# Chiral Virasoro algebra from a single wavefunction

Isaac H. Kim1, Xiang Li2, Ting-Chun Lin2, John McGreevy2, Bowen Shi2

###### Abstract

Chiral edges of 2+1D systems can have very robust emergent conformal symmetry.
When the edge is purely chiral, the Hilbert space of low-energy edge
excitations can form a representation of a single Virasoro algebra. We propose
a method to systematically extract the generators of the Virasoro algebra from
a single ground state wavefunction, using entanglement bootstrap and an input
from the edge conformal field theory. We corroborate our construction by
numerically verifying the commutation relations of the generators. We also
study the unitary flows generated by these operators, whose properties (such
as energy and state overlap) are shown numerically to agree with our
analytical predictions.

1Department of Computer Science, University of California, Davis, CA 95616,
USA
2Department of Physics, University of California San Diego, La Jolla, CA
92093, USA

###### Contents

  1. 1 Introduction
  2. 2 Primer on entanglement bootstrap
  3. 3 Good modular flows
    1. 3.1 Definition and properties of good modular flows
    2. 3.2 Fidelity tests
  4. 4 Good modular flow generators and CFT
    1. 4.1 CFT Perspective
    2. 4.2 Vector fixed-point equation
    3. 4.3 Analytical evidence for Hypothesis 1
  5. 5 Coherent states
    1. 5.1 Virasoro algebra and coherent states
    2. 5.2 State overlap
    3. 5.3 Energy of the coherent states
  6. 6 Construction of approximate Virasoro generators
    1. 6.1 Explicit construction of $\widetilde{L}_{n}$
    2. 6.2 Quantitative properties
  7. 7 Numerical tests
    1. 7.1 Tests in $p+{\bf i}p$ superconductor
    2. 7.2 Tests in chiral semion wavefunction
  8. 8 Discussion
    1. 8.1 Summary
    2. 8.2 Further remarks
    3. 8.3 Future directions
    4. 8.4 Preview of results about irregular edges
  9. A Notation
  10. B Smoothness and decay of Fourier coefficients
  11. C Improvement of the approximate Virasoro generators
    1. C.1 Improvement procedure
    2. C.2 Numerical tests of the improvement
  12. D Semion Setup
  13. E The cylinder argument
    1. E.1 Review of 1+1D CFT fixed-point equation
    2. E.2 Vector fixed-point equation for chiral edge
  14. F Fidelity near the edge under good modular flow

##  1 Introduction

A remarkable fact about many-body quantum systems is that a variety of simple
and elegant mathematical structures can emerge as a description of their low-
energy physics. For instance, the low-energy physics of gapped quantum many-
body systems — aside from the notable example of fracton phases [1, 2, 3, 4] —
are expected to be described by topological quantum field theory (TQFT) [5],
which successfully predicts many universal features of the underlying system,
such as ground state degeneracy [6], low-energy excitations [7], and
entanglement properties [8, 9].

Over the past few decades, it was realized that fingerprints of these
mathematical structures often manifest themselves in the entanglement
properties of ground state wavefunctions. Early studies discovered that the
entanglement entropy often encodes universal information about the underlying
quantum phases, both in gapped [8, 9] and gapless systems [10, 11].

More recently, new progress is being made under the moniker of “entanglement
bootstrap,” which aims to recover the universal properties of the underlying
phase of matter from ground state entanglement. For instance, the list of
anyon types, their fusion rules [12] and braiding data [13] (including the
data associated with gapped interfaces [14]) can be extracted from the ground
state wavefunction; similar statements apply in general dimensions [15, 16].
Moreover, a new entanglement measure known as the modular commutator was used
to extract the chiral central charge of the edge theory [17]; in the presence
of a $\textsf{U}(1)$ symmetry, a similar formula exists for the Hall
conductivity [18]. These results are strongly suggestive of the basic dogma
that underlies all these studies: that all of the universal information about
a state of matter is encoded in the local reduced density matrix of a
representative wavefunction.

In fact, there is nontrivial evidence for this dogma even in more general
physical contexts. As a basic example, equal-time correlators of local
operators in gapped ground states decay exponentially [19], whereas in gapless
ground states they do not. In conformal field theory (CFT) ground states,
renormalization group monotones (called $c,a,$ or $F$ depending on the
dimension) can be extracted from the entanglement entropy of a round ball [20,
10, 21].

More ambitiously, one may posit that structural features, that is,
mathematical relationships between these quantities, can emerge from the
universal entanglement properties of the ground state wavefunctions. In
systems with liquid topological order, there is already strong evidence that
supports this expectation. For instance, in two [12] and higher [15, 16]
dimensions, various identities relating superselection sectors, their quantum
dimensions, and fusion multiplicities emerge from simple universal properties
of gapped ground states. In the presence of a gapped boundary or domain wall,
similar progress can be made [14]. In fact, in the context of three space
dimensions, fusion spaces associated with knotted excitations (and partons for
domain wall excitations) are identified, making the existing puzzle of “what
is the minimal set of data?” more pronounced in such physical contexts.

The focus of this paper is a simple but nontrivial physical setup that differs
from these previous studies, namely the chiral gapless edge of a 2+1D system.
From the field theory perspective, the key mathematical structure that should
be present in this system is the Virasoro algebra:

$[\mathscr{L}_{m},\mathscr{L}_{n}]=(m-n)\mathscr{L}_{m+n}+{c\over
12}(m^{3}-m)\delta_{m+n,0},$ (1.1)

where $c$ is the central charge and $\\{\mathscr{L}_{n}\\}_{n\in\mathbb{Z}}$
is the set of Virasoro generators. The Virasoro algebra and its unitary
representations appear in 1+1D CFT, 1+1D critical lattice models111See [22]
for a concrete example of how Virasoro algebra could appear in critical
lattice models., such as the critical transverse field Ising model, and the
edge theories of 2+1D topological orders, such as the $\nu=2/3$ quantum Hall
state [23] and critical points describing phase transitions between two gapped
edges [24]. In general, these theories contain both the left and right moving
parts which are the representations of two separate copies of Virasoro
algebra. However, in a 2+1D system with a _purely chiral_ edge, such as the
edge of $\nu=1/3$ Laughlin state [25] or the most natural ($c=8$) edge of the
$E_{8}$ bosonic invertible phase [26], there is only one copy of the Virasoro
algebra, simplifying the analysis.

Compared to the case of liquid gapped ground states, for gapless systems,
precisely how these mathematical objects (such as the Virasoro algebra) emerge
from entanglement is much more wide open. Before diving into this problem
directly, we need to do exploratory work and test various conjectures. This is
the main aim of this paper.

In this paper, we identify generators of such a Virasoro algebra in terms of
linear combinations of modular Hamiltonians (entanglement Hamiltonian) of a
purely chiral ground state supported near the edge. These form a part of a
larger class of modular flows called good modular flows that can create
excitations at the edge whilst preserving the bulk local density matrices. (By
‘bulk local density matrices’ here we mean reduced density matrices of regions
that do not touch the gapless edge. Throughout the paper, we will refer to
such regions as ‘bulk regions’.) We provide analytical and numerical arguments
that support our claim in Section 3. The arguments presented in this Section
are independent of any CFT assumption and heavily rely on the techniques from
entanglement bootstrap [12].

In Section 4, we put forward a hypothesis [Hypothesis 1] that relates the
generators of certain subsets of the good modular flows to quantities in CFT.
This hypothesis will be the foundation upon which the results of the ensuing
sections rely. While proving this hypothesis is beyond the scope of this
paper, we provide several pieces of nontrivial evidence in support of it.
First, we provide several locally checkable consequences of the hypothesis in
Section 4.2, including a generalization of the recently-discovered vector
fixed point equation of [27]. We verify these numerically in Section 7.
Secondly, we prove a special case of this conjecture under a reasonable
physical assumption: that a 2+1D gapped ground state with a chiral gapless
edge on a strip can be formally viewed as a 1+1D CFT ground state under
dimensional reduction [Section 4.3].

In Section 5, we define a carrier space for our representation of the chiral
Virasoro algebra from purely chiral states on a two-dimensional disk. It is
swept out by the generators of good modular flow.

We then construct the Virasoro generators modulo our hypothesis and explore
their properties in Section 6 and 7. In Section 6, we devise a systematic
method to extract the Virasoro generators, which are shown to be a special
class of good modular flow generators. We numerically verify their commutation
relations in Section 7 using $p+ip$ superconductor and chiral semion lattice
ground states, which both show good agreement with the predictions.

We remark that there is a large literature on the construction of Virasoro
algebras in 1+1D systems, starting from [22] and more recently in [28, 29, 30,
31]. Here, we make some comparisons: Firstly, the idea of these works is to
perform an exact Fourier transformation of the Hamiltonian density. The
starting point of their construction is very different from ours: [22] starts
from the partition function, [28] directly starts from the local Hamiltonian
and [30, 31] starts from a tensor network representation of the partition
function. So all of these begin with a time evolution operator, either
transfer matrix or Hamiltonian. Our construction, instead, is directly from a
single quantum state. Taking advantage of the form of a certain linear
combination of modular Hamiltonians near the edge, designed in such a way that
it acts nontrivially only on the edge, we can perform an approximate222We also
design a systematic process to make the approximation converge to the exact
Fourier transformation. Fourier transformation to the stress-energy tensor,
even without the construction of the Hamiltonian density beforehand. Secondly,
previous work focuses on 1+1D systems, which involve two copies of Virasoro
algebra, while our construction is for 2+1D purely chiral systems, where there
is only one copy of Virasoro algebra. Therefore, there is no need to separate
the two copies of Virasoro algebra as in the 1+1D construction.

##  2 Primer on entanglement bootstrap

In this Section, we provide a brief review of the basics of entanglement
bootstrap [12]. In the absence of edges, there are two axioms (A0 and A1) that
constitute the core of entanglement bootstrap. These axioms pertain to the
entanglement property of the global state, often referred to as the _reference
state_. These are described in Figure 1. Here we use the convention that the
subscript $\sigma$ appearing after the parenthesis represents the underlying
global state used for the evaluation of the entanglement entropies; for
instance, in expressions like $(S_{AB}+\ldots)_{\sigma}$, one can simply
replace $S_{AB}$ with $S(\sigma_{AB})$. While the only assumptions we make are
that A0 and A1 hold true on every constant-sized ball, as a logical
consequence, one can show that the analogous set of assumptions holds true at
larger scales [12]. Therefore, without loss of generality, we can assume that
these axioms hold on any disks, independent of their size.

$C$$B$A0: $\Delta(B,C)_{\sigma}=0$$B$$C$$D$A1: $\Delta(B,C,D)_{\sigma}=0$
Figure 1: Axioms of the entanglement bootstrap. Here $\sigma$ is the global
state, often referred to as the _reference state_. Here
$\Delta(B,C)_{\sigma}:=S(\sigma_{BC})+S(\sigma_{C})-S(\sigma_{B})$ and
$\Delta(B,C,D)_{\sigma}:=S(\sigma_{BC})+S(\sigma_{CD})-S(\sigma_{B})-S(\sigma_{D})$.

The axioms A0 and A1 are intimately related to quantum information-theoretic
measures of correlations, such as mutual information and conditional mutual
information. Mutual information with respect to a state $\rho$ is defined as

$I(A:B)_{\rho}:=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}).$ (2.1)

Conditional mutual information is a generalization of mutual information,
defined as

$I(A:C|B)_{\rho}:=S(\rho_{AB})+S(\rho_{BC})-S(\rho_{B})-S(\rho_{ABC}).$ (2.2)

If the global state is pure, one can verify that A0 and A1 reduce to
$I(A:C)_{\sigma}=0$ (choosing $A$ as a purifying system) and
$I(A:C|B)_{\sigma}=0$ (choosing $A$ as a purifying system). Therefore, A0 and
A1 imply that certain mutual information and conditional mutual information
are zero.

Intuitively, the fact that the mutual and conditional mutual information are
zero can be understood as follows. Because mutual information is a measure of
the correlation between two subsystems, in gapped systems we can expect
$I(A:C)$ to approach $0$ as the distance between $A$ and $C$ diverges.
Therefore, A0 can be thought of as a condition that we expect to hold at a
long-distance scale, in particular, at the fixed point of some coarse-graining
procedure. We can arrive at the same conclusion for A1, but using a different
reasoning. At long distance, the entanglement entropies of the subsystems are
expected to be approximated well by a leading ultraviolet-divergent term,
proportional to the area, and a subleading universal term, known as the
topological entanglement entropy [8, 9]. It is an easy exercise to show that
this scaling law implies A1. Therefore, A0 and A1 are the conditions that we
can expect to hold in gapped systems at long distances. 333Note that for
chiral systems with finite-dimensional Hilbert spaces, we expect the axioms to
only hold approximately.

In the context of this paper, the main power of A0 and A1 comes from the fact
that they let us decompose the _modular Hamiltonian_ [32, 33] (also known as
the entanglement Hamiltonian, defined as $-\ln\rho_{A}$ for the subsystem $A$)
of any region into terms that act locally within that region. For any quantum
state, the following relations are known:

$\displaystyle I(A:B)_{\rho}=0$
$\displaystyle\iff\ln\rho_{AB}=\ln\rho_{A}+\ln\rho_{B},$ (2.3) $\displaystyle
I(A:C|B)_{\rho}=0$
$\displaystyle\iff\ln\rho_{ABC}=\ln\rho_{AB}+\ln\rho_{BC}-\ln\rho_{B}.$

Therefore, by iteratively applying these decompositions, one can often
decompose the modular Hamiltonian of a region to a linear combination of
modular Hamiltonians acting on smaller regions. In the presence of edges, one
should not expect these axioms to hold. However, one can still assume that the
axioms hold at any location that is sufficiently far away from the edge; we
numerically verified this fact, the result of which is presented in Section
3.2. The fact that the axioms are still satisfied in the bulk comes into play
in our analysis, letting us decompose the linear combination of modular
Hamiltonians over regions that are anchored on the edge. However, the
remaining piece of the puzzle is what kind of entanglement-based assumptions
one ought to make at the edge. This is what we discuss and explore in the rest
of the paper.

##  3 Good modular flows

In this Section, we introduce and study a special class of unitary
flows444i.e. flows generated by unitary operators, referred to as _good
modular flows_. A good modular flow creates only edge excitations, and this is
the intuitive reason that their study can be related to an (anomalous) 1+1D
system. The central hypothesis of this paper [Hypothesis 1] is that certain
generators of the good modular flows can be identified as the Virasoro
generators, in the sense we make precise in Section 6.

###  3.1 Definition and properties of good modular flows

In order to motivate the definition of good modular flows, it will be
instructive to consider a general linear combination of modular Hamiltonians:

$\mathbb{L}=\sum_{A}\lambda_{A}K_{A},\quad\lambda_{A}\in\mathbb{R},$ (3.1)

where $K_{A}=-\ln\rho_{A}$ is the modular Hamiltonian of the subsystem $A$,
$\rho_{A}=\text{Tr}_{\bar{A}}\left({|\Psi\rangle}{\langle\Psi|}\right)$ is the
reduced density matrix over $A$, and $\lambda_{A}$ is a real number. Upon
evolving the global state by a unitary flow $e^{-{\bf
i}\mathbb{L}t},t\in\mathbb{R}$, (such that $|\Psi(t)\rangle=e^{-{\bf
i}\mathbb{L}t}|\Psi\rangle$), we can ask how the reduced density matrices of
different subsystems change.

In general, such a flow can change the reduced density matrices in the bulk.
For instance, consider a flow generated by $\mathbb{L}=K_{BCD}$ for the
subsystems shown in Figure 2. For small $\delta t$, $e^{-{\bf i}K_{BCD}\delta
t}$ changes the entanglement entropy of the state on $AB$ by an amount
${\pi\over 3}c_{-}\delta t$, where $c_{-}$ is the chiral central charge [17,
34], which is generally nonzero. Therefore, this particular flow can change
reduced density matrices in general. Indeed, for a chiral state555Throughout
this paper we use “chiral” to mean $c_{-}\neq 0$ and “purely chiral” to mean
$|c_{-}|=c_{\text{tot}}$, where $c_{\text{tot}}\equiv c_{L}+c_{R}$ is the
total central charge., by which we mean $c_{-}\neq 0$, the density matrix on
$AB$ must change in the first order of $\delta t$.

$A$$B$$C$$D$ Figure 2: A modular flow generated by $K_{BCD}$ changes the
density matrix on $AB$. The outermost boundary in this diagram is the physical
edge.

However, unlike these more general modular flows, good modular flows are
constructed in such a way that they do not change the bulk reduced density
matrices. Here is a simple rule that we advocate.

######  Definition 3.1 (Good modular flows).

Let $\mathbb{L}=\sum_{A}\lambda_{A}K_{A}$ (with $\lambda_{A}\in\mathbb{R}$) be
a modular flow generator. This generator is _good_ if for any cut segment $e$,

$\sum_{e\subset\partial A}\lambda_{A}=0,$ (3.2)

where $e\subset\partial A$ means $e$ is part of the entanglement boundary of
$A$666By entanglement boundary $\partial A$ (sometimes called entanglement
cut) we mean the boundary of $A$ inside the bulk.. Throughout this paper, we
denote the class of good modular flow generators as $\mathfrak{g}$. We call
the modular flow $e^{-{\bf i}\mathbb{L}t}$ _good_ if
$\mathbb{L}\in\mathfrak{g}$.

Any linear combination of modular Hamiltonians that does not satisfy Eq. (3.2)
shall be referred to as a _bad_ modular flow generator. See Table 1 for
examples of good and bad modular flow generators.

