supplementalMaterials

# Bisognano-Wichmann Hamiltonian for the entanglement spectroscopy of
fractional quantum Hall states

A. Nardin<EMAIL_ADDRESS>Université Paris-Saclay,
CNRS, LPTMS, 91405 Orsay, France  R. Lopes  Laboratoire Kastler Brossel,
Collège de France, CNRS, ENS-PSL University,
Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France  M.
Rizzi  Institute for Theoretical Physics, University of Cologne, D-50937 Köln,
Germany  Forschungszentrum Jülich GmbH, Institute of Quantum Control,
Peter Grünberg Institut (PGI-8), 52425 Jülich, Germany  L. Mazza  Université
Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France  S. Nascimbene  Laboratoire
Kastler Brossel, Collège de France, CNRS, ENS-PSL University,
Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
<EMAIL_ADDRESS>Université Paris-Saclay, CNRS,
LPTMS, 91405 Orsay, France  Laboratoire Kastler Brossel, Collège de France,
CNRS, ENS-PSL University,
Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
Institute for Theoretical Physics, University of Cologne, D-50937 Köln,
Germany  Forschungszentrum Jülich GmbH, Institute of Quantum Control,
Peter Grünberg Institut (PGI-8), 52425 Jülich, Germany  Université Paris-
Saclay, CNRS, LPTMS, 91405 Orsay, France  Laboratoire Kastler Brossel, Collège
de France, CNRS, ENS-PSL University,
Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France

###### Abstract

We study the Bisognano-Wichmann Hamiltonian for fractional quantum Hall states
defined on a sphere and explore its relationship with the entanglement
Hamiltonian associated to the state. We present results for several examples,
namely the bosonic Laughlin state stabilized by contact two-body interactions
and the bosonic Moore-Read state by either three- or two-body interactions.
Our findings demonstrate that the Bisognano-Wichmann Hamiltonian provides a
reliable approximation of the entanglement Hamiltonian as a fully-local
operator that can be written without any prior knowledge of the specific state
under consideration.

Introduction.—Exotic phases of matter exhibiting a fractional quantum Hall
(FQH) effect evade the standard paradigm of symmetry breaking [1]. They can
instead be classified according to a topological order encoded in the ground
state’s patterns of long-range entanglement [2]. Various signatures of
topological order have been identified, including a robust ground-state
degeneracy [3], the existence of fractionalized excitations [4] the presence
of chiral edge-modes [5, 6], and a quantized topological entanglement entropy
[7, 8].

In order to gain a deeper understanding of the structure of a FQH state, Li
and Haldane emphasized the importance of the spectrum of the reduced density
matrix ${\rho}_{A}$, obtained after tracing out any degree of freedom that
does not belong to the spatial region $A$. This defines the real-space
entanglement Hamiltonian (EH) as

${H}_{A}=-\log{\rho}_{A},$ (1)

whose spectrum, also called the real-space entanglement spectrum (RSES), was
conjectured to exhibit the same structure as a topological edge mode,
suggesting that the spatial partition acts as a virtual edge in the
entanglement Hamiltonian [9, 10].

Since measuring the entanglement spectrum presents a considerable experimental
challenge [11, 12, 13, 14, 15, 16, 17, 18], an interesting alternative
consists in simulating the EH as a physical Hamiltonian defined in subregion
$A$. This physical realization of the EH maps the entanglement spectrum to an
energy spectrum, which is more easily accessible for experimental measurement
[19]. This approach relies on the Bisognano-Wichmann (BW) theorem [20, 21],
which provides an explicit form for $H_{A}$ in terms of a local deformation of
the original Hamiltonian $H$. By introducing the Hamiltonian density
$\mathcal{H}(\mathbf{r})$, such that
$H=\int\mathcal{H}(\mathbf{r})d\mathbf{r}$, the exact form of $H_{A}$ reads

$H^{\rm BW}_{A}=\int_{\mathbf{r}\in
A}\beta(\mathbf{r})\mathcal{H}(\mathbf{r})d\mathbf{r},$ (2)

where the deformation factor $\beta(\mathbf{r})$ is a function of the distance
from the partition cut. In the simplest case of a straight-line bipartition,
$\beta(\mathbf{r})$ is a linear function of the distance from the cut.

Rigorously proven in the context of Lorentz-invariant local quantum field
theories, the BW Hamiltonian ${H}_{A}^{\text{BW}}$ is not generically exact in
condensed matter physics [22, 23, 24, 25, 26]. Despite this limitation, it has
led to the physical realization of the EH of a non-interacting quantum Hall
system with ultracold atomic gases using a synthetic dimension encoded in the
internal spin of an atom; in this context, the spatial deformation
$\beta(\mathbf{r})$ of the BW Hamiltonian has been engineered through the
inhomogeneity of spin couplings [27]. Yet, the use of the BW Hamiltonian to
characterize the EH of FQH state in conjunction with the Li-Haldane conjecture
has not been addressed so far.

In this article, we study the link between the entanglement Hamiltonian and
the BW Hamiltonian for strongly-interacting FQH states. We first show that the
BW theorem applies to the single-particle dynamics within the lowest Landau
level (LLL). We then focus on systems of bosonic atoms with contact two- or
three-body interactions, which are known to exhibit bosonic Laughlin and
Moore-Read states. Through a combination of analytical and numerical
calculations, we establish that the lowest-energy-branch of the BW Hamiltonian
is gapped and matches the conformal-field-theory counting of a topological
edge mode. For model wavefunctions, the BW Hamiltonian and the EH share the
same eigenvectors, whereas in realistic cases this is true with a high level
of accuracy. Overall, our findings provide valuable insights into the spatial
entanglement properties of FQH systems.

Single-particle BW Hamiltonian.—Before delving into the FQH interacting
problem and the Li-Haldane conjecture, we first focus on the real-space EH for
the single-particle problem. To mitigate any edge effects, we conduct our
investigations using a spherical geometry. Note that the sphere can be viewed
as a compactified plane, and a stereographic projection would allow us to map
our results to the planar geometry.

We examine the dynamics of a particle of charge $q$ and mass $M$ as it moves
on the surface of a sphere with radius $R$. The particle is subjected to a
radial magnetic field $B$ generated by a Dirac magnetic monopole of charge $Q$
positioned at the centre of the sphere. The magnetic flux through the sphere
is an integer multiple $2Q\in\mathbb{N}$ of flux quanta $h/q$, so that the
magnetic field is $B=\hbar Q/(qR^{2})$. Using the latitude gauge for the
vector potential $\mathbf{A}=-\hbar Q/qR\,\cot(\theta){\bm{e}_{\phi}}$, where
$(\theta,\phi)$ are the spherical angles, the single particle Hamiltonian
reads $H_{0}={\Lambda}^{2}/(2MR^{2})$, with
$\mathbf{\Lambda}=\mathbf{R}\times\left(-i\hbar\bm{\nabla}-q\mathbf{A}\right)$
[28, 29].

Owing to rotational symmetry, the eigenstates can be labelled by the angular
momentum quantum numbers $L$ and $m$. The spectrum organises in families of
degenerate levels, the Landau levels,
$E_{L}=\hbar\omega_{c}(L(L+1)-Q^{2})/2Q$, where $\omega_{c}=qB/M$ is the
cyclotron frequency, and the angular momentum is restricted to $L=Q+n$
($n\in\mathbb{N}$). The LLL corresponds to $n=0$ and the single particle wave-
functions $\psi_{Q,m}\propto u^{Q+m}v^{Q-m}$ are usually written in terms of
spinor variables, $u(\theta,\phi)=\cos(\theta/2)e^{i\phi/2}$ and
$v(\theta,\phi)=\sin(\theta/2)e^{-i\phi/2}$, with $-L\leq m\leq L$ [30].

Figure 1: Single-particle RSES $\xi_{A}(m)$ [yellow circles] and spectrum of
the BW Hamiltonian in Eq. (3) [black crosses] for a bipartition at the equator
and $Q=9$: the lowest branch of the latter spectrum corresponds to the long-
wavelength linearized limit of the EH [dashed red line]. The three
wavefunctions plotted on top are defined on region $A$ and are eigenfunctions
of both the EH and the BW Hamiltonian.

We now consider a bipartition of the sphere into two regions $A$ and $B$,
where $A$ corresponds to the northern hemisphere $0\leq\theta\leq\pi/2$. This
partition preserves the rotational symmetry around the $z$-axis, so that both
the reduced density matrix $\rho_{A}=e^{-H_{A}}$ and the entanglement
Hamiltonian $H_{A}$ can be decomposed over spin projections $m$. The EH of the
single-particle problem reads
$H_{A}=\sum_{m}\xi_{A}(m)\left|\psi^{A}_{Q,m}\right>\left<\psi_{Q,m}^{A}\right|$.
The wavefunctions $\psi^{A}_{Q,m}$ are obtained by truncating at
$\theta=\pi/2$ and normalising, $\psi_{Q,m}^{A}\propto\psi_{Q,m}\Theta_{\rm
H}(\pi/2-\theta)$, where $\Theta_{\rm H}(x)$ is the Heaviside function (see
Fig. 1 and [31]). Like in the original LLL, the spin projection $m$ takes
integer values in $-L\leq m\leq L$.

