# Fractional disclination charge and discrete shift in the Hofstadter
butterfly

Yuxuan Zhang  Department of Physics and Joint Quantum Institute, University of
Maryland, College Park, Maryland 20742, USA  Condensed Matter Theory Center,
University of Maryland, College Park, Maryland 20742, USA  Naren Manjunath
Department of Physics and Joint Quantum Institute, University of Maryland,
College Park, Maryland 20742, USA  Condensed Matter Theory Center, University
of Maryland, College Park, Maryland 20742, USA  Gautam Nambiar  Department of
Physics and Joint Quantum Institute, University of Maryland, College Park,
Maryland 20742, USA  Maissam Barkeshli  Department of Physics and Joint
Quantum Institute, University of Maryland, College Park, Maryland 20742, USA
Condensed Matter Theory Center, University of Maryland, College Park, Maryland
20742, USA

###### Abstract

In the presence of crystalline symmetries, topological phases of matter
acquire a host of invariants leading to non-trivial quantized responses. Here
we study a particular invariant, the discrete shift $\mathscr{S}$, for the
square lattice Hofstadter model of free fermions. $\mathscr{S}$ is associated
with a $\mathbb{Z}_{M}$ classification in the presence of $M$-fold rotational
symmetry and charge conservation. $\mathscr{S}$ gives quantized contributions
to (i) the fractional charge bound to a lattice disclination, and (ii) the
angular momentum of the ground state with an additional, symmetrically
inserted magnetic flux. $\mathscr{S}$ forms its own ‘Hofstadter butterfly’,
which we numerically compute, refining the usual phase diagram of the
Hofstadter model. We propose an empirical formula for $\mathscr{S}$ in terms
of density and flux per plaquette for the Hofstadter bands, and we derive a
number of general constraints. We show that bands with the same Chern number
may have different values of $\mathscr{S}$, although odd and even Chern number
bands always have half-integer and integer values of $\mathscr{S}$
respectively.

††preprint: APS/123-QED

Topological phases of matter are characterized by invariants that give rise to
quantized physical responses, such as the Chern number and associated
quantized Hall conductivity. In the presence of spatial symmetries, additional
invariants also arise, such as the Wen-Zee shift [1] in clean isotropic
continuum quantum Hall systems, which characterizes the response to geometric
curvature [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In order to fully understand
systems where lattice effects play an important role, we must develop a
complete understanding of invariants and their associated quantized responses
for topological phases with charge conservation and crystalline symmetries.
Recently, [12, 13] developed such a systematic theory in two dimensions by
analyzing topological terms for crystalline gauge fields and found several
additional quantized invariants. Some of these invariants have no continuum
analog, while others give a crystalline analog of invariants known from the
setting of continuum spatial symmetries.

In this paper, we study one such invariant, the discrete shift $\mathscr{S}$,
and its physical consequences through numerical studies of the square lattice
Hofstadter model [14] of spinless free fermions. $\mathscr{S}$ is an invariant
that depends on a discrete $\mathbb{Z}_{M}$ rotational symmetry and $U(1)$
charge conservation, and is a discrete analog of the Wen-Zee shift arising in
continuum systems. For invertible fermionic topological states [15, 16, 17],
$2\mathscr{S}$ is an integer defined mod $2M$; we show that for fixed Chern
number, $\mathscr{S}$ can in principle take one of $M$ distinct values, and
odd (even) Chern numbers must have half-integer (integer) values of
$\mathscr{S}$.111For invertible bosonic topological states, $\mathscr{S}$ must
be integer, while for fractionalized topological states, $\mathscr{S}$ can be
fractional.

Remarkably, $\mathscr{S}$ refines the known phase diagram of the Hofstadter
model, leading to a new Hofstadter butterfly (Fig. 1), which we numerically
compute. As we study numerically in detail, $\mathscr{S}$ has a physical
manifestation in terms of a quantized contribution to the fractional charge
bound to lattice disclinations (see Eq. (2)) and, dually, the fractional
angular momentum bound to magnetic flux (see Eq.(4)). We theoretically justify
several properties of $\mathscr{S}$ that are evident from Fig. 1, and also
propose an empirical formula for $\mathscr{S}$ (Eq. (5)).

Since the Hofstadter model has now been effectively realized in moiré
superlattice systems [18, 19, 20, 21], ultracold atoms [22, 23, 24], and
photonics [25, 26], an experimental verification of our results may be
possible.

We note that [27, 28, 29, 30, 31, 32, 33] also study some aspects of the shift
in lattice settings, with limited results when Chern number $C\neq 0$, as
discussed below and in Appendix A.

Figure 1:  $\mathscr{S}$ for Hofstadter model, from Eq. (5).

##### Model and response theory.

We consider a system of fermions hopping on a lattice with a discrete
$\mathbb{Z}_{4}$ rotational symmetry, a background magnetic flux $\phi$ per
unit cell, and filling $\nu_{0}$ charge per unit cell. We focus on the
Hofstadter model on a square lattice, with the Hamiltonian $H=-t\sum_{\langle
i,j\rangle}c_{i}^{\dagger}c_{j}e^{-iA_{bgd;ij}}+h.c$. This describes spinless
free fermions $c_{i}$ coupled to a background $U(1)$ gauge field $A_{bgd}$,
whose holonomy around each plaquette is $\phi$. When $\phi=2\pi\frac{p}{q}$
with coprime integers $p,q$, the system has $q$ bands. When $r$ bands are
filled, $\nu_{0}=r/q$. At any gapped point in the parameter space
$(\nu_{0},\phi)$, the total Chern number $C$ of the filled bands is determined
by the conditions $\nu_{0}=\frac{p}{q}C\mod 1;\quad|C|\leq\frac{q}{2}$ [34].
Each connected gapped region in this parameter space has a fixed value of $C$.
Simply connected gapped regions with Chern number $C$ (referred to as Chern
number $C$ lobes) are separated by special values of $\frac{\phi}{2\pi}$ which
lie in the Farey sequence of order $2|C|$ [35]. The continuum limit of $n$
filled Landau levels is obtained by taking $\frac{p}{q}\rightarrow 0^{+},C=n$.

$H$ has a symmetry group $G$ which is a central extension of the wallpaper
group p4 = $\mathbb{Z}^{2}\rtimes\mathbb{Z}_{4}$ by $U(1)$. This means that
the magnetic translations are generated by the many-body operators
$T_{\mathbf{x}},T_{\mathbf{y}}$, which satisfy
$T_{\mathbf{x}}T_{\mathbf{y}}=T_{\mathbf{y}}T_{\mathbf{x}}e^{i\phi\hat{N}}$,
where $\hat{N}$ is the total particle number operator. The Hamiltonian is
invariant under a “magnetic” rotation operator
$\tilde{C}_{4,\lambda}\equiv\hat{C_{4}}e^{i\sum_{j}\lambda_{j}c_{j}^{\dagger}c_{j}}$
where $\lambda_{j}$ is a gauge transformation at site $j$ which is fixed up to
an overall constant by $A_{bgd}$. The usual rotation operator $\hat{C}_{4}$
acts as $\hat{C}_{4}c_{j}\hat{C}_{4}^{\dagger}=c_{R(j)}$, where $R(j)$ is a
vertex-centered $\pi/2$ rotation of site $j$.

The quantized universal properties can be encoded by a topological response
theory involving a background $G$ gauge field $B=(\delta A,\vec{R},\omega)$,
which is nonabelian [12]. Here $\delta A=A-A_{bgd}$ is the deviation of the
total $U(1)$ gauge field $A$ relative to $A_{bgd}$, while $\omega$, $\vec{R}$
are the crystalline gauge fields (Appendix G.1 gives additional details).
$\omega$ is a background gauge field for the $\mathbb{Z}_{4}$ rotational
symmetry; in the continuum limit, it is identified with the spin connection.
$\vec{R}$ is a two-component gauge field for the $\mathbb{Z}^{2}$
translational symmetry. In terms of $\vec{R}$ and $\omega$ we also define an
area element $A_{XY}$ which counts the number of unit cells [12]. The integral
$\delta\Phi_{W}\equiv\int_{W}d\delta A$ over a two-dimensional region $W$
gives the total excess magnetic flux within $W$, not including the background
flux, while $\int_{W}d\omega$ gives the total disclination angle of
disclinations within $W$. $\omega$, $\vec{R}$ are taken to be real-valued
fields with quantized periods, since they are $\mathbb{Z}_{4}$ and
$\mathbb{Z}^{2}$ gauge fields respectively.222One can also work in a
simplicial formulation where $\omega$, $\vec{R}$ are taken to have discrete
values [12].

We can write all terms in the topological response theory which depend on $A$
or $\omega$ [12] (note that the response theory involves $A$ and not just
$\delta A$):

$\displaystyle\mathcal{L}$ $\displaystyle=\frac{C}{4\pi}A\wedge
dA+\frac{\mathscr{S}}{2\pi}A\wedge
d\omega+\frac{\vec{\mathscr{P}}_{c}}{2\pi}\cdot
A\wedge\vec{T}+\frac{k_{0}}{2\pi}A\wedge A_{XY}$
$\displaystyle+\frac{\tilde{\ell}_{s}}{4\pi}\omega\wedge
d\omega+\frac{\vec{\mathscr{P}}_{s}}{2\pi}\cdot\omega\wedge\vec{T}+\frac{k_{s}}{2\pi}\omega\wedge
A_{XY}.$ (1)

See Appendix G.1 for a discussion.
$\vec{T}=d\vec{R}+i\sigma_{y}\omega\wedge\vec{R}$ is the torsion 2-form. Here
$C,2\mathscr{S}$ must be quantized to integers. For fermionic invertible
phases there are some additional terms in the theory, discussed in Appendix
G.1. The first term defines the Hall conductivity
$\overline{\sigma}_{H}=\frac{C}{2\pi}$ in natural units and assigns charge to
flux. The second term assigns a fractional $U(1)$ charge
$\mathscr{S}\frac{\Omega}{2\pi}$ to a defect with disclination angle $\Omega$.
On a square lattice, it is topologically trivial if a $\frac{\pi}{2}$
disclination is assigned an integer charge, which can be removed by applying
local operators at the disclination core. Thus only $\mathscr{S}\mod 4$ is a
symmetry-protected invariant, in contrast to the continuum shift, which is a
$\mathbb{Z}$ invariant.333Note that the conventional definition of shift in
the quantum Hall literature is
$S\equiv\chi\frac{\mathscr{S}}{\overline{\sigma}_{H}}$, with $\chi$ the Euler
characteristic of the space. The second term can also be written as
$\frac{\mathscr{S}}{2\pi}\omega\wedge dA$, which assigns angular momentum
$\mathscr{S}\frac{dA}{2\pi}$ to flux $dA$. In Appendix H we show the
nontrivial result that for spinless fermions in free or interacting systems,
$\mathscr{S}$ is quantized to a half-integer if $C$ is odd, and to integers if
$C$ is even. The numerical values of $\mathscr{S}$ in Fig. 1 agree with this
result.

$\nu_{0}=C\frac{\phi}{2\pi}+k_{0}$ and
$\nu_{s}=\mathscr{S}\frac{\phi}{2\pi}+k_{s}$ are the charge and angular
momentum per unit cell, with $k_{0}$, $k_{s}$ integers. The terms with
$\vec{T}$ in Eq. (Model and response theory.) can be detected by inserting
defects with nontrivial dislocation Burgers vector, but we do not consider
such defects in this work. Hereafter we ignore these, as well as the
$\omega\wedge d\omega$ term.

##### Fractional disclination charge.

Eq. (Model and response theory.) predicts a contribution to the charge
localized at a $\frac{\pi}{2}$ disclination coming from $\mathscr{S}$. Here we
compare the field theory prediction to microscopic calculations. The
discussion below applies to general lattices with p4 space group symmetry.

We construct a $\frac{\pi}{2}$ disclination at the point $o$ by deleting a
quadrant from the infinite plane and reconnecting sites using the operator
$\tilde{C}_{4,\lambda}$ (see Appendix C). In particular, if each unit cell in
the disclination lattice has the same flux, we show that $\lambda_{o}=0$. Now
consider a region $W$ enclosing the disclination, such that the distance
between the disclination and the boundary $\partial W$ is much greater than
the correlation length. The total charge is

$Q_{W}=\int_{W}\frac{\delta\mathcal{L}}{\delta
A_{0}}=C\frac{\delta\Phi_{W}}{2\pi}+\mathscr{S}\frac{\Omega_{W}}{2\pi}+\nu_{0}n_{u.c.,W},$
(2)

where $n_{u.c.,W}$, $\Omega_{W}$ and $\delta\Phi_{W}$, are the number of unit
cells in $W$, disclination angle, and excess magnetic flux (on top of the
background flux $n_{u.c.,W}\phi$) respectively. Here we use that
$\int_{W}dA=\delta\Phi_{W}+n_{u.c.,W}\phi$ and $\nu_{0}=C\phi/2\pi+k_{0}$. In
order to use Eq. (2), $W$ should enclose a definite integer number of unit
cells. Furthermore, in order to ensure that the condition
$Q_{W}+Q_{W^{\prime}}=Q_{W\cup W^{\prime}}$ holds microscopically, as in the
field theory, we define $Q_{W}\equiv\sum_{i\in W}\text{wt}(i)Q_{i}$ where the
weight $\text{wt}(i)=1$ for interior points and
$\text{wt}(i)=\frac{1}{4},\frac{2}{4},\frac{3}{4}$ if the interior of $W$
subtends an angle $\frac{\pi}{2},\pi,\frac{3\pi}{2}$ at site $i$ (see Fig. A.
Lattice disclination with disclination angle $\Omega=\frac{\pi}{2}$. The blue
region $W$ covers 11 unit cells. $Q_{i}$ is weighted by the indicated amount
when calculating $Q_{W}$ or $\bar{Q}_{W}$ . B. Standard deviation of
$\mathscr{S}$ as a function of bond disorder $\sigma_{\text{bond}}$ or onsite
disorder $\sigma_{\text{onsite}}$ for $C=1$ and $C=2$ main Landau level
(average hopping is 1).). Note our definition of disclination charge differs
from previous work. [29]

Figure 2: A. Lattice disclination with disclination angle
$\Omega=\frac{\pi}{2}$. The blue region $W$ covers 11 unit cells. $Q_{i}$ is
weighted by the indicated amount when calculating $Q_{W}$ or $\bar{Q}_{W}$ .
B. Standard deviation of $\mathscr{S}$ as a function of bond disorder
$\sigma_{\text{bond}}$ or onsite disorder $\sigma_{\text{onsite}}$ for $C=1$
and $C=2$ main Landau level (average hopping is 1).

The charge on a site $Q_{i}$ is simply
$\expectationvalue{c_{i}^{\dagger}c_{i}}$ in the ground state. We choose $W$
such that it encloses a single disclination with $\Omega_{W}=\frac{\pi}{2}$.
We set the excess flux $\delta\Phi_{W}=0$ in our numerics. $\nu_{0}$ can be
defined as the filling of a corresponding clean lattice on a torus with the
same flux per unit cell $\phi$. On a lattice with disclinations, $\nu_{0}$ is
also the charge per unit cell far away from the disclination. Suppose we
define the excess charge in $W$ as $\bar{Q}_{W}\equiv
Q_{W}-\nu_{0}n_{u.c.,W}$. We can then extract the shift $\mathscr{S}$ to be
$\frac{\mathscr{S}}{4}=\bar{Q}_{W}$. Numerically, we find that $\bar{Q}_{W}$,
and hence the computed $\mathscr{S}$ is indeed independent of the size of $W$
for large enough $W$. We show this by explicitly plotting $Q_{i}$ and the size
dependence of $\bar{Q}_{W}$ for three representative sets of parameters in the
Hofstadter butterfly (Fig. 3). We can thus use this procedure to calculate
$\mathscr{S}$ throughout the Hofstadter butterfly; this is shown in Fig. 1. In
the Landau level limit, we numerically recover the result
$\mathscr{S}=\frac{C^{2}}{2}$ [1].

