representations by studying the (anti-)commutation relations as shown in
Appendix F.

##  Appendix D  Triangular lattice symmetries in a magnetic field

$g$ | $T(\Delta_{1}\hat{\mathbf{x}}_{1})$ | $T(\Delta_{2}\hat{\mathbf{x}}_{2})$ | $C_{2}(\bar{r}_{1},\bar{r}_{2})$ | $C_{6}(\bar{r}_{1},\bar{r}_{2})$ | $C_{3}(\bar{r}_{1},\bar{r}_{2})$
---|---|---|---|---|---
$\hat{g}=\\{R_{g}|\tau_{g}\\}$ | $\\{0|(\Delta_{1},0)\\}$ | $\\{0|(0,\Delta_{2})\\}$ | $\\{\hat{C}_{2}|(2\bar{r}_{1},2\bar{r}_{2})\\}$ | $\\{\hat{C}_{6}|(\bar{r}_{1}+\bar{r}_{2},-\bar{r}_{1})\\}$ | $\\{\hat{C}_{3}|(2\bar{r}_{1}+\bar{r}_{2},\bar{r}_{2}-\bar{r}_{1})\\}$
$\lambda(r_{1},r_{2})$ | 0 | $-\phi\Delta_{2}r_{1}$ | $-2\phi\bar{r}_{2}(r_{1}-\bar{r}_{1})$ | $-\phi((r_{1}-\bar{r}_{1})(r_{2}-\bar{r}_{2})+\frac{(r_{1}-\bar{r}_{1})^{2}}{2})$ | $-\phi((r_{1}-\bar{r}_{1})(r_{2}-\bar{r}_{2})+\frac{(r_{2}-\bar{r}_{2})^{2}}{2})$
|  |  |  | $+\phi\bar{r}_{2}(r_{2}-\bar{r}_{2})$ | $-\phi\bar{r}_{2}((r_{1}-\bar{r}_{1})-(r_{2}-\bar{r}_{2}))$
Table 7: The gauge transformation $\lambda_{g}(x,y)$ for symmetries of the
six-fold symmetric triangular lattice in Landau gauge. For each symmetry $g$
in the first row, the second row lists the symmetry in the notation
$\\{R_{g}|\tau_{g}\\}$ and the third row provides $\lambda_{g}$ from Eq. (6).

The Landau gauge in triangular and hexagonal lattices is defined differently
than for the square lattice. Therefore, we discuss the magnetic symmetries of
these lattices separately here.

We consider the lattice vectors $\mathbf{a}_{1}=(1,0)$,
$\mathbf{a}_{2}=(\frac{1}{2},\frac{\sqrt{3}}{2})$ in Cartesian coordinates.
The reciprocal lattice vectors are $\mathbf{b}_{1}=(1,-1/\sqrt{3})$,
$\mathbf{b}_{2}=(0,2/\sqrt{3})$ in Cartesian coordinates, where
$\mathbf{a}_{i}\cdot\mathbf{b}_{j}=\delta_{ij}$. The vector potential for the
magnetic field in Landau gauge is
$\mathbf{A}(\mathbf{r})=-\phi\mathbf{b}_{1}(\mathbf{b}_{2}\cdot\mathbf{r})$,
where $\phi$ is the flux per primitive unit cell. It is helpful to note the
gradient in these coordinates is
$\nabla=\mathbf{b}_{1}\partial_{r_{1}}+\mathbf{b}_{2}\partial_{r_{2}}$.

From Eq. (6) we can compute the gauge transformation terms
$\lambda(r_{1},r_{2})$, where
$\mathbf{r}=r_{1}\mathbf{a}_{1}+r_{2}\mathbf{a}_{2}$. The results for several
spatial symmetries $\hat{g}$ compatible with the triangular lattice are
summarized in Table 7.

##  Appendix E Equivalent representations for the $q$-by-$1$ and $q$-by-$q$
unit cells

In this appendix, we argue that the symmetry operators for the $q$-by-$1$ and
$q$-by-$q$ unit cells form equivalent $q^{2}\times q^{2}$-dimensional
representations. We study $C_{4}$ and $T(\hat{\mathbf{y}})$ as examples.

###  E.1 $q$-by-$1$ unit cell

As shown in Sec. III.1.1, in the $q$-by-$1$ unit cell the $C_{4}$ symmetry
mixes a Bloch eigenstate at momentum $(k_{x},k_{y})$ into a linear combination
of eigenstates at momenta $(k_{y}+2\pi l/q,-k_{x})$, $l=0,1,\dots q-1$. Each
momentum can be classified by the number of other momenta it mixes with under
$C_{4}$: a generic $\mathbf{k}$ mixes into $4q$ distinct momenta; $(\pi,0)$
mixes into $2q$ momenta (including $(0,\pi/q)$); and $(0,0)$ or
$(\pi/q,\pi/q)$ mix into $q$ distinct momenta.

We now compute the symmetry operators for the highest symmetry momenta,
$(0,0)$ and $(\pi/q,\pi/q)$. The $T(\hat{\mathbf{y}})$ and $C_{4}$ symmetries
will be implemented by a $q^{2}\times q^{2}$ matrix: the first factor of $q$
comes from the $q$ original unit cells in the magnetic unit cell, while the
second factor comes from $q$ different momenta that mix into each other under
$C_{4}$. Specifically, following Eqs. (25) and (26), at the high-symmetry
points $\mathbf{k}=\Gamma$ or $M$, we choose the basis
$\\{c^{\dagger}_{(l\phi,0),j}|0\rangle\\}$ or
$\\{c^{\dagger}_{(\pi/q+l\phi,\pi/q),j}|0\rangle\\}$, respectively, where
$l,j=0,1,\dots,q-1$.

As a concrete example, we write the matrix form of the symmetry operators at
$\phi=\pi$ flux, i.e., $q=2$. A symmetry operator $g$ defined in the above
basis takes the general form

$g=\frac{q}{(2\pi)^{2}}\int_{0}^{\frac{2\pi}{q}}\int_{0}^{\frac{2\pi}{q}}d\mathbf{k}\sum_{l,j,l^{\prime},j^{\prime}=0,1}c^{\dagger}_{\mathbf{k},l,j}D(g,\mathbf{k})_{l,j;l^{\prime},j^{\prime}}c_{\mathbf{k},l^{\prime},j^{\prime}}$
(81)

where $c_{\mathbf{k},l,j}=c_{\mathbf{k}+l(2\pi/q,0),j}$ and
$D(g,\mathbf{k})_{l,j;l^{\prime},j^{\prime}}$ is a $4\times 4$ matrix. (This
form does not apply at momenta other than $(0,0)$ or $(\pi/2,\pi/2)$; at other
momenta the $C_{4}$ symmetry requires a larger matrix.)

