# Dynamics of quasiholes and quasiparticles at the edges of small lattices

Xikun Li  School of Physics and Optoelectronic Engineering, Anhui University,
Hefei, Anhui 230601, China  Max-Planck-Institut für Physik komplexer Systeme,
D-01187 Dresden, Germany  Błażej Jaworowski  Department of Physics and
Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C  Masudul
Haque  Institut für Theoretische Physik, Technische Universität Dresden,
D-01062 Dresden, Germany  Max-Planck-Institut für Physik komplexer Systeme,
D-01187 Dresden, Germany  Department of Theoretical Physics, Maynooth
University, County Kildare, Ireland  Anne E. B. Nielsen  Department of Physics
and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C  Max-
Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany

###### Abstract

We study quench dynamics of bosonic fractional quantum Hall systems in small
lattices with cylindrical boundary conditions and low particle density. The
states studied have quasiholes or quasiparticles relative to the bosonic
Laughlin state at half filling. Pinning potentials are placed at edge sites
(or sites close to the edges) and are then turned off. Because the edges of
fractional quantum Hall systems host chiral edge modes, we expect chiral
dynamics, with motion in one direction for positive potentials pinning
quasiholes, and motion in the other direction for negative potentials pinning
quasiparticles. We numerically show that chiral motion of the density
distribution is observed and robust for the case with positive potentials
(quasiholes), but that there is no noticeable chiral motion for negative
potentials (quasiparticles). The comparison of the numerical ground states
with model lattice Laughlin wavefunctions suggests that both positive and
negative potentials do create and pin anyons that are not necessarily well-
separated on small lattices. Initializing the dynamics with the model state
also shows the lack of chiral dynamics of quasiparticles. Our results suggest
that, in small lattices with low particle density, quasiparticles are strongly
adversely affected in dynamical processes, whereas quasiholes are dynamically
robust.

##  I Introduction

A fascinating aspect of topologically ordered phases of matter is that they
support anyonic quasiparticles with fractional exchange statistics [1]. Anyons
appear, e.g., as charged excitations of fractional quantum Hall (FQH) systems
in two-dimensional electron gases subject to strong magnetic fields [2]. The
robustness of exchange statistics against local noise makes anyons an
interesting platform for topological quantum computation, which has motivated
much work towards realizing anyons in different physical systems [3, 4].
Various models hosting FQH physics and anyons have been proposed for different
physical platforms [5, 6, 7, 8, 9, 10]. Both the fractional charge [11, 12,
13, 14] and the fractional statistics [15] of anyons have been observed
experimentally.

Ultracold atoms in optical lattices provide a versatile setup to study
collective quantum physics [16, 17, 18, 19]. Several lattice models displaying
FQH physics and schemes for implementations in ultracold atoms have been
proposed [20, 21, 22, 23, 24, 25, 26, 27], and recently a FQH system with two
particles on sixteen sites was realized experimentally [28]. Moreover, it has
been claimed that anyons can be created and trapped with pinning potentials at
specific positions [29, 30, 31, 32], and that the braiding of anyons can be
realized by adiabatically moving the potentials [33, 34].

It is a crucial question whether the characteristic features of FQH systems in
two-dimensional electron gases still hold for small lattice systems with low
particle densities, or what features should be chosen to characterize FQH
states in such systems [31, 32]. Small lattice systems are ideal testing
grounds from both experimental and computational viewpoints: Experimentally,
the manipulation of single sites and single atoms [19, 35] provides tools to
observe the FQH effect in small lattices (if we know what to observe), and
computationally, the exponentially growing size of the Hilbert space puts a
limit on the system sizes that can be studied. Proposals for probing the
fractional charge and fractional statistics in small optical lattices have
been made [33, 31, 34]. It is, however, an open question whether known results
for solid-state systems with macroscopic numbers of electrons are still true
for small (or even medium) size lattice systems with dozens of sites and few
particles. One issue is that the overlap between anyons can be significant in
small lattices, so that the behavior can differ from the ideal properties of
well separated anyons. Another issue is that, due to low site occupancies, the
process of pinning or localizing quasiparticles could excite an uncontrolled
amount of excitations.

Here, we investigate whether quasiholes (quasiparticles) can be pinned by
positive (negative) potentials in small lattices and whether chiral dynamics
along the edge is present. We consider hardcore bosonic FQH systems with
filling factor $\nu=1/2$ and place pinning potentials at or near the edges of
the lattice. This results in the creation of density depletions (increases),
which we interpret as quasiholes (quasiparticles) due to the accumulated
excess density being close to $\pm 0.5$ and due to high overlap with model
wavefunctions. Because the FQH states support chiral edge modes [36], chiral
motion of these density inhomogenities is expected to be observed along edges
once the pinning potentials are turned off, i.e., in the quench dynamics of
FQH systems [37, 38, 39, 32]. For the lattice sizes considered in this work,
i.e. dozens of sites, we observe chiral motion of quasiholes. We also find
that the chiral motion is robust in the sense that it is observed for a range
of lattice sizes and for various strengths and locations of the potentials. On
the contrary, we do not observe chiral dynamics for the quasiparticles.

We find that the ground states with potentials (both positive and negative)
have high overlaps with model lattice Laughlin states with anyons, suggesting
that in both cases we do create anyons. This suggests that the absence of
chiral motion for negative potentials stems from complexities arising in the
dynamics of quasiparticles, presumably involving a large number of highly
excited states. Thus, we show that in these setups (small lattices and low
particle densities) there are substantial differences between the dynamics of
quasiparticles and the dynamics of quasiholes.

The paper is structured as follows. In Sec. II, we introduce the model. In
Sec. III, we consider the case with positive potentials and demonstrate chiral
motion of the density distribution in the quench dynamics. In Sec. IV, we
study the case with negative pinning potentials and find no chiral motion of
the density distribution. In Sec. V we compare the ground state of the
Hamiltonian with potentials with model lattice Laughlin states with anyons,
showing that the overlap is high. In Sec. VI we expand on the proposed
explanation of our main observations and conclude the paper by pointing out
implications of our results. Appendix A explains the model lattice
wavefunctions used in Sec. V, and Appendix B shows how to relate them to
continuum Laughlin wavefunctions.

##  II Model

We consider an interacting Hofstadter model with hardcore bosons on a two-
dimensional square lattice. The sites are at the positions
$\vec{r}_{i}=a(x_{i},y_{i})$, where $a$ is the lattice constant,
$x_{i}\in\\{1,2,\ldots,N_{x}\\}$, and $y_{i}\in\\{1,2,\ldots,N_{y}\\}$. The
Hamiltonian takes the form

$\displaystyle H_{0}=-\hbar J\sum_{\langle
k,j\rangle}c^{\dagger}_{k}c_{j}e^{\frac{2\pi
i}{\phi_{0}}\int^{\vec{r}_{k}}_{\vec{r}_{j}}\vec{A}\cdot\mathrm{d}\vec{r}},$
(1)

where the sum is over all $k$ and $j$ for which the sites at $\vec{r}_{k}$ and
$\vec{r}_{j}$ are nearest neighbors. Note that each pair of neighbors appears
twice in the sum. The operator $c_{k}$ annihilates a boson at site
$\vec{r}_{k}$, $\phi_{0}=h/e$ is the flux quantum, $h$ is Planck’s constant,
$e$ is the elementary charge, and $\vec{A}$ is the vector potential
corresponding to a uniform magnetic field $B$ perpendicular to the plane of
the lattice. The strength of the nearest neighbor hopping is set to $\hbar
J=1$ for simplicity. The interaction between bosons is included implicitly by
enforcing the hardcore condition, i.e. allowing at most one boson per site.