Figure 3:  (A-C): $Q_{i}$ for each site $i$ in a clean system with parameters
A. $\frac{\phi}{2\pi}=\epsilon,C=1$; B.
$\frac{\phi}{2\pi}=\frac{1}{2}-\epsilon$, $C=-2$; C.
$\frac{\phi}{2\pi}=\frac{1}{3}+\epsilon,C=3$. The colorbar is in units of
$\nu_{0}$. $\epsilon$ is a small fraction which opens up the band gap. D.
$\bar{Q}_{W}$ for (A-C) is quantized at $\frac{\mathscr{S}}{4}$ when $R\geq
9$. $R$ is the distance from the disclination center to $\partial W$. (total
side length is $L=24$ unit cells). Here we have cropped out the edges to show
the bulk features more clearly; the full figures with edges are given Appendix
.E

It is instructive to apply Eq. (2) when $W$ is the entire surface of a cube,
which has 8 $\frac{\pi}{2}$ disclinations corresponding to the 8 corners. In
this case, we obtain

$Q_{\text{cube}}=2\mathscr{S}+\nu_{0}n_{u.c.,\text{cube}}.$ (3)

Thus, in order to be in the same gapped phase as a state on a torus with
filling $\nu_{0}$ and identical $\phi$ and Chern number $C$, the total charge
on the cube is shifted from the naïve expectation of
$\nu_{0}n_{u.c.,\text{cube}}$ by $2\mathscr{S}$. Numerically, this agrees with
the number of extra single particle states we need to fill.

Note that the weighting procedure and the numerical result
$\bar{Q}_{W}=\frac{\mathscr{S}}{4}$ generalize to any $C_{4}$ symmetric
lattice. The details are described in Appendix E.

Let us now introduce bond and on-site potential disorder, which break the
crystalline symmetry. In this case, the value of $\mathscr{S}$ extracted from
$\bar{Q}_{W}$ through $\mathscr{S}=4\bar{Q}_{W}$ deviates from its quantized
value for each disorder realization, although remains quantized after disorder
averaging. The standard deviation $\sigma_{\mathscr{S}}$ computed from
$\bar{Q}_{W}$ grows to order 1 with an increase in disorder strength, as shown
for two representative lobes in Fig. A. Lattice disclination with disclination
angle $\Omega=\frac{\pi}{2}$. The blue region $W$ covers 11 unit cells.
$Q_{i}$ is weighted by the indicated amount when calculating $Q_{W}$ or
$\bar{Q}_{W}$ . B. Standard deviation of $\mathscr{S}$ as a function of bond
disorder $\sigma_{\text{bond}}$ or onsite disorder $\sigma_{\text{onsite}}$
for $C=1$ and $C=2$ main Landau level (average hopping is 1).. The value of
$\mathscr{S}$ extracted from $Q_{\text{cube}}$ is much more robust (i.e. much
smaller standard deviation) as long as the chemical potential is far from the
band edge, since in this case $Q_{\text{cube}}$ can change only if a given
disorder configuration moves a single particle state across the chemical
potential.

##### Angular momentum due to flux

Since $\omega$ is a rotation gauge field, the angular momentum is the charge
under rotations, given by
$l=\int\frac{\delta\mathcal{L}}{\delta\omega_{0}}=\mathscr{S}(\frac{\delta\Phi}{2\pi}+\frac{\phi}{2\pi}n_{u.c.})+k_{s}n_{u.c.}\mod
4$. Below we compare this topological field theory (TFT) prediction to
microscopic calculations. Consider the Hofstadter Hamiltonian on an $L\times
L$ torus, with $L$ even and $\phi$ flux per plaquette. By turning on $\delta
A$, we add a flux $2\pi\Delta m$, also distributed uniformly. If the system
has Chern number $C$, we fill $\Delta mC$ additional single particle states in
order to get a gapped many body state with the same $C$. This state has
$m=\frac{\phi L^{2}}{2\pi}+\Delta m$ flux quanta.

We use the same magnetic rotation operator $\tilde{C}_{4,\lambda}$ that we
used to define the disclination. This means that we require $\lambda_{o}=0$,
where $o$ is a fixed point of the rotation.

Note that the holonomies $e^{-i\oint(\delta A+A_{bgd})\cdot dl}$ along the two
non-contractible cycles of the torus are position dependent. Translation by
one site changes the holonomy by a factor $e^{i2\pi\frac{m}{L}}$, and is
therefore an exact symmetry only for the infinite system; for any finite
system, translation by one site can only be an approximate symmetry. On a
finite size torus with even $L$, there are two points, $o_{1}$ and $o_{2}$,
distinguished by having holonomy $1$ or $e^{i\pi m}$ along both directions.
The vertex-centered $\pi/2$ rotational symmetry is only exact for a finite
size system when $o_{1}$ and $o_{2}$ are both fixed points of this rotation.

Since we have two distinct fixed points $o_{1}$ and $o_{2}$, there are two
distinct choices of gauge satisfying the above condition, either
$\lambda_{o_{1}}=0$ and $\lambda_{o_{2}}=m\frac{\pi}{2}$, or
$\lambda_{o_{2}}=0$ and $\lambda_{o_{1}}=-m\frac{\pi}{2}$. In what follows we
pick the first choice, denoted $\tilde{C}_{4}$; the second is related by an
overall $U(1)$ rotation $e^{-i\hat{N}m\frac{\pi}{2}}$, as will be discussed in
Appendix F.

The many-body ground state $|\Psi_{m}\rangle$ satisfies
$\tilde{C}_{4}\ket{\Psi_{m}}=e^{il(m)\frac{\pi}{2}}\ket{\Psi_{m}}$. Since
$\tilde{C}_{4}^{4}=1$, $l(m)$ is an integer mod 4.

In this setup, we require each rotation center to be a vertex. On a torus,
this forces $L$ to be even: if $L$ were odd, any rotation would leave two
points invariant, one at a vertex, and the other at a plaquette center. If the
rotation center was at a plaquette center, then the original $\tilde{C}_{4}$
rotation would be modified by a lattice translation. The associated eigenvalue
would receive a contribution from the $A\wedge\vec{T}$ term in Eq. (Model and
response theory.), which we do not wish to consider here. Indeed, the
numerical result with plaquette centred rotations is not consistent with Fig.
1.

We find from direct numerical calculation:

$l(m)=\mathscr{S}m+C\frac{m^{2}}{2}+K(C,L)\mod 4.$ (4)

The quadratic dependence on $m$ is beyond the TFT description; while it is
well-known from the continuum Landau level problem, it has not been derived
using effective field theory.444[28] accounted for the $m^{2}$ contribution in
terms of the angular momentum of the electromagnetic field, however physically
this term arises from the electron fluid itself. For a given lobe, $K(C,L)$ is
an integer which does not depend on $m$, but does depend on $C$ and $L$. We
can obtain $\mathscr{S}$ by subtracting the quadratic term and taking the
difference: $\mathscr{S}=l(m+1)-l(m)-Cm-\frac{C}{2}\mod 4.$ For each lobe with
a given $C$, the value of $\mathscr{S}$ obtained from Eq. (4) matches the
result using Eq. (3), confirming the expected duality.

Instead of inserting additional $\Delta m$ flux uniformly everywhere, we can
insert it locally in a smaller region $W$ symmetrically around the rotation
center. We find that the value of $\mathscr{S}$ remains constant for different
sizes of $W$, even in the limit when $W$ contains just 4 plaquettes (when $C$
is small enough).

We can also extract $\mathscr{S}$ using partial rotations [30, 36]. That is,
in a system with background flux $2\pi m_{0}$ we insert a local flux of
$2\pi\Delta m$ in a region $W$, and we compute $\langle\Psi_{\Delta
m,m_{0}}|\tilde{C}_{4}|_{D}|\Psi_{\Delta m,m_{0}}\rangle$, where
$\tilde{C}_{4}|_{D}$ is the restriction of $\tilde{C}_{4}$ to a region $D$
containing $W$ and $|\Psi_{\Delta m,m_{0}}\rangle$ is the ground state. Let us
fix $W$ and $D$ to be centered on $o_{2}$ and continue with our previous gauge
choice $\lambda_{o_{1}}=0$. The ground state expectation value can be written
as $\bra{\Psi_{\Delta m,m_{0}}}\tilde{C}_{4}|_{D}\ket{\Psi_{\Delta
m,m_{0}}}=e^{-\gamma_{D}(\Delta m,m_{0})+il_{D}(\Delta m,m_{0})\pi/2}$. The
magnitude $e^{-\gamma_{D}(\Delta m,m_{0})}$ has an exponentially decaying
envelope as the perimeter $\partial D$ increases, as expected, in addition to
an oscillatory behavior that we do not study in detail. We empirically find
$l_{D}(\Delta m,m_{0})=l_{D}(0,m_{0})+\mathscr{S}\Delta m+Cm\Delta
m+\frac{C(\Delta m)^{2}}{2}\mod 4$. This matches the expectation from Eq. (4).
One can also perform the partial rotation computation in the case where the
total system is defined with open boundary conditions, as discussed in
Appendix F.

On the torus we find that the formula for $l_{D}(\Delta
m,m_{0})-l_{D}(0,m_{0})$ is sensitive in complicated ways to the gauge choice
$\lambda_{o_{1}}=0$ or $\lambda_{o_{2}}=0$, and whether $D$ is centered on
$o_{1}$ or $o_{2}$; these dependencies are not fully understood. We discuss
this and related issues arising for open boundary conditions in Appendix F.

##### Theoretical analysis.

As a function of $\mu$ and $\phi$, $\mathscr{S}$ and $C$ have the following
general properties (note $\mathscr{S}$ is defined $\mod 4$):

  1. 1.

$\mathscr{S}\text{ mod }1=\frac{C}{2}\text{ mod 1}$.

  2. 2.

$\mathscr{S}(\mu,\phi)=\mathscr{S}(\mu,2\pi-\phi)$, i.e. $\mathscr{S}$ is
invariant under time-reversal.

  3. 3.

For the bands with the same Chern number $C$,
$\mathscr{S}(\mu,\phi)=1-\mathscr{S}(-\mu,\phi)$.

  4. 4.

When $\mathscr{S}$ changes, $\frac{\phi}{2\pi}$ must lies in the Farey
sequence of order $|C|$.

  5. 5.

$\mathscr{S}(\mu,0^{+})=\frac{C^{2}}{2}$ for $C>0$.

Properties (1-3) will be justified in Appendix H. We explain (1) also for
general interacting systems, using the classification of invertible
topological phases in Ref. [15]. (2) follows for general interacting systems
from the time-reversal invariance of the field theory term $\omega\wedge dA$,
while (5) reproduces the known results in the continuum Landau level limit
[1].

Now let us explain property (4). Consider all possible fractions $\frac{p}{q}$
with $0\leq\frac{p}{q}\leq 1$, $1\leq q\leq 2|C|$ and $p,q$ coprime. Arrange
them in increasing order, with 0 being the first element and 1 being the last.
The resulting sequence is called the Farey sequence of order $2|C|$. Now, the
different lobes with Chern number $C$ are uniquely specified by the intervals
of $\frac{\phi}{2\pi}$ obtained from this sequence [35]. Moreover, each lobe
has a constant value of shift. In our numerics, when $\mathscr{S}$ jumps,
$\phi/2\pi$ must lie at fractions $\frac{p}{q}$ in a smaller set, namely the
Farey sequence of order $|C|$: see Appendix B.

##### Empirical formula for $\mathscr{S}$.

Suppose we fix a $C>0$ and consider $\mathscr{S(\phi)}$ as
$\frac{\phi}{2\pi}=\frac{p}{q}$ is increased from 0 to 1. As stated above,
$\mathscr{S}(\phi)$ jumps by integers at specific values of $\frac{p}{q}$,
where $q\leq|C|$. At a given $\frac{p}{q}$, the total jump, defined as
$\lim\limits_{\epsilon\rightarrow
0^{+}}\mathscr{S}(2\pi\frac{p}{q}+\epsilon)-\mathscr{S}(2\pi\frac{p}{q}-\epsilon)$,
is the sum of two contributions: A contribution of $-C-1$ whenever $q$ divides
$C$, and another contribution of $2\lfloor\frac{C+q}{2q}\rfloor$ whenever $q$
is odd. Both contributions are automatically 0 if $q>|C|$. The observed jumps
are tabulated in Appendix B up to $C=12$. From these observations, we propose
the following empirical formula by summing over all jumps that occur at
$2\pi\frac{p}{q}<\phi$. For $C>0$,

$\mathscr{S}(\phi)=\frac{C^{2}}{2}-(C+1)\left\lfloor\frac{C\phi}{2\pi}\right\rfloor+2\sum_{\begin{subarray}{c}\frac{p}{q}<\frac{\phi}{2\pi}\\\
\text{odd }q\end{subarray}}\left\lfloor\frac{C+q}{2q}\right\rfloor\mod 4,$ (5)

where in the third term we sum over all $\frac{p}{q}$ in the Farey sequence of
order $C$ that satisfy $\frac{p}{q}<\frac{\phi}{2\pi}$ and $q$ odd.
$\mathscr{S}$ for $C<0$ can be obtained from the symmetry transformation
$\mathscr{S}(\mu,\phi)=1-\mathscr{S}(-\mu,\phi)$ which flips the sign of $C$.
We numerically checked Eq. (5) for all $|C|\leq 7$ lobes using Eq. (3) on a
cube of side length $L=28$. We also checked all $|C|\leq 12$ lobes using Eq.
(4) on a torus with side length $L=140$, and we checked some representative
$|C|=45$ lobes with $L=180$. We use (5), together with an eigenvalue database
[35], to generate Fig. 1.

Lobes with the same $(C,k_{0})$ can have distinct values of $\mathscr{S}$. For
example when $(C,k_{0})=(4,-1)$ we can take
$\frac{\phi}{2\pi}=1/3\mp\epsilon$, and find $\mathscr{S}=3,1\mod 4$
respectively. Also note that in Fig. 1, we do not see any bands with total
shift $\mathscr{S}=3/2$ or 7/2. However, there is no theoretical obstruction
to realizing this in a system with odd $C$. Indeed, there are several examples
of single excited bands that have odd $C$ and $\mathscr{S}=3/2$ or $7/2$ in
this model.

##### Acknowledgements.

We thank M. Hafezi and S. Das Sarma for comments on the draft, and V. Galitski
and D. Bulmash for discussions on related projects. This work is supported by
the Laboratory for Physical Sciences through the Condensed Matter Theory
Center, NSF CAREER (DMR- 1753240) (MB, NM), ARO W911NF-20-1-0232 (GN).