Taking the specific symmetries $g=T(\hat{\mathbf{y}})$ or $C_{4}$,

$D(T(\hat{\mathbf{y}}),\mathbf{k})=e^{ik_{y}}\begin{pmatrix}&&&1\\\ &&1&\\\
&1&&\\\ 1&&&\end{pmatrix},$ (82)

whose eigenvalues are $e^{ik_{y}}\\{+1,+1,-1,-1\\}$;

$D(C_{4},\Gamma)=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\ 1&-1&-1&1\\\
1&-1&1&-1\\\ 1&1&-1&-1\end{pmatrix},$ (83)

whose eigenvalues are $\\{+1,+1,-1,-1\\}$; and

$D(C_{4},(\pi/2,\pi/2))=\frac{1}{2}\begin{pmatrix}1&-i&1&-i\\\ -i&1&i&-1\\\
1&i&1&i\\\ -i&-1&i&1\end{pmatrix},$ (84)

whose eigenvalues are $\\{+1,+1,+i,-i\\}$.

###  E.2 $q$-by-$q$ unit cell

For the $q$-by-$q$ unit cell, the $C_{4}$ operator behaves similar to the non-
magnetic case, in that it mixes each Bloch wave function into a single other
Bloch wave function. The highest symmetry points are $\Gamma=(0,0)$ and
$M=(\pi/q,\pi/q)$.

As in the previous section, we consider the flux $\phi=\pi$ as an example. An
operator $g$ takes the general form

$g=(\frac{q}{2\pi})^{2}\int
d\mathbf{k}\sum_{j_{x},j_{y},j_{x}^{\prime},j_{y}^{\prime}}c^{\dagger}_{\mathbf{k},j_{x},j_{y}}D(g,\mathbf{k})_{j_{x},j_{y};j_{x}^{\prime},j_{y}^{\prime}}c_{\mathbf{k},j_{x}^{\prime},j_{y}^{\prime}}$
(85)

This form of the matrix representation $D(g,\mathbf{k})$ is valid for
$T(\hat{\mathbf{y}})$ at any $\mathbf{k}$ and for $C_{4}$ at $\Gamma=(0,0)$
and $M=(\pi/q,\pi/q)$. From Eqs. (29) and (30),

$D(T(\hat{\mathbf{y}}),\mathbf{k})=e^{ik_{y}}\begin{pmatrix}&&1&\\\ &&&-1\\\
1&&&\\\ &-1&&\end{pmatrix}$ (86)

whose eigenvalues are $e^{ik_{y}}\\{+1,+1,-1,-1\\}$;

$D(C_{4},(0,0))=\begin{pmatrix}1&&&\\\ &&1&\\\ &1&&\\\ &&&-1\end{pmatrix},$
(87)

whose eigenvalues are $\\{+1,+1,-1,-1\\}$; and

$D(C_{4},(\pi/2,\pi/2))=\begin{pmatrix}1&&&\\\ &&-1&\\\ &1&&\\\
&&&1\end{pmatrix},$ (88)

whose eigenvalues are $\\{+1,+1,+i,-i\\}$.

These eigenvalues are identical to the eigenvalues of the symmetry operators
derived for the $q$-by-1 unit cell in Eqs. (82), (83) and (84), which implies
that the representations of the symmetry operators are the same for a
$q$-by-$1$ unit cell and a $q$-by-$q$ unit cell.

In conclusion: the two natural choices of magnetic unit cell yield unitarily
equivalent symmetry representations, but the $q$-by-$q$ unit cell allows the
symmetry operators to act in a more familiar way because their action on
$\mathbf{k}$ is identical to that in zero field.

##  Appendix F Irreps at high symmetry momenta for $p2,p4,p4/m^{\prime}$

The irreps of little co-groups at high symmetry momenta for $p2$, $p4$ and
$p4/m^{\prime}$ with $\pi$ flux are calculated in detail in this section. The
results are summarized in Tables 2, 3, and 4 in Sec. III.2.

For simplicity, we denote $T_{x}\equiv T(\hat{\mathbf{x}})$, $T_{y}\equiv
T(\hat{\mathbf{y}})$ in this section.

In this appendix, we will repeatedly use the gauge invariant commutation
relation

$C_{n}T(\mathbf{a})=T(R_{n}\mathbf{a})C_{n}$ (89)

which is derived in Appendix B and

$T_{x}T_{y}=T_{y}T_{x}e^{i\phi}=-T_{y}T_{x}$ (90)

which is a consequence of the Aharonov-Bohm phase. We will also use
$T_{x}^{2}=e^{2ik_{x}}$, $T_{y}^{2}=e^{2ik_{y}}$ to derive the
(anti-)commutation relations of rotation and translation symmetries.

###  F.1 $p2$

We first study the $p2$ with $\phi=\pi$ and a $2$-by-$1$ unit cell. The
magnetic lattice translation group is
$\mathbb{T}_{M}=\\{(n_{1},2n_{2})|n_{1},n_{2}\in\mathbb{Z}\\}$. The little co-
group is a subgroup of $p2/{\mathbb{T}}_{M}$, which is generated by $C_{2}$
and $T_{y}$ symmetries.

For the $2$-by-$1$ unit cell, the Brillouin zone is
$[-\pi,\pi]\times[-\pi/2,\pi/2]$. In momentum space, $T_{y}$ maps
$(k_{x},k_{y})\mapsto(k_{x}+\pi,k_{y})$ because Eq. (90) implies

$\displaystyle T_{x}T_{y}|k_{x},k_{y}\rangle$
$\displaystyle=-T_{y}T_{x}|k_{x},k_{y}\rangle$
$\displaystyle=e^{i(k_{x}+\pi)}T_{y}|k_{x},k_{y}\rangle$ (91)

Therefore, given a Bloch wave function $|0,0;\xi\rangle$ at $(0,0)$ with
$C_{2}$ eigenvalue $\xi$, we can construct a state $T_{y}|0,0;\xi\rangle$ at
$(\pi,0)$ with the same $C_{2}$ eigenvalue:

$\displaystyle C_{2}T_{y}|0,0;\xi\rangle$
$\displaystyle=T_{y}T_{y}^{-2}C_{2}|0,0;\xi\rangle$ $\displaystyle=\xi
T_{y}^{-1}|0,0;\xi\rangle,$ (92)

where the first equality is due to the commutation relation Eq. (89) and the
second equality follows because the $T_{y}^{2}$ eigenvalue of a Bloch wave
function at $k_{y}$ is $e^{2ik_{y}}$. Similarly, for each Bloch state at
$(0,\pi/2)$ with $C_{2}$ eigenvalue $\xi$, there is a state at $(\pi,\pi/2)$
with eigenvalue $-\xi$:

$\displaystyle C_{2}T_{y}|0,\pi/2;\xi\rangle$
$\displaystyle=T_{y}T_{y}^{-2}C_{2}|0,\pi/2;\xi\rangle$ $\displaystyle=\xi
T_{y}T_{y}^{-2}|0,\pi/2;\xi\rangle$ $\displaystyle=-\xi
T_{y}|0,\pi/2;\xi\rangle$ (93)

It follows that although there are four $C_{2}$ symmetric momenta, only two of
them have independent eigenvalues, $\Gamma=(0,0)$ and $Y=(0,\pi/2)$. Each
Bloch wave function at those points can have $C_{2}$ eigenvalue $+i$ or $-i$.

