# Complete crystalline topological invariants from partial rotations in (2+1)D
invertible fermionic states and Hofstadter’s butterfly

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

###### Abstract

The theory of topological phases of matter predicts invariants protected only
by crystalline symmetry, yet it has been unclear how to extract these from
microscopic calculations in general. Here we show how to extract a set of
many-body invariants $\\{\Theta_{\text{o}}^{\pm}\\}$, where o is a high
symmetry point, from partial rotations in (2+1)D invertible fermionic states.
Our results apply in the presence of magnetic field and Chern number $C\neq
0$, in contrast to previous work. $\\{\Theta_{\text{o}}^{\pm}\\}$ together
with $C$, chiral central charge $c_{-}$, and filling $\nu$ provide a complete
many-body characterization of the topological state with symmetry group
$G=\text{U}(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{M}]$. Moreover,
all these many-body invariants can be obtained from a single bulk ground
state, without inserting additional defects. We perform numerical computations
on the square lattice Hofstadter model. Remarkably, these match calculations
from conformal and topological field theory, where $G$-crossed modular $S,T$
matrices of symmetry defects play a crucial role. Our results provide
additional colorings of Hofstadter’s butterfly, extending recently discovered
colorings by the discrete shift and quantized charge polarization.

There has been intense work over the past decade in developing a comprehensive
characterization and classification of topological phases of matter with
internal [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] and crystalline symmetries [12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Despite much progress, the
problem of how to fully characterize such systems, in particular by
numerically extracting a complete set of invariants, remains partially solved
[24, 25, 26, 27, 28].

In this paper, we study (2+1)D topological states of fermions with symmetry
group $G=\text{U}(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{M}]$, which
consists of $\text{U}(1)$ charge conservation, discrete magnetic translation
and point group rotation symmetries. We focus on invertible fermionic phases,
which have a unique ground state on all manifolds [9, 10]. We show how to
numerically extract a set of many-body invariants
$\\{\Theta^{\pm}_{\text{o}}\\}$ associated to the crystalline symmetries by
computing expectation values of partial rotations centered at high symmetry
points o of the unit cell. For example, if o is preserved by rotations of
order $M_{\text{o}}$ where $M_{\text{o}}$ is even, then for a fixed Chern
number, $\Theta^{+}_{\text{o}}$ and $\Theta^{-}_{\text{o}}$ define
$\mathbb{Z}_{M_{\text{o}}/2}$ and $\mathbb{Z}_{2M_{\text{o}}}$ invariants
respectively, for each o. Our numerical results are in remarkable agreement
with analytical calculations from conformal and topological field theory;
specifically, the invariants $\\{\Theta^{\pm}_{\text{o}}\\}$ encode the
$G$-crossed modular $S,T$ matrices of symmetry defects [8, 23].

Figure 1: $\Theta_{\beta}^{+}\mod 2$ for Hofstadter model, plotted using an
empirical formula Eq. (7) and Eq. (17) in App. A. Figure 2:
$\Theta_{\alpha}^{+}\mod 2$ for Hofstadter model, where $\alpha$ is a
plaquette center, plotted using an empirical formula Eq. (7) and Eq. (16) in
App. A. The choice of square lattice unit cell is shown on the right.

For square lattice and $\text{U}(1)$ charge conservation symmetries, the
expected theoretical classification of topological invariants is given by the
group
$\mathbb{Z}^{3}\times\mathbb{Z}_{8}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}^{2}$
[20, 29]. We find that the invariants $\\{\Theta^{\pm}_{\text{o}}\\}$,
together with the Chern number $C$, the chiral central charge $c_{-}$ and the
filling $\nu$, give a complete characterization of these invariants. In
particular, the discrete shift $\mathscr{S}_{\text{o}}$ and quantized charge
polarization $\vec{\mathscr{P}}_{\text{o}}$ studied in Refs. [26, 27] can be
obtained from $\\{\Theta^{\pm}_{\text{o}},C,c_{-},\nu\\}$. Our methods to
extract $\\{\Theta_{\text{o}}^{\pm}\\}$ require only a single bulk ground
state on a disk, without needing to insert additional defects, adding to a
line of work that extracts topological invariants from single bulk ground
states [30, 31, 24, 32, 33, 34, 35, 36].

As an application, we study the square lattice Hofstadter model and obtain a
number of additional colorings of Hofstadter’s butterfly, extending the
colorings by $\mathscr{S}_{\text{o}}$ and $\vec{\mathscr{P}}_{\text{o}}$
discovered in [26, 27]. Analogous classification results hold for point group
rotations of order 2,3 and 6, and will be mentioned at the end of the paper.

Partial rotations and partial translations on a cylinder have been studied in
prior works and have been shown to yield invariants such as topological spins
of anyons, central charges [37, 38, 39], and invariants of symmetry-protected
topological (SPT) states [40, 24, 41, 42, 43]. In contrast our results apply
both in the presence of arbitrary rational magnetic flux, $C\neq 0$,
demonstrate a complete set of invariants for crystalline symmetries, and also
incorporate several subtleties missed in prior work. These include the proper
use of $G$-crossed modular matrices, the quantization of
$\\{\Theta^{\pm}_{\text{o}}\\}$, and their dependence on o.

##### Symmetries and model.

We study gapped phases of matter with the symmetry group
$G=\text{U}(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{M}]$, where
$\mathbb{Z}^{2}$ denotes magnetic lattice translations, $\mathbb{Z}_{M}$ for
$M=2,3,4,6$ denotes point group rotations. We assume $M=4$ throughout, and
briefly comment on $M=2,3,6$ at the end of the paper.

The symbol $\times_{\phi}$ implies that the magnetic translation operators,
generated by $\tilde{T}_{\bf x},\tilde{T}_{\bf y}$, obey the algebra
$\tilde{T}_{\bf y}^{-1}\tilde{T}_{\bf x}^{-1}\tilde{T}_{\bf y}\tilde{T}_{\bf
x}=e^{i\phi\hat{N}}$ where $\hat{N}$ is the total fermion number. The tilde
indicates that the definition of the operator involves a $\text{U}(1)$ gauge
transformation. Note $e^{i\pi\hat{N}}=(-1)^{F}$, which is fermion parity.

Let $2\pi/M_{\text{o}}$ be the smallest angle of rotation which preserves the
rotation center o. The possible high symmetry points o of the square lattice
are shown in Fig. 2; they are $\alpha,\beta$ (plaquette center and vertex
respectively, with $M_{\alpha}=M_{\beta}=4$) and $\gamma$ (edge center, with
$M_{\gamma}=2$) 111Note we will abuse language somewhat and use o both as the
rotation center and as a maximal Wyckoff position.. As shown in [26, 27] and
reviewed in App. A, there is a canonical choice of magnetic point group
rotation operators $\tilde{C}_{M_{\text{o}}}^{+}$ which are centered at o and
satisfy $(\tilde{C}_{M_{\text{o}}}^{+})^{M_{\text{o}}}=+1$. We also define
another set of operators
$\tilde{C}_{M_{\text{o}}}^{-}:=e^{i\frac{\pi}{M_{\text{o}}}\hat{N}}\tilde{C}_{M_{\text{o}}}^{+}$,
and
$\tilde{C}_{M_{\text{o}},\chi}^{\pm}:=e^{i\chi\frac{2\pi}{M_{\text{o}}}\hat{N}}\tilde{C}_{M_{\text{o}}}^{\pm}$.
Note that $\tilde{C}_{M_{\text{o}}}^{-}=\tilde{C}_{M_{\text{o}},1/2}^{+}$ and
$(\tilde{C}_{M_{\text{o}}}^{-})^{M_{\text{o}}}=(-1)^{F}$.

We study the spinless free fermion Hofstadter model with flux $\phi$ per unit
cell, with Hamiltonian $H=-\sum_{\langle
ij\rangle}e^{-iA_{ij}}c_{i}^{\dagger}c_{j}+\text{h.c.}$. Our numerical methods
and theoretical analysis hold even if we consider further neighbor hopping
and/or interaction terms.

Figure 3: Raw data for $\Theta_{\text{o}}^{-}$. $\Theta_{\alpha}^{-}$ and
$\Theta_{\gamma}^{-}$ are calculated on a $30\times 30$ open disc;
$\Theta_{\beta}^{-}$ is calculated on a $30\times 30$ torus. The diameter of
the partial rotation region $D$ is taken to be roughly half the system size.
In the $C=\\{1,2,3\\}$ main Landau levels, $\Theta_{\beta}^{-}$ quantizes to
$\\{15/4\pm 0.003,1\pm 0.022,3/4\pm 0.051\\}$.

Partial rotation calculation. Consider a state $|\Psi\rangle$, which can be
defined on a system with either open or periodic boundary conditions. Let
$\tilde{C}_{M_{\text{o}}}^{\pm}|_{D}$ be the restriction of
$\tilde{C}_{M_{\text{o}}}^{\pm}$ to an invariant subregion $D$ centered at o.

We find that
$\bra{\Psi}\tilde{C}_{M_{\text{o}},\chi}^{\pm}|_{D}\ket{\Psi}=e^{\frac{2\pi
i}{M_{\text{o}}}l^{\pm}_{D,\text{o},\chi}-\gamma^{\pm}_{D,\text{o},\chi}}$.
$\gamma^{\pm}_{D,\text{o},\chi}\propto|\partial D|$ sets the real-valued
amplitude. When $\chi$ is an integer, we find

$l^{\pm}_{D,\text{o},\chi}=\frac{C\chi^{2}}{2}+\mathscr{S}^{\pm}_{\text{o}}\chi+K^{\pm}_{\text{o}}\mod
M_{\text{o}}$ (1)

$\mathscr{S}^{+}_{\text{o}}$ is the discrete shift, which was studied in
Refs.[26, 27] and was referred to there as $\mathscr{S}_{\text{o}}$. Here we
define $\mathscr{S}^{-}_{\text{o}}$ analogously, as the term linear in $\chi$
in Eq. 1. Empirically we observe that $K^{\pm}_{\text{o}}\mod M_{\text{o}}$
may not be invariant throughout a given lobe in the Hofstadter model:
$K^{+}_{\text{o}}$ can jump by multiples of $M_{\text{o}}/2$ within a single
lobe, but there are no jumps in $K^{-}_{\text{o}}$. For an intuitive
explanation of these jumps, which are system size dependent, see App. A. We
thus define

$\displaystyle\Theta_{\text{o}}^{+}:=K^{+}_{\text{o}}\mod\frac{M_{\text{o}}}{2},\;\;\;\;\Theta_{\text{o}}^{-}:=K^{-}_{\text{o}}\mod
M_{\text{o}}.$ (2)

The main result of this paper is that $\\{\Theta_{\text{o}}^{\pm}\\}$ are
topological invariants protected by point group rotational symmetry about o.
$\Theta_{\text{o}}^{+}$ is in general an integer or half-integer, while
$\Theta_{\text{o}}^{-}$ is a multiple of $1/4$. For a fixed Chern number, they
can take $M_{\text{o}}/2$ and $2M_{\text{o}}$ different values respectively
and thus define $\mathbb{Z}_{M_{\text{o}}/2}$ and $\mathbb{Z}_{2M_{\text{o}}}$
invariants, respectively. We find both empirically and analytically using Eq.
10 that $\Theta_{\gamma}^{+}=0\mod 1$, and so it will henceforth be ignored.
When $C=0$, $\\{\Theta_{\text{o}}^{\pm}\\}$ are closely related to the many-
body real-space indicators defined in [28].

