eqs

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

# Higher Hall conductivity from a single wave function:
Obstructions to symmetry-preserving gapped edge of (2+1)D topological order

Ryohei Kobayashi<EMAIL_ADDRESS>Department of Physics, Condensed Matter Theory
Center, and Joint Quantum Institute, University of Maryland, College Park,
Maryland 20742, USA  Taige Wang  Department of Physics, University of
California, Berkeley, California 94720 USA  Material Science Division,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA  Kavli
Institute for Theoretical Physics, University of California, Santa Barbara,
California 93106, USA  Tomohiro Soejima (副島智大) Department of Physics, Harvard
University, Cambridge, MA 02138, USA  Roger S. K. Mong (蒙紹璣)  Department of
Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA
Shinsei Ryu  Department of Physics, Princeton University, Princeton, New
Jersey, 08544, USA

###### Abstract

A (2+1)D topological ordered phase with U(1) symmetry may or may not have a
symmetric gapped edge state, even if both thermal and electric Hall
conductivity are vanishing. It is recently discovered that there are “higher”
versions of Hall conductivity valid for fermionic fractional quantum Hall
(FQH) states, which obstructs symmetry-preserving gapped edge state beyond
thermal and electric Hall conductivity. In this paper, we show that one can
extract higher Hall conductivity from a single wave function of an FQH state,
by evaluating the expectation value of the “partial rotation” unitary which is
a combination of partial spatial rotation and a U(1) phase rotation. This
result is verified numerically with the fermionic Laughlin state with
$\nu=1/3$, $1/5$, as well as the non-Abelian Moore-Read state. Together with
topological entanglement entropy, we prove that the expectation values of the
partial rotation completely determines if a bosonic/fermionic Abelian
topological order with U(1) symmetry has a symmetry-preserving gappable edge
state or not. We also show that thermal and electric Hall conductivity of
Abelian topological order can be extracted by partial rotations. Even in non-
Abelian FQH states, partial rotation provides the Lieb-Schultz-Mattis type
theorem constraining the low-energy spectrum of the bulk-boundary system. The
generalization of higher Hall conductivity to the case with Lie group symmetry
is also presented.

##  I Introduction

One of the most striking properties in topological phases of matter is the
appearance of non-trivial edge states Halperin (1982); Wen (1990); Stone
(1991); Frohlich and Kerler (1991). The most well-known example is the Integer
Quantum Hall (IQH) effect, where non-trivial bulk Chern number leads to the
presence of gapless edge state with U(1) symmetry Hatsugai (1993), which can
be probed by the Hall conductance $\sigma_{H}$. Even in the absence of U(1)
symmetry, systems with nonzero chiral central charge $c_{-}$, which signals
nonzero thermal Hall conductivity, have a gapless edge state Kane and Fisher
(1997). The thermal or electric Hall conductivity can be characterized through
the bulk effective theory, as well as extracted from a microscopic
wavefunction, through various entanglement measures such as modular
commutator, through randomized measurement, and so on Kitaev (2006); Mitchell
_et al._ (2018); Tu  _et al._ (2013); Zaletel  _et al._ (2013); Kim  _et al._
(2022a, b); Zou  _et al._ (2022); Fan (2022); Fan  _et al._ (2022); Shiozaki
_et al._ (2018); Dehghani  _et al._ (2021); Cian  _et al._ (2021).

When the bulk hosts a topologically ordered state, the edge state is sometimes
enforced to be gapless even if electric and thermal Hall conductivity are both
vanishing. This happens when one cannot condense a sufficient set of anyons at
the boundary enough to gap out the edge theory Kapustin and Saulina (2011);
Levin (2013). When the U(1) symmetry is taken into account, the anyon
condensation on the boundary further has to respect the global symmetry to
host a symmetric edge state.

Recently, it is discovered that a set of quantities called the higher central
charge Ng  _et al._ (2019, 2020); Kaidi  _et al._ (2021) and the higher Hall
conductivity Kobayashi (2022) partially give obstructions to having a gapped
edge state beyond thermal and electric Hall conductivity. Concretely, for
bosonic topological order in (2+1)D, having non-trivial higher central charge
$\zeta_{n}$ for some integer $n$ prohibits us from gapping out the edge state
even if $c_{-}=0$. Similarly, let us consider fermionic topological order with
$\mathrm{U}(1)^{f}$ symmetry in (2+1)D, where $\mathrm{U}(1)^{f}$ denotes
$\mathrm{U}(1)$ symmetry that charges a local fermion. This is a global
symmetry of fermionic fractional quantum Hall (FQH) states. In that case,
having non-trivial higher Hall conductivity again prohibits us from gapping
the edge state while respecting $\mathrm{U}(1)^{f}$ symmetry, even if
$c_{-}=\sigma_{H}=0$.

While both the higher central charge and higher Hall conductivity are
characterized through effective topological field theory, it is an important
question to ask if these quantities allow microscopic characterizations. In
the prior work by the authors Kobayashi  _et al._ (2024a), it was found that
the higher central charge can be extracted from a single wavefunction, by
evaluating the expectation value of a unitary called the partial rotation. The
extracted higher central charge then allows us to constrain the low-energy
spectrum of the bulk-boundary system, e.g., completely determine if the given
wavefunction bosonic of Abelian topological order has a gappable edge state or
not. In this work, we study microscopic characterization of the higher Hall
conductivity in the presence of $\mathrm{U}(1)^{f}$ symmetry, and explore its
physical consequences including constraints on the low energy spectrum of edge
states enforced by symmetry.

###  I.1 Summary of results

Here we summarize the main results of this paper. One of the main results is
that one can extract the higher Hall conductivity $\\{\zeta_{n}^{H}\\}$, which
is a set of invariants labeled by an integer $n$ satisfying certain
conditions.

Let us consider a single wave function of a fermionic FQH state on a cylinder
$\ket{\Psi_{a}}$ with a fixed topological sector, which is labeled by a simple
anyon of the bulk topological order $a$. We then take the expectation value of
the “partial rotation” unitary, which is the product of the partial spatial
rotation $T_{\theta,\mathrm{A}}$ associated with the partial $\mathrm{U}(1)$
transformation $U_{\phi,\mathrm{A}}$ on the wave function,

$\displaystyle\mathcal{T}_{a}\left(\theta,\phi\right):=\bra{\Psi_{a}}U_{\phi,\mathrm{A}}T_{\theta,\mathrm{A}}\ket{\Psi_{a}}.$
(2)

In a slight abuse of notation, we also refer to the expectation values as
partial rotation when it is clear from context. The case of $\phi=0$ reduces
to the partial rotation operator introduced in Ref. Kobayashi  _et al._ ,
2024a. We show that a nontrivial phase of the above quantity
$\mathcal{T}_{a}\left(\theta,\phi\right)$ with specific rotation angle is an
obstruction to having a gapped edge state while preserving $\mathrm{U}(1)^{f}$
symmetry. That is, for integers $n$ satisfying the condition
$\gcd(n,{N_{\mathrm{FS}}})=1$ with ${N_{\mathrm{FS}}}$ an integer called the
Frobenius-Schur exponent, we demonstrate that

$\displaystyle\mathcal{T}_{1}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$
$\displaystyle\propto\exp[\frac{2\pi
i}{n}\bigg{(}\frac{\sigma_{H}}{8}-\Big{(}\frac{n^{2}+2}{24}\Big{)}c_{-}\bigg{)}\biggr{]}\zeta_{n}^{H}.$
(3)

This quantity is the combination of $\sigma_{H}$, $c_{-}$, and
$\zeta_{n}^{H}$. The latter gives an obstruction to the presence of a
symmetric gapped edge state beyond $\sigma_{H}$, $c_{-}$.

Since the non-trivial higher Hall conductivity prohibits the symmetry-
preserving gapped edge state of the given wave function, the expectation
values of the partial rotation can be used to determine or constrain the low-
energy spectrum of the edge state enforced by symmetry. In particular, when
the topological order is Abelian, one can completely determine if the given
wavefunction admits a symmetry-preserving gapped edge or not. Our result is
supported by an analytical computation in terms of conformal field theory
(CFT), as well as numerics on fermionic Laughlin states and the non-Abelian
Moore-Read FQH state.

Next, let us describe our result valid for Abelian topological order. We will
first see that both electric and thermal Hall conductivity can be extracted by
partial rotation and topological entanglement entropy of a single
wavefunction. This is valid in either fermionic or bosonic Abelian phases, and
we numerically verify the result on a bosonic Laughlin states. Even when
$\sigma_{H}=c_{-}=0$, we show that once the topological entanglement entropy
of the given wave function is known, the partial rotations completely
determine if a given wave function for Abelian topological order admits a
gapped edge state preserving $\mathrm{U}(1)$ symmetry or not. This statement
is again valid for either bosonic or fermionic Abelian phases.

While partial rotation does not span complete obstructions to symmetric gapped
edge state for non-Abelian topological order, it is still useful for
constraining the low-energy spectrum of the bulk-boundary system. For (2+1)D
fermionic non-Abelian topological order with $\mathrm{U}(1)^{f}$ symmetry, we
show that the suitable set of expectation values of partial rotation enforces
the bulk-boundary spectrum to be 1: gapless, 2: symmetry broken, or 3:
symmetry-preserving gapped edge state with non-trivial lower bound on the
number of anyons in the bulk. The third possibility requires an extensive
number of bulk ground state degeneracy to realize the symmetry-preserving
gapped boundary. This is reminiscent of the Lieb-Schultz-Mattis type theorem
Lieb  _et al._ (1961); Oshikawa (2000); Hastings (2004), which constrains the
low-energy spectrum of a quantum many-body system with given input of symmetry
action on the state.

Finally, while higher Hall conductivity has been defined only for topological
order with U(1) symmetry, we propose its generalization to the case with Lie
group symmetry. In that case, higher Hall conductivity again gives
obstructions to gapped edge state preserving the Lie group symmetry, and can
be extracted from partial rotation.

This paper is organized as follows. In Sec. II, we review the definitions and
properties of higher Hall conductivity and higher central charge. In Sec. III,
we show that the higher Hall conductivity can be extracted from a single wave
function through partial rotation, and also that $c_{-}$ and $\sigma_{H}$ of
Abelian topological order can be extracted through partial rotation and
topological entanglement entropy. In Sec. IV, we provide numerical results
from DMRG for FQH states on a cylinder, and verify our analytical results. In
Sec. V, we explicitly show that one can completely determine the edge
gappability of Abelian topological order with U(1) symmetry. In Sec. VI, we
show the Lieb-Schultz-Mattis type theorem for bulk-boundary systems of
fermionic non-Abelian topological order with $\mathrm{U}(1)^{f}$ symmetry. In
Sec. VII, we describe a generalization of the higher Hall conductivity for the
phases with Lie group symmetry. In Sec. VIII, we conclude our paper with
possible future work.

##  II Definition and properties of higher Hall conductivity

###  II.1 Higher central charge of bosonic topological order

For completeness, let us first review higher central charge of (2+1)D bosonic
topological order that obstructs the presence of a gapped boundary. Given a
modular category $\mathcal{C}$, ${N_{\mathrm{FS}}}$ is the Frobenius-Schur
exponent defined as the smallest positive integer satisfying
$\theta_{a}^{N_{\mathrm{FS}}}=1$ for all anyons $a\in\mathcal{C}$.

The higher central charge $\zeta_{n}$ is labeled by a positive integer $n$.
When $n$ satisfies $\gcd(n,{N_{\mathrm{FS}}})=1$, higher central charge is
defined as

$\displaystyle\zeta_{n}:=\frac{\sum_{a\in\mathcal{C}}d_{a}^{2}\theta_{a}^{n}}{\bigl{|}\sum_{a\in\mathcal{C}}d_{a}^{2}\theta_{a}^{n}\bigr{|}}$
(4)

where $d_{a}$, $\theta_{a}$ are the quantum dimension and the topological
twist of the anyon $a$, respectively. The sum runs over all the anyons $a$ of
the bosonic phase, which are simple objects of the modular tensor category
$\mathcal{C}$.

The trivial higher central charge $\zeta_{n}=1$ for all $n$ such that
$\gcd(n,{N_{\mathrm{FS}}})=1$ gives necessary conditions to gapped boundary of
bosonic topological order, which are generally independent of the vanishing
chiral central charge condition $c_{-}=0$. This implies that the set of
quantities $\\{\zeta_{n}\\}$ prohibits the existence of the gapped edge state
even if $c_{-}$ is vanishing. In the special case with $n=1$, $\zeta_{1}$ just
gives the Gauss-Milgram formula for $c_{-}$ mod 8; $\zeta_{1}=e^{2\pi
ic_{-}/8}$. In this sense, $\zeta_{n}$ with $n>1$ can naturally be regarded as
a “higher” version of $c_{-}$, hence the name of higher central charge.

When the topological order is Abelian, $\zeta_{n}=1$ for all $n$ satisfying
$\gcd\bigl{(}n,\frac{{N_{\mathrm{FS}}}}{\gcd(n,{N_{\mathrm{FS}}})}\bigr{)}=1$
gives necessary and sufficient conditions for the gapped boundary Kaidi  _et
al._ (2021), so $\\{\zeta_{n}\\}$ completely determines the gappability of the
edge state of Abelian topological order. 111Though we defined $\zeta_{n}$ only
when $\gcd(n,{N_{\mathrm{FS}}})=1$ in Eq. (4), we can define $\zeta_{n}$ by
the same formula as (4) for $n$ satisfying
$\gcd(n,\frac{{N_{\mathrm{FS}}}}{\gcd(n,{N_{\mathrm{FS}}})})=1$ in the Abelian
case.