## References

  * Wen and Zee [1992] X. G. Wen and A. Zee, Shift and spin vector: New topological quantum numbers for the hall fluids, Phys. Rev. Lett. 69, 953 (1992).
  * Avron  _et al._ [1995] J. E. Avron, R. Seiler, and P. G. Zograf, Viscosity of quantum hall fluids, Phys. Rev. Lett. 75, 697 (1995).
  * Read [2009] N. Read, Non-abelian adiabatic statistics and hall viscosity in quantum hall states and ${p}_{x}+i{p}_{y}$ paired superfluids, Phys. Rev. B 79, 045308 (2009).
  * Read and Rezayi [2011] N. Read and E. H. Rezayi, Hall viscosity, orbital spin, and geometry: Paired superfluids and quantum hall systems, Phys. Rev. B 84, 085316 (2011).
  * Haldane [2009] F. D. M. Haldane, ”hall viscosity” and intrinsic metric of incompressible fractional hall fluids (2009), arXiv:0906.1854 [cond-mat.str-el] .
  * Haldane [2011] F. D. M. Haldane, Geometrical description of the fractional quantum hall effect, Phys. Rev. Lett. 107, 116801 (2011).
  * Abanov and Gromov [2014] A. G. Abanov and A. Gromov, Electromagnetic and gravitational responses of two-dimensional noninteracting electrons in a background magnetic field, Phys. Rev. B 90, 014435 (2014).
  * Bradlyn and Read [2015] B. Bradlyn and N. Read, Low-energy effective theory in the bulk for transport in a topological phase, Phys. Rev. B 91, 125303 (2015).
  * Gromov  _et al._ [2015] A. Gromov, G. Y. Cho, Y. You, A. G. Abanov, and E. Fradkin, Framing anomaly in the effective theory of the fractional quantum hall effect, Phys. Rev. Lett. 114, 016805 (2015).
  * Schine  _et al._ [2016] N. Schine, A. Ryou, A. Gromov, A. Sommer, and J. Simon, Synthetic landau levels for photons, Nature 534, 671 (2016).
  * Wu  _et al._ [2017] Y.-H. Wu, H.-H. Tu, and G. J. Sreejith, Fractional quantum hall states of bosons on cones, Phys. Rev. A 96, 033622 (2017).
  * Manjunath and Barkeshli [2021] N. Manjunath and M. Barkeshli, Crystalline gauge fields and quantized discrete geometric response for abelian topological phases with lattice symmetry, Phys. Rev. Research 3, 013040 (2021).
  * Manjunath and Barkeshli [2020] N. Manjunath and M. Barkeshli, Classification of fractional quantum hall states with spatial symmetries, arXiv preprint arXiv:2012.11603 (2020).
  * Hofstadter [1976] D. R. Hofstadter, Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976).
  * Barkeshli  _et al._ [2021] M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and N. Manjunath, Classification of (2+1)d invertible fermionic topological phases with symmetry (2021), arXiv:2109.11039 [cond-mat.str-el] .
  * Freed and Hopkins [2016] D. S. Freed and M. J. Hopkins, arXiv:1604.06527 (2016).
  * Aasen  _et al._ [2021] D. Aasen, P. Bonderson, and C. Knapp, Characterization and classification of fermionic symmetry enriched topological phases, (2021), arXiv:2109.10911 [cond-mat.str-el] .
  * Dean  _et al._ [2013] C. R. Dean _et al._ , Hofstadter’s butterfly and the fractal quantum hall effect in moiré superlattices, Nature 497, 598 (2013).
  * Hunt  _et al._ [2013] B. Hunt _et al._ , Massive dirac fermions and hofstadter butterfly in a van der waals heterostructure, Science 340, 1427 (2013).
  * Saito  _et al._ [2021] Y. Saito _et al._ , Hofstadter subband ferromagnetism and symmetry-broken chern insulators in twisted bilayer graphene, Nature Physics 17, 478 (2021).
  * Spanton  _et al._ [2018] E. M. Spanton _et al._ , Observation of fractional chern insulators in a van der waals heterostructure, Science 360, 62 (2018).
  * Aidelsburger  _et al._ [2013] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Realization of the hofstadter hamiltonian with ultracold atoms in optical lattices, Phys. Rev. Lett. 111, 185301 (2013).
  * Miyake  _et al._ [2013] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Realizing the harper hamiltonian with laser-assisted tunneling in optical lattices, Phys. Rev. Lett. 111, 185302 (2013).
  * Kennedy  _et al._ [2015] C. J. Kennedy, W. C. Burton, W. C. Chung, and W. Ketterle, Observation of bose–einstein condensation in a strong synthetic magnetic field, Nature Physics 11, 859 (2015).
  * Hafezi  _et al._ [2013] M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Imaging topological edge states in silicon photonics, Nature Photonics 7, 1001 (2013).
  * Ozawa  _et al._ [2019] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91, 015006 (2019).
  * Biswas and Son [2016] R. R. Biswas and D. T. Son, Fractional charge and inter-landau–level states at points of singular curvature, Proceedings of the National Academy of Sciences 113, 8636 (2016).
  * Liu  _et al._ [2019] S. Liu, A. Vishwanath, and E. Khalaf, Shift insulators: Rotation-protected two-dimensional topological crystalline insulators, Phys. Rev. X 9, 031003 (2019).
  * Li  _et al._ [2020] T. Li, P. Zhu, W. A. Benalcazar, and T. L. Hughes, Fractional disclination charge in two-dimensional ${C}_{n}$-symmetric topological crystalline insulators, Phys. Rev. B 101, 115115 (2020).
  * You  _et al._ [2020] Y. You, J. Bibo, and F. Pollmann, Higher-order entanglement and many-body invariants for higher-order topological phases, Phys. Rev. Research 2, 033192 (2020).
  * May-Mann and Hughes [2021] J. May-Mann and T. L. Hughes, Crystalline responses for rotation-invariant higher-order topological insulators (2021), arXiv:2108.00008 .
  * Peterson  _et al._ [2021] C. W. Peterson, T. Li, W. Jiang, T. L. Hughes, and G. Bahl, Trapped fractional charges at bulk defects in topological insulators, Nature 589, 376 (2021).
  * Han  _et al._ [2019] B. Han, H. Wang, and P. Ye, Generalized wen-zee terms, Phys. Rev. B 99, 205120 (2019).
  * Thouless  _et al._ [1982] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
  * Osadchy and Avron [2001] D. Osadchy and J. E. Avron, Hofstadter butterfly as quantum phase diagram, Journal of Mathematical Physics 42, 5665 (2001), https://doi.org/10.1063/1.1412464 .
  * Shiozaki  _et al._ [2017] K. Shiozaki, H. Shapourian, and S. Ryu, Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries, Phys. Rev. B 95, 205139 (2017).
  * Barkeshli  _et al._ [2019] M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B 100, 115147 (2019).
  * Kapustin [2014] A. Kapustin, Bosonic topological insulators and paramagnets: a view from cobordisms, (2014), arXiv:1404.6659 .
  * Witten [1989] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121, 351 (1989).
  * Thorngren and Else [2018] R. Thorngren and D. V. Else, Gauging spatial symmetries and the classification of topological crystalline phases, Phys. Rev. X 8, 011040 (2018).
  * Debray [2021] A. Debray, Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle (2021), arXiv:2102.02941 [math-ph] .
  * Wang and Gu [2020] Q.-R. Wang and Z.-C. Gu, Construction and classification of symmetry-protected topological phases in interacting fermion systems, Phys. Rev. X 10, 031055 (2020).
  * Else and Thorngren [2019] D. V. Else and R. Thorngren, Crystalline topological phases as defect networks, Phys. Rev. B 99, 115116 (2019).
  * Zhang  _et al._ [2020] J.-H. Zhang, S. Yang, Y. Qi, and Z.-C. Gu, Real-space construction of crystalline topological superconductors and insulators in 2d interacting fermionic systems, (2020), arXiv:2012.15657 [cond-mat.str-el] .

##  Appendix A Relation to prior work

Here we summarize how our results relate to those in several prior works where
the physical manifestation of the Wen-Zee shift is studied in a lattice
setting.

Our contributions include:

  1. 1.

A method to unambiguously extract the discrete shift by computing the charge
$Q_{W}=\sum_{i\in W}\text{wt}(i)Q_{i}$ in a region $W$ containing a
disclination, or through the angular momentum $l(m)$. Crucially, one must use
the same rotation operator to define the disclination Hamiltonian and to
compute the angular momentum. An important ingredient in our formula which was
not introduced previously is the fractional weighting
$\text{wt}(i)=1/4,1/2,3/4$ for sites $i$ on the boundary $\partial W$. This
fractional weighting is crucial to obtaining a simple formula that matches the
topological field theory result.

  2. 2.

We have shown that the shift $\mathscr{S}$ is a half-integer defined modulo
$M$ in the case of $M$-fold rotational symmetry, and that $\mathscr{S}$ is
half-integer or integer depending on whether $C$ is odd or even, and we have
derived these statements even in the case of interacting fermions.

  3. 3.

We have also provided an empirical formula for $\mathscr{S}$ everywhere in the
Hofstadter butterfly for the square lattice Hofstadter model.

Ref. [28] discusses disclination charge in a model on the honeycomb lattice
with $\mathbb{Z}_{6}$ rotational symmetry and particle-hole symmetry in the
absence of a background magnetic field. They exploited the particle-hole
symmetry to derive an analytical formula for disclination charge, and
therefore the shift. We studied the disclination charge on the square lattice
numerically in the presence of a background magnetic field (allowing for a
tunable Chern number). In order to do so, we arrived at the prescription
$Q_{W}=\sum_{i}\text{wt}(i)Q_{i}=\mathscr{S}/4+\nu_{0}n_{u.c.,W}$ that allows
us to isolate the shift, given a very general charge distribution in a state.

Ref. [28] also studied the angular momentum response using global plaquette
centered $2\pi/6$ rotations. We note that in our case, i.e., for 4-fold
rotational symmetries, there exist both vertex- and plaquette-centred
rotations, which are related to each other by a lattice translation; the
latter may also introduce contributions from the field theory term
$\vec{\mathscr{P}}_{c}\cdot A\wedge\vec{T}$. However, the model considered in
Ref. [28] has only plaquette rotation centers, so this distinction does not
arise in their case. The extra field theory term $\vec{\mathscr{P}}_{c}\cdot
A\wedge\vec{T}$ is also expected to be trivial for systems with $6$-fold
rotational symmetry [12, 13]. We also note that gauge transformations required
to accompany the rotation operator are nontrivial on the square lattice,
because we cannot use a symmetric gauge (w.r.t. a vertex rotation center) to
insert $2\pi$ flux. But this is not an issue in Ref. [28], as a symmetric
gauge can be used for $C_{6}$ symmetric lattices, in which case the rotation
operator does not require any additional gauge transformations.

Ref. [27] also performs a disclination charge calculation for Landau levels in
detail, especially studying the contributions from inter-LL states. The
analytical derivations of disclination charge are done in the continuum limit,
and numerically verified for lattice systems. The calculations are all
performed on closed surfaces with corners, so that the charge of a single
disclination is deduced by symmetry, analogous to the $Q_{\text{cube}}$
results discussed in this paper. This work is effectively restricted to parts
of the Hofstadter butterfly that reduce to Landau levels in the continuum
limit.

Ref. [29] derives a formula for disclination charge when Chern number $C=0$,
in terms of the charge distribution at high-symmetry points of the unit cell.
Their definition of disclination charge is distinct from ours. In the $C_{4}$
symmetric case, it is equivalent to choosing a region $W$ in the dual lattice,
and evaluating $\bar{Q}_{W}$ for this $W$ as we have outlined. If we denote
their disclination charge result as $Q_{\text{disc}}^{\text{TZBH}}$, we find
that $\frac{\mathscr{S}}{4}-Q_{\text{disc}}^{\text{TZBH}}=\frac{\nu_{0}}{4}$
for $C_{4}$ symmetric lattices with zero Chern number. We note that our
definition of disclination charge, $\bar{Q}_{W}$ with $W$ defined on the
physical lattice, is directly proportional to $\mathscr{S}$, which also agrees
with the coefficient of the linear contribution to the angular momentum
associated to vertex-centred rotations. In this sense $\bar{Q}_{W}$ as defined
in the main text is more directly related to the Wen-Zee coefficient and the
prediction from the topological field theory.

In the $C\neq 0$ case, Ref. [29] writes down the Wen-Zee term with
$\mathscr{S}=C$ for specific models. It also proposes a formula for
disclination charge in the $C\neq 0$ case, in terms of band invariants at
high-symmetry points of the Brillouin zone. The values of disclination charge
from this definition are different from ours, although we have not studied
this relationship in detail when $C\neq 0$. This paper does not make an
explicit connection between this formula and the Wen-Zee coefficient
$\mathscr{S}$. In our work we have shown that $\mathscr{S}$, which is
proportional to $\bar{Q}_{W}$, depends sensitively on the Chern number through
the relation $\mathscr{S}=\frac{C}{2}\mod 1$.

In related work, Ref. [31] discussed a spinless fermion system on the square
lattice with $C=0$, and calculated the fractional charge bound to a
disclination at half filling. Finally, Refs. [36, 30] study the angular
momentum response using partial rotations in bosonic and fermionic models
respectively. The specific models studied numerically have Chern number $C=0$.

##  Appendix B Jumps of $\mathscr{S}(\phi)$

In this section we tabulate the data used to empirically obtain Eq.(5) in the
main text. Starting from the zero-flux value
$\mathscr{S}(\phi=0^{+})=C^{2}/2$, Fig. 4 shows how $\mathscr{S}(\phi)$ jumps
as $\phi$ is increased keeping $C$ fixed. The Hofstadter lobes corresponding
to Chern number $C$ are separated by the Farey sequence of order $2|C|$. But
from the figure, we see that $\mathscr{S}(\phi)$ only jumps at fractions
$\frac{p}{q}$ lying in the Farey sequence of order $|C|$, which is a smaller
set. We have confirmed this up to $C=12$, see Fig. 4. For example, at $C=4$,
the Farey sequence of order 8 is
$0,\frac{1}{8},\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{1}{4},\frac{2}{7},\frac{1}{3},\frac{3}{8},\frac{2}{5},\frac{3}{7},\frac{1}{2},\dots$,
with the rest of the sequence obtained by reflecting about $\frac{1}{2}$.
However, $\mathscr{S}$ only jumps at
$\frac{\phi}{2\pi}=\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4}$,
all of which lie in the Farey sequence of order 4.

Figure 4:  Jumps in $\mathscr{S}(\phi)$ for fixed $C$, as $\frac{\phi}{2\pi}$
increases from 0 to $\frac{1}{2}$.

##  Appendix C Definition of disclination Hamiltonian using
$\tilde{C}_{4,\lambda}$

In this section we first define $\tilde{C}_{4,\lambda}$ on the infinite plane;
$\lambda$ is fixed up to a constant $\lambda_{o}$ by the background vector
potential $A$, where $o$ is the rotation center. We then construct a
disclination Hamiltonian $H_{\text{disc}}$ by removing a quadrant and
connecting the open edges by defining new hopping terms. This is essentially
an adaptation of the prescription to construct symmetry defects for on-site
symmetries given in Ref. [37]. The values of the new hopping terms are
determined by $\tilde{C}_{4,\lambda}$. We show that requiring the disclination
to not insert any additional $U(1)$ flux, fixes $\lambda_{o}=0$. For a fixed
$A$ on the infinite plane, this completely fixes $\lambda$.

###  C.1 Definition of $\tilde{C}_{4,\lambda}$ and $H$ on an infinite plane

We define the rotation operator on the infinite plane as

$\tilde{C}_{4,\lambda}=\hat{C}_{4}e^{i\sum_{j}\lambda_{j}c_{j}^{\dagger}c_{j}}$
(6)

where $\hat{C}_{4}$ implements a counterclockwise $\pi/2$ spatial rotation
about some origin $o$. $\lambda$ depends on our definition of $A$, see below.
In what follows we will simplify our notation by using $A_{ij}$ as
$A_{ij}=\delta A_{ij}+A_{bgd,ij}$. We also assume that the rotation center at
$o\equiv(0,0)$ coincides with the disclination center.

