# The Composite Particle Duality: A New Class of Topological Quantum Matter

Gerard Valentí-Rojas<EMAIL_ADDRESS>Joel Priestley  Patrik Öhberg  SUPA,
Institute of Photonics and Quantum Sciences, Heriot-Watt University,
Edinburgh, EH14 4AS, United Kingdom

###### Abstract

The composite particle duality extends the notions of both flux attachment and
statistical transmutation in spacetime dimensions beyond 2+1D. It constitutes
an exact correspondence that can be understood either as a theoretical
framework or as a dynamical physical mechanism. The immediate implication of
the duality is that an interacting quantum system in arbitrary dimensions can
experience a modification of its statistical properties if coupled to a
certain gauge field. In other words, commutation relations of quantum fields
can be effectively modified by a dynamical physical process. For instance, an
originally bosonic quantum fluid in $d$ spatial dimensions can feature
composite fermionic (or anyonic) excitations when coupled to a statistical
gauge field. We compute the explicit form of the aforementioned synthetic
gauge fields in $\text{D}\leq 3+1$. This opens the door to a new realm of
topological phases across dimensions both in lattice and continuum systems.

###### Contents

  1. I Introduction
  2. II Changing the Identity of Quantum Matter
    1. II.1 To Remove or Not To Remove… Gauge Fields
    2. II.2 Bose-Fermi is Bare-Composite
  3. III The Composite Particle Duality
    1. III.1 Generalised Statistical Transmutation by a Statistical Gauge Potential
    2. III.2 Generalised Flux Attachment by a Topological Gauge Field
    3. III.3 Discussion
    4. III.4 Jordan-Wigner Transformation and Duality
  4. IV A Recipe for Novel Topological Quantum Matter
    1. IV.1 In Continuum.
    2. IV.2 On the Lattice.
  5. V Closing Remarks
    1. V.1 Discussion and Extensions
  6. A Snippets of $d=1,2,3$
  7. B The Composite Particle Duality in a Nutshell
  8. C Extension to Relativistic Systems

##  I Introduction

We explore the underpinnings and implications of the recently discovered
composite particle duality [1], introduced as a universal framework for the
transmutation of statistics in interacting quantum matter in arbitrary
dimensions. This framework clarifies long-standing and obscure issues
regarding Bose-Fermi correspondences, provides a reinterpretation of well-
established results, enables the introduction of entire new classes of
topologically ordered phases of matter in several dimensions, puts some old
controversies to rest and, might even provide some understanding for the
dynamical generation of topological order in quantum matter. In this work, we
discuss in detail several of the statements made in Ref. [1].

We start by motivating the problem in Section II. We introduce the composite
particle duality in Section III after separately generalising flux attachment,
statistical transmutation and the Jordan-Wigner transformation. We examine in
detail the cases of one, two and three spatial dimensions. In light of this,
in Section IV we are able to introduce new classes of topological quantum
matter with a particularly simple composite dual description both in continuum
and on the lattice. We conclude this work in Section V by emphasising some of
the universal aspects of our proposed framework and highlighting the potential
for future work and extensions, especially for relativistic systems. For the
sake of clarity, we have included grey box environments, which summarise or
highlight key concepts; and blue box environments, in which examples are
worked out and comparison with well-established models is made. Spacetime
dimensionality is denoted as $D=d+1$ throughout this work.

##  II Changing the Identity of Quantum Matter

It was Leinaas and Myrheim [2] who challenged the symmetrisation postulate
approach in Quantum Mechanics and identified important loopholes in the
allowed quantum statistics of identical particles. In proving that there was
no need for a symmetrisation postulate in order to obtain Bose and Fermi
statistics for point particles in $d\geq 3$, they found that intermediate
statistics were possible in $d<3$. In addition, for spatial dimension greater
than one, they argued that a flat vector potential would give rise to a gauge
transformation that could alter, what they called, the “many-body nature” of
the system. Nowadays, we call this transformation statistical transmutation
[3]. The work of Leinaas and Myrheim was, in essence, an existence proof
allowing possibilities that had been overlooked since the very birth of
Quantum Mechanics. Their work, however, was framed in a seemingly academic
setup, with no reference to a physical mechanism, nor concrete examples of
such a transmutation. It was not until the advent of Chern-Simons theory [4,
5] and the discovery of the fractional quantum Hall effect (FQHE) [6, 7] that
this was given a physical meaning in the form of an emergent phenomenon in a
strongly-correlated system [8, 9, 10]. The low-energy excitations of a solid
could become anyons despite the fundamental constituents being electrons (i.e.
fermions). Parallel, and superficially independent developments with regards
to unusual behaviours of quantum statistics, appeared also in other contexts.
Theoretical work in high energy physics suggested that spin-statistics
properties of dyons could be altered [11, 12] in the presence of a
$\theta-$vacuum. On the other hand, works on the Luttinger liquid and the
development of bosonization techniques seemed to point out that there is a
direct non-local mapping between bosonic and fermionic fields [13, 14, 15, 16,
17], a Bose-Fermi correspondence. All the previous phenomena happen in
different spacetime dimensions and look disconnected from one another, should
it not be for the sole fact that there is statistical transmutation at play.

At the time of writing, it appears to be early days for a general theory and
universal understanding of fractionalisation, although some aspects of it,
like charge fragmentation, seem to be fairly cornered [18, 19, 20, 21, 22, 23,
24, 25, 26, 27]. For instance, there seems to be consensus in that Dirac
fermions coupled to topological defects often fractionalise fermion number.
This consensus is not present in the case of statistical transmutation, which
does not seem to be discussed much beyond Chern-Simons theory. We now
recognise statistical transmutation as a manifestation of fractionalisation of
statistics. Moreover, there seems to be enough evidence to infer that the
effective statistics of matter is not an immutable property, but can actually
be altered and maybe even controlled [28]. Therefore, we raise the question of
whether there is a physical mechanism encoded in a mathematical mapping
driving effective Bose-Fermi correspondences regardless of dimensionality. In
addition, how can one physically obtain anyons in different dimensions and
what do they look like? In a nutshell, how can bosons be transformed into
effective fermion-like fields, or even anyons, in any dimension? And, what is
the role of gauge fields and topology in the process?

Here, we approach these questions based on the success of flux attachment [29,
30] in explaining and providing a mechanism for statistical transmutation in
the FQHE. Both these concepts, together with the associated composite particle
picture [27, 31, 32], we argue, are notions that transcend the context in
which they were originally conceived. In fact, we introduce a composite
particle duality as an extended framework that includes the previous notions
and brings together several related topics such as bosonization, Jordan-Wigner
transformation, Bose-Fermi correspondences, topological solitons, disorder
operators and topological gauge theories.

Initial Hypothesis

Polyakov’s Bose-Fermi correspondence [3] in 2+1D can be understood as a
statistical transmutation performed by a Chern-Simons gauge field. We argue
that this is the 2+1D incarnation of a physical process that can happen in any
dimension.

###  II.1 To Remove or Not To Remove… Gauge Fields

A gauge transformation connects locally different, but equivalent, gauge
connections $A(x)$ and $A^{\prime}(x)$ giving rise to the same curvature or
field strength $F(x)$. In practice, this amounts to adding a total derivative
to the gauge field to find a gauge-transformed configuration without altering
the Physics. Certain configurations or gauge orbits are equivalent — or can be
deformed — to the identity. For an Abelian 1-form gauge field in $D=d+1$
spacetime dimensions, a conventional gauge transformation is given by

$A_{\mu}(x)\;\;\;\longrightarrow\;\;\;A_{\mu}^{\prime}(x)=\mathcal{U}(x)\,A_{\mu}(x)\,\mathcal{U}^{-1}(x)-\frac{i}{g}[\partial_{\mu}\,\mathcal{U}(x)]\;\mathcal{U}^{-1}(x)=A_{\mu}(x)+\frac{1}{g}\,\partial_{\mu}\xi(x)$
(1)

where $\mu=0,\dots,d\,$. We find the field strength
$F_{\mu\nu}(x)=\partial_{\mu}A_{\nu}(x)-\partial_{\nu}A_{\mu}(x)=F^{\prime}_{\mu\nu}(x)\,$
to be gauge-invariant. Furthermore, matter couples to gauge fields via a
gauge-covariant derivative, which transforms like

$D_{\mu}\Psi(x)\;\;\;\longrightarrow\;\;\;D^{\prime}_{\mu}\Psi^{\prime}(x)=\Big{[}\partial_{\mu}-igA_{\mu}^{\prime}(x)\Big{]}\,\Psi^{\prime}(x)=\mathcal{U}(x)\,\Big{[}\partial_{\mu}-igA_{\mu}(x)\Big{]}\,\Psi(x)$
(2)

where the matter field transforms like
$\Psi^{\prime}(x)=\mathcal{U}(x)\,\Psi(x)$, while for the conjugate field
$\bar{\Psi}^{\prime}(x)=\bar{\Psi}(x)\,\mathcal{U}^{\dagger}(x)$. Now, since
the gauge transformation is given by a unitary matrix
$\mathcal{U}^{-1}(x)=\mathcal{U}^{\dagger}(x)$, kinetic terms in the
associated Lagrangians and Hamiltonians are invariant to gauge
transformations. If the gauge field can be expressed in the form111Here
$\tilde{\mathcal{U}}$ is simply used to indicate that it is a particular case
of the general case $\mathcal{U}$. This is merely a stylistic point.

$A_{\mu}(x)=-i\,[\partial_{\mu}\,\mathcal{\tilde{U}}(x)]\;\mathcal{\tilde{U}}^{-1}(x)=\partial_{\mu}\phi(x)$
(3)

where $\phi(x)$ is some well-defined scalar field, $A_{\mu}$ is referred to as
a topologically trivial pure gauge configuration. The associated curvature is
$F_{\mu\nu}=0$, so it does not give rise to field strengths — e.g.
electromagnetic fields — and the connection is said to be flat. In addition,
it can be gauge transformed to $A_{\mu}^{\prime}(x)=0$. That is, it can be
removed or trivialised via a gauge transformation.

Imagine now that the field $\phi(x)$ is not well-defined everywhere. It, for
instance, can have a singularity at some spacetime point $x=x_{0}$, which we
signal by $\phi(x)\equiv\phi_{s}(x;x_{0})$. Then, the above mechanism still
holds everywhere except for precisely at $x_{0}$ as the gauge field and
transformation become ill-defined. Here, we say that the gauge field is
locally, but not globally, a pure gauge, or that the connection is non-
trivially flat. The gauge field becomes “topological” as a consequence of
these ill-defined points, lines, etc. being seeds for topological invariants,
i.e. countable objects to wrap around. They are defects that cannot be naïvely
lifted, so they provide obstructions to the removal of the gauge field. Fully
removing these gauge fields requires a large gauge transformation parametrised
by $\mathcal{W}(x)$, which does not connect physically equivalent states, but
configurations with different homotopic properties yielding the same
observable consequences. This means that matter fields transform like
$\Psi^{\prime}(x)=\mathcal{W}(x)\,\Psi(x)\;,$ but
$\mathcal{W}(x)\equiv\mathcal{W}(x;x_{0})=e^{i\phi_{s}(x;x_{0})}$ now carries
defects, as they are present in the scalar field $\phi(x)$. This has important
implications as these features are transferred to the transformed matter field
$\Psi^{\prime}(x)\equiv\Psi^{\prime}(x,x_{0})$, implying that such a gauge
transformation has the ability to add (or remove) defects. Hence, even though
we believe we remove topologically non-trivial pure gauge fields from a
system, this comes at a cost of altering the matter field, so it is not truly
removed but hidden. This redefinition of the matter field can also be thought
of as a dressing process. This type of transformation is recurrently found in
literature, not as a gauge transformation, but as a mere identity or mapping
of matter fields. In this context, $\mathcal{W}(x)$ might be referred to as a
disorder field [34]. For quantised theories, it is promoted to a disorder
operator. Depending on the specific type of defect it contains, it is
typically known as a soliton, kink, vortex or monopole operator, amongst
others. If $\Psi(x)$ is an order parameter or operator, we say that the
mapping has an order-disorder operator structure [35, 36, 37]. This is a
recurrent structure in Bose-Fermi dualities [27]. The reason is relatively
simple and rarely spelled out in literature. If we consider the equal-time
(anti)commutation relations $[\bullet,\bullet]_{\,\mp}$ of initially bosonic
(fermionic) matter fields, we can apply said transformation to obtain

$\displaystyle\big{[}\hat{\Psi}^{\prime}(x),\hat{\Psi}^{\prime}(x^{\prime})\big{]}_{\,\mp}=\big{[}\hat{\mathcal{W}}(x)\,\hat{\Psi}(x),\hat{\mathcal{W}}(x^{\prime})\,\hat{\Psi}(x^{\prime})\big{]}_{\,\mp}=\hat{\mathcal{W}}(x)\,\big{[}\hat{\Psi}(x),\hat{\mathcal{W}}(x^{\prime})\big{]}_{\,-}\,\hat{\Psi}(x^{\prime})$
(4)
$\displaystyle+\hat{\mathcal{W}}(x)\hat{\mathcal{W}}(x^{\prime})\,\big{[}\hat{\Psi}(x),\hat{\Psi}(x^{\prime})\big{]}_{\,-}+\big{[}\hat{\mathcal{W}}(x),\hat{\mathcal{W}}(x^{\prime})\big{]}_{\,-}\,\hat{\Psi}(x^{\prime})\hat{\Psi}(x)+\hat{\mathcal{W}}(x^{\prime})\,\big{[}\hat{\mathcal{W}}(x),\hat{\Psi}(x^{\prime})\big{]}_{\,\mp}\hat{\Psi}(x)$
(5)

and

$\displaystyle\big{[}\hat{\Psi}^{\prime}(x),\hat{\bar{\Psi}}^{\prime}(x^{\prime})\big{]}_{\,\mp}=\big{[}\hat{\mathcal{W}}(x)\,\hat{\Psi}(x),\hat{\bar{\Psi}}(x^{\prime})\,\hat{\mathcal{W}}^{\dagger}(x^{\prime})\big{]}_{\,\mp}=\hat{\mathcal{W}}(x)\,\big{[}\hat{\Psi}(x),\hat{\bar{\Psi}}(x^{\prime})\big{]}_{\,-}\,\hat{\mathcal{W}}^{\dagger}(x^{\prime})$
(6)
$\displaystyle+\hat{\mathcal{W}}(x)\hat{\bar{\Psi}}(x^{\prime})\,\big{[}\hat{\Psi}(x),\hat{\mathcal{W}}^{\dagger}(x^{\prime})\big{]}_{\,-}+\big{[}\hat{\mathcal{W}}(x),\hat{\bar{\Psi}}(x^{\prime})\big{]}_{\,-}\,\hat{\mathcal{W}}^{\dagger}(x^{\prime})\hat{\Psi}(x)+\hat{\bar{\Psi}}(x^{\prime})\,\big{[}\hat{\mathcal{W}}(x),\hat{\mathcal{W}}^{\dagger}(x^{\prime})\big{]}_{\,\mp}\hat{\Psi}(x)\;.$
(7)

Therefore, we observe that the statistics of the original fields are
transformed and depend on the statistics of the disorder operator. We also
notice that these reduce to the usual relations when
$\hat{\mathcal{W}}(x)=\hat{\mathbb{I}}$. If the disorder operators trivially
commute with themselves and with the matter fields, the original commutation
relations are preserved up to re-scaling of the matter fields. It is now no
mystery how to get a fermionic $\hat{\Psi}^{\prime}$ theory out of a bosonic
$\hat{\Psi}$ one (or vice versa), a suitable $\hat{\mathcal{W}}(x)$ will do
the job222We emphasise once again that such a transformation can be defined
without any reference to it being a gauge transformation, but merely a local
canonical transformation.. Note that we have not assumed anything about the
dimensionality of our system, nor even a type of model.

The previous argument on gauge transformation will allow us to design a
physical mechanism out of the mathematical mapping that we just witnessed. The
disorder operator will become a large gauge transformation that will introduce
or remove a minimally-coupled non-trivial pure gauge field. This idea will be
formalised and extended under the name of the composite particle duality [1].

###  II.2 Bose-Fermi is Bare-Composite

Inspired by our knowledge on disorder operators, we notice that certain
canonical transformations can alter the statistical properties of quantum
fields. In particular, these transformations can be large gauge
transformations but, for now, we consider them as some mathematical mapping
between fields. Let us compute a particular example for two particles in first
quantised formulation. This amounts to finding a transformation
$\mathcal{W}(x_{1},x_{2})=\exp{[i\phi(x_{1},x_{2})]}$, so that the transformed
two-body wave function reads

$\Psi^{\prime}(x_{1},x_{2})=e^{i\phi(x_{1},x_{2})}\Psi(x_{1},x_{2})=\pm
e^{i\phi(x_{1},x_{2})}\Psi(x_{2},x_{1})\;.$ (8)

Now, upon exchange of particle position labels we might consider some
properties for the scalar field; for instance, symmetry under exchange
$\phi(x_{1},x_{2})=\phi(x_{2},x_{1})$. The previous expression then becomes

$\Psi^{\prime}(x_{1},x_{2})=\pm
e^{i\phi(x_{2},x_{1})}\Psi(x_{2},x_{1})=\pm\Psi^{\prime}(x_{2},x_{1})\;,$ (9)

so we recover the original symmetry under exchange of particles. Nevertheless,
we could have considered a slightly more exotic property for the scalar field
such as $\phi(x_{1},x_{2})=\alpha+\phi(x_{2},x_{1})$ with $\alpha\in[0,2\pi)$.
We then see that Eq. (8) becomes

$\Psi^{\prime}(x_{1},x_{2})=\pm
e^{i\,[\alpha+\phi(x_{2},x_{1})]}\Psi(x_{2},x_{1})=\pm
e^{i\alpha}\Psi^{\prime}(x_{2},x_{1})\;,$ (10)

where the transformed field picks up an additional phase $\alpha$ with respect
to the original symmetry properties under exchange of the two-body
wavefunction. In particular, for $\alpha=\pi$ this reverses the statistical
properties, so an originally bosonic two-body state transforms into a two-body
fermionic state and vice versa. More profoundly, a bare field transforms into
a composite one with altered — e.g. anyonic — statistics. A subclass of the
possible values that the transformed or composite field can acquire are the
standard bosonic and fermionic ones.