The values of $\Theta_{\text{o}}^{\pm}$ are plotted for the square lattice
Hofstadter model in Figs. 1, 2, and 3. The values of $\Theta_{\text{o}}^{+}$
satisfy the empirical relation

$\Theta_{\text{o}}^{+}=k_{3,\text{o}}^{+}-\frac{C}{2}\mod
2,\;\;\text{o}=\alpha,\beta$ (3)

where $k^{+}_{3,\text{o}}$ is an integer. The subscript 3 is used to match a
convention from [22]. $\Theta_{\text{o}}^{-}$ satisfies

$\Theta_{\text{o}}^{-}=\begin{cases}k^{-}_{3,\text{o}}+\frac{k_{s,\text{o}}}{2}-\frac{3C}{4}\mod
4,&\text{o}=\alpha,\beta\\\
k_{3,\text{o}}^{-}+\frac{k_{s,\text{o}}}{2}-\frac{C}{4}\mod
2,&\text{o}=\gamma,\end{cases}$ (4)

where $k_{s,\text{o}}\in\\{0,1\\}$ and $k_{3,\text{o}}^{-}$ is an integer.
Eqs. (3), (4) are in excellent agreement with a conformal field theory
calculation, which predicts the basic quantization and $C$ dependence of
$\Theta^{\pm}_{\text{o}}$ exactly, but does not give the actual values of
$k_{3,\text{o}}^{\pm}$ and $k_{s,\text{o}}$ for a particular model. The CFT
prediction is summarized by Eq. (10).

If we view $\Theta^{\pm}_{\text{o}}$ in the square lattice Hofstadter model as
a function of $\phi$, $\Theta^{\pm}_{\beta}$ has period $2\pi$ while
$\Theta^{\pm}_{\alpha}$ has period $8\pi$. A similar periodicity depending on
origin was observed for the discrete shift in Ref. [27]. Mathematically, the
lack of periodicity in $2\pi$ arises because $\tilde{C}_{M_{\text{o}}}$ is not
a gauge-invariant operator, and $H$ is only periodic (without a gauge
transformation) under $\phi\rightarrow\phi+8\pi$ (For details see App. A). The
lack of $2\pi$ periodicity is physically meaningful and arises because $\phi$
must be defined as a real number to consistently specify generic perturbations
of $H$ [27].

The continuum limit of the Hofstadter model is obtained by taking
$\phi\rightarrow 0^{+},\nu\rightarrow 0^{+}$. In this limit states with Chern
number $C$ are equivalent to $C$ filled Landau levels (LLs), and we have

$\displaystyle\Theta^{+}_{\text{o},\text{LL}}$
$\displaystyle=\frac{C^{3}}{6}-\frac{2C}{3}\mod 2,\;\;\text{o}=\alpha,\beta$
(5) $\displaystyle\Theta^{-}_{\text{o},\text{LL}}$
$\displaystyle=\begin{cases}\frac{C^{3}}{6}+\frac{C^{2}}{4}-\frac{2C}{3}\mod
4&\text{o}=\alpha,\beta\\\ \frac{C^{3}}{6}+\frac{C^{2}}{4}-\frac{C}{6}\mod
2&\text{o}=\gamma\end{cases}.$ (6)

As explained below, we can derive Eqs. 5, 6 analytically using results from
CFT and TQFT, and also verify them empirically in our numerical calculations.
$\Theta^{\pm}_{\text{o}}$ are closely related to the topological term
$\frac{\ell_{s,\text{o}}^{\pm}}{4\pi}\omega\wedge d\omega$ which appears in
the effective response theory of continuum LLs coupled to an $\text{SO}(2)$
spin connection $\omega$ [45].

The empirical data for $\Theta^{+}_{\text{o}}$ can be fit to precise formulas
throughout the square lattice Hofstadter model, which we use to fully color
the butterflies in Fig. 1,2. These equations all have the general form

$\Theta^{\pm}_{\text{o}}=\Theta^{\pm}_{\text{o},\text{LL}}+\Theta^{\pm}_{\text{o},\text{diff}}.$
(7)

Eq. (7) indicates that we can express any given state with $C\neq 0$ as a
stack of $C$ filled LLs (which contribute $\Theta^{\pm}_{\text{o},\text{LL}}$)
and a state with zero Chern number (which contributes
$\Theta^{\pm}_{\text{o},\text{diff}}$). Thus,
$\Theta^{\pm}_{\text{o},\text{diff}}$ specifies how the given state differs
from the limit of $C$ filled LLs. The empirical formulas for
$\Theta^{+}_{\text{o},\text{diff}}$ throughout the butterfly are summarized in
App. A. We have not obtained empirical formulas for
$\Theta^{-}_{\text{o},\text{diff}}$ as these are substantially more
complicated.

##### Verification using conformal field theory.

Eqs. (3), (4) can be analytically derived by using the cut-and-glue approach
established in [46], which describes the entanglement spectrum of the disk
subregion in the long wavelength limit by that of the (1+1)D CFT on its edge.
That is, the reduced density matrix for the disk subregion $D$ is effectively
given by $\rho_{D}=\rho_{\mathrm{CFT}}$, where $\rho_{\mathrm{CFT}}$ denotes
the CFT on the edge of the disk. The edge of the disk entangled with the
complement subsystem is described by a thermal density matrix of a perturbed
edge CFT within a fixed topological sector [47]. The form of the perturbation
in the entanglement Hamiltonian is not universal. In the following, we assume
that the entanglement Hamiltonian is that of the unperturbed CFT:
$\rho_{\mathrm{CFT}}=e^{-\beta H}$, and check the validity of this assumption
with our numerics.

In accordance with the crystalline equivalence principle [18, 48], the
$\tilde{C}_{M_{\text{o}}}$ rotation symmetry acts as a translation symmetry
combined with an internal $\mathbb{Z}_{M_{\text{o}}}$ symmetry of the CFT on
the boundary of $D$. The expectation value of $\tilde{C}_{M_{\text{o}}}$ for
the disk $D$ without any additional flux insertion is hence evaluated in terms
of the symmetry generators in the CFT as follows:

$\begin{split}&\bra{\Psi}\tilde{C}_{M_{\text{o}}}^{\pm}|_{D}\ket{\Psi}=\frac{\mathrm{Tr}[e^{iQ_{M_{\text{o}}}\frac{\pi}{M_{\text{o}}}}e^{i\tilde{P}\frac{L}{M_{\text{o}}}}e^{-\frac{\xi}{v}H}]}{\mathrm{Tr}[e^{-\frac{\xi}{v}H}]}\\\
&=e^{-\frac{2\pi
i}{24M_{\text{o}}}c_{-}}\frac{\sum_{a=1,\psi}\chi_{a}(\frac{i\xi}{L}-\frac{1}{M_{\text{o}}};[\mathrm{AP},0],[\mathrm{AP},1])}{\sum_{a=1,\psi}\chi_{a}(\frac{i\xi}{L};[\mathrm{AP},0],[\mathrm{AP},0])}\end{split}$
(8)

where we introduced the velocity $v$ of the CFT, finite temperature
correlation length of the edge theory $\xi:=\beta v$, the length of the
boundary $L=|\partial D|$, and $\\{\text{AP,P}\\}$ denotes the boundary
condition with respect to $\mathbb{Z}_{2}^{f}$ fermion parity symmetry. The
action of the $M_{\text{o}}$-fold rotation on the CFT is expressed as the
combination
$e^{iQ_{M_{\text{o}}}\frac{\pi}{M_{\text{o}}}}e^{i\tilde{P}\frac{L}{M_{\text{o}}}}$,
where $Q_{M_{\text{o}}}$ generates an internal
$\mathbb{Z}_{2M_{\text{o}}}^{f}$ (resp. $\mathbb{Z}_{M_{\text{o}}}$) symmetry
when we take the rotation symmetry to be $\tilde{C}_{M_{\text{o}}}^{+}$ (resp.
$\tilde{C}_{M_{\text{o}}}^{-}$). $\tilde{P}$ is the normalized translation
operator

$\displaystyle\tilde{P}:=\frac{1}{v}(H-E_{0})=\frac{2\pi}{L}\left[L_{0}-\frac{c_{-}}{24}-\langle
L_{0}-\frac{c_{-}}{24}\rangle\right]$ (9)

so that $\tilde{P}\ket{\mathrm{vac}}=0$ on the vacuum state
$\ket{\mathrm{vac}}$ of the CFT.
$\chi_{a}(\tau;[s,j],[s^{\prime},j^{\prime}])$ with
$s,s^{\prime}\in\\{\mathrm{AP},\mathrm{P}\\}$,
$j,j^{\prime}\in\mathbb{Z}_{M_{\text{o}}}$ is the CFT character that
corresponds to the partition function on a torus equipped with spin structure
and $\mathbb{Z}_{M_{\text{o}}}$ gauge field. As shown in App. C, the above CFT
characters at high temperature $\xi\ll L$ can be evaluated using the modular
$S,T$ matrices of the $G$-crossed braided fusion category [8, 23] describing
the invertible phase. Our calculation uses the precise defect sectors of the
$G$-crossed modular $S,T$ matrices that correspond to the boundary conditions
which define the CFT character; the use of $G$-crossed modularity allows us to
readily treat any symmetry action on the state encoded in the classification
of fermionic invertible phases in (2+1)D [9, 10], some of whose details are
missed in Ref. [24]. The results for even $M_{\text{o}}$ are to leading order
given by

$\displaystyle\bra{\Psi}\tilde{C}_{M_{\text{o}}}^{+}|_{D}\ket{\Psi}_{\text{CFT}}$
$\displaystyle\propto e^{-\frac{2\pi
i}{24}(M_{\text{o}}-\frac{1}{M_{\text{o}}})c_{-}}\mathcal{I}_{M_{\text{o}}}^{+}=e^{\frac{2\pi
i}{M_{\text{o}}}\Theta_{\text{o}}^{+}}$
$\displaystyle\bra{\Psi}\tilde{C}_{M_{\text{o}}}^{-}|_{D}\ket{\Psi}_{\text{CFT}}$
$\displaystyle\propto e^{-\frac{2\pi
i}{24}(M_{\text{o}}+\frac{2}{M_{\text{o}}})c_{-}}\mathcal{I}_{M_{\text{o}}}^{-}=e^{\frac{2\pi
i}{M_{\text{o}}}\Theta_{\text{o}}^{-}}$ (10)

where

$\displaystyle\mathcal{I}_{M_{\text{o}}}^{\pm}:=e^{\frac{2\pi
i}{M_{\text{o}}}\frac{\ell_{s,\text{o}}^{\pm}}{2}}=e^{\frac{2\pi
i}{M_{\text{o}}}\left((1\mp 1)\frac{k_{s,\text{o}}}{4}+(1\pm
1)\frac{c_{-}}{16}+k_{3,\text{o}}^{\pm}\right)}$ (11)

for integers $k^{\pm}_{3,\text{o}}\in\mathbb{Z}_{M_{\text{o}}}$ and
$k_{s,\text{o}}\in\mathbb{Z}_{2}$. Eqs. (10) and (11) directly give Eqs. (3),
(4). Moreover, we have an equivalence $k^{+}_{3,\text{o}}\simeq
k^{+}_{3,\text{o}}+M_{\text{o}}/2$ which comes from relabelling symmetry
defects in the CFT by fermions, but no such equivalence for
$k^{-}_{3,\text{o}}$. Combining Eqs. (10) and (11) gives