Due to the unusual behavior of $T_{y}$ in Eq. (91), the four momenta invariant
under $T_{y}C_{2}$ are $(\pm\pi/2,0)$ and $(\pm\pi/2,\pi/2)$. Similarly only
two of them are independent. We choose the two points to be $X=(\pi/2,0)$ and
$M=(\pi/2,\pi/2)$. Since $(T_{y}C_{2})^{2}=-1$, there are two irreps for each
point and each irrep has $T_{y}C_{2}$ eigenvalue $+i$, $-i$.

We summarize the irreps at all independent high symmetry points for the group
$p2$ in Table 2. Notice that we have used $C_{2}^{2}=-1$, corresponding to
spinful electrons, in the above derivations. If $C_{2}^{2}=+1$, the $C_{2}$
eigenvalues will be $+1$, $-1$.

###  F.2 $p4$

We now study the group $p4$ group with $\phi=\pi$ and a $2$-by-$2$ unit cell.
The lattice translation group is
$\mathbb{T}_{M}=\\{(2n_{1},2n_{2})|n_{1},n_{2}\in\mathbb{Z}\\}$. The little
co-group is a subgroup of $p4/{\mathbb{T}}_{M}$ generated by $C_{4}$ and
$T_{x}$, $T_{y}$ symmetries.

For the $2$-by-$2$ unit cell, the Brillouin zone is
$[-\pi/2,\pi/2]\times[-\pi/2,\pi/2]$. $T_{x}$ and $T_{y}$ both map
$(k_{x},k_{y})$ to itself. Since $\\{T_{x},T_{y}\\}=0$, each irrep at generic
$\mathbf{k}$ is at least two-dimensional.

The two high symmetry points invariant under $C_{4}$ are $(0,0)$ and
$(\pi/2,\pi/2)$, while the two high symmetry points invariant under $C_{2}$
but not $C_{4}$ are $(\pi/2,0)$ and $(0,\pi/2)$. We study each point
separately.

At $X=(\pi/2,0)$, the (anti-)commutation relations derived from Eqs. (89) and
(90), $[C_{2},T_{y}]=0$, $\\{C_{2},T_{x}\\}=0$, $\\{T_{x},T_{y}\\}=0$ lead to
two two-dimensional irreps $X_{1}$, $X_{2}$ as shown in Table 3.

At $Y=(0,\pi/2)$, the (anti-)commutation relations derived from Eqs. (89) and
(90), $[C_{2},T_{x}]=0$, $\\{C_{2},T_{y}\\}=0$, $\\{T_{x},T_{y}\\}=0$ lead to
two two-dimensional irreps $Y_{1}$, $Y_{2}$ as shown in Table 3.

At $\Gamma=(0,0)$, $[C_{4}T_{x},T_{x}T_{y}]=0$ and $\\{T_{x}T_{y},T_{x}\\}=0$
due to Eqs. (89) and (90). For a Bloch eigenstate $|\xi,\eta\rangle$ with
eigenvalue $\xi=\pm 1$ or $\pm i$ of $C_{4}T_{x}$ and eigenvalues $\eta=\pm i$
of $T_{x}T_{y}$, $T_{x}$ maps it to an eigenstate of $C_{4}T_{x}$ with
eigenvalue $\eta\xi$ and an eigenstate of $T_{x}T_{y}$ with eigenvalue
$-\eta$:

$\displaystyle C_{4}T_{x}T_{x}|\xi,\eta\rangle$
$\displaystyle=T_{y}C_{4}T_{x}|\xi,\eta\rangle$
$\displaystyle=T_{x}T_{x}^{-2}T_{x}T_{y}C_{4}T_{x}|\xi,\eta\rangle$
$\displaystyle=T_{x}e^{-2ik_{x}}\eta\xi|\xi,\eta\rangle$
$\displaystyle=\eta\xi T_{x}|\xi,\eta\rangle$ (94)

where the first equality follows Eq. (89) and

$\displaystyle T_{x}T_{y}T_{x}|\xi,\eta\rangle$
$\displaystyle=-T_{x}T_{x}T_{y}|\xi,\eta\rangle$ $\displaystyle=-\eta
T_{x}|\xi,\eta\rangle$ (95)

due to Eq. (90). The two equations pair up eigenvalues and lead to four two-
dimensional irreps $\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$, $\Gamma_{4}$ as
shown in Table 3.

At $M=(\pi/2,\pi/2)$, there are (anti-)commutation relations
$[C_{4},T_{x}T_{y}]=0$ and $\\{T_{x},T_{x}T_{y}\\}=0$. For an eigenstate
$|\xi,\eta\rangle$ with eigenvalue $\xi=e^{i\pi/2},e^{3i\pi/2},e^{-3i\pi/2},$
or $e^{-i\pi/2}$ of $C_{4}$ and eigenvalue $\eta=+i,-i$ of $T_{x}T_{y}$,
$T_{x}$ maps it to an eigenstate of $C_{4}$ and $T_{x}T_{y}$ with eigenvalues
$-\eta\xi$ and $-\eta$, respectively:

$\displaystyle C_{4}T_{x}|\xi,\eta\rangle$
$\displaystyle=T_{y}C_{4}|\xi,\eta\rangle$
$\displaystyle=T_{x}T_{x}^{-2}T_{x}T_{y}C_{4}|\xi,\eta\rangle$
$\displaystyle=T_{x}e^{-2ik_{x}}\eta\xi|\xi,\eta\rangle$
$\displaystyle=-\eta\xi T_{x}|\xi,\eta\rangle$ (96)

where the first line comes from Eq. (89) and

$\displaystyle T_{x}T_{y}T_{x}|\xi,\eta\rangle$
$\displaystyle=-T_{x}T_{x}T_{y}|\xi,\eta\rangle$ $\displaystyle=-\eta
T_{x}|\xi,\eta\rangle$ (97)

due to Eq. (90). The commutation relations imply four two-dimensional irreps
$M_{1}$, $M_{2}$, $M_{3}$, $M_{4}$ as shown in Table. 3.