Inspired by the BW theorem, we write a BW Hamiltonian for the single-particle
problem as

$H_{A}^{\rm
BW}=\frac{4\sqrt{Q}}{\sqrt{\pi}}\,\left\\{\frac{H_{0}-\epsilon}{\hbar\omega_{c}},\cos\theta\right\\},\quad\text{for
}0\leq\theta\leq\frac{\pi}{2}.$ (3)

The spatial deformation factor $\cos\theta$, which acts as a regularised
distance from the equator, is the essence of the BW approach. It disconnects
the regions $A$ and $B$, so that it is legitimate to consider the restriction
of $H_{A}^{\rm BW}$ to subregion $A$. The anticommutator is required in order
to properly define a Hermitian operator, and the energy offset $\epsilon$
reflects the fact that $H_{0}$ is defined up to a constant.

By selecting an energy offset $\epsilon=\hbar\omega_{c}(1+1/2Q)$, which lies
midway between the lowest and first excited LL, the truncated LLL
wavefunctions $\psi_{Q,m}^{A}$ are exactly the eigenfunctions of the lowest-
lying energy band of the BW Hamiltonian in Eq. (3), with eigenvalues

$\xi^{\rm BW}_{A}(m)=-\frac{4}{\sqrt{\pi\,Q}}\,m,\qquad\text{for }m\leq Q.$
(4)

The full spectrum of the BW Hamiltonian is plotted in Fig. 1. The lowest band
aligns with the EH exact result in the thermodynamic limit
($Q=R^{2}/l_{B}^{2}\to\infty$) [32, 33, 34, 35, 36, 37], thus providing its
long-wavelength limit. Note that it extends all the way to $m\to-\infty$, in
contrast to the restricted range for the RSES. Indeed, the limitation
$m\geq-Q$ for LLL orbitals arises due to divergences at the south pole for
$m<-Q$. The BW Hamiltonian $H_{A}^{\rm BW}$, defined on the northern
hemisphere only, does not experience this problem. Interestingly, the BW
Hamiltonian offers an efficient approach to investigating infinite and
precisely linear spectra on finite systems, which may prove useful in various
contexts.

This analysis shows that the single-particle EH in the long-wavelength limit
can be expressed as a fully local operator. Note that we are not presenting
here an effective boundary Hamiltonian, as it has already been discussed [35,
36, 38], but a true bulk Hamiltonian defined on the entire region $A$. Prior
to this study, it was not immediately apparent that such a formulation was
feasible.

Figure 2:  (a) RSES of a bosonic Laughlin state of $N=10$ particles
(corresponding to $Q=9$), with $N_{A}=5$. (b) Spectrum of the BW Hamiltonian
$\Pi_{\rm BW}V_{A}^{\rm BW}\Pi_{\rm BW}$ for $N_{A}=5$ bosons. The lowest
lying zero-energy band has the same degeneracy of the RSES at the same
$L_{z}^{A}$. The inset shows a close-up small portion (highlighted by the blue
rectangle) of such a band; numbers with the counting of degenerate states have
been added for clarity.

BW Hamiltonian for model FQH wavefunctions.—One notable feature of Eq. (3) is
that it does not require prior diagonalization of the Hamiltonian $H$ in order
to express the BW Hamiltonian $H_{A}^{\rm BW}$. As we will see, this appealing
aspect holds true even in the FQH case, where the interactions between
particles play a major role.

We now consider the spatial partition of the ground state of $N$ interacting
particles evolving on the sphere. The reduced density matrix $\rho_{A}$ (and
hence the EH) can be decomposed into symmetry sectors with a fixed number of
particles $N_{A}$ and angular momentum $L_{z}^{A}$. In general, the study of
the EH in a given sector with $N_{A}$ particles requires computing the ground
state for a larger atom number $N>N_{A}$. In contrast, the direct study of the
BW Hamiltonian $H_{A}^{\rm BW}$ only requires computing the spectrum for
$N_{A}$ particles, which is less computationally demanding.

We begin our investigation by examining the bosonic Laughlin state at filling
fraction $\nu=1/2$, which is the exact and gapped ground-state of a two-body
contact interaction Hamiltonian
$V_{2}(\mathbf{r}_{1},\mathbf{r}_{2})=g_{2}\,\delta^{(2)}(\mathbf{r}_{1}-\mathbf{r}_{2})$
[39, 40, 41]; on the sphere, its stabilization requires $Q=N-1$ [42]. Its
RSES, shown in Fig. 2(a), can be characterized by the quantum numbers $N_{A}$
and $L_{z}^{A}$. It consists of a finite branch of states whose counting in
the thermodynamic limit corresponds to that of a chiral boson, in agreement
with the Li-Haldane conjecture [31, 43, 44, 45, 46, 35, 36, 34].

The generalization of the BW ansatz of Eq. (3) to the many-body case with
contact interactions involves a combination of the single-particle Hamiltonian
given in Eq. (3) and an interaction term

$\displaystyle V_{A}^{\rm
BW}\\!=\\!\frac{4\sqrt{Q}}{\sqrt{\pi}}\sum_{i<j}^{N_{A}}\left\\{\frac{g_{2}\,\delta(\cos\theta_{i}-\cos\theta_{j})\delta(\phi_{i}-\phi_{j})}{R^{2}\hbar\omega_{c}},\cos\theta_{i}\right\\}\\!.$
(5)

In the FQH limit, where the cyclotron energy $\hbar\omega_{c}$ dominates over
the interactions, the low-lying features of $H_{A}^{\rm BW}+V_{A}^{\rm BW}$
can be determined using the projector $\Pi_{\rm BW}$ onto the truncated
wavefunctions in $\psi_{Q,m}^{A}$, analogously to the standard treatment of
FQH effects. Since the BW single-particle Hamiltonian has an exactly linear
spectrum, after projection the single-particle BW Hamiltonian becomes trivial:
$\Pi_{\rm BW}H_{A}^{\rm BW}\Pi_{\rm BW}=-\frac{4}{\sqrt{\pi Q}}{L_{z}^{A}}$.
We thus only focus on the interacting component $\Pi_{\rm BW}V_{A}^{\rm
BW}\Pi_{\rm BW}$, and show the results of its numerical diagonalisation in
Fig. 2(b) for $N_{A}=5$. A branch of zero-energy states extends up to
$L_{z}^{A}=N_{A}(Q+1-N_{A})$, where a single state is seen. This is exactly
like the RSES in Fig. 2(a). As we move towards smaller values of $L_{z}^{A}$,
the degeneracy of the states increases, precisely matching the expected
counting for a chiral boson [47, 31]. To further support this result, we
provide exact analytical expressions for the zero-energy eigenstates of
$\Pi_{\rm BW}V_{A}^{\rm BW}\Pi_{\rm BW}$ [31], drawing parallels to the well-
known construction of excitations in a Laughlin fluid using totally-symmetric
polynomials [47]. We can conclude that the interacting BW Hamiltonian exhibits
a spectrum with a low-energy branch that satisfies the Li-Haldane conjecture,
which proves the interest of performing a quantum simulation of this
Hamiltonian.

Note that, for the numerical calculation, the single-particle spectrum, which
in principle extends to $m\to-\infty$, is truncated to a finite interval
$-Q\leq m\leq Q$. Although the excluded single-particle states could
potentially contribute additional zero-energy states to the many-body
spectrum, the truncation has no effect on the region
$L_{z}^{A}>(N_{A}-2)(Q+1-N_{A})$. Within this interval, our results
demonstrate that the subspace spanned by the zero-energy branch of $H_{A}^{\rm
BW}$ coincides with the one spanned by the eigenvectors of the EH.

In practice, the experimental implementation of spatially-deformed
interactions as given by Eq. 5 may be challenging. However, the fact that the
relevant many-body states of the BW Hamiltonian have zero interaction energy
makes the spatial deformation unnecessary. We have verified this by examining
the case of pure contact interactions $V_{2}$, finding the same zero-energy
band of Fig. 2(b), but a different excited-band structures. [31].

Figure 3:  (a) RSES of a bosonic Moore-Read state of $N=14$ particles
(corresponding to $Q=6$), with $N_{A}=7$. (b) Spectrum of the projected BW
Hamiltonian $V$ for $N_{A}=7$ bosons, without the linear dispersion term (Eq.
(3)), in the case of three-body contact interactions $V_{3}$. The lowest lying
zero-energy band has the same degeneracy of the RSES at the same $L_{z}^{A}$.
The inset shows a close-up small portion (highlighted by the blue rectangle)
of such a band; numbers with the counting of degenerate states have been added
for clarity.

Next, we consider the bosonic Moore-Read state at filling $\nu=1$ as a second
case study. This state corresponds to the exact ground state of a three-body
contact interaction
$V_{3}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3})=g_{3}\,\delta^{(2)}(\mathbf{r}_{1}-\mathbf{r}_{2})\delta^{(2)}(\mathbf{r}_{1}-\mathbf{r}_{3})$
[48, 49, 50]. On the sphere, its stabilization requires $2Q=N-2$ [42]. The
extension of the BW Hamiltonian to the contact three-body interaction is
straightforward. In this case as well, the interacting BW Hamiltonian exhibits
a zero-energy band that shares the same eigenvector structure as the exact
entanglement Hamiltonian (see Fig. 3). In agreement with the Li-Haldane
conjecture, the state degeneracies matches the edge conformal field theory,
which consists for the Moore-Read state of a chiral boson and a chiral fermion
[31].

BW Hamiltonian for numerical FQH states.—The exactness of the results
presented so far for analytically-known model wavefunctions opens up the
question on whether the BW Hamiltonian can be used to study the RSES of
generic FQH states. In this final section we present a positive answer by
considering the state in the Moore-Read phase stabilized by two-body contact
interactions [51, 42, 52, 41, 53]–a state relevant for cold-atom experiments
[54] that cannot be expressed analytically.

