#  Beyond braid anyons: A lattice model for one-dimensional anyons with a
Galilean invariant continuum limit

Sebastian Nagies  Technische Universität Berlin, Institut für Theoretische
Physik, Hardenbergstr. 36, 10623 Berlin  Pitaevskii BEC Center and Department
of Physics, University of Trento, Via Sommarive 14, 38123 Trento, Italy  INFN-
TIFPA, Trento Institute for Fundamental Physics and Applications, Trento,
Italy  Botao Wang  Technische Universität Berlin, Institut für Theoretische
Physik, Hardenbergstr. 36, 10623 Berlin  CENOLI, Université Libre de
Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium  A.C. Knapp
University of Florida, Gainesville, FL 32610, USA  André Eckardt  eckardt@tu-
berlin.de Technische Universität Berlin, Institut für Theoretische Physik,
Hardenbergstr. 36, 10623 Berlin  N.L. Harshman<EMAIL_ADDRESS>Physics
Department, American University, Washington, DC 20016, USA

###### Abstract

Anyonic exchange statistics can emerge when the configuration space of quantum
particles is not simply-connected. Most famously, anyon statistics arises for
particles with hard-core two-body constraints in two dimensions. Here, the
exchange paths described by the braid group are associated to non-trivial
geometric phases, giving rise to abelian braid anyons. Hard-core three-body
constraints in one dimension (1D) also make the configuration space of
particles non-simply connected, and it was recently shown that this allows for
a different form of anyons with statistics given by the traid group instead of
the braid group. In this article we propose a first concrete model for such
traid anyons. We first construct a bosonic lattice model with number-dependent
Peierls phases which implement the desired geometric phases associated with
abelian representations of the traid group and then define anyonic operators
so that the Hamiltonian becomes local and quadratic with respect to them. The
ground-state of this traid-anyon-Hubbard model shows various indications of
emergent approximate Haldane exclusion statistics. The continuum limit results
in a Galilean invariant Hamiltonian with eigenstates that correspond to
previously constructed continuum traid-anyonic wave functions. This provides
not only an a-posteriori justification of our model, but also shows that our
construction serves as an intuitive approach to traid anyons. Moreover, it
contrasts with the non-Galilean invariant continuum limit of the anyon-Hubbard
model [Keilmann et al., Nat. Comm. 2, 361 (2011)] describing braid anyons on a
discrete 1D configuration space. We attribute this difference to the fact that
(unlike braid anyons) traid anyons are well defined also in the continuum in
1D.

##  I Introduction

How do wave functions transform when indistinguishable particles are
exchanged? In three dimensions and higher, this question has two answers:
symmetric under pairwise exchanges like bosons or antisymmetric under pairwise
exchanges like fermions. In either case, the transformation does not depend on
the path taken by the exchange, only on the permutation. All fundamental
particles in the Standard Model are bosons or fermions. However in lower
dimensions, the quasiparticles that emerge in interacting systems can possess
different forms of exchange statistics. The explanation is topological: unlike
in three dimensions, particle interactions in one and two dimensions (2D)
create defects that make configuration space not simply connected. Different
exchange paths for the same permutation can be topologically distinguished by
how they wind around these defects. Anyons are particles with path-dependent
exchange transformations, and they have statistics that lie intermediate
between bosons and fermions.

Anyonic exchange statistics were first predicted for particles in 2D with
hard-core two-body constraints [1, 2]. Unlike in three dimensions, in 2D the
hard-core two-body interactions create defects in configuration space, making
it not simply-connected. The same particle permutation can be accomplished by
topologically distinct exchange paths, and the braid group describes the
topology of those exchange paths [3, 4, 5, 6]. Representations of the braid
group determine how wave functions transform. The simplest abelian
representations of the braid group give fractional exchange statistics
described by a statistical angle $\theta\in[0,2\pi)$ that interpolates between
bosons $\theta=0$ and fermions $\theta=\pi$. Two-dimensional anyons obeying
fractional statistics can be realized as quasiparticles of topologically
ordered states of matter, such as fractional quantum Hall states [7, 8]. The
anyonic braiding statistics with $\theta=2\pi/3$ have been measured in the
fractional quantum Hall effect through interference and scattering
measurements [9, 10].

In one dimension (1D), interactions are even more disruptive to the topology
of configuration space; particles must pass through each other in order to
exchange. Hard-core two-body interactions make exchange impossible in 1D.
However, it was recently demonstrated that a different form of anyonic
exchange statistics is possible in 1D if the two-body hard-core relation is
relaxed but a three-body hard-core constraint is enforced [11, 12]. Hard-core
three-body constraints in one-dimension also make the configuration space not
simply-connected, and this allows for multi-valued wave functions and non-
trivial exchange phases. Such a system is characterized by the so-called traid
group which, like the braid group, can be visualized by strand diagrams with
equivalence relations 111The strand group $T_{N}$ was independently discovered
by several groups of mathematicians and goes by various other names, including
the doodle group, planar braid group, and twin group [72, 73, 74, 75, 76]..
Similar to fractional statistics and the braid group, abelian traid statistics
are also intermediate between bosons and fermions, but in a different manner.
The traid group breaks the Yang-Baxter relations, so that pairwise exchanges
of some neighboring particle pairs can be symmetric and antisymmetric for
others, for instance in a staggered fashion with respect to the particle order
(as is explained below and in Refs. [11, 12]).

So far traid-anyonic exchange statistics has been studied only for quantum
states in the continuum, where they are described by multi-valued wave
functions. In this paper, we construct the first lattice model for traid
anyons. For this purpose, we start from a one-dimensional bosonic lattice
Hamiltonian with non-local, number-dependent tunneling phases. The phases
embody the exchange properties of abelian representations of the traid group
in a way reminiscent of bosonic descriptions of two-dimensional anyons via
flux tube attachments and magnetic potentials. Using a non-local Jordan-
Wigner-like gauge transformation, the bosonic model is then transformed to a
local Hubbard-type Hamiltonian of anyonic particles obeying non-trivial, non-
local commutation relations. The resulting construction bears similarities to
that of the anyon-Hubbard model [14, 15, 16, 17, 18, 19]. The continuum limit
of the bosonic representation of our model is found to give rise to solutions
that correspond to the previously constructed traid anyonic wave functions,
justifying our construction. We also find that the continuum limit of our
traid-anyon-Hubbard model is Galilean invariant, unlike the continuum limit of
the anyon-Hubbard model [18].

The motivation for our work is threefold: (i) We want to demonstrate that
traid exchange statistics can be realized in a concrete model system. (ii) By
directly engineering the required geometric phases associated with particle
exchange, we want to provide an intuitive approach to the physics of traid
anyons. (iii) Finally, we hope that our model might be a first step towards an
experimental implementation of traid anyons, using techniques of quantum
engineering in artificial quantum systems (such as the recent implementation
of the anyon-Hubbard model with ultracold atoms in optical lattices [19]).
Apart from (i-iii), we also make the interesting observation that our model,
which by construction describes particles with non-trivial traid-anyonic
_exchange_ statistics, also shows indications of approximate fractional
Haldane-type _exclusion_ statistics [20, 21, 22].

The paper is organized as follows: In Sec. II, we review how non-trivial
exchange statistics can be described by strand diagrams governed by rules that
emerge from the topology of configuration space. In doing so, we compare
abelian braid and traid anyons to particles obeying standard (bosonic or
fermionic) statistics as well as to each other. In Sec. III, we then construct
the traid-anyon-Hubbard model by starting from a bosonic model with number-
dependent Peierls phases. Numerical results presented in Sec. IV then allow us
to study the ground-state properties of the model. We find generalized Friedel
oscillations as well as other indications for fractional exclusion statistics
in both the dependence of the chemical potential on the total particle number
and the occupations of the natural orbits. In Sec. V we finally take the
continuum limit and show that in this limit the solutions of our model
coincide with the previously predicted wave functions for traid anyons. The
concluding section puts our work into a broader context and points towards
future research.

##  II Exchange statistics in continuum models

As a prelude to the lattice models in the next section, in this section we
consider exchange statistics in continuum systems in first quantization. We
first review the Symmetrization Postulate and how the symmetric group
describes permutations of fermions and bosons. Then we show how topological
defects created by hard-core interactions in low dimensions can break the
relations of the symmetric group, leading to the braid group and traid group.
The abelian representations of these groups provide multi-valued anyonic wave
functions that, unlike the symmetric group, have path-dependent exchange
transformations. These results can be rigorously derived using the so-called
intrinsic approach to topological exchange statistics; see Appendix A for an
overview and some references.

###  II.1 Standard statistics

Bosons and fermions, whether single-component or multi-component, have
_standard_ exchange statistics. Unlike anyons, there is no distinction
required between particle permutations and particle exchanges; all exchange
paths leading to the same permutation are equivalent. For standard exchange
statistics, the Symmetrization Postulate is sufficient to describe wave
functions of bosons and fermions. To implement the Symmetrization Postulate,
bosons or fermions are given arbitrary particle labels, temporarily breaking
indistinguishability. Then the artificial labels are made unobservable by
symmetrizing or antisymmetrizing the wave function over these labels,
effectively restoring indistinguishibility.

In more detail, consider $N$ particles with labels $i=1,2,\ldots,N$ and
positions $x_{i}\in{\mathcal{M}}$ summarized in the vector ${\bf
x}=(x_{1},\ldots,x_{N})$. When there are no points excluded by hard-core
interactions then the configuration space is
${\mathcal{X}}={\mathcal{M}}^{N}$, otherwise
${\mathcal{X}}\subset{\mathcal{M}}^{N}$. The permutation
$p=\\{p_{1}p_{2}\cdots p_{N}\\}$ of these labels form the symmetric group
$S_{N}$ and correspond to coordinate transformations on ${\mathcal{X}}$ that
maps the point ${\bf x}=(x_{1},x_{2},\ldots,x_{N})$ to the point ${\bf
x^{\prime}}=p({\bf x})=(x_{p_{1}},x_{p_{2}},\ldots,x_{p_{N}})$. Any
permutation $p\in S_{N}$ can be expressed as a sequence of pairwise exchanges
$s_{i}$ with $i=1,2\ldots,N-1$, each of which swaps the particle labelled $i$
with the particle labelled $(i+1)$. The transpositions $s_{i}$ form a
generating set for the symmetric group and obey the relations

$\displaystyle\mbox{Self-inverse:}\ s_{i}^{2}=1,$ (1a)
$\displaystyle\mbox{Yang-Baxter:}\ s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1},$
(1b) $\displaystyle\mbox{Locality:}\ s_{i}s_{j}=s_{j}s_{i}\ \mbox{when}\
|i-j|>1.$ (1c)

A way to visualize these permutations is in terms of strand diagrams as shown
in Fig. 1. Instead of a permutation of labels or a coordinate transformation
on ${\mathcal{X}}$, the strand diagrams depict particle exchanges as $N$
continuous paths. The horizontal direction represents space and the vertical
direction represents time (or a control parameter) increasing from bottom to
top. The rules (1) can be understood intuitively as the equivalences
achievable by continuously transforming the strands (moving and stretching in
space, while maintaining forward progress in time) and allowing strands to
cross and overlap arbitrarily. For the symmetric group, every strand diagram
leading to the same permutation is equivalent under the rules (1); this is not
true for braid and traid strand diagrams below.

Figure 1: Examples of strand diagrams depicting equivalency relations (1).
These are read from bottom to top, with particle $1$ furthest left. (a) self-
inverse $s_{1}^{2}=1$; (b) Yang-Baxter $s_{1}s_{2}s_{1}=s_{2}s_{1}s_{2}$; (c)
locality $s_{1}s_{3}=s_{3}s_{1}$.

Applying the structural rules (1) to first-quantized wave functions for $N$
indistinguishable particles gives only two possibilities: bosons and fermions.
To see this, consider the single-component (e.g., spinless or spin polarized)
wave function $\psi({\bf x})$ defined on the configuration space
${\mathcal{X}}$. The probability $|\psi({\bf x})|^{2}$ to find particles at
the positions ${\bf x}$ must be independent of the particle labels, so under
the permutation $p$ of particle labels the wave function can only change by a
phase factor $\eta_{p}$. Let the operator $\hat{U}$ be a representation of
$S_{N}$ on the Hilbert space of wave functions on ${\mathcal{X}}$ defined by

$\hat{U}_{p}\psi({\bf x})=\psi(p({\bf x}))=\eta_{p}\psi({\bf x}).$ (2)

The phase factors $\eta_{p}$ provide an abelian representation of the
symmetric group, so they must obey the constraints related to Eqs. 1. Denoting
$\sigma_{i}$ as the phase associated with the permutation $s_{i}$, then an
exchange of adjacent pairs gives $\hat{U}_{s_{i}}\psi({\bf
x})=\sigma_{i}\psi({\bf x})$. From Eq. (1a), we have that $\psi(s_{i}^{2}({\bf
x}))$ is equal to both $\sigma_{i}^{2}\psi({\bf x})$ and $\psi({\bf x})$,
implying that $\sigma_{i}=\pm 1$. In turn, from the Yang-Baxter relation (1b),
we obtain $\psi(s_{i}s_{i+1}s_{i}({\bf
x}))=\sigma_{i}^{2}\sigma_{i+1}\psi({\bf x})$ is equal to
$\psi(s_{i+1}s_{i}s_{i+1}({\bf x}))=\sigma_{i}\sigma_{i+1}^{2}\psi({\bf x})$,
implying that $\sigma_{i}=\sigma_{i+1}$ for all $i$. For abelian
representations, locality (1c) provides no constrains on the phases. All
together, we find that all phases are equal:

$\sigma_{i}=\sigma,\quad\text{with}\quad\sigma=+1\quad\text{or}\quad\sigma=-1,$
(3)

where the two possible choices correspond to bosons and fermions, respectively
222Note that a similar argument also holds for multi-component wave functions
but requires non-abelian representations of the symmetric group. For a
detailed exposition on the Symmetrization Postulate in 1D, see [77, 78]..