###  II.2 Higher Hall conductivity of fermionic FQH states

Here we review the definition and properties of the higher Hall conductivity,
which is valid for the (2+1)D fermionic topological phase with
$\mathrm{U}(1)^{f}$ symmetry Kobayashi (2022).

For a super-modular category $\mathcal{C}$ that describes the fermionic
topological order, we again define the Frobenius-Schur exponent
${N_{\mathrm{FS}}}$, defined as the smallest positive integer satisfying
$\theta_{a}^{N_{\mathrm{FS}}}=1$ for all anyons $a\in\mathcal{C}$.
${N_{\mathrm{FS}}}$ must be an even integer since the fermion has topological
spin $1/2$.

The higher Hall conductivity $\zeta_{n}^{H}$ is again labeled by positive
integers $n$ satisfying $\gcd(n,{N_{\mathrm{FS}}})=1$. We define
$\zeta_{n}^{H}$ as the following quantity computed from the data of anyons:

$\displaystyle\zeta^{H}_{n}:=\dfrac{\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}^{n}}{\bigl{|}\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}^{n}\bigr{|}}$ (5)

The sum runs over all the anyons $a$ of the fermionic phase, which are
formally regarded as simple objects of the super-modular tensor category
$\mathcal{C}$. $Q_{a}$ denotes the fractional $\mathrm{U}(1)^{f}$ charge of
the anyon $a$.

Importantly, $\zeta_{n}^{H}=1$ for all $n$ gives necessary conditions to
having a symmetry-preserving gapped edge state, generally independent of the
most fundamental conditions $c_{-}=\sigma_{H}=0$. This implies that the set of
quantities $\\{\zeta^{H}_{n}\\}$ prohibits the existence of the symmetry-
preserving gapped edge states even if both $c_{-}$ and $\sigma_{H}$ are
vanishing. In that sense, $\zeta^{H}_{n}$ is regarded as a “higher” version of
thermal and electric Hall conductivity, hence the name of higher Hall
conductivity.

We note that $c_{-}$ and $\sigma_{H}$ are also computed modulo 1 from the data
of anyons as Kobayashi and Barkeshli (2023)

$\displaystyle e^{-2\pi ic_{-}}$
$\displaystyle=(\sqrt{2}\mathcal{D})^{8}\cdot\frac{\sum_{a\in\mathcal{C}}e^{3i\pi
Q_{a}}d_{a}^{2}\theta_{a}}{\left(\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}\right)^{9}}~{},$ (6) $\displaystyle e^{-2\pi
i{\sigma}_{H}}$ $\displaystyle=\frac{\sum_{a\in\mathcal{C}}e^{3i\pi
Q_{a}}d_{a}^{2}\theta_{a}}{\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}}~{},$ (7)

where $\mathcal{D}$ is the total quantum dimension
$\mathcal{D}^{2}=\sum_{a}d_{a}^{2}$.

The higher Hall conductivity $\zeta_{n}^{H}$ cannot generally be expressed as
a linear combination of $c_{-},\sigma_{H}$. Meanwhile, in the special case
with $n=1$, we have the relation

$\displaystyle\zeta_{1}^{H}=e^{\frac{2\pi i}{8}(c_{-}-\sigma_{H})}~{}.$ (8)

##  III Extracting higher Hall conductivity from a single wavefunction

###  III.1 Partial rotation for the fermionic phase

In this section, we show that higher Hall conductivity $\zeta_{n}^{H}$ can be
extracted from a single wave function, by evaluating the expectation value of
the partial spatial rotation followed by partial $\mathrm{U}(1)^{f}$ symmetry
transformation.

Let us consider a (2+1)D fermionic topological ordered state with
$\mathrm{U}(1)^{f}$ symmetry on a cylinder. The state on the cylinder is
labeled by the anyon $a$, which corresponds to a quasiparticle obtained by
shrinking the puncture at the end of the cylinder. We denote the ground state
$\ket{\Psi_{a}}$ on the cylinder with the anti-periodic boundary condition for
${\mathbb{Z}}_{2}^{f}$ fermion parity symmetry. In that case, the state is
labeled by the anyon $a$ of the theory, which is the object of the super-
modular category $\mathcal{C}$.

Let us take a bipartition of the cylinder into the two subsystems labeled by A
and B. We write the translation operator for the A subsystem by the angle
$\theta$ along the circumference as $T_{\theta;\mathrm{A}}$, and also the
“partial” $\mathrm{U}(1)^{f}$ transformation by the angle $\phi$ that solely
acts on the A subsystem as $U_{\phi,\mathrm{A}}$, where $\phi=\pi$ corresponds
to the fermion parity. We then consider the expectation value of the
combination of the partial symmetry transformations as

$\displaystyle\mathcal{T}^{f}_{a}\left(\theta,\phi\right):=\bra{\Psi_{a}}U_{\phi,\mathrm{A}}T_{\theta,\mathrm{A}}\ket{\Psi_{a}},$
(9)

see Fig. 1 for a schematic illustration.

Figure 1: The invariant $\mathcal{T}_{a}\left(\theta,\phi\right)$ is defined
as an expectation value of a unitary
$U_{\phi,\mathrm{A}}T_{\theta,\mathrm{A}}$ acting within the A subsystem of
the cylinder.

Let us assume that $T_{\mathrm{A};\theta}$ induces the trivial symmetry
fractionalization on the underlying topological order, meaning that the phase
is trivial as a symmetry-enriched topological phase with translation symmetry.
We then find that this quantity with $(\theta,\phi)=(2\pi/n,\pi/n)$ solely
gives the constant universal value independent of the system size; when $n$ is
odd, we get

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&\propto
e^{\textstyle\frac{2\pi
i}{n}(\frac{\sigma_{H}}{8}-\frac{n^{2}+2}{24}c_{-})}e^{\textstyle\frac{2\pi
i}{n}h_{a}}\\\
&\qquad\times\sum_{b\in\mathcal{C}}d_{b}S_{ab}\theta_{b}^{n}e^{i\pi
Q_{b}}.\end{split}$ (10)

where $S_{ab}$ is the modular $S$ matrix of the bulk topological order, and
$h_{a}$ is the spin carried by the vacuum in the topological sector of CFT
labeled by the anyon $a$. In particular, when $a=1$ and $n$ is odd,

$\displaystyle\mathcal{T}^{f}_{1}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$
$\displaystyle\propto e^{\frac{2\pi
i}{n}(\frac{\sigma_{H}}{8}-(\frac{n^{2}+2}{24})c_{-})}\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}^{n},$ (11)

which is proportional to higher Hall conductivity $\zeta_{n}^{H}$ when
$\gcd(n,{N_{\mathrm{FS}}})=1$, and defines the obstruction to
$\mathrm{U}(1)^{f}$-preserving gapped edge state beyond $c_{-}$ and
$\sigma_{H}$.

One can do a simple consistency check of the formula (11) by taking $n=1$. In
that case, the unitary acting on the subsystem A is given by
$U_{\pi,\mathrm{A}}T_{2\pi,\mathrm{A}}$. $U_{\pi}$ generates
${\mathbb{Z}}_{2}^{f}$ fermion parity symmetry, while $T_{2\pi}$ generates the
$2\pi$ rotation of the cylinder. This cylinder is equipped with the anti-
periodic boundary condition for ${\mathbb{Z}}_{2}^{f}$ symmetry, as this
corresponds to the trivial sector $a=1$. Due to this boundary condition,
$T_{2\pi}$ again amounts to the action of ${\mathbb{Z}}_{2}^{f}$ symmetry. As
a whole, the combined operator $U_{\pi,\mathrm{A}}T_{2\pi,\mathrm{A}}$ gives
an identity, hence one should have
$\mathcal{T}^{f}_{1}\left(2\pi,\pi\right)=1$. This is satisfied with Eq. (11)
thanks to $\zeta_{1}^{H}=e^{\frac{2\pi i}{8}(c_{-}-\sigma_{H})}$.

One can derive the above results of partial rotations using the cut-and-glue
approach, which states that the reduced density matrix at the A subsystem is
effectively described by the thermal density matrix of the edge CFT at high
temperature near the bipartition Qi  _et al._ (2012); Kitaev and Preskill
(2006). This allows us to identify the expectation value of the partial
symmetry operators in the bulk Hilbert space as that evaluated in the Hilbert
space of edge CFT, and compute the quantity
$\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ at the level of
effective edge CFT. See Appendix A for the detailed descriptions.

For completeness, let us also present the results when $n$ is even. We get

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&\propto
e^{\frac{2\pi
i}{n}(\frac{\sigma_{H}}{8}-(\frac{n^{2}+2}{24})c_{-})}\sum_{v^{\prime}}e^{\frac{2\pi
i}{n}(h_{a}+h_{v^{\prime}})}\\\
&\quad\times\sum_{b\in\mathcal{C}}S_{ab}\theta_{b}^{n}e^{i\pi
Q_{b}}S_{bv^{\prime}},\end{split}$ (12)

where we sum over the labels of the ${\mathbb{Z}}_{2}^{f}$ vortex $v^{\prime}$
carrying the smallest scaling dimension. $v^{\prime}$ is regarded as an object
of the minimal modular extension of the super-modular category $\mathcal{C}$.

While the above formula involves the sum over ${\mathbb{Z}}_{2}$ vortices
$v^{\prime}$, one gets a simpler formula valid for even $n$ when we do not
include the partial U(1) transformation,

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},0\right)&\propto
e^{-\frac{2\pi i}{24}(n+\frac{2}{n})c_{-}}e^{\frac{2\pi
i}{n}h_{a}}\sum_{b\in\mathcal{C}}S_{ab}\theta_{b}^{n}d_{b}.\end{split}$ (13)

We note the viability of measuring the partial rotation. If the wavefunction
is put on a universal quantum computer, we can evaluate the expectation value
of the unitary operator through the Hadamard test.

####  III.1.1 Extraction of Hall conductivity via partial rotation in
fermionic Abelian phases

When the topological order is Abelian, one can extract both the electric and
thermal Hall conductivity from the partial rotation and topological
entanglement entropy. For fermionic phases, the algebraic theory of anyons
$\mathcal{C}$ is a super-modular category. In the case of Abelian phases, one
can write the super-modular tensor category as
$\mathcal{C}=\mathcal{C}_{0}\boxtimes\\{1,\psi\\}$ with a modular tensor
category $\mathcal{C}_{0}$. Here $\boxtimes$ denotes the Deligne product,
which physically implies that at the level of effective theory, fermionic
Abelian phase is regarded as stacking of bosonic topological order
$\mathcal{C}_{0}$ with fermionic invertible phase $\\{1,\psi\\}$.

With this in mind, let us illustrate the way of extracting $\sigma_{H}$ and
$c_{-}$ of a fermionic Abelian phase. We assume we know the total quantum
dimension $\mathcal{D}$ of $\mathcal{C}_{0}$ through e.g. computing the
topological entanglement entropy Kitaev and Preskill (2006).

When the phase is Abelian, the Frobenius-Schur exponent ${N_{\mathrm{FS}}}$
must divide $2\mathcal{D}^{2}$. By using this fact and
$\zeta_{1}^{H}=e^{\frac{2\pi i}{8}(c_{-}-\sigma_{H})}$, we can obtain $c_{-}$
and $\sigma_{H}$ from the following two quantities

$\displaystyle\begin{split}&\mathcal{T}^{f}_{1}\left(\frac{2\pi}{2\mathcal{D}^{2}+1},\frac{\pi}{2\mathcal{D}^{2}+1}\right)\\\
&\propto\exp\left[\tfrac{2\pi
i}{2\mathcal{D}^{2}+1}\left(\tfrac{\sigma_{H}}{8}-\tfrac{(2\mathcal{D}^{2}+1)^{2}+2}{24}c_{-}\right)+\tfrac{2\pi
i}{8}(c_{-}{-}\sigma_{H})\right],\end{split}$ (14)
$\displaystyle\begin{split}&\mathcal{T}^{f}_{1}\left(\frac{2\pi}{2\mathcal{D}^{2}-1},\frac{\pi}{2\mathcal{D}^{2}-1}\right)\\\
&\propto\exp\left[\tfrac{2\pi
i}{2\mathcal{D}^{2}-1}\left(\tfrac{\sigma_{H}}{8}-\tfrac{(2\mathcal{D}^{2}-1)^{2}+2}{24}c_{-}\right)-\tfrac{2\pi
i}{8}(c_{-}{-}\sigma_{H})\right].\end{split}$ (15)

That is, by obtaining the above two partial rotations and solving the linear
equation with respect to $\sigma_{H}$, $c_{-}$ enables us to determine both
electric and thermal Hall conductivity.

###  III.2 Bosonic phase

One can obtain similar results for (2+1)D bosonic topological phases with
$\mathrm{U}(1)$ symmetry. In that case, the quantity
$\mathcal{T}^{b}_{a}\left(\theta,\phi\right):=\bra{\Psi_{a}}U_{\phi,\mathrm{A}}T_{\theta,\mathrm{A}}\ket{\Psi_{a}}$
gives a constant universal value for $(\theta,\phi)=(2\pi/n,2\pi m/n)$ with
$n,m\in{\mathbb{Z}}$,

$\displaystyle\begin{split}\mathcal{T}^{b}_{a}\left(\frac{2\pi}{n},\frac{2\pi
m}{n}\right)&\propto e^{\frac{2\pi
i}{n}(\frac{m^{2}}{2}\sigma_{H}-(\frac{n^{2}+2}{24})c_{-})}e^{\frac{2\pi
i}{n}h_{a}}\\\
&\qquad\times\sum_{b\in\mathcal{C}}d_{b}S_{ab}\theta_{b}^{n}e^{2i\pi
mQ_{b}}.\end{split}$ (16)

When $m=0$ and $a=1$, it reduces to the higher central charge of the bosonic
topological phase Kobayashi  _et al._ (2024a)

$\displaystyle\begin{split}\mathcal{T}^{b}_{a}\left(\frac{2\pi}{n},0\right)&\propto
e^{-\frac{2\pi i}{24}(n+\frac{2}{n})c_{-}}\zeta_{n}.\end{split}$ (17)

####  III.2.1 Extraction of Hall conductivity via partial rotation in bosonic
Abelian phases

As we have done in the fermionic Abelian phases, one can also extract
$\sigma_{H}$ and $c_{-}$ of the bosonic Abelian topological order from partial
rotations and topological entanglement entropy.