Figure 4:  (a) RSES ($N_{A}=7$) of a state of $N=14$ bosons, $Q=6$, stabilized
by two-body contact interactions $V_{2}$, which is expected to be in the
Moore-Read universality class. (b) Spectrum of the BW Hamiltonian $\Pi_{\rm
BW}V_{A}^{\rm BW}\Pi_{\rm BW}$ for $N_{A}=7$ bosons. In both cases the points
have been coloured according to the expectation value of the three-body
contact interaction $V_{3}$. For clarity reasons an arbitrary linear
dispersion term has been introduced in the BW case.

It has been shown that the RSES of this state, plotted in Fig. 4(a), displays
a low-lying branch analogous to that of the analytical Moore-Read state in
Fig. 3(a), as well as a novel high-lying branch [55]. While the two branches
are separated at low energy they merge at higher energy, limiting the
identification of the levels that belong to the low-energy branch.In the
specific example of Fig. 4(a), we are able to unequivocally isolate the ground
energy branch for 7 values of $L_{z}^{A}$, from $18$ to $24$.

We now compare this entanglement spectrum with the energy spectrum of the
corresponding BW Hamiltonian, shown in Fig. 4(b). Just like the RSES, a single
state emerges at angular momentum $L_{z}^{A}=24$. As we move towards lower
angular momentum states, we observe the development of a branch with the same
number of states as the RSES, namely $\\{1,2,4\\}$ from right to left. This
branch eventually merges at angular momentum $L_{z}^{A}=21$ with the high-
lying branch. We conclude that, even for this non-model wavefunction, the low-
energy spectrum of the BW Hamiltonian reproduces the Li-Haldane conjecture. In
the SM we present additional numerical results for different bipartitions,
highlighting the even-odd $N_{A}$ effect on the edge-state counting [31].

We also compare the eigenvectors of the EH and BW Hamiltonians. We first
consider the expectation value of the three-body contact interaction potential
$V_{3}$, which is encoded in the colour of the markers of Fig. 4. This local
observable quantifies the characteristic three-body anti-bunching expected in
the Moore-Read state. This observable takes very similar values in both EH and
BW Hamiltonians, and a more detailed comparison is presented in [31],
demonstrating relative differences of a few percent points. We also computed
the overlap matrix [56] between the EH and BW Hamiltonian eigenvectors, within
the low-energy branch, at fixed quantum numbers $N_{A}$ and $L_{z}^{A}$. We
expect this matrix to describe a change of basis if the BW Hamiltonian
captures the EH physics. We found a unitary matrix at the $10^{-3}$ level
[31]. We thus conclude that the EH and BW Hamiltonians can be approximately
put in direct correspondance in their low-energy branch.

Conclusions.—We have revisited the fundamental problem of the RSES of FQH
states proposing a study based on the BW Hamiltonian; our results show that it
correctly captures the features of the entanglement Hamiltonian, satisfying
several properties among which the Li-Haldane conjecture. These positive
results open the exciting perspective of establishing an accurate mathematical
connection beyond the numerical results that we have presented.

A remarkable result is that the BW Hamiltonian can be constructed
straightforwardly from the knowledge of the physical Hamiltonian only, without
the need of computing the ground state to be studied. This in particular means
that one does not even need to know what specific form of topological order is
displayed by the ground state, which is typically necessary for constructing
the EH as an effective boundary conformal field theory. For this reason, we
expect the BW Hamiltonian to become a useful tool in numerical studies.

Finally, given the recent realisation of a non-interacting BW Hamiltonian in a
cold-atom quantum simulator, this work opens new paths in the experimental
study of the RSES of the paradigmatic topological phases of the FQH effects.

11footnotetext: See Supplemental Material for the eigenvalues and eigenvectors
of the single-particle entanglement Hamiltonian, the analytical expressions
for the eigenstates of zero interaction energy of the BW Hamiltonian, the
many-body spectrum of the BW Hamiltonian with uniform contact interactions, a
detailed comparison between EH and BW Hamiltonians for the numerical Moore-
Read state, and a detailed explanation of the CFT edge mode counting for
Laughlin and Moore-Read states.

Acknowledgements.—We acknowledge fruitful discussions with N. Regnault. We
thank I. Carusotto for careful reading of the manuscript. This work is
supported by European Union (grants TOPODY 756722 (Paris) and LoCoMacro 805252
(Orsay) from the European Research Council) and by Region Ile-de-France in the
framework of the DIM Sirteq. M. R. acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG) with project Grant No. 277101999 within the CRC
network TR 183 (subproject B01) and under Germany’s Excellence Strategy -
Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1
- 390534769. A. N. acknowledges support by Fondazione Angelo dalla Riccia and
thanks LPTMS for warm hospitality. S. N. acknowledges support from Institut
Universitaire de France.

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Supplemental Materials to
Bisognano-Wichmann Hamiltonian for the entanglement spectroscopy
of fractional quantum Hall states A. Nardin

R. Lopes

M. Rizzi

L. Mazza

S. Nascimbene

###### Contents

  1. I Lowest band of the single-particle Bisognano-Wichmann Hamiltonian
  2. II Single-particle entanglement spectrum
  3. III Diagonalization of the single-particle Bisognano-Wichmann Hamiltonian
  4. IV Diagonalization of the many-particles Bisognano-Wichmann Hamiltonian
  5. V Exact zero-energy eigenstates of the Bisognano-Wichmann Hamiltonian for the Laughlin state
  6. VI Contact two-body interaction as a proxy for the Bisognano-Wichmann Hamiltonian
  7. VII State counting
  8. VIII Numerical analysis for the contact interacting Moore-Read state

##  I Lowest band of the single-particle Bisognano-Wichmann Hamiltonian

In this first section we show that the sharply-truncated lowest Landau level
wavefunctions on the sphere are eigenvectors of the Bisognano-Wichmann
Hamiltonian. In this supplemental materials, we discuss a general latitude
angle $\theta_{0}$ for the bipartition, whereas in the main text we focused
only on the half-bipartition of the sphere, $\theta_{0}=\pi/2$.

We rewrite the Bisognano-Wichmann Hamiltonian as

$H_{BW}=\frac{4}{\sqrt{\pi\,Q}}\left\\{\frac{H_{0}-\epsilon}{\hbar\omega_{c}},\mathcal{F}(\theta)\right\\},\qquad\text{with}\quad
0\leq\theta\leq\theta_{0}\quad\text{and}\quad\mathcal{F}(\theta)=Q(\cos(\theta)-\cos(\theta_{0}))\geq
0;$ (6)

the coordinates are restricted to the upper spherical cap and we introduced
the function $\mathcal{F}(\theta)$ measuring the distance from the bipartition
cut. In the fractional quantum Hall context, this distance function can
interpreted as measuring the “orbital-distance” between lowest-Landau level
orbitals on the sphere, which are peaked at $\cos(\theta_{M})=M/Q$. Notice
that such a form appeared previously in Ref. [57] within a different context.

Here $\epsilon$ is in principle an arbitrary energy shift, which we will
choose by requiring the lowest Landau level truncated wavefunctions to be
eigenfunction of the Bisognano-Wichmann Hamiltonian, and $H_{0}$ is the free
single particle Hamiltonian [29]

$H_{0}=\frac{|\mathbf{\Lambda}|^{2}}{2MR^{2}}=\frac{\hbar\omega_{c}}{2Q}\left(-\frac{1}{\sin(\theta)}\frac{\partial}{\partial\theta}\sin(\theta)\frac{\partial}{\partial\theta}+\left(Q\cot(\theta)+\frac{i}{\sin(\theta)}\frac{\partial}{\partial\phi}\right)^{2}\right).$
(7)

The (normalized) eigenvectors of $H_{0}$ are so-called monopole harmonics [29,
58, 59]

$Y_{Q,l,m}=\sqrt{\frac{2l+1}{4\pi}\,\frac{(l-m)!}{(l-Q)!}\frac{(l+m)!}{(l+Q)!}}\,2^{-m}\,(1-\cos(\theta))^{\frac{-Q+m}{2}}(1+\cos(\theta))^{\frac{Q+m}{2}}\,P_{l-m}^{-Q+m,Q+m}(\cos(\theta))e^{im\phi},$
(8)

where
$P_{n}^{(\alpha,\beta)}(x)=2^{-n}\sum_{s=0}^{n}\binom{n+\alpha}{s}\binom{n+\beta}{n-s}(x-1)^{n-s}(x+1)^{s}$
are the Jacobi polynomials. Here, $2Q\in\mathbb{Z}$ is the flux through the
sphere, $l\geq Q$ is the total angular momentum eigenvalue and $-l\leq m\leq
l$ the eigenvalue of its projection along the $z$ axis. The eigenvalue
associated to $Y_{Q,l,m}$ is $E_{Q,l}=\hbar\omega_{c}\frac{l(l+1)-Q^{2}}{2Q}$;
the lowest Landau level corresponds to $l=|Q|$, i.e.
$E_{Q,Q}=\frac{1}{2}\hbar\omega_{c}$.