###  II.2 Braid anyons

In 2D with hard-core two-body interactions, the equivalence between
permutations of particle labels and particle exchanges no longer holds.
Instead of the symmetric group $S_{N}$, particle exchanges for $N$
indistinguishable particles are described by the braid group $B_{N}$ [1, 2, 4,
5]. Because particles cannot coincide, the configuration space is not simply-
connected and multi-valued wave functions are allowed; see Appendix B for more
details on how the braid group derives from topological exchange statistics.

Figure 2: Strand diagrams depicting self-inverse relations for the symmetric
group and traid group and the broken self-inverse relations for the braid
group. Subfigure (a) and (d) depicts the the self-inverse relations for
$s_{i}$ and $t_{i}$, the pairwise exchange of particle $i$ and $i+1$.
Subfigures (b) and (c) depict the broken self-inverse relation for the
pairwise exchanges $b_{i}\neq b_{i}^{-1}$. The two-body hard-core constraint
means that strands cannot overlap, forcing distinction between pairwise
exchanges that are right-handed $b_{i}$ and left-handed $b_{i}^{-1}$. If the
particles live in three or more dimensions, these strands representing
particle exchanges that could be continuously deformed into one another. In
just one dimension, paths cannot go over or under, and so pairwise exchanges
are topologically forbidden if there are hard-core two-body interactions. As
we argue below, the anyon-Hubbard model allows to effectively define braid
anyons also without hard-core constraint in 1D.

The key structural features of the braid group can be understood by
considering strand diagrams. Unlike the symmetric group, when two paths in the
diagram cross, the particles cannot coincide and so we must indicate which
particle is passing in front. As a result, the exchange of two particles is no
longer self-inverse; the braids $b_{i}$ and $b_{i}^{-1}$ are the distinct
over- and under-braided exchange of the particles labeled $i$ and $i+1$; see
Fig. 2(b) and (c). With this distinction, the diagrams of Fig. 1 have to be
modified as shown in Fig. 3. However, both the Yang-Baxter and locality
relations are preserved as they correspond to continuous transformations of
strands that are not possible also in the presence of two-body hardcore
interactions. Note that in three spatial dimensions also for hard-core
interactions no distinction between over and under can be made since both
diagrams can be transformed into each other using the third dimension. In one
dimension, in turn, two-body hardcore interactions prevent any particle
exchange. Thus, in the continuum, braid anyons are naturally expected to occur
only in 2D. However, we will argue below that the discrete configuration space
of a lattice allows for the definition braid anyons also in 1D.

Figure 3: Strand diagrams depicting the braid group. Subfigure (a) depicts the
consequence $b_{1}^{2}\neq 1$ of the broken self-inverse relation $b_{i}\neq
b_{i}^{-1}$. Subfigures (b) depicts the Yang-Baxter
$b_{1}b_{2}b_{1}=b_{2}b_{1}b_{2}$ relation and subfigure (c) depicts the
locality relation $b_{1}b_{3}=b_{3}b_{1}$, both unchanged from (1).

Every particle exchange $\gamma\in B_{N}$ can be represented as the product of
$b_{i}$ and $b_{i}^{-1}$’s. These generators obey the relations

$\displaystyle\mbox{Braid inequivalence:}\ b_{i}^{2}\neq 1,$ (4a)
$\displaystyle\mbox{Yang-Baxter:}\ b_{i}b_{i+1}b_{i}=b_{i+1}b_{i}b_{i+1},$
(4b) $\displaystyle\mbox{Locality:}\ b_{i}b_{j}=b_{j}b_{i}\ \mbox{when}\
|i-j|>1.$ (4c)

Note that unlike the symmetric group, the same particle permutation can be
enacted by multiple inequivalent exchange paths $\gamma\in B_{N}$. For
example, the braids $b_{1}$, $b_{1}^{-1}$, $b_{1}^{3}$, $b_{1}^{-3}$, etc.,
all enact the same particle permutation of the first and second particles. The
multi-valuedness of anyon wave functions on ${\mathcal{X}}$ originates from
the multiplicity of topologically distinct exchange paths that execute the
same permutation. Unlike the Symmetrization Postulate, elements of the braid
group do not correspond to coordinate transformations on ${\mathcal{X}}$.
However, we can define an operator $\hat{U}_{\gamma}$ that represents a
particle exchange $\gamma\in B_{N}$, and for an abelian representation,
$\hat{U}_{b_{i}}$ multiples the wave function by a phase factor $\beta_{i}$:

$\hat{U}_{b_{i}}\psi({\bf x})=\beta_{i}\psi({\bf x}).$ (5)

Because the generators are no longer self-inverse, the phase $\beta_{i}$ can
take any value. As a result of the Yang-Baxter relation, as before we have
$\beta_{i+1}=\beta_{i}$ and all the $\beta_{i}$ are required to be equal.
Therefore, we obtain the continuous family of possible choices for abelian
representations of $B_{N}$

$\beta_{i}=e^{i\theta}\quad\text{with}\quad\theta\in[0,2\pi),$ (6)

labeled by the exchange phase $\theta$. The special cases $\theta=0$ and
$\theta=\pi$ correspond to bosons and fermions, respectively, and lead to
single-valued wave functions on ${\mathcal{X}}$. All other $\theta$ correspond
to braid anyons with fractional exchange statistics and have wave functions
that are multi-valued on ${\mathcal{X}}$ [6].

###  II.3 Traid anyons

The equivalence between permutations of particle labels and particle exchanges
also no longer holds when there are three-body hard-core interactions in 1D.
The excluded points in configuration space are defects that result in
topological exchange statistics described by the traid group $T_{N}$ [11, 12].
See App. C for how the traid group is derived from topological exchange
statistics.

To understand the resulting exchange statistics, we can again modify the
strand diagrams of Fig. 1, as shown in Fig. 4. Since we do not assume a two-
body hard-core constraint, two strands can cross, just like in Fig. 1, and
there is no “over” or “under” distinction possible. We can immediately verify
that the exchange of particles is self-inverse and local. However, the Yang-
Baxter relation is broken; there is no continuous deformation of the strand
diagram on the left-hand side of 1(b) to that on the right-hand side of 1(b).
Such a deformation would necessarily involve crossing of three strands in one
point, violating the three-body hard-core constraint.

Figure 4: Strand diagrams depicting the traid group. Subfigures (a) and (c)
depict the self-inverse $t_{1}^{2}=1$ and locality relations
$t_{1}t_{3}=t_{3}t_{1}$, both unchanged from (1). Subfigure (b) depicts the
broken Yang-Baxter relation $t_{1}t_{2}t_{1}\neq t_{2}t_{1}t_{2}$. The absence
of a two-body hard-core constraint means that two strands can overlap, but the
three-body hard-core constraint implies that three strands cannot. Therefore,
a continuous deformation from the strand diagram on the left to the strand
diagram on the right is impossible, and these two exchange paths are
topologically distinguishable and may acquire different phases, despite
leading to the same permutation.

As before, we can define generators $t_{i}$ for the exchange of the $i$th and
$(i+1)$th particles that obey relations that we can read off Fig. 4:

$\displaystyle\mbox{Self-inverse:}\ t_{i}^{2}=1,$ (7a)
$\displaystyle\mbox{Traid inequivalence:}\ t_{i}t_{i+1}t_{i}\neq
t_{i+1}t_{i}t_{i+1},$ (7b) $\displaystyle\mbox{Locality:}\
t_{i}t_{j}=t_{j}t_{i}\ \mbox{when}\ |i-j|>1.$ (7c)

With these relations, the $N-1$ generators $t_{i}$ define the traid group
$T_{N}$ for $N$ strands. Note that while, for the symmetric group, the labels
$i$ were an arbitrary assignment and, for the braid group, the labels were
based on an arbitrary positioning of the particles, in 1D there is a natural
way to assign labels based on their ordering on the line; i.e. the particle
furthest to the left (or equivalently, right) is the particle labeled $i=1$,
then the next particle in order is labeled $i=2$, and so on. This ordering
scheme is invariant under homeomorphisms of the line, and _ordinality_ (so
defined) is a topological property specific to one-dimensional systems.
Therefore, unlike the pairwise exchange generators of $S_{N}$ or $B_{N}$, the
generator $t_{i}$ exchanges _neighboring_ particles in real space, not just in
label space. This topological feature of ordinality is intimately connected
with the breaking of the Yang-Baxter relation (7b).

Irreducible abelian representations of the traid group are specified by the
phases $\tau_{i}$ picked up during the exchange of neighboring particles

$\hat{U}_{t_{i}}\psi({\bf x})=\tau_{i}\psi({\bf x}).$ (8)

where the operator $\hat{U}_{t_{i}}$ represents the generator $t_{i}$. The
self-inverse relation (7a) ensures that, like for standard $S_{N}$ statistics,
$\tau_{i}=\pm 1$. However, unlike both standard and braid statistics, where
the Yang-Baxter relation (7b) enforces equality of the phases, for the traid
group the phase factors $\tau_{i}$ are independent (generally,
$\tau_{i}\neq\tau_{j}$). Thus, we have

$\tau_{i}=+1\quad\text{or}\quad\tau_{i}=-1,$ (9)

leading to to $2^{N-1}$ abelian representations of the traid group,
corresponding to the $N-1$ choices of signs $\tau_{i}$. Like braid anyons,
traid anyons contain bosons and fermions as special cases, here corresponding
to all $\tau_{i}=+1$ or all $\tau_{i}=-1$, respectively. Unlike braid anyons,
traid anyon representations do not continuously interpolate between these two
extreme cases, but instead comprise a discrete set with non-standard exchange
statistics.

To better understand traid anyon representations, consider the simplest non-
trivial case of $N=3$ particles. Besides bosons
$(\tau_{1},\tau_{2})=(+1,+1)\equiv(++)$ and fermions $(--)$, there are two
additional abelian representations $(+-)$ and $(-+)$. For the representation
$(+-)$, the first pair of particles exchange symmetrically like bosons and the
second pair exchange antisymmetrically like fermions. For either of these
cases, the products $\tau_{1}\tau_{2}\tau_{1}$ and $\tau_{2}\tau_{1}\tau_{2}$
have opposite signs, showing that the same permutation of the first and third
particle can be accomplished though topologically distinct exchange paths.

##  III Lattice model for traid anyons

The path-dependence of exchange statistics in one-dimensional tight-binding
lattice models is in some ways more intuitive and easier to visualize than the
continuum case. The configuration space becomes a discrete Fock space and
exchange paths become a series of discrete hops with dynamical phases.
Exchange paths can be understood as loops in the discrete configuration space
given by the Fock states, and the total dynamical phase can be visualized as a
flux through this loop.

By engineering these phases and fluxes, bosonic lattice models can implement
non-standard exchange statistics with density-dependent tunneling phases. In
the first subsection we first recapitulate how this works for the anyon-
Hubbard model, which was introduced to simulate fractional exchange statistics
with bosons on a one-dimensional lattice [14]. In the next subsection, we
reverse engineer the argument that led to the anyon-Hubbard model to create a
Hubbard model with abelian traid-anyon statistics.

###  III.1 Anyon-Hubbard model with braid statistics

To introduce the anyon-Hubbard model, we follow Ref. [14] and begin from the
deformed commutation relations hypothesized for fractional exchange
statistics:

$\displaystyle a_{j}a_{k}-a_{k}e^{i\theta\text{sgn}(j-k)}a_{j}=0,$ (10)
$\displaystyle
a_{j}a_{k}^{\dagger}-a_{k}^{\dagger}e^{-i\theta\text{sgn}(j-k)}a_{j}=\delta_{jk},$

where $a_{j}$ is the annihilation operator for an anyon obeying fractional
exchange statistics described by exchange phase $\theta$. In terms of these
operators, the anyon-Hubbard Hamiltonian is then

$H=-J\sum_{j}\left(a^{\dagger}_{j+1}a_{j}+\textrm{h.c.}\right)+\frac{U}{2}\sum_{j}n_{j}(n_{j}-1),$
(11)

where $n_{j}=a_{j}^{\dagger}a_{j}$ is the number operator on site $j$, $J$ is
the tunneling strength, and $U$ is the two-body interaction strength.

The anyonic creation and annhihilation operators with commutation relations
(10) can be expressed in terms of bosonic ones $b_{j}$ via a generalized gauge
transformation called the fractional Jordan-Wigner transformation:

$a_{j}=e^{i\theta\sum_{k\leq j}n_{k}}b_{j},$ (12)

where also $n_{j}=b_{j}^{\dagger}b_{j}$. This density-dependent gauge
transformation (12) is _non-local_ because the phase for the operator $a_{j}$
depends on all occupation numbers to the left of site $j$. Although the
fractional Jordan-Wigner gauge transformation is non-local, the deformed
commutation relations (10) remain local.

Figure 5: (a) Sketch of density-dependent hopping processes in the anyon-
Hubbard model (Eq. 13). (b) Four configurations of two particles on three
lattice sites. The parity-breaking density-dependent hopping phase means that
the transition from the leftmost configuration to the rightmost configuration
that takes place without coincidence accumulates a different phase than the
lower path. Mapping this back to Fig. 3a, a clockwise loop returning to the
same configuration corresponds to the braid generator $b_{i}$ and a
counterclockwise to $b_{i}^{-1}$.

Expressed in terms of boson operators, the anyon-Hubbard Hamiltonian (11)
becomes

$H\\!=\\!-J\sum_{j}\left(b^{\dagger}_{j+1}e^{-i\theta
n_{j+1}}b_{j}\\!+\\!\textrm{h.c.}\\!\right)+\frac{U}{2}\\!\sum_{j}\\!n_{j}(n_{j}-1).$
(13)

In the bosonic form, there are operator-valued, occupation-number-dependent
Peierls phases attached to the tunneling matrix elements. These phases break
parity, and they occur when a particle hops to the right onto an already
occupied lattice site [Fig. 5(a)] or in the hermitian conjugate process when a
particle hops to the left from a multiply occupied lattice site.

These Peierls phases can give rise to non-trivial geometric phases associated
with closed paths in the many-body configuration space of indistinguishable
particles, given by all Fock states with sharp site occupation numbers. Such
geometric phases can be viewed as generalized magnetic fluxes. In Fig. 5(b),
we show an example of a Fock-space loop associated with a nontrivial phase of
$\theta$ ($-\theta$), when encircled in clockwise (anticlockwise) direction.
Recently, the bosonic representation of the anyon-Hubbard model (13) was
realized experimentally with ultracold atoms in an optical lattice, where the
non-trivial geometric phases were probed directly via interference of the
upper and the lower path of Fig. 5(b) during quantum walks [19].

