# The Monte Carlo simulation of the topological quantities in FQH systems

Yi Yang  Zi-Xiang Hu<EMAIL_ADDRESS>Department of Physics and Chongqing Key
Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing
401331, People’s Republic of China

###### Abstract

Generally speaking, for a fractional quantum Hall (FQH) state, the electronic
occupation number for each Landau orbit could be obtained from numerical
methods such as exact diagonalization, density matrix renormalization group or
algebraic recursive schemes (Jack polynomial). In this work, we apply a
Metroplis Monte Carlo method to calculate the occupation numbers of several
FQH states in cylinder geometry. The convergent occupation numbers for more
than 40 particles are used to verify the chiral bosonic edge theory and
determine the topological quantities via momentum polarization or dipole
moment. The guiding center spin, central charge and topological spin of
different topological sectors are consistent with theoretical values and other
numerical studies. Especially, we obtain the topological spin of $e/4$
quasihole in Moore-Read and 331 states. At last, we calculate the electron
edge Green’s functions and analysis position dependence of the non-Fermi
liquid behavior.

###### pacs:

73.43.Lp, 71.10.Pm

##  I Introduction

Since the discovery of fractional quantum Hall effect (FQHE), its rich
physical connotations and novel topological properties have attracted the
extensive attention of physicists. DCTHLSACG ; RBL  Different from integer
quantum Hall state (IQHE), FQH state is embedded with quantum topological
order which manifests novel properties include fractional charge excitation,
fractional statistics, topological ground state degeneracy, gapless chiral
edge excitation and topological entanglement entropy, etc. XGWQN ; WXG3 ;
DAJRSFW ; FDMHEHR  The quantum Hall bulk is an incompressible insulator which
is difficult to produce a signal in experimental measurements. Therefore, its
conducting gapless edge mode provides a window to detect the topological
properties due to the mechanism of bulk-edge correspondence. XGW ; XGWQN
Early in 1990s, the physics of edge excitation was considered extremely
important for studying the FQHE. AHMD ; WXG1 ; WXG2 ; WXG3  It is known that
most of the FQH edges can be treated as a chiral Luttinger liquid ($\chi LL$)
instead of a non-interacting Fermi-liquid. MHDCTMGLNPKWW  In experiments, one
can measure the non-Fermi liquid behavior via nonlinear $I\propto V^{\alpha}$
relation in the tunneling experiment from Fermi-liquid to FQH liquid. The
Tomonage-Luttinger (TL) exponent $\alpha$ could be calculated from the edge
Green’s function
$G(|\vec{r}_{1}-\vec{r}_{2}|)=\langle\psi^{\dagger}(\vec{r}_{1})\psi(\vec{r}_{2})\rangle\propto|\vec{r}_{1}-\vec{r}_{2}|^{-\alpha}$.
The edge electron propagator also can describe the entanglement of two
particles on the edge. Wen’s effective theory WXG3  demonstrated that the
spatial decay of electron propagator involves a non-Fermi-liquid exponent
$\alpha=q$ for $\nu=1/q$ Laughlin state, $\alpha=3$ for MR state and $331$
state. For a realistic system with Coulomb interaction, the values of $\alpha$
are found not that universal. This has attracted a lot of theoretical and
experimental attentions, MGDCTLNPKWWAMC ; AMCMKWCCCLNPKWW ; AMCLNPKWW ; AMC ;
MG ; VJGEVT ; SSMJKJ1 ; XWFEEHR ; Chamon94 ; XWKYEHR ; KY ; YNJHKNGM ;
MHDCTMGLNPKWW  such as the influence of the edge reconstruction, the sample
qualities and the emergence of neutral mode. Recently, it was verified that
the FQH in suspended graphene could avoid those obstructive factors and
realize the universal edge physics. ZXHRNBXWKY ; Kumar21 ; Kumar22a ; Kumar22b
Similarly, the occupation numbers near the edge obey
$\displaystyle\lim_{k\rightarrow edge}n_{k}\propto k^{\beta}$ in continuum
limit, as predicted by chiral boson edge theory. WXG4  At the same time, the
information of the bulk magnetoroton excitation has been claimed to be
embodied in the oscillation of the occupation numbers near the edge. Shibata

In a correlated FQH system, the density deviates from the bulk filling
$\rho=\frac{\nu}{2\pi l_{B}^{2}}$ near the edge and thus results in an extra
“intrinsic dipole moment” which is related to the guiding-center Hall
viscosity. JEARSPGZ ; Park ; F. D. M. Haldane  It is worth mentioning that the
Hall viscosity is characterized by a rational number and a metric tensor that
defines distances on an “incompressibility length-scale”, and its magnitude
provides a lower bound to the coefficient of the $O(q^{4})$ small-$q$ limit of
the guiding center structure factor. The Hall viscosity is also related to the
momentum polarization Read09 ; HHTYZXLQ  of the system while rotating half of
the system and keeping another half invariant. In fact, the momentum
polarization is the sub-leading term of the average value of a “partial
translation operator”. Therefore, the calculation of the intrinsic dipole
moment, or the momentum polarization, is averaging the momentum operator of a
subsystem in a bipartition. The interesting thing is that the topological
quantities of the FQH state, such as the guiding-center spin, central charge
and the topological spin of the quasiparticle excitation could also be
determined from the coefficients and corrections of the momentum polarization.
MPZRSKMFP ; Park ; LDHWZ ; LHZLDNSFDMHWZ  The guiding-center spin is related
to the non-dissipative response of the metric perturbation in FQH liquids. Its
coupling with the geometric curvature of the underlying manifold gives the
topological shift of the FQH states in spherical geometry. Then topological
spin and central charge are the elements of modular-$T$ matrix which are used
to describe the topological order of the FQH state. EKVXGW  Meanwhile, the
central charge determines the heat current $I_{E}=\frac{\pi}{6}cT^{2}$ at a
given temperature IA  and is also related to the gravitational anomaly of the
edge. LAGEW

In numerical calculation, the density fluctuations of the quantum Hall edge
affect several Landau orbitals and the scope of its influence becomes larger
for small bulk density. For example, the edge of $\nu=1/5$ Laughlin state
affects more orbits than that of the $\nu=1/3$ Laughlin state. Moreover, in
case of the realistic long range Coulomb interaction, the edge oscillates
deeper into the bulk than the short range model interaction. A criteria to get
a full profile of the edge state is the density in the bulk should be stable
at the filling factor $\nu/2\pi l_{B}^{2}$. This is usually out of the reach
of exact diagonalization or the Jack polynomial which are limited by the small
size of the Hilbert space. The inaccurate momentum polarization calculation in
small system size can not give the convergence of the physical quantities and
even wrong results at some time. The developments to solve this problem have
been done are the density-matrix renormalization group Shibata ; LHZLDNSFDMHWZ
and Matrix product state description MPZRSKMFP . In this work, we develop the
Monte Carlo simulation method in cylinder geometry to calculate the occupation
numbers for several FQH states. From these occupation numbers, we explore the
momentum polarization and its related topological quantities in a high
accuracy. The edge Green’s function is also calculated for large system and
the parameter of the chiral Lutteringer liquid theory is determined in a
higher accuracy than the previous studies.

The rest of the paper is organized as follows. In Sec. II, we calculate the
occupation numbers for several FQH states in cylinder geometry and revisit the
exponents of the chiral boson theory (CBT) of the edge. In Sec. III, we
calculate the topological quantities from edge dipole moment and momentum
polarization. In Sec. IV, we acquire the TL exponent $\alpha$ from equal time
Green’s function and discuss the validity of the TL theory. The MR and 331
states are also considered. Sec. V gives the conclusions and discussions.

##  II The occupation number and its scaling behaviours

Figure 1:  The sketch map of cylinder model with $L_{x}$ and $L_{y}$ in two
directions. $L_{y}=2\pi R$ is the circumference of cylinder, and the
reciprocal of the radius is defined as $\gamma=1/R$.

