eqs

$\begin{split}\BODY\end{split}$ (1)

# Crystalline invariants of fractional Chern insulators

Ryohei Kobayashi  Department of Physics, Joint Quantum Institute, and
Condensed Matter Theory Center, University of Maryland, College Park, Maryland
20742, USA  Yuxuan Zhang  Department of Physics, Joint Quantum Institute, and
Condensed Matter Theory Center, University of Maryland, College Park, Maryland
20742, USA  Naren Manjunath  Perimeter Institute for Theoretical Physics,
Waterloo, Ontario N2L 2Y5, Canada  Maissam Barkeshli  Department of Physics,
Joint Quantum Institute, and Condensed Matter Theory Center, University of
Maryland, College Park, Maryland 20742, USA

###### Abstract

In the presence of crystalline symmetry, topologically ordered states can
acquire a host of symmetry-protected invariants. These determine the patterns
of crystalline symmetry fractionalization of the anyons in addition to
fractionally quantized responses to lattice defects. Here we show how ground
state expectation values of partial rotations centered at high symmetry points
can be used to extract crystalline invariants. Using methods from conformal
field theory and G-crossed braided tensor categories, we develop a theory of
invariants obtained from partial rotations, which apply to both Abelian and
non-Abelian topological orders. We then perform numerical Monte Carlo
calculations for projected parton wave functions of fractional Chern
insulators, demonstrating remarkable agreement between theory and numerics.
For the topological orders we consider, we show that the Hall conductivity,
filling fraction, and partial rotation invariants fully characterize the
crystalline invariants of the system. Our results also yield invariants of
continuum fractional quantum Hall states protected by spatial rotational
symmetry.

A fundamental question in condensed matter physics is to understand the role
of crystalline symmetry in distinguishing phases of matter. In the context of
topological phases of matter, crystalline symmetry can significantly expand
the set of distinct possible phases Wen (2002, 2004); Hasan and Kane (2010);
Barkeshli and Qi (2012); Essin and Hermele (2013, 2014); Barkeshli  _et al._
(2019a); Benalcazar  _et al._ (2014); Ando and Fu (2015); Chiu  _et al._
(2016); Hermele and Chen (2016); Barkeshli  _et al._ (2019b); Zaletel  _et
al._ (2017); Song  _et al._ (2017); Huang  _et al._ (2017); Shiozaki  _et al._
(2017a); Kruthoff  _et al._ (2017); Bradlyn  _et al._ (2017); Thorngren and
Else (2018); Liu  _et al._ (2019); Manjunath and Barkeshli (2021, 2020);
Manjunath  _et al._ (2023); Herzog-Arbeitman  _et al._ (2022); Zhang  _et al._
(2022a, 2023); Manjunath  _et al._ (2024); Sachdev (2023); Kobayashi  _et al._
(2024a). For example, discrete translation and rotation symmetries might
permute anyons, causing lattice defects to become non-Abelian Barkeshli and Qi
(2012); Barkeshli  _et al._ (2019a). Anyons can themselves carry fractional
quantum numbers under the crystalline symmetry, giving rise to crystalline
symmetry fractionalization Jalabert and Sachdev (1991); Cheng  _et al._
(2016); Sachdev (2018); Sachdev and Vojta (1999); Essin and Hermele (2013,
2014); Qi and Fu (2015); Zaletel  _et al._ (2017); Barkeshli  _et al._
(2019b); Manjunath and Barkeshli (2021, 2020). Finally, the system can have
fractional quantized responses to lattice defects Van Miert and Ortix (2018);
Liu  _et al._ (2019); Li  _et al._ (2020); Manjunath and Barkeshli (2021,
2020); Zhang  _et al._ (2022a, b), in analogy to fractional quantized Hall
conductivity. Despite significant progress over the last several decades,
there are still important open questions about how to define and extract
topological invariants that arise due to crystalline symmetry, particularly in
fractionalized topologically ordered phases with anyons. This question has
gained renewed urgency, given the experimental discovery of fractional Chern
insulators (FCIs) Kol and Read (1993); Hafezi  _et al._ (2007); Tang  _et al._
(2011); Regnault and Bernevig (2011); Sun  _et al._ (2011); Parameswaran  _et
al._ (2013) in crystalline two-dimensional materials Spanton  _et al._ (2018);
Cai  _et al._ (2023); Lu  _et al._ (2024) and ultracold atoms Semeghini  _et
al._ (2021).

In this paper we develop a theory of how to extract many-body topological
invariants protected by crystalline symmetry from ground state wave functions
of topologically ordered systems. The results presented here apply to
fractional Chern insulators (FCIs) of bosons and quantum spin liquids with
crystalline symmetry, and also have implications for fractional quantum Hall
states with continuous spatial symmetry. In particular, we develop a theory of
ground state expectation values of partial rotations centered at high symmetry
points for systems with intrinsic topological order. We show how such partial
rotation expectation values can be used to completely characterize the
crystalline invariants, and in particular determine the crystalline symmetry
fractionalization and defect responses. This significantly extends recent work
Zhang  _et al._ (2022a, 2023); Manjunath  _et al._ (2024) that shows how
partial rotations can be used to completely characterize invertible
topological phases Barkeshli  _et al._ (2022); Aasen  _et al._ (2021), which
have no anyons. It also adds to a line of work showing how topological
invariants are encoded in bulk ground state wave functions Levin and Wen
(2006); Kitaev and Preskill (2006); Shiozaki  _et al._ (2017b); Dehghani  _et
al._ (2021); Cian  _et al._ (2021, 2022); Kim  _et al._ (2022); Fan  _et al._
(2022); Kobayashi  _et al._ (2024b); Zhang  _et al._ (2023); Kobayashi  _et
al._ (2024a, c).

We focus on systems with $U(1)$ charge conservation symmetry and orientation-
preserving wallpaper group symmetries, corresponding to the symmetry group
$G=U(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{M}]$, for $M=1,2,3,4,6$,
where $\phi$ denotes a rational magnetic flux $\phi=2\pi p/q$ per unit cell.
Topological orders with such symmetries were systematically characterized and
classified in Manjunath and Barkeshli (2021, 2020) by utilizing the
theoretical framework of G-crossed braided tensor categories (BTCs) for
symmetry-enriched topological (SET) phases Barkeshli  _et al._ (2019a). The
results of this paper show how the invariants obtained in Manjunath and
Barkeshli (2021, 2020) can be extracted from partial rotations.

We further use the parton construction Baskaran and Anderson (1988); Jain
(1989); Wen (1991, 1992, 1999); Lee  _et al._ (2006); Wen (2002); Barkeshli
and Wen (2010); McGreevy  _et al._ (2011); Wen (2004); Sachdev (2023) to
obtain trial wave functions for crystalline symmetry-enriched FCIs by
projecting wave functions of free fermion Chern insulators. Using the parton
effective field theory, we predict the crystalline invariants of the projected
FCI wave functions in terms of the crystalline invariants of the parton free
fermion states. Utilizing large-scale numerical Monte Carlo calculations, we
then compare the numerical results of the partial rotation expectation values
to the theoretical predictions, finding remarkable agreement.

Most of the main text is focused on discussing at length a paradigmatic
example, the $1/2$-Laughlin topological order on the square lattice. We
provide theoretical predictions for partial rotations for general bosonic
topological orders in Sec. IV.

##  I $1/2$-Laughlin on square lattice

The $1/2$-Laughlin topological order is described by $U(1)_{2}$ Chern-Simons
(CS) theory, and arises in the description of bosonic fractional quantum Hall
states, chiral spin liquids Kalmeyer and Laughlin (1987); Wen  _et al._
(1989); Bauer  _et al._ (2014), and FCIs. The topological order consists of
two topologically distinct charges, $\\{[0],[1]\\}$, where $[1]$ labels the
semion with topological twist $\theta_{[1]}=e^{i\pi/2}$. We have the fusion
rules $[a]\times[b]=[a+b]$, where the brackets imply mod $2$ reduction.

###  I.1 $G=U(1)\times\mathbb{Z}_{4}$

For now let us assume the global symmetry is $G=U(1)\times\mathbb{Z}_{4}$,
where $\mathbb{Z}_{4}$ is a $4$-fold spatial rotational symmetry (also denoted
$C_{4}$). We will include lattice translational symmetries afterwards.
Following Manjunath and Barkeshli (2021), the symmetry fractionalization class
can be specified by two anyons, $[v],[s]\in\\{[0],[1]\\}$. $v$ is the charge
vector (vison) while $s$ is a crystalline analog of the spin vector Wen and
Zee (1992). $v$ determines Hall conductivity via
$\sigma_{H}=\frac{1}{2\pi}(\frac{v^{2}}{2}+2k_{1})$ in natural units. $v$ also
determines the fractional charge of the anyon $a\in\\{[0],[1]\\}$,
$Q_{a}=av/2\mod 1$. $s$ determines the discrete shift, which specifies the
charge bound to lattice disclinations and fractional orbital angular momentum
of the anyons, $L_{a}=as/2\mod 1$. The topological quantum numbers are further
specified by three integers $k_{1},k_{2},k_{3}\in\mathbb{Z}$. These invariants
appear as coefficients in an effective CS theory:

$\displaystyle\mathcal{L}=\frac{2}{4\pi}ada-\frac{v}{2\pi}Ada-\frac{s}{2\pi}\omega
da+\mathcal{L}_{\text{SPT}},$ (2)

where
$\mathcal{L}_{\text{SPT}}=-(\frac{k_{1}}{2\pi}AdA+\frac{k_{2}}{2\pi}Ad\omega+\frac{k_{3}}{2\pi}\omega
d\omega)$. 111We note that the Hall conductivity appears in the response
action as $-\frac{\sigma_{H}}{4\pi}AdA$ in our convention, which leads to the
formula $\sigma_{H}=\frac{1}{2\pi}(\frac{v^{2}}{2}+2k_{1})$. Here $a$ is a
dynamical $U(1)$ gauge field, $A$ is the background $U(1)$ gauge field, and
$\omega$ is a background $\mathbb{Z}_{4}$ gauge field of the crystalline
symmetry. Mathematically $\omega$ is treated as a real-valued 1-form with
quantized holonomies $\oint\omega\in\frac{2\pi}{4}\mathbb{Z}$. Note we have
equivalences $(s,k_{2},k_{3})\sim(s-2,k_{2}+v,k_{3}+s-1)$ and
$(v,k_{1},k_{2})\sim(v-2,k_{1}+v-1,k_{2}+s-1)$, which arise by relabeling
$a\rightarrow a+\omega$ and $a\rightarrow a+A$ respectively in Eq. (2).

####  I.1.1 Partial rotations

Next, we consider the $4$-fold rotational symmetry operator $\hat{C}_{4}$, and
we define $\hat{C}_{4,l}\equiv e^{i\frac{2\pi l}{4}\hat{N}}\hat{C}_{4}$ for
integer $l$, with $\hat{N}$ the total $U(1)$ number operator. We pick a 4-fold
symmetric subregion $D$ whose length $L\gg\xi$, with $\xi$ the correlation
length. Our main result, derived in general in Sec. IV, is that the rotation
restricted to $D$, $\hat{C}_{4,l}|_{D}$, satisfies

$\displaystyle\mathcal{T}^{b}\left(\frac{2\pi}{4};\frac{2\pi
l}{4}\right):=\langle\Psi|\hat{C}_{4,l}|_{D}|\Psi\rangle\approx
e^{-\gamma|\partial D|}e^{i\frac{2\pi}{4}K_{l}},$ (3)

where $|\Psi\rangle$ is the ground state. Eq. (3) is expected to hold up to
non-universal corrections exponentially small in $L/\xi$. Our theory predicts
invariants $\Theta_{l}$ by taking appropriate modular reductions, assuming $D$
encloses an integer number of magnetic flux quanta:

$\Theta_{l}:=\begin{cases}K_{l}\mod 2,&\quad\text{if }s+vl=1\mod 2\\\
K_{l}\mod 4,&\quad\text{if }s+vl=0\mod 2\end{cases}$ (4)

These modular reductions can be understood in several ways. One is terms of
certain redundancies in the G-crossed BTC description Manjunath and Barkeshli
(2020); Barkeshli  _et al._ (2019a) as explained in App. D.3 and summarized by
Eq. (19). The other is from a real space construction, which we explain in
Sec. III. We further find

$\displaystyle
K_{l}=-\frac{3}{4}+K_{l}^{\text{frac}}+K_{l}^{\text{SPT}}+A_{l},$ (5)

with
$K_{l}^{\text{frac}}=l^{2}\frac{v^{2}}{4}+l\frac{vs}{2}+\frac{s^{2}}{4}\mod
4$, $K_{l}^{\text{SPT}}=l^{2}k_{1}+lk_{2}+k_{3}\mod 4$, and
$A_{l}=\frac{1}{4}\delta([lv+s+1]_{2})$, where $\delta([x]_{2})=1$ when $x=0$
mod 2, otherwise 0. Note that $K_{l}^{\text{SPT}}$ is an integer. By computing
the above expectation value for generic $l$, together with $\sigma_{H}$, one
can completely determine the crystalline invariants $v,s,k_{1},k_{2},k_{3}$.