$\displaystyle\ell^{\pm}_{s,\text{o}}$ $\displaystyle=\begin{cases}\frac{11\mp
1}{8}C+2\Theta^{\pm}_{\text{o}}\mod 4\quad\text{o}=\alpha,\beta\\\ \frac{1\mp
1}{4}C+2\Theta^{\pm}_{\text{o}}\mod 2\quad\text{o}=\gamma.\end{cases}$ (12)

##### Relation to topological action.

$\ell^{\pm}_{s,\text{o}}$ appear in a general topological action which was
derived for (2+1)D bosonic topological phases with symmetry
$G=\text{U}(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{M}]$ in Refs.
[22, 23], and extended to invertible fermionic systems in the appendix of Ref.
[26]. It is written in terms of a $\text{U}(1)$ gauge field $A$, and
crystalline gauge fields $(\vec{R},\omega)$. $\omega$ is a background
‘rotation’ gauge field, which is treated as a real field with quantized
holonomies. We can define $\omega$ as a $\mathbb{Z}_{M}$ gauge field or a
$\mathbb{Z}_{2M}$ gauge field corresponding to the subgroups of $G$ generated
by $\tilde{C}_{M_{\text{o}}}^{+}$ and $\tilde{C}_{M_{\text{o}}}^{-}$
respectively. We use $+$ and $-$ superscripts for the coefficients that appear
in the action in each case. The coefficients with supercript $-$ can be
obtained by replacing $A\rightarrow A+\omega/2$ in the action with only $+$
superscripts.

The full action includes a term
$\frac{\ell_{s,\text{o}}^{\pm}-c_{-}/12}{4\pi}\omega\wedge d\omega$, where
$\ell^{\pm}_{s,\text{o}}$ is quantized mod $M_{\text{o}}$, while
$-\frac{c_{-}}{12}$ is a contribution from the framing anomaly [49, 50]. The
quantities $\ell^{\pm}_{s,\text{o}}$ determine the invariant of invertible
phases protected purely by rotations about o [19], and arise in the CFT
computation above through the $G$-crossed modular $S,T$ matrices.
$\ell^{\pm}_{s,\text{o}}$ are directly related to $\Theta^{\pm}_{\text{o}}$,
by Eq. (12). We defer a discussion of the remaining topological response
coefficients, which include an angular momentum polarization
$\vec{\mathscr{P}}_{s,\text{o}}^{\pm}$ and angular momentum filling
$\nu_{s}^{\pm}$, to Ref. [29].

For continuum Landau levels, $\ell^{\pm}_{s,\text{LL}}$ is an origin-
independent $\mathbb{Z}$ invariant associated to continuous $\text{SO}(2)$
rotational symmetry. Now $\ell^{+}_{s,\text{LL}}$ is the coefficient of the
term $\frac{1}{4\pi}\omega\wedge d\omega$ that arises in the effective action
upon integrating out the fermion fields [45],
$\mathcal{L}=\frac{1}{4\pi}\sum_{n=1}^{C}(A+s_{n}\omega)\wedge
d(A+s_{n}\omega)$, where $s_{n}=\frac{2n-1}{2}$ is the spin of the fermion in
the $n$th LL; and $\ell^{-}_{s,\text{LL}}$ is similarly obtained after
replacing $A\rightarrow A+\omega/2$. Moreover, in the continuum limit of the
Hofstadter model, we have
$\ell^{\pm}_{s,\text{o},\text{LL}}=\ell^{\pm}_{s,\text{LL}}\mod M_{\text{o}}$.
After simplification (see App. D), we obtain

$\displaystyle\ell^{+}_{s,\text{o},\text{LL}}$
$\displaystyle=\frac{C^{3}}{3}-\frac{C}{12}\mod M_{\text{o}}$
$\displaystyle\ell^{-}_{s,\text{o},\text{LL}}$
$\displaystyle=\frac{C^{3}}{3}+\frac{C^{2}}{2}+\frac{C}{3}\mod M_{\text{o}}.$
(13)

Combining Eq. (Relation to topological action.) and Eq. (12) provides the
analytical derivation of Eqs. (5) and (6).

##### Classification.

There are various relationships involving $\\{\Theta^{\pm}_{\text{o}}\\}$:

$\displaystyle 2(\Theta^{-}_{\text{o}}-\Theta^{+}_{\text{o}})$
$\displaystyle=\mathscr{S}^{+}_{\text{o}}\mod M_{\text{o}},$ (14)
$\displaystyle\Theta_{\alpha}^{+}+\Theta_{\beta}^{+}$
$\displaystyle=\Theta_{\alpha}^{-}+\Theta_{\beta}^{-}+2\Theta_{\gamma}^{-}-\frac{\kappa}{2}+C\mod
2.$ (15)

Eqs. (14), (15) can be derived in the LL limit (where $\kappa=0$ but $C\neq
0$) using Eq. (5), (6) and in the case of $C=0$ using a real-space Wannier
function argument [29]; combining the two results using linearity under
stacking gives (14), (15). Eq. (15) implies that the 5 invariants
$\\{\Theta^{+}_{\alpha},\Theta^{+}_{\beta},\Theta^{-}_{\alpha},\Theta^{-}_{\beta},\Theta^{-}_{\gamma}\\}$
define 4 independent invariants.

Eqs. (3), (4) imply that $I_{1}:=2\Theta_{\alpha}^{-}+C/2$,
$I_{2}:=\Theta_{\alpha}^{+}+C/2$, and $I_{3}:=2\Theta_{\gamma}^{-}+C/2$ are
integers modulo $8$, $2$, and $4$, respectively. Additionally,
$I_{4}:=\Theta_{\alpha}^{-}+\Theta_{\beta}^{-}+2\Theta_{\gamma}^{-}-\kappa/2$
is an integer modulo $4$. This can also be established in the LL limit using
(5), (6) and in the case of $C=0$ using a real-space Wannier function
argument, and finally by combining the two using linearity under stacking.

The above results imply that
$\\{I_{1},I_{2},I_{3},I_{4}\\}\in\mathbb{Z}_{8}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}$.
But note that the overall classification of invertible fermionic phases with
$G=\text{U}(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{4}]$ is a group
$\mathbb{Z}^{3}\times\mathbb{Z}_{8}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}^{2}$
[20, 29]. The three $\mathbb{Z}$ invariants are $c_{-}$, $C$,
$\kappa=\nu-\frac{C\phi}{2\pi}$. A subset of this classification which assumes
$c_{-}=0,C=0$ was derived in Ref. [20] through a real space construction. The
full derivation will be explained in detail in a forthcoming work [29], which
also shows that $I_{1},I_{2},I_{3},I_{4}$ are independent. From this, we can
conclude that $\\{c_{-},C,\kappa,I_{1},I_{2},I_{3},I_{4}\\}$, or equivalently
$\\{c_{-},C,\kappa,\Theta^{\pm}_{\text{o}}\\}$ fully characterize invertible
fermionic states with symmetry group $G$.

##### Discussion.

We have obtained the remarkable result that we can extract a complete set of
invariants (apart from $C,c_{-},\nu$) from partial rotations, without
inserting any extra magnetic flux or lattice defects. As a corollary, we can
extract the previously studied invariants
$\mathscr{S}^{+}_{\text{o}},\vec{\mathscr{P}}^{+}_{\text{o}}$ in terms of
$\\{\Theta^{\pm}_{\text{o}},C,c_{-},\nu\\}$ as well, using Eq. (14) and the
relationships between $\mathscr{S}^{+}_{\text{o}}$ and
$\vec{\mathscr{P}}^{+}_{\text{o}}$ derived in [27]. We also showed how to
understand the topological response theory coefficients
$\ell^{\pm}_{s,\text{o}}$ in terms of $\Theta^{\pm}_{\text{o}}$.

We can instead study the eigenvalues of a global rotation operator on a torus;
this was used in [26, 27] to find
$\mathscr{S}^{+}_{\text{o}},\vec{\mathscr{P}}^{+}_{\text{o}}$, and also
discussed in Ref. [28]. Some preliminary results for global rotations are
shown in App. B; it is unclear how much of the classification can be obtained
in this way.

It is worth highlighting that we have extracted
$\\{\Theta^{\pm}_{\text{o}}\\}$ from a single wave function without requiring
the insertion of additional magnetic flux or lattice defects. Note that one
can also extract from a single wave function the quantities $c_{-}$ [47, 38,
37, 35], $C$ [24, 32, 33, 36], and $\nu$. The above discussion then implies
that one can in principle extract all the many-body invariants characterizing
an invertible state with
$G=\text{U}(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{M}]$ from a
single wavefunction.

Above, we focussed on the square lattice, with $M=4$ and $M_{\text{o}}=2$ or
4. For any orientation presrving space group symmetry, we have the following
results [29]. If $M_{\text{o}}=2,4,6$, $\Theta^{+}_{\text{o}}$ defines a
$\mathbb{Z}_{M_{\text{o}}/2}$ invariant, while $\Theta^{-}_{\text{o}}$ defines
a $\mathbb{Z}_{2M_{\text{o}}}$ invariant. (This means that if we know $C$,
$\Theta^{+}_{\text{o}}$ and $\Theta^{-}_{\text{o}}$ can take $M_{\text{o}}/2$
and $M$ values respectively, although their precise quantization will depend
on $C$.) On the other hand, if $M_{\text{o}}=3$, both $\Theta^{+}_{\text{o}}$
and $\Theta^{-}_{\text{o}}$ define $\mathbb{Z}_{M_{\text{o}}}$ invariants. In
either case, $\\{\Theta^{\pm}_{\text{o}}\\}$ along with $C,\nu$ and $c_{-}$,
fully characterize the many-body crystalline invariants of systems with
symmetry $\text{U}(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{M}]$, and
these can be extracted using a single wave function.

##### Acknowledgements.

This work is supported by NSF CAREER grant (DMR- 1753240), and the Laboratory
for Physical Sciences through the Condensed Matter Theory Center. RK is
supported by a JQI postdoctoral fellowship at the University of Maryland.