Modular flow generators and their types (Good/Bad)
---
$\includegraphics[width=85.35826pt,valign={c},margin={0.2cm}]{fig-vir-
submit/flow-gen1.pdf}\;\;\begin{aligned} &K_{ABC}+K_{B}+K_{AC}&&\text{Bad}\\\
&K_{AB}+K_{BC}+K_{A}+K_{C}&&\text{Bad}\\\
&\hat{\Delta}(AC,B,\emptyset)=K_{ABC}+K_{B}-K_{AC}&&\text{Good}\\\
&\hat{\Delta}(A,B,C)=K_{AB}+K_{BC}-K_{A}-K_{C}&&\text{Good}\end{aligned}$
$\includegraphics[width=85.35826pt,valign={c},margin={0.2cm}]{fig-vir-
submit/flow-gen2.pdf}\;\;\begin{aligned}
&\hat{I}(A:C|B)=K_{AB}+K_{BC}-K_{ABC}-K_{B}&&\text{Bad}\\\
&K_{AB}+K_{BCD}+K_{CDA}-K_{A}-K_{B}-K_{CD}-K_{ABCD}&&\text{Good}\end{aligned}$
$\includegraphics[width=85.35826pt,valign={c},margin={0.2cm}]{fig-vir-
submit/flow-gen3.pdf}\;\;\begin{aligned} &K_{A}+K_{C}+K_{AC}&&\text{Bad}\\\
&K_{AB}+K_{BC}+K_{ABC}+K_{B}&&\text{Bad}\\\
&\hat{I}(A:C)=K_{A}+K_{C}-K_{AC}&&\text{Good}\\\
&\hat{I}(A:C|B)=K_{AB}+K_{BC}-K_{ABC}-K_{B}&&\text{Good}\end{aligned}$
$\includegraphics[width=85.35826pt,valign={c},margin={0.2cm}]{fig-vir-
submit/flow-gen4.pdf}\;\;\begin{aligned}
&K_{AA^{\prime}B}+K_{BCC^{\prime}}+K_{AA^{\prime}}+K_{CC^{\prime}}&&\text{Bad}\\\
&K_{AB}+K_{BC}+K_{ABC}+K_{B}&&\text{Bad}\\\
&\hat{\Delta}(AA^{\prime},B,CC^{\prime})=K_{AA^{\prime}B}+K_{BCC^{\prime}}-K_{AA^{\prime}}-K_{CC^{\prime}}&&\text{Good}\\\
&\hat{I}(A:C|B)=K_{AB}+K_{BC}-K_{ABC}-K_{B}&&\text{Good}\end{aligned}$
$\includegraphics[width=170.71652pt,valign={c},margin={0.2cm}]{fig-vir-
submit/flow-gen5.pdf}\quad\begin{aligned}
&L=K_{A}-K_{B}+K_{C}-K_{D}&&\text{Bad}\\\
&L^{\prime}=K_{A^{\prime}}-K_{B^{\prime}}+K_{C^{\prime}}-K_{D^{\prime}}&&\text{Bad}\\\
&L+L^{\prime}&&\text{Good}\end{aligned}$
Table 1: Examples of good and bad modular flow generators. The setups are
shown in the figure on the left. Each line of the equation is a modular flow
generator, with its type (good or bad) marked at the end. The two figures in
the last row are two ways to partition the same disk, where the colored dots
are identified.

While we do not have a rigorous proof for the invariance of the bulk reduced
density matrices under good modular flows, we provide an argument that
supports the validity of this claim. As a starter, we prove this claim for an
infinitesimally small flow. Let $\mathbb{L}$ be a good modular flow generator.
We compute the reduced density matrices in the bulk under a flow generated by
$\exp(-{\bf i}\mathbb{L}\delta t)$, for an infinitesimally small $\delta t$.
We show that the linear-order contribution in $\delta t$ vanishes.

Without loss of generality, consider a local bulk region $S$, by which we mean
$S$ is a subset of a bulk disk. Under the good modular flow generated by
$\mathbb{L}$, the first order contribution can be computed as

$\rho_{S}(\delta t)=\rho_{S}+{\bf i}\delta
t\text{Tr}_{\bar{S}}\left([{|\Psi\rangle}{\langle\Psi|},\mathbb{L}]\right).$
(3.3)

The generator $\mathbb{L}$ is a linear combination of modular Hamiltonians.
Our goal is to show that the linear order contribution vanishes,
$\text{Tr}_{\bar{S}}\left([{|\Psi\rangle}{\langle\Psi|},\mathbb{L}]\right)=0$.

By the cyclicity of the trace, for any modular Hamiltonian associated with a
region included in $\bar{S}$, its contribution is identically zero. Similarly,
by noting the identity $K_{X}|\Psi\rangle=K_{\overline{X}}|\Psi\rangle$ for
any subsystem $X$, any contribution coming from the region that includes $S$
is also identically zero. Therefore, for computing the linear-order
contribution, it suffices to compute the contributions from the modular
Hamiltonians associated with regions that have nontrivial intersections with
both $S$ and $\overline{S}$. Put another way, only the modular Hamiltonians
whose entanglement cut passes through $S$ contribute.

To show that these contributions vanish, we will deform the cuts away from the
local bulk region $S$. In particular, we will show that

$\mathbb{L}|\Psi\rangle=\mathbb{L}^{\prime}|\Psi\rangle,$ (3.4)

where $\mathbb{L}^{\prime}$ is a good modular flow generator whose
entanglement cut does not pass through $S$. This implies

$\displaystyle{{\rm
Tr}}_{\overline{S}}\left([{|\Psi\rangle}{\langle\Psi|},\mathbb{L}]\right)={{\rm
Tr}}_{\overline{S}}\left([{|\Psi\rangle}{\langle\Psi|},\mathbb{L}^{\prime}]\right)=0,$
(3.5)

where the second equality applies the argument from the previous paragraph.

Therefore, if we were to show that the entanglement cuts present in the
definition of $\mathbb{L}$ can be deformed away from the local bulk region
$S$, we would be done. More precisely, if we can show that

$\mathbb{L}|\Psi\rangle=\mathbb{L}^{\prime}|\Psi\rangle,$ (3.6)

where $\mathbb{L}^{\prime}$ is a good modular flow generator whose
entanglement cut does not pass through $S$, we have

$\displaystyle{{\rm
Tr}}_{\overline{S}}\left([{|\Psi\rangle}{\langle\Psi|},\mathbb{L}]\right)$
$\displaystyle={{\rm
Tr}}_{\overline{S}}\left([{|\Psi\rangle}{\langle\Psi|},\mathbb{L}^{\prime}]\right)$
(3.7) $\displaystyle=0.$

To prove the deformability of the entanglement cut [Eq. (3.6)], let us
consider a special toy example, which will be an instructive exercise that
ultimately leads to the proof of this claim. Consider the bipartition of the
entire system into two disks, $A$ and $B$ [Figure 3]. Our claim is that the
action of $K_{A}-K_{B}$ on the reference state is invariant under the
deformation of the entanglement cut [Eq. (3.6)]. Upon deforming $A$ to $A\cup
c$ and $B$ to $B\setminus c$, the change in the good modular flow generator is
$K_{A\cup c}-K_{B\setminus c}-K_{A}+K_{B}$. It is known that this quantity is
zero if and only if $S(\rho_{A\cup c})+S(\rho_{B})-S(\rho_{B\setminus
c})-S(\rho_{A})=0$ [35]. Because of strong subadditivity (SSA) [36],

$S(\rho_{A\cup c})+S(\rho_{c})-S(\rho_{B\setminus c})-S(\rho_{A})\leq
S(\rho_{a\cup c})+S(\rho_{c})-S(\rho_{b\setminus c})-S(\rho_{a}).$ (3.8)

The right hand side of Eq. (3.8) is zero by A1 [Section 2]. On the other hand,
the left-hand side of Eq. (3.8) is at least zero by the weak monotonocity
[36]. Therefore, $S(\rho_{A\cup c})+S(\rho_{c})-S(\rho_{B\setminus
c})-S(\rho_{A})=0$, yielding

$(K_{A\cup c}-K_{B\setminus c}-K_{A}+K_{B})|\Psi\rangle=0,$ (3.9)

where $|\Psi\rangle$ is the global state.

$A$$B$$A\cup c$$B\setminus c$ Figure 3: Deformation of the subsystem $A$ and
$B$ (left) by a half-disk $c$ (right).

This argument can be straightforwardly extended to a more general setup in
which $A$ and $B$ do not necessarily partition the entire system. As long as
the subsystem $c$ is a disk in the bulk lying in a small neighborhood of an
entanglement cut shared by $A$ and $B$, our argument works exactly the same
way. Furthermore, we remark that $A$ and $B$ need not lie on opposite sides of
the entanglement cut. A similar argument applies to the case in which the
subsystems $A$ and $B$ lie on the same side of the cut. In this case, while
the overall structure of the argument is similar, strong subadditivity plays
the role of weak monotonicity.

Now we are in a position to prove our main claim: that the action of good
modular flow generators on the reference state are invariant under
deformations of the entanglement cut. Without loss of generality, consider a
modular flow generator $\mathbb{L}$ and a small line segment $e$ along an
entanglement cut. We shall show that one can always deform the entanglement
cut $e$ in a way similar to Figure 3. Let $K_{A_{i}},i=1,\cdots,n$ be the
modular Hamiltonians in $\mathbb{L}$ whose entanglement boundaries contain
$e$, i.e. $e\subset\partial A_{i}$, and let $\lambda_{i}$ be the coefficient
of $K_{A_{i}}$ in $\mathbb{L}$. The idea behind this proof is simply to “pair
up” these modular Hamiltonians in $\mathbb{L}$ as

$\begin{split}\sum_{i=1}^{n}\lambda_{A_{i}}K_{A_{i}}&=\sum_{i=1}^{n}\lambda_{A_{i}}(K_{A_{i}}-K_{A_{1}})+\left(\sum_{i=1}^{n}\lambda_{A_{i}}\right)K_{A_{1}}\\\
&=\sum_{i=1}^{n}\lambda_{A_{i}}(K_{A_{i}}-K_{A_{1}}),\end{split}$ (3.10)

where the first line is simply a rearrangement of the terms in the left hand
side and the second line follows because $\sum_{i=1}^{n}\lambda_{A_{i}}=0$ by
the definition of the good modular flow generators [Definition 6]. Now we can
see that the action of $\mathbb{L}$ on $e$ is simply the action of a sum of
$n$ pairs $K_{A_{i}}-K_{A_{1}}$. Notice each pair is just the simple case we
described in the previous paragraph (or the version with both regions on the
same side of the cut). Therefore we can deform the entanglement cut $e$ for
each pair as in Figure 3, without changing the action on the reference state.
Thus, the action of a good modular flow generator on the reference state is
invariant under deformations near any segment of an entanglement cut.
Repeatedly applying this argument, we conclude that the action of a good
modular flow generator remains invariant under any smooth deformation of the
entanglement cuts.

Therefore, we have shown that for any good modular flow generator, its
entanglement cuts can be deformed [Eq. (3.6)]. In particular, for any local
bulk region, we can deform the generator so that none of its entanglement cuts
pass through it. Therefore, for an infinitesimally small flow, the reduced
density matrix of any local bulk region777An example of what we are excluding
is a cylinder, where the modular flow generators involve regions touching both
boundaries, and the bulk region in question wraps around the cylinder. remains
invariant [Eq. (3.7)]. We will comment elsewhere on the effects of good
modular flow on density matrices of more general regions.

$A$$B$$A^{\prime}$$B^{\prime}$ Figure 4: Here we consider a simple example of
a good modular flow generated by $K_{A}-K_{B}$, where the relevant subsystems
are shown in the left figure. The entanglement cut of this good modular flow
generator can be deformed close to the physical edge (right). In particular,
$(K_{A}-K_{B})|\Psi\rangle=(K_{A^{\prime}}-K_{B^{\prime}})|\Psi\rangle,$ where
$|\Psi\rangle$ is the state from which the modular Hamiltonians are defined.

What if the flow is no longer infinitesimal? In this case, we can
heuristically argue for the invariance of the bulk reduced density matrices by
making a nontrivial assumption: that Eq. (3.9) can be promoted to an operator
identity acting on an appropriate subspace. Specifically, let $\mathbb{L}$ be
a good modular flow generator. Let $\mathbb{L}^{\prime}$ be a deformation of
$\mathbb{L}$ such that $(\mathbb{L}-\mathbb{L}^{\prime})|\Psi\rangle=0$. (For
instance, one may consider $\mathbb{L}=K_{A}-K_{B}$ and
$\mathbb{L}^{\prime}=K_{A^{\prime}}-K_{B^{\prime}}$ in Figure 4.) We posit
that

$(\mathbb{L}-\mathbb{L}^{\prime})|\phi_{\text{edge}}\rangle=0$ (3.11)

for any $|\phi_{\text{edge}}\rangle$ which is indistinguishable from the
reference state in the bulk. This includes the reference state $|\Psi\rangle$
but also includes states that may have excitations at the boundary.

We claim that
$\mathbb{L}^{n}|\Psi\rangle={\mathbb{L}^{\prime}}^{n}|\Psi\rangle$ for any
$n\in\mathbb{N}$. For $n=1$, the claim is what we showed above [Eq. (3.6)].
For $\mathbb{L}^{2}|\Psi\rangle$, note that
$\mathbb{L}^{2}|\Psi\rangle=\mathbb{L}\mathbb{L}^{\prime}|\Psi\rangle$, where
$\mathbb{L}^{\prime}|\Psi\rangle$ is a state with excitations only at the
edge. Therefore, using Eq. (3.11), we get
$\mathbb{L}\mathbb{L}^{\prime}|\Psi\rangle={\mathbb{L}^{\prime}}^{2}|\Psi\rangle$.
The same argument can be repeated recursively, establishing
$\mathbb{L}^{n}|\Psi\rangle={\mathbb{L}^{\prime}}^{n}|\Psi\rangle$. In
particular, using Taylor expansion we conclude that $e^{-{\bf
i}\mathbb{L}t}|\Psi\rangle=e^{-{\bf i}\mathbb{L}^{\prime}t}|\Psi\rangle$ for
any $t\in\mathbb{R}$.

Now we claim that the reduced density matrix of a disk in the local bulk
region (denoted as $C$) is invariant under $e^{-{\bf i}\mathbb{L}^{\prime}t}$
provided that the entanglement cut associated with $\mathbb{L}^{\prime}$ does
not pass through $C$. This is due to the existence of a disentangling unitary
acting on a region surrounding $C$ (denoted as $D$) [Figure 5]. More
concretely, the entanglement bootstrap axioms imply that A0 is satisfied on
this disk; it is known that [12]

$\left(S_{CD}+S_{C}-S_{D}\right)_{\sigma}=0.$ (3.12)

Therefore, for any extension of $\sigma_{CD}$ to a larger system, say,
$\rho_{BCD}$, there is a disentangling unitary acting on $D$ (denoted as
$U_{D}$) such that

$U_{D}\rho_{BCD}U_{D}^{\dagger}={|\phi\rangle_{CD_{L}}}{\langle\phi|}\otimes\rho_{D_{R}B},$
(3.13)

for some pure state $|\phi\rangle_{CD_{L}}$, where
$\mathcal{H}_{D}\cong\mathcal{H}_{D_{L}}\otimes\mathcal{H}_{D_{R}}$. What is
more, $|\phi\rangle_{CD_{L}}$ depends only on $\sigma_{CD}$, independent of
the choice of extension $\rho_{BCD}$. In particular, the _same_ $U_{D}$
disentangles the global state $|\Psi\rangle$ as well as its reduced density
matrices over $A^{\prime}$.

Therefore, all the modular Hamiltonians appearing in $\mathbb{L}$ associated
with supersets of $CD$ can be disentangled using $U_{D}$. Without loss of
generality, let $X$ be such a subsystem.

$K_{X}=\ln\left(|\phi\rangle_{CD_{L}}\langle\phi|\right)+\ln\rho_{D_{R}\cup(X\setminus
CD)}.$ (3.14)

Therefore, $\mathbb{L}$ can be expressed as a sum of terms that act
nontrivially on the complement of $CD_{L}$ and the terms that act on $CD_{L}$.
The latter term is a multiple of
$\ln\left(|\phi\rangle_{CD_{L}}\langle\phi|\right)$, which acts trivially on
$|\Psi\rangle$. Therefore, the reduced density matrix of $|\Psi\rangle$ over
$CD_{L}$ is invariant under the good modular flow. In particular, the reduced
density matrix over $C$ is invariant under the flow. Because $e^{-{\bf
i}\mathbb{L}t}|\Psi\rangle=e^{-{\bf i}\mathbb{L}^{\prime}t}|\Psi\rangle$ and
we showed that the bulk reduced density matrix of $e^{-{\bf
i}\mathbb{L}^{\prime}t}|\Psi\rangle$ is independent of $t$, we can also
conclude that the bulk reduced density matrix of $e^{-{\bf
i}\mathbb{L}t}|\Psi\rangle$ is invariant.

$C$$D$ Figure 5: For any bulk disk ($C$) there is a region surrounding $C$
(denoted as $D$) on which a disentangling unitary exists. Upon applying this
disentangling unitary, we obtain a state in which the disk $C$ is fully
entangled with part of $D$.

We remark that our argument for the invariance of the bulk reduced density
matrices for non-infinitesimal flow is not rigorous. There are two key
mathematical issues that need to be dealt with. First, Eq. (3.11) may not hold
even if Eq. (3.9) is true. Second, Eq. (3.14) is unbounded because we are
taking the logarithm of the rank-one projector
$|\phi\rangle_{CD_{L}}\langle\phi|$. Making our argument into a rigorous
argument that takes care of these subtleties is left for future work. We
complement this shortcoming of our argument by providing strong numerical
evidence for our claim in Section 3.2.

In summary, the argument we present in this section says that if we apply a
good modular flow to the ground state, the resulting state

$\left|\Psi(t)\right\rangle=e^{-{\bf
i}\mathbb{L}t}\left|\Psi\right\rangle,\quad\mathbb{L}\in\mathfrak{g}$ (3.15)

will have only edge excitations, without any bulk excitations. This is
because, as we argued, the bulk density matrix should not change. We will
denote the subspace spanned by these states as $\mathcal{H}_{\text{edge}}$
from now on.

###  3.2 Fidelity tests

In this subsection, we provide numerical evidence for our claim that the bulk
reduced density matrices are invariant under good modular flow.

####  3.2.1 Fidelity as a diagnostic of change under modular flow

In order to assess how close two density matrices are, we employed the
fidelity $F(\rho,\sigma)\equiv{{\rm
Tr}}\left(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)$. For the ensuing
discussion, recall the basic properties of fidelity: that $0\leq
F(\rho,\sigma)\leq 1$ and $F(\rho,\sigma)=1$ if and only if $\rho=\sigma$.