####  I.1.2 Parton construction: projected wave functions and effective field
theory

We can obtain model ground state wave functions of these crystalline SET
phases by utilizing the parton construction. We write the boson operator as
$b=f_{1}f_{2}$, which are fermionic partons. This introduces a $U(1)$ gauge
symmetry, $f_{1}\rightarrow e^{i\theta}f_{1}$, $f_{2}\rightarrow
e^{-i\theta}f_{2}$, with associated $U(1)$ gauge field $\alpha$. We assume a
mean-field state where each fermionic parton forms a free fermion state with
Chern number $C_{1}=C_{2}=1$. The bosonic wave function is obtained by
projecting the partons to the same location:

$\displaystyle\Psi_{b}(\\{\vec{r}_{i}\\})=\psi_{1}(\\{\vec{r}_{i}\\})\psi_{2}(\\{\vec{r}_{i}\\})$
(6)

The many-body state of each parton is further labeled by crystalline
invariants. With $G=U(1)^{f}\times C_{4}$ symmetry, the additional crystalline
invariants form a $\mathbb{Z}_{8}\times\mathbb{Z}_{2}$ classification and can
be characterized by two invariants $(\mathscr{S},\ell_{s})$ Manjunath  _et
al._ (2024). These contribute terms
$\frac{\mathscr{S}}{2\pi}Ad\omega+\frac{\ell_{s}}{4\pi}\omega d\omega$ to the
response theory. Below we will show that the invariants
$v,s,k_{1},k_{2},k_{3}$ of the $1/2$ Laughlin state can be obtained from the
crystalline invariants of the two parton states:
$(\mathscr{S}_{1},\ell_{s,1},\mathscr{S}_{2},\ell_{s,2})$. We find:

$\displaystyle v$
$\displaystyle=1,\;\;\;\;\;s=\mathscr{S}_{1}+\mathscr{S}_{2},\;\;k_{1}=k_{2}=0$
$\displaystyle k_{3}$
$\displaystyle=\ell_{s,1}+\ell_{s,2}-\mathscr{S}_{1}^{2}-\mathscr{S}_{2}^{2}.$
(7)

A more general choice of invariants (e.g. $v=0$ or any choice of
$k_{1},k_{2}$) requires a more sophisticated parton construction. To derive
Eq. (I.1.2), first note that each parton state is described by an effective
field theory,
$\mathcal{L}_{f}=\frac{1}{4\pi}K_{IJ}a^{I}da^{J}-\frac{v_{f}^{I}}{2\pi}Ada^{I}-\frac{s_{f}^{I}}{2\pi}\omega
da^{I}$, where repeated indices are implicitly summed. To describe an
invertible fermionic state with Chern number $C=1$ and
$(\mathscr{S},\ell_{s})$, we take $K=1\oplus\sigma_{x}$, $v_{f}=(1,0,0)$,
$s_{f}=(\mathscr{S},\ell_{s}-\mathscr{S}^{2},1)$, where $\sigma_{x}$ is a
Pauli matrix. The effective field theory of the bosonic state then is obtained
by coupling the two parton theories (labeled with indices $i=1,2$) to the
$U(1)$ gauge field $\alpha$:

$\displaystyle\mathcal{L}=$
$\displaystyle\frac{1}{4\pi}K_{IJ}a^{I}_{(i)}da^{J}_{(i)}-\frac{s_{f,i}^{I}}{2\pi}\omega
da^{I}_{(i)}-\frac{v_{f,1}^{I}}{2\pi}Ada_{(1)}^{(I)}$
$\displaystyle+\frac{1}{2\pi}\alpha
d(v^{I}_{f,1}a_{(1)}^{I}-v^{I}_{f,2}a_{(2)}^{I}).$ (8)

Integrating out $\alpha$ enforces $a_{(1)}^{1}=a_{(2)}^{1}=a$ up to a gauge
transformation. Integrating out the remaining gauge fields then gives Eq. (2)
with the couplings of Eq. (I.1.2). Performing the construction with $C=-1$
parton states instead gives projected wave functions for $U(1)_{-2}$
topological order, see App. B for partial rotations in $U(1)_{-2}$.

###  I.2 Including translational symmetry:
$G=U(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{4}]$

Figure 1: Left. Maximal Wyckoff positions for $C_{4}$ symmetry (colored
circles). The high symmetry points $\gamma_{1},\gamma_{2}$ both belong to the
$\gamma$ maximal Wyckoff position, but are inequivalent under lattice
translations. In our convention, the sites and plaquette centers are at
$\beta$ and $\alpha$ respectively. Right. Partial rotation around
${\text{o}}=\beta$. The blue region is the partial rotation disk $D$, the red
arrows represent the partial rotation operator $\hat{C}_{4,\beta}|_{D}$.

In the presence of translation symmetry and magnetic flux $\phi$ per unit
cell, the symmetry becomes
$G=U(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{4}]$. We take $\phi=2\pi
p/q$, with $p,q$ coprime. One can then consider rotations about distinct high
symmetry points o in the unit cell. The possible choices of o are shown in
Fig. 1. On a square lattice we have two high symmetry points with $C_{4}$
symmetry, which are the vertex and plaquette centers denoted
$\text{o}=\alpha,\beta$ respectively. We also have the bond centers denoted
$\text{o}=\gamma$ with $C_{2}$ symmetry. The rotational symmetry operator
depends on o, so we have $\hat{C}_{4,\text{o}}$ and $\hat{C}_{4,\text{o},l}$
for $\text{o}=\alpha,\beta$, and $\pi$ rotations $\hat{C}_{2,\text{o}}$, with
$\text{o}=\alpha,\beta,\gamma$, and
$\hat{C}_{2,\text{o},l}:=e^{i\pi\hat{N}l}\hat{C}_{2,\text{o}}$. The background
gauge field $\omega$ also becomes origin dependent, $\omega_{\text{o}}$.
$\omega_{\alpha}$, $\omega_{\beta}$ are $\mathbb{Z}_{4}$ gauge fields,
$\omega_{\gamma}$ is a $\mathbb{Z}_{2}$ gauge field.

For $\text{o}=\alpha,\beta$, we now have
$\mathcal{T}^{b}_{\text{o}}(\frac{2\pi}{4},\frac{2\pi l}{4})$ and
$K_{\text{o},l}$, and $\Theta_{\text{o},l}$ defined analogously to Eqs. (3),
(4). For $\text{o}=\gamma$ we define $\mathcal{T}_{\gamma}^{b}(\pi,\pi
l):=\langle(\hat{C}_{2,\gamma,l})_{D}\rangle=e^{-\gamma|\partial D|}e^{i\pi
K_{\gamma,l}}$.

Including translation symmetry has two main effects. First, the invariants
that depend on rotational symmetry acquire an extra subscript o. Second, one
can define a filling fraction (charge per unit cell) $\nu$. Specifically,
symmetry fractionalization can be fully specified by
$([v],[s_{\alpha}],[s_{\beta}],[s_{\gamma}])$. There is significant physics in
the dependence of $[s_{\text{o}}]$ on o. In particular, there is an “anyon per
unit cell” $[m]$ characterizing the fractionalization of the translation
algebra. There is also a discrete torsion anyon $[t_{\alpha}]$, which
specifies the fractionalization of linear momentum and gives rise to a
fractional quantized charge polarization Manjunath and Barkeshli (2021, 2020).
As we show in Sec. III and App. D.2, these can be written as

$\displaystyle[m]=[s_{\alpha}]\times[s_{\beta}]\times[s_{\gamma}]^{2},\;\;\;[t_{\text{o}}]=[s_{\text{o}}]\times[s_{\gamma}],\;o=\alpha,\beta.$
(9)

The filling fraction (charge per unit cell) $\nu$ obeys a Lieb-Schultz-Mattis
type constraint Lu  _et al._ (2020); Manjunath and Barkeshli (2020)

$\displaystyle e^{2\pi i\nu}=e^{i2\pi(vm/2+[\phi]_{2\pi}\sigma_{H})},$ (10)

where $[\phi]_{2\pi}:=\phi\mod 2\pi$. Fractional filling requires $v=[1]$.
Fractional filling at zero flux $\phi$ requires $[v]=[m]=1$.

A novel consequence of the above formulas is that when $\nu$ is fractional and
the flux $\phi=0$, then $[m]$ must be non-trivial, which in turn implies that
at least one of $[s_{\text{o}}]$ must be non-trivial. Therefore a $1/2$
Laughlin FCI at zero field on a square lattice necessarily must have
fractionalization of the rotation symmetry about some high symmetry point o.
This is in addition to the well-known fractionalization of the translation
algebra dictated by $[m]$.

The SPT indices also become origin dependent, $k_{2,\text{o}}$ and
$k_{3,\text{o}}$, for $\text{o}=\alpha,\beta,\gamma$. For
$\text{o}=\alpha,\beta$, these are mod 4 invariants, while for
$\text{o}=\gamma$ these are mod 2 invariants. The dependence of
$k_{2,\text{o}}$ and $k_{3,\text{o}}$ on o also contains significant physics,
including a notion of angular momentum filling and angular momentum
polarization Zhang  _et al._ (2023); Manjunath  _et al._ (2024).

Our theory predicts the partial $C_{2}$ rotation around the origin $\gamma$ to
be

$\displaystyle\mathcal{T}^{b}\left(\frac{2\pi}{2};\frac{2\pi
l}{2}\right)\approx e^{-\gamma|\partial D|}e^{i\frac{2\pi}{2}K_{l,\gamma}},$
(11)

with

$\displaystyle
K_{l,\gamma}=-\frac{1}{4}+K_{l,\gamma}^{\text{frac}}+K_{l,\gamma}^{\text{SPT}}+A_{l,\gamma},$
(12)

where
$K_{l,\gamma}^{\text{frac}}=l^{2}\frac{v^{2}}{4}+l\frac{vs}{2}+\frac{s^{2}}{4}\mod
2$, $K_{l,\gamma}^{\text{SPT}}=l^{2}k_{1}+lk_{2}+k_{3}\mod 2$, and
$A_{l,\gamma}=\frac{1}{4}\delta([lv+s]_{2})$. Finally we have

$\Theta_{\gamma,l}:=\begin{cases}K_{\gamma,l}\mod 1,&\quad\text{if }s+vl=0\mod
2\\\ K_{\gamma,l}\mod 2,&\quad\text{if }s+vl=1\mod 2\end{cases}.$ (13)

This modular reduction is also derived using Eq. (19). The invariants defined
above can now be used in the effective field theory. If we only consider the
rotation gauge field $\omega_{\text{o}}$, we recover Eq. (2), just with
$s,k_{2},k_{3},\omega$ replaced by $s_{\text{o}}$, $k_{2,\text{o}}$,
$k_{3,\text{o}}$, $\omega_{\text{o}}$. Note that a more complete effective
field theory would also include background translation gauge fields Manjunath
and Barkeshli (2021, 2020) and additional coupling constants would directly
specify the dependence of the invariants on o.

Now we wish to describe the theory above in terms of the parton construction.
Invertible fermionic states with
$G=U(1)\times_{\phi}[\mathbb{Z}^{2}\rtimes\mathbb{Z}_{4}]$ symmetry have a
$\mathbb{Z}^{3}\times\mathbb{Z}_{8}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}$
classification Zhang  _et al._ (2023); Manjunath  _et al._ (2024). As shown in
Ref. Manjunath  _et al._ (2024), the torsion part can be completely
characterized by $\\{\mathscr{S}_{\text{o}},\ell_{s,\text{o}}\\}$ for
$\text{o}=\alpha,\beta,\gamma$. The results of the preceding section directly
imply:

$\displaystyle v$
$\displaystyle=1,\;\;\;\;\;s_{\text{o}}=\mathscr{S}_{\text{o},1}+\mathscr{S}_{\text{o},2},\;\;k_{1}=k_{2,\text{o}}=0$
$\displaystyle k_{3,\text{o}}$
$\displaystyle=\ell_{s,\text{o},1}+\ell_{s,\text{o},2}-\mathscr{S}_{\text{o},1}^{2}-\mathscr{S}_{\text{o},2}^{2}.$
(14)

Note that for the projection to survive, each parton state must have the same
filling, $\nu_{p}=\frac{\phi_{p}}{2\pi}+\kappa_{i}$, for $i=1,2$, where
$\kappa_{i}$ is an integer, $\phi_{p}$ is the flux seen by each parton, and we
have assumed each parton state has Chern number 1. The projection can survive
only if the filling $\nu$ of the boson $b$ is equivalent to the filling of
each parton state: $\nu=\nu_{p}$. Furthermore, $b$, being a composite of two
partons, sees a flux per unit cell $\phi=2\phi_{p}$. Therefore, $e^{2\pi
i\nu}=e^{2\pi i\frac{1}{2}\frac{\phi}{2\pi}}=e^{2\pi
i(mv/2+[\phi]_{2\pi}\sigma_{H})}$, where
$m=\frac{1}{2\pi}(\phi-[\phi]_{2\pi})$.

