Before we present the results, we emphasize that by “ _comparison_ ” of $x$
with $y$, we explicitly mean $\left|{x-y\over y}\right|$ when we list the data
below.

We first examine some selected commutation relations
$\left\langle\Psi\right|[\widetilde{L}_{n},\widetilde{L}_{-n}]\left|\Psi\right\rangle$
for $n=1,2,\cdots 6$, and make comparisons with the CFT predictions

$\text{CFT prediction:
}\left\langle\Psi\right|[\widetilde{L}_{n},\widetilde{L}_{-n}]\left|\Psi\right\rangle={\pi
c\over 24}n(n^{2}-1)^{2}{\sin^{2}(\pi/(4n))\over\cos(\pi/(2n))}$ (7.12)

with the central charge $c=1/2$, which is computed in Section (6.38) in
Section 6.2.2 under the Hypothesis 1. The results are listed in Table 3. The
pretty small discrepancy ($10^{-5}$ in magnitude) between the numerical
results and CFT expectation indicates the reference state
$\left|\Psi\right\rangle$ we used in numerics satisfies the hypothesis pretty
well on the length scale we chose. We also compare the data with results from
exact Virasoro generators

$\text{Vir:
}\left\langle\Psi\right|[L_{n},L_{-n}]\left|\Psi\right\rangle=\left\langle\Omega\right|[\mathscr{L}_{n},\mathscr{L}_{-n}]\left|\Omega\right\rangle={c\over
12}(n^{3}-n),\quad c={1\over 2},$ (7.13)

which are computed under Hypothesis 1. The dominating source for the deviation
is because we ignore the higher harmonic terms in $\widetilde{L}_{n}$ even
though we know how to compute them. The purpose of such a comparison is to
test the quality of the approximation of $\widetilde{L}_{n}$ to $L_{n}$. The
fact that the “error” is still small, indicates that higher harmonics only
contribute a small correction and supports the name “approximate Virasoro
generators” for $\widetilde{L}_{n}$. Later in appendix C, we shall improve the
construction and do similar comparisons. As we shall see the “approximation
error” is reduced as expected.

$n$ | $\langle\Psi|[\widetilde{L}_{n},\widetilde{L}_{-n}]\Psi\rangle$ |  CFT predictions |  Virasoro algebra $\left\langle\Omega\right|[\mathscr{L}_{n},\mathscr{L}_{-n}]\left|\Omega\right\rangle$ |  Comparison with CFT prediction |  Comparison with Vir
---|---|---|---|---|---
1 | $O(10^{-8})$ | 0 | 0 | NA | NA
2 | 0.24399045 | 0.24399193 | 0.25 | $6.05\times 10^{-6}$ | 2.404%
3 | 0.97200578 | 0.97201215 | 1 | $6.55\times 10^{-6}$ | 2.799%
4 | 2.42663026 | 2.42665060 | 2.5 | $8.38\times 10^{-6}$ | 2.935%
5 | 4.85014000 | 4.85020037 | 5 | $1.245\times 10^{-5}$ | 2.997%
6 | 8.48474652 | 8.48491574 | 8.75 | $1.994\times 10^{-5}$ | 3.031%
Table 3: Results for expectation values of some seleccted commutators among
approximate Virasoro generators $\widetilde{L}_{n}$, and their comparison with
CFT predictions and with results from exact Virasoro generator.

Secondly, we test the expectation values of double commutators
$\left\langle\Psi\right|[[\widetilde{L}_{m},\widetilde{L}_{n}],\widetilde{L}_{k}]\left|\Psi\right\rangle$
among the approximate Virasoro generators. As above, the tests are summarized
into two groups. The first group is the computation of
$\left\langle\Psi\right|[[\widetilde{L}_{n},\widetilde{L}_{0}],\widetilde{L}_{-n}]\left|\Psi\right\rangle,n=1,\cdots,6$.
The results are listed in Table 4. We first compare the results with the CFT
prediction, as computed in Eq. (6.42) in Section 6.2.3:

$\text{CFT prediction:
}\left\langle\Psi\right|[[\widetilde{L}_{n},\widetilde{L}_{0}],\widetilde{L}_{-n}]\left|\Psi\right\rangle={\pi
c\over 48}n(n^{2}-1)^{2}\tan\left({\pi\over 2n}\right),$ (7.14)

as well as the results from the exact Virasoro generators:

$\text{Vir:
}\left\langle\Psi\right|[[L_{n},L_{0}],L_{-n}]\left|\Psi\right\rangle=\left\langle\Omega\right|[[\mathscr{L}_{n},\mathscr{L}_{0}],\mathscr{L}_{-n}]\left|\Omega\right\rangle={c\over
12}(n^{4}-n^{2}).$ (7.15)

The central charge in Eq. (6.42) and Eq. (7.15) are both $c=1/2$. The results
are listed in Table 4. The sources of the discrepancy shown in both
comparisons are similar to those we discussed in the commutator test above.
The second group is
$\left\langle\Psi\right|[[\widetilde{L}_{m},\widetilde{L}_{n}],\widetilde{L}_{k}]\left|\Psi\right\rangle$
of some selected non-zero $m,n,k$, and we compare the results with those
obtained by Virasoro algebra:

$\text{Vir:
}\left\langle\Psi\right|[[L_{m},L_{n}],L_{k}]\left|\Psi\right\rangle=\left\langle\Omega\right|[[\mathscr{L}_{m},\mathscr{L}_{n}],\mathscr{L}_{k}]\left|\Omega\right\rangle={c\over
12}(m-n)(k^{3}-k)\delta_{m+n,-k},$ (7.16)

with $c=1/2$. We also computed the results using “improved” approximate
Virasoro algebra in Appendix C, and the approximation error is reduced as
expected.

$n$ | $\left\langle\Psi\right|[[\widetilde{L}_{n},\widetilde{L}_{0}],\widetilde{L}_{-n}]\left|\Psi\right\rangle$ |  CFT prediction |  $\left\langle\Omega\right|[[\mathscr{L}_{n},\mathscr{L}_{0}],$ $\mathscr{L}_{-n}]\left|\Omega\right\rangle$ |  Comparison with CFT prediction |  Comparison with Vir
---|---|---|---|---|---
1 | $O(10^{-5})$ | 0 | 0 | NA | NA
2 | 0.58734 | 0.58905 | 0.5 | 0.290% | 14.981%
3 | 3.60489 | 3.62776 | 3 | 0.626% | 15.834%
4 | 12.07894 | 12.19960 | 10 | 0.989% | 15.006%
5 | 30.25392 | 30.62296 | 25 | 1.205% | 13.150%
6 | 63.53049 | 64.44933 | 52.5 | 1.426% | 11.298%
Table 4: Results of $\left\langle\Psi\right|[[\widetilde{L}_{n},\widetilde{L}_{0}],\widetilde{L}_{-n}]\left|\Psi\right\rangle$ and comparison with CFT prediction and results of Virasoro algebra generators. $(m,n,k)$ | $\left\langle\Psi\right|[[\widetilde{L}_{m},\widetilde{L}_{n}],\widetilde{L}_{k}]\left|\Psi\right\rangle$ | $\left\langle\Omega\right|[[\mathscr{L}_{m},\mathscr{L}_{n}],\mathscr{L}_{k}]\left|\Omega\right\rangle$ |  Comparison with Virasoro algebra
---|---|---|---
$(2,1,-3)$ | $1.00376+2.0\times 10^{-5}{\bf i}$ | 1 | 0.376%
$(3,1,-4)$ | $4.95632+1.3\times 10^{-4}{\bf i}$ | 5 | 0.874%
$(3,2,-5)$ | $4.98279+0.0{\bf i}$ | 5 | 0.344%
$(4,2,-6)$ | $17.40158-6\times 10^{-4}{\bf i}$ | 17.5 | 0.562%
Table 5: Tests of
$\left\langle\Psi\right|[[\widetilde{L}_{m},\widetilde{L}_{n}],\widetilde{L}_{k}]\left|\Psi\right\rangle$
and comparison with the Virasoro generators.