Consider a flow (one-parameter family of unitary) $U(t)=e^{{\bf i}Gt}$ on a
state $\left|\Psi\right\rangle$ generated by a Hermitian operator $G$. We
compute $F(\rho_{X}(0),\rho_{X}(t))$, where $\rho_{X}(t)={{\rm
Tr}}_{\overline{X}}\left|\Psi(t)\right\rangle\left\langle\Psi(t)\right|$ and
$\left|\Psi(t)\right\rangle=U(t)\left|\Psi\right\rangle$ to quantify the
extent to which the local reduced density matrix of a region $X$ has been
changed under the flow.

We will also be interested in the change of the state under the flow
$U(t)=e^{{\bf i}Gt}$ for infinitesimal $t$. Provided that
$F(\rho_{X}(0),\rho_{X}(t))$ is twice differentiable around $t=0$, we define

$\alpha(G,X)\equiv-\left.{d^{2}\over
dt^{2}}F(\rho_{X}(0),\rho_{X}(t))\right|_{t=0}.$ (3.16)

The Taylor expansion around $t$ becomes:

$F(\rho_{X}(0),\rho_{X}(t))=1-{1\over 2}\alpha(G,X)t^{2}+O(t^{3}).$ (3.17)

The absence of the linear term comes from the fact that
$F(\rho_{X}(0),\rho_{X}(t))\leq 1$. Otherwise, under an infinitesimal time
$t$, $F(\rho_{X}(0),\rho_{X}(t))$ can exceed $1$. (For a similar reason, we
can also conclude that $\alpha(G,X)\geq 0$.) Thus the single number
$\alpha(G,X)$ accurately captures the small $t$ behavior of the
fidelity888This is related to the fact that $\sqrt{1-F}$ is a metric; see Ref.
[37] and _Bures distance_ [38] for related discussions. under the flow.

In the following section, we shall numerically study a specific model and
utilize these quantities, namely fidelity and $\alpha(G,X)$, to quantitatively
study the change of local reduced density matrices various under modular
flows.

####  3.2.2 Numerical tests

The main goal of our numerical tests is to verify our prediction in Section
3.1: that good modular flow at finite time preserves the bulk local density
matrices.999Note that this statement holds in the limit in which the size of
the involved subsystems are sufficiently large compared to the correlation
length. For realistic systems, the invariance of the bulk local density matrix
is only approximate. For this purpose, we shall use the fidelity [Section
3.2.1] to quantify the degree to which the bulk local density matrix changes.
For concreteness, we will study the ground state of a $p+{\bf i}p$
superconductor on a square lattice:

$H=\sum_{\vec{r},\vec{a}}\left[-ta_{\vec{r}}^{\dagger}a_{\vec{r}+\vec{a}}+\Delta
a_{\vec{r}}^{\dagger}a_{\vec{r}+\vec{a}}^{\dagger}e^{{\bf
i}\vec{a}\cdot\vec{A}}+h.c.\right]-\sum_{\vec{r}}(\mu-4t)a_{\vec{r}}^{\dagger}a_{\vec{r}},$
(3.18)

where $\vec{r}=(x,y)$ represents a site on the square lattice, and $\vec{a}$
is the lattice vector, taking values of $(1,0)$ and $(0,1)$ for positive $x$
and $y$ direction respectively. We set $\vec{A}=(0,\pi/2)$, so that $e^{{\bf
i}\vec{a}\cdot\vec{A}}$ is either $1$ or ${\bf i}$. In the following test, we
choose $t=1.0,\Delta=1.0,$ and $\mu=1.3$; this choice of parameter yields a
relatively small correlation length (approximately $1.2$ lattice spacings). We
impose an anti-periodic boundary condition on the $x$-direction so that no
flux is threaded.

We have numerically observed that the ground state of this model satisfies
entanglement bootstrap axioms A0 and A1 approximately with error decaying with
subsystem sizes; see Section 7.1.1 for details. We thus anticipate the
conclusions made in the previous section to continue to hold, at least
approximately. In particular, we expect the local bulk reduced density
matrices to be approximately invariant under good modular flows.

Let us now discuss our setup [Figure 6]. We considered two types of good and
bad modular flows, each shown on the left and the right side in Figure 6; we
shall refer to these two different setups as “test 1” and “test 2,”
respectively. In both cases, we chose three disk-like regions, indexed by
$i=1,2,3.$ As $i$ increases, the region gets progressively farther away from
the physical edge. In particular, $i=1$ corresponds to a region anchored at
the physical edge. We computed $F(\rho_{i}(0),\rho_{i}(t))$ for $i=1,2,3$ in
Figure 6, where $\rho_{i}(t)$ is the reduced density matrix of region $i$ at
time $t$.

Figure 6: Setups for testing the good and bad modular flows. The left and
right figure represents the setup for test 1 and test 2, respectively. Each
test involves a choice of good (blue) and bad (pink) modular flow generators.
The sizes and the locations of the subsystems can be inferred from the figure
by noting that each dot represents a site of a square lattice. Figure 7: The
fidelity tests associated with the regions shown in Fig. 6. For each test
(each row), we have $F(\rho_{i}(0),\rho_{i}(t))$ for modular flow with a long
evolution time (left) and a zoom-in to the short evolution time (right). For
the short time data, we perform linear fit of
$\ln\left(1-F(\rho_{i}(0),\rho_{i}(t))\right)$ vs $\ln t$. From the
intercepts, we compute the $\alpha$s for the associated bad modular flows: For
test 1: $\alpha_{1}=1.10,\alpha_{2}=2.27,\alpha_{3}=2.25$. For test 2:
$\alpha_{1}=13.41,\alpha_{2}=8.45,\alpha_{3}=3.23$.

The fidelities are plotted in Figure 7. As expected, the reduced density
matrices of the bulk regions (regions 2 and 3) remain invariant under the good
modular flow, though they can change under the bad modular flow. On the other
hand, for the reduced density matrices anchored at the physical edge (region
1), the good modular flow can change the fidelity. We remark that, for test 1,
the fidelity of region 1 appears unchanged under the good modular flow. The
origin of this invariance is unclear to us and we leave it as an open problem
to explain this unexpected behavior.

At small $t$, the fidelity can be approximated as

$F(\rho_{i}(0),\rho_{i}(t))\approx 1-{1\over 2}\alpha_{i}\,t^{2},$ (3.19)

where $\alpha_{i}$ is equal to the $\alpha(G,X)$ are defined in Eq. (3.16),
with $G$ being the generators of the flow and $X$ being the disk-like region
$i$. The coefficient $\alpha_{i}$ can be inferred from the intercept of a
linear fit between $\ln(1-F(\rho_{i}(0),\rho_{i}(t)))$ and $\ln(t)$. We list
the results of $\alpha_{i}$ in the caption of Figure 7. (For these tests, we
estimated $\alpha_{i}$ only for the bad modular flows, since good modular
flows did not change the density matrices for small $t$.)

##  4 Good modular flow generators and CFT

The numerical results in Section 3.2 showed that good modular flows do not
alter the bulk reduced density matrices, corroborating our argument in Section
3.1. This suggests that there is an invariant subspace associated with good
modular flows, which is a set of states that are locally indistinguishable
from the reference state in the bulk. This subspace, $\mathcal{H}_{\rm edge}$,
can therefore be thought of as a subspace formed by states with excitations
that are localized at the edge.

Now the natural question is how the good modular flows act on
$\mathcal{H}_{\rm edge}$. To make progress, in the following, we shall employ
an extra assumption (motivated by the chiral CFT description of the edge)
which allows us to establish a dictionary between the good modular flow
generators and the Virasoro generators of a 1+1D chiral CFT. More
specifically, we apply these concepts to gapped ground states with purely-
chiral gapless edges, laying the groundwork for the construction of Virasoro
generators in Section 6.

We shall study the following subclass of good modular flow generators, denoted
as $\mathfrak{g}^{\bullet}$:

######  Definition 4.1.

We define $\mathfrak{g}^{\bullet}$ as the subclass of good modular flow
generators [Definition 6] of the form
$\mathbb{L}=\sum_{A}\lambda_{A}K_{A},\lambda_{A}\in\mathbb{R}$, where every
subsystem appearing in the summation is _a disk-like region that intersects
with at most one single interval on the edge._

By definition, $\mathfrak{g}^{\bullet}\subset\mathfrak{g}$. In Table 2, we
shall see that $\mathfrak{g}^{\bullet}\neq\mathfrak{g}$. There we list
examples of good modular flow generators, delineating the difference between
the generators that are in $\mathfrak{g}^{\bullet}$ and those that are not.

Good modular flow generators and their types
---
$\includegraphics[width=85.35826pt,margin={0.2cm},valign={c}]{fig-vir-
submit/flow-gen3.pdf}\quad\begin{aligned}
&\hat{I}(A:C)=K_{A}+K_{C}-K_{AC}&&\notin\mathfrak{g}^{\bullet}\\\
&\hat{I}(A:C|B)=K_{AB}+K_{BC}-K_{ABC}-K_{C}&&\notin\mathfrak{g}^{\bullet}\end{aligned}$
$\includegraphics[width=85.35826pt,margin={0.2cm},valign={c}]{fig-vir-
submit/flow-gen4.pdf}\quad\begin{aligned}
&\hat{I}(A:C)=K_{A}+K_{C}-K_{AC}&&\notin\mathfrak{g}^{\bullet}\\\
&\hat{I}(A:C|B)=K_{AB}+K_{BC}-K_{ABC}-K_{C}&&\in\mathfrak{g}^{\bullet}\\\
&\hat{\Delta}(AA^{\prime}CC^{\prime},B,\emptyset)=K_{AA^{\prime}CC^{\prime}B}+K_{B}-K_{AA^{\prime}BCC^{\prime}}&&\notin\mathfrak{g}^{\bullet}\\\
&\hat{\Delta}(AA^{\prime},B,CC^{\prime})=K_{AA^{\prime}B}+K_{CC^{\prime}B}-K_{AA^{\prime}}-K_{CC^{\prime}}&&\in\mathfrak{g}^{\bullet}\end{aligned}$
Table 2: Examples of good modular flow generators. Some of them are in
$\mathfrak{g}^{\bullet}$ and some are not.

At a high level, the main motivation for defining $\mathfrak{g}^{\bullet}$ is
to focus on the good modular flow generators that can be identified with the
modular Hamiltonians of CFT on an interval, for which we have a good
understanding [39]. For the good modular flow generators outside of
$\mathfrak{g}^{\bullet}$, it is less clear if such an identification holds
because the corresponding subsystems in the CFT would contain more than one
interval. The CFT modular Hamiltonians of such subsystems are not local [40,
41, 42], posing a challenge for making such an identification.

As we shall soon see, by hypothesizing a certain relationship between the good
modular flow generators and the modular Hamiltonians of CFT, we can identify
the good modular flow generators as weighted integrals of the stress-energy
tensor in CFT. Moreover, this will lead to several testable predictions which
we verify, thereby supporting our hypothesis.

###  4.1 CFT Perspective

We shall focus on the low-energy physics of (2+1)D systems which are gapped
and chiral in the bulk. Such a system has robust gapless edge, which is
expected to be described by a chiral CFT [43, 44, 45]. Due to the
gravitational anomaly of the chiral CFT, this theory cannot be realized on a
stand-alone 1D system with a tensor product Hilbert space structure [44, 46].
However, the chiral CFT is well-defined as an edge of a 2+1D system due to the
cancellation of the anomaly [47, 44].

From the CFT perspective, it is natural to expect the modular Hamiltonian of
the edge to have a particularly simple form. It is well-known that for a 1+1D
CFT (without gravitational anomaly) on a circle101010The circle can have some
radius $R$. The form of the modular Hamiltonian in Eq. (4.1) in angular
coordinates doesn’t depend on $R$, because of conformal symmetry., the modular
Hamiltonian of the ground state $|\Omega\rangle$ over an interval
$[\theta_{1},\theta_{2}]$ is of the following form:

$\mathscr{K}_{[\theta_{1},\theta_{2}]}=\int_{0}^{2\pi}d\theta\beta^{[\theta_{1},\theta_{2}]}(\theta)\left(\mathscr{T}(\theta)+\bar{\mathscr{T}}(\theta)\right)+\alpha_{[\theta_{1},\theta_{2}]}\mathbb{1},$
(4.1)

where $\alpha_{[\theta_{1}.\theta_{2}]}$ is a UV-dependent constant,
$\beta^{[\theta_{1},\theta_{2}]}(\theta)$ is

$\beta^{[\theta_{1},\theta_{2}]}(\theta)=2\Theta(\theta-\theta_{1})\Theta(\theta_{2}-\theta){\sin({\theta-\theta_{1}\over
2})\sin({\theta_{2}-\theta\over 2})\over\sin({\theta_{2}-\theta_{1}\over
2})},$ (4.2)

and $\mathscr{T}(\theta),\bar{\mathscr{T}}(\theta)$ are the holomorphic and
anti-holomorphic components of the stress-energy tensor, respectively. (Both
$\mathscr{T}(\theta)$ and $\bar{\mathscr{T}}(\theta)$ are hermitian.)
$\Theta(x)$ is the Heaviside function, namely $\Theta(x)=1$ for $x\geq 0$, and
$\Theta(x)=0$ for $x<0$. We will refer to
$\beta^{[\theta_{1},\theta_{2}]}(\theta)$ as the _coolness function_
111111Here we consider the CFT is on a circle. If the CFT is on an infinite
line, for an interval $[x_{1},x_{2}]$, the coolness function takes the form
$\beta^{[x_{1},x_{2}]}(x)=2\Theta(x-x_{1})\Theta(x_{2}-x){(x-x_{1})(x_{2}-x)\over
x_{2}-x_{1}}$..

For a _purely-chiral_ CFT, by which we mean a 1+1D CFT with only holomorphic
components, we expect the modular Hamiltonian to take the form:

$\mathscr{K}_{[\theta_{1},\theta_{2}]}^{\chi}=\int_{0}^{2\pi}d\theta\beta^{[\theta_{1},\theta_{2}]}(\theta)\mathscr{T}(\theta)+\alpha^{\prime}_{[\theta_{1},\theta_{2}]}\mathbb{1},$
(4.3)

where the upper index $\chi$ signifies the fact that we are talking about a
purely-chiral CFT.

While Eq. (4.3) is a well-defined object in CFT, as we mentioned, there is an
obstruction to defining a chiral CFT on a stand-alone 1+1D system. However,
there are certain 2+1D states with an energy gap in the bulk, whose gapless
edge is described by such purely-chiral CFTs. We shall also refer to these
2+1D states as “purely-chiral”. Next, we put forward a hypothesis that
quantitatively connects good modular flow generators in
$\mathfrak{g}^{\bullet}$ from a purely-chiral 2+1D state to the modular
Hamiltonians in a purely-chiral CFT. For concreteness, let
$\mathcal{H}_{\text{edge}}$ be the low-energy subspace which is spanned by the
states obtained from good modular flows [Section 3].

######  Hypothesis 1 (Action of good modular flow generators for purely-chiral
states).

Consider a purely chiral reference state $\left|\Psi\right\rangle$ on a two-
dimensional disk. Let $\mathbb{L}=\sum_{A}\lambda_{A}K_{A}\in g^{\bullet}$.
There exists an isometry $\mathbb{V}:\mathcal{H}_{\rm CFT}\to\mathcal{H}_{\rm
edge}$ (i.e., $\mathbb{V}^{\dagger}\mathbb{V}=\mathbb{1}$) such that

$\mathbb{L}\mathbb{V}=\mathbb{V}\left(\sum_{A}\lambda_{A}\mathscr{K}_{b(A)}^{\chi}+\alpha^{\prime\prime}(\mathbb{L})\mathbb{1}\right)\quad\text{
and }\quad|\Psi\rangle=\mathbb{V}|\Omega\rangle$ (4.4)

where $\mathscr{K}_{b(A)}^{\chi}$ is the modular Hamiltonian of the CFT ground
state $\left|\Omega\right\rangle$ for the interval $b(A)$ obtained from the
intersection of $\partial A$ with the edge [see Figure 8].
$\alpha^{\prime\prime}(\mathbb{L})$ is a constant.

We will also use the following short-hand notation:

$\mathbb{L}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\sum_{A}\lambda_{A}\mathscr{K}_{b(A)}^{\chi}+\alpha^{\prime\prime}(\mathbb{L})\mathbb{1}\quad\textrm{is
short
for}\quad\mathbb{L}\mathbb{V}=\mathbb{V}\left(\sum_{A}\lambda_{A}\mathscr{K}_{b(A)}^{\chi}+\alpha^{\prime\prime}(\mathbb{L})\mathbb{1}\right).$
(4.5)

$A$$b(A)$ Figure 8: The edge interval $b(A)$ (red) corresponding to a disk $A$
(blue).

A few remarks are in order. First, although the hypothesis may seem to be
limited to expressions linear in $\mathbb{L}$, this is actually not the case.
Because $\mathbb{V}$ is an isometry, the hypothesis immediately implies the
following identity:

$\left\langle\Psi\right|f(\\{\mathbb{L}_{i}\in\mathfrak{g}^{\bullet}\\})\left|\Psi\right\rangle=\left\langle\Omega\right|f(\\{L^{\chi\mathrm{CFT}}_{i}\\})\left|\Omega\right\rangle,\quad\mathfrak{g}^{\bullet}\ni\mathbb{L}_{i}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}L^{\chi\mathrm{CFT}}_{i}$
(4.6)

for any (multi-variable) polynomial $f$. Second, we note that the hypothesis
does not say anything about how the individual $K_{A}$ are related to the CFT
modular Hamiltonians. One may think that the hypothesis is demanding that
$K_{A}\buildrel\mathbb{V}?\over{=}\mathscr{K}_{b(A)}^{\chi}+\alpha^{\prime\prime}(A)\mathbb{1}$
for every disk $A$, but this is not the case. Rather, the hypothesis tells us
only how to map elements of $g^{\bullet}$ 121212A stronger hypothesis we can
make is to replace $\mathfrak{g}^{\bullet}$ with $\mathfrak{g}$. We will not
need this stronger version in this paper. to a linear combination of CFT
modular Hamiltonians. Thirdly, the constant
$\alpha^{\prime\prime}(\mathbb{L})$ of (4.4) is necessary and non-trivial in
some cases. One example of a good modular flow generator with such non-zero
$\alpha^{\prime\prime}$ is shown in the second row in Table 1, where the
constant term $\alpha^{\prime\prime}$ comes from the topological entanglement
entropy [8, 9] (see also [12]) of bulk disks.131313We suspect the only source
of contribution to $\alpha^{\prime\prime}$ is topology dependent. Lastly, by
applying the good modular flow generators in $\mathfrak{g}^{\bullet}$ to
$|\Psi\rangle$, one can obtain states of the form of
$\mathbb{V}|\Omega^{\prime}\rangle$, where $|\Omega^{\prime}\rangle$ is some
excited state of the CFT. Physically, we can expect $\mathbb{V}$ to map states
of the CFT to $\mathcal{H}_{\rm edge}$. In particular, because the CFT Hilbert
space is infinite-dimensional, this isometry can only exist in the limit in
which the $\mathcal{H}_{\rm edge}$ is an infinite-dimensional Hilbert space.
If $\mathcal{H}_{\rm edge}$ is finite-dimensional, such as some purely-chiral
reference state realized in a finite lattice system we shall numerically study
later, we anticipate $\mathbb{V}$ to map a certain regularized subspace of
$\mathcal{H}_{\rm CFT}$ to $\mathcal{H}_{\rm edge}$.