##  II Numerical Monte Carlo results

Figure 2: Raw data from QMC calculations. The system is defined on an
$L_{x}\times L_{y}=60\times 60$ torus with the partial rotation disk $D$ of
size $30\times 30$ and the rotation center ${\text{o}}=\alpha$ (plaquette
center). We use the projected parton state with
{$C_{1},\kappa_{1},\mathscr{S}_{\alpha,1},\ell_{s,\alpha,1}$}={$C_{2},\kappa_{2},\mathscr{S}_{\alpha,2},\ell_{s,\alpha,2}$}={1,0,1/2,1/4}.
(a) $l=0$. Each data point represents the expectation value
$\mathcal{T}_{\alpha}^{b}(2\pi/M,0)$ plotted in the complex plane and computed
using $\approx 5\times 10^{6}$ Metropolis-Hastings steps, referred to as a
batch. Points with the same color represent different batches with the same
flux and filling. (b) Each data point is an average over 20 batches, with
standard error shown as error bars. The slope $m$ of the fitted line gives
$K_{\alpha,0}=\tan^{-1}(m)\times 4/(2\pi)$. (c)(d)
$\text{Abs}(\mathcal{T}^{b}_{\alpha}(2\pi/M,0))$ and $K_{\alpha,0}$ for each
flux. Note $K_{\alpha,0}$ can jump by $2$ as $\phi_{p}$ is varied. (e,f,g,h)
are the same as (a,b,c,d) but with $l=1$. $K_{\alpha,1}$ does not show jumps.
Square Lattice $U(1)_{\pm 2}$
---
| $\ket{\psi_{C,\kappa}^{2}}$ | $\Theta_{{\text{o}},l}$
o | $~{}C~{}$ | $~{}\kappa~{}$ | $~{}\mathscr{S}_{{\text{o}}}~{}$ | $\ell_{s,{\text{o}}}$ | $~{}l=0~{}$ | $~{}l=1~{}$ | $~{}l=2~{}$ | $~{}l=3~{}$
$\beta$ | 1 | 0 | $1/2$ | 1/4 | -0.247 | 0.253 | -0.217 | -0.746
$\beta$ | -1 | 1 | $1/2$ | 15/4 | 0.236 | 0.750 | 0.227 | -0.251
$\alpha$ | 1 | 0 | $1/2$ | 1/4 | -0.232 | 0.251 | -0.227 | -0.750
$\alpha$ | -1 | 1 | $3/2$ | 7/4 | 0.208 | -0.251 | 0.228 | 0.748
$\alpha$ | 1 | -1 | $7/2$ | 1/4 | -0.220 | -0.749 | -0.233 | 0.253
$\alpha$ | -1 | 2 | $5/2$ | 7/4 | 0.245 | 0.747 | 0.218 | -0.251
$\alpha$ | 1 | -2 | $5/2$ | 9/4 | -0.233 | 0.252 | -0.223 | -0.750
$\alpha$ | -1 | 3 | $7/2$ | 15/4 | 0.213 | -0.249 | 0.241 | 0.748
$\alpha$ | 1 | -3 | $3/2$ | 9/4 | -0.226 | -0.748 | -0.227 | 0.250
$\alpha$ | -1 | 4 | $1/2$ | 15/4 | 0.231 | 0.750 | 0.211 | -0.251
$\gamma$ | 1 | 0 | $1/2$ | 1/4 | 0.000 | 0.000 | N/A | N/A
$\gamma$ | -1 | 1 | $1/2$ | 7/4 | 0.000 | 0.000 | N/A | N/A
Table 1: QMC results of $\Theta_{{\text{o}},l}$ for the squared projected
parton ground states $\ket{\psi_{C,\kappa}^{2}}$, which denote the cases
$\\{C_{1},\kappa_{1}\\}=\\{C_{2},\kappa_{2}\\}$. For
${\text{o}}=\alpha,\beta$, $\Theta_{{\text{o}},l}$ is defined modulo 2 when
$l$ is even, and mod 4 when $l$ is odd. $\Theta_{\gamma,0}$ is defined mod 2,
and $\Theta_{\gamma,1}$ is defined mod 1. Values of $\mathscr{S}_{{\text{o}}}$
for the Hofstadter states are calculated using methods in Zhang  _et al._
(2022b). Values of $\ell_{s,{\text{o}}}$ for the Hofstadter states are
calculated in Zhang  _et al._ (2023).  Square Lattice $U(1)_{\pm 2}$
---
$\ket{\psi_{C_{1},\kappa_{1}}}$ | $\ket{\psi_{C_{2},\kappa_{2}}}$ | $\Theta_{\alpha,l}$
$C_{1}$ | $\kappa_{1}$ | $\mathscr{S}_{\alpha,1}$ | $\ell_{s,\alpha,1}$ | $C_{2}$ | $\kappa_{2}$ | $\mathscr{S}_{\alpha,2}$ | $\ell_{s,\alpha,2}$ | $l=0$ | $l=1$ | $l=2$ | $l=3$
1 | 0 | 1/2 | 1/4 | 1 | -1 | 7/2 | 1/4 | $-0.751\mod 4$ | $-0.228\mod 2$ | $0.253\mod 4$ | $-0.235\mod 2$
1 | 0 | 1/2 | 1/4 | 1 | -2 | 5/2 | 9/4 | $-0.230\mod 2$ | $-0.745\mod 4$ | $-0.226\mod 2$ | $0.252\mod 4$
1 | 0 | 1/2 | 1/4 | 1 | -3 | 3/2 | 9/4 | $0.252\mod 4$ | $-0.223\mod 2$ | $-0.749\mod 4$ | $-0.230\mod 2$
1 | -1 | 7/2 | 1/4 | 1 | -2 | 5/2 | 9/4 | $0.255\mod 4$ | $-0.215\mod 2$ | $-0.749\mod 4$ | $-0.224\mod 2$
1 | -1 | 7/2 | 1/4 | 1 | -3 | 3/2 | 9/4 | $-0.229\mod 2$ | $0.251\mod 4$ | $-0.217\mod 2$ | $-0.747\mod 4$
1 | -2 | 5/2 | 9/4 | 1 | -3 | 3/2 | 9/4 | $-0.747\mod 4$ | $-0.229\mod 2$ | $0.251\mod 4$ | $-0.231\mod 2$
Table 2: QMC result of $\Theta_{\alpha,l}$ for the projected parton ground
states $\ket{\psi_{C_{1},\kappa_{1}}}\ket{\psi_{C_{2},\kappa_{2}}}$.
$\mathscr{S}_{{\text{o}}},\ell_{s,{\text{o}}}$ are calculated as stated in
Table 1. The theory in Eq. (4)(5) is aligned with the numerics and predict
that $\Theta_{\alpha,l}$ to be the nearest quarter integer of this table.

We consider the parton wave function of Eq. (6) and compute the partial
rotation expectation value using numerical Monte Carlo calculations. For the
parton wave functions, we use the ground state wave functions of the free
fermion Harper-Hofstadter model on a square lattice with magnetic flux
$\phi_{p}$ per unit cell. The states $\ket{\psi_{C,\kappa}}$ of this model are
labeled by the Chern number $C$ and $\kappa\equiv\nu-C\phi_{p}/(2\pi)$, where
$\nu$ is the filling, and these determine the other invariants
$\\{\mathscr{S}_{{\text{o}}},\ell_{s,{\text{o}}}\\}$ Zhang  _et al._ (2022a,
b, 2023).

In Fig. 2 we show numerical results for $l=\\{0,1\\}$, ${\text{o}}=\alpha$.
Both parton states have invariants
$\\{C,\mathscr{S}_{\alpha},\ell_{\alpha}\\}=\\{1,1/2,1/4\\}$.
$\frac{2\pi}{4}K_{\alpha,l}$ is then extracted from the partial rotation
expectation value through Eq. (3). The complex phase of
$\bra{\Psi}(\hat{C}_{4,l})_{D}\ket{\Psi}$ can jump by $\pi$ as $\phi_{p}$ is
varied. For example, in Fig. 2 (b), $K_{\alpha,0}$ could be either
$\approx-0.25\mod 4$ or $\approx 1.75\mod 4$ depending on the flux. Therefore,
to get an invariant, we need to define $\Theta_{\alpha,0}=K_{\alpha,0}\mod 2$
in this case. In Fig. 2 (e), $K_{\alpha,1}\approx 0.25\mod 4$ for all flux
$\phi_{p}$. Consequently, $\Theta_{\alpha,1}=K_{\alpha,1}\mod 4$ is an
invariant. These are consistent with the modular reductions predicted by our
theory in Sec. I.

The calculation has two sources of error. The first is the Monte Carlo error,
shown as the error bars in Fig. 2(b,f), and which can be reduced by taking
more Monte Carlo steps. The other source of error is finite size effects,
which causes $\arg(\mathcal{T}_{{\text{o}}}^{b})$ to deviate from the
theoretical prediction.

We perform the above calculations using several different parton states that
realize $U(1)_{\pm 2}$ topological order. For projected wave functions
constructed with two identical parton states, we summarize the value of
$\Theta_{{\text{o}},l}$ for all ${\text{o}}=\alpha,\beta,\gamma$ and
$l=0,1,2,3$ in Table 1. For $C=1$ entries, the theoretical predictions are
summarized in the preceding section. For the $C=-1$ entries, the theoretical
predictions are given in App. B. In all cases the theory is aligned with the
numerics and predict $\Theta_{{\text{o}},l}$ to be the nearest quarter
integer. The Monte Carlo sampling error is of the order of $0.001$, therefore
the main source of error is expected to be due to finite size effects.

For projected wave functions constructed with two different parton states, we
summarize the value of $\Theta_{{\text{o}},l}$ for ${\text{o}}=\alpha$ and
$l=0,1,2,3$ in Table 2. The numerical values demonstrate remarkable agreement
with the theoretical predictions. Additional details of the calculations are
summarized in App. A.

##  III Real space construction

To gain additional insight into the formulas above, here we present a model
for each symmetry fractionalization pattern by utilizing a real space
construction Huang  _et al._ (2017); Else and Thorngren (2019); Zhang  _et
al._ (2022c); Manjunath  _et al._ (2024).

Consider starting with the ground state of a $U(1)_{2}$ topological order with
$[v]=1$, but with trivial crystalline symmetry fractionalization. We can then
construct a distinct symmetric state by placing an Abelian anyon
$[s_{\text{o}}]$ at each high symmetry point $\text{o}=\alpha,\beta,\gamma$ of
the square lattice unit cell. The triple
$([s_{\alpha}],[s_{\beta}],[s_{\gamma}])$ now determines a new SET state,
relative to the original state we started with. For $U(1)_{2}$ on the square
lattice, we get a $(\mathbb{Z}_{2})^{3}$ classification of the possible
triples $([s_{\alpha}],[s_{\beta}],[s_{\gamma}])$. In general one needs to mod
out by equivalences that move anyons between high symmetry points (see App.
D), but these are trivial for $U(1)_{2}$.

We can interpret a $2\pi$ disclination centered at o as inducing the anyon
$[s_{\text{o}}]$. To see this, consider the system to be at the surface of a
cube. Each of the 8 corners can be interpreted as a $\pi/2$ disclination
centered at $\text{o}=\beta$, so we effectively have $2$ $2\pi$ disclinations.
For $F$ faces, there will be $F+2$ vertices, so we have two extra
$[s_{\beta}]$ anyons as compared to being on flat space, for which the number
of faces are in one-to-one correspondence. In this sense, we can assign an
anyon $[s_{\beta}]$ to a $2\pi$ disclination centered at $\beta$. Similar
arguments apply for $\text{o}=\alpha$ and $\gamma$.

The results above agree with the same classification obtained in Refs.
Manjunath and Barkeshli (2021, 2020) using a different approach based on
crystalline gauge fields. In particular, for a given $C_{4}$ symmetric origin
o, the classification is given up to equivalences in terms of an anyon
$s_{\text{o}}$ (“spin vector”), a pair of anyons
$\vec{t}_{{\text{o}}}=(t_{{\text{o}},x},t_{{\text{o}},y})$ (“torsion vector”),
and an ‘anyon per unit cell’, denoted $m$. The individual classifications of
these anyons were given in Ref. Manjunath and Barkeshli (2021). In terms of
the real space construction, $[m]$ is given in Eq. 9. This follows because
each unit cell has one $\alpha$ and $\beta$ site each, and two $\gamma$ sites.
Furthermore, the real-space construction shows that there can be a topological
charge (anyonic) polarization. As discussed in App. D.2, this can be used to
give a heuristic understanding of $\vec{t}_{{\text{o}}}$ in terms of
differences of the form $[s_{{\text{o}}+\hat{x}(\hat{y})}-s_{{\text{o}}}]$.

The real space construction helps us understand why a partial rotation
centered at o can distinguish $s_{\text{o}}$. Relative to the state with
trivial crystal symmetry fractionalization, the partial $\hat{C}_{4,\text{o}}$
rotation in the state decorated with $\\{[s_{{\text{o}}}]\\}$ should pick up
an extra phase of $e^{i\frac{2\pi}{4}h_{s_{\text{o}}}}$, where
$h_{s_{\text{o}}}=s_{\text{o}}^{2}/4$ is the topological spin of
$s_{\text{o}}$. To understand the partial $\hat{C}_{4,\text{o},l}$ rotation,
note that $[s_{\text{o}}]$ can also be thought of as the anyon induced by four
$C_{4}$ defects, which are lattice disclinations. Four
$\hat{C}_{4,\text{o},l}$ symmetry defects differ from four
$\hat{C}_{4,\text{o}}$ symmetry defects by a $2\pi l$ $U(1)$ flux, which
induces $l$ copies of the vison $[v]$. Thus the result for
$\hat{C}_{4,\text{o},l}$ can be obtained from the result for
$\hat{C}_{4,\text{o}}$ by taking $[s_{\text{o}}]\rightarrow[s_{\text{o}}+lv]$,
giving the phase
$h_{s_{\text{o}}+lv}=s_{\text{o}}^{2}/4+lvs_{\text{o}}/2+l^{2}v^{2}/4$. This
explains the $K_{\text{frac}}$ contribution in Eq. (5). The $K_{\text{SPT}}$
contribution can be understood by including a real space construction for
bosonic SPTs Huang  _et al._ (2017); Song  _et al._ (2020). The additional
term involving $\delta(..)$ requires a more sophisticated analysis. From the
edge CFT computation presented in Sec. IV, we can see that this is an
additional contribution arising fundamentally because the entanglement
spectrum is described by the CFT at high temperatures.

The real space construction also shows why we need to take the modular
reductions of Eq. (4). Suppose that we increase the size of the disk $D$ to
include more anyons. Since $D$ has to be 4-fold symmetric, it can include an
integer multiple of 4 copies of an anyon $[a]$. Now under the partial
$C_{4,\text{o},l}$ rotation, we consider four $[a]$ anyons making a $2\pi/4$
rotation around $[s_{\text{o}}]$, which is equivalent to a single $[a]$
rotating by $2\pi$ around $[s_{\text{o}}]$, giving a phase $e^{2\pi
ias_{\text{o}}/2}$. Furthermore, if $[a]$ has a charge of $av/2\mod 1$, the
$U(1)$ rotation by $2\pi l/4$ picks up an additional phase of $e^{2\pi
iavl/2}$. Therefore, we have an ambiguity $K_{l}\rightarrow
K_{l}+2as_{\text{o}}+2avl$. To get an invariant for all possible $a$ then
leads us to Eq. (4). A more formal explanation is given in Sec. IV and App.
D.3 based on relabeling symmetry defects in the $G$-crossed BTC description.

##  IV General bosonic topological orders

Here we evaluate the ground state expectation value of the partial rotation by
an angle $2\pi/M$ restricted to a disk subregion $D$. Our result applies to
general SET states of bosons, including both Abelian and non-Abelian states
and to the case where symmetries may permute anyon types. We assume the ground
state in the disk has trivial topological charge, integer magnetic flux, and
no lattice defects. The choice of high symmetry point o is implicit.