The geometric phase resulting from the density-dependent tunneling can be
directly related to the braiding of particles. In Fig. 5(b), we can associate
paths that encircle the depicted loop, starting (say) from the leftmost
configuration, with the exchange of the two particles. Encircling the loop in
clockwise direction, so that the left particle tunnels rightwards twice,
passing an occupied site, can be associated with $b_{i}$ depicted in the
strand diagram Fig. 2(b), where the left particle moves over the right one. In
turn, encircling the loop in anticlockwise direction, so that the right
particle tunnels leftwards twice, passing an occupied site, can be associated
with $b_{i}^{-1}$ depicted in the strand diagram Fig. 2(c), where the right
particle moves over the left one. The corresponding phase factors are
$\beta_{i}=\beta=e^{-i\theta}$ and $\beta_{i}^{-1}=\beta^{*}=e^{i\theta}$. In
this sense, the anyon-Hubbard model describes braid anyons. Therefore, and in
order to distinguish it from the traid-anyon-Hubbard model that we construct
in the next subsection, we will also refer to it as the braid-anyon-Hubbard
model in the following.

###  III.2 Hubbard model with traid exchange statistics

The goal of this section is to introduce a lattice model of interacting bosons
that captures traid anyonic exchange statistics. For this purpose, we will
reverse the strategy used for the braid-anyon-Hubbard model above to implement
these extreme cases. Namely, we first construct a bosonic model with non-local
number-dependent tunnelling phases that give rise to the desired exchange
phases. Then we find the corresponding traid anyon field operators and their
commutation relations by constructing a generalized Jordan-Wigner
transformation, so that the Hamiltonian becomes quadratic and local with
respect to these new operators. We will first focus on the case of staggered
abelian representations of the traid group, with alternating exchange phases
given by $(+-+-\cdots)$ and $(-+-+\cdots)$. These are of specific interest,
since they are furthest away both from the bosonic and (pseudo)fermionic
cases, characterized by the homogeneous exchange phases $(++\cdots+)$ and
$(--\cdots-)$, respectively. Subsequently, we will also contruct a model for
general abelian representations $(\tau_{1},\tau_{2},\ldots,\tau_{N-1})$ with
$\tau_{i}=+1$ or $-1$ of the traid group.

A model that realizes the two abelian representations of the traid group with
alternating exchange phases is achieved by the following lattice model using
bosons with number-dependent hopping phases,

$\displaystyle H$ $\displaystyle=$
$\displaystyle-J\sum_{j}^{L}\left(b^{\dagger}_{j+1}e^{i\pi\left(N_{j}+I\right)n_{j+1}}b_{j}+\mathrm{h.c.}\right)$
(14) $\displaystyle+\frac{U}{2}\sum^{L}_{j}n_{j}(n_{j}-1),$

In contrast to the braid-anyon-Hubbard model (13), the number-dependent
hopping phase is now defined in terms of the operator

$N_{j}\equiv\sum_{k\leq j}n_{k}.$ (15)

that counts the number of particles to the left of (and including) lattice
site $j$. The integer $I\in\\{0,1\\}$ that appears in the phase determines
which of the two possible abelian representations of the traid group with
alternating exchange phases the model realizes: For $I=0$ we have
$(+-+-+\cdots)$, meaning that the two leftmost particles exchange like bosons,
the second and third particles from the left exchange like fermions, and so
on. For $I=1$ on the other hand, the pattern $(-+-+\cdots)$ gets flipped, i.e.
the two leftmost particles now exchange like fermions.

The symmetries of the lattice traid anyon model can be inferred from the
bosonic form of the Hamiltonian (14). Time reversal invariance
$\mathcal{T}H\mathcal{T}=H$ results by observing that $\exp(i\pi k)=\exp(-i\pi
k)$ for any integer $k$, so the Peierls phase is invariant under complex
conjugation, as are all other terms. Spatial inversion $\mathcal{P}$
transforms $j$ to $L-j$ and $N_{j}$ to $N-N_{j}$. After reindexing the sum,
the only difference between $\mathcal{P}H\mathcal{P}$ and $H$ that remains is
an additional phase $\exp(i\pi N)$ in the hopping terms. Therefore, for $N$
even, we have $\mathcal{P}H(I)\mathcal{P}=H(I)$ with the same $I$ whereas for
$N$ odd we have $\mathcal{P}H(I=0)\mathcal{P}=H(I=1)$. In comparison, neither
$\mathcal{T}$ for $\mathcal{P}$ are symmetries of the braid-anyon-Hubbard
model, although a gauge-transformed combination of them is [17].

Figure 6: (a-c) Sketch of some possible hopping processes and associated phase
factors in our lattice model for traid anyons (Eqs. 14). Depicted is a system
of $4$ particles with an even integer $I$, corresponding to the abelian
representation ($+-+$). (d-e) Two loops in configuration space for the $N=3$
traid anyon model representation $(+-)$, analgous to Fig. 5. Unlike the braid-
anyon-Hubbard model, all hopping parameters are real, so there is no
difference between clockwise and counterclockwise loops, i.e., these exchanges
are self-inverse. In (d), the first and second particles undergo an exchange
process. All hopping phases have the same sign and the product of all the
phase factors is $+1$. In (e), the second and third particles undergo an
exchange process. The sign of lower left hops is reverse, so the product of
all thephase factors is $-1$.

Figs. 6(a-c) depict these alternating hopping phases for the case of 4
particles and $I=0$: When the leftmost particle hops to the right onto an
already occupied lattice site [Fig. 6(a)], a phase factor of $+1$ gets picked
up. After a particle then tunnels from the doubly occupied lattice site to the
empty site on the right, the particles have swapped positions and the overall
exchange phase factor is $+1$; tunneling to the right onto an empty site
always gives a trivial phase factor. Exchanging the third and fourth particle
[Fig. 6(b)] gives the same result, while a swap of the central two particles
[Fig. 6(c)] results in an overall phase factor of $-1$, i.e. the second and
third particles exchange like fermions. Figs. 6 (d,e) show how these phases
accumulate during particle exchanges. We can see that, like for the braid-
anyon-Hubbard model, the number-dependent Peierls phases give rise to the
desired exchange phases.

Note that our lattice model in Eq. 14 uses bosons and thus allows an arbitrary
number of particles to occupy the same lattice site. However, a three-body
hard-core constraint is essential for the existence of genuine traid anyons in
the continuum. Indeed, if we consider hopping processes on the lattice
involving three or more particles, the desired pattern of alternating exchange
phases breaks down for our model. We can address this problem theoretically in
two ways: we can add a three-body interaction term to the Hamiltonian (14) and
take the limit of the coupling coefficient to infinity, or we can implement
the three-body hard-core constraint algebraically by requiring $b_{j}^{3}=0$.
For numerical convenience, we have chosen the latter approach by simply
truncating the state space so that at most two particles can occupy the same
site. Note that if we restrict ourselves to the low-energy, dilute limit, then
three-body occupation of the same site becomes increasingly unlikely to occur
and should be negligible for the calculation of most observables. Our
numerical results in the next section show that this is indeed the case for
the low-energy dilute limit. Further, in the following section on the
continuum limit we show that three-body, hard-core interactions emerge
naturally from the number-dependent tunneling phases for the alternating
representations.

We now define annihilation and creation operators, $t_{i}$ and
$t_{j}^{\dagger}$, for the traid anyons. For that purpose, we define a
generalized Jordan-Wigner transformation, so that the Hamiltonian becomes
local and quadratic with respect to these new operators:

$\displaystyle H$ $\displaystyle=$
$\displaystyle-J\sum_{j}^{L}\left(t^{\dagger}_{j+1}t_{j}+\mathrm{h.c.}\right)$
(16) $\displaystyle+\frac{U}{2}\sum^{L}_{j}n_{j}(n_{j}-1).$

For the staggered traid anyon model (14), this is achieved by:

$\displaystyle t_{j}$ $\displaystyle\equiv$ $\displaystyle e^{-i\pi
M_{j}}b_{j},$ (17a) $\displaystyle M_{j}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left[\sum_{k\leq
j}n_{k}\left(N_{j}-n_{k}\right)\right]+IN_{j},$ (17b)

where $M_{j}$ depends on the $N_{j}$ and on the integer $I$. Like the braid-
anyon-Hubbard model, the transformation (17) leaves the number operator on
site $j$ invariant

$t^{\dagger}_{j}t_{j}=b^{\dagger}_{j}b_{j}=n_{j}.$ (18)

The crucial difference is that when expressed in the bosonic form, the hopping
phases of the traid anyon model (14) are non-local and depend on the notion of
ordinality, whereas hopping phases for the braid-anyon-Hubbard model are
local.

This non-local property is noticeable also in the commutation relations for
the traid operators $t_{j}$. From Eq. (17) and the bosonic commutation
relations, we obtain:

$\displaystyle
t_{j}t_{k}-t_{k}e^{i\pi\text{sgn}(j-k)\left[N_{jk}+I\right]}t_{j}$
$\displaystyle=$ $\displaystyle 0,$ (19a) $\displaystyle
t_{j}t_{k}^{\dagger}-t_{k}^{\dagger}e^{-i\pi\text{sgn}(j-k)\left[N_{jk}+I\right]}t_{j}$
$\displaystyle=$ $\displaystyle\delta_{jk},$ (19b)

where $N_{jk}$ are new operators defined as

$\displaystyle N_{jk}\equiv\begin{cases}N_{j}-n_{k},\quad\text{if $j>k$}\\\
N_{k}-n_{j},\quad\text{if $j<k$}\\\ N_{j-1},\;\enspace\quad\enspace\text{if
$j=k$}.\end{cases}$ (20)

From Eqs. 19, we see that the commutation relations of the traid anyon
operators reduce to the bosonic case when acting on the same lattice site.
However for operators acting on different sites, the commutation relations are
either fermionic or bosonic, depending on the configuration of all the other
particles on the lattice.

The traid anyon lattice model (14) generalizes to arbitrary abelian
representations of the traid group in the following way:

$\displaystyle H$ $\displaystyle=$
$\displaystyle-J\sum_{j}^{L}\left(b^{\dagger}_{j+1}e^{i\pi
F(N_{j})n_{j+1}}b_{j}+h.c.\right)$ (21)
$\displaystyle+\frac{U}{2}\sum^{L}_{j}n_{j}(n_{j}-1),$

where $F(N_{j})$ is a polynomial of $N_{j}$ (Eq. 15) that takes on even or odd
integer values for integer inputs. $F(N_{j})$ differs for each abelian
representation and takes on the simplest form for the alternating cases:
$F_{\text{alt}}(N_{j})=N_{j}+I$. For example, if one wanted to construct a
model to realize the representation $(++-)$ for 4 traid anyons, we would need
to find a polynomial that takes on the values $F_{++-}(0)=0$, $F_{++-}(1)=0$
and $F_{++-}(2)=1$ (or equivalently other even/odd integer values). In
practice, for our numerical simulations in Sec. IV, we do not construct the
polynomial explicitly and only set the desired inputs/outputs.

Analogously, we can find a Jordan-Wigner-like transformation that brings the
generalized Hamiltonian of the traid anyon model (21) to theform (11) with a
quadratic nearest neighbor tunneling term:

$\displaystyle t_{j}$ $\displaystyle\equiv$ $\displaystyle e^{-i\pi
f(N_{j})}b_{j},$ (22)

where $f(N_{j})$ is a polynomial of $N_{j}$ (Eq. 15), taking on odd or even
integer values depending on the traid group representation which is to be
realized. In particular, $f(N_{j+1})-f(N_{j})=F(N_{j})n_{j+1}$ should be
fulfilled, so that Eq. 21 is quadratic when expressed in terms of these
general traid anyon operators. For example, to construct traid anyon operators
for the representation $(++-)$, we can set $f(0)=f(1)=f(2)=0$ and $f(3)=1$. It
is easy to check that this yields the desired pattern of tunneling phases,
provided we exclude three-body coincidences. Similarly, the generalized
transformation (22) leads to generalized commutations relations for the
$t_{j}$.

We note that the transformation defined in Eqs. 17 for the alternating abelian
representations does not match the general form given by Eq. 22: the operator
$M_{j}$ is not just a polynomial of $N_{j}$ but also depends on the
configuration of the particles to the left of site $j$ (and not just their
number). For instance, a two-body coincidence to the left of site $j$ would
result in an extra factor of $-1$ in $e^{-i\pi M_{j}}$. These potential
additional factors of $-1$ cancel out for terms of the form
$t_{j+1}^{\dagger}t_{j}$ or $n_{j}=t_{j}^{\dagger}t_{j}$ and have thus no
influence on the Hamiltonian in Eq. 16. On the other hand, for observables
like the two-point correlation function $\braket{t_{i}^{\dagger}t_{j}}$, it
can indeed make a difference, e.g. if there is a two-body coincidence between
lattice sites $i$ and $j$. The factor of $-1$ then only occurs for one of the
traid anyon operators and does not cancel out. The advantage of using the
traid anyon operators defined in Eqs. 17 for the alternating abelian
representations instead of the general form given by Eq. 22, is that the
transformation to boson operators always results in the conceptually simple
Hamiltonian in Eq. 14, even if we do not enforce a three-body hard-core
constraint.

Finally, we would like to point out that the representation of traid anyons
using bosons with number dependent tunneling matrix elements resembles the
composite charged-particle/flux tube picture of braid anyons in 2D (see, e.g.
Ref. [6]). In the latter case, braid anyons are mapped to charged bosons (or
fermions) to which a magnetic flux is attached to implement the geometric
phases picked up when particles are moved around each other. In the former
case of our traid anyon lattice model, the number dependent tunneling matrix
elements also give rise to generalized magnetic fluxes in the discrete
configuration space of our system that are determined by the position of the
particles and implement the geometric phases associated with the exchange of
traid anyons; [See e.g. Fig. 5(b) and Fig. 6(e)].

