# Bosonization duality in 2+1 dimensions and critical current correlation
functions in Chern-Simons $U(1)\times U(1)$ Abelian Higgs model

Vira Shyta  Institute for Theoretical Solid State Physics, IFW Dresden,
Helmholtzstr. 20, 01069 Dresden, Germany  Flavio S. Nogueira  Institute for
Theoretical Solid State Physics, IFW Dresden, Helmholtzstr. 20, 01069 Dresden,
Germany  Jeroen van den Brink  Institute for Theoretical Solid State Physics,
IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany  Institute for
Theoretical Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, TU
Dresden, 01069 Dresden, Germany

###### Abstract

While the phase structure of the $U(1)\times U(1)$-symmetric Higgs theory is
still under debate, a version of this theory with an additional Chern-Simons
term was recently shown to undergo a second-order phase transition [V. Shyta,
J. van den Brink, and F. S. Nogueira, Phys. Rev. Lett. 127, 045701 (2021)].
This theory is dual to a topological field theory of massless fermions
featuring two gauge fields. Here we elaborate on several aspects of this
duality, focusing on the critical current correlators and on the nature of the
critical point as reflected by the bosonization duality. The current
correlators associated to the $U(1)\times U(1)$ symmetry and the topological
current are shown to coincide up to a universal prefactor, which we find to be
the same for both $U(1)$ and $U(1)\times U(1)$ topological Higgs theories. The
established duality offers in addition another way to substantiate the claim
about the existence of a critical point in the bosonic Chern-Simons
$U(1)\times U(1)$ Higgs model: a Schwinger-Dyson analysis of the fermionic
dual model shows that no dynamical mass generation occurs. The same cannot be
said for the theory without the Chern-Simons term in the action.

##  I Introduction

The study of bosonization in dimensions higher than 1+1 has been around for a
long time Lüscher (1989); Polyakov (1988); Fradkin and Schaposnik (1994).
While bosonization in 1+1 dimension is a well developed field Giamarchi
(2003); Zinn-Justin (2002), this does not hold for the (2+1)-dimensional case,
in spite of the intense recent activity in the field Seiberg  _et al._ (2016);
Aharony (2016); Karch and Tong (2016); Wang  _et al._ (2017); Metlitski
(2015); Nastase and Núñez (2018); Chen  _et al._ (2018); Türker  _et al._
(2020); Shyta  _et al._ (2021a). Typically, in 1+1 dimensions scalar fields
are expressed locally in terms of fermion bilinears Coleman (1975); Witten
(1984). One such fermion bilinear is the current
$j_{\mu}=i\bar{\psi}\gamma_{\mu}\psi$, but other local forms are also
possible. A well known example is the bosonization of the Thirring model,
where the fermion current is expressed as
$j_{\mu}\sim\epsilon_{\mu\nu}\partial_{\nu}\varphi$ Coleman (1975). Since
$j_{\mu}$ is a vector, going one dimension higher implies necessarily that
bosonization in 2+1 dimensions should involve a gauge field, $a_{\mu}$,
therefore, one can guess,
$j_{\mu}\sim\epsilon_{\mu\nu\lambda}\partial_{\nu}a_{\lambda}$. Of course,
this intuitive picture has to be substantiated by actual calculations and it
becomes immediately clear that unlike the (1+1)-dimensional case, more fields
have to be involved, so the gauge field has to be minimally coupled to a
scalar field in order to match the symmetries of fermionic and bosonic
theories Seiberg  _et al._ (2016). As a consequence, in the case of $U(1)$
symmetries in 2+1 dimensions the bosonization manifests itself as a kind of
particle-vortex duality Peskin (1978); Thomas and Stone (1978); Dasgupta and
Halperin (1981); Kleinert (1982). One important feature of dualities is that
they provide mappings between strongly and weakly coupled theories, thus
presenting a non-perturbative tool to analyze quantum field theories.
Dualities between bosonic $U(1)$-symmetric theories on the lattice have been
established exactly in several dimensions Banks  _et al._ (1977); José  _et
al._ (1977); Savit (1980); Peskin (1978); Cardy (1982, 1980). Since
bosonization dualities in 2+1 dimensions involve Chern-Simons (CS) terms
Seiberg  _et al._ (2016); Karch and Tong (2016), it is considerably more
difficult to perform the procedure on the lattice, although it has been
considered in some cases more rigorously for gapped fermions Chen  _et al._
(2018); Türker  _et al._ (2020).