To perform the calculation, we utilize the correspondence between the
entanglement spectrum and edge CFT Li and Haldane (2008); Qi  _et al._ (2012).
Let the length of the boundary be $L=|\partial D|$. The partial rotation is
then evaluated as a translation by $\frac{L}{M}$ acting in the edge CFT. This
translation acts in the edge CFT by a combination of an internal
${\mathbb{Z}}_{M}$ symmetry and the Lorentz translation symmetry in the 1d
space:

$\displaystyle\begin{split}\mathcal{T}^{b}\left(\frac{2\pi}{M};\frac{2\pi
l}{M}\right)&=\bra{\Psi}\hat{C}_{M,l}|_{D}\ket{\Psi}\\\
&=\frac{\mathrm{Tr}[e^{i\frac{2\pi
l}{M}\hat{N}}e^{i\hat{Q}_{M}\frac{2\pi}{M}}e^{i\tilde{P}\frac{L}{M}}e^{-\xi
H}]}{\mathrm{Tr}[e^{-\xi H}]},\end{split}$ (15)

where the trace is taken in the trivial sector of CFT. The internal
${\mathbb{Z}}_{M}$ symmetry is generated by the operator $\hat{Q}_{M}$. $H$ is
the CFT Hamiltonian density $H=\frac{2\pi}{L}(L_{0}-c_{-}/24)$, and
$\tilde{P}$ is a (normalized) translation operator $\tilde{P}=2\pi L_{0}/L$.
Note the CFT is effectively at a high temperature because $\xi/L\ll 1$ is
effectively an inverse temperature. In App. C we present a general formula for
$\mathcal{T}^{b}$ in terms of G-crossed BTC data that characterizes the
crystalline SET. In the special case where the $C_{M}$ symmetry does not
permute anyons, the result simplifies to the following form:

$\displaystyle\begin{split}\mathcal{T}^{b}\left(\frac{2\pi}{M};\frac{2\pi
l}{M}\right)\propto&{e^{-2\pi
i(M+\frac{2}{M})\frac{c_{-}}{24}}}\mathcal{I}_{M,l}\sum_{a}d_{a}^{2}\theta_{a}^{M}e^{2\pi
iQ_{a,l}}\\\ &\times\sum_{b}S_{ab}e^{\frac{2\pi i}{M}h_{b}}e^{-\frac{2\pi
L}{M^{2}\xi}(h_{b}-\frac{c_{-}}{24})},\end{split}$ (16)

where $\propto$ means being proportional up to a real positive number. We sum
over all possible anyons $a,b$. $d_{a}$ is the quantum dimension of $a$,
$\theta_{a}=e^{2\pi ih_{a}}$ is the topological twist of $a$, $h_{a}$ is the
topological spin, and $S$ is the modular $S$-matrix. For the sum over the $b$,
the trivial anyon has the leading order contribution, with the other terms
exponentially suppressed. When the contribution of the trivial anyon vanishes,
which happens in the $U(1)_{2}$ case, the leading term is determined by the
non-trivial anyon $a$ with the smallest scaling dimension in the chiral edge
CFT. We have also defined the quantities $\mathcal{I}_{M,l}$ and $Q_{a,l}$,
which are invariants determined by the response of the
$U(1)\times{\mathbb{Z}}_{M}$ symmetry. Using some technical results from
Manjunath and Barkeshli (2020), in App. C we show that

$\displaystyle e^{2\pi iQ_{a,l}}:=M_{a,s\times v^{l}},$ (17)
$\displaystyle\mathcal{I}_{M,l}=e^{\frac{2\pi
i}{M}(k_{3}+h_{s}+l(k_{2}+m_{v,s})+l^{2}(k_{1}+h_{v}))},$ (18)

with $M_{a,b}$ the mutual braiding between anyons $a,b$. $v$ and $s$ are
Abelian anyons; $v$ is the vison, which determines the fractional $U(1)$
charge of anyons, while $s$ is the generalization of the spin vector, which
determines the fractional orbital angular momentum of anyons Manjunath and
Barkeshli (2020). $k_{1},k_{2},k_{3}$ are the integers parameterizing the
$\mathcal{H}^{3}(G,U(1))$ freedom of the SET Barkeshli  _et al._ (2019a). In
the case of $U(1)_{2}$ CS theory, the sum over $a,b$ in Eq. (16) accounts for
the additional $A_{l}$ term involving $\delta(...)$ in Eq. (5). In App. C.4,
we present a simplification of Eq. (18) for general Abelian topological orders
using the $K$-matrix of Abelian CS theory.

Finally, $\mathcal{T}^{b}$ only gives an invariant modulo a $U(1)$ phase.
There are certain equivalences on $\mathcal{I}_{M}$ obtained by relabelling
symmetry defects with anyons. As we explain in App. D.3, such relabellings
induce the transformation

$\mathcal{I}_{M,l}\rightarrow\mathcal{I}_{M,l}\times M_{s\times
v^{l},a}\times\theta_{a}^{M},$ (19)

for any anyon $a$. From this result we recover the modular reductions of Eqs.
(4) and (13), which give the invariants $\Theta_{{\text{o}},l}$.

We note that the results above also apply to partial rotations in continuum
fractional quantum Hall states with continuous $SO(2)$ rotational symmetry,
under the assumption of large rotation angles, $2\pi/M\gg 2\pi\sqrt{\xi/L}$.
For small rotation angles, $2\pi/M\ll 2\pi\sqrt{\xi/L}$, a different analysis
is required, as discussed in App. C.5.

##  V Acknowledgements

We thank Brayden Ware for collaboration during the initial stages of this
work. This work is supported by NSF DMR-2345644 and the Laboratory for
Physical Sciences through the Condensed Matter Theory Center. Research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Economic Development and by the
Province of Ontario through the Ministry of Colleges and Universities.

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##  Appendix A Numerical Details

###  A.1 Review: Partial Rotations in Hofstadter Model

Figure 3: Square lattice Hofstadter model around the partial rotation disk
(blue region) for different origins. The blue dots represent sites, the bonds
represent hoppings, and the blue arrows represents the vector potential
$A_{ij}$. We insert $\phi_{p}$ flux per plaquette. (a) ${\text{o}}=\alpha$ and
each blue arrow represents a hopping phase of $A_{ij}=\phi_{p}/4$, (b)
${\text{o}}=\beta$ and each blue arrow represents a hopping phase of
$A_{ij}=\phi_{p}/2$, (a) ${\text{o}}=\gamma$ and each arrow represents a
hopping phase of $A_{ij}=\phi_{p}$. Under these symmetric gauges, the
canonical partial rotation operators $\tilde{C}_{M_{{\text{o}}}}$ reduce to
the bare partial rotation operator $\tilde{C}_{M_{{\text{o}}}}$ which does not
involve any $U(1)$ phase. $\tilde{C}_{M_{{\text{o}}}}$ is just a permutation
of sites shown as the red arrows.

In this section, we review the definition of the rotation operators and the
methods in Zhang  _et al._ (2023) which are used to calculate the partial
rotation invariant $\Theta_{{\text{o}},l}$ and the associated field theory
coefficient $\ell_{s,{\text{o}}}$ in the free fermion Hofstadter model. Note
that $\ell_{s,{\text{o}}}$ is needed to compute the invariants for the parton
mean-field states that are projected to give FCI wave functions.

As shown in Zhang  _et al._ (2022a, b, 2023), the magnetic point group
rotation operator is defined as $\hat{C}_{M_{{\text{o}}}}\equiv
e^{\sum_{j}i\lambda_{j}\hat{N}_{j}}\hat{C}_{M_{{\text{o}}}}^{\text{bare}}$,
where $\hat{C}_{M_{{\text{o}}}}^{\text{bare}}$ is a many body rotation
operator which only moves points, and $\lambda_{j}$ is the $U(1)$ gauge
transformation at site $j$ required to make the operator commute with the
Hamiltonian. $M_{{\text{o}}}$ is the order of the point group symmetry at o.
On the square lattice, $M_{\alpha}=M_{\beta}=4,M_{\gamma}=2$. There are
$M_{{\text{o}}}$ different choices of $\hat{C}_{M_{{\text{o}}}}$ that satisfy
$(\hat{C}_{M_{{\text{o}}}})^{M_{{\text{o}}}}=+1$. Among these, there is a
canonical choice we denote $\hat{C}_{M_{{\text{o}}}}^{+}$; a disclination
created using this operator does not carry any extra magnetic flux at the
disclination core. The partial rotation operator
$\hat{C}_{M_{{\text{o}}}}^{+}|_{D}$ is defined as
$\hat{C}_{M_{{\text{o}}}}^{+}$ restricted to the disk $D$ and acting as an
identity outside of $D$.

The Hofstadter model is defined by the Hamiltonian:

$H=-t\sum_{<ij>}e^{-iA_{ij}}c_{i}^{\dagger}c_{j}+h.c.$ (20)

with $i,j$ site indexes. The vector potential $A_{ij}$ threads $\phi_{p}$ flux
per plaquette. For each different origin, we can always find a choice of
symmetric vector potential $A_{ij}$ such that the gauge transformation is
unnecessary and $\hat{C}_{M_{{\text{o}}}}^{+}|_{D}$ reduces to the bare
partial rotation $\hat{C}_{M_{{\text{o}}}}^{\text{bare}}|_{D}$. The choices of
$A_{ij}$ we use are shown in Fig. 3. The expectation value of
$\hat{C}_{M_{{\text{o}}}}^{+}|_{D}$ for Hofstadter states $\ket{\Psi}$ is
defined similar to Eq. (3):

$\displaystyle\langle\Psi|\hat{C}_{M_{{\text{o}}}}^{+}|_{D}|\Psi\rangle\approx
e^{-\gamma|\partial D|}e^{i\frac{2\pi}{4}K_{l}}.$ (21)

The invariants $\Theta_{{\text{o}}}^{+}$ satisfy the following modular
reduction:

$\displaystyle\Theta_{{\text{o}}}^{+}:=K^{+}_{{\text{o}}}\mod\frac{M_{{\text{o}}}}{2}.$
(22)

Figure 4: $\Theta_{\alpha}^{+}\mod 2$ for Hofstadter model. $\\{C,\kappa\\}$
is labeled to the relevant lobes of the Monte Carlo calculation.

As an example $\Theta_{\alpha}^{+}$ is shown in Fig. 4 (incorporated from
Zhang  _et al._ (2023)). We label $C$ and $\kappa$ in Fig. 4 to indicate the
relevant bands which will be used in the Monte Carlo calculation.

On the square lattice $\ell_{s,{\text{o}}}$ is related to
$\Theta_{\text{o}}^{+}$ through the following equation222Here, we suppressed
the $+$ superscript of $\ell_{s,{\text{o}}}^{+}$ and corrected an error in Eq.
12 of Zhang  _et al._ (2023). The correct equation is
$\displaystyle\ell^{\pm}_{s,{\text{o}}}$
$\displaystyle=\begin{cases}\frac{11\mp 1}{8}C+2\Theta^{\pm}_{{\text{o}}}\mod
4\quad{\text{o}}=\alpha,\beta\\\ \frac{3\mp
1}{8}C+2\Theta^{\pm}_{{\text{o}}}\mod 2\quad{\text{o}}=\gamma.\end{cases}$ :

$\displaystyle\ell_{s,{\text{o}}}$
$\displaystyle=\begin{cases}\frac{5}{4}C+2\Theta^{+}_{{\text{o}}}\mod
4,\quad{\text{o}}=\alpha,\beta\\\ \frac{1}{4}C+2\Theta^{+}_{{\text{o}}}\mod
2,\quad{\text{o}}=\gamma.\end{cases}$ (23)

We use Eq. (23) along with the values in Fig. 4 to generate the
$\ell_{s,{\text{o}}}$ entries in Table 1,2.

###  A.2 Constructing FCI wave functions from Hofstadter model ground states

The Hofstadter model offers us a plethora of ground states classified by the
topological invariants $\\{c_{-},C,\kappa,\Theta_{{\text{o}}}^{\pm}\\}$,
making it a useful starting point to build states with topological order. For
$C=\pm 1$ states, $\\{C,\kappa\\}$ uniquely defines the topological phase of
matter associated with a Hofstadter ground state (for higher $|C|$ this is
generally not true) and they are used to label the ground state as
$\ket{\psi_{C,\kappa}}$.

In the $U(1)_{\pm 2}$ parton construction, suppose there are $N$ hard core
bosons arranged in the configuration $\ket{r_{1},r_{2},\dots r_{N}}$, with
$r_{i}$ denoting their positions. In the $U(1)_{\pm 2}$ theory we consider the
boson operator

$\displaystyle
b^{\dagger}_{r_{i}}=f_{1,r_{i}}^{\dagger}f_{2,r_{i}}^{\dagger},$ (24)

such that

$\displaystyle\ket{r_{1},r_{2},\dots
r_{N}}=b_{r_{1}}^{\dagger}b_{r_{2}}^{\dagger}\dots
b_{r_{N}}^{\dagger}\ket{0}.$ (25)

The bosonic many-body ground state wave function for a given configuration is
defined by multiplying the free fermion wavefunctions for the individual
partons (the projection implies that the partons which form
$b^{\dagger}_{r_{i}}$ are forced to be at the same location $r_{i}$). For
example, consider the case where the two partons have equal $C$ and $\kappa$.
The above procedure gives the “squared state”
$\ket{\Psi}=\ket{\psi_{C,\kappa}^{2}}$:

$\displaystyle\bra{\psi_{C,\kappa}^{2}}\ket{r_{1},r_{2}\dots r_{N}}$
$\displaystyle=\left(\frac{1}{\sqrt{N!}}\text{Det}(\mathcal{M})\right)^{2}$
(26) $\displaystyle\mathcal{M}_{ij}$ $\displaystyle=\chi_{i}(r_{j}),$ (27)

where $\chi_{i}$ is a single particle eigenstate for one of the parton states.
$\chi_{i}$ has energy $\epsilon_{i}$ below the chemical potential.