## References

  * Kitaev [2009] A. Kitaev, Periodic table for topological insulators and superconductors, AIP Conference Proceedings 1134, 22 (2009), https://aip.scitation.org/doi/pdf/10.1063/1.3149495 .
  * Ryu  _et al._ [2010] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010).
  * Chen  _et al._ [2013] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87, 155114 (2013).
  * Kapustin [2014] A. Kapustin, Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology, arXiv preprint arXiv:1403.1467 (2014).
  * Kapustin  _et al._ [2015] A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, Fermionic symmetry protected topological phases and cobordisms, Journal of High Energy Physics 2015, 1 (2015).
  * 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).
  * Senthil [2015] T. Senthil, Symmetry Protected Topological phases of Quantum Matter, Ann. Rev. Condensed Matter Phys. 6, 299 (2015), arXiv:1405.4015 [cond-mat.str-el] .
  * 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), arXiv:1410.4540 .
  * Barkeshli  _et al._ [2022] M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and N. Manjunath, Classification of $(2+1)$d invertible fermionic topological phases with symmetry, Phys. Rev. B 105, 235143 (2022).
  * 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] .
  * Bulmash and Barkeshli [2022] D. Bulmash and M. Barkeshli, Fermionic symmetry fractionalization in $(2+1)$ dimensions, Phys. Rev. B 105, 125114 (2022).
  * Chiu  _et al._ [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  * Cheng  _et al._ [2016] M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, and P. Bonderson, Translational symmetry and microscopic constraints on symmetry-enriched topological phases: A view from the surface, Phys. Rev. X 6, 041068 (2016).
  * Bradlyn  _et al._ [2017] B. Bradlyn, L. Elcoro, and e. a. Jennifer Cano, Topological quantum chemistry, Nature 547, 298 (2017).
  * Po  _et al._ [2017] H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nature Communications 8, 10.1038/s41467-017-00133-2 (2017).
  * Kruthoff  _et al._ [2017] J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Topological classification of crystalline insulators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017).
  * Huang  _et al._ [2017] S.-J. Huang, H. Song, Y.-P. Huang, and M. Hermele, Building crystalline topological phases from lower-dimensional states, Phys. Rev. B 96, 205106 (2017).
  * 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).
  * Cheng and Wang [2022] M. Cheng and C. Wang, Rotation symmetry-protected topological phases of fermions, Phys. Rev. B 105, 195154 (2022).
  * Zhang  _et al._ [2022a] 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, Phys. Rev. Res. 4, 033081 (2022a).
  * Elcoro  _et al._ [2021] L. Elcoro, B. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, Magnetic topological quantum chemistry, Nature Communications 12, https://doi.org/10.1038/s41467-021-26241-8 (2021).
  * 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 (2020), arXiv:2012.11603 [cond-mat.str-el] .
  * 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).
  * Shapourian  _et al._ [2017] H. Shapourian, K. Shiozaki, and S. Ryu, Many-body topological invariants for fermionic symmetry-protected topological phases, Phys. Rev. Lett. 118, 216402 (2017).
  * Zhang  _et al._ [2022b] Y. Zhang, N. Manjunath, G. Nambiar, and M. Barkeshli, Fractional disclination charge and discrete shift in the hofstadter butterfly, Phys. Rev. Lett. 129, 275301 (2022b).
  * Zhang  _et al._ [2022c] Y. Zhang, N. Manjunath, G. Nambiar, and M. Barkeshli, Quantized charge polarization as a many-body invariant in (2+1)d crystalline topological states and hofstadter butterflies (2022c), 2211.09127 .
  * Herzog-Arbeitman  _et al._ [2022] J. Herzog-Arbeitman, B. A. Bernevig, and Z.-D. Song, Interacting topological quantum chemistry in 2d: Many-body real space invariants (2022), arXiv:2212.00030 [cond-mat.str-el] .
  * Manjunath  _et al._ [2023a] N. Manjunath, V. Calvera, and M. Barkeshli, to appear (2023a).
  * Levin and Wen [2006] M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).
  * Kitaev and Preskill [2006] A. Kitaev and J. Preskill, Topological entanglement entropy, Physical review letters 96, 110404 (2006).
  * Dehghani  _et al._ [2021] H. Dehghani, Z.-P. Cian, M. Hafezi, and M. Barkeshli, Extraction of the many-body chern number from a single wave function, Physical Review B 103, 075102 (2021).
  * Cian  _et al._ [2021] Z.-P. Cian, H. Dehghani, A. Elben, B. Vermersch, G. Zhu, M. Barkeshli, P. Zoller, and M. Hafezi, Many-body chern number from statistical correlations of randomized measurements, Physical Review Letters 126, 050501 (2021).
  * Cian  _et al._ [2022] Z.-P. Cian, M. Hafezi, and M. Barkeshli, Extracting wilson loop operators and fractional statistics from a single bulk ground state (2022), arXiv:2209.14302 [cond-mat.str-el] .
  * Kim  _et al._ [2022] I. H. Kim, B. Shi, K. Kato, and V. V. Albert, Chiral central charge from a single bulk wave function, Phys. Rev. Lett. 128, 176402 (2022).
  * Fan  _et al._ [2022] R. Fan, R. Sahay, and A. Vishwanath, Extracting the quantum hall conductance from a single bulk wavefunction (2022), arXiv:2208.11710 .
  * Tu  _et al._ [2013] H.-H. Tu, Y. Zhang, and X.-L. Qi, Momentum polarization: An entanglement measure of topological spin and chiral central charge, Physical Review B 88, 10.1103/physrevb.88.195412 (2013), arXiv:1212.6951 .
  * Zaletel  _et al._ [2013] M. P. Zaletel, R. S. K. Mong, and F. Pollmann, Topological characterization of fractional quantum hall ground states from microscopic hamiltonians, Phys. Rev. Lett. 110, 236801 (2013).
  * Kobayashi  _et al._ [2023] R. Kobayashi, T. Wang, T. Soejima, R. S. K. Mong, and S. Ryu, Extracting higher central charge from a single wave function (2023), arXiv:2303.04822 [cond-mat.str-el] .
  * Zaletel [2014] M. P. Zaletel, Detecting two-dimensional symmetry-protected topological order in a ground-state wave function, Physical Review B 90, 10.1103/physrevb.90.235113 (2014).
  * Shiozaki  _et al._ [2018] K. Shiozaki, H. Shapourian, K. Gomi, and S. Ryu, Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries, Physical Review B 98, 10.1103/physrevb.98.035151 (2018).
  * Kobayashi and Shiozaki [2019] R. Kobayashi and K. Shiozaki, Anomaly indicator of rotation symmetry in (3+1)d topological order (2019), arXiv:1901.06195 .
  * 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).
  * Note [1] Note we will abuse language somewhat and use o both as the rotation center and as a maximal Wyckoff position.
  * 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).
  * Qi  _et al._ [2012] X.-L. Qi, H. Katsura, and A. W. W. Ludwig, General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states, Physical Review Letters 108, 10.1103/physrevlett.108.196402 (2012), arXiv:1103.5437 [cond-mat.mes-hall] .
  * Li and Haldane [2008] H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states, Phys. Rev. Lett. 101, 010504 (2008).
  * Manjunath  _et al._ [2023b] N. Manjunath, V. Calvera, and M. Barkeshli, Non-perturbative constraints from symmetry and chirality on majorana zero modes and defect quantum numbers in (2+1)d (2023b), arXiv:2210.02452 [cond-mat.str-el] .
  * Witten [1989] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121, 351 (1989).
  * 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).

##  Appendix A Further details on the numerics

###  A.1 Definition of vector potential

In this section we provide a definition of the vector potential $A$ for the
square lattice Hofstadter model on a torus. Some of the following details have
previously appeared in Ref. [26]. We first define our vector potential on a
$L\times L$ torus with $L$ even. Considering even $L$ is sufficient for our
calculations on the torus. An example of the gauge choice is shown in Fig. 4.
We can use a similar gauge to define $A$ on an open disc.

Figure 4: $L\times L=6\times 6$ square lattice with periodic boundary
condition indicated by the dotted bonds. We insert $2\pi m$ flux uniformly
through the torus, where $m=\frac{\phi}{2\pi}L^{2}$. Each blue arrow
represents a hopping phase $A_{ij}=\frac{2\pi m}{L^{2}}$; each red arrow
represents a hopping phase $A_{ij}=\frac{2\pi m}{L}$.
$\text{o}_{1},\text{o}_{2}$ are the rotation centers. The $3\times 3$ partial
rotation region is defined around $\text{o}_{2}$ and is colored blue.

###  A.2 Choice of global rotation operator

In systems with $\text{U}(1)$ and point group rotation symmetries, we have the
ambiguity that any rotation symmetry operator can be multiplied by a phase
$e^{i\theta\hat{N}}$ to give another valid symmetry operator. It is necessary
to fix this ambiguity in order to uniquely define a rotation operator. Below
we illustrate how to do this on the infinite plane, and then on the torus.

####  A.2.1 On the infinite plane

Ref. [26] showed how to canonically define one such operator
($\tilde{C}_{M_{\text{o}}}^{+}$ in the notation of this paper) on the infinite
plane, which assigns zero excess flux around a disclination of angle
$2\pi/M_{\text{o}}$ created at o using $\tilde{C}_{M_{\text{o}}}^{+}$. Since
the defect Hamiltonian depends on the specific choice of $\text{U}(1)$ gauge
transformation defining $\tilde{C}_{M_{\text{o}}}^{+}$, considering a
different operator $e^{i\chi\hat{N}}\tilde{C}_{M_{\text{o}}}^{+}$ modifies the
flux around the disclination by $\chi\mod 2\pi$, and therefore we can
physically distinguish it from $\tilde{C}_{M_{\text{o}}}^{+}$. See Ref. [26]
for the explicit construction of a disclination Hamiltonian with fourfold
rotational symmetry.

Let us specialize to the case $M_{\text{o}}=4$ and
$(\tilde{C}^{+}_{M_{\text{o}}})^{4}=+1$. Assume each plaquette on the infinite
plane has flux $\phi\mod 2\pi$. If $\text{o}=\beta$ (a vertex), the four
possible operators assign equal flux $\phi\mod 2\pi$ to all plaquettes in the
lattice with a $\pi/2$ disclination, except one that lies next to the
disclination. In this plaquette the assigned flux is $\phi+k\frac{\pi}{2}\mod
2\pi$, where $k=0,1,2,3$. Thus there is a canonical choice of
$\tilde{C}^{+}_{M_{\beta}}$, corresponding to $k=0$ (meaning zero excess
flux). When $\text{o}=\alpha$ (a plaquette center), there is a triangular
plaquette at a $\pi/2$ disclination which is assigned a flux
$\frac{3\phi}{4}+k\frac{\pi}{2}\mod 2\pi$. The operator
$\tilde{C}^{+}_{M_{\alpha}}$ for which $k=0$ is also canonical, because it
assigns a flux at the disclination that is proportional to $\phi$.

To extract the desired topological invariants in this paper, it turns out to
be insufficient to use the operators $\tilde{C}_{M_{\text{o}}}^{+}$ alone,
because they only generate groups of order $M_{\text{o}}$. Thus we also define
$\tilde{C}_{M_{\text{o}}}^{-}:=e^{i\frac{\pi}{M_{\text{o}}}\hat{N}}\tilde{C}_{M_{\text{o}}}^{+}$,
which is of order $2M_{\text{o}}$ and satisfies
$(\tilde{C}_{M_{\text{o}}}^{-})^{M_{\text{o}}}=(-1)^{F}$. The two sets of
operators, together denoted $\tilde{C}_{M_{\text{o}}}^{\pm}$, are indeed
sufficient for our purpose. Arbitrary $\text{U}(1)$ rotations can be
considered by defining
$\tilde{C}_{M_{\text{o}},\chi}^{\pm}=e^{i\chi\frac{2\pi}{M_{\text{o}}}\hat{N}}\tilde{C}_{M_{\text{o}}}^{\pm}$.
Note that
$\tilde{C}_{M_{\text{o}},\chi}^{+},\tilde{C}_{M_{\text{o}},\chi}^{-}$ insert
excess flux $2\pi\chi/M_{\text{o}}$ at a $2\pi/M_{\text{o}}$ disclination,
compared to $\tilde{C}_{M_{\text{o}}}^{+}$ and $\tilde{C}_{M_{\text{o}}}^{-}$
respectively.