The Hamiltonian $H$ on the infinite plane is given by

$H=-t\sum_{<ij>}e^{-iA_{ij}}c_{i}^{\dagger}c_{j}+h.c.$ (7)

with $i,j\in\mathbb{Z}^{2}$. We assume a general vector potential $A_{ij}$ on
the infinite plane that adds constant flux $\frac{\phi}{2\pi}$ per plaquette.
$\lambda$ satisfies the equation

$A_{\hat{C}_{4}i,\hat{C}_{4}j}=A_{ij}+\lambda_{j}-\lambda_{i}$ (8)

due to the condition that $\tilde{C}_{4,\lambda}$ commutes with $H$.
Therefore, given a gauge choice $A_{ij}$, $\lambda_{j}$ is completely fixed up
to a global constant as follows. Consider a connected path $(j_{0}\rightarrow
j_{N})\equiv j_{0},j_{1},j_{2},\dots,j_{N}$, where $j_{0}\equiv o$ and
$j_{N}\equiv f$. Let’s define $\int_{j_{0}\rightarrow
j_{N}}A\equiv\sum_{k=1}^{N}A_{j_{k-1}j_{k}}$. Then $\lambda_{f}$ is determined
by

$\lambda_{f}=\lambda_{o}+\int_{\hat{C}_{4}j_{0}\rightarrow\hat{C}_{4}j_{N}}A-\int_{j_{0}\rightarrow
j_{N}}A.$ (9)

###  C.2 Definition of $H_{\text{disc}}$ through cut and glue procedure

Figure 5:  Cut and glue procedure of constructing a disclination. A. Original
lattice on an open plane; B. Cutting; C. Gluing, with two different ways
(green and orange path) of constructing the red bonds. This creates new
plaquettes $\zeta_{i}$; D. Reorganizing.

Now we discuss how the definition of $\tilde{C}_{4,\lambda}$ is used in
constructing a $\pi/2$ disinclination. We first cut out a $\pi/2$ quadrant of
the infinite plane, centred at point $o$. In general, the quadrant removed
need not have edges parallel to the $x$ and $y$ directions. For ease of
visualization, we show an example in Fig. 5 where we cut out the bottom right
quadrant, by deleting all points $(x,y)$ with $x>0,y\leq 0$. In this case, the
Hilbert space of the disclination Hamiltonian $H_{\text{disc}}$ only includes
the local Hilbert spaces at sites
$(x,y)\in\Gamma=\mathbb{Z}^{2}\backslash\\{(x,y)|x>0,y\leq 0\\}$.

The cutting procedure results in pairs of severed bonds related by
$\hat{C}_{4}$ which should be glued together. Consider one such pair of bonds
in $H$ (see Fig. Cut and glue procedure of constructing a disclination. A.
Original lattice on an open plane; B. Cutting; C. Gluing, with two different
ways (green and orange path) of constructing the red bonds. This creates new
plaquettes $\zeta_{i}$; D. Reorganizing.) involving points $J$ and
$K\equiv\hat{C}_{4}J$: these are
$c_{J^{\prime}}^{\dagger}c_{J}e^{-iA_{J^{\prime}J}}$ (green dotted line) and
$c_{K}^{\dagger}c_{K^{\prime}}e^{-iA_{KK^{\prime}}}$ (orange dotted line).
They get severed because points $J$ and $K^{\prime}$ are removed from
$H_{\text{disc}}$. Thus, $J^{\prime}$ and $K$ should now be joined in
$H_{\text{disc}}$ with a hopping coefficient determined by
$\tilde{C}_{4,\lambda}$ as follows.

We start with the term $c_{J^{\prime}}^{\dagger}c_{J}e^{-iA_{J^{\prime}J}}$ in
$H$, and conjugate only the operator $c_{J}$ with $\tilde{C}_{4,\lambda}$
(green arrow in Fig. Cut and glue procedure of constructing a disclination. A.
Original lattice on an open plane; B. Cutting; C. Gluing, with two different
ways (green and orange path) of constructing the red bonds. This creates new
plaquettes $\zeta_{i}$; D. Reorganizing.), i.e.

$\displaystyle c_{J^{\prime}}^{\dagger}c_{J}e^{-iA_{J^{\prime}J}}$
$\displaystyle\rightarrow
c_{J^{\prime}}^{\dagger}\pqty{\tilde{C}_{4,\lambda}c_{J}\tilde{C}_{4,\lambda}^{\dagger}}e^{-iA_{J^{\prime}J}}$
(10)
$\displaystyle=e^{-i(A_{J^{\prime}J}+\lambda_{J})}c_{J^{\prime}}^{\dagger}c_{K}$

where we have used
$\tilde{C}_{4,\lambda}c_{J}\tilde{C}^{\dagger}_{4,\lambda}=e^{-i\lambda_{J}}c_{J}$.
Therefore, the gauge potential for the newly formed bond $J^{\prime}K$ is
given by $\tilde{A}_{J^{\prime}K}=A_{J^{\prime}J}+\lambda_{J}$.

There is an alternative construction in which we start with the term
$c_{K}^{\dagger}c_{K^{\prime}}e^{-iA_{KK^{\prime}}}$ in $H$ and conjugate only
$c_{K^{\prime}}$ with $\tilde{C}_{4,\lambda}^{\dagger}$ (orange arrow in Fig.
Cut and glue procedure of constructing a disclination. A. Original lattice on
an open plane; B. Cutting; C. Gluing, with two different ways (green and
orange path) of constructing the red bonds. This creates new plaquettes
$\zeta_{i}$; D. Reorganizing.). This would give

$\displaystyle c_{K^{\prime}}^{\dagger}c_{K}e^{-iA_{K^{\prime}K}}\mapsto$
$\displaystyle\left(\tilde{C}_{4,\lambda}^{\dagger}c_{K^{\prime}}^{\dagger}\tilde{C}_{4,\lambda}\right)c_{K}e^{-iA_{K^{\prime}K}}$
(11)
$\displaystyle=e^{-i(A_{K^{\prime}K}+\lambda_{J^{\prime}})}c_{J^{\prime}}^{\dagger}c_{K}$

This procedure gives
$\tilde{A}_{J^{\prime}K}=A_{KK^{\prime}}+\lambda_{J^{\prime}}$ which is the
same as obtained previously, because of Eq. (8).

Figure 6:  A $\hat{C}_{4}$ symmetric region $D$ with boundary points $J,K,L,M$
that are arranged $\hat{C}_{4}$-symmetrically about $o$. To construct a
disclination, all points in the quadrant shaded red are deleted, and the
gluing procedure in Fig. 5 is applied.

###  C.3 Derivation of $\lambda_{o}=0$ on infinite plane

As we saw above, $\lambda_{o}$ determines the global $U(1)$ transformation
that accompanies spatial rotation. We should keep in mind that conjugating
only one side of a cut with a global $U(1)$ transformation inserts a $U(1)$
flux through the origin $o$ of the cut. This is the intuition behind why the
choice of $\lambda_{o}$ determines the excess flux $\delta\Phi$ around $o$.
Consider a region $D_{\text{disc}}$ that contains a disclination at $o$. It is
obtained from a $\hat{C}_{4}$-symmetric region $D$ on the infinite plane (see
Fig. 6) by deleting the points in the red quadrant and applying the gluing
procedure described above. Then the extra flux at $o$ is

$\delta\Phi=\Phi_{D_{\text{disc}}}-\frac{3}{4}\Phi_{D}$ (12)

where $\Phi_{D}$ and $\Phi_{D_{\text{disk}}}$ are the total flux through $D$
and $D_{\text{disc}}$ respectively. We will now show that
$\lambda_{o}=\delta\Phi$.

Consider 4 points $J,K,L,M$ on the boundary of $D$, such that
$J\xrightarrow{\hat{C}_{4}}K\xrightarrow{\hat{C}_{4}}L\xrightarrow{\hat{C}_{4}}M$.
We break up the boundary of $D$ into four $\hat{C}_{4}$ symmetric segments
$K\rightarrow L$, $L\rightarrow M$, $M\rightarrow N$, $N\rightarrow J$. Then,

$\displaystyle\Phi_{D}$ $\displaystyle=\int_{K\rightarrow
L}A+\int_{L\rightarrow M}A+\int_{M\rightarrow J}A+\int_{J\rightarrow K}A$ (13)
$\displaystyle\Phi_{D_{\text{disc}}}$ $\displaystyle=\int_{K\rightarrow
L}A+\int_{L\rightarrow M}A+\int_{M\rightarrow J}A+\lambda_{J}$

where we have used Eq. (10) in the second equation. We now use Eq. (8)
repeatedly to relate each segment to the first segment $K\rightarrow L$:

$\displaystyle\int_{L\rightarrow M}A$ $\displaystyle=(\int_{K\rightarrow
L}A)+\lambda_{L}-\lambda_{K}$ (14) $\displaystyle\int_{M\rightarrow J}A$
$\displaystyle=(\int_{K\rightarrow L}A)+\lambda_{M}-\lambda_{K}$
$\displaystyle\int_{J\rightarrow K}A$ $\displaystyle=(\int_{K\rightarrow
L}A)+\lambda_{J}-\lambda_{K}$

Then, we get $\Phi_{D}=4(\int_{K\rightarrow
L}A)+\lambda_{L}+\lambda_{M}+\lambda_{J}-3\lambda_{K}$ and
$\Phi_{D_{\text{disc}}}=3(\int_{K\rightarrow
L}A)+\lambda_{L}+\lambda_{M}+\lambda_{J}-2\lambda_{K}$. Thus, the extra flux
$\delta\Phi$ is (using Eq. 12)

$\delta\Phi=(\lambda_{J}+\lambda_{K}+\lambda_{L}+\lambda_{M})/4.$ (15)

Now we relate $\lambda_{J},\lambda_{K},\lambda_{L},\lambda_{M}$ to
$\lambda_{o}$ using Eq. (9). For this, we use an arbitrary path $o\rightarrow
M$ (see Fig. 6). Then the other three paths are related via $\hat{C}_{4}$ as:
$o\rightarrow J\equiv\hat{C}_{4}(o\rightarrow M)$, $o\rightarrow
K\equiv\hat{C}_{4}(o\rightarrow J)$ and $o\rightarrow
L\equiv\hat{C}_{4}(o\rightarrow K)$. So,

$\displaystyle\lambda_{J}$ $\displaystyle=\lambda_{o}+\int_{o\rightarrow
K}A-\int_{o\rightarrow J}A$ (16) $\displaystyle\lambda_{K}$
$\displaystyle=\lambda_{o}+\int_{o\rightarrow L}A-\int_{o\rightarrow K}$
$\displaystyle\lambda_{L}$ $\displaystyle=\lambda_{o}+\int_{o\rightarrow
M}A-\int_{o\rightarrow L}A$ $\displaystyle\lambda_{M}$
$\displaystyle=\lambda_{o}+\int_{o\rightarrow N}A-\int_{o\rightarrow M}A$
$\displaystyle\implies$
$\displaystyle(\lambda_{J}+\lambda_{K}+\lambda_{L}+\lambda_{M})/4=\lambda_{o}.$

Therefore

$\delta\Phi=\lambda_{o}.$ (17)

Demanding $\delta\Phi=0$ fixes $\lambda_{o}=0$, as desired.

##  Appendix D Dependence of $\mathscr{S}$ on definition of rotation operator

For a fixed $\tilde{C}_{4,\lambda}$, we expect from field theory that we
should obtain the same result for $\mathscr{S}$ from either the disclination
charge or from the angular momentum. We confirm this in two steps. First we
fix $\lambda_{o}=0$ on a system with open boundary conditions. We can
numerically compute $\mathscr{S}$ using both methods, and the results agree.
The second step is to check that $\mathscr{S}$ changes in the same way using
either method if we redefine $\lambda$. This is shown below. We redefine
$\tilde{C}_{4}$ by a transformation
$\lambda_{j}\rightarrow\lambda_{j}+\frac{\pi}{2}\beta$. $\beta$ must be an
integer, in order for $\tilde{C}_{4}$ to have order $4$. We show that it takes
$\mathscr{S}\rightarrow\mathscr{S}+C\beta$ in both the disclination charge
calculation and clean lattice angular momentum calculation, confirming the
duality expected from the field theory.

###  D.1 Change in $\mathscr{S}$ from disclination charge calculation

Applying Eq. (17) to a small region $D$, we see that if $\lambda_{o}\neq 0$,
then there is an extra local flux $\delta\Phi=\lambda_{o}$ in $D$. (In fact
this flux is localized to the plaquette marked $\zeta_{1}$ in Fig. Cut and
glue procedure of constructing a disclination. A. Original lattice on an open
plane; B. Cutting; C. Gluing, with two different ways (green and orange path)
of constructing the red bonds. This creates new plaquettes $\zeta_{i}$; D.
Reorganizing.).

Since the system has Chern number $C$, any region $W$ containing this
plaquette will have an excess charge
$\delta\bar{Q}_{W}=\frac{\pi\beta}{2}\frac{C}{2\pi}$ that must be attributed
to the shift. This means that
$\bar{Q}_{W}=\frac{\mathscr{S}}{4}\rightarrow\frac{\mathscr{S}+C\beta}{4}$.
This implies our claim.

###  D.2 Change in $\mathscr{S}$ from angular momentum calculation

Now we show that $\mathscr{S}\rightarrow\mathscr{S}+C\beta$ under
$\lambda_{j}\rightarrow\lambda_{j}+\frac{\pi}{2}\beta$ in the angular momentum
calculation as well. The many-body ground state with Chern number $C$ and flux
$\phi+\Delta m/L^{2}$ per unit cell has $\nu_{0}L^{2}+C\Delta m$ filled
single-particle states, with $m=m_{0}+\Delta m$ total flux quanta. Redefining
the rotation operator by $\beta$ would then shift $l$ to

$l(m)\rightarrow l(m)+(\nu_{0}L^{2}+C\Delta m)\beta\mod 4.$ (18)

Recall that

$-k_{0}\equiv\frac{C\phi}{2\pi}-\nu_{0}=\frac{Cm_{0}}{L^{2}}-\nu_{0}.$ (19)

Note that $k_{0}$ is always an integer. Using $m_{0}=\frac{\phi L^{2}}{2\pi}$,
we find after a short calculation that

$l\rightarrow l+Cm\beta-k_{0}L^{2}\beta\mod 4.$ (20)

Since on a torus $L$ is an even integer, the last term vanishes. $\mathscr{S}$
is the coefficient of the linear term in $l(m)$, therefore if $\beta$ is a
constant, we would have $\mathscr{S}\rightarrow\mathscr{S}+C\beta$, as before.

###  D.3 Field theory analysis

The same transformation appears in the effective response theory. Let
$\mathcal{L}=\frac{C}{4\pi}AdA+\frac{\mathscr{S}}{2\pi}Ad\omega$. Take
$A\rightarrow A+\beta\omega$. The point of this transformation is that it
sends $dA\rightarrow dA+\beta d\omega$. Thus the magnetic flux at the
disclination is shifted by an amount proportional to the disclination angle.
This interpretation agrees with the microscopic analysis of disclination
charge in the preceding paragraphs. We can directly check that this
transformation shifts $\mathscr{S}\rightarrow\mathscr{S}+C\beta$ but keeps
$C$, the $AdA$ coefficient, fixed.

##  Appendix E Formula for disclination charge in a general unit cell
configuration

Figure 7:  Raw data for $\mathscr{S}$ from Eq. (3). The cube has $16$ unit
cells per side. The noise in regions with higher Chern number is reduced by
taking a larger system size.  Figure 8:  General square lattice unit cell with
$A$ sites at the corners, $B$ sites inside the unit cell, and $C$ sites on the
edges.

Here we assume that the disclination Hamiltonian is already given, and that
every unit cell has the same flux. This fixes the rotation operator
$\tilde{C}_{4,\lambda}$, as discussed in Appendix C.