We could also consider the case where the first parton state is labeled by
$\ket{\psi_{C_{1},\kappa_{1}}}$, and the second parton state by
$\ket{\psi_{C_{2},\kappa_{2}}}$. The wave function for
$\ket{\Psi}=\ket{\psi_{C_{1},\kappa_{1}}\psi_{C_{2},\kappa_{2}}}$ is

$\displaystyle\bra{\psi_{C_{1},\kappa_{1}}\psi_{C_{2},\kappa_{2}}}\ket{r_{1},r_{2}\dots
r_{N}}$
$\displaystyle=\frac{1}{N!}\text{Det}(\mathcal{M}_{1})\text{Det}(\mathcal{M}_{2})$
(28) $\displaystyle\mathcal{M}_{1,ij}=\chi_{1,i}(r_{j})\qquad$
$\displaystyle\mathcal{M}_{2,ij}=\chi_{2,i}(r_{j}),$ (29)

where $\chi_{1,i}$ and $\chi_{2,i}$ are single particle eigenstates for the
two parton states.

We also define

$\displaystyle\hat{C}_{M_{\text{o}},l}|_{D}:=(e^{\sum_{j}i\frac{2\pi
l}{M_{\text{o}}}\hat{N}_{j}}\hat{C}_{M_{\text{o}}}^{+})|_{D}.$ (30)

For the squared states, we have the amplitude

$\displaystyle\bra{\psi_{C,\kappa}^{2}}\hat{C}_{M_{\text{o}},l}|_{D}\ket{r_{1},r_{2}\dots
r_{N}}$ $\displaystyle=\bra{\psi_{C,\kappa}^{2}}\ket{R(r_{1}),R(r_{2})\dots
R(r_{N})}e^{i\frac{2\pi l}{4}n_{D}}$ $\displaystyle=e^{i\frac{2\pi
l}{4}n_{D}}(\frac{1}{\sqrt{N!}}\text{Det}(\mathcal{M}^{\prime}))^{2}.$ (31)
$\displaystyle\qquad\mathcal{M}^{\prime}_{ij}=\chi_{i}\left(R(r_{j})\right)$

Here, $R(r_{j})=r_{j}$ if $r_{j}\not\in D$, and $R(r_{j})=r_{k}$ if $r_{j}\in
D$ and $r_{j}$ rotates into $r_{k}$ under a $M_{\text{o}}$ fold
counterclockwise rotation around o. $n_{D}$ is the number of bosons within
region $D$ in the configuration $\ket{r_{1},r_{2}\dots r_{N}}$.

Similarly, for the ground states
$\ket{\Psi}=\ket{\psi_{C_{1},\kappa_{1}}\psi_{C_{2},\kappa_{2}}}$,

$\displaystyle\bra{\psi_{C_{1},\kappa_{1}}\psi_{C_{2},\kappa_{2}}}\hat{C}_{M_{\text{o}},l}|_{D}\ket{r_{1},r_{2}\dots
r_{N}}$ $\displaystyle=e^{i\frac{2\pi
l}{4}n_{D}}\frac{1}{N!}\text{Det}(\mathcal{M}^{\prime}_{1})\text{Det}(\mathcal{M}^{\prime}_{2}).$
(32)
$\displaystyle~{}\mathcal{M}^{\prime}_{1,ij}=\chi_{1,i}\left(R(r_{j})\right)\quad\mathcal{M}^{\prime}_{2,ij}=\chi_{2,i}\left(R(r_{j})\right)$

###  A.3 Variational Quantum Monte Carlo

In this section we introduce the variational Quantum Monte Carlo method to
calculate $\bra{\Psi}\hat{C}_{M_{\text{o}},l}|_{D}\ket{\Psi}$ for the bosonic
ground state $\ket{\Psi}$ constructed in the section above. The expectation
value is expressed as:

$\displaystyle\bra{\Psi}\hat{C}_{M_{\text{o}},l}|_{D}\ket{\Psi}$
$\displaystyle=$ $\displaystyle\sum_{r_{1},r_{2},\dots
r_{N}}\langle\Psi|\hat{C}_{M_{\text{o}},l}|_{D}|r_{1},r_{2},\dots
r_{N}\rangle\langle r_{1},r_{2},\dots r_{N}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\sum_{r_{1},r_{2},\dots
r_{N}}\frac{\bra{\Psi}\hat{C}_{M_{\text{o}},l}|_{D}\ket{r_{1},r_{2},\dots
r_{N}}}{\bra{\Psi}\ket{r_{1},r_{2},\dots r_{N}}}P(r_{1},r_{2},\dots r_{N}),$
(33)

where $P(r_{1},r_{2},\dots r_{N}):=|\bra{\Psi}\ket{r_{1},r_{2},\dots
r_{N}}|^{2}$ is the probability of the configuration $\ket{r_{1},r_{2},\dots
r_{N}}$. This expectation value can be sampled and calculated through the
Metropolis-Hastings algorithm Hastings  _et al._ (2010), which is summarized
the the following 3 steps:

  1. 1.

Begin with a random initial configuration $\ket{r_{1},r_{2}\dots r_{N}}$

  2. 2.

Propose a new configuration
$\ket{\tilde{r}_{1},\tilde{r}_{2}\dots\tilde{r}_{N}}$ by moving one particle
to a random unfilled position. Update the configuration to this new
configuration if

$\displaystyle\left|\frac{\bra{\Psi}\ket{\tilde{r}_{1},\tilde{r}_{2}\dots\tilde{r}_{N}}}{\bra{\Psi}\ket{r_{1},r_{2},\dots
r_{N}}}\right|^{2}>q,$

where $q$ is a random real number from 0 to 1; Reject the new configuration
and revert to $\ket{r_{1},r_{2}\dots r_{N}}$ if the above condition is false.

  3. 3.

Repeat Step 2. The expectation value in Eq. (A.3) is
$\frac{\bra{\Psi}\hat{C}_{M_{\text{o}},l}|_{D}\ket{r_{1},r_{2},\dots
r_{N}}}{\bra{\Psi}\ket{r_{1},r_{2},\dots r_{N}}}$ averaged over the trajectory
(with the first few iterations removed as they are not converged to the
desired distribution).

####  A.3.1 Parameters and discussions

We now define the parameters used in the calculations. Based on the
periodicity of the free fermion invariants $\Theta_{\text{o}}^{\pm}$ in
$\phi_{p}$ Zhang  _et al._ (2023), we set $C$ and $\kappa$ to the values shown
in Table 2. We first diagonalize the Hofstadter model and obtain the parton
states $\ket{\psi_{C_{1},\kappa_{1}}}$, $\ket{\psi_{C_{2},\kappa_{2}}}$ on an
$L_{x}\times L_{y}$ torus.

The parameters $\phi_{p},N$ for each parton state satisfy the following
relations with $\kappa$ and charge per unit cell $\nu$:

$\displaystyle\nu$ $\displaystyle=C\frac{\phi_{p}}{2\pi}+\kappa$ (34)
$\displaystyle\implies N$
$\displaystyle=(C\frac{\phi_{p}}{2\pi}+\kappa)L_{x}L_{y}.$ (35)

For ${\text{o}}=\alpha$, we use $L_{x}\times L_{y}=60\times 60$, for
${\text{o}}=\beta$, $L_{x}\times L_{y}=62\times 62$, and for
${\text{o}}=\gamma$, $L_{x}\times L_{y}=62\times 60$. We choose these
parameters so that the linear size of $D$ can be exactly half of the system
size which empirically works well. Though we have checked that having slightly
larger or smaller $D$ or performing the calculation on a open disk does not
change $\Theta_{{\text{o}},l}$ in a significant way. Note that when
calculating on an open disk, instead of filling $n$ particles, we place the
chemical potential $\mu$ in the middle of the band gap and fill every energy
level below $\mu$. Filling more or less edge states by moving $\mu$ within the
band gap does not change $\Theta_{{\text{o}},l}$ significantly in the
numerical calculation.

In the CFT calculation discussed below, $\mathcal{T}^{b}_{{\text{o}}}$ is a
sum of terms containing different powers of $e^{-L/\xi}$, where the exponents
are fixed by the scaling dimensions of the anyons in the CFT. $L/\xi$ is the
ratio between the size of the partial rotation disk boundary $L=|\partial D|$
and the correlation length. In the ‘sparse’ limit where the particle number is
very small, $\xi$ is proportional to the magnetic length $l_{B}$ which is the
only other length scale in the system. This means that for a fixed $L$,
$L/\xi\propto 1/l_{B}\propto\sqrt{\phi}\propto\sqrt{N}$. For some choices of
$l$, higher order CFT contributions (involving larger powers of $e^{-L/\xi}$)
identically vanish, and in these cases the phase of
$\mathcal{T}^{b}_{{\text{o}}}$ remains a constant when changing $L/\xi$.
However, for general choices of $l$ higher order contributions are
nonvanishing, and in these cases we need to pick $L/\xi$ to be large enough to
suppress the higher-order terms and measure a quantized phase of
$\mathcal{T}^{b}_{{\text{o}}}$. There is a numerical trade-off here: Though a
larger $L/\xi$ suppresses higher order contributions, it also reduces the
overall amplitude $\text{Abs}(\mathcal{T}^{b}_{{\text{o}}})$, thereby reducing
the accuracy of $\text{Arg}(\mathcal{T}^{b}_{{\text{o}}})$ from the Monte
Carlo calculation. This is because we have errors
$\sigma_{\text{Re}},\sigma_{\text{Im}}$ of $\mathcal{T}^{b}_{{\text{o}}}$ in
the imaginary plane dictated by the amount of Monte Carlo steps and the amount
of batches we take. The angular error
$\sigma_{\text{Arg}}\propto\frac{1}{\text{Abs}(\mathcal{T}^{b}_{{\text{o}}})}\sigma_{\text{Re}}$.

In our numerics for the partial rotation with ${\text{o}}=\alpha,\beta$, we
choose to fill $N$ particles where $20\leq N\leq 45$; the corresponding
$\phi_{p}$ is calculated through Eq. (34). Empirically, we find this choice
suppresses the higher order contributions from the CFT to a reasonable amount,
and allows us to measure a quantized $K_{{\text{o}},l}$. For the $C_{2}$
partial rotation calculation with ${\text{o}}=\gamma$, the amplitude of
$\mathcal{T}^{b}_{{\text{o}}}(\pi,\pi)$ nearly vanishes in the filling range
$20\leq N\leq 45$, therefore we choose a more sparse limit with the filling
range $3\leq N\leq 15$, where there is a finite
$\text{Abs}\left(\mathcal{T}^{b}_{{\text{o}}}(\pi,\pi)\right)$.

We run step 2 of the Metropolis-Hastings calculation 5,000,000 times in 20
separate batches of calculations. For ${\text{o}}=\alpha$, $l=0,1$, the raw
data of $\mathcal{T}_{{\text{o}},l}^{b}$ are shown in Fig. 2.

###  A.4 Calculation of topological entanglement entropy

Figure 5: An $L_{x}\times L_{y}$ cylinder with periodic boundary condition in
y direction bi-partitioned into region $A$ and $B$. This cylinder has
$(L_{x}+1)\times L_{y}$ sites.

We calculate the topological entanglement entropy (TEE) to give evidence that
our ground state $\ket{\Psi}$ from parton construction has topological order.
Consider a bi-partition of a cylinder into regions $A$ and $B$ as shown in
Fig. 5. Assuming $L_{x}$ is fixed, the Renyi entropy $S_{2}$ is predicted to
followKitaev and Preskill (2006); Levin and Wen (2006); Jiang  _et al._ (2012)

$\displaystyle S_{2}=aL_{y}-S_{\text{topo}},$ (36)

where $a$ is a constant. $S_{\text{topo}}$ is the TEE that encodes the total
quantum dimension of the anyons. For a state without anyons, such as a integer
quantum Hall state, $S_{\text{topo}}=0$, while for $U(1)_{2}$ topological
order, $S_{\text{topo}}=\frac{1}{2}\text{ln}(2)\approx 0.347$. In this
section, we numerically calculate $S_{\text{topo}}$ for a integer quantum Hall
state and a squared parton state to show that the results are aligned with the
prediction of Eq. (36).

We first prepare the Hofstadter state $\ket{\psi_{C,\kappa}}$ with $C=1$ and
$\kappa=0$ on a $L_{x}\times L_{y}$ cylinder (see Fig. 5), fixing $L_{x}=13$.
We symmetrically partition the system into regions $A$ and $B$.

When defining the vector potential on the cylinder, we make sure that the
total flux through the holes of the cylinder is trivial; that is, we demand
that the holonomy computed at the two boundaries of the cylinder are integers.
Since $\phi$ is a constant, this constrains the total flux
$\phi_{\text{tot}}=\phi L_{x}L_{y}$ through the surface of the cylinder to be
an integer multiple of $2\pi$, for any $L_{y}$. Thus we must have
$\phi\in(2\pi/L_{x}){\mathbb{Z}}$. We pick $\phi=2\pi/L_{x}$ in our numerics
and calculate $S_{2}$ for different $L_{y}$.

We prepare two configurations $\ket{r_{1},r_{2},\dots r_{n}}$ and
$\ket{r_{1}^{\prime},r_{2}^{\prime},\dots r_{n}^{\prime}}$ on two different
layers. The Renyi entropy can be calculated as expectation value of a partial
layer swap operator Cirac and Sierra (2010), and is expressed as

$\displaystyle e^{-S_{2}}=$
$\displaystyle\sum_{r_{A},r_{B},r_{A}^{\prime},r_{B}^{\prime}}\frac{\bra{\Psi}\ket{r_{A},r_{B}^{\prime}}\bra{\Psi}\ket{r_{A}^{\prime},r_{B}}}{\bra{\Psi}\ket{r_{A},r_{B}}\bra{\Psi}\ket{r_{A}^{\prime},r_{B}^{\prime}}}P(r_{A},r_{B})P(r_{A}^{\prime},r_{B}^{\prime})$
(37)

where $r_{A},r_{B}$ is shorthand for the set of electron positions in the
$A,B$ subregions. We sample the probability $P(r_{A},r_{B})$ and
$P(r_{A},r_{B})P(r_{A}^{\prime},r_{B}^{\prime})$ as before, but in each
Metropolis-Hastings step we pick randomly whether to update
$\ket{r_{A},r_{B}}$ or $\ket{r_{A}^{\prime},r_{B}^{\prime}}$. Note that if the
swapped configuration $\ket{r_{A},r_{B}^{\prime}}$ or
$\ket{r_{A}^{\prime},r_{B}}$ is not of the same filling as $\ket{\Psi}$, it is
not in the projected Hilbert space and contributes nothing to $e^{-S_{2}}$.