We consider a square lattice with open boundary conditions in the $x$
direction and periodic boundary conditions in the $y$ direction. This cylinder
topology provides two parallel edges without corners, and hence is
particularly suitable for studying chiral dynamics along the edges. We use the
Landau gauge

$\vec{A}=B(0,x-x_{0},0)$ (2)

where $x_{0}=(N_{x}+1)/2$, which ensures that the vector potential vanishes at
the center of the system. Note that $a^{2}B=\alpha\phi_{0}$, where $\alpha$ is
the number of flux quanta per plaquette.

We consider half filling $\nu=1/2$ in this paper, where the filling factor
$\nu$ is the number of particles divided by the number of flux quanta
penetrating the lattice. (Note that $\nu=1/2$ is the filling fraction of the
lowest Hofstadter band, and not the lattice filling fraction or average
particle density, which is considerably smaller in our case.) If there are
quasiholes (quasiparticles) present, they each count for $\nu$ particles
($-\nu$ particles) when computing $\nu$. The number of flux quanta is $\alpha$
times the number of plaquettes $N_{\mathrm{plaq}}$ in the lattice. Altogether,
we hence have

$\displaystyle\nu=\frac{M+\nu\sum_{k}p_{k}}{\alpha N_{\mathrm{plaq}}},$ (3)

where $M$ is the number of particles and $p_{k}=+1$ ($p_{k}=-1$) if the $k$th
anyon is a quasihole (quaisparticle). At the edge appearing due to the open
boundary conditions in the $x$ direction, there are ambiguities in how to
count the amount of flux through the lattice, and this ambiguity is
significant for small lattices. It was suggested in [31] that the choice
$N_{\mathrm{plaq}}=(N_{x}-1)N_{y}$ is best for creating and stabilizing a FQH
droplet in small lattice systems, and we hence also use this way of counting
the flux here. We compute $\alpha$ from (3) with $\nu=1/2$, and this gives us
the vector potential $\vec{A}$ appearing in the Hamiltonian.

Since we are considering the $\nu=1/2$ Laughlin state, a quasihole corresponds
to half a particle missing in a local region, and a quasiparticle corresponds
to half a particle extra in a local region. We would like the anyons to be
pinned at particular positions to begin with and local pinning potentials are
a standard approach to do that. Adding a potential term $V_{j}n_{j}$ to the
Hamiltonian, where $n_{j}=c_{j}^{\dagger}c_{j}$ is the particle number
operator, results in an energy penalty for a particle to sit at the $j$th site
if $V_{j}>0$ and the opposite for $V_{j}<0$. A natural way to try to trap two
quasiholes is hence to reduce the number of particles $M$ by one and add
positive potentials on two sites. Similarly, one could try to trap two
quasiparticles by increasing the number of particles $M$ by one and putting
negative potentials on two sites. Note that $M+\nu\sum_{k}p_{k}$ is unchanged
in this process, and hence $\alpha$ is also unchanged. The Hamiltonian with
potentials is

$H=H_{0}+\sum_{j}V_{j}n_{j},$ (4)

where only two of the $V_{j}$ are nonzero.

Below we shall investigate quench dynamics. We take the initial state at time
zero to be the ground state $|\psi(0)\rangle\equiv|\mathcal{E}_{0}\rangle_{H}$
of the Hamiltonian (4) with trapping potentials present. At time zero, we turn
off the trapping potentials to start the quench dynamics, and after a time
$t$, the state has evolved into

$|\psi(t)\rangle=\exp{(-\mathrm{i}H_{0}t/\hbar)}|\psi(0)\rangle.$ (5)

With the purpose of investigating properties of $|\psi(t)\rangle$, we define
the density distribution at time $t$ as

$\displaystyle\rho_{j}(t)=\langle\psi(t)|n_{j}|\psi(t)\rangle-\frac{M+\nu\sum_{k}p_{k}}{N}.$
(6)

The first term $\langle\psi(t)|n_{j}|\psi(t)\rangle$ on the right hand side is
the expectation value of the number of particles on site $j$ in the state
$|\psi(t)\rangle$, and the second term is the average number of particles per
site when there are $M+\nu\sum_{k}p_{k}$ particles in the system. Note that
$\sum_{j}\rho_{j}(t)=-\nu\sum_{k}p_{k}$, so $\sum_{j}\rho_{j}(t)=-1$ for two
quasiholes and $\sum_{j}\rho_{j}(t)=+1$ for two quasiparticles at half
filling.

FQH systems tend to have uniform density in the bulk, and hence nonzero
$\rho_{j}(t)$ can appear due to edge variations or density disturbances due to
a potential. To eliminate the edge variations, we shall also sometimes
consider the excess particle density

$\displaystyle\tilde{\rho}_{j}(t)=\langle\psi(t)|n_{j}|\psi(t)\rangle-\langle
n_{j}\rangle_{0},$ (7)

where $\langle n_{j}\rangle_{0}$ is the expectation value of $n_{j}$ computed
in the ground state of $H_{0}$ with $M+\nu\sum_{k}p_{k}$ particles.

Our computations involve exact diagonalization. We consider $3$ particles for
the case without anyons to ensure that the dimension of the Hilbert space is
not too large. When trapping potentials are present, we have
$M+\nu\sum_{k}p_{k}=3$, i.e., $M=2$ bosons for the case of positive potentials
(two quasiholes), and $M=4$ bosons for the case of negative potentials (two
quasiparticles). We present results for the real-time dynamics of the lattice
system with $N=N_{x}\times N_{y}=7\times 7$ sites. We have numerically checked
that the dynamics is qualitatively the same in other lattices of comparable
size, e.g. $N=7\times 11$, $N=11\times 9$, etc. We concentrate on initial
states for which the trapping potentials are located at particular sites. Two
cases are considered: case (i), in which the two sites are placed on the
middle of the left and right edges, i.e. the sites $(1a,4a)$ and $(7a,4a)$ for
the $7\times 7$ lattice; case (ii), in which the two sites are moved one site
into the bulk from the left and right edges, i.e. $(2a,4a)$ and $(6a,4a)$ for
the $7\times 7$ lattice.

##  III Quasiholes

Figure 1: Effects of the pinning potential in the quasihole case with two
particles. (a) Particle density $\langle n_{i}\rangle_{V_{0}}$ for the site
$(1a,4a)$ (at the edge) versus potential strength $V_{0}$. (b) Squared overlap
$O_{V_{0}}^{2}$ between the ground state of the total Hamiltonian $H$ with
potentials and the lowest eigenstates of the Hamiltonian $H_{0}$ without
potentials.

We first consider the case of quasiholes, with positive potentials used to pin
(localize) them. We choose the following configuration: the number of sites is
$N=7\times 7$, the number of particles in the system with positive potentials
is two, the number of fluxes per plaquette is $\alpha=1/7$, and positive
potentials with strength $V_{0}$ are introduced on the sites $(1a,4a)$ and
$(7a,4a)$. In contrast to the case of quasiparticles, which is shown and
discussed in the following section, the chiral motion along the $y$-axis is
clearly demonstrated in the case with positive potentials. Moreover, the
observed chiral motion is quite robust in the sense that it does not depend on
the strength of the potentials or their precise location on the edge. For
example, we obtain similar results if we instead put the potentials on the
sites $(2a,4a)$ and $(6a,4a)$. We also find that the chiral motion persists
for a very long time.