We firstly introduce the occupation number calculation by Metropolis Monte
Carlo which was previously implemented in disk geometry. Morf ; SMAHMD ;
UKMGYG  The disk geometry in symmetric gauge has unequally spaced orbits which
evolves much more orbitals for edge profile and induces to slow the
convergence of the bulk density. In this work, as shown in the sketch of Fig.
1, we use the cylinder geometry which has advantages that the space between
adjacent Landau orbits is homogeneous and the length of the edge on the two
ends is tunable by varying the aspect ratio $L_{x}/L_{y}$ with keeping the
surface area invariant. The normalized $N$-electron Laughlin wave function
$|\psi^{c}_{1/q}\rangle$ at filling $\nu=1/q$ is SJEHLRS ; D. J Thouless

$\displaystyle|\psi^{c}_{1/q}\rangle=\frac{1}{\sqrt{N!}}\frac{1}{(2\pi\gamma^{-1}\sqrt{\pi})^{N/2}}\exp(-\frac{9}{2}\gamma^{2}\sum_{j=0}^{N-1}j^{2})$
$\displaystyle\prod_{j<k}(e^{\gamma\mathbf{z}_{j}}-e^{\gamma\mathbf{z}_{k}})^{q}e^{-\frac{1}{2}\sum_{i=1}^{N}x_{i}^{2}}e^{-\sum_{i}\frac{q\gamma}{2}(N-1)\mathbf{z_{i}}}$
(1)

in which $\mathbf{z}_{i}=(x_{i}+iy_{i})/l_{B}$ is the coordinate of the $i$’th
particle, $l_{B}$ is the magnetic length $l_{B}=\sqrt{\hbar/eB}$ which we set
to one and the Landau orbial space $\gamma=2\pi/L_{y}=1/R$ where $R$ is the
radius of the cylinder. The last term is a global shift which lets the FQH
state be symmetric around the center of cylinder at $x=0$. The average
occupation of the $m$th single-paricle state is

$\begin{split}\langle
c_{m}^{\dagger}c_{m}\rangle_{1/q}&=\frac{\langle\psi^{c}_{1/q}|c_{m}^{\dagger}c_{m}|\psi^{c}_{1/q}\rangle}{\langle\psi^{c}_{1/q}|\psi^{c}_{1/q}\rangle}\\\
&=\int
d^{2}z_{1}d^{2}z_{2}\rho_{1/q}(\mathbf{z_{1}},\mathbf{z_{2}})\phi_{m}^{c*}(\mathbf{z_{1}})\phi^{c}_{m}(\mathbf{z_{2}})\end{split}$
(2)

where $\rho_{1/q}$ is the one-particle density matrix Jain  and
$\phi^{c}_{m}(\mathbf{z})=\frac{1}{\sqrt{\pi^{1/2}L_{y}}}e^{iky}e^{-(x-k)^{2}/2}$
is the wave package of the lowest Landau level in Landau gauge with wave
vector $k=\frac{2\pi m}{L_{y}}$. Now the $y$ direction translation momentum
quantum number $m$’s are symmetrically distributed in range
$[-\frac{q(N-1)}{2},\frac{q(N-1)}{2}]$. The one-particle density matrix is

$\displaystyle\rho_{1/q}(\mathbf{z_{a}},\mathbf{z_{b}})$ $\displaystyle=$
$\displaystyle N\int d^{2}\mathbf{z_{2}}\cdots\int
d^{2}\mathbf{z_{N}}\psi^{c}_{1/q}(\mathbf{z_{a},z_{2}\cdots z_{N}})$ (3)
$\displaystyle\psi_{1/q}^{c*}(\mathbf{z_{b},z_{2}\cdots
z_{N}})/\int\prod_{i=1}^{N}d^{2}\mathbf{z_{i}}|\psi^{c}_{1/q}|^{2}.$

Because $\phi^{c}_{m}$ and $|\psi^{c}_{1/q}\rangle$ conserve the translation
momentum operator along $y$, the one-particle density matrix could be written
in second quantized form

$\rho_{1/q}(\mathbf{z_{a}},\mathbf{z_{b}})=\sum_{m}\langle
c_{m}^{\dagger}c_{m}\rangle_{1/q}\phi^{c}_{m}(\mathbf{z_{a}})\phi_{m}^{c*}(\mathbf{z_{b}})$
(4)

In the special case of $\mathbf{z}_{a}=x+iy$ and
$\mathbf{z}_{b}=x+i(y+y_{j})$, namely $\mathbf{z}_{a}$ and $\mathbf{z}_{b}$
have the same $x$ and a shift $y_{i}$ in $y$,

$\rho_{1/q}(\mathbf{z},y_{j},\mathbf{z})=\sum_{k}\langle
c_{k}^{\dagger}c_{k}\rangle_{1/q}|\phi^{c}_{k}(\mathbf{z})|^{2}e^{iky_{j}}.$
(5)

Since $\langle c_{k}^{\dagger}c_{k}\rangle_{1/q}$ is non-zero over a
contiguous, finite and known range
$k\in[-\frac{2\pi}{L_{y}}\frac{q(N-1)}{2},\frac{2\pi}{L_{y}}\frac{q(N-1)}{2}]$,
the summation over $k$ can be restricted to this range without any
uncertainty. Then the above relation could be explained as a discrete Fourier
transformation from momentum space $k$ to real space conjugate $y$. The
inverse transformation has the following form

$\langle
c_{k}^{\dagger}c_{k}\rangle_{1/q}|\phi^{c}_{k}(\mathbf{z})|^{2}=\frac{1}{N_{orb}}\sum_{j=0}^{q(N-1)}e^{-iky_{j}}\rho_{1/q}(\mathbf{z},y_{j},\mathbf{z})$
(6)

where $y_{j}=\frac{L_{y}}{N_{orb}}j$ and $N_{orb}=qN-q+1$ is the number of
orbits. Note that Eq. (6) is only true for $-\frac{q(N-1)}{2}\leq
m\leq\frac{q(N-1)}{2}$. In principle Eq. (6) is valid for any value of $z$,
but practically the resulting uncertainty in the occupation number will be a
minimum when $r$ is near the maximum in $|\phi^{c}_{k}(\mathbf{z})|^{2}$ which
occurs at $z\sim m\gamma l_{B}$. We evaluate the occupation number by
integrating Eq. (6) over $\mathbf{z}$ to get,

$\langle
c_{k}^{\dagger}c_{k}\rangle_{1/q}=\frac{1}{N_{orb}}\sum_{j=0}^{N_{orb}-1}e^{-iky_{j}}\rho_{1/q}(y_{j})$
(7)

where $\rho_{j}=\rho_{1/q}(y_{j})=\int
d^{2}z\rho_{1/q}(\mathbf{z},y_{j},\mathbf{z})$. Then the occupation at any $k$
(within the appropriate range) can be found after evaluating $\rho_{j}$ for
all $j=0,\cdots,N_{orb}-1$. From Eq. (3), we have,

$\rho_{j}=\frac{N\int\prod_{i=1}^{N}d^{2}\mathbf{z_{i}}\psi^{c}_{1/q}(\mathbf{z_{1}-iy_{j},\cdots
z_{N}})\psi_{1/q}^{c*}(\mathbf{z_{1},\cdots
z_{N}})}{\int\prod_{i=1}^{N}d^{2}\mathbf{z_{i}}|\psi^{c}_{1/q}|^{2}}$ (8)

Ignoring the normalization factor, the Eq. (II) becomes,

$\psi^{c}_{1/q}(\mathbf{z_{1}-iy_{j}},\mathbf{z_{2}},\cdots)=\psi^{c}_{1/q}(\mathbf{z_{i}})Z_{1}(y_{j},\mathbf{z})$
(9)

where

$Z_{b}(y_{j},\mathbf{z})=\prod_{k\neq
b}\frac{(e^{\gamma(z_{b}-iy_{j})}-e^{\gamma z_{k}})^{q}}{(e^{\gamma
z_{b}}-e^{\gamma z_{k}})^{q}}e^{i\gamma\frac{q(N-1)}{2}y_{j}}$ (10)

thus we have

$\rho_{j}=\frac{N\int\prod_{i=1}^{N}d^{2}\mathbf{z_{i}}|\psi^{c}_{1/q}|^{2}Z_{1}(y_{j},\mathbf{z})}{\int\prod_{i=1}^{N}d^{2}\mathbf{z_{i}}|\psi^{c}_{1/q}|^{2}}$
(11)

In the final, the $\rho_{j}$ can be expressed as

$\rho_{j}=\frac{\int\prod_{i=1}^{N}d^{2}\mathbf{z_{i}}|\psi^{c}_{1/q}|^{2}\sum_{b=1}^{N}Z_{b}(y_{j},\mathbf{z})}{\int\prod_{i=1}^{N}d^{2}\mathbf{z_{i}}|\psi^{c}_{1/q}|^{2}}$
(12)

where we have symmetrized $Z_{b}$ over all particles index to increase the
rate of convergence without loss of generality. The above expression can be
evaluated through Metropolis sampling with a high accuracy. Then we can obtain
the average occupation number of $1/q$ Laughlin state on cylinder after going
back to Eq. (7). In the similar scheme, we can obtained the occupation numbers
for other FQH states, such as Moore-Read Pfaffian state and two-component
Halperin 331 state. The technical details for these states are in the Appendix
A and B.