The action of $H_{BW}$ on these states can be computed

$\frac{H_{BW}}{4/\sqrt{\pi
Q}}Y_{Q,l,m}=\frac{1}{2Q}\left(-\frac{\partial^{2}\mathcal{F}}{\partial\theta^{2}}Y_{Q,l,m}-2\frac{\partial\mathcal{F}}{\partial\theta}\,\frac{\partial
Y_{Q,l,m}}{\partial\theta}-\frac{\cos(\theta)}{\sin(\theta)}\frac{\partial\mathcal{F}}{\partial\theta}Y_{Q,l,m}\right)+2\mathcal{F}\left(\frac{l(l+1)-Q^{2}}{2Q}-\frac{\epsilon}{\hbar\omega_{c}}\right)Y_{Q,l,m}.$
(9)

Notice that introducing Heaviside theta functions in the monopole harmonics
(i.e. restricting them to the proper Bisognano-Wichmann domain) does not (as
can be checked explicitly) change the above equation since the two domains
$\theta>\theta_{0}$ and $\theta<\theta_{0}$ are disconnected by
$\mathcal{F}(\theta)$, i.e. $\mathcal{F}(\theta_{0})=0$. Provided we set
$\epsilon=\hbar\omega_{c}\frac{2Q+1}{2Q}$ (midway between the lowest LL
$E_{Q,Q}$ and the first LL $E_{Q,Q+1}$), it is easy to check (to this purpose,
see Eq. (29)) that $Y_{Q,Q,m}$ (the lowest-Landau level wavefunctions with
$l=Q$) are eigenvectors

$H_{BW}Y_{Q,Q,m}=-\frac{4}{\sqrt{\pi
Q}}\left(m-(Q+1)\cos(\theta_{0})\right)Y_{Q,Q,m}.$ (10)

Analogously to lowest Landau level wavefunctions on the sphere, the normalized
eigenvectors of the Bisognano-Wichmann Hamiltonian are most simply written in
terms of spinor-variables $u(\theta,\phi)=\cos(\theta/2)e^{i\phi/2}$ and
$v(\theta,\phi)=\sin(\theta/2)e^{-i\phi/2}$ as

$\psi_{Q,m}^{A}=\frac{u^{Q+m}(\theta,\phi)v^{Q-m}(\theta,\phi)}{\sqrt{4\pi\mathrm{B}_{\sin^{2}\left(\frac{\theta_{0}}{2}\right)}(Q+1-m,Q+1+m)}}\Theta_{\rm
H}(\theta_{0}-\theta),$ (11)

where $\mathrm{B}_{x}(a,b)$ is the incomplete Beta function,
$\mathrm{B}_{x}(a,b)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,dt$. Notice that we
introduced the Heaviside theta function $\Theta_{\rm H}(\theta_{0}-\theta)$,
which sharply separates the two spherical caps.

##  II Single-particle entanglement spectrum

In this section we analytically describe the structure of the entanglement
Hamiltonian in the case of a single-particle state occupying a well defined
orbital $Y_{Q,Q,m}$ in the lowest Landau level. The reduced density matrix can
be decomposed as

$\rho_{A}=\left(1-\mathcal{N}_{A}\right)\left|v_{A}\right>\left<v_{A}\right|+\mathcal{N}_{A}\left|\psi_{A}\right>\left<\psi_{A}\right|$
(12)

where $A$ is the spherical cap defined by $\theta<\theta_{0}$,
$\left|v_{A}\right>$ is the state with no particles in $A$ and $\psi_{A}$ the
one with a single particle restricted to this region. Finally

$\mathcal{N}_{A}=\int
d\phi\int_{0}^{\theta_{0}}\sin(\theta)d\theta\,|Y_{Q,Q,m}|^{2}=\frac{\mathrm{B}_{\sin^{2}\left(\theta_{0}/2\right)}\left(Q+1-m,Q+1+m\right)}{\mathrm{B}(Q+1-m,Q+1+m)}.$
(13)

Here, $\mathrm{B}(a,b)=\mathrm{B}_{1}(a,b)$ is the Beta function. Since the
particle either is the $A$ region or it is not, the entanglement Hamiltonian,
defined by $\rho_{A}=e^{-H_{A}}/\text{Tr}[e^{-H_{A}}]$, can be written as

$H_{A}=E_{v_{A}}\left|v_{A}\right>\left<v_{A}\right|+E_{\psi_{A}}\left|\psi_{A}\right>\left<\psi_{A}\right|$
(14)

with $E_{v_{A}}$ and $E_{\psi_{A}}$ being dimensionless quantities. With some
algebra it can be shown that the previous relations are compatible provided

$\mathcal{N}_{A}=\frac{1}{2}\left[1-\tanh\left(\frac{E_{\psi_{A}}-E_{v_{A}}}{2}\right)\right]$
(15)

which characterizes the entanglement energies up to a global additive
constant, which can be fixed by requiring $E_{v_{A}}=0$.

Inverting Eq. (15) and using Eq. (13) we get

$\begin{split}E_{\psi_{A}}=&-2\text{atanh}\left(2\mathcal{N}_{A}-1\right)\\\
=&-2\,\text{atanh}\left(2\,\frac{\mathrm{B}_{\sin^{2}\left(\theta_{0}/2\right)}\left(Q+1-m,Q+1+m\right)}{\mathrm{B}(Q+1-m,Q+1+m)}-1\right).\end{split}$
(16)

When $\theta_{0}=\pi/2$, Eq. (13) can be linearized for $m/Q\ll 1$ using the
Beta function integral representation and Laplace’s method, yielding

$\mathcal{N}_{A}\approx\frac{1}{2}+\frac{m}{\sqrt{\pi Q}}$ (17)

which, when inserted in Eq. (16) gives

$E_{\psi_{A}}\approx-\frac{4m}{\sqrt{\pi Q}}=-\frac{4\,kl_{B}}{\sqrt{\pi}}.$
(18)

where the momentum $k=\frac{2\pi}{\mathcal{C}}\,m$ and $\mathcal{C}=2\pi R$ is
the length of the bipartition cut. This last expression is the same one can
find for an integer quantum Hall system on the plane [37], which is locally
indistinguishable from a sphere when its radius goes to infinity.

##  III Diagonalization of the single-particle Bisognano-Wichmann Hamiltonian

In this section, we detail how the full spectrum of the Bisognano-Wichmann
Hamiltonian is obtained.

We begin by introducing $\lambda_{Q,l}=l(l+1)-Q^{2}-(2Q+1)$ and rewriting Eq.
(9) as

$\frac{H_{BW}}{4/\sqrt{\pi
Q}}Y_{Q,l,m}=(\lambda_{Q,l}+1)\,\cos(\theta)Y_{Q,l,m}+\sin(\theta)\,\frac{\partial
Y_{Q,l,m}}{\partial\theta}-\lambda_{Q,l}\cos(\theta_{0})Y_{Q,l,m}.$ (19)

We now want to diagonalize $H_{BW}$ by expanding its general eigenvector on
the basis provided by the monopole harmonics. With the choice of bipartition
$\mathcal{F}=\mathcal{F}(\theta)$ (independent of $\phi$), the $z$-projection
of the angular momentum is still a good quantum number. We can therefore write

$H_{BW}\Psi_{Q,\alpha,m}=E_{Q,\alpha,m}\Psi_{Q,\alpha,m}$ (20)

and expand

$\Psi_{Q,\alpha,m}=\sum_{l}C_{l,\alpha}^{Q,m}Y_{Q,l,m}.$ (21)

multiplying by $Y^{*}_{Q,l^{\prime},m}$ and integrating over the spherical cap
$A$ we are considering we end up with a generalized eigenvalue problem (the
states $Y_{Q,l,m}$ are not normalized nor orthogonal on the bipartition)

$\sum_{l}\mathcal{H}_{l^{\prime},l}^{Q,m}C_{l,\alpha}^{Q,m}=E_{Q,\alpha,m}\sum_{l}\mathcal{M}_{l^{\prime},l}^{Q,m}C_{l,\alpha}^{Q,m}$
(22)

where

$\begin{cases}\mathcal{H}_{l^{\prime},l}^{Q,m}=\int_{A}Y^{*}_{Q,l^{\prime},m}H_{BW}Y_{Q,l,m}d\Omega\\\
\mathcal{M}_{l^{\prime},l}^{Q,m}=\int_{A}Y^{*}_{Q,l^{\prime},m}Y_{Q,l,m}d\Omega.\end{cases}$
(23)

The integrals could be performed numerically, but it is not difficult to
express them as finite summations, which allow for a more straightforward
numerical evaluation. We here sketch the calculation of these matrix elements,
giving only the main results.