An Illuminating Realisation

The initial question of going from Bose to Fermi, is truly a question of going
from Bare to Composite. But

A composite field is a dressed bare field.

For consistency, we might want to check the exchange properties when
$x_{1}=x_{2}$ such that, under exchange of labels, the field returns the
identity. For a property of the form
$\phi(x_{1},x_{2})=\alpha+\phi(x_{2},x_{1})$ when $x_{1}=x_{2}$, the
associated phase factor is the same if $\alpha=2\pi N$ for $N\in\mathbb{Z}$.
More generally, $\phi$ can be ill-defined at this point while retaining the
desired structure elsewhere, i.e. when $x_{1}\neq x_{2}$. This implies a
delicate treatment should be taken in order to determine whether or not there
is a Pauli exclusion principle. Examples of fields satisfying these conditions
are, once again, topological defects with singularities at $x_{1}=x_{2}$. For
instance, in 2d, a non-trivial function can be a vortex
$\phi(x_{1},x_{2})=\text{arg}(\mathbf{x}_{1}-\mathbf{x}_{2})$. This
corresponds to an Aharonov-Bohm-type vector potential, which is not surprising
taking into account it gives rise to flux attachment and statistical
transmutation. In the present context we see that this just comes in the
explicit functional form of a scalar field $\phi(x)$, while the rest of the
discussion is rather general. At this point, we believe to be on firm ground
to motivate a framework that brings the above ideas together.

##  III The Composite Particle Duality

The previous discussion illustrates how some exact equivalences can
effectively modify the identity of charged particles, i.e. transmute them. We
now argue that these transformations come naturally from the coupling to some
type of gauge fields with topological content. Naïvely, there is nothing
preventing such a notion from being general333Here we are not debating whether
they transmute to anyons or only to fermions/bosons. The specific statistical
nature of the composite fields will depend on dimensionality and locality, but
the ability to go from bare to composite will not., regardless of
dimensionality, the original statistics of the system, and whether it is found
on the lattice or in continuum. This is, moving from a mathematical
equivalence to a dynamical physical process taking place in a quantum system
of many particles.

Here, we postulate that an interacting many-body system can self-generate,
amongst others, effective statistical gauge fields. These gauge fields can,
not only couple to matter, but also bind to it and, in doing so, “dress it” so
that it appears to change its intrinsic properties. This should be no
different than an interaction-driven topological many-body Berry phase effect
[28]. In the following, we provide evidence portraying how such a physical
mechanism can work. For that, it becomes necessary to disentangle the notions
of statistical transmutation and flux attachment, which will be generalised
independently to be then connected and interpreted in a broader framework. The
former is the ability for fields to dynamically change their statistics, while
the latter is the mechanism by which matter nucleates gauge flux.

###  III.1 Generalised Statistical Transmutation by a Statistical Gauge
Potential

We understand statistical transmutation [3] as the ability to dynamically
change the properties under exchange or, equivalently, the (anti)commutation
relations of quantum matter fields. A well-known example appears in the
context of FQHE, where it is stated as a theorem [40] for the particular case
of a Chern-Simons vector potential in 2+1D.

In the following, we show444This is a Physicist’s proof and unlikely to
satisfy a Mathematician’s standards by any means. statistical transmutation as
an equivalence in arbitrary dimensions for non-interacting non-relativistic
systems and then extend the proof to interacting ones. This is formulated in
first quantised formalism to be later translated to second-quantised language.
Thus, we make the following claim: Claim

There exists an exact equivalence between theories B (Bare) and C (Composite)
sharing the same eigenvalues; but, while theory B satisfies bosonic/fermionic
factors under exchange of particle labels, theory C satisfies generalised
exchange relations. Therefore, B and C give rise to the same Physics and have
different statistical properties.

In layman’s terms, a system of locally interacting charged particles in the
presence of a gauge field is equivalent to a different system of charged
particles where no gauge field is present, but whose many-body wavefunction
has changed along with some of its properties. We work with $\hbar=c=q=1$ and
consider theory B in $d$ spatial dimensions as

$\mathcal{H}_{\text{\,B}}\,\Psi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=E\,\Psi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\;\;\;\;\;\;\;\;\;\;\;\;\text{with}\;\;\;\;\;\;\mathcal{H}_{\text{\,B}}=\sum_{i=1}^{N}\frac{\big{[}\mathbf{p}\,(\mathbf{x}_{i})-\bm{a}\,(\mathbf{x}_{i})\big{]}^{\,2}}{2m}\;,$
(11)

where
$\bm{a}(\mathbf{x}_{i})\equiv\bm{a}(\mathbf{x}_{i})\,[\mathbf{x}_{1},\dots,\mathbf{x}_{N}]$
is an Abelian gauge potential, whose form is to be specified, such that a
particle located at point $\mathbf{x}_{i}=(x^{1}_{i},\dots,x^{d}_{i})$
experiences the presence of the rest of particles. Consequently, its value can
depend on the positions of all $N$ particles in the system, and therefore is a
many-body gauge field. The wavefunction satisfies

$\Psi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{N})=\pm\,\Psi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{N})$
(12)

under the exchange of two particles, where the sign depends on the
bosonic/symmetric ($+$) or fermionic/antisymmetric ($-$) nature of the
wavefunction. We claim the existence of another theory C, equivalent to the
previous one, of the form

$\tilde{\mathcal{H}}_{\text{\,C}}\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=E\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\;\;\;\;\;\;\;\;\;\;\;\;\text{with}\;\;\;\;\;\;\tilde{\mathcal{H}}_{\text{\,C}}=\sum_{i=1}^{N}\frac{\tilde{\mathbf{p}}\,(\mathbf{x}_{i})^{\,2}}{2m}$
(13)

where the wavefunction satisfies

$\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{N})=\pm\,e^{\,i\gamma}\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{N})\;,$
(14)

and where $\gamma$ is a statistical factor (or angle). We shall prove that it
is possible to obtain $\tilde{\mathcal{H}}_{C}$ from $\mathcal{H}_{B}$, and
that such a transformation alters the exchange properties of the many-body
wavefunction, giving rise to the $\gamma$ factor. This is known as statistical
transmutation. Let us then introduce a duality (gauge) transformation of the
form

$\Psi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=\mathcal{W}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\;\;\;\;\;\;\;\;\;\;\;\;\text{with}\;\;\;\;\;\;\mathcal{W}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=e^{\,i\alpha\,\Phi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})}$
(15)

where we keep $\Phi$ unspecified and $\alpha\in S^{1}\,$. Substituting in Eq.
(11) and using
$\mathbf{p}\,(\mathbf{x}_{i})=-i\,\bm{\nabla}_{\mathbf{x}_{i}}$, we are left
with

$\mathcal{H}_{\text{\,B}}\,\Psi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=\frac{1}{2m}\,\Big{[}\sum_{i=1}^{N}\big{(}-i\,\bm{\nabla}_{\mathbf{x}_{i}}-\bm{a}\,(\mathbf{x}_{i})\big{)}^{2}\Big{]}\cdot\Big{[}\mathcal{W}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\Big{]}\;.$
(16)

Using the notation $\bm{a}_{\mathbf{x}_{i}}\equiv\bm{a}\,(\mathbf{x}_{i})$,
$\mathcal{W}$, $\Phi$ and $\Psi_{c}$, as well as the property
$\bm{\nabla}_{\mathbf{x}_{i}}\mathcal{W}=i\alpha\,\mathcal{W}\,\bm{\nabla}_{\mathbf{x}_{i}}\Phi\,$,
we obtain

$\frac{\mathcal{W}}{2m}\,\bigg{\\{}\sum_{i=1}^{N}\Big{[}-\bm{\nabla}_{\mathbf{x}_{i}}^{2}\Psi_{c}+2i\,\Big{(}\,\bm{a}_{\mathbf{x}_{i}}-\alpha\bm{\nabla}_{\mathbf{x}_{i}}\Phi\Big{)}\,\bm{\nabla}_{\mathbf{x}_{i}}\Psi_{c}+\Big{(}(\bm{a}_{\mathbf{x}_{i}}-\alpha\,\bm{\nabla}_{\mathbf{x}_{i}}\Phi)^{2}-i\alpha\bm{\nabla}_{\mathbf{x}_{i}}^{2}\Phi+i\,\bm{\nabla}_{\mathbf{x}_{i}}\bm{a}_{\mathbf{x}_{i}}\Big{)}\,\Psi_{c}\Big{]}\bigg{\\}}\;.$
(17)

We observe that the second ($\sim\nabla_{\mathbf{x}_{i}}\Psi_{c}$ ) and third
($\sim\Psi_{c}$ ) terms in brackets have to vanish in order to recover

$\frac{\mathcal{W}}{2m}\sum_{i=1}^{N}\Big{[}-\nabla_{\mathbf{x}_{i}}^{2}\Psi_{c}\Big{]}=\mathcal{W}\,\tilde{\mathcal{H}}_{C}\Psi_{c}=E\,\Psi\;,\;\;\;\;\;\;\;\;\;\;\;\text{and
thus}\;\;\;\;\;\;\tilde{\mathcal{H}}_{\text{\,C}}\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=E\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\;.$
(18)

Therefore, solving the above constraints as differential equations for
$\bm{a}$ and $\Phi$ allows this mapping to be true. However, we observe a
cancellation of both terms for the constraint

$\bm{a}_{\mathbf{x}_{i}}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=\alpha\,\bm{\nabla}_{\mathbf{x}_{i}}\Phi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\;.$
(19)

This tells us that the vector potential has the form of a pure gauge and
proves that this mapping works regardless of the value of $\Phi$. Recall both
the vector potential and the transformation have been kept as free parameters.
This concludes the first part of the proof, which neither contains
approximations, nor does it depend on fixing an a priori dimensionality of the
system. It is solely the statement that pure gauges can be removed by an
appropriate gauge transformation which, in this case, is given by
$\mathcal{W}$.

It remains then to be shown that Eq. (14) is satisfied upon substitution of
the duality transformation. In essence, that in the process of removing the
gauge potential, the new many-body wave function acquires exotic exchange
properties. For that, we will define
$\mathcal{W}^{-1}=\exp\,(-i\alpha\,\Phi^{*})$ and use properties
$\mathcal{W}^{-1}\mathcal{W}=1$, $\mathcal{W}^{-1}=\mathcal{W^{*}}$ and
$\Phi=\Phi^{*}$, to find

$\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{N})=\pm\,e^{\,-i\alpha\,\Phi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{N})}e^{\,i\alpha\,\Phi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{N})}\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{N})\;.$
(20)

Let us assume for now that the duality transformation can be decomposed in a
sum of two-body functions in the form
$\Phi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=\sum_{m<l}\phi\,(\mathbf{x}_{m},\mathbf{x}_{l})\,$.
This allows writing

$\mathcal{W}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=\prod_{m<l}e^{i\alpha\,\phi\,(\mathbf{x}_{m},\mathbf{x}_{l})}\equiv\prod_{m<l}w\,(\mathbf{x}_{m},\mathbf{x}_{l})\equiv\prod_{m<l}w_{ml}\;.$
(21)

Note that if both $m$ and $l$ are different from $i$ or $j$, the phase factors
cancel out and we are left with

$\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{N})=\pm\,e^{-\,i\alpha\,\big{(}\sum_{m<l}^{{}^{\prime}}\phi_{ml}\,-\,\sum_{p<q}^{{}^{\prime}}\phi_{pq}\big{)}}\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{j},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{N})\;,$
(22)

where we use the notation
$\phi_{ml}\equiv\phi\,(\mathbf{x}_{m},\mathbf{x}_{l})$. The primes (′) denote
that sums are taken over $l$ and $q$ indices respectively that are equal to
$i$ or $j$. Also, for every $\phi_{ml}$ there is a $\phi_{pq}=\phi_{lm}\,$.
Furthermore, we see that the decomposition in terms of two-body function is a
sufficient but not a necessary condition, i.e. it is chosen for simplicity. A
similar logic would apply if we considered a more general three-body
decomposition like
$\Phi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=\sum_{m<l<k}\phi\,(\mathbf{x}_{m},\mathbf{x}_{l},\mathbf{x}_{k})$
or $n$-body decomposition for $n<N$, so that
$\Phi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=\sum_{m<l<k<\dots<n}\phi\,(\mathbf{x}_{m},\mathbf{x}_{l},\mathbf{x}_{k},\dots,\mathbf{x}_{n})$
, but the treatment rapidly becomes tedious.

Exchange Example

It is illuminating to discuss a concrete example that illustrates these
points. We imagine a system of $N=6$ particles in $d$ spatial dimensions. We
now consider the exchange of particle at position $\mathbf{X}_{2}$ and
particle at position $\mathbf{X}_{5}$, hence $(i\leftrightarrow j)$ becomes
$(2\leftrightarrow 5)$ for this example. More formally,

$\Psi_{c}\,(\mathbf{x}_{1},\mathbf{X}_{2},\mathbf{x}_{3},\mathbf{x}_{4},\mathbf{X}_{5},\mathbf{x}_{6})=\pm\,\Big{(}\prod_{m<l}w^{-1}_{ml}\Big{)}\Big{(}\prod_{p<q}w_{pq}\Big{)}\,\Psi_{c}\,(\mathbf{x}_{1},\mathbf{X}_{5},\mathbf{x}_{3},\mathbf{x}_{4},\mathbf{X}_{2},\mathbf{x}_{6})\;.$
(23)

We can explicitly write

$\displaystyle\prod_{m<l=1}^{N}w^{-1}_{ml}=$
$\displaystyle\big{(}w^{-1}_{12}\cdot w^{-1}_{13}\cdot w^{-1}_{14}\cdot
w^{-1}_{15}\cdot w^{-1}_{16}\big{)}\cdot\big{(}w^{-1}_{23}\cdot
w^{-1}_{24}\cdot w^{-1}_{25}\cdot w^{-1}_{26}\big{)}$ (24)
$\displaystyle\cdot\big{(}w^{-1}_{34}\cdot w^{-1}_{35}\cdot
w^{-1}_{36}\big{)}\cdot\big{(}w^{-1}_{45}\cdot w^{-1}_{46}\big{)}\cdot
w^{-1}_{56}$ (25)

and

$\displaystyle\prod_{p<q=1}^{N}w_{pq}=$ $\displaystyle\big{(}w_{15}\cdot
w_{13}\cdot w_{14}\cdot w_{12}\cdot w_{16}\big{)}\cdot\big{(}w_{53}\cdot
w_{54}\cdot w_{52}\cdot w_{56}\big{)}$ (26)
$\displaystyle\cdot\big{(}w_{34}\cdot w_{32}\cdot
w_{36}\big{)}\cdot\big{(}w_{42}\cdot w_{46}\big{)}\cdot w_{26}\;,$ (27)

where we have simply applied the exchange $(2\leftrightarrow 5)$. We see all
the pairs of the form $w^{-1}_{ab}\cdot w_{ab}=1\,$ but $w^{-1}_{ab}\cdot
w_{ba}\neq 1\,$, so Eq. (23) becomes

$\displaystyle\Psi_{c}\,(\mathbf{x}_{1},\mathbf{X}_{2},\mathbf{x}_{3},\mathbf{x}_{4},\mathbf{X}_{5},\mathbf{x}_{6})=$
$\displaystyle\pm\,\big{(}w^{-1}_{23}\cdot
w_{32}\big{)}\cdot\big{(}w^{-1}_{24}\cdot
w_{42}\big{)}\cdot\big{(}w^{-1}_{25}\cdot w_{52}\big{)}$ (28)
$\displaystyle\cdot\big{(}w^{-1}_{35}\cdot
w_{53}\big{)}\cdot\big{(}w^{-1}_{45}\cdot
w_{54}\big{)}\,\Psi_{c}\,(\mathbf{x}_{1},\mathbf{X}_{5},\mathbf{x}_{3},\mathbf{x}_{4},\mathbf{X}_{2},\mathbf{x}_{6})\;.$
(29)

Hence, the change in statistics reduces to computing $w^{-1}_{ab}\cdot w_{ba}$
for the “string” of particle labels, the ends of which are the positions of
the exchanged particles $(2\leftrightarrow 5)$. At this point we can already
identify theories B and C as equivalent provided the pure gauge constraint
[Eq.(19)] is satisfied, and where we identify the statistical factor as

$\gamma=-\alpha\,\Big{(}\sum_{m<l}^{\prime}\phi_{ml}\,-\,\sum_{p<q}^{\prime}\phi_{pq}\Big{)}$
(30)

with $\phi_{ml}$ being real-valued functions yet to be specified. We can now
consider further properties of $\phi_{ml}$ — or, equivalently $w_{ml}$ — in
order to find explicit values for $\gamma$. Natural choices are:

  * •

Symmetric exchange. We consider even parity upon exchange of labels
$\phi_{ml}=\phi_{lm}\,$. This choice yields $\gamma_{s}=0$, and it corresponds
to a trivial removal of a pure gauge. Statistics are not altered.

  * •

Antisymmetric exchange. We consider odd parity upon exchange of labels
$\phi_{ml}=-\phi_{lm}\,$. Here,
$\gamma_{a}\equiv\gamma_{a}(\mathbf{x}_{m},\mathbf{x}_{l})=-2\alpha\sum_{m<l}^{\prime}\phi_{ml}\,$.
There is explicit dependence on the pure gauge function.