The bosonization duality in relativistic field theories in 2+1 dimensions
Seiberg  _et al._ (2016); Karch and Tong (2016) finds its precursors Chen  _et
al._ (1993) in actual condensed matter systems where quasiparticles with
fractional statistics (anyons) arise, like for example in the fractional
quantum Hall effect Ezawa (2013); Nayak  _et al._ (2008). Recall that in
quantum Hall systems statistical transmutation occurs via an emergent gauge
field associated to the Berry phase of the Laughlin wave function Arovas  _et
al._ (1984). The idea is that bosons can be transmuted into fermions via flux
attachment, which changes the (intrinsic) angular momentum from integer to
half-integer via a mechanism reminiscent of the Aharonov-Bohm effect Stern
(2008). The emergent gauge field is typically not governed by an
electrodynamics à la Maxwell, as in this case the flux alone cannot guarantee
the change of statistics via the angular momentum, since a Maxwell-like
electrodynamics produces a field contribution to the angular momentum that
generally cancels the mechanical contribution arising by letting the particle
go around another particle with a flux attached to it Jackiw (1990); Forte
(1992). That is the reason why a topological action in the form of a CS term
is required: the energy-momentum tensor being given by the functional
derivative of the action with respect to the metric, a CS term does not
contribute to it, since it is topological (i.e., it does not depend on the
metric) Jackiw (1990), so no field contribution to the total angular momentum
arises from it.

Things become considerably more subtle when gauge fields are periodic, i.e.,
the $U(1)$ group is compact. In this case even the pure Abelian CS action,

$S_{CS}=\frac{i\kappa}{2}\int
d^{3}xa_{\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}a_{\lambda},$ (1)

features nontrivial aspects, despite the theory being free. This situation
provides one of the simplest realizations of a so-called topological order Wen
(2004). The compactness of the gauge field $a_{\mu}$ forces the CS coupling to
be quantized, $\kappa=k/(2\pi)$, where $k\in\mathbb{Z}$ is the so-called CS
level. This way eigenstates of the Wilson loop have a degeneracy that depends
on the genus $g$ of the manifold where the CS form is defined, being given by
$k^{g}$. This fact is not only important in a purely abstract sense, but has
far reaching consequences in condensed matter systems in 2+1 dimensions Wen
(2004); Fradkin (2013). There are other, mathematically deeper ways of
understanding the quantization of the CS action, and this involves concepts
like spin manifolds and connections Seiberg and Witten (2016).

The Wilson loops and the topological quantization, which they ensue in the
pure compact CS action, acquire an even more interesting meaning when the
loops are actually closed vortex lines due to the coupling to matter fields.
Recall that in the (bosonic) particle-vortex duality Peskin (1978); Thomas and
Stone (1978); Dasgupta and Halperin (1981); Kleinert (1982) particle
worldlines in a theory are expressed in terms of an ensemble of vortex lines
in a dual theory, with the duality transformation mapping the strong coupling
of a theory into the weak coupling of the other one, and vice-versa. In this
case the topologically nontrivial gauge transformation involves the phase
$\theta$ of the complex Higgs scalar, so $a_{\mu}\to
a_{\mu}+\partial_{\mu}\theta$ causes the CS action to be transformed into
$S_{CS}^{\prime}[a]=S_{CS}[a+d\theta]$, but periodicity of $\theta$ implies
that $\epsilon_{\mu\nu\lambda}\partial_{\nu}\partial_{\lambda}\theta$ does not
vanish at vortex singularities Kleinert (1989). Considering a single vortex
loop for simplicity,

$\epsilon_{\mu\nu\lambda}\partial_{\nu}\partial_{\lambda}\theta(x)=2\pi
n\oint_{L}dy_{\mu}\delta^{3}(x-y),$ (2)

where $n\in\mathbb{Z}$ is the winding number defined via
$\oint_{L}d\theta=2\pi n$, we obtain from the CS action a field equation for
the gauge field in the form,

$\epsilon_{\mu\nu\lambda}\partial_{\nu}a_{\lambda}=-2\pi
kn\oint_{L}dy_{\mu}\delta^{3}(x-y).$ (3)

The quantity
$J_{\mu}=i\epsilon_{\mu\nu\lambda}\partial_{\nu}a_{\lambda}/(2\pi)$ defines a
topological current. The role of topology is apparent also in the sense of the
vortex loop as a topological defect, since the RHS turns out to have a zero
divergence independently of the shape of the loop.

