eqs

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

# Extracting higher central charge from a single wave function

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, CA 94720, USA  Tomohiro
Soejima (副島智大)  Department of Physics, University of California, Berkeley,
California 94720 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 topologically ordered phase may or may not have a gappable edge, even
if its chiral central charge $c_{-}$ is vanishing. Recently, it is discovered
that a quantity regarded as a “higher” version of chiral central charge gives
a further obstruction beyond $c_{-}$ to gapping out the edge. In this Letter,
we show that the higher central charges can be characterized by the
expectation value of the partial rotation operator acting on the wavefunction
of the topologically ordered state. This allows us to extract the higher
central charge from a single wavefunction, which can be evaluated on a quantum
computer. Our characterization of the higher central charge is analytically
derived from the modular properties of edge conformal field theory, as well as
the numerical results with the $\nu=1/2$ bosonic Laughlin state and the non-
Abelian gapped phase of the Kitaev honeycomb model, which corresponds to
$\mathrm{U}(1)_{2}$ and Ising topological order respectively. The letter
establishes a numerical method to obtain a set of obstructions to the gappable
edge of (2+1)D bosonic topological order beyond $c_{-}$, which enables us to
completely determine if a (2+1)D bosonic Abelian topological order has a
gappable edge or not. We also point out that the expectation values of the
partial rotation on a single wavefunction put a constraint on the low-energy
spectrum of the bulk-boundary system of (2+1)D bosonic topological order,
reminiscent of the Lieb-Schultz-Mattis type theorems.

Introduction – (2+1)D topological phases with bulk energy gap host various
intriguing physical phenomena Wen (2004). One of the most striking is the
bulk-edge correspondence, where the property of the bulk heavily constrains
dynamics at its boundary. The most celebrated example is the Integer Quantum
Hall effect, where nonzero bulk Chern number implies the presence of gapless
charged edge modes Hatsugai (1993). Even without charge conservation, systems
with nonzero chiral central charge $c_{-}$, which signals nonzero thermal Hall
conductance, has gapless edge modes Kane and Fisher (1997). We have a good
theoretical understanding of these quantities through coarse-grained Chern-
Simons theory, and we can extract them from microscopic wavefunctions Kitaev
(2006); Mitchell  _et al._ (2018); 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).

In the presence of anyonic excitations, there are properties beyond $c_{-}$
that enforces the presence of gapless edge modes. In many cases, nontrivial
braiding statistics between anyons can present an obstruction to gapping out
all anyonic degrees of freedom simultaneously at the boundary Kapustin and
Saulina (2011); Levin (2013). Such phases of matter are said to have an
ungappable edge. Recently, it is discovered that a quantity called higher
central charge can partially capture “ungappability” of the edge Ng  _et al._
(2019, 2020). In particular, higher central charges of an Abelian topological
order completely determines whether it has an ungappable edge Kaidi  _et al._
(2021). However, so far the quantity has been characterized purely through the
topological quantum field theory (TQFT) framework, and a microscopic
understanding of higher central charges has been lacking.

In this Letter, we show that the expectation value of the partial rotation
operator – rotation operator that acts only on a part of the system – can be
used to reliably extract higher central charges of topologically ordered
systems. This is the first proposal that relates the wavefunction of a
topological ordered state to its higher central charges, and our operational
definition even allows its evaluation on a quantum computer. Our finding is
supported by an analytical conformal field theory (CFT) calculation, as well
as numerics on the non-Abelian phase of the Kitaev honeycomb model and
$\nu=1/2$ bosonic Laughlin state. This Letter establishes a general numerical
method to obtain obstructions to a gappable edge of a bosonic topological
order beyond $c_{-}$, which enables us to completely determine if bosonic
Abelian topological order has a gappable edge.

Figure 1: (a) The setup for which we considered the gappability problem. The
obstruction can be captured by $c_{-}$ and higher central charge $\zeta_{n}$.
(b) Schematics of the partial rotation of a cylinder bisected into A and B
subsystems.

Definition and properties of higher central charge – The higher central
charges $\zeta_{n}$ are complex numbers characterizing a topologically ordered
state, labeled by a positive integer $n$. $\zeta_{n}$ can be easily computed
from the properties of anyons in the topological order; for a given bosonic
(2+1)D topologically order, $\zeta_{n}$ is defined as the following phase

$\displaystyle\zeta_{n}=\frac{\sum_{a}d_{a}^{2}\theta_{a}^{n}}{\mathopen{\big{missing}}\lvert\sum_{a}d_{a}^{2}\theta_{a}^{n}\big{\rvert}}~{},$
(2)

where the sum is over all anyons in the topological order. $d_{a}$ is quantum
dimension, and $\theta_{a}$ is topological twist of an anyon $a$. When $n=1$,
$\zeta_{1}$ reduces to the Gauss sum formula for chiral central charge modulo
8, $\zeta_{1}=e^{\frac{2\pi i}{8}c_{-}}$, hence $\zeta_{n}$ formally provides
a generalization of $c_{-}$.

Higher central charges put a constraint on the gappability of the edge; it was
proven in Ng  _et al._ (2019) that $\zeta_{n}=1$ for all $n$ such that
$\gcd(n,N_{\mathrm{FS}})=1$ give necessary conditions for a gappable edge.
Here, $N_{\mathrm{FS}}$ is called the Frobenius-Schur exponent, defined as the
smallest positive integer satisfying $\theta_{a}^{N_{\mathrm{FS}}}=1$ for all
anyons $a$. For example, $\mathrm{U}(1)_{2}\times\mathrm{U}(1)_{-4}$ Chern-
Simons theory has $\zeta_{3}=-1$, which shows that the topological order has
an ungappable edge even though $c_{-}=0$. For (2+1)D bosonic Abelian
topological phases one can also derive sufficient conditions: the higher
central charges $\\{\zeta_{n}\\}$ for
$\gcd(n,\frac{N_{\mathrm{FS}}}{\gcd(n,N_{\mathrm{FS}})})=1$ give both
necessary and sufficient conditions for a gappable boundary Kaidi  _et al._
(2021).

Main result – To extract higher central charges from a single wavefunction, we
consider a (2+1)D topological ordered state located 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.
Suppose we have realized a ground state $\ket{\Psi}$ on the cylinder labeled
by the trivial anyon $1$. Let us take a bipartition of the cylinder into the
two subsystems labeled by A and B, and write the translation operator for the
A subsystem by the angle $\theta$ along the circumference as
$T_{\mathrm{A};\theta}$ (see Fig. 1). We then find that the following quantity
extracts $\zeta_{n}$,

$\displaystyle\begin{split}\mathcal{T}_{1}\left(\frac{2\pi}{n}\right)&:={\bra{\Psi}T_{\mathrm{A};\frac{2\pi}{n}}\ket{\Psi}}\propto
e^{-2\pi
i(\frac{2}{n}+n)\frac{c_{-}}{24}}\times\sum_{a}d_{a}^{2}\theta_{a}^{n}~{},\end{split}$
(3)

where $\propto$ in this Letter always means being proportional up to a
positive real number. In the special case where $n=1$, the rhs becomes 1 since
$\sum_{a}d_{a}^{2}\theta_{a}\propto e^{\frac{2\pi i}{8}c_{-}}$, consistent
with the fact that the $2\pi$ rotation of the cylinder A gives the identity.
For $n>1$ and $\gcd(n,N_{\mathrm{FS}})=1$, the above rhs becomes proportional
to $\zeta_{n}$ and gives a non-trivial obstruction to gapped boundary beyond
$c_{-}$. Since $c_{-}$ can be extracted from a single wavefunction Kim  _et
al._ (2022a, b), our method allows a complete characterization of all higher
central charges.

For (2+1)D bosonic Abelian topological order one can show that partial
rotation, together with topological entanglement entropy Kitaev and Preskill
(2006); Levin and Wen (2006), fully determines if its edge is gappable. See
Supplemental Materials for an explicit algorithm determining gappability. We
also note that partial rotation, which is unitary, can be easily evaluated on
a quantum computer using methods such as the Hadamard test.

Analytic derivation – Eq. (3) can be derived by employing the cut-and-glue
approach established in Kitaev and Preskill (2006); Qi  _et al._ (2012), which
describes the entanglement spectrum of the A subsystem at long wavelength by
that of the (1+1)D CFT on its edges 111While the argument in Qi  _et al._
(2012) is valid for chiral (i.e., holomorphic) edge CFT, one can utilize the
cut-and-glue approach for non-chiral edge CFT as well, as demonstrated in
Supplemental Materials.. Namely, the reduced density matrix for the A
subsystem is effectively given by
$\rho_{\mathrm{A}}=\rho_{\mathrm{A};l}\otimes\rho_{\mathrm{A};r}$, where
$\rho_{\mathrm{A};l},\rho_{\mathrm{A};r}$ denote the CFTs on the left and
right edges respectively. The left edge lies at the end of the whole cylinder
realizing the ground state of CFT; the right edge of the A subsystem entangled
with the B subsystem is described by a thermal density matrix of a perturbed
edge CFT Li and Haldane (2008). The form of the perturbation in the
entanglement Hamiltonian is not universal. In the following, we assume that
the entanglement Hamiltonian is that of the unperturbed CFT:
$\rho_{\mathrm{A};r}=e^{-\beta_{r}H_{r}}$, and check the validity of this
assumption with our numerics.

Since the operator $T_{\mathrm{A};\theta}$ acts as the translation of the edge
CFT, the partial rotation is expressed as the expectation values of
translation operators within the edge CFT as

$\displaystyle\begin{split}\mathcal{T}_{1}\left(\frac{2\pi}{n}\right)&=\frac{\mathrm{Tr}\big{[}e^{i{P}_{l}\frac{L}{n}}e^{-\frac{\xi_{l}}{v}H_{l}}\big{]}\mathrm{Tr}\big{[}e^{i{P}_{r}\frac{L}{n}}e^{-\frac{\xi_{r}}{v}H_{r}}\big{]}}{\mathrm{Tr}\big{[}e^{-\frac{\xi_{l}}{v}H_{l}}\big{]}\mathrm{Tr}\big{[}e^{-\frac{\xi_{r}}{v}H_{r}}\big{]}}\\\
&=\frac{\chi_{1}\big{(}\frac{i\xi_{l}}{L}+\frac{1}{n}\big{)}\,\chi_{1}\big{(}\frac{i\xi_{r}}{L}-\frac{1}{n}\big{)}}{\chi_{1}\big{(}\frac{i\xi_{l}}{L}\big{)}\,\chi_{1}\big{(}\frac{i\xi_{r}}{L}\big{)}},\end{split}$
(4)

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
${P}_{l}=-\frac{1}{v}H_{l}$, ${P}_{r}=\frac{1}{v}H_{r}$. $\chi_{1}(\tau)$ is
the CFT character of the trivial sector with modular parameter $\tau$. In our
setup where $L\ll\xi_{l}$, the characters for the left edge are approximated
as

$\displaystyle\chi_{1}\left(\frac{i\xi_{l}}{L}\right)\approx
e^{\frac{2\pi\xi_{l}}{L}\frac{c_{-}}{24}},\
\chi_{1}\left(\frac{i\xi_{l}}{L}+\frac{1}{n}\right)\approx
e^{\frac{2\pi\xi_{l}}{L}\frac{c_{-}}{24}}e^{-\frac{2\pi
i}{n}\frac{c_{-}}{24}}.$ (5)

Meanwhile, the edge CFT at the right edge cutting the system has high
temperature $L\gg\xi_{r}$. These characters can be approximately computed by
performing proper modular $S,T$ transformations as Shiozaki  _et al._ (2017)

$\displaystyle\begin{split}\chi_{1}\left(\frac{i\xi_{r}}{L}\right)&=\sum_{a}S_{1,a}\chi_{a}\left(\frac{iL}{\xi_{r}}\right)\approx\frac{1}{\mathcal{D}}e^{\frac{2\pi
L}{\xi_{r}}\frac{c_{-}}{24}},\end{split}$ (6)
$\displaystyle\begin{split}\chi_{1}\left(\frac{i\xi_{r}}{L}-\frac{1}{n}\right)&=\sum_{a}(ST^{n}S)_{1,a}\chi_{a}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n}\right)\\\
&\approx(ST^{n}S)_{1,1}e^{-\frac{2\pi i}{n}\frac{c_{-}}{24}}e^{\frac{2\pi
L}{n^{2}\xi_{r}}\frac{c_{-}}{24}}\\\ &=\frac{1}{\mathcal{D}^{2}}e^{-2\pi
i(n+\frac{1}{n})\frac{c_{-}}{24}}e^{\frac{2\pi
L}{n^{2}\xi_{r}}\frac{c_{-}}{24}}\sum_{a}d_{a}^{2}\theta_{a}^{n},\end{split}$
(7)

where $n$ is assumed to be small satisfying $n^{2}\ll L/\xi_{r}$. The sum is
over the anyons $a$ that labels the conformal block of the edge CFT, and
$\mathcal{D}=\sqrt{\sum_{a}d_{a}^{2}}$ is the total quantum dimension. By
combining the above approximations of the characters,
$\mathcal{T}_{1}\left(2\pi/n\right)$ in Eq. (4) is expressed as Eq. (3).

A similar computation can be performed when the ground state lives in a
generic topological sector,

$\displaystyle\begin{split}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)\propto
e^{\frac{2\pi i}{n}h_{a}-2\pi
i(\frac{2}{n}+n)\frac{c_{-}}{24}}\times\zeta_{n,a}~{},\end{split}$ (8)

where
$\mathcal{T}_{a}\left({2\pi}/{n}\right):=\bra{\Psi_{a}}T_{\mathrm{A};{2\pi}/{n}}\ket{\Psi_{a}}$
with $\ket{\Psi_{a}}$ being the ground state in the topological sector labeled
by an anyon $a$. We defined twisted higher central charge

$\displaystyle\zeta_{n,a}:=\sum_{b}S_{ab}d_{b}\theta_{b}^{n}~{},$ (9)

which is proportional to $\zeta_{n}$ when $a=1$. The derivation of Eq. (8) is
relegated to Supplemental Materials.