####  7.1.3 Tests about coherent states

After testing the constructions of approximate Virasoro generators, we now
test some selected properties of coherent states, defined as
$\left|\Psi(t)\right\rangle=e^{i\mathbb{L}t}\left|\Psi\right\rangle$ for some
good modular flow generator $\mathbb{L}\in\mathfrak{g}^{\bullet}$ [Section 5].

To be more explicit, we use $p+{\bf i}p$ SC with the same parameters as we
used before and put the system on a cylinder with $L_{x}=72$, $L_{y}=18$. The
boundary condition is the same as before: open boundary condition in $y$
direction and anti-periodic boundary condition in $x$-direction. We construct
$\mathbb{L}^{(o)}_{2}$ and $\mathbb{L}^{(o)}_{3}$ to generate the good modular
flow acting on the reference state. The construction is similar to what we did
in the previous subsubsection shown in Figure 22, in which the size parameters
for the subsystems used in the construction are $l_{x}=L_{x}/(4n),l_{y}=6$.

We focus on testing the “difference” between the coherent state
$\left|\Psi(t)\right\rangle$ and the reference state
$\left|\Psi\right\rangle=\left|\Psi(0)\right\rangle$, which is characterized
by the fidelity $F(\rho_{X}(0),\rho_{X}(t))$ between density matrices
$\rho_{X}(0),\rho_{X}(t)$ from $\left|\Psi\right\rangle$ and
$\left|\Psi(t)\right\rangle$. We choose the region $X$ as the entire system
and the entire edge region. Explicitly, the edge region in the test is a
cylinder that includes the entire gapless edge on which the modular flows are
acting and extends 5 lattice spacings into the bulk.

Based on the idea that good modular flow preserves the bulk state, we expect
the two choices of $X$ to give the same results [Section 5.2]252525Since the
system is on a cylinder, the proof of this statement requires full boundary
A0. As we showed in Section 7.1.1, indeed, with the current choice of
Hamiltonian parameters and boundary conditions, the full boundary A0 is
satisfied with considerably small errors. Therefore, the proof is applicable
here..

The change of fidelity with the evolution time $\Delta
F(\rho_{X}(0),\rho_{X}(t))\equiv
F(\rho_{X}(0),\rho_{X}(0))-F(\rho_{X}(0),\rho_{X}(t))$ is plotted in Figure
23. We also compute $\alpha(\mathbb{L}^{(o)}_{2},X),\alpha(\mathbb{L}_{3},X)$
defined in Eq. (3.16) in Section 3.2.1, by fitting the data of

$\Delta F(\rho_{X}(0),\rho_{X}(t))\approx{1\over 2}\alpha(\mathbb{L},X)t^{2},$
(7.17)

for the modular flow generated by
$\mathbb{L}=\mathbb{L}^{(o)}_{2},\mathbb{L}^{(o)}_{3}$ as a function of $t$ in
the small $t$ regions. Based on the argument in Section 5.2, we shall expect
they are equal to $\sigma(\mathbb{L}^{(o)}_{2})^{2}$,
$\sigma(\mathbb{L}^{(o)}_{3})^{2}$. Recall from Section 6.2.1 that we can
analytically compute $\sigma(\mathbb{L}^{(o)}_{n})^{2}$ using Eq. (6.28) under
the Hypothesis 1. Therefore, we shall compare the results of
$\alpha(\mathbb{L}^{(o)}_{2}),\alpha(\mathbb{L}^{(o)}_{3})$ with the CFT
prediction of
$\sigma(\mathbb{L}^{(o)}_{2})^{2},\sigma(\mathbb{L}^{(o)}_{3})^{2}$ from Eq.
(6.28). Moreover, we also compare the numerical evaluation of
$\sigma(\mathbb{L}^{(o)}_{2})^{2},\sigma(\mathbb{L}^{(o)}_{3})^{2}$ with the
CFT prediction, which estimates how well the reference state from the lattice
system satisfies the Hypothesis 1. The data and comparison results are listed
in Table 6.

Figure 23: Change of fidelity $\Delta F(\rho_{X}(0),\rho_{X}(t))\equiv F(\rho_{X}(0),\rho_{X}(0))-F(\rho_{X}(0),\rho_{X}(t))$ between reduced density matrices $\rho_{X}(0)$ and $\rho_{X}(t)$ under good modular flow $e^{{\bf i}\mathbb{L}t}$, with flow generator $\mathbb{L}$ taken to be $\mathbb{L}^{(o)}_{2},\mathbb{L}^{(o)}_{3}$. $\rho_{X}(t)$ is the reduced density matrix of $\left|\Psi\right\rangle=e^{{\bf i}\mathbb{L}t}\left|\Psi\right\rangle$ of region $X$. The region $X$ is chosen to be the entire system and the entire edge. $\mathbb{L}$ |  $\sigma(\mathbb{L})^{2}$ CFT prediction |  $\sigma(\mathbb{L})^{2}$ numerical results | $\alpha(\mathbb{L},\text{entire system})$ | $\alpha(\mathbb{L},\text{entire edge})$
---|---|---|---|---
$\mathbb{L}=\mathbb{L}^{(o)}_{2}$ | 0.065322 | 0.065013 (0.05%) | 0.077222 (18.22%) | 0.070044 (7.23%)
$\mathbb{L}=\mathbb{L}^{(o)}_{3}$ | 0.262994 | 0.261825 (0.04%) | 0.271779 (3.34%) | 0.264401 (0.53%)
Table 6: Results of $\sigma(\mathbb{L})^{2}$ and $\alpha(\mathbb{L},X)$.
$\mathbb{L}$ is chosen to be $\mathbb{L}^{(o)}_{2},\mathbb{L}^{(o)}_{3}$ and
$X$ is chosen to be the entire system and entire edge. The data in the second
row is the CFT prediction. The rest of the data are numerical results. The
comparison between numerical results with CFT prediction is given in the
parenthesis after each data.

The agreement in Fig. 23 and Table 6 supports the theoretical prediction that
the action of good modular flow generators is only on the gapless edge.
Furthermore, the agreement between the numerical results for the variance and
the CFT prediction further supports the validity of Hypothesis 1.

###  7.2 Tests in chiral semion wavefunction

We also test the construction of approximate Virasoro generators starting from
a chiral semion wavefunction. The chiral semion model describes a chiral
topologically ordered phase with two abelian anyons $\mathbf{1}$ (vacuum) and
$s$ (semion). The wavefunction we use is the discrete version of $\nu=1/2$
bosonic Laughlin ground state introduced by Nielsen et al [61], with no semion
excitations inserted. Such a wavefunction is expected to have chiral central
charge $c_{-}=1$ in the bulk and a purely-chiral CFT ground state on the edge
with total central charge $c_{\text{tot}}=1$. Therefore, it will satisfy
Hypothesis 1 and be suitable to apply our construction of Virasoro generators.

The wavefunction is on a cylinder and the underlying degrees of freedom are
arranged as “ladders” of qubits [Figure 24]. The explicit expression of the
wavefunction $\left|\Psi\right\rangle$ is introduced in Appendix D.

Figure 24: A finite cylinder on which we define the semion wavefunction. The
horizontal direction is glued periodically. (a) on a 16-qubit 2-leg ladder,
and (b) on a 24-qubit 3-leg ladder. See Appendix D for the details of the
wavefunction. We shall also need regions $Y$s, which are obtained from $X$s by
a shift by one site to the right.