The integration over the region $A$ can be seen as an integral over the whole
space with a characteristic function $\Theta_{\rm H}(\theta_{0}-\theta)$ which
can be expanded in terms of monopole harmonics

$\begin{cases}\Theta_{\rm H}(\theta_{0}-\theta)=\sum_{l}\zeta_{l}Y_{0,l,0}\\\
\zeta_{l}=2\pi\sqrt{\frac{2l+1}{4\pi}}\,\frac{P_{l-1}(\cos(\theta_{0})-P_{l+1}(\cos(\theta_{0}))}{2l+1}\end{cases}$
(24)

where $P_{n}(x)$ are Legendre polynomials. Products of monopole harmonics can
be simplified in terms of Wigner 3j symbols [59]

$\begin{split}Y_{Q,l,m}Y_{Q^{\prime},l^{\prime},m^{\prime}}=&\sum_{l^{\prime\prime}}(-1)^{l+l^{\prime}+l^{\prime\prime}}(-1)^{Q+Q^{\prime}}(-1)^{m+m^{\prime}}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2l^{\prime\prime}+1)}{4\pi}}\,\\\
&\begin{pmatrix}l&l^{\prime}&l^{\prime\prime}\\\
m&m^{\prime}&-(m+m^{\prime})\end{pmatrix}\begin{pmatrix}l&l^{\prime}&l^{\prime\prime}\\\
Q&Q^{\prime}&-(Q+Q^{\prime})\end{pmatrix}Y_{Q+Q^{\prime},l^{\prime\prime},m+m^{\prime}}\end{split}$
(25)

and as a consequence of orthonormality integrals involving three of them can
be written as [59]

$\int
Y_{Q,l,m}Y_{Q^{\prime},l^{\prime},m^{\prime}}Y_{Q^{\prime\prime},l^{\prime\prime},m^{\prime\prime}}d\Omega=(-1)^{l+l^{\prime}+l^{\prime\prime}}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2l^{\prime\prime}+1)}{4\pi}}\,\begin{pmatrix}l&l^{\prime}&l^{\prime\prime}\\\
-m&-m^{\prime}&-m^{\prime\prime}\end{pmatrix}\begin{pmatrix}l&l^{\prime}&l^{\prime\prime}\\\
Q&Q^{\prime}&Q^{\prime\prime}\end{pmatrix}.$ (26)

Using $Y^{*}_{Q,l,m}=(-1)^{Q-m}Y_{Q,l,m}$ [59], the overlaps matrix elements
can be written as

$\mathcal{M}_{l^{\prime},l}^{Q,m}=(-1)^{Q-m}\sum_{l^{\prime\prime}=|l^{\prime}-l|}^{l+l^{\prime}}\zeta_{l^{\prime\prime}}(-1)^{l+l^{\prime}+l^{\prime\prime}}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2l^{\prime\prime}+1)}{4\pi}}\,\begin{pmatrix}l&l^{\prime}&l^{\prime\prime}\\\
-m&m&0\end{pmatrix}\begin{pmatrix}l&l^{\prime}&l^{\prime\prime}\\\
Q&-Q&0\end{pmatrix}.$ (27)

Using $\cos(\theta)=\sqrt{\frac{4\pi}{3}}Y_{0,1,0}$ together with Eq. (25) and
Eq. (26), the first integral appearing inside
$\mathcal{H}_{l,l^{\prime}}^{Q,m}$ can be calculated

$\begin{split}I_{l^{\prime},l}^{Q,m}=\int_{A}Y^{*}_{Q,l^{\prime},m}\cos(\theta)Y_{Q,l,m}d\Omega=&(-1)^{Q-m}\sum_{l^{\prime\prime\prime}=|l^{\prime}-l|}^{l+l^{\prime}}(2l^{\prime\prime\prime}+1)\begin{pmatrix}l&l^{\prime}&l^{\prime\prime\prime}\\\
-m&m&0\end{pmatrix}\begin{pmatrix}l&l^{\prime}&l^{\prime\prime\prime}\\\
Q&-Q&0\end{pmatrix}\\\
&\sum_{l^{\prime\prime}=|l^{\prime\prime\prime}-1|}^{l^{\prime\prime\prime}+1}(-1)^{1+l+l^{\prime}+l^{\prime\prime}}\zeta_{l^{\prime\prime}}\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2l^{\prime\prime}+1)}{4\pi}}\begin{pmatrix}1&l^{\prime\prime\prime}&l^{\prime\prime}\\\
0&0&0\end{pmatrix}^{2}.\end{split}$ (28)

Using the results of Ref. [60] it is not difficult to show that

$\sin(\theta)\,\frac{\partial
Y_{Q,l,m}}{\partial\theta}=\left(l\cos(\theta)-\frac{Q}{l}m\right)Y_{Q,l,m}-\frac{1}{l}\sqrt{\frac{2l+1}{2l-1}(l^{2}-m^{2})(l^{2}-Q^{2})}\,\,Y_{Q,l-1,m}.$
(29)

The only contribution to $\mathcal{H}_{l,l^{\prime}}^{Q,m}$ with a functional
form different from the ones already analysed is seen to be

$\begin{split}J_{l^{\prime},l}^{Q,m}=\int_{A}Y^{*}_{Q,l^{\prime},m}Y_{Q,l-1,m}d\Omega=&(-1)^{Q-m}\sum_{l^{\prime\prime}=|l-l^{\prime}-1|}^{l+l^{\prime}-1}\zeta_{l^{\prime\prime}}(-1)^{l-1+l^{\prime}+l^{\prime\prime}}\sqrt{\frac{(2l-1)(2l^{\prime}+1)(2l^{\prime\prime}+1)}{4\pi}}\,\\\
&\begin{pmatrix}l-1&l^{\prime}&l^{\prime\prime}\\\
-m&m&0\end{pmatrix}\begin{pmatrix}l-1&l^{\prime}&l^{\prime\prime}\\\
Q&-Q&0\end{pmatrix}.\end{split}$ (30)

Therefore, putting everything together we get

$\begin{split}\frac{\mathcal{H}_{l^{\prime},l}^{Q,m}}{4/\sqrt{\pi
Q}}=&\int_{A}Y^{*}_{Q,l^{\prime},m}H_{BW}Y_{Q,l,m}d\Omega=(\lambda_{Q,l}+1)\,I_{l^{\prime},l}^{Q,m}+\left(l\,I_{l^{\prime},l}^{Q,m}-\frac{Q}{l}m\,\mathcal{M}_{l^{\prime},l}^{Q,m}\right)-\\\
&\frac{1}{l}\sqrt{\frac{2l+1}{2l-1}(l^{2}-m^{2})(l^{2}-Q^{2})}\,\,J_{l^{\prime},l}^{Q,m}-\lambda_{Q,l}\cos(\theta_{0})\,\mathcal{M}_{l^{\prime},l}^{Q,m}.\end{split}$
(31)

The matrix elements of both the Bisognano-Wichmann Hamiltonian
$\mathcal{H}_{l^{\prime},l}^{Q,m}$ and the overlaps matrix
$\mathcal{M}_{l^{\prime},l}^{Q,m}$ are henceforth obtained from Eq. (31), Eq.
(28), Eq. (30) and Eq. (27); the generalized eigenvalue problem Eq. (22) can
then be numerically solved.

Notice that the ground-band (see Eq. (10)) can be recovered in this formalism
by setting $C_{l,\alpha_{gb}}^{Q,m}\propto\delta_{l,Q}$, meaning that
$\Psi_{Q,\alpha_{gb},m}\propto Y_{Q,Q,m}$. This ansatz is indeed a valid
solution for the generalized eigenvalue problem Eq. (22)

$\mathcal{H}_{l^{\prime},Q}^{Q,m}=E_{Q,\alpha_{gb},m}\mathcal{M}_{l^{\prime},Q}^{Q,m}$
(32)

provided $E_{Q,\alpha_{gb},m}$ are those given by Eq. (10). Indeed after
setting $l=Q$, the Hamiltonian matrix elements Eq. (31) simplify considerably

$\frac{\mathcal{H}_{l^{\prime},Q}^{Q,m}}{4/\sqrt{\pi
Q}}=-\left(m-(Q+1)\cos(\theta_{0})\right)\,\mathcal{M}_{l^{\prime},Q}^{Q,m}$
(33)

leading to

$E_{Q,\alpha_{gb},m}=-\frac{4}{\sqrt{\pi
Q}}\left(m-(Q+1)\cos(\theta_{0})\right)$ (34)

which as anticipated matches Eq. (10).

##  IV Diagonalization of the many-particles Bisognano-Wichmann Hamiltonian

In this section we briefly detail how the many-body problem has been
diagonalized. We consider, as we did in the main text, the two- and three-
body contact interaction
$V_{2}(\mathbf{r}_{i},\mathbf{r}_{j})=g_{2}\delta^{(2)}(\mathbf{r}_{i}-\mathbf{r}_{j})$
and
$V_{3}(\mathbf{r}_{i},\mathbf{r}_{j},\mathbf{r}_{k})=g_{3}\delta^{(2)}(\mathbf{r}_{i}-\mathbf{r}_{j})\delta^{(2)}(\mathbf{r}_{i}-\mathbf{r}_{k})$.
The spatially deformed Bisognano-Wichmann interactions read

$\displaystyle V_{A,2}^{\rm BW}$
$\displaystyle=\frac{8\sqrt{Q}}{\sqrt{\pi}}\,\frac{g_{2}}{R^{2}\hbar\omega_{c}}\,\frac{\delta(\theta_{i}-\theta_{j})}{\sin(\theta_{i})}\delta(\phi_{i}-\phi_{j})\,\left(\cos(\theta_{i})-\cos(\theta_{0})\right)$
(35) $\displaystyle V_{A,3}^{\rm BW}$
$\displaystyle=\frac{8\sqrt{Q}}{\sqrt{\pi}}\,\frac{g_{3}}{R^{4}\hbar\omega_{c}}\,\frac{\delta(\theta_{i}-\theta_{j})}{\sin(\theta_{i})}\frac{\delta(\theta_{i}-\theta_{k})}{\sin(\theta_{i})}\delta(\phi_{i}-\phi_{j})\delta(\phi_{i}-\phi_{k})\,\left(\cos(\theta_{i})-\cos(\theta_{0})\right).$
(36)

Since the magnetic field is assumed to be large, we project these deformed
interactions onto the lowest level of the Bisognano-Wichmann Hamiltonian, Eq.
(11). In second-quantization language the deformed interaction Hamiltonian
therefore reads