While the definition of the quantity Eq. (3) is akin to that of the momentum
polarization in the large $n$ limit Tu  _et al._ (2013); Zaletel  _et al._
(2013), we emphasize that the partial rotation by the finite angle
$\mathcal{T}_{a}(2\pi/n)$ extracts a completely different universal quantity
from the momentum polarization. Indeed, the momentum polarization with
$n\to\infty$ does not give the higher central charge, which is expressed as

$\lim_{n\to\infty}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)\propto\exp\left[\frac{2\pi
i}{n}\left(h_{a}-\frac{c_{-}}{24}-\frac{c_{-}}{24}\frac{L^{2}}{\xi_{r}^{2}}\right)\right].$
(10)

Remarkably, while Eq. (10) depends on the circumference $L$ and the non-
universal correlation length $\xi_{r}$, Eq. (3) solely gives a constant
universal value as the combination of $c_{-}$ and $\zeta_{n}$. In Supplemental
Materials, we describe how the behavior of the partial rotation interpolates
between higher central charge and momentum polarization.

Figure 2: (a) Geometry of the Kitaev model on a cylinder. Red, blue and yellow
lines correspond to $X$, $Y$ and $Z$ type Ising interactions, respectively.
The lattice is periodic in the $y$ direction, and has the zigzag boundary
condition in the $x$ direction. (b) The partial rotations
$\mathcal{T}_{a}(2\pi/n)$ evaluated in the Ising topological phase of the
Kitaev model at $n=3,4$. The $\sigma$ sector at $n=4$ is not shown since it
evaluates zero. We used $J_{x}=J_{y}=J_{z}=1,\kappa=0.1$ for computation.

Numerical results – We demonstrate the validity of the formula Eq. (3) for two
examples: the Ising TQFT realized by the Kitaev honeycomb model, and the
$\mathrm{U}(1)_{2}$ TQFT realized by the $\nu=1/2$ bosonic Laughlin state.
Their $\zeta_{n,a}$ and expected values of the partial rotation
$\mathcal{T}_{a}({2\pi}/{n})$ are summarized in Table 1. For some of the $n$’s
in a given topological sector, the magnitude of $\mathcal{T}_{1}$ vanishes.
However, this could only occur when $\gcd(n,N_{\mathrm{FS}})\neq 1$, which
therefore does not obscure the examination of whether the topological order
has a gappable boundary.

The Kitaev honeycomb model is defined on a honeycomb lattice with a qubit on
each vertex, with the Hamiltonian

$\displaystyle\begin{split}H&=J_{x}\sum_{\langle ij\rangle\in\text{R
edge}}X_{i}X_{j}+J_{y}\sum_{\langle ij\rangle\in\text{B edge}}Y_{i}Y_{j}\\\
&\quad+J_{z}\sum_{\langle ij\rangle\in\text{Y
edge}}Z_{i}Z_{j}+\kappa\sum_{\langle ijk\rangle}X_{i}Y_{j}Z_{k},\end{split}$
(11)

where the last term is introduced by turning on magnetic field, which realizes
the non-Abelian gapped phase Kitaev (2006). The non-Abelian phase is known to
host Ising TQFT with anyons $1$, $\sigma$, $\psi$ with topological twists
$\theta_{1}=1$, $\theta_{\sigma}=e^{2\pi i/16}$, $\theta_{\psi}=-1$.

| sector $a$ | $\zeta_{n,a}$ | $\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)$
---|---|---|---
Ising | Trivial $1$ | $e^{\frac{2\pi i}{16}},e^{\frac{2\pi i}{16}},e^{\frac{6\pi i}{16}},e^{\frac{4\pi i}{16}}$ | $1,1,e^{\frac{2\pi i}{9}},e^{\frac{\pi i}{16}}$
$\sigma$ | $1,0,1,0$ | $1,0,e^{-\frac{\pi i}{9}},0$
$\mathrm{U}(1)_{2}$ | Trivial $1$ | $e^{\frac{2\pi i}{8}},0,e^{-\frac{2\pi i}{8}},1$ | $1,0,e^{\frac{13\pi i}{9}},e^{\frac{13\pi i}{8}}$
Semion $s$ | $e^{-\frac{2\pi i}{8}},1,e^{\frac{2\pi i}{8}},0$ | $1,1,e^{\frac{\pi i}{9}},0$
Table 1: The phases of $\zeta_{n,a}$ and the partial rotation
$\mathcal{T}_{a}(\frac{2\pi}{n})$ for $n=1,2,3,4$ in each topological sector
of Ising and $\mathrm{U}(1)_{2}$. We write $0$ when the magnitude is
vanishing.

To compute partial rotation, we employ a cylinder geometry terminated with
zigzag boundary condition on both ends as depicted in Fig. 2, and we act on
the left half of system with partial rotation.

The model is equivalent to a system of free Majorana fermions coupled to
${\mathbb{Z}}_{2}$ gauge field, by rewriting the qubits using Majorana fermion
operators $c$, which act as dynamical free fermions, and $b$, which describes
the $\mathbb{Z}_{2}$ gauge field. As demonstrated in Supplemental Materials,
the partial rotation for the state on the cylinder lying in the trivial sector
can be expressed as

$\displaystyle\mathcal{T}_{1}\left(\frac{2\pi}{n}\right)\propto\mathrm{Tr}\left(\frac{1+(-1)^{F}}{2}e^{-H_{E}}T_{\mathrm{A};\frac{2\pi}{n}}\right),$
(12)

where $H_{E}$ is the entanglement Hamiltonian for the free fermion state in
the A subsystem with the fixed flat ${\mathbb{Z}}_{2}$ gauge field, with the
boundary condition in $y$ direction taken to be anti-periodic. The operator
$(1+(-1)^{F})/2$ gives a projector onto the Hilbert space with even fermion
parity. Following Tu  _et al._ (2013), one can further evaluate it from the
entanglement spectrum of the free Majorana fermions:

$\begin{split}\mathcal{T}_{1}\left(\frac{2\pi}{n}\right)&\propto\prod_{m,k_{y}}\left[\frac{1+e^{ik_{y}L_{y}/n}}{2}+\frac{1-e^{ik_{y}L_{y}/n}}{2}\tanh\frac{\xi_{mk_{y}}}{2}\right]\\\
&+\prod_{m,k_{y}}\left[\frac{1-e^{ik_{y}L_{y}/n}}{2}+\frac{1+e^{ik_{y}L_{y}/n}}{2}\tanh\frac{\xi_{mk_{y}}}{2}\right]\end{split}$
(13)

where $\xi_{mk_{y}}$ is the entanglement spectrum for $H_{E}$, carried by a
quasiparticle with momentum $k_{y}$ in $y$ direction. Analogously, the partial
rotation for the $\sigma$ sector is expressed in terms of the entanglement
Hamiltonian $H_{E}^{\sigma}$ given by setting the periodic boundary condition
in $y$ direction,
$\mathcal{T}_{\sigma}(2\pi/n)\propto\mathrm{Tr}(e^{-H^{\sigma}_{E}}T_{\mathrm{A};2\pi/n})$,
which can also be computed from entanglement spectrum of $H_{E}^{\sigma}$.

We show the result of this evaluation for $1,\sigma$ sectors in Fig. 2. We see
that $\mathrm{Arg}\left(\mathcal{T}_{a}\left({2\pi}/{n}\right)\right)$
converges to predicted values. We only present for $n\geq 3$ and
$\left|\mathcal{T}_{a}\left({2\pi}/{n}\right)\right|>0$.
$\mathcal{T}_{a}(2\pi/n)$ is always real (no phase) for $n=1$ and $2$ since
the phase part exactly cancels.

Figure 3: (a) A schematics of the infinite cylinder geometry and the LLL
orbital basis of the MPS. Partial rotation along a real-space cut can be
accomplished by acting a unitary operator on the auxiliary bond of the MPS
obtained by the RSES algorithm. (b)
$\operatorname{Arg}\mathcal{T}_{a}(2\pi/n)$ of the $\nu=1/2$ bosonic Laughlin
state extracted using Eq. (14). The dotted lines are the CFT predictions given
in Table 1.

The second example is the $\nu=1/2$ bosonic Laughlin state, which realizes the
$\mathrm{U}(1)_{2}$ Chern-Simons theory. Its only non-trivial anyon is the
semion $s$ with $\theta_{s}=i$.

The model we study is a half-filled lowest Landau level (LLL) of two-
dimensional bosons with a contact interaction $V_{0}=1$ plus a small
perturbation $\delta V_{2}=0.1$, where $V_{m}$ are the Haldane
pseudopotentials Rezayi and Haldane (1994); Halperin  _et al._ (2020). We
consider an infinite cylinder geometry (Fig. 3 (a)), and use infinite density
matrix renormalization group (iDMRG) calculations Zaletel  _et al._ (2015) to
obtain the infinite matrix product state (iMPS) representation of the ground
state $|\Psi\rangle$.

Compared to other numerical methods, the MPS representation is advantageous
for evaluating the action of partial rotation. If rotation is a good symmetry,
the Schmidt states $|\alpha\rangle_{\mathrm{A/B}}$ across subsystems A and B
have definite momentum $k_{y}^{\alpha}$ along the circumference. Thus, the
action of partial rotation can be evaluated by

$\mathcal{T}_{a}(\theta)=\sum_{\alpha}\lambda_{\alpha}^{2}e^{ik_{y}^{\alpha}L_{y}\theta},$
(14)

where $\lambda_{\alpha}$ is the corresponding Schmidt value. We can easily
obtain both $k_{y}^{\alpha}$ and $\lambda_{\alpha}$ from the momentum label
$\bar{K}_{\bar{n}_{\mathrm{B}};\alpha}$ and the Schmidt value
$\lambda_{\bar{n}_{\mathrm{B}};\alpha}$ of the auxiliary bond
$\bar{n}_{\mathrm{B}}$ across subsystems A and B.

For the $\nu=1/2$ bosonic Laughlin state, we work in the Landau gauge and the
corresponding LLL orbital basis. To accelerate the calculation and obtain the
momentum label mentioned above, we incorporate both particle number
$\hat{C}=\sum_{n}\hat{C}_{n}\equiv\sum_{n}(\hat{N}_{n}-\nu)$ and momentum
$\hat{K}=\sum_{n}\hat{K}_{n}\equiv\sum_{n}n(\hat{N}_{n}-\nu)$ conservation,
where $\hat{N}_{n}$ is the number operator at site $n$. We find that
$\mathcal{T}_{a}(2\pi/n)$ converges at bond dimension $\chi=3200$, cylinder
circumference $L_{y}=40\ell_{B}$ and onsite boson number cutoff
$N_{\mathrm{boson}}=5$.

We note there are a few technical complications in applying Eq. (14) to
compute $\mathcal{T}_{a}(\theta)$, which we will scratch here and readers can
find more details in Supplemental Materials. Firstly, there are a few
ambiguities in extracting the physical momentum $k_{y}^{\alpha}$ from the
momentum label $\bar{K}_{\bar{n}_{\mathrm{A}};\alpha}$. For iMPS, there is an
overall ambiguity of momentum labels on auxiliary bonds. The magnetic
translation symmetry in quantum Hall systems further tangles the momentum
label
$\bar{K}_{\bar{n};\alpha}=\langle\sum_{n<\bar{n}}\hat{K}_{n}\rangle_{\alpha}$
with the charge label
$\bar{C}_{\bar{n};\alpha}=\langle\sum_{n<\bar{n}}\hat{C}_{n}\rangle_{\alpha}$
Zaletel and Mong (2012). These ambiguities can be fixed by matching the
entanglement spectrum and the edge CFT spectrum as elaborated in Supplemental
Materials.

Secondly, which topological sector subsystem A, B belongs to depends on the
cut. The $\nu=1/2$ bosonic Laughlin state has a two-fold ground state
degeneracy, characterized by root configuration (pattern of zeros) $[01]$ and
$[10]$ Bernevig and Haldane (2008); Wen and Wang (2008). It turns out that
cutting through the LLL orbital center that corresponds to the $0$’s ($1$’s)
bisects the system into two trivial sectors (semion) sectors. Finally, when we
work in the LLL orbital basis, an auxiliary bond divides the system into two
sets of LLL orbitals instead of two regions of physical space. This problem
can be resolved using the real-space entanglement spectrum (RSES) algorithm
developed in Zaletel and Mong (2012). We note that many of the technicalities
discussed here are not specific to the $\nu=1/2$ bosonic Laughlin state, but
provide a general procedure for computing higher central charge of arbitrary
wavefunction in the MPS form.

Finally we present the result of $\mathcal{T}_{a}(2\pi/n)$ in both the trivial
and the semion sectors. As shown in Fig. 3 (b), $\mathcal{T}_{a}(2\pi/n)$
always converges to the expected phase as shown in Table 1 at sufficiently
large $L_{y}$.

Discussion – In this Letter, we characterize the higher central charges
$\\{\zeta_{n}\\}$ in terms of the partial rotation evaluated on a wavefunction
of the (2+1)D bosonic topological order, and confirmed the prediction using
the Kitaev honeycomb model and the $\nu=1/2$ bosonic Laughlin state. Partial
rotation can be implemented easily in quantum computing architectures with
cheap SWAP gates, such as Rydberg atom arrays, which opens up another avenue
to studying topological order directly on a quantum computer. Together with
topological entanglement entropy, partial rotation allows us to fully
determine edge gappability of Abelian topological order.

It would be interesting to study applications of partial rotation to generic
non-Abelian topological phases. Remarkably, even for non-Abelian phases,
numerical results of $\\{\mathcal{T}_{1}(2\pi/n)\\}$ put a tight constraint on
the possible low-energy spectrum of the bulk-boundary system. For instance,
suppose that we observed $\\{\mathcal{T}_{1}(2\pi/p_{j})\\}$ is 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 is ungappable, or 2. the edge is
gappable, where $N_{\mathrm{FS}}$ must be divisible by $\prod_{j}p_{j}$. If
the minimal $N_{\mathrm{FS}}$ required for a gappable edge is large and
physically unrealistic, one can essentially determine that the boundary must
be ungappable.

Notably, the lower bound $N_{\mathrm{FS}}\geq\prod_{j}p_{j}$ for a gappable
edge implies the lower bound for the number of anyons $r$ given by $r\geq
r_{0}$, with $r_{0}$ the smallest integer satisfying
$2^{2r_{0}/3+8}3^{2r_{0}/3}\geq\prod_{j}p_{j}$. This is derived from the fact
that $N_{\mathrm{FS}}$ of the bosonic topological order with $r$ distinct
anyons has the upper bound $N_{\mathrm{FS}}\leq 2^{2r/3+8}3^{2r/3}$ Bruillard
_et al._ (2015). It implies that the ground state on a torus must carry at
least $r_{0}$-fold degeneracy in order to realize a gappable edge. This
argument 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.