First, we again compute topological entanglement entropy to obtain the total
quantum dimension $\mathcal{D}$. Next, we obtain the chiral central charge
$c_{-}$. By noticing that $2\mathcal{D}^{2}$ is divisible by
${N_{\mathrm{FS}}}$ in Abelian phases, one can extract $c_{-}$ by the partial
rotation

$\displaystyle\mathcal{T}^{b}_{1}\left(\frac{2\pi}{2\mathcal{D}^{2}},0\right)\propto
e^{-2\pi i(\frac{2}{2\mathcal{D}^{2}}+2\mathcal{D}^{2})\frac{c_{-}}{24}}~{},$
(18)

from which we get $c_{-}$.

Then, we obtain the electric Hall conductivity $\sigma_{H}$ via partial
rotation. To do this, pick an integer $k$ coprime with $2\mathcal{D}^{2}$,
$\gcd(k,2\mathcal{D}^{2})=1$. The partial rotation with
$(\theta,\phi)=(2\pi/k,2\pi/k)$ is then given by

$\displaystyle\begin{split}\mathcal{T}^{b}_{1}\left(\frac{2\pi}{k},\frac{2\pi}{k}\right)&\propto
e^{\frac{2\pi
i}{k}(\frac{\sigma_{H}}{2}-(\frac{k^{2}+2}{24})c_{-})}\sum_{b\in\mathcal{C}}\theta_{b}^{k}M_{b,v}\\\
&=e^{\frac{2\pi
i}{k}(\frac{\sigma_{H}}{2}-(\frac{k^{2}+2}{24})c_{-})}\sum_{b\in\mathcal{C}}\theta_{b}^{k}M_{b,{\tilde{v}}^{k}}\\\
&=e^{\frac{2\pi
i}{k}(\frac{\sigma_{H}}{2}-(\frac{k^{2}+2}{24})c_{-})}\sum_{b\in\mathcal{C}}\left(\frac{\theta_{b\times\tilde{v}}}{\theta_{\tilde{v}}}\right)^{k}\\\
&=e^{\frac{2\pi
i}{k}(\frac{\sigma_{H}}{2}-(\frac{k^{2}+2}{24})c_{-})}\frac{\zeta_{n}}{(\theta_{\tilde{v}})^{k}}~{},\end{split}$
(19)

where $M_{a,b}$ denotes the mutual braiding between anyons $a,b$, and $v$ is a
specific anyon called a vison satisfying $M_{b,v}=e^{2i\pi Q_{a}}$, and the
anyon $\tilde{v}$ satisfies $v=\tilde{v}^{k}$ which exists since
$\gcd(k,2\mathcal{D}^{2})=1$. We then have

$\displaystyle\frac{\mathcal{T}^{b}_{1}\left(\frac{2\pi}{k},\frac{2\pi}{k}\right)}{\mathcal{T}^{b}_{1}\left(\frac{2\pi}{k},0\right)}=\frac{e^{\frac{2\pi
i}{k}\sigma_{H}}}{(\theta_{\tilde{v}})^{k}}.$ (20)

Let us take an integer $l$ satisfying $kl=1$ mod $2\mathcal{D}^{2}$. Noticing
that $\theta_{v}=e^{i\pi\sigma_{H}}$, we can extract $\sigma_{H}$ from the
ratio of partial rotations

$\displaystyle\frac{\mathcal{T}^{b}_{1}\left(\frac{2\pi}{k},\frac{2\pi}{k}\right)}{\mathcal{T}^{b}_{1}\left(\frac{2\pi}{k},0\right)}=e^{\pi
i\sigma_{H}(\frac{1}{k}-l)}~{},$ (21)

from which we get $\sigma_{H}$.

##  IV Numerical results

We first briefly describe the procedure to compute the partial $\mathrm{U}(1)$
rotation given a matrix product state (MPS). The technology we need here is
very similar to the one used in Ref. Kobayashi  _et al._ , 2024a, whereas the
only difference is that we also need the particle number label for the
$\mathrm{U}(1)$ rotation. For the quantum Hall systems we consider here, we
always work with an infinite cylindrical geometry, and Landau gauge orbitals
around the cylinder. We conserve both momentum around the cylinder and
particle number, allowing us to reach bond dimension $\chi=3200$ and cylinder
circumference $L_{y}=40\ell_{B}$ such that the partial rotation in all Abelian
phases studied is well converged. We use the real-space entanglement spectrum
(RSES) algorithm described in Ref. Zaletel and Mong, 2012 to cut the cylinder
in two halves in physical space, along which we perform the partial
$\mathrm{U}(1)$ rotation. The final expectation value we need can be evaluated
at the virtual bond corresponding to the cut,

$\mathcal{T}_{a}(\theta,\phi)=\sum_{\alpha}\lambda_{\alpha}^{2}e^{ik_{y}^{\alpha}L_{y}\theta}e^{in^{\alpha}\phi}$
(22)

where $\lambda_{\alpha}$ is the Schmidt value, and $k_{y}^{\alpha}$ and
$n^{\alpha}$ are the corresponding physical momentum and particle number. Here
we resolve the ambiguity in the momentum label by matching the entanglement
spectrum and the edge CFT spectrum, such that (a) the highest weight state has
zero momentum and charge, and (b) that sectors with opposite charges have the
same momentum labels.

Now we present $\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$
of the fermionic $\mathrm{U}(1)_{m}$ topological order, which have $2m$
Abelian anyons $\\{[j]_{2m}\\}$ for $0\leq j\leq 2m-1$ with
$\theta_{j}=\exp(\pi i\,j^{2}/m)$, $Q_{j}=j/m$, following the
${\mathbb{Z}}_{2m}$ fusion rule. The anyon $[m]_{2m}$ corresponds to a
transparent fermion $\psi$. In the expression of
$\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ for even $n$ in
Eq. (12), we need to sum over the ${\mathbb{Z}}_{2}$ vortices $v^{\prime}$
with the smallest spin, which are $v^{\prime}=([1]_{4m}),([4m-1]_{4m})$ with
$h_{v^{\prime}}=1/8m$ in $\mathrm{U}(1)_{4m}$. Here, note that the
${\mathbb{Z}}_{2}$ vortices are described by the object of the minimal modular
extension of $\mathrm{U}(1)_{m}$, which is $\mathrm{U}(1)_{4m}$.

We show the expectation values of
$\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ for small $n$
in the $\nu=1/3,1/5$ Laughlin state in Table 1 as microscopic realization of
the $\mathrm{U}(1)_{m}$ topological order. The model we study here is two-
dimensional fermions in the lowest Landau level (LLL) at $\nu=1/m$ filling
with an interaction $V_{n<m}=1$ plus a small perturbation $\delta
V_{n=m}=0.1$, where $V_{n}$ are the Haldane pseudopotentials Rezayi and
Haldane (1994); Halperin  _et al._ (2020). We obtained the ground state MPS
from infinite density matrix renormalization group (iDMRG) calculations
Zaletel  _et al._ (2015). As shown in Fig. 2 (b),
$\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ always
converges to the expected phase in Table 1 at sufficiently large $L_{y}$.

Figure 2: $\operatorname{Arg}\mathcal{T}_{1}(2\pi/n,\pi/n)$ of the $\nu=1/3$
and $\nu=1/5$ Laughlin states. The dotted lines are the CFT predictions in
Table 1.

The non-Abelian phases, however, present more numerical challenges. Here we
show $\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ of the
Moore-Read state in Fig. 3. The effective field theory for the Moore-Read
state is obtained by condensing a fermion $\psi$ in the
$\mathrm{U}(1)_{8}\times\mathrm{Ising}$ topological order. The anyons are
labeled by $([j]_{8},a)$ with $a\in\\{1,\sigma,\epsilon\\}$, where the fermion
is labeled by $\psi=([4]_{8},\epsilon)$. The fractional U(1) charge is given
by $Q_{([j]_{8},a)}=j/4$. See Table 3 for spins of the anyons in the Moore-
Read state, and Table 1 for the values of
$\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ for small $n$.
In the expression of
$\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ for even $n$ in
Eq. (12), we need to sum over the ${\mathbb{Z}}_{2}$ vortices $v^{\prime}$
with the smallest spin, which are
$v^{\prime}=([1]_{8},1),([7]_{8},1),([0]_{8},\sigma)$ with $h_{m}=1/16$ in
$\mathrm{U}(1)_{8}\times\mathrm{Ising}$.

We first construct the MPS for the Moore-Read model wavefunction from the
corresponding $1+1$D chiral CFT, a tensor product of a chiral boson and a
Majorana mode Zaletel and Mong (2012). Here we keep $17$ CFT Virasoro levels
in the MPS construction. As shown in Fig. 3 (a),
$\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ is converging,
although oscillatory, to the expected phase in Table 1. We cannot reach full
convergence even at the largest system size we reach. We attribute the slow
convergence to the longer correlation length of the Moore-Read state. To
explore the Moore-Read phase, we also study a microscopic model of half-filled
$N=1$ Landau level of two-dimensional fermions with long-range Coulomb
interaction using iDMRG. As shown in Fig. 3 (b),
$\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ is also
converging to the expected values with this more generic state in the Moore-
Read phase.

Figure 3: $\operatorname{Arg}\mathcal{T}_{1}(2\pi/n,\pi/n)$ of the $\nu=1/2$
Moore-Read state. We obtain the ground state from (a) CFT construction and (b)
iDMRG calculation respectively. The dotted lines are the CFT predictions given
in Table 1.

Finally, we show results of bosonic Abelian phases and explore the possibility
to extract electric Hall conductance from partial rotation. Here we present
$\mathcal{T}_{1}^{b}\left(\frac{2\pi}{n},\frac{2\pi}{n}\right)$ of the bosonic
$\nu=1/2$ Laughlin state, with a single non-trivial anyon being the semion. We
follow Ref. Kobayashi  _et al._ , 2024a to obtain the ground state MPS by
solving the half-filled LLL of two-dimensional bosons with a contact
interaction $V_{0}=1$ plus a small perturbation $\delta V_{2}=0.1$ with iDMRG.
As shown in Fig. 4 (a),
$\mathcal{T}_{1}^{b}\left(\frac{2\pi}{n},\frac{2\pi}{n}\right)$ always
converges to the expected phase as shown in Table 2 at sufficiently large
$L_{y}$. Combining with $\mathcal{T}_{1}^{b}\left(\frac{2\pi}{n},0\right)$
obtained in Ref. Kobayashi  _et al._ , 2024a, we can extract the electric Hall
conductance using Eq. (21). As shown in Fig. 4 (b), the Hall conductance
converges to the expected value $\sigma_{H}=1/2$. The numerical success of Eq.
(21) presents partial rotation as an appealing method to extract boson Hall
conductance.

Figure 4: (a) $\operatorname{Arg}\mathcal{T}_{1}(2\pi/n,2\pi/n)$ of the $\nu=1/2$ bosonic Laughlin state. The dotted lines are the CFT predictions given in Table 2. (b) Extracted electric Hall conductivity using Eq. (21). | $\mathcal{T}_{1}^{f}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$
---|---
$\mathrm{U}(1)_{3}$ | $1,1,0,-i$
$\mathrm{U}(1)_{5}$ | $1,1,e^{\frac{13\pi i}{45}}(4+2e^{2\pi i/5}+4e^{4\pi i/5})\sim 4.20+1.53i,e^{\pi i/4}$
MR | $1,1,e^{\pi i/3},0$
Table 1: The phases of the partial rotation $\mathcal{T}^{f}_{1}(\frac{2\pi}{n},\frac{\pi}{n})$ for $n=1,2,3,4$ in the trivial sector of $\mathrm{U}(1)_{3}$, $\mathrm{U}(1)_{5}$ and Moore-Read. We write $0$ when the magnitude is vanishing. | $\mathcal{T}_{1}^{b}\left(\frac{2\pi}{n},\frac{2\pi}{n}\right)$ | $\mathcal{T}_{1}^{b}\left(\frac{2\pi}{n},0\right)$
---|---|---
$\mathrm{U}(1)_{2}$ | $1,1,e^{\pi i/9},0$ | $1,0,e^{\frac{13\pi i}{9}},e^{\frac{13\pi i}{8}}$
Table 2: The phases of the partial rotation $\mathcal{T}^{b}_{1}(\frac{2\pi}{n},\frac{2\pi}{n}),\mathcal{T}_{1}^{b}\left(\frac{2\pi}{n},0\right)$ for $n=1,2,3,4$ in the trivial sector of the bosonic Laughlin state $\mathrm{U}(1)_{2}$. We write $0$ when the magnitude is vanishing. | [0]8 | [1]8 | [2]8 | [3]8 | [4]8 | [5]8 | [6]8 | [7]8
---|---|---|---|---|---|---|---|---
$1$ | 0 |  | $\frac{1}{4}$ |  | 0 |  | $\frac{1}{4}$ |
$\sigma$ |  | $\frac{1}{8}$ |  | $\frac{5}{8}$ |  | $\frac{5}{8}$ |  | $\frac{1}{8}$
$\epsilon$ | $\frac{1}{2}$ |  | $\frac{3}{4}$ |  | $\frac{1}{2}$ |  | $\frac{3}{4}$ |
Table 3: Spins of anyons in the Moore-Read state.