  * •

Linear exchange. We assume a property under exchange of the form
$\phi_{ml}=\beta_{1}+\beta_{2}\,\phi_{lm}$ where $\beta_{1},\beta_{2}\in
S^{1}$. The particular case of $\beta_{2}=+1$ yields
$\gamma_{l}=\alpha\beta_{1}\in S^{1}$, meaning that boson-fermion (fermion-
boson) transmutation is achieved for the particular case
$\alpha\beta_{1}=\pi+2\pi k$ with $k\in\mathbb{Z}$. More generally,
$\alpha\beta_{1}$ define an anyonic-type exchange, with the caveat that
additional constraints to these values can come from further considerations of
physical consistency, e.g. a conventional spin-statistics relation for point-
like objects in $d=3$.

In all the cases above, the choice is not unique, as we are only specifying
the exchange properties of $\phi_{ml}$, but not their explicit functional
form. This implies that the route to statistical transmutation from bare to
composite is also non-unique. In other words, there are several vector
potentials that give rise to the same statistical factor. This can be
understood as gauge freedom and such a family of vector potentials forms a
statistical gauge orbit.

As a closing remark of this section, we have shown that an Abelian vector
potential can be removed in a non-interacting system in $d$ spatial dimensions
at the expense of changing the two-particle exchange relations of the many-
body wavefunction, thus connecting two superficially different theories. It is
sufficient for the total vector potential to be

$\bm{a}\,(\mathbf{x})=\sum_{i=1}^{N}\bm{a}(\mathbf{x}_{i})=\alpha\,\sum_{i=1}^{N}\bm{\nabla}_{\mathbf{x}_{i}}\,\Big{[}\sum_{a<b=1}^{N}\phi\,(\mathbf{x}_{a},\mathbf{x}_{b})\Big{]}\;,$
(31)

to perform statistical transmutation. It is therefore called a statistical
gauge potential. The statistical properties of the composite theory manifestly
depend on the exchange properties of the many-body pure gauge function
$\phi_{ab}\,$. The interested reader is referred to Appendix A for a
discussion of practical cases. Later on we will see how this picture is
preserved also for interacting systems.

###  III.2 Generalised Flux Attachment by a Topological Gauge Field

We define flux attachment in 2+1D as a local constraint or Gauss’s law between
magnetic field and charge (or number) density
$\bm{\nabla}\times\hat{\bm{a}}\,(t,\mathbf{x})=\gamma\,\hat{n}\,(t,\mathbf{x})$.
We can also define it globally over a closed surface $\Sigma$ as a
proportionality relation between magnetic flux and electric charge (or number
of particles) contained in that region as
$\hat{\Phi}_{b}^{\Sigma}=\gamma\,\hat{Q}_{\Sigma}$. It is evident that flux
attachment is to be expected in 2n+1D for $n\in\mathbb{Z}^{+}$, from the mere
existence of a Chern-Simons term in odd spacetime dimensions. However, the
global formulation of the constraint suggests that it might be universal
across dimensions $d\geq 2$, since magnetic flux and charges can be defined in
all generality. By this token, having bare electric charges that act as a
source of magnetic flux is something that transcends dimensions. We can also
wonder about the $d=1$ case. In essence, what is left of flux attachment when
a magnetic field cannot even be defined? Can there be statistical
transmutation?

The success of Chern-Simons theory has taught us that when such a topological
field theory minimally couples to matter — i.e. a conserved Noether current
$\mathcal{J}^{\mu}(x)$ —, flux attachment emerges in the form of equations of
motion. Solving them for the associated Chern-Simons gauge field gives us its
functional form, in a given gauge, as a function of matter. This gauge field
acts as a statistical gauge field begetting the transmutation of statistics.
Hence, flux attachment not only generates electric-magnetic composites through
Gauss’s law, but also fixes the form of the statistical gauge field. In other
words, it provides a physical mechanism that naturally manufactures the
unknown gauge field that we need to insert in the previous section and that,
up until now, had to be “guessed”. Our point of departure to extend flux
attachment to arbitrary dimensions is precisely the Abelian Chern-Simons
theory. In covariant form, the flux attachment constraint is manifestly a
topological current of the form
$\mathcal{J}^{\mu}(t,\mathbf{x})\propto\epsilon^{\,\mu\nu\lambda}\,\partial_{\nu}a_{\lambda}(t,\mathbf{x})$.
It is not difficult to formulate a $p$-form extension of this topological
current law, namely

$\star\mathcal{J}^{\,(p)}(x)=\mathcal{F}^{(D-p)}(x)=\text{d}\mathcal{A}^{(D-p-1)}(x)\;.$
(32)

This is a Hodge duality mapping $p-$forms in D dimensions to
$(\text{D}-p)-$forms through the Hodge star operator $(\star)\,$. We can check
that, in $D=3$ and for a $p=1$ gauge form, this precisely coincides with the
Chern-Simons result. Expression (32) can be found rather systematically by
assuming the existence of a conserved Noether current satisfying
$\star\,\text{d}\mathcal{J}=0$. Poincaré’s lemma guarantees the existence of a
form $\mathcal{A}$ that solves this conservation equation yielding (32) as
both the solution and the definition of a topological current. Furthermore it
ensures the existence of solutions upon transformations of the form

$\mathcal{A}^{(p)}(x)\rightarrow\mathcal{A}^{(p)}(x)+\text{d}\xi^{(p-1)}(x)$
(33)

which we now interpret as proof of gauge redundancy, and Eq.(33) as a higher
form of antisymmetric gauge transformation. We can write the
$(\text{D}-2)-$form field as $\mathcal{A}=\text{d}\alpha$ at the expense of
violating the corresponding Bianchi identity $\text{d}(\text{d}\alpha)\neq 0$
due to non-trivial topology of $\alpha\,$. This motivates the introduction of
$\mathcal{A}(x)$ as a $p$-form topological gauge connection or, more loosely,
a topological gauge field. We claim that Eq. (32) constitutes the
$D$-dimensional extension of conventional flux attachment. Now, we might
insist that the Noether current remains a $D$-dimensional Lorentz vector,
which constrains $\mathcal{A}$ to be a $(D-2)$ gauge-dependent form. In
covariant field theoretical language, and introducing a coupling constant
$\gamma$, the expression reads Generalised Flux Attachment

$\mathcal{J}^{\mu}(t,\mathbf{x})=\gamma\,\epsilon^{\,\mu\nu\lambda\alpha\beta\dots}\,\partial_{\nu}\mathcal{A}_{\lambda\alpha\beta\dots}(t,\mathbf{x})\;,$
(34)

where $\mathcal{A}\,$ is a rank-$(\text{D}-2)$ antisymmetric $\text{U}(1)$
gauge field that locally transforms like

$\mathcal{A}_{\mu\nu\lambda\alpha\beta\dots}(t,\mathbf{x})\longrightarrow\mathcal{A}_{\mu\nu\lambda\alpha\beta\dots}(t,\mathbf{x})+\partial_{\,[\mu}\,\xi_{\,\nu\lambda\alpha\beta\dots]}(t,\mathbf{x})\;\;,$
(35)

and $\gamma$ is a real constant whose value may or may not be constrained
through Dirac quantisation. We observe that the central object is not the
Chern-Simons gauge field, but in general, an antisymmetric tensorial
topological gauge field. The rank $p$ of the topological gauge field is tied
to the dimensionality $D$ of the problem. This means that:

  * •

In $D=1+1$ the topological gauge field is a $0$-form, i.e. a scalar
$\mathcal{A}^{(0)}=\phi(t,\mathbf{x})$.

  * •

In $D=2+1$ it is a $1$-form, i.e. a vector — a.k.a. a Chern-Simons gauge field
— of the form $\mathcal{A}^{(1)}_{\mu}=A_{\mu}(t,\mathbf{x})$.

  * •

In $D=3+1$ it becomes a $2$-gauge-form, i.e. an antisymmetric rank-2 tensor —
a.k.a. a Kalb-Ramond gauge field — of the form
$\mathcal{A}^{(2)}_{\mu\nu}=B_{\mu\nu}(t,\mathbf{x})$.

  * •

In $D=d+1$ for $d\geq 4$, it is an antisymmetric rank-$p$ tensor
$\mathcal{A}^{(p)}_{\mu_{1}\dots\mu_{p}}=C_{\mu_{1}\dots\mu_{p}}(t,\mathbf{x})$.

This is a remarkably general result that includes the 1+1D case, for which we
initially had no expectations. Interestingly, the topological current
expression reduces to the familiar Abelian bosonisation current formula, for
which no gauge redundancy is present. Conversely, we can think of the general
$D$ expression as a higher-dimensional Abelian bosonisation, also known as
functional bosonisation [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Now, we
understand it also describes generalised flux attachment.

The statistical ($a_{\mu}$) and the topological
($\mathcal{A}_{\mu\nu\lambda\dots}$) gauge fields are rather different
objects. Conventional charged matter couples to electromagnetic fields through
minimal coupling to a vector potential
$\mathbf{p}\rightarrow\mathbf{p}-q\bm{a}$ or, in covariant form, as a source
term $\mathcal{J}^{\mu}a_{\mu}\,$. This motivates the assertion that the
statistical gauge field is a physical gauge field that will couple to matter.
It is the relevant degree of freedom Nature and experiments will care about.
As opposed to this, the topological gauge field will be auxiliary or internal.
This implies it will transfer the structural information of flux attachment to
the statistical gauge field and disappear in the process. In a way, it is
integrated out or used to enforce a local constraint. We aim for this to be
enough to obtain an explicit flux attachment expression in different
dimensions for the statistical gauge field, which allows us to compute its
functional form. Once again, we take inspiration from Chern-Simons theory and
aim to find a parent topological gauge action that gives us the topological
current in Eq.(34) as an equation of motion when the statistical gauge field
is minimally coupled to matter. We demand the following:

  1. 1.

It has to exist in arbitrary spacetime dimension $D$.

  2. 2.

The gauge action has to be topological, local, with a gauge-invariant
partition function.

  3. 3.

It has to be low order in derivatives in order to dominate the effective long-
wavelength Physics of quantum many-body systems.

  4. 4.

It has to contain both the statistical and topological gauge fields.

  5. 5.

It has to yield two sets of equations of motion:

    1. (a)

A current bosonisation relation.

    2. (b)

A functional relation between the topological gauge and statistical gauge
fields.

In addition, further inspection of the problem suggests that:

  1. 1.

The action is likely to be non-unique.

  2. 2.

The terms should imprint torsion or helicity — e.g. a twist — onto the gauge
connection and not merely curvature.

  3. 3.

The action is likely to be related to Chern-Simons terms and $\theta$-terms in
odd and even spacetime dimensions respectively.

  4. 4.

The action is likely to break Parity ($\mathcal{P}$) and time-reversal
($\mathcal{T}$) symmetries.

With all the above, we postulate the action $S=\int
d^{D}x\;\big{\\{}\,\mathcal{L}_{\text{top.}}[a,\mathcal{A}]-\mathcal{J}^{\mu}a_{\mu}\,\big{\\}}$
with

$\mathcal{L}_{\text{top.}}[a,\mathcal{A}]=\frac{\kappa}{2\pi}\,\epsilon^{\,\mu\nu\lambda\dots\alpha\beta}\mathcal{A}_{\mu\nu\lambda\dots}\partial_{\alpha}a_{\beta}+\mathcal{L}\,[\mathcal{A}]$
(36)

where we have used $\gamma=\kappa/(2\pi)\,$. The first term is a Background
Field (BF) term that ensures flux attachment when extremising the action for
the statistical gauge field. The term $\mathcal{L}\,[\mathcal{A}]$, on the
other hand, gives topological dynamics to the topological gauge field. We,
however, have not found its expression in all generality as it is changing
across dimensions. Nevertheless, we restrict to writing its form in spatial
dimensions $d=1$, $d=2$, and $d=3$. Namely

$\displaystyle
1+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{L}_{\text{top.}}[a_{\mu},\Phi]=\frac{\kappa}{2\pi}\,\epsilon^{\,\mu\nu}\Phi\partial_{\mu}a_{\nu}+\frac{\kappa}{4\pi}\,\epsilon^{\,01}\partial_{0}\Phi\,\partial_{1}\Phi\;,$
(37) $\displaystyle
2+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{L}_{\text{top.}}[a_{\mu},A_{\mu}]=\frac{\kappa}{2\pi}\,\epsilon^{\,\mu\nu\lambda}A_{\mu}\partial_{\nu}a_{\lambda}+\frac{\kappa}{4\pi}\,\epsilon^{\,\mu\nu\lambda}A_{\mu}\partial_{\nu}A_{\lambda}\;,$
(38) $\displaystyle
3+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{L}_{\text{top.}}[a_{\mu},B_{\mu\nu}]=\frac{\kappa}{2\pi}\,\epsilon^{\,\mu\nu\lambda\alpha}B_{\mu\nu}\partial_{\lambda}a_{\alpha}+\frac{\kappa}{4\pi}\,\epsilon^{\,\mu\nu\lambda\alpha}B_{\mu\nu}B_{\lambda\alpha}\;.$
(39)

whose Euler-Lagrange equations of motion can be computed for both gauge
fields, leading to the double identities

$\displaystyle
1+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{J}^{\,\mu}_{\,\text{1+1D}}=\frac{\kappa}{2\pi}\epsilon^{\,\mu\nu}\partial_{\nu}\Phi=\frac{\kappa}{2\pi}\epsilon^{\,\mu\nu}a_{\nu}\;,$
(40) $\displaystyle
2+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{J}^{\,\mu}_{\,\text{2+1D}}=\frac{\kappa}{2\pi}\epsilon^{\,\mu\nu\lambda}\partial_{\nu}A_{\lambda}=\frac{\kappa}{2\pi}\epsilon^{\,\mu\nu\lambda}\partial_{\nu}a_{\lambda}\;,$
(41) $\displaystyle
3+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{J}^{\,\mu}_{\,\text{3+1D}}=\frac{\kappa}{2\pi}\epsilon^{\,\mu\nu\lambda\alpha}\partial_{\nu}B_{\lambda\alpha}=\frac{\kappa}{2\pi}\epsilon^{\,\mu\nu\lambda\alpha}\partial_{\nu}\partial_{\lambda}a_{\alpha}\,,$
(42)

where the conserved Noether matter current
$\partial_{\mu}\,\mathcal{J}^{\mu}(x)=0$ becomes a topological current defined
in terms of the statistical gauge field. These are the covariant expressions
for generalised flux attachment or, alternatively, the functional forms for a
$\text{U}(1)$ statistical gauge field in $d\leq 3$. We can write down these
results in vector notation. We find the corresponding flux attachment Gauss’s
laws

$\displaystyle
1+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;a_{x}\,(t,x)=\frac{2\pi}{\kappa}\,n\,(t,x)\;,$
(43) $\displaystyle
2+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;b\,(t,\mathbf{x})=\frac{2\pi}{\kappa}\,n\,(t,\mathbf{x})\;,$
(44) $\displaystyle
3+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\bm{\nabla}\cdot\bm{b}\,(t,\mathbf{x})=\frac{2\pi}{\kappa}\,n\,(t,\mathbf{x})\;,$
(45)

and quantum Hall-like responses

$\displaystyle
1+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;J_{x}\,(t,x)=-\frac{\kappa}{2\pi}\,a_{t}\,(t,x)\;,$
(46) $\displaystyle
2+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathbf{J}\,(t,\mathbf{x})=-\frac{\kappa}{2\pi}\,\hat{\mathbf{e}}_{z}\times\bm{\varepsilon}\,(t,\mathbf{x})\;,$
(47) $\displaystyle
3+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathbf{J}\,(t,\mathbf{x})=-\frac{\kappa}{2\pi}\,\big{[}\partial_{t}\bm{b}\,(t,\mathbf{x})+\bm{\nabla}\times\bm{\varepsilon}\,(t,\mathbf{x})\big{]}\;,$
(48)

where $\bm{b}=b^{i}=\epsilon^{\,ijk}\partial_{j}a_{k}$ and
$\bm{\varepsilon}=\varepsilon^{i}=f^{0i}=\partial^{0}a^{i}-\partial^{i}a^{0}$
correspond to the “magnetic” and “electric” fields, respectively. For clarity
of notation, we have also used $\mathcal{J}^{\mu}=(n,\mathbf{J})$,
$\partial_{i}=\bm{\nabla}$, $\partial_{t}=\partial_{0}$, and $a_{0}=a_{t}\,$.

It is worth noting that we can integrate out the topological gauge field and
obtain the effective action in terms of the physical statistical gauge field
only $S_{\text{eff}}=\int
d^{D}x\;\big{\\{}\,\mathcal{L}_{\text{eff}}[a]-\mathcal{J}^{\mu}a_{\mu}\,\big{\\}}$
with effective topological Lagrangian density

$\displaystyle
1+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{L}_{\text{eff}}[a_{\mu}]=\frac{\kappa}{2\pi}\,\epsilon^{\,\mu\nu}\bar{a}\,\partial_{\mu}a_{\nu}+\frac{\kappa}{4\pi}\,\epsilon^{\,01}a_{0}\,a_{1}=\frac{\kappa}{2\pi}\,\bar{a}\tilde{f}+\frac{\kappa}{4\pi}\,\epsilon^{\,01}\partial_{0}\bar{a}\,\partial_{1}\bar{a}\;,$
(49) $\displaystyle
2+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{L}_{\text{eff}}[a_{\mu}]=\frac{\kappa}{4\pi}\,\epsilon^{\,\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}=\frac{\kappa}{8\pi}\,\epsilon^{\,\mu\nu\lambda}a_{\mu}f_{\nu\lambda}\;,$
(50) $\displaystyle
3+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\mathcal{L}_{\text{eff}}[a_{\mu}]=\frac{\kappa}{16\pi}\,\epsilon^{\,\mu\nu\lambda\alpha}\partial_{\mu}a_{\nu}\partial_{\lambda}a_{\alpha}=\frac{\kappa}{32\pi}\,f_{\mu\nu}\tilde{f}^{\mu\nu}\;,$
(51)

where we have defined the dual field strengths in 3+1D as
$\tilde{f}^{\mu\nu}=\frac{1}{2}\,\epsilon^{\,\mu\nu\lambda\alpha}f_{\lambda\alpha}$
and in 1+1D as $\tilde{f}=\epsilon^{\,\mu\nu}\partial_{\mu}a_{\nu}$.
Furthermore, we introduce the scalar field

$\bar{a}\,(x)=\epsilon_{\alpha\beta}\int
d^{2}x^{\prime}\,\partial^{\,\alpha}_{x}G(x-x^{\prime})\,a^{\,\beta}(x^{\prime})\;.$
(52)

Hence, we naturally recover the familiar Chern-Simons term for 2+1D and the
$\theta$-term for 3+1D. We are inclined to think that this pattern will repeat
itself giving rise to higher-dimensional versions of these terms for odd
($2n+1D$) and even ($2nD$) spacetime dimensions respectively. Notice this is
rather poetic, as it constitutes a dimensional cascade of topological terms,
provided that the $\theta$-term evaluates to a total derivative of a Chern-
Simons term at the boundary. This also suggestive of a cascade of Callan-
Harvey anomaly inflow mechanisms.