We first investigate the effect of the potential strength $V_{0}$ on the
system by considering the particle density

$\langle n_{i}\rangle_{V_{0}}=\langle\psi(0)|n_{i}|\psi(0)\rangle$ (8)

at site $i$ and the overlap

$O_{V_{0}}=|\langle E_{n}|{\psi(0)\rangle}|$ (9)

between the ground state $|\psi(0)\rangle$ of the total Hamiltonian $H$ with
potentials, and the $n$th eigenstate $|E_{n}\rangle$ of the Hamiltonian
$H_{0}$ with the same number of particles. If $H_{0}$ has several degenerate
eigenstates $|E_{n}\rangle_{m}$ at a given energy $E_{n}$, their total overlap
is given by

$O_{V_{0}}=\sqrt{\sum_{m}|\langle\psi(0)|E_{n}\rangle_{m}|^{2}}.$ (10)

The squared overlap $O_{V_{0}}^{2}$ measures the percentage that the
eigenstate $|E_{n}\rangle$ (or, in case of degeneracy, all the states at
energy $E_{n}$) contributes during the quench dynamics given the initial state
is $|\psi(0)\rangle$.

In Fig. 1(a) we plot the particle density $\langle n_{i}\rangle_{V_{0}}$ at
site $(1a,4a)$. Due to the symmetry under $\pi$ rotations (inversion), the
density at this site is equal to the density at site $(7a,4a)$. We observe
that the particle density at site $(1a,4a)$ is quite small even when the
potential is not strong, say, $V_{0}=0.5$. In addition, $\langle
n_{i}\rangle_{V_{0}}$ drops quickly as $V_{0}$ increases. This is expected,
since the particle density on the site should go to zero when there is a large
positive potential on the site.

In Fig. 1(b) we show the squared overlap between the initial state
$|\psi(0)\rangle$ and the zeroth, first, and second excited state of $H_{0}$.
The ground state $|E_{0}\rangle$ of $H_{0}$ is non-degenerate, while the first
excited state is two-fold degenerate $|E_{1}\rangle_{1,2}$. This degeneracy is
not lifted even with a very strong potential, say, $V_{0}=10000$. We can see
that the overlap of the ground state dominates over all other overlaps of the
excited states. We find that $O_{V_{0}}^{2}>0.9604$ for the ground state for
all $V_{0}>0$. We note that this means that there is a large overlap between
states with pinned and unpinned quasiholes, as the ground state with two
particles has two unpinned quasiholes.

Figure 2: (a) Quench dynamics of the density distribution (6) for the
quasihole case. The size of the lattice is $7\times 7$, and the number of
particles is two. The trapping potentials are located at the sites $(1a,4a)$
and $(7a,4a)$, i.e. at the left and right edges, and have strength
$V_{0}=100$. The flux density is $\alpha=1/7$. (b) The density distribution of
the second column of sites, counted from the left, as a function of time for
the quench dynamics shown in (a). A staircase pattern is seen, which means
that there is chiral motion.

In Fig. 2 we show the quench dynamics of the system with positive potentials.
In Fig. 2(a) we observe that the density distribution moves chirally around
the edge (the $y$-axis) in addition to the dispersion along the edge.
Meanwhile, there is little density spreading into the bulk sites. This is
different from the dynamics observed in the models studied in Refs. [40, 38]
where the particles spread into larger regions in the bulk on longer
timescales. In our work, however, the density depletion hardly spreads into
the bulk anymore after a short time $t=1$. The leftmost and rightmost columns
(edge columns) do not show much visible variation in density, so it is
convenient to use the second column (column next to the edge) to track and
visualize the density dynamics. In Fig. 2(b) we take snapshots of the density
distribution of the second column of sites counted from the left, and
concatenate snapshots at different instants to form a spacetime plot. This
visualizes the chiral motion of the density distribution which is seen as a
‘staircase’ pattern in such a plot. The observed chiral dynamics suggests that
we have a quasihole propagating along each of the two edges.

This picture is supported by monitoring the excess density along the edges. We
observe that even during long periods of quench dynamics the sum of (7) over
the sites along the two leftmost (rightmost) columns of sites equals $1/2$
with a good accuracy, $\tilde{\rho}_{\mathrm{edge}}=\sum_{i:x_{i}\leq
2}\tilde{\rho}_{i}(t)\approx-0.49$. The total excess density corresponding to
half a particle near each edge is consistent with the expectation that
quasiholes are created and trapped. We note however that the system is
inversion-symmetric, and there is one particle missing compared to the case
without potentials, so a similar density pattern could in principle also arise
in non-topological systems. The interpretation of the density depletions as
quasiholes is further corroborated by the analysis of model wavefunctions in
Sec. V. The nearly-constant value of $\tilde{\rho}_{\mathrm{edge}}$ over time
suggests that the quasiholes stay confined to the edges as they evolve in
time.

We find similar results as above when we examine the quench dynamics for
various sizes of the lattice, different locations of the trapping potentials
(at the edges or one site away from the edges) and various potential strengths
$V_{0}$. Moreover, the chiral motion is also observed for values of the flux
density, say, $\alpha=6/49$, which deviate slightly from the correct value
$\alpha=1/7$. (In fact, if we replace $N_{\textrm{plaq}}$ with $N$ in Eq. (3)
we get the filling factor $1/2$ for $\alpha=6/49$. The ambiguity in defining
the magnetic flux through the lattice, when edges are present, is thus not a
severe problem for the considered systems.) These results indicate that the
chiral dynamics of quasiholes pinned by the positive potentials is quite
robust in small lattices. In addition, we observe that the chiral motion of
the quasiholes persists for a large timescale $t>700$.

Figure 3: (a) Snapshots of the time-evolving density distribution (6) for the
system with quasiholes and a flux density $\alpha=3/10$, significantly
different from the ‘correct’ value $\alpha=1/7$. The trapping potentials
($V_{0}=100$) are at the edges, $(1a,4a)$ and $(7a,4a)$. (b) The density
distribution of the second column of sites, counted from the left, as a
function of time for the quench dynamics shown in (a). The lack of a clear
staircase pattern signals the absence of chiral motion.

If the flux density deviates much from the correct value $\alpha=1/7$, one
would expect that the chiral motion is absent. In Fig. 3, we show the quench
dynamics of the system with the same configurations in Fig. 2, but with the
flux density $\alpha=3/10$. As expected, no chiral motion is observed.

Figure 4: (a) Snapshots of the density distribution (6) for the quasihole case
in the absence of the magnetic field, i.e. $\alpha=0$. The trapping potentials
($V_{0}=100$) are located at the edge sites $(1a,4a)$ and $(7a,4a)$. (b) The
density distribution of the second column of sites, counted from the left, as
a function of time for the quench dynamics shown in (a). When the magnetic
field is turned off, the model has mirror symmetry both between up and down
and between left and right, and hence there is no chiral motion.

Furthermore, we consider an extreme case in which the magnetic field is turned
off, $\alpha=0$. The result is shown in Fig. 4. In this case, there is no
chiral motion. Instead, we observe that the density distribution at the sites
of the left (and right) half of the lattices split into two parts with equal
weight, and two regions of depleted density move with opposite chirality
(upward and downward directions).

##  IV Quasiparticles

In this section, we consider the case with four particles. We use negative
potentials for creating the initial state. In this way we attempt to spatially
localize quasiparticles. We find that the trapped objects have total excess
particle density close to 0.5, as quasiparticles are expected to, but that
they do not display clear chiral dynamics.