As a consequence of this hypothesis, we can establish a relationship between
good modular flow generators in $\mathfrak{g}^{\bullet}$ and the stress-energy
tensor of the purely-chiral CFT:

$\mathbb{L}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\sum_{A}\lambda_{A}\,\mathscr{K}^{\chi}_{b(A)}+\alpha^{\prime\prime}(\mathbb{L})\mathbb{1}=\int_{0}^{2\pi}f(\theta)\mathscr{T}(\theta)+\kappa(\mathbb{L})\mathbb{1},$
(4.7)

where $\kappa(\mathbb{L})$ is a real constant (depending on $\mathbb{L}$) and
$f(\theta)$, which will be called a _weight function_ , is a linear
combination of the coolness functions $\beta^{a}(\theta)$ of intervals $a$
with the coefficients appearing in the good modular flow generator:

$f(\theta)=\sum_{A}\lambda_{A}\beta^{b(A)}(\theta).$ (4.8)

Eq. (4.7) is the main starting point of our ensuing analysis. This is an
important identity that establishes a correspondence between good modular flow
generators and the CFT stress-energy tensor for purely chiral states.

In the remainder of this paper, we will study many good modular flow
generators, whose corresponding weight functions $f(\theta)$ are of special
interest to us. One particularly simple case is when the weight function in
Eq. (4.7) vanishes. This will result in an equation (which we call the _vector
fixed-point equation_) that we can verify using means independent of
Hypothesis 1. We discuss this equation in Section 4.2 and provide an
alternative argument for its derivation in Section 4.3.

######  Remark.

Hypothesis 1 implies that the behavior of a good modular flow for any
$t\in\mathbb{R}$ matches a certain CFT prediction. Note, however, Hypothesis 1
is intended for systems in the thermodynamic limit. For systems with finite
sizes (and with finite onsite Hilbert dimensions), we expect the match to hold
approximately only within a finite time range $t\in(-t^{\ast},t^{\ast})$,
where $t^{\ast}$ depends on the system size and parameters. At larger $t$, the
modular flow behavior depends on the UV and is not universal. We will not
study such UV phenomena in this paper.

###  4.2 Vector fixed-point equation

A straightforward consequence of Hypothesis 1 is that there is a nontrivial
identity that certain good modular flow generators ought to satisfy. Consider
a 1+1D CFT ground state $|\Omega\rangle$. Let $a,b,$ and $c$ be three
contiguous intervals. It was shown in [27] that

$\eta\hat{\Delta}^{\rm CFT}(a,b,c)+(1-\eta)\hat{I}^{\rm
CFT}(a:c|b)\propto\mathbb{1},$ (4.9)

where
$\hat{\Delta}^{\mathrm{CFT}}(a,b,c)\equiv\mathscr{K}_{ab}+\mathscr{K}_{bc}-\mathscr{K}_{a}-\mathscr{K}_{c},\hat{I}^{\mathrm{CFT}}(a:c|b)\equiv\mathscr{K}_{ab}+\mathscr{K}_{bc}-\mathscr{K}_{abc}-\mathscr{K}_{b}$,
and $\eta$ is the cross-ratio141414Explicitly, $\eta={|a||c|\over|ab||bc|}$,
where on an infinite line $|a|$ is the length of edge interval $a$, and on a
circle $|a|$ is the chordal distance associated with arc $a$.. Eq. (4.9)
follows from the form of the coolness function, as observed in [27]. In
particular, the derivation of this equation can be generalized
straightforwardly to purely chiral CFTs.

Then by Hypothesis 1, we obtain a vector fixed-point equation for a purely
chiral 2+1D state $|\Psi\rangle$; see Eq. (4.13). Explicitly, for the regions
$AA^{\prime}BCC^{\prime}$ shown in Figure 9, we consider a particular linear
combination of modular Hamiltonians:

${\mathcal{K}}_{\mathbb{D}}(\eta)\equiv\eta\hat{\Delta}(AA^{\prime},B,CC^{\prime})+(1-\eta)\hat{I}(A:C|B),$
(4.10)

where $\hat{\Delta}(AA^{\prime},B,CC^{\prime})\equiv
K_{AA^{\prime}B}+K_{CC^{\prime}B}-K_{AA^{\prime}}-K_{CC^{\prime}}$, and
$\hat{I}(A:C|B)\equiv K_{AB}+K_{BC}-K_{ABC}-K_{B}$. Here $\eta$ is the cross-
ratio computed from the length of the edge interval. Note that

$\hat{\Delta},\hat{I}\in\mathfrak{g}^{\bullet}\quad\Rightarrow\quad{\mathcal{K}}_{\mathbb{D}}(\eta)\in\mathfrak{g}^{\bullet}.$
(4.11)

In other words, ${\mathcal{K}}_{\mathbb{D}}(\eta)$ is a good modular flow
generator. It is a very special one, because by Hypothesis 1 we have

${\mathcal{K}}_{\mathbb{D}}(\eta)\overset{\mathbb{V}}{\propto}\mathbb{1},$
(4.12)

implying that ${\mathcal{K}}_{\mathbb{D}}(\eta)$ acts trivially on the
invariant subspace $\mathcal{H}_{\rm edge}$, that is,

$\boxed{{\mathcal{K}}_{\mathbb{D}}(\eta)|\Psi\rangle\propto|\Psi\rangle.}$
(4.13)

Eq. (4.13) is the _vector fixed-point equation_ (for a chiral edge), which we
study in the rest of this Section.

$A$$B$$C$$A^{\prime}$$C^{\prime}$ Figure 9: Subsystems used in the
construction of $\hat{I}$ and $\hat{\Delta}$ appearing in Eq. (4.10). The
yellow region represents the bulk and the thick bold line represents the
physical edge.

While Eq. (4.13) is a priori only a _necessary condition_ for Hypothesis 1 to
be true, it has the advantage of being _locally checkable_. This is due to the
equivalence of the following two statements:

${\mathcal{K}}_{\mathbb{D}}(\eta)\left|\Psi\right\rangle\propto\left|\Psi\right\rangle\quad\Leftrightarrow\quad{\mathcal{K}}_{\mathbb{D}}(\eta)\rho_{AA^{\prime}BCC^{\prime}}\propto\rho_{AA^{\prime}BCC^{\prime}},$
(4.14)

where $\rho_{AA^{\prime}BCC^{\prime}}$ is the reduced density matrix of
$|\Psi\rangle$. Note that the right-hand side of Eq. (4.14) is formulated
entirely in terms of local reduced density matrices. As such, this condition
can be verified locally.

Now let us provide the proof of Eq. (4.14). The $\Rightarrow$ direction is
obvious as one can simply trace out the complement of
$AA^{\prime}BCC^{\prime}$ on both hand side of
${\mathcal{K}}_{\mathbb{D}}(\eta)\left|\Psi\right\rangle\left\langle\Psi\right|\propto\left|\Psi\right\rangle\left\langle\Psi\right|$.
For $\Leftarrow$ direction, one can first consider a purification of
$\rho_{AA^{\prime}BCC^{\prime}}$, denoted as
$\left|\Psi^{\prime}\right\rangle$, to get
${\mathcal{K}}_{\mathbb{D}}(\eta)\left|\Psi^{\prime}\right\rangle\propto\left|\Psi^{\prime}\right\rangle$.
By Uhlmann’s theorem [48], there is an isometry acting on the complement of
$AA^{\prime}BCC^{\prime}$ that maps $\left|\Psi^{\prime}\right\rangle$ from
$\left|\Psi\right\rangle$. Because this unitary commutes with
${\mathcal{K}}_{\mathbb{D}}(\eta)$ by construction, we obtain the original
vector equation. Therefore, the vector fixed-point equation can be verified
from the local reduced density matrix of $|\Psi\rangle$ over
$AA^{\prime}BCC^{\prime}$.

We note that our Hypothesis 1 assumes that the bulk state is purely chiral.
The vector fixed-point equation, however, applies more generally. We expect
that a suitable generalization of Hypothesis 1 to non-purely-chiral states
would also imply the vector fixed-point equation.

Currently, we do not know if the vector fixed-point equation implies
Hypothesis 1 in the case of purely-chiral states. While we leave this question
for future work, we make some further remarks on it in Section 4.3.

###  4.3 Analytical evidence for Hypothesis 1

The purpose of this section is to give evidence for Hypothesis 1. We do this
by first proving the vector fixed point equation under an assumption that we
call the “cylinder assumption”. Then we ask about the extent to which the
vector fixed point equation and its operator version imply Hypothesis 1.

Analytical evidence for the vector fixed-point equation. In Section 4.2, we
introduced a straightforward consequence of Hypothesis 1, which we referred to
as the vector fixed-point equation. In this subsection, we use physics
arguments to derive the vector fixed-point equation. A certain reformulation
of this condition will be used centrally in our upcoming work [49]. We then
discuss the strength of such fixed-point equations, and the extent to which
they in turn imply Hypothesis 1. This subsection thereby provides nontrivial
evidence for Hypothesis 1.

Because we will use an independent physics argument to derive the vector
fixed-point equation, we emphasize that the assumptions used in this Section
(and its associated Appendix E) are different from the rest of the paper. Our
starting point is the idea that a (possibly chiral) 2+1D topological phase on
a cylinder is a (non-chiral) 1+1D CFT. This assumption lets us use the
entanglement structure of a 1+1D CFT ground state and the bulk assumptions
[Section 2] to learn about the entanglement structure of the 2+1D state
altogether. We will call this line of thought “the cylinder argument”.

Consider a system with a large bulk gap on a cylinder. It is natural to regard
the whole system as a 1+1D CFT [Figure 10]: we dimensionally reduce along the
direction normal to the boundary, meaning that we regard the degrees of
freedom with the same value of the position along the boundary as lying at the
same site of the 1D system. The basic assumption is that the resulting system
satisfies the vector fixed-point equation of [27], reviewed in Appendix E, for
regions of the form indicated on the right side of Figure 10. We will call
this statement (that the dimensionally reduced state satisfies the 1+1D vector
fixed-point equation) the cylinder assumption.

$A$$C$$B$$\longrightarrow$$A$$C$$B$ Figure 10: An illustration of the cylinder
assumption. One can view a 2+1D gapped system with a chiral gapless edge mode
(red) as a 1+1D CFT using dimensional reduction. For illustration purposes we
draw a purely-chiral 2+1D system where we have either left or right moving
mode for each edge (not both). The cylinder assumption works more generally.

Taking the cylinder assumption as a starting point, with entanglement
bootstrap axioms A0 and A1 plus two global assumptions, we indeed partially
verify the “dictionary” provided by Hypothesis 1. Here we state the
assumptions and results and defer proof details to Appendix E.

We introduce the two technical assumptions (similar to the axioms A0 and A1)
to guarantee that no nefarious person has distributed a Bell pair between the
two boundaries of the cylinder [Figure 11] so as to break the ensuing
argument. These can be thought of as versions of A0 and A1 over regions that
include the inner boundary of the cylinder. If the inner boundary is
contracted to a point, this reduces to the usual A0 and A1. Nonetheless, we
note that these are extra assumptions that do not follow from A0 and A1. To
see why these assumptions are necessary, we note the following fact. If we
impose conditions on disks that only include the bulk regions, we cannot
exclude the possibility of having an end-point of a twist defect [50] (or
Majorana zero modes in fermionic models [51]) in the inner boundary. It is
well-known that this leads to long-range mutual information [52, 53]. We are
ruling out such pathological cases by imposing the full-boundary version of
the axioms A0 and A1.

Full boundary axioms$C$$B$A0: $\Delta(B,C)_{\sigma}=0$$B$$C$$D$A1:
$\Delta(B,C,D)_{\sigma}=0$ Figure 11: Full-boundary versions of the axioms A0
and A1. The solid lines represent the physical boundary and the dashed lines
represent the entanglement cut.

The main result of this Section is the following proposition.

######  Proposition 4.2 (Cylinder assumption gives boundary vector fixed-point
equation).

Given the cylinder assumption, bulk A1, full-boundary A0 and full-boundary A1,
the 2+1D vector fixed-point equation (4.13) holds near each boundary
independently.

We defer the proof of this Proposition to Appendix E.

Vector fixed-point equation hints at Hypothesis 1. Having just explained that
the vector fixed-point equation near the physical boundary follows from the
cylinder assumption, we now discuss to what extent the vector fixed-point
equation supports Hypothesis 1. We shall give a physical discussion here. Work
on the question of the strength of the vector fixed-point equation (as well as
its operator version) will be reported elsewhere [54].

In fact, the vector fixed-point equation largely determines the form of the
modular Hamiltonian of a disk $A$ that touches the gapless boundary along an
interval $b(A)$, at least when it acts on the reference state
$\left|\Psi\right\rangle$. Recall that a modular Hamiltonian $K_{A}$ only has
non-trivial action on $\left|\Psi\right\rangle$ along its entanglement cut
$\partial A$151515Explicitly, $K_{A}\left|\Psi\right\rangle=O_{\partial
A}\left|\Psi\right\rangle$. This only requires bulk A0.. If the reference
state satisfies the vector fixed point equation for every set of three
contiguous intervals, it has the following three consequences. (1) Vector
fixed-point equations suggest that

$K_{A}\left|\Psi\right\rangle=\left(\int_{b(A)}d\theta\beta(\theta)\mathfrak{h}(\theta)+{\mathcal{B}}_{\partial
A}\right)\left|\Psi\right\rangle,$ (4.15)

where ${\mathcal{B}}_{\partial A}$ acts on $\partial A$ (the entanglement cut
of $A$) inside the bulk, and will be canceled in a linear combination of
modular Hamiltonians that gives a good modular flow generator. (2) Moreover,
the vector fixed-point implies that the weight function $\beta(x)$ takes the
form of the coolness function (4.2). (3) Finally, the operator
$\mathfrak{h}(\theta)$ in (4.15) is the “reconstructed Hamiltonian density”
[27] that governs the edge excitations.

We make three further remarks regarding the reconstructed Hamiltonian density.
Firstly, $\mathfrak{h}(\theta)$ is a suitable limit of a particular linear
combination of modular Hamiltonians of small disks that touch the gapless edge
at infinitesimal intervals near the point labelled $\theta$. The reconstructed
Hamiltonian from the integral of $\mathfrak{h}(\theta)$ can be regarded as the
continuous version of the reconstructed Hamiltonian in [27]. Secondly, if one
considers a lattice realization of $\left|\Psi\right\rangle$, then the
Hamiltonian density shall be supported on the thickened edge. The thickness is
inevitable at least in chiral state, since otherwise, one can obtain a lattice
realization of chiral CFT, which is forbidden by gravitational anomaly.
Lastly, we should point out that from the vector fixed-point equation (which
only involves a linear action of modular Hamiltonians), it is not clear that
$\mathfrak{h}(\theta)$ satisfies the operator product expansion (OPE) relation
required for a stress-energy tensor of a CFT.

##  5 Coherent states

In the previous section, we introduced Hypothesis 1, relating the action of a
good modular flow generator in $\mathfrak{g}^{\bullet}$ on the reference state
to a weighted integral of the stress-energy tensor on the CFT ground state. In
this section, we discuss the general properties of states obtained from good
modular flows. These states have the form $\left|\Psi(t)\right\rangle=e^{{\bf
i}\mathbb{L}t}\left|\Psi\right\rangle$, where
$\mathbb{L}\in\mathfrak{g}^{\bullet}$ is Hermitian. We shall refer to these
states as _coherent states_ for the reasons we discuss in Section 5.1. The
vector space spanned by such coherent states gives a representation of the
Virasoro algebra. This representation is known as the “identity module” in
literature [55].

###  5.1 Virasoro algebra and coherent states

Consider a good modular flow generator $\mathbb{L}\in\mathfrak{g}^{\bullet}$:

$\mathbb{L}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int
d\theta\,f(\theta)\mathscr{T}(\theta).$ (5.1)

Define $|\Psi(t)\rangle\equiv e^{i\mathbb{L}t}|\Psi\rangle$. According to
Hypothesis 1, this state is of the following form:

$\left|\Psi(t)\right\rangle=e^{{\bf
i}\mathbb{L}t}\left|\Psi\right\rangle=\mathbb{V}\exp\left({\bf i}t\int
d\theta\,f(\theta)\mathscr{T}(\theta)\right)\left|\Omega\right\rangle,$ (5.2)

where $\left|\Omega\right\rangle$ is the CFT ground state. The state
$\exp\left({\bf i}t\int
d\theta\,f(\theta)\mathscr{T}(\theta)\right)\left|\Omega\right\rangle$ is
known as a _coherent state_ in the CFT literature [56]. As such, we shall
refer to $|\Psi(t)\rangle$ also as a coherent state. (We remark that
$|\Psi(t)\rangle$ is a state in 2+1 dimensions, though
$\left|\Omega\right\rangle$ describes a state in 1+1 dimensions.)