(a) Kink

(b) Vortex

(c) Monopole

Figure 1: Profiles for statistical field topological defects. Blue (red)
colouring at the bottom row denotes high (low) intensity of the vector field.
Observe a jump discontinuity at the origin for the kink, a singular point for
the vortex and a singular Dirac string on the positive $z-$axis for the
monopole.

From the above, we might infer the rough form of a statistical gauge field in
general dimensions. It should look like

$a^{\,\mu}(t,\mathbf{x})\sim\frac{1}{\kappa}\,\epsilon^{\,\mu\nu\lambda\dots\alpha}\int
d^{d}\mathbf{x}^{\prime}\;\bm{\nabla}_{\mathbf{x}}\,\omega_{\nu\lambda\dots}(\mathbf{x}-\mathbf{x}^{\prime})\,\mathcal{J}_{\alpha}(t,\mathbf{x}^{\prime})\;,$
(53)

where the $\omega_{\mu\nu\dots}(\mathbf{x})$ is a topological — and
potentially multivalued — kernel containing non-smooth points and changing
with dimension. This might be compactly expressed as a non-trivial pure gauge
of the form
$a_{\mu}(t,\mathbf{x})=\frac{1}{\kappa}\partial_{\mu}\,\Omega(t,\mathbf{x})$.
It is non-trivial provided $\Omega(t,\mathbf{x})$ is a topological defect (see
Figure 1). We can compare this expression with the constraint (19) we obtained
in first-quantised non-covariant formulation. Furthermore, Eq. (53) tells us
that the statistical vector potential in general dimensions is a density-
dependent — or covariant-current-dependent — gauge field. Hence, it becomes
trivial $a_{\mu}=\text{const.}$ in absence of matter. Notwithstanding, the
dependence on matter is, with counted exceptions, non-local. This implies
that, even though we might evaluate the field in a region of null covariant
current $\mathcal{J}_{\mu}(t,\mathbf{x})\approx 0$, the statistical gauge
field at that same point is $a_{\mu}(t,\mathbf{x})\napprox 0$ as it depends on
the integrated value of the covariant current throughout the whole spatial
manifold we are considering, i.e.
$\forall\mathbf{x}^{\prime}\in\text{System}$. This is not unusual as it is
analogous to retarded potentials or Jefimenko’s equations in conventional
electrodynamics.

###  III.3 Discussion

We have come a long and abstract way to end up with relatively compact
expressions (see Summary Table 1). We have generalised both flux attachment
and statistical transmutation to arbitrary dimensions. The overall picture is
that conventional quantum matter can become statistically anomalous or exotic
if coupled to an Abelian gauge field. The particular form of this gauge field
changes with dimension, but the physical mechanism is universal. Let us
briefly comment in more detail on the results we just found.

Spatial Dimension | $d=1$ | $d=2$ | $d=3$
---|---|---|---
Flux Attachment | $a_{x}\propto n$ | $b\propto n$ | $\bm{\nabla}\cdot\bm{b}\propto n$
Current Response | $-\,J_{x}\propto a_{t}$ | $-\,\mathbf{J}\propto\hat{\mathbf{e}}_{z}\times\bm{\varepsilon}$ | $-\,\mathbf{J}\propto\partial_{t}\bm{b}+\bm{\nabla}\times\bm{\varepsilon}$
Topological Quantisation | No | Yes | Yes
Topological Soliton | Kink | Vortex | Monopole
Topological Gauge Field | Chiral Scalar | Chern-Simons | Kalb-Ramond
Table 1: Main features of the electromagnetic response for matter coupled to
statistical gauge fields.

In 2+1D we recover the familiar results of conventional flux attachment in the
long-wavelength limit of FQHE [9, 40]. We now see it in new light as we
explicitly introduced a topological gauge field, thus making the parent action
a sum of a mixed and a pure Chern-Simons terms. In the FQHE context, the pure
Chern-Simons term is found for the statistical gauge field $a_{\mu}$ — as
opposed to Eq. (38) — , which is considered as an internal degree of freedom
to be eliminated in favour of the true electromagnetic gauge potential
$A_{\mu}$, which comes in as an initially background field. Upon elimination,
the Chern-Simons level or, equivalently the filling fraction, is fractional.
We instead consider the statistical gauge field as a physical field and
consider $A_{\mu}$ as a topological gauge field. Flux attachment happens for
the topological gauge field and it is then effectively transferred to the
statistical gauge field. The Chern-Simons level becomes fractional as in the
conventional formulation. This slight modification of the conventional
effective field theory for FQHE in 2+1D is our control case, while the other
results in other dimensions are new555We notice that it is in itself
remarkable that there exist results in even spacetime dimensions. It is often
mentioned in literature that it only makes sense to talk about fractional
quantum Hall effect in 2d, 4d, 6d and so on. We challenge that view..

The 3+1D case immediately catches the eye as Eq. (45) is recognised as the
magnetic or Dirac’s monopole Gauss’s law. Only… not quite! In extending
electrodynamics to include the existence of hypothetical magnetic monopoles as
sources of the magnetic field, we would write
$\bm{\nabla}\cdot\mathbf{B}(t,\mathbf{x})\propto\tilde{\rho}_{m}(t,\mathbf{x})$.
We say then that the (hypothetic) magnetic charge density $\tilde{\rho}_{m}$
acts as a source of magnetic field. In Eq. (45), however, $n\,(t,\mathbf{x})$
is the “electric” — as opposed to magnetic — charge density. In 2+1D that same
“electric” charge density is proportional to the magnetic field, but in 3+1D
is a true source of magnetic field. This means that matter satisfying flux
attachment in 3+1D is locally both electric and magnetically charged. Thus, we
say that a gauge-matter composite is formed. These electric-magnetic
composites have historically been called dyons [54, 55, 56, 12] and have
relied on the existence of magnetic monopoles. In fact, Witten [57] found that
a magnetic monopole in the presence of a $\theta$-term can bind quantised
electric charge. This is known as the Witten effect. Hence, a magnetic
monopole found in a medium with $\theta\neq 0$ is, effectively, a dyon. Here
we see the reciprocal picture of that, and hence, not relying on the existence
of magnetic monopoles. We observe from Eq.(45) that 3+1D flux attachment is an
inverse (or reciprocal) Witten effect in which a charged particle is endowed
with magnetic charge when $\theta\neq 0$. To distinguish the newly found
composites from the historically fundamental ones, we might call these
emergent or dynamically-generated dyons. Emergent dyons are, thus, the 3d
analogue of Laughlin quasiparticles (or quasiholes). Their effective magnetic
charge will be quantised through conventional Dirac quantisation arguments
working in reverse. Integrating Eq. (45) over a 3d spatial volume $\Sigma$
with closed boundary $\partial\Sigma$, we find

$Q_{\Sigma}(t)=\int_{\Sigma}d^{3}\mathbf{x}\;n\,(t,\mathbf{x})=\frac{\kappa}{2\pi}\int_{\Sigma}d^{3}\mathbf{x}\;\bm{\nabla}\cdot\bm{b}\,(t,\mathbf{x})=\frac{\kappa}{2\pi}\int_{\partial\Sigma}d\mathbf{S}\cdot\bm{b}\,(t,\mathbf{x})\equiv\frac{\kappa}{2\pi}\,\Phi_{b}^{\Sigma}(t)\;,$
(54)

which explicitly corresponds to the global expression for flux attachment.
Further evidence, of the dyonic behaviour comes from Eq. (48) which plays the
role of a 3d Hall current, but it also exactly corresponds to Faraday’s law of
induction in the presence of a magnetic current. This electro-magnetic
response is also related in spirit to the usual Topological Magnetoelectric
Effect (TME) discussed in axionic topological insulators but explicitly
differs in the expressions [58].

Emergent Electro-Magnetic Hadrons

Dyons were conceived by Julian Schwinger as an alternative to quarks [54]. He
proposed that Dirac charge quantisation condition imposed by the existence of
magnetic monopoles could give rise to bound-states of dyons, namely hadrons,
which were magnetically neutral, but had the usual $\frac{1}{3}e$ quantised
electric charges. Our framework allows us to conceive a new and emergent view
on “hadrons” in terms of dyons understood, not as Schwinger’s fundamental
subatomic entities, but as quasiparticles in, for instance, a 3d quantum
fluid. This means a bound-state of two or more emergent dyonic composites,
i.e. a composite of composites. These would hypothetically be, in addition to
composite bosons/fermions, also electrically-charged (but magnetically
neutral) — essentially “electric hadrons” —, magnetically-charged (but
electrically neutral) — essentially “magnetic hadrons” —, or a mixture of
both, meaning “electro-magnetic hadrons”.

The 1+1D case is, if anything, the most exotic of the above. No Chern-Simons
term can be defined in a strict sense. In fact, neither can the magnetic
field, as there is no sense of the curl of a vector potential in space, so
there is no notion of a magnetic flux unless one considers an annular
geometry, i.e. periodic boundary-conditions. Yet, Eq. (43) appears as the
remnant of flux attachment in 1+1D. This is, despite the non-existence of a
magnetic field, a statistical gauge potential can still be defined. In this
case the vector potential becomes not only local, but also linear in density.
All the apparent simplicity in the form of the gauge field turns into
complexity of a parent action containing the analogue of a dynamical $\theta$
or axion term endowed with chiral dynamics. This appears as a consequence of
our construction. One can see that in 1+1D a similar argument to the 3+1D case
does not work, as upon integrating out the topological gauge field we do not
recover a 1+1D version of a $\theta$-term, but Eq. (49). In fact, a mere
$\theta$-vacuum in 1+1D is just a background electric field666The axion and BF
terms coincide in 1+1D. If not subsidised with further dynamics, the axion
field decouples from matter and does not play other role than a constant
background., there is nothing dynamical about it, so it cannot possibly give
rise to flux attachment.

Some further general remarks are in order. While the present formulation of
generalised statistical transmutation is valid for non-relativistic systems,
the generalisation of flux attachment in covariant form [Eq. (34)] does not
assume any particular matter field. This means that such a formulation is
valid for general fermionic/bosonic bare fields regardless of whether they are
relativistic or not. In fact, it holds for any covariantly conserved Abelian
current. It is likely that the arguments for generalised statistical
transmutation can be extended also to fully relativistic matter field
theories.

With regards to the proposed parent gauge theories, we find Eqs. (37) to (39)
to be in close relation, if not coincident, with Dijkgraaf-Witten-type
topological field theories. This unexpected connection deserves further study
and will be a subject for future work. Quantisation of the gauge theories is
not discussed nor pursued. The reduction of the gauge action to a mere
constraint for the gauge field fully determined by matter is used as a proxy
to demand that eventual quantisation of the gauge field must be consistent
with that of matter.

Another aspect of this extended framework is the possibility of fractional
charges. We find that the global flux-charge relation guarantees that in the
presence of a quantised magnetic flux, electric charge within the same region
will be fractionally quantised by Dirac’s argument in $d\geq 2$. We recall
that flux is quantised if the region encloses topological defects, and whose
presence is ensured by our construction. In $d=1$ there is no sense of
magnetic flux, so the above argument does not hold unless an exception is
found if we externally change the topology of the system by considering it in
a $S^{1}$ manifold, i.e. using periodic boundaries in space. Then, the
previous arguments are restored since the notion of magnetic flux can be
defined on a circle.

Figure 2: (Left) Retraction argument for a point particle in three-dimensional
parameter space. (Right) Obstruction to the retraction map in an analogous
scenario when the particles are extended in space. The string may be a line of
singularities akin of a Dirac string.

We have shown that statistical transmutation is possible in any dimension,
however, we have deliberately prevented ourselves from specifying whether they
implied the existence of anyons. In $d\leq 2\,$, anyons are allowed according
to the common physical lore. Whereas in $d=3\,$, the spin-statistics relation
prevents point-like objects from acquiring any statistics that are neither
Bose nor Fermi. This is not necessarily true any more if some of the
postulates leading to such stringent result are relaxed. In particular, if
particles are now extended — for instance string-like — objects. The rough
topological argument is that for a point-like singular object at
$\mathbf{x}_{0}$ in a 2d plane one cannot retract a closed line contour
enclosing the defect to a point. In 3d, one can make use of the third spatial
dimension to avoid the obstruction, and thus retract to a point. If the defect
is now not point-like but string-like (see Figure 2), retraction is not
possible anymore in 3d, but only in 4d and higher. From this heuristic
argument, we see the anyons considered in this view will necessarily be
extended in higher dimensions. Bose and Fermi statistics and their
transmutation are, however, allowed in arbitrary $d$.

###  III.4 Jordan-Wigner Transformation and Duality

We have witnessed the construction of statistical gauge fields that can be
written as topologically non-trivial pure gauges. These gauge fields both
couple and bind to matter. Upon performing large gauge transformations,
statistical gauge fields can be removed at the expense of changing the
statistical nature of matter fields. The transformation is a duality that
physically connects the bare fields to the homotopically distinct, but
equivalent, composite fields. This operation reads as

$\Psi_{\,C}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=e^{\,-i\alpha\,\sum_{m<l}\phi\,(\mathbf{x}_{m},\mathbf{x}_{l})}\,\Psi_{\,B}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\;,$
(55)

where $\phi$ is a disorder scalar field with non-trivial exchange of pairs of
labels, i.e. it is topological. In a sense, such an expression is nothing but
the $d$-dimensional generalisation of the classic Jordan-Wigner
transformation, expressed in continuum and first-quantised language. In
second-quantised formalism, we can obtain an equivalent expression for it,
namely

$\hat{\Psi}_{\,C}\,(t,\mathbf{x})=e^{-\frac{i}{\hbar}\hat{\Phi}(t,\mathbf{x})}\,\hat{\Psi}_{\,B}\,(t,\mathbf{x})\;,$
(56)

where $\hat{\Psi}_{\,B}$ and $\hat{\Psi}_{\,C}$ are, respectively, the bare
and composite matter field operators, and

$\hat{\Phi}(t,\mathbf{x})=\alpha\int
d^{d}\mathbf{x}^{\prime}\;\omega\,(\mathbf{x}-\mathbf{x}^{\prime})\;\hat{n}\,(t,\mathbf{x}^{\prime})$
(57)

is a real scalar field that sums the contributions of the number density at
all positions, weighted by a topological kernel. This reduces to a generalised
Jordan-Wigner string object for $d=1$ and a Jordan-Wigner “membrane” for
$d=2$. Borrowing terminology from string theory, we might call the general
case for Eq. (57) a $d$-dimensional Jordan-Wigner brane (or JW $d$-brane for
short). We can now define a disorder operator
$\hat{\mathcal{W}}(t,\mathbf{x})=\exp{[\frac{i}{\hbar}\hat{\Phi}(t,\mathbf{x})]}$
as the object performing the transformation. This operator creates (removes)
topological defects777These defects can, in principle, live freely in the
system. Here, we consider the case in which they couple and bind to matter. of
codimension $d$ in an, otherwise ordered, configuration when applied to a
state. Once again, these defects are a kink in $d=1$, a vortex in $d=2$, a
monopole in $d=3$, or a general topological soliton in arbitrary $d$. This
familiar object, that we see appear often in some embodiment in duality
contexts, plays now a clear role. It “gauge dresses” the bare matter field and
creates a topologically non-trivial composite. We see that this also brings
new meaning to the usual Jordan-Wigner transformation, as we know now it is a
large gauge transformation connecting bare and composite fields. The
conventional Jordan-Wigner transformation is nothing but the remnant of flux
attachment and statistical transmutation in 1+1D. Conversely, the Jordan-
Wigner transformation in any given dimension is a Bose-Fermi correspondence
understood as a particular instance of the composite particle duality, and
thus, intimately related to statistical gauge fields and topological field
theory.

Coming back to Eq. (56), we can call $\hat{\Psi}_{B}$ an order operator as an
extension of the notion of an order parameter, typically defined as the
expectation value of the relevant quantum field. Then, Eq. (56) reads
symbolically as a recipe: Composite Operator = Disorder Operator $\times$
Order Operator . This order-disorder structure has been highlighted in
literature since the early works of Kadanoff [34], and has continued to appear
recursively [11, 35, 37, 61, 62, 36, 27, 63, 64, 65]. We support the view of
several authors of the order-disorder structure being the seed of Bose-Fermi
dualities. We see that this is not by chance, but the manifestation of flux
attachment and statistical transmutation in higher degree of generality.
Understood as a recipe it is tempting to conclude that this order-disorder
structure is extremely robust, likely beyond our limited study here.