Also, it would be interesting to extract the higher Hall conductivity proposed
in Kobayashi (2022), which gives an obstruction to $\mathrm{U}(1)$ symmetry-
preserving gapped boundary of the fermionic topological order with
$\mathrm{U}(1)$ symmetry beyond electric Hall conductivity and $c_{-}$.

Acknowledgements – We thank Michael Zaletel, Roman Geiko, and Tianle Wang for
helpful discussions. RK is supported by the JQI postdoctoral fellowship at the
University of Maryland. TW is supported by the U.S. DOE, Office of Science,
Office of Basic Energy Sciences, Materials Sciences and Engineering Division,
under Contract No. DE-AC02-05-CH11231, within the van der Waals
Heterostructures Program (KCWF16). TS is supported by a fellowship from the
Masason foundation. RM is supported by the National Science Foundation under
Award No. DMR-1848336. SR is supported by the National Science Foundation
under Award No. DMR-2001181, and 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. This research used the Lawrencium computational cluster resource
provided by the IT Division at the Lawrence Berkeley National Laboratory
(Supported by the Director, Office of Science, Office of Basic Energy
Sciences, of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231).

## References

  * Wen (2004) X.-G. Wen, “Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons,” (Oxford University Press, 2004).
  * Hatsugai (1993) Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993).
  * Kane and Fisher (1997) C. L. Kane and M. P. A. Fisher, Phys. Rev. B 55, 15832 (1997).
  * Kitaev (2006) A. Kitaev, Annals of Physics 321, 2 (2006), arXiv:cond-mat/0506438 .
  * Mitchell  _et al._ (2018) N. P. Mitchell, L. M. Nash, D. Hexner, A. M. Turner, and W. T. M. Irvine, Nature Physics 14, 380 (2018).
  * Kim  _et al._ (2022a) I. H. Kim, B. Shi, K. Kato, and V. V. Albert, Physical Review Letters 128 (2022a), 10.1103/physrevlett.128.176402.
  * Kim  _et al._ (2022b) I. H. Kim, B. Shi, K. Kato, and V. V. Albert, Physical Review B 106 (2022b), 10.1103/physrevb.106.075147.
  * Zou  _et al._ (2022) Y. Zou, B. Shi, J. Sorce, I. T. Lim, and I. H. Kim, Physical Review Letters 129 (2022), 10.1103/physrevlett.129.260402.
  * Fan (2022) R. Fan, Physical Review Letters 129 (2022), 10.1103/physrevlett.129.260403.
  * Fan  _et al._ (2022) R. Fan, R. Sahay, and A. Vishwanath, “Extracting the quantum hall conductance from a single bulk wavefunction,” (2022), arXiv:2208.11710 [cond-mat.str-el] .
  * Shiozaki  _et al._ (2018) K. Shiozaki, H. Shapourian, K. Gomi, and S. Ryu, Physical Review B 98 (2018), 10.1103/physrevb.98.035151.
  * Dehghani  _et al._ (2021) H. Dehghani, Z.-P. Cian, M. Hafezi, and M. Barkeshli, Physical Review B 103 (2021), 10.1103/physrevb.103.075102.
  * Cian  _et al._ (2021) Z.-P. Cian, H. Dehghani, A. Elben, B. Vermersch, G. Zhu, M. Barkeshli, P. Zoller, and M. Hafezi, Physical Review Letters 126 (2021), 10.1103/physrevlett.126.050501.
  * Kapustin and Saulina (2011) A. Kapustin and N. Saulina, Nucl. Phys. B845, 393 (2011), arXiv:1008.0654 [hep-th] .
  * Levin (2013) M. Levin, Physical Review X 3 (2013), 10.1103/physrevx.3.021009, arXiv:1301.7355 [cond-mat.str-el] .
  * Ng  _et al._ (2019) S.-H. Ng, A. Schopieray, and Y. Wang, Selecta Mathematica 25 (2019), 10.1007/s00029-019-0499-2, arXiv:1812.11234 [math.QA] .
  * Ng  _et al._ (2020) S.-H. Ng, E. C. Rowell, Y. Wang, and Q. Zhang, “Higher central charges and witt groups,”  (2020), arXiv:2002.03570 [math.QA] .
  * Kaidi  _et al._ (2021) J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, “Higher central charges and topological boundaries in 2+1-dimensional tqfts,”  (2021), arXiv:2107.13091 [hep-th] .
  * Kitaev and Preskill (2006) A. Kitaev and J. Preskill, Physical Review Letters 96 (2006), 10.1103/physrevlett.96.110404, hep-th/0510092 .
  * Levin and Wen (2006) M. Levin and X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).
  * Qi  _et al._ (2012) X.-L. Qi, H. Katsura, and A. W. W. Ludwig, Physical Review Letters 108 (2012), 10.1103/physrevlett.108.196402, arXiv:1103.5437 [cond-mat.mes-hall] .
  * Note (1) While the argument in Qi  _et al._ (2012) is valid for chiral (i.e., holomorphic) edge CFT, one can utilize the cut-and-glue approach for non-chiral edge CFT as well, as demonstrated in Supplemental Materials.
  * Li and Haldane (2008) H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
  * Shiozaki  _et al._ (2017) K. Shiozaki, H. Shapourian, and S. Ryu, Physical Review B 95 (2017), 10.1103/physrevb.95.205139, arXiv:1609.05970 [cond-mat.str-el] .
  * Tu  _et al._ (2013) H.-H. Tu, Y. Zhang, and X.-L. Qi, Physical Review B 88 (2013), 10.1103/physrevb.88.195412, arXiv:1212.6951 .
  * Zaletel  _et al._ (2013) M. P. Zaletel, R. S. K. Mong, and F. Pollmann, Phys. Rev. Lett. 110, 236801 (2013).
  * Rezayi and Haldane (1994) E. H. Rezayi and F. D. M. Haldane, Phys. Rev. B 50, 17199 (1994).
  * Halperin  _et al._ (2020) B. I. Halperin, J. K. Jain, and N. R. Cooper, “Fractional quantum hall states of bosons: Properties and prospects for experimental realization,” (World Scientific, 2020) p. 487–521.
  * Zaletel  _et al._ (2015) M. P. Zaletel, R. S. K. Mong, F. Pollmann, and E. H. Rezayi, Phys. Rev. B 91, 045115 (2015).
  * Zaletel and Mong (2012) M. P. Zaletel and R. S. K. Mong, Phys. Rev. B 86, 245305 (2012).
  * Bernevig and Haldane (2008) B. A. Bernevig and F. D. M. Haldane, Phys. Rev. Lett. 100, 246802 (2008).
  * Wen and Wang (2008) X.-G. Wen and Z. Wang, Phys. Rev. B 78, 155109 (2008).
  * Bruillard  _et al._ (2015) P. Bruillard, S.-H. Ng, E. Rowell, and Z. Wang, Journal of the American Mathematical Society 29, 857 (2015), arXiv:1310.7050 .
  * Lieb  _et al._ (1961) E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961).
  * Oshikawa (2000) M. Oshikawa, Physical Review Letters 84, 1535 (2000), arXiv:cond-mat/9911137 .
  * Hastings (2004) M. Hastings, Physical Review B 69, 104431 (2004), arXiv:cond-mat/0305505 .
  * Kobayashi (2022) R. Kobayashi, Physical Review Research 4 (2022), 10.1103/physrevresearch.4.033137, arXiv:2203.08156 .
  * Drinfeld  _et al._ (2010) V. Drinfeld, S. Gelaki, D. Nikshych, and V. Ostrik, “On braided fusion categories i,”  (2010), arXiv:0906.0620 [math.QA] .
  * Wen (2003) X.-G. Wen, Phys. Rev. Lett. 90, 016803 (2003).
  * Geraedts  _et al._ (2017) S. D. Geraedts, C. Repellin, C. Wang, R. S. K. Mong, T. Senthil, and N. Regnault, Phys. Rev. B 96, 075148 (2017).
  * Soejima  _et al._ (2020) T. Soejima, D. E. Parker, N. Bultinck, J. Hauschild, and M. P. Zaletel, Phys. Rev. B 102, 205111 (2020).
  * Read (2009) N. Read, Phys. Rev. B 79, 045308 (2009).

Supplemental Materials

##  I Complete determination of edge gappability for bosonic Abelian phases

Here, we show that the partial rotation
$\\{\mathcal{T}_{1}\left(2\pi/n\right)\\}$, together with topological
entanglement entropy Kitaev and Preskill (2006); Levin and Wen (2006), can
completely determine if the boundary of the (2+1)D bosonic Abelian topological
phase is gappable or not. Concretely, a complete set of the gappability
conditions can be obtained from a single wavefunction by the following steps:

  1. 1.

First, compute the topological entanglement entropy of the wavefunction to
obtain the total quantum dimension $\mathcal{D}$ of the Abelian TQFT. Since
the theory is Abelian, $\mathcal{D}^{2}$ gives the number of distinct anyons
including the trivial one.

  2. 2.

Next, we want to extract the chiral central charge $c_{-}$. To do this, we
notice that the Frobenius-Schur exponent $N_{\mathrm{FS}}$ of the Abelian TQFT
must divide $2\mathcal{D}^{2}$ as explained in the next step. One can then
extract $c_{-}$ by computing the partial rotation with $n=2\mathcal{D}^{2}$,

$\displaystyle\mathcal{T}_{1}\left(\frac{2\pi}{2\mathcal{D}^{2}}\right)\propto
e^{-2\pi i(\frac{2}{2\mathcal{D}^{2}}+2\mathcal{D}^{2})\frac{c_{-}}{24}}~{}.$
(15)

  3. 3.

Finally, we want to extract the higher central charges to cover all
obstructions to a gapped edge beyond $c_{-}$. For convenience, let us write
the prime factorization of $\mathcal{D}^{2}$ as

$\displaystyle\mathcal{D}^{2}=p_{1}^{k_{1}}\times
p_{2}^{k_{2}}\times\dots\times p_{M}^{k_{M}}$ (16)

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

$\displaystyle\mathcal{C}=\mathcal{C}_{p_{1}}\boxtimes\mathcal{C}_{p_{2}}\boxtimes\dots\boxtimes\mathcal{C}_{p_{M}}.$
(17)

Here, $\boxtimes$ denotes the Deligne product of modular tensor categories
that physically represents stacking of the theories. Each Abelian TQFT
$\mathcal{C}_{p_{j}}$ contains the $p_{j}^{k_{j}}$ anyons including the
trivial one. The Frobenius-Schur exponent also naturally admits the
factorization as

$\displaystyle N_{\mathrm{FS}}=N_{1}N_{2}\dots N_{M},$ (18)

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}$ (19)

This explains why $N_{\mathrm{FS}}$ must divide $2\mathcal{D}^{2}$ which is
used in the previous step.

Now we are ready to describe the all obstructions to a gapped edge in terms of
the partial rotation. For this purpose, we recall that the Abelian TQFT admits
a gapped edge if and only if all the theories $\mathcal{C}_{p_{j}}$ do Kaidi
_et al._ (2021). Then, each theory $\mathcal{C}_{p_{j}}$ admits a gapped edge
if and only if the higher central charge $\zeta_{n}(\mathcal{C}_{p_{j}})=1$
for all $1\leq n\leq N_{j}$ such that $\gcd(n,p_{j})=1$ Drinfeld  _et al._
(2010). One can cover all the above higher central charges of
$\mathcal{C}_{p_{j}}$ by computing the partial rotation
$\mathcal{T}_{1}(2\pi/n)$ with

$\displaystyle n=\frac{m\times 2\mathcal{D}^{2}}{p_{j}^{\tilde{k}_{j}}},$ (20)

where $m$ scans all integers $1\leq m\leq p_{j}^{\tilde{k}_{j}}$ such that
$\gcd(m,p_{j})$ = 1. Here, we define the integers $\tilde{k}_{j}$ as

$\displaystyle\begin{cases}\tilde{k}_{1}=k_{1}+1&\text{if $p_{1}=2$,}\\\
\tilde{k}_{j}=k_{j}&\text{otherwise}.\end{cases}$ (21)

Let us check that above partial rotations exhaust all desired higher central
charges of $\mathcal{C}_{p_{j}}$. 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

$\displaystyle\begin{split}\mathcal{T}_{1}\left(\frac{2\pi}{n}\right)&\propto
e^{-2\pi
i(\frac{2}{n}+n)\frac{c_{-}}{24}}\times\sum_{a\in\mathcal{C}}\theta_{a}^{n}\\\
&=e^{-2\pi
i(\frac{2}{n}+n)\frac{c_{-}}{24}}\times\left(\sum_{a\in\mathcal{C}_{p_{1}}}\theta_{a}^{n}\right)\times\dots\times\left(\sum_{a\in\mathcal{C}_{p_{M}}}\theta_{a}^{n}\right)\\\
&\propto e^{-2\pi
i(\frac{2}{n}+n)\frac{c_{-}}{24}}\times\sum_{a\in\mathcal{C}_{p_{j}}}\theta_{a}^{n}\\\
&\propto e^{-2\pi
i(\frac{2}{n}+n)\frac{c_{-}}{24}}\times\zeta_{n}(\mathcal{C}_{p_{j}}).\end{split}$
(22)

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 $c_{-}$, by computing the partial rotation
$\mathcal{T}_{1}(2\pi/n)$ with

$\displaystyle n=\frac{m\times 2\mathcal{D}^{2}}{p_{j}^{\tilde{k}_{j}}},$ (23)

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.