####  A.2.2 On the torus

Defining a unique rotation operator on a torus with fourfold rotational
symmetry is slightly more subtle than on the infinite plane, because any order
4 rotation preserves 2 points $\text{o}_{1},\text{o}_{2}$. If the side length
$L$ of the torus is even, the two fixed points are either both at $\alpha$ or
both at $\beta$. If $L$ is odd, we have one fixed point each at $\alpha$ and
$\beta$.

$\tilde{C}_{M_{\text{o}}}^{+}$ can be defined for $\text{o}=\text{o}_{1}$ or
$\text{o}_{2}$ by restricting to a disc $D$ centered around o, and creating a
$\pi/2$ disclination within $D$ centered at o, using
$\tilde{C}_{M_{\text{o}}}^{+}|_{D}$. As in the case of the infinite plane
discussed above, we canonically define $\tilde{C}_{M_{\text{o}}}^{+}$ such
that $\tilde{C}_{M_{\text{o}}}^{+}|_{D}$ creates a disclination with zero
excess flux. We then define
$\tilde{C}_{M_{\text{o}}}^{-}:=e^{i\frac{\pi}{M_{\text{o}}}\hat{N}}\tilde{C}_{M_{\text{o}}}^{+}$
and
$\tilde{C}_{M_{\text{o}},\chi}^{\pm}=e^{i\chi\frac{2\pi}{M_{\text{o}}}\hat{N}}\tilde{C}_{M_{\text{o}}}^{\pm}$.

###  A.3 Partial rotation on a torus

Given the global rotation operators $\tilde{C}_{M_{\text{o}},\chi}^{\pm}$, we
wish to restrict them to a region $D$, hence obtaining
$\tilde{C}_{M_{\text{o}},\chi}^{\pm}|_{D}$. Note that the chosen operator
inserts excess flux $2\pi\chi/M_{\text{o}}$ in a $2\pi/M_{\text{o}}$
disclination centered at o relative to $\tilde{C}_{M_{\text{o}}}^{\pm}|_{D}$.
We find that the result for $l^{\pm}_{D,\text{o},\chi}$ follows Eq. (1).

###  A.4 Partial rotation on an open disc

We can also perform the partial rotation calculation on an open disc. This can
be done by first defining the vector potential similarly as in Fig. 4, using a
gauge which inserts $\phi$ flux per plaquette in some region. As shown in Fig.
5, It is not necessary that the vector potential has support on the whole
disc: the flux needs to be inserted only in an invariant subregion around o.
Besides, $D$ does not have to be defined around the center of the subregion
with flux, as long as $D$ is enclosed by this subregion. The center of $D$,
which we define as o, can be at any maximal Wyckoff point.

On the torus, we have a global rotation symmetry operator
$\tilde{C}_{M_{\text{o}}}^{\pm}$, and we define
$\tilde{C}_{M_{\text{o}}}^{\pm}|_{D}$ to be the restriction of
$\tilde{C}_{M_{\text{o}}}^{\pm}$ to region $D$. On the disc, this cannot be
done as there might not be a global symmetry operator to start with.
Therefore, we first choose $D$ and then define an operator
$\tilde{C}_{M_{\text{o}}}^{\pm}|_{D}$, which commutes with the Hamiltonian
restricted to $D$. We find that choosing linear size $L_{D}$ of $D$ to be too
large ($L_{D}\geq L-1$) or too small ($L_{D}\leq 2$) gives unquantized
$K^{\pm}_{\text{o}}$, which is tested this for $L\leq 30$. As before, we need
to impose a constraint on $\tilde{C}_{4,\text{o}}|_{D}$ to fix the ambiguity
by a global $\text{U}(1)$ phase $e^{i\theta\hat{N}}$. This can be done by
fixing the excess flux inserted around a disclination created using the
$\tilde{C}_{M_{\text{o}}}^{\pm}|_{D}$ operator as explained in the previous
section. We define $\tilde{C}_{M_{\text{o}},\chi}^{+}$ (respectively
$\tilde{C}_{M_{\text{o}},\chi}^{-}$) so that it inserts an excess flux of
$2\pi\chi/M_{\text{o}}$ at a $2\pi/M_{\text{o}}$ disclination, relative to
$\tilde{C}_{M_{\text{o}}}^{+},\tilde{C}_{M_{\text{o}}}^{-}$.

Note that, we obtain the same result, Eq. (1), on the torus or the open disc,
as long as we choose o at the same maximal Wyckoff position, and choose a
partial rotation operator that inserts the same flux around a disclination at
o.

Figure 5: Calculation on an open disc. Red shading indicates the sub-region
where the flux is inserted. Blue shading inside the red region indicates
different choices of maximal Wyckoff point as the partial rotation center.

###  A.5 Details on the jumps of $K_{\text{o}}^{+}$

Figure 6: Raw data for $K_{\beta}$, calculated on a $30\times 30$ open disc.
Partial rotation region $D$ is a $15\times 15$ square. $K_{\beta}$ jumps by 2
within a single lobe whenever the color switches between bright and dim
shades.

As stated in the main text, if we use the operators
$\tilde{C}_{M_{\text{o}}}^{+}$, $K^{+}_{\text{o}}$ sometimes jumps by
$M_{\text{o}}/2$ inside a single Hofstadter lobe, as seen in Fig. 6. We define
$\Theta_{\text{o}}^{+}\equiv K^{+}_{\text{o}}\mod M_{\text{o}}/2$ in order to
eliminate these jumps. In general, the position of these jumps will depend on
the system size and the size of the partial rotation region, but the value of
these jumps is always $M_{\text{o}}/2$. We can understand these jumps as
follows. The following arguments are equivalent to those given in [28] using
the many-body real space invariants defined there.

Suppose $M_{\text{o}}=4$ in a system where all orbitals are filled ($\nu=1$).
First consider $K^{+}_{\text{o}}$. Let $K_{\text{o}}^{+}|_{\nu=1}$ be the
value of $K^{+}_{\text{o}}$ at $\nu=1$. Now suppose we minimally enlarge the
partial rotation region $D$ by enclosing $4$ more sites which rotate into each
other Under $\tilde{C}_{M_{\text{o}}}^{+}$. The states at these 4 sites form
an orbit of size 4 under $\tilde{C}_{M_{\text{o}}}^{+}$ and contribute extra
phases of $\\{1,e^{i2\pi/4},e^{i4\pi/4},e^{i6\pi/4}\\}$, which are fourth
roots of 1. The expectation value of $\tilde{C}_{M_{\text{o}}}^{+}$ will gain
a total extra phase of $e^{i\pi}$ (which is the product of the 4 phases), and
thus $K_{\text{o}}^{+}|_{\nu=1}$ changes by 2 mod 4. But a change of the size
of the partial rotation region should be regarded as trivial. Therefore only
$K^{+}_{\text{o}}\mod 2$ is an invariant quantity.

On the other hand, considering $\tilde{C}_{M_{\text{o}}}^{-}$ in the same
argument, the states at the 4 extra sites in the enlarged region contribute
phases of $\\{e^{i\pi/4},e^{i3\pi/4},e^{i5\pi/4},e^{i7\pi/4}\\}$, which are
fourth roots of -1. The total phase contribution from these four sites is 1,
and thus $K_{\text{o}}^{-}$ is invariant upon changing $D$.

Since changing $D$ is trivial, the above example suggests that
$\Theta_{\text{o}}^{+}$ is an invariant mod $M_{\text{o}}/2$, and
$\Theta_{\text{o}}^{-}$ is an invariant mod $M_{\text{o}}$. In the latter case
we indeed verify numerically that there are no jumps of $K^{-}_{\text{o}}$
inside a given Hofstadter lobe.

###  A.6 Periodicities of $\Theta_{\text{o}}^{\pm}$

Figure 7: Symmetric gauge defined with respect to (a) $\alpha$; (b) $\beta$.
Each blue arrow represents the vector potential $\phi/4$.

We find that as a function of the background flux per unit cell in the
Hofstadter model, $\Theta^{\pm}_{\beta}$ has periodicity $2\pi$, while
$\Theta^{\pm}_{\alpha}$ has periodicity $8\pi$. We can straightforwardly see
why these invariants need not be $2\pi$ periodic in $\phi$. Consider a system
on the infinite plane, with flux $\phi$ per unit cell. For a given origin o,
it is most convenient to define the vector potential in symmetric gauge around
o; in this case the operator that trivially rotates points without any
$\text{U}(1)$ gauge transformation is also a symmetry operator that commutes
with the Hamiltonian.

As shown in Fig. 7, if $\text{o}=\beta$, the links are assigned a vector
potential which is an integer multiple of $\pi/2$, while if $\text{o}=\beta$,
the links are assigned a vector potential which is an integer multiple of
$\pi/4$. Therefore $H$ is $4\pi$ periodic in $\phi$ when $\text{o}=\beta$, and
$8\pi$ periodic when $\text{o}=\alpha$. The actual periodicity of
$\Theta^{\pm}_{\beta},\Theta^{\pm}_{\alpha}$ in $\phi$ must therefore be
divisible by $4\pi$ and $8\pi$ respectively. The fact that
$\Theta^{\pm}_{\beta}$ has a smaller periodicity $2\pi$ while
$\Theta^{\pm}_{\alpha}$ has the maximal allowed periodicity of $8\pi$ is a
more subtle result that we see empirically, but do not have a simple
justification for on a lattice without defects.

However, we can consider the flux inserted at a $\pi/2$ disclination created
using $\tilde{C}_{M_{\beta}}^{\pm}$ and $\tilde{C}_{M_{\alpha}}^{\pm}$
respectively. As seen previously, for $\text{o}=\beta$ the flux is always
expressed in terms of integer multiples of $\phi$ (which is $2\pi$ periodic),
while for $\text{o}=\alpha$ the flux is expressed in terms of the quantity
$3\phi/4$ (which is $8\pi$ periodic). Therefore defects of
$\tilde{C}_{M_{\beta}}^{\pm}$ and $\tilde{C}_{M_{\alpha}}^{\pm}$ do have the
same periodicities in $\phi$ as $\Theta^{\pm}_{\beta},\Theta^{\pm}_{\alpha}$.
This gives additional justification for the observed periodicities.

###  A.7 Obtaining empirical formulas for $\Theta_{\text{o}}^{+}$

In this section, we explain how to obtain empirical formula for
$\Theta_{\alpha}^{+}$ and $\Theta_{\beta}^{+}$. The result is in Eqs. (16),
(17).

In Fig. 3 and 8 we plot the raw Hofstadter butterflies for
$\Theta_{\text{o}}^{\pm}$. For a fixed Chern number $C$, the different
Hofstadter lobes are separated by the so called Farey sequence of order
$2|C|$, which consists of ordered irreducible fractions $\frac{p}{q}$ with
$0<p\leq q\leq 2|C|$. For example, the Farey sequence of order 4 is
$\\{\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{1}{1}\\}$.
If we track the lobes at a fixed $C$ as $\phi/2\pi$ is increased from 0 to 1,
$\Theta_{\text{o}}^{\pm}$ may change its value between lobes which meet at the
Farey seq of order $|C|$. (A similar behaviour was observed for the discrete
shift $\mathscr{S}_{\text{o}}^{+}$ in Ref. [26].)