Beyond the implications already stated, the transformation in Eq. (56) has
several peculiarities. Upon evaluation of conjugate matter fields in arbitrary
dimensions, we find

$\hat{\Psi}^{\dagger}_{\,C}(t,\mathbf{x}^{\prime})\hat{\Psi}_{\,C}(t,\mathbf{x})=\hat{\Psi}^{\dagger}_{\,B}(t,\mathbf{x}^{\prime})\,\hat{\mathcal{W}}(t,\mathbf{x}^{\prime})\,\hat{\mathcal{W}}^{\dagger}(t,\mathbf{x})\,\hat{\Psi}_{\,B}(t,\mathbf{x})\;.$
(58)

We observe that for $\mathbf{x}=\mathbf{x}^{\prime}\,$, disorder operators
evaluate to the identity
$\hat{\mathcal{W}}^{\dagger}(\mathbf{x})\hat{\mathcal{W}}(\mathbf{x})=\hat{\mathbb{I}}\,$,
i.e. opposite defects annihilate. The immediate implication is that the
density operator
$\hat{n}\,(t,\mathbf{x})=\hat{\Psi}_{\,B}^{\dagger}(t,\mathbf{x})\,\hat{\Psi}_{\,B}(t,\mathbf{x})=\hat{\Psi}_{\,C}^{\dagger}(t,\mathbf{x})\,\hat{\Psi}_{\,C}(t,\mathbf{x})\,$
is preserved by the transformation. Intuitively, as it happens in conventional
Jordan-Wigner transformations, the strings cancel each other at the same
position. Here, we see that the the JW $d$-branes must cancel each other by
construction when evaluated at the same position. This means that any operator
that can be expressed as local powers of the density operator will be
invariant under the mapping. This is important as it includes certain local
interactions at the level of the field theoretical Lagrangian or Hamiltonian.
Thus, our generalisation of flux attachment, which was proved by systems free
of interparticle interaction, holds at the exact same level of rigour if
interactions are of the form

$\hat{\mathcal{H}}_{\,\text{int}}=g_{0}(t,\mathbf{x})+g_{1}(t,\mathbf{x})\,\hat{n}(t,\mathbf{x})+g_{2}(t,\mathbf{x})\,\hat{n}(t,\mathbf{x})\hat{n}(t,\mathbf{x})+\,\dots\,+g_{n}(t,\mathbf{x})\,\big{[}\hat{n}(t,\mathbf{x})\big{]}^{n}\;.$
(59)

The cancellation of JW $d$-branes also implies that the statistical properties
of the composite field reduce to those of the bare field at
$\mathbf{x}=\mathbf{x}^{\prime}$. In other words, similar to what happens with
conventional anyons or Jordan-Wigner transformed fields, the composite field
will have a hard or soft core depending on whether the bare field is a fermion
or a boson. Hence, the composite fields can be hard-core or soft-core bosons,
fermions or anyons. For $\mathbf{x}\neq\mathbf{x}^{\prime}\,$, the product
$\hat{\mathcal{W}}^{\dagger}(\mathbf{x}^{\prime})\hat{\mathcal{W}}(\mathbf{x})\neq\hat{\mathbb{I}}$
is crucially not equal to the identity. The mapping still holds, but the
system is no longer described locally as there are strings connecting these
defects, which are spatially separated. The fact that these disorder operators
have non-trivial commutation is precisely what allows for the transmutation of
statistics.

##  IV A Recipe for Novel Topological Quantum Matter

Armed with our knowledge of the composite particle duality we can now
introduce models of quantum matter that become topologically ordered by
construction upon coupling to a statistical gauge field. These models,
interacting and highly nonlinear in the bare basis, have a particularly simple
and local composite dual description, which allows for exact or simplified
solutions. This family of models serves as an illustration and proof-of-
principle, meaning it is not limited to those explicitly discussed. Some
previously known examples of topologically ordered matter may appear as
particular instances of our framework. They can, in principle, exist in $d$
dimensions, although limited explicit computation of the required gauge field
restricts our discussion to $d<4\,$.

###  IV.1 In Continuum.

It is time to get us back to the study of quantum matter. We have seen in
previous chapters that a fractional quantum Hall fluid is a rather exotic
topological quantum liquid. On the other hand, we regard as conventional
quantum matter those systems that follow the usual Landau symmetry breaking
paradigm. For non-relativistic $D$-dimensional systems, a Lagrangian density
of the form

$\mathcal{L}=i\hbar\,\hat{\Psi}^{\dagger}(x)\partial_{t}\hat{\Psi}(x)-\frac{\hbar^{2}}{2m}\bm{\nabla}\hat{\Psi}^{\dagger}(x)\cdot\bm{\nabla}\hat{\Psi}(x)-V(x)\,\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)-U(x)\,\hat{\Psi}^{\dagger}(x)\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)\hat{\Psi}(x)$
(60)

with notation $x\equiv(t,\mathbf{x})$ and satisfying (anti)commutation
relations

$\displaystyle\big{[}\hat{\Psi}(x),\hat{\Psi}(x^{\prime})\big{]}_{\,\mp}=\big{[}\hat{\Psi}^{\dagger}(x),\hat{\Psi}^{\dagger}(x^{\prime})\big{]}_{\,\mp}=0\;$
(61)
$\displaystyle\big{[}\hat{\Psi}(x),\hat{\Psi}^{\dagger}(x^{\prime})\big{]}_{\,\mp}=\delta^{(D)}(x-x^{\prime})\;,$
(62)

describes a conventional Bose or Fermi quantum fluid interacting locally in
spacetime. Matter fields in this theory interact pairwise via a short-range
potential of the form $U(x,x^{\prime})=U(x)\,\delta^{(D)}(x-x^{\prime})$.
Three-body or higher-body interactions are neglected for simplicity. Such a
model describes not one, but a family of systems, approximate incarnations of
which can be found in atomic condensates or electron fluids with screened
interactions. This is clearly conventional quantum matter. Our study of the
composite particle duality suggests that the mere coupling to a statistical
gauge field, drastically alters the system and automatically turns it into a
topological quantum fluid, as in topologically ordered or “beyond Landau”.
This new family of models will be described by the Lagrangian density

$\displaystyle\mathcal{L}$
$\displaystyle=i\hbar\,\hat{\Psi}^{\dagger}(x)\,\Big{(}\partial_{t}-\frac{i}{\hbar}\,\hat{a}_{0}(x)\Big{)}\,\hat{\Psi}(x)-\frac{\hbar^{2}}{2m}\,\Big{[}\Big{(}\bm{\nabla}-\frac{i}{\hbar}\,\hat{\bm{a}}(x)\Big{)}\,\hat{\Psi}(x)\Big{]}^{\dagger}\cdot\Big{(}\bm{\nabla}-\frac{i}{\hbar}\,\hat{\bm{a}}(x)\Big{)}\,\hat{\Psi}(x)$
(63)
$\displaystyle-V(x)\,\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)-U(x)\,\hat{\Psi}^{\dagger}(x)\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)\hat{\Psi}(x)$
(64)

with the statistical gauge field $\hat{a}_{\mu}(x)$ satisfying the
corresponding $D$-dimensional flux attachment constraint in Eqs. (40–42).
Hence, the statistical gauge field will in general be a non-local function of
the matter current $\hat{a}_{\mu}=\hat{a}_{\mu}[\hat{J}^{\mu}]$. This implies
that the previous theory can be encoded or re-expressed in matter as an
interacting quantum matter theory, meaning Eq. (63) can be rewritten as

$\displaystyle\mathcal{L}$
$\displaystyle=i\hbar\,\hat{\Psi}^{\dagger}(x)\partial_{t}\hat{\Psi}(x)-\frac{\hbar^{2}}{2m}\bm{\nabla}\hat{\Psi}^{\dagger}(x)\cdot\bm{\nabla}\hat{\Psi}(x)-V(x)\,\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)-U(x)\,\hat{\Psi}^{\dagger}(x)\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)\hat{\Psi}(x)-\hat{J}^{\mu}(x)\,\hat{a}_{\mu}\,[\hat{J}^{\mu}(x)]$
(65)

where the last term should be thought of as a statistical interaction. In
other words, it is the normal minimal coupling, but since the gauge field has
no free dynamics it can be fully encoded in matter and appear
indistinguishable to conventional interactions888This should be considered as
opposed to more conventional Maxwell-like theories with propagating modes that
exist even in the absence of matter, namely photons. There is no free photons
in the theories proposed.. We expect statistical interactions to significantly
alter the groundstate of the system as well as its dynamics and excitations.
Even at weak couplings nonlinearities will develop and are likely to dominate
the dynamics. As a consequence, we expect exotic soliton solutions for the
matter field $\hat{\Psi}$ to be found. Quantum fluids with these kind of
interactions will be topological and accept a dual description

$\displaystyle\tilde{\mathcal{L}}_{C}$
$\displaystyle=i\hbar\,\hat{\tilde{\Psi}}_{C}^{\dagger}(x)\partial_{t}\hat{\tilde{\Psi}}_{C}(x)-\frac{\hbar^{2}}{2m}\bm{\nabla}\hat{\tilde{\Psi}}_{C}^{\dagger}(x)\cdot\bm{\nabla}\hat{\tilde{\Psi}}_{C}(x)-V(x)\,\hat{\tilde{\Psi}}_{C}^{\dagger}(x)\hat{\tilde{\Psi}}_{C}(x)-U(x)\,\hat{\tilde{\Psi}}_{C}^{\dagger}(x)\hat{\tilde{\Psi}}_{C}^{\dagger}(x)\hat{\tilde{\Psi}}_{C}(x)\hat{\tilde{\Psi}}_{C}(x)$
(66)

in terms of a composite field $\hat{\tilde{\Psi}}_{C}$ that will no longer
satisfy relations (61) and (62).

###  IV.2 On the Lattice.

The previously introduced model and arguments can also be considered in a
lattice setup. This is common in quantum solids. For simplicity, we restrict
ourselves to work in the tight-binding approximation for a regular lattice. In
this case, we consider the lattice Hamiltonian

$\hat{H}=\sum_{\mathbf{j}=1}^{N}\Big{[}\sum_{\mu=1}^{\alpha}\Big{(}\hat{c}^{\dagger}_{\mathbf{j}}\,J_{\mathbf{j},\mathbf{j}+\hat{\mathbf{e}}_{\mu}}(t)\,\hat{c}_{\mathbf{j}+\hat{\mathbf{e}}_{\mu}}+\text{H.c.}\Big{)}+V_{\mathbf{j}}(t)\,\hat{n}_{\mathbf{j}}+U_{\mathbf{j}}(t)\,\hat{n}_{\mathbf{j}}\hat{n}_{\mathbf{j}}+\sum_{\mu=1}^{\alpha}\Big{(}\hat{n}_{\mathbf{j}}\,W_{\mathbf{j},\mathbf{j}+\hat{\mathbf{e}}_{\mu}}(t)\,\hat{n}_{\mathbf{j}+\hat{\mathbf{e}}_{\mu}}+\text{H.c.}\Big{)}\Big{]}$
(67)

with (anti)commutation relations

$\displaystyle\big{[}\hat{c}_{\mathbf{i}},\hat{c}_{\mathbf{j}}\big{]}_{\,\mp}=\big{[}\hat{c}^{\dagger}_{\mathbf{i}},\hat{c}^{\dagger}_{\mathbf{j}}\big{]}_{\,\mp}=0\;$
(68)
$\displaystyle\big{[}\hat{c}_{\mathbf{i}},\hat{c}^{\dagger}_{\mathbf{j}}\big{]}_{\,\mp}=\delta_{\,\mathbf{ij}}\;,$
(69)

where $\mathbf{j}$ is the lattice coordinate, $\alpha$ is the number of
nearest neighbours and $\hat{\mathbf{e}}_{\mu}$ is a unit vector connecting
neighbouring sites along direction $\mu$. This model describes spinless bosons
or fermions in a periodic lattice which are allowed to tunnel between
neighbouring sites with rate $J$, that experience an on-site potential $V$,
and that can interact on-site with strength $U$, or between neighbouring sites
with $W$. This is nothing but an extended $d$-dimensional spinless Hubbard
model. Equation (67) provides a rather general toy model of a quantum solid.
Here again, we might couple the system to a statistical gauge field. We can do
this explicitly by working in the temporal gauge $a_{0}=0$ and minimally
coupling the vector potential. Hence, the coupling is achieved performing
Peierls substitution, yielding

$J_{\mathbf{j},\mathbf{j}+\hat{\mathbf{e}}_{\mu}}(t)\longrightarrow
J_{\mathbf{j},\mathbf{j}+\hat{\mathbf{e}}_{\mu}}(t)\;\hat{\mathcal{U}}_{\mu}(\mathbf{j})\;,$
(70)

where the operator $\hat{\mathcal{U}}_{\mu}$ is a parallel transporter, also
known as a Wilson line or link variable, over link $\mu$ of the lattice. For
an Abelian group we define it as

$\hat{\mathcal{U}}_{\mu}(\mathbf{j})\equiv
e^{i\hat{a}_{\mu}(\mathbf{j})}\;\;\;\;\;\;\;\;\;\;\text{and}\;\;\;\;\;\;\;\;\;\;\;\;\hat{a}_{\mu}(\mathbf{j})\equiv\hat{a}\,(\mathbf{j},\mathbf{j}+\hat{\mathbf{e}}_{\mu})=\frac{1}{\hbar}\int_{\mathbf{j}}^{\mathbf{j}+\hat{\mathbf{e}}_{\mu}}d\mathbf{x}\cdot\hat{\bm{a}}(t,\mathbf{x})\;,$
(71)

where $\hat{a}_{\mu}$ is the statistical vector potential and appears as a
$\text{U}(1)$ operator-valued Peierls phase. In fact, it is a density-
dependent Peierls phase. It is important to notice that matter fields live on
the direct lattice sites $\mathbf{j}$, gauge fields live on the links of the
lattice $\mu$, and disorder operators live on the sites of the dual lattice
$\mathbf{\tilde{\mathbf{j}}}$. Links $\mu$ and dual sites $\tilde{j}$ are
coincident for $d=1$, but differ substantially in $d>1$ (see Figure 3).

Figure 3: Direct (red) and dual (blue) bipartite lattices in 1d and 2d.

The form of the gauge potential can be explicitly implemented by the flux
attachment constraints in the corresponding dimension. Alternatively, one can
implement these constraints directly on the lattice, so that the model becomes
a lattice gauge theory. This is defining a Gauss’s law operator
$\hat{G}_{\mathbf{j}}$ restricting the Hilbert space of the system to a
physical subset $\mathscr{H}_{\,\text{Phys}}\subset\mathscr{H}$ obeying the
local flux attachment constraint

$\hat{G}_{\mathbf{j}}\ket{\text{Phys}}\equiv\Big{[}\hat{\beta}_{\,\mathbf{j}}-\exp{\big{(}\gamma\hat{n}_{\mathbf{j}}\big{)}}\Big{]}\,\ket{\text{Phys}}=0\;,$
(72)

where $\gamma$ is the statistical factor. Note that $\ket{\text{Phys}}$
represent the physical states of the system and that
$[\hat{G}_{\mathbf{j}}\,,\hat{H}]=0$ is satisfied at any point in spacetime.
The operator $\hat{\beta}_{\mathbf{j}}$ is a scalar field that changes with
dimension and is a function of the statistical gauge fields. It plays the role
of a “magnetic flux” on the dual lattice (e.g. blue lines in Figure 3) or,
alternatively, in the links lattice (e.g. dashed lines in Figure 3). We shall
call it the statistical flux operator999It is important to distinguish this
operator from conventional disorder (i.e. kink, vortex, monopole) operators we
find in conventional literature. from now on. Thus, it is defined with local
reference to the links $\mu$ coming out of the direct lattice site
$\mathbf{j}$. More explicitly, it is the product of links associated to a
given direct lattice site

$\hat{\beta}_{\,\mathbf{j}}=\prod_{\braket{\mu}}\hat{\mathcal{U}}_{\mu}(\mathbf{j})\;,$
(73)

where $\braket{\bullet}$ signals that the product is taken for all the link
variables associated to location $\mathbf{j}$. It is the lattice equivalent to
computing the magnetic flux centred at $\mathbf{j}$, which ensures the charge-
flux binding. That is with the exception of $d=1$, provided no magnetic flux
can be defined and the statistical flux operator reduces to

$\hat{\beta}_{\,\mathbf{j}}=\hat{\mathcal{U}}_{\mu}(\mathbf{j})\;,$ (74)

in the most convenient gauge. The discrete version of the composite dual
mapping (56) at a given lattice site becomes a $d$-dimensional Jordan-Wigner
transformation

$\hat{f}_{\mathbf{j}}\,(\Gamma)=\hat{\mathcal{W}}^{\dagger}_{\tilde{\mathbf{j}}}(\Gamma)\,\hat{c}_{\mathbf{j}}\;,\;\;\;\;\;\;\;\text{with}\;\;\;\;\;\;\;\;\;\;\;\;\hat{\mathcal{W}}^{\dagger}_{\tilde{\mathbf{j}}}(\Gamma)\,\hat{\mathcal{W}}_{\,\tilde{\mathbf{j}}}\,(\Gamma)=\hat{\mathbb{I}}\;,$
(75)

where $\Gamma$ denotes a ’t Hooft “magnetic” line, which is inherited by the
composite field. The number density operator is locally invariant under large
gauge transformations (e.g. Eq.(75)), yielding
$\hat{n}_{\mathbf{j}}=\hat{c}^{\dagger}_{\mathbf{j}}\hat{c}_{\mathbf{j}}=\hat{f}^{\dagger}_{\mathbf{j}}\hat{f}_{\mathbf{j}}\,$,
as expected from continuum. Provided most of the terms in the model (67) —
minimally coupled through Eq.(70) and constrained by Eq.(72) — are expressed
in powers of the number density, only the kinetic term will be altered in the
composite dual picture.