##  II Partial rotation of a cylinder in a twisted sector

Here we evaluate the partial rotation in the case where the state lies in the
non-trivial topological sector labeled by an anyon $a$ of the topological
order. Denoting the ground state as $\ket{\Psi_{a}}$, the partial rotation is
again expressed as the Virasoro characters

$\displaystyle\begin{split}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)&=\frac{\mathrm{Tr}_{a}[e^{i{P}_{l}\frac{L}{n}}e^{-\frac{\xi_{l}}{v}H_{l}}]\mathrm{Tr}_{a}[e^{i{P}_{r}\frac{L}{n}}e^{-\frac{\xi_{r}}{v}H_{r}}]}{\mathrm{Tr}_{a}[e^{-\frac{\xi_{l}}{v}H_{l}}]\mathrm{Tr}_{a}[e^{-\frac{\xi_{r}}{v}H_{r}}]}\\\
&=\frac{\chi_{a}(\frac{i\xi_{l}}{L}+\frac{1}{n})\chi_{a}(\frac{i\xi_{r}}{L}-\frac{1}{n})}{\chi_{a}(\frac{i\xi_{l}}{L})\chi_{a}(\frac{i\xi_{r}}{L})}~{},\end{split}$
(24)

where $\mathrm{Tr}_{a}$ denotes the trace in the twisted Hilbert space labeled
by $a$, and $\chi_{a}(\tau)$ is the CFT character for the chiral primary $a$.
The above quantity can be evaluated in the same logic as the main text; in our
setup where $L\ll\xi_{l}$, the CFT characters for the left edge are
approximated as

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{l}}{L}\right)&\approx
e^{-\frac{2\pi\xi_{l}}{L}(h_{a}-\frac{c_{-}}{24})},\\\
\chi_{a}\left(\frac{i\xi_{l}}{L}+\frac{1}{n}\right)&\approx
e^{-\frac{2\pi\xi_{l}}{L}(h_{a}-\frac{c_{-}}{24})}e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}.\end{split}$ (25)

Meanwhile, the characters for the right edge can be approximately computed by
performing modular transformations as

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{r}}{L}\right)&=\sum_{b}S_{ab}\chi_{b}\left(\frac{iL}{\xi_{r}}\right)\approx\frac{d_{a}}{\mathcal{D}}e^{\frac{2\pi
L}{\xi_{r}}\frac{c_{-}}{24}},\end{split}$ (26)
$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{r}}{L}-\frac{1}{n}\right)&=\sum_{b}(ST^{n}S)_{ab}\chi_{b}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n}\right)\\\
&\approx(ST^{n}S)_{a,1}e^{-\frac{2\pi i}{n}\frac{c_{-}}{24}}e^{\frac{2\pi
L}{n^{2}\xi_{r}}\frac{c_{-}}{24}}\\\ &=\frac{1}{\mathcal{D}}e^{-2\pi
i(n+\frac{1}{n})\frac{c_{-}}{24}}e^{\frac{2\pi
L}{n^{2}\xi_{r}}\frac{c_{-}}{24}}\sum_{b}S_{ab}d_{b}\theta_{b}^{n},\end{split}$
(27)

where we assumed $n$ to be small, $n^{2}\ll L/\xi_{r}$, and used the
approximation of the CFT characters for the right edge as

$\displaystyle\begin{split}\chi_{b}\left(\frac{iL}{\xi_{r}}\right)&\approx
e^{-\frac{2\pi L}{\xi_{r}}(h_{b}-\frac{c_{-}}{24})}\\\
\chi_{b}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n}\right)&\approx e^{\frac{2\pi
i}{n}(h_{b}-\frac{c_{-}}{24})}e^{-\frac{2\pi
L}{n^{2}\xi_{r}}(h_{b}-\frac{c_{-}}{24})}\\\ \end{split}$ (28)

Combining the above results, we obtain

$\displaystyle\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)\propto e^{\frac{2\pi
i}{n}h_{a}-2\pi
i(\frac{2}{n}+n)\frac{c_{-}}{24}}\times\sum_{b}S_{ab}d_{b}\theta_{b}^{n}~{}.$
(29)

##  III Interpolating between higher central charge and momentum polarization

In the main text, we have seen that the partial rotation
$\mathcal{T}_{1}({2\pi}/{n})$ is proportional to higher central charge for a
small positive integer $n$, while when $n\to\infty$ it rather gives the
momentum polarization $\lambda$ in Eq. (10). Here, we describe how these two
different behaviors are interpolated by changing the value of $n$ from a small
integer to infinity. Following Eq. (24) and Eq. (25), we express the partial
rotation as

$\displaystyle\begin{split}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)&=e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}\times\frac{\chi_{a}(\frac{i\xi_{r}}{L}-\frac{1}{n})}{\chi_{a}(\frac{i\xi_{r}}{L})}.\end{split}$
(30)

The CFT characters on the right edge is approximated depending on the scale of
$n$ compared with $L/\xi_{r}$. Below we compute it in cases of the value of
$n$.

  * •

$n^{2}\ll L/\xi_{r}$. The approximation in Eq. (27) is valid for such $n$ and
we have

$\displaystyle\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)\propto e^{\frac{2\pi
i}{n}h_{a}-2\pi
i(\frac{2}{n}+n)\frac{c_{-}}{24}}\times\sum_{b}S_{ab}d_{b}\theta_{b}^{n}~{}.$
(31)

  * •

$L/\xi_{r}\ll n^{2}$. When $n$ satisfies $L/\xi_{r}\ll n^{2}$, Eq. (27) is no
longer valid, where the alternative approximation is available

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{r}}{L}-\frac{1}{n}\right)&=\sum_{b}S_{ab}\chi_{b}\left(-\frac{1}{\frac{i\xi_{r}}{L}-\frac{1}{n}}\right)\\\
&\approx\sum_{b}S_{ab}\exp\left({-2\pi
i\left(h_{b}-\frac{c_{-}}{24}\right)\frac{1}{\frac{i\xi_{r}}{L}-\frac{1}{n}}}\right)\\\
&\approx\frac{d_{a}}{\mathcal{D}}\exp\left({2\pi
i\frac{c_{-}}{24}\frac{1}{\frac{i\xi_{r}}{L}-\frac{1}{n}}}\right)~{}.\end{split}$
(32)

The partial rotation is then given by

$\displaystyle\begin{split}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)=&e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}\exp\left({2\pi
i\frac{c_{-}}{24}\frac{1}{\frac{i\xi_{r}}{L}-\frac{1}{n}}}-\frac{2\pi
L}{\xi_{r}}\frac{c_{-}}{24}\right)\\\ =&\exp\left(\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})-2\pi
i\frac{c_{-}}{24}\frac{\frac{1}{n}}{\frac{\xi_{r}^{2}}{L^{2}}+\frac{1}{n^{2}}}\right)\\\
&\times\exp\left(2\pi\frac{c_{-}}{24}\frac{\frac{\xi_{r}}{L}}{\frac{\xi_{r}^{2}}{L^{2}}+\frac{1}{n^{2}}}-2\pi\frac{c_{-}}{24}\frac{L}{\xi_{r}}\right)~{}.\end{split}$
(33)

By taking the limit $n\to\infty$, it reproduces the expression of momentum
polarization in Eq. (10).

##  IV Relationship between Entanglement spectrum and edge theory in non-
chiral topological phases

Here, we describe the correspondence between entanglement spectrum and
physical edge CFT in the (2+1)D bosonic topological phases, following the cut-
and-glue approach introduced in Ref. Qi  _et al._ (2012). While the
correspondence is discussed for chiral topological phases whose edge CFT is
holomorphic in Ref. Qi  _et al._ (2012), we slightly generalize their argument
to the case of possibly non-chiral topological phases, verifying the cut-and-
glue approach for generic non-chiral phases which are of our interest.

We consider a (possibly non-chiral) bosonic topological ordered state in
(2+1)D. Let us take bipartition of the state into subsystems A and B. The
physical Hamiltonian can be described as
$H_{\mathrm{A}}+H_{\mathrm{B}}+\lambda H_{\mathrm{AB}}$, where
$H_{\mathrm{AB}}$ denotes the term that couples the subsystem on A and B
subregions. We want to study the reduced density matrix $\rho_{\mathrm{A}}$
for the A subsystem, where the entanglement Hamiltonian is given by
$\rho_{\mathrm{A}}=e^{-H_{E}}$. We will see that $\rho_{\mathrm{A}}$ can be
described by the thermal density matrix of the edge CFT at high temperature.

If the coupling $\lambda$ is taken much smaller than the bulk energy gap, the
main effect of the term $H_{\mathrm{AB}}$ is to induce a perturbation to the
edge theories on the cut. Let us assume that the entanglement property of the
system can be well-described by taking the small coupling $\lambda$, which is
a natural assumption when $H_{\mathrm{AB}}$ induces a relevant perturbation to
CFT that can gap out the edge spectrum with small coupling. In that case, the
entanglement spectrum can be effectively described by the edge theories on the
cut,

$\displaystyle
H=H_{\mathrm{A}}^{\mathrm{CFT}}+H_{\mathrm{B}}^{\mathrm{CFT}}+H_{\mathrm{int}},$
(34)

where $H_{\mathrm{A}}^{\mathrm{CFT}},H_{\mathrm{B}}^{\mathrm{CFT}}$ describe
the edge CFT at the cut, and $H_{\mathrm{int}}$ introduces the inter-edge
perturbation. We consider the case where
$H_{\mathrm{A}}^{\mathrm{CFT}},H_{\mathrm{B}}^{\mathrm{CFT}}$ are not
necessarily chiral. To study the ground state of this perturbed Hamiltonian
$H$, we recall that the relevant perturbation $H_{\mathrm{int}}$ corresponds
to a conformally invariant boundary condition of the CFT
$H_{\mathrm{A}}^{\mathrm{CFT}}+H_{\mathrm{B}}^{\mathrm{CFT}}$, where the
boundary condition can be explicitly obtained by turning on the perturbation
on the half region of the 1d space. This perspective allows us to describe the
ground state of the perturbed Hamiltonian $H$ using the toolkit of boundary
conformal field theory (BCFT).

To do this, let us consider a “quenching” process of the system, whose
dynamics is given by the Hamiltonian $H$ at the time $t<0$, and we suddenly
turn off the perturbation as $\lambda=0$ at $t=0$. In the (1+1)D spacetime,
this quench is regarded as having the boundary condition of the unperturbed
CFT at $t=0$, so the initial state at $t=0$ can be effectively described by
the boundary state for the boundary condition that corresponds to
$H_{\mathrm{int}}$.

Let us now describe this boundary state. Suppose that the state of
$H_{\mathrm{A}}^{\mathrm{CFT}}$ lies in the topological sector of the CFT
labeled by the anyon $a$ in the bulk. Since we want to work on a non-chiral
topological phase in general, we assume that the anyon $a$ in the bulk
corresponds to a generic non-chiral primary field $(\alpha,\overline{\beta})$
on the edge CFT, where $\alpha$ (resp. $\overline{\beta}$) labels chiral
(resp. anti-chiral) primary. Accordingly, the state of
$H_{\mathrm{B}}^{\mathrm{CFT}}$ lies in the topological sector
$(\beta,\overline{\alpha})$. The state of the combined theory
$H_{\mathrm{A}}^{\mathrm{CFT}}+H_{\mathrm{B}}^{\mathrm{CFT}}$ is defined in
the Hilbert space
$(\mathcal{H}_{\alpha}\otimes\mathcal{H}_{\beta})\otimes(\mathcal{H}_{\overline{\alpha}}\otimes\mathcal{H}_{\overline{\beta}})$,
where $\mathcal{H}_{\alpha}$ denotes the holomorphic CFT Hilbert space labeled
by the chiral primary $\alpha$. Note that
$\mathcal{H}_{\alpha}\otimes\mathcal{H}_{\overline{\beta}}$ (resp.
$\mathcal{H}_{\beta}\otimes\mathcal{H}_{\overline{\alpha}}$) corresponds to
the Hilbert space supported on the A (resp. B) subsystem. In this Hilbert
space, the boundary state of the combined theory
$H_{\mathrm{A}}^{\mathrm{CFT}}+H_{\mathrm{B}}^{\mathrm{CFT}}$ can be described
by the Ishibashi state

$\displaystyle\ket{a_{\mathrm{bdry}}}=\sum_{N}\ket{(\alpha,\beta),N}\otimes\ket{\overline{(\alpha,\beta)},N}$
(35)

where $\ket{(\alpha,\beta),N}$ denotes the state in the holomorphic Hilbert
space $\mathcal{H}_{\alpha}\otimes\mathcal{H}_{\beta}$ with the label $N$, and
the sum of $N$ runs all the orthonormal basis of
$\mathcal{H}_{\alpha}\otimes\mathcal{H}_{\beta}$. Since each basis state can
be written as $\ket{V_{\alpha}}\otimes\ket{V_{\beta}}$, the sum is rewritten
as

$\displaystyle\ket{a_{\mathrm{bdry}}}=\sum_{N_{\alpha},N_{\beta}}\ket{\alpha,N_{\alpha}}\ket{\beta,N_{\beta}}\otimes\ket{\overline{\alpha},N_{\alpha}}\ket{\overline{\beta},N_{\beta}}$
(36)

using the label $N_{\alpha},N_{\beta}$ for the orthonormal basis of
$\mathcal{H}_{\alpha},\mathcal{H}_{\beta}$ respectively. To obtain the reduced
density matrix $\rho_{\mathrm{A}}$, we trace out the Hilbert space
$\mathcal{H}_{\beta}\otimes\mathcal{H}_{\overline{\alpha}}$ for the B
subsystem. We then have

$\displaystyle\rho_{\mathrm{A}}=\sum_{N_{\alpha},N_{\beta}}[\ket{\alpha,N_{\alpha}}\ket{\overline{\beta},N_{\beta}}][\bra{\alpha,N_{\alpha}}\bra{\overline{\beta},N_{\beta}}]$
(37)

which is regarded as the thermal density matrix of CFT for the A subsystem at
infinite temperature. This demonstrates that the entanglement spectrum is
described by that of the physical edge CFT at high temperature, including the
non-chiral topological phases.

##  V Preparing the trivial sector of Kitaev honeycomb model on a cylinder

To compute higher central charge of the topologically ordered state, one needs
to prepare a low-energy state on the cylinder that lies in the trivial sector
labeled by the trivial anyon $1$. In this section, we describe how to obtain
such a state for the non-Abelian gapped phase of the Kitaev honeycomb model on
a cylinder.

The Kitaev honeycomb model is described in terms of a free fermion model
coupled to a dynamical ${\mathbb{Z}}_{2}$ gauge field. The ${\mathbb{Z}}_{2}$
vortex of the gauge field is regarded as a single quasiparticle excitation
$\sigma$, and the low-energy state is realized by flat ${\mathbb{Z}}_{2}$
gauge field without a vortex on each plaquette. The distinct low-energy states
on a cylinder then correspond to the different configurations of flat
${\mathbb{Z}}_{2}$ gauge field labeled by the boundary conditions of the free
fermion model. In particular, the anti-periodic boundary condition (APBC) in
the $y$ direction (i.e., circumference) of the cylinder corresponds to 1 or
$\psi$ sector, while the periodic (PBC) corresponds to the $\sigma$ sector. To
further obtain the trivial sector $1$ instead of $\psi$, we need to carefully
study the holonomy of ${\mathbb{Z}}_{2}$ gauge field in $x$ direction. As we
will see below, fixing the holonomy of the ${\mathbb{Z}}_{2}$ gauge field in
$x$ direction gives the state labeled by a non-simple anyon $1+\psi$. To
obtain a state in the trivial sector $1$, it turns out that we need to take a
superposition over the two distinct states that correspond to different
${\mathbb{Z}}_{2}$ holonomy in $x$ direction.