Figure 8: Raw data for $\Theta_{\text{o}}$. $\Theta_{\alpha}^{+}$,
$\Theta_{\beta}^{+}$, and $\Theta_{\gamma}^{+}$ are calculated on a $30\times
30$ open disc. $\Theta_{\gamma}^{+}=0\mod 1$ everywhere in the Hofstadter
butterfly, and is therefore not plotted here. The diameter of the partial
rotation region $D$ is taken to be roughly half the system size.

The jumps for $\Theta_{\alpha}^{+}$ and $\Theta_{\beta}^{+}$ are tabulated in
Figs. 9, 10 and can be categorized into two contributions: A jump of 1 at
every odd denominator $q$ for $q<|C|$; and a possible jump of 1 when $q$
divides $C$. When $C>0$ we find that after summing the contribution from each
jump point,

$\displaystyle\Theta^{+}_{\alpha,\text{diff}}$
$\displaystyle=\left(\sum_{\begin{subarray}{c}\frac{p}{q}<\frac{\phi}{2\pi}\\\
\text{odd
}q\end{subarray}}\left\lfloor\frac{C+q}{2q}\right\rfloor\right)+\begin{cases}\left\lfloor\frac{C\phi}{2\pi}\right\rfloor\text{if
}C\mod 4=2\\\ \left\lfloor\frac{C\phi}{4\pi}\right\rfloor\text{if }C\mod
4=3\\\ 0\text{ if }C\mod 4=0\\\
\left\lfloor\frac{1}{2}+\frac{C\phi}{4\pi}\right\rfloor\text{if }C\mod
4=1\end{cases}$ (16) $\displaystyle\Theta^{+}_{\beta,\text{diff}}$
$\displaystyle=\left(\sum_{\begin{subarray}{c}\frac{p}{q}<\frac{\phi}{2\pi}\\\
\text{odd
}q\end{subarray}}\left\lfloor\frac{C+q}{2q}\right\rfloor\right)+\begin{cases}\left\lfloor\frac{C\phi}{2\pi}\right\rfloor\text{if
}C\mod 4=1\\\ \left\lfloor\frac{C\phi}{4\pi}\right\rfloor\text{if }C\mod
4=2\\\ 0\text{ if }C\mod 4=3\\\
\left\lfloor\frac{1}{2}+\frac{C\phi}{4\pi}\right\rfloor\text{if }C\mod
4=0\end{cases}$ (17) $\displaystyle\Theta^{+}_{\gamma,\text{diff}}$
$\displaystyle=0.$ (18)

These equations are all taken mod $M_{\text{o}}/2$. $\Theta^{+}_{\alpha}$ in
the $C<0$ lobes can be obtained by the symmetry transformation
$\Theta^{+}_{\alpha}(\mu,\phi)=\Theta^{+}_{\alpha}(-\mu,\phi)\mod 2$;
$\Theta^{+}_{\beta}$ in the $C<0$ lobes can be obtained by the symmetry
transformation
$\Theta^{+}_{\beta}(\mu,\phi)=-\Theta^{+}_{\beta}(-\mu,\phi)\mod 2$. We have
not found the analogous formulas for $\Theta^{-}_{\text{o}}$, as the jump
patterns are different and much more complicated in this case.

Figure 9: Jumps in $\Theta_{\alpha}^{+}$ for fixed $C$, as $\frac{\phi}{2\pi}$
increases from 0 to $\frac{1}{2}$. Figure 10: Jumps in $\Theta_{\beta}^{+}$
for fixed $C$, as $\frac{\phi}{2\pi}$ increases form 0 to $\frac{1}{2}$

##  Appendix B Numerical results for global rotations

In this section we present some preliminary numerical results for the
invariants of the Hofstadter model obtained from the eigenvalue under global
rotations, $\bra{\Psi}\tilde{C}_{M_{\text{o}}}^{\pm}\ket{\Psi}$, on the ground
state $\ket{\Psi}$ on an even length torus.

Define $\tilde{C}_{M_{\text{o}}}^{\pm}$ as in the previous section. We define
a set of invariants $\Phi_{\text{o}}^{\pm}$ as follows:

$\bra{\Psi}\tilde{C}_{M_{\text{o}}}^{\pm}\ket{\Psi}=e^{il^{\pm}_{\text{o},\text{global}}},$
(19)

and we empirically find that

$\displaystyle l^{\pm}_{\text{o}_{2},\text{global}}$
$\displaystyle=\frac{Cm^{2}}{2}+\mathscr{S}^{\pm}_{\text{o}_{2}}m+K_{\text{o}_{2},\text{global}}^{\pm}\mod
4$ (20) $\displaystyle l^{\pm}_{\text{o}_{1},\text{global}}$
$\displaystyle=K_{\text{o}_{1},\text{global}}^{\pm}\mod 4.$ (21)

If we fix a given Hofstadter lobe, the quantities
$K_{\text{o},\text{global}}^{+}$ jump by multiples of $M_{\text{o}}/2$ as the
total length of the torus is changed. (Recall that for partial rotations in a
region $D$, the quantities $K_{\text{o}}^{+}$ similarly jump by multiples of
$M_{\text{o}}/2$ for the same system, as the size of $D$ is changed.) However
there is no change in the value of $K_{\text{o},\text{global}}^{-}$. So we
define

$\displaystyle\Phi_{\text{o}}^{+}$
$\displaystyle=K^{+}_{\text{o},\text{global}}\mod M_{\text{o}}/2$
$\displaystyle\Phi_{\text{o}}^{-}$
$\displaystyle=K^{-}_{\text{o},\text{global}}\mod M_{\text{o}}.$ (22)

The raw Hofstadter butterflies for $\\{\Phi_{\text{o}}^{\pm}\\}$ are shown in
Fig. 11. We see that for either $\text{o}=\text{o}_{1}$ or
$\text{o}=\text{o}_{2}$,

$\displaystyle\Phi^{+}_{\alpha}=\Phi^{+}_{\beta}=\Phi^{+}_{\gamma}$
$\displaystyle=\Phi^{-}_{\gamma}=-C\mod 2$ (23)
$\displaystyle\Phi^{-}_{\alpha}=\Phi^{-}_{\beta}$ $\displaystyle=-C\mod 4.$
(24)

Ref. [28] also studied global rotations on an even length torus in terms of a
many-body real space invariant which appears closely related to
$\Phi_{\text{o}}^{\pm}$, and also found that this invariant only depends on
$C$. In the above cases, these invariants do not give any additional
information beyond the Chern number. It is not clear how much more information
can be obtained by also considering odd length tori.

Figure 11: Raw data for $\Phi^{-}_{\alpha}$ and $\Phi^{+}_{\gamma}$,
calculated on a $30\times 30$ torus. They follow Eq. 24 and Eq. 23
respectively.

##  Appendix C CFT calculation

In this appendix, we derive the expression Eq. (10) for the partial rotation
by evaluating the CFT character,

$\begin{split}&\bra{\Psi}\tilde{C}^{\pm}_{M}|_{D}\ket{\Psi}=\frac{\mathrm{Tr}[e^{iQ_{M}\frac{\pi}{M}}e^{i\tilde{P}\frac{L}{M}}e^{-\frac{\xi}{v}H}]}{\mathrm{Tr}[e^{-\frac{\xi}{v}H}]}\\\
&=e^{-\frac{2\pi
i}{24M}c_{-}}\frac{\sum_{a=1,\psi}\chi_{a}(\frac{i\xi}{L}-\frac{1}{M};[\mathrm{AP},0],[\mathrm{AP},1])}{\sum_{a=1,\psi}\chi_{a}(\frac{i\xi}{L};[\mathrm{AP},0],[\mathrm{AP},0])},\end{split}$
(25)

where we set $M$ to be even. For ease of notation, we will use $M$ in this
appendix and suppress the o subscript, but we assume that some o has been
fixed. While comparing with numerics we take $M=M_{\text{o}}=4$ for
$\text{o}=\alpha,\beta$ and $M=2$ for $\text{o}=\gamma$. Here,
$\chi_{a}(\tau;[s,j],[s^{\prime},j^{\prime}])$ with
$s,s^{\prime}\in\\{\mathrm{AP},\mathrm{P}\\}$, $j,j^{\prime}\in\mathbb{Z}_{M}$
is the CFT character that corresponds to the partition function on a torus
equipped with a spin structure and a background $\mathbb{Z}_{M}$ gauge field.
For example,

$\displaystyle\chi_{a}(\tau;[\mathrm{AP},j],[\mathrm{AP},j^{\prime}])=\mathrm{Tr}_{a,[\mathrm{AP},j]}[e^{iQ_{M}\frac{j^{\prime}\pi}{M}}e^{2\pi
i\tau(L_{0}-\frac{c_{-}}{24})}]$ (26)

where $a$ labels the quasiparticle within the AP (antiperiodic) sector, with
the twisted boundary condition corresponding to $j\in\mathbb{Z}_{n}$ along the
spatial cycle.

In order to evaluate the above CFT character, we need to study different cases
of the action of the rotation operator $\tilde{C}^{\pm}_{M}$, i.e., whether
$(\tilde{C}^{+}_{M})^{M}=+1$ or $(\tilde{C}^{-}_{M})^{M}=(-1)^{F}$. This
choice of the symmetry action amounts to considering the twisted or untwisted
spin structure respectively for the $\mathbb{Z}_{M}$ gauge field coupled to
the CFT. That is, the rotation symmetry $\tilde{C}_{M}^{\pm}$ at long
wavelengths is expressed as a combination of an internal $\mathbb{Z}_{M}$
symmetry and the translation symmetry of the CFT:
$\tilde{C}_{M}^{\pm}=e^{iQ_{M}\frac{\pi}{M}}e^{i\tilde{P}\frac{L}{n}}$. Note
that the symmetry action of $\tilde{C}_{M}^{+}$ (resp. $\tilde{C}_{M}^{-}$) is
realized by an internal symmetry satisfying $e^{iQ_{M}\pi}=(-1)^{F}$ (resp.
$e^{iQ_{M}\pi}=1$). The former case corresponds to a $\mathbb{Z}_{2M}^{f}$
symmetry where the $\mathbb{Z}_{M}$ is nontrivially extended by the fermion
parity $(-1)^{F}$, which corresponds to the twisted spin structure, or
equivalently a spin${}^{\mathbb{Z}_{2M}}$ structure. Meanwhile, the latter
corresponds to the untwisted spin structure together with a flat
$\mathbb{Z}_{M}$ gauge field.

The above identification of the spatial symmetry
$\tilde{C}_{M}^{+}(\tilde{C}_{M}^{-})$ with an internal symmetry
$\mathbb{Z}_{2M}^{f}(\mathbb{Z}_{M}\times\mathbb{Z}_{2}^{f})$ can be thought
of a consequence of the crystalline equivalence principle [18, 48]. As we will
see below, these two choices of the symmetry action give rise to different
modular properties of the CFT characters, and hence distinct values of
$\bra{\Psi}\tilde{C}^{\pm}_{M}|_{D}\ket{\Psi}$.