Following the same method as above, we construct

$\widetilde{L}_{\pm 2}=\mathbb{L}^{(e)}_{2}\pm{\bf i}\mathbb{L}^{(o)}_{2},$
(7.18)

where

$\mathbb{L}^{(o)}_{2}={A_{2}\over
2}\sum_{k=1}^{4}(-1)^{k}(K_{X_{k}^{R}}+K_{X_{k}^{L}}),\quad\mathbb{L}^{(e)}_{2}={A_{2}\over
2}\sum_{k=1}^{4}(-1)^{k}(K_{Y_{k}^{R}}+K_{Y_{k}^{L}}).$ (7.19)

The prefactor $A_{2}=3/8$ follows from Eq. (6.11). Due to the limitation of
system size allowed by the single value decomposition (SVD) method, we cannot
construct approximate Virasoro generators of higher mode numbers.

With the construction $\mathbb{L}^{(o)}_{2},\mathbb{L}^{(e)}_{2}$ and
$\widetilde{L}_{\pm 2}$, we compute the expectation values of commutators
$\left\langle\Psi\right|[\widetilde{L}_{2},\widetilde{L}_{-2}]\left|\Psi\right\rangle$
and the variance $\sigma(\mathbb{L}^{(o)}_{2})^{2}$, and make comparisons with
the theoretical prediction based on Hypothesis 1. We also compare the
commutators of approximate Virasoro generators $\widetilde{L}_{\pm 2}$ with
the results from exact Virasoro generators $L_{\pm 2}$ to test the quality of
the approximation.

From Table 7, we see that the finite-size corrections to the approximate
Virasoro algebra are large for the 16-site system. However, the agreement with
CFT predictions is already quite good for the system with 24 sites.

Tests | $\left\langle\Psi\right|[\widetilde{L}_{2},\widetilde{L}_{-2}]\left|\Psi\right\rangle$ |  CFT prediction | $\left\langle\Omega\right|[\mathscr{L}_{2},\mathscr{L}_{-2}]\left|\Omega\right\rangle$ |  Comparison with CFT prediction |  Comparison with Vir
---|---|---|---|---|---
16 sites | 0.242078 | 0.487984 | 0.5 | 50.39% | 51.58%
24 sites | 0.442392 | 0.487984 | 0.5 | 9.34% | 11.5%
Table 7: Same as Table 3, but for the semion wavefunction. Tests |  $\sigma(\mathbb{L}^{(o)}_{2})^{2}$ CFT prediction |  $\sigma(\mathbb{L}^{(o)}_{2})^{2}$ numerical results | Comparison
---|---|---|---
16 sites | 0.129965 | 0.110759 | 14.78%
24 sites | 0.129965 | 0.122377 | 5.84%
Table 8: Comparison of the variance of second Virasoro mode in the semion
wavefunction with CFT predictions from Hypothesis 1.

##  8 Discussion

###  8.1 Summary

We have described a way to construct Virasoro generators from a purely-chiral
wavefunction. The argument is based on entanglement bootstrap axioms within
the bulk and a hypothesis [Hypothesis 1] about the chiral CFT description of
the physical edge. The theoretical predictions are tested against numerical
data. A notable concept we introduce is the good modular flow; the idea is
that such flows preserve the bulk density matrix, but create edge excitations
that form a representation of the Virasoro algebra.

We have provided theoretical arguments (based on a hypothesis) and numerical
evidence that our method to extract the Virasoro algebra works for both
invertible phases and chiral topological orders. In particular, we have tested
$p+{\bf i}p$ superconductor ground states and found an excellent match with
the theoretical prediction. We also tested our prediction on a semion
wavefunction, and the error is consistent with what we should expect on
systems of small sizes, thus supporting our prediction.

###  8.2 Further remarks

Our construction of Virasoro generators is effective for purely-chiral systems
but not for systems with both left and right movers. This is because the
vector fixed point equation (4.13) holds only when the cross-ratios associated
with the left- and right-moving modes are the same $\eta_{L}=\eta_{R}$.

This issue can be resolved by separating the left- and right-moving modes
using the commutators of the modular Hamiltonians, similar to the construction
of the global conformal generators in [27]. Alternatively, another approach
involves performing a Fourier transformation in space followed by a Fourier
transformation in time.

Another difference between the purely-chiral system and the systems with both
left and right movers is how they change under a modular flow. We start from a
state that satisfies the vector fixed-point equation (4.13) and apply a finite
time modular flow. For a system with both left and right movers, since the
modular flow acts differently to the left- and right-movers, the cross-ratios
become different $\eta_{L}(t)\neq\eta_{R}(t)$ and the vector fixed-point
equation no longer holds.

On the other hand, for a purely-chiral theory, where there is only one set of
modes and one corresponding cross-ratio, we can always set $\eta$ in the
vector fixed-point equation (4.13) to be the cross-ratio. This ensures that
the vector fixed-point equation holds, which enables us to generate the family
of states utilized in the section on coherent states, §5.

###  8.3 Future directions

One might think that good modular flows only exhibit nontrivial features near
the edge. Nevertheless, we believe that similar results can also be extracted
directly from the bulk. In pursuit of this, we will introduce the notion of an
“instantaneous modular flow” and generalize the vector fixed-point equation,
Eq. (4.13).

We would like to further explore the relation between the conformal data of
the 1+1D chiral edge and the anyon data of the 2+1D bulk under the framework
of entanglement bootstrap. Such results would support the axioms widely
believed in the category theory approach to this problem [62, 63, 64].

One may hope to extend our formalism to include Kac-Moody algebra, which
arises in CFT with continuous global symmetry. Presumably, this involves
replacing the entanglement Hamiltonian with the logarithm of the flux-
insertion operator $U_{A}(\theta)=\prod_{x\in
A}u_{x}(\theta)\stackrel{{\scriptstyle\mathbb{V}}}{{=}}e^{{\bf
i}\theta\int_{b(A)}dxj_{0}(x)}$, where $\mathbb{V}$ is a generalization of the
isometry in Hypothesis 1 to include other degrees of freedom.

Our definition of good modular flow allows the inclusion of regions with
disconnected components (this is the distinction between $\mathfrak{g}$ and
$\mathfrak{g}^{\bullet}$). The modular Hamiltonians of such regions in 1+1D
CFT are no longer local [40, 39, 41, 42], and depend on the detailed operator
content of the CFT (not just $c$ and the Hamiltonian density). We thus expect
that such generators involve operators beyond the Virasoro generators, but
still preserve the bulk density matrix. This provides a possible route towards
generalizing the isometry $\mathbb{V}$ in Hypothesis 1 to access other CFT
operators besides the stress tensor. Note that, in contrast to the good
modular flow studied in this paper, this type of modular flow with
disconnected components likely does not preserve the vector fixed-point
equation.

We have expressed the elements of the Virasoro algebra as quantum information
quantities from the chiral ground state. It is known that for a system with
CFT behaviors (e.g. Hypothesis 1), the Virasoro algebra forms a Lie algebra,
which is a vector space with commutator relations. However, the current
construction only implies that the constructed Virasoro algebra forms a vector
space. Can we find a checkable condition on the chiral state which implies
these commutator relations?

###  8.4 Preview of results about irregular edges

In Nature, many chiral edges appear to be irregular, such as those in
experiments (e.g. [65, 66, 67, 68, 69, 70, 71]). The theoretical reason for
the persistence of edge modes in such situations is that chiral edges are
robust and cannot be gapped out if we add disorder (locally perturbing the
Hamiltonian) to the edge. However, as we shall discuss in [49], with some
general considerations, such irregular edges are secretly regular in a broader
sense: they have a well-defined cross-ratio, measured using information-
theoretic quantities. We believe that our way of extracting the Virasoro
algebra can also work in the context of irregular edges and give evidence for
this in the forthcoming paper.

Acknowledgments. This work was supported in part by funds provided by the U.S.
Department of Energy (D.O.E.) under cooperative research agreement DE-
SC0009919, by the University of California Laboratory Fees Research Program,
grant LFR-20-653926, and by the Simons Collaboration on Ultra-Quantum Matter,
which is a grant from the Simons Foundation (652264, JM). IK acknowledges
supports from NSF under award number PHY-2337931. JM received travel
reimbursement from the Simons Foundation; the terms of this arrangement have
been reviewed and approved by the University of California, San Diego, in
accordance with its conflict of interest policies. The bureaucrats there are
very proud of this fact and insist that we tell you about their
accomplishments.