In Ref. Peskin (1978) the correlation functions of vortex loops are shown to
be directly related to the current correlation function associated to the
global (Noether) $U(1)$ symmetry. There this result was established on the
lattice in the context of a bosonic particle-vortex duality. Such a connection
is expected to exist also in the case of other dualities in 2+1 dimensions,
including the ones associated to bosonization Goldman and Fradkin (2018).
Particularly relevant in this context is the approach to the critical point,
where the theory is a conformal field theory (CFT) and, in addition, the fixed
points on both sides of the duality coincide. This example highlights well the
power of the duality approach, since it establishes the existence of a
critical point (and therefore a second-order phase transition) in the Abelian
Higgs model in 2+1 dimensions for one Higgs scalar, something that is
otherwise very hard to assert by perturbative means, unless a large enough
number of scalars is available Halperin  _et al._ (1974); Hikami (1979); Ihrig
_et al._ (2019). The duality result obtained in Ref. Peskin (1978) in the
lattice and elaborated further via Monte Carlo simulations in Refs. Dasgupta
and Halperin (1981); Hove and Sudbø (2000) allows to unequivocally conclude
that a second-order phase transition indeed takes place. That this result can
also be obtained within a field theoretical approach via bosonization is a
remarkable achievement Seiberg  _et al._ (2016); Karch and Tong (2016).

Regarding the phase structure, the bosonization duality approach is
considerably less conclusive in the case of a $U(1)\times U(1)$-symmetric
Higgs theory in 2+1 dimensions Wang  _et al._ (2017). In this case numerical
results using lattice gauge theory models favor a first-order phase transition
Kuklov  _et al._ (2006); Kragset  _et al._ (2006). However, other lattice
models with $U(1)\times U(1)$ symmetry lead to a second-order phase transition
Hove and Sudbø (2007); Herland  _et al._ (2013); Zhang  _et al._ (2018). In a
recent letter Shyta  _et al._ (2021a) we have pointed out yet another route
towards quantum criticality in $U(1)\times U(1)$ Higgs systems, namely, one
where a CS term is present in the Lagrangian. There we focused on a derivation
of the duality in terms of interactions between vortex loops, which in the
case of a CS Higgs theory naturally leads to a sum over sign factors
$(-1)^{N_{ab}}$ in the partition function, where $N_{ab}=\sum_{a\neq
b}n_{a}n_{b}Lk_{ab}$, with integers $n_{a}$ and $n_{b}$ being the quanta
(winding numbers) from given vortex loops $a$ and $b$. These sign factors are
weighted by a phase factor $e^{i\pi\sum_{a}n_{a}\mathcal{W}_{a}}$, where
$\mathcal{W}_{a}$ is the writhe Călugăreanu (1959, 1961); Moffatt and Ricca
(1992). The sign factors are here identified to integrated out massless
fermions. The duality and a renormalization group (RG) analysis provided
strong evidence for the existence of a quantum critical point in this theory.
Key in the RG analysis was the vanishing of certain Feynman diagrams at zero
momenta as a consequence of the CS term Shyta  _et al._ (2021a).

In this paper we discuss the $U(1)\times U(1)$ CS Higgs theory further using a
point of view more focused on local field theory actions rather than directly
considering vortex loop configurations. The basic idea is to take advantage of
the flux attachment with CS terms in the bosonic particle-vortex duality for
the $U(1)$ theory Karch and Tong (2016). In spite of the absence of a CS term
in the Higgs theory of this bosonic duality, a CS $U(1)$ Higgs model is used
in the intermediate steps via the bosonization technique in 2+1 dimensions
Seiberg  _et al._ (2016); Karch and Tong (2016). Here we will discuss some
important differences between the bosonization duality for the usual
$U(1)\times U(1)$ Higgs theory discussed earlier Karch and Tong (2016); Wang
_et al._ (2017) and the case of a CS $U(1)\times U(1)$ Higgs theory. One main
focus of our analysis is to obtain results that elucidate further the quantum
critical behavior discussed in our previous work Shyta  _et al._ (2021a). We
will address with considerable level of detail the correspondence via the
duality between the conserved current associated to the $U(1)\times U(1)$
symmetry and the topological current from the dual model. This point of view
adheres to the logic of bosonization in (1+1)-dimensional systems Coleman
(1975) where a precise correspondence is derived between the currents of the
fermionic and bosonic theories. For this reason we will start by briefly
reviewing some basic facts about the (1+1)-dimensional case in Section II.