##  IV Ground-state properties and signatures of fractional exclusion
statistics

After introducing our lattice model for traid anyons, let us investigate its
ground state properties in this section. To that end we numerically calculate
the particle densities, energies and natural orbitals via exact
diagonalization of Eqs. 14 and 21 (supplemented by DMRG simulations for higher
filling factors in Fig. 9). These results show how the abelian traid
representations lie between fermions and bosons, and also hint at an
intriguing connection to Haldane’s fractional exclusion statistics [20] (c.f.
App. D).

For all our simulations in this section, we set the on-site interactions to
zero, $U=0$. We also enforced a three-body hard-core constraint to reduce
computation times and to always preserve the characteristic pattern of
exchange phases, which breaks down for tunneling processes involving three or
more particles in our traid anyon lattice model Eq. 14. Because we stay in the
low-energy, dilute regime for most of our simulations, three-body processes
can be neglected on the calculated observables (see also the discussion in
section III.2). Consequently, the signatures of traid anyons reported below do
not rely on this additional constraint.

###  IV.1 Particle density: generalized Friedel oscillations

Figure 7:  Ground state particle densities for the traid anyon lattice model
(Eq. 14) for 3-6 particles on 20 lattice sites and $U=0$. Each plot shows the
densities corresponding to the two possible abelian representations of the
traid group with alternating exchange phases of a given number of particles,
i.e. the blue lines correspond to the $I=0$ model where the first and second
particles exchange like bosons, the second and third like fermions and so on.
The $I=1$ exchange phases are flipped.  Figure 8:  Ground state particle
densities for the generalized traid anyon lattice model (Eq. 21) on 20 lattice
sites for $U=0$. Shown are all 8 possible abelian representations of the traid
group for a system of 4 particles.

As a first physical observable, we consider the ground state particle
densities. The particle density is independent of whether we formulate the
model in terms of boson or traid anyon operators:
$\braket{n_{j}}=\braket{b_{j}^{\dagger}b_{j}}=\braket{t_{j}^{\dagger}t_{j}}$,
i.e., the number of bosons and traid anyons on each lattice site is identical.
Fig. 7 shows the ground state densities of 3 to 6 traid anyons on $20$ lattice
sites. We depict the two possible abelian representations with alternating
exchange phases ($+-+-...$) and ($-+-+...$) for each number of particles. As
expected, the reflection symmetry of the densities about the midpoint of the
lattice embodies the symmetry of the arrangement of the exchange phases: While
the individual density profiles for the representations with $I=0$ and $I=1$
are not symmetric for odd $N$, they are mirror images of each other. In turn
for even particle numbers, we find distinct, individually symmetric densities.

Examples of non-alternating abelian representations of the traid group are
shown in Fig. 8, where the densities of all $8$ possible abelian
representations (not just the alternating case) for $4$ particles are
represented. The uppermost plot depicts the trivial case of bosons and pseudo-
fermions, where all exchange phases are identical. The bottom two plots show 4
representations which can only be realized by the generalized version of our
lattice model (Eq. 21).

The density profiles depicted in Figs. 7 and 8 show that for each minus sign
in the chosen traid-group representation, there is a minimum separating two
local density maxima. Intuitively, depending on the exchange phases on both
sides of the minus sign, these maxima are associated with either single
particles resembling fermions, or two particles forming a ‘bosonic’ pair
(separated by fermionic exchanges from its neighbors). For non-alternating
traid group representations, when there are several plus signs next to each
other, also ‘bosonic’ triples, quadruples, etc. can be observed (see Fig. 8).
In the case of pseudo-fermions, i.e. the representations $(---\cdots)$ with
only minus signs (see the uppermost plot in Fig. 8), these oscillations of the
density correspond to Friedel oscillations [24], as they have been predicted
also for the braid-anyon-Hubbard model [16]. These oscillations are a result
of the fermionic two-particle correlations (reflecting Pauli exclusion), as
they become visible in the density distribution close to a local defect, which
is here given by the edge of our finite system with open boundary conditions.
Thus, for the non-trivial representations, containing both plus and minus
signs, the observed oscillations can be viewed as generalized Friedel
oscillations.

###  IV.2 Ground state energy and chemical potential

Figure 9:  Change in ground state energy (chemical potential) in units of $J$
after adding the $N$th particle to a system with $N-1$ particles on $30$
lattice sites and no on-site interaction ($U=0$). Traid anyons (Eq. 14) with
alternating exchange phases ($+-+-...$) or ($-+-+...$) are compared to bosons,
fermions, pseudo-fermions and the braid-anyon-Hubbard model with
$\theta=\frac{\pi}{2}$. Except for bosons and fermions, a maximum site
occupation of 2 particles was enforced.

Like the particle densities discussed above, the ground-state energy (as well
as the full spectrum) does not depend on whether we consider bosons with
number-dependent tunneling or traid anyons, since the Hamiltonians for both
pictures are equivalent and just expressed in terms of different operators. We
numerically calculate the ground state energies $E_{0}$ of our lattice model
and consider the chemical potential, i.e. the ground state energy difference
$E_{0}(N)-E_{0}(N-1)$ of a system with $N$ and $N-1$ respective particles. The
results for a system with $30$ lattice sites, up to $10$ particles and no on-
site interactions ($U=0$) are shown in Fig. 9. We compare traid anyons with
alternating exchange phases (14) to (spinless) bosons and fermions, as well as
to pseudofermions and the braid-anyon-Hubbard model (13) with exchange phase
$\theta=\frac{\pi}{2}$.

The ground state energies of free fermions and free bosons can be calculated
analytically from the single-particle energies $E(k)=-2J\cos(dk)$ of a system
with $M$ lattice sites, lattice spacing $d$, and allowed quasi-momenta
$k=\frac{\pi\nu}{d(M+1)}$ with $\nu=1,2,...,M$. For the bosonic ground state
all bosons occupy the single-particle ground-state, thus
$E_{0}(N)-E_{0}(N-1)=-2J\cos(\pi/(M+1))$. For (spinless) fermions on the other
hand, the Pauli exclusion principle implies that
$E_{0}(N)-E_{0}(N-1)=-2J\cos(N\pi/(M+1))$. The ground state energies of the
braid-anyon-Hubbard (13) model were obtained numerically. For completeness, we
have also included pseudo-fermions corresponding to the $(--\cdots-)$
representation of the traid-anyon Hubbard model as well as to the braid-anyon-
Hubbard model with $\theta=\pi$. Their chemical potential agrees nicely with
that of free fermions in the dilute regime (roughly below quarter filling or
$N=7$).

We chose the braid-anyon-Hubbard model with exchange phase
$\theta=\frac{\pi}{2}$ for comparison, because it provides a different notion
of particles exhibiting behaviour ‘right in the middle’ between fermions and
bosons: while traid anyons with representations $(+-+-...)$ or $(-+-+...)$
alternate between exchanging fermion-like (corresponding to an exchange phase
$\theta=\pi$) and boson-like ($\theta=0$), the anyons in the braid-anyon-
Hubbard model always exchange with the same phase $\theta=\frac{\pi}{2}$,
averaging between bosonic and fermionic behavior.

These different notions of interpolation between fermions and bosons are also
reflected in the chemical potentials in Fig. 9. While the chemical potential
lies approximately in the middle of the fermionic and bosonic case for both
traid anyons and the braid-anyon-Hubbard model with $\theta=\frac{\pi}{2}$,
the behaviour is nonetheless strikingly different: We observe a smooth
increase of the potential for the braid-anyon-Hubbard model, while the traid
anyons exhibit a step-like behaviour, where the change in the chemical
potential increases or decreases in an alternating fashion.

The step-wise increase of the chemical potential, observed for the staggered
traid anyons, can be interpreted as a signature of approximate Haldane-like
_semionic_ fractional exclusion statistics, i.e. of a generalized Pauli
exclusion principle, where single-particle states can be occupied by up to two
particles. Ideally, such behaviour might be associated with the chemical
potential increasing for every other particle added and staying constant
otherwise. We, however, find that rather than staying constant in the latter
case, the chemical potential decreases (hinting towards an effectively induced
attractive interaction). In the next subsection, where we investigate the
natural orbitals and their occupations, we find further evidence for the
emergence of approximate fractional exclusion statistics and an explanation
for the observed decrease of the chemical potential in every other step.

The emergence of semionic _exclusion_ statistics can be understood intuitively
as a consequence of the staggered _exchange_ statistics of the traid anyons.
Namely, when the rightmost particles form a ‘bosonic’ pair, a newly added
particle comes with a minus sign and thus plays the role of a new single
‘fermion’ on the right edge. In turn, when the rightmost particle resembles a
single ‘fermion’, the added particle comes with a plus sign, so that it can
occupy the same space as the former rightmost particle (both form a new
‘bosonic’ pair).

It is straightforward to generalize these ideas to non-staggered
representations. We then expect to find scenarios where 3 (or more)
neighboring particles to behave like bosons to each other, e.g. for
$(++-++-...)$. Also mixed fractional statistics can be imagined, e.g. for
$(+-++-+-...)$ pairs and triples of particles are expected to be in the same
state. Such cases will be discussed in the next subsection.

###  IV.3 Natural orbitals

Figure 10:  Ground state natural orbitals (insets) and their occupation
numbers (main plots) for our traid anyon lattice model (Eqs. 14 and 21) on
$20$ lattice sites without on-site interaction ($U=0$), obtained from
diagonalizing the single-particle density matrix
$\braket{t_{i}^{\dagger}t_{j}}$ with the traid anyon operators defined in Eqs.
17 and 22.

The natural orbitals are defined as the eigenvectors of the single-particle
density matrix (SPDM), with the corresponding eigenvalues being the orbital
occupation numbers (which represent how many particles occupy each natural
orbital on average). Like other operators, the SPDM can be expressed in terms
of either boson or traid anyon operators defined in Eqs. 17 and 22,
respectively. In contrast to the observables discussed above, the SPDM is not
equivalent for both pictures, i.e.

$\displaystyle\braket{t_{i}^{\dagger}t_{j}}=\braket{b_{i}^{\dagger}e^{i\pi\left(M_{i}-M_{j}\right)}b_{j}}\neq\braket{b_{i}^{\dagger}b_{j}}.$
(23)

Since we are interested in the properties of the traid anyons, we have to
consider the SPDM $\braket{t_{i}^{\dagger}t_{j}}$. The signatures obtained in
this way are unique to the traid anyon picture and not visible for the bosonic
quantity $\braket{b_{i}^{\dagger}b_{j}}$.

Our numerical results are depicted in Fig. 10. We display the occupation
numbers of the natural orbitals (main plots), as well as the one-particle
densities of the first three natural orbitals ordered by their occupation
numbers (insets). While Figs. 10(a-d) show examples of traid anyons with
alternating hopping signs, specified by Eqs. 14 and 17a, Figs. 10(e,f)
additionally show two different nontrivial abelian representation for 4
particles, namely $(+--)$ and $(++-)$, defined by the generalized traid anyon
lattice model (Eqs. 21 and 22). In analogy to the densities and energies
discussed above, the densities of the natural orbitals corresponding to
mirrored abelian representations [i.e., $(+-)$ and $(-+)$] are also mirrored
with identical occupation numbers.

The most striking feature of the natural orbitals in the traid anyon picture
is their (near) integer-valued occupation numbers. In particular, for the
three-particle case shown in Fig. 10(a), we have occupation numbers of 1.984
and 1.009 for the first two natural orbitals, which suggests that there can be
two traid anyons occupying the first orbital. This behaviour provides further
indication of an approximate emergent Haldane-like fractional exclusion
statistics, as we have discussed in Sec. IV.2. We again emphasize that this is
a feature of the traid anyons, and it is apparently different from bosons or
fermions. We find a qualitative agreement for systems with more particles,
although the near integer values are not always of the same degree of
accuracy.

The distribution of natural orbitals also helps us to interpret the ground
state density distribution. For example, as shown by the orange line in Fig.
7(a), two traid anyons occupying the first orbital corresponds to the dominate
density distribution on the right of the lattice (i.e. a ‘bosonic pair’), and
a single particle occupying the second orbital is distributed to the left of
the lattice. We observe similar behavior in other cases like four particles
with $(-+-)$ [Fig. 10(b)], and even with $(+--)$ defined by our generalized
traid anyon model [Fig. 10(e)].

Despite an almost perfect occupation of the first orbitals with two traid
anyons in the case with one single ‘bosonic pair’ ($+$), there appears some
differences when it comes to occupy higher orbitals. Taking the cases of
$(+-+)$ and $(-+-+)$ as examples, where the ground state density profiles
exhibit two large maxima [see Figs. 7(b,c)], we do not find two spatially
separated ‘bosonic pairs’. The corresponding natural orbitals in Figs. 10(c)
and 10(d) reveal that, instead the particles occupy natural orbitals which are
delocalized over the whole lattice. Likewise, both single ‘fermions’ in
Fig.10(b) are delocalized.

Last but not least, the shape of the natural orbitals also provides a possible
explanation for the observation that in Fig. 9, the chemical potential
decreases every other time when a particle is added (to form a ‘bosonic’ pair
with a previously fermion-like single particle). Namely, in Fig. 10, one can
clearly see that the shape of the natural orbitals depends on the number of
particles in the system. In particular, we find that the width of the natural
orbitals tends to increase with their occupation, so that the kinetic energy
is lowered for multiply-occupied orbitals relative to that of singly-occupied
ones, suggesting a mechanism for the reduction of the chemical potential. This
behaviour is clearly different from that found for the ground state of both
free bosons and fermions, for which the natural orbitals are not number
dependent and simply given by the single-particle eigenstates. This broadening
of multiply-occupied orbitals can be interpreted as a signature of some form
of (effective) attractive interactions in the system. It provides a likely
explanation for the observation that the chemical potential decreases every
other time a particle is added to a system for staggered traid anyons [orange
and blue lines in Fig. 9]. It might also explain the deviations from perfectly
quantized occupations of the natural orbitals visible in Fig. 10). The origin
of these interactions must be related to the form of the commutation relations
for the traid operators (17). Different from the standard bosonic and
fermionic ones, they involve ‘interaction’ terms, i.e. terms that involve
products of more than two annihilation and creation operators. One immediate
consequence is, that the commutation relations become basis-dependent. That
is, transformations $t_{\alpha}=\sum_{i}\braket{\alpha}{t}t_{i}$ to a basis of
single particle states $|\alpha\rangle$ (for instance the basis of natural
orbitals), other than the local Wannier basis $|i\rangle$ defining the lattice
sites, change the form of the commutation relations.