Making sense of the Statistical Flux Operator

It is worth comparing morphology and effect of the statistical flux operator
with respect to more familiar electromagnetic operators on the lattice. Let us
discuss a particular case for a square lattice in $d=2$ and a $\mathbb{Z}_{2}$
gauge group. We define the link variable to be
$\hat{\mathcal{U}}_{\mu}(\mathbf{j})\equiv\hat{\sigma}^{x}_{\mu}(\mathbf{j})$.
In this context, abusing language, one might refer to link variables as “gauge
fields”. In addition, “electric fields” are given by some other link
variables, namely
$\hat{\mathcal{E}}_{\mu}(\mathbf{j})\equiv\hat{\sigma}^{z}_{\mu}(\mathbf{j})$.
Finally, “magnetic fields” or “fluxes” are given by the plaquette operator

$\hat{\mathcal{B}}_{\,\tilde{\mathbf{j}}}=\prod_{\braket{\tilde{\mu}}}\hat{\mathcal{U}}_{\tilde{\mu}}(\tilde{\mathbf{j}})\equiv\prod_{\mathbf{j}\,\in\,\square}\hat{\sigma}^{x}_{\mu}(\mathbf{j})\;,$
(76)

which is the product of dual links for a given dual lattice site, or
equivalently, the product of gauge fields over a real-space plaquette. We can
also define a star or vertex operator accounting for the electric flux in a
similar manner

$\hat{\mathcal{A}}_{\,\mathbf{j}}=\prod_{\braket{\mu}}\hat{\mathcal{E}}_{\mu}(\mathbf{j})=\prod_{\mathbf{j}\,\in\,\bm{+}}\hat{\sigma}^{z}_{\mu}(\mathbf{j})\;.$
(77)

In other words, electric fluxes live on the real lattice while magnetic fluxes
live on the dual lattice. See Figures 3 and 4 for a graphical representation
of the notation so far. Then, quantum double models such as the Toric Code are
simply written as “electromagnetism in the absence of matter”, meaning

$\hat{H}=-\sum_{\mathbf{j}}\hat{\mathcal{A}}_{\,\mathbf{j}}-\sum_{\tilde{\mathbf{j}}}\hat{\mathcal{B}}_{\,\tilde{\mathbf{j}}}=-\sum_{\mathbf{j}}\bigg{(}\prod_{\mathbf{j}\,\in\,\bm{+}}\hat{\sigma}^{z}_{\mu}(\mathbf{j})+\prod_{\mathbf{j}\,\in\,\square}\hat{\sigma}^{x}_{\mu}(\mathbf{j})\bigg{)}\;.$
(78)

We now clearly see the difference with the statistical flux operator as it is

$\hat{\beta}_{\,\mathbf{j}}\equiv\prod_{\braket{\mu}}\hat{\mathcal{U}}_{\mu}(\mathbf{j})=\prod_{\mathbf{j}\,\in\,\bm{+}}\hat{\sigma}^{x}_{\mu}(\mathbf{j})\;$
(79)

in this context. It is the dual magnetic field or the twisted (or magnetic)
analogue of the electric vertex operator. It counts the magnetic flux in the
direct lattice sites and yet, it is different to the conventional magnetic
flux operator. In the presence of matter on the sites then, we see that Eq.
(72) is nothing but the magnetic Gauss’ law counting monopoles. In other
words, it is lattice flux attachment or the incarnation of Chern-Simons gauge
theory for a discrete system and local $\mathbb{Z}_{2}$ symmetry. Notice that
the statistical flux operator is gauge invariant in the same way the
conventional magnetic flux is.

#### Excitations.

Furthermore, we can study excitations for such a system. Before launching into
the discussion it is worth stressing that there is no matter in this example,
so all the possible excitations are “electromagnetic” in nature. In order to
find those, we must define the electric and magnetic string operators, also
known as Wilson lines

$\hat{W}^{(e)}_{\Gamma}=\prod_{\mu\,\in\,\Gamma}\hat{\sigma}^{x}_{\mu}$ (80)

and ’t Hooft lines

$\hat{W}^{(m)}_{\tilde{\Gamma}}=\prod_{\mu\,\in\,\tilde{\Gamma}}\hat{\sigma}^{z}_{\mu}$
(81)

respectively. In a given vacuum or ordered background, if lines are open, a
pair of electric $e$ (magnetic $m$) excitations are found at the ends of the
Wilson (’t Hooft) strings. Hence a disorder operator, namely a
$\mathbb{Z}_{2}$ magnetic vortex-antivortex linear combination or vison can be
defined as

$\hat{\tilde{\mathcal{V}}}^{\dagger}(\tilde{\mathbf{j}})\equiv\hat{\tau}^{x}(\tilde{\mathbf{j}})=\prod_{(\tilde{\mu},\tilde{\mathbf{k}})\,\in\,\tilde{\Gamma}<\tilde{\mathbf{j}}}\hat{\sigma}^{z}_{\tilde{\mu}}(\tilde{\mathbf{k}})\;,$
(82)

which defines the excitation $m$. This works as a dual Jordan-Wigner string
extending from the edge of the system up to $\tilde{\mathbf{j}}$ and following
some path $\tilde{\Gamma}$. We note that an open string of the form

$\prod_{\tilde{\mathbf{i}}\leq(\tilde{\mu},\tilde{\mathbf{k}})\,\in\,\tilde{\Gamma}<\tilde{\mathbf{j}}}\hat{\sigma}^{z}_{\tilde{\mu}}(\tilde{\mathbf{k}})\;.$
(83)

will feature two visons at its endpoints, located at $\tilde{\mathbf{i}}$ and
$\tilde{\mathbf{j}}\,$ (see Figure 4). Analogously, the associated ’t Hooft
line for the excitations of the statistical flux operator is

$\hat{W}^{(\tilde{m})}_{\Gamma}=\prod_{\mu\,\in\,\Gamma}\hat{\sigma}^{z}_{\mu}\;,$
(84)

in terms of which we can define a twisted or statistical vison

$\hat{\mathcal{W}}^{\dagger}(\mathbf{j})\equiv\prod_{(\mu,\mathbf{k})\,\in\,\Gamma<\mathbf{j}}\hat{\sigma}^{z}_{\mu}(\mathbf{k})\;,$
(85)

which defines the excitation $\tilde{m}$. This is nothing but our conventional
Jordan-Wigner string, which we observe to be a “gauge string”. It is worth
pursuing this example a little further and realise that in the presence of
matter fields living in the direct lattice sites $\hat{c}_{\mathbf{j}}$, a
configuration such as

$\hat{c}_{\mathbf{i}}^{\dagger}\;\Bigg{(}\,\prod_{\mathbf{i}\leq(\mu,\mathbf{k})\,\in\,\Gamma<\mathbf{j}}\hat{\sigma}^{z}_{\mu}(\mathbf{k})\Bigg{)}\;\hat{c}_{\mathbf{j}}$
(86)

describes a pair of composite particles formed by matter attached to,
respectively $\mathbb{Z}_{2}$ statistical gauge vortices and antivortices, and
located at sites $\mathbf{i}$ and $\mathbf{j}$. These composites interact non-
locally through the gauge string $\Gamma$. The string tension causes an energy
penalty and, in the usual sense, allows these composites to be either confined
or free. We see they are, actually, the lattice $\mathbb{Z}_{2}$ closest
relatives to Zhang-Hansson-Kivelson (ZHK) vortices [9, 40, 68]. This is flux-
attachment considerations at play or, more generally, composite-particle-
duality ones. It is worth noting that these configurations are excitations of
the system are similar to the so-called $e-m$ pairs or “dyons” from the Toric
Code, which are composite fermions according to the fusion rule

$e\;\times\;m=f\;.$ (87)

These are related in spirit to the composites just discussed but, as it is
palpable, they are different objects. However, we could envision, using a
similar language, some new $e-\tilde{m}$ composites that could be “emergent
dyons”, in reference to previous Section III.2.

Then, what is the relation of quantum double models like the Toric Code and
the models we propose? The toric code is a twisted doubled version of the type
of models we introduce. Schematically, this can be seen provided that the
Toric Code is nothing but the lattice pure Chern-Simons limit of a doubled or
mutual $\text{U}(1)\times\text{U}(1)$ Maxwell-Chern-Simons theory coupled to
matter [69, 70]. When the matter field condenses, it enters a $\mathbb{Z}_{2}$
spin liquid phase and the $\text{U}(1)$ symmetry breaks into $\mathbb{Z}_{2}$
[71]. This corresponds to the pure doubled-Chern-Simons term

$S_{\text{2CS}}=\frac{1}{4\pi}\int
d^{3}x\;K_{IJ}\,\epsilon^{\mu\nu\lambda}a^{I}_{\mu}\partial_{\nu}a^{J}_{\lambda}$
(88)

for an off-diagonal K-matrix
$K_{IJ}\equiv\hat{\mathbf{K}}=2\hat{\sigma}^{x}\,$. The presence of two mutual
Chern-Simons dynamics translates into $e$ and $m$ being mutual semions. Our
model, on the contrary, there are no mutual semions, but there are anyonic
composites. The theory is trivial in the absence of matter, and it is the
lattice limit of a single conventional Chern-Simons term. In a sense, a
“twisted square-root” of the Toric Code. Yet, the generation of composites in
both our model and the Toric Code follow an order-disorder structure. A final
comment is that the discussion here can be extended in all generality to
different symmetry groups $G$, lattices and dimensionalities. This, once again
will correspond to different avatars of flux-attached gauge theories.

Figure 4: Gauge-theoretic operators on a square lattice. Observe the
statistical operators as opposed to the more conventional electromagnetic
operators.

In summary, we have introduced a family of interacting tight-binding models in
$d$ spatial dimensions that admit a simple composite dual description

$\hat{\tilde{H}}_{C}=\sum_{\mathbf{j}=1}^{N}\Big{[}\sum_{\mu=1}^{\alpha}\Big{(}\hat{f}^{\dagger}_{\mathbf{j}}(\Gamma)\,J_{\mathbf{j},\mathbf{j}+\hat{\mathbf{e}}_{\mu}}(t)\,\hat{f}_{\mathbf{j}+\hat{\mathbf{e}}_{\mu}}(\Gamma)+\text{H.c.}\Big{)}+V_{\mathbf{j}}(t)\,\hat{n}_{\mathbf{j}}+U_{\mathbf{j}}(t)\,\hat{n}_{\mathbf{j}}\hat{n}_{\mathbf{j}}+\sum_{\mu=1}^{\alpha}\Big{(}\hat{n}_{\mathbf{j}}\,W_{\mathbf{j},\mathbf{j}+\hat{\mathbf{e}}_{\mu}}(t)\,\hat{n}_{\mathbf{j}+\hat{\mathbf{e}}_{\mu}}+\text{H.c.}\Big{)}\Big{]}\,.$
(89)

Notice that the composite dual of any spinless tight-binding system can always
be obtained by performing the transformation (75), provided it is an operator
identity. This statement is nothing but that of a, now generalised, Jordan-
Wigner transformation being always valid, although not always useful. We see
that, despite the transformation being extremely non-local, the dual model is,
in general, reasonably local if no anomalous terms101010An example can be
Cooper-pairing like terms such as $\hat{c}_{\mathbf{j}}\hat{c}_{\mathbf{j}}$
or $\hat{c}^{\dagger}_{\mathbf{j}}\hat{c}^{\dagger}_{\mathbf{j}}$. are present
due to cancellation of JW $d$-branes. Note that this is a type of duality in
matter, meaning that the bare and the composite fields “live” on the same
point of the lattice, contrary to classic electric-magnetic, Kramers-Wannier
or order-disorder dual models, which move between the direct and dual spatial
lattices. The latter notion, however, is contained in the flux attachment
constraint (72), provided gauge fluxes are defined on the dual lattice but
with reference to matter living on the direct lattice. The dual lattice, and
correspondingly the notion of magnetic flux, change with dimension. All
composite models are lattice gauge theories in disguise. This includes the
usual Jordan-Wigner transformed systems in 1d, as illustrated in the example
box. We observe now, that the Jordan-Wigner transformation is emergent and not
known a priori.

Quantum Spins Revisited: Hidden Gauge Theories.

#### On the quantum Heisenberg Chain.

Let us illustrate that the composite dual of the spin–$\frac{1}{2}$ quantum
Heisenberg model in $d=1$ is, secretly, a lattice gauge theory. We consider
the Heisenberg Hamiltonian with anisotropy or transverse field $\gamma$ with

$\hat{H}=J\sum_{j=1}^{N}\Big{(}\hat{S}^{x}_{j}\hat{S}^{x}_{j+1}+\hat{S}^{y}_{j}\hat{S}^{y}_{j+1}\Big{)}+\gamma
J\sum_{j=1}^{N}\hat{S}^{z}_{j}\hat{S}^{z}_{j+1}\;.$ (90)

Defining $\hat{S}^{\pm}=\hat{S}^{x}\pm i\hat{S}^{y}$ as the spin ladder
operators for $\hat{S}^{a}=\frac{1}{2}\hat{\sigma}^{a}$, we find relations

$\displaystyle\hat{S}^{+}\hat{S}^{-}=\frac{1}{2}+\hat{S}^{z}\;,$ (91)
$\displaystyle\hat{S}^{-}\hat{S}^{+}=\frac{1}{2}-\hat{S}^{z}$ (92)

We can now re-express the Hamiltonian as

$\hat{H}=\frac{J}{2}\sum_{j=1}^{N}\Big{(}\hat{S}^{+}_{j}\hat{S}^{-}_{j+1}+\hat{S}^{-}_{j}\hat{S}^{+}_{j+1}\Big{)}+\gamma
J\sum_{j=1}^{N}\Big{(}\hat{S}^{+}_{j}\hat{S}^{-}_{j}-\frac{1}{2}\Big{)}\,\Big{(}\hat{S}^{+}_{j+1}\hat{S}^{-}_{j+1}-\frac{1}{2}\Big{)}\;.$
(93)

We can now define our bare operators $\hat{c}_{j}\equiv\hat{S}^{-}_{j}$ and
$\hat{c}^{\dagger}_{j}\equiv\hat{S}^{+}_{j}$. The disorder operator is a kink
operator and the Jordan-Wigner brane is the usual string. In $d=1$ the ’t
Hooft line and JW string are essentially the same object. Explicitly,

$\hat{\mathcal{W}}_{\tilde{j}}\,(\Gamma)=\exp{\big{[}i\,\hat{\Phi}_{\tilde{j}}\,(\Gamma)\big{]}}=\exp{\Big{(}i\alpha\sum_{l=1}^{N}\Theta(j-l)\,\hat{n}_{l}\Big{)}}=\exp{\Big{(}i\alpha\sum_{l=1}^{j-1}\hat{n}_{l}\Big{)}}\;,$
(94)

where the links or dual lattice sites are encapsulated in
$\tilde{j}\equiv(j,\mu)\equiv(j-l)$ in the lattice Heaviside step function
$\Theta$. We thus see that the hopping term transforms according to

$\hat{S}^{+}_{j}\hat{S}^{-}_{j+1}=\hat{c}^{\dagger}_{j}\hat{c}_{j+1}=\hat{f}^{\dagger}_{j}\,\hat{\mathcal{W}}^{\dagger}_{\tilde{j}}\,\hat{\mathcal{W}}_{\tilde{j}+1}\,\hat{f}_{j+1}=\hat{f}^{\dagger}_{j}e^{i\alpha\big{(}\sum_{l=1}^{j-1}\hat{n}_{l}-\sum_{m=1}^{j}\hat{n}_{m}\big{)}}\hat{f}_{j+1}=\hat{f}^{\dagger}_{j}e^{-i\alpha\hat{n}_{j}}\hat{f}_{j+1}\;.$
(95)

This defines an operator-valued Peierls phase

$\hat{a}_{x}(j)=\hat{a}\,(j,j+\hat{\text{e}}_{x})=\frac{1}{\hbar}\int_{j}^{j+1}dx\;\hat{a}_{x}(x)=-\alpha\hat{n}_{j}\;,$
(96)

which is nothing but the lattice statistical gauge field satisfying the
Gauss’s law

$\hat{G}_{j}\ket{\text{Phys}}\equiv\hat{a}_{x}(j)\,\ket{\text{Phys}}=-\alpha\hat{n}_{j}\,\ket{\text{Phys}}=0\;.$
(97)

Hence, the composite dual model to the Heisenberg chain is the constrained
Hamiltonian

$\displaystyle\begin{cases}&\hat{H}=\frac{J}{2}\sum_{j=1}^{N}\big{(}\hat{f}^{\dagger}_{j}e^{i\hat{a}_{x}(j)}\hat{f}_{j+1}+\hat{f}_{j}\,e^{-i\hat{a}_{x}(j)}\hat{f}^{\dagger}_{j+1}\big{)}+\gamma
J\sum_{j=1}^{N}\big{(}\hat{f}^{\dagger}_{j}\hat{f}_{j}-\frac{1}{2}\big{)}\,\big{(}\hat{f}^{\dagger}_{j+1}\hat{f}_{j+1}-\frac{1}{2}\big{)}\;,\\\
\\\
&\big{[}\hat{a}_{x}(j)+\alpha\hat{n}_{j}\big{]}\,\ket{\text{Phys}}=0\;.\end{cases}$
(98)

This is a lattice gauge theory of interacting composite matter fields
$\hat{f}$ coupled to a statistical gauge potential. They are “gauge-dressed”
by the Jordan-Wigner string. It is worth noting that $\hat{f}$ fields are
anyonic in nature, meaning that their statistics is governed by $\alpha$, and
have a hard core since they anticommute on the same site. Hence, the
Heisenberg chain is equivalent to a theory of linear Abelian anyons in the
presence of effective dynamical electric field $\sim\partial_{t}\hat{n}_{j}$.
As it turns out, this model has a particularly simple expression for
$\alpha=\pi$, since the composite fields $\hat{f}$ can be identified with
hard-core spinless fermions, meaning that they satisfy

$\displaystyle\\{\hat{f}_{i},\hat{f}_{j}\\}=\\{\hat{f}^{\dagger}_{i},\hat{f}^{\dagger}_{j}\\}=0\;,$
(99) $\displaystyle\\{\hat{f}_{i},\hat{f}^{\dagger}_{j}\\}=\delta_{ij}\;.$
(100)

Then, the operator-valued Peierls phase can be expressed in a series expansion

$e^{i\hat{a}_{x}(j)}=e^{-i\pi\hat{f}^{\dagger}_{j}\hat{f}_{j}}=1-2\,\hat{f}^{\dagger}_{j}\hat{f}_{j}$
(101)

so that the original hopping (95) now becomes

$\hat{S}^{+}_{j}\hat{S}^{-}_{j+1}=\hat{f}^{\dagger}_{j}e^{-i\pi\hat{n}_{j}}\hat{f}_{j+1}=\hat{f}^{\dagger}_{j}(1-2\,\hat{f}^{\dagger}_{j}\hat{f}_{j})\hat{f}_{j+1}=\hat{f}^{\dagger}_{j}\hat{f}_{j+1}\;.$
(102)

In this case, the Peierls phase has no effect. Hence, the original Heisenberg
model is mapped to an interacting theory of spinless fermions

$\hat{H}=\frac{J}{2}\sum_{j=1}^{N}\big{(}\hat{f}^{\dagger}_{j}\hat{f}_{j+1}+\text{H.c.}\big{)}+\gamma
J\sum_{j=1}^{N}\big{(}\hat{n}_{j}-\frac{1}{2}\big{)}\,\big{(}\hat{n}_{j+1}-\frac{1}{2}\big{)}\;\;\;\;\;\;+\;\;\text{Boundary
Term},$ (103)

it is “fermionised” and can be solved via Bethe ansatz. In the absence of
anisotropy ($\gamma=0$), the system reduces to the quantum 1d XY model and
fermions become free. We can then diagonalise the Hamiltonian in momentum
space as usual.