##  V Complete set of obstructions to U(1)-preserving gapped edge in Abelian
topological order

In this section, we show that together with topological entanglement entropy,
the invariants $\mathcal{T}^{f}_{1},\mathcal{T}^{b}_{1}$ completely determine
the gappability of the edge state in the case of Abelian topological order
with $\mathrm{U}(1)$ symmetry. This result holds for both fermionic and
bosonic Abelian phases, and we provide an explicit algorithm for determining
gappability in both cases.

###  V.1 Bosonic Abelian phase

Let us start with bosonic topological order with $\mathrm{U}(1)$ symmetry.
Together with topological entanglement entropy, one can see that the partial
rotations
$\mathcal{T}^{b}_{1}\left(\frac{2\pi}{n},0\right),\mathcal{T}^{b}_{1}\left(\frac{2\pi}{n},\frac{2\pi}{n}\right)$
completely determines if the bosonic Abelian topological order admits a
$\mathrm{U}(1)$-preserving gapped edge state or not.

To see this, let us recall that higher central charges $\\{\zeta_{n}\\}$ with
$n>1$ together with the electric and thermal Hall conductivity
$\sigma_{H},c_{-}$ can fully determine the U(1)-preserving edge gappability
Kobayashi (2022).

It was already shown in Ref. Kobayashi  _et al._ , 2024a that
$\\{\zeta_{n}\\}$ can all be extracted from the partial rotation. In addition,
both $\sigma_{H}$ and $c_{-}$ are shown to be extracted from partial rotation
and topological entanglement entropy in Sec. III.2.1. These span all the
obstructions to $\mathrm{U}(1)$-preserving gapped edge state mentioned above.

###  V.2 Fermionic Abelian phase

One can also completely determine if a fermionic Abelian topological order
with U(1)f symmetry admits a symmetry-preserving gapped edge state or not,
based on the partial rotation
$\mathcal{T}^{f}_{1}\left(\frac{2\pi}{n},\frac{\pi}{n}\right),\mathcal{T}^{f}_{1}\left(\frac{2\pi}{n},0\right)$
and topological entanglement entropy.

For fermionic Abelian phases, one can write the super-modular tensor category
as $\mathcal{C}=\mathcal{C}_{0}\boxtimes\\{1,\psi\\}$ with a modular tensor
category $\mathcal{C}_{0}$. One can then show that $c_{-},\sigma_{H}$ and set
of higher central charges $\\{\zeta_{n}(\mathcal{C}_{0})\\}$ for
$\gcd(n,\frac{{N_{\mathrm{FS}}}(\mathcal{C}_{0})}{\gcd(n,{N_{\mathrm{FS}}}(\mathcal{C}_{0}))})=1$
give the complete obstructions to the U(1)-preserving gapped edge state. See
Appendix B for a proof.

In Sec. III.1.1, we have already seen that $\mathcal{D}$, $c_{-},\sigma_{H}$
are extracted from partial rotation and topological entanglement entropy.
Hence, the remaining task for determining gappability is to extract higher
central charges $\\{\zeta_{n}(\mathcal{C}_{0})\\}$ when $\sigma_{H}=c_{-}=0$.
Below, we provide an explicit algorithm for extracting
$\\{\zeta_{n}(\mathcal{C}_{0})\\}$ in Abelian fermionic phase, when
$\sigma_{H}=c_{-}=0$ is understood and $\mathcal{D}$ is already known.

For convenience, let us write the prime factorization of $\mathcal{D}^{2}$ as

$\displaystyle\mathcal{D}^{2}=p_{1}^{k_{1}}\times
p_{2}^{k_{2}}\times\dots\times p_{M}^{k_{M}}$ (23)

with $p_{1}<p_{2}<\dots<p_{M}$ prime numbers, and $k_{1},k_{2},\dots,k_{M}$
positive integers. Then, the modular tensor category $\mathcal{C}_{0}$ also
factorizes Kaidi  _et al._ (2021) as

$\displaystyle\mathcal{C}_{0}=\mathcal{C}_{p_{1}}\boxtimes\mathcal{C}_{p_{2}}\boxtimes\dots\boxtimes\mathcal{C}_{p_{M}}.$
(24)

Each Abelian theory $\mathcal{C}_{p_{j}}$ contains the $p_{j}^{k_{j}}$ anyons
including the trivial one. The Frobenius-Schur exponent also naturally admits
the factorization as

$\displaystyle{N_{\mathrm{FS}}}(\mathcal{C}_{0})=N_{1}N_{2}\dots N_{M},$ (25)

where $N_{j}$ gives the Frobenius-Schur exponent of the theory
$\mathcal{C}_{p_{j}}$. Each $N_{j}$ is expressed by a positive integer power
of $p_{j}$, $N_{j}=p_{j}^{l_{j}}$, where $l_{j}$ satisfies

$\displaystyle\begin{cases}1\leq l_{1}\leq k_{1}+1&\text{if $p_{1}=2$,}\\\
1\leq l_{j}\leq k_{j}&\text{otherwise}.\end{cases}$ (26)

In particular, the above relations explain why ${N_{\mathrm{FS}}}$ must divide
$2\mathcal{D}^{2}$.

Further, a vison $v$ encoding the U(1)f symmetry fractionalization also admits
a factorization in $\mathcal{C}=\mathcal{C}_{0}\boxtimes\\{1,\psi\\}$ as

$\displaystyle v=v_{1}\times v_{2}\dots\times v_{M}\times v_{\psi},$ (27)

where $v_{j}\in\mathcal{C}_{p_{j}}$, and $v_{\psi}$ is a vortex in the modular
extension of $\\{1,\psi\\}$ satisfying $M_{v_{\psi},\psi}=-1$.

Now we are ready to describe all higher central charges in terms of the
partial rotation. For this purpose, we note that having
$\zeta_{n}(\mathcal{C}_{0})=1$ for all $n$ such that
$\gcd\bigl{(}n,\frac{{N_{\mathrm{FS}}}(\mathcal{C}_{0})}{\gcd(n,{N_{\mathrm{FS}}}(\mathcal{C}_{0}))}\bigr{)}=1$
is equivalent to having the higher central charge
$\zeta_{n}(\mathcal{C}_{p_{j}})=1$ for all $j$ and $1\leq n\leq N_{j}$ such
that $\gcd(n,p_{j})=1$ Kaidi  _et al._ (2021).

One can cover all the above $\zeta_{n}(\mathcal{C}_{p_{j}})$ by computing
either $\mathcal{T}^{f}_{1}(2\pi/n,\pi/n)$ or $\mathcal{T}^{f}_{1}(2\pi/n,0)$
with

$\displaystyle n=\frac{m\times 2\mathcal{D}^{2}}{p_{j}^{\tilde{k}_{j}}},$ (28)

where $m$ scans all integers $1\leq m\leq p_{j}^{\tilde{k}_{j}}$ such that
$\gcd(m,p_{j})$ = 1. The integers $\tilde{k}_{j}$ are defined as

$\displaystyle\begin{cases}\tilde{k}_{1}=k_{1}+1&\text{if $p_{1}=2$,}\\\
\tilde{k}_{j}=k_{j}&\text{otherwise}.\end{cases}$ (29)

Here, depending on the value of $m$ we compute
$\mathcal{T}^{f}_{1}(2\pi/n,\pi/n)$ when $n$ is odd, and compute
$\mathcal{T}^{f}_{1}(2\pi/n,0)$ when $n$ is even.

Let us check that above partial rotations exhaust all desired higher central
charges of $\mathcal{C}_{p_{j}}$, when $c_{-}=\sigma_{H}=0$. Since
$2\mathcal{D}^{2}/p_{j}^{\tilde{k}_{j}}$ is coprime with $p_{j}$ and divisible
by $N_{i}$ for all $i$ such that $i\neq j$, one can write the partial rotation
with $n=m\times 2\mathcal{D}^{2}/p_{j}^{\tilde{k}_{j}}$ as

  * •

when $n$ is odd and $c_{-}=\sigma_{H}=0$,

$\displaystyle\begin{split}&\mathcal{T}^{f}_{1}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)\propto\sum_{a\in\mathcal{C}_{0}}e^{i\pi
Q_{a}}\theta_{a}^{n}\\\ \qquad&=\prod_{1\leq i\leq
M}\left(\sum_{a\in\mathcal{C}_{p_{i}}}e^{i\pi
Q_{a}}\theta_{a}^{n}\right)\propto\sum_{a\in\mathcal{C}_{p_{j}}}e^{i\pi
Q_{a}}\theta_{a}^{n}\\\
\qquad&=\sum_{a\in\mathcal{C}_{p_{j}}}M_{a,v_{j}}\theta_{a}^{n}=\sum_{a\in\mathcal{C}_{p_{j}}}M_{a,\tilde{v}_{j}^{n}}\theta_{a}^{n}\\\
\qquad&=\sum_{a\in\mathcal{C}_{p_{j}}}\left(\frac{\theta_{a\times\tilde{v}_{j}}}{\theta_{\tilde{v}_{j}}}\right)^{n}\propto\frac{\zeta_{n}(\mathcal{C}_{p_{j}})}{\theta_{v_{j}}^{l}}.\end{split}$
(30)

where $\tilde{v}_{j}\in\mathcal{C}_{p_{j}}$ satisfies
$\tilde{v}_{j}^{n}=v_{j}$, and we take an integer $l$ satisfying $ln=1$ mod
$p_{j}$. Since $\theta_{v_{j}}=1$ when $\sigma_{H}=0$,
$\mathcal{T}^{f}_{1}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$ gives
$\zeta_{n}(\mathcal{C}_{p_{j}})$ when $c_{-},\sigma_{H}$ are vanishing.

  * •

when $n$ is even and $c_{-}=0$,

$\displaystyle\begin{split}\mathcal{T}^{f}_{1}\left(\frac{2\pi}{n},0\right)&\propto\sum_{a\in\mathcal{C}_{0}}\theta_{a}^{n}\\\
&=\prod_{1\leq i\leq
M}\left(\sum_{a\in\mathcal{C}_{p_{i}}}\theta_{a}^{n}\right)\propto\sum_{a\in\mathcal{C}_{p_{j}}}\theta_{a}^{n}\\\
&\propto\zeta_{n}(\mathcal{C}_{p_{j}}).\end{split}$ (31)

One can then see that $\zeta_{n}(\mathcal{C}_{p_{j}})$ covers all the desired
higher central charges by scanning $m$, since $n$ mod $N_{j}$ covers all
$1\leq n\leq N_{j}$ such that $\gcd(n,p_{j})=1$. Since we do not know
${N_{\mathrm{FS}}}$ or $N_{j}$ in advance, we want to take the range of $m$ as
$1\leq m\leq p_{j}^{\tilde{k}_{j}}$ so that $1\leq m\leq N_{j}$ is included.
This shows that the above $\mathcal{T}_{1}\left(2\pi/n\right)$ covers all
gappability obstructions of $\mathcal{C}_{p_{j}}$.

Summarizing, we can obtain all gappability obstructions of the whole theory
$\mathcal{C}$ beyond $c_{-}$ and $\sigma_{H}$, by computing the partial
rotation ($\mathcal{T}^{f}_{1}(2\pi/n,\pi/n)$ or
$\mathcal{T}^{f}_{1}(2\pi/n,0)$ depending on odd/even $n$) with

$\displaystyle n=\frac{m\times 2\mathcal{D}^{2}}{p_{j}^{\tilde{k}_{j}}},$ (32)

where $j$ scans all $1\leq j\leq M$, and $m$ scans all integers $1\leq m\leq
p_{j}^{\tilde{k}_{j}}$ such that $\gcd(m,p_{j})$ = 1.

##  VI Lieb-Schultz-Mattis type constraint

In this section, we consider the setup where the topological order is not
necessarily Abelian. We argue that the partial rotation can still be utilized
to constrain the low-energy spectrum of the bulk-boundary system with
$\mathrm{U}(1)^{f}$ global symmetry, even for non-Abelian topological order.

Suppose that we observed $\\{\mathcal{T}_{1}(2\pi/p_{j},\pi/p_{j})\\}$ has a
non-trivial phase for a set of distinct prime numbers $\\{p_{j}\\}$. One can
see that this leaves us two possibilities: 1. the edge cannot be gapped while
preserving $\mathrm{U}(1)^{f}$ symmetry, or 2. the edge is gappable preserving
$\mathrm{U}(1)^{f}$ symmetry, where the bulk must have multiple ground states
on a torus with the degeneracy non-trivially bounded from below. This
statement is reminiscent of the Lieb-Schultz-Mattis type theorems Lieb  _et
al._ (1961); Oshikawa (2000); Hastings (2004), which constrains the low-energy
spectrum for a given input of the symmetry action on the ground state.
Concretely, one can show the following theorem:

Theorem. For a (2+1)D topological order with $\mathrm{U}(1)^{f}$ symmetry,
suppose that the partial rotation
$\\{\mathcal{T}_{1}(2\pi/p_{j},\pi/p_{j})\\}$ has a non-trivial phase for a
set of distinct prime numbers $\\{p_{j}\\}$. Then, the topological order
admits a $\mathrm{U}(1)^{f}$ symmetry-preserving gapped edge state only if the
number of anyons $r$ satisfies $r\geq r_{0}$, where $r_{0}$ is the smallest
integer satisfying $\prod_{j}p_{j}\leq
2^{\frac{4}{3}r_{0}+8}3^{\frac{4}{3}r_{0}}$.