With the occupation numbers for large system, including $\nu=1/3,1/5$ Laughlin
states, $\nu=5/2$ Moore-Read and Halperin $331$ state, we verify the behavior
of $n_{k}$ near the edge with comparing to the chiral boson theory in a high
accuracy. The magnetoroton minimum could also be fitted in a large range.
Specifically, for $1/5$ and $331$ state, since the size of Hilbert space is
extremely large in exact diagonalization. The bulk density is difficult to
reach the uniform density $\rho=\nu/2\pi l_{B}^{2}$ and thus many of the
physical quantities are obscured by finite size effects. Fig. 2 shows half of
the occupation numbers for these states due to the central symmetry. The
occupation numbers are plotted as a function of the wave vector $k=\frac{2\pi
m}{L_{y}}$ rather than orbital index $m$. By properly choosing the Fermi
points Park  to assure there are $N/\nu$ orbitals between two Fermi points,
i.e., the momentum of the first non-vanishing occupation number is $m_{0}=3/2$
for $\nu=1/3,5/2$ and $331$ states and $m_{0}=5/2$ for $\nu=1/5$ state, the
data for different circumferences $L_{y}$ (The $L_{y}$ takes value in the
range $(a)[19l_{B},30l_{B}]$, $(b)[25l_{B},40l_{B}]$, $(c)[15l_{B},23l_{B}]$
and $(d)[20l_{B},35l_{B}]$ to make sure the two edges are well separated)
collapses into a perfectly smooth curve which manifests the universality of
the FQH edge. Since we now have much more data near the Fermi points and no
breakpoints as comparing to the results from Jack polynomials. Park  The
$n(k)$ near the edge clearly demonstrates that the FQH edge is described by
the CBT with $n_{k}\propto k^{r}$ in which $r=\nu^{-1}-1$ for Laughlin states.
Thus we take the linear fit of $\log n(k)$ versus $\log k$ with the first non-
vanishing occupation number of different $L_{y}$. For the four FQH states we
considered, the CBT predicts their exponents being $r=2,4,2,2$ respectively.
Our simulation gives these fitting values as $r=1.99541\pm
0.008402,3.968181\pm 0.03806,1.9779\pm 0.03377,1.98843\pm 0.04556$ as shown in
the insets of each figure in Fig. 2. They are exactly the same as the expected
value within the statistical error.

Figure 2:  Occupation numbers at the edge of (a) $\nu=1/3$ state for 50
particles, (b) $\nu=1/5$ state for 40 particles, (c) $\nu=5/2$ MR state for 40
particles and (d) $331$ state for 80 particles(40 in each layer). The inset
plots are the linear fit in logarithmic scale for the data near the Fermi
point(labelled in red circles in $n(k)$). The slopes are $r_{1/3}=1.99541\pm
0.008402$, $r_{1/5}=3.96818\pm 0.03806$, $r_{MR}=1.9779\pm 0.03377$ and
$r_{331}=1.98843\pm 0.04556$ which are exactly the same as predicted in CBT.

On the other hand, as in Ref. Shibata, , we fit the oscillations of the
occupation numbers by
$f_{\nu}(x)=c_{\nu}\exp(-x/\epsilon_{\nu})\cos(k_{\nu}x+\theta_{\nu})+\nu$. It
was claimed that $k_{\nu}$ is in good agreement with the wave number of the
bulk magnetoroton minimum and $\epsilon_{\nu}$ is proportional to the bulk
excitation gap. The density oscillation at the edge reflecting the bulk
excitation is a good example of the bulk-edge correspondence in topological
ordered phase. From our simulations of the model wave functions, the fitting
parameters are $\epsilon_{1/3}=1.357$, $k_{1/3}=1.526$;
$\epsilon_{1/5}=2.415$, $k_{1/5}=1.212$, $\epsilon_{5/2}=1.185$,
$k_{5/2}=1.357$ and $\epsilon_{331}=1.08$, $k_{331}=1.304$ respectively. Here
we should note that our results are for the model wave functions which
corresponding to the eigenstates of the model Hamiltonian, such as $V_{1}$
Haldane pseudopotential Hamiltonian for $\nu=1/3$ Laughlin state. The
realistic Coulomb interaction naturally gives different results, especially
the energies. Comparing to the result of density matrix renormalization group
with Coulomb interaction Shibata , the $k_{1/3}$ is quite close and
$\epsilon_{1/3}$ is very different as expected since the wavefunction is quite
close and energy should be different. Therefore, we expect that the
magnetoroton minimums for other three FQH states($1/5$, MR and $331$) in
Coulomb Hamiltonian are almost the same as the $k_{\nu}$ we obtained.

##  III Topological Quantities from Momentum Polarization

It is known that the quasihole mutual exchange in the FQH liquids contains
rich information of its topological order. Suppose we have a quasihole on each
edge of cylinder, the rotation along $y$ direction will not give any
information since the rotational symmetry of this manifold. However, if one
can rotate half of the cylinder (subsystem A) and keep another half (subsystem
B) unchanged, the many-body wave function will have a phase containing the
information of the quasihole in subsystem A. This phase is called momentum
polarization, HHTYZXLQ  which contains important topological quantities such
as the Hall viscosity, Read09  guiding-center spin, central charge and the
topological spin (conformal dimension) of the quasihole excitation. Momentum
polarization has previously been studied using the entanglement entropy in
cylinder geometry HHTYZXLQ  and modular transformation in torus geometry.
MPZRSKMFP ; LDHWZ  It could also be studied in the entanglement spectrum at
the bipartite boundary in the bulk and the intrinsic dipole moment from the
density profile on the edge. Park ; LDHWZ

Here we firstly employ the occupation numbers to calculate momentum
polarization. It can be acquired by

$\langle\Delta M_{A}\rangle=\sum_{m\in A}m\langle n_{m}\rangle-M_{A}^{0}$ (13)

where $M_{A}^{0}$ just depends on the root occupation number, such as
$1001001001\cdots$, $1000010000100001\cdots$, $1100110011\cdots$.
Theoretically, the momentum polarization contains three leading terms as
follows

$\langle\Delta
M_{A}\rangle=\frac{\eta_{H}}{2\pi\hbar}L_{y}^{2}-h_{\alpha}+\frac{\gamma}{24}$
(14)

where the first term is from the contribution of guiding-center Hall
viscosity. The second term $h_{\alpha}=M_{A}^{0}-\bar{M_{A}}$ is called
topological spin MPZRSKMFP ; HHTYZXLQ  or conformal spin of the elementary
excitations which corresponds to quasihole sector $\alpha$ and depends on the
position of the bipartition in the occupation space. It can be calculated for
different model FQH states by using the root configuration pattern in the Jack
polynomial description or the conformal field correlator of the quasihole
operators as shown in the Appendix D. The third leading term
$\gamma=\widetilde{c}-\nu$ is the difference between the (signed) conformal
anomaly ($\widetilde{c}=c-\bar{c}$) and the chiral charge anomaly (filling
factor) $\nu$, which are the two fundamental quantum anomalies of the FQH
fluids. The theoretical values are as follows: $c=1$ for Laughlin states,
$c=3/2$ for the $5/2$ Moore-Read state and $c=2$ for the bilayer $331$ state,
and all chiral states have $\bar{c}=0$. Notice that $\gamma$ vanishes in
integer quantum Hall states, which are topological trivial.