###  F.3 $p4/m^{\prime}$

We study the group $p4/m^{\prime}$ with $\phi=\pi$ and a $2$-by-$2$ unit cell.
The lattice translation group and Brillouin zone are the same as for the $p4$
case. We first prove that each band is four-fold degenerate: since
$T_{x}^{2}=e^{2ik_{x}}$, $T_{x}$ has eigenvalues $\eta e^{ik_{x}}$, $\eta=\pm
1$. A Bloch eigenstate at $k$, $|k;\eta\rangle$, is mapped by $\cal TI$ to
another state at $k$ with the same eigenvalue:

$\displaystyle T_{x}{\cal TI}|k;\eta\rangle$ $\displaystyle={\cal
TI}T_{x}^{-2}T_{x}|k;\eta\rangle$ $\displaystyle={\cal TI}\eta
e^{-ik_{x}}|k;\eta\rangle$ $\displaystyle=\eta e^{ik_{x}}{\cal
TI}|k;\eta\rangle$ (98)

where the first line can be derived from Table 1. In the spinful case where
$({\cal TI})^{2}=-1$, there is a Kramers degeneracy, i.e. ${\cal
TI}|k;\eta\rangle$ and $|k;\eta\rangle$ are two different states. The
anticommutation $\\{T_{x},T_{y}\\}=0$ implies that $T_{y}$ flips the
eigenstate of $T_{x}$:

$\displaystyle T_{x}T_{y}|k;\eta\rangle$
$\displaystyle=-T_{y}T_{x}|k;\eta\rangle$ $\displaystyle=-\eta
e^{ik_{x}}T_{y}|k;\eta\rangle$ (99)

Combining Eqs. (F.3) and (F.3), we conclude that each eigenstate is at least
four-fold degenerate.

We now study the irreps at the high symmetry points $\Gamma$, $X$, $Y$, $M$.
The role of $\cal TI$ is to pair the two-dimensional irreps of the $p4$ group.
Following the irrep lebelling scheme used for $p4$ in the previous section,
the irreps are summarized in Table 4 and justified as follows:

At $X=(\pi/2,0)$, the commutation relations $[C_{2},{\cal TI}]=0$ and
$[T_{y},{\cal TI}]=0$ lead to one irrpe $X_{1}X_{2}$, and similarly at $Y$ for
$Y_{1}Y_{2}$. At $\Gamma=(0,0)$, the commutation relations $[C_{4},{\cal
TI}]=0$ leads to one irrep $\Gamma_{1}\Gamma_{2}$. Finally, at
$M=(\pi/2,\pi/2)$, the relations $[C_{4},{\cal TI}]=0$ and
$\\{T_{x}T_{y},{\cal TI}\\}=0$ lead to three four-dimensional irreps
$M_{1}M_{1}$, $M_{3}M_{3}$, and $M_{2}M_{4}$.

##  Appendix G  Deriving the symmetry indicator

In this appendix, we describe how to find the symmetry indicator
classification of a space group and how to apply it to a group of bands to
determine in which topological class the bands belong.

At the crux of the theory is the “EBR matrix” for the space group. Each row of
the matrix corresponds to a particular choice of $\mathbf{q}$ and an irrep
$\rho$ of $G_{\mathbf{q}}$ (Sec. IV.2). Each column corresponds to a
particular irrep of the little co-group at a particular high symmetry momentum
(Sec. III.2). The entry in the matrix indicates the number of times the irrep
appears in the band representation $\rho_{G}$ induced from $\rho$, as we will
define in Eq. (100) [12, 42, 43].

Let $A$ be an integer EBR matrix of the symmetry group under consideration.
Since a group of topologically trivial bands transforms identically to a sum
of Wannier functions, its irreps at high symmetry points satisfy

$v=A{n},$ (100)

where $v_{j}$ is the number of times the $j^{\text{th}}$ irrep appears in the
band structure [6].

Let the Smith normal form of $A$ be given by

$A=U^{-1}DV^{-1},$ (101)

where $D$ is a diagonal positive integer matrix with diagonal entries
$(d_{1},\dots,d_{M},0,\dots 0)$, i.e., the first $M$ entries are positive and
the remaining entries are zero, and $U,V$ are integer matrices invertible over
the integers. The stable topological classification for the space group is
given by

$\mathbb{Z}_{d_{1}}\times\cdots\times\mathbb{Z}_{d_{M}}.$ (102)

We seek a formula that expresses the topological invariant (i.e., the element
of $\mathbb{Z}_{d_{m}}$ of a particular group of bands) in terms of the little
co-group irreps at high symmetry points. This index is given by [14, 44, 42,
43, 12]

$\text{index}=\left(Uv\right)_{m}\mod d_{m}$ (103)

where $1\leq m\leq M$, and $d_{m}>1$.

The number of Wannier functions that are centered at a particular maximal
Wyckoff position $w$ can be determined by the following formula [29, 34]:

$e_{w}=\sum_{i\in w}\text{dim}(\rho_{i})\left[VD^{p}Uv\right]_{i}\\\
\mod\text{gcd}\\{\left(\sum_{i\in
w}\text{dim}(\rho_{i})V_{ij}\right)|_{j>M}\\}.$ (104)

The sum over ${i\in w}$ indicates the sum over EBRs induced from a
representation $\rho_{i}$ of the site symmetry group of the Wyckoff position
$w$; $D^{p}$ is the pseudo-inverse of $D$, a diagonal matrix with diagonal
entries $(d_{1}^{-1},...,d_{M}^{-1},0,...,0)$; and gcd indicates the greatest
common divisor.

In the following, we compute the EBR matrix and Smith decomposition for
relevant groups discussed in the main text.

###  G.1 $p2$

The basis for band representations (columns) and the basis for coefficients of
EBRs (rows) are

$\left(\Gamma_{1}^{(p2)},\Gamma_{2}^{(p2)},Y_{1}^{(p2)},Y_{2}^{(p2)},X_{1}^{(p2)},X_{2}^{(p2)},M_{1}^{(p2)},M_{2}^{(p2)}\right),$
(105)

where $\Pi^{(p2)}_{i}$ is defined in Table 2, and

$\left({}^{1}\bar{E}^{4a},^{2}\bar{E}^{4a},^{1}\bar{E}^{4b},^{1}\bar{E}^{4b},^{1}\bar{E}^{4c},^{2}\bar{E}^{4c},^{1}\bar{E}^{4d},^{2}\bar{E}^{4d}\right).$
(106)

where ${}^{j}\bar{E}^{nw}$ is an irrep of the site symmetry group $C_{2}$ of
the Wyckoff position $nw$. ${}^{1}\bar{E}$ and ${}^{2}\bar{E}$ have $C_{2}$
eigenvalue $+i$ and $-i$ respectively. In this basis, the EBR matrix is