##  V Continuum limit

In section III we constructed a lattice model (14) using bosons with non-
local, density-dependent Peierls phases that corresponds to abelian
representations of the traid group. Here we derive the continuum limit of this
model with staggered exchange phases (Eqs. 14). This approach can easily
generalized also to general abelian representations of the traid group.
Similar to the Tonks-Girardeau model in which one-dimensional fermions can be
mapped to hard-core bosons, we find that in the continuum limit abelian traid
anyons can be expressed in terms of bosons with hard-core contact
interactions. Importantly, these interactions will depend on the oridinality
and, therefore, be non-local.

We find that the eigenstates of the continuum model can be directly mapped to
traid-anyon wave-functions, as they were constructed previously from general
considerations [11, 12]. This mapping is similar to that from hard-core bosons
to fermions. Namely, in our model, hard-core interactions occur whenever two
particles meet and exchange like fermions. As a result, the many-body wave-
function becomes zero at this coincidence, allowing to flip the sign of the
wave function, when mapping between bosons and traid anyons, without
increasing the energy. The fact that we recover previous results from the
continuum limit of our model system provides an a-posteriori justification of
the construction of our model.

###  V.1 Continuum limit of lattice model

We take the continuum limit of our lattice model by letting the lattice
spacing go to zero,

$d\to 0,$ (24)

while keeping the physical length $l=Ld$ constant. We will also see that the
effective mass of the continuum particles $m^{*}$ is defined by
$Jd^{2}=\hbar^{2}/(2m^{*})$ 333This result follows also directly from
expanding the dispersion relation of free bosons with respect to the lattice
spacing,
$\varepsilon(k)=-2J\cos(dk)\simeq-2J+Jd^{2}k^{2}\equiv\frac{\hbar^{2}k^{2}}{2m^{*}}+\text{const.}$.
and that the Hubbard interactions of strength $U$ give rise to a contact
integration potential $g\delta(x)$ of strength $g=Ud$. Thus, we have to scale
$L$, $J$, and $U$ like

$L=\frac{l}{d},\qquad J=\frac{\hbar^{2}}{2m^{*}}\frac{1}{d^{2}},\qquad
U=\frac{g}{d},$ (25)

with $m^{*}$ and $g$ held constant. Moreover, we have to replace sums by
integrals,

$\sum_{j=1}^{L}\to\frac{1}{d}\int_{0}^{l}\\!dx,$ (26)

and bosonic annihilation and creation operators $b_{j}$ by bosonic field
operators $\psi(x)$, according to [26]

$\displaystyle\psi^{(\dagger)}(x)\to\frac{b^{(\dagger)}_{j}}{\sqrt{d}},$ (27)

where $[\psi(x),\psi^{\dagger}(x^{\prime})]=\delta(x-x^{\prime})$.

It is straightforward to obtain the continuum limit of the on-site Hubbard
interactions

$\displaystyle
H_{\text{os}}=\frac{U}{2}\sum_{j}b_{j}^{\dagger}b_{j}^{\dagger}b_{j}b_{j}$
$\displaystyle\to$
$\displaystyle\frac{g}{2}\int\\!dx\,\psi^{\dagger}(x)\psi^{\dagger}(x)\psi(x)\psi(x)$
(28) $\displaystyle=$ $\displaystyle\frac{g}{2}\int\\!dx\,:\rho^{2}(x):,$

where we have introduced the density operator
$\rho(x)=\psi^{\dagger}(x)\psi(x)$ as well as the usual convention that
$:\bullet:$ means normal ordering of all operators between the colons, i.e.
all creation operators are shifted to the left of the annihilation operators.

Because of the number-dependent Peierls phases, the continuum limit of the
tunneling term in Eq. (14) is more involved. Restricting ourselves to the case
$I=0$, it reads

$H_{\text{tun}}=-J\sum_{j}^{L}\left(b^{\dagger}_{j+1}e^{i\pi
N_{j}n_{j+1}}b_{j}+\mathrm{h.c.}\right)$ (29)

At the end, we will reintroduce the dependence on $I$, simply via the
substitution $N_{j}\to N_{j}+I$. Calculating the continuum limit of the braid-
anyon-Hubbard model, Bonkhoff et al. managed to preserve the crucial symmetry
of the anyonic exchange phase under the transformation $\theta\to\theta+2\pi$
in the continuum, by normal ordering the operators in the complex phase factor
before taking the limit [18]. We closely follow their approach in deriving the
continuum limit of the traid anyon lattice model. In our case, we want to
preserve the symmetry of the complex phase factor under the transformation
$N_{i}\to N_{i}+2$. Thus, we calculate the continuum limit of the ‘number-to-
the-left’ operator $N_{i}$ independently and then write all other operators in
the number dependent hopping phase in normal order before taking the limit.
The continuum limit of this operator is given by

$\displaystyle N_{j}=\sum_{k\leq j}b^{\dagger}_{k}b_{k}$ $\displaystyle\to$
$\displaystyle\int_{0}^{x}\\!dx^{\prime}\,\psi^{\dagger}(x^{\prime})\psi(x^{\prime})$
(30) $\displaystyle=$
$\displaystyle\int_{0}^{x}\mathop{}\\!\mathrm{d}x^{\prime}\rho(x^{\prime})\equiv
N(x),$

where $x\equiv jd$. Analogous to the discrete case, the operator $N(x)$ (Eq.
30) counts the number of particles to the left of position $x$. In particular,
one has $N(l)=N$.

Next, we expand the exponential phase factor:

$H_{\text{tun}}=-J\sum_{j}b^{\dagger}_{j+1}\left(\sum_{q=0}^{\infty}\frac{(i\pi)^{q}}{q!}N_{j}^{q}n_{j+1}^{q}\right)b_{j}+\text{h.c.}.$
(31)

The powers of the number operators $n_{j+1}$ can now be written in the normal
ordered form

$\displaystyle
n_{j+1}^{q}=\sum_{m=0}^{\infty}S(q,m)d^{m}\left(\frac{b^{\dagger}_{j+1}}{\sqrt{d}}\right)^{m}\left(\frac{b_{j+1}}{\sqrt{d}}\right)^{m},$
(32)

with the Stirling numbers of the second kind $S(q,m)$ [27, 18]. We now take
the continuum limit, approximate the field operators at $x+d$ as:

$\displaystyle\psi(x+d)$ $\displaystyle\simeq$
$\displaystyle\psi(x)+d\frac{\partial}{\partial
x}\psi(x)+\frac{d^{2}}{2}\frac{\partial^{2}}{\partial x^{2}}\psi(x)$ (33)
$\displaystyle\equiv$ $\displaystyle\psi+d\psi_{x}+\frac{d^{2}}{2}\psi_{xx},$

(note that higher order terms will not contribute in the continuum limit) and
finally sort all resulting terms by their power in the lattice spacing $d$.
The terms of zeroth order in $d$ are

$\displaystyle-J\int\mathop{}\\!\mathrm{d}x\left(\psi^{\dagger}\psi+\mathrm{h.c.}\right)=-2JN.$
(34)

This amounts to a constant energy contribution, which we can neglect in the
final continuum model.

For the first-order terms we find

$\displaystyle-2Jd\int\mathop{}\\!\mathrm{d}x(\cos(\pi
N(x))-1)\psi^{\dagger}\psi^{\dagger}\psi\psi$
$\displaystyle=\frac{\hbar^{2}}{m^{*}}\int\mathop{}\\!\mathrm{d}x\frac{1-\cos(\pi
N(x))}{d}:\rho^{2}:,$ (35)

This term describes a diverging (i.e. hard-core) two-body contact interactions
for even values of $N(x)$ due to the $1/d$ dependence.

Finally, the second-order terms read

$-\frac{\hbar^{2}}{m^{*}}\int\\!dx\Big{[}\left(1-\cos(\pi
N(x))\right):\rho^{3}:+\frac{1}{2}\left(\psi^{\dagger}_{xx}\psi+\psi^{\dagger}\psi_{xx}\right)\Big{]},$
(36)

corresponding to a kinetic energy term and a three-body interaction term. All
other terms are of order $\mathcal{O}(d^{3})$ and vanish in the continuum
limit.

Summing up all terms (and including the continuum limit of the on-site
interaction term in Eq. 14a), we obtain the final continuum Hamiltonian

$H_{\mathrm{cont.}}=\int\mathop{}\\!\mathrm{d}x\left[-\frac{\hbar^{2}}{2m^{*}}\psi^{\dagger}\partial_{x}^{2}\psi+V_{2}:\rho^{2}:+\enspace
V_{3}:\rho^{3}:\right],$ (37)

with

$\displaystyle V_{2}(x)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{m^{*}d}\left(1-\cos\left[\pi(N(x)+I)\right]\right)+\frac{g}{2},$
(38) $\displaystyle V_{3}(x)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{m^{*}}\big{(}\cos\left[\pi(N(x)+I)\right]-1\big{)},$
(39)

where we reintroduced the index $I$, determining which of the two abelian
representations of the traid group with alternating exchange phases the model
realizes.

From the two-body interaction coefficient $V_{2}$ (Eq. 38), we can see that
two particles that exchange like fermions in the lattice model (i.e., where
$N(x)+I$ is odd), now exhibit a hard-core interaction in the continuum limit.
On the other hand, particles that exchange like bosons ($N(x)+I$ is even) do
not interact at all with one another in the continuum model (if $g=0$). The
hard-core two-body interaction between alternating pairs of particles also
implies a three-body hard-core constraint, which is characteristic of the
traid group. Thus, it emerges naturally in the continuum limit of the traid
anyon-Hubbard model (14), even if we do not impose it by additional potentials
or constraints. As a result, we can neglect the finite-strength three-body
interaction term $V_{3}(x)$ in our alternating traid anyon model and simply
write

$\displaystyle H_{\mathrm{cont.}}$ $\displaystyle=$
$\displaystyle\int\mathop{}\\!\mathrm{d}x\left[-\frac{\hbar^{2}}{2m^{*}}\psi^{\dagger}\partial_{x}^{2}\psi+V_{2}(x):\rho^{2}:\right].$
(40)

The emergence of the contact interactions with infinite (zero) strength for
two particles that behave like fermions (bosons) to each other is also
consistent with previous results of Refs. [28, 15]. There it was shown that
for the braid-anyon-Hubbard model that the combination of number dependent
tunneling with anyonic phase $\theta$ and on-site Hubbard interactions $U$
gives rise to an effective contact interaction, whose strength, which diverges
for $\theta\to\pi$ and vanishes when both $\theta=0$ and $U=0$.

Comparing the continuum limit (37) of the traid-anyon-Hubbard model to the
continuum limit of the braid-anyon-Hubbard model [18], we see that we recover
the same structure of the Hamiltonian, when we make the identification
$\theta\to\pi(N(x)+I)$. The crucial difference between the two models is that
in the case of the traid anyons the coefficients $V_{2}$ and $V_{3}$ are
promoted to operators, resulting in non-local interactions between the
particles. Moreover the current interaction term that arises in the continuum
limit of the braid-anyon-Hubbard model vanishes for our model. Thus, in
contrast to the continuum model of the braid-anyon-Hubbard model, the traid-
anyon-Hubbard model constructed here gives rise to a Galilean invariant
Hamiltonian.

A possible explanation for this difference is that the breaking of Galilean
invariance via a dependence on the current operator originates from the
requirement to distinguish exchange processes corresponding to the left
particle moving over or under the right one, as depicted in Figs.2(b) and (c).
While in the braid-anyon-Hubbard model these processes can be distinguished
via different paths that form a non-contractable loop in the discrete
configuration space [see Fig. 5], this is not possible in the continuum limit.
Thus, the dependence on the current operator (as it is also found in the
chiral BF and the Kundu models) can be interpreted as a way to “translate” the
distinction between these two processes to the continuum. A similar
observation on the necessity for breaking Galilean invariance to implement
fractional exchange statistics in 1D models has been made for excitons in the
Haldane-Shastry model [29, 30]. However, for traid anyons this is not the
case, suggesting that they correspond to a more natural definition of anyons
in one dimension in the continuum (as is also suggested by topology of the
continuum strand diagrams discussed above.).

###  V.2 Continuum model in first quantization

From (37), we can derive the bosonic continuum model in first quantization.
First consider the Hamiltonian restricted to the sector $\mathcal{X}_{12\cdots
N}\subset{\mathbb{R}}^{N}$ where the particle labels coincide with particle
order $x_{1}\leq x_{2}\leq\cdots\leq x_{N}$, the Hamiltonian that realizes the
alternating representations characterized by $I$ has the form

$H_{\mathrm{cont.}}^{\text{FQ}}=\sum_{j}\left(-\frac{\hbar^{2}}{2m^{*}}\frac{\partial^{2}}{\partial
x_{j}^{2}}+\tilde{V}_{2}(j)\delta(x_{j}-x_{j+1})\right)$ (41a) where the two-
body interaction coefficient $\tilde{V}_{2}(j)$ is no longer an operator but
an order-dependent number
$\tilde{V}_{2}(j)=\frac{\hbar^{2}}{m^{*}d}(1+\cos(\pi(j+I)))+\frac{g}{2}.$
(41b)

The specific form chosen for the coefficient $\tilde{V}_{2}(j)$ is somewhat
arbitrary; any function that alternates between $0$ and $2$ depending on the
particle order index $j$ would work for the expression in parentheses. In
another ordering sector $\mathcal{X}_{p_{1}p_{2}\cdots p_{N-1}}$, where the
labelled particles are in the order $x_{p_{1}}\leq x_{p_{2}}\leq\cdots\leq
x_{p_{N}}$, the only change to $H_{\mathrm{cont.}}$ is that the delta function
in (41a) becomes $\delta(x_{p_{j}}-x_{p_{j+1}})$.