Figure 5: Effects of pinning potential in the quasiparticle case with four
particles. (a) Particle density $\langle n_{i}\rangle_{V_{0}}$ at site
$(1a,4a)$ versus potential strength $-V_{0}$. (b) Squared overlap
$O_{V_{0}}^{2}$ between the ground state of the total Hamiltonian $H$ with
potentials and the lowest eigenstates of the Hamiltonian $H_{0}$ without
potentials.

In Fig. 5(a), we show the particle density at site $(1a,4a)$ as a function of
the potential strength $-V_{0}$. As $-V_{0}$ increases, $\langle
n_{i}\rangle_{V_{0}}$ grows and eventually converges to its maximum possible
value, which is $1$. This is expected because the stronger the on-site
negative potential is, the more the energy is reduced by having a particle on
that site. We avoid potentials close to zero as $|\psi(0)\rangle$ has a
degeneracy for $V_{0}=0$.

Figure 6: (a) Snapshots of the density distribution (6) in the quasiparticle
case. The trapping potentials (strength $V_{0}=-100$) are at the edge sites
$(1a,4a)$ and $(7a,4a)$. A different color scale has been used in this plot,
because the particles spread into the bulk sites on a longer timescale. (b)
The density distribution of the leftmost column of sites as a function of time
for the quench dynamics shown in (a). No staircase pattern is seen, which
signals the absence of chiral motion.

In Fig. 5(b) we show the squared overlap $O_{V_{0}}^{2}$ between the initial
state $|\psi(0)\rangle$ and the lowest eigenstates of $H_{0}$ with four
particles as a function of $|V_{0}|=-V_{0}$. The ground states of $H_{0}$ are
doubly degenerate, and thus the overlap is calculated using (10). As $-V_{0}$
increases, more and more higher excited states have appreciable overlap and
thus contribute to the quench dynamics. We observe that the overlaps with
respect to $|E_{n}\rangle$ depend substantially on $|V_{0}|$, and many of them
have comparable magnitude. The total overlap with respect to the doubly
degenerate ground state $|E_{0}\rangle_{1,2}$ decreases rapidly with $-V_{0}$;
its magnitude does not dominate over other eigenstates for $-V_{0}\gtrsim
1.5$. This behavior is very different from the case of quasiholes trapped by
positive potentials, shown in Fig. 1(b). In that case the ground state
dominates even for large potentials.

In Fig. 6 we demonstrate the quench dynamics for negative potentials located
at the edges with strength $V_{0}=-100$ which is huge compared to the hopping
strength. In Fig. 6(a) we show the dynamics of the density distribution, and
in Fig. 6(b) we plot the density distribution of the first column of sites,
counted from the left, as a function of time. The particle density (7) in the
initial state has an excess density of
$\tilde{\rho}_{\mathrm{edge}}(0)=\sum_{i:x_{i}\leq
2}\tilde{\rho}_{i}(0)\approx 0.51$ in the two leftmost (two rightmost) columns
of sites. That is, at the beginning of the dynamics we have an excess density
of approximately a half of a particle (this time with opposite sign) located
near each edge, consistent with the expectation that we trapped
quasiparticles. Again we note that the system has inversion symmetry, and
there is one particle extra compared to the case without potentials, so a
similar excess density could also appear without the presence of topology.
Further support that quasiparticles are trapped is presented in Sec. V.

A significant difference from the results in Sec. III appears when the density
evolves in time. From Figs. 6(a) and 6(b), we observe no chiral motion of the
density distribution, but patterns that are different from the case of
quasiholes. We find that, during quench dynamics the excess density roughly
splits into two parts and each of them move in opposite directions. Meanwhile,
in contrast to the quasihole case, the excess density spreads into the bulk
with much weight in the beginning, but stops spreading more into the bulk
sites after some time.

In addition, we have varied $V_{0}$ and also varied the positions of the
trapping potentials, e.g., at the edges, or one or two sites away from the
edge. We have not observed clear chiral motion in any of these cases.
Furthermore, we find that the patterns of quench dynamics for negative
potentials vary substantially as we change the strength and/or locations of
the pinning potentials, which was not the case for quasiholes trapped by
positive potentials. We also tried to use pinning potentials over a block of
sites to create quasiparticles as in [31]. Also in that case, we have not seen
clear chiral motion.

##  V Model quasiparticle state

Could the difference between the dynamics for positive and negative potentials
arise because the former succeeds in pinning anyons, and the latter does not?
To further study the question whether the trapped objects are quasiholes and
quasiparticles, we here discuss a model wavefunction with anyonic excitations
[41, 42, 43].

We write the position of site $j$ as a complex number $\xi_{j}=x_{j}+iy_{j}$.
Then, we define a mapping from a cylinder to a complex plane:
$z_{j}=\exp(2\pi\xi_{j}/L_{y})$, where $L_{y}$ is the circumference of the
cylinder (for a square lattice with unit lattice constant we have
$L_{y}=N_{y}$). For brevity, we write all $z_{j}$s as a vector
$\mathbf{z}=[z_{1},z_{2},\dots,z_{N}]$. Like any state of a hardcore boson
system, the model lattice Laughlin state $|\psi_{\mathrm{model}}\rangle$ can
be written in the occupation number basis,

$|\psi_{\mathrm{model}}\rangle=\frac{1}{C}\sum_{\mathbf{n}}\psi_{\mathrm{model}}(\mathbf{z},\mathbf{n})|\mathbf{n}\rangle,$
(11)

where $\mathbf{n}=[n_{1},n_{2},\dots,n_{N}]$ is the vector of the occupation
numbers of each site, with $n_{j}\in\\{0,1\\}$, $|\mathbf{n}\rangle$ is the
corresponding basis state, $\psi_{\mathrm{model}}(\mathbf{z},\mathbf{n})$ are
the unnormalized wavefunction coefficients and $C$ is the normalization
constant. For a model lattice Laughlin state without anyons on a cylinder
[42], the coefficients are

$\psi_{\mathrm{model}}(\mathbf{z},\mathbf{n})=\delta(qM+p_{+}+p_{-}-N\eta)\prod_{j}\chi_{j}^{n_{j}}\times\\\
\times\prod_{j}z_{j}^{p_{-}n_{j}}\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}}.$
(12)

Here, $\delta(qM+p_{+}+p_{-}-N\eta)$ is the Kronecker delta fixing the charge
neutrality, i.e. the number of particles $M$ (with $M=\sum_{j}n_{j}$) and
$\gamma_{j}=(qn_{j}-\eta)/\sqrt{q}$, with $\eta$ being the flux assigned to
each site (in our case, we put $\eta=\alpha$). The integer $q$ determines the
topological order of the wavefunction and in our case $q=2$. The gauge factors
$\chi_{j}$ are arbitrary complex numbers with $|\chi_{j}|=1$. We note that in
this work we use the word “gauge” in two meanings: the Landau gauge (2) of the
hoppings of the Hofstadter model (1), and the factors $\chi_{j}$ of the model
wavefunctions. To avoid confusion, we will always refer to the latter as “the
gauge factors $\chi_{j}$”.

The real numbers $p_{-}$, $p_{+}$, have to fulfill the relation

$p_{+}=p_{-}-q+2\eta.$ (13)

We note that the formulation in [41, 42] of the lattice Laughlin states on a
cylinder did not contain $p_{-}$ and $p_{+}$. The origin and meaning of these
parameters is described in Appendix A, while the relation of (12) to the
continuum states is described in Appendix B.

In our calculations, we want to use a state with a given $M$, $N$ and
$\eta=\alpha$. Thus, we determine $p_{-}$ from a combination of the charge
neutrality condition and (13),

$p_{-}=\frac{1}{2}\left(\left(N-2\right)\eta-q\left(M-1\right)\right),$ (14)

and then $p_{+}$ from (13).