###  C.1 The case with $\tilde{C}_{M}^{+}$

Let us first take the rotation symmetry $\tilde{C}_{M}^{+}$, which has an
equivalent internal $\mathbb{Z}_{2M}^{f}$ symmetry satisfying
$e^{iQ_{M}\pi}=(-1)^{F}$. The CFT character on the edge can be evaluated by
the modular $S,T$ transformations as follows (note that $M$ is even):

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi}{L}-\frac{1}{M};[\mathrm{AP},0],[\mathrm{AP},1]\right)&=S_{ab}\chi_{b}\left(-\frac{1}{\frac{i\xi}{L}-\frac{1}{M}};[\mathrm{AP},1],[\mathrm{AP},0]\right)\\\
&=(ST^{M})_{ab}\chi_{b}\left(\frac{-iM\frac{\xi}{L}}{\frac{i\xi}{L}+\frac{1}{M}};[\mathrm{AP},1],[\mathrm{P},0]\right),\end{split}$
(27)

where we used the fact that $T$ exchanges Spin${}^{\mathbb{Z}_{2M}}$ structure
as follows:

$\displaystyle
T:\quad\begin{cases}([\mathrm{AP},j],[\mathrm{AP},j^{\prime}])&\to([\mathrm{AP},j],[\mathrm{P}+\frac{[j]_{M}+[j^{\prime}]_{M}-[j+j^{\prime}]_{M}}{M},[j+j^{\prime}]_{M}),\\\
([\mathrm{AP},j],[\mathrm{P},j^{\prime}])&\to([\mathrm{AP},j],[\mathrm{AP}+\frac{[j]_{M}+[j^{\prime}]_{M}-[j+j^{\prime}]_{M}}{M},[j+j^{\prime}]_{M}).\\\
\end{cases}$ (28)

Here, $[]_{M}$ denotes the mod $M$ operation, and the spin structure changes
under $\mathbb{Z}_{2}$ action as $\mathrm{AP}+1=\mathrm{P}$,
$\mathrm{P}+1=\mathrm{AP}$.

To evaluate the action of the $S,T$ modular matrices in the presence of a
background $\mathbb{Z}_{2M}^{f}$ gauge field, we note that the modular
transformation in the defect Hilbert space of the CFT can be determined from
the modular $S,T$ matrices of the $G_{b}$-crossed braided fusion category that
describes the bosonic shadow of the invertible phase of interest with bosonic
global symmetry $G_{b}$ in the bulk [8]. In our case, the bosonic symmetry
group is $G_{b}=\mathbb{Z}_{M}$. Using the data of the
$\mathbb{Z}_{M}$-crossed braided fusion category, the $T$-matrix element is
expressed as [8]

$\displaystyle
T^{(\mathbf{g},\mathbf{h})}_{a_{\mathbf{g}},b_{\mathbf{g}}}=e^{-\frac{2\pi
i}{24}c_{-}}\cdot\theta_{a_{\mathrm{g}}}\cdot\eta_{a}(\mathbf{g},\mathbf{h})\cdot\delta_{a_{\mathbf{g}},b_{\mathbf{g}}}$
(29)

where $\mathbf{g},\mathbf{h}\in\mathbb{Z}_{M}$, and
$\eta_{a}(\mathbf{g},\mathbf{h})$ is a phase that describes the symmetry
fractionalization of the bulk topological order [8]. The character is then
further rewritten as

$\displaystyle\begin{split}&\chi_{a}\left(\frac{i\xi}{L}-\frac{1}{M};[\mathrm{AP},0],[\mathrm{AP},1]\right)\\\
&=\sum_{b\in\mathcal{C}_{1}}(ST^{M})_{ab}\chi_{b}\left(\frac{-iM\frac{\xi}{L}}{\frac{i\xi}{L}+\frac{1}{M}};[\mathrm{AP},1],[\mathrm{AP},0]\right)\\\
&=e^{-\frac{2\pi
iM}{24}c_{-}}\sum_{b\in\mathcal{C}_{1}}S_{ab}\theta_{b}^{n}\prod_{j=0}^{M-1}\eta_{b}(1,j)\times\chi_{b}\left(\frac{-iM\frac{\xi}{L}}{\frac{i\xi}{L}+\frac{1}{M}};[\mathrm{AP},1],[\mathrm{P},0]\right)\\\
&=e^{-\frac{2\pi
iM}{24}c_{-}}\sum_{b\in\mathcal{C}_{{1}}}\sum_{c\in\mathcal{C}_{\mathrm{P}}}S_{ab}\theta_{b}^{M}\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right)\times
S_{bc}\chi_{c}\left(\frac{iL}{M^{2}\xi}+\frac{1}{M};[\mathrm{P},0],[\mathrm{AP},1]\right),\end{split}$
(30)

and

$\displaystyle\begin{split}&\chi_{a}\left(\frac{i\xi}{L};[\mathrm{AP},0],[\mathrm{AP},0]\right)=\sum_{b\in\mathcal{C}_{0}}S_{ab}\chi_{b}\left(\frac{iL}{\xi};[\mathrm{AP},0],[\mathrm{AP},0]\right),\end{split}$
(31)

where $\mathcal{C}_{0}$ (resp. $\mathcal{C}_{1}$) is the untwisted (resp.
twisted by $1\in\mathbb{Z}_{M}$) sector in $G_{b}$-crossed extension of super-
modular category that consists of defects in the $[\mathrm{AP},0]$ sector,
namely the identity particle and the fermion (resp. the $[\mathrm{AP},1]$
defects, namely the elementary $\mathbb{Z}_{M}$ fluxes). Meanwhile,
$\mathcal{C}_{\mathrm{P}}$ denotes the defects in the $[\mathrm{P},0]$ sector
that are labelled by fermion parity.

The character in Eq. (31) can further be approximated as

$\displaystyle\begin{split}\chi_{b}\left(\frac{iL}{\xi};[\mathrm{AP},0],[\mathrm{AP},0]\right)&\approx
e^{-\frac{2\pi L}{\xi}(h_{b}-\frac{c_{-}}{24})}~{}.\end{split}$ (32)

The phase of the partial rotation can then be written as

$\displaystyle\begin{split}\bra{\Psi}\tilde{C}_{M}^{+}|_{D}\ket{\Psi}&\propto{e^{-2\pi
i(M+\frac{1}{M})\frac{c_{-}}{24}}}\sum_{b\in\mathcal{C}_{1}}\sum_{c\in\mathcal{C}_{\mathrm{P}}}(S_{1b}+S_{\psi
b})\theta_{b}^{n}\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right)\times
S_{bc}\chi_{c}\left(\frac{iL}{M^{2}\xi}+\frac{1}{M};[\mathrm{P},0],[\mathrm{AP},1]\right).\end{split}$
(33)

While the above expression requires the sum over quasiparticles in the twisted
sector $b\in\mathcal{C}_{1}$, it is more illuminating to express it in terms
of the anyons in the untwisted sector. This is done by using the consistency
equations and Verlinde formula of the $\mathbb{Z}_{M}$-crossed braided fusion
category, given by [8]

$\displaystyle\begin{split}\theta_{b_{1}}&=\theta_{b_{0}}\theta_{0_{1}}\cdot(R^{b_{0},0_{1}}R^{0_{1},b_{0}}),\\\
\eta_{b_{1}}\left(1,j\right)&=\eta_{b_{0}}\left(1,j\right)\eta_{0_{1}}\left(1,j\right),\\\
S_{a,b_{1}}&=\frac{1}{d_{a}}S_{a,b_{0}}(R^{a,0_{-1}}R^{0_{-1},a}),\end{split}$
(34)

where $0_{j}\in\mathcal{C}_{j}$ denotes a $j\in\mathbb{Z}_{M}$ defect. From
the general solution outlined in Ref. [8], one can work in the gauge where
$R^{b,0_{1}}=1$, $R^{c,0_{1}}=1$ for $b\in\mathcal{C}_{0}$,
$c\in\mathcal{C}_{\mathrm{P}}$. We then obtain

$\displaystyle\begin{split}\bra{\Psi}\tilde{C}^{+}_{M}|_{D}\ket{\Psi}\propto&e^{-\frac{2\pi
i}{24}(M+\frac{1}{M})c_{-}}\theta_{0_{1}}^{M}\prod_{j=0}^{M-1}\eta_{0_{1}}\left(1,j\right)\sum_{b\in\mathcal{C}_{0}}\sum_{c\in\mathcal{C}_{\mathrm{P}}}d_{b}\theta_{b}^{M}\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right)\times
S_{bc}\chi_{c}\left(\frac{iL}{M^{2}\xi}+\frac{1}{M};[\mathrm{P},0],[\mathrm{AP},1]\right).\end{split}$
(35)

Here we observe invariants given by the following combination of symmetry
fractionalization data:

$\displaystyle\mathcal{I}_{M}^{\pm}:=\theta_{0_{1}}^{M}\prod_{j=0}^{M-1}\eta_{0_{1}}\left(1,j\right),$
(36)

where the $\pm$ superscript is used depending on whether we consider
$\tilde{C}_{M}^{+}$ or $\tilde{C}_{M}^{-}$, and

$\displaystyle e^{i\pi Q_{b}}:=\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right).$
(37)

They can be interpreted as a $\mathbb{Z}_{M}$ analog of the Hall conductivity,
and a fractional $\mathbb{Z}_{M}$ charge respectively [23]. For
$\frac{L}{\xi}\gg 1$, we approximate the CFT character in terms of the highest
weight state $\ket{h_{b}}$:

$\displaystyle\begin{split}\chi_{c}\left(\frac{iL}{M^{2}\xi}+\frac{1}{M};[\mathrm{P},0],[\mathrm{AP},1]\right)&\approx
e^{\frac{2\pi i}{M}(h_{c}-\frac{c_{-}}{24})}e^{-\frac{2\pi
L}{M^{2}\xi}(h_{c}-\frac{c_{-}}{24})}~{}.\end{split}$ (38)

We then have

$\displaystyle\begin{split}\bra{\Psi}\tilde{C}^{+}_{M}|_{D}\ket{\Psi}\propto&e^{-\frac{2\pi
i}{24}(M+\frac{2}{M})c_{-}}e^{\frac{2\pi
i}{M}h_{v}}\mathcal{I}^{+}_{M}\sum_{b\in\mathcal{C}_{0}}d_{b}\theta_{b}^{M}e^{i\pi
Q_{b}}S_{bv},\end{split}$ (39)

where $v\in\mathcal{C}_{\mathrm{P}}$ is the quasiparticle in the periodic
sector (fermion parity flux) with the lowest value of spin. For the Chern
insulator with chiral central charge $c_{-}$, there are two such defects
$v,v\times\psi$ with equal topological spin $c_{-}/8$. The result with
$\mathcal{C}_{0}=\\{1,\psi\\}$ is then given by

$\displaystyle\begin{split}\bra{\Psi}\tilde{C}^{+}_{M}|_{D}\ket{\Psi}\propto&e^{-\frac{2\pi
i}{24}(M-\frac{1}{M})c_{-}}\mathcal{I}^{+}_{M}.\end{split}$ (40)

###  C.2 The case with $\tilde{C}_{M}^{-}$

Here we consider the case where we take $\tilde{C}_{M}^{-}$ symmetry, which
corresponds to the internal $\mathbb{Z}_{M}$ symmetry satisfying
$e^{iQ_{M}\pi}=1$. The CFT character on the edge can be evaluated by the
modular $S,T$ transformation as (note that $M$ is even)