In the limit $d\to\infty$, the coefficient $\tilde{V}_{2}(j)$ forces a hard-
core condition between particles that have antisymmetric exchanges (i.e. that
behave like fermions to each other). The alternating hard-core two-body
interactions exclude triple coincidences and so again we choose to leave out
the three-body interaction term from (37) in (41a). Note that the total
Hamiltonian only depends on the relative coordinates and the interactions are
invariant under Galilean transformations and permutations of particle labels.
Time reversal invariance is immediate from the real form of (41), and spatial
inversion leads to the same results as for the lattice model: invariance for
$N$ even and mapping between $I=0$ and $I=1$ for $N$ odd.

The ground state of this model can be easily calculated numerically, e.g. with
a finite difference method. This allows us to compare the continuum model
directly to the lattice model for traid anyons (Eq. 14). We compute the ground
state densities for both models and plot them in Fig. 11 (rescaling them for
comparison). We see that the densities of both models match exactly,
indicating that the continuum model (Eq. 41) accurately captures the traid
anyon in the low-energy, dilute limit and its characteristics that we observed
in the lattice model.

Figure 11: Comparison of the ground state densities of a system with 3
particles and representation ($-+$) for both the traid anyon lattice model
(Eq. (14)) and its continuum limit (Eq. (41)). The lattice system contains
$L=20$ sites. The continuum model was solved with a finite difference method.
The lattice model densities were rescaled for comparison. Figure 12:  Dynamics
of two identical particles on a lattice that behave like pseudofermions to
each other. The position of the left and right particle are labeled by
$\ell_{\text{L}}$ and $\ell_{\text{R}}$, respectively, with
$\ell_{\text{L}}\leq\ell_{\text{R}}$ (inset). The configuration space is
plotted with dark (light) blue bullets corresponding to states, where both
particles occupy the same (different) lattice site(s). Vertical (horizontal)
lines represent tunneling processes of the left (right) particle. Thick lines
correspond to tunneling matrix element to or from a state with a doubly
occupied site; these are enhanced by a factor of $\sqrt{2}$. Tunneling along
the red lines is associated with an extra phase of $\pi$, resulting from
number-dependent tunneling and giving rise to the pseudo-fermionic nature of
the two particles. This leads to a flux of $\pi$ through the yellow-shaded
plaquettes. In the continuum limit, the wave function can at most vary
infinitesimally from lattice site to lattice site, because otherwise the
kinetic energy would diverge. Thus, tunneling processes to a dark blue state
with two particles at the same site will destructively interfere. In this way,
two-particle coincidences are suppressed.

###  V.3 Origin of emergent hard-core interactions

The appearance of the delta-function-type contact interactions of infinite
strength between two particles that behave like pseudofermions to each other
can be understood intuitively from the lattice model. Let us consider a
reduced model for two particles only, which locally captures the state space
of the $N$-particle traid-anyon-Hubbard model close to a coincidence of two
particles that are pseudofermions to each other. This situation is plotted in
Fig. 12 (see caption for a description) and also captures the pseudofermion
limit $\theta=\pi$ of the braid-anyon Hubbard model. The destructive
interference prohibiting two particles to occupy the same site when
approaching the continuum limit is associated with the appearance of
plaquettes with $\pi$-flux at the edge of configuration space. These are
indicated by the yellow shading.

In the continuum limit, the wave function can vary at most infinitesimally
from site to site, because otherwise the kinetic energy would diverge.
Therefore, tunneling processes that would create a doubly occupied site (dark
blue bullets) are prohibited by destructive interference, as a result of the
fact that tunneling matrix elements represented by red and black lines have
opposite sign. In the continuum limit two-particle coincidences are, thus,
suppressed completely. This effect is taken care of by the delta-function
interaction of infinite strength. Note that for two particles that behave like
bosons to each other, where tunneling matrix elements described by red lines
have the same sign as all the others, so that no plaquette flux is induced,
this effect vanishes and no interactions appear in the continuum limit.

###  V.4 General abelian representations for traid anyons

The derivation of the continuum limit can be generalized to arbitrary abelian
representations of the traid group. As discussed in section III for the traid
anyon lattice model, other representations are realized by constructing more
complicated polynomials of the operator $N_{j}$ (Eq. 15) in the number
dependant hopping phase: $e^{i\pi f(N_{j})n_{j+1}}$. Analogous to the case of
alternating exchange phases, we can then preserve the symmetry under the
transformation $f(N_{j})\to f(N_{j})+2$ by normal ordering the other operators
in the exponential before taking the limit. The resulting Hamiltonian has the
same structure as Eq. 37, with the new interaction coefficients

$\displaystyle V_{2}(x)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{m^{*}d}\big{(}1-\cos[\pi
f(N(x))]\big{)}+\frac{g}{2},$ (42a) $\displaystyle V_{3}(x)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{m^{*}}\big{(}\cos[\pi f(N(x))]-1\big{)}.$ (42b)

The polynomial $f(N)$ is constructed in the lattice model so that it takes on
even integer values if a pair of particles exchanges like bosons and odd
integer values if a pair of particles exchanges like fermions. Therefore,
analogous to the case of alternating exchange phases, bosonic pairs do not
interact in the continuum limit, whereas pairs of particles behaving like
fermions experience hard-core repulsion. Alternatively, the first quantization
equivalent of (42a) for the general abelian traid representation
$\tau=(\tau_{1}\cdots\tau_{N-1})$ also could be expressed in the ordering
sector ${\mathcal{X}}_{12\cdots N}$ as

$\tilde{V}_{2}(j)=\frac{\hbar^{2}}{m^{*}d}(1-\tau_{j})+\frac{g}{2}.$ (43)

For abelian representations of the traid group, with more than two phases
$\tau_{i}=+1$ next to each other, i.e. $\tau_{i}=\tau_{i+1}=+1$ or
$(\cdots,+,+,\cdots)$, one finds sequences of three or more neighboring
particles that behave like bosons to each other. In these cases, the continuum
limit of our traid-anyon-Hubbard model does not possess emergent three-body
hard-core interactions within these bosonic sequences, unless we include such
a three-body hard-core constraint by hand. A proper continuum limit is
approached even if we do not enforce three-body hard-core interactions on the
level of the lattice model. This raises the interesting question, whether the
definition of traid anyons in continuum actually requires three-body hard-core
interactions between all particles, or whether they are required only among
those triples of neighboring particles that do not all behave like bosons to
each other. The answer is that the latter is true. An alternative approach is
to include the constraint $b_{i}^{3}=0$, as has been considered previously for
bosons in a lattice in Refs. [31, 32]. This leads to a continuum limit with a
three-body hard-core constraint among all triplets that is also consistent
with wave functions corresponding to traid anyon statistics. The differences
between these approaches should appear in the signatures for exclusion
statistics, not exchange statistics, and could be a subject for future
investigation.

###  V.5 Mapping between bosonic and traid anyon wave functions

Figure 13: The relative configuration space for three particles in a
coordinate system (V.5). The configuration space is divided into six sectors
depending on the order of the particles, indicated by inequalities of the form
$x_{p_{1}}<x_{p_{2}}<x_{p_{3}}$. Two-body coincidences $x_{i}-x_{j}=0$ occur
at the colored, dashed lines separating the sectors at
$\varphi\in\\{-2\pi/3,-\pi/3,0,\pi/3,2\pi/3\\}$, where
$\tan\varphi=z_{2}/z_{1}$. The dashed red half-lines at $\varphi=0$, $2\pi/3$,
and $-2\pi/3$ correspond to configurations where the leftmost and center
particles coincide. The green half-lines at $\varphi=\pi/3$, $\pi$, and
$-2\pi/3$ correspond to configurations where the center and rightmost
particles coincide. A hard-core three-body constraint excludes the origin of
this coordinate system.

In this last subsection on the continuum limit, we discuss the mapping of
bosonic states that evolve according to the Hamiltonian (41) and anyonic
states that obey the symmetries given by the traid group [11, 12]. For
simplicity, we will consider the case of three particles with $g=0$ as the
simplest non-trivial scenario. As shown in Fig. 13, the structure of
interactions is revealed by transforming from the particle coordinates ${\bf
x}=(x_{1},x_{2},x_{3})\in{\mathbb{R}}^{3}$ to Jacobi coordinates ${\bf
z}=(z_{0},z_{1},z_{2})$:

$\displaystyle z_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left(x_{1}+x_{2}+x_{3}\right)$ $\displaystyle
z_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{6}}(2x_{3}-x_{1}-x_{2})$
$\displaystyle z_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(x_{2}-x_{1}),$ (44)

where $z_{0}$ is proportional to the center-of-mass, $z_{2}$ proportional to
the relative position of the particles labelled $1$ and $2$, and $z_{1}$
proportional to the relative position of particle $3$ with the center-of-mass
of particles $1$ and $2$, and all of these have been normalized so that the
transformation ${\bf x}\to{\bf z}$ is orthogonal and
$\sum_{j=1}^{3}\partial^{2}/\partial
x_{j}^{2}=\sum_{j=0}^{2}\partial^{2}/\partial z_{j}^{2}$. Converting to polar
coordinates in the relative plane,

$\displaystyle\tan\varphi=\frac{z_{2}}{z_{1}},$ (45)
$\displaystyle\rho=\sqrt{z_{1}^{2}+z_{2}^{2}},$ (46)

gives the relative radius $\rho$, describing the size of the three-body
configuration, and the angular coordinate $\varphi$ describing the relative
shape [33].

The relative coordinate plane is depicted in Fig. 13 and the symmetry revealed
there explains why Jacobi coordinates are convenient. As $\varphi$ is varied
from $-\pi$ to $\pi$ for fixed $\rho$, the three particles execute a cyclic
exchange through six two-body coincidences. Three angles denoted $\alpha_{1}$
correspond to coincidences of the first two particles ($x_{i}=x_{j}<x_{k}$),

$\alpha_{1}\in\left\\{-\frac{2\pi}{3},0,\frac{2\pi}{3}\right\\},$ (47a) and
three angles denoted $\alpha_{2}$ correspond to coincidences of the second two
particles ($x_{k}<x_{i}=x_{j}$),
$\alpha_{2}\in\left\\{-\frac{\pi}{3},\frac{\pi}{3},\pi\right\\},$ (47b)

with $i\neq j\neq k$. Expressed in Jacobi cylindrical coordinates, any bosonic
wave function $\Psi(z_{0},\rho,\varphi)$ must satisfy the following boundary
conditions at both $\alpha_{1}$ and $\alpha_{2}$ to be symmetric under any
two-particle exchange:

$\displaystyle\lim_{\epsilon\to 0}\Psi(z_{0},\rho,\alpha_{i}+\epsilon)$
$\displaystyle=$ $\displaystyle\lim_{\epsilon\to
0}\Psi(z_{0},\rho,\alpha_{i}-\epsilon)$ (48a) $\displaystyle\lim_{\epsilon\to
0}\Psi^{\prime}(z_{0},\rho,\alpha_{i}+\epsilon)$ $\displaystyle=$
$\displaystyle-\lim_{\epsilon\to
0}\Psi^{\prime}(z_{0},\rho,\alpha_{i}-\epsilon),$ (48b)

where $\Psi^{\prime}(z_{0},\rho,\varphi)=\partial\Psi/\partial\varphi$ and the
extra minus sign in (48b) accounts for the reflection of $\varphi$ in the
derivative.

In the Jacobi relative coordinates, the Hamiltonian (41) for the sector
$x_{1}\leq x_{2}\leq x_{3}$ has the form

$\displaystyle H_{\mathrm{cont.}}$ $\displaystyle=$
$\displaystyle-\sum_{j=0}^{2}\frac{\hbar^{2}}{2m^{*}}\frac{\partial^{2}}{\partial
z_{j}^{2}}+(1-\tau_{1})\frac{\hbar^{2}}{\sqrt{2}m^{*}d}\frac{\delta(\varphi)}{\rho}$
(49)
$\displaystyle+(1-\tau_{2})\frac{\hbar^{2}}{\sqrt{2}m^{*}d}\frac{\delta(\varphi-\pi/3)}{\rho},$

and symmetric forms in other sectors 444Note that in the intrinsic approach to
topological exchange statistics (see App. C), indistinguishability effectively
restricts the configuration space to only one of the six sectors depicted in
Fig. 13.. For traid representations with $\tau_{1}=-1$, the red dashed half-
lines at $\alpha_{1}$ in Fig. 13 represent infinite-strength, contact
interactions between the first and second particle in the continuum limit
$d\to 0$. This forces an additional condition on the wave function

$\Psi(z_{0},\rho,\alpha_{1})=0\quad\text{for}\quad\tau_{1}=-1.$ (50a)
Similarly, for traid representations with $\tau_{2}=-1$, the green half-lines
at $\alpha_{2}$ represent infinite-strength delta functions between the second
and third particle and this forces the condition
$\Psi(z_{0},\rho,\alpha_{2})=0\quad\text{for}\quad\tau_{2}=-1.$ (50b)

Because the many-body wave function vanishes at certain coincidences, it costs
no extra energy to flip the sign of the wave function on one side of such a
coincidence. This allows us to construct maps from bosonic wave functions of
Eq. 49 to fermionic or traid-anyonic wave functions
$\tilde{\Psi}(z_{0},\rho,\varphi)$ that are antisymmetric with respect to the
exchange of the two leftmost particles, if $\tau_{1}=-1$, and/or the two
rightmost particles, if $\tau_{2}=-1$:

$\displaystyle\lim_{\epsilon\to
0}\tilde{\Psi}(z_{0},\rho,\alpha_{i}+\epsilon)$ $\displaystyle=$
$\displaystyle\tau_{i}\lim_{\epsilon\to
0}\tilde{\Psi}(z_{0},\rho,\alpha_{i}-\epsilon)$ (51a)
$\displaystyle\lim_{\epsilon\to
0}\tilde{\Psi}^{\prime}(z_{0},\rho,\alpha_{i}+\epsilon)$ $\displaystyle=$
$\displaystyle-\tau_{i}\lim_{\epsilon\to
0}\tilde{\Psi}^{\prime}(z_{0},\rho,\alpha_{i}-\epsilon)..$ (51b)

In the fermionic case $(--)$, the analogy to the Tonks-Girardeau gas is
evident; infinite-strength two-body interactions mimic the exclusion provided
by fermion statistics. In contrast, for the anyonic representations $(+-)$ and
$(-+)$, the two-body hard-core interactions depend on the relative location of
the third particle. We will see below that, as a result, the mapping from
bosonic to traid-anyon wave functions is not one-to-one.