Some properties of (12), such as the particle density and the topological
order, are independent from the gauge factors $\chi_{j}$. Its overlap with the
ground state of the Hofstadter model, however, strongly depends on $\chi_{j}$.
Therefore, we optimize these factors numerically to maximize the overlap. For
the $7\times 7$ system with $M=3$ and $\alpha=1/7$, we obtain the squared
overlap $|\langle\psi(0)|\psi_{\mathrm{model}}\rangle|^{2}\approx 0.984$ after
optimization.

We can add anyonic excitations to the state (12). They are characterized by
the integers $p_{k}$ (see Eq. (3)) and the complex positions
$\omega_{k}\in\mathbb{C}$. Note that the complex positions are not required to
coincide with lattice sites [43]. After mapping to the complex plane, these
locations transform into $w_{k}=\exp(2\pi\omega_{k}/L_{y})$. The wavefunction
with anyons is then

$\psi_{\mathrm{model}}(\mathbf{z},\mathbf{n},\mathbf{w},\mathbf{p})=\\\
\delta(qM+p_{+}+p_{-}+\sum_{k}p_{k}-N\eta)\prod_{j}\chi_{j}^{n_{j}}\prod_{j}z_{j}^{p_{-}n_{j}}\times\\\
\times\prod_{j,k}(z_{j}-w_{k})^{p_{k}n_{j}}\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}}.$
(15)

The $w_{k}$ have the braiding properties expected for anyons in systems with
Laughlin type topology when the $w_{k}$ remain well-separated during the
entire braiding process [43]. On small lattices with low densities, the
changes in the particle density caused by the anyons may not be fully
separated.

We consider two examples in a $7\times 7$ system with $\eta=\alpha=1/7$: two
quasiholes ($M=2$) and two quasiparticles ($M=4$). In both cases, we place the
anyons at the same positions as the pinning potentials in Secs. III and IV
(plus a small shift to avoid infinities in (15)). We use the same $\chi_{j}$
as in the $M=3$ case (i.e. we do not re-optimize them). The resulting squared
overlaps with ground states of the Hofstadter model with potentials
($V_{0}=\pm 100$) are
$|\langle\psi(0)|\psi_{\mathrm{model}}\rangle|^{2}\approx 0.976$ and
$|\langle\psi(0)|\psi_{\mathrm{model}}\rangle|^{2}\approx 0.977$ for
quasiholes and quasiparticles, respectively.

Therefore, the ground states of the Hofstadter model states are similar to the
model states, but not perfectly equal to them. To check whether the
discrepancy is the reason for the lack of chiral dynamics in the quasiparticle
case, we also compute the quench dynamics generated by the hopping Hamiltonian
$H_{0}$ (Eq. (1)) with the model state (15) with two quasiparticles as the
initial state. The results are quite similar to Fig. 6, i.e. we do not observe
clear chiral dynamics.

Figure 7: The locations and strengths of the potentials that we optimize to
increase the overlap of the ground state of the Hamiltonian with negative
$V_{0}$ with the model wavefunction (15) with two quasiparticles. During the
optimization, $V_{0}=-100$ stays constant, while $V_{A}$ and $V_{B}$ are
adjusted.

We note that we can increase the overlap between the quasiparticle state of
the Hofstadter model and the model lattice Laughlin quasiparticle state by
introducing auxiliary potentials $V_{A}$ and $V_{B}$ at the nearest-
neighboring sites of the original potentials (see Fig. 7), and optimize them
to maximize the overlap (in the spirit of Ref. [44]). In such a way, we
obtained the squared overlap
$|\langle\psi(0)|\psi_{\mathrm{model}}\rangle|^{2}=0.993$. However, using the
ground state with the optimized potentials as the initial state also does not
lead to chiral dynamics.

The results presented in this section suggest that the difference between the
cases of positive and negative potentials is not related to the ability to pin
anyons, as both quasiholes and quasiparticles seem to be successfully created
and pinned by the pinning potentials.

##  VI Discussion and Conclusions

We have studied quench dynamics in small lattices with cylindrical boundary
conditions, using interacting (hardcore) bosons subjected to the Hofstadter
Hamiltonian. We considered particle numbers appropriate to both two quasiholes
($M=2$) and two quasiparticles ($M=4$), relative to a FQH state with filling
factor $\nu=1/2$. In the two cases we used positive and negative pinning
potentials so that the initial positions of the quasiholes and quasiparticles
are localized near the edges of the lattices. The quench dynamics begins when
the trapping potentials are turned off. One expects to observe chiral motion
of the excess density due to the presence of chiral edge modes. Our results
for the two cases are very different. While it is generally easy to observe
chiral motion of quasiholes (negative excess density), and the observed chiral
dynamics are quite robust, we have not found clear chiral motion of
quasiparticles (positive excess density).

The absence of the chiral motion of quasiparticles was further verified by
investigating the quench dynamics with the model lattice Laughlin state as the
initial state. Moreover we introduced auxiliary potentials in the Hofstadter
model to optimize the overlap between the ground state of the Hofstadter
Hamiltonian and the model lattice Laughlin state. The chiral dynamics was
still absent.

Why is there such a pronounced difference between the quasihole and
quasiparticle cases? Due to the small particle densities, the details of
pinning are quite different in the two cases — locally decreasing the density
is a very different process compared to locally increasing the density. In the
former case (quasiholes, negative excess density) there is a natural limit to
how far the local density $\langle n_{i}\rangle_{V_{0}}$ can be reduced, since
$\langle n_{i}\rangle_{V_{0}}$ is small already for $V_{0}=0$. In the latter
case (quasiparticles, positive excess density), the site density can increase
enormously, all the way up to $1$, leading to the excitation of many high-
energy modes whose nature might be unrelated to FQH physics. These high-energy
contributions can be expected to mask or disturb the topological and chiral
physics that FQH quasiparticles should show. This difference between the two
cases is mirrored in the overlaps shown in Figs. 1(b) and 5(b).

This difference raises the possibility that the pinning of quasiparticles
might be disrupted, unlike the pinning of quasiholes. However, the comparison
with model quasiparticle states (Sec. V) indicates that both positive and
negative potentials pin the anyons successfully. We therefore conclude that
the difference must come from the contributions of excited states in the
dynamics — the quasihole dynamics is dominated by the ground state, while the
quasiparticle dynamics involves a number of highly excited states, as
indicated in Figs. 1(b) and 5(b). The disruption of chiral dynamics is a
_dynamical_ effect.

The absence of chiral motion suggests that the dynamics of quasiparticles in
small lattices with low particle densities, of the sizes considered in this
work, is different from that in large lattices, and from that in two-
dimensional electron gases. This has important implications for the study of
FQH physics in novel platforms such as that of Ref. [28], where the particle
density and system size might be naturally small. Our results indicate that,
in such situations, quasiparticle dynamics is fragile as there is a natural
tendency to create substantial additional (possibly unwanted) excitations when
localized quasiparticles are created. Remarkably, the difficulty with
observing quasiparticle features shows up primarily in the dynamics, and not
necessarily in creating or pinning them.

This work opens up a number of questions: (1) Our conjectured explanation
above suggests that quasiparticles and quasiholes would behave more similarly
if the average particle density (lattice filling) were close to $1/2$. It
would be worthwhile to check this explicitly; however in our setup this is
computationally too challenging. (2) A detailed understanding of how the
dynamical behaviors change with system size (e.g., with constant density or
with constant particle number) is currently lacking. An intriguing question is
whether chiral motion of quasiparticles become readily observable for large
lattice sizes even when the particle density remains small. (3) It would also
be interesting to investigate how the presence or absence of inversion
symmetry affects the excess density and dynamics in small systems. (4) We here
considered the case of Abelian anyons. In future work it would be interesting
to study the quench dynamics of FQH systems which host non-Abelian anyons.