##  Appendix A Notation

Notations | Meanings
---|---
$c$ | Holomorphic central charge
$\bar{c}$ | Anti-holomporphic central charge
$c_{\mathrm{tot}}$ | Total central charge: $c_{\mathrm{tot}}=c+\bar{c}$
$c_{-}$ | Chiral central charge: $c_{-}=c-\bar{c}$
$K_{A}$ | Modular Hamiltonian of region $A$ in 2+1D state $\left|\Psi\right\rangle$
$\mathscr{K}_{a}$ | Modular Hamiltonian of an interval $a$ in 1+1D CFT ground state
$\mathscr{K}_{a}^{\chi}$ | Holomorphic component of $\mathscr{K}_{a}$
$\mathbb{V}$ | An isometry defined in Hypothesis 1
$\mathbb{L}$ | Good modular flow generator [Definition 6]
$\mathscr{L}$ |  Linear combination of chiral CFT modular Hamiltonians, with $\mathbb{L}\mathbb{V}=\mathscr{L}\mathbb{V}$
$\mathscr{L}_{n}$ | Virasoro generator from a purely chiral CFT
$L_{n}$ |  Representation of Virasoro generators on 2+1D edge, with $L_{n}\mathbb{V}=\mathscr{L}_{n}\mathbb{V}$
$\mathbb{L}^{(o)}_{n},\mathbb{L}^{(e)}_{n}$ |  Two particular good modular flow generators, defined in Eq. (6.15) and Eq. (6.16)
$\widetilde{L}_{n}$ |  Approximate Virasoro generators, defined in Eq. (6.18)

Some remarks on these notations:

  * •

If a CFT is purely chiral, then $\bar{c}=0$, therefore
$c_{\mathrm{tot}}=c=|c_{-}|$.

  * •

Usually, the curly symbols, such as $\mathscr{K},\mathscr{L}$ are operators
for a CFT, while others are on the Hilbert space of the 2+1D state.

##  Appendix B Smoothness and decay of Fourier coefficients

Here, we prove a statement that relates the smoothness of a function and the
asymptotic decay properties of its Fourier coefficients.

Let $f(x)$ be a real-valued function and $f^{(m)}(x)$ denote its $m$-th
derivative. We call a function $f(x)$ ‘piecewise $C^{m}$’ if $|f^{(m)}(x)|$ is
bounded and piecewise continuous with a finite number of finite jump
discontinuities, while $f^{(m+1)}(x)$ is unbounded. By $f(x)$ having a finite
jump discontinuity at $x_{*}$, we mean $0<|\lim_{x\to
x_{*}^{+}}f(x)-\lim_{x\to x_{*}^{-}}f(x)|<\infty$.

######  Lemma B.1.

Let $f(\theta)$ be a real-valued $2\pi$-periodic function. If $f(\theta)$ is
piecewise $C^{m}$, then its Fourier coefficient

$\tilde{f}_{n}\equiv{1\over 2\pi}\int_{0}^{2\pi}d\theta
f(\theta)e^{-in\theta},$ (B.1)

has the property $|\tilde{f}_{n}|\sim O(1/n^{m+1})$. That is, to be more
explicit, $\lim_{n\to\infty}n^{m+1}|\tilde{f}_{n}|$ is bounded.

###### Proof.

Since $f(\theta)=\sum_{n}\tilde{f}_{n}e^{{\bf i}n\theta}$,

$f^{(k)}(\theta)=\sum_{n}{{\bf i}}^{k}n^{k}\tilde{f}_{n}e^{{\bf i}\theta
n}\quad\Rightarrow\quad\tilde{f}_{n}\cdot n^{k}={(-{\bf i})^{k}\over
2\pi}\int_{0}^{2\pi}d\theta f^{(k)}(\theta)e^{-{\bf i}n\theta}.$ (B.2)

$f(\theta)$ is piecewise $C^{m}$. Therefore, let
$\theta_{1}<\theta_{2}<\cdots<\theta_{N}$ be the discontinuous points of
$f^{(m)}(\theta)$, then $f^{(m)}(\theta)$ can be written as

$f^{(m)}(\theta)=\sum_{k=1}^{N}\Theta(\theta-\theta_{k})\Theta(\theta_{k+1}-\theta)F_{k}(\theta),$
(B.3)

where $\theta_{N+1}\equiv\theta_{1}$, $\Theta(x)$ is the Heaviside step
function. $F_{k}(\theta)$ is some smooth function, which is identical to
$f^{(m)}(\theta)$ on the interval $[\theta_{k},\theta_{k+1}]$. The derivative
of $f^{(m)}(\theta)$ involves various $\delta$ functions:

$\begin{split}f^{(m+1)}(\theta)=\sum_{k=1}^{N}\Big{[}&\delta(\theta-\theta_{k})\Theta(\theta_{k+1}-\theta)F_{k}(\theta)-\Theta(\theta-\theta_{k})\delta(\theta_{k+1}-\theta)F_{k}(\theta)\\\
&+\Theta(\theta-\theta_{k})\Theta(\theta_{k+1}-\theta){dF_{k}(\theta)\over
d\theta}\Big{]}.\end{split}$ (B.4)

Therefore, even though the function $f^{(m+1)}(\theta)$ itself is not upper
bounded, the absolute value of its integral

$\left|{1\over 2\pi}\int_{0}^{2\pi}d\theta f^{(m+1)}(\theta)e^{-{\bf
i}n\theta}\right|$ (B.5)

shall be a upper bounded by some finite number $M$. This is because there are
finite number of $\delta$ functions, and each $\delta$ function only
contribute a finite number to the result of the integral.

Then utilizing Eq. (B.2), one can see that

$|\tilde{f}_{n}|\cdot|n^{m+1}|=\left|{1\over 2\pi}\int_{0}^{2\pi}d\theta
f^{(m+1)}(\theta)e^{-{\bf i}n\theta}\right|\leq M,$ (B.6)

Hence $|\tilde{f}_{n}|\sim O(1/n^{m+1})$.262626Some careful readers might
wonder, is it possible $|\tilde{f}_{n}|\sim O(n^{-(m+2)})$? The answer is no
because if so, $f^{(m)}(\theta)=\sum_{n}i^{m}n^{m}\tilde{f}_{n}e^{i\theta n}$
will be absolutely summable, and as a result, it will absolutely and uniformly
converge to a continuous function, so the function would actually be piecewise
$C^{m+1}$. ∎

We can then apply this lemma to the weight function $f(\theta)$ of a good
modular flow generator

$\mathfrak{g}^{\bullet}\ni\mathbb{L}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}f(\theta)\mathscr{T}(\theta).$
(B.7)

As $f(\theta)$ is piecewise $C^{2}$, with a finite number of jump
discontinuities, we can conclude

$\tilde{f}_{n}={1\over 2\pi}\int_{0}^{2\pi}d\theta f(\theta)e^{-in\theta}\sim
O(n^{-3}).$ (B.8)

For bad modular flow generators, if we assume its action on the edge is of a
similar form as Eq. (B.7), the weight function will only be piecewise $C^{1}$,
and therefore $|\tilde{f}_{n}|\sim O(|n|^{-2})$.

##  Appendix C Improvement of the approximate Virasoro generators

We introduced the construction of the approximate Virasoro generators
$\widetilde{L}_{n}$ and their expansions in terms of Virasoro generators

$\widetilde{L}_{n}=\sum_{m\in\mathbb{Z}}\lambda_{n,m}L_{m},$ (C.1)

with $\lambda_{n,m}$ given in Eq. (6.23). In this section, we introduce a
procedure to reduce the contribution of the higher harmonic terms in
$\widetilde{L}_{n}$. As a result of the procedure, we shall obtain the
Virasoro generator as a linear combination of the approximate Virasoro
generators $\widetilde{L}_{n}$. Furthermore, this indicates that
$\\{\widetilde{L}_{n}\left|\Psi\right\rangle\\}_{n\in\mathbb{Z}}$ forms a
(non-orthorgonal) basis of the vector space spanned by
$\\{L_{n}\left|\Psi\right\rangle\\}_{n\in\mathbb{Z}}$.