To make this mapping more explicit, consider the special case when there is
either no trapping potential or a harmonic trapping potential and there are no
additional two-body interactions. Then, we can find exact solutions by
recognizing that in the $d\to 0$ limit the Hamiltonian (49) separates in
$z_{0}$, $\rho$, and $\varphi$ coordinates. Call $F(\varphi)$ the angular
factor of the three-body wave function, i.e.
$\Psi(z_{0},\rho,\varphi)=\chi(z_{0},\rho)F(\varphi)$. In between
coincidences, there is no angular potential, so the angular wave function has
the free form $F(\varphi)=A\exp(i\mu\varphi)+B\exp(-i\mu\varphi)$ for $\mu\geq
0$. Combining the bosonic requirement (48) with the representation-dependent
two-body coincidence conditions (50), the lowest energy, unnormalized
solutions $F_{(\tau_{1},\tau_{2})}(\varphi)$ for the the representations
$(\tau_{1},\tau_{2})$ are

$\displaystyle F_{(++)}(\varphi)$ $\displaystyle=$ $\displaystyle 1$ (52a)
$\displaystyle F_{(--)}(\varphi)$ $\displaystyle=$
$\displaystyle|\sin(3\varphi)|,$ (52b) $\displaystyle F_{(+-)}(\varphi)$
$\displaystyle=$ $\displaystyle|\cos(3\varphi/2)|,$ (52c) $\displaystyle
F_{(-+)}(\varphi)$ $\displaystyle=$ $\displaystyle|\sin(3\varphi/2)|,$ (52d)

corresponding to bosons [$(+,+)$], fermions [$(-,-)$], and traid anyons
[$(+,-)$, $(-,+)$]. The last three wave functions (52b-52d) are depicted by
the solid lines in Fig. 14.

For the representation $(--)$, the antisymmetric boson-fermion map of
Girardeau [35, 36, 37] maps the bosonic wave function (52b) depicted in Fig.
14(a) to the corresponding fermionic wave function by flipping the sign at
each two-body coincidence. The resulting wave function
$\tilde{F}_{(--)}(\varphi)=\sin(3\varphi)$, depicted by a dashed line in Fig.
14(a), has the symmetries under exchange of particle labels that we expect for
fermions: $\tilde{F}(\alpha_{i}+\epsilon)=-\tilde{F}(\alpha_{i}-\epsilon)$ and
$\tilde{F}^{\prime}(\alpha_{i}+\epsilon)=\tilde{F}^{\prime}(\alpha_{i}-\epsilon)$
for $\epsilon\to 0$. Note that it is possible to satisfy these requirements
with single-valued functions on the interval $\varphi\in(-\pi,\pi]$.

In contrast, when we attempt to construct a similar map from the bosonic to
the anyonic wave functions for the $(+-)$ and $(-+)$ representations, we find
the function that satisfies (51) must be double-valued. For these
representations, traid anyon exchange statistics impose the symmetries at
$\alpha_{1}$ and $\alpha_{2}$:
$\tilde{F}(\alpha_{i}+\epsilon)=\tau_{i}\tilde{F}(\alpha_{i}-\epsilon)$ and
$\tilde{F}^{\prime}(\alpha_{i}+\epsilon)=-\tau_{i}\tilde{F}^{\prime}(\alpha_{i}-\epsilon)$
for $\epsilon\to 0$. Unlike the fermionic case, flipping the sign of the wave
functions at the hard-core two-body coincidences no longer yields a single-
valued map; the sign of the resulting function depends on where we begin the
sequence of sign-flipping exchanges. For example, consider the function
$F_{(+-)}$ (52c). One possibility for flipping the signs is depicted by the
dashed line in Fig. 14(b), where $F_{(+-)}$ has been sign-flipped at the two-
body hard-core coincidences at $\varphi=-\pi/3$ and $\pi/3$, giving the
function $\tilde{F}_{(+-)}(\varphi)=\cos(3\varphi/2)$. However, choosing to
start in the domain $\varphi\in[-\pi,-\pi/3]$ and antisymmetrizing across
$\varphi=-\pi$ and $-\pi/3$ gives the opposite sign
$\tilde{F}_{(+-)}(\varphi)=-\cos(3\varphi/2)$. This indicates that to
implement the correct exchange statistics, the traid anyon wave function for
the $(+-)$ representation must be double-valued functions on the interval
$\varphi\in(-\pi,\pi]$.

Note that the three-body wavefunctions $\tilde{F}(\varphi)$ that we have
obtained here by taking the continuum limit of our lattice model, directly
correspond to the continuum traid-anyon wavefunctions constructed previously
in Ref. [11] from general properties of the traid group. This provides another
justification of the intuitive construction of the lattice model presented
here.

In the presence of non-quadratic trapping potentials or additional
interactions between the particles, the above reasoning remains valid, except
that separability is lost and the wave functions do not have a simple form
like (52). We can also generalize Eqs. (48-51) to the case of more than three
particles by requiring corresponding symmetries dictated by $\tau_{i}$ for
each of the relative coordinates of neighboring particles, while keeping all
other coordinates fixed. This mapping between bosons and traid anyons provides
traid-anyon wavfunctions in the continuum and corresponds to the generalized
Jordan-Wigner transformation that we employed for the traid-anyon-Hubbard
model.

Figure 14: This figure depicts the angular factor
$F_{(\tau_{1}\tau_{2})}(\varphi)$ of bosonic three-body wave functions solving
the Hamiltonian (49) for the abelian traid group representations $(--)$,
$(+-)$, and $(-+)$, respectively. The solid lines correspond to the lowest
energy solutions. The thicker dotted lines show corresponding traid-anyon wave
functions $\tilde{F}_{(\tau_{1}\tau_{2})}(\varphi)$. In the pseudofermion case
depicted in subfigure (a), infinite-strength contact interactions force nodes
at every two-body coincidence for the black, solid wave function. The gray,
thicker dotted line represents the angular wave function of the lowest energy
free fermion state. This function has the same energy as its bosonic
counterpart and is single-valued on the interval $\varphi\in(-\pi,\pi]$. It
can be related to the bosonic wave function by the totally antisymmetric
boson-fermion map of Girardeau [35].The solid lines in subfigures (b) and (c)
depict the bosonic wave functions correspond to traid anyon representations
$(+-)$ and $(-+)$, respectively. These have nodes at alternating two-body
coincidences. The dotted lines represent one sheet of the corresponding multi-
valued anyonic wave functions.

##  VI Conclusions

The main result of our paper is the construction of a lattice model for
abelian traid anyons in one dimension. The starting point of this construction
is the observation that the discreteness of the lattice’s configuration space
(which is given by Fock states with sharp site occupation numbers) allows to
associate the exchange of two particles on neighboring sites with different
paths in that space. The exchange can either be accomplished by the left
particle hopping to the right onto the coincidence followed by a second hop to
the right or by the right particle hopping to the left onto the coincidence
followed by a second hop the left. One can argue that distinguishing both
processes, and associating them with the left or the right particle passing
“in front” of the other one, allows to define the braiding of particles in one
dimension and that this physics is realized by the (braid) anyon Hubbard model
[14]. The phase picked up by the wave function under such particle exchange
processes directly corresponds to a geometric phase around (or flux through)
closed loops in configuration space. Abelian representations for braid anyons
are characterized by a single angle $\theta$ that determines the exchange
phase and smoothly interpolates between bosons ($\theta=0$) and fermions
($\theta=\pi$). For abelian traid anyons, in turn, the phase $\theta_{i}$
associated with the exchange of the $i$th and $(i+1)$st particle from the left
depends on the ordinality $i$, but can take two possible values, $0$ or $\pi$
only, giving rise to a countable number of abelian representations
$(\tau_{1},\tau_{2},\dots,\tau_{N-1})$ with $\tau_{i}=\exp(i\theta_{i})$.

In order to construct a lattice model exhibiting these traid anyon statistics,
we first consider bosonic particles with non-local occupation-number-dependent
hopping (or Peierls) phases. These are chosen to give rise to the geometric
phases in Fock space corresponding to the desired abelian representation of
traid anyons. This approach resembles the composite-particle description of
braid anyons in two dimensions, where anyons are mapped to charged bosons (or
fermions) to which a flux tube is attached that gives rise to the geometric
phase $\theta$ when one particle is moved around another one. In a second
step, we then introduce traid-anyon annihilation and creation operators, with
respect to which the Hamiltonian becomes quadratic and local, and derive their
commutation relations. This construction provides an intuitive approach to
traid anyons.

Investigating the ground-state properties of this traid-anyon-Hubbard model,
we find various indications also of an approximate Haldane-type exclusion
statistics. For instance, we observe generalized Friedel oscillations, nearly
integer occupations of the natural orbits, and a step-wise change of the
chemical potential with particle number.

Finally, we take the continuum limit of our lattice model and find a
Hamiltonian, whose eigenstates indeed correspond to traid-anyon wavefunctions,
as they were constructed previously from the properties of the traid group
[11]. This justifies _a posteriori_ the construction of our model. Moreover,
the continuum limit of the traid-anyon Hubbard model is found to be Galilean
invariant. This is in stark contrast to the continuum limit of the braid-
anyon-Hubbard model, which breaks Galilean invariance by a Hamiltonian that,
like the Kundu model [38, 39, 40, 41, 42, 43] and chiral BF model [44, 45, 46,
47, 48], depends on the current-density operator [18]. We attribute this
difference to the fact that traid anyons are naturally defined also in the
continuum in 1D . For braid anyons, on the other hand, this is not the case
(as they require to distinguish exchange processes with the left particle
passing over or under the right one during an exchange, which in a 1D
continuum system can only be mimicked dynamically by a velocity dependence).

Our results indicate that the possibilities for non-standard statistics might
in some sense be even richer in one dimension than in two dimensions, at least
when considering discrete lattice models. We have shown how 1D bosonic lattice
models with density-dependent Peierls phases can implement abelian anyons that
possess two completely different forms of exchange statistics, each
corresponding to a different way of breaking the symmetric group: braid
statistics and traid statistics. In both cases, the theoretical link between
the interacting bosonic lattice Hamiltonian and the ‘free’ anyonic Hamiltonian
is provided by a Jordan-Wigner-like gauge transformation that can be expressed
in the form

$a_{j}=e^{i\Theta(N_{j})}b_{j},$ (53)

where $\Theta(N_{j})$ is a so-called non-local string operator that depends on
$N_{j}$, the number of particles to the left of (and including) site $j$. For
the braid-anyon Hubbard model, the function has the simple form
$\Theta(N_{j})=\theta N_{j}$ and the bosonic Hamiltonian possesses local
interactions but has a continuum limit that is not Galilean-invariant and may
not lead to a consistent boson-anyon mapping [37]. For the traid-anyon Hubbard
model, the function has a more complicated form $\Theta(N_{j})=-\pi f(N_{j})$
where $f(N_{j})$ is an integer function that depends on the traid group
representation. The resulting bosonic Hamiltonian posseses non-local
interactions that depend on $N_{j}$. It has a Galilean-invariant continuum
limit and a boson-anyon mapping that is multi-valued but well-defined. The
precise form of $f(N_{j})$ depends on whether we allow or exclude three-body
(or more) coincidences on the lattice. Since the specific choice of $f(N_{j})$
results in differing continuum limits, the role of the three-body constraints
merits further investigation.

These two choices for $\Theta(N_{j})$ were inspired by analogy with anyons in
the continuum, and there are certainly other possible functions to consider.
Whether these alternate forms would have physically-meaningful exchange
statistics, locality properties, and continuum limits is an interesting
question. One can also imagine more general string operators that depend not
just on the total number of particles to the left $N_{j}$, but on their
specific occupation-number configuration. However, as noted before, in the
continuum limit the ordinality operator $N_{j}\to N(x)$ has a special
topological property: it is invariant under homeomorphisms of a line. More
generally, even indistinguishable particles can be given ‘natural’ labels on a
line based on their ordinality, unlike particles in higher dimensions. While
the notion of ordinality on a line may seem trivial, its topological origin
may explain why 1D quantum models support such a wealth of integrable and
statistical structures.

The chiral BF model as well the braid-anyon Hubbard model have recently been
realized experimentally with ultracold atoms [48, 19]. The construction of the
traid-anyon Hubbard model presented here constitutes an important step also
towards a first experimental realization of traid anyons in the laboratory. It
is an interesting task for future research to design experimentally accessible
schemes for the implementation of this (or related) traid-anyon models.

##  VII Acknowledgments

This research was funded by the Deutsche Forschungsgemeinschaft (DFG) via the
Research Unit FOR 2414 under project No. 277974659. Moreover, N.H.
acknowledges the support from the National Science Foundation under Grant No.
NSF PHY-1748958, B.W. from the ERC grant LATIS and the EOS project CHEQS, and
S.N. from the Provincia Autonoma di Trento, Q@TN, and the BMBF through the
project ’MAGICApp’. We thank Martin Bonkhoff and Philip Johnson for useful
discussions.

##  Appendix A Topological exchange statistics

Topological exchange statistics is an alternate approach to the Symmetrization
Postulate for treating indistinguishable particles in first quantization.
Instead of being imposed at the outset, the transformation properties of wave
functions under particle exchanges are derived from the topology of
configuration space. Topological exchange statistics can reproduce the results
of the Symmetrization Postulate, but can also provide additional anyonic
solutions in low-dimensional system with hard-core interactions.

The key observation that underlies this approach to exchange statistics is
that, when the classical configuration space ${\mathcal{Q}}$ for a physical
system is not simply-connected, i.e. not all loops can be continuously
deformed to a point, then in addition to single-valued wave functions on
${\mathcal{Q}}$, multi-valued wave functions may be necessary to describe the
full Hilbert space of functions of ${\mathcal{Q}}$. These multi-valued
functions can be formulated in several different frameworks [49, 50, 51, 52,
53].