$\displaystyle\Pi_{\rm BW}V_{A,2}^{\rm BW}\Pi_{\rm BW}$
$\displaystyle=\frac{1}{2!}\sum_{\begin{subarray}{c}m_{1},m_{2}\\\
m_{3},m_{4}\end{subarray}}\left<m_{1},m_{2}\right|V_{2}\left|m_{3},m_{4}\right>\,a^{\dagger}_{m_{1}}a^{\dagger}_{m_{2}}a^{\vphantom{\dagger}}_{m_{3}}a^{\vphantom{\dagger}}_{m_{4}}$
(37) $\displaystyle\Pi_{\rm BW}V_{A,3}^{\rm BW}\Pi_{\rm BW}$
$\displaystyle=\frac{1}{3!}\sum_{\begin{subarray}{c}m_{1},m_{2},m_{3}\\\
m_{4},m_{5},m_{6}\end{subarray}}\left<m_{1},m_{2},m_{3}\right|V_{3}\left|m_{4},m_{5},m_{6}\right>\,a^{\dagger}_{m_{1}}a^{\dagger}_{m_{2}}a^{\dagger}_{m_{3}}a^{\vphantom{\dagger}}_{m_{4}}a^{\vphantom{\dagger}}_{m_{5}}a^{\vphantom{\dagger}}_{m_{6}}$
(38)

where $\left|m\right>=a^{\dagger}_{m}\left|0\right>$ and $a^{\dagger}_{m}$
creates a particle with angular momentum $m$ in the lowest-lying band of the
Bisognano-Wichmann Hamiltonian; namely,
$\left<\mathbf{r}|m\right>=\psi_{Q,m}^{A}(\mathbf{r})$ (see Eq. (11)). Since
we consider bosons, annihilation/creation operators at the same angular
momentum do not commute and rather obey
$[a^{\vphantom{\dagger}}_{m},a^{\dagger}_{m^{\prime}}]=\delta_{m,m^{\prime}}$.

The relevant matrix elements can be computed explicitly as

$\displaystyle\left<m_{1},m_{2}\right|V_{2}\left|m_{3},m_{4}\right>$
$\displaystyle=\frac{8\sqrt{Q}}{\sqrt{\pi}}\,\frac{g_{2}}{R^{2}\hbar\omega_{c}}\,\frac{\delta_{m_{1}+m_{2},m_{3}+m_{4}}\,F_{\sin^{2}(\theta_{0}/2)}\left(Q,\frac{m_{1}+m_{2}+m_{3}+m_{4}}{2}\right)}{4\pi\sqrt{\prod_{i=1}^{4}\mathrm{B}_{\sin^{2}(\theta_{0}/2)}\left(Q+1-m_{i},Q+1+m_{i}\right)}}$
(39)
$\displaystyle\left<m_{1},m_{2},m_{3}\right|V_{3}\left|m_{4},m_{5},m_{6}\right>$
$\displaystyle=\frac{8\sqrt{Q}}{\sqrt{\pi}}\,\frac{g_{3}}{R^{4}\hbar\omega_{c}}\,\frac{\delta_{m_{1}+m_{2}+m_{3},m_{4}+m_{5}+m_{6}}\,G_{\sin^{2}(\theta_{0}/2)}\left(Q,\frac{m_{1}+m_{2}+m_{3}+m_{4}+m_{5}+m_{6}}{2}\right)}{(4\pi)^{2}\sqrt{\prod_{i=1}^{6}\mathrm{B}_{\sin^{2}(\theta_{0}/2)}\left(Q+1-m_{i},Q+1+m_{i}\right)}}$
(40)

where

$\begin{split}F_{x}\left(Q,M\right)=\mathrm{B}_{x}\left(2Q-M+1,2Q+M+3\right)&-\mathrm{B}_{x}\left(2Q-M+3,2Q+M+1\right)-\\\
&-\cos(\theta_{0})\,\mathrm{B}_{x}\left(2Q-M+1,2Q+M+1\right)\end{split}$ (41)

and

$\displaystyle\begin{split}G_{x}\left(Q,M\right)=\mathrm{B}_{x}\left(3Q-M+1,3Q+M+3\right)&-\mathrm{B}_{x}\left(3Q-M+3,3Q+M+1\right)-\\\
&-\cos(\theta_{0})\,\mathrm{B}_{x}\left(3Q-M+1,3Q+M+1\right).\end{split}$ (42)

Using standard exact diagonalization techniques the projected Bisognano-
Wichmann interaction Hamiltonians given by Eq. (37) and Eq. (38) can then be
diagonalized.

Figure 5: Deviations from unitarity of the matrix of overlaps between real-
space entanglement Hamiltonian eigenvectors with finite entanglement energy
and the Bisognano-Wichmann Hamiltonian eigenvectors lying below the
entanglement gap. The image on the left is for the case of a bosonic Laughlin
state with $N=10$ particles ($Q=9$), bipartitioned at the equator in two equal
hemispheres ($N_{A}=5$, $\theta_{0}=\pi/2$). The image on the right is for the
case of a bosonic Moore-Read state with $N=14$ particles ($Q=6$),
bipartitioned at the equator in two equal hemispheres ($N_{A}=7$,
$\theta_{0}=\pi/2$).

##  V Exact zero-energy eigenstates of the Bisognano-Wichmann Hamiltonian for
the Laughlin state

In this section, we analytically write down the eigenstates of the many-body
Bisognano-Wichmann in the case of the projected two-body contact interaction,
$\Pi_{\rm BW}V_{A}^{\rm BW}\Pi_{\rm BW}$. It is not difficult in this case to
write down the analytical expressions for the zero-energy eigenstates of the
Bisognano-Wichmann Hamiltonian, using the fact that the single-particle levels
are restricted to the lowest Bisognano-Wichmann band. We stereographically
project the sphere onto the complex plane
$z_{i}=2R\,v_{i}/u_{i}=2R\,\tan(\theta_{i}/2)e^{-i\phi_{i}}$ [61]. We require
that the many-body wavefunction has the good symmetry for a system of $N_{A}$
bosons, and that each particle sees a wavefunction-zero at the position of the
other ones. In this case the many-body states will lie in the kernel of the
interaction Hamiltonian. We can therefore mimic the well-known construction of
excitations in a Laughlin fluid [47], by writing

$\Psi(\\{z_{i}\\})\sim\Phi(\\{z_{i}\\})\,\,\prod_{i<j}(z_{i}-z_{j})^{2}\,\prod_{i}\,\frac{\Theta_{\rm
H}\left(\tan\left(\frac{\theta_{0}}{2}\right)-\frac{|z_{i}|}{2R}\right)}{\left(1+\frac{|z_{i}|^{2}}{4R^{2}}\right)^{1+Q}}.$
(43)

Here the Heaviside function comes from the restriction of the $N_{A}$
particles to the northern spherical cap, and $\Phi(\\{z_{i}\\})$ is a
symmetric polynomial of homogeneous degree $\ell$. If, as we did in the main
text, we restrict to truncated wavefunctions with $-Q\leq m\leq Q$, the
maximal degree at which each $z_{i}$ can appear must be positive and cannot
exceed $2Q-2(N_{A}-1)$. In each angular momentum sector $L_{z}^{A}$, the
counting of these polynomials exactly matches the one of the real-space
entanglement spectrum at fixed $N_{A}$ and $L_{z}^{A}$, and in the scaling
limit approaches the chiral-boson one. It is important to notice that these
states exactly match the eigenvectors of the real-space entanglement
Hamiltonian.

On the other hand, if the full Bisognano-Wichmann lowest lying band is
considered, there is no such constraint on the degree of $\Phi(\\{z_{i}\\})$.
The two countings will be the same provided $\ell\leq 2(Q+1-N_{A})$, i.e.
$L_{z}^{A}>(N_{A}-2)(Q+1-N_{A})$.

We numerically verified that the vector-space spanned by these eigenvectors at
fixed $N_{A}$, $L_{z}^{A}$ spans exactly the same vector-space spanned by the
eigenvectors of the entanglement Hamiltonian having finite entanglement
energy. We did so by computing the matrix of overlaps between the eigenvectors
$\left|\psi^{\rm BW}_{i}\right>$ of the Bisognano-Wichmann Hamiltonian those
of the entanglement one, $\left|\phi^{\rm RSES}_{\alpha}\right>$. We then
tested whether such a matrix represented a change of basis by testing its
unitarity computing the Hilbert-Schmidt norm of
$\mathcal{O}^{\dagger}\mathcal{O}^{\vphantom{\dagger}}-\mathbb{1}$. The
results are shown in Fig. 5. It can be seen that, to numerical accuracy, the
two set of eigenvectors indeed span the same vector-subspace.

##  VI Contact two-body interaction as a proxy for the Bisognano-Wichmann
Hamiltonian

From the previous section it is clear that, since the relevant many-body
states of the Bisognano-Wichmann Hamiltonian have zero interaction-energy, the
spatial deformation associated to the Bisognano-Wichmann interaction
Hamiltonian is not necessary when dealing with model wavefunctions. This
simple observation is highly relevant for quantum-simulation platforms, where
in practice the experimental implementation of spatially-deformed interactions
is not straightforward: provided the deformed single particle Bisognano-
Wichmann can be implemented, the “bare” two- or three-body interactions will
stabilize states which correspond to the eigenstates of the entanglement
Hamiltonian of a Laughlin and Moore-Read state, respectively.

This is illustrated in Fig. 6, both for the case of the Laughlin (left panel)
and Moore-Read (right panel) states. Even though the excited (non-zero energy)
states in the two cases are different (and notice that the Bisognano-Wichmann
Hamiltonian makes the entanglement gap larger), the zero-energy branch shares
exactly the same eigenvectors (and therefore it exhibits the same state
counting).