In purely chiral CFTs, the group generated by $\left\\{\exp\left({\bf i}t\int
f(\theta)\mathscr{T}(\theta)\right)|f(\theta)\right\\}$ forms a projective
representation of the orientation-preserving diffeomorphisms over $S^{1}$,
denoted as $\mathrm{Diff}_{+}(S^{1})$ from now on. As such, in the context of
2+1D purely chiral state, we expect the good modular flow $e^{i\mathbb{L}t}$
to be an element of a (projective) representation of
$\mathrm{Diff}_{+}(S^{1})$. A natural Hilbert space to consider is
${\mathcal{H}}^{\bullet}_{\mathrm{edge}}$, defined as:

${\mathcal{H}}^{\bullet}_{\mathrm{edge}}\equiv\mathrm{Span}\left\\{e^{{\bf
i}\mathbb{L}t}\left|\Psi\right\rangle|\mathbb{L}\in\mathfrak{g}^{\bullet}\right\\}.$
(5.3)

This is our carrier space for $\mathrm{Diff}_{+}(S^{1})$.

We now formally define the representation of the Virasoro generators with
respect to ${\mathcal{H}}^{\bullet}_{\mathrm{edge}}$. We define $L_{n}$ as an
operator satisfying

$L_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\mathscr{L}_{n},\quad
n\in\mathbb{Z},$ (5.4)

where $\mathbb{V}$ is the isometry appearing in Hypothesis 1. We shall discuss
constructions of $L_{n}$ in Section 6. Following from Hypothesis 1, $L_{n}$
shall satisfy the Virasoro algebraic relation in
${\mathcal{H}}^{\bullet}_{\mathrm{edge}}$, with the same charge $c$ as
$\mathscr{L}_{n}$.

Let us make a few remarks. First, Eq. (5.4) does not define $L_{n}$ uniquely
in the original Hilbert space of the 2+1D chiral system. This is because there
can be different operators that satisfy Eq. (5.4) for the same value of $n$.
However, within the subspace ${\mathcal{H}}^{\bullet}_{\mathrm{edge}}$, these
different choices act in an identical way. Specifically, consider $L_{n}$ and
$L_{n}^{\prime}$ such that
$L_{n}\mathbb{V}=L_{n}^{\prime}\mathbb{V}=\mathbb{V}\mathscr{L}_{n}$. Then for
any $|\Psi(t)\rangle=\mathbb{V}e^{{\bf i}\mathscr{L}t}|\Omega\rangle$,

$L_{n}|\Psi(t)\rangle=L_{n}^{\prime}|\Psi(t)\rangle=\mathbb{V}\mathscr{L}_{n}e^{{\bf
i}\mathscr{L}t}|\Omega\rangle.$ (5.5)

Second, $\mathbb{L}^{k}|\Psi\rangle\in{\mathcal{H}}^{\bullet}_{\mathrm{edge}}$
for any $\mathbb{L}\in\mathfrak{g}^{\bullet}$. This can be seen by taking the
$n$’th order derivative of $e^{{\bf i}\mathbb{L}t}|\Psi\rangle$ at $t=0$. In
Section 6, we shall see that $L_{n}$ can be approximated arbitrarily well by a
linear combination of various good modular flow generators. Therefore,
${\mathcal{H}}^{\bullet}_{\mathrm{edge}}$ is closed under applications of
$L_{n}$s.

Now, we can understand the action of a good modular flow generator in
$\mathfrak{g}^{\bullet}$ in terms of the Virasoro algebra (up to the isometry
$\mathbb{V}$). Recall the mode expansion of the stress-energy tensor:

$\mathscr{L}_{n}={1\over 2\pi}\int_{0}^{2\pi}d\theta e^{{\bf
i}n\theta}\mathscr{T}(\theta)\quad\text{and}\quad\mathscr{T}(\theta)=\sum_{n\in\mathbb{Z}}e^{-{\bf
i}n\theta}\mathscr{L}_{n}.$ (5.6)

By Hypothesis 1, any good modular flow generator in $\mathfrak{g}^{\bullet}$
can be expressed in terms of a weighted integral of the stress-energy tensor:

$\mathbb{L}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}d\theta\,f(\theta)\mathscr{T}(\theta)=\sum_{n\in\mathbb{Z}}\lambda_{n}\mathscr{L}_{n},\quad\lambda_{n}=\int_{0}^{2\pi}d\theta\,f(\theta)e^{-{\bf
i}n\theta}.$ (5.7)

Recalling $L_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\mathscr{L}_{n}$, we
can write $\mathbb{L}$ in Eq. (5.7) as

$\mathbb{L}=\sum_{n\in\mathbb{Z}}\lambda_{n}L_{n}.$ (5.8)

Therefore, the action of $\mathbb{L}$ on $\left|\Psi\right\rangle$ is
essentially a linear combination of Viraoro generators acting on the gapless
edge. Moreover, a coherent state can also be expressed as

$\left|\Psi(t)\right\rangle=e^{{\bf
i}\mathbb{L}t}\left|\Psi\right\rangle=\exp\left({\bf
i}t\sum_{n\in\mathbb{Z}}\lambda_{n}L_{n}\right)\left|\Psi\right\rangle.$ (5.9)

We wish to remark on an important property of the Fourier coefficient
$\lambda_{n}$, namely, $|\lambda_{n}|$ decays as $O(|n|^{-3})$. This follows
because the weight function $f(\theta)$ is piecewise161616The precise
definition of piecewise $C^{m}$ is given in Appendix B. $C^{2}$ by the
construction of good modular flow. By contrast, if the action of a bad modular
flow on the edge could be written as a weighted integral of the stress tensor
(as in Hypothesis 1 for good modular flow), the weight function would be
piecewise $C^{1}$, which leads to $O(|n|^{-2})$ decay of its Fourier
coefficients. (See Lemma B.1 for the detailed proof.) As we shall discuss
below, the Fourier coefficients of the modular flow generators play an
important role in computing several quantities, such as the variance of the
generator and the energy of the edge excitations. These quantities are finite
for good modular flow, thanks to the $O(|n|^{-3})$ decay of the Fourier
coefficients for the generators. However, these quantities diverge for bad
modular flow, since the Fourier coefficients of the generators are
$O(|n|^{-2})$ decay.

Using this expression Eq (5.9), one can obtain several analytic properties of
the coherent states, which we discuss in the following two subsections.

###  5.2 State overlap

In this subsection, we compare $\left|\Psi(t)\right\rangle=e^{{\bf
i}\mathbb{L}t}\left|\Psi\right\rangle,\mathbb{L}\in\mathfrak{g}^{\bullet}$ and
$\left|\Psi\right\rangle$. The difference between $\left|\Psi\right\rangle$
and $\left|\Psi(t)\right\rangle$ can be characterized by the overlap
$|\langle\Psi(t)|\Psi\rangle|^{2}$. The behavior for small $t$ is

$|\langle\Psi(t)|\Psi\rangle|^{2}=1-\sigma(\mathbb{L})^{2}t^{2}+O(t^{4}),$
(5.10)

where $\sigma(\mathbb{L})^{2}$ is the variance of the operator $\mathbb{L}$ on
$\left|\Psi\right\rangle$:

$\sigma(\mathbb{L})^{2}\equiv\left\langle\Psi\right|\mathbb{L}^{2}\left|\Psi\right\rangle-\left\langle\Psi\right|\mathbb{L}\left|\Psi\right\rangle^{2}.$
(5.11)

$\sigma$ quantifies the change of $\left|\Psi\right\rangle$ under the good
flow $e^{{\bf i}\mathbb{L}t}$ for small $t$.

Moreover, the states $\left|\Psi(t)\right\rangle$ and
$\left|\Psi\right\rangle$ only differ near the edge region where
$\mathbb{L}\in\mathfrak{g}^{\bullet}$ has non-trivial action. To be more
precise, we utilize the fidelity $F(\rho_{A}(0),\rho_{A}(t))$ to compare
reduced density matrices from $\left|\Psi\right\rangle$ and
$\left|\Psi(t)\right\rangle$ on a region $A$. (By fidelity, we mean
$F(\rho,\lambda)={{\rm Tr}}\sqrt{\sqrt{\rho}\lambda\sqrt{\rho}}$ as in Section
3.2.) Denoting the total system as $\mathcal{X}$, we can show that

$F(\rho_{\mathrm{edge}}(0),\rho_{\mathrm{edge}}(t))=F(\rho_{\mathcal{X}}(0),\rho_{\mathcal{X}}(t))=|\langle\Psi(t)|\Psi\rangle|,$
(5.12)

where $\rho_{\mathrm{edge}}(0)$ and $\rho_{\mathrm{edge}}(t)$ are reduced
density matrices on the thickened edge region. The first equality is proven in
Appendix F. The idea is that one can deform the support of
$\mathbb{L}\in\mathfrak{g}^{\bullet}$ to a region near the gapless edge
without changing the action on the reference state, and hence, only the
reduced density matrix near the edge is changed. In the circumstances where
there are multiple gapless edges, such as when the system is on a cylinder,
the proof requires the full-boundary A0 condition171717If the system is on a
disk (i.e. it has only one gapless edge), this full-boundary A0 condition
follows from the bulk A0 condition., which is explained previously in Figure
11 in Section 4.3. The role of full boundary A0 can be understood as a
constraint that there is no long-ranged entanglement between the gapless
edges. The second equality in Eq. (5.12) follows from the definition of
fidelity.

As a result of Eq. (5.12), we obtain

$F(\rho_{\mathcal{X}}(0),\rho_{\mathcal{X}}(t))=F(\rho_{\text{edge}}(0),\rho_{\text{edge}}(t))=1-{1\over
2}\sigma(\mathbb{L})^{2}t^{2}+O(t^{3}).$ (5.13)

and therefore

$\boxed{\sigma(\mathbb{L})^{2}=\alpha(\mathbb{L},\mathcal{X})=\alpha(\mathbb{L},\text{edge
region}),\quad\mathbb{L}\in\mathfrak{g}^{\bullet}},$ (5.14)

where $\alpha$ is defined in Eq. (3.16) in Section 3.2.1. Later, we shall
numerically test Eq. (5.14) for some selected good modular flow generators. We
shall also compare the results with theoretical predictions, which are
obtained by the Fourier coefficients in the Virasoro mode expansion
$\mathfrak{g}^{\bullet}\ni\mathbb{L}=\sum_{n\in\mathbb{Z}}\lambda_{n}L_{n}$

$\sigma(\mathbb{L})^{2}={c\over 12}\sum_{n\geq 0}|\lambda_{n}|^{2}(n^{3}-n).$
(5.15)

It’s worth noting that $\sigma(\mathbb{L})^{2}$ converges due to the decaying
property $|\lambda_{n}|\sim|n|^{-3}$ for the good modular flow. In contrast,
for the bad modular flow, where $|\lambda_{n}|\sim|n|^{-2}$,
$\sigma(\mathbb{L})^{2}\sim\sum_{n=1}^{\infty}1/n\to\infty$, which diverges.

###  5.3 Energy of the coherent states

Since the Virasoro algebra contains a natural notion of Hamiltonian, we can
also compute the “energy” of the edge excitations. In Section 6.1.1, we will
construct

$L_{0}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\mathscr{L}_{0},$ (5.16)

and the energy of the edge excitation created by the good modular flow
$e^{{\bf i}\mathbb{L}t}$ with small $t$ is

$\left\langle\Psi(t)\right|L_{0}\left|\Psi(t)\right\rangle={c\,t^{2}\over
12}\sum_{m\geq 0}|\lambda_{m}|^{2}(m^{4}-m^{2})+O(t^{4}).$ (5.17)

Interestingly, for small $t$, the leading order of this average energy is
equal to the leading order (i.e. the $t^{2}$ order) of the Schwarzian action
of an element $\varphi_{t}\in\rm{Diff}(S^{1})$

$I[\varphi_{t}]={2\pi c\over
24}\int_{0}^{2\pi}d\theta\left(\varphi^{\prime}_{t}(\theta)^{2}+\left({\varphi^{\prime\prime}_{t}(\theta)\over\varphi^{\prime}_{t}(\theta)}\right)^{2}-1\right).$
(5.18)

Specifically, the transformation $\varphi_{t}(\theta)$ is generated by the
good modular flow $e^{{\bf i}\mathbb{L}t}$. The weight function $f(\theta)$ of
$\mathbb{L}$ in Eq. (5.7) is the vector field of the transformation, i.e.
$\varphi_{t}(\theta)=\theta+tf(\theta)$ for small $t$, with which one can show
Eq. (5.17) and Eq. (5.18) agrees at the leading order.

The Schwarzian action (5.18) governs the dynamics of various theories of low-
dimensional quantum gravity and appears to be an effective description of
certain 0+1-dimensional approximate CFTs like the low-energy limit of the SYK
model (see e.g. [57]). It also gives the relative entropy of coherent states
in chiral CFT [56].

##  6 Construction of approximate Virasoro generators

In the previous section, we formally defined the Virasoro generators $L_{n}$
by requiring $L_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\mathscr{L}_{n}$.
We now begin to discuss their constructions from a single 2+1D purely-chiral
wavefunction $\left|\Psi\right\rangle$.

In Section 6.1, we shall use good modular flow generators to construct
approximate Virasoro generators $\widetilde{L}_{n}$, assuming the state
$\left|\Psi\right\rangle$ satisfies Hypothesis 1. For $n=0,\pm 1$, the
construction is exact $\widetilde{L}_{0}=L_{0},\widetilde{L}_{\pm 1}=L_{\pm
1}$181818For this reason, we will occasionally drop the “$\sim$” for
$\widetilde{L}_{0}$ and $\widetilde{L}_{\pm 1}$., while for the other
generators, $\widetilde{L}_{\pm n}$ take the form

$\widetilde{L}_{\pm n}=L_{\pm n}+\sum_{|k|\geq 2}\lambda_{kn}L_{kn},\quad
n\geq 2.$ (6.1)

The first term $L_{\pm n}$ gives the leading-order contribution and the
remaining higher-harmonic terms are sub-leading in the sense that
$|\lambda_{kn}|\sim|k|^{-3}$. We shall discuss the exact version of Eq. (6.1)
at the end of the next subsection.

In Section 6.2, we analytically discuss several properties of
$\widetilde{L}_{n}$. Using the result of Eq. (6.1), we compute several
meaningful quantities that are related to $\widetilde{L}_{n}$. The purposes of
studying these quantities are not only for their own physical meaningfulness
but also for comparisons. Later in Section 7, we first compare the theoretical
predictions for these quantities with numerical results. The good agreement
between them supports the validity of Hypothesis 1. Secondly, we compare the
results for these quantities from $\widetilde{L}_{n}$ and those from $L_{n}$,
from which we will see that the approximation “error” of $\widetilde{L}_{n}$
to $L_{n}$ is quite small.

We remark that one can further improve the construction by a procedure that
reduces the discrepancy between $\widetilde{L}_{n}$ and $L_{n}$. The procedure
is introduced in Appendix C. As the limit of such a procedure, one can obtain
exact $L_{n}$.

###  6.1 Explicit construction of $\widetilde{L}_{n}$

In this subsection, we introduce the explicit construction of
$\widetilde{L}_{n}$ using good modular flow generators from a 2+1D purely-
chiral state $\left|\Psi\right\rangle$. The general idea of the construction
can be understood as an approximate spatial Fourier transformation. Recall the
Virasoro generator $L_{n}$ is

$L_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\mathscr{L}_{n}={1\over
2\pi}\int_{0}^{2\pi}d\theta e^{{\bf i}n\theta}\mathscr{T}(\theta).$ (6.2)

Based on Hypothesis 1, a good modular flow generator in
$\mathfrak{g}^{\bullet}$ under the isometry $\mathbb{V}$ is a weighted
integral of stress-energy tensors, i.e. a linear combination of them. That is:

$\sum_{i}\lambda_{i}\mathbb{L}_{i}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}f(\theta)\mathscr{T}(\theta),\quad\mathbb{L}_{i}\in\mathfrak{g}^{\bullet},\quad\lambda_{i}\in\mathbb{C}.$
(6.3)

Therefore, to approximate $L_{n}$, one can find a certain linear combination
of suitable good modular flow generators in $\mathfrak{g}^{\bullet}$ such that
the weight function $f(\theta)$ is approximately $e^{{\bf i}n\theta}$.

We shall introduce the construction of $L_{0}$ and $L_{n\neq 0}$ separately,
as their construction schemes are different. For $L_{0}$, it can be obtained
directly from a single good modular flow generator [Section 6.1.1]. However,
the construction for $L_{n},n\neq 0$ takes more steps [Section 6.1.2]. First
of all, since $L_{n}$ for $n\neq 0$ are not Hermitian while good modular flow
generators are, a certain linear combination of good modular flow generators
with imaginary coefficients is necessary. Secondly, in the construction of
$\widetilde{L}_{n\neq 0}$, we utilize a method to which we shall refer as
“twisting trick” (illustrated in Figure 14). The twisting trick enables us to
obtain a certain set of good modular flow generators with desirable weight
functions whose linear combinations can approximate $e^{{\bf i}n\theta}$.

####  6.1.1 Construction of $L_{0}$

We begin with the construction of the Virasoro generator $L_{0}$. We shall
construct $L_{0}$ as a good modular flow generator in
$\mathfrak{g}^{\bullet}$. Such a generator needs to give
$f(\theta)\equiv{1\over 2\pi}$ for $f(\theta)$ appearing in Eq. (6.3). As
discovered in 1+1D CFT [27], if the system is on a circle divided into $N$
equal-sized intervals (labeled by $i=1,\cdots,N$), a constant weight function
can be obtained by summing over all the coolness functions on two successive
intervals $i,i+1$ and then subtracting the sum over all the coolness functions
on single intervals:

$\sum_{i}\left(\mathscr{K}^{\chi}_{i,i+1}-\mathscr{K}^{\chi}_{i}\right)\propto\int_{0}^{2\pi}d\theta\mathscr{T}(\theta)+\kappa\mathbb{1},$
(6.4)

where $\mathscr{K}^{\chi}_{a}$, as a reminder, is the modular Hamiltonian of
interval $a$ from a chiral CFT ground state, defined in Eq. (4.3). The
constant $\kappa$ can be computed by taking the expectation value on the
ground state.