###  E.1 Testing $\bar{Q}_{W}$ for general unit cells

In the main text, we defined $Q_{W}=\sum_{i\in W}\text{wt}(i)Q_{i}$ through
Eq. (2). Motivated by the field theory, we chose $\text{wt}(i)$ as in Fig. A.
Lattice disclination with disclination angle $\Omega=\frac{\pi}{2}$. The blue
region $W$ covers 11 unit cells. $Q_{i}$ is weighted by the indicated amount
when calculating $Q_{W}$ or $\bar{Q}_{W}$ . B. Standard deviation of
$\mathscr{S}$ as a function of bond disorder $\sigma_{\text{bond}}$ or onsite
disorder $\sigma_{\text{onsite}}$ for $C=1$ and $C=2$ main Landau level
(average hopping is 1).; this led us to the result $\bar{Q}_{W}\equiv
Q_{W}-\nu_{0}n_{u.c.W}=\frac{\mathscr{S}}{4}$ when $W$ encloses a single
disclination. Choosing instead $W$ to be the entire surface of a cube gives us
Eq. (3), and we plot the raw data of the extracted $\mathscr{S}$ in Fig. 7. In
this section we provide some more details on the derivation and its numerical
verification, assuming the most general possible unit cell configuration
compatible with the square lattice symmetry.

We note that the most general unit cell can be thought of as having $A$ sites
on the corners, multiple flavors of $B$ sites within the plaquettes, and
multiple flavors of $C$ sites on the edges, see Fig. 8. In general there can
be several types of $B$ and $C$ sites, but for the purpose of counting charge,
we can treat all sites lying in the interior of the plaquette as a single
interior site $B$ lying at the plaquette center. Similarly, we can treat all
sites lying on the edge in the actual microscopic model as a single edge-
centered site $C$. We also define the quantities
$Q_{\text{refA}},Q_{\text{refB}},Q_{\text{refC}}$, which are the average
charges at an $A,B$ and $C$ site respectively far away from any defects. Note
that the average charge per unit cell is
$Q_{\text{refA}}+Q_{\text{refB}}+2Q_{\text{refC}}=\nu_{0}$. This is because
each unit cell contains one $A$ site (4 corner sites, each with a weight of
1/4), one $B$ site, and two $C$ sites (4 edge sites, each with a weight of
1/2).

Below we show that $\bar{Q}_{W}=\frac{\mathscr{S}}{4}$ for any region $W$
which encloses a $\pi/2$ disclination, as long as the edge of $W$ is
sufficiently far away from the disclination. Let us start by considering a
special region $W_{0}$ such that 8 copies of $W_{0}$ exactly tile the cube.
For this $W_{0}$, we would have
$8\bar{Q}_{W_{0}}=\bar{Q}_{\text{cube}}=2\mathscr{S}$. Thus
$\bar{Q}_{W_{0}}=\frac{\mathscr{S}}{4}$, and this result holds for a general
unit cell configuration. Finally we argue that $\bar{Q}_{W}$ does not change
if $W$ is deformed into some region $W^{\prime}$ which includes an additional
unit cell, as long as the deformation occurs far away from the enclosed
disclinations. An example of a deformation is given in Fig. 10. Note that
$\bar{Q}_{W^{\prime}}=\bar{Q}_{W}+\bar{Q}_{Z}$ where $Z$ encloses just one
unit cell, and moreover $Z$ is far away from any lattice disclination, so that
the $A,B,C$ sites associated to $Z$ have charge
$Q_{\text{refA}},Q_{\text{refB}},Q_{\text{refC}}$ respectively. Thus we have
$Q_{Z}=Q_{\text{refA}}+Q_{\text{refB}}+2Q_{\text{refC}}=\nu_{0}$. As a result
$\bar{Q}_{Z}=Q_{Z}-\nu_{0}=0$. This implies that
$\bar{Q}_{W^{\prime}}=\bar{Q}_{W}=\frac{\mathscr{S}}{4}$, i.e. $\bar{Q}_{W}$
is invariant under deformations of $W$ far away from the disclination. Thus,
starting from $W_{0}$, we can change $W$ while preserving
$\bar{Q}_{W}=\frac{\mathscr{S}}{4}$.

If the disclination charge calculation is done on an open disk, we also
require that $W$ is defined far away from the boundary. This is because the
boundary can host edge states as shown in Fig. 9.

Figure 9:  The uncropped charge density distribution of Fig. 2 in the main
text. The parameters are A. $\frac{\phi}{2\pi}=\epsilon,C=1$; B.
$\frac{\phi}{2\pi}=\frac{1}{2}-\epsilon$, $C=-2$; C.
$\frac{\phi}{2\pi}=\frac{1}{3}+\epsilon,C=3$. The colorbar is in units of
$\nu_{0}$. $\epsilon$ is a small fraction which opens up the band gap. The
open boundary can host edge states. Figure 10:  An $AB$ sublattice with red
$A$ sites and grey $B$ sites. Black solid lines are unit cell boundaries and
blue dashed lines represent hoppings. Adding a new unit cell $Z$ (red region)
to $W$ (blue region) will change the weights by 1/4 at each corner of $Z$ and
by 1 at the center of $Z$.

In Fig. 11, we show the result of our numerical simulations with $AB$
sublattices, i.e. there are no $C$ sites. We consider a cube with 9 unit cells
per side. We find that in the main Landau levels, as long as the edge of $W$
is $\approx 3$ lattice units away from the disclination,
$\mathscr{S}=4\bar{Q}_{W}$ is invariant and equals $\frac{C^{2}}{2}$.

Figure 11:  Numerical result for $\mathscr{S}$ extracted from $\bar{Q}_{W}$.
Data is taken at $(C,\frac{\phi}{2\pi})=(1,\frac{1}{4})\text{ and
}(2,\frac{1}{8})$ in the main Landau levels. We have included different values
of an on-site energy $U$ applied only to even ($A$) sites of the square
lattice, which gives the square lattice an $AB$ sublattice structure with two
sites per unit cell. $W$ encloses a single disclination centered on an $A$
site. $\mathscr{S}$ converges to a quantized value when $R\geq 3$.

###  E.2 Lieb lattice

Figure 12: A. $C_{4}$ symmetric unit cell for the Lieb lattice. Solid lines
represent hoppings (with unit amplitude) as well as unit cell boundaries. B.
$\Omega=\frac{\pi}{2}$ pure disclination for the Lieb lattice. The blue region
represents $W$, and the weightings $\text{wt}(i)$ are marked on each relevant
site.

In this subsection we use the Hofstadter model on the Lieb lattice as an
example to show that our result generalizes to arbitrary $C_{4}$ symmetric
lattices. We pick the $C_{4}$ symmetric unit cell of the Lieb lattice shown in
Fig. A. $C_{4}$ symmetric unit cell for the Lieb lattice. Solid lines
represent hoppings (with unit amplitude) as well as unit cell boundaries. B.
$\Omega=\frac{\pi}{2}$ pure disclination for the Lieb lattice. The blue region
represents $W$, and the weightings $\text{wt}(i)$ are marked on each relevant
site., where there is one $A$ site and two $C$ sites. A disclination in the
Lieb lattice is shown in Fig. A. $C_{4}$ symmetric unit cell for the Lieb
lattice. Solid lines represent hoppings (with unit amplitude) as well as unit
cell boundaries. B. $\Omega=\frac{\pi}{2}$ pure disclination for the Lieb
lattice. The blue region represents $W$, and the weightings $\text{wt}(i)$ are
marked on each relevant site., and there is a background flux $\phi$ through
each unit cell. There is no on site potential on any sites.

We define $W$ such that the boundary of $W$ is aligned with the boundary of
the unit cell. As defined in the main text, the weighting is $\text{wt}(i)=1$
for interior points and $\text{wt}(i)=\frac{1}{4},\frac{2}{4},\frac{3}{4}$ if
the interior of $W$ subtends an angle $\frac{\pi}{2},\pi,\frac{3\pi}{2}$ at
site $i$. For a disclination on a Lieb lattice, this is explicitly shown in
Fig. A. $C_{4}$ symmetric unit cell for the Lieb lattice. Solid lines
represent hoppings (with unit amplitude) as well as unit cell boundaries. B.
$\Omega=\frac{\pi}{2}$ pure disclination for the Lieb lattice. The blue region
represents $W$, and the weightings $\text{wt}(i)$ are marked on each relevant
site.. We can again use the equation

$\bar{Q}_{W}=Q_{W}-\nu_{0}n_{u.c.W}=\frac{\mathscr{S}}{4}$ (21)

to extract $\mathscr{S}$. An explicit numerical calculation gives the
$\mathscr{S}$ butterfly shown in Fig. 13.

Figure 13: Numerical data for the Lieb lattice: $\mathscr{S}$ is extracted
using Eq.(21). The calculation is done on an $R=18$ open disk ($R$ is the
radius in units of lattice constant). The region $W$ is chosen to be an open
disk with $R=9$.

###  E.3 Alternative definition of disclination charge

It is also natural to use an alternative definition of the disclination
charge, different from the one used in the main text, as follows.

Consider a general unit cell configuration in which the disclination is
centered at an $A$ site. We define $\check{W}$ to contain a subset of sites on
the lattice surrounding the disclination with an extra condition that the
number of $A,B,C$ sites in $\check{W}$ excluding the disclination $A$ site be
in the same ratio as the number of $A,B,C$ sites in a single unit cell. We do
not require $\check{W}$ to enclose an integer number of unit cells.

We also propose a different definition of charge, namely

$\check{Q}_{\check{W}}\equiv\sum_{i\in\check{W}}Q_{i},$ (22)

where the weight $\text{wt}(i)=1$ for all sites $i\in\check{W}$. Since the
number of unit cells inside $\check{W}$ is not necessarily an integer, we
cannot define a normalized charge $\bar{Q}_{W}$ as in the main text. Instead,
we define the quantity $\overline{\check{Q}}_{\check{W}}$:

$\overline{\check{Q}}_{\check{W}}\equiv\check{Q}_{\check{W}}-n_{s,\check{W}}Q_{\text{ref}},$
(23)

where $n_{s,W}$ counts the number of sites in $W$ and
$Q_{\text{ref}}=\frac{\nu_{0}}{k_{u.c.}}$ is the average charge per site on a
clean lattice. $k_{u.c.}$ is the number of sites inside a unit cell. Using
this definition, we can use symmetry and counting arguments to evaluate
$\overline{\check{Q}}$. We find that if $\check{W}$ encloses a $\frac{\pi}{2}$
disclination,

$\overline{\check{Q}}_{\check{W}}=\frac{\mathscr{S}}{4}-\frac{\nu_{0}}{k_{u.c.}}+\frac{3}{4}Q_{\text{ref}A},$
(24)

and this agrees with numerical checks (see Fig. 14).

It is more complicated to extract $\mathscr{S}$ using this equation since it
also requires us to know $Q_{\text{ref}A}$. Combined with the fact that
$\check{Q}_{\check{W}}$ is not additive in the present case, in contrast to
what we expect from the field theory, we adopted the definition $Q_{W}$
instead in the main text.

##  Appendix F Angular momentum calculations

Figure 14:  Numerical result for $\overline{\check{Q}}_{\check{W}}$ with
$\check{W}$ enclosing a $\pi/2$ disclination along with the prediction from
Eq. (24). We use an $AB$ sublattice, setting
$C=1,\frac{\phi}{2\pi}=\frac{1}{2}-\epsilon$, and on-site energy $-10<U<10$.
$\epsilon$ is a small fraction which keeps the band gap open.

###  F.1 Choice of $\lambda$ on the torus

In Appendix C, we argued that we have to define the magnetic rotation operator
with a gauge transformation $\lambda$ that vanishes at the origin, which is
the fixed point of the rotation. This choice ensures that the disclination
Hamiltonian has the same flux in each unit cell. Now on the torus, there are
two fixed points $o_{1}$ and $o_{2}$. Thus we have two choices for the
definition of the global rotation operator, corresponding to setting
$\lambda_{o_{1}}=0$ or $\lambda_{o_{2}}=0$.

Empirically, we find that if we pick one of the two choices discussed above,
$\lambda_{o_{1}}=0$ or $\lambda_{o_{2}}=0$, then the coefficient of the linear
term in $l(m)$ gives an $\mathscr{S}$ that is consistent with the fractional
disclination charge result and with Fig. 1.

Note that we can in particular consider redefining $\tilde{C}_{4}$ with
$\lambda_{j}\rightarrow\lambda_{j}\pm\frac{\pi}{2}\beta(m)$ with
$\beta(m)=-m$. This converts between the two natural choices
$\lambda_{o_{1}}=0$ and $\lambda_{o_{2}}=0$. Using Eq. (20), we see that this
also changes the quadratic term in Eq. (4) from $\frac{Cm^{2}}{2}$ to
$-\frac{Cm^{2}}{2}$.

###  F.2 Partial rotation

In the main text, we showed that there is a particular definition of the
partial rotation operator $\tilde{C}_{4}|_{D}$ which gives the result for
$\mathscr{S}$ that is consistent with Fig. 1. Now we discuss some additional
subtleties that arise only in the definition of $\tilde{C}_{4}|_{D}$.

Suppose we insert $2\pi\Delta m$ flux in a local $L^{\prime}\times L^{\prime}$
region $D$. We use a Landau-like gauge, which is shown in Fig. 15. We can
insert the remaining $m_{0}$ flux quanta globally by inserting $m_{0}$ flux
quanta locally around $o_{2}$ using this procedure, and taking the limit
$L^{\prime}=L$.

Figure 15:  Insertion of $2\pi\Delta m$ flux distributed uniformly in a local
$L^{\prime}\times L^{\prime}=4\times 4$ region around the partial rotation
center $o$. Each blue arrow represents a hopping phase
$A_{ij}=\frac{2\pi\Delta m}{L^{\prime 2}}$; each red arrow represents a
hopping phase $A_{ij}=\frac{2\pi\Delta m}{L^{\prime}}$.

In defining the partial rotation operator, we have freedom in choosing whether
$D$ is centered around $o_{1}$ or $o_{2}$. In either case, there are two
natural choices for $\lambda$, as discussed above. Thus in total we have 4
natural choices for $\tilde{C}_{4,\lambda}|_{D}$. We find empirically that
only when we enclose $o_{2}$ with $D$ and define $\lambda_{o_{1}}=0$, do we
get a shift that is consistent with the global rotation result. In this case,
we can calculate $\mathscr{S}$ using the following equation,

$\mathscr{S}=l_{D}(\Delta m=1,m_{0})-l_{D}(0,m_{0})-Cm_{0}-\frac{C}{2}\mod 4.$
(25)

For the other 3 choices, we have not found a meaningful way of extracting
$\mathscr{S}$. More specifically, let us assume that $\mathscr{S}$ is given by
$\mathscr{S}=l_{D}(\Delta m=1,m_{0})-l_{D}(0,m_{0})+\sum_{j}a_{j}m_{0}^{j}\mod
4$. If $a_{j}$ are arbitrary half-integers independent of $m_{0}$, we have the
identity $\sum_{j}a_{j}m_{0}^{j}=\sum_{j}a_{j}(m_{0}+8)^{j}\mod 4$. This
requires that $l_{D}(\Delta m=1,m_{0})-l_{D}(0,m_{0})$ be invariant under
$m_{0}\rightarrow m_{0}+8$. However, empirically we do not observe this when
we consider the remaining 3 choices. Thus there appears to be no polynomial
fit with half-integer coefficients that can extract $\mathscr{S}$ in these
cases. We do not have a complete theoretical explanation for this, and leave a
more detailed study for future work.

We have tested Eq. 25 on an $L=24$ torus. $D$ is chosen to be a $14\times 14$
square region centered at $o_{2}$ and the flux is inserted uniformly in a
$6\times 6$ region. The resulting values of $\mathscr{S}$ match Fig. 1.

###  F.3 Partial rotation on space with open boundary conditions

We can repeat the partial rotation calculations on a system with open
boundaries. Now there is only one fixed point of rotation $o$. Naïvely we
might wish to take $\lambda_{o}=0$ for all $m$. However this choice does not
work, since we are unable to obtain a uniform $\mathscr{S}$ in a single lobe
using a polynomial fit defined as above.