Figure 6: The images compare the spectra of the original interaction (red
circles) with the Bisognano-Wichmann one (black crosses) projected onto the
lowest branch of the single-particle Bisognano-Wichmann Hamiltonian, for the
two cases of contact two-body interactions at $N=10$ and $Q=9$ (left image)
and contact three-body ones at $N=14$ and $Q=6$ (right image). The two cases
mimic the entanglement Hamiltonian of the Laughlin and Moore-Read states,
respectively.

##  VII State counting

Before discussing the analysis we performed for the case of non-model
wavefunctions, in this section we briefly review the counting of edge modes
for the two topological orders we analysed, the Laughlin and the Moore-Read
states.

### Laughlin edge-states counting

A simple physical way of accounting for the edge excitations of a Laughlin
state is the hydrodynamical approach. This fractional quantum Hall state can
be seen as a single-component incompressible fluid, which contains no low-
energy bulk excitations. On the other hand, gapless chiral modes are localized
at the system’s boundary. These surface deformations at low-energy and long-
wavelengths can be completely described by a $\text{U}(1)$ Kac-Moody algebra
[47]

$\begin{array}[]{l}H=\frac{2\pi}{L}\hbar\frac{v}{\nu}\sum_{k>0}\rho_{-k}\,\rho_{k}\\\
\left[\rho_{k},\rho_{-k^{\prime}}\right]=\nu\frac{kL}{2\pi}\,\delta_{k,k^{\prime}}\end{array}$
(44)

where $\rho_{k}$ are the modes of the 1-dimensional density at the edge of the
system. We consider a compact edge of length $L$ and label the momentum
$k=\frac{2\pi}{L}\,n$ by an integer $n$. The edge of a Laughlin state can
therefore be described by a bosonic field theory, with the Laughlin ground
state being the vacuum of the theory $\left|0\right>$. The modes at a given
momentum $K$ can then be written as
$\prod_{n\in\bm{\lambda}}a^{\dagger}_{n}\left|0\right>$, where $\bm{\lambda}$
is an integer partition of $K/(2\pi/L)\equiv M$ and
$a^{\dagger}_{n}\propto\rho_{-k}$. We list the lowest lying excitations in
Table 1.

| Momentum
---
$\Delta M$
| Partition
---
$\bm{\lambda}$
States |  | Number
---
of states
| Energy
---
$\Delta E$
$0$ | $\\{\\}$ | $\left|0\right>$ | 1 | $0$
$1$ | $\\{1\\}$ | $a^{\dagger}_{1}\left|0\right>$ | 1 | $e_{0}$
$2$ | $\\{2\\}$ | $a_{2}^{\dagger}\left|0\right>$ | 2 | $2e_{0}$
| $\\{1,1\\}$ | $(a^{\dagger}_{1})^{2}\left|0\right>$ |  |
$3$ | $\\{3\\}$ | $a_{3}^{\dagger}\left|0\right>$ | 3 | $3e_{0}$
| $\\{2,1\\}$ | $a_{2}^{\dagger}a_{1}^{\dagger}\left|0\right>$ |  |
| $\\{1,1,1\\}$ | $(a_{1}^{\dagger})^{3}\left|0\right>$ |  |
$4$ | $\\{4\\}$ | $a_{4}^{\dagger}\left|0\right>$ | 5 | $4e_{0}$
| $\\{3,1\\}$ | $a_{3}^{\dagger}a_{1}^{\dagger}\left|0\right>$ |  |
| $\\{2,2\\}$ | $(a_{2}^{\dagger})^{2}\left|0\right>$ |  |
| $\\{2,1,1\\}$ | $a_{2}^{\dagger}(a_{1}^{\dagger})^{2}\left|0\right>$ |  |
| $\\{1,1,1,1\\}$ | $(a_{1}^{\dagger})^{4}\left|0\right>$ |  |
Table 1:  List of the lowest lying edge excitations on top of the Laughlin
ground state. We defined $e_{0}=v\,\hbar\frac{2\pi}{L}$. Momentum and energy
are measured with respect to the one of the ground state, $\left|0\right>$.

### Moore-Read edge-states counting

The edge excitations of the Moore-Read state contain, in addition to the
$\text{U}(1)$ Kac-Moody algebra associated to a branch of bosonic edge
excitations analogous to the one of the Laughlin liquid described above, a
chiral Majorana fermion [62, 63], which can have both periodic or anti-
periodic boundary conditions. The low-energy and long-wavelength description
of this mode reads

$\begin{array}[]{l}H=\frac{2\pi}{L}\,\hbar v^{\prime}\sum_{n\geq
0}n\,c_{-n}\,c_{n}\\\
\left\\{c_{n},c_{-n^{\prime}}\right\\}=\delta_{n,n^{\prime}},\end{array}$ (45)

where $n-\frac{1}{2}\in\mathbb{N}$ ($n\in\mathbb{N}$) for anti-periodic
(periodic) boundary conditions on the fermion field. The edge-excitations of
the Moore-Read ground state turn out to correspond to the choice of anti-
periodic boundary conditions. The theory has two additional independent
sectors, according to the (conserved) parity of the number of fermions excited
along the edge. This parity has to match the one of the number of electrons in
the system. We list the lowest lying excitations in Table 3 and 3.

Momentum $\Delta M$ States Number of states Energy $\Delta E$ $0$
$\left|0\right>$ 1 $0$ $1$ 0 $e_{0}^{\prime}$ $2$
$c_{3/2}c_{1/2}\left|0\right>$ 1 $2e_{0}^{\prime}$ $3$
$c_{5/2}c_{1/2}\left|0\right>$ 1 $3e_{0}^{\prime}$ $4$
$c_{7/2}c_{1/2}\left|0\right>$ 2 $4e_{0}^{\prime}$
$c_{5/2}c_{3/2}\left|0\right>$ Table 2: List of the lowest lying edge
excitations of a Majorana fermion with antiperiodic boundary conditions and
even number of fermions. We defined $e_{0}^{\prime}=\hbar
v^{\prime}\,\frac{2\pi}{L}$. Momentum and energy are measured with respect to
the zero-momentum ground state, $\left|0\right>$. Momentum $\Delta M$ States
Number of states Energy $\Delta E$ $0$ $c_{1/2}\left|0\right>$ 1 $0$ $1$
$c_{3/2}\left|0\right>$ 1 $e_{0}^{\prime}$ $2$ $c_{5/2}\left|0\right>$ 1
$2e_{0}^{\prime}$ $3$ $c_{7/2}\left|0\right>$ 1 $3e_{0}^{\prime}$ $4$
$c_{9/2}\left|0\right>$ 2 $4e_{0}^{\prime}$
$c_{5/2}c_{3/2}c_{1/2}\left|0\right>$ Table 3: List of the lowest lying edge
excitations of a Majorana fermion with antiperiodic boundary conditions and
odd number of fermions. We defined $e_{0}^{\prime}=\hbar
v^{\prime}\,\frac{2\pi}{L}$. Momentum and energy are measured with respect to
the ground state, $c_{1/2}\left|0\right>$.

The total number of excitations of the Moore-Read state is then obtained by
taking the product between the Majorana fermion sector and the bosonic charge
sector. We list the number of degeneracies in Table 5 with a few selected
examples for the states at $\Delta M=3$ in Table 5.

Momentum $\Delta M$ Number of states (even $N$) Number of states (odd $N$) $0$
1 1 $1$ 1 2 $2$ 3 4 $3$ 5 7 $4$ 10 13 Table 4:  Number of gapless edge modes
at the boundary of a Moore-Read topological order with either even or odd
particle numbers, $N$. Momentum $\Delta M$ States (even $N$) States (odd $N$)
3 $a_{3}^{\dagger}\left|0\right>$
$a_{3}^{\dagger}c^{\vphantom{\dagger}}_{1/2}\left|0\right>$
$a_{2}^{\dagger}a_{1}^{\dagger}\left|0\right>$
$a_{2}^{\dagger}a_{1}^{\dagger}c^{\vphantom{\dagger}}_{1/2}\left|0\right>$
$(a_{1}^{\dagger})^{3}\left|0\right>$
$(a_{1}^{\dagger})^{3}c^{\vphantom{\dagger}}_{1/2}\left|0\right>$
$a_{1}^{\dagger}c^{\vphantom{\dagger}}_{5/2}c^{\vphantom{\dagger}}_{1/2}\left|0\right>$
$a_{2}^{\dagger}c^{\vphantom{\dagger}}_{3/2}\left|0\right>$
$c_{5/2}^{\vphantom{\dagger}}c_{1/2}^{\vphantom{\dagger}}\left|0\right>$
$(a_{1}^{\dagger})^{2}c^{\vphantom{\dagger}}_{3/2}\left|0\right>$
$a_{1}^{\dagger}c^{\vphantom{\dagger}}_{5/2}\left|0\right>$
$c^{\vphantom{\dagger}}_{7/2}\left|0\right>$ Table 5:  Edge-states of a Moore-
Read topological order with either even or odd particle numbers, $N$ in the
$\Delta M=3$ momentum sector.

##  VIII Numerical analysis for the contact interacting Moore-Read state

In this final section we present additional numerical data for the relevant
case of the Moore-Read topological order stabilized by two-body contact
interactions. We start by discussing bipartitions with particles numbers
$N_{A}\neq N/2$, which allow us to investigate more thoroughly the connection
between the Bisognano-Wichamann and the real-space entanglement Hamiltonian.
We then show that the eigenvectors of the Bisognano-Wichmann Hamiltonian are,
to a good level of approximation, the same as those of the real-space
entanglement spectrum, thereby proving the validity of the Bisognano-Wichmann
ansatz.