#### On the quantum Ising model.

The above reasoning has also implications for the classical 2d Ising model,
which is equivalent to the 1d quantum (or transverse) Ising model through
transfer matrix formalism [27, 73, 74]. The corresponding Hamiltonian is

$\hat{H}=-J\sum_{j}\hat{\sigma}^{z}_{j}\hat{\sigma}^{z}_{j+1}-J\gamma\sum_{j}\hat{\sigma}^{x}_{j}\;.$
(104)

This model is electric-magnetic lattice self-dual
$\hat{\sigma}_{j}\leftrightarrow\hat{\mathcal{W}}_{\,\tilde{j}}\;$. Here, it
is convenient111111This is because
$\hat{\sigma}^{x}=\hat{c}^{\dagger}+\hat{c}\;$, which is highly non-local.
Alternatively, we can consider a non-rotated Jordan-Wigner transformation but
a rotated Ising model
$\hat{H}^{\prime}=-J\sum_{j}\hat{\sigma}^{z}_{j}-J\gamma\sum_{j}\hat{\sigma}^{x}_{j}\hat{\sigma}^{x}_{j+1}\,$
[98]. to perform a $\pi/2$ spin rotation or global transformation around the
$y$-axis, which yields $\hat{\sigma}^{z}\rightarrow\hat{\sigma}^{x}$ and
$\hat{\sigma}^{x}\rightarrow-\hat{\sigma}^{z}$. So the raising and lowering
operators are redefined as

$\hat{\sigma}^{\pm}=-\frac{1}{2}(\hat{\sigma}^{z}\mp i\hat{\sigma}^{y})$ (105)

The rotated mapping now yields a non-local operator
$\hat{\sigma}^{z}_{j}=-(\hat{\sigma}^{+}_{j}+\hat{\sigma}^{-}_{j})\equiv-(\hat{c}^{\dagger}_{j}+\hat{c}_{j})\;$.
Writing in the composite basis gives

$\displaystyle\hat{\sigma}^{x}_{j}=1-2\,\hat{n}_{j}\;,$ (106)
$\displaystyle\hat{\sigma}^{z}_{j}=(\hat{f}^{\dagger}_{j}+\hat{f}_{j})\,\hat{\mathcal{W}}_{\tilde{j}}\,(\Gamma)\;.$
(107)

The quantum Ising model can be re-expressed in the composite dual anyonic
picture, yielding again a lattice gauge theory, albeit a non-local one due to
the non-conservation of anyon number. This is, non-conserving anyon terms give
rise to JW strings, which we identify as ’t Hooft lines that extend to
infinity or the edge of the system. When these anyons coincide with spinless
hard-core fermions, the system is local and quadratic in (composite) fermion
fields, namely

$\hat{H}=-J\sum_{j}\Big{[}\hat{f}^{\dagger}_{j}\hat{f}_{j+1}+\hat{f}^{\dagger}_{j+1}\hat{f}_{j}+\hat{f}^{\dagger}_{j}\hat{f}^{\dagger}_{j+1}+\hat{f}_{j+1}\hat{f}_{j}-2\gamma\,\Big{(}\hat{n}_{j}-\frac{1}{2}\Big{)}\Big{]}\;\;\;\;\;\;+\;\;\text{Boundary
Term},$ (108)

but with anomalous BCS-pairing-like terms. In fact, the composite dual becomes
an instance of the Kitaev p-wave topological superconducting chain, written in
more generality as

$\hat{H}_{\text{Kitaev}}=\sum_{j}\Big{[}-t\,\Big{(}\hat{f}^{\dagger}_{j}\hat{f}_{j+1}+\hat{f}^{\dagger}_{j+1}\hat{f}_{j}\Big{)}+\Delta\,\hat{f}^{\dagger}_{j}\hat{f}^{\dagger}_{j+1}+\Delta^{*}\,\hat{f}_{j+1}\hat{f}_{j}-\mu\,\Big{(}\hat{n}_{j}-\frac{1}{2}\Big{)}\Big{]}\;,$
(109)

for $t=-\Delta$ and $\Delta\in\mathbb{R}$. This model, in turn, can be re-
expressed in terms of Majorana (real) operators. Hence, either in terms of
Majorana or conventional Dirac (complex) fermion operators, the model can be
diagonalised in momentum space by means of conventional Bogoliubov
transformation. Hence, the resulting quasiparticle operators will be composite
bogoliubons, namely Majoranas. Study of the energy spectrum indicates a
topological (i.e. gap-closing) phase transition. There is a well-known
correspondence between topological/trivial phases in the composite dual model
and the ordered/disorder symmetry-breaking phases in the original Ising model.
This is not by chance, but because we have performed the 1+1D equivalent to
flux attachment by means of the Jordan-Wigner transformation. It is worth
noting that the system does not preserve the number of composite fermions
$[\hat{N},\hat{H}]\neq 0$ due to the anomalous terms, but it preserves
fermionic parity operator
$\hat{P}\equiv(-1)^{\hat{N}}=\exp{\big{(}i\pi\sum_{j}\hat{n}_{j}\big{)}}$,
meaning $[\hat{P},\hat{H}]=0$ since fermion operators come in pairs. The
number of Majoranas is not conserved but the fact that they come in pairs is,
i.e. they are the excitations at the edges of strings. In our current view,
the conservation of fermion parity is a global by-product of gauge invariance,
since $\hat{P}=\prod_{j}\hat{G}_{j}\,$, something which has not been
acknowledged in literature. Careful attention to boundary conditions [27]
indicates that for each theory of spins there is a fermionic parity
$\mathbb{Z}_{2}$–ambiguity on the boundary conditions for the composite
counterpart, meaning the composite $\longrightarrow$ bare mapping is 2
$\longrightarrow$ 1\. In fact, this shows up in the form of spin domain wall
pairs in the Ising model, or as a pair of Majorana zero-modes at the edges in
the composite picture. Now, it is worth thinking about this model in its
classical 2d version, as the presence of defects can be seen as a classical
$\mathbb{Z}_{2}$ gauge theory, as pointed out by Kadanoff and Ceva [34]. In
this case, Onsager’s “classical composite fermions” are nothing but a
consequence of classical flux attachment. This 2-to-1 ambiguity appears there
provided the Onsager fermion is an order-disorder composite. Hence two links —
a.k.a. two dual sites — and, in turn, two corresponding disorder variables
living there — a.k.a. magnetic vortices —, can be attached to each spin matter
field living on the sites. This $\mathbb{Z}_{2}$ local ambiguity is a
$\mathbb{Z}_{2}$ gauge invariance. All the above, has a $\mathbb{Z}_{N}$
counterpart [36], where the Onsager fermions now become parafermions, and the
Majorana zero-modes are now parafermion zero-modes.

Similar remarks apply to the classical Ising lattice gauge theory in 3d, which
is equivalent to a quantum Ising (or $\mathbb{Z}_{2}$) lattice gauge theory in
2d, and electric-magnetic lattice dual
$\hat{\sigma}_{\mathbf{j}}\leftrightarrow\hat{\mathcal{W}}_{\,\tilde{\mathbf{j}}}\,$
to a 2d quantum Ising model. In the quantum $\mathbb{Z}_{2}$ LGT, visons
appear at the ends of an open ’t Hooft line. Composite fields there are the
result of attaching of such visons to matter, namely “charges” or spin fields
residing on the sites, which provides a quantum version of Onsager’s composite
fermions.

The family of proposed models are, by construction, topological lattice gauge
theories and topologically ordered. However, as previously stated in the
continuum case, they can also be fully encoded in matter. This will lead to
non-local interactions in high dimensions, but can reduce to standard local or
semi-local interactions in low dimensions, making the system indistinguishable
from a conventional model of interacting matter fields. Hence, a new realm of
topological models is expected, and well-known examples might emerge as
particular cases of this proposal. Particularly intuitive limits of the above
families [Eq. (89)] are the spinless anyon-Hubbard models in 1d ($\alpha=1$)
and 2d ($\alpha=2$) in either their hard-core or soft-core avatars, which
correspond to the simple case of $U=W=0$ and
$J_{\mathbf{j},\mathbf{j}+\hat{\mathbf{e}}_{\mu}}=J=\text{const.}$ in the
composite dual basis. In fact, examples of these particular cases might be
hidden in real materials.

##  V Closing Remarks

The quantum statistics of matter can be altered in interacting many-body
systems by means of statistical gauge fields. We have extended this familiar
notion in the context of FQHE in 2+1D to arbitrary dimensions. We have
introduced and discussed the composite particle duality, which provides an
exact mapping from conventional quantum matter to topologically-ordered
quantum matter. The duality gives as an output the type of gauge field and
transformation needed to transform an ordinary system into an statistically
exotic one. To be more concrete, we provide a recipe and a physical mechanism
to dynamically turn ordinary (Landau-like) quantum solids and fluids into
topologically-ordered (beyond-Landau) quantum matter. This enables us to write
down entire families of interacting models in $d$ spatial dimensions that have
a local composite dual model but can be expressed in terms of conventional
systems coupled to a fluctuating statistical gauge field. The anyon-Hubbard
models, anyonic Lieb-Liniger gases or certain FQH fluids are found as simple
instances of this universal prescription. This provides a first-of-its-kind
way to understand and modify the statistical properties of phases of matter
across dimensions.

Statistical gauge fields are found to be the key to change the identity of
matter. The form of these gauge fields is set by a generalised flux attachment
law acting as a local constraint, and whose explicit form changes depending on
dimensionality. These gauge fields are peculiar as they admit a re-expression
solely in terms of matter121212This is not the case for conventional gauge
fields like the one in quantum electrodynamics as there are non-trivial
solutions even in the absence of matter.. This means they can be rewritten as
interaction terms. Thus, statistical gauge fields are matter interactions.
Read in reverse, certain interactions in many-body systems can be rewritten as
statistical gauge fields, which give rise to statistical transmutation and
topological order.

Ability to engineer and tune these interactions or gauge fields would allow
manipulating and transforming ordinary quantum systems into exotic quantum
matter. Recent experiments in 1+1D, performed at ICFO [77] for a Bose-Einstein
Condensate and at Harvard [78] with ultracold bosons in an optical lattice,
have already shown that said manipulation is attainable with the current
state-of-the-art technology. We expect these experiments to provide new
avenues for the understanding of topological phases of matter and the
implementation of topological gauge theories. The composite particle duality
[1] is a consistent and insightful framework for the phenomenological
understanding of those systems.

###  V.1 Discussion and Extensions

Statistical transmutation is a purely quantum-mechanical effect, as the notion
of quantum statistics does not have a classical counterpart. This means that
statistical gauge fields lose their interpretation as dressing agents altering
quantum numbers when in a classical setup. However, the notion of flux
attachment and its generalisation is not necessarily related to quantum
mechanics. Hence, we can indeed consider classical flux attachment, in which
matter fields are bound to gauge fields and create unusual composites, but one
should adapt the interpretation.

Classical systems are essentially determined by their dynamics, their
equations of motion. Then, their binding to gauge fields that have a matter
self-coupling — which are the ones we studied here — will create
nonlinearities in all cases. These nonlinearities are an effect of self-
interaction or self-interference, and will grow in range as we consider higher
dimensional systems. At least for the Abelian cases considered, this gauge
fields can be completely eliminated in favour of matter. There is no guarantee
this should be true if one extends the composite particle duality or
generalised flux attachment to non-Abelian gauge fields. It would be extremely
interesting to consider examples of the latter in low dimensions and get an
intuition for what features can the richer internal gauge structure
offer131313We expect the differences to be analogous, for instance, to those
between the Dirac and the ’t Hooft-Polyakov monopoles.. In a quantum context,
this possibility would, by common sense, give rise to non-Abelian anyons for
the composites141414In the models in which this is allowed., but this remains
to be shown explicitly.

We should also take the opportunity to comment on the comparison with
literature. We find the exciting results reported by Valiente [81, 82] to be
an extension of the ones discussed here in 1+1D and, for that matter, also of
conventional flux attachment [29, 40]. The starting point of the analysis is
based on Shirokov’s algebra, this provides a rigorous mathematical treatment
of the ill-defined points $\mathbf{x}=\mathbf{x}^{\prime}$ in the formalism
and avoids the rather artificial definitions of hard and soft cores. In other
words, it is a potential way to include (or not) Pauli’s exclusion principle
in the mathematical formalism. At this point it is not exactly clear what
would be the phenomenological relevance of that.

Following on the connections with established literature, we have claimed the
Bose-Fermi dualities can, for the most part, be understood as composite
particle dualities. It is tempting to conjecture on the relation with
dualities of Kramers-Wannier type, which are Bose-Bose. There, topological
gauge constraints appear also as a key ingredient [83], and admit order-
disorder interpretations or particle-soliton correspondences involving gauge
fields. Similarities between FQHE-like and $\mathbb{Z}_{N}$ models and the
formation of an electric-magnetic composite condensate were acknowledged in
Ref. [84]. Further evidence from this potential relation, seems to come from
the connection with the $\mathbb{Z}_{N}$ Dijkgraaf-Witten type theories
discussed in Ref. [85]. Further study should clarify this connection.

If we are to make any claim about the duality studied here being fundamental
and ubiquitous, we should aim to extend all the previous discussion to
relativistic systems. A quick literature survey shows that duality webs have
appeared in 1+1D [65], 2+1D [86, 87] and 3+1D [88] as, in part, the extension
of the concept of statistical transmutation. This is not surprising, provided
that the initial duality that started this discussion, that of Polyakov [3],
was manifestly relativistic. In fact, Polyakov’s Bose-Fermi duality was
extended to 3+1D [89] shortly after its initial proposal. Should our formalism
generalise, it should capture all the above notions. In practice, the
generalisation to a complex Klein-Gordon field is trivial. However, extending
this framework to real scalars and Dirac spinors is clearly not. We refer the
reader to Appendix C for additional comments on this issue.

###### Acknowledgements.

We warmly thank A. Celi, B. Bakkali-Hassani, J. Kwan, J. Pachos, G. Palumbo
and L. Tarruell for useful discussions. G.V-R. acknowledges financial support
from EPSRC CM-CDT Grant No. EP/L015110/1.

##  Appendix A Snippets of $d=1,2,3$

In order to see the previous concepts at work, let us compute the statistical
pure gauge scalar function. In particular, we discuss the case of linear
exchange $\phi_{ab}=\beta_{1}+\beta_{2}\,\phi_{ba}$ of labels. We postulate
that the $d$-dimensional angular phase of the $d$-dimensional vector
$\mathbf{\tilde{x}}_{ab}=\mathbf{x}_{a}-\mathbf{x}_{b}$ with respect to some
arbitrary reference $\hat{\text{e}}_{\mathbf{n}}$ direction, is parametrised
by functions with the aforementioned exchange property. We want to compare
this picture with the field theoretical intuition from generalised flux
attachment.