Empirically, we find that to obtain a value of $\mathscr{S}$ that is constant
within a lobe through the above equation, we have to pick the choice
$\lambda_{o}=\frac{\pi}{2}m_{0}$ before local flux insertion and
$\lambda_{o}=\frac{\pi}{2}m_{0}+\frac{\pi}{2}\Delta m$ after local flux
insertion. Using this, $\mathscr{S}$ can be extracted using Eq. (25), and the
result matches Fig. 1. This is numerically tested using the same system size
dimensions as on the torus. For both open and closed boundary conditions, we
have also varied $L,L^{\prime}$ and the size of $D$. The equation for
$\mathscr{S}$ is independent of these parameters as long as they are
sufficiently large, and the local flux is completely covered by $D$.

##  Appendix G Crystalline gauge theory

###  G.1 Review

Here we discuss how to write down topological response actions for systems
with $U(1)$ charge conservation, discrete magnetic translation and
$\mathbb{Z}_{M}$ rotational symmetries on a spacetime 3-manifold
$\mathcal{M}^{3}$. We do this by defining background ‘crystalline gauge
fields’ for the symmetry and then demanding that the action written in terms
of these gauge fields obey a topological invariance condition.

In order to construct actions from discrete gauge fields, it is convenient to
work in terms of simplicial cohomology and simplicial calculus (see Appendix A
of Ref [38] for a review). The basic idea is to triangulate $\mathcal{M}^{3}$
and define a discrete background gauge field for the symmetry group $G$ on the
links of the triangulation. In the simplicial formulation we demand that the
gauge fields be flat; this means that the product of group elements around a
2-simplex (triangle) with 3 edges equals the identity whenever the simplex is
contractible. The holonomy of the gauge field through non-contractible cycles
can be non-trivial, and the integral of the flux through any closed
2-dimensional submanifold must be quantized.

To more easily relate our results to the existing literature on the Wen-Zee
shift in the context of continuum systems, here we instead consider a
continuum formulation. We take the gauge fields to be real-valued differential
$1$-forms on $\mathcal{M}^{3}$. We define

$\displaystyle\delta A,X,Y,\omega\in\Omega^{1}(\mathcal{M}^{3},\mathbb{R}),$
(26)

where $\Omega^{k}(\mathcal{M}^{3},\mathbb{R})$ denotes the space of real-
valued differential $k$-forms. $\delta A$, $\vec{R}=(X,Y)$, and $\omega$ are
the $U(1)$, translation, and rotation components of the full gauge field $B$,
now defined as differential $1$-forms. In the continuum formulation, the
analog of requiring that the gauge field be flat is that the flux of the gauge
fields can have delta function sources of curvature, such that the holonomies
always evaluate to trivial group elements in $G$. This mirrors the fact that
the flux through a 2-simplex in the simplicial formulation can be an integer
multiple of $2\pi$ for a flat gauge field. Next, we impose the same
quantization conditions for the holonomies of the gauge field through non-
contractible cycles and the total flux through closed 2-manifolds as in the
simplicial case.

We note that the topological action for the background gauge fields is derived
by assuming the gauge fields are flat. Nevertheless, we are treating the
topological action as a response theory, which determines the universal, long
wavelength response of the system to non-flat gauge field configurations. To
some extent it is an assumption that we can utilize the topological action for
non-flat gauge field configurations and obtain physically correct universal
results. Indeed our numerical results involving various types of flux
insertion can be thought of as testing the topological response theory beyond
the domain where it is originally derived.

The physical meaning of the gauge fields is the following. As mentioned in the
main text, if $A_{bgd}$ is the part of the vector potential which assigns flux
$\phi$ to each unit cell, then the total vector potential is given by
$A:=A_{bgd}+\delta A$, i.e. $\delta A$ is the deviation of the vector
potential relative to the uniform background. Physically, $\vec{R}$ measures
the number of lattice units traversed along a given path in $\mathcal{M}^{3}$,
while $\omega$ measures the rotation in the local coordinate axes along a
path. $\vec{R},\omega$ can thus be identified with the coframe fields and the
$SO(2)$ spin connection associated to the spacetime metric on
$\mathcal{M}^{3}$. For a fuller discussion, see Ref. [12].

Now the triplet $B=(\delta A,\vec{R},\omega)$ is a gauge field for the non-
Abelian magnetic space group symmetry. This means that the three components
transform according to a specific group law under composition or under gauge
transformations. Let ${\bf g}=(e^{2\pi iz},{\bf r},e^{2\pi i\frac{h}{M}})$ be
an element of the magnetic space group, where $e^{2\pi iz}\in U(1),{\bf
r}\in\mathbb{Z}^{2},h\in\mathbb{Z}_{M}$. The group law is

$(e^{2\pi iz_{1}},{\bf r}_{1},e^{2\pi i\frac{h_{1}}{M}})(e^{2\pi iz_{2}},{\bf
r}_{2},e^{2\pi i\frac{h_{2}}{M}})=(e^{2\pi
i(z_{1}+z_{2}+\frac{\phi}{2\pi}\frac{{\bf r}_{1}\times U(h_{1}){\bf
r}_{2}}{2})},{\bf r}_{1}+U(h_{1}){\bf r}_{2},e^{2\pi
i\frac{h_{1}+h_{2}}{M}}),$ (27)

where $U(h)$ is the $2\times 2$ rotation matrix corresponding to the angle
$\frac{2\pi h}{M}$. This group law must be obeyed by $B$. In particular, the
integral of $B$ along a cycle $\gamma$ enclosing a region $W$ equals

$\displaystyle\oint_{\gamma}B$ $\displaystyle=\left(\oint_{\gamma}\delta
A+\phi\int_{W}A_{XY},\oint_{\gamma}U(\smallint_{x_{0}}^{x}\omega)\vec{R},\oint_{\gamma}\omega\right)$
(28) $\displaystyle=\int_{W}\left(d\delta
A+\frac{\phi}{2\pi}A_{XY},d\vec{R}+i\sigma_{y}\omega\wedge\vec{R},d\omega\right)$
(29)

Here the differential 2-form $A_{XY}$ counts the number of unit cells within
$W$. When $\omega=0$, $A_{XY}=\frac{1}{4\pi}(X\wedge Y-Y\wedge X)$; when
$\omega\neq 0$ it is difficult to write an expression in this continuum
formulation, but we can still do so in the simplicial formulation [12]. Note
that the $U(1)$ component of the above equation measures the total $U(1)$ flux
in $W$, because $F:=d\delta A+\frac{\phi}{2\pi}A_{XY}=d\delta A+dA_{bgd}=dA$
where $A$ is the full vector potential. For each $x\in\gamma$,
$U(\smallint_{x_{0}}^{x}\omega)$ represents the net rotation of the coordinate
axes between some chosen origin $x_{0}\in\gamma$ and $x$, obtained by
integrating $\omega$ along $\gamma$. This rotation is then applied to
$\vec{R}$ at $x$. $U(\smallint_{x_{0}}^{x}\omega)\vec{R}$ is thus a parallel-
transport of $\vec{R}$, which accounts for how $\vec{R}$ transforms due to
$\omega$. The quantity $\vec{T}:=d\vec{R}+i\sigma_{y}\omega\wedge\vec{R}$ is
precisely the torsion 2-form whose integral gives the total Burgers vector of
dislocations within $W$. The matrix $i\sigma_{y}$ is the generator of fourfold
rotational symmetries, and appears naturally when $M=4$; if $M$ is changed we
must use the corresponding rotation generator in defining $\vec{T}$.

Given any 2-dimensional submanifold $D^{2}\subset\mathcal{M}^{3}$, the
continuum version of the flatness condition is

$\displaystyle\vec{T}(\vec{x})$ $\displaystyle=2\pi(1-U({\bf
h}))\sum_{j}\vec{l}_{j}\delta^{(2)}(\vec{x}-\vec{x}_{j});$ $\displaystyle
A(\vec{x})$
$\displaystyle=2\pi\sum_{j}m_{j}\delta^{(2)}(\vec{x}-\vec{x}_{j});$
$\displaystyle\omega(\vec{x})$
$\displaystyle=2\pi\sum_{j}n_{j}\delta^{(2)}(\vec{x}-\vec{x}_{j})$ (30)

for $\vec{x},\vec{x}_{j}\in D^{2}$. Importantly, $m_{j}$, $n_{j}\in\mathbb{Z}$
and $\vec{l}_{j}\in\mathbb{Z}^{2}$. This corresponds to delta function sources
at the points $\vec{x}_{j}\in D^{2}$ which carry integer multiples of the
appropriate flux quanta. $U({\bf h})$ is the matrix generator of
$\mathbb{Z}_{M}$ rotations. The factor of $(1-U({\bf h}))$ in the above
equation is imposed to match the simplicial result in which the flux due to a
dislocation Burgers vector is treated as trivial if the flux of the form
$2\pi(1-U({\bf h}))\vec{n}$, for integer $\vec{n}$ [12].

To describe non-trivial symmetry defects, we want to allow non-flat gauge
field configurations. In the continuum theory, this means we need to allow the
total flux associated to the delta function sources above to be suitable
fractions of the above flux quanta on open manifolds. This is equivalent to
saying that

$\displaystyle\vec{T}(\vec{x})$
$\displaystyle=2\pi\sum_{j}\vec{l}_{j}\delta^{(2)}(\vec{x}-\vec{x}_{j})$ (31)
$\displaystyle\omega(\vec{x})$
$\displaystyle=\frac{2\pi}{M}\sum_{j}n_{j}\delta^{(2)}(\vec{x}-\vec{x}_{j})$
(32)

There is no constraint on $dA$, since the magnetic flux is $U(1)$-valued.

Even when we allow non-flat gauge field configurations, we still require that
the integral through any closed $2$d submanifold $D^{2}$ is appropriately
quantized:

$\displaystyle\frac{1}{2\pi}\int_{D^{2}}\vec{T}$
$\displaystyle=\sum_{j}\vec{l}_{j}=(1-U({\bf h}))\vec{l}_{\text{tot}};$
$\displaystyle\frac{1}{2\pi}\int_{D^{2}}dA$
$\displaystyle=\frac{1}{M}\sum_{j}m_{j}=m_{\text{tot}}$
$\displaystyle\frac{1}{2\pi}\int_{D^{2}}d\omega$
$\displaystyle=n_{\text{tot}},$ (33)

where $m_{\text{tot}},n_{\text{tot}}\in\mathbb{Z}$ and
$\vec{l}_{\text{tot}}\in\mathbb{Z}^{2}$.

Gauge transformations correspond to conjugating $B$ by gauge variables $b$,
i.e. $B\rightarrow b^{-1}Bb$. The gauge transformations are also real-valued.
In particular large gauge transformations for $A$ and $\omega$ must be
quantized in units of $2\pi$. By construction, the holonomy of $B$ is gauge-
invariant.

We now discuss how to construct topological response theories in terms of $B$.

####  G.1.1 Derivation for bosonic SPTs

We first show the general calculation for bosonic SPT phases, in which the
details are simpler. It is known that bosonic SPTs with symmetry $G$ are
classified by the group $\mathcal{H}^{3}(G,U(1))$, and that any allowed
response action is related to a 3-cocycle representative within this group.
The response coefficients are quantized by a 3-cocycle condition, along with
the flux quantization conditions on the gauge fields introduced above.

Let $\mathcal{L}$ be a 3-form written in terms of the components of $B$. For
bosonic SPT phases, the condition that $\mathcal{L}$ be a topological action
is that the partition function of the theory be invariant under a cobordism
that deforms $\mathcal{M}^{3}$ to some other 3-manifold $\mathcal{M}^{\prime
3}$ [38]. This cobordism is given by a 4-manifold $\mathcal{W}^{4}$ such that
$\partial\mathcal{W}^{4}=\mathcal{M}^{3}\cup\overline{\mathcal{M}^{\prime
3}}$. We have

$e^{i\int_{\mathcal{M}^{3}}\mathcal{L}}=e^{i\int_{\mathcal{M}^{\prime
3}}\mathcal{L}}.$ (34)

when $\mathcal{M}^{3},\mathcal{M}^{\prime 3}$ are cobordant. Using Stokes’
theorem, we can write this condition as
$e^{i\int_{\mathcal{W}^{4}}d\mathcal{L}}=1$. By gluing different manifolds
with boundary, we can extend this condition to arbitrary $\mathcal{W}^{4}$,
including closed manifolds. We now consider $\mathcal{W}^{4}=D_{1}^{2}\times
D_{2}^{2}$, where $D_{1}^{2},D_{2}^{2}$ are both closed. The integrals of
$F,\vec{T},d\omega,A_{XY}$ over $D_{1}^{2},D_{2}^{2}$ are all quantized, due
to Eq. (G.1). By taking products of these 2-forms, we can guess a general
expression for $d\mathcal{L}$:

$\displaystyle d\mathcal{L}_{\text{}}$ $\displaystyle=\frac{C/2}{2\pi}F\wedge
F+\frac{\mathscr{S}}{2\pi}F\wedge
d\omega+\frac{\vec{\mathscr{P}}_{c}}{2\pi}\cdot
F\wedge\vec{T}+\frac{k_{0}}{2\pi}F\wedge A_{XY}$
$\displaystyle+\frac{\ell_{s}/2}{2\pi}d\omega\wedge
d\omega+\frac{\vec{\mathscr{P}}_{s}}{2\pi}\cdot
d\omega\wedge\vec{T}+\frac{k_{s}}{2\pi}d\omega\wedge
A_{XY}+\frac{\Pi_{ij}/2}{2\pi}T_{i}\wedge
T_{j}+\frac{\vec{k}_{p}}{2\pi}\cdot\vec{T}\wedge
A_{XY}+\frac{\alpha/2}{2\pi}A_{XY}\wedge A_{XY}.$ (35)

The coefficients in the numerator of each term are quantized by virtue of Eqs.
(34) and (G.1). For example, $C/2,\mathscr{S},\ell_{s}/2$ must be integers,
because the integral of $F$ and $d\omega$ over any closed 2-manifold is always
a multiple of $2\pi$. Since the integral of $T$ over any 2-manifold lies in
$2\pi(1-U({\bf h}))\mathbb{Z}^{2}$, we additionally obtain that
$\vec{\mathscr{P}}_{c},\vec{\mathscr{P}}_{s}\in(1-U({\bf
h}))^{-1}\mathbb{Z}^{2}$. The remaining coefficients can be quantized by
similar arguments. By integrating the above expression, we obtain the desired
response theory

$\displaystyle\mathcal{L}_{\text{}}$ $\displaystyle=\frac{C/2}{2\pi}A\wedge
dA+\frac{\mathscr{S}}{2\pi}A\wedge
d\omega+\frac{\vec{\mathscr{P}}_{c}}{2\pi}\cdot
A\wedge\vec{T}+\frac{k_{0}}{2\pi}A\wedge A_{XY}$
$\displaystyle+\frac{\ell_{s}/2}{2\pi}\omega\wedge
d\omega+\frac{\vec{\mathscr{P}}_{s}}{2\pi}\cdot\omega\wedge\vec{T}+\frac{k_{s}}{2\pi}\omega\wedge
A_{XY}+d^{-1}\left(\frac{\Pi_{ij}}{2\pi}(T_{i}\wedge
T_{j})+\frac{\vec{\nu}_{p}}{2\pi}\cdot\vec{T}\wedge
A_{XY}+\frac{\alpha}{2\pi}A_{XY}\wedge A_{XY}\right),$ (36)

The total charge in a region can be obtained by computing
$\frac{\delta\mathcal{L}}{\delta A_{0}}=\frac{C}{2\pi}d\delta
A+\frac{C\phi/2\pi+k_{0}}{2\pi}A_{XY}$. By identifying the term proportional
to $A_{XY}/2\pi$ with the filling per unit cell $\nu_{0}$, we obtain
$\nu_{0}=C\frac{\phi}{2\pi}+k_{0}$. By similarly varying $\mathcal{L}$ with
respect to $\omega_{0}$, we can show that
$\nu_{s}=\mathscr{S}\frac{\phi}{2\pi}+k_{s}$. Note that this action
automatically encodes known relations such as $\nu_{0}=C\frac{\phi}{2\pi}\mod
1$, since $k_{0}$ is an integer.