### Spherical-cap bipartition

We here introduce a new cut in order to extensively study the connection
between the Bisognano-Wichmann ansatz and the real-space entanglement
Hamiltonian as a function not only of the particle number $N$, but also as a
function of the bipartition particle number sector $N_{A}$. A natural choice
is to move the longitudinal bipartition, so that the probability of having
$N_{A}$ particles within the spherical cap defined by
$0\leq\theta\leq\theta_{0}$ (see the insets of Fig. 7) does not change
drastically as $N_{A}$ is varied.

Such an angle can be determined by the following simple physical
consideration. A fractional quantum Hall ground state is an incompressible
featureless liquid which in the thermodynamic limit has uniform density
$\rho_{0}=\nu/2\pi l_{B}^{2}$, where $\nu$ is the filling fraction and
$l_{B}=\sqrt{\hbar/qB}$ the magnetic length. We require

$N_{A}=\int_{0}^{2\pi}R\,d\phi\int_{0}^{\theta_{0}}R\sin(\theta)d\theta\,\rho_{0}$
(46)

i.e. that the number of particles in the spherical-cap is precisely $N_{A}$.
Since $R^{2}/l_{B}^{2}=Q$, this leads to $N_{A}=2\nu
Q\sin^{2}\left(\frac{\theta_{0}}{2}\right)$. Since
$2Q=N_{\phi}=\nu^{-1}N-\mathcal{S}\approx\nu^{-1}N$ ($\mathcal{S}$ being the
so-called “shift” [64]) we finally get

$\theta_{0}=2\arcsin\sqrt{\frac{N_{A}}{N}}.$ (47)

In the following, we will vary the cut longitude $\theta_{0}$ with $N$ and
$N_{A}$ as prescribed by the relation Eq. (47). Notice that with this choice
the length $\mathcal{L}$ of the cut does not change as the number of particles
$N$ is increased and $N_{A}$ is kept constant, for $\mathcal{L}=2\pi
R\sin(\theta_{0})\approx\pi\sqrt{N_{A}/\nu}\,l_{B}$.

Figure 7: The three images show the real-space entanglement spectra for
$N=16$, $Q=7$ bosonic Moore-Read state at filling $\nu=1$, stabilized by two-
body contact interactions (panels (a)), compared to the Bisognano-Wichmann
ansatz for the given bipartiton (panels (b)), which, from left to right, are
given by $N_{A}=4,6,8$. In the lower panel, the dashed elliptical shape serves
as a guide to the eye for the lower branch of states and the numbers indicate
the number of states it comprises at a given $L_{z}^{A}$, for the values that
are in agreement with the real-space entanglement spectrum. The insets show
the density of the lowest-lying state of the Bisognano-Wichmann Hamiltonian as
a heat-map, illustrating how the spherical cap at longitude $\theta_{0}$ is
varied as a function of $N_{A}$. In all cases the points have been coloured
according to the expectation value of the three-body contact interaction
$V_{3}$. For clarity reasons an arbitrary linear dispersion term has been
introduced in the Bisognano-Wichmann case.

Comparisons between the real-space entanglement spectrum and the Bisognano-
Wichmann one are presented in Fig. 7 for $N=16$, $Q=7$ and three different
values of $N_{A}=4,6,8$. The cuts at different values of $\theta_{0}$ are
clearly visible in the insets, which display the ground-state density of the
Bisognano-Wichmann Hamiltonian.

More importantly, it can be seen that the counting of the lowest lying
Bisognano-Wichmann states matches the one of the real-space entanglement
spectrum. In particular, these plots complement the data presented in the main
text, where we considered $N_{A}$ to be odd: it can be seen that for both the
even and odd cases the Bisognano-Wichmann Hamiltonian reproduces the correct
counting for a Moore-Read topological order, and that these value approach the
conformal field theory countings.

It can be noted moreover that the Li-Haldane counting for smaller values of
$N_{A}$ extends to smaller values of $L_{z}^{A}$, covering a larger region.
Furthermore, the Bisognano-Wichmann entanglement gap looks to be more robust
for smaller $N_{A}$. We do not have an explanation for this behaviour, which
deserves further investigation.

Finally it can be seen that the expectation values of the three-body contact
interaction (plotted as a colour-scale) computed using the real-space
entanglement Hamiltonian eigenvectors and the Bisognano-Wichmann ones are in
qualitative agreement. In the case of the “entanglement ground-state”, we
verified the relative error committed by the Bisognano-Wichmann ansatz is
within some percent points, see Fig. 8. We did not analyse states at different
values of $L_{z}^{A}$, because the analysis is complicated by the presence of
additional states at the same value of $L_{z}^{A}$ which can in principle mix
among each other.

Figure 8: Absolute value of the relative error on the expectation value of the
three-body contact interaction computed using either the eigenvectors of the
real space entanglement Hamiltonian or the Bisognano-Wichmann one, in the case
of a bosonic $\nu=1$ Moore-Read-like phase at $2Q=N-2$ stabilized by two-body
contact interactions. On the $x$-axis, various values for the number of
particles $N$ (which fix the value of $Q$) are shown, while on the $y$-axis we
plot various values for the particles $N_{A}$ in the northern spherical-cap.

### Overlaps

We already discussed that in the case of model wavefunctions the eigenvectors
of the Bisognano-Wichmann Hamiltonian are the same as those of the real-space
entanglement one. One does not expect this to be true in general for non-model
states. Therefore, a relevant question which we address here is to what extent
the eigenvectors $\left|\psi^{\rm BW}_{i}\right>$ of the Bisognano-Wichmann
Hamiltonian provide a good description of those of the entanglement one,
$\left|\phi^{\rm RSES}_{\alpha}\right>$.

Instead of blindly computing the overlaps between eigenvectors, as already
discussed above we study the matrix of the overlaps
$\mathcal{O}_{\alpha,i}=\left<\phi^{\rm RSES}_{\alpha}|\psi^{\rm
BW}_{i}\right>$ as a function of $N_{A}$ and $L_{z}^{A}$; we already discussed
in the main text that, at least for a certain range of values of $L_{z}^{A}$,
the counting of states below the so-called “entanglement-gap” match for the
two approaches, satisfying the Li-Haldane conjecture. As a consequence, the
overlap matrix $\mathcal{O}$ restricted to such a subspace is a square matrix,
which we expect to be unitary if the Bisognano-Wichmann Hamiltonian captures
the same features as the entanglement one. In particular this means that the
two low-entanglement-energy subspaces are expected to span the same vector
subspace within the whole Hilbert space.

To test the unitarity we compute the (normalized) Hilbert-Schmidt norm of
$\delta=\mathcal{O}^{\dagger}\mathcal{O}^{\vphantom{\dagger}}-\mathbb{1}$

$||\delta||=\frac{1}{d}\,\sum_{i,j}^{d}|\delta_{i,j}|^{2}$ (48)

where the sum runs over the $d^{2}$ coefficients of the difference matrix
$\delta$, namely the Li-Haldane counting of the low-entanglement-energy modes.
Normalizing by $d$ ensures that $||\mathbb{1}||=1$, i.e. that the identity has
unit norm. Moreover, it accounts for finite-size effects in the Li-Haldane
counting: at fixed $N_{A}$ and $L_{z}^{A}$, the conformal field theory
counting is recovered only when $N\rightarrow\infty$.

Figure 9: Deviations from unitarity of the matrix of overlaps between real-
space entanglement and Bisognano-Wichmann Hamiltonian eigenvectors lying below
the entanglement gap. Data are presented as a function of the angular-momentum
variation $\Delta L_{z}^{A}$ with respect to the one of the entanglement
Hamiltonian ground state, for different values of the number of particles $N$
(which fix the value of the flux through the sphere, $2Q=N-2$). The image on
the left considers $N_{A}=4$ particles in the northern spherical cap, the one
on the right $N_{A}=6$.

The results are presented in Fig. 9 for $N_{A}=4$ and $N_{A}=6$, for different
values of $L_{z}^{A}$ measured with respect to the “entanglement ground-state”
value. It can be seen that the Bisognano-Wichmann Hamiltonian provides indeed
a good description of the entanglement structure of fractional quantum Hall
ground states, even away from the model-wavefunctions limit. No apparent
scaling is visible, though. A possible reason could lie in the choice of the
spherical geometry, which was chosen so as to avoid edge effects, but may be
affected by Gaussian curvature effects. The analysis of a planar geometry,
which is free of curvature effects but may be plagued by edge ones, is
reserved for a future study. A second possibility could be that a system
sufficiently large so that the long-wavelength regime in which the Bisognano-
Wichmann can be expected to give better results was not reached with the small
sizes exact-diagonalization puts at our disposal. A hint of this fact can be
seen even in the density plots in Fig. 7, where density oscillations at the
bipartition cut prevent the bulk from reaching the incompressible
“featureless” limit $\rho_{0}$. Finally, a third possibility could be rooted
in the fact that overlaps are not well defined in the thermodynamic limit,
even though the proper long-wavelength features of the topological order have
been captured. This would actually be analogous to what happens for model
wavefunctions: their overlap with “real” topologically ordered ground states
is expected to drop to zero in the thermodynamic limit, even if the model
wavefunction captures all of its essential details. A better indicator could
be to analyse expectation values of local observables, analogously to what was
done in Fig. 8, but this again is reserved for future more systematic studies.