For the 2+1D purely chiral state $\left|\Psi\right\rangle$ with a gapless
edge, we will be using a linear combination of modular Hamiltonians from
$\left|\Psi\right\rangle$, which takes the form of $\hat{\Delta}(A,B,C)\equiv
K_{AB}+K_{BC}-K_{A}-K_{C}$ for the region $(A,B,C)$ near the edge [Figure 12].
Note that $\hat{\Delta}(A,B,C)\in\mathfrak{g}^{\bullet}$ is already a good
modular flow generator by construction. We expect

$\hat{\Delta}(A,B,C)\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\mathscr{K}_{ab}^{\chi}+\mathscr{K}_{bc}^{\chi}-\mathscr{K}_{a}^{\chi}-\mathscr{K}_{c}^{\chi}$
(6.5)

for intervals $a,b,$ and $c$ [Figure 12]. While we believe equation (6.5)
holds exactly, Hypothesis 1 allows a possible additive multiple of the
identity operator. This additive constant is irrelevant since (in the next
paragraph) we will express $L_{0}$ as a sum of $\hat{\Delta}$s with their
expectation values subtracted.

$A$$B$$C$$a$$b$$c$ Figure 12: Subsystems for $\hat{\Delta}(A,B,C)$. The
gapless edge is indicated by the thick black line. The construction of
Virasoro generator $L_{0}$ utilizes a similar choice of subsystems with
$(A,B,C)$ taken to be $(A_{i},B_{i},C_{i})$ that touches the gapless edge at
interval $(i-1,i,i+1)$.

We now introduce the construction from the reference state
$\left|\Psi\right\rangle$ that satisfies Hypothesis 1 near the gapless edge.
We divide the edge into $N$ equal-sized intervals, labeled by
$i=1,2,\cdots,N$. Let $A_{i},B_{i},C_{i}$ be regions touching the edge at the
$i-1$, $i$, and $i+1$th intervals respectively, and they are arranged in way
topologically equivalent to the $A,B,C$ shown in Figure 12. We see that

$L_{0}\equiv{1\over
4\pi\tan(\pi/N)}\sum_{j=1}^{N}\left(\hat{\Delta}(A_{i},B_{i},C_{i})-\Delta(A_{i},B_{i},C_{i})\right)$
(6.6)

satisfies the desired property
$L_{0}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}{1\over
2\pi}\int_{0}^{2\pi}d\theta\,\mathscr{T}(\theta)=\mathscr{L}_{0}$, and it is
the generator we are looking for. Here, the lower indices of the regions
should be understood modulo $N$ and
$\Delta(A_{i},B_{i},C_{i})=\left\langle\Psi\right|\hat{\Delta}(A_{i},B_{i},C_{i})\left|\Psi\right\rangle$.

####  6.1.2 Construction of $\widetilde{L}_{n},n\neq 0$ with the twisting
trick

In this subsection, we introduce a scheme to construct the approximate
Virasoro generator $\widetilde{L}_{n}$ which is generically applicable for all
$n\neq 0$. Our construction for $n=\pm 1$ shall give the exact $L_{\pm 1}$,
i.e. $\widetilde{L}_{\pm 1}=L_{\pm 1}$. On the other hand, $\widetilde{L}_{n}$
for $n\geq 2$ is equal to $L_{n}$ plus a sum of sub-leading higher-harmonic
terms $L_{kn},|k|\geq 2$, as we mentioned in Eq. (6.1).

As we mentioned at the beginning of this subsection, to construct an
approximate Virasoro generator, our goal is to design a suitable set of good
modular flow generators which realize approximate spatial Fourier
transformations. We shall later introduce a _twisting trick_ [Figure 14] to
achieve this goal. This trick leads to a judicious choice of linear
combination of modular Hamiltonians that only create excitations on the edge.
Below we explain this idea in detail with an example, followed by the
discussion on the more general cases.

The main task of the construction is to find two types of good modular flow
generators, denoted as $\mathbb{L}^{(o)}_{n},\mathbb{L}^{(e)}_{n}$, such that

$\mathbb{L}^{(o)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{\approx}}{1\over
2\pi}\int_{0}^{2\pi}d\theta\sin(n\theta)\mathscr{T}(\theta),\quad\mathbb{L}^{(e)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{\approx}}{1\over
2\pi}\int_{0}^{2\pi}d\theta\cos(n\theta)\mathscr{T}(\theta).$ (6.7)

Once $\mathbb{L}^{(o)}_{n},\mathbb{L}^{(e)}_{n}$ are constructed, one can
obtain the approximate Virasoro generators $\widetilde{L}_{\pm n}$ as

$\widetilde{L}_{\pm n}=\mathbb{L}^{(e)}_{n}\pm{\bf
i}\mathbb{L}^{(o)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{\approx}}{1\over
2\pi}\int_{0}^{2\pi}d\theta e^{\pm{\bf
i}n\theta}\mathscr{T}(\theta)=\mathscr{L}_{\pm n},\quad n\geq 1,$ (6.8)

since $e^{\pm{\bf i}n\theta}=\cos(n\theta)\pm{\bf i}\sin(n\theta)$.

Now we discuss the construction of $\mathbb{L}^{(o)}_{n}$ and
$\mathbb{L}^{(e)}_{n}$. Let us use $\beta^{(o)}_{n}(\theta)$ and
$\beta^{(e)}_{n}(\theta)$ to denote their weight function:

$\mathbb{L}^{(o)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}d\theta\beta^{(o)}_{n}(\theta)\mathscr{T}(\theta),\quad\mathbb{L}^{(e)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}d\theta\beta^{(e)}_{n}(\theta)\mathscr{T}(\theta).$
(6.9)

We aim to choose $\beta^{(o)}_{n}(\theta),\beta^{(e)}_{n}(\theta)$ in Eq.
(6.9) so that they approximate $\sin(n\theta)$ and $\cos(n\theta)$. Based on
Hypothesis 1, the weight function
$\beta^{(o)}_{n}(\theta),\beta^{(e)}_{n}(\theta)$ shall be a linear
combination of the coolness functions
$\beta^{[\theta_{1},\theta_{2}]}(\theta)$, introduced previously in Eq. (4.2),
on various intervals on a circle. Therefore, to obtain the desired
$\mathbb{L}^{(o)}_{n},\mathbb{L}^{(e)}_{n}$, one needs to answer the following
two questions: (1) Which linear combination of coolness functions gives a good
approximation of $\sin(n\theta),\cos(n\theta)$? (2) Which good modular flow
generator can result in such a linear combination in its weight function?

Figure 13: Comparison between $\beta_{n}^{(o,e)}(\theta)$ and
$\sin(n\theta)/(2\pi),\cos(n\theta)/(2\pi)$ respectively.

For the first question, dividing a circle into $2n$ equal-size intervals

$\left[(k-1){2\pi\over 2n},k{2\pi\over 2n}\right],~{}k=1,\cdots,2n$

and adding up the coolness function of these intervals with alternating signs
will provide an approximation of $\sin(n\theta)$, up to a normalization factor
(denoted as $A_{n}$). Similarly, the approximation of $\cos(n\theta)$ can be
obtained by a shift of $\pi/(2n)$ angle along the circle.202020The important
part is the $\pi/(2n)$ phase difference between $\beta^{(o)}_{n}$ and
$\beta^{(e)}_{n}$, It’s also okay to call $\beta^{(o)}_{n}$ the approximation
for $\cos(n\theta)$ and $\beta^{(e)}_{n}$ for $\sin(n\theta)$. That just
amounts to a $\pi/(2n)$ overall coordinate translation. Explicitly,

$\begin{split}&\beta^{(o)}_{n}(\theta)=A_{n}\sum_{k=1}^{2n}(-1)^{k-1}\beta^{\left[{(k-1)\pi\over
n},{k\pi\over n}\right]}(\theta)\approx{1\over 2\pi}\sin(n\theta),\\\
&\beta^{(e)}_{n}(\theta)=\beta^{(o)}_{n}\left(\theta+{\pi\over
2n}\right)=A_{n}\sum_{k=1}^{2n}(-1)^{k-1}\beta^{\left[{(k-3/2)\pi\over
n},{(k-1/2)\pi\over n}\right]}(\theta)\approx{1\over
2\pi}\cos(n\theta),\end{split}$ (6.10)

where normalization factor $A_{n}$ is defined as

$A_{n}=\left\\{\begin{aligned} &{1\over 2\pi}&&n=1\\\ &\tan\left({\pi\over
2n}\right)(n^{2}-1)/8,&&n\neq 1\end{aligned}\right.~{}.$ (6.11)

This factor $A_{n}$ in the equation above is obtained by demanding

$\int_{0}^{2\pi}\beta^{(o)}_{n}(\theta)\sin(n\theta)={1\over
2},\quad\int_{0}^{2\pi}\beta^{(e)}_{n}(\theta)\cos(n\theta)={1\over 2}.$
(6.12)

See Figure 13 for a comparison between $\beta^{(o,e)}_{n}$ and
$\sin(n\theta)/(2\pi),\cos(n\theta)/(2\pi)$ respectively. We will provide a
quantitative comparison in Appendix C, discussing the error in this
approximation in detail.

Now, we need to identify a certain _good_ modular flow generator that results
in such weight function Eq. (6.10). To that end, we utilize the following
“twisting trick" [Figure 14]. For illustration purposes, let us consider the
construction for $n=2$. We divide the chiral gapless edge into four intervals
of equal size, labeled by $x_{k}\equiv[(k-1)\pi/2,k\pi/2],k=1,2,3,4$. We wish
to obtain a good modular flow generator, denoted as $\mathbb{L}^{(o)}_{2}$,
whose weight function is
$\beta^{x_{1}}(\theta)-\beta^{x_{2}}(\theta)+\beta^{x_{3}}(\theta)-\beta^{x_{4}}(\theta)$,
that is:

$\mathbb{L}^{(o)}_{2}\stackrel{{\scriptstyle\mathbb{V}}}{{\propto}}\mathscr{L}=\mathscr{K}_{x_{1}}^{\chi}-\mathscr{K}_{x_{2}}^{\chi}+\mathscr{K}_{x_{3}}^{\chi}-\mathscr{K}_{x_{4}}^{\chi},$
(6.13)

where $\mathscr{K}_{a}^{\chi}$ is the modular Hamiltonian of an interval $a$
for chiral CFT ground state, defined in Eq. (4.3).

Figure 14: Illustration of the twisting trick. The color (green, red) of each
region stands for the sign $(+,-)$ of the modular Hamiltonian on that region.
Between the two plots, the colored dots are identified.
$L=K_{X_{1}}-K_{X_{2}}+K_{X_{3}}-K_{X_{4}}$,
$L^{\mathrm{twist}}=L_{X^{\prime}_{1}}-L_{X^{\prime}_{2}}+K_{X^{\prime}_{3}}-L_{X^{\prime}_{4}}$.
Neither $L$ nor $L^{\mathrm{twist}}$ is a good modular flow generator, as
there are “excitations” near the entanglement cut. Once we add them, such
“excitations” cancel.

One approach to achieve this would be to choose four regions $X_{k}$ anchored
at the interval $x_{k}$, $k=1,2,3,4$ along the gapless edge shown in the left
column of Figure 14. The linear combination
$L=K_{X_{1}}-K_{X_{2}}+K_{X_{3}}-K_{X_{4}}$, with alternating signs, acting on
$\left|\Psi\right\rangle$ shall create the same edge excitations as
$\mathscr{L}$ in Eq. (6.13) does on the chiral CFT ground state. However, this
$L$ is not a good modular flow generator: $L$ will create bulk excitations
along the entanglement cuts $b_{k}$ shown in the Figure 14 due to the action
of $K_{X_{k}}$ on $\left|\Psi\right\rangle$. To cancel these bulk excitations,
we introduce a _twisting trick_ to construct another linear combinations of
modular Hamiltonian $L^{\rm
twist}=K_{X^{\prime}_{1}}-K_{X^{\prime}_{2}}+K_{X^{\prime}_{3}}-K_{X^{\prime}_{4}}$.
Each region $X^{\prime}_{k}$ is obtained by a twisting of $X_{k}$ in the bulk,
such that the boundary of $X^{\prime}_{k}$ includes the entanglement cut
$b_{k+1}$ from its neighboring untwisted region $X_{k+1}$, i.e.
$b_{k+1}\subset\partial X^{\prime}_{k}$; see the right column of Figure 14 for
the illustration of the twisted regions. Notice $X^{\prime}_{k}$ anchors at
the same interval at the gapless edge as $X_{k}$. Therefore $L^{\rm twist}$
will create the same edge excitations as $L$. Now we can simply add up

$\mathbb{L}^{(o)}_{2}\equiv{A_{2}\over 2}\left(L+L^{\rm twist}\right),$ (6.14)

where $A_{2}$ is the prefactor defined in Eq. (6.11). Notice
$\mathbb{L}^{(o)}_{2}$ will create the same edge excitations as $L$ and
$L^{\rm twist}$, but create no excitations in the bulk. Indeed, we can see
that the excitations in $b_{k}$ created by $K_{X_{k}}$ is canceled by
$K_{X^{\prime}_{k-1}}$, as they have opposite sign in $\mathbb{L}^{(o)}_{2}$.
As a result, we obtain a good modular flow generator $\mathbb{L}^{(o)}_{2}$
that satisfies Eq. (6.13). To obtain $\mathbb{L}^{(e)}_{2}$, we can simply
shift the whole setup by $\pi/4$ clockwise, such that $x_{k}\to
y_{k}=[(k-3/2)\pi/2,(k-1/2)\pi/2],X_{k}\to Y_{k},X^{\prime}_{k}\to
Y^{\prime}_{k}$, and then use the same summation using
$K_{Y_{k}},K_{Y^{\prime}_{k}}$ as we did for $K_{X_{k}},K_{X^{\prime}_{k}}$.

The idea we described above works for the construction of
$\mathbb{L}^{(o)}_{n},\mathbb{L}^{(e)}_{n}$ with $n\geq 1$, defined in Eq.
(6.9). We now explicitly describe the algorithm using Figure 15:

  * •

_Step 1_ : For a 2+1D purely-chiral state $\left|\Psi\right\rangle$, we first
choose a gapless edge of length $L$ and divide it into $2n$ equal-size
intervals, labeled by $x_{j},j=1,\cdots,2n$. We remark that the starting point
of the choice of intervals does not affect the construction due to the
translation symmetry along the edge.

  * •

_Step 2_ : Then we make a set of twisted squares $X^{L}_{j},X^{R}_{j}$
anchored at the interval $x_{j}$. They are twisted to the left $(L)$ and right
$(R)$ such that $X^{L}_{j}$ meets $X^{R}_{j+1}$ at the top edge of the
squares, see Figure 15.

Figure 15: Construction of $\mathbb{L}^{(o)}_{n}$. The color indicates the
sign of the modular Hamiltonian of that region. Each of the figures depicts a
connected component of a boundary. Left and right sides of the figure are
identified. We can be agnostic about what happens farther into the bulk; for
example, the whole geometry could be a disk. The regions used in the
construction of $\mathbb{L}^{(e)}_{n}$ are called $Y_{k}^{R}$ and $Y_{k}^{L}$;
they are obtained by shifting $X_{k}^{R}$ and $X_{k}^{L}$ to the left by
$\pi/2n$.

  * •

_Step 3:_ With this setup, we construct

$\mathbb{L}^{(o)}_{n}={A_{n}\over
2}\sum_{j=1}^{2n}(-1)^{j-1}\left(K_{X_{j}^{R}}+K_{X_{j}^{L}}\right).$ (6.15)

  * •

_Step 4:_ To obtain $\mathbb{L}^{(e)}_{n}$, we can shift the intervals and
squares by $L/(4n)$ distance to the left direction: $x_{i}\to y_{i}$,
$(X^{L}_{j},X^{R}_{j})\to(Y^{L}_{j},Y^{R}_{j}),j=1,\cdots,2n$. Then

$\mathbb{L}^{(e)}_{n}={A_{n}\over
2}\sum_{j=1}^{2n}(-1)^{j-1}\left(K_{Y_{j}^{R}}+K_{Y_{j}^{L}}\right).$ (6.16)

In summary, in this way, we obtain good modular flow generators
$\mathbb{L}^{(o)}_{n},\mathbb{L}^{(e)}_{n}$:

$\begin{split}&\mathbb{L}^{(o)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}\beta_{n}^{(o)}(\theta)\mathscr{T}(\theta)\approx{1\over
2\pi}\int_{0}^{2\pi}\sin(n\theta)\mathscr{T}(\theta)\\\
&\mathbb{L}^{(e)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}\beta_{n}^{(e)}(\theta)\mathscr{T}(\theta)\approx{1\over
2\pi}\int_{0}^{2\pi}\cos(n\theta)\mathscr{T}(\theta)\end{split}$ (6.17)

At last, by making linear combinations of $\mathbb{L}^{(o)}_{n}$ and
$\mathbb{L}^{(e)}_{n}$,

$\widetilde{L}_{\pm n}=\mathbb{L}^{(e)}_{n}\pm{\bf i}\mathbb{L}^{(o)}_{n},$
(6.18)

we can approximate $e^{\pm{\bf i}n\theta}/(2\pi)$ in the weight function,

$\begin{split}&\widetilde{L}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}\left(\beta^{(e)}_{n}+{\bf
i}\beta^{(o)}_{n}\right)\mathscr{T}(\theta)\approx{1\over
2\pi}\int_{0}^{2\pi}e^{{\bf i}n\theta}\mathscr{T}(\theta)=\mathscr{L}_{n}\\\
&\widetilde{L}_{-n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}\left(\beta^{(e)}_{n}-{\bf
i}\beta^{(o)}_{n}\right)\mathscr{T}(\theta)\approx{1\over
2\pi}\int_{0}^{2\pi}e^{-{\bf
i}n\theta}\mathscr{T}(\theta)=\mathscr{L}_{-n},\end{split}$ (6.19)

and as a result we obtain the approximate Virasoro generators
$\widetilde{L}_{\pm
n}\stackrel{{\scriptstyle\mathbb{V}}}{{\approx}}\mathscr{L}_{\pm n},n\geq 1$.