Certain choices of the coefficients may be nonzero but topologically trivial.
For example, if we choose $\mathscr{S}$ to be a multiple of $M$, the resulting
partition function equals 1 for any 3-manifold, and thus only $\mathscr{S}\mod
M$ is a topological invariant, classified by the group $\mathbb{Z}_{M}$. The
classification of the other terms is obtained for bosonic symmetry-enriched
topological phases in Refs. [12, 13].

We can prove that this is the most general action possible for the given
magnetic space group symmetry by checking that it describes all elements of
the cohomology group $\mathcal{H}^{3}(G,U(1))$, which classifies (2+1)D
bosonic SPT phases. This is verified in Ref. [13] using the simplicial
formulation. For example, in bosonic SPTs the three terms in parentheses can
be written as coboundaries, so they do not represent nontrivial topological
responses.

####  G.1.2 From bosonic to fermionic SPT phases

For fermionic SPT phases, Eq. (34) cannot be used; the correct topological
invariance condition also needs to account for certain additional topological
data related to the fermionic degrees of freedom [15]. We show the required
analysis for a simplified case with only charge conservation and rotational
symmetries in Appendix H.2. We have also done the more general calculation for
the magnetic space group symmetry, which we do not show here. Below we discuss
how the general response theory for fermionic SPT phases compares with the
bosonic SPT action in Eq. (G.1.1).

The terms in Eq. (G.1.1) are also present in the fermionic SPT response
theory, but the coefficients are quantized differently. For example, the three
terms in parentheses are not trivial, since the coefficients can be
fractional. However these coefficients are completely fixed if we know the
previous seven terms. Moreover, they vanish if we set $X,Y$ to zero, so we
ignore them in Eq. (Model and response theory.).

We also note that in the bosonic SPT case, $C/2$ and $\mathscr{S}$ were only
required to be independent integers. However, in the fermionic case they can
both be half-integers. In Appendix H we prove this quantization explicitly.

####  G.1.3 Chiral central charge and framing anomaly

One important additional ingredient in the Hofstadter model is the chiral
central charge $c_{-}$, which is proportional to the thermal Hall conductance
(and in our case equals the Chern number $C$). For bosonic and fermionic SPTs,
$c_{-}=0$. However, we need to consider more general fermionic invertible
phases, in which $c_{-}$ can be any integer or half-integer; the
classification of invertible phases depends on the value of $c_{-}\mod 8$. A
system with nonzero $c_{-}$ always has a framing anomaly, which must be
cancelled by adding an extra term
$\mathcal{L}_{\text{anom}}=-\frac{c_{-}}{96\pi}\text{Tr}\left(\Omega\wedge
d\Omega+\frac{2}{3}\Omega\wedge\Omega\wedge\Omega\right)$ to the response
theory [9, 39]. Here $\Omega$ is the Levi-Civita spin connection associated to
the metric on $\mathcal{M}^{3}$. In order to describe the spatial symmetry the
rotation gauge field $\omega$ must be pinned to $\Omega$. Therefore we
identify the relevant components: $\Omega^{a}_{0,\mu}=\Omega^{0}_{b,\mu}=0$,
and $\Omega^{1}_{2,\mu}=\omega_{\mu}$. The framing anomaly can then be
rewritten as $\mathcal{L}_{\text{anom}}=-\frac{c_{-}}{48\pi}\omega\wedge
d\omega$. In Eq. (Model and response theory.), we collected the $\omega\wedge
d\omega$ terms by defining $\tilde{\ell}_{s}=\ell_{s}-\frac{c_{-}}{12}$.

Finally, we note that when $c_{-}\neq 0$, several response coefficients,
including $C$ and $\mathscr{S}$, can depend on $c_{-}$. This gives rise to
nontrivial relationships among the coefficients. In particular, in Appendix H
we argue that $\mathscr{S}=\frac{C}{2}\mod 1$ for spinless fermions.

###  G.2 Deriving the linear term in Eq. (4) using crystalline gauge theory

In the empirical result Eq. (4) in the main text, the total angular momentum
$l(m)$ of a state with $m$ flux quanta has three terms. Here we show that the
first term $\mathscr{S}m$ can be explained entirely within the TFT
description, after correctly accounting for the magnetic translation symmetry.
It is important that $m$ be the total number of magnetic flux quanta in the
state, including the uniform background flux per unit cell $\phi$ as well as
any additional flux. We do not have a complete derivation for the $Cm^{2}/2$
term, although we observe it empirically in our numerics.

In the field theory, $\mathscr{S}$ appears through the term

$\mathcal{L}_{\mathscr{S}}:=\frac{\mathscr{S}}{2\pi}\omega\wedge dA,$ (37)

where $dA=d\delta A+\frac{\phi}{2\pi}A_{XY}$ is the total magnetic flux
density. It is a sum of two terms: (i) the flux of $\delta A$, which measures
the excess magnetic flux in addition to the background, and (ii) the uniform
background flux, which equals $\phi\int_{D}A_{XY}$ within any region $D$.
Suppose $\int_{D}A_{XY}=L^{2}$ and $\int_{D}d\delta A=\Delta m$, with total
flux quanta $m=\frac{\phi L^{2}}{2\pi}+\Delta m$. The total angular momentum
in the ground state then equals

$\int\frac{\delta\mathcal{L}_{\mathscr{S}}}{\delta\omega_{0}}=\mathscr{S}(\Delta
m+\frac{\phi}{2\pi}L^{2})=\mathscr{S}m.$ (38)

The linear term in Eq. (4) can thus be straightforwardly explained by
accounting for the magnetic translation symmetry, which couples $\omega$ to
the background flux and not just the excess flux.

##  Appendix H Derivation of constraints on $\mathscr{S}$

Here we prove the constraints on $\mathscr{S}$ mentioned in the main text.

###  H.1 Derivation of $\mathscr{S}=\frac{C}{2}\mod 1$ for free fermions

To argue this result in the free fermion setting, we make the following chain
of assertions:

  1. 1.

When $C=0$, $\mathscr{S}$ is an integer. To see this, note that when $C=0$,
the system can be adiabatically deformed into a set of localized Wannier
orbitals at the high symmetry points of the square lattice. This corresponds
to an $ABX$ sublattice configuration with $A$ sites on the vertices of the
unit cell, $B$ sites on the face centres and $X$ sites at the edge centres.
(We temporarily use the notation $X$ instead of $C$ for edge centred sites to
avoid confusion with the Chern number $C$.) We assign the sites a charge
$n_{A},n_{B}$ or $n_{X}$, all of which are crucially integers. We have
$\nu_{0}=n_{A}+n_{B}+2n_{X}$. (Note that $\nu_{0}=C\frac{\phi}{2\pi}\mod 1$
implies that $\nu_{0}\in\mathbb{Z}$ when $C=0$.)

If we consider a cube with the above unit cell structure, the total number of
$A,B,X$ sites equals $n_{u.c.,\text{cube}}+2$, $n_{u.c.,\text{cube}}$,
$2n_{u.c.,\text{cube}}$ respectively. Therefore we should have

$\displaystyle Q_{\text{cube}}$ $\displaystyle=(n_{u.c.,\text{cube}}+2)n_{A}$
$\displaystyle+n_{u.c.,\text{cube}}n_{B}+2n_{u.c.,\text{cube}}n_{X}.$ (39)

Comparing with Eq. (3), we see that $\mathscr{S}=n_{A}$, so it must be an
integer for $C=0$.

  2. 2.

When the spinless system is composed of $C$ integer filled Landau levels,
$\mathscr{S}=C^{2}/2$ (see eg. Ref. [1] for a proof). This is indeed an
integer or a half-integer when $C$ is even or odd, respectively.

  3. 3.

A system with general $C$ can always be thought of as a stack of a system
composed of $C$ filled Landau levels with another system that has $C=0$. Since
$\mathscr{S}$ is additive under stacking, our main claim follows.

###  H.2 Argument for $\mathscr{S}=\frac{C}{2}\mod 1$ for interacting fermions

We now present an argument which is valid for interacting fermion systems,
using a more sophisticated theory developed in Ref. [15] along with the
general formalism of crystalline gauge fields reviewed in Appendix G.1. We
explain the result on physical grounds, and then give a more formal argument.

####  H.2.1 Physical argument

We will first derive the result assuming a $\mathbb{Z}_{4}$ rotational
symmetry, and then discuss how to generalize the result to a $\mathbb{Z}_{M}$
rotational symmetry. Consider a system with $U(1)$ charge conservation and a
$\mathbb{Z}_{4}$ rotational symmetry under which the fermion is spinless (i.e.
a $2\pi$ rotation acts on fermions with a plus sign). The symmetry acting on
bosons is thus $G_{b}=U(1)\times\mathbb{Z}_{4}$; the fermionic symmetry is
denoted as $G_{f}=U(1)^{f}\times\mathbb{Z}_{4}$. Our argument relies on
gauging the fermion parity and studying the $U(1)$ charge at the
$\mathbb{Z}_{4}$ symmetry defects. Strictly speaking, this argument applies
only to defects of internal symmetries. However, we can still make progress
using the ‘crystalline equivalence principle’ for fermions (fCEP), discussed
in Refs. [40, 41].

For our purposes, the fCEP states that the classification of Chern insulating
phases with the above spatial symmetry is identical to that of Chern
insulating phases with an internal bosonic symmetry
$G_{b}=U(1)\times\mathbb{Z}_{4}$, where the fermion now has spin-1/2 under the
internal $\mathbb{Z}_{4}$ (i.e. a complete rotation under $\mathbb{Z}_{4}$
acts on fermions with a minus sign). That is,
$G_{f}^{\text{int}}=[U(1)^{f}\times\mathbb{Z}_{8}^{f}]/\mathbb{Z}_{2}$.
Therefore we now consider the latter internal symmetry, and use the fCEP to
claim that the conclusions we obtain will also hold for the spatial symmetry
of interest.

Let us gauge the fermion parity. We obtain a topologically ordered system with
symmetry $G_{b}=U(1)\times\mathbb{Z}_{4}$: the topological order depends on
the Chern number $C=c_{-}$, where $c_{-}$ is the chiral central charge [15].
The fact that the fermion has spin-1/2 means that encircling a $2\pi$ flux of
the $\mathbb{Z}_{4}$ symmetry (i.e. a composite of four elementary
$\mathbb{Z}_{4}$ defects) gives a minus sign. This sign can be thought of as
the braiding phase of the fermion with a fermion parity flux, which is an
anyon in the gauged theory, denoted as $m$. Thus inserting a $2\pi$ defect of
$\mathbb{Z}_{4}$ induces the anyon $m$. If we write the elementary defect as
$D$, then we have the fusion rule $D^{4}=m$.

Let the $U(1)$ charge at $D$ be denoted as $Q_{D}=\frac{\mathscr{S}}{4}$, in
analogy with the result for the original spatial symmetry. Then,
$4\times\frac{\mathscr{S}}{4}=\mathscr{S}$ must be the $U(1)$ charge of 4
copies of $D$. This in turn equals $Q_{m}$, up to fusion with a fermion. That
is,

$\mathscr{S}=Q_{m}\mod 1.$ (40)

The crucial observation is that $Q_{m}$ depends on the fusion rules of the
topological order, which in turn depend on $c_{-}=C$. When $c_{-}$ is odd,
$m\times m=\psi$, therefore $Q_{m}=\frac{1}{2}\mod 1$. When $c_{-}$ is even,
$m\times m=I$ (a trivial bosonic particle, which has even charge), therefore
$Q_{m}=0\mod 1$. This implies that

$Q_{m}=\frac{c_{-}}{2}=\frac{C}{2}\mod 1,$ (41)

from which the claim follows.

The above argument can be generalized to any even $M$ without any
difficulties. However, if $M$ is odd, for example $M=3$, we have the following
problem. If $D_{1}$ is an elementary $\mathbb{Z}_{M}$ disclination satisfying
$D_{1}^{M}=1$, then there is another disclination $D_{2}=D_{1}\times m$, which
is a bound state of $D_{1}$ with a fermion parity defect ($\pi$ flux). It
satisfies $D_{2}^{M}=D_{1}^{M}\times m^{M}=m$. We would like to define
$\mathscr{S}\mod 1$ as the charge of either $D_{1}^{M}=1$ or $D_{2}^{M}=m$.
But when $M$ is odd, the theory does not tell us which definition to use in
general.

Even though the $M=4$ derivation cannot be used, there are still strong
arguments that $\mathscr{S}=\frac{C}{2}\mod 1$ is the correct relation even
when $M$ is odd. The argument is based on the result that the shift has a
$\mathbb{Z}_{M}$ classification for each $M$, which can be derived from the
general theory of Ref. [15]

For example, take $M=3$. If the $\mathbb{Z}_{3}$ rotational symmetry in the
microscopic model can be viewed as a subgroup of a $\mathbb{Z}_{6}$ symmetry,
then we must have $\mathscr{S}=\frac{C}{2}\mod 1$ in order to be consistent
with the $\mathbb{Z}_{6}$ result. We can always find a system with this
property, for any $C$. Now suppose there is a system $A$ for which
$\mathscr{S}=\frac{C}{2}+\frac{1}{2}$. We can stack $A$ with a system $B$ that
has Chern number $-C$ and $\mathscr{S}=\frac{C}{2}\mod 1$ (say $B$ has a
$\mathbb{Z}_{6}$ rotational symmetry). The stacked system $AB$ will have Chern
number zero but $\mathscr{S}=\frac{1}{2}\mod 1$. But this would imply that
there are $6$ distinct values of $\mathscr{S}$ when $C=0$, generated by $AB$.
But this is inconsistent with the result that $\mathscr{S}$ has a
$\mathbb{Z}_{3}$ classification. Similar arguments apply to all odd $M$.

Below we formalize the physical argument in the language of topological field
theory, assuming $M$ is even.

####  H.2.2 Symmetry definition

Consider a fermionic system in which the symmetry that acts on the bosonic
operators is given by $G_{b}=U(1)_{b}\times\mathbb{Z}_{M}$ for some even $M$.
(The subscript $b$ emphasizes that this is a bosonic symmetry.) We choose $M$
even so that the spin of the fermion under rotations is well-defined. For the
moment $G_{b}$ is taken to be an internal symmetry, however we will eventually
account for the spatial nature of the $\mathbb{Z}_{M}$ rotations using the
‘crystalline equivalence principle’ [40]. We do not assume any translation
symmetry for simplicity, because it is not required to define $C,\mathscr{S}$.

The symmetry group $G_{f}$ which acts on fermionic operators is given by a
group extension of $G_{b}$ by the fermion parity $\mathbb{Z}_{2}^{f}$,
corresponding to a representative cocycle $\omega_{2}$ in the group
$\mathcal{H}^{2}(G_{b},\mathbb{Z}_{2})$. For $G_{b}=U(1)\times\mathbb{Z}_{M}$,
we have

$\displaystyle\mathcal{H}^{2}(G_{b},\mathbb{Z}_{2})=\mathcal{H}^{2}(U(1),\mathbb{Z}_{2})\times\mathcal{H}^{2}(\mathbb{Z}_{M},\mathbb{Z}_{2})=\mathbb{Z}_{2}\times\mathbb{Z}_{2}.$
(42)

Let
$\alpha_{c}\in\mathcal{H}^{2}(U(1),\mathbb{Z}),\alpha_{s}\in\mathcal{H}^{2}(\mathbb{Z}_{M},\mathbb{Z})$
be representative cocycles generating their respective groups. Denote each
element of $G_{b}$ by ${\bf g}_{i}=(e^{2\pi iz_{i}},{\bf h}_{i})$. Then, the
physical system has

$\omega_{2}({\bf g}_{1},{\bf g}_{2}):=\alpha_{c}(e^{2\pi iz_{1}},e^{2\pi
iz_{2}})+k\alpha_{s}({\bf h}_{1},{\bf h}_{2})\mod 2.$ (43)

The first term on the rhs indicates that a $2\pi$ rotation in $U(1)_{b}$ acts
trivially on any bosonic operator, but transforms any fermionic operator by a
minus sign. The second term indicates that a $2\pi$ rotation in
$\mathbb{Z}_{M}$ acts trivially on any bosonic operator, but transforms any
fermionic operator by a sign $(-1)^{k}$. The case $k=0$ corresponds to
fermions that are ‘spinless’ under the internal $\mathbb{Z}_{M}$ symmetry; in
this case, $G_{f}=U(1)^{f}\times\mathbb{Z}_{M}$. The case $k=1$ corresponds to
‘spin-1/2’ fermions under the internal $\mathbb{Z}_{M}$ symmetry; in this
case, $G_{f}=(U(1)^{f}\times\mathbb{Z}_{2M}^{f})/\mathbb{Z}_{2}$.