Proof. Let us assume that the topological order admits a
$\mathrm{U}(1)^{f}$-preserving gapped edge, while the phases of
$\\{\mathcal{T}_{1}(2\pi/p_{j})\\}$ are non-trivial.
$\mathcal{T}_{1}(2\pi/p_{j},0)$ obstructs the $\mathrm{U}(1)^{f}$-preserving
gapped edge when $\gcd({N_{\mathrm{FS}}}(\mathcal{C}),p_{j})=1$, where
$\mathcal{C}$ is the super-modular tensor category that describes the anyons
of the topological order. We hence must have
$\gcd({N_{\mathrm{FS}}}(\mathcal{C}),p_{j})>1$. This implies that
${N_{\mathrm{FS}}}(\mathcal{C})$ must be divisible by $\prod_{j}p_{j}$, so we
have $\prod_{j}p_{j}\leq{N_{\mathrm{FS}}}(\mathcal{C})$. Meanwhile, let us
consider a modular tensor category $\breve{\mathcal{C}}$ obtained by gauging
${\mathbb{Z}}_{2}^{f}$ symmetry of the fermionic topological phase
$\mathcal{C}$. Mathematically, $\breve{\mathcal{C}}$ is the minimal modular
extension of the super-modular category $\mathcal{C}$. For a modular category,
it is known that ${N_{\mathrm{FS}}}(\breve{\mathcal{C}})\leq
2^{\frac{2\breve{r}}{3}+8}3^{\frac{2}{3}\breve{r}}$, where $\breve{r}$ is the
number of anyons in $\breve{\mathcal{C}}$ Bruillard  _et al._ (2015). In the
presence of $\mathrm{U}(1)^{f}$ symmetry, there is an Abelian vison $v$ that
carries ${\mathbb{Z}}_{2}^{f}$ vortex. In that case, one can write
$\breve{r}=2r$, where $r$ is the number of anyons in $\mathcal{C}$. Using
${N_{\mathrm{FS}}}(\mathcal{C})\leq{N_{\mathrm{FS}}}(\breve{\mathcal{C}})$, we
have

$\displaystyle\prod_{j}p_{j}\leq{N_{\mathrm{FS}}}(\mathcal{C})\leq{N_{\mathrm{FS}}}(\breve{\mathcal{C}})\leq
2^{\frac{2\breve{r}}{3}+8}3^{\frac{2}{3}\breve{r}}=2^{\frac{4}{3}r+8}3^{\frac{4}{3}r}$
(33)

which finishes the proof. $\square$

##  VII Obstructions to gapped edge preserving Lie group symmetry

Finally, let us provide a generalization of higher Hall conductivity to the
case where the global symmetry is a generic connected Lie group. The idea for
the generalization is straightforward; it is done by restricting the symmetry
group to $\mathrm{U(1)}^{f}\subset G_{f}$ and defining higher Hall
conductivity for each inequivalent choice of the subgroup
$\mathrm{U}(1)^{f}\subset G_{f}$. This generalized version of higher Hall
conductivity again gives a set of obstructions to gapped edge state preserving
the Lie group symmetry, and can be extracted from partial rotation.

###  VII.1 Review: symmetry fractionalization of the Lie group

To start with, let us outline how the Lie group symmetry generally acts on the
topological order. Since the higher Hall conductivity and its generalizations
are defined for fermionic systems, we consider the (2+1)D fermionic
topological order with $G_{f}$ global symmetry. At the level of effective
field theory, the anyons are thought to be simple objects of a super-modular
category $\mathcal{C}$. Also, the anyons and ${\mathbb{Z}}_{2}^{f}$ vortices
are the objects of a minimal modular extension $\breve{\mathcal{C}}$ of the
category $\mathcal{C}$. Physically, $\breve{\mathcal{C}}$ is the bosonic
theory obtained by gauging ${\mathbb{Z}}_{2}^{f}$ symmetry of the fermionic
theory $\mathcal{C}$.

The symmetry group $G_{f}$ has the structure of the group extension

$\displaystyle{\mathbb{Z}}_{2}^{f}\to G_{f}\to G_{b},$ (34)

with ${\mathbb{Z}}_{2}^{f}$ the fermion parity symmetry. The bosonic symmetry
group $G_{b}$ is assumed to be a connected Lie group, and the above group
extension is specified by the element
$[\omega_{2}]\in\mathcal{H}^{2}(G_{b},{\mathbb{Z}}_{2})$.

In general, it is convenient to specify $\omega_{2}$ by a choice of a group
homomorphism $\pi_{f}:K\to{\mathbb{Z}}_{2}^{f}$ with $K:=\pi_{1}(G_{b})$. That
is, there is a group extension to the universal cover $\widetilde{G}_{b}$ of
$G_{b}$ given by $[w_{2}]\in\mathcal{H}^{2}(G_{b},K)$,

$\displaystyle K\to\widetilde{G}_{b}\to G_{b},$ (35)

and then $\omega_{2}$ can be specified by

$\displaystyle\omega_{2}({\bf g},{\bf h})=\pi_{f}(w_{2}({\bf g},{\bf h})),$
(36)

with ${\bf g},{\bf h}\in G_{b}$.

The action of the global symmetry on topological order is then determined by
the symmetry fractionalization pattern Barkeshli  _et al._ (2019).

The connected Lie group symmetry cannot permute the labels of the anyons. In
that case, the equivalence class of the symmetry fractionalization is
specified by the second cohomology
$[\mathfrak{t}]\in\mathcal{H}^{2}(G_{b},\mathcal{A})$ Bulmash and Barkeshli
(2022a), which encodes the assignment of fractional charge on anyons. Here,
$\mathcal{A}$ is the Abelian group for the fusion algebra of Abelian objects
in $\breve{\mathcal{C}}$, i.e., the group of Abelian anyons and Abelian
${\mathbb{Z}}_{2}^{f}$ vortices. This cohomology class $[\mathfrak{t}]$ is
again characterized by the choice of a group homomorphism
$\pi_{\mathcal{A}}:K\to\mathcal{A}$ via

$\displaystyle\mathfrak{t}({\bf g},{\bf h})=\pi_{\mathcal{A}}(w_{2}({\bf
g},{\bf h})),$ (37)

which is subject to a constraint Bulmash and Barkeshli (2022a)

$\displaystyle M_{\psi,\mathfrak{t}({\bf g},{\bf h})}=(-1)^{\omega_{2}({\bf
g},{\bf h})},$ (38)

where $M_{a,b}$ is the mutual braiding between $a,b$, and $\psi$ is a
transparent fermion. This condition enforces that the symmetry
fractionalization class $[\mathfrak{t}]$ is compatible with the fermionic
symmetry group $G_{f}$, where the $G_{b}$ symmetry acts projectively on the
transparent fermion $\psi$.

###  VII.2 Review: anomaly and Hall conductivity of topological order with Lie
group symmetry

Now we have set the symmetry action on the theory, then we can talk about the
’t Hooft anomaly and generalized version of Hall conductivity of the
topological order. Suppose that the Abelian group $K$ is written as
$K=\prod_{j}{\mathbb{Z}}_{n_{j}}$, and let ${\bf k}_{j}\in K$ be the generator
of each ${\mathbb{Z}}_{n_{j}}$ group. In that case, the symmetry
fractionalization is fully characterized by a set of anyons
$\\{\pi_{\mathcal{A}}({\bf k}_{j})\\}$; it fixes a map $\pi_{\mathcal{A}}$, so
characterizes a symmetry fractionalization class $[\mathfrak{t}]$ accordingly.
Physically, the anyon $\\{\pi_{\mathcal{A}}({\bf k}_{j})\\}$ corresponds to
the vortex whose vorticity is given by ${\bf k}_{j}\in\pi_{1}(G_{b})$. When an
anyon moves around the vortex, it is fractionally charged by the phase
$M_{a,\pi_{\mathcal{A}}({\bf k}_{j})}$ due to the Aharonov-Bohm effect.

Then, the Hall conductivity and anomalies of the theory can be characterized
through the spins and braiding of the vortices $\\{\pi_{\mathcal{A}}({\bf
k}_{j})\\}$ Benini  _et al._ (2019); Cheng  _et al._ (2023). Concretely, the
(2+1)D topological order has the ’t Hooft anomaly of $G_{f}$ symmetry
characterized by the theta angle in (3+1)D (which may either be continuous or
discrete),

$\displaystyle\begin{split}&\sum_{j}2\pi h(\pi_{\mathcal{A}}({\bf
k}_{j}))\int_{M^{4}}\mathcal{P}(\pi_{\mathcal{A}}(w_{2}^{j}))\\\
&+\sum_{j<k}2\pi m(\pi_{\mathcal{A}}({\bf k}_{j}),\pi_{\mathcal{A}}({\bf
k}_{k}))\int_{M^{4}}\pi_{\mathcal{A}}(w_{2}^{j})\cup\pi_{\mathcal{A}}(w_{2}^{k})~{},\end{split}$
(39)

where $e^{2\pi ih(a)}=\theta_{a}$, $e^{2\pi im(a,b)}=M_{a,b}$, and
$\mathcal{P}$ is the Pontryagin square.
$[w_{2}^{j}]\in\mathcal{H}^{2}(G_{b},{\mathbb{Z}}_{n_{j}})$ labels the second
Stiefel-Whitney class $[w_{2}]\in\mathcal{H}^{2}(G_{b},K)$. Since the global
symmetry $G_{b}$ is extended by fermion parity symmetry, the background gauge
field of $G_{b}$ follows the shifted Dirac quantization condition Thorngren
(2019)

$\displaystyle\sum_{j}\int_{C}\pi_{f}(g_{b}^{\ast}w_{2}^{j})=\int_{C}w_{2}(TM^{4})\quad\bmod
2,$ (40)

for any 2-cycle $C$. $w_{2}(TM^{4})$ is the second Stiefel-Whitney class of
the tangent bundle of the spacetime $M^{4}$ on which the response action is
supported.

When the ’t Hooft anomaly (39) defines a non-trivial SPT phase with $G_{f}$
symmetry, this means that our (2+1)D topological order has a non-trivial ’t
Hooft anomaly. We do not try to define the Hall conductivity in this case.
Also, non-trivial ’t Hooft anomaly automatically implies that the edge state
of the (2+1)D topological order cannot be $G_{f}$ symmetric Thorngren and Wang
(2021). Hence, we cannot talk about the generalization of higher Hall
conductivity as an obstruction to $G_{f}$ symmetric gapped boundary. Since we
are interested in the $G_{f}$ symmetric edge state, let us consider the case
where ’t Hooft anomaly of $G_{f}$ symmetry is absent.

Then, let us assume that the ’t Hooft anomaly in Eq. (39) gives a trivial SPT
phase. In that case, the action (39) corresponds to a continuous theta angle
rather than a discrete theta angle. In that case, the (3+1)D action (39) is
identified as having the Chern-Simons type response action in (2+1)D, and
describes the generalized version of Hall conductivity for $G_{f}$ symmetry.
By fixing a suitable normalization, we can define the Hall conductivity
through the level of the Chern-Simons action that corresponds to the theta
angle (39). As we will see below, we can also define a generalization of
higher Hall conductivity, which is again associated with partial rotation of
the wave function on a cylinder.

###  VII.3 Higher Hall conductivity and partial rotation of the topological
order

As mentioned above, let us assume that the ’t Hooft anomaly of the (2+1)D
topological order is trivial. In this setup, one can again compute the partial
rotation on a cylinder using the same logic as the case
$G_{f}=\mathrm{U}(1)^{f}$. We write the partial rotation associated with
internal $G_{f}$ transformation as

$\displaystyle\mathcal{T}_{a}\left(\theta;{\bf
g}_{f}\right):=\bra{\Psi_{a}}U_{{\bf
g}_{f},\mathrm{A}}T_{\theta,\mathrm{A}}\ket{\Psi_{a}}$ (41)

where ${\bf g}_{f}\in G_{f}$.

Suppose that $n$-th power of ${\bf g}_{f}$ becomes the fermion parity, ${\bf
g}_{f}^{n}=(-1)^{F}$. When $n$ is odd, one can then compute the partial
rotation for the ${\bf g}_{f}$ as

$\displaystyle\begin{split}\mathcal{T}_{1}\left(\frac{2\pi}{n};{\bf
g}_{f}\right)&\propto e^{-\frac{2\pi i}{24}(n+\frac{2}{n})c_{-}}e^{\frac{2\pi
i}{8n}\sigma_{H}}\\\
&\qquad\times\sum_{b\in\mathcal{C}_{0}}M_{b,\pi_{\mathcal{A}}({\bf
k})}d_{b}^{2}\theta_{b}^{n}~{},\end{split}$ (42)

where ${\bf k}\in K$ is defined as follows. First let us define $\mathbf{g}\in
G_{b}$ as the projection of ${\bf g}_{f}$ onto the bosonic symmetry group
$G_{b}$. When $G_{b}$ is extended to $\tilde{G}_{b}$, ${\bf g}^{n}$ generally
defines an element of $K$, and ${\bf k}\in K$ is defined as ${\bf g}^{n}={\bf
k}\in K$.