###### Acknowledgements.

This work has been supported by Danmarks Frie Forskningsfond under grant
number 8049-00074B.

##  Appendix A Inversion-symmetric lattice Laughlin wavefunctions on a
cylinder

In this Appendix, we derive the expression (12) and condition (13) from the
known expressions for lattice Laughlin wavefunctions [41, 42] by demanding
inversion symmetry.

Let us start by considering a planar system of $N$ sites at locations
described by complex numbers $z_{j}=x_{j}+iy_{j}$, with $j=1,2,\dots N$,
populated by $M$ particles: either by hardcore bosons or fermions. To each
site, we assign a flux $\eta\in\mathbb{R}$ in units of the flux quantum, so
the total number of flux quanta in the system is $N\eta$. We also place a
compensating charge of $\eta/q$ (in units where a single particle has charge
$-1$) at each site, in analogy to a continuum Landau level where a uniform
background charge is used.

According to [41, 42], the coefficients of a lattice Laughlin wavefunction on
a plane (see Eq. (11)) are

$\psi_{\mathrm{model}}(\mathbf{z},\mathbf{n})=\\\
\delta(qM-N\eta)\prod_{j}\chi_{j}^{n_{j}}\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}},$
(16)

(see below Eq. (12) for an explanation of the notation). One can add one or
more anyonic excitations to (16), [42, 43] which transforms it into

$\psi_{\mathrm{model}}(\mathbf{z},\mathbf{n},\mathbf{w},\mathbf{p})=\\\
=\delta(qM+\sum_{k}p_{k}-N\eta)\prod_{j}\chi_{j}^{n_{j}}\prod_{j,k}(z_{j}-w_{k})^{p_{k}n_{j}}\times\\\
\times\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}},$ (17)

where, similarly to (15), $p_{k}\in\mathbb{Z}$ and $p_{k}/q$ is the charge of
the $k$th anyon. The coordinate $w_{k}\in\mathbb{C}$ is its position. Both
$p_{k}$ and $w_{k}$ are external parameters of the wavefunctions. Note that
$w_{k}$ is not required to be equal to one of the $z_{j}$, but can take any
value in the complex plane.

Equations (16) and (17) can be generalized to a cylinder [41, 42]. Let us
consider a plane of size $L_{x}\times L_{y}$, whose upper and lower edges are
glued to make a cylinder (i.e. $y$ is the periodic direction, and $L_{y}$ is
the circumference of the cylinder). If $\xi_{j}$ and $\omega_{i}$ are the
coordinates of sites and anyons, respectively, within the $L_{x}\times L_{y}$
rectangle, then the substitution $z_{j}=\exp(2\pi\xi_{j}/L_{y})$,
$w_{j}=\exp(2\pi\omega_{j}/L_{y})$ turns (16) and (17) into cylinder
wavefunctions [41].

It was shown that these cylinder wavefunctions have Laughlin-type topological
order [41, 42]. However, while the particle density in the continuum cylinder
wavefunction with no anyons seems to be invariant under inversion [45], in the
lattice cylinder wavefunction it is not (except from the special case
$\eta=q/2$). An example for a $5\times 5$ square lattice with $q=2$ and $M=2$
is shown in Fig. 8(a). The figure shows the average particle number at a given
$x$ coordinate, defined as
$n(x)=\sum_{j}\delta(x-x_{j})\langle\psi_{\mathrm{model}}|n_{j}|\psi_{\mathrm{model}}\rangle$.

Figure 8: (a) The average particle number as a function of the (non-periodic)
coordinate $x$ for a $5\times 5$ square lattice, for the “naive”
implementation of a $q=2$, $M=2$ lattice Laughlin wavefunction on a cylinder,
i.e. Eq. (16) with $z_{j}=\exp(2\pi\xi_{j}/L_{y})$ and $\eta=4/25$. (b) The
same quantity for the modified wavefunction (12) with $p_{-}$ given by (14),
and $q$, $M$ and $\eta$ the same as in (a).

The lack of inversion symmetry is alarming for two reasons. First, in real
quantum Hall systems exchanging the left and right side of the cylinder should
not matter. Second, on the plane, the lattice wavefunctions approach the
continuum wavefunction in the limit of $\eta\rightarrow 0$ (i.e. infinite
number of sites per flux quantum) [41]. Similar arguments should apply for a
cylinder [42], and if this is indeed the case, we should explain why the
inversion invariance does not exist for general $\eta$ and is restored at
$\eta=0$.

In the following, we will show that the inversion invariance of (16) on a
cylinder can be restored at any $\eta$ by choosing the numbering of the sites
appropriately and adding appropriate charges, described by the numbers
$p_{-},p_{+}\in\mathbb{R}$, at the coordinates $w_{-}=0$ and $w_{+}=\infty$,
respectively. They should be understood as infinitely thin, immobile flux
tubes. If we plot the complex numbers $z_{j}$ on the plane, they form rings of
different radii. The flux tube at ($w_{-}=0$) is at the center of the rings,
while the flux tube at ($w_{+}=\infty$) is located outside the rings,
infinitely far away. As a result, the effects of the charge $p_{+}$ appears
only in the charge neutrality condition [42]. The wavefunction is thus given
by (12) (but so far, we have not determined the values of $p_{-}$ and
$p_{+}$).

In our considerations, we will permute and modify the $z_{j}$ and $n_{j}$.
Therefore, we will write the individual arguments explicitly, i.e.

$\psi_{\mathrm{model}}(\mathbf{z},\mathbf{n})=\\\
\psi_{\mathrm{model}}(z_{1},z_{2},\dots,z_{N};n_{1},n_{2},\dots,n_{N}).$ (18)

Inversion of the coordinates $\xi_{j}$ around the point $\xi_{c}$ transforms
$\xi_{j}-\xi_{c}\to-(\xi_{j}-\xi_{c})$ and hence

$z_{j}\rightarrow c/z_{j},$ (19)

where $c=e^{4\pi\xi_{c}/L_{y}}$. For now, we will consider a general lattice,
which is not necessarily inversion invariant. As an intermediate step, we will
investigate when

$\psi_{\mathrm{model}}\left(z_{1},z_{2},\dots,z_{N};n_{1},n_{2},\dots,n_{N}\right)\propto\\\
\psi_{\mathrm{model}}\left(\frac{c}{z_{1}},\frac{c}{z_{2}},\dots,\frac{c}{z_{N}};n_{1},n_{2},\dots,n_{N}\right).$
(20)

We demand proportionality instead of equality because the wavefunctions are
unnormalized.