In the case of FQH fluid, the edge density deviates from the uniform density
$\nu/2\pi l_{B}^{2}$ due to the electron-electron correlation. This nonuniform
occupation distribution gives a quantized dipole moment $p_{x}$, which is
related to the guiding-center Hall viscosity (the expectation value of area-
preserving deformation generators). Park ,LDHWZ  The essential physics here is
the intrinsic dipole momentum coupling with the gradient of the electric field
from the Coulomb interaction and the confining potential. This coupling
results in an electric force which is balanced by the guiding center Hall
viscosity $\eta_{H}$. Moreover, the guiding center Hall viscosity was found to
have a relation to a topological quantity named as the guiding-center spin.
$\eta_{H}^{ab}=-\frac{\hbar}{4\pi l_{B}^{2}}\frac{s}{q}g^{ab}$ where $g^{ab}$
is the guiding-center metric in Haldane’s geometric description of FQH liquid
HaldanePRL11  and the guiding-center spin $s$ coupling with the curvature
gives the topological shift on sphere. Finally, we have the relation

$\eta_{H}=-\frac{p_{x}}{L_{y}}B=\frac{\hbar}{4\pi l_{B}^{2}}\frac{-s}{q}$ (15)

where $B$ is the strength of magnetic field and $q$ is the flux quantua number
attached by a “composite boson” which is made of $p$ particles with $q$ flux
quanta for $\nu=p/q$. After a simple substitution, we have

$\langle\Delta M_{A}\rangle=-\frac{1}{2}(\frac{L_{y}}{2\pi
l_{B}})^{2}\frac{s}{q}-h_{\alpha}+\frac{c-\nu}{24}$ (16)

The edge dipole moment per length could be calculated from the occupation
numbers as follows,

$\frac{p_{x}(k)}{L_{y}}=-\frac{e}{2\pi}\int_{0}^{k}dk^{\prime}k^{\prime}l_{B}^{2}[n(k^{\prime})-\nu].$
(17)

and for finite $L_{y}$, comparing to Eq. (16), the integration is approximated
by the sum with corrections

$\frac{p_{x}(k)}{L_{y}}=-\frac{2\pi
l_{B}^{2}e}{L_{y}^{2}}(\sum_{m^{\prime}}[n(m^{\prime})-\nu]m^{\prime}+h_{\alpha}-\frac{c-\nu}{24}).$
(18)

Here we should note that the origin paper of Eq. (37) in Ref. Park,  does not
have the correction terms and the difference is also discussed detailedly in a
recent work. LDHWZ  It is due to the equivalence of intrinsic dipole moment
and momentum polarization, which can be considered as the same topological
quantity. On the other hand, we point out that this may also be the reason why
Fig. 16-20 in Park,  is less convergent. A slight shift between theoretical
and numerical values is clearly observed there which is clearly not a finite
size effect. The quantities $s,h_{\alpha},c$ contains very rich information.
Guiding center spin $s$ is related to the non-dissipative response of the
metric perturbation. LHZLDNSFDMHWZ  Topological spin $h_{\alpha}$ and central
charge $c$ are the elements of modular-$T$ matrix, which is the unitary
transformation of the ground state manifold under modular transformation.
EKVXGW

In our simulation process, we use a self-consistent test method to determine
these topological quantities, namely we set the other quantities at their
respective theoretical values when calculating one of them. For example, while
calculating the central charge $c$, the guiding center spin and topological
spin are predetermined by their CFT values and thus

$c=24\langle\Delta
M_{A}\rangle+24\cdot\frac{L_{y}^{2}}{8\pi^{2}}(-\frac{1}{3})+24h_{\alpha}+\frac{1}{3}$
(19)

for $\nu=1/3$ Laughlin state. The results of the topological spin strongly
depend on how many quasiparticle the subsystem has. This could be tuned by
shifting the bipartite position in the root configuration of the Jack
description. Bernevig08a ; Bernevig08b  Basically, there are three topological
sectors for $1/3$ Laughlin state with root $010010\cdots 010010$. One is the
equal bipartition with $\cdots 10|01\cdots$ named vacuum cut in which case the
subsystem of $N_{A}$ particles exactly occupy $N_{A}/\nu=3N_{A}$ orbitals and
thus no quasiparicle (quasihole) excitation. If one more (less) orbital is
allocated to subsystem, such as $\cdots 100|1\cdots$ ( $\cdots 1|001\cdots$),
a quasihole (quasiparticle) is created in the left subsystem. The different
bipartitions and their corresponding topological spins in other FQH states are
discussed in Appendix D. Finally, for a specific system with fixed number of
electrons, we calculated these topological quantities by varying the aspect
ratio of cylinder, or changing the $L_{y}$ with keeping the area invariant.
Therefore, each $L_{y}$ gives one set of results as shown in all of the
following results.

Figure 3:  (a) $\Delta n(k)$ and $-s/q$ in half of the cylinder for $1/3$
Laughlin state with 50 particles. The $\Delta n(k)$ converges to $0$ verifies
the electric neutrality condition and convergence of the simulation. (b) The
$-s/q$ converges perfectly to the expected value and thus $s=-1$. Figure 4:
Same as Fig. 3 for $1/5$ with 40 particles. The guiding center spin converges
to $s=-2$. Figure 5:  Same as Fig. 3 for MR state with 40 particles. The
guiding center spin converges to $s=-2$.  Figure 6:  Same as Fig. 3 for 331
state with 80 particles. The guiding center spin converges to $s=-2$.

Combining Eq. (15) and (18), considering the contribution of central charge
$c$ and topological spin $h_{\alpha}$ HHTYZXLQ  to the guiding-center spin,
and discretizing the momentum, we have

$-\frac{s}{q}=\frac{8\pi^{2}}{L_{y}^{2}}(\sum_{m_{1}}(n(m_{1})-\nu)m_{1}+h_{\alpha}-\frac{c-\nu}{24})$
(20)

Before calculating guiding-center spin, we need to validate the Luttinger’s
sum rule, J.M. Luttinger  i.e. charge neutral conditions
$\sum_{m_{1}}[n(m_{1})-\nu]=0$. Our numerical results are shown in Fig. 3-6.
Firstly we verify the Luttinger’s sum rule, the difference $\Delta n_{k}$
between the occupation number and the uniform occupation $\nu$ converges to
zero. Then, the $-s/q$ converges to $1/3,2/5,1/2,1/2$ respectively, which
gives us the guiding-center spin for $1/3,1/5$ Laughlin state, $5/2$ MR state
and 331 state as $s=-1,-2,-2,-2$ respectively.

Now we go back to Eq. (16) to calculate the other topological quantities. We
extract these topological quantities numerically of different FQH states, the
results are shown in Figure 7-9. First of all, we observe that the Monte Carlo
simulation of the large systems indeed gives us much more accurate topological
quantities of the FQH states. Those values are in good agreement with the
theoretical predictions from the CFT. For example, we get $c=1,1,3/2,2$,
$s=-1,-2,-2,-2$ for $1/3,1/5$ Laughlin states, Moore-Read state and $331$
state respectively. As for topological spin, all theoretical predictions are
presented in Appendix D. In comparison with the previous study by matrix
product state (MPS) with a low truncation level, Park  the accuracy of our
method is prominent and the computational cost is effective. Especially for
the $1/5$ Laughlin state, it is found that the convergence of these
topological quantities are very slow comparing to the $1/3$ Laughlin state.
For example, Fig. 7(c) shows that the central charge has apparent larger
fluctuations than that in Fig. 7(a). The slow convergence in $1/5$ is clearly
shown in the scope of the edge density fluctuation in Fig. 4.