$A=\begin{pmatrix}2&0&1&1&1&1&1&1\\\ 0&2&1&1&1&1&1&1\\\ 1&1&2&0&1&1&1&1\\\
1&1&0&2&1&1&1&1\\\ 1&1&1&1&2&0&1&1\\\ 1&1&1&1&0&2&1&1\\\ 1&1&1&1&1&1&2&0\\\
1&1&1&1&1&1&0&2\end{pmatrix}$ (107)

The Smith normal form matrices are

$D=\begin{pmatrix}1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\
0&0&0&1&0&0&0&0\\\ 0&0&0&0&4&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ \end{pmatrix}$ (108)

$U=\begin{pmatrix}2&2&-1&0&-1&0&-1&0\\\ 1&2&-1&0&-1&0&0&0\\\
-1&-1&1&0&0&0&1&0\\\ -1&-1&0&0&1&0&1&0\\\ -3&-5&2&0&2&0&2&0\\\
-1&-1&0&0&1&1&0&0\\\ -1&-1&1&1&0&0&0&0\\\ -1&-1&0&0&0&0&1&1\\\ \end{pmatrix}$
(109)

$V=\begin{pmatrix}1&0&0&0&1&-1&-1&-1\\\ 0&0&0&0&-1&-1&-1&-1\\\
0&-1&1&0&-2&0&1&0\\\ 0&0&0&0&0&0&1&0\\\ 0&-1&0&1&-2&1&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&1&0&0&2&0&0&1\\\ 0&0&0&0&0&0&0&1\\\ \end{pmatrix}$ (110)

The stable indicator is given by Eq. (42) in the main text:

$\text{index}=\\#\Gamma_{1}-\\#\Gamma_{2}+\\#X_{1}-\\#X_{2}+\\#Y_{1}-\\#Y_{2}\\\
+\\#M_{1}-\\#M_{2}-N\mod 4.$ (111)

The symmetry indicators for Wannier centers are

$\displaystyle e_{2a}$ $\displaystyle=2N-\\#X_{1}-\\#Y_{1}-\\#M_{1}\mod 2$
(112) $\displaystyle e_{2b}$ $\displaystyle=-N/2+\\#Y_{1}\mod 2$ (113)
$\displaystyle e_{2c}$ $\displaystyle=-N/2+\\#X_{1}\mod 2$ (114)
$\displaystyle e_{2d}$ $\displaystyle=-N/2+\\#M_{1}\mod 2$ (115)

###  G.2 $p4$

The basis for band representations (columns) and the basis for coefficients of
EBRs (rows) are

$\left(\Gamma_{1},\Gamma_{2},\Gamma_{3},\Gamma_{4},M_{1},M_{2},M_{3},M_{4},X_{1},X_{2}\right),$
(116)

where $\Pi_{i}$ is defined in Table 3, and

$\left({}^{1}\bar{E}^{4a}_{1},^{1}\bar{E}^{4a}_{2},^{2}\bar{E}^{4a}_{2},^{2}\bar{E}^{4a}_{1},E^{4b}_{1},^{1}\bar{E}^{4b}_{2},^{2}\bar{E}^{4b}_{2},^{2}\bar{E}^{4b}_{1},^{1}\bar{E}^{8c},^{2}\bar{E}^{8c}\right).$
(117)

where ${}^{j}\bar{E}^{nw}$ is an irrep of the site symmetry group of the
Wyckoff position labelled by $nw$. ${}^{1}\bar{E}$ and ${}^{2}\bar{E}$ have
$C_{2}$ eigenvalues $+i$ and $-i$ respectively, while ${}^{1}\bar{E}_{1}$,
${}^{1}\bar{E}_{2}$, ${}^{2}\bar{E}_{2}$, and ${}^{2}\bar{E}_{1}$ have $C_{4}$
eigenvalues $e^{i\pi/4}$, $e^{i3\pi/4}$, $e^{-3i\pi/4}$, $e^{-i\pi/4}$
respectively. In this basis, the EBR matrix is

$A=\begin{pmatrix}1&0&1&0&1&0&0&1&1&1\\\ 0&1&0&1&1&1&0&0&1&1\\\
1&0&1&0&0&1&1&0&1&1\\\ 0&1&0&1&0&0&1&1&1&1\\\ 1&0&0&1&1&0&1&0&1&1\\\
1&1&0&0&0&1&0&1&1&1\\\ 0&1&1&0&1&0&1&0&1&1\\\ 0&0&1&1&0&1&0&1&1&1\\\
1&1&1&1&1&1&1&1&3&1\\\ 1&1&1&1&1&1&1&1&1&3\\\ \end{pmatrix}$ (118)

The Smith normal form matrices are

$D=\begin{pmatrix}1&0&0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&8&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\ \end{pmatrix}$ (119)

$U=\begin{pmatrix}0&-1&-1&0&1&1&0&1&0&0\\\ -1&-1&-1&0&1&1&1&1&0&0\\\
0&-1&-1&0&1&1&1&2&-1&0\\\ -1&-1&-1&0&2&1&1&2&-1&0\\\ 0&1&1&0&0&-1&0&-1&0&0\\\
0&1&1&0&-1&-1&-1&-1&1&0\\\ -1&0&1&0&0&-1&0&-1&1&0\\\
6&4&2&0&-9&-3&-5&-7&4&0\\\ 1&1&1&1&-1&-1&-1&-1&0&0\\\
0&0&0&0&-1&-1&-1&-1&1&1\\\ \end{pmatrix}$ (120)

$V=\begin{pmatrix}1&0&0&-3&0&0&-1&-5&-1&-1\\\ 0&1&0&-2&0&0&-1&-3&-1&-1\\\
0&0&1&-1&0&0&0&-1&-1&-1\\\ 0&0&0&1&0&0&0&1&-1&-1\\\ 0&0&0&1&1&0&0&2&1&0\\\
0&0&0&-2&0&1&-1&-4&1&0\\\ 0&0&0&-1&0&0&0&-2&1&0\\\ 0&0&0&0&0&0&0&0&1&0\\\
0&0&0&2&0&0&1&4&0&1\\\ 0&0&0&0&0&0&0&0&0&1\\\ \end{pmatrix}$ (121)

The stable indicators is given by Eq. (44) in the main text,

$\text{index}=2\\#\Gamma_{1}+4\\#\Gamma_{2}-2\\#\Gamma_{3}+\\#M_{1}+3\\#M_{2}\\\
-3\\#M_{3}-\\#M_{4}+4\\#X_{1}\mod 8,$ (122)

which corresponds to the $8$ in the diagonal of $D$.