Under the transformation (19), we have

$z_{j}-z_{k}\rightarrow\frac{c}{z_{j}}-\frac{c}{z_{k}}=-c\,\frac{z_{j}-z_{k}}{z_{j}z_{k}},$
(21)

and

$\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}}\rightarrow\\\
\rightarrow\prod_{j<k}(-c)^{\gamma_{j}\gamma_{k}}\prod_{j<k}\left(z_{j}z_{k}\right)^{-\gamma_{j}\gamma_{k}}\prod_{j<k}\left(z_{j}-z_{k}\right)^{\gamma_{j}\gamma_{k}}.$
(22)

Let us look closer at the arising factors. We have

$\prod_{j<k}(-c)^{\gamma_{j}\gamma_{k}}=\prod_{j\neq
k}(-c)^{\gamma_{j}\gamma_{k}/2}=\\\
=\prod_{j,k}(-c)^{\gamma_{j}\gamma_{k}/2}\prod_{j}(-c)^{-\gamma_{j}^{2}/2}=\\\
=(-c)^{\frac{\left(p_{+}+p_{-}\right)^{2}-\left(q^{2}-2q\eta\right)M-\eta^{2}N}{2q}},$
(23)

where we used the facts that $\sum_{j}\gamma_{j}=-(p_{+}+p_{-})/\sqrt{q}$
(from the charge neutrality condition) and that $n_{j}^{2}=n_{j}$ as
$n_{j}\in\\{0,1\\}$. Because $M$ and $N$ are constant, this term can be
absorbed into normalization, so we can ignore it. Similarly,

$\prod_{j<k}(z_{j}z_{k})^{-\gamma_{j}\gamma_{k}}=\prod_{j,k}(z_{j}z_{k})^{-\gamma_{j}\gamma_{k}/2}\prod_{j}z_{j}^{\gamma_{j}^{2}}=\\\
=\prod_{j,k}z_{j}^{-\gamma_{j}\gamma_{k}/2}\prod_{j,k}z_{k}^{-\gamma_{j}\gamma_{k}/2}\prod_{j}z_{j}^{qn_{j}^{2}-2\eta
n_{j}+\eta^{2}/q}=\\\
=\prod_{j}z_{j}^{\gamma_{j}\frac{p_{+}+p_{-}}{2\sqrt{q}}}\prod_{k}z_{k}^{\gamma_{k}\frac{p_{+}+p_{-}}{2\sqrt{q}}}\prod_{j}z_{j}^{(q-2\eta)n_{j}+\eta^{2}/q}=\\\
=\prod_{j}z_{j}^{(q-2\eta+p_{+}+p_{-})n_{j}-\eta(p_{+}+p_{-}-\eta)/q}.$ (24)

Thus, the transformed wavefunction coefficients are proportional to

$\psi_{\mathrm{model}}\left(\frac{c}{z_{1}},\frac{c}{z_{2}},\dots,\frac{c}{z_{N}};n_{1},n_{2},\dots,n_{N}\right)\propto\\\
\delta(qM+p_{+}+p_{-}-N\eta)\prod_{j}\chi_{j}^{n_{j}}\prod_{j}z_{j}^{(q-2\eta+p_{+})n_{j}}\times\\\
\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}}.$ (25)

By comparing the wavefunctions before and after the transformation, we find
that (20) is true if the condition (13) is enforced.

Equation (20) says that the wavefunction is the same if we put the $j$th site
at $z_{j}$ for all $j$ or we put the $j$th site at $c/z_{j}$ for all $j$.
Here, we are, of course, interested in the case, where the lattice itself has
inversion symmetry such that the inversion maps the set of lattice coordinates
to itself. We choose the numbering of the sites such that the inversion maps
site $j$ into site $N+1-j$, i.e.,

$z_{j}=\frac{c}{z_{N+1-j}},$ (26)

to avoid complications with branch cuts. We have not yet shown that the
wavefunction is invariant under inversion, as the wavefunction does not map to
itself under inversion, but rather to the wavefunction in which the sites are
numbered from $N$ to $1$ rather than from $1$ to $N$.

The only part of the wavefunction that is affected by the ordering of the
coordinates is the factor $\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}}$,
which transforms as

$\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}}\to\\\
\to\prod_{j>k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}}=\prod_{j<k}(-1)^{\gamma_{j}\gamma_{k}}\prod_{j<k}(z_{j}-z_{k})^{\gamma_{j}\gamma_{k}}$
(27)

when we reverse the ordering of the numbering. It follows from (23) with $-c$
replaced by $e^{i\pi}$ that $\prod_{j<k}(-1)^{\gamma_{j}\gamma_{k}}$ is a
global phase factor, and hence reversing the ordering of the numbering leaves
the wavefunction unchanged up to a global phase factor. Combining this result
with (20), we conclude that if we number the sites such that inversion maps
$z_{j}$ to $z_{N+1-j}$, then the wavefunction is invariant under inversion up
to a global factor that can be absorbed in the normalization constant.

This reasoning can be generalized to the case of a wavefunction with anyons,
if the anyons are distributed in an inversion-symmetric way (that is, there
are pairs of anyons with the same $p_{k}$s, located at $w_{k}$s being related
by inversion symmetry).

Finally, we comment on different choices of $\eta$, $p_{+}$ and $p_{-}$. When
(13) is enforced, the charge neutrality relation $qM+p_{+}+p_{-}-N\eta=0$ of
the wavefunction without anyons becomes

$q(M-1)+2p_{-}=N\eta-2\eta.$ (28)

For example, if $p_{-}=0$, then, as $\eta\rightarrow$ 0 at constant $N\eta$,
we obtain the relation

$q(M-1)=N\eta.$ (29)

This is similar to the relation between particle number and flux in the
continuum Laughlin state on the sphere, where “$-1$” is related to the “shift”
quantum number [46].

To compare the model lattice Laughlin state to the exact diagonalization
results, we set $\eta=\alpha$. Then, (29) is typically not fulfilled, i.e.
$p_{-}\neq 0$. If we have a given $\eta$, $M$ and $N$, the correct $p_{-}$ can
be determined from (28), and is equal to (14).

We note that in the above calculations the total flux $N\eta$ is different
from the total flux used in Eq. (3), i.e. $N\eta\neq\alpha N_{\mathrm{plaq}}$.
This is because in this Appendix we assign a flux $\eta$ to a given site,
while in Eq. (3) the flux $\alpha$ is assigned to a given plaquette (and we
have to set $\eta=\alpha$ so that the flux encircled when going around a
plaquette is the same in both cases).

Figure 9: The relation between continuum and lattice systems for the example
with $q=2$, $M=2$, and $L_{y}^{\prime}=2\sqrt{\pi}l_{B}^{\prime}$ described in
the main text. The many-body wavefunction, with $x$-dependent particle density
shown schematically in (a), is composed out of three single-particle orbitals
with $m=0,1,2$ – the blue ones in (b). The cylinder is drawn schematically in
(c), with blue and grey lines denoting the Gaussian centers of the orbitals.
The blue region of the cylinder located between the centers of the 0th and 2nd
orbitals contains $q(M-1)=2$ flux quanta. The dots “$\dots$” in (b) and (c)
mean that the cylinder extends infinitely in both positive and negative $x$
directions. The blue and green regions contain a total number of 6 flux
quanta. In (d) we show the blue and green regions unwound to a plane. The
circumference of the cylinder is chosen in such a way that the blue region is
a square. In (e) and (f) we show the site positions for the analogous lattice
systems with total flux $N\eta=2$ and $N\eta=6$, respectively, for an
$N_{x}\times N_{y}=10\times 10$ lattice. Figure 10: Comparison of the rescaled
particle density for the example with $q=2$, $M=2$, and
$L_{y}^{\prime}=2\sqrt{\pi}l_{B}^{\prime}$ described in the main text for the
total flux of (a) two flux quanta and (b) six flux quanta. The $x$ coordinate
is measured in units of the magnetic length of the given system.

##  Appendix B Relation between lattice and continuum particle densities

In the following, we will compare the particle density in the lattice
wavefunction (12) to the case of the continuum Laughlin wavefunction. This can
serve as a sanity check for (12), as in the limit $\eta\rightarrow 0$ we
expect it to converge to the continuum wavefunction. Also, it connects (12) to
the commonly used concepts such as single-particle Landau level orbitals.