$\sigma_{H}$ is the electric Hall conductivity defined by restricting the
symmetry group to $\mathrm{U}(1)^{f}$ symmetry that contains ${\bf g}_{f}$,
and then consider the electric Hall conductivity of the $\mathrm{U}(1)^{f}$
symmetric topological order.

When $\gcd(n,{N_{\mathrm{FS}}})=1$ is satisfied, this partial rotation defines
an obstruction to gapped boundary preserving $G_{f}$ symmetry. One can define
the above higher Hall conductivity for each choice of a loop $U(1)\subset
G_{b}$ which corresponds to an element $\mathbf{k}\in K=\pi_{1}(G_{b})$
satisfying $\pi_{f}(\mathbf{k})=1\in{\mathbb{Z}}_{2}^{f}$. Homotopically
different choices of such loops (i.e., choices of $\mathbf{k}\in K$) generally
give independent obstructions to a symmetric gapped edge state.

The computation of the partial rotation can be performed in the same fashion
as the $\mathrm{U}(1)^{f}$ case in terms of edge CFT (Appendix A), since we
only use the specific $\mathrm{U}(1)^{f}$ subgroup of the whole Lie group
symmetry. We note that the CFT computation assumes the entanglement spectrum
of the edge state preserving $G_{f}$ symmetry, which makes sense only if the
bulk topological order does not have an ’t Hooft anomaly Thorngren and Wang
(2021).

###  VII.4 Example: $G_{f}=\mathrm{SU}(2)^{f}$

Let us consider the case where $G_{f}=\mathrm{SU}(2)^{f}$,
$G_{b}=\mathrm{SO}(3)$. In that case, we have $K={\mathbb{Z}}_{2}$, and the
vortex $\pi_{\mathcal{A}}({\bf k})$ for SO(3) symmetry follows the
${\mathbb{Z}}_{2}$ fusion rule. The spin of $\pi_{\mathcal{A}}({\bf k})$ is
given by $k/4$ with $k\in{\mathbb{Z}}_{4}$, and the bulk response action Eq.
(39) is given by

$\displaystyle\frac{2\pi k}{4}\int_{M^{4}}\mathcal{P}(w_{2}(\mathrm{SO}(3))).$
(43)

This response action of $\mathrm{SU}(2)^{f}$ symmetry actually corresponds to
a trivial SPT, when the Dirac quantization condition (40) is understood;
$w_{2}(\mathrm{SO}(3))=w_{2}(TM^{4})$. In that case we have
$\int\mathcal{P}(w_{2}(TM^{4}))=\sigma(M^{4})$ mod 4, with $\sigma(M^{4})$ the
signature of a 4-manifold $M^{4}$. Hence, the above response action (43) is
identified as the continuous gravitational theta angle, and topological order
does not carry ’t Hooft anomaly of SU(2)f symmetry.

Then, the partial rotation is given by Eq. (42). We have $k/4=\sigma_{H}/8$
mod 1, since the vison $\pi_{\mathcal{A}}(\mathbf{k})$ characterizes the
fractionalization of $\mathrm{U}(1)^{f}\in\mathrm{SU}(2)^{f}$ symmetry, and
its spin is proportional to electric Hall conductivity as
$h_{\pi_{\mathcal{A}}(\mathbf{k})}=\sigma_{H}/8$ mod 1 Manjunath and Barkeshli
(2020). Eq. (42) defines an obstruction to $\mathrm{SU}(2)^{f}$ preserving
gapped edge state beyond $c_{-}$.

##  VIII Discussions

In this paper, we characterized the higher Hall conductivity $\zeta_{n}^{H}$
on a single wave function of fermionic topological order with
$\mathrm{U}(1)^{f}$ symmetry. $\zeta_{n}^{H}$ is extracted by evaluating the
expectation value of the partial rotation followed by the partial
$\mathrm{U}(1)$ transformation. In the case of Abelian topological order, the
partial rotation also enables us to extract $c_{-}$ and $\sigma_{H}$ from a
single wave function, in both the bosonic and fermionic cases. Moreover, using
partial rotations one can completely determine if the given Abelian
topological order admits a symmetry-preserving gapped edge state. The higher
Hall conductivity from partial rotation can generally be used to constrain the
low-energy spectrum of the bulk-boundary system in possibly non-Abelian
topological order with $\mathrm{U}(1)^{f}$ symmetry, and its natural
generalizations to the case with Lie group symmetry are also presented.

Let us conclude our paper with several open questions to be addressed in
future work. First of all, in a bosonic non-Abelian topological order, it is
known that $\zeta_{n}$ only gives necessary conditions for gapped boundary and
in general not sufficient. For instance, 20 copies of Fibonacci topological
order (stacked with $\overline{E}_{8}$ states) does not admit a gapped edge
state, even though it carries $c_{-}=0$ as well as $\zeta_{n}=1$ for all $n$
Neupert  _et al._ (2016); Kaidi  _et al._ (2021). This calls for an easily
computable quantity that can detect ingappability of edge state beyond higher
central charges.

Second, while we only discussed the fermionic phases with $\mathrm{U}(1)^{f}$
symmetry, it was recently found that the non-Abelian fermionic topological
order without any additional global symmetry has a set of invariants which
gives obstructions to a gapped edge state You (2023). Similar to higher
central charges of Abelian topological order, these invariants are also
computable in terms of the modular $S,T$ matrices of a given topological
order. A microscopic characterization of these invariants on a single many-
body wave function is a fruitful future direction.

Finally, the partial operations also allow us to characterize the invariants
of crystalline topological phases with point group symmetry Shiozaki  _et al._
(2017); Zhang  _et al._ (2023); Kobayashi  _et al._ (2024b). It would be
interesting to develop a theory for the crystalline invariants of topological
order, which is relevant to various invariants characterizing the crystalline
spin liquids.

###### Acknowledgements.

R.K. is supported by JQI postdoctoral fellowship at the University of
Maryland, and by National Science Foundation QLCI grant OMA-2120757. T.W. is
supported by the U.S. Department of Energy, Office of Science, Office of Basic
Energy Sciences, Materials Sciences and Engineering Division under Contract
No. DE-AC02-05-CH11231 (Theory of Materials program KC2301). T.W. is also
supported by the Heising-Simons Foundation, the Simons Foundation, and NSF
grant No. PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP).
This research used the Lawrencium computational cluster provided by the
Lawrence Berkeley National Laboratory (Supported by the U.S. Department of
Energy, Office of Basic Energy Sciences under Contract No. DE-
AC02-05-CH11231). This research is funded in part by the Gordon and Betty
Moore Foundation’s EPiQS Initiative, Grant GBMF8683 to T.S. R.M. is supported
by National Science Foundation grant DMR-1848336. S.R. is supported by a
Simons Investigator Grant from the Simons Foundation (Award No. 566116). This
work is supported by the Gordon and Betty Moore Foundation through Grant
GBMF8685 toward the Princeton theory program.

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##  Appendix A Detailed computations of the partial rotation

###  A.1 Fermionic case

Here we compute the phase of the quantity given by

$\displaystyle\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right):=\bra{\Psi_{a}}U_{\frac{\pi}{n},\mathrm{A}}T_{\frac{2\pi}{n},\mathrm{A}}\ket{\Psi_{a}}$
(44)

in terms of edge CFT with $\mathrm{U}(1)^{f}$ symmetry by utilizing the cut-
and-glue approach Qi  _et al._ (2012). For simplicity, let us consider the
case where the edge CFT is described in terms of chiral rational CFT; the
generalization to non-chiral case is rather straightforward, see Ref.
Kobayashi  _et al._ , 2024a for the discussions in bosonic non-chiral phases.

This approach describes the entanglement spectrum of the bipartition at low
energy by that of the (1+1)D edge CFT on its entangling surface. In our case,
the reduced density matrix for the A subsystem is effectively given by
$\rho_{\mathrm{A}}=\rho^{l}_{\mathrm{CFT}}\otimes\rho^{r}_{\mathrm{CFT}}$,
where $\rho^{l}_{\mathrm{CFT}}$ (resp. $\rho^{r}_{\mathrm{CFT}}$) denotes the
CFT on the left (resp. right) edge of the cylinder.

The state $\rho^{l}_{\mathrm{CFT}}$ at the edge of the total subsystem
realizes the ground state of the edge CFT, i.e., low temperature
$\beta_{l}\to\infty$. Meanwhile, the state $\rho^{r}_{\mathrm{CFT}}$ lies at
the entangling surface and corresponds to high temperature $\beta_{r}\to 0$ of
perturbed edge CFT. In the following, we assume that the entanglement
Hamiltonian is that of the unperturbed CFT:
$\rho_{\mathrm{A};r}=e^{-\beta_{r}H_{r}}$, and check the validity of this
assumption with our numerics.

The partial rotation is evaluated in terms of the expectation values of
$U_{\frac{\pi}{L},\mathrm{A}}T_{\frac{2\pi}{L},\mathrm{A}}$ acting within the
CFT Hilbert space. This can be expressed as the CFT character for each edge,

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&=\frac{\sum_{\alpha=\\{a,a\psi\\}}\mathrm{Tr}_{\alpha}[e^{iQ_{l}\frac{\pi}{n}}e^{iP_{l}\frac{L}{n}}e^{-\frac{\xi_{l}}{v}H_{l}}]\sum_{\alpha=\\{a,a\psi\\}}\mathrm{Tr}_{\alpha}[e^{iQ_{r}\frac{\pi}{n}}e^{iP_{r}\frac{L}{n}}e^{-\frac{\xi_{r}}{v}H_{r}}]}{\sum_{\alpha=\\{a,a\psi\\}}\mathrm{Tr}_{a}[e^{-\frac{\xi_{l}}{v}H_{l}}]\sum_{\alpha=\\{a,a\psi\\}}\mathrm{Tr}_{a}[e^{-\frac{\xi_{r}}{v}H_{r}}]}\\\
&=\frac{\sum_{\alpha=\\{a,a\psi\\}}\chi_{\alpha}(\frac{i\xi_{l}}{L}+\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}])\sum_{\alpha=\\{a,a\psi\\}}\chi_{\alpha}(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}])}{\sum_{\alpha=\\{a,a\psi\\}}\chi_{\alpha}(\frac{i\xi_{l}}{L};[\mathrm{AP},0],[\mathrm{AP},0])\sum_{\alpha=\\{a,a\psi\\}}\chi_{\alpha}(\frac{i\xi_{r}}{L};[\mathrm{AP},0],[\mathrm{AP},0])}\end{split}$
(45)

where we introduced the velocity $v$, correlation length $\xi_{l}=v\beta_{l}$,
$\xi_{r}=v\beta_{r}$, and the circumference of the cylinder $L$. ${P}_{l}$ and
${P}_{r}$ are translation operators on the left and right edge
${P}_{l}=-\frac{1}{v}H_{l}$, ${P}_{r}=\frac{1}{v}H_{r}$.
$\chi_{\alpha}(\tau;[s,\theta],[s^{\prime},\theta])$ with
$s,s^{\prime}\in\\{\mathrm{AP},\mathrm{P}\\}$,
$\theta,\theta^{\prime}\in[0,\pi)$ is the CFT character that corresponds to
the partition function on a torus with modular parameter $\tau$ equipped with
Spinc structure. For example,

$\displaystyle\chi_{\alpha}(\tau;[\mathrm{AP},\theta],[\mathrm{AP},\theta^{\prime}])=\mathrm{Tr}_{\alpha,[\mathrm{AP},\theta]}[e^{iQ\theta^{\prime}}e^{2\pi
i\tau(L_{0}-\frac{c_{-}}{24})}]$ (46)

where $\alpha$ labels the topological sector within the anti-periodic sector,
with the $\mathrm{U}(1)$ twisted boundary condition by $\theta$ in the spatial
cycle. In our setup where $L\ll\xi_{l}$, the CFT characters for the left edge
are approximated as

$\displaystyle\begin{split}\chi_{\alpha}\left(\frac{i\xi_{l}}{L};[\mathrm{AP},0],[\mathrm{AP},0]\right)&\approx
e^{-\frac{2\pi\xi_{l}}{L}(h_{\alpha}-\frac{c_{-}}{24})},\\\
\chi_{\alpha}\left(\frac{i\xi_{l}}{L}+\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}]\right)&\approx
e^{-\frac{2\pi\xi_{l}}{L}(h_{\alpha}-\frac{c_{-}}{24})}e^{\frac{2\pi
i}{n}(h_{\alpha}-\frac{c_{-}}{24})}.\end{split}$ (47)

Due to the exponential decay of the characters with respect to its spin, only
the sector with the lowest spin carried by its vacuum becomes dominant in the
sum over $\alpha=\\{a,a\psi\\}$. For simplicity of the notation, let us assume
that $h_{a}\leq h_{a\psi}$ (otherwise one can just relabel $a\rightarrow
a\psi$). By plugging the above expressions into Eq. (45), we obtain

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&=e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}\frac{\sum_{\alpha=\\{a,a\psi\\}}\chi_{\alpha}(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}])}{\sum_{\alpha=\\{a,a\psi\\}}\chi_{\alpha}(\frac{i\xi_{r}}{L};[\mathrm{AP},0],[\mathrm{AP},0])}\end{split}$
(48)

####  A.1.1 The case where $n$ is odd

From now on, let us work in cases of even/odd of $n$. When $n$ is odd, the CFT
character on the right edge can be evaluated by the modular $S,T$
transformation as