####  6.1.3 Expansion of $\widetilde{L}_{n}$ in Virasoro modes

With the approximate $\widetilde{L}_{n}$ constructed, we now discuss the exact
form of $\widetilde{L}_{n}$ in terms of Virasoro generators $L_{m}$. Such an
expression of $\widetilde{L}_{n}$ is useful in analytically computing several
quantities that are related to them, such as those discussed in Section 6.2.
The explicit expression of $\widetilde{L}_{n}$ also reveals the approximation
“error” of $\widetilde{L}_{n}$ to $L_{n}$.

For $n=0$, as we mentioned before in Section 6.1.1, the construction gives
$\widetilde{L}_{0}=L_{0}$. Therefore, we shall focus on $\widetilde{L}_{n\neq
0}$. We remark that even though the construction of $\widetilde{L}_{\pm 1}$
also gives exactly $L_{\pm 1}$, we still include $n=\pm 1$ into the discussion
below, because the construction for all $n\neq 0$ follows the same procedure,
so the analysis is applicable to all $n\neq 0$.

Based on the construction Eq. (6.19), the approximate Virasoro generator takes
the form

$\widetilde{L}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\int_{0}^{2\pi}d\theta\beta_{n}(\theta)\mathscr{T}(\theta),\quad
n\neq 0$ (6.20)

where

$\beta_{\pm n}(\theta)=\beta^{(e)}_{n}(\theta)\pm{\bf
i}\beta^{(o)}_{n}(\theta),\quad n\geq 1.$ (6.21)

Employing $\mathscr{T}(\theta)=\sum_{m\in\mathbb{Z}}L_{m}e^{-{\bf i}m\theta}$
in Eq. (6.20), one can obtain the exact form of $\widetilde{L}_{n}$ in terms
of exact Virasoro generators $L_{n}$

$\widetilde{L}_{n}=\sum_{m\in\mathbb{Z}}\lambda_{n,m}L_{m},$ (6.22)

where the $\lambda_{n,m}$ are the Fourier coefficients of $\beta_{n}(\theta)$:

$\lambda_{n,m}=\int_{0}^{2\pi}d\theta\beta_{n}(\theta)e^{-{\bf
i}m\theta}=\left\\{\begin{aligned} &1,&&\text{ if }|n|=1,m=n\\\ &{n^{3}-n\over
m^{3}-m},&&\text{ if }|n|\geq 2,m=(-1)^{k}(2k+1)n,k\in\mathbb{N}\\\
&0,&&\text{ otherwise}\end{aligned}\right.$ (6.23)

We make several remarks regarding the exact form $\widetilde{L}_{n},n\neq 0$.

  * •

For $|n|=1$, we emphasize that $\widetilde{L}_{n}$ is exactly $L_{n}$, i.e.
$\widetilde{L}_{\pm 1}=L_{\pm 1}$. This can be seen directly by checking
$\beta_{\pm 1}(\theta)=e^{\pm{\bf i}\theta}$.

  * •

For $|n|\geq 2$, we can see that the Virasoro mode labels appearing in
$\widetilde{L}_{n}$ are $n$ multiplied by odd integers with alternating signs:
$L_{n},L_{-3n},L_{5n}$, etc. Among these modes, the leading order contribution
is $L_{n}$, as $\lambda_{n,n}=1$. As $|m/n|$ increases, the magnitude of its
coefficient $|\lambda_{n,m}|$ decay as $O(|m/n|^{-3})$, which ensures the
contribution from these sub-leading terms are considerably small.

Moreover, having obtained the exact form of $\widetilde{L}_{n},n\neq 0$, we
can also design a procedure to reduce the subleading terms in
$\widetilde{L}_{n}$, so that it becomes a better approximation to $L_{n}$. The
procedure is explicitly introduced in Appendix C. As a limit of such a
procedure, one can obtain the _exact_ Virasoro generator $L_{n}$ as a linear
combination of good modular flow generators.

###  6.2 Quantitative properties

In this subsection, we discuss several quantitative properties related to
$\widetilde{L}_{n}$, which will be tested in numerics [Section 7]. Based on
Hypothesis 1, these proprieties are analytically expressed in terms of the
Fourier coefficients $\widetilde{L}_{n}$ in Eq. (6.22). Therefore the
excellent agreement we shall see in Section 7 provides support for the
correctness of the exact form of $\widetilde{L}_{n}$ in Eq. (6.22), as well as
to the validity of Hypothesis 1. These quantities of $\widetilde{L}_{n}$ will
also be compared with those from $L_{n}$. Such a comparison gives a measure of
the quality of the approximation of $\widetilde{L}_{n}$ to $L_{n}$.

####  6.2.1 Variance

The first quantity we consider is the variance of the good modular flow
generators $\mathbb{L}^{(o)}_{n},\mathbb{L}^{(e)}_{n}$ that are involved in
the construction of $\widetilde{L}_{n}$. We will focus on
$\mathbb{L}^{(o)}_{n}$; the results for the other one $\mathbb{L}^{(e)}_{n}$
can be obtained similarly.

Based on Eq. (6.18), $\mathbb{L}^{(o)}_{n},n\geq 1$ is of the form

$\mathbb{L}^{(o)}_{n}={-{\bf i}\over
2}(\widetilde{L}_{n}-\widetilde{L}_{-n}).$ (6.24)

Taking the exact form of $\widetilde{L}_{\pm n}$ from Eq. (6.22), we can
express $\mathbb{L}^{(o)}_{n}$ in terms of Virasoro generators

$\mathbb{L}^{(o)}_{n}=\sum_{n\in\mathbb{Z}}\lambda^{(o)}_{n,m}L_{m},$ (6.25)

where

$\lambda^{(o)}_{n,m}=-{{\bf i}\over 2}(\lambda_{n,m}-\lambda_{-n,m}),\quad
n\geq 1.$ (6.26)

As we mentioned in Section 6.2.1, the variance of a good modular flow
generator can be expressed in terms of its Fourier coefficient as Eq. (5.15).
Therefore, for $\mathbb{L}^{(o)}_{n}$, we obtain

$\sigma(\mathbb{L}^{(o)}_{n})^{2}={c\over 12}\sum_{m\geq
0}|\lambda^{(o)}_{n,m}|^{2}(m^{3}-m).$ (6.27)

This quantity can be explicitly computed by making use of the analytic formula
of $\lambda^{(o)}_{n,m}$ obtained from Eq. (6.26) and Eq. (6.23):

$\sigma(\mathbb{L}^{(o)}_{n})^{2}={c\over
192}n(n^{2}-1)^{2}\left[2\psi\left({1\over 2}\right)-\psi\left({n+1\over
2n}\right)-\psi\left({n-1\over 2n}\right)\right],$ (6.28)

where $\psi(z)$ is the polygamma function, defined as
$\psi(z)\equiv\Gamma^{\prime}(z)/\Gamma(z)$ from the gamma function
$\Gamma(z)$ and its first derivative $\Gamma^{\prime}(z)$. Thus, at large $n$,
$\sigma(\mathbb{L}^{(o)}_{n})^{2}\sim n^{3}$.

####  6.2.2 Expectation values of commutators

We are also interested in expectation values of commutators among
$\widetilde{L}_{n},n\neq 0$ since they encode information about the Virasoro
algebra relations. Explicitly, we shall consider
$\left\langle\Psi\right|[\widetilde{L}_{n},\widetilde{L}_{-n}]\left|\Psi\right\rangle,n\geq
1$. Recall $\widetilde{L}_{\pm n}=\mathbb{L}^{(e)}_{n}\pm{\bf
i}\mathbb{L}^{(o)}_{n}$, therefore the expectation value of the commutator can
be expressed as

$\left\langle\Psi\right|[\widetilde{L}_{n},\widetilde{L}_{-n}]\left|\Psi\right\rangle=2{\bf
i}\left\langle\Psi\right|[\mathbb{L}^{(e)}_{n},\mathbb{L}^{(o)}_{n}]\left|\Psi\right\rangle,\quad
n\geq 1.$ (6.29)

Before we explicitly compute ${\bf
i}\left\langle\Psi\right|[\mathbb{L}^{(e)}_{n},\mathbb{L}^{(o)}_{n}]\left|\Psi\right\rangle,n\geq
1$, we first discuss the general case, namely, the expectation value ${\bf
i}\left\langle\Psi\right|[\mathbb{L}^{1},\mathbb{L}^{2}]\left|\Psi\right\rangle$
for any two good modular flow generators
$\mathbb{L}^{1},\mathbb{L}^{2}\in\mathfrak{g}^{\bullet}$. There are, in
general, two methods to compute it.

One way is to follow the same idea we used to compute the variance and use
Fourier expansion of $\mathbb{L}^{1},\mathbb{L}^{2}$

$\mathbb{L}^{1}=\sum_{m\in\mathbb{Z}}\lambda^{1}_{m}L_{m},\quad\mathbb{L}^{2}=\sum_{m\in\mathbb{Z}}\lambda^{2}_{m}L_{m},$
(6.30)

then the expectation value of their commutators is

$\left\langle\Psi\right|[\mathbb{L}^{1},\mathbb{L}^{2}]\left|\Psi\right\rangle={c\over
12}\sum_{m\in\mathbb{Z}}\lambda_{m}^{1}\lambda_{-m}^{2}(m^{3}-m).$ (6.31)

The other way is to utilize the result in [58]: The expectation value of the
commutator between two chiral CFT modular Hamiltonian on interval $ab,bc$ is

${\bf
i}{\left\langle\Omega\right|}[\mathscr{K}_{ab}^{\chi},\mathscr{K}_{bc}^{\chi}]\left|\Omega\right\rangle={\pi
c_{-}\over 6}(2\eta_{a,b,c}-1),$ (6.32)

where $\eta_{a,b,c}={|a|\cdot|c|\over|ab|\cdot|bc|}$ is the cross-ratio of the
successive intervals $(a,b,c)$, and $c_{-}$ is the chiral central charge. With
Eq. (6.32) at hand, we can, in general, compute expectation values between any
two good modular flow generators

$\displaystyle\mathfrak{g}^{\bullet}\ni\mathbb{L}^{1}=\sum_{A^{1}}\lambda_{A}^{1}K_{A^{1}}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\sum_{a^{1}}\lambda_{A^{1}}\mathscr{K}^{\chi}_{a^{1}},$
(6.33)
$\displaystyle\mathfrak{g}^{\bullet}\ni\mathbb{L}^{2}=\sum_{A^{2}}\lambda_{A}^{2}K_{A^{2}}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\sum_{a^{2}}\lambda_{A^{2}}\mathscr{K}^{\chi}_{a^{2}}.$

In the equation above, $a^{1},a^{2}$ are the intervals where $A^{1},A^{2}$
intersect with the gapless edge. Using the isometry relation in Hypothesis 1,
the expectation value of the commutator $[\mathbb{L}^{1},\mathbb{L}^{2}]$ is

${\bf
i}\left\langle\Psi\right|[\mathbb{L}^{1},\mathbb{L}^{2}]\left|\Psi\right\rangle=\sum_{A^{1},A^{2}}\lambda_{A^{1}}\lambda_{A^{2}}\left({\bf
i}\left\langle\Omega\right|[\mathscr{K}_{a^{1}},\mathscr{K}_{a^{2}}]\left|\Omega\right\rangle\right),$
(6.34)

where each commutator ${\bf
i}\left\langle\Omega\right|[\mathscr{K}_{a^{1}},\mathscr{K}_{a^{2}}]\left|\Omega\right\rangle$
in the right hand side can be computed via Eq. (6.32).

We shall use the second method to compute ${\bf
i}\left\langle\Psi\right|[\mathbb{L}^{(e)}_{n},\mathbb{L}^{(o)}_{n}]\left|\Psi\right\rangle$,
as it provides a succinct analytic formula of the result. Based on the
construction of $\mathbb{L}^{(e)}_{n},\mathbb{L}^{(o)}_{n}$ in Eqs. (6.15) and
(6.16),

$\mathbb{L}^{(o)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}A_{n}\sum_{k}(-1)^{k-1}\mathscr{K}_{x_{k}},\quad\mathbb{L}^{(e)}_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}A_{n}\sum_{k}(-1)^{k-1}\mathscr{K}_{y_{k}},$
(6.35)

where $x_{k}=\left[{(k-1)\pi\over n},{k\pi\over n}\right]$ and
$y_{k}=\left[{(k-3/2)\pi\over n},{(k-1/2)\pi\over n}\right]$ are sequences of
intervals on a circle, as explained in Section 6.1.2. By counting the
intersection of intervals, we can obtain

${\bf
i}\left\langle\Psi\right|[\mathbb{L}^{(e)}_{n},\mathbb{L}^{(o)}_{n}]\left|\Psi\right\rangle=A_{n}^{2}\cdot
4n\cdot{\pi c_{-}\over 6}(1-2\eta_{n}),$ (6.36)

where

$\eta_{n}={\sin(\pi/(4n))^{2}\over\sin(\pi/(2n))^{2}}.$ (6.37)

Therefore, plugging in the expression of $A_{n}$ in Eq. (6.11), we can obtain
the final expression of
$\left\langle\Psi\right|[\widetilde{L}_{n},\widetilde{L}_{-n}]\left|\Psi\right\rangle$.

$\left\langle\Psi\right|[\widetilde{L}_{n},\widetilde{L}_{-n}]\left|\Psi\right\rangle={\pi
c_{-}\over 24}n(n^{2}-1)^{2}{\sin^{2}(\pi/(4n))\over\cos(\pi/(2n))},\quad
c_{-}=c.$ (6.38)

where the chiral central charge $c_{-}$ is equal to the holomorphic central
charge $c$, since we are working with purely chiral CFT212121$\bar{c}=0$,
therefore $c_{-}=c-\bar{c}=c$..

Later we will numerically compute the commutators
$\left\langle\Psi\right|[\widetilde{L}_{n},\widetilde{L}_{-n}]\left|\Psi\right\rangle$
and compare them with Eq. (6.38). Such tests examine the validity of the
Hypothesis 1. We are also going to compare the numerical results with the
results obtained by Virasoro algebra generators. These tests demonstrate the
quality of $\widetilde{L}_{n}$ as an approximation to Virasoro generator
$L_{n}$.

As a mathematical aside, from the two ways of computing
$\left\langle\Psi\right|[\mathbb{L}^{1},\mathbb{L}^{2}]\left|\Psi\right\rangle$
we are able to show a number of interesting relations between their Fourier
coefficients and the set of cross-ratios. For example, in the computation of
$\left\langle\Psi\right|[\mathbb{L}^{(e)}_{n},\mathbb{L}^{(o)}_{n}]\left|\Psi\right\rangle$,
we have shown

$\sum_{m\in\mathbb{Z}}|\lambda_{n,m}|^{2}(m^{3}-m)={\pi\over
2}n(n^{2}-1)^{2}{\sin^{2}(\pi/(4n))\over\cos(\pi/(2n))},\quad n\geq 1$ (6.39)

where $\lambda_{n,m}$ is given in Eq. (6.23).

####  6.2.3 Expectation value of double commutators

Besides the expectation values of commutators, we shall also consider
expectation values of double commutators
$\left\langle\Psi\right|[[\widetilde{L}_{m},\widetilde{L}_{n}],\widetilde{L}_{k}]\left|\Psi\right\rangle$.
The results can be explicitly computed by utilizing the Fourier expansion of
$\widetilde{L}_{n}$ (Eq. (6.22)) in terms of Virasoro generators. Such
quantities will allow us to test further the closure of our realization of the
Virasoro algebra.

One particular double commutator that we will compute is
$\left\langle\Psi\right|[[\widetilde{L}_{n},L_{0}],\widetilde{L}_{-n}]\left|\Psi\right\rangle$,
$n\geq 1$. By making use of the Fourier expansion of $\widetilde{L}_{\pm n}$
as follows

$\widetilde{L}_{n}=\sum_{m\in\mathbb{Z}}\lambda_{n,m}L_{m},\quad\widetilde{L}_{-n}=(\widetilde{L}_{n})^{\dagger}=\sum_{m\in\mathbb{Z}}\lambda_{n,m}^{*}L_{-m},\quad
n\geq 1,$ (6.40)

and the Virasoro algebraic relations, we can derive

$\left\langle\Psi\right|[[\widetilde{L}_{n},L_{0}],\widetilde{L}_{-n}]\left|\Psi\right\rangle={c\over
12}\sum_{m\in\mathbb{Z}}|\lambda_{n,m}|^{2}(m^{4}-m^{2}).$ (6.41)

With the analytic formula of $\lambda_{n,m}$ given in Eq. (6.23), we are able
to explicitly compute Eq. (6.41) and the result is

$\left\langle\Psi\right|[[\widetilde{L}_{n},L_{0}],\widetilde{L}_{-n}]\left|\Psi\right\rangle={\pi
c\over 48}n(n^{2}-1)^{2}\tan\left({\pi\over 2n}\right).$ (6.42)

The result shall be compared with numerical results. Such a test examines not
only the Fourier expansion of $\widetilde{L}_{\pm n}$ but also the
construction of $L_{0}$.

Note that Eq. (6.41) is the ${\cal O}(t^{2})$ term in the quantity
$\left\langle\Psi(t)\right|L_{0}\left|\Psi(t)\right\rangle$, with
$\left|\Psi(t)\right\rangle=e^{{\bf
i}t\widetilde{L}_{-n}}\left|\Psi\right\rangle$, and therefore as explained in
Section 5.3 is the leading contribution to a Schwarzian functional.

##  7 Numerical tests

In the previous section, we explicitly described the procedures to construct
approximate Virasoro algebra $\widetilde{L}_{n}$ and made several predictions
based on Hypothesis 1. In this section, we study two specific reference
states, namely $p+{\bf i}p$ superconductor (SC) ground state and semion ground
state, to test the construction of $\widetilde{L}_{n}$ via the predictions we
discussed in the previous section. We will also compare results obtained by
$\widetilde{L}_{n}$ and $L_{n}$, to test the quality of the approximation
$\widetilde{L}_{n}\approx L_{n}$.

$p+{\bf i}p$ SC is a free fermion system. By the means introduced by [59], we
are able to reach pretty large system sizes. Therefore, we will perform a
thorough test in $p+{\bf i}p$ SC. One may wonder, however, whether the
conclusions rely on the free fermion property. Therefore, we also perform
tests on a ground state with chiral semion topological order, which has no
free-fermion representation. For the semion ground state, due to the
limitation of the computation power, we won’t do as many tests as in $p+{\bf
i}p$ SC, but the results nevertheless indicate that the construction works and
the predictions are valid.