The continuum Laughlin wavefunction [45] is given by

$\Psi(\Xi_{1}^{\prime},\Xi_{2}^{\prime},\dots\Xi_{M}^{\prime})=\\\
=\prod_{j<k}(Z_{j}^{\prime}-Z_{k}^{\prime})^{q}\exp\left(-\frac{\sum_{j}X_{j}^{\prime
2}}{2l_{B}^{\prime 2}}\right),$ (30)

where $\Xi_{j}^{\prime}=X_{j}^{\prime}+iY_{j}^{\prime}$ are the coordinates of
the particles, $Z_{j}^{\prime}=\exp(2\pi\Xi_{j}^{\prime}/L_{y}^{\prime})$, and
$l_{B}^{\prime}$ is the magnetic length. We use primed symbols for the lengths
and coordinates in the continuum system to stress that in principle they can
be measured in different units than in the lattice case. A schematic plot of
particle density as a function of $X^{\prime}$ in the case of $q=2$, $M=2$ and
$L_{y}^{\prime}=2\sqrt{\pi}l_{B}^{\prime}$ is shown in Fig. 9(a).

It is instructive to view (30) in terms of single-particle lowest Landau level
orbitals on a cylinder [45]. These orbitals are labeled by an integer $m$ and
given by

$\phi_{m}(X^{\prime},Y^{\prime})=\exp\left(2\pi
i\frac{mY^{\prime}}{L_{y}^{\prime}}\right)\times\\\
\times\exp\left(-\frac{1}{2l_{B}^{\prime 2}}\left(X^{\prime}-\frac{2\pi
ml_{B}^{\prime 2}}{L_{y}^{\prime}}\right)^{2}\right),$ (31)

which consist of a plane wave in the $y$ direction and a Gaussian centered at
$\frac{2\pi ml_{B}^{\prime 2}}{L_{y}^{\prime}}$ in the $x$ direction. The
density profiles of these orbitals, as a function of the $X^{\prime}$
coordinate, are shown schematically in Fig. 9(b), and the position of their
Gaussian centers on the cylinder (the grey and blue rings) in Fig. 9(c). The
basis for (30) is constructed out of orbitals with $m=0,1,\dots,q(M-1)$.

The region between the Gaussian centers of the nearest-neighbor orbitals
always contains one flux quantum. Therefore, the $x$ distance between the
Gaussian centers depends on $L_{y}^{\prime}$. As the circumference of the
cylinder decreases, the orbitals get more separated, and the nearly-uniform
quantum Hall state transforms into a charge density wave [45] (in the example
in Fig. 9(a), the inhomogenity of the density is already quite developed, with
high density on the $m=0,2$ orbitals and low density on the $m=1$ orbital).

We also note that (30) is in fact defined on an infinite cylinder. However, we
can also restrict (30) to a finite cylinder by truncating the domain of
$\Xi_{j}^{\prime}$ to some finite range and normalizing the wavefunction
appropriately. One possible choice is to set the cylinder borders to the
centers of $m=0$ and $m=q(M-1)$ orbitals – in such a way the system is pierced
by $q(M-1)$ flux quanta, in accordance with (29). In our $M=2$ example, this
is the blue region in Fig. 9(c), also shown in Fig. 9(d) after unwinding to a
plane. It contains two flux quanta. However, we can also use a bigger system,
which would allow us to observe the Gaussian “tails” at the edges. For
example, we can study a system thrice as large (six flux quanta), denoted by
combined green and blue regions in Fig. 9(c), (d). Since we focus on the
systems where the particle density is inversion-invariant, we consider only
cases where the lengths added before the center of $m=0$ orbital and after the
center of $m=q(M-1)$ orbitals are the same (i.e. that the two green regions in
Fig. 9(c), (d) have the same size). We do not restrict the total flux to be an
integer, i.e. the edges of the domain do not have to lie at the Gaussian
centers as in Fig. 9.

Now, knowing the meaning of various parameters of the continuum wavefunction,
we propose that to match the particle densities of continuum and lattice
wavefunctions, we have to do the following:

  1. 1.

obtain the magnetic length $l_{B}$ of the lattice system to establish a common
length scale,

  2. 2.

match the circumferences $L_{y}^{\prime}/l_{B}^{\prime}=L_{y}/l_{B}$ (i.e. the
distance between the Gaussian centers),

  3. 3.

match the total flux through the system (i.e. the extent of the system in the
$x$ direction),

  4. 4.

match the center of symmetry (i.e. shift the $x$ coordinates so that the
center of both systems is at $x=0$),

not necessarily in this order (it depends on which parameters of which system
we want to vary).

As an example, let us consider finding lattice systems corresponding to the
$M=2$ continuum wavefunction from Fig. 9 defined on the blue region (two flux
quanta). We will consider a rectangular lattice of a fixed size $N_{x}\times
N_{y}$ in terms of unit cells, with a unit cell of size $a\times 1$, where $a$
will be adjusted in the process. We have $N=N_{x}N_{y}$ and $L_{y}=N_{y}$.
Thus, by doing step 2 (i.e. demanding
$L_{y}^{\prime}/l_{B}^{\prime}=L_{y}/l_{B}$), and recalling that for the
system from Fig. 9 we had $L_{y}^{\prime}=2\sqrt{\pi}l_{B}^{\prime}$, we find
$l_{B}=N_{y}/(2\sqrt{\pi})$. From step 3 we get that $N\eta=2$, and hence
$\eta=2/N$. Now, let us look again at the magnetic length. Eq. (12) does not
refer to it explicitly, but we can infer the relation of $l_{B}$ to the unit
cell size by considering a loop and comparing its area to the encircled flux
(for simplicity, we treat the flux as uniformly distributed in space). A unit
cell of size $a\times 1$ corresponds to $\eta$ flux quanta, so we must have
$a=2\pi\eta l_{B}^{2}$. In this way, we determined the shape of the lattice.

Next, we find $p_{-}$ from (14). Figure 10(a) shows the comparison of the
particle densities for continuum and lattice for various $N_{x}\times N_{y}$.
In the plot, the $x$ coordinate is plotted in units of magnetic length of the
respective system and shifted so that the center of the system is at $x=0$.
The particle density is normalized so that its integral is equal to $M$ (in
the case of lattice systems, we plot $\rho(x)=\langle n(x)\rangle l_{B}/a$).
The densities are calculated exactly for sufficiently small systems and using
Monte Carlo for the largest ones. It can be seen that there is a good match
between the lattice and continuum wavefunctions, especially in the center.
Notable mismatches, especially near the edges, exist only for the cases with
small $N_{y}$. As $N_{y}$ approaches $N_{x}$, the mismatch vanishes almost
completely.

Similarly, we can repeat the procedure for the combined blue and green regions
of Fig. 9 (six flux quanta), with the result shown in Fig. 10(b). Again we
obtain a good match, with small but visible mismatches for small $N_{x}$.
While in the case of two flux quanta and small $N_{y}$ the density near the
edges significantly departed from the continuum values, here we do not observe
such an effect, probably because the system is more elongated, or because the
density at the edges is very small.

If one considers a Hofstadter model (1) with the gauge (2), then one can
compare the particle densities in the ground state and the model wavefunction
in an analogous way. Matching the wavefunctions to compute the overlap is also
possible, but more complicated. The model wavefunction has to be centered at
the $m=0$ orbital, which can be achieved by a shift of the $X^{\prime}_{j}$
coordinates together with a modification of the phase of (30) (the phase
transformation does not affect the particle density).

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