Let us move to the explicit description of the states of the Kitaev honeycomb
model. We assume that the system size $L_{y}$ in the $y$ direction is even.
Recall that to solve this model, we introduce four Majorana fermion operators
$b^{x}_{j},b^{y}_{j},b^{z}_{j},c_{j}$ on each site $j$, and the model then
becomes equivalent to the free fermion model for the Majorana fermions $c$,
coupled to the dynamical ${\mathbb{Z}}_{2}$ gauge field $u_{ij}:=ib_{i}b_{j}$
Kitaev (2006). Let us fix one configuration of flat ${\mathbb{Z}}_{2}$ gauge
field of the Kitaev honeycomb model $\\{u_{ij}\\}$ that corresponds to the
APBC in the $y$ direction. For this given configuration of the
${\mathbb{Z}}_{2}$ gauge field $\\{u_{ij}\\}$, the ground state of the Kitaev
honeycomb model is given in the form of

$\displaystyle\ket{\Psi}=P\left[\ket{\Psi_{F}(\\{u_{ij}\\})}\otimes\ket{\\{u_{ij}\\}}\right],$
(38)

where the state is given by the tensor product of the state of the free
fermion $\ket{\Psi_{F}(\\{u_{ij}\\})}$ and the state $\ket{\\{u_{ij}\\}}$ for
the dynamical ${\mathbb{Z}}_{2}$ field. $P:=\prod_{j}\frac{1+D_{j}}{2}$
denotes a projection onto the gauge invariant Hilbert space, where
$D_{j}:=ib^{x}_{j}b^{y}_{j}b^{z}_{j}c$ generates the gauge transformation at
the vertex $j$.

The distinct state with different ${\mathbb{Z}}_{2}$ holonomy in the $x$
direction can be obtained by considering a straight line $l$ extended in the
$y$ direction cutting the links, and then shifting the ${\mathbb{Z}}_{2}$
gauge field $\\{u_{ij}\\}$ along the links cut by the line $l$. Let us denote
the shifted ${\mathbb{Z}}_{2}$ gauge field as $\\{u^{\prime}_{ij}\\}$, and the
corresponding state of the Kitaev honeycomb model as

$\displaystyle\ket{\Psi^{\prime}}=P\left[\ket{\Psi_{F}(\\{u^{\prime}_{ij}\\})}\otimes\ket{\\{u^{\prime}_{ij}\\}}\right].$
(39)

We aim to show that the two states $\ket{\Psi}\pm\ket{\Psi^{\prime}}$
correspond to the states in the sector 1 and $\psi$, though we do not try to
specify which of $\ket{\Psi}\pm\ket{\Psi^{\prime}}$ corresponds to the trivial
sector. To see this, we evaluate the partial rotation for the states
$\ket{\Psi}\pm\ket{\Psi^{\prime}}$, since the partial rotation can diagnose
the distinct topological sector.

To perform the partial rotation, we need to take the bipartition of the
cylinder into A and B subsystems. For convenience, let us perform the
bipartition along the line $l$ that we used for shifting the
${\mathbb{Z}}_{2}$ gauge field from $\\{u_{ij}\\}$ to $\\{u^{\prime}_{ij}\\}$.
Without loss of generality, we can take the gauge where $u_{ij}=1$ on each
link cut by the line $l$, while $u^{\prime}_{ij}=-1$ on these links.

Since the link variables $u_{ij}$, $u^{\prime}_{ij}$ cut by the line $l$
overlap the Hilbert space of A and B subsystems, it is convenient to redefine
link variables near the cut so that the newly defined ones lie solely in the A
or B subsystem. This can be done by following the argument in Appendix A of
Ref. Tu  _et al._ (2013). We recall that the link variable is given by the
doublet of Majorana fermions as $u_{ij}=ib_{i}b_{j}$, and let us write the
Majorana fermions at the plaquette on the cut as $b_{1},b_{2},b_{3},b_{4}$ as
shown in Fig. 4. The newly defined link variables are then given by
$w_{13}:=ib_{1}b_{3},w_{42}=ib_{4}b_{2}$. The local Hilbert space spanned by
these four Majorana fermions is identified as the Hilbert space of two qubits,
where we have the identification of operators as

$\displaystyle u_{12}=Z_{1},\quad u_{34}=Z_{2},\quad w_{13}=X_{1}X_{2},\quad
w_{42}=-Y_{1}Y_{2}.$ (40)

Some of the eigenstates of $w_{13},w_{42}$ are expressed as

$\displaystyle\ket{++}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11}),\quad\ket{--}=\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})$
(41)

where $+,-$ denotes eigenvalues of $w_{13},w_{42}$, while $0,1$ denotes those
of $u_{12},u_{34}$.

Figure 4: The link variables near the bipartition. The thick dots denote the
Majorana fermions $b_{j}$ for $1\leq j\leq 4$. The link variables
$u_{12}=ib_{1}b_{2},u_{34}=ib_{3}b_{4}$ are cut by the line $l$, and we define
the new ones $w_{13}=ib_{1}b_{3},w_{42}=ib_{4}b_{2}$.

Noting that $u_{ij}=1$ on the edges cut by the bipartition, the state for the
${\mathbb{Z}}_{2}$ gauge field $\ket{\\{u_{ij}\\}}$ is decomposed into the
Hilbert spaces of A, B subsystems as

$\displaystyle\ket{\\{u_{ij}\\}}=2^{-\frac{L_{y}}{4}}\sum_{w_{ij}=\pm
1}\ket{\\{u^{\mathrm{A}}_{ij},w_{ij}\\}}_{\mathrm{A}}\otimes\ket{\\{u^{\mathrm{B}}_{ij},w_{ij}\\}}_{\mathrm{B}}~{}.$
(42)

Meanwhile, since $u^{\prime}_{ij}=-1$ on the edges cut by the bipartition, the
state for the shifted ${\mathbb{Z}}_{2}$ gauge field
$\ket{\\{u^{\prime}_{ij}\\}}$ is decomposed as

$\displaystyle\ket{\\{u^{\prime}_{ij}\\}}=2^{-\frac{L_{y}}{4}}\sum_{w_{ij}=\pm
1}(-1)^{|w|}\ket{\\{u^{\mathrm{A}}_{ij},w_{ij}\\}}_{\mathrm{A}}\otimes\ket{\\{u^{\mathrm{B}}_{ij},w_{ij}\\}}_{\mathrm{B}}~{},$
(43)

where $|w|$ is the mod 2 number of links on the A subsystem with $w_{ij}=-1$.
Note that $u_{ij}=u^{\prime}_{ij}$ away from the cut, so each Schmidt state in
the A, B subsystems are the same in the expression of $\ket{\\{u_{ij}\\}}$ and
$\ket{\\{u^{\prime}_{ij}\\}}$.

Let us also write the Schmidt decomposition of the free fermion state
$\ket{\Psi_{F}(\\{u_{ij}\\})}$ as

$\displaystyle\ket{\Psi_{F}(\\{u_{ij}\\})}=\sum_{N}\alpha_{N}\ket{\Psi^{\mathrm{A}}_{N}(\\{u^{\mathrm{A}}_{ij}\\})}\otimes\ket{\Psi^{\mathrm{B}}_{N}(\\{u^{\mathrm{B}}_{ij}\\})}~{}.$
(44)

Next, we assume that the low-energy state for the free fermion
$\ket{\Psi_{F}(\\{u^{\prime}_{ij}\\})}$ under the shifted ${\mathbb{Z}}_{2}$
gauge field $\\{u^{\prime}_{ij}\\}$ can be chosen to be identical to that
under the trivial ${\mathbb{Z}}_{2}$ gauge field $\\{u_{ij}\\}$,

$\displaystyle\ket{\Psi_{F}(\\{u^{\prime}_{ij}\\})}=\ket{\Psi_{F}(\\{u_{ij}\\})},$
(45)

and hence has the same Schmidt decomposition. Below, we show that
$\ket{\Psi}\pm\ket{\Psi^{\prime}}$ with the above $\ket{\Psi^{\prime}}$
corresponds to the 1 or $\psi$ sector.

The reduced density matrix of $\ket{\Psi}$ for the A subsystem is given by Tu
_et al._ (2013)

$\displaystyle\rho_{\mathrm{A}}:=\mathrm{Tr}_{\mathrm{B}}(\ket{\Psi}\bra{\Psi})\propto\sum_{w_{ij}=\pm
1}\sum_{N}|\alpha_{N}|^{2}P_{\mathrm{A}}\left[\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\mathrm{A}}_{ij}\\})}\ket{\\{u^{\mathrm{A}}_{ij},w_{ij}\\}}\right]P_{\mathrm{A}}\left[\bra{\Psi_{N}^{\mathrm{A}}(\\{u^{\mathrm{A}}_{ij}\\})}\bra{\\{u^{\mathrm{A}}_{ij},w_{ij}\\}}\right],$
(46)

where $P_{\mathrm{A}}=\prod_{j\in\mathrm{A}}\frac{1+D_{j}}{2}$ denotes the
projector supported on the A subsystem. Noting that the product of gauge
transformations in the A subsystem $\prod_{j\in\mathrm{A}}D_{j}$ leaves the
link variables $\\{u^{\mathrm{A}}_{ij},w_{ij}\\}$ invariant, one can rewrite
the expression of the state as

$\displaystyle
P_{\mathrm{A}}\left[\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\mathrm{A}}_{ij}\\})}\ket{\\{u^{\mathrm{A}}_{ij},w_{ij}\\}}\right]\propto\sum_{\\{u^{\prime\mathrm{A}}_{ij},w^{\prime}_{ij}\\}\approx\\{u^{\mathrm{A}}_{ij},w_{ij}\\}}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\prime\mathrm{A}}_{ij}\\})}\ket{\\{u^{\prime\mathrm{A}}_{ij},w^{\prime}_{ij}\\}}$
(47)

where the sum is taken over gauge fields
$\\{u^{\prime\mathrm{A}}_{ij},w^{\prime}_{ij}\\}$ that are equivalent to
$\\{u^{\mathrm{A}}_{ij},w_{ij}\\}$ up to gauge transformations. The reduced
density matrix is then expressed as

$\displaystyle\rho_{\mathrm{A},\ket{\Psi}}\propto\sum_{w_{ij}=\pm
1}\sum_{\begin{subarray}{c}\\{u^{\prime\mathrm{A}}_{ij},w^{\prime}_{ij}\\}\approx\\{u^{\mathrm{A}}_{ij},w_{ij}\\}\\\
\\{u^{\prime\prime\mathrm{A}}_{ij},w^{\prime\prime}_{ij}\\}\approx\\{u^{\mathrm{A}}_{ij},w_{ij}\\}\end{subarray}}\sum_{N}|\alpha_{N}|^{2}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\prime\mathrm{A}}_{ij}\\})}\ket{\\{u^{\prime\mathrm{A}}_{ij},w^{\prime}_{ij}\\}}\bra{\Psi_{N}^{\mathrm{A}}(\\{u^{\prime\prime\mathrm{A}}_{ij}\\})}\bra{\\{u^{\prime\prime\mathrm{A}}_{ij},w^{\prime\prime}_{ij}\\}}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}$
(48)

Also, the reduced density matrix for the state
$\ket{\Psi}\pm\ket{\Psi^{\prime}}$ is given by

$\displaystyle\begin{split}\rho_{\mathrm{A},\ket{\Psi}\pm\ket{\Psi^{\prime}}}\propto&\sum_{w_{ij}=\pm
1}\frac{1\pm(-1)^{|w|}}{2}\sum_{\begin{subarray}{c}\\{u^{\prime\mathrm{A}}_{ij},w^{\prime}_{ij}\\}\approx\\{u^{\mathrm{A}}_{ij},w_{ij}\\}\\\
\\{u^{\prime\prime\mathrm{A}}_{ij},w^{\prime\prime}_{ij}\\}\approx\\{u^{\mathrm{A}}_{ij},w_{ij}\\}\end{subarray}}\sum_{N}\\\
&|\alpha_{N}|^{2}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\prime\mathrm{A}}_{ij}\\})}\ket{\\{u^{\prime\mathrm{A}}_{ij},w^{\prime}_{ij}\\}}\bra{\Psi_{N}^{\mathrm{A}}(\\{u^{\prime\prime\mathrm{A}}_{ij}\\})}\bra{\\{u^{\prime\prime\mathrm{A}}_{ij},w^{\prime\prime}_{ij}\\}}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}.\end{split}$
(49)

The partial rotation of these reduced density matrices for
$\ket{\Psi},\ket{\Psi}\pm\ket{\Psi^{\prime}}$ are evaluated as

$\displaystyle\begin{split}\mathcal{T}_{\ket{\Psi}}\left(\frac{2\pi}{n}\right)&\propto\sum_{w_{ij}=\pm
1}\sum_{N}|\alpha_{N}|^{2}\bra{\Psi_{N}^{\mathrm{A}}(\\{T_{\frac{L_{y}}{n}}u^{\mathrm{A}}_{ij}\\})}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}T_{\mathrm{A};\frac{L_{y}}{n}}\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\mathrm{A}}_{ij}\\})}~{},\\\
\mathcal{T}_{\ket{\Psi}\pm\ket{\Psi^{\prime}}}\left(\frac{2\pi}{n}\right)&\propto\sum_{w_{ij}=\pm
1}\sum_{N}|\alpha_{N}|^{2}\bra{\Psi_{N}^{\mathrm{A}}(\\{T_{\frac{L_{y}}{n}}u^{\mathrm{A}}_{ij}\\})}\frac{1\pm(-1)^{|w|}}{2}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}T_{\mathrm{A};\frac{L_{y}}{n}}\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\mathrm{A}}_{ij}\\})}~{},\end{split}$
(50)

where we assume $L_{y}$ is the integer multiple of $n$, and
$T_{\mathrm{A};\frac{L_{y}}{n}}$ is the translation along the $y$ direction by
$L_{y}/n$ lattice units. $\\{T_{\frac{L_{y}}{n}}u^{\mathrm{A}}_{ij}\\}$
denotes the gauge field obtained by acting the translation
$T_{\mathrm{A};\frac{L_{y}}{n}}$ on the configuration $\\{u_{ij}\\}$. The
above expressions for the partial rotation can further be simplified by
writing the operator $\prod_{j\in\mathrm{A}}D_{j}$ as