The symmetry indicators for Wannier centers are

$\displaystyle e_{4a}$
$\displaystyle=2\left(2\\#\Gamma_{2}+2\\#\Gamma_{3}+\\#M_{2}+\\#M_{4}+2\\#X_{2}\right)\mod
8$ (123) $\displaystyle e_{4b}$
$\displaystyle=2\left(2\\#\Gamma_{2}+2\\#\Gamma_{3}-\\#M_{2}-\\#M_{4}-N/2\right)\mod
8$ (124) $\displaystyle e_{8c}$ $\displaystyle=\\#X_{1}-\\#X_{2}\mod 4$ (125)

###  G.3 $p4/m^{\prime}$

The basis for band representations (columns) and the basis for coefficients of
EBRs (rows) are

$\left(\Gamma_{1}\Gamma_{4},\Gamma_{2}\Gamma_{3},M_{1}M_{1},M_{3}M_{3},M_{2}M_{4}\right),$
(126)

where $\Pi_{i}$ is defined in Table 4, and

$\left(\bar{E}^{4a}_{1/2},\bar{E}^{4a}_{3/2},{}^{1}\bar{E}^{4b}_{1/2}{}^{2}\bar{E}^{4b}_{3/2},{}^{1}\bar{E}^{4b}_{3/2}{}^{1}\bar{E}^{4b}_{3/2},{}^{2}\bar{E}^{4b}_{1/2}{}^{2}\bar{E}^{4b}_{1/2},\bar{E}^{8c}\right).$
(127)

where $\bar{E}^{nw}$ is an irrep of the site symmetry group of the Wyckoff
position labelled by $nw$. $\bar{E}^{4a}_{j_{z}}$ has $C_{4}$ eigenvalues
$e^{\pm i\frac{\pi}{2}j_{z}}$, while ${{}^{1}}\bar{E}^{4b}_{j_{z}}$ and
${{}^{2}}\bar{E}^{4b}_{j_{z}}$ correspond to eigenvalues
$e^{i\frac{\pi}{2}j_{z}}$ and $e^{-i\frac{\pi}{2}j_{z}}$ of
$C_{4}(\frac{1}{2},\frac{1}{2})$ separately. For the site-symmetry group of
the $4a$ position, there are two irreps $\bar{E}^{4a}_{1/2}$ and
$\bar{E}^{4a}_{3/2}$, while for the site-symmetry group of the $4b$ position,
there are three irreps, ${}^{1}\bar{E}^{4b}_{1/2}{}^{2}\bar{E}^{4b}_{3/2}$,
${}^{1}\bar{E}^{4b}_{3/2}{}^{1}\bar{E}^{4b}_{3/2}$ and
${}^{2}\bar{E}^{4b}_{1/2}{}^{2}\bar{E}^{4b}_{1/2}$. The difference between
these comes from the unusual pairing of irreps with
$C_{4}(\frac{1}{2},\frac{1}{2})$ eigenvalues $\xi$ and $-i\xi^{*}$ as derived
in Eq. (21). For the site-symmetry group of the $8c$ position, there is only
one irrep, $\bar{E}^{8c}$, with $C_{2}(\frac{1}{2},0)$ eigenvalues $\pm i$.

In this basis, the EBR matrix is

$A=\begin{pmatrix}1&1&1&2&2\\\ 1&1&1&0&2\\\ 1&0&0&1&1\\\ 0&1&0&1&1\\\
1&1&2&0&2\\\ \end{pmatrix}$ (128)

The Smith normal form matrices are

$D=\begin{pmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&2&0\\\
0&0&0&0&0\\\ \end{pmatrix}$ (129)

$U=\begin{pmatrix}0&0&1&0&0\\\ 0&0&0&1&0\\\ 1&0&-1&-1&0\\\ 1&-1&0&0&0\\\
-1&-1&1&1&1\\\ \end{pmatrix}$ (130)

$V=\begin{pmatrix}1&0&0&-1&-1\\\ 0&1&0&-1&-1\\\ 0&0&1&0&0\\\ 0&0&0&1&0\\\
0&0&0&0&1\\\ \end{pmatrix}$ (131)

The stable indicator is given by Eq. (48) in the main text,

$\displaystyle\text{index}_{1}$ $\displaystyle=N/4\mod 2.$ (132)

The symmetry indicators for Wannier centers are

$\displaystyle e_{4a}$
$\displaystyle=\frac{N}{2}+2(\\#M_{1}M_{1}+\\#M_{3}M_{3})\mod 4$ (133)
$\displaystyle e_{4b}$
$\displaystyle=-\frac{N}{4}+2(\\#M_{1}M_{1}+\\#M_{3}M_{3})\mod 4$ (134)
$\displaystyle e_{8c}$ $\displaystyle=0\mod 2$ (135)

##  Appendix H  Symmetry indicator for the Chern number

The Chern number in a four-fold symmetric system is given by Eq. (45) [18]:

$e^{i\frac{\pi}{2}C}=(-)^{2SN}w_{C_{4}}^{\Gamma}w_{C_{2}}^{X}w_{C_{4}}^{M},$
(136)

where $S=1/2$ for spinful systems and $w_{g}^{\Pi}$ is the product of
eigenvalues of the symmetry $g$ for filled bands at momentum $\Pi$. We now
rewrite this formula in terms of the irreps of the little co-groups for $p4$.

From the irreps of $p4$ listed in Table 3, it is evident that
$w_{C_{2}}^{X}\equiv+1$ for both irreps $X_{1}$ and $X_{2}$. For the four
irreps at $M$, $w_{C_{4}}^{M_{1}}=w_{C_{4}}^{M_{3}}=+1$ and
$w_{C_{4}}^{M_{2}}=w_{C_{4}}^{M_{4}}=-1$, where the superscript is now
labeling the irrep.

We now find the $C_{4}$ eigenvalues of irreps at $\Gamma$. We know from
Appendix F that for an eigenstate $|\xi,\eta\rangle$ with an eigenvalue $\xi$
of $C_{4}T_{x}$, and $\eta$ of $T_{x}T_{y}$, there is a degenerate state
$T_{x}|\xi,\eta\rangle$ with eigenvalues:

$\displaystyle C_{4}T_{x}~{}T_{x}|\xi,\eta\rangle$ $\displaystyle=\xi\eta
T_{x}|\xi,\eta\rangle$ (137) $\displaystyle
T_{x}T_{y}~{}T_{x}|\xi,\eta\rangle$ $\displaystyle=-\eta
T_{x}|\xi,\eta\rangle$ (138)

Neither of these states is separately an eigenstate of $C_{4}$, but we can
find a linear combination that is an eigenstate by solving the eigenvalue
equation

$\displaystyle C_{4}\left(\alpha|\xi,\eta\rangle+\beta
T_{x}|\xi,\eta\rangle\right)=\lambda\left(\alpha|\xi,\eta\rangle+\beta
T_{x}|\xi,\eta\rangle\right)$ (139)

Using the equations above and $T_{x}^{2}=1$, one finds
$\lambda=\pm\sqrt{\xi^{2}\eta}$. Therefore, for the four irreps at $\Gamma$
listed in Table 3, $w_{C_{4}}^{\Gamma_{1}}=w_{C_{4}}^{\Gamma_{3}}=+i$ and
$w_{C_{4}}^{\Gamma_{2}}=w_{C_{4}}^{\Gamma_{4}}=-i$.