#### Two spatial dimensions.

In two spatial dimension we shall recover the physics of conventional flux
attachment and statistical transmutation as in the Abelian fractions of FQHE.
This is, recovering the physics of the many-body Aharonov-Bohm effect. The
natural angular variable is the polar angle, which can take values in $S^{1}$
and has a singularity at $\mathbf{x}=0$, so the relevant homotopy group is
$\pi_{1}(S^{1})=\mathbb{Z}$. This can be formalised by means of the argument
function
$\phi_{ml}\equiv\arg\,(\mathbf{\tilde{x}}_{ab})=\arg\,(\mathbf{x}_{a}-\mathbf{x}_{b}\,;\hat{\text{e}}_{\mathbf{x}})$,
where the angle is taken with respect to some arbitrary reference, in this
case, the $x$-axis. The exchange property for this function reads
$\phi_{ab}=\pm\pi+\phi_{ba}\,$. We can now compute the gauge potential as

$\displaystyle\bm{a}\,(\mathbf{x}_{i})$
$\displaystyle=\alpha\,\nabla_{\mathbf{x}_{i}}\Phi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{i},\dots,\mathbf{x}_{N})=\alpha\,\bm{\nabla}_{\mathbf{x}_{i}}\bigg{(}\sum_{a<b}\phi_{ab}\bigg{)}=\alpha\,\bm{\nabla}_{\mathbf{x}_{i}}\Big{[}\sum_{i<b}\phi_{ib}+\sum_{a<i}\phi_{ai}\Big{]}$
(110)
$\displaystyle=\alpha\,\bm{\nabla}_{\mathbf{x}_{i}}\Big{[}\sum_{i<b}\phi_{ib}+\sum_{a<i}\big{(}\pm\pi+\phi_{ia}\big{)}\Big{]}=\alpha\,\bm{\nabla}_{\mathbf{x}_{i}}\Big{[}\sum_{j\neq
i}\phi_{ij}\pm\sum_{a<i}\pi\Big{]}$ (111) $\displaystyle=\alpha\,\sum_{j\neq
i}\bm{\nabla}_{\mathbf{x}_{i}}\arg\,(\mathbf{x}_{i}-\mathbf{x}_{j}\,;\hat{\text{e}}_{\mathbf{x}})=\alpha\,\sum_{j\neq
i}\bm{\nabla}_{\mathbf{x}_{i}}\arg\,(\tilde{\mathbf{x}}_{ij})\;,$ (112)

for which one can define a magnetic field and verify that there exist a
relation with the charge density $n\,(\mathbf{x}_{i})$ of the form

$b\,(\mathbf{x}_{i})=\bm{\nabla}_{\mathbf{x}_{i}}\times\bm{a}\,(\mathbf{x}_{i})=\alpha\sum_{j\neq
i}2\pi\,\delta^{\,(2)}\,(\mathbf{x}_{i}-\mathbf{x}_{j})\equiv
2\pi\alpha\,n\,(\mathbf{x}_{i})\,.$ (113)

In other words, we naturally recover local flux attachment. The associated
gauge transformation in this case is commonly known in literature as a
statistical or singular transformation, and reduces to

$\Psi\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})=e^{\,i\alpha\,\sum_{m<l}\text{arg}(\mathbf{x}_{m}-\mathbf{x}_{l}\,;\,\hat{\text{e}}_{\mathbf{x}})}\,\Psi_{c}\,(\mathbf{x}_{1},\dots,\mathbf{x}_{N})\;,$
(114)

where the sum in the exponent is over all particles. Hence, for a given
pairwise exchange of particles $i\leftrightarrow j\,$, where $1\leq i<j\leq
N$, a corresponding $\pi$ phase from the argument function is collected by the
composite wavefunction for every $w_{ab}^{-1}w_{ba}$ term $i\leq a<b\leq j$.
This yields a statistical factor $\gamma_{ij}=\mp\alpha\pi\eta$, where
$\eta\in\mathbb{Z}$ is the number of $a\leftrightarrow b$ possible pairs. This
is nothing but a many-particle Aharonov-Bohm phase for flux $\alpha$.

Alternatively, such a peculiar choice of gauge potential is provided by
construction, if the correct term is incorporated at the level of a field
theoretical Lagrangian. We make use of the quantised Abelian Chern-Simons term
at level $1/\alpha$ with $\alpha\in\mathbb{Z}$, minimally coupled to matter
via source term

$S=\frac{1}{4\pi\alpha}\int
dt\,d^{2}\mathbf{x}\;\epsilon^{\,\mu\nu\lambda}\hat{a}_{\mu}\partial_{\nu}\hat{a}_{\lambda}-\int
dt\,d^{2}\mathbf{x}\;\hat{J}^{\mu}\hat{a}_{\mu}\;.$ (115)

Computing the Euler-Lagrange equations for the gauge field in the presence of
the matter source we are left with

$\hat{J}^{\mu}=\frac{1}{2\pi\alpha}\,\epsilon^{\,\mu\nu\lambda}\partial_{\nu}\hat{a}_{\lambda}\;,\;\;\;\;\;\;\;\;\;\;\partial_{\mu}\,\hat{J}^{\mu}=0\;,$
(116)

where the current time component becomes nothing but a constraint equation or
Gauss’s law of the form

$\bm{\nabla}\times\hat{\bm{a}}\,(t,\mathbf{x})=2\pi\alpha\,\hat{n}\,(t,\mathbf{x})$
(117)

which we can attempt to solve in the Coulomb gauge
$\bm{\nabla}\cdot\bm{a}=0\,$. This allows us to write the vector potential as
$\bm{a}=\bm{\nabla}\times\varphi\,$, so that the Gauss’s law becomes

$n\,(t,\mathbf{x})=\frac{1}{2\pi\alpha}\bm{\nabla}^{2}\varphi\,(t,\mathbf{x})$
(118)

that can be solved using conventional Green’s function methods to find that
$\hat{\bm{a}}\,(t,\mathbf{x})=\alpha\,\bm{\nabla}\,\hat{\Phi}\,(t,\mathbf{x})$
and

$\hat{\Phi}\,(t,\mathbf{x})=\int
d^{2}\mathbf{x}^{\prime}\;\varphi\,(t,\mathbf{x}-\mathbf{x}^{\prime})\,\hat{n}\,(t,\mathbf{x}^{\prime})\;,$
(119)

where $\varphi\,(t,\mathbf{x})=\tan^{-1}\,(y/x)$ is the conventional polar
angle, and where we have assumed that the “charge” density is point-like so
that the Chern-Simons gauge potential can be written as a pure gauge. This is
nothing but the field theoretical version of flux attachment previously found
in first-quantised language. The corresponding singular gauge transformation
is

$\hat{\Psi}\,(t,\mathbf{x})=e^{\,i\alpha\,\hat{\Phi}(t,\mathbf{x})}\,\hat{\Psi}_{C}\,(t,\mathbf{x})\;.$
(120)

#### One spatial dimension.

When particles move on a line, there are only two possible choices for
direction, left and right. Should one define the analogue of an angle variable
or phase given a reference direction, we would find that the only allowed
angles are $0$ or $\pi\,$. There is an intrinsic $\mathbb{Z}_{2}$ symmetry and
the relevant homotopy group is $\pi_{0}(\mathbb{Z}_{2})$. This degree of
freedom can be encoded in a sign ($\pm$) or a binary variable ($0$ or $1$).
Hence, a natural choice we can make is the Heaviside step function
$\phi_{ml}=\Theta\,(x_{m}-x_{l})=1-\Theta\,(x_{l}-x_{m})=1-\phi_{lm}\,$. The
corresponding statistical factor can then be obtained explicitly from Eq.
(22), yielding $\gamma=\alpha\sum_{m<l}\text{sgn}\,(x_{m}-x_{l})\,$. The
statistical vector potential is computed as

$\displaystyle
a_{x}\,(x_{i})=\alpha\,\nabla_{x_{i}}\Phi\,(x_{1},\dots,x_{i},\dots,x_{N})=\alpha\,\sum_{a<b=1}^{N}\nabla_{x_{i}}\phi_{ab}=\alpha\,\Big{(}\sum_{b>i}\nabla_{x_{i}}\phi_{ib}+\sum_{a<i}\nabla_{x_{i}}\phi_{ai}\Big{)}$
(121)
$\displaystyle=\alpha\,\Big{[}\sum_{b>i}\delta\,(x_{i}-x_{b})-\sum_{a<i}\delta\,(x_{a}-x_{i})\Big{]}=\alpha\,\sum_{j\neq
i}\text{sgn}\,(j-i)\;\delta\,(x_{i}-x_{j})\equiv\alpha\,n\,(x_{i})\;,$ (122)

where we have used properties $\delta\,(x_{i})=\nabla_{x_{i}}\Theta\,(x_{i})$
and $\delta\,(x_{i}-x_{j})=\delta\,(x_{j}-x_{i})\,$. Notice that the
definition for the density is of the form $n(x_{i})=\sum_{j\neq
i}q_{j}\,\delta(x_{i}-x_{j})$ with a charge $q$ that depends not on label
position, but on label ordering. It is with this example that we realise that
the phase $\Phi$ acquires the form of a generalised Jordan-Wigner string in
which the topological content and a certain nonlocality are encoded. The gauge
transformation becomes

$\Psi\,(x_{1},\dots,x_{N})=e^{\,i\alpha\,\sum_{m<l}\Theta(x_{m}-x_{l})}\,\Psi_{c}\,(x_{1},\dots,x_{N})\;.$
(123)

By analogy with the Chern-Simons case, it is expected that some topological
gauge theory will naturally give rise to such form of $\Phi$. Preserving the
same structure, we see that the field theoretical constraint analogous to Eq.
(121) would be

$\hat{a}_{x}\,(t,x)=\alpha\,\nabla_{x}\,\hat{\Phi}\,(t,x)=\alpha\,\hat{n}\,(t,x)$
(124)

with

$\hat{\Phi}\,(t,x)=\int_{-\infty}^{\infty}\text{d}x^{\prime}\;\Theta\,(x-x^{\prime})\,\hat{n}\,(t,x^{\prime})=\int_{-\infty}^{x}\text{d}x^{\prime}\;\hat{n}\,(t,x^{\prime})\;,$
(125)

being the kink or Jordan-Wigner string. The transformation then becomes

$\hat{\Psi}\,(t,x)=e^{\,i\alpha\,\hat{\Phi}(t,x)}\,\hat{\Psi}_{C}\,(t,x)\;.$
(126)

Hence, we have provided an example for which, guessing or determining the form
of $\phi_{ml}$ allows the explicit computation of the statistical vector
potential, the statistical factor and the duality transformation. In fact, the
field theoretical formulation allows us to interpret the Jordan-Wigner string
as a kink.

#### Three spatial dimensions.

We shall generalise the previous prescription in the group of three-
dimensional rotations, parametrised now by two angles or an axis-angle
complex. Here we are looking for an analogue of the argument function, so the
natural choice to make is the solid angle
$\Omega(\mathbf{x}_{m}-\mathbf{x}_{l})$. We want this function to acquire
values in $[0,4\pi)$. This is slightly more convoluted than previous cases as
not only do we need a reference axis $\hat{\text{e}}_{\mathbf{n}}$ with
reference to $\mathbf{x}_{m}$ in order to define an angle, but also a
reference closed contour $\Gamma$ around which $\mathbf{x}_{l}$ is transported
in order to define a 3d section (or slice), and the relative orientation
between references. A simple choice can be $\hat{e}_{z}$, $\Gamma$ to lie in
the $x-y$ plane (i.e. perpendicular to the axis), and for it to be a circle,
so that the resulting solid angle measures the area subtended by a conical
section. Despite this discussion, we have not been able to construct this
object in closed and local form, but we point to Refs. [37, 90] as potential
sources of inspiration for the interested reader.

##  Appendix B The Composite Particle Duality in a Nutshell

The minimal coupling of charged matter to a $\text{U}(1)$ statistical gauge
field $a_{\mu}$ in $\text{D}=d+1$ spacetime dimensions is equivalent to the
formation of electric-magnetic entities identified as gauge-charge composites.
In some instances, the latter may be regarded as anyons. This correspondence
is summarised as

$\mathcal{H}_{\text{\,B}}=\sum_{i=1}^{N}\frac{\bm{\pi}_{i}^{\,2}}{2m}+\mathcal{H}_{\text{int}}\;\;\longleftrightarrow\;\;\tilde{\mathcal{H}}_{\text{\,C}}=\sum_{i=1}^{N}\frac{\tilde{\bm{p}}_{i}^{\,2}}{2m}+\tilde{\mathcal{H}}_{\text{int}}\;\;,$
(127)

where $\bm{\pi}_{i}=\bm{p}_{i}-\bm{a}(t,\mathbf{x}_{i})\,$, $N$ is the number
of particles, and interactions are short-ranged. We postulate that the
statistical gauge potential is a topologically non-trivial pure gauge
configuration

$\bm{a}(t,\mathbf{x}_{i})=\bm{\nabla}_{\mathbf{x}_{i}}\Phi\,(t,\mathbf{x}_{1},\dots,\mathbf{x}_{N})\;,$
(128)

where
$\bm{a}(t,\mathbf{x}_{i})\equiv\bm{a}(t,\mathbf{x}_{i}\,;\,\mathbf{x}_{1},\dots,\mathbf{x}_{N})$
refers to the gauge potential being evaluated at the location of particle
$\mathbf{x}_{i}$, although it might be a function of the position of all the
particles in the system and, as we will find later, also of matter density
$\lvert\Psi_{\text{B}}\,(t,\mathbf{x}_{1},\dots,\mathbf{x}_{N})\rvert^{2}\,$.
We identify a many-body Hamiltonian in the bare (B) basis, and another
corresponding to the composite (C) basis. Both sides of the duality are
related by a large gauge transformation which removes/introduces a minimally
coupled statistical gauge field. In doing so, it connects homotopically
distinct states with the same physical properties. This transformation
corresponds to the naïve generalised continuum version of the well-known
Jordan-Wigner transformation which, in second-quantised language, reads

$\hat{\Psi}_{\text{C}}\,(t,\mathbf{x};\Gamma_{\mathbf{x}})=\hat{\mathcal{W}}^{\,\dagger}(t,\mathbf{x};\,\Gamma_{\mathbf{x}})\,\hat{\Psi}_{\text{B}}\,(t,\mathbf{x})\;.$
(129)

The operator

$\hat{\mathcal{W}}\,(t,\mathbf{x};\Gamma_{\mathbf{x}})=e^{\,\frac{i}{\hbar}\,\hat{\Phi}\,(t,\mathbf{x};\,\Gamma_{\mathbf{x}})}$
(130)

is identified as a disorder operator, $\hat{\Phi}$ is a
$(\text{D}-1)-$dimensional Jordan-Wigner brane, and $\Gamma_{\mathbf{x}}$ is a
reference contour centred at $\mathbf{x}$ in the sense of [37] but physically
identified as a Dirac string or open ’t Hooft line. This is meant to
generalise the concept of a Jordan-Wigner string to higher dimensions. For low
dimensions the brane, and thus the composite operator, are local, i.e. they
can be defined at a given point in space. However, for $\text{D}\geq 3$ this
is not the case and these objects become intrinsically non-local, a feature
captured by $\Gamma_{\mathbf{x}}$.

Correspondence (129) can be understood as an operator identity valid
regardless of the underlying Hamiltonian. Considering the bare species
$\hat{\Psi}_{\text{B}}$ to be a bosonic (fermionic) field, and hence
satisfying ordinary equal-time (anti)commutation relations, then,
$\hat{\Psi}_{\text{C}}$ constitutes a composite field obeying generalised
commutation relations, except for at the point
$\mathbf{x}=\mathbf{x^{\prime}}$, where relations reduce to those of bare
species due to cancellation of branes. The density operator is
$\hat{n}\,(t,\mathbf{x})=\hat{\Psi}_{\text{B}}^{\dagger}(t,\mathbf{x})\,\hat{\Psi}_{\text{B}}(t,\mathbf{x})=\hat{\Psi}_{\text{C}}^{\dagger}(t,\mathbf{x};\Gamma_{\mathbf{x}})\,\hat{\Psi}_{\text{C}}(t,\mathbf{x};\Gamma_{\mathbf{x}})\,$,
so all possible local interaction terms which are functions of the density are
identical on both sides of the duality. The correspondence also dictates the
functional form of the statistical gauge field. In $d\leq 3$ and considering a
temporal gauge, they result from solving

$\displaystyle
1+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{2\pi}{\kappa}\,\hat{n}\,(t,x)=\hat{a}_{x}\,(t,x)$
(131) $\displaystyle
2+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{2\pi}{\kappa}\,\hat{n}\,(t,\bm{x})=\bm{\nabla}\times\hat{\bm{a}}\,(t,\bm{x})$
(132) $\displaystyle
3+1D\;:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{2\pi}{\kappa}\,\hat{n}\,(t,\bm{x})=\bm{\nabla}\cdot\bm{\nabla}\times\hat{\bm{a}}\,(t,\bm{x})\;,$
(133)

where the matter current
$\partial_{t}\hat{n}\,(t,\mathbf{x})+\bm{\nabla}\cdot\hat{\mathbf{J}}\,(t,\mathbf{x})=0$
is conserved. Evidence from the composite particle duality enables us to make
the following general observations:

  * •

The composite particle duality naturally generalises those of conventional
flux attachment and statistical transmutation. It survives in any dimension,
its origin is geometric, it is physically enforced by topological terms, and
satisfies an order-disorder operator structure.

  * •

Bose-Fermi correspondences can be seen as statistical transmutation. Hence,
they can be probed in experiments gauge-coupling quantum matter and tuning the
coupling constant of the statistical gauge field. This will correspond to a
physical interpolation between faces of the duality or, equivalently, changing
the statistical parameter. We also expect the presence of exotic gauge-charge
composite quasiparticles.

A crucial implication of our formalism is that Eq. (127) not only signals an
identification between theories, but also a physical process of gauge-dressing
and transmutation of matter fields. This means that some “conventional”
interaction between matter fields can be interpreted as a “statistical”
interaction. In other words, statistical gauge fields can be just a
relabelling of certain conventional interactions in quantum matter. This
provides new meaning to the emergence of gauge fields in quantum many-body
systems.

##  Appendix C Extension to Relativistic Systems

In 3+1D, there seems to be consensus in the existence of extended excitations
forming strings or loops. While the statistics between point particles should
be conventional according to the spin-statistics relation [91], the postulate
of locality might break down for extended objects for which anyonic relations
can be found [92]. Bose-Fermi transmutation of point objects was discussed in
the early literature on dyons [12, 11, 18, 93, 94], while the possibility of