###  C.1 Improvement procedure

We first discuss the improvement of $\widetilde{L}_{n}$ to produce a better
approximation to $L_{n}$. Throughout this section, we shall focus on $|n|\geq
2$ cases, as $\widetilde{L}_{0},\widetilde{L}_{\pm 1}$ are already equal to
$L_{0},L_{\pm 1}$ by their constructions.

Upon a close examination of the Fourier coefficients $\lambda_{n,m}$ in Eq.
(6.23), we can see that the mode numbers of Virasoro generators in
$\widetilde{L}_{n}$ are $(-1)^{j}(2j+1),j\in\mathbb{N}$. To be more explicit,

$\widetilde{L}_{n}=L_{n}+\sum_{j\in\mathbb{N}^{*}}\lambda_{n,(-1)^{j}(2j+1)n}L_{(-1)^{j}(2j+1)n},$
(C.2)

where $\mathbb{N}^{*}$ is defined as $\mathbb{N}\setminus\\{0\\}$, and

$\lambda_{n,(-1)^{j}(2j+1)n}=(-1)^{j}{n^{2}-1\over(2j+1)^{3}n^{2}-(2j+1)}.$
(C.3)

Moreover, it is also these modes
$\left\\{L_{(-1)^{j}(2j+1)n}\right\\}_{j\in\mathbb{N}^{*}}$ that contribute in
the Virasoro mode expansion of
$\widetilde{L}_{(-1)^{k}(2k+1)n},k\in\mathbb{N}^{*}$. This can be seen by

$\begin{split}&(-1)^{j}(2j+1)n\cdot(-1)^{k}(2k+1)\\\
=&(-1)^{j+k}[2(2jk+j+k)+1]n\\\ =&(-1)^{m}(2m+1)n,\quad\text{with
}m=2jk+j+k.\end{split}$ (C.4)

Therefore, it’s plausible that one can use the approximate Virasoro generators
$\left\\{\widetilde{L}_{(-1)^{j}(2j+1)n}\right\\}_{j\in\mathbb{N}^{*}}$
themselves to cancel the contribution of
$\left\\{L_{(-1)^{j}(2j+1)n}\right\\}_{j\in\mathbb{N}^{*}}$ in
$\widetilde{L}_{n}$, as it won’t introduce new Virasoro modes.

The exact expression for $L_{n}$ in terms of
$\left\\{\widetilde{L}_{(-1)^{j}(2j+1)n}\right\\}_{j\in\mathbb{N}}$ can be
obtained by the following procedure. Starting from $\widetilde{L}_{n}$, we can
see the first sub-leading order contribution is from $L_{-3n}$. This term can
be canceled by the replacement

$\widetilde{L}_{n}\equiv(\widetilde{L}_{n})^{(0)}\to(\widetilde{L}_{n})^{(1)}\equiv\widetilde{L}_{n}-\lambda_{-3n}\widetilde{L}_{-3n}.$
(C.5)

Then we can cancel the first sub-leading order contribution in
$(\widetilde{L}_{n})^{(1)}$ (which is from $L_{5n}$) by $\widetilde{L}_{5n}$,
and so on. Denoting the improved approximate Virasoro generator after the
$k$-th correction as $(\widetilde{L}_{n})^{(k)}$, we can summarize the
procedure as the following two steps: (1) Find the first sub-leading order
$L_{m}$ in $(\widetilde{L}_{n})^{(k)}$ and its coefficient
$\lambda^{(k)}_{n,m}$. (2) Cancel this term by

$(\widetilde{L}_{n})^{(k)}\to(\widetilde{L}_{n})^{(k+1)}=(\widetilde{L}_{n})^{(k)}-\lambda^{(k)}_{n,m}\widetilde{L}_{m}.$
(C.6)

Then the Virasoro generator $L_{n}$ can be obtained as the limit

$L_{n}=\lim_{k\to\infty}(\widetilde{L}_{n})^{(k)},$ (C.7)

in the sense that

$\lim_{k\to\infty}\lambda_{n,m}^{(k)}=0,\quad\forall m\neq n.$ (C.8)

###  C.2 Numerical tests of the improvement

We can test the success of the procedure elaborated in the previous subsection
numerically in the $p+{\bf i}p$ SC ground state. In the construction of
$\widetilde{L}_{\pm n}$, the next leading order contribution is $L_{\mp 3n}$
[Eq. (C.2)]. We can thus subtract this term, yielding the following
construction:

$\widetilde{L}_{\pm n}\to(\widetilde{L}_{\pm n})^{\prime}=\widetilde{L}_{\pm
n}+{n^{2}-1\over 27n^{2}-3}\widetilde{L}_{\mp 3n}\\\ $ (C.9)

Although there are additional terms being introduced from $\widetilde{L}_{\pm
3n}$, we find the total error is reduced [Table 9]. One can repeat the process
to further improve the approximation.

For example, we can improve $\widetilde{L}_{\pm 2}$ and $\widetilde{L}_{\pm
4}$ as follows, using the setups and construction in Section 7.1.2. For
$\widetilde{L}_{\pm 2}$, the next leading order is $\mathscr{L}_{\pm 6}$. Then
we use $\widetilde{L}_{\pm 6}$ to subtract off $L_{\pm 6}$. For
$\widetilde{L}_{\pm 4}$, we use $\widetilde{L}_{\pm 12}$ to cancel the next
leading term $L_{\pm 12}$:

$\begin{split}&\widetilde{L}_{\pm 2}\to(\widetilde{L}_{\pm
2})^{\prime}=\widetilde{L}_{\pm 2}+{2^{2}-1\over 27\times
2^{2}-3}\widetilde{L}_{\mp 6}\\\ &\widetilde{L}_{\pm 4}\to(\widetilde{L}_{\pm
4})^{\prime}=\widetilde{L}_{\pm 4}+{4^{2}-1\over 27\times
4^{2}-3}\widetilde{L}_{\mp 12}.\end{split}$ (C.10)

As expected, we found improvements; see Table 9

Tests | Original data with $\widetilde{L}_{\pm n}$ | Replacing $\widetilde{L}_{\pm n}$ with $(\widetilde{L}_{\pm n})^{\prime}$
---|---|---
Results |  Comparison with Virasoro algebra | Results |  Comparison with Virasoro algebra
$\left\langle\Psi\right|[\widetilde{L}_{2},\widetilde{L}_{-2}]\left|\Psi\right\rangle$ | 0.2440 | 2.404% | 0.2509 | 0.367%
$\left\langle\Psi\right|[\widetilde{L}_{4},\widetilde{L}_{-4}]\left|\Psi\right\rangle$ | 2.4266 | 2.935% | 2.5113 | 0.454%
$\left\langle\Psi\right|[[\widetilde{L}_{2},\widetilde{L}_{0}]\widetilde{L}_{-2}]\left|\Psi\right\rangle$ | 0.5749 | 14.98% | 0.5272 | 5.44%
$\left\langle\Psi\right|[[\widetilde{L}_{4},\widetilde{L}_{0}]\widetilde{L}_{-4}]\left|\Psi\right\rangle$ | 11.5006 | 15.01% | 10.4140 | 4.14%
Table 9: Commutator tests for the constructed Virasoro generators, with and
without improvements.