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}]\right)&=S_{ab}\chi_{b}\left(-\frac{1}{\frac{i\xi_{r}}{L}-\frac{1}{n}};[\mathrm{AP},\frac{\pi}{n}],[\mathrm{AP},0]\right)\\\
&=(ST^{n})_{ab}\chi_{b}\left(\frac{-in\frac{\xi_{r}}{L}}{\frac{i\xi_{r}}{L}+\frac{1}{n}};[\mathrm{AP},\frac{\pi}{n}],[\mathrm{AP},0]\right)\end{split}$
(49)

where we used $T$ exchanges Spinc structure as

$\displaystyle
T:\begin{cases}([\mathrm{AP},\theta],[\mathrm{AP},\theta^{\prime}])\to([\mathrm{AP},\theta],[\mathrm{P}+\frac{\theta+\theta^{\prime}-[\theta+\theta^{\prime}]_{\pi}}{\pi},[\theta+\theta^{\prime}]_{\pi}]),\\\
([\mathrm{AP},\theta],[\mathrm{P},\theta^{\prime}])\to([\mathrm{AP},\theta],[\mathrm{AP}+\frac{\theta+\theta^{\prime}-[\theta+\theta^{\prime}]_{\pi}}{\pi},[\theta+\theta^{\prime}]_{\pi}])\\\
\end{cases}$ (50)

Here, $[]_{\pi}$ denotes the mod $\pi$ operation and the spin structure is
acted by ${\mathbb{Z}}_{2}$ action as $\mathrm{AP}+1=\mathrm{P}$,
$\mathrm{P}+1=\mathrm{AP}$. To evaluate the action of $S,T$ modular matrices,
we note that the modular transformation in the twisted Hilbert space of CFT
can be read by $S,T$ matrices of $G$-crossed braided fusion category that
describes the topological order with global symmetry in the bulk Barkeshli
_et al._ (2019). In our case, the bosonic symmetry group is
$G_{b}=\mathbb{R}/(\pi\mathbb{Z})$, and the relevant subgroup for the
operation $U_{\frac{\pi}{n}}$ is the ${\mathbb{Z}}_{n}$ subgroup of $G_{b}$.
Using the data of ${\mathbb{Z}}_{n}$-crossed braided fusion category, the
$T$-matrix element is expressed as Barkeshli  _et al._ (2019)

$\displaystyle
T^{(\mathbf{g},\mathbf{h})}_{a_{\mathbf{g}},b_{\mathbf{g}}}=e^{-\frac{2\pi
i}{24}c_{-}}\cdot\theta_{a_{\mathrm{g}}}\cdot\eta_{a}(\mathbf{g},\mathbf{h})\cdot\delta_{a_{\mathbf{g}},b_{\mathbf{g}}}$
(51)

where $\mathbf{g},\mathbf{h}\in G_{b}$. Hence, the character is further
rewritten as

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}]\right)&=\sum_{b\in\mathcal{C}_{\mathrm{AP},\frac{\pi}{n}}}(ST^{n})_{ab}\chi_{b}\left(\frac{-in\frac{\xi_{r}}{L}}{\frac{i\xi_{r}}{L}+\frac{1}{n}};[\mathrm{AP},\frac{\pi}{n}],[\mathrm{AP},0]\right)\\\
&=e^{-\frac{2\pi
in}{24}c_{-}}\sum_{b\in\mathcal{C}_{\mathrm{AP},\frac{\pi}{n}}}S_{ab}\theta_{b}^{n}\prod_{j=0}^{n-1}\eta_{b}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\cdot\chi_{b}\left(\frac{-in\frac{\xi_{r}}{L}}{\frac{i\xi_{r}}{L}+\frac{1}{n}};[\mathrm{AP},\frac{\pi}{n}],[\mathrm{AP},0]\right)\\\
&=e^{-\frac{2\pi
in}{24}c_{-}}\sum_{b\in\mathcal{C}_{\mathrm{AP},\frac{\pi}{n}}}\sum_{c\in\mathcal{C}_{\mathrm{AP},0}}S_{ab}\theta_{b}^{n}\prod_{j=0}^{n-1}\eta_{b}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\cdot
S_{bc}\chi_{c}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}]\right)\end{split}$
(52)

The partial rotation is then expressed as

$\displaystyle\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)=e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}\frac{e^{-\frac{2\pi
in}{24}c_{-}}\sum_{b,c}(S_{a,b}+S_{a\psi,b})\theta_{b}^{n}\prod_{j=0}^{n-1}\eta_{b}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\cdot
S_{bc}\chi_{c}(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}])}{\sum_{b}(S_{a,b}+S_{a\psi,b})\chi_{b}(\frac{iL}{\xi_{r}};[\mathrm{AP},0],[\mathrm{AP},0])}$
(53)

Then, this quantity can be evaluated in the same logic as the bosonic case.
For $\frac{L}{\xi_{r}}\gg 1$, we use the approximation that allows us to
compute the CFT character only in terms of the highest weight state
$\ket{h_{b}}$

$\displaystyle\begin{split}\chi_{b}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}]\right)&\approx
e^{\frac{2\pi i}{n}(h_{b}-\frac{c_{-}}{24})}e^{-\frac{2\pi
L}{n^{2}\xi_{r}}(h_{b}-\frac{c_{-}}{24})}\\\
\chi_{b}\left(\frac{iL}{\xi_{r}};[\mathrm{AP},0],[\mathrm{AP},0]\right)&\approx
e^{-\frac{2\pi L}{\xi_{r}}(h_{b}-\frac{c_{-}}{24})}\end{split}$ (54)

Also, due to this approximation only the state with the lowest spin $h=0$
contributes to the sum for $\frac{L}{\xi_{r}}\gg 1$. Hence we have

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&\propto\frac{e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}e^{-\frac{2\pi
in}{24}c_{-}}\cdot\sum_{b}(S_{a,b}+S_{a\psi,b})\theta_{b}^{n}\prod_{j=0}^{n-1}\eta_{b}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\cdot
d_{b}\chi_{1}(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},\frac{\pi}{n}])}{d_{a}\chi_{1}(\frac{iL}{\xi_{r}};[\mathrm{AP},0],[\mathrm{AP},0])}\\\
&\propto e^{\frac{2\pi i}{n}h_{a}}e^{-\frac{2\pi
i}{24}(n+\frac{2}{n})c_{-}}\sum_{b\in\mathcal{C}_{\mathrm{AP},\frac{\pi}{n}}}\theta_{b}^{n}\prod_{j=0}^{n-1}\eta_{b}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\cdot
d_{b}\end{split}$ (55)

where $\mathcal{C}_{\mathrm{AP},\frac{\pi}{n}}$ is the twisted sector in
$G_{b}$-crossed extension of super-modular category that consists of anyons in
the $[\mathrm{AP},\frac{\pi}{n}]$ sector. To relate this quantity to
$\zeta^{H}_{n}$, we express the formula in terms of the anyons in the
untwisted sector, instead of the twisted sector by $\frac{\pi}{n}$. This is
done by using the consistency equations and the Verlinde formula of
$G$-crossed braided fusion category given by Barkeshli  _et al._ (2019)

$\displaystyle\begin{split}\theta_{b_{\frac{\pi}{n}}}&=\theta_{b_{0}}\theta_{0_{\frac{\pi}{n}}}\cdot(R^{b_{0},0_{\frac{\pi}{n}}}R^{0_{\frac{\pi}{n}},b_{0}})\\\
\eta_{b_{\frac{\pi}{n}}}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)&=\eta_{b_{0}}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\eta_{0_{\frac{\pi}{n}}}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\\\
S_{a,b_{\frac{\pi}{n}}}&={S_{a,b_{0}}S_{a,0_{\frac{\pi}{n}}}}\frac{\mathcal{D}}{d_{a}}=\frac{1}{d_{a}}S_{a,b_{0}}(R^{a,0_{-\frac{\pi}{n}}}R^{0_{-\frac{\pi}{n}},a})\end{split}$
(56)

where we work on the gauge with $U_{\mathbf{g}}(a,b;c)=1$. Also, one can set
the gauge where $R^{a,0_{\mathbf{g}}}=R^{0_{\mathbf{g}},a}=1$ Barkeshli  _et
al._ (2019). We then have

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&\propto
e^{\frac{2\pi i}{n}h_{a}}e^{-\frac{2\pi
i}{24}(n+\frac{2}{n})c_{-}}\left(\theta_{0_{\frac{\pi}{n}}}^{n}\prod_{j=0}^{n-1}\eta_{0_{\frac{\pi}{n}}}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\right)\sum_{b\in\mathcal{C}_{\mathrm{AP},0}}\prod_{j=0}^{n-1}\eta_{b}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\cdot
S_{ab}d_{b}\theta_{b}^{n}\end{split}$ (57)

where the sum is over the AP anyons in the untwisted sector. We then note that
the $\mathrm{U}(1)$ fractional charge is expressed as Manjunath and Barkeshli
(2020); Bulmash and Barkeshli (2022b)

$\displaystyle e^{i\pi
Q_{b}}=\prod_{j=0}^{n-1}\eta_{b}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)$
(58)

Also, we observe the following gauge invariant quantity in the expression

$\displaystyle\mathcal{I}_{n}:=\theta_{0_{\frac{\pi}{n}}}^{n}\prod_{j=0}^{n-1}\eta_{0_{\frac{\pi}{n}}}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)$
(59)

We then have

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&\propto
e^{\frac{2\pi i}{n}h_{a}}e^{-\frac{2\pi
i}{24}(n+\frac{2}{n})c_{-}}\mathcal{I}_{n}\sum_{b\in\mathcal{C}_{\mathrm{AP},0}}e^{i\pi
Q_{b}}S_{ab}d_{b}\theta_{b}^{n}~{}.\end{split}$ (60)

The gauge invariant quantity $\mathcal{I}_{n}$ is further computed by plugging
the symmetry fractionalization data into its expression, which was performed
in Manjunath and Barkeshli (2020). To compute this, we can work on the
specific gauge where $\theta_{0_{\bf g}}=1$, and

$\displaystyle\begin{split}\eta_{0_{\bf g}}(\bf h,\bf k)&=(F^{0_{\bf g}0_{\bf
h}0_{\bf k}})^{-1},\\\ F^{0_{\bf g}0_{\bf h}0_{\bf k}}&=\exp\left(-2\pi
i(h_{v}+k)(z_{1}(z_{2}+z_{3}-[z_{2}+z_{3}]_{1}))\right),\end{split}$ (61)

where we labeled ${\bf g}=\pi z_{1},{\bf h}=\pi z_{2},{\bf k}=\pi z_{3}$ with
$z_{1},z_{2},z_{3}\in[0,1)$, and $v\in\mathcal{C}_{\mathrm{P}}$ is the vison
defined as the relation $M_{b,v}=e^{i\pi Q_{b}}$ for all anyons
$b\in\mathcal{C}_{0},\mathcal{C}_{\mathrm{P}}$. The parameter
$k\in{\mathbb{Z}}$ corresponds to stacking the bosonic IQH phase atop the
state (here, the bosonic IQH is equivalent to eight copies of fermionic IQH
state stacked with $\overline{E}_{8}$ state, so carries $\sigma_{H}=8$). The
combination $h_{v}+k$ is identified with the electric Hall conductivity of the
bosonic topological order as Manjunath and Barkeshli (2020)

$\displaystyle h_{v}+k=\frac{\sigma_{H}}{8}.$ (62)

The quantity $\mathcal{I}_{n}$ is then computed as

$\displaystyle\mathcal{I}_{n}=\prod_{j=1}^{n-1}\left(F^{0_{\frac{\pi}{n}}0_{\frac{\pi}{n}}0_{\frac{j\pi}{n}}}\right)^{-1}=\exp\left(2\pi
i\frac{h_{v}+k}{n}\right)=e^{2\pi i\frac{\sigma_{H}}{8n}}.$ (63)

The partial rotation is then simply expressed as

$\displaystyle\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$
$\displaystyle\propto e^{\frac{2\pi i}{n}h_{a}}e^{\frac{2\pi
i}{n}(\frac{\sigma_{H}}{8}-(\frac{n^{2}+2}{24})c_{-})}\sum_{b\in\mathcal{C}}e^{i\pi
Q_{b}}S_{ab}d_{b}\theta_{b}^{n}.$ (64)

In particular, when $a=1$,

$\displaystyle\mathcal{T}^{f}_{1}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$
$\displaystyle\propto e^{\frac{2\pi
i}{n}(\frac{\sigma_{H}}{8}-(\frac{n^{2}+2}{24})c_{-})}\zeta_{n}^{H}$ (65)

which gives an obstruction to symmetry-preserving gapped boundary.