$\displaystyle\prod_{j\in\mathrm{A}}D_{j}=(-1)^{F}(-1)^{|w|}\prod
u^{\mathrm{A}}_{ij},$ (51)

where $(-1)^{F}$ is the fermion parity operator of the A subsystem acting only
on the free fermion state. The product over link variables are dependent on
the orientations of the links, but its detailed definition is not discussed
here. Since the projector $\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}$ enforces
the relation $(-1)^{F}=(-1)^{|w|}\prod u^{\mathrm{A}}_{ij}$ and we sum over
all possible choices of $w_{ij}=\pm 1$, the half of the projectors in the
summand of $w_{ij}$ becomes $\frac{1+(-1)^{F}}{2}$, while the other half
becomes $\frac{1-(-1)^{F}}{2}$. Hence we have

$\displaystyle\sum_{w_{ij}=\pm
1}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}\propto\frac{1+(-1)^{F}}{2}+\frac{1-(-1)^{F}}{2}=1.$
(52)

Meanwhile, for the operator
$\frac{1\pm(-1)^{|w|}}{2}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}$, the
contribution of either $\frac{1+(-1)^{F}}{2}$ or $\frac{1+(-1)^{F}}{2}$ is
suppressed by the presence of the $\frac{1\pm(-1)^{|w|}}{2}$ factor. We hence
have

$\displaystyle\sum_{w_{ij}=\pm
1}\frac{1\pm(-1)^{|w|}}{2}\frac{1+\prod_{j\in\mathrm{A}}D_{j}}{2}\propto\quad\frac{1+(-1)^{F}}{2}\quad\text{or}\quad\frac{1-(-1)^{F}}{2}.$
(53)

Eventually, one can express the partial rotation of each state in the form
only involving the free fermion state as

$\displaystyle\begin{split}\mathcal{T}_{\ket{\Psi}}\left(\frac{2\pi}{n}\right)&\propto\sum_{N}|\alpha_{N}|^{2}\bra{\Psi_{N}^{\mathrm{A}}(\\{T_{\frac{L_{y}}{n}}u^{\mathrm{A}}_{ij}\\})}T_{\mathrm{A};\frac{L_{y}}{n}}\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\mathrm{A}}_{ij}\\})}~{},\end{split}$
(54)

$\displaystyle\mathcal{T}_{\ket{\Psi}\pm\ket{\Psi^{\prime}}}\left(\frac{2\pi}{n}\right)\propto\begin{cases}&\sum_{N}|\alpha_{N}|^{2}\bra{\Psi_{N}^{\mathrm{A}}(\\{T_{\frac{L_{y}}{n}}u^{\mathrm{A}}_{ij}\\})}\frac{1+(-1)^{F}}{2}T_{\mathrm{A};\frac{L_{y}}{n}}\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\mathrm{A}}_{ij}\\})}~{},\\\
&\quad\text{or}\\\
&\sum_{N}|\alpha_{N}|^{2}\bra{\Psi_{N}^{\mathrm{A}}(\\{T_{\frac{L_{y}}{n}}u^{\mathrm{A}}_{ij}\\})}\frac{1-(-1)^{F}}{2}T_{\mathrm{A};\frac{L_{y}}{n}}\ket{\Psi_{N}^{\mathrm{A}}(\\{u^{\mathrm{A}}_{ij}\\})}~{}.\end{cases}$
(55)

Let us write the entanglement Hamiltonian of the free fermion state
$\ket{\Psi_{F}(\\{u^{\mathrm{A}}_{ij}\\})}$ as $H_{E}$. The above partial
rotations are then simply expressed as

$\displaystyle\mathcal{T}_{\ket{\Psi}}\left(\frac{2\pi}{n}\right)\propto\mathrm{Tr}\left(e^{-H_{E}}T_{\mathrm{A};\frac{L_{y}}{n}}\right),$
(56)
$\displaystyle\mathcal{T}_{\ket{\Psi}\pm\ket{\Psi^{\prime}}}\left(\frac{2\pi}{n}\right)\propto\quad\mathrm{Tr}\left(\frac{1+(-1)^{F}}{2}e^{-H_{E}}T_{\mathrm{A};\frac{L_{y}}{n}}\right)\quad\text{or}\quad\mathrm{Tr}\left(\frac{1-(-1)^{F}}{2}e^{-H_{E}}T_{\mathrm{A};\frac{L_{y}}{n}}\right)~{}.$
(57)

At the CFT level, the projector $(1+(-1)^{F})/2$ (resp. $(1-(-1)^{F})/2$)
corresponds to the projector onto the fermion parity even (resp. odd) Hilbert
space, which corresponds to the sector 1 (resp. $\psi$). We hence have

$\displaystyle\mathcal{T}_{1}\left(\frac{2\pi}{n}\right)\propto\mathrm{Tr}\left(\frac{1+(-1)^{F}}{2}e^{-H_{E}}T_{\mathrm{A};\frac{L_{y}}{n}}\right),\quad\mathcal{T}_{\psi}\left(\frac{2\pi}{n}\right)\propto\mathrm{Tr}\left(\frac{1-(-1)^{F}}{2}e^{-H_{E}}T_{\mathrm{A};\frac{L_{y}}{n}}\right).$
(58)

In the main text, we directly confirm the first relation for
$\mathcal{T}_{1}\left(\frac{2\pi}{n}\right)$ by comparing the CFT results for
LHS and the numerical results for RHS.

##  VI Extracting higher central charge of the Kitaev honeycomb model

Figure 5: (a) Higher central charge $\zeta_{n}$ of the toric code phase of the
Kitaev model, extracted from $\mathcal{T}_{a}(2\pi/n)$ in Fig. 2 (b) and
chiral central charge $c_{-}=0$ using the main result Eq. (3), computed using
$J_{x}=2.5,J_{y}=J_{z}=1,\kappa=0$. The dotted lines are the CFT predictions
as usual. (b) Extrapolating $\zeta_{n}$ as a function of $1/L_{y}^{2}$ (c)
Twisted higher central charge $\zeta_{n,a}$ of the Ising phase of the Kitaev
model, extracted using chiral central charge $c_{-}=1/2$. The computation is
done with $J_{x}=J_{y}=J_{z}=1,\kappa=0.1$. (d) Extrapolating $\zeta_{n,a}$ as
a function of $1/L_{y}^{2}$. (e) Phase of partial rotation
$\mathcal{T}_{1}(2\pi/n)$ as it goes through a phase transition from the Ising
phase to the toric code phase. $J_{z}=2$ line marks the phase transition, and
the horizontal dashed (dotted) lines are CFT predictions for Ising (toric
code) phase. The computation is done with $J_{x}=J_{y}=1$, $\kappa=0.1$,
$L_{x}=100,L_{y}=120$.

The Kitaev honeycomb model supports two types of topological orders Kitaev
(2006): the toric code phase with $c_{-}=0$ and the Ising topological order
with $c_{-}=1/2$. In this appendix, we show the higher central charges of
these two phases, and see how they interpolate at finite $L_{y}$.

The toric code phase is obtained when one of $J_{x},J_{y},J_{z}$ is much
larger than other e.g. $|J_{x}|>|J_{y}|+|J_{z}|$. Perturbation theory
calculation confirms the effective theory for this corresponds to Wen’s
plaquette model Wen (2003). Let us now consider the phase with
$|J_{z}|>|J_{y}|+|J_{x}|$. With the zigzag boundary condition given in Fig. 2,
this corresponds to a edge critical theory between the $m$ and $e$ type
boundaries. However, since all topological sectors of toric code has trivial
$\zeta_{n,a}$, this does not affect the extraction of partial rotation.
Therefore, we will simply reuse the formula Eq.(13) and its $\sigma$ phase
counterpart to compute partial rotation, even though it no longer corresponds
to the trivial or $\sigma$ sector of the Ising topological order.

In Fig. 5(a)-(d), we show the convergence of higher central charge with
circumference $L_{y}$, both for the toric code phase and the Ising phase. The
toric code phase has four Abelian anyons with topological spins
$\theta_{1}=1,\theta_{m}=1,\theta_{e}=1,\theta_{f}=-1$, and correspondingly
the higher central charge is always real and positive. We see $\zeta_{n}$
converges quickly to their theoretical values. In Fig. 5 (e), we show the
partial rotation $\mathcal{T}_{a}(2\pi/n)$. At this fixed $L_{y}$, phase angle
smoothly interpolates between the Ising phase and the toric code phase.

##  VII Higher central charge of fermionic $\nu=2/3$ FQH state: non-chiral
example

In this appendix, we numerically evaluate the partial rotation on a simple
example of non-chiral topological order; $\nu=2/3$ fractional quantum Hall
(FQH) state. This is a fermionic topological order described by the
$\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{-3}$ Chern-Simons theory and carries
$c_{-}=0$. We will see that the numerical results for the partial rotations
with $n=2,3,4$ has an excellent agreement with the CFT predictions, supporting
the validity of the cut-and-glue approach to evaluate the partial rotation
$\mathcal{T}_{a}(2\pi/n)$ even for non-chiral phases.

###  VII.1 Higher central charge and partial rotation on Abelian fermionic
topological order

Since we are interested in fermionic FQH states instead of bosonic states
studied in the main text, let us first compute the partial rotation of
fermionic topologically ordered states and derive the formula for
$\mathcal{T}_{a}(2\pi/n)$.

For simplicity, let us restrict ourselves to the case of Abelian fermionic
topological order. In that case, the anyons are described by a super-modular
category $\mathcal{C}$ with the form of

$\displaystyle\mathcal{C}=\mathcal{C}_{0}\boxtimes\\{1,\psi\\}$ (59)

where $\mathcal{C}_{0}$ is a modular tensor category, and the operation
$\boxtimes$ is Deligne product that physically means stacking two theories. We
show in the leading order that

$\displaystyle\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)\propto\begin{cases}e^{\frac{2\pi
i}{n}h_{a}-2\pi
i(\frac{2}{n}+n)c_{-}}\cdot\sum_{b\in\mathcal{C}_{0}}S_{ab}d_{b}\theta_{b}^{n}&\text{when
$n$ is even},\\\ e^{\frac{2\pi i}{n}h_{a}-2\pi
i(\frac{2}{n}+n)c_{-}}\cdot\sum_{v}\sum_{b\in\mathcal{C}_{0}}S_{ab}d_{b}\theta_{b}^{n}S_{bv}e^{\frac{2\pi
i}{n}h_{v}}&\text{when $n$ is odd},\end{cases}$ (60)

where $a\in\mathcal{C}$ specifies the topological sector, and $v$ is a vortex
of ${\mathbb{Z}}_{2}^{f}$ fermion parity symmetry carrying the smallest spin.
If there are multiple labels of ${\mathbb{Z}}_{2}^{f}$ vortices with the
smallest spin we sum over such vortices in the expression, which is meant by
the sum $\sum_{v}$. Formally, $v$ is a simple object of the modular extension
of the super-modular category $\mathcal{C}$ denoted as
$\breve{\mathcal{C}}=\mathcal{C}\oplus\mathcal{C}_{v}$. $\breve{\mathcal{C}}$
is physically regarded as a bosonic theory obtained by gauging
${\mathbb{Z}}_{2}^{f}$ symmetry of the fermionic theory $\mathcal{C}$, where
$\mathcal{C}_{v}$ describes the defects carrying the vortex of
${\mathbb{Z}}_{2}^{f}$ symmetry. The above formula can be obtained by
following the logic of the analytic computation described in the main text.
Namely, by employing the cut-and-glue approach, we first write the partial
rotation in terms of the CFT character as

$\displaystyle\begin{split}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)&=\frac{\mathrm{Tr}_{a}[e^{i{P}_{l}\frac{L}{n}}e^{-\frac{\xi_{l}}{v}H_{l}}]\mathrm{Tr}_{a}[e^{i{P}_{r}\frac{L}{n}}e^{-\frac{\xi_{r}}{v}H_{r}}]}{\mathrm{Tr}_{a}[e^{-\frac{\xi_{l}}{v}H_{l}}]\mathrm{Tr}_{a}[e^{-\frac{\xi_{r}}{v}H_{r}}]}\\\
&=\frac{\chi_{a}(\frac{i\xi_{l}}{L}+\frac{1}{n};[\mathrm{AP,AP}])\chi_{a}(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP,AP}])}{\chi_{a}(\frac{i\xi_{l}}{L};[\mathrm{AP,AP}])\chi_{a}(\frac{i\xi_{r}}{L};[\mathrm{AP,AP}])}~{},\end{split}$
(61)

where [AP,AP] denotes the boundary condition of ${\mathbb{Z}}_{2}^{f}$
symmetry on a torus, labeled by anti-periodic (AP) or periodic (P) in each
cycle of the torus. Due to $L\ll\xi_{l}$, the contribution from the left edge
can be approximated as

$\displaystyle\begin{split}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)&=e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}\frac{\chi_{a}(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP,AP}])}{\chi_{a}(\frac{i\xi_{r}}{L};[\mathrm{AP,AP}])}~{}.\end{split}$
(62)

The denominator does not contribute to the phase of
$\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)$ since

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{r}}{L};[\mathrm{AP,AP}]\right)&=\sum_{b}S_{ab}\chi_{b}\left(\frac{iL}{\xi_{r}};[\mathrm{AP,AP}]\right)\approx\frac{d_{a}}{\mathcal{D}}e^{\frac{2\pi
L}{\xi_{r}}\frac{c_{-}}{24}},\end{split}$ (63)

so we have

$\displaystyle\begin{split}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)&=e^{\frac{2\pi
i}{n}(h_{a}-\frac{c_{-}}{24})}{\chi_{a}\left(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP,AP}]\right)}~{}.\end{split}$
(64)

In the following let us derive Eq. (60) by cases of even or odd $n$.