##  Appendix D Semion Setup

In this appendix, we explain the detailed setup of the semion wavefunction
used in Section 7.2. This model wavefunction, introduced by Nielsen _et al_
[61], describes a chiral topological order with semion statistics. It is also
known as a discrete version of the $\nu={1\over 2}$ bosonic Laughlin state
[25, 72]. The semion topological order has $c_{-}=1$ and two species of
Abelian anyons $\\{1,s\\}$. The central charge of the chiral edge is expected
to be $c=1$, which should be the same as its total central charge since the
state is purely chiral.

The construction of the semion wavefunction in Ref. [61] is as follows.
Consider a pair $(N,\\{z_{j}\\}_{j=1}^{N})$, where $N$ is an even integer
denoting the number of qubits in the system, and $z_{j}\in\mathbb{C}$ for
$j=1,\cdots,N$ are distinct complex numbers. For each $z_{j}$ on the complex
plane, we assign a qubit. The system is bosonic, and thus the total Hilbert
space is a tensor product of these qubits. The wavefunction (not normalized)
is

$|\Psi(N,\\{z_{j}\\}_{j=1}^{N})\rangle=\sum_{\\{s_{i}\\}_{i=1}^{N}}w(\\{s_{i}\\}_{i=1}^{N})|\\{s_{i}\\}_{i=1}^{N}\rangle,\quad
s_{i}=\pm 1.$ (D.1)

The complex coefficients $w(\\{s_{i}\\}_{i=1}^{N})$ are given by

$w(\\{s_{i}\\}_{i=1}^{N})=\delta_{\\{s_{i}\\}}\prod_{n<m}^{N}(z_{n}-z_{m})^{{1\over
2}s_{n}s_{m}},$ (D.2)

where $\delta_{\\{s_{i}\\}}=1$ for $\sum_{i}s_{i}=0$ and
$\delta_{\\{s_{i}\\}}=0$ otherwise.

It is convenient to map the coordinates $z_{j}$ to the sphere via the
stereographic projection. Namely, we assign a sphere coordinate
$(\theta_{j},\phi_{j})$ to $z_{j}$ according to

$z_{j}={\sin\theta_{j}\over
1+\cos\theta_{j}}\exp\left(\phi_{j}\right),\qquad\theta_{j}\in[0,\pi]\textrm{
and }\phi_{j}\in[0,2\pi).$ (D.3)

The unit vector in 3-dimensional Euclidean space, corresponding to a sphere
coordinate $(\theta,\phi)$ is
$\vec{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, as explained in
Fig. 25(c). As an aside, the sphere coordinate is convenient in this problem
because sphere rotation keeps the wavefunction invariant [61].

Figure 25: The sphere coordinates for the semion 2 and 3-leg ladders used in
Section 7.2.

Now, we are ready to specify the wavefunctions used in Section 7.2. This is
done by specifying the coordinates on the sphere.

  * •

For the 2-leg ladder in Figure 25(a), the 16 points correspond to the
coordinates

$\text{ladder 1:}\left({\pi\over 2}-{\pi\over 8},{k\pi\over
4}\right),\quad\text{ladder 2:}\left({\pi\over 2}+{\pi\over 8},{k\pi\over
4}\right),$ (D.4)

with $k=1,2,\cdots,8$.

  * •

For the 3-leg ladder in Figure 25(b), the 24 points correspond to the
coordinates

$\text{ladder 1:}\left({\pi\over 2}-{\pi\over 8},{k\pi\over
4}\right),\quad\text{ladder 2:}\left({\pi\over 2},{k\pi\over 4}+{\pi\over
8}\right),\quad\text{ladder 3:}\left({\pi\over 2}+{\pi\over 8},{k\pi\over
4}\right),$ (D.5)

with $k=1,2,\cdots,8$.

##  Appendix E The cylinder argument

###  E.1 Review of 1+1D CFT fixed-point equation

Here, we briefly review a local condition satisfied by 1+1D conformal field
theory ground states $\left|\Omega\right\rangle$ on a real line in [27], which
will play an important role in our analysis. The entanglement entropy of an
interval of length $\ell$ in the ground state, $S(\ell)={c_{\mathrm{tot}}\over
6}\ln{\ell\over\epsilon}$ [20, 10], is by itself not strong enough to imply
that the state is a CFT ground state. For the ground state of a CFT, the
modular Hamiltonian $K_{A}$, $\rho_{A}=e^{-K_{A}}$, for a single interval on
the line is [39]:

$\displaystyle\mathscr{K}_{[x_{1},x_{2}]}=\int_{x_{1}}^{x_{2}}dx\beta_{[x_{1},x_{2}]}(x)h(x)+S_{[x_{1},x_{2}]}\hbox{$1\hskip-1.2pt\vrule
depth=0.0pt,height=6.58746pt,width=0.7pt\vrule
depth=0.0pt,height=0.3pt,width=1.19995pt$},\text{ where }$ (E.1)
$\displaystyle\beta_{[x_{1},x_{2}]}(x)=2\Theta(x-x_{1})\Theta(x_{2}-x){(x-x_{1})(x_{2}-x)\over
x_{2}-x_{1}}.$

Here $T(x)$ is the Hamiltonian density of the CFT, and the $\Theta(x)$ is the
Heaviside step function.

A certain linear combination of these locally-quadratic functions is
identically zero. Therefore the corresponding combination of modular
Hamiltonians is proportional to the identity. For any three contiguous
intervals $A,B$ and $C$,

$\displaystyle\mathcal{K}_{\Delta}$
$\displaystyle\equiv\mathscr{K}_{AB}+\mathscr{K}_{BC}-\eta(\mathscr{K}_{A}+\mathscr{K}_{C})-(1-\eta)(\mathscr{K}_{B}+\mathscr{K}_{ABC})={c_{\mathrm{tot}}\over
6}h(\eta)\hbox{$1\hskip-1.2pt\vrule
depth=0.0pt,height=6.58746pt,width=0.7pt\vrule
depth=0.0pt,height=0.3pt,width=1.19995pt$}$ (E.2)
$\displaystyle\equiv\eta\hat{\Delta}(A,B,C)+(1-\eta)\hat{I}(A:C|B)~{}.$ (E.3)

Here

$\hat{\Delta}(A,B,C)\equiv\mathscr{K}_{AB}+\mathscr{K}_{BC}-\mathscr{K}_{A}-\mathscr{K}_{C},~{}~{}~{}\hat{I}(A:C|B)\equiv\mathscr{K}_{AB}+\mathscr{K}_{BC}-\mathscr{K}_{B}-\mathscr{K}_{ABC}~{}.$
(E.4)

The function $h(\eta)$ on the right-hand side of Eq. (E.2) is determined by
taking expectation values to be the binary entropy function
$h(\eta)=-\eta\ln\eta-(1-\eta)\ln(1-\eta)$. In a lattice wavefunction (as
opposed to continuum field theory), the density matrix may have a kernel, and
so $\mathscr{K}$ may be singular; as a result, the operator form of this
equation is less robust than the following vector fixed-point equation:

$\mathcal{K}_{\Delta}\left|\Omega\right\rangle={c_{\mathrm{tot}}\over
6}h(\eta)\left|\Omega\right\rangle.$ (E.5)

###  E.2 Vector fixed-point equation for chiral edge

In this Section, we derive several nontrivial identities for the modular
Hamiltonians localized near the edge. Let us first recall the setup. We shall
consider a finite cylinder hosting a 2+1D reference state
$\left|\Psi\right\rangle$ with chiral gapless edges on both the top and the
bottom edge; see Figure 26. It can be overall viewed as a 1+1D non-chiral CFT
[Figure 10] by the _cylinder assumption_ : if we dimensionally reduce along
the direction normal to the boundaries, we obtain a 1+1D field theory with
conformal symmetry. Thus for the regions in Figure 10, we get

$\left(\eta\hat{\Delta}(A,B,C)+(1-\eta)\hat{I}(A:C|B)\right)\left|\Psi\right\rangle={c_{\mathrm{tot}}\over
6}h(\eta)\left|\Psi\right\rangle,$ (E.6)

where $\eta$ is the cross-ratio computed form the circle and $h(\eta)$ is the
binary entropy function as above. The total central charge is
$c_{\mathrm{tot}}=c_{L}+c_{R}$, with $c_{L},c_{R}$ being the total central
charge of the top and bottom edge. For purely chiral state
$c_{L}=c_{R}=|c_{-}|$, where $c_{-}$ is the chiral central charge of the
state.