We perform the Monte Carlo calculation for the integer quantum Hall state
$\Psi=\ket{\psi_{1,0}}$ and the squared state $\Psi=\ket{\psi_{1,0}^{2}}$ for
$3\leq L_{y}\leq 30$. The resulting Renyi entropy is shown in Fig. 6. The TEE
is extracted as the negative intercept, which demonstrates excellent agreement
with the prediction Eq. (36) ( within $1.6\%$ error). This gives further
evidence that the ground states from parton construction are indeed 1/2
Laughlin states.

Figure 6: Renyi entropy $S_{2}$ for $L_{x}=14$ and different $L_{y}$ (error
bars included). $S_{2}$ is fitted in the linear regime $14\leq L_{y}\leq 30$,
and the TEE is calculated as the negative $L_{y}\rightarrow 0$ intercept

##  Appendix B The partial rotation in $U(1)_{-2}$ topological order

In the main text, we mainly studied the partial rotation in the 1/2-Laughlin
state described by the $U(1)_{2}$ topological order. Here we provide the
formulae for partial rotations valid for the $U(1)_{-2}$ topological order,
which we also compare to our numerical Monte Carlo calculations. The partial
rotation is given in the form

$\displaystyle\mathcal{T}^{b}\left(\frac{2\pi}{4};\frac{2\pi
l}{4}\right):=\langle\Psi|(\hat{C}_{4,l})_{D}|\Psi\rangle\approx
e^{-\gamma|\partial D|}e^{i\frac{2\pi}{4}K_{l}},$ (38)

with

$\displaystyle
K_{l}=\frac{3}{4}-K_{l}^{\text{frac}}+K_{l}^{\text{SPT}}-A_{l},$ (39)

where
$K_{l}^{\text{frac}}=l^{2}\frac{v^{2}}{4}+l\frac{vs}{2}+\frac{s^{2}}{4}\mod
4$, $K_{l}^{\text{SPT}}=l^{2}k_{1}+lk_{2}+k_{3}\mod 4$, and
$A_{l}=\frac{1}{4}\delta([lv+s+1]_{2})$, where $\delta([x]_{2})=1$ when $x=0$
mod 2, otherwise 0.

The partial $C_{2}$ rotation along the bond center $\gamma$ is given by

$\displaystyle\mathcal{T}^{b}\left(\frac{2\pi}{2};\frac{2\pi
l}{2}\right)\approx e^{-\gamma|\partial D|}e^{i\frac{2\pi}{2}K_{l,\gamma}},$
(40)

with

$\displaystyle
K_{\gamma,l}=\frac{1}{4}-K_{l,\gamma}^{\text{frac}}+K_{l,\gamma}^{\text{SPT}}-A_{l,\gamma},$
(41)

with
$K_{l,\gamma}^{\text{frac}}=l^{2}\frac{v^{2}}{4}+l\frac{vs}{2}+\frac{s^{2}}{4}\mod
2$, $K_{l,\gamma}^{\text{SPT}}=l^{2}k_{1}+lk_{2}+k_{3}\mod 2$, and
$A_{l,\gamma}=\frac{1}{4}\delta([lv+s]_{2})$.

When the $U(1)_{-2}$ topological order is formed by a pair partons forming
Chern insulators with $C=-1$, its topological invariants are expressed in
terms of those of the Chern insulators as

$\displaystyle v$
$\displaystyle=1,\;\;\;\;\;s_{\text{o}}=-\mathscr{S}_{\text{o},1}-\mathscr{S}_{\text{o},2},\;\;k_{1}=k_{2,\text{o}}=0$
$\displaystyle k_{3,\text{o}}$
$\displaystyle=-\ell_{s,\text{o},1}-\ell_{s,\text{o},2}-\mathscr{S}_{\text{o},1}^{2}-\mathscr{S}_{\text{o},2}^{2}.$
(42)

##  Appendix C CFT calculations

###  C.1 Derivation of partial rotation formula with anyon permutations

In this section, we perform the detailed CFT analysis outlined in Sec. IV.
Following Ref. Shiozaki  _et al._ (2017a), the quantity in Eq. (15) is
expressed in terms of CFT partition function on the edge. When $l=0$, the
partial rotation is expressed as

$\displaystyle\begin{split}\mathcal{T}^{b}\left(\frac{2\pi}{M};0\right)&=\frac{\mathrm{Tr}[e^{i\hat{Q}_{M}\frac{2\pi}{M}}e^{i\tilde{P}\frac{L}{M}}e^{-\xi
H}]}{\mathrm{Tr}[e^{-\xi H}]}\\\ &=e^{-\frac{2\pi
i}{24M}c_{-}}\frac{\chi_{1}(\frac{i\xi}{L}-\frac{1}{M};0,1)}{\chi_{1}(\frac{i\xi}{L};0,0)}\end{split}$
(43)

Here, $\chi_{1}(\tau;j,k)$ with $j,k\in{\mathbb{Z}}_{M}$ is the CFT character
that corresponds to the partition function on a torus equipped with
${\mathbb{Z}}_{M}$ gauge field; $j,k$ denotes the ${\mathbb{Z}}_{M}$ twist
along the spatial and temporal direction respectively. $\tau$ is the modular
parameter and the subscript $1$ means the trivial sector of the Hilbert space.

In the presence of the ${\mathbb{Z}}_{M}$ twisted boundary condition on the
torus, the CFT character on the edge transforms under the modular $S,T$
transformations according to the ${\mathbb{Z}}_{M}$-crossed modularity of the
${\mathbb{Z}}_{M}$-crossed BTC Barkeshli  _et al._ (2019a). To evaluate the
CFT character, let us write the CFT character as

$\displaystyle\begin{split}\chi_{1}\left(\frac{i\xi}{L}-\frac{1}{M};0,1\right)&=\sum_{b}S_{1b}\chi_{b}\left(-\frac{1}{\frac{i\xi}{L}-\frac{1}{M}};1,0\right)\\\
&=\sum_{b}(ST^{M})_{1b}\chi_{b}\left(\frac{-iM\frac{\xi}{L}}{\frac{i\xi}{L}+\frac{1}{M}};1,0\right)\end{split}$
(44)

where $S,T$ are modular $S,T$ matrices of ${\mathbb{Z}}_{M}$-crossed BTC. They
implicitly depend on the ${\mathbb{Z}}_{M}$ twist on a torus
$(\mathbf{g},\mathbf{h})$ with $\mathbf{g},\mathbf{h}\in{\mathbb{Z}}_{M}$, and
the $T$ matrix has the form of

$\displaystyle
T^{(\mathbf{g},\mathbf{h})}_{a_{\mathbf{g}},b_{\mathbf{g}}}=e^{-\frac{2\pi
i}{24}c_{-}}\cdot\theta_{a_{\mathrm{g}}}\cdot\eta_{a}(\mathbf{g},\mathbf{h})\cdot\delta_{a_{\mathbf{g}},b_{\mathbf{g}}}$
(45)

where the dependence on $(\mathbf{g},\mathbf{h})$ is made explicit. The
${\mathbb{Z}}_{M}$-crossed BTC $\mathcal{C}_{{\mathbb{Z}}_{M}}$ has the
structure of
$\mathcal{C}_{{\mathbb{Z}}_{M}}=\bigoplus_{\mathbf{g}\in{\mathbb{Z}}_{M}}\mathcal{C}_{\mathbf{g}}$,
and a simple object $a_{\mathbf{g}}\in\mathcal{C}_{\mathbf{g}}$ represents a
vortex carrying $\mathbf{g}\in{\mathbb{Z}}_{M}$ in the bulk SET phase. The
phase $\eta_{b}(\mathbf{g},\mathbf{h})$ describes the symmetry
fractionalization of the vortex $b$ Barkeshli  _et al._ (2019a). The vortex
$b$ in Eq. (44) carries the twist of $1\in{\mathbb{Z}}_{M}$ which is a
generator of ${\mathbb{Z}}_{M}$, so $b\in\mathcal{C}_{1}$. Using the above
${\mathbb{Z}}_{M}$-crossed modularity, the character is further rewritten as

$\displaystyle\begin{split}&\chi_{1}\left(\frac{i\xi}{L}-\frac{1}{M};0,1\right)=\sum_{b\in\mathcal{C}_{1}}(ST^{M})_{1b}\chi_{b}\left(\frac{-iM\frac{\xi}{L}}{\frac{i\xi}{L}+\frac{1}{M}};1,0\right)\\\
&=e^{-\frac{2\pi
iM}{24}c_{-}}\sum_{b\in\mathcal{C}_{1}}S_{1b}\theta_{b}^{M}\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right)\cdot\chi_{b}\left(\frac{-iM\frac{\xi}{L}}{\frac{i\xi}{L}+\frac{1}{M}};1,0\right)\\\
&=e^{-\frac{2\pi
iM}{24}c_{-}}\sum_{b\in\mathcal{C}_{1}}\sum_{c\in\mathcal{C}_{0}}S_{1b}\theta_{b}^{M}\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right)\\\
&\quad\times
S_{bc}\chi_{c}\left(\frac{iL}{M^{2}\xi}+\frac{1}{M};0,1\right)\end{split}$
(46)

The above $ST^{M}S$ transformation makes the imaginary factor of the modular
parameter $\tau=iL/(M^{2}\xi)+1/M$ large, assuming $\frac{L}{M^{2}\xi}\gg 1$.
Recalling that $\chi_{c}=\mathrm{Tr}(e^{2\pi i\tau(L_{0}-\frac{c_{-}}{24})})$,
the CFT character in the last expression can be approximated by

$\displaystyle\begin{split}\chi_{c}\left(\frac{iL}{M^{2}\xi}+\frac{1}{M};0,1\right)&\approx
e^{\frac{2\pi i}{M}(h_{c}-\frac{c_{-}}{24})}e^{-\frac{2\pi
L}{M^{2}\xi}(h_{c}-\frac{c_{-}}{24})}\end{split}$ (47)

One can then express the partial rotation as

$\displaystyle\mathcal{T}^{b}\left(\frac{2\pi}{M};0\right)\propto$
$\displaystyle{e^{-2\pi
i(M+\frac{2}{M})\frac{c_{-}}{24}}}\sum_{b\in\mathcal{C}_{1}}d_{b}\theta_{b}^{M}\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right)$
$\displaystyle\times\sum_{c\in\mathcal{C}_{0}}S_{bc}e^{\frac{2\pi
i}{M}h_{c}}e^{-\frac{2\pi L}{M^{2}\xi}(h_{c}-\frac{c_{-}}{24})}~{}.$ (48)

We note that this expression is valid even when the $C_{M}$ symmetry permutes
anyons.

###  C.2 Including partial $U(1)$ charge rotation

In the presence of $U(1)$ symmetry with generic $l$, one can also evaluate the
partial rotation associated with the partial $U(1)$ transformation within the
disk,

$\displaystyle\begin{split}\mathcal{T}^{b}\left(\frac{2\pi}{M};\frac{2\pi
l}{M}\right)&:=\bra{\Psi}\hat{C}_{M,l}|_{D^{2}}\ket{\Psi}\\\
&=\frac{\mathrm{Tr}[e^{i\hat{N}\frac{2\pi
l}{M}}e^{i\hat{Q}_{M}\frac{2\pi}{M}}e^{i\tilde{P}\frac{L}{M}}e^{-\xi
H}]}{\mathrm{Tr}[e^{-\xi H}]}\end{split}$ (49)

The above quantity can also be computed by the same logic as above, and given
by

$\displaystyle\mathcal{T}^{b}\left(\frac{2\pi}{M};\frac{2\pi l}{M}\right)$
$\displaystyle\propto{e^{-2\pi
i(M+\frac{2}{M})\frac{c_{-}}{24}}}\sum_{b\in\mathcal{C}_{{2\pi
l/M,1}}}d_{b}^{2}\theta_{b}^{M}\prod_{j=0}^{M-1}\eta_{b}\left((\left[\frac{2\pi
l}{M}\right]_{2\pi},1),(\left[\frac{2\pi
jl}{M}\right]_{2\pi},j)\right)\sum_{c\in\mathcal{C}_{0}}S_{bc}e^{\frac{2\pi
i}{M}h_{c}}e^{-\frac{2\pi L}{M^{2}\xi}(h_{c}-\frac{c_{-}}{24})}$ (50)

where $\mathcal{C}_{{\theta,1}}$ is the twisted sector with $(\theta,1)\in
U(1)\times{\mathbb{Z}}_{M}$.