####  H.2.3 Overview of argument

The main result of Ref. [15] is that each invertible phase can be described by
a set of data $(c_{-},n_{1},n_{2},\nu_{3})$ satisfying various consistency
conditions and equivalences. $c_{-}$ is the chiral central charge, defined mod
8.

$n_{1}\in\mathcal{H}^{1}(G_{b},\mathbb{Z}_{2})$ determines whether a ${\bf
g}$-defect localizes a Majorana zero mode. One can show that $n_{1}$ non-
trivial is incompatible with $U(1)^{f}$ symmetry, and therefore we can set
$n_{1}=0$ in the present discussion.

Let $C^{n}(G_{b},M)$ be the space of $n$-variable functions ($n$-cochains)
$f_{n}:G_{b}\rightarrow M$. The parameter $n_{2}\in
C^{2}(G_{b},\mathbb{Z}_{2})$ fixes the fusion rules of the symmetry defects in
the theory, while $\nu_{3}\in C^{3}(G_{b},U(1))$ is analogous to a local
counterterm that modifies the defect $F$-symbols of the invertible phase,
ensuring that they satisfy the pentagon equation for associativity. It is
constrained by $\omega_{2},c_{-},n_{2}$.

We will be particularly interested in $\nu_{3}$, because in the case of
bosonic SPT phases, it is known [37] that the response action is obtained by
pulling back $\nu_{3}$ using the background symmetry gauge field $B$. If we
think of $B$ as a map from the space-time manifold $\mathcal{M}^{3}$ to the
classifying space $BG$, and let $B^{*}$ denote its pullback, then the
topological action is determined by the Lagrangian $\mathcal{L}=2\pi
B^{*}\nu_{3}$. More concretely, this means that if we triangulate space-time
and label the 1-simplices of the triangulation with group elements, the action
integrated on a single 3-simplex $\Delta^{3}$ evaluates to $e^{2\pi
i\nu_{3}({\bf g},{\bf h},{\bf k})}$, where ${\bf g},{\bf h},{\bf k}$ determine
the labelings on the 1-simplices of $\Delta^{3}$.

For bosonic SPTs we have the condition $d\nu_{3}=0\mod 1$, which is equivalent
to Eq. (34) and which is the topological invariance condition. Thus knowing
$\nu_{3}$ is equivalent to knowing $\mathcal{L}$.

For invertible fermionic topological phases, we continue to identify the
topological response action in terms of the Lagrangian $\mathcal{L}=2\pi
B^{*}\nu_{3}$. This is familiar in the case of the usual Hall response, as we
will see below.

We will make use of the following facts. If we consider our space-time
manifold $\mathcal{M}^{3}$ to be the boundary of a $4$-manifold
$\mathcal{W}^{4}$, i.e. $\partial\mathcal{W}^{4}=\mathcal{M}^{3}$, and we
extend the background gauge field $B$ over $\mathcal{W}^{4}$, then we can
write

$\displaystyle d\mathcal{L}=2\pi B^{*}d\nu_{3}.$ (44)

In the general theory of invertible fermionic topological phases [15],
$\nu_{3}$ satisfies (when $n_{1}=0$),

$\displaystyle
d\nu_{3}=\mathcal{O}_{4}=\frac{1}{2}n_{2}\cup(n_{2}+\omega_{2})+\frac{c_{-}}{8}\omega_{2}\cup\omega_{2}\mod
1.$ (45)

####  H.2.4 Background gauge fields

We define the bosonic $U(1)_{b}$ and $\mathbb{Z}_{M}$ gauge fields $\delta
A,\omega$ as in Appendix G.1. The response theory in the fermionic case can be
written in terms of $G_{b}$ gauge fields if we first gauge the fermion parity
symmetry and then consider the resulting anyon theory, which now has $G_{b}$
symmetry [15]. In addition to $A,\omega$, this gauged theory has ‘1-form’
symmetries which are generated by the fermions and the fermion parity fluxes.
This simply means that the closed fermion loops and fermion parity flux loops
can be smoothly deformed without affecting the topological path integral.

We define a background 2-form $\mathbb{Z}_{2}$ gauge field $B_{2}^{(2)}$ for
the 1-form symmetry generated by the fermion parity fluxes. This means that to
each 2-simplex of the triangulated space-time manifold we assign an element of
$\mathbb{Z}_{2}$; if $B_{2}^{(2)}$ is non-trivial on a 2-simplex, this means
that a fermion parity flux is piercing the 2-simplex. This in turn means that
a fermion traversing the boundary of such a 2-simplex will acquire a minus
sign. Thus, $B_{2}^{(2)}$ encodes $\omega_{2}$: if we consider a triangular
spacetime region $D\in\mathcal{M}^{3}$ with gauge fields corresponding to the
group elements ${\bf g}_{1},{\bf g}_{2},({\bf g}_{1}{\bf g}_{2})^{-1}$ on the
sides, the integral of $B_{2}^{(2)}$ over $D$ equals $\pi\omega_{2}({\bf
g}_{1},{\bf g}_{2})$, i.e. 0 or $\pi$. A fermion traversing the boundary of
$D$ acquires a sign $(-1)^{\omega_{2}({\bf g}_{1},{\bf g}_{2})}$, as expected.

The definition of $\omega_{2}$ implies that

$B_{2}^{(2)}=\frac{1}{2}(dA+kd\omega)=\pi B^{*}\omega_{2}.$ (46)

We write $B_{2}^{(2)}=\pi B^{*}\omega_{2}$ to emphasize that $B_{2}^{(2)}$
encodes $\omega_{2}$ defined in Eq. (43) in terms of $B$.

There is also a 1-form symmetry generated by the fermion, which encodes some
additional data $n_{2}\in C^{2}(G_{b},\mathbb{Z}_{2})$. Proceeding as above,
we define a 2-form gauge field

$\displaystyle B_{2}^{(1)}=\pi B^{*}n_{2}$ (47)

for this 1-form symmetry, which encodes $n_{2}$. Physically, the triangular
region $D$ introduced above hosts an extra fermionic degree of freedom when
$n_{2}({\bf g}_{1},{\bf g}_{2})=1$ [42].

####  H.2.5 Quantization of $\mathscr{S}$

The most general topological field theory for invertible fermion phase with
symmetry $G_{f}$ (with either choice of $k$) is defined as

$\mathcal{L}=\frac{C}{4\pi}A\wedge dA+\frac{\mathscr{S}}{2\pi}A\wedge
d\omega+\frac{\ell}{4\pi}\omega\wedge d\omega$ (48)

for some coefficients $C,\mathscr{S},\ell$ to be determined. As discussed
above, the main conceptual step is to relate $\mathcal{L}$ to $\nu_{3}$, by
defining $\mathcal{L}:=2\pi\times B^{*}\nu_{3}$, where $\nu_{3}\in
C^{3}(G_{b},\mathbb{R}/\mathbb{Z})$. Assuming this, the quantization of
$C,\mathscr{S}$ and $\ell$ is fixed by $c_{-}$, as we now discuss. For an
invertible fermion phase with $U(1)^{f}$ symmetry, we must have (i) $c_{-}$ be
an integer, and (ii) no Majorana zero modes in the system, i.e. $n_{1}=0$.
These two conditions force $dn_{2}=0\mod 2$ (see Ref. [15] for a proof).

The most general choice of $n_{2}$ is then

$\displaystyle n_{2}$ $\displaystyle:=k_{c}\alpha_{c}+k_{s}\alpha_{s}$
$\displaystyle\implies B_{2}^{(1)}$
$\displaystyle:=\frac{1}{2}(k_{c}dA+k_{s}d\omega)$ (49)

for some $k_{c},k_{s}\in\\{0,1\\}$. Since there is an equivalence $n_{2}\sim
n_{2}+\omega_{2}$ in the theory [15], we can set $k_{c}=0$ without loss of
generality.

To constrain $\mathcal{L}$ we now use Eqs. (44), (45), (46), (47). We then
have

$\displaystyle\frac{d\mathcal{L}}{2\pi}$
$\displaystyle=\frac{1}{\pi^{2}}\left(\frac{1}{2}B_{2}^{(1)}\wedge(B_{2}^{(1)}+B_{2}^{(2)})+\frac{c_{-}}{8}B_{2}^{(2)}\wedge
B_{2}^{(2)}\right)$ $\displaystyle\mod 1.$ (50)

(For arbitrary $G_{f}$, there is an additional term [15], however for our
choice of $G_{f}$, this term is trivial.) Roughly speaking, this formula
indicates that the full theory is topologically invariant under deformations
of $\mathcal{M}^{3}$ if the transformation of the counterterm $\mathcal{L}$
cancels out a corresponding transformation of the data associated to the
1-form symmetries. In terms of $(A,\omega)$, this equation can be written as
follows:

$\displaystyle 2\pi d\mathcal{L}$ $\displaystyle=\frac{c_{-}}{8}dA\wedge dA$
$\displaystyle+\left(\frac{kc_{-}}{4}+\frac{k_{s}}{2}\right)dA\wedge d\omega$
$\displaystyle+\left(\frac{c_{-}k^{2}}{8}+\frac{k_{s}(k_{s}+k)}{2}\right)d\omega\wedge
d\omega.$ (51)

We cannot directly compare these coefficients with $C,\mathscr{S}$ and $\ell$.
In the usual action, we interpret $A$ as a $U(1)^{f}$ gauge field. However,
$A$ has thus far been defined as a $U(1)_{b}$ gauge field. Now fermions with
charge 1 under $U(1)^{f}$ have charge 1/2 under $U(1)_{b}$, so the above field
theory assigns them half the correct charge. We can fix this problem by
replacing $A$ with $2A$ in the expression for $\mathcal{L}$. Upon doing so, we
can readily compare the above expression to the definitions of
$C,\mathscr{S},\ell$. This gives

$\displaystyle C$ $\displaystyle=c_{-}+8k_{1}$ (52) $\displaystyle\mathscr{S}$
$\displaystyle=\frac{kc_{-}}{2}+k_{s}+2k_{2}\mod M$ (53) $\displaystyle\ell$
$\displaystyle=\frac{c_{-}k^{2}}{4}+k_{s}(k_{s}+k)+2k_{3}\mod M$ (54)

The coefficients $k_{1},k_{2},k_{3}$ correspond to additional bosonic SPT
terms that satisfy $d\nu_{3}=0\mod 1$ and can be freely added to
$\mathcal{L}$.

####  H.2.6 Crystalline equivalence principle

We now address one final subtlety. The physical $\mathbb{Z}_{M}$ symmetry of
the square lattice (for $M=4$) is a spatial symmetry. However, the result in
Eq. (52) applies strictly only to internal symmetries. In order to describe
spatial symmetries, we need to transform $\omega_{2}$ according to certain
rules which are formalized as the crystalline equivalence principle for
fermions (fCEP) [40, 43, 44].

When the point group consists only of rotations, the fCEP states that the
classification of fermionic topological phases with the spatial
$\mathbb{Z}_{M}$ symmetry and with integer spin fermions is identical to the
classification of fermionic topological phases with an internal
$\mathbb{Z}_{M}$ symmetry but with spin-1/2 fermions, and vice versa. Thus in
our formalism, we can describe a square lattice with spinless fermions by
setting $k=1$, appropriate for the Hofstadter model. Setting $k=0$ would
describe the square lattice with spin-1/2 fermions. Taking $k=1$, we obtain
our final result for spinless fermions:

$\displaystyle C$ $\displaystyle=c_{-}+8k_{1}$ (55) $\displaystyle\mathscr{S}$
$\displaystyle=\frac{c_{-}}{2}+k_{s}+2k_{2}\mod 4$
$\displaystyle\implies\mathscr{S}$ $\displaystyle=\frac{C}{2}\mod 1$ (56)
$\displaystyle\ell$ $\displaystyle=\frac{c_{-}}{4}+k_{s}(k_{s}+1)+2k_{3}\mod
4.$ (57)

This proves our claim that $\mathscr{S}=\frac{C}{2}\mod 1$ for spinless
fermions. Note that the assumption of $k=1$ (spinless fermions with respect to
the spatial symmetry) is important; if we set $k=0$, $\mathscr{S}$ would be
independent of $C$, and be forced to take integer values.

###  H.3 Derivation of properties (2)-(3)

Next we argue property (2). We notice that the Wen-Zee term
$\frac{1}{2\pi}\epsilon^{\mu\nu\lambda}A_{\mu}\partial_{\nu}\omega_{\lambda}$
is invariant under time-reversal, which takes $\phi\rightarrow 2\pi-\phi$. $A$
and $\omega$ transform as follows:

$\displaystyle A_{0}\rightarrow A_{0},$ $\displaystyle A_{i}\rightarrow-A_{i}$
(58) $\displaystyle\omega_{0}\rightarrow-\omega_{0},$
$\displaystyle\omega_{i}\rightarrow\omega_{i}.$ (59)

The transformation of $\omega$ is obtained as follows. The angular momentum
$\vec{r}\times\vec{p}$ is odd under time-reversal, and in field theory it is
computed by varying the action with respect to $\omega_{0}$. Thus $\omega_{0}$
should also be odd under time reversal. Moreover, a disclination (whose
disclination angle is given by the spatial integral of
$\partial_{1}\omega_{2}-\partial_{2}\omega_{1}$) is time-reversal invariant.
Thus, $\omega_{1},\omega_{2}$ should also be time-reversal invariant. Using
this, we can now check that
$\epsilon^{\mu\nu\lambda}A_{\mu}\partial_{\nu}\omega_{\lambda}$ is time-
reversal invariant. Thus we conclude that $\mathscr{S}$ should also be time-
reversal invariant.

We will argue property (3) using Eq. (3). When $\mu$ changes sign, we have
$\nu_{0}\rightarrow 1-\nu_{0}$ in the Hofstadter butterfly. Suppose
$\mathscr{S}\rightarrow\mathscr{S}^{\prime}$. For convenience, consider a
system on a cube, in which the total number of sites is
$n_{u.c.,\text{cube}}+2$, i.e. assume there is no sublattice splitting. Then,
the transformation also takes $Q_{\text{cube}}\rightarrow
n_{u.c.,\text{cube}}+2-Q_{\text{cube}}$. The original and transformed versions
of Eq. (3) are thus

$\displaystyle Q_{\text{cube}}$
$\displaystyle=\nu_{0}n_{u.c.,\text{cube}}+2\mathscr{S}$ (60) $\displaystyle
n_{u.c.,\text{cube}}+2-Q_{\text{cube}}$
$\displaystyle=(1-\nu_{0})n_{u.c.,\text{cube}}+2\mathscr{S}^{\prime}.$ (61)

This gives $\mathscr{S}^{\prime}=1-\mathscr{S}$, as claimed.