  * •

Even $n$. We perform the $ST^{n}S$ modular transformation on the CFT
character, which transforms the spin structure of the torus as $ST^{n}S:$
[AP,AP]$\to$[AP,AP] for even $n$. We can then approximate the CFT character by
the contribution from the vacuum as

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP,AP}]\right)&=\sum_{b}(ST^{n}S)_{ab}\chi_{b}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{AP,AP}]\right)\\\
&\approx(ST^{n}S)_{a,1}e^{-\frac{2\pi i}{n}\frac{c_{-}}{24}}e^{\frac{2\pi
L}{n^{2}\xi_{r}}\frac{c_{-}}{24}}\\\ &=\frac{1}{\mathcal{D}}e^{-2\pi
i(n+\frac{1}{n})\frac{c_{-}}{24}}e^{\frac{2\pi
L}{n^{2}\xi_{r}}\frac{c_{-}}{24}}\sum_{b\in\mathcal{C}}S_{ab}d_{b}\theta_{b}^{n}\\\
&\propto e^{-2\pi i(n+\frac{1}{n})\frac{c_{-}}{24}}e^{\frac{2\pi
L}{n^{2}\xi_{r}}\frac{c_{-}}{24}}\sum_{b\in\mathcal{C}_{0}}S_{ab}d_{b}\theta_{b}^{n}.\end{split}$
(65)

In the last equation, we used that the transparent fermion $\psi$ braids
trivially with other anyons and has $S_{a\psi}\theta_{\psi}^{n}\propto 1$ for
even $n$. We can then obtain the expression in Eq. (60).

  * •

Odd $n$. We again perform the $ST^{n}S$ modular transformation on the CFT
character, which now transforms the spin structure of the torus as $ST^{n}S:$
[AP,AP]$\to$[P,AP] for odd $n$. We then have

$\displaystyle\begin{split}\chi_{a}\left(\frac{i\xi_{r}}{L}-\frac{1}{n};[\mathrm{AP,AP}]\right)&=\sum_{b}(ST^{n}S)_{ab}\chi_{b}\left(\frac{iL}{n^{2}\xi_{r}}+\frac{1}{n};[\mathrm{P,AP}]\right)\\\
&=\sum_{v\in\mathcal{C}_{v}}(ST^{n}S)_{a,v}e^{\frac{2\pi
i}{n}(h_{v}-\frac{c_{-}}{24})}e^{-\frac{2\pi
L}{n^{2}\xi_{r}}(h_{v}-\frac{c_{-}}{24})}\\\ &\propto e^{-2\pi
i(n+\frac{1}{n})\frac{c_{-}}{24}}\sum_{v\in\mathcal{C}_{v}}e^{-\frac{2\pi
L}{n^{2}\xi_{r}}h_{v}}\sum_{b\in\mathcal{C}_{0}}S_{ab}d_{b}\theta_{b}^{n}S_{bv}e^{\frac{2\pi
i}{n}h_{v}}.\end{split}$ (66)

The expression now involves the sum over the defects $v$ carrying the
${\mathbb{Z}}_{2}^{f}$ vortex, which labels the states in the periodic sector
$\mathcal{C}_{v}$. Due to the exponentially dropping factor $e^{-\frac{2\pi
L}{n^{2}\xi_{r}}h_{v}}$, the sectors carrying the lowest spin give the leading
contributions. We then arrive at the expression in Eq. (60) to the leading
order.

###  VII.2 $\nu=2/3$ FQH state

Here let us consider the partial rotation on the $\nu=2/3$ FQH state,
described by $\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{-3}$ Chern-Simons theory.
Since this theory is Abelian, it allows the expression in terms of
$\mathcal{C}_{0}\boxtimes\\{1,\psi\\}$ with a modular category
$\mathcal{C}_{0}$; we have $\mathcal{C}_{0}=\\{1,p,p^{2}\\}$ with $p$ an
Abelian anyon carrying spin $h_{p}=1/3$. The partial rotation
$\mathcal{T}_{1}(2\pi/n)$ with small even $n$ are computed as
$\mathcal{T}_{1}(2\pi/2)\propto-i,\mathcal{T}_{1}(2\pi/4)\propto i$.

To compute the partial rotation for odd $n$, we need to look at the periodic
sector $\mathcal{C}_{v}$ carrying the ${\mathbb{Z}}_{2}^{f}$ vortex. The
modular extension $\breve{\mathcal{C}}$ of the $\nu=2/3$ state is described by
$\mathrm{U}(1)_{4}\times\mathrm{U}(1)_{-12}$ Chern-Simons theory with the
diagonal fermion $([2]_{4},[6]_{12})$ condensed. Here, we label the anyon of
$\mathrm{U}(1)_{\pm m}$ theory as $[j]_{m}$ with $j\in{\mathbb{Z}}_{m}$. There
are six simple objects in $\mathcal{C}_{v}$ carrying the
${\mathbb{Z}}_{2}^{f}$ vortex, given by $v_{j}:=([1]_{4},[2j-1]_{12})$ with
$1\leq j\leq 6$. The spins are
$h_{v_{1}}=h_{v_{5}}=h_{v_{7}}=h_{v_{11}}=1/12,h_{v_{3}}=h_{v_{9}}=3/4$.

Let us study the simplest case at $n=1$. We have
$\sum_{b}\theta_{b}S_{b,v}=\sum_{b}\theta_{b\times
v^{*}}/\theta_{v^{*}}=\theta_{v}^{-1}$, so we get $\mathcal{T}_{1}(2\pi)=1$.
We then consider the partial rotation at $n=3$, $\mathcal{T}_{1}(2\pi/3)$. For
$v=v_{1},v_{5},v_{7},v_{11}$, one can see that $\sum_{b}\theta_{b}^{3}S_{b,v}$
gives zero, so in the leading order the partial rotation vanishes. Then, the
main contribution comes from the subleading order with $v=v_{3},v_{9}$. In
that case we have $\sum_{b}\theta_{b}^{3}S_{b,v}\propto 1$, so we get
$\mathcal{T}_{1}(2\pi/3)=e^{\frac{2\pi i}{3}h_{v}}=i$. We summarize the
results of the partial rotation for small $n$ in Table 2.

| $\mathcal{T}_{1}\left(\frac{2\pi}{n}\right)$
---|---
$\nu=2/3$ | $1,-i,i,i$
Table 2: The phases of the partial rotation $\mathcal{T}_{1}(\frac{2\pi}{n})$
for $n=1,2,3,4$ in $\nu=2/3$ state.

Following the methodology detailed in the main text, we numerically evaluate
the partial rotation of a $\nu=2/3$ filled LLL of fermions, with model
interaction $V_{1}=1$ plus a small perturbation $\delta V_{3}=0.1$. From the
orbital space entanglement spectrum, we confirm that the obtained ground state
is described by $\mathrm{U}(1)_{1}\times\mathrm{U}(1)_{-3}$ Chern-Simons
theory. The result of $\mathcal{T}_{1}(\frac{2\pi}{n})$ in the trivial sector
is presented in Fig. 6, which converges to the expected phase shown in Table
2.

Figure 6: Partial rotation $\operatorname{Arg}\>\mathcal{T}_{1}(2\pi/n)$ of
the fermionic $\nu=2/3$ FQH state. The dotted lines are the CFT predictions
given in Table 2.

###  VII.3 Partial rotation of Abelian fermionic states as obstructions to
gapped edge

Finally, let us comment that the partial rotation of an Abelian fermionic
topological order can also define an obstruction to gapped edge state. When
the fermionic theory $\mathcal{C}=\mathcal{C}_{0}\boxtimes\\{1,\psi\\}$ admits
a gapped edge, it can be shown that the modular category $\mathcal{C}_{0}$
must have a Lagrangian subgroup, i.e., a bosonic theory described by
$\mathcal{C}_{0}$ must admit a bosonic gapped boundary Kobayashi (2022). This
implies that the higher central charge of $\mathcal{C}_{0}$

$\displaystyle\zeta_{n}(\mathcal{C}_{0})=\frac{\sum_{a\in\mathcal{C}_{0}}\theta_{a}^{n}}{|\sum_{a\in\mathcal{C}_{0}}\theta_{a}^{n}|}\quad\text{with
$\gcd(n,\frac{N_{\mathrm{FS}}(\mathcal{C}_{0})}{\gcd(n,N_{\mathrm{FS}}(\mathcal{C}_{0}))})=1$}$
(67)

define obstructions to gapped boundary of an Abelian fermionic theory. So,
comparing with the result of the partial rotation described above,
$\mathcal{T}_{1}(2\pi/n)$ defines an obstruction to a gapped edge for even $n$
satisfying
$\gcd(n,\frac{N_{\mathrm{FS}}(\mathcal{C}_{0})}{\gcd(n,N_{\mathrm{FS}}(\mathcal{C}_{0}))})=1$.
For the $\nu=2/3$ state described above, the partial rotation
$\mathcal{T}_{1}(2\pi/n)$ with $n=2,4$ becomes an obstruction to gapped edge.

##  VIII Evaluating the action of partial rotation of an MPS wave function

Here we explain how to evaluate the action of partial rotation of an MPS wave
function, with quantum Hall systems as an example. In the following we will
first review the quantum Hall DMRG setup developed in Ref. Zaletel and Mong,
2012; Zaletel  _et al._ , 2013; Geraedts  _et al._ , 2017. We consider an
infinite cylinder geometry with $y$ running around the circumference, and $x$
running along the infinite direction (see Fig. 1). We work in the Landau gauge
and the corresponding LLL orbital takes the form (see Fig. 3 (a))

$\varphi_{n}(x,y)=\frac{e^{ik_{n}y-\frac{1}{2\ell_{B}^{2}}\left(x-k_{y}^{n}\ell_{B}^{2}\right)^{2}}}{\sqrt{L_{y}\ell_{B}\pi^{1/2}}},\quad
k_{y}^{n}=\frac{2\pi n}{L_{y}}$ (68)

where $n\in\mathbb{Z}$ and orbital $n$ localizes at $k_{n}\ell_{B}^{2}=2\pi
n\ell_{B}/L_{y}$. We note that in order to perform the partial rotation, it is
important to work in an eigenbasis in momentum $k_{y}$, which implicitly
assumes translation symmetry, continuous or discrete, along the $y$ direction.
For a lattice model or a $k$-space continuum model, such a basis can be
obtained by hybrid Wannier localization in the $y$ direction, which maps the
original model to a cylinder in mixed $(x,k_{y})$ basis Soejima  _et al._
(2020).

There is a natural one-dimensional iMPS representation of quantum Hall
wavefunction in this basis,

$|\Psi[B]\rangle=\sum_{\left\\{j_{n}\right\\}}\left[\cdots
B^{[0]j_{0}}B^{[1]j_{1}}\cdots\right]\left|\ldots,j_{0},j_{1},\ldots\right\rangle$
(69)

where $B^{[n]j_{n}}$ are $\chi\times\chi$ matrices and
$\left|j_{n}\right\rangle,j_{n}\in\\{0,1,\cdots,N_{\mathrm{boson}}\\}$
represent the boson occupancy at orbital $n$. Both bond dimension $\chi$ and
onsite boson number cutoff $N_{\mathrm{boson}}$ constrain the representability
of the iMPS. However, as we will show in later sections of the Supplemental
Materials, the partial rotation $\mathcal{T}_{a}(2\pi/n)$ quickly saturates to
its true value at relatively small $\chi$ and $N_{\mathrm{boson}}$.

One key ingredient to accelerate the DMRG algorithm and allow the action of
partial rotation is to implement both particle number and momentum
conservation,

$\hat{C}=\sum_{n}\hat{C}_{n}\equiv\sum_{n}\left(\hat{N}_{n}-\frac{p}{q}\right),\quad\hat{K}=\sum_{n}\hat{K}_{n}\equiv\sum_{n}n\left(\hat{N}_{n}-\frac{p}{q}\right)$
(70)

where $\hat{N}_{n}$ is the number operator at site $n$ and $\nu=p/q$. Now for
a generic conserved $\mathrm{U}(1)$ charge $\hat{Q}=\hat{C}$ or $\hat{K}$, we
can decompose the symmetry operator $\hat{Q}$ into
$\hat{Q}=\sum_{n<\bar{n}}\hat{Q}_{n}+\sum_{n>\bar{n}}\hat{Q}_{n}=\hat{Q}_{\mathrm{A}}+\hat{Q}_{\mathrm{B}}$,
where $\bar{n}$ is the auxiliary bond across A and B. For the moment we will
assume A, B represent two sets of LLL orbitals and discuss the difference
between orbital-space cut and real-space cut later. Then for a Schimidt
decomposition across bond $\bar{n}$,
$\ket{\Psi}=\sum_{\alpha}\lambda_{\alpha}^{2}\ket{\alpha}_{\mathrm{A}}\otimes\ket{\alpha}_{\mathrm{B}}$,
we can require the Schimidt states to be eigenstates of
$\hat{Q}_{\mathrm{A/B}}$,

$\hat{Q}_{\mathrm{B}}\ket{\alpha}_{\mathrm{B}}=\bar{Q}_{\bar{n};\alpha}\ket{\alpha}_{\mathrm{B}}$
(71)

At the level of MPS, local charge conservation on site $n$ takes the form,

$\left[\bar{Q}_{\bar{n};\beta}-\bar{Q}_{\bar{n}+1;\alpha}-\hat{Q}_{n;j}\right]B_{\alpha\beta}^{[n]j}=0$
(72)

where $\bar{n}<n<\bar{n}+1$. This conservation law naturally generalizes to
the corresponding symmetry action. For example, pictorially, momentum
conservation pictorially means

(73)

where $\hat{U}=e^{i\theta\hat{K}}$ is the symmetry action of rotating site $n$
along the $y$ direction by angle $\theta$.

###  VIII.1 Partially rotate an MPS

Now we explain how to exploit the implemented momentum conservation to perform
the partial rotation. Naively,
$\hat{K}_{\mathrm{B}}=\sum_{n>\bar{n}}\hat{K}_{n}$ has the interpretation of
total momentum to the right of bond $\bar{n}$, so one would expect the action
of partial rotation of angle $\theta$ to be

$e^{-i\theta\hat{K}_{\mathrm{B}}}\ket{\Psi}=e^{-i\theta\hat{K}_{\mathrm{B}}}\sum_{\alpha}\lambda_{\alpha}\ket{\alpha}_{\mathrm{A}}\otimes\ket{\alpha}_{\mathrm{B}}=\sum_{\alpha}\lambda_{\alpha}^{2}e^{-i\theta\bar{K}_{\bar{n};\alpha}}\ket{\alpha}_{\mathrm{A}}\otimes\ket{\alpha}_{\mathrm{B}}$
(74)

This would be obvious for a finite MPS whose momentum eigenvalues
$\bar{K}_{\bar{n};\alpha}$ are finite. However, for iMPS, a more rigorous
proof presented below would be appreciated.