From Fig. 5, Fig. 6 and Fig. 8, we can see that except the central charge $c$,
all the other topological quantities are the same for Moore-Read state and
$331$ state. It is known that their $e/4$ quasihole excitations are very
different in the anynonic statistics. The $e/4$ quasihole in the Moore-Read
state is non-Abelian since it contains Majorana mode and that in the $331$
state is trivial Abelian. Here we model these quasiholes at the edge of the
cylinder in Monte Carlo (details are shown in Appendix E) and calculate their
topological quantities as shown in Fig. 9. From which we find that the central
charge and guiding-center spin are the same as the ground state. However, for
topological spin of the $e/4$ quasihole, the numerical results show that
$h_{\alpha}=1/8$ for Moore-Read state and $h_{\alpha}=3/16$ for 331 state.
These values are in good agreement with their theoretical predictions as shown
in Appendix D which demonstrates their different topological properties.

Figure 7:  Topological quantities for $\nu=1/3$ and $\nu=1/5$ Laughlin states.
In (a) and (c), we present the result of $-s/q$, $c$ and $h_{\alpha}$ in the
vacuum cut. (b) and (d) show the $h_{\alpha}$ for different bipatitions. The
corresponding theoretical results are labelled by horizontal dash lines.
Figure 8:  Same as Fig. 7. Topological quantities for MR and 331 states. Here
the quasihole/quasiparticle is $e/2$ charged since one orbit is shifted in the
occupation number configuration. Figure 9:  Topological quantities for (a)
Moore-Read state and (b) 331 state with $e/4$ quasihole excitation on the
edge.

##  IV Edge Green’s function

Owing to the existence of gapless edge states in the FQH liquids with open
boundaries, current exists between two contacts connected by an edge channel,
as electrons can be injected into or removed from the FQH edge with costing
zero energy. The standard theory for the FQH edge physics is the $\chi LL$
theory. XGW ; XGWQN  The theory predicts that a FQH droplet exhibits a power-
law behavior in the electric current-voltage characteristics ($I\propto
V^{\alpha}$) when electrons are tunneling through a barrier into the FQH edge
from a Fermi liquid. XGW ; XGWQN ; AMCLNPKWW ; AMC  Generally speaking,
$\alpha$ is also a topological quantity which is related to the topological
order of the FQH liquid and immune from the perturbations. For the celebrated
$\nu=1/3$ Laughlin state, the $\chi LL$ theory predicts a tunneling exponent
$\alpha=3$ though it has controversial in realistic system as we mentioned in
the introduction. The $\alpha$ measured in experiments is sample dependent
with a value mostly smaller than 3. AMCLNPKWW ; MGDCTLNPKWWAMC ;
AMCMKWCCCLNPKWW ; AMC  One of the possible causes of this discrepancy is
existence of counterpropagating edge modes, which result from edge
reconstruction. Chamon94 ; XWFEEHR ; VJGEVT ; XWKYEHR ; KY  The $\chi LL$
theory WXG1 ; WXG2  also predicts the $\alpha=5$ for $1/5$ Lauglin state,
$\alpha=3$ for both of the Moore-Read and $331$ states. The related
experimental and theoretical values for Laughlin states are in Ref. WXG3, ;
MGDCTLNPKWWAMC, ; XLCDMAKLNPKWW, ; HLFPJWPJSLXLNPKWMAKXL, .

Numerically, we can obtain $\alpha$ by calculating the electron edge Green’s
function which is the electron propagator along the edge of the FQH droplet.
The scaling behavior of edge Green’s function has been studied in disk
geometry. SSMJKJ1 ; XWFEEHR ; XWKYEHR  As we claimed previously, the disk
geometry has inhomogeneous Landau orbital space and thus the edge density
profile is always incomplete in small system size. Another problem is the edge
distance is limited by the circumference of disk and the scaling behavior
suffers strong finite size effects. The Monte Carlo simulation in cylinder
geometry could overcome these weaknesses since the lengthscale of the edge
could be tuned by the aspect ratio. In cylinder geometry, the edge Green’s
function can be defined as

$G_{edge}(|\vec{z}-\vec{z^{\prime}}|)=\frac{N\int\prod_{j=1}^{N-1}d^{2}z_{j}\psi^{*}(\vec{z},\\{\vec{z_{j}}\\})\psi(\vec{z^{\prime}},\\{\vec{z_{j}}\\})}{\int\prod_{k=1}^{N}d^{2}z_{k}\psi^{*}(\\{\vec{z}_{k}\\})\psi(\\{\vec{z}_{k}\\})}$
(21)

where $\vec{z}$ and $\vec{z^{\prime}}$ are on the same edge of cylinder, and
they have the same value of $x$ coordinates. In the limit of large distance
($|\vec{z}-\vec{z^{{}^{\prime}}}|\gg 1$), the Green’s function behaves as

$G_{edge}(|\vec{z}-\vec{z^{\prime}}|)\sim|\vec{z}-\vec{z^{\prime}}|^{-\alpha}$
(22)

From Appendix C, the equal-time edge Green’s function on cylinder can be
written as

$G_{edge}(|\frac{2}{\gamma}\sin(\frac{Y\gamma}{2})|)=\sum_{k}\frac{1}{\pi^{1/2}L_{y}}e^{-ikY}e^{-(X-k)^{2}}n_{k}$
(23)

The chord distance is $|\frac{2}{\gamma}\sin(\frac{Y\gamma}{2})|$ where
$Y=|y_{1}-y_{2}|$ is the arc length between $\vec{z}$ and $\vec{z^{\prime}}$
on the surface of cylinder and $\gamma=2\pi/L_{y}=1/R$ is the inverse of the
radius or the space between two continuous Landau orbits.

Figure 10:  The density profiles (a)(b) and scaling exponents of the edge
Green’s functions at different $\tilde{X}$ (c)(d). Here the $x$-axis is set to
$\tilde{X}=X-N\gamma/(2\nu)$, where $N\gamma/(2\nu)$ is the physical edge on
the right. The theoretical predictions of the respective FQH states are marked
as horizontal dashed lines.

For a $N$-particle FQH liquid at filling $\nu$, the number of Landau orbits is
$N/\nu$ and thus the length of the cylinder is $\frac{2\pi N}{\nu
L_{y}}=N\gamma/\nu$. Two edges locate at $\pm N\gamma/2\nu$. As shown in Fig.
10(a) and (b), we plot half of the density profile for the four states. Here
we set the $x$ coordinate as $\tilde{X}=X-N\gamma/2\nu$ and then the edge on
the right side locates at the position of $\tilde{X}=0$. First of all, as we
mentioned previously, the $1/5$ state has a deeper density oscillation than
that of the $1/3$ state. Comparing the Moore-Read state and the 331 state with
the same electron number and $L_{y}$, the bulk density are the same since both
of them are candidates of the $\nu=5/2$ FQH state. However, it is shown that
the edge density has certain differences which demonstrates they belong to
different topological phases, or have different topological quantities such as
the central charge as that in previous section. For the Green’s function along
the edge, we fix the $x$ position and calculate Eq. (21) in $y$ direction.
Because the density profile always has a tail near the edge, we sweep the
position of $\tilde{X}$ around $\tilde{X}=0$. For each $\tilde{X}$, we
calculate the Green’s function and extrapolate the exponent $\alpha$ by the
data of the large distance. The results are shown in Fig. 10(c) and (d).
Overall, we find the exponent $\alpha$ has a dependence on $\tilde{X}$. The
interesting thing is that the values of $\alpha$ for all the four states reach
to their respective theoretical value at around $\tilde{X}\simeq 1.2l_{B}$
which indicated by a star in Fig. 10(a) and (b). When $\tilde{X}>1.2l_{B}$, we
find the $\alpha$ always decays and becomes smaller than the theoretical
value. We understand this result as that the edge of the FQH liquid always has
a width in order of one magnetic length $l_{B}$. The Luttinger liquid exponent
has its exact value at the tail of the realistic edge where the electron
density is close to zero as shown in Fig. 10(a) and (b). This is acceptable
since only the electrons at the tail of the edge are on the Fermi points and
have gapless excitation. The electrons that are away from the Fermi points
need a finite energy to excite and thus can not be strictly described as the
gapless edge excitation, or $\chi LL$ theory.