The topological approach to exchange statistics was pioneered by researchers
in path integration methods [54, 55, 49]. In 1977, Leinaas and Myrheim [1]
first recognized the possibility for novel exchange statistics in low-
dimensional systems; Goldin et al. independently rediscovered this result a
few years later using a related approach based on current algebras [2]. The
topological approach to exchange statistics was extended to orbifold
configuration spaces in [11, 12].

The topological approach is also called the _intrinsic approach_ to exchange
statistics because it does not introduce arbitrary particle labels for
indistinguishable particles [56]. Whereas the Symmetrization Postulate
quantizes the distinguishable particle configuration space
${\mathcal{X}}\subseteq{\mathcal{M}}^{N}$, the intrinsic approach quantizes
the quotient space ${\mathcal{Q}}={\mathcal{X}}/S_{N}$. In the quotient map
from ${\mathcal{X}}$ to ${\mathcal{Q}}$, points of ${\mathcal{X}}$ that differ
by a permutation of labels are collapsed to same point in ${\mathcal{Q}}$.
Unless ${\mathcal{X}}$ excludes all two-body coincidences, the quotient space
${\mathcal{Q}}$ is not naturally a manifold and is better described as an
orbifold [56, 57, 11, 58, 12, 59]. An advantage of the intrinsic approach is
that, unlike a symmetry, true indistinguishability cannot be broken; the
intrinsic approach treats particle labels as a gauge structure imposed on
indistinguishable particles, not as a symmetry of configuration space [22,
60].

In the intrinsic approach, particle exchanges are concrete continuous
trajectories interchanging the particles, not abstract label permutations.
These exchange trajectories start and end at configurations that differ at
most by particle labels and so, in the indistinguishable configuration space
${\mathcal{Q}}$, they form based loops. Under continuous deformation,
equivalence classes of based loops form the fundamental (or first homotopy)
group $\pi_{1}$ on a manifold; for orbifolds, such as ${\mathcal{Q}}$, this
can be generalized to the orbifold fundamental group $\pi_{1}^{*}$ [61, 62].
Unitary irreducible representations of this group classify different
quantizations of the configuration space, or in the case of particle
exchanges, different superselection sectors.

When ${\mathcal{M}}$ is simply-connected (i.e., $\pi_{1}({\mathcal{M}})$ is
trivial) and there are no hard-core interactions, then
$\pi_{1}^{*}({\mathcal{Q}})=S_{N}$ for any $\dim{\mathcal{M}}$ and the group
of possible exchanges is equivalent to the group of particle permutations. In
this case, the Symmetrization Postulate holds and there are only bosons and
fermions with single-valued wave functions on ${\mathcal{X}}$ and standard
exchange statistics.

##  Appendix B Braid group and fractional exchange statistics

In 2D, hard-core two-body interactions create topological defects that make
the configuration space not simply-connected. The paths of particle exchanges
can wind around these defects, meaning that multiple _topologically
inequivalent_ exchanges may lead to the same permutation. The configuration
space for $N$ indistinguishable particles on a plane ${\mathbb{R}}^{2}$ is

${\mathcal{Q}}_{B}\equiv\frac{{\mathbb{R}}^{2N}-\Delta_{2}}{S_{N}},$ (54)

where $\Delta_{2}$ is the set of points in
${\mathcal{M}}^{N}={\mathbb{R}}^{2N}$ where at least two coordinates coincide.
The space ${\mathcal{Q}}_{B}$ is a manifold; all coincidences $\Delta_{2}$
have been removed and so ${\mathcal{Q}}_{B}$ contains no singular orbifold
points. In the simplest case of $N=2$, the intrinsic configuration space
${\mathcal{Q}}_{B}$ is the direct product of a plane ${\mathbb{R}}^{2}$ for
the center-of-mass coordinates and a cone with opening angle $\pi/3$ with an
excluded tip for the relative coordinates [1].

The resulting fundamental group for $N$ particles is the braid group
$\pi_{1}({\mathcal{Q}}_{B})=B_{N}$ [4]. As described in the main text,
fractional exchange statistics are given by the abelian representations of the
braid group (5) with phase $b_{i}\to\beta_{i}=\exp(i\theta_{i})$. This
exchange phase $\theta$ can be associated to the Aharonov-Bohm phase in the
composite charged particle/flux tube model of Wilczek [63] that gave anyons
their name. Fractional exchange statistics emerge for quasiparticle
excitations in certain quantum liquid and quantum spin liquid models like the
fractional quantum Hall effect [8] or the Haldane-Shastry spin-chain model
[29, 30]. In addition to motivating the braid-anyon-Hubbard model [14],
implementing fractional exchange statistics in one-dimensional systems
provided the original motivation for several continuum models, including the
recently experimentally-realized chiral BF model [44, 45, 46, 47, 48], the
closely-related Kundu model (also called Lieb-Liniger anyons) [38, 39, 40, 41,
42, 43], the Calogero-Sutherland anyon model [64, 22, 65], and hard-core anyon
models [66, 67, 68]. Note that we have only discussed abelian braid anyons;
non-abelian representations of $B_{N}$ are carried by multi-component, multi-
valued wave functions, c.f. [69, 70].

In some of this literature fractional exchange statistics, there exists an
alternate definition to (5) for fractional exchange statistics where the
transformation law for wave functions $\Psi({\bf x})$ defined on
${\mathcal{X}}$ instead of ${\mathcal{Q}}$. In the simplest case for two
particles, this alternate choice for the transformation has the form
$\Psi(x_{1},x_{2})=\exp(i\theta)\Psi(x_{2},x_{1})$, and this generalizes to
more particles (c.f. [38, 68]). Iterating this form of the transformation
again, we find $\Psi(x_{1},x_{2})=\exp(2i\theta)\Psi(x_{1},x_{2})$, showing
this wave function is indeed multi-valued as one expects for an anyonic wave
function. However, there is not a mechanism for implementing the inverse
transformation (i.e, an ‘under’ path with phase $\exp(-i\theta)$ to match the
‘over’ path). Effectively, this equates both $b_{i}$ and $b_{i}^{-1}$ with the
same permutation on ${\mathcal{X}}$, so this alternate transformation rule
does not describe abelian representations of the braid group.

Anyonic wave functions $\Psi({\bf q})$ can be lifted to functions on
${\mathcal{X}}$, but they will be multi-valued except for the special cases of
fermions $\theta=\pi$ and bosons $\theta=0$. Alternatively, one can work with
single-valued functions on ${\mathcal{X}}$ for arbitrary $\theta$ at the cost
of a gauge transformation. Famously, anyons in two-dimensions with fractional
exchange statistics can be transmuted into bosons or fermions with single-
valued wave functions by introducing a magnetic gauge potential [8, 6].

##  Appendix C Traid group and abelian traid anyon exchange statistics

In analogy to the braid group, hard-core three-body interactions in 1D also
introduce topological defects to configuration space that lead to a
generalization of the symmetric group. The relevant intrinsic configuration
space is

${\mathcal{Q}}_{T}=\frac{{\mathbb{R}}^{N}-\Delta_{3}}{S_{N}},$ (55)

where $\Delta_{3}$ is the locus of points in
${\mathcal{M}}^{N}={\mathbb{R}}^{N}$ where at least three particle coordinates
coincide. The orbifold fundamental group
$\pi_{1}^{*}({\mathcal{Q}}_{T})=T_{N}$ for $N\geq 3$ indistinguishable
particles is the traid group. Note that ${\mathcal{Q}}_{T}$ is an orbifold,
not a manifold, because not all orbifold points in $\Delta_{2}$ have been
excluded [11, 12].

The configuration space ${\mathcal{Q}}_{T}$ for the simplest case of $N=3$ is
represented in Fig. 15. In this depiction, it looks like a wedge $q_{1}\leq
q_{2}\leq q_{3}$ with $q_{i}\in{\mathbb{R}}$ with an opening angle of $\pi/3$.
Note that this picture is misleading at the orbifold points, as their non-
trivial topology cannot be represented in this embedding. The two half-plane
faces of the wedge correspond to configurations where the leftmost and
rightmost pairs of particles coincide. The line of intersection of these half-
planes is the excluded locus $\Delta_{3}$.

Figure 15: Subfigure (a) shows the the configuration space for three
indistinguishable particles on a line
${\mathcal{Q}}_{T}=({\mathbb{R}}^{2}-\Delta_{3})/S_{3}$. See text for
description; the coordinates for the sector are $q_{1}\leq q_{2}\leq q_{3}$
where $q_{i}$ is the the $i$th particle from the left. The yellow boundary
half-plane at the bottom is the orbifold locus $q_{1}=q_{2}$; the cyan
boundary half-plane at the top is the orbifold locus $q_{2}=q_{3}$. The black
diagonal is the three-body coincidence locus $\Delta_{3}$. Subfigure (b) shows
the _relative_ configuration space for ${\mathcal{Q}}_{T}$ along with two
representative exchange paths for loops $t_{1}$ that touch the orbifold locus
$q_{1}=q_{2}$ and loops $t_{2}$ that touch the orbifold locus $q_{2}=q_{3}$.
In the relative wedge, the angular coordinate $\varphi$ ranges from
$\varphi=0$ to $\varphi=\pi/3$.

For the case of $N=3$, the orbifold fundamental group is the traid group
$T_{3}$ generated by two pairwise exchanges $t_{1}$ and $t_{2}$. In Fig. 15,
these looks like paths that start at a base point in ${\mathcal{Q}}_{T}$ and
touch the orbifold singularities at $q_{1}=q_{2}$ and $q_{2}=q_{3}$,
respectively. Because $t_{i}^{2}=1$, a loop that touches the same orbifold
singularity twice can be contracted to a loop that does not touch the
boundary. Therefore, all loops are either trivial, corresponding to the
identity element of $T_{3}$, or are alternating sequences of $t_{1}$ and
$t_{2}$. Aficionados of discrete groups will recognize that $T_{3}$ is
isomorphic to the infinite dihedral group $D_{\infty}$, which has four one-
dimensional irreducible representations corresponding to the four we have
already described $(++)$, $(+-)$, $(-+)$, and $(--)$; c.f. Table. 1. There is
also a 1-parameter family of two-dimensional irreducible representations
characterized by the angle $\alpha$ between the two ‘reflections’ $t_{1}$ and
$t_{2}$; these can be parameterized as

$t_{1}\rightarrow\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),t_{2}\rightarrow\left(\begin{array}[]{cc}\cos\alpha&\sin\alpha\\\
\sin\alpha&-\cos\alpha\end{array}\right).$ (56)

The single two-dimensional representation of $S_{3}$ occurs as one of these
representations of $T_{3}\sim D_{\infty}$ corresponding to the specific angle
of $\alpha=2\pi/3$.

irrep | $e$ | $t_{1}$ | $t_{2}$ | statistics
---|---|---|---|---
$(++)$ | $1$ | $+1$ | $+1$ | bosons
$(--)$ | $1$ | $-1$ | $-1$ | fermions
$(+-)$ | $1$ | $+1$ | $-1$ | abelian traid anyon
$(-+)$ | $1$ | $-1$ | $+1$ | abelian traid anyon
$0<\alpha<\pi$ | $2$ | $0$ | $0$ | non-abelian traid anyons
Table 1: Character table for identity $e$ and generators $t_{1}$ and $t_{2}$
of $T_{3}\sim D_{\infty}$. The generators $t_{1}$ and $t_{2}$ describe loops
in ${\mathcal{Q}}_{3}$ that realize particle exchanges and their
representations describe possible topological exchange statistics.

Single-component wave functions $\Psi({\bf q})$ in the representation
$(\tau_{1}\tau_{2}\ldots\tau_{N-1})$ are defined on ${\bf
q}\in{\mathcal{Q}}_{T}$ and transform under an exchange path constructed from
$M$ pairwise exchanges $\gamma=t_{i_{M}}\cdots t_{i_{2}}t_{i_{1}}\in T_{N}$
like

$\hat{U}_{\gamma}\Psi({\bf
q})=(\tau_{i_{M}}\cdots\tau_{i_{2}}\tau_{i_{1}})\Psi({\bf q}).$ (57)

In the cases of bosons or fermions, this agrees with the transformation law
(2) and $\Psi({\bf q})$ can be lifted to a single-valued function on
${\mathcal{X}}_{T}={\mathbb{R}}^{N}-\Delta_{3}$. However, if not all
$\tau_{i}$ are the same, then the same permutation can be executed by
topologically distinguishable exchange paths. Lifting the wave function for
one of these abelian mixed representations to ${\mathcal{X}}_{T}$ so that
every exchange $\gamma\in T_{N}$ is represented correctly results in a multi-
valued function, as described in the main text.

##  Appendix D Haldane exclusion statistics

We have seen one type of the statistics of identical particles mentioned
above, namely the topological exchange statistics which is usually
characterized by their properties under exchange. Nevertheless, the fractional
exclusion statistics arises from the generalised Pauli principle, which is
valid in arbitrary space dimensions [30, 20]. As formulated by Haldane, the
generalization of Pauli principle implies that the dimension $d(N)$ of the
available states with $N$ identical particles satisfies

$d(N+\Delta N)=d(N)-g\Delta N,$ (58)

where $g$ is independent of particle number and is thus regarded as a
statistical parameter. For instance, $g=0$ corresponds to bosons as more added
bosons can occupy the same number of states $d(N+\Delta N)=d(N)$; $g=1$ gives
rise to fermionic statistics and a single added fermion can occupy any of the
$d(N+1)=d(N)-1$ available states due to Pauli exclusion. As $g$ may be
arbitrary in general, the above equation (58) represents a generalization of
Pauli principle. Fractional statistics $g=p/q=\theta/\pi$ indicates that the
creation of $q$ particles will reduce $p$ orbitals available for other
particles [30]. The abelian anyonic excitations of fractional quantum Hall
states are found to obey also fractional exclusion statistics. In 1D, the
Haldane-Shastry model obeys fractional exclusion statistics, but not
fractional exchange statistics [20, 71]. It would therefore be very
interesting to find also an example obeying both fractional exchange and
exclusion statistics in 1D.

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