Plug these $w_{C_{n}}^{\Pi_{i}}$ of irreps into Eq. (136), we get Eq. (IV.3.2)

$\displaystyle C=2N$
$\displaystyle+\\#\Gamma_{1}+\\#\Gamma_{3}-\\#\Gamma_{2}-\\#\Gamma_{4}$
$\displaystyle+2(\\#M_{2}+\\#M_{4})\mod 4.$ (140)

##  Appendix I  Symmetry analysis at other fluxes

$\phi/2\pi$ | $\\#\Gamma_{\frac{1}{2}}$ | $\\#M_{\frac{1}{2}}$ | $N$ | $e_{1a^{\prime}}\mod 4$ | $e_{1b^{\prime}}\mod 4$ | $a_{1a^{\prime}}$ | $\eta\mod 4$ | Phase
---|---|---|---|---|---|---|---|---
$0$ | 0 | 1 | 2 | 0 | 2 | 2 | 2 | OAL
$1/2$ | 2 | 3 | 8 | 2 | 2 | 2 | 0 | Trivial
Table 8: Evaluation of symmetry indicators in Eqs. (141) and (142) at half filling of the OAL model at $\phi=0$ (middle row) and $\phi=\pi$ (last row). ${\phi}/{2\pi}$ | $[X_{2}]$ | $[M_{1}+M_{3}]$ | $[M_{2}]$ | $N$ | $e_{1a^{\prime}}\mod 4$ | $e_{1b^{\prime}}\mod 4$ | $e_{2c^{\prime}}\mod 2$ | $a_{1a^{\prime}}$ | $\eta\mod 4$ | Phase
---|---|---|---|---|---|---|---|---|---|---
$1/5$ | 0 | 0 | 1 | 50 | 0 | 2 | 0 | 2 | 2 | OAL
$2/5$ | 0 | 0 | 0 | 50 | 2 | 0 | 0 | 2 | 0 | Trivial
Table 9: Evaluation of symmetry indicators Eqs. (143), (144), and (145) at
half filling of the OAL model at $\phi=2\pi/5$ (middle row) and $\phi=4\pi/5$
(last row).

In this section, we apply the non-magnetic symmetry indicators (i.e. ignoring
the sublattice symmetries) to analyze fluxes $\phi/2\pi=0,1/5,2/5$. The
Wyckoff positions are defined in Fig. 6(b).

For $\phi=0$, the layer group is $p4/m^{\prime}mm$. The symmetry indicators of
Wannier centers are [29]

$\displaystyle e_{1a^{\prime}}$ $\displaystyle=N-2[M_{\frac{1}{2}}]\mod 4$
(141) $\displaystyle e_{1b^{\prime}}$ $\displaystyle=2[M_{\frac{1}{2}}]\mod 4$
(142)

where $N$ is the number of filled bands and
$[M_{\frac{1}{2}}]=\\#M_{\frac{1}{2}}-\\#\Gamma_{\frac{1}{2}}$, where
$\\#\Pi_{\frac{1}{2}}$ indicates the number of times the two-dimensional irrep
$E_{\frac{1}{2}}$ ($C_{4}$ eigenvalues $e^{i\pi/4}$, $e^{-i\pi/4}$) appears in
the valence bands at the high-symmetry point $\Pi=\Gamma,M$.

For $\phi=\frac{2\pi}{5},~{}\frac{4\pi}{5}$, the layer group is $p4$. The
symmetry indicators are [29]

$\displaystyle e_{1a^{\prime}}$
$\displaystyle=N-[X_{2}]+\frac{3}{2}([M_{1}]+[M_{3}])+2[M_{2}]\mod 4$ (143)
$\displaystyle e_{1b^{\prime}}$
$\displaystyle=[X_{2}]-\frac{1}{2}([M_{1}]+[M_{3}])-2[M_{2}]~{}\mod 4$ (144)
$\displaystyle e_{2c^{\prime}}$
$\displaystyle=-\frac{1}{2}([M_{1}]+[M_{3}])~{}\mod 2$ (145)

Here we use the notation $[M_{j}]=\\#M_{j}-\\#\Gamma_{j}$, where $j=1,2,3,4$
corresponds to the irrep with $C_{4}$ eigenvalue
$\exp(i\frac{\pi}{2}(j-\frac{1}{2}))$, and
$[X_{1}]=\\#X_{1}-\\#\Gamma_{1}-\\#\Gamma_{3}$,
$[X_{2}]=\\#X_{2}-\\#\Gamma_{2}-\\#\Gamma_{4}$, where $X_{1,2}$ corresponds to
the irrep with $C_{2}$ eigenvalues $+i$, $-i$. $\\#\Pi_{j}$ indicates the
number of times the irrep $\Pi_{j}$ appears in the valence bands at the high-
symmetry point $\Pi$.

The OAL phase and corner charges are indicated by the filling anomaly $\eta$.
The filling anomaly is defined as the electron number difference between
neutral and symmetric configurations for a symmetric finite system [22]. In
the absence of polarization, nonzero $\eta$ implies that the ion charge at
$1a^{\prime}$ Wyckoff position is not equal to the electron charge
corresponding to the Wannier functions centered at $1a^{\prime}$ [29, 45]

$\displaystyle\eta$
$\displaystyle=N_{e,\text{neutral}}-N_{e,\text{symmetric}}\mod 4$
$\displaystyle=a_{1a^{\prime}}-e_{1a^{\prime}}\mod 4$ (146)

where $a_{1a^{\prime}}$ and $e_{1a^{\prime}}$ are the ion charge and electron
charge in units of $|e|$ at $1a^{\prime}$ position. The modulus $4$ is
specific for our model. In the absence of polarization, the corner charge is
given by $Q_{c}=\eta/4\mod 1$.

The non-magnetic symmetry indicators of the model described in Sec. V at $0$
and $\pi$-flux are shown in Table 8. The indicators at $2\pi/5$ and
$4\pi/5$-flux are in Table 9. The symmetry indicators show that the system at
$0$ and $2\pi/5$-flux is an OAL and at $4\pi/5$ and $\pi$ flux is trivial.
These results are consistent agree with the open-boundary Hofstadter spectrum
in Fig. 5.