Figure 11:  The edge Green’s function at $\tilde{X}=1.2l_{B}$ for (a) 1/3
Laughlin state, (b) 1/5 Laughlin state, (c) MR state and (d) 331 state.

At the place of $\tilde{X}=1.2l_{B}$, we show the Green’s function as a
function of the chord distance in logarithmic plot in Fig. 11. Similar to the
density profile, the edge Green’s function for different $L_{y}$ (aspect
ratio) collapses into one curve with a small finite size fluctuation. The
$\alpha$ displayed in the figure is obtained by fitting the data for large
chord distance. The Moore-Read state and $331$ state share the same $\alpha$
which illustrates the electron tunneling, such as in the strong tunneling
limit of quantum point contact (QPC) experiment, can not distinguish the two
inequable states. However, since their $e/4$ quasiholes have different
topological quantities as calculated in the momentum polarization, we expect
the quasihole tunneling, such as in the weak tunneling limit of QPC could give
their distinctions.

##  V Summaries and Discussions

In this work, we have applied a Metroplis Monte Carlo method to calculate the
electron occupation numbers of the Landau orbits for FQH model wave functions
in cylinder geometry. We consider large systems with more than 40 electrons of
the $\nu=1/3,1/5$ Laughlin states and two candidates for the $\nu=5/2$ FQH
states, namely the Moore-Read Pfaffian state and the Halperin bilayer 331
state. With smooth date near the edge, the full density profiles of the edge
states are obtained and the chiral boson theory of the FQH edge has been
verified in a high accuracy. As a first inspection of the effectiveness of
this method, we numerically determine the topological quantities via the
dipole moment and momentum polarization calculations. The guiding-center spin,
central charge and topological spin of the quasihole all exactly converge to
their respective theoretical values. Notably, since the non-Abelian nature of
the MR $e/4$ quasihole excitation, its topological spin is very different from
its Abelian counterpart in 331 state. We model the $e/4$ quasihole excitation
both these two states and identify their topological spins, which are
consistent to the CFT predictions. With the occupation numbers of large
system, another quantity we recalculated is the non-Fermi liquid behaviors of
the electron Green’s function along the edge. As sweeping the locations, we
find only the electrons near the physical boundary have the theoretically
predicted $\alpha$. Therefore, we conclude that the $\chi LL$ theory is an
idealized description of the boundary of the FQH liquid. This could be another
possible mechanism that the $\alpha$ is not quantized and sample dependent in
realistic experiments even in the absence of the edge reconstruction. This
method could be easily generalized to other FQH states or the interface
between different FQH states.

###### Acknowledgements.

Z-X. Hu thanks H-H. Tu for helping us to explain the 331 state in CFT
description. Y. Yang thanks C-X. Jiang for numerical skill discussions. The
Pfaffian polynomial calculation was implemented by using the algorithm in Ref.
Michael Wimmer, . This work was supported by National Natural Science
Foundation of China Grant No. 11974064 and 12147102, the Chongqing Research
Program of Basic Research and Frontier Technology Grant No.
cstc2021jcyjmsxmX0081, Chongqing Talents: Exceptional Young Talents Project
No. cstc2021ycjh-bgzxm0147, and the Fundamental Research Funds for the Central
Universities Grant No. 2020CDJQY-Z003.

##  Appendix A Moore-Read State

The Moore-Read Pfaffian model wave function in cylinder geometry could be
written as

$\begin{split}|\psi^{c}_{5/2}\rangle&=\text{Pf}[M(e^{\gamma
Z})]\prod_{j<k}[\exp(\gamma\mathbf{z}_{j})-\exp(\gamma\mathbf{z}_{k})]^{2}e^{-\frac{1}{2}\sum_{i=1}^{N}x_{i}^{2}}e^{-\sum_{i}\frac{(2N-3)\gamma}{2}\mathbf{z_{i}}}\end{split}$
(24)

where $\text{Pf}[M(e^{\gamma Z})]$ is the Pfaffian polynomial of the
antisymmetric matrix $M(e^{\gamma Z})$. For 4 particles, it is

$\begin{split}\text{Pf}[M(e^{\gamma
Z})]&=\frac{1}{\exp(\gamma\mathbf{z_{1}})-\exp(\gamma\mathbf{z_{2}})}\frac{1}{\exp(\gamma\mathbf{z_{3}})-\exp(\gamma\mathbf{z_{4}})}-\frac{1}{\exp(\gamma\mathbf{z_{1}})-\exp(\gamma\mathbf{z_{3}})}\frac{1}{\exp(\gamma\mathbf{z_{2}})-\exp(\gamma\mathbf{z_{4}})}\\\
&+\frac{1}{\exp(\gamma\mathbf{z_{1}})-\exp(\gamma\mathbf{z_{4}})}\frac{1}{\exp(\gamma\mathbf{z_{2}})-\exp(\gamma\mathbf{z_{3}})}\end{split}$
(25)

In the Metroplis algorithm, we just need $|\text{Pf}(M(e^{\gamma
Z}))|^{2}=\det(M(e^{\gamma Z}))$. Using the same calculation method as $1/q$
Laughlin state, we have

$\rho_{j}=\frac{\int\prod_{i=1}^{N}d^{2}z_{i}|\psi^{c}_{5/2}|^{2}\sum_{b=1}^{N}Z_{b}(y_{j},\mathbf{z})}{\int\prod_{i=1}^{N}d^{2}z_{i}|\psi^{c}_{5/2}|^{2}}$
(26)

where

$Z_{b}(y_{j},\mathbf{z})=\prod_{k\neq
b}\frac{(e^{\gamma(\mathbf{z_{b}}-iy_{j})}-e^{\gamma\mathbf{z_{k}}})^{2}}{(e^{\gamma\mathbf{z_{b}}}-e^{\gamma\mathbf{z_{k}}})^{2}}e^{i\gamma(N-3/2)y_{j}}\frac{\text{Pf}[M(e^{\gamma
R(\mathbf{z_{b}}\rightarrow\mathbf{z_{b}}-iy_{j})})]}{\text{Pf}[M(e^{\gamma
Z})]}$ (27)

where $y_{j}=\frac{L_{y}}{N_{orb}}j$ with $j=0\cdots 2N-3$ and $k$ is from
$-\frac{2\pi}{L_{y}}\frac{2N-3}{2}$ to $\frac{2\pi}{L_{y}}\frac{2N-3}{2}$.
Similarly, we can obtain the average occupation number of $5/2$ MR state on
cylinder by Metropolis sampling. We use the algorithm of Ref. Michael Wimmer,
to implement the Pfaffian polynomial.

##  Appendix B Halperin $331$ State

For the bilayer Halperin 331 state on cylinder, we assume there are
$N_{1}(N_{2})$ electrons in upper(lower) layer. The unnormalized wave function
is

$\displaystyle|\psi_{331}^{c}\rangle$ $\displaystyle=$
$\displaystyle\prod_{i<j,i,j\in
N_{1}}(\exp(\gamma\mathbf{z}_{i})-\exp(\gamma\mathbf{z}_{j}))^{3}\prod_{k<l,k,l\in
N_{2}}(\exp(\gamma\mathbf{z}_{k})-\exp(\gamma\mathbf{z}_{l}))^{3}\prod_{i\in
N_{1},k\in N_{2}}(\exp(\gamma\mathbf{z}_{i})-\exp(\gamma\mathbf{z}_{k}))$ (28)
$\displaystyle e^{-\frac{1}{2}\sum_{i=1}^{N}x_{i}^{2}}e^{-\sum_{i\in
N_{1}}\frac{\gamma}{2}(4N_{1}-3)\mathbf{z_{i}}}e^{-\sum_{i\in
N_{2}}\frac{\gamma}{2}(4N_{2}-3)\mathbf{z_{i}}}$

where $N=N_{1}+N_{2}$ is total number of electrons. The total momentum is
$\frac{3N_{1}(N_{1}-1)}{2}+\frac{3N_{2}(N_{2}-1)}{2}+N_{1}N_{2}$. When
$N_{1}=N_{2}$, the total momentum is $M_{tot}=\frac{N(2N-3)}{2}$ which is the
same as that of the Moore-Read state. Each layer has filling $1/4$. Similarly,
we have