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi}{L}-\frac{1}{M};[\mathrm{AP},0],[\mathrm{AP},1]\right)&=S_{ab}\chi_{b}\left(-\frac{1}{\frac{i\xi}{L}-\frac{1}{M}};[\mathrm{AP},1],[\mathrm{AP},0]\right)\\\
&=(ST^{M})_{ab}\chi_{b}\left(\frac{-iM\frac{\xi}{L}}{\frac{i\xi}{L}+\frac{1}{M}};[\mathrm{AP},1],[\mathrm{AP},0]\right),\end{split}$
(41)

where we used $T$ exchanges spin$\times{\mathbb{Z}_{M}}$ structure as

$\displaystyle
T:\begin{cases}([\mathrm{AP},j],[\mathrm{AP},j^{\prime}])\to([\mathrm{AP},j],[\mathrm{P},[j+j^{\prime}]_{M}]),\\\
([\mathrm{AP},j],[\mathrm{P},j^{\prime}])\to([\mathrm{AP},j],[\mathrm{AP},[j+j^{\prime}]_{M}]).\\\
\end{cases}$ (42)

Using a similar discussion as the previous subsection, one can write the phase
of the partial rotation as

$\displaystyle\begin{split}\bra{\Psi}\tilde{C}^{-}_{M}|_{D}\ket{\Psi}&\propto{e^{-2\pi
i\frac{c_{-}}{24}(M+\frac{1}{M})}}\mathcal{I}^{-}_{M}\sum_{b,c\in\mathcal{C}_{0}}d_{b}\theta_{b}^{M}\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right)\times
S_{bc}\chi_{c}\left(\frac{iL}{M^{2}\xi}+\frac{1}{M};[\mathrm{AP},0],[\mathrm{AP},1]\right),\end{split}$
(43)

where we work on the gauge $R^{b,0_{\frac{\pi}{n}}}=1$ for
$b\in\mathcal{C}_{0}$. The dominant contribution for the sum over anyons $c$
in the trivial sector comes from $c=1$, where the CFT character is
approximated as

$\displaystyle\begin{split}\chi_{1}\left(\frac{iL}{M^{2}\xi}+\frac{1}{M};[\mathrm{AP},0],[\mathrm{AP},1]\right)&\approx
e^{-\frac{2\pi i}{M}\frac{c_{-}}{24}}e^{\frac{2\pi
L}{M^{2}\xi}\frac{c_{-}}{24}}~{}.\end{split}$ (44)

We then obtain

$\displaystyle\begin{split}\bra{\Psi}\tilde{C}_{M}^{-}|_{D}\ket{\Psi}\propto&e^{-\frac{2\pi
i}{24}(M+\frac{2}{M})c_{-}}\mathcal{I}^{-}_{M}\sum_{b\in\mathcal{C}_{0}}d^{2}_{b}\theta_{b}^{M}e^{i\pi
Q_{b}}~{}.\end{split}$ (45)

For Chern insulators, in which $\mathcal{C}_{0}=\\{1,\psi\\}$, the partial
rotation is simply given by

$\displaystyle\begin{split}\bra{\Psi}\tilde{C}_{M}^{-}|_{D}\ket{\Psi}\propto&e^{-\frac{2\pi
i}{24}(M+\frac{2}{M})c_{-}}\mathcal{I}^{-}_{M}.\end{split}$ (46)

###  C.3 Calculation of $\mathcal{I}^{\pm}_{M}$

In Eq. (40), (46), we observed that the partial rotation is proportional to
the phase $\mathcal{I}^{\pm}_{M}$ that should be determined by
$\tilde{C}^{\pm}_{M}$ symmetry action on the state. Here, we derive the
general expression for $\mathcal{I}^{\pm}_{M}$ in terms of the data of a given
fermionic invertible phase with $G_{b}=\mathbb{Z}_{M}$ symmetry and chiral
central charge $c_{-}$.

The bosonic shadow of this phase is a 16-fold way $G_{b}$-enriched topological
phase, also with chiral central charge $c_{-}$. In our setup, the symmetry
does not permute anyons. Let the symmetry fractionalization data be given by

$\displaystyle\omega_{2}({\bf g},{\bf h})$ $\displaystyle=k_{\pm}\frac{[{\bf
g}]_{M}+[{\bf h}]_{M}-[{\bf gh}]_{M}}{M}\mod 2$ (47) $\displaystyle n_{2}({\bf
g},{\bf h})$ $\displaystyle=k_{s}\frac{[{\bf g}]_{M}+[{\bf h}]_{M}-[{\bf
gh}]_{M}}{M}\mod 2.$ (48)

$\omega_{2}$ describes the extension of the internal symmetry $\mathbb{Z}_{M}$
by fermion parity. From the fermionic crystalline equivalence principle,
$k_{+}=1\mod 2$ while $k_{-}=0\mod 2$. For a longer discussion of this point,
see for example Ref. [48]. The two cases can be combined by defining
$k_{\pm}=(1\pm 1)\frac{1}{2}$.

If ${\bf g}_{0}$ is the generator of $G_{b}$, we wish to compute

$\mathcal{I}^{\pm}_{M}:=\theta_{0_{{\bf
g}_{0}}}^{M}\prod_{j=0}^{M-1}\eta_{0_{{\bf g}_{0}}}\left({\bf g}_{0},{\bf
g}_{0}^{j}\right).$ (49)

We use the general solution outlined in Ref. [8] combined with the general
theory of invertible fermionic phases in Ref. [9], which provides explicit
expressions for the $\eta$ and $\theta$ symbols in terms of
$n_{2},\omega_{2},c_{-}$. In particular, for this symmetry we can choose a
gauge in which $\theta_{0_{{\bf g}_{0}}}=1$ and

$\eta_{0_{\bf k}}\left({\bf g},{\bf h}\right)=\nu_{3}^{-1}({\bf g},{\bf
h},{\bf k})$ (50)

where

$\displaystyle d\nu_{3}$ $\displaystyle=e^{-2\pi i\mathcal{O}_{4}},$ (51)
$\displaystyle\mathcal{O}_{4}$
$\displaystyle=\frac{1}{2}n_{2}(n_{2}+\omega_{2})+\frac{c_{-}}{8}\omega_{2}^{2}\mod
1.$ (52)

After plugging in the functional forms of $\omega_{2},n_{2}$ and integrating,
we get the following solution for $\nu_{3}$:

$\begin{split}\nu_{3}({\bf g},{\bf h},{\bf k})=&\exp\left(-2\pi
i\left[\frac{k_{s}(k_{\pm}+k_{s})}{2}+\frac{c_{-}k_{\pm}^{2}}{8}+k_{3}\right]\frac{[{\bf
g}]_{M}}{M}\frac{[{\bf h}]_{M}+[{\bf k}]_{M}-[{\bf
hk}]_{M}}{M}\right).\end{split}$ (53)

Here $k_{3}$ is a bosonic SPT index. Now a computation gives the desired
result

$\mathcal{I}^{\pm}_{M}=\exp\left(\frac{2\pi
i}{M}\left[\frac{k_{s}(k_{\pm}+k_{s})}{2}+\frac{c_{-}k_{\pm}^{2}}{8}+k_{3}\right]\right).$
(54)

Combining Eqs. (40), (46) with Eq. (54), and restoring the o subscripts, we
obtain Eq. (10) in the main text. Note that there is a phase proportional to
$c_{-}$ coming from $\mathcal{I}^{+}_{M}$ but not $\mathcal{I}_{M}^{-}$.

##  Appendix D Computation of $\ell_{s,\text{o},\text{LL}}$

Here we state the effective response theory for a continuum system of $C$
filled Landau levels with symmetry $\text{U}(1)\times\text{SO}(2)$ (where
$\text{SO}(2)$ signifies spatial rotations), and find its relationship to the
continuum limit of the Hofstadter model.

In terms of a $\text{U}(1)$ gauge field $A$ and an $\text{SO}(2)$ gauge field
$\omega$ (which can be identified with the components of the spin connection
on the underlying spatial manifold), the continuum response theory in the case
where a $2\pi$ spatial rotation acts trivially on fermions is

$\mathcal{L}_{eff}=\sum_{n=1}^{C}(\frac{1}{4\pi}(A+s_{n}\omega)\wedge
d(A+s_{n}\omega)-\frac{1}{48\pi}\omega\wedge d\omega)$ (55)

where $s_{n}=\frac{2n-1}{2}$ is the orbital spin per particle in the n-th
Landau level. The Chern number of each filled Landau level is 1. The last term
represents the gravitational anomaly. Expanding the above Lagrangian, we
obtain

$\mathcal{L}_{eff}=\frac{1}{4\pi}A\wedge
dA+\frac{1}{2\pi}\frac{C^{2}}{2}A\wedge
d\omega+\frac{1}{4\pi}\frac{2C^{3}-C}{6}\omega\wedge d\omega.$ (56)

The coefficient of $\frac{1}{2\pi}A\wedge d\omega$ is identified with the Wen-
Zee shift $\mathscr{S}^{+}_{\text{LL}}$. It has no origin dependence, and
takes the value $\mathscr{S}^{+}_{\text{LL}}=C^{2}/2$. The coefficient
$\tilde{\ell}^{+}_{s,\text{LL}}$ of the term $\frac{1}{4\pi}\omega\wedge
d\omega$ also has no origin dependence, and takes the values

$\tilde{\ell}_{s,\text{LL}}^{+}=\frac{2C^{3}-C}{6}=\\{\frac{1}{6},\frac{7}{3},\frac{17}{2},\frac{62}{3},\frac{245}{6},\dots\\}$
(57)

We define $\ell_{\text{LL}}=\tilde{\ell}_{s}+C/12$, which is the coefficient
of $\frac{1}{4\pi}\omega\wedge d\omega$ if we ignore the framing anomaly.

In the continuum limit of the Hofstadter model, assuming that a $2\pi$
rotation acts trivially on fermions (that is, the rotation operator is given
by $\tilde{C}_{M_{\text{o}}}^{+}$), the analogous response coefficient is
denoted $\ell^{+}_{s,\text{o},\text{LL}}=\ell^{+}_{s,\text{LL}}\mod
M_{\text{o}}/2$; it acquires an origin dependence and is quantized mod
$M_{\text{o}}/2$ where $M_{\text{o}}$ is the order of rotations that preserve
o. Here we assume $M_{\text{o}}$ is even. For the square lattice,

$\ell^{+}_{s,\text{o},\text{LL}}=\frac{2C^{3}-C}{6}+\frac{C}{12}=\\{\frac{1}{4},\frac{5}{2},\frac{3}{4},1,\frac{5}{4}\dots\\}\mod
M_{\text{o}}/2.$ (58)

To derive the expression for $\ell^{-}_{s,\text{o},\text{LL}}$ when we instead
use the operators $\tilde{C}_{M_{\text{o}}}^{-}$, we can simply replace
$A\rightarrow A+\omega/2$ in Eq. (55) (this corresponds to choosing a
reference rotation operator $\tilde{C}_{M_{\text{o}}}^{-}$ that inserts
$\pi/M_{\text{o}}$ flux at a disclination of angle $2\pi/M_{\text{o}}$) and
again compute the $\frac{1}{4\pi}\omega\wedge d\omega$ coefficient, after
subtracting off the framing anomaly. The final result is

$\ell^{-}_{s,\text{o},\text{LL}}=\frac{2C^{3}-C}{6}+\frac{C^{2}}{2}+\frac{C}{4}+\frac{C}{12}=\\{1,1,2,2,3,3,0,0,1,\dots\\}\mod
M_{\text{o}}.$ (59)

The above equations reproduce Eqs. (5), (6) appear in the main text.