######  Theorem 1.

For an infinite MPS $\ket{\Psi}$, the expectation value
$\mathcal{T}_{a}(\theta)$ of partial rotating all sites in B by angle $\theta$
is equivalent to inserting a $e^{-i\theta\bar{K}_{\bar{n};\alpha}}$ factor to
the bond $\bar{n}$ across A, B.

$\bra{\Psi}\prod_{n>\bar{n}}e^{i\theta\hat{K}_{n}}\ket{\Psi}=\sum_{\alpha}\lambda_{\alpha}^{2}\bra{\alpha}_{\mathrm{A}}\otimes\bra{\alpha}_{\mathrm{B}}e^{-i\theta\bar{K}_{\bar{n};\alpha}}\ket{\alpha}_{\mathrm{A}}\otimes\ket{\alpha}_{\mathrm{B}}=\sum_{\alpha}\lambda_{\alpha}^{2}e^{-i\theta\bar{K}_{\bar{n};\alpha}}$
(75)

Pictorially,

(76)

###### Proof.

To prove this theorem, we introduce the formulation of transfer matrices. For
simplicity, we assume the unit cell size to be one in the following proof. The
transfer matrix $M_{U}$ is defined to be
$M_{U}=\sum_{j,j^{\prime}}U_{j,j^{\prime}}B^{j}\otimes\bar{B}^{j^{\prime}}$.
The most important property of transfer matrix $M_{1}$ (with no operator
insertion) is that we can approximate $M_{1}^{N}$ by its dominant eigenvector
$\rho$ with eigenvalue $\lambda=1$,

(77)

since all eigenvectors with eigenvalue $\lambda^{\prime}<1$ would be
negligible in the iMPS limit $N\to\infty$. The theorem can be rewritten in a
more precise form using the language of transfer matrices,

$M_{U}^{N}\approx e^{i\phi}\bar{U}^{-1}M_{1}^{N},\quad\text{where
}U=e^{i\theta\hat{K}_{n}},\;\bar{U}^{-1}=e^{-i\theta\bar{K}_{\bar{n};\alpha}}\rho_{\alpha,\alpha^{\prime}}$
(78)

where $\phi$ is an overall phase ambiguity. Using the momentum conservation in
Eq. (72), we can show
$e^{-i\theta\bar{K}_{\bar{n};\alpha}}\rho_{\alpha,\alpha^{\prime}}$ is an
eigenvector of $M_{U}$ with eigenvalue $\lambda=1$, which therefore must be
the dominant eigenvector of $M_{U}$,

(79)

We note that
$e^{i\phi}e^{-i\theta\bar{K}_{\bar{n};\alpha}}\rho_{\alpha,\alpha^{\prime}}$
is also a dominant eigenvector, so there is an overall phase ambiguity that
cannot be determined by this argument. We will see the consequence of this
ambiguity in the next section. Combined with the property of transfer matrix
in Eq. (77), we have established $M_{U}^{N}\approx
e^{i\phi}\bar{U}^{-1}M_{1}^{N}$ and therefore proved the theorem. ∎

###  VIII.2 Matching iMPS momentum labels to CFT momentums

The most subtle but important aspect in the evaluation of partial rotation is
to resolve some of the gauge degrees of freedom in matching the momentum
labels $\bar{K}_{\alpha}$ in the iMPS representation to the CFT momentum
labels of a state. The conserved-charges (i.e., charge and momentum) labels
for particles on each site are given by Eq. (70). The conserved-charge labels
for the Schmidt states across any bond is the accumulations of the those to
the left of the cut; which is sensitive even to electrons and dipole that are
far away from the entanglement cut. As we “grow” the MPS chain in the iDMRG
algorithm, such electrons/dipoles may be “stuck” based on the initialization
of the DMRG. For example, an electron far away may shift the conserved-charge
labels via $\hat{C}\to\hat{C}+q$ and
$\hat{K}\to\hat{K}+d_{\text{electron}}\hat{C}$, where $d_{\text{electron}}$ is
the distance (proportional to number of DMRG steps) of the charge. Therefore
the momentum label $\bar{K}_{\bar{n};\alpha}$ has two integer ambiguity
$x_{0}$ and $x_{1}$ to the determined,

$\bar{K}_{\alpha}\mapsto\bar{K}_{\alpha}^{\prime}=\bar{K}_{\alpha}+x_{1}\bar{C}_{\alpha}+x_{0}.$
(80)

To reproduce the CFT calculation in Eq. (3), we can match the entanglement
spectrum and the (1+1)D edge CFT spectrum to fix the momentum labels. We
demand, for the ground state corresponding to the vacuum, that (a) the highest
weight state (corresponding to the lowest energy state of the entanglement
Hamiltonian) has zero momentum and charge, and (b) that sectors with opposite
charges have the same momentum labels. (For the semion ground state, its
entanglement spectrum has two highest weight state–we assign them to have
charges $\pm 1/2$ and zero momentum.)

###  VIII.3 Orbital cut v.s. real-space cut

Now we are ready to talk about the difference between the orbital cut and
real-space cut. The CFT prediction in Eq. (3) assumes a real space partition.
Before we explain the numerical realization, we first discuss the physical
implication of a real-space cut at different locations. In quantum Hall
systems, the entanglement spectrum depends on where the real-space cut is
made. In particular, we want to make the cut such that anyon $a$ can appear
right on the cut, then the entanglement spectrum would agree best with the 1D
edge CFT spectrum. For fermions, this happens when the cut is in the middle of
two LLL orbital centers; for bosons, this happens when the cut is right at the
LLL orbital center.

This is most obvious in the thin cylinder limit, where the $\nu=1/2$ bosonic
Laughlin state takes the form of

$\ket{\Psi}=\ket{\cdots 010101\cdots}$ (81)

in the occupation basis. This is usually referred as the root configuration of
the state (or pattern of zeros). For the entanglement spectrum corresponding
to the vacuum sector, we want to choose the boundaries of the unit cell such
that the electron (i.e., the ‘1’ of the root configuration) is in the middle
of the unit cell. (This ensures that entanglement spectrum has a unique
highest weight state and a symmetry between positive and negative charge
selection sectors.) The real-space cut corresponding to boundaries of the unit
cell.

Therefore, cutting through the $0$’s gives the trivial sector and cutting
through the $1$’s (related by threading one flux quanta through the cylinder)
gives the semion sector. Furthermore, this also explains why for fermionic
Laughlin states ($q$ odd), the unit cell cuts between a pair of neighboring
sites. This identification remains true at finite circumference $L$.

Finally, we briefly explain the numerical realization of the real-space cut.
We take the most straightforward approach based on the real-space entanglement
spectrum (RSES) algorithm developed in Ref. Zaletel and Mong, 2012. For a
real-space partition at $x=x_{c}$, RSES algorithm reconstructs the orbital
space MPS $\ket{\Psi[B]}$ obtained in iDMRG into a new one
$\ket{\psi[B]_{\mathrm{RSES}}}$ such that a particular bond $\bar{n}_{c}$
gives the real-space Schmidt decomposition at $x_{c}$. This is accomplished by
splitting each matrix $B^{[n]}$ into two $B^{[n]}_{\mathrm{A}}$ and
$B^{[n]}_{\mathrm{B}}$ based on the weight of the orbital $\varphi_{n}$
supported on A, B, and then swap all $B^{[n]}_{\mathrm{A}}$ to the left of
bond $\bar{n}_{c}$ and $B^{[n]}_{\mathrm{B}}$ to the right. Then we can easily
evaluate the action of partial rotation with the techniques discussed above.
We note that the difference between orbital cut and real-space cut is quite
general in systems where band projection was used, since the Wannier function
usually span across a few sites.

##  IX Extracting higher central charge of the $\nu=1/2$ bosonic Laughlin
state

###  IX.1 Extracting higher central charge from scratch

Figure 7: (a) Infinitesimal rotation $\lim_{\theta\to
0}\>\operatorname{Arg}\>\mathcal{T}_{a}(2\pi/n)$ of the $\nu=1/2$ bosonic
Laughlin state extracted using Eq. (14). (b) The corresponding momentum
polarization $p_{a}=h_{a}-c_{-}/24$ extracted using Eq. (10). The dotted lines
are the CFT predictions.

In the main text Fig. 3, we only present the numerical result of
$\mathcal{T}_{a}(2\pi/n)$. We note that $\mathcal{T}_{a}(2\pi/n)$ itself is
sufficient in determining the gappability of the edge, without referring to
the actual value of the higher central charge $\zeta_{n}$. However, if one
wants to extract $\zeta_{n}$, one also needs to know about the chiral central
charge $c_{-}$, which could be obtained from either some prior knowledge about
the topological order, or the momentum polarization Tu  _et al._ (2013);
Zaletel  _et al._ (2013), or certain entanglement measures Kim  _et al._
(2022a, b); Fan (2022); Fan  _et al._ (2022). Our numerical algorithm can
easily reproduce the momentum polarization by taking the large $n$ limit,
though the physics is completely different as explained in the main text. For
completeness, we show the momentum polarization, and the resulting higher
central charge in this section.

In quantum Hall systems, the momentum polarization takes the universal form
Zaletel  _et al._ (2013)

$\lim_{n\to\infty}\mathcal{T}_{a}\left(\frac{2\pi}{n}\right)=\exp\left[\frac{2\pi
i}{n}\left(h_{a}-\frac{c_{-}}{24}-\frac{\eta_{H}}{2\pi\hbar}L^{2}\right)\right]$
(82)

where $\eta_{H}=\hbar\nu\mathcal{S}/8\pi\ell_{B}^{2}$ is the universal Hall
viscosity with $\mathcal{S}$ being the topological shift Read (2009). We first
evaluate the action of infinitesimal rotation and then fit the result to Eq.
(82) (see Fig. 7 (a)), where we find $\mathcal{S}=1.998$ for both topological
sectors, in excellent agreement of the CFT prediction $\mathcal{S}=2$.
Subtracting off the the Hall viscosity contribution, we plot the resulting
momentum polarization $p_{a}=h_{a}-c_{-}/24$ in Fig. 7 (b), which again gives
the expected $c_{-}=1$ and $h_{s}=1/4$. We note that we can only reach
$L_{y}=22\ell_{B}$ using the largest practical onsite boson cutoff
$N_{\mathrm{boson}}$, whereas for partial rotation we can easily reach
$L_{y}>40\ell_{B}$. This is because a much higher precision is required in
evaluating $\mathcal{T}_{a}\left(\theta\right)$ in order to reliably extract
$p_{a}$, a point we will return to in the next section.

Figure 8: (a) (Twisted) higher central charge $\zeta_{n,a}$ extracted from
$\mathcal{T}_{a}(2\pi/n)$ in Fig. 3 (b) and chiral central charge $c_{-}=1$ in
Fig. 7 using the main result Eq. (3). The dotted lines are the CFT predictions
as usual. (b) Extrapolating $\zeta_{n,a}$ as a function of
$\ell_{B}^{2}/L_{y}^{2}$.

Using $c_{-}=1$ and $h_{s}=1/4$ extracted from momentum polarization, we can
extract the (twisted) higher central charge $\zeta_{n,a}$ using the main
result Eq. (3). As shown in Fig. 8, $\zeta_{n,a}$ all converge to the expected
values as shown in Table 1 at sufficiently large $L_{y}$.

###  IX.2 Numerical performance of partial rotation

Figure 9: Partial rotation $\mathcal{T}_{a}(2\pi/n)$ as a function of bond
dimension $\chi$ and onsite boson number cutoff $N_{\mathrm{Boson}}$ at
$L_{y}=40\ell_{B}$. $\mathcal{T}_{a}(2\pi/n)$ saturates at relatively small
$\chi=1600$ and $N_{\mathrm{Boson}}=5$.

Now we briefly discuss the numerical performance of our protocol to extract
higher central charge from partially rotating an MPS wavefunction. Compared to
free fermion models, e.g. the Kitaev model, the MPS representation of a
topological state wavefunction is generically not exact. Both the bond
dimension $\chi$ and the onsite boson number cutoff $N_{\mathrm{Boson}}$ put
constraints on the representability of the MPS. However, the quantity we are
interested in, i.e. the partial rotation $\mathcal{T}_{a}(2\pi/n)$ rapidly
saturates to its true value up to the largest system size we reach (see Fig.
9). In practice, for a relatively small $\chi$ and $N_{\mathrm{Boson}}$, we
can already obtain $\mathcal{T}_{a}(2\pi/n)$ and therefore $\zeta_{n,a}$ to
rather high precision.

For readers who are familiar with the momentum polarization, we note that the
precision required in the evaluation of $\mathcal{T}_{a}(\theta)$ is much
higher in order to reliably extract the momentum polarization $p_{a}$ due to
the Hall viscosity term that is quadratic in cylinder circumference $L_{y}$.
Specifically, the error in $\zeta_{n,a}$ and $p_{a}$ are related to the error
in $\operatorname{Arg}\mathcal{T}_{a}(\theta)$ by

$\Delta\zeta_{n,a}\sim\frac{\Delta\operatorname{Arg}\mathcal{T}_{a}(\theta)}{\operatorname{Arg}\mathcal{T}_{a}(\theta)},\quad\Delta
p_{a}\sim
L_{y}^{2}\frac{\Delta\operatorname{Arg}\mathcal{T}_{a}(\theta)}{\operatorname{Arg}\mathcal{T}_{a}(\theta)}$
(83)

We note that $\zeta_{n,a}$ and $p_{a}$ are both $O(1)$, but $L_{y}^{2}$ can be
fairly large to avoid finite size effect. Therefore, a much smaller $\chi$ and
$N_{\mathrm{Boson}}$ is required to reliable extract the higher central charge
compared to that one would use for momentum polarization.

The main numerical challenge in our protocol to extract the higher central
charge is the interpolating between higher central charge and momentum
polarization discussed in an earlier section. We need to reach system size
$L_{y}>n^{2}\xi_{r}$ to reliably extract higher central charge $\zeta_{n,a}$.
In quantum Hall systems, by comparing Eq. (10) and Eq. (82), we find

$\xi_{r}=\sqrt{\frac{2\pi^{2}}{3}}\frac{c_{-}}{\nu\mathcal{S}}\ell_{B}=\sqrt{\frac{2}{3}}\pi\ell_{B}$
(84)

where the last equality assumes the $\nu=1/2$ bosonic Laughlin state. In order
to obtain $\zeta_{n,a}$ with $n\geq 5$, we need to reach $L_{y}\geq
25\sqrt{\frac{2}{3}}\pi\ell_{B}\approx 64\ell_{B}$, which can be challenging
in practice.