###  C.3 Simplified expression for non-permuting $C_{M}$ symmetries

The above formula (48) with $l=0$ is further simplified when the $C_{M}$
symmetry does not permute anyons. Let us employ a general expression of these
quantities valid for the case without permutation action Barkeshli  _et al._
(2019a),

$\displaystyle\begin{split}\theta_{b_{1}}&=\theta_{b_{0}}\theta_{0_{1}}\cdot(R^{b_{0},0_{1}}R^{0_{1},b_{0}}),\\\
\eta_{b_{1}}\left(1,j\right)&=\eta_{b_{0}}\left(1,j\right)\eta_{0_{1}}\left(1,j\right),\\\
S_{a,b_{1}}&=\frac{1}{d_{a}}S_{a,b_{0}}(R^{a,0_{-1}}R^{0_{-1},a}).\end{split}$
(51)

Here $R^{ab}$ is the R-symbol of the G-crossed theory that specifies the
algebraic braiding properties of anyons and defects Barkeshli  _et al._
(2019a). One can choose the gauge where $R^{b,0_{1}}=1$, $R^{c,0_{1}}=1$ for
$b,c\in\mathcal{C}_{0}$. We then obtain

$\displaystyle\begin{split}\mathcal{T}^{b}\left(\frac{2\pi}{M};0\right)\propto&{e^{-2\pi
i(M+\frac{2}{M})\frac{c_{-}}{24}}}\mathcal{I}_{M}\sum_{b\in\mathcal{C}_{0}}d_{b}\theta_{b}^{M}e^{2\pi
iL_{b}}\\\ &\sum_{c\in\mathcal{C}_{0}}S_{bc}e^{\frac{2\pi
i}{M}h_{c}}e^{-\frac{2\pi L}{M^{2}\xi}(h_{c}-\frac{c_{-}}{24})}\end{split}$
(52)

which shows Eq. (16). We defined the invariants

$\displaystyle\mathcal{I}_{M}$
$\displaystyle:=\theta_{0_{1}}^{M}\prod_{j=0}^{M-1}\eta_{0_{1}}\left(1,j\right),$
$\displaystyle e^{2\pi iL_{b}}$
$\displaystyle:=\prod_{j=0}^{M-1}\eta_{b}\left(1,j\right).$ (53)

The $U(1)$ phase $\eta_{a}(\mathbf{g},\mathbf{h})$ describes the symmetry
fractionalization of the topological charge $a$. $L_{b}$ is the fractional
$\mathbb{Z}_{M}$ charge of $b$. The gauge invariant quantity $\mathcal{I}_{M}$
is further computed by plugging the symmetry fractionalization data
$\theta_{0_{\bf g}},\eta_{0_{\bf g}}(\bf h,\bf k)$ into its expression, which
was performed in Manjunath and Barkeshli (2020). To do this, we can work in
the specific gauge where $\theta_{0_{\bf g}}=1$, and

$\displaystyle\begin{split}\eta_{0_{\bf g}}(\bf h,\bf k)&=\exp\left(2\pi
i(h_{s}+k_{3})\frac{[{\bf g}]_{M}}{M}\frac{[{\bf h}]_{M}+[{\bf k}]_{M}-[{\bf
h}+{\bf k}]_{M}}{M}\right),\end{split}$ (54)

where we define $s\in\mathcal{C}_{0}$ as the relation $M_{b,s}=e^{2\pi
iL_{b}}$ for all anyons $b\in\mathcal{C}_{0}$. The parameter
$k_{3}\in{\mathbb{Z}}_{M}$ corresponds to the label of bosonic SPT phase with
$C_{M}$ symmetry. The quantity $\mathcal{I}_{M}$ is then computed as

$\displaystyle\mathcal{I}_{M}=e^{\frac{2\pi i}{M}(h_{s}+k_{3})}.$ (55)

In the absence of the permutation action, the formula (50) with generic $l$
can also be simplified by the same logic. It can be written as

$\displaystyle\begin{split}\mathcal{T}^{b}\left(\frac{2\pi}{M};\frac{2\pi
l}{M}\right)\propto&{e^{-2\pi
i(M+\frac{2}{M})\frac{c_{-}}{24}}}\mathcal{I}_{M,l}\sum_{a}d_{a}^{2}\theta_{a}^{M}e^{2\pi
iQ_{a,l}}\\\ &\times\sum_{b}S_{ab}e^{\frac{2\pi i}{M}h_{b}}e^{-\frac{2\pi
L}{M^{2}\xi}(h_{b}-\frac{c_{-}}{24})},\end{split}$ (56)

with

$\displaystyle\mathcal{I}_{M,l}:=\theta_{0_{\frac{2\pi
l}{M},1}}^{M}\prod_{j=0}^{M-1}\eta_{0_{\frac{2\pi
l}{M},1}}\left((\left[\frac{2\pi l}{M}\right]_{2\pi},1),(\left[\frac{2\pi
jl}{M}\right]_{2\pi},j)\right),$ $\displaystyle e^{2\pi
iQ_{a,l}}:=\prod_{j=0}^{M-1}\eta_{b}\left((\left[\frac{2\pi
l}{M}\right]_{2\pi},1),(\left[\frac{2\pi jl}{M}\right]_{2\pi},j)\right),$ (57)

which reduces to Eq. (C.3) when $l=0$. By plugging the form of $\theta,\eta$
of defects in Manjunath and Barkeshli (2020) into the above expression, we
obtain

$\displaystyle e^{2\pi iQ_{a,l}}:=M_{a,s\times v^{l}},$ (58)
$\displaystyle\mathcal{I}_{M,l}=e^{\frac{2\pi
i}{M}(k_{3}+h_{s}+l(k_{2}+m_{v,s})+l^{2}(k_{1}+h_{v}))},$ (59)

where
$(k_{1},k_{2},k_{3})\in{\mathbb{Z}}\times{\mathbb{Z}}_{M}\times{\mathbb{Z}}_{M}$
characterizes the bosonic SPT index, and $M_{v,s}=e^{2\pi im_{v,s}}$.

###  C.4 K-matrix theory

For Abelian topological orders, we can use the more familiar language of
K-matrix CS theory coupled to a $U(1)\times{\mathbb{Z}}_{M}$ crystalline gauge
field $(A,\omega)$:

$\displaystyle\begin{split}\mathcal{L}=&\frac{1}{4\pi}K_{IJ}a_{I}da_{J}-v_{I}a_{I}\frac{dA}{2\pi}-s_{I}a_{I}\frac{d\omega}{2\pi}\\\
&-\frac{k_{3}}{2\pi}\omega
d\omega-\frac{k_{2}}{2\pi}Ad\omega-\frac{k_{1}}{2\pi}AdA,\end{split}$ (60)

where $k_{1},k_{2},k_{3}\in{\mathbb{Z}}$. $v_{I},s_{I}$ are integer vectors
that determine the fractionalization of $U(1)$ and $C_{M}$ symmetry
respectively. In this case, when the leading contribution $a=0$ is not
vanishing in Eq. (56), the partial rotation is given to leading order by

$\displaystyle\begin{split}&\mathcal{T}^{b}\left(\frac{2\pi}{M};\frac{2\pi
l}{M}\right)\\\ &\propto{e^{-2\pi
i(M+\frac{2}{M})\frac{c_{-}}{24}}}e^{\frac{2\pi
i}{M}(k_{3}+h_{s}+l(k_{2}+m_{v,s})+l^{2}(k_{1}+h_{v}))}\\\ &\sum_{{\bf
m}\in{\mathbb{Z}}^{|\bf K|}/\bf K{\mathbb{Z}}^{|\bf K|}}e^{M\pi i{\bf
m}^{\mathrm{T}}{\bf K}^{-1}{\bf m}}e^{2\pi i{\bf m}^{\mathrm{T}}{\bf
K}^{-1}({l\bf q+\bf s})}.\end{split}$ (61)

In Tables 3, 4, we list the partial rotation invariants of bosonic $U(1)_{2}$
states with different symmetry fractionalization classes, which can be
compared to the numerical results in Tables 1, 2.

$U(1)_{2}$ | $v$ | $s$ |  $\mathcal{T}^{b}\left(\frac{2\pi}{4},\frac{2\pi l}{4}\right)$ ($l=0,1,2,3$)
---|---|---|---
| $[0]_{2}$ | $[0]_{2}$ | $e^{-3\pi i/8},e^{-3\pi i/8},e^{-3\pi i/8},e^{-3\pi i/8}$
| $[0]_{2}$ | $[1]_{2}$ | $e^{-\pi i/8},e^{-\pi i/8},e^{-\pi i/8},e^{-\pi i/8}$
| $[1]_{2}$ | $[0]_{2}$ | $e^{-3\pi i/8},e^{-\pi i/8},e^{\pi i/8},e^{7\pi i/8}$
| $[1]_{2}$ | $[1]_{2}$ | $e^{-\pi i/8},e^{\pi i/8},e^{7\pi i/8},e^{-3\pi i/8}$
Table 3: $\mathcal{T}^{b}\left(\frac{2\pi}{4},\frac{2\pi l}{4}\right)$ for different symmetry fractionalization classes of $U(1)_{2}$, computed using Eq. (61). $U(1)_{2}$ | $v$ | $s$ |  $\mathcal{T}^{b}\left(\frac{2\pi}{2},\frac{2\pi l}{2}\right)$ ($l=0,1$)
---|---|---|---
| $[0]_{2}$ | $[0]_{2}$ | $1,1$
| $[0]_{2}$ | $[1]_{2}$ | $1,1$
| $[1]_{2}$ | $[0]_{2}$ | $1,1$
| $[1]_{2}$ | $[1]_{2}$ | $1,-1$
Table 4: $\mathcal{T}^{b}\left(\frac{2\pi}{2},\frac{2\pi l}{2}\right)$ for
different symmetry fractionalization classes of $U(1)_{2}$, computed using Eq.
(61).

###  C.5 Comparison between small and large rotation angle

In the main text, we mainly studied the partial rotation
$\mathcal{T}^{b}(2\pi/M)$ where the rotation is associated with the point
group symmetry of the lattice system with $M=1,2,3,4,6$. Meanwhile, if we
instead consider a continuous system for topological order such as the
Laughlin wave function, the system has continuous rotation symmetry and one
can study the partial rotation $\mathcal{T}^{b}(\theta)$ with generic rotation
angle $\theta\in\mathbb{R}/2\pi{\mathbb{Z}}$. In this appendix, we study the
behavior of the partial rotation $\mathcal{T}^{b}(2\pi/M,2\pi l/M)$ with
generic rotation angle $\theta=2\pi/M$ with $M\in{\mathbb{Z}}$. We will see
that the partial rotation with the large rotation angle $M^{2}\ll L/\xi$
behaves quite differently from the small rotation angle $L/\xi\ll M^{2}$.
Unlike the case with the large rotation angle, $\mathcal{T}^{b}(2\pi/M,2\pi
l/M)$ with the small rotation angle gives a non-universal value that depends
on the correlation length of the system. At the same time, it also depends on
the universal responses such as Hall conductivity.

  * •

$M^{2}\ll L/\xi$. This case corresponds to the analysis in Sec. C.1, and the
partial rotation with $l=0$ is given by Eq. (50). When the rotation symmetry
does not permute anyons, the expression simplifies as Eq. (56).

  * •

$L/\xi\ll M^{2}$. Let us assume that the rotation symmetry does not permute
anyons.

When $M$ satisfies $L/\xi\ll M^{2}$, one cannot follow the logic in Sec. C.1
since the approximation of the character (47) is no longer valid. Instead of
the modular $ST^{M}S$ transformation to the CFT character performed in App. C,
let us do $S$ transformation with generic nonzero $l$

$\displaystyle\begin{split}&\chi_{1}\left(\frac{i\xi}{L}-\frac{1}{M};(0,0),(1,\frac{2\pi
l}{M})\right)\\\
&=\sum_{a}S_{1a}\chi_{a}\left(-\frac{1}{\frac{i\xi}{L}-\frac{1}{M}};(1,\frac{2\pi
l}{M}),(0,0)\right)\end{split}$ (62)

where $(0,0),(1,\frac{2\pi l}{M}))\in{\mathbb{Z}}_{M}\times U(1)$ represents
the twist along each direction of the torus. One can then use the alternative
approximation

$\displaystyle\begin{split}&\sum_{a}S_{1a}\chi_{a}\left(-\frac{1}{\frac{i\xi}{L}-\frac{1}{M}};(1,\frac{2\pi
l}{M}),(0,0)\right)\\\ &\approx\frac{1}{\mathcal{D}}\exp\left(-{2\pi
i(h_{0_{(1,\frac{2\pi
l}{M})}}-\frac{c_{-}}{24})\frac{1}{\frac{i\xi}{L}-\frac{1}{M}}}\right)\end{split}$
(63)

where $h_{0_{(1,\frac{2\pi l}{M})}}$ is the lowest energy of the
$(1,{\frac{2\pi l}{M}})$-twisted sector. When the CFT is chiral, the spin of
the $\mu\in U(1)$ vortex $h_{\mu}$ is given by Schwimmer and N. (1987); Fan
_et al._ (2022)

$\displaystyle 2\pi h_{0_{\mu}}=\frac{k}{2}\left(\frac{\mu}{2\pi}\right)^{2},$
(64)

where $k$ is the level of the holomorphic $U(1)$ current algebra. It is
related to the electric Hall conductivity as $\sigma_{H}=k/2\pi$. Using the
above relation, one can write the ${\mathbb{Z}}_{M}\times U(1)$ vortex as

$\displaystyle h_{0_{(1,\frac{2\pi
l}{M})}}=\frac{\sigma_{H}l^{2}}{2M^{2}}+\frac{\mathscr{S}l}{M^{2}}+\frac{\ell_{s}}{2M^{2}},$
(65)

where $\sigma_{H},\mathscr{S},\ell_{s}$ are coefficients of the response
action

$\displaystyle-\frac{\sigma_{H}}{4\pi}AdA-\frac{\mathscr{S}}{2\pi}Ad\omega-\frac{\ell_{s}}{4\pi}\omega
d\omega$ (66)

where $\omega$ is a $SO(2)=U(1)$ gauge field for continuous spatial rotation
symmetry. The partial rotation is then given by