The filling anomaly cannot jump when $C_{4}$ symmetry is preserved, unless the
bulk gap or the surface gap closes. In this model, the bulk gap closes between
$0$ and $\pi$ flux, as shown in Fig. 5. The gap closing corresponds to the
transition between two distinct atomic insulating phases that have different
Wannier centers.

##  Appendix J  Wilson loop and nestfed Wilson loop

Figure 7: Wilson loop spectrum for the valence bands at half-filling, directed
in $x$-direction and in $y$-direction, as defined in Eq. (147). The upper row
corresponds to the spectrum of $W_{x}(k_{y})$, while the lower two row
corresponds to the spectrum of $W_{y}(k_{x})$. The critical flux is between
$2\pi/5$ and $4\pi/5$.

An alternative topological invariant that characterizes the model in Sec. V is
the quadrupole moment $Q_{xy}$, or equivalently the Wilson loop of Wilson loop
(nested Wilson loop) [21].

For discrete tight binding models, the Wilson loop is defined as [46, 47, 21]

$[W_{\cal C}(\mathbf{k}_{0})]_{mn}=\langle
u_{m}(\mathbf{k}_{0})|\prod_{\mathbf{k}\in{\cal C}}{\cal
P}(\mathbf{k})|u_{n}(\mathbf{k}_{0})\rangle$ (147)

where ${\cal
P}(\mathbf{k})=\sum_{l=1}^{n_{occ}}|u_{l}(\mathbf{k})\rangle\langle
u_{l}(\mathbf{k})|$ is the projector onto occupied bands, ${\cal C}$ is a loop
in the Brillouin zone and $\mathbf{k}_{0}$ is the base point. We consider two
Wilson loops: $W_{x}(k_{x}^{0},k_{y})$ with the path $k_{x}:0\rightarrow 2\pi$
and $W_{y}(k_{x},k_{y}^{0})$ with the path $k_{y}:0\rightarrow 2\pi$. The
Wilson loop matrices are not gauge invariant since they depend on the gauge
choice of Bloch wavefunctions. However, the eigenvalues of the Wilson loop
matrices are gauge invariant. Since Wilson loops are unitary, their
eigenvalues are unit complex number of the form $e^{2\pi i\nu}$. We plot
$\nu_{x}(k_{y})$ for $W_{x}(k_{y})$ and $\nu_{y}(k_{x})$ for $W_{y}(k_{x})$
for the half filling gap for several fluxes in Fig. 7. The eigenvalue
$\nu_{x}(k_{y})$ has a physical meaning, namely the $x$-coordinate of the
hybrid Wannier function $|w(x,k_{y})\rangle$. The polarization is determined
by the trace of the Wilson loop matrix $\nu$ [22, 48]

$p_{x}=\int\frac{k_{y}}{2\pi}\frac{1}{2\pi}\text{Im}\left(~{}tr\log
W_{x}(k_{y})\right),$ (148)

which is equal to the sum of Wilson loop eigenvalues. In our model,
polarization vanishes at half filling at any $\phi$. A similar expression for
the quadrupole moment has been studied in Refs. 49. However we do not use this
approach. Instead we study the nested Wilson loop.

When the Wilson loop spectrum $\nu_{x}(k_{y})$ is gapped, one can compute the
nested Wilson loop of the Wilson loop eigenstates that have gaps with other
states. The eigenvalues of the nested Wilson loop are denoted as
$\nu_{y}^{\nu_{x}}$ and $\nu_{x}^{\nu_{y}}$. For the selected gapped Wilson
loop eigenstates, the sum of the nested Wilson loop eigenvalues determines its
quadrupole moment [22]

$Q_{xy}=\sum_{i=1}^{n_{occ}^{\nu_{x}}}\nu_{y}^{\nu_{x}}=\sum_{i=1}^{n_{occ}^{\nu_{y}}}\nu_{x}^{\nu_{y}}$
(149)

We choose $q$-by-$1$ unit cell for flux $\phi=\frac{2\pi p}{q}$ to compute the
Wilson loop spectrum. In Fig. 7 we show the Wilson loop spectrum of the $2q$
valence bands below half filling of the energy spectrum.

The Wilson loop spectrum $\nu_{x}(k_{y})$ has a gap at half filling.
Therefore, we study the nested Wilson loop $\nu_{y}^{\nu_{x}}$ for the $q$
states below this gap. We find that the quadrupole moment Eq. (149) for the
chosen bands is $0.5$ when $0\leq\phi<\phi^{*}$, and $0$ when
$\phi^{*}<\phi<\pi$, where $\phi^{*}$ is the critical flux where the
Hofstadter spectrum shows the bulk gap closing at half filling.

We now turn to the Wilson loop spectrum $\nu_{y}(k_{x})$ which has two gaps
that separate one band on the top, one band on the bottom and $2q-2$ bands
around zero. Therefore, we study the nested Wilson loop $\nu_{x}^{\nu_{y}}$
for the single band on the bottom. It turns out that the quadrupole moment Eq.
(149) for the chosen bands is again $0.5$ when $0\leq\phi<\phi^{*}$, and $0$
when $\phi^{*}<\phi<\pi$.

##  Appendix K Magnetic Wannier functions and Balian-Low theorem

In this appendix, we explain the Balian-Low obstruction of exponentially
localized Wannier functions in two dimensional systems in a magnetic field.
The key ingredient of this theorem is the projective translation group, which
bridges our two dimensional magnetic systems and the quantum phase space of
one dimensional quantum mechanics where the theorem was originally introduced.

Two-dimensional system under uniform magnetic field have magnetic translation
operators that are projective representations of the translation group. The
operators satisfy the following multiplication rule

$T(\mathbf{a}_{1})T(\mathbf{a}_{2})=T(\mathbf{a}_{1}+\mathbf{a}_{2})e^{\frac{i}{2}\mathbf{B}\cdot(\mathbf{a}_{1}\times\mathbf{a}_{2})}$
(150)

The magnetic unit cell encloses an integer multiple of $2\pi$ flux, which
ensures that lattice translations commute. Then the momentum space and Bloch
wavefunctions are defined. The Bloch wavefunctions in momentum space Fourier
transform into the magnetic Wannier functions in real space.

The quantum phase space of one dimensional quantum mechanics is labeled by two
dependent variables, position $x$ and wave vector $k=p/\hbar=-i\frac{d}{dx}$.
Now consider the translation operators in the quantum phase space,
$T_{x}(\Delta_{x})=e^{-ik\Delta_{x}}$, $T_{k}(\Delta_{k})=e^{ix\Delta_{k}}$.
The two translations satisfy commutation relation