####  A.1.2 The case where $n$ is even

A similar computation can also be done when $n$ is even. We start with the
expression in Eq. (53),

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&=e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}\frac{e^{-\frac{2\pi
in}{24}c_{-}}\sum_{b,c}(S_{a,b}+S_{a\psi,b})\theta_{b}^{n}\prod_{j=0}^{n-1}\eta_{b}\left(\frac{\pi}{n},\frac{j\pi}{n}\right)\cdot
S_{bc}\chi_{c}(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{P},0],[\mathrm{AP},\frac{\pi}{n}])}{\sum_{b}(S_{a,b}+S_{a\psi,b})\chi_{b}(\frac{iL}{\xi_{r}};[\mathrm{AP},0],[\mathrm{AP},0])}\end{split}$
(66)

where
$b\in\mathcal{C}_{\mathrm{AP},\frac{\pi}{n}},c\in\mathcal{C}_{\mathrm{P},0}$.
By setting the gauge where $R^{a,0_{\mathbf{g}}}=R^{0_{\mathbf{g}},a}=1$ and
repeating the same logic as the case with odd $n$, we further get the
expression in the leading order

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&\propto
e^{\frac{2\pi i}{n}h_{a}}e^{-\frac{2\pi
i}{24}(n+\frac{1}{n})c_{-}}\mathcal{I}_{n}\sum_{b\in\mathcal{C}_{\mathrm{AP},0}}\sum_{c\in\mathcal{C}_{\mathrm{P},0}}S_{ab}\theta_{b}^{n}e^{i\pi
Q_{b}}S_{bc}\chi_{c}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{P},0],[\mathrm{AP},\frac{\pi}{n}]\right)\\\
&\approx e^{\frac{2\pi i}{n}h_{a}}e^{\frac{2\pi
i}{n}h_{v^{\prime}}}e^{-\frac{2\pi
i}{24}(n+\frac{2}{n})c_{-}}\mathcal{I}_{n}\sum_{b\in\mathcal{C}_{\mathrm{AP},0}}\sum_{v^{\prime}}S_{ab}\theta_{b}^{n}e^{i\pi
Q_{b}}S_{bv^{\prime}},\end{split}$ (67)

where we sum over anyons $v^{\prime}\in\mathcal{C}_{\mathrm{P},0}$ in the
periodic sector with the smallest spin $h_{v^{\prime}}$. The final expression
is given by

$\displaystyle\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)$
$\displaystyle\propto e^{\frac{2\pi i}{n}h_{a}}e^{\frac{2\pi
i}{n}h_{v^{\prime}}}e^{\frac{2\pi
i}{n}(\frac{\sigma_{H}}{8}-(\frac{n^{2}+2}{24})c_{-})}\sum_{b\in\mathcal{C}_{\mathrm{AP},0}}\sum_{v^{\prime}}S_{ab}\theta_{b}^{n}e^{i\pi
Q_{b}}S_{bv^{\prime}}$ (68)

Finally, let us also compute the case with $n$ is even, and we do not perform
partial U(1) transformation. In that case, we have an analogue of Eq. (53)
which becomes

$\displaystyle\begin{split}\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},0\right)&=e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}\frac{e^{-\frac{2\pi
in}{24}c_{-}}\sum_{b,c}(S_{a,b}+S_{a\psi,b})\theta_{b}^{n}\cdot
S_{bc}\chi_{c}(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{AP},0],[\mathrm{AP},0])}{\sum_{b}(S_{a,b}+S_{a\psi,b})\chi_{b}(\frac{iL}{\xi_{r}};[\mathrm{AP},0],[\mathrm{AP},0])}.\end{split}$
(69)

In this case, we have

$\displaystyle\mathcal{T}^{f}_{a}\left(\frac{2\pi}{n},0\right)$
$\displaystyle\propto e^{\frac{2\pi i}{n}h_{a}}e^{-\frac{2\pi
i}{24}(n+\frac{2}{n})c_{-}}\sum_{b\in\mathcal{C}_{\mathrm{AP},0}}d_{b}S_{ab}\theta_{b}^{n}$
(70)

###  A.2 Bosonic case

One can carry out a similar computation in the bosonic topological phase as
well. Here we compute

$\displaystyle\mathcal{T}^{b}_{a}\left(\frac{2\pi}{n},\frac{2\pi
m}{n}\right):=\bra{\Psi_{a}}U_{\frac{2\pi
m}{n},\mathrm{A}}T_{\frac{2\pi}{n},\mathrm{A}}\ket{\Psi_{a}}$ (71)

The computation is basically the same as the fermionic case, except that the
CFT character now does not depend on the spin structure of a torus. We can
obtain the expression analogous to Eq. (53),

$\displaystyle\mathcal{T}^{b}_{a}\left(\frac{2\pi}{n},\frac{2m\pi}{n}\right)=e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}\frac{e^{-\frac{2\pi
in}{24}c_{-}}\sum_{b,c}S_{ab}\theta_{b}^{n}\prod_{j=0}^{n-1}\eta_{b}\left(\frac{2m\pi}{n},\frac{2mj\pi}{n}\right)\cdot
S_{bc}\chi_{c}(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};(0,\frac{2m\pi}{n}))}{\sum_{b}S_{ab}\chi_{b}(\frac{iL}{\xi_{r}};(0,0))}$
(72)

where $b\in\mathcal{C}_{\frac{2m\pi}{n}},c\in\mathcal{C}_{0}$, where
$\mathcal{C}_{\mathbf{g}}$ denotes $\mathbf{g}\in{\mathbb{Z}}_{n}$ twisted
sector of ${\mathbb{Z}}_{n}$-twisted category, where ${\mathbb{Z}}_{n}$ is
understood as being embedded in $\mathrm{U}(1)$.
$\chi(\tau;(\mathbf{g},\mathbf{h}))$ denotes the torus with the
${\mathbb{Z}}_{n}$ background gauge field. By setting the gauge where
$R^{a,0_{\mathbf{g}}}=R^{0_{\mathbf{g}},a}=1$ and repeating the same logic as
the fermionic case, we further get the expression in the leading order

$\displaystyle\begin{split}\mathcal{T}^{b}_{a}\left(\frac{2\pi}{n},\frac{\pi}{n}\right)&\propto
e^{\frac{2\pi i}{n}h_{a}}e^{-\frac{2\pi
i}{24}(n+\frac{1}{n})c_{-}}\mathcal{I}_{n;m}\sum_{b,c\in\mathcal{C}_{0}}S_{ab}\theta_{b}^{n}e^{2m\pi
iQ_{b}}S_{bc}\chi_{c}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};\left(0,\frac{2m\pi}{n}\right)\right)\\\
&\approx e^{\frac{2\pi i}{n}h_{a}}e^{-\frac{2\pi
i}{24}(n+\frac{2}{n})c_{-}}\mathcal{I}_{n;m}\sum_{b\in\mathcal{C}_{0}}S_{ab}d_{b}\theta_{b}^{n}e^{2m\pi
iQ_{b}},\end{split}$ (73)

where

$\displaystyle\mathcal{I}_{n;m}:=\theta_{0_{\frac{2m\pi}{n}}}^{n}\prod_{j=0}^{n-1}\eta_{0_{\frac{2m\pi}{n}}}\left(\frac{2m\pi}{n},\frac{2mj\pi}{n}\right)$
(74)

Following Manjunath and Barkeshli (2020), we work on the specific gauge where
$\theta_{0_{\bf g}}=1$, and

$\displaystyle\begin{split}\eta_{0_{\bf g}}(\bf h,\bf k)&=(F^{0_{\bf g}0_{\bf
h}0_{\bf k}})^{-1},\\\ F^{0_{\bf g}0_{\bf h}0_{\bf k}}&=\exp\left(-2\pi
i(h_{v}+k)(z_{1}(z_{2}+z_{3}-[z_{2}+z_{3}]_{1}))\right),\end{split}$ (75)

where we labeled ${\bf g}=2\pi z_{1},{\bf h}=2\pi z_{2},{\bf k}=2\pi z_{3}$
with $z_{1},z_{2},z_{3}\in[0,1)$, and $v\in\mathcal{C}_{0}$ is the vison
defined as the relation $M_{b,v}=e^{2\pi iQ_{b}}$ for all anyons
$b\in\mathcal{C}_{0}$. The parameter $k\in{\mathbb{Z}}$ corresponds to
stacking the bosonic IQH phase atop the state, which carries $\sigma_{H}=2$.
The combination $h_{v}+k$ is identified with the electric Hall conductivity of
the bosonic topological order as $h_{v}+k=\sigma_{H}/2$. One can then compute
$\mathcal{I}_{n;m}$ as

$\displaystyle\mathcal{I}_{n;m}=\exp\left(2\pi
im^{2}\frac{h_{v}+k}{n}\right)=e^{2\pi i\frac{m^{2}}{2n}\sigma_{H}}.$ (76)

We then obtain the final expression

$\displaystyle\begin{split}\mathcal{T}^{b}_{a}\left(\frac{2\pi}{n},\frac{2\pi
m}{n}\right)&\propto e^{\frac{2\pi
i}{n}(\frac{m^{2}}{2}\sigma_{H}-(\frac{n^{2}+2}{24})c_{-})}e^{\frac{2\pi
i}{n}h_{a}}\times\sum_{b\in\mathcal{C}}d_{b}S_{ab}\theta_{b}^{n}e^{2i\pi
mQ_{b}}.\end{split}$ (77)

##  Appendix B Obstructions to symmetry-preserving gapped edge state in (2+1)D
fermionic Abelian FQH state

In this appendix, we study obstructions to symmetry-preserving gapped edge
state in (2+1)D fermionic Abelian topological order with $\mathrm{U}(1)$
symmetry. The super-modular tensor category for fermionic Abelian phase is
expressed as $\mathcal{C}=\mathcal{C}_{0}\boxtimes\\{1,\psi\\}$ with a modular
tensor category $\mathcal{C}_{0}$.

We find that $c_{-}=\sigma_{H}=0$, and vanishing higher central charges
$\zeta_{n}(\mathcal{C}_{0})=1$ for
$\gcd(n,\frac{{N_{\mathrm{FS}}}(\mathcal{C}_{0})}{\gcd(n,{N_{\mathrm{FS}}}(\mathcal{C}_{0}))})=1$
give both necessary and sufficient conditions for symmetry-preserving gapped
boundary.

First, let us check the “sufficient” part of the statement.
$\zeta_{1}(\mathcal{C}_{0})=1$ implies that chiral central charge carried by
the bosonic theory $\mathcal{C}_{0}$ is 0 mod 8, so the fermionic invertible
phase $\\{1,\psi\\}$ lies in the trivial class of sixteen fold way to realize
$c_{-}=0$ in total. This means that once we gauge fermion parity, the
resulting bosonic phase is given by untwisted ${\mathbb{Z}}_{2}$ gauge theory
with anyons $D({\mathbb{Z}}_{2})=\\{1,\psi,m,e\\}$ with $m,e$ bosons, where
$m$ carries the vortex of ${\mathbb{Z}}_{2}^{f}$ fermion parity. Since
$\sigma_{H}=0$, the vison $v$ for U(1) symmetry fractionalization must be a
boson, and written as $v=v_{0}\times m$ with a boson
$v_{0}\in\mathcal{C}_{0}$. So far, we have seen that our fermionic phase is
regarded as stacking of the following two theories:

  * •

Bosonic SET phase with U(1) symmetry, where anyons are given by a modular
category $\mathcal{C}_{0}$, carries $c_{-}=0$ and U(1) symmetry
fractionalization given by a vison $v_{0}\in\mathcal{C}_{0}$. Since $v_{0}$ is
a boson, we have $\sigma_{H}=0$.

  * •

Trivial fermionic invertible phase with $\mathrm{U}(1)^{f}$ symmetry carrying
$c_{-}=\sigma_{H}=0$.

Then, it has been shown in Ref. Kobayashi, 2022 that the bosonic phase with
U(1) symmetry satisfying $c_{-}=\sigma_{H}=0$, together with the conditions
$\zeta_{n}(\mathcal{C}_{0})=1$ for
$\gcd(n,\frac{{N_{\mathrm{FS}}}(\mathcal{C}_{0})}{\gcd(n,{N_{\mathrm{FS}}}(\mathcal{C}_{0}))})=1$,
admits a symmetry-preserving gapped boundary. This boundary condition for a
bosonic phase $\mathcal{C}_{0}$ directly gives a $\mathrm{U}(1)^{f}$ symmetry-
preserving gapped boundary of our fermionic theory, so it establishes the
proof of the “sufficient” part.

Next, let us check the “necessary” part of the statement. Since the
requirement of $c_{-}=\sigma_{H}=0$ is obvious, we only have to show that
having $\zeta_{n}(\mathcal{C}_{0})=1$ for
$\gcd(n,\frac{{N_{\mathrm{FS}}}(\mathcal{C}_{0})}{\gcd(n,{N_{\mathrm{FS}}}(\mathcal{C}_{0}))})=1$
is necessary. When the fermionic theory $\mathcal{C}$ has a gapped boundary,
one can show that a bosonic theory given by minimal modular extension
$\breve{\mathcal{C}}$ of $\mathcal{C}$ has a gapped interface with the
untwisted ${\mathbb{Z}}_{2}$ gauge theory
$D({\mathbb{Z}}_{2})=\\{1,\psi,m,e\\}$ Kobayashi (2022). Here,
$\breve{\mathcal{C}}$ physically describes a bosonic theory obtained by
gauging fermion parity of the fermionic theory $\mathcal{C}$, and must have a
gapped interface with the trivial fermionic phase $D({\mathbb{Z}}_{2})$ whose
fermion parity has been gauged. Since
$\breve{\mathcal{C}}=\mathcal{C}_{0}\boxtimes D({\mathbb{Z}}_{2})$, this means
$\mathcal{C}_{0}\boxtimes D({\mathbb{Z}}_{2})$ has a bosonic gapped interface
with $D({\mathbb{Z}}_{2})$, hence $\mathcal{C}_{0}\boxtimes
D({\mathbb{Z}}_{2})\boxtimes D({\mathbb{Z}}_{2})$ has a gapped boundary (by
folding along the interface). Since $D({\mathbb{Z}}_{2})\boxtimes
D({\mathbb{Z}}_{2})$ has bosonic gapped boundary, one can see that
$\mathcal{C}_{0}$ has a bosonic gapped boundary. This is equivalent to having
$\zeta_{n}(\mathcal{C}_{0})=1$ for
$\gcd(n,\frac{{N_{\mathrm{FS}}}(\mathcal{C}_{0})}{\gcd(n,{N_{\mathrm{FS}}}(\mathcal{C}_{0}))})=1$,
so it establishes the “necessary” part of the proof.