$\rho_{j}=\frac{\int\prod_{i=1}^{N}d^{2}z_{i}|\psi^{c}_{331}|^{2}\sum_{b=1}^{N}Z_{b}(y_{j},\mathbf{z})}{\int\prod_{i=1}^{N}d^{2}z_{i}|\psi^{c}_{331}|^{2}}$
(29)

where

$Z_{b}(y_{j},\mathbf{z})=\begin{cases}\prod_{k\neq b,k\in
N_{1}}\frac{(e^{\gamma(z_{b}-iy_{j})}-e^{\gamma z_{k}})^{3}}{(e^{\gamma
z_{b}}-e^{\gamma z_{k}})^{3}}\prod_{k\in
N_{2}}\frac{e^{\gamma(z_{b}-iy_{j})}-e^{\gamma z_{k}}}{e^{\gamma
z_{b}}-e^{\gamma
z_{k}}}e^{i\gamma\frac{4N_{1}-3}{2}y_{j}},~{}\text{if}~{}z_{b}\in N_{1}.\\\
\prod_{k\neq b,k\in N_{2}}\frac{(e^{\gamma(z_{b}-iy_{j})}-e^{\gamma
z_{k}})^{3}}{(e^{\gamma z_{b}}-e^{\gamma z_{k}})^{3}}\prod_{k\in
N_{1}}\frac{e^{\gamma z_{k}}-e^{\gamma(z_{b}-iy_{j})}}{e^{\gamma
z_{k}}-e^{\gamma
z_{b}}}e^{i\gamma\frac{4N_{2}-3}{2}y_{j}},~{}\text{if}~{}z_{b}\in
N_{2}.\end{cases}$ (30)

and $y_{j}=\frac{L_{y}}{N_{orb}}j$.

##  Appendix C The Edge Green’s Function on Cylinder

Since the single particle wave function is
$\phi_{k}(z)=\frac{1}{\sqrt{\pi^{1/2}L_{y}}}e^{iky}e^{-(x-k)^{2}/2}$, the edge
Green’s function can be transfer to

$\langle\phi^{\dagger}(\vec{z_{1}})\phi(\vec{z_{2}})\rangle=\sum_{k}\frac{1}{\pi^{1/2}L_{y}}e^{ik(y_{2}-y_{1})}e^{-(x_{1}-k)^{2}/2}e^{-(x_{2}-k)^{2}/2}\langle
a_{k}^{\dagger}a_{k}\rangle$ (31)

the coordinates $\vec{z_{1}}$ and $\vec{z_{2}}$ are chosen with the same
position $x_{1}=x_{2}=X$ near the edge and have a shift in $y$ direction
$Y=y_{1}-y_{2}$. So the chord distance could be expressed as
$|\frac{2}{\gamma}\sin(\frac{Y\gamma}{2})|$. In the final, the edge Green’s
function on the cylinder could be calculated by the occupation numbers as

$\langle\phi^{\dagger}(\vec{z_{1}})\phi(\vec{z_{2}})\rangle=\sum_{k}\frac{1}{\pi^{1/2}L_{y}}e^{-ikY}e^{-(X-k)^{2}}n_{k}.$
(32)

##  Appendix D Topological Spin

The topological spin $h_{\alpha}$ could be calculated from the root
configuration in the occupation space as

$h_{\alpha}=M_{A}^{0}-\bar{M}_{A}.$ (33)

The subscript $\alpha$ represents different topological sectors and depends on
the location of the bipartition for subsystem A. $\bar{M_{A}}=-\frac{1}{2}\nu
m_{F}^{2}$ is the total momentum for the subsystem A with uniform occupation
density $\nu$, where $m_{F}$ is the orbital number in A.

Taking $4$-particle system as an example, for $1/3$ Laughlin state, there are
three topological sectors, vacuum cut sector: 010010$|$010010, quasiparticle
cut sector: 01001$|$0010010 and quasihole cut sector: 0100100$|$10010\. So we
have

$\begin{split}&010010|010010~{}~{}~{}h_{\alpha}=0\\\
&01001|0010010~{}~{}~{}h_{\alpha}=1/6\\\
&0100100|10010~{}~{}~{}h_{\alpha}=1/6\end{split}$ (34)

where we only consider subsystem A on the left. The first momentum near the
cut is $-1/2$. For example, for quasiparticle cut sector,
$M_{A}^{0}=-\frac{1+7}{2}=-4$, and
$\bar{M_{A}}=\frac{1}{3}(-\frac{1+3+5+7+9}{2})=-\frac{25}{6}$, so
$h_{\alpha}=-4+\frac{25}{6}=\frac{1}{6}$.

For $1/5$ Laughlin state, there are five topological sectors: vacuum cut:
0010000100$|$0010000100, two quasiparticles cut: 00100001$|$000010000100, one
quasiparticle cut: 001000010$|$00010000100, one quasihole cut:
00100001000$|$010000100, two quasiholes cut: 001000010000$|$10000100\. So we
have

$\begin{split}&0010000100|0010000100~{}~{}~{}h_{\alpha}=0\\\
&00100001|000010000100~{}~{}~{}h_{\alpha}=2/5\\\
&001000010|00010000100~{}~{}~{}h_{\alpha}=1/10\\\
&00100001000|010000100~{}~{}~{}h_{\alpha}=1/10\\\
&001000010000|10000100~{}~{}~{}h_{\alpha}=2/5\end{split}$ (35)

Then for MR state, there are four topological sectors: vacuum cut:
0110$|$0110, isolated fermion cut: 01$|$100110, $e/2$ quasiparticle cut:
011$|$00110 and $e/2$ quasihole cut 01100$|$110\. We have

$\begin{split}&0110|0110~{}~{}~{}h_{\alpha}=0\\\
&01|100110~{}~{}~{}h_{\alpha}=1/2\\\ &011|00110~{}~{}~{}h_{\alpha}=1/4\\\
&01100|110~{}~{}~{}h_{\alpha}=1/4\end{split}$ (36)

Since the $e/2$ quasihole/quasiparticle is just one more/less flux attached by
the electrons and both of them are Abelian, we assume the $e/2$ excitation in
331 state has the same properties as that in MR. This has been verified in the
calculation of the topological spins as shown Fig. 8.

For the $e/4$ excitation, the MR state and 331 state are distinct. Here we
consider each of the edge has one $e/4$ quasi-hole. Then the root
configuration is $\cdots 010101010\cdots$. In this case, there are two
topological sectors: $\bar{0}101|0101\bar{0}$ and $\bar{0}1010|101\bar{0}$. we
have

$\begin{split}&\bar{0}101|0101\bar{0}~{}~{}~{}h_{\alpha}=1/8\\\
&\bar{0}1010|101\bar{0}~{}~{}~{}h_{\alpha}=1/8\\\ \end{split}$ (37)

For quasi-hole MR state, since the total number of orbits is odd, the Fermi
points are on top of the first orbit which is labelled by $\bar{0}$. It means
only half of this orbit belongs to the subsystem. For example, we consider the
first sector of quasi-hole MR state, $M_{A}^{0}=-\frac{1+5}{2}=-3$, and
$\bar{M_{A}}=\frac{1}{2}(-\frac{1+3+5+7/2}{2})=-\frac{25}{8}$, so
$h_{\alpha}=-3+\frac{25}{8}=\frac{1}{8}$. As a comparison, from the CFT
description, the $e/4$ quasihole operator is expressed as $\sigma
e^{i\sqrt{2}\phi/4}$ where $\sigma$ is the Majorana fermion field of Ising CFT
and $\phi$ is the free chiral boson field. The $\sigma$ operator has conformal
dimension $h=1/16$ and thus the total dimension is
$h_{\alpha}=1/16+\frac{(\sqrt{2}/4)^{2}}{2}=1/8$.