$\displaystyle\begin{split}\mathcal{T}_{1}\left(\frac{2\pi}{M},\frac{2\pi
l}{M}\right)=&e^{-\frac{2\pi i}{24M}c_{-}}\exp\left(-{2\pi
i\left(\frac{\sigma_{H}l^{2}}{2M^{2}}+\frac{\mathscr{S}l}{M^{2}}+\frac{\ell_{s}}{2M^{2}}-\frac{c_{-}}{24}\right)\frac{1}{\frac{i\xi}{L}-\frac{1}{M}}}-\frac{2\pi
L}{\xi}\frac{c_{-}}{24}\right)\\\ =&e^{-\frac{2\pi
i}{24M}c_{-}}\exp\left({2\pi
i\left(\frac{\sigma_{H}l^{2}}{2M^{2}}+\frac{\mathscr{S}l}{M^{2}}+\frac{\ell_{s}}{2M^{2}}-\frac{c_{-}}{24}\right)\frac{\frac{1}{M}}{\frac{\xi^{2}}{L^{2}}+\frac{1}{M^{2}}}}\right)\\\
&\times\exp\left({-2\pi\left(\frac{\sigma_{H}l^{2}}{2M^{2}}+\frac{\mathscr{S}l}{M^{2}}+\frac{\ell_{s}}{2M^{2}}-\frac{c_{-}}{24}\right)\frac{\frac{\xi}{L}}{\frac{\xi^{2}}{L^{2}}+\frac{1}{M^{2}}}}-\frac{2\pi
L}{\xi}\frac{c_{-}}{24}\right)\\\ \end{split}$ (67)

##  Appendix D More on the classification of symmetry-enriched topological
phases

###  D.1 Equivalences on $([s_{\alpha}],[s_{\beta}],[s_{\gamma}])$

As stated in the main text, for a general topological order on the square
lattice, the assignment $([s_{\alpha}],[s_{\beta}],[s_{\gamma}])$ can be
adiabatically modified. For example, we can create four pairs of the Abelian
anyons $a,a^{-1}$ at $\alpha$ and move the $a$ anyons symmetrically to $\beta$
or $\gamma$ (See Fig. 7). This changes the assignment, giving us the
equivalences

$\displaystyle([s_{\alpha}],[s_{\beta}],[s_{\gamma}])$
$\displaystyle\simeq([s_{\alpha}\times a^{-4}],[s_{\beta}\times
a^{4}],[s_{\gamma}])$ (68)
$\displaystyle([s_{\alpha}],[s_{\beta}],[s_{\gamma}])$
$\displaystyle\simeq([s_{\alpha}\times a^{-4}],[s_{\beta}],[s_{\gamma}\times
a^{2}]).$ (69)

The full symmetry fractionalization classification is given by the number of
possible triples $([s_{\alpha}],[s_{\beta}],[s_{\gamma}])$, modulo the above
equivalences. Since $a^{2}$ is trivial in $U(1)_{\pm 2}$, this does not affect
our case. But more generally, if the Abelian anyons form a group $\mathcal{A}$
under fusion, the final classification from this procedure turns out to be
$\mathcal{A}\times(\mathcal{A}/4\mathcal{A})\times(\mathcal{A}/2\mathcal{A})$,
where $n\mathcal{A}$ denotes the group of Abelian anyons $a^{n}$, where
$a\in\mathcal{A}$. By comparing with Ref. Manjunath and Barkeshli (2021), we
see that this equals the cohomology group
$\mathcal{H}^{2}(\text{p4},\mathcal{A})$, which is the expected mathematical
classification of symmetry fractionalization from the G-crossed braided tensor
category approach.

Note that the anyon per unit cell $m$ is invariant under the equivalences on
$s_{\alpha},s_{\beta},s_{\gamma}$ in general. Therefore the $\mathcal{A}$
factor in the classification can be understood as the possible choices of $m$.
The $\mathcal{A}/4\mathcal{A}$ factor corresponds to a choice of
$s_{{\text{o}}}$ for ${\text{o}}=\alpha$ or $\beta$ up to the equivalences
above, while the remaining $\mathcal{A}/2\mathcal{A}$ factor corresponds to a
choice of either $s_{\gamma}$, or the torsion anyon $t$, which we define
below.

Figure 7: Equivalences in the real-space symmetry fractionalization data.
Left.  We place abelian anyons $s_{\text{o}}$ at the high symmetry points and
create four pairs of anyons $(a,a^{-1})$ (orange and purple bubbles
respectively) at $\alpha$. Middle. We symmetrically move only the $a$ anyons
to $\beta$, implementing Eq. (68). Right. We symmetrically move only the $a$
anyons to $\gamma$, implementing Eq. (69).

###  D.2 Discrete torsion vector in terms of real space construction

Refs. Manjunath and Barkeshli (2021, 2020) introduced a discrete torsion
vector $\vec{t}=(t_{x},t_{y})$, which is a pair of Abelian anyons that
partially characterizes the crystalline symmetry fractionalization. Here we
give an intuitive understanding of $\vec{t}_{{\text{o}}}$ using the notion of
anyonic polarization.

According Refs. Manjunath and Barkeshli (2021, 2020) to, on the square
lattice, $\vec{t}_{{\text{o}}}$ assigns an anyon
$t_{{\text{o}},x}^{a}t_{{\text{o}},y}^{b}$ to a region with Burgers vector
$(a+b,a-b)$, up to an equivalence relation. Braiding another anyon $c$ around
a region containing such a defect then gives a phase given by the mutual
braiding between $c$ and $t_{{\text{o}},x}^{a}t_{{\text{o}},y}^{b}$. In
particular, a dislocation with Burgers vector $(2,0)$ is assigned an anyon
$t_{{\text{o}}}:=t_{{\text{o}},x}t_{{\text{o}},y}$. In the present example,
$t_{{\text{o}}}$ is an invariant under the equivalence relations of Manjunath
and Barkeshli (2021, 2020) and completely characterizes the inequivalent
choices of $\vec{t}_{{\text{o}}}$. We can understand this heuristically in the
real-space construction as follows. Observe that $s_{\alpha}$, $s_{\beta}$,
$s_{\gamma}$ formally defines a topological charge (anyonic) polarization
$\vec{p}_{o}$. $\vec{p}_{\beta}$ is formally defined as
$\vec{p}_{\beta}=([s_{\gamma}+s_{\alpha}]^{1/2},[s_{\gamma}+s_{\alpha}]^{1/2})$,
and the topological charge polarization with respect to $\alpha$ is
$\vec{p}_{\alpha}=([s_{\gamma}+s_{\beta}]^{1/2},[s_{\gamma}+s_{\beta}]^{1/2})$.
The fundamental property of polarization is that a dislocation with Burgers
vector $\vec{b}$ is assigned a charge $\vec{p}_{\beta}\cdot\vec{b}$ Zhang  _et
al._ (2022b). Thus for $\vec{b}=(2,0)$, the region is assigned the anyon
$[s_{\alpha}+s_{\gamma}]$. This motivates us to define
$t_{\alpha}=s_{\alpha}s_{\gamma}$, thus explaining the relationship between
$t_{o}$ and $s_{o}$.

The above analysis was heuristic, as $[s]^{1/2}$ is not a well-defined object
in the mathematical theory. A more technical explanation that is in line with
the G-crossed BTC description is given below. Consider a general space group
operation ${\bf g}=(\vec{r},{\bf h})$, where ${\bf h}$ is an elementary
rotation by the angle $2\pi/M_{{\text{o}}}$ about some origin o, and ${\bf r}$
is a lattice translation. We have $M_{{\text{o}}}=4$ for
${\text{o}}=\alpha,\beta$ and $M_{{\text{o}}}=2$ for ${\text{o}}=\gamma$. Note
that

${\bf g}=(\vec{r},0)\times(\vec{0},{\bf h})=(\vec{0},{\bf
h})\times(^{\bf\bar{h}}\vec{r},0)$ (70)

where ${{}^{\bf\bar{h}}}\vec{r}$ is the vector ${\bf r}$ rotated by ${\bf
h}^{-1}$. This implies that we can obtain a ${\bf g}$ defect in two ways: by
fusing an ${\bf h}$-disclination at o to a dislocation with Burgers vector
$\vec{r}$ (we will call this an $\vec{r}$-dislocation) where the dislocation
is to the left, or fusing the ${\bf h}$-defect to an
${{}^{\bf\bar{h}}}\vec{r}$-dislocation where the dislocation is to the right.
The main point is that when there is nontrivial symmetry fractionalization,
the two fusion processes will differ by the anyon
$\vec{t}_{{\text{o}}}\cdot\vec{r}:=t_{{\text{o}},x}^{r_{x}}\times
t_{{\text{o}},y}^{r_{y}}$. Equivalently, $\vec{t}_{{\text{o}}}\cdot\vec{r}$ is
the residual anyon left behind at an ${\bf h}$-disclination, upon dragging an
${\bf r}$-dislocation through it from left to right.

Let us denote a reference ${\bf g}$ defect by $0_{\bf g}$. The subscript
denotes the group element associated to the defect, and for Abelian
topological order without anyon-permuting symmetries, the number of ${\bf
g}$-defects is given by the number of Abelian anyons, for any ${\bf g}$
Barkeshli  _et al._ (2019a). When ${\bf g}$ generates a discrete subgroup, we
can pick any of the above defects as our $0_{\bf g}$. The other ${\bf
g}$-defects are related to it as $a_{\bf g}:=a\times 0_{\bf g}$ where $a$ is
an Abelian anyon. Now from the above discussion it follows that

$0_{\vec{r}}\times 0_{\bf h}=\vec{t}_{{\text{o}}}\cdot\vec{r}\times 0_{\bf
h}\times 0_{{{}^{\bf\bar{h}}}\vec{r}}.$ (71)

For the square lattice, we will now derive the relationship

$t_{\alpha(\beta)}=s_{\alpha(\beta)}s_{\gamma}\mod a^{2},a\in\mathcal{A}.$
(72)

The derivation is as follows. Let ${\text{o}}=\alpha$, and ${\bf h}$ be a
$\pi/2$ rotation about $\alpha$. We know that $s_{\alpha}$ is the anyon
induced by fusing four ${\bf h}$ disclinations:

$s_{\alpha}=0_{\bf h}^{4}\mod a^{4}.$ (73)

But note that $(\hat{x},0)\times{\bf h}^{2}$ is actually a $\pi$ rotation
about $\gamma$. Therefore $s_{\gamma}$ is induced by fusing two
$(\hat{x},0)\times{\bf h}^{2}$ defects:

$\displaystyle s_{\gamma}=(0_{\vec{x}}\times 0_{\bf h}^{2})^{2}\mod a^{2}.$
(74)

But we can drag the $\vec{x}$ dislocation defects through the ${\bf
h}$-defects and obtain a relation between $s_{\gamma}$ and $s_{\alpha}$:

$\displaystyle s_{\gamma}$ $\displaystyle=0_{\vec{x}}\times 0_{\bf
h}^{2}\times 0_{\vec{x}}\times 0_{\bf h}^{2}$
$\displaystyle=t_{\alpha,x}\times 0_{\bf h}\times 0_{\vec{y}}\times 0_{\bf
h}\times 0_{\vec{x}}\times 0_{\bf h}^{2}$
$\displaystyle=t_{\alpha,x}t_{\alpha,y}\times 0_{\bf h}^{2}\times
0_{-\vec{x}}\times 0_{\vec{x}}\times 0_{\bf h}^{2}$
$\displaystyle=t_{\alpha,x}t_{\alpha,y}\times 0_{\bf h}^{4}$
$\displaystyle=t_{\alpha,x}t_{\alpha,y}\times s_{\alpha}=:t_{\alpha}\times
s_{\alpha}\mod a^{2}.$ (75)

Here we chose a gauge in which we can trivially fuse defects which are not
important for symmetry fractionalization, such as $0_{\vec{x}}$ and
$0_{-\vec{x}}$. This does not affect the final result. Moreover, since
$t_{\alpha}$ is defined mod $a^{2}$, we can equivalently write the last line
as $t_{\alpha}\times s_{\alpha}^{-1}$, for any topological order in which the
symmetry does not permute anyons. This gives the claimed result.

###  D.3 Relabeling and modular reduction

Let ${\bf h}$ be the group element associated to $\hat{C}_{4,l}$. Note that
the invariant $\mathcal{I}_{M,\theta=2\pi l/M}$ in Eq. (18) computes the
$\hat{C}_{4,l}$ charge associated to a reference $\hat{C}_{4,l}$ defect which
we denote by $0_{\bf h}$. But as we emphasized in the previous section, there
is no canonical choice of this reference defect; we can always redefine it by
fusing an Abelian anyon $a$, so that $0_{\bf h}\rightarrow 0_{\bf h}\times
a=:a_{\bf h}$. This will change the value of $\mathcal{I}_{M,\theta=2\pi l/M}$
by various braiding phases associated to $a$. Ref. Manjunath and Barkeshli
(2020) showed that the invariants before and after relabelling the defect are
related as follows:

$\frac{\mathcal{I}_{M,\theta=2\pi l/M}(a_{\bf h})}{\mathcal{I}_{M,\theta=2\pi
l/M}(0_{\bf h})}=M_{s^{*},a}\times\theta_{a}^{M}$ (76)

where $s^{*}=0_{\bf h}^{M}$ is an anyon that we determine below from the
symmetry fractionalization data (charge/spin vectors). Since there is no
canonical choice of an elementary $C_{4,l}$ defect, two systems in which the
partial rotation phases differ by this amount should be treated as equivalent
under relabellings.

We denote by $s^{*}$ the anyon induced upon inserting $M$ elementary $C_{4,l}$
defects, that is, disclinations with total angle $2\pi$ together with $2\pi l$
flux of $U(1)$. This implies $s^{*}=s\times v^{l}$. For $U(1)_{2}$ topological
order, we further have $s^{*}=a^{s+vl}$ where $a$ is a semion. The right-hand
side of Eq. (76) now reduces to

$M_{s^{*},a}\theta_{a}^{M}=\begin{cases}(-1)^{s+vl}\times
i^{4}=(-1)^{s+vl}&{\text{o}}=\alpha,\beta\\\
(-1)^{1+s+vl}&{\text{o}}=\gamma.\end{cases}$ (77)

After modding out by these quantities, the partial rotation invariant is
reduced modulo $\gcd(4,2(s+vl))$ for ${\text{o}}=\alpha,\beta$ (see Eq. (4) of
the main text), and mod $\gcd(2,1+s+vl)$ for ${\text{o}}=\gamma$ (see Eq.
(13)).