Our goal is to derive Eq. (4.13), a version of the vector fixed-point equation
near the chiral edge. For the reader’s convenience, we restate this below. For
the regions shown in Figure 9, the fixed-point equation is

$\left(\eta\hat{\Delta}(AA^{\prime},B,CC^{\prime})+(1-\eta)\hat{I}(A:C|B)\right)\left|\Psi\right\rangle\propto\left|\Psi\right\rangle.~{}~{}$
(E.7)

The assumptions behind this derivation are the cylinder assumption [Section
4.3], axioms of the entanglement bootstrap in the bulk [Section 2], and the
full boundary axioms introduced in Section 4.3 [Figure 11].

See 4.2

###### Proof of 4.2.

We wish to show:

$\displaystyle{\mathcal{K}}_{\mathbb{D}}^{L}\left|\psi\right\rangle=\Gamma^{L}\left|\psi\right\rangle\quad\text{and}\quad{\mathcal{K}}_{\mathbb{D}}^{R}\left|\psi\right\rangle$
$\displaystyle=\Gamma^{R}\left|\psi\right\rangle,$ (E.8)

where, for $\alpha=L,R$,
${\mathcal{K}}_{\mathbb{D}}^{\alpha}\equiv\left(\eta\Delta(A_{\uparrow}^{\alpha},B^{\alpha},C_{\uparrow}^{\alpha})+(1-\eta)I(A^{\alpha}:C^{\alpha}|B^{\alpha})\right)$,
with $\eta$ is the geometric cross-ratio for the regions $abc$ on the
boundary, and $C_{\uparrow}^{L}\equiv C^{L}C^{\prime},A_{\uparrow}^{L}\equiv
A^{L}A^{\prime}$, $C_{\uparrow}^{R}\equiv
C^{R}C^{\prime},A_{\uparrow}^{R}\equiv A^{R}A^{\prime}$ [Figure 26].
Furthermore $\Gamma^{L}+\Gamma^{R}={c_{\mathrm{tot}}\over 6}h(\eta)$.

$A^{R}$$A^{M}$$A^{L}$$B^{R}$$B^{M}$$B^{L}$$C^{R}$$C^{M}$$C^{L}$$a$$b$$c$$A^{R}$$A^{\prime}$$A^{L}$$B^{R}$$B^{L}$$C^{R}$$C^{\prime}$$C^{L}$$a$$b$$c$
Figure 26: Two decompositions of a strip connecting two physical edges.

Here is the argument, in two steps. First, we can disconnect the two
boundaries in the following sense. We show that the LHS of the equation (E.7)
is

${\mathcal{K}}_{\mathbb{D}}\left|\Psi\right\rangle=({\mathcal{K}}_{\mathbb{D}}^{L}(\eta)+{\mathcal{K}}_{\mathbb{D}}^{R}(\eta))\left|\Psi\right\rangle$
(E.9)

where ${\mathcal{K}}_{\mathbb{D}}^{L/R}$ is made from modular Hamiltonians of
regions that only touch the $L/R$ boundary.

To show Eq. (E.9), we use the full-boundary A1 [Figure 11] to conclude that
$I(A^{L}:A^{R}|A^{M})_{\left|\Psi\right\rangle}=0$; see Figure 26 for the
description of these regions. Therefore,

$K_{A}\left|\Psi\right\rangle=(K_{A^{ML}}+K_{A^{MR}}-K_{A^{M}})\left|\Psi\right\rangle,$
(E.10)

and similarly for the other vertical strips. Therefore

$\displaystyle\hat{I}(A:C|B)\left|\psi\right\rangle$
$\displaystyle=\left(\hat{I}(A^{LM}:C^{LM}|B^{LM})+\hat{I}(A^{RM}:C^{RM}|B^{RM})\right.$
$\displaystyle\left.-\hat{I}(A^{M}:C^{M}|B^{M})\right)\left|\psi\right\rangle~{}.$
(E.11)

But bulk A1 says $\hat{I}(A^{M}:C^{M}|B^{M})\left|\psi\right\rangle=0$.
Therefore, using bulk A1 to deform the regions, we get

$\hat{I}(A:C|B)\left|\Psi\right\rangle=\left(\hat{I}(A^{L}:C^{L}|B^{L})+\hat{I}(A^{R}:C^{R}|B^{R})\right)\left|\Psi\right\rangle.$
(E.12)

$AB$$C$$\widetilde{AB}$$\widetilde{C}$ Figure 27: Because of bulk A1,
$(K_{AB}-K_{C})\left|\Psi\right\rangle=(K_{\widetilde{AB}}-K_{\widetilde{C}})\left|\Psi\right\rangle$.
Comparing this figure with Figure 26, we find $\widetilde{AB}=AB\setminus
B^{M}$ and $\widetilde{C}=C\cup B^{M}$.

Now we use bulk A1 to deform the regions as in Figure 27:

$\begin{split}\hat{\Delta}(A,B,C)\left|\Psi\right\rangle&=\left(K_{AB}-K_{C}\right)+\left(K_{BC}-K_{A}\right)\left|\Psi\right\rangle\\\
&=\left(K_{\widetilde{AB}}-K_{\widetilde{C}}\right)+\left(K_{BC}-K_{A}\right)\left|\Psi\right\rangle\end{split}~{}.$
(E.13)

Then using (E.10) to decompose each of these pieces vertically, we have

$\hat{\Delta}(A,B,C)\left|\Psi\right\rangle=\left(\hat{\Delta}(A_{\uparrow}^{L},B^{L},C^{L}_{\uparrow})+\hat{\Delta}(A_{\uparrow}^{R},B^{R},C^{R}_{\uparrow})\right)\left|\Psi\right\rangle.$
(E.14)

Combining (E.12) with (E.14) gives the sum of the criticality conditions for
the two boundaries:

$\left({\mathcal{K}}_{\mathbb{D}}^{L}+{\mathcal{K}}_{\mathbb{D}}^{R}\right)\left|\Psi\right\rangle={c_{\mathrm{tot}}\over
6}h(\eta)\left|\Psi\right\rangle.$ (E.15)

Note that (E.15) implies

$\left({\mathcal{K}}_{\mathbb{D}}^{L}+{\mathcal{K}}_{\mathbb{D}}^{R}\right)\rho^{LR}={c_{\mathrm{tot}}h(\eta)\over
6}\rho^{LR},$ (E.16)

where $\rho^{LR}$ is the reduced density matrix for $\psi$ reduced to the
union of the regions $L,R$ of support of ${\mathcal{K}}_{\mathbb{D}}^{L}$ and
${\mathcal{K}}_{\mathbb{D}}^{R}$ respectively.