###  7.1 Tests in $p+{\bf i}p$ superconductor

We choose the $p+{\bf i}p$ SC Hamiltonian as Eq. (3.18), with the same
parameters used in Section 3.2. Again, we choose the anti-periodic boundary
condition in the $x$-direction and the open boundary condition in the
$y$-direction so that there is no flux threaded inside the cylinder, and full-
boundary A0, A1 introduced in Section 4.3 are satisfied. We will specify the
size of the cylinder $L_{x}\times L_{y}$ later.

The ground state of $p+{\bf i}p$ SC for this choice of parameters is in its
chiral gapped phase. In the bulk, its entanglement entropy satisfies the area
law and therefore, the entanglement bootstrap axioms A0 and A1 are
approximately satisfied. It has chiral central charge $c_{-}=1/2$ in the bulk
and has chiral gapless edge, which is robust against any local perturbations.
Next we check these expectations.

####  7.1.1 Tests of the ground state

Before we test the construction of the approximate Virasoro generators, we
first check whether $p+{\bf i}p$ ground state is indeed a suitable state to do
the constructions. Therefore, we will check the following: (1) The
entanglement bootstrap axioms A0 and A1 in the bulk as well as the full
boundary A0 and A1. Recall this property is crucial in our argument that the
bulk reduced density matrices are invariant under good modular flow. (2)
Modular commutators. This quantity indicates the chirality of the state and
computes the chiral central charge $c_{-}=c-\bar{c}$ from the bulk. (3) Vector
fixed-point equation: As argued in [27, 54], the satisfaction of the vector
fixed-point equation near the edge is a strong indication that the edge is
described by a CFT. Moreover, the proportionality factor in the vector fixed-
point equation contains the total central charge $c_{\mathrm{tot}}=c+\bar{c}$
of the edge CFT. If the total central charge $c_{\mathrm{tot}}$ is equal to
the absolute value of chiral central charge $c_{-}$, then this indicates the
edge CFT is purely chiral. Therefore, this test provides strong evidence that
Hypothesis 1 is satisfied by the state.

Figure 16: Subsystems used in numerical tests. $BCD$ in regions 1, 2, and 3 in
the left column are used to test bulk entanglement bootstrap axioms A0 and A1.
$ABC$ in regions 1, 2, 3 of the middle column are used to test the modular
commutator formula. The region $ABCD$ in the right column, near the edge, is
used to test the vector fixed-point equation. The location of the subsystems
can be inferred from the plot. We omit the $x$-coordinate of each region
because the ground state is translational invariant in that direction.

We choose the subsystems shown in Figure 16 on a cylinder with size
$L_{x}=80,L_{y}=80$. The tests are explicitly enumerated as follows:

  1. 1.

For A0 and A1, we compute

$\Delta(BD,C,\emptyset)=S_{BCD}+S_{C}-S_{BD},\quad\Delta(B,C,D)=S_{BD}+S_{CD}-S_{B}-S_{D}$
(7.1)

respectively, and see if they are close to zero. The subsystems are regions
1,2,3 shown in the left column in in Figure 16, which are progressively
farther away from the gapless edge. We also compute the full-boundary A0 and
A1, using the setup shown in Figure 17.

Figure 17: Subsystems used in the test of full boundary axioms A0 and A1.

  2. 2.

For the modular commutator, we computed

$J(A,B,C)={\bf
i}\left\langle\Psi\right|[K_{AB},K_{BC}]\left|\Psi\right\rangle$ (7.2)

for regions 1,2,3 shown in the middle column, which are also progressively
farther away from the gapless edge. We compare the results with $c_{-}\pi/3$
with $c_{-}=1/2$.

  3. 3.

For the vector fixed-point equation (4.13), the setup is shown in the right
column in Figure 16. We construct
$\hat{\Delta}(AD,B,C)=K_{ABD}+K_{BC}-K_{AD}-K_{C}$ and
$\hat{I}(A:C|B)=K_{AB}+K_{BC}-K_{ABC}-K_{B}$ to obtain

${\mathcal{K}}_{\mathbb{D}}(\eta)=\eta\hat{\Delta}(AD,B,C)+(1-\eta)\hat{I}(A:C|B),$
(7.3)

with $\eta$ being the cross-ratio of the intervals $(a,b,c)$

$\eta={\sin(\pi|a|/L_{x})\sin(\pi|c|/L_{x})\over\sin(\pi|bc|/L_{x})\sin(\pi|ab|/L_{x})}.$
(7.4)

To test
${\mathcal{K}}_{\mathbb{D}}(\eta)\left|\Psi\right\rangle\propto\left|\Psi\right\rangle$,
we compute the square root of its variance:

$\sigma({\mathcal{K}}_{\mathbb{D}}(\eta))=\sqrt{\left\langle\Psi\right|{\mathcal{K}}_{\mathbb{D}}(\eta)^{2}\left|\Psi\right\rangle-\left\langle\Psi\right|{\mathcal{K}}_{\mathbb{D}}(\eta)\left|\Psi\right\rangle^{2}}$
(7.5)

and see if it vanishes. We also compute the proportionality factor

$\left\langle\Psi\right|{\mathcal{K}}_{\mathbb{D}}(\eta)\left|\Psi\right\rangle={c_{\mathrm{tot}}\over
6}h(\eta),$ (7.6)

with $h(\eta)=-\eta\ln(\eta)-(1-\eta)\ln(1-\eta)$, and test whether
$c_{\mathrm{tot}}=1/2$, which is an indication that the edge CFT is purely
chiral.

The linear sizes of the subsystems for these tests are labeled by $\ell$, as
shown in Figure 18. We now describe the numerical results.

(1) Entanglement bootstrap axioms: The results for axioms A0, A1 are shown in
Figure 18.

Figure 18: Numerical results for the violation of entanglement bootstrap
axioms A0 and A1 in the $p+{\bf i}p$ SC ground state as a function of the
linear size of the subsystem $\ell$, as shown in Figure 16. From the slope of
the linear fit result of A0, one can extract an upper bound of the correlation
length, as explained in Eq. (7.8)

For the parameters we chose, both $\Delta(BD,C,\emptyset)$ and $\Delta(B,C,D)$
decay to zero exponentially as the linear size $\ell$ of the subsystem
increases. For some different choices of parameters,222222One instance of such
parameter is $\Delta=0.9,\mu=1.3$ in Eq. (3.18). Such an algebraic decay
feature persists if one varies these parameters around this choice. as far as
we observed, A1 and modular commutator could decay algebraically, while A0
still decays exponentially. It is interesting that the means of approach to
the fixed point seems not to be a universal property of the phase. The reason
for such a distinction is a topic for further study.

As an aside, the rate of exponential decay of $\Delta(BD,C,\emptyset)$ (i.e.
the entanglement entropy combination that appears in A0) can be used as a
characterization of correlation length, as it upper bounds any correlation
functions in the following way:

$\Delta(BD,C,\emptyset)\geq I(A:C)\geq{1\over
2}\left({\langle{\mathcal{O}}_{A}{\mathcal{O}}_{C}\rangle_{c}\over\|{\mathcal{O}}_{A}\|\cdot\|{\mathcal{O}}_{C}\|}\right)^{2},$
(7.7)

where $A$ is any region outside $BCD$, for $BCD$ shown in the left column in
Figure 16, and ${\mathcal{O}}_{A},{\mathcal{O}}_{C}$ are any two operators
supported on $A$ and $C$ respectively. The first inequality follows from SSA.
The second inequality is a well-known result in quantum information theory
[60], where
$\langle{\mathcal{O}}_{A}{\mathcal{O}}_{C}\rangle_{c}\equiv\langle{\mathcal{O}}_{A}{\mathcal{O}}_{C}\rangle-\langle{\mathcal{O}}_{A}\rangle\langle{\mathcal{O}}_{C}\rangle$
stands for the connected correlation function and $\|{\mathcal{O}}\|$ is
defined as
$\mathrm{sup}\left\\{\left.\sqrt{\langle\psi|{\mathcal{O}}^{\dagger}{\mathcal{O}}|\psi\rangle}\right|\forall\left|\psi\right\rangle\text{
w.t. }|\langle\psi|\psi\rangle|=1\right\\}$. Therefore, from the exponential
decaying violation of A0, i.e. $\Delta(BD,C,\emptyset)\propto
e^{-\ell/\xi_{*}}$, one can obtain an upper bound on the correlation length
$\xi$ of the correlation function
$\langle{\mathcal{O}}_{A}{\mathcal{O}}_{C}\rangle_{c}\propto e^{-\ell/\xi}$:

$\Delta(BD,C,\emptyset)=\alpha e^{-\ell/\xi_{*}}\geq{1\over
2}\left({\langle{\mathcal{O}}_{A}{\mathcal{O}}_{C}\rangle_{c}\over\|{\mathcal{O}}_{A}\|\cdot\|{\mathcal{O}}_{C}\|}\right)^{2}=\beta
e^{-2\ell/\xi}\quad\Rightarrow\quad\xi\leq 2\xi_{*},$ (7.8)

where $\alpha,\beta$ are two prefactors which are not important in the current
discussion. As a result, from the linear fit of $\ln(\Delta(BD,C,\emptyset))$
versus $\ell$ shown in the first plot in Figure 18, we can conclude that the
correlation length $\xi$ of any correlation function satisfies $\xi\leq
2/1.66\approx 1.2$ lattice spacings, for the choice of $p+{\bf i}p$
Hamiltonian parameters used in the numerics.

We also computed the violation of full-boundary A0, A1 in the setup shown in
Figure 17. The results are of the order $O(10^{-8})$. Recall such full-
boundary axioms play an important role in the cylinder argument [Section 4.3],
as well as in the discussion of relation between state overlap and fidelity of
reduced density matrices on the edge region for comparison between the
reference state and coherent states [Section 5.2]. The latter shall be tested
in the following subsubsection 7.1.3.

(2) Modular commutators: The results for modular commutators are shown in
Figure 19. With this choice of parameters, the modular commutators $J(A,B,C)$
approach the expected value ${\pi c_{-}\over 3}$ with $c_{-}=1/2$
exponentially, which agrees with the fact that the non-trivial phase of the
$p+{\bf i}p$ superconductor has a chiral central charge $1/2$. As for the test
of A1, we also observe that $J(A,B,C)$ can approach $\pi/6$ algebraically for
certain other choices of parameters. The fact that the data for regions 1, 2
and 3 largely overlap corroborates the fact that the state satisfies bulk A1
with pretty small violation, because bulk A1 enables one to deform or move the
regions of the subsystems used in the modular commutator in the bulk without
changing its value [17]232323One might be concerned that region 1 for modular
commutator test touches the gapless edge. We comment that the proof using A1
still goes through, as long as the triple intersection point is far away
(compared with the correlation length) from the edge and $A,B,C$ are not all
intersecting with the gapless edge..

Figure 19: Difference of the numerical results of the modular commutator from
its theoretical expectation in the $p+{\bf i}p$ SC ground state, for various
regions, as a function of the linear size of the region $\ell$ shown in Figure
16.

(3) Vector fixed-point equation: The results of tests of the vector fixed-
point equation are shown in Figure 20. The left column shows the results for
testing the proportionality via $\sigma({\mathcal{K}}_{\mathbb{D}}(\eta))$,
and the right column shows the results for the total central charge computed
from the proportionality factor. Both results satisfy our expectations with
algebraically decreasing discrepancy. As the system size grows,
$\sigma({\mathcal{K}}_{\mathbb{D}}(\eta))\to 0$ indicates the CFT nature of
the gapless edge. And $c_{\mathrm{tot}}$ approaches $1/2$, the expected chiral
central charge, indicating that the CFT is purely chiral.

Figure 20: Left: The amount of violation of the vector fixed-point equation in
the $p+{\bf i}p$ superconductor ground state, as a function of linear region
size $\ell$, as shown in Fig. 16. The $\eta$ used in the fixed point equation
is the geometrical cross-ratio. Right: The deviation of the central charge
inferred from the RHS from its expected value. The fact that we find
$c_{\text{tot}}=1/2$ is evidence that the state is indeed purely chiral.

In summary, the numerical data indicates that: (1) The $p+{\bf i}p$ SC ground
state $\left|\Psi\right\rangle$ from the Hamiltonian Eq. (3.18) with such
choice of parameters satisfies EB axioms A0, A1 (both bulk version and full
boundary version) and is in a non-trivial chiral phase. (2) It has a purely
chiral CFT on the gapless edge, whose total central charge agrees with the
chiral central charge computed from the bulk wavefunction. Therefore, such a
state is ideal for the construction of approximate Virasoro generators.

####  7.1.2 Tests of constructions of Virasoro generators

In the previous subsubsection, we concluded that the $p+{\bf i}p$ SC ground
state from the Hamiltonian Eq. (3.18) with such a choice of parameters listed
in [Section 3.2] is suitable for the construction of approximate Virasoro
generators $\widetilde{L}_{n}$. We now put the system on a cylinder of size
$L_{x}=240,L_{y}=24$ and test the construction of $\widetilde{L}_{n}$ from its
ground state $\left|\Psi\right\rangle$. Such a choice of total system size is
large enough so that the linear size of the smallest subsystem used in the
construction of $\widetilde{L}_{n}$ is much larger than the correlation
length, as discovered in the previous subsubsection. The gapless edge is along
the $x$-direction of the cylinder. In order to construct $\widetilde{L}_{n}$
with several choices of $n$, we choose a pretty large $L_{x}$. This is because
the construction of $\widetilde{L}_{n}$ with large $n$ requires dividing the
gapless edge into many intervals, and a large $L_{x}$ can make sure that the
length of the smallest interval is much larger than the correlation length. In
the $y$-direction, the construction doesn’t demand many lattice sites, so
$L_{y}$ doesn’t have to be very large.

Following the instructions given in Section 6, we construct

$\begin{split}&\widetilde{L}_{0}={1\over 4\pi\tan\left({\pi\over
N}\right)}\sum_{i=1}^{N}\left(\hat{\Delta}_{i}-\left\langle\Psi\right|\hat{\Delta}_{i}\left|\Psi\right\rangle\right)\\\
\textrm{where}\quad&\hat{\Delta}_{i}=K_{X_{i}X_{i+1}Y_{i}}+K_{X_{i+2}X_{i+1}Y_{i+1}}-K_{X_{i}Y_{i}}-K_{X_{i+2}Y_{i+1}}\\\
\end{split}$ (7.9)

for regions $X_{i},Y_{i}$, $i=1,\cdots,N$ shown in Figure 21. For $n\geq 1$,
we construct approximate Virasoro generators $\widetilde{L}_{n}$ and
$\widetilde{L}_{-n}$ as

$\begin{split}&\widetilde{L}_{n}=\mathbb{L}^{(e)}_{n}+i\mathbb{L}^{(o)}_{n},\quad\widetilde{L}_{-n}=\mathbb{L}^{(e)}_{n}-i\mathbb{L}^{(o)}_{n}\\\
\textrm{with}\quad&\mathbb{L}^{(o)}_{n}={A_{n}\over
2}\sum_{j=1}^{2n}(-1)^{j-1}\left(K_{X_{j}^{R}}+K_{X_{j}^{L}}\right)\\\
\textrm{and}\quad&\mathbb{L}^{(e)}_{n}={A_{n}\over
2}\sum_{j=1}^{2n}(-1)^{j-1}\left(K_{Y_{j}^{R}}+K_{Y_{j}^{L}}\right),\end{split}$
(7.10)

where regions $X_{j}^{R},X_{j}^{L},Y_{j}^{R},Y_{j}^{L}$ are shown in Figure 22
and

$A_{n}=\left\\{\begin{aligned} &{1\over 2\pi}&&n=1\\\ &\tan\left({\pi\over
2n}\right)(n^{2}-1)/8,&&n\neq 1\end{aligned}\right..$ (7.11)

Figure 21:  Regions used in the construction of $\widetilde{L}_{0}$. The
locations of equal-sized ($2l_{x}\times l_{y}$) regions $X_{i},Y_{i}$ can be
inferred from the figure. In the numerical tests, we used
$L_{x}=240,L_{y}=24,l_{x}=20,l_{y}=6$. Each gray dot in the picture is not
quite a coarse-grained site for a literal representation of our numerical
setup. Figure 22: Construction of $\mathbb{L}^{(o)}_{n}$ (and
$\mathbb{L}^{(e)}_{n}$). The location and size of $X_{j}^{R},X_{j}^{L}$ can be
inferred from the diagram. The color indicates the sign of the modular
Hamiltonian of that region used in $\mathbb{L}^{(o)}_{n}$: Green ($+$), Red
($-$). In the numerical tests, we used $L_{x}=240,L_{y}=24,l_{x}={N_{x}\over
4n},l_{y}=6$. The regions used in the construction of $\mathbb{L}^{(e)}_{n}$
are called $Y_{k}^{R}$ and $Y_{k}^{L}$, which are regions obtained from
$X_{k}^{R}$ and $X_{k}^{L}$ by shifting to the left by $l_{x}$.

To test the construction, we will numerically compute several quantities
involving $\widetilde{L}_{n}$, and make the following two comparisons:

  * •

We compare the numerical results with theoretical predictions discussed in
Section 6.2. The theoretical prediction is made under the assumption that
$\left|\Psi\right\rangle$ satisfies Hypothesis 1 is exactly, so that one can
do the computation under the CFT ground state $\left|\Omega\right\rangle$. We
shall call these theoretical results as “CFT prediction”. This comparison
tests the validity of the Hypothesis 1.

  * •

We compare the numerical results with those quantities from exact Virasoro
generator $L_{n}$. The latter is computed also under the assumption that
Hypothesis 1 is satisfied242424When we talk about $L_{n}$, we should assume
Hypothesis 1 is satisfied, as it’s defined by
$L_{n}\stackrel{{\scriptstyle\mathbb{V}}}{{=}}\mathscr{L}_{n}$ which utilize
$\mathbb{V}$ in the hypothesis. Note we are not constructing $L_{n}$ in the
lattice system and computing the results from them.. These results are titled
with “Virasoro algebra” (or “Vir” for short) when we list the data. Such a
comparison estimates the quality of approximation of $\widetilde{L}_{n}$ to
$L_{n}$.