For the $e/4$ excitation in 331 state, we obtain its conformal dimension from
the CFT correlator. The ground state wave function could be written as MR

$\displaystyle\psi_{331}(\\{z^{\uparrow}\\},\\{z^{\downarrow}\\})$
$\displaystyle=$ $\displaystyle\langle V^{+}(\\{z_{1}^{\uparrow}\\})\cdots
V^{+}(\\{z_{N}^{\uparrow}\\})V^{-}(\\{z_{1}^{\downarrow}\\})\cdots
V^{-}(\\{z_{N}^{\downarrow}\\})\rangle_{\text{spin}}\left\langle\prod_{i=1}^{2N}e^{i\sqrt{\alpha}\phi_{c}(z_{i})}\mathcal{O}_{bg}\right\rangle_{\text{charge}}$
(38) $\displaystyle\simeq$ $\displaystyle\prod_{1\leq i<j\leq
N}(z_{i}^{\uparrow}-z_{j}^{\uparrow})^{\beta}\prod_{1\leq i<j\leq
N}(z_{i}^{\downarrow}-z_{j}^{\downarrow})^{\beta}\prod_{i,j}^{N}(z_{i}^{\uparrow}-z_{j}^{\downarrow})^{-\beta}\prod_{1\leq
i<j\leq 2N}(z_{i}-z_{j})^{\alpha}$ $\displaystyle\simeq$
$\displaystyle\prod_{1\leq i<j\leq
N}(z_{i}^{\uparrow}-z_{j}^{\uparrow})^{\alpha+\beta}\prod_{1\leq i<j\leq
N}(z_{i}^{\downarrow}-z_{j}^{\downarrow})^{\alpha+\beta}\prod_{i,j}^{N}(z_{i}^{\uparrow}-z_{j}^{\downarrow})^{\alpha-\beta}$

where the spin vertex operators are $V^{\pm}(z)=e^{\pm
i\sqrt{\beta}\phi_{s}(z)}$ and $\mathcal{O}_{bg}$ is the background charge.
Here the Gaussian factor has been neglected. For 331 state, $\alpha=2,\beta=1$
and thus the electron operator (carrying charge $e$ and spin 1/2) is

$\displaystyle V^{\pm}(z)e^{i\sqrt{\alpha}\phi_{c}(z)}=e^{\pm
i\phi_{s}(z)}e^{i\sqrt{2}\phi_{c}(z)}$ (39)

The Abelian $e/4$ quasihole is written as $e^{\pm
i\frac{1}{2}\phi_{s}(z)}e^{i\frac{\sqrt{2}}{4}\phi_{c}(z)}$ which has
conformal dimension

$\displaystyle
h=\frac{1}{2}(\frac{1}{2})^{2}+\frac{1}{2}(\frac{\sqrt{2}}{4})^{2}=\frac{3}{16}.$
(40)

The $e/4$ quasihole wave function (with a Lauglin quasihole in $\uparrow$
layer) can be written with chiral CFT correlator

$\displaystyle\psi_{331}(w,\\{z^{\uparrow}\\},\\{z^{\downarrow}\\})$
$\displaystyle=$ $\displaystyle\left\langle
e^{i\frac{1}{2}\phi_{s}(w)}e^{i\frac{\sqrt{2}}{4}\phi_{c}(w)}V^{+}(\\{z_{1}^{\uparrow}\\})\cdots
V^{+}(\\{z_{N}^{\uparrow}\\})V^{-}(\\{z_{1}^{\downarrow}\\})\cdots
V^{-}(\\{z_{N}^{\downarrow}\\})\prod_{i=1}^{2N}e^{i\sqrt{2}\phi_{c}(z_{i})}\mathcal{O}_{bg}\right\rangle$
(41) $\displaystyle\simeq$
$\displaystyle\prod_{i=1}^{N}(w-z_{i}^{\uparrow})^{1/2}\prod_{i=1}^{N}(w-z_{i}^{\downarrow})^{-1/2}\prod_{j=1}^{2N}(w-z_{j})^{1/2}\prod_{1\leq
i<j\leq N}(z_{i}^{\uparrow}-z_{j}^{\uparrow})^{3}\prod_{1\leq i<j\leq
N}(z_{i}^{\downarrow}-z_{j}^{\downarrow})^{3}\prod_{i,j=1}^{N}(z_{i}^{\uparrow}-z_{j}^{\downarrow})$
$\displaystyle=$ $\displaystyle\prod_{i=1}^{N}(w-z_{i}^{\uparrow})\prod_{1\leq
i<j\leq N}(z_{i}^{\uparrow}-z_{j}^{\uparrow})^{3}\prod_{1\leq i<j\leq
N}(z_{i}^{\downarrow}-z_{j}^{\downarrow})^{3}\prod_{i,j=1}^{N}(z_{i}^{\uparrow}-z_{j}^{\downarrow})$

which demonstrates that a Laughlin quasihole in the upper layer has been
created.

##  Appendix E $e/4$ Quasi-hole State on Cylinder

Because of the pairing nature of the Majorana mode, we can only create even
number of $e/4$ quasiholes in the MR state. For MR state, creating one $e/4$
at $w$ means putting another one at infinity. Its wave function is

$\begin{split}|\psi^{c}_{5/2}\rangle&=\text{Pf}(\frac{\exp(\mathbf{\gamma
z_{i})}-\exp(\mathbf{\gamma w)}+\exp(\mathbf{\gamma
z_{j}})-\exp(\mathbf{\gamma w)}}{\exp(\mathbf{\gamma
z_{i}})-\exp(\mathbf{\gamma
z_{j}})})\prod_{j<k}(\exp(\gamma\mathbf{z}_{j})-\exp(\gamma\mathbf{z}_{k}))^{2}e^{-\frac{1}{2}\sum_{i=1}^{N}x_{i}^{2}}e^{-\sum_{i}(N-1)\gamma\mathbf{z_{i}}}\end{split}$
(42)

If we consider a pair $e/4$ quasiholes at $w_{1}$ and $w_{2}$, the wave
function is

$\begin{split}|\psi^{c}_{5/2}\rangle&=\text{Pf}(\frac{(\exp(\gamma\mathbf{z_{i}})-\exp(\gamma\mathbf{w_{1}}))(\exp(\gamma\mathbf{z_{j}})-\exp(\gamma\mathbf{w_{2}}))+(\exp(\gamma\mathbf{z_{i}})-\exp(\gamma\mathbf{w_{2}}))(\exp(\gamma\mathbf{z_{j}})-\exp(\gamma\mathbf{w_{1}}))}{\exp(\mathbf{\gamma
z_{i})-\exp(\gamma z_{j}})})\\\
&\prod_{j<k}(\exp(\gamma\mathbf{z}_{j})-\exp(\gamma\mathbf{z}_{k}))^{2}e^{-\frac{1}{2}\sum_{i=1}^{N}x_{i}^{2}}e^{-\sum_{i}(N-1)\gamma\mathbf{z_{i}}}\end{split}$
(43)

For $e/4$ quasihole in the $\uparrow$ layer of the 331 state, the wave
function is

$\begin{split}|\psi_{331}\rangle&=\prod_{i\in
N_{1}}(\exp(\gamma\mathbf{z}_{i})-\exp(\gamma\mathbf{w}))\prod_{i<j,i,j\in
N_{1}}(\exp(\gamma\mathbf{z}_{i})-\exp(\gamma\mathbf{z}_{j}))^{3}\prod_{k<l,k,l\in
N_{2}}(\exp(\gamma\mathbf{z}_{k})-\exp(\gamma\mathbf{z}_{l}))^{3}\\\
&\prod_{i\in N_{1},k\in
N_{2}}(\exp(\gamma\mathbf{z}_{i})-\exp(\gamma\mathbf{z}_{k}))e^{-\frac{1}{2}\sum_{i=1}^{N}x_{i}^{2}}\cdot
e^{-\sum_{i\in N_{1}}\frac{\gamma}{2}(4N_{1}-2)\mathbf{z_{i}}-\sum_{i\in
N_{2}}\frac{\gamma}{2}(4N_{2}-2)\mathbf{z_{i}}}\end{split}$ (44)

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