Using full-boundary A0, we conclude that $I(L:R)=0$ and therefore
$\rho^{LR}=\rho^{L}\otimes\rho^{R}$. Since the state factorizes, the
contributions of ${\mathcal{K}}_{\mathbb{D}}^{L}$ and
${\mathcal{K}}_{\mathbb{D}}^{R}$ to (E.15) are independent. More precisely, we
can take the trace of (E.16) over $R$ to get

${\mathcal{K}}_{\mathbb{D}}^{L}\rho^{L}+{\rm
tr}_{R}({\mathcal{K}}_{\mathbb{D}}^{R}\rho^{R})\rho^{L}={c_{\mathrm{tot}}h(\eta)\over
6}\rho^{L},$ (E.17)

which says

${\mathcal{K}}_{\mathbb{D}}^{L}\rho^{L}\propto\rho^{L}.$ (E.18)

Therefore, each of them acting on $\left|\Psi\right\rangle$ gives a c-number,
which is denoted as $\Gamma^{L},\Gamma^{R}$ respectively. One can purify the
equation above to obtain

${\mathcal{K}}_{\mathbb{D}}^{L}\left|\Psi\right\rangle=\Gamma^{L}\left|\Psi\right\rangle.$
(E.19)

The one for $R$ can be obtained similarly. The two c-numbers add up to

$\Gamma^{L}+\Gamma^{R}={c_{\mathrm{tot}}\over 6}h(\eta).$ (E.20)

∎

If we assume the two edges contribute equally to the proportionality factor,
then we obtain

$\Gamma^{L}={c_{L}\over 6}h(\eta),\quad\Gamma^{R}={c_{R}\over 6}h(\eta)$
(E.21)

where $c_{L}=c_{R}$ and $c_{\mathrm{tot}}=c_{L}+c_{R}$. $c_{R},c_{L}$ can be
interpreted as the holomorphic and anti-holomoarphic central charge of the
1+1D CFT.

We note that the assumptions in Proposition 4.2 (i.e., cylinder assumption,
bulk A1, full-boundary A0, and full-boundary A1), together with the extra
assumption that the bulk is purely chiral, leads to the following equation:

$\left(\hat{\Delta}(A_{\uparrow},B,C_{\uparrow})-\Delta(A_{\uparrow},B,C_{\uparrow})\right)\left|\Psi\right\rangle\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int
dxf_{A,B,C}(x){\mathfrak{h}}(x)\left|\Omega\right\rangle$ (E.22)

with $A_{\uparrow}\equiv AA^{\prime},C_{\uparrow}\equiv CC^{\prime}$ as in the
figure in (E.7). Here ${\mathfrak{h}}(x)$ is the Hamiltonian density for the
edge CFT and $f_{A,B,C}(x)$ is the $C^{1}$ bump function implied by Hypothesis
1. This is an equation that we use throughout the body of results that follow
from Hypothesis 1. Thus, our argument below can be yet another justification
for these derivations that do not rely on Hypothesis 1.

The argument for Eq. (E.22) follows a similar chain of reasoning as in
Proposition 4.2. Consider again the regions $ABC$ in Fig. 10. First, we have
(E.14). On the other hand, the cylinder assumption implies that

$\left(\hat{\Delta}(A,B,C)-\Delta(A,B,C)\right)\left|\Psi\right\rangle\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int
dxf_{A,B,C}(x)\left({\mathfrak{h}}_{L}(x)+{\mathfrak{h}}_{R}(x)\right)\left|\Omega\right\rangle$
(E.23)

where $f_{A,B,C}(x)$ is the $C^{1}$ bump function implied by Hypothesis 1, and
${\mathfrak{h}}_{L}$ and ${\mathfrak{h}}_{R}$ are the right-moving and left-
moving parts of the CFT Hamiltonian. In the case where the CFT is purely
chiral, these contributions are localized at the two ends of the cylinder.
That is, full-boundary A0 implies that the two terms on the RHS of (E.14) are
independent, and therefore

$\displaystyle\left(\hat{\Delta}(A_{\uparrow}^{L},B^{L},C_{\uparrow}^{L})-\Delta(A_{\uparrow}^{L},B^{L},C_{\uparrow}^{L})\right)\left|\Psi\right\rangle$
$\displaystyle\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int
dxf_{A,B,C}(x){\mathfrak{h}}_{L}(x)\left|\Omega\right\rangle,~{}~{}~{}$
$\displaystyle\left(\hat{\Delta}(A_{\uparrow}^{R},B^{R},C_{\uparrow}^{R})-\Delta(A_{\uparrow}^{R},B^{R},C_{\uparrow}^{R})\right)\left|\Psi\right\rangle$
$\displaystyle\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int
dxf_{A,B,C}(x){\mathfrak{h}}_{R}(x)\left|\Omega\right\rangle.$ (E.24)

##  Appendix F Fidelity near the edge under good modular flow

This appendix explains the relation
$F(\rho_{\Omega}(0),\rho_{\Omega}(t))=F(\rho_{\text{edge}}(0),\rho_{\text{edge}}(t))$
based on the full-boundary A0 condition. This result is used in Section 5.2.
Consider a good modular flow generated by $\mathbb{L}$, intersecting with only
one gapless edge272727If a good modular flow generator covers more than one
edge, then one can first decompose the generator and then deform it to all the
edges, similar as we did in the cylinder argument in E.. We partition the
entire system $\Omega$ into $ABC$ shown in Figure 28. We let $C$ be the
annulus that covers the particular gapless edge on which $\mathbb{L}$ has
nontrivial action, and $B$ be another annulus that surrounds $C$ and separates
$C$ from the remaining region $A$282828We do not make any further assumption
about the topology of $A$. Without loss of generality, one can think the
support of $\mathbb{L}$ is within $C$, as its support can be deformed in the
bulk such that it’s localized near the edge [Section 3]. We claim that

$F(\rho_{ABC}(0),\rho_{ABC}(t))=F(\rho_{BC}(0),\rho_{BC}(t)).$ (F.1)

$C$$B$$A$ Figure 28: The partition used in the proof of Eq. (F.1). Here we
don’t assume the topology of $A$, which could have holes that are not shown.

This is plausible because the good modular flow does not change the state in
the bulk $AB$.

The proof of Eq. (F.1) follows from a well-known fact: that fidelity between
any two states $\rho$ and $\sigma$ is non-increasing under any quantum channel
${\mathcal{E}}$, i.e., $F(\rho,\sigma)\leq
F({\mathcal{E}}(\rho),{\mathcal{E}}(\sigma))$. First, note that
$F(\rho_{ABC}(0),\rho_{ABC}(t))\leq F(\rho_{BC}(0),\rho_{BC}(t))$ by choosing
the quantum channel to be the partial trace over $A$. On the other hand, we
have $F(\rho_{BC}(0),\rho_{BC}(t))\leq F(\rho_{ABC}(0),\rho_{ABC}(t))$. The
reason is as follows: Firstly, due to full-boundary A0,

$0\leq
I(A:C|B)_{\left|\Psi\right\rangle}\leq\Delta(B,C,\emptyset)_{\left|\Psi\right\rangle}=0\quad\Rightarrow\quad
I(A:C|B)_{\left|\Psi\right\rangle}=0,$ (F.2)

where the first two “$\leq$” signs are due to strong subadditivity [36].
Secondly, under the good modular flow
$\left|\Psi(t)\right\rangle=e^{i\mathbb{L}t}\left|\Psi\right\rangle$ still
satisfies the Markov condition $I(A:C|B)_{\left|\Psi(t)\right\rangle}$. To see
this, notice $\left|\Psi(t)\right\rangle$ still satisfies the full-boundary
A0:

$\Delta(B,C,\emptyset)_{\left|\Psi(t)\right\rangle}=(S_{A}+S_{AB}-S_{B})_{\left|\Psi(t)\right\rangle}=(S_{A}+S_{AB}-S_{B})_{\left|\Psi\right\rangle}=0,$
(F.3)

where the first equal sign is due to the purity of
$\left|\Psi(t)\right\rangle$ and the second equal sign is due to the support
of $\mathbb{L}$ is within $C$ and as a result $\rho_{AB}(0)=\rho_{AB}(t)$. As
a result of Markov structure of $\rho_{ABC}(0)$ and $\rho_{ABC}(t)$, one can
use Petz recovery map ${\mathcal{E}}_{B\to AB}^{\rho_{AB}(0)}$292929The same
channel for recovery because $\rho_{AB}(0)=\rho_{AB}(t)$. [73] to recover both
$\rho_{ABC}(t)$ from $\rho_{BC}(t)$ for any $t\in\mathbb{R}$. Therefore, we
can conclude $F(\rho_{ABC}(0),\rho_{ABC}(t))=F(\rho_{BC}(0),\rho_{BC}(t))$.

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