Besides the duality and critical behavior of the $U(1)\times U(1)$ CS theory,
this paper will additionally explore several aspects of the particle-vortex
duality in 2+1 dimensions that reflect the interplay between current
correlation functions featuring both Noether and topological currents. As
mentioned, we will mainly focus on theories at the quantum critical point. In
order to set the stage, after briefly recapping in Section II well known
results from bosonization in 1+1 dimensions, our discussion of bosonization in
2+1 dimensions in Section III starts by first considering a toy example in 0+1
dimensions (Section III.1). The interest in such an example is not only driven
by the fact that it provides exact results within a path integral treatment.
In fact, the (0+1)-dimensional case exhibits many features similar to those
encountered in the (2+1)-dimensional case, but in a much simpler setting. One
of the similarities being the generation of a CS term in 0+1 dimensions after
exactly performing the path integral over fermions Deser  _et al._ (1997);
Dunne  _et al._ (1997). Also the effect of the parity anomaly in the partition
function and its role in the bosonization is made explicit. Given that the
Thirring model in 1+1 dimensions is one of the best known examples of
bosonization Coleman (1975), we briefly consider it in both $0+1$ (Section
III.1.2) and $2+1$ (Section III.2) dimensions. Sections IV, V, VI, and VII
constitute the core results of this paper. Section IV focuses mainly on the
role of criticality in particle-vortex duality, offering new insights from a
field theoretic perspective on the well known bosonic particle-vortex duality
in 2+1 dimensions Peskin (1978); Thomas and Stone (1978); Dasgupta and
Halperin (1981). There the importance of the critical point is emphasized in
the discussion we give of this duality (Section IV.2). Next, establishing the
bosonization duality for the $U(1)$-symmetric Higgs theory, in Section V.2 we
obtain the relation between topological and Noether current correlators at
criticality. Section V serves as a warm-up for the discussion of the
$U(1)\times U(1)$ case in Sections VI and VII. Section VI gives a derivation
of both the bosonic and fermionic dual models as well as the calculation of
the corresponding current correlators. In Section VII we discuss the phase
structure and compare it to the one of the $U(1)\times U(1)$ Higgs
electrodynamics without the CS term. Section VIII summarizes our results and
gives an outlook.

##  II Bosonization in 1+1 dimensions

Let us briefly recapitulate a number of results from (1+1)-dimensional
bosonization that will be relevant within the context of this paper.

###  II.1 The Thirring model

The Thirring model of interacting Dirac fermions represents a primary example
of bosonization in field theory Coleman (1975). The Lagrangian for the
Thirring model differs from the one for QED in the way the $U(1)$ Noether
currents interact. Instead of featuring a long-range, photon-mediated
interaction between currents, the interaction between currents in the Thirring
model is short-ranged. The Lagrangian is thus given by,

$\mathcal{L}_{\rm Th}=\bar{\psi}(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial+M)\psi-\frac{\lambda}{2}j_{\mu}j_{\mu},$
(4)

where the current, $j_{\mu}=i\bar{\psi}\gamma_{\mu}\psi$. In 1+1 dimensions
the Thirring model is known to be dual to the bosonic sine-Gordon model,

$\mathcal{L}_{\rm
SG}=\frac{1}{2\kappa^{2}}(\partial_{\mu}\varphi)^{2}-\alpha\cos\varphi,$ (5)

where $\alpha\sim M$. The conserved current $j_{\mu}$ of the Thirring
Lagrangian is related to the scalar field $\varphi$ in the sine-Gordon
Lagrangian via Coleman (1975),

$j_{\mu}=\frac{1}{2\pi}\epsilon_{\mu\nu}\partial_{\nu}\varphi,$ (6)

provided the following relation between couplings holds,

$\kappa^{2}=\frac{4\pi^{2}}{\pi+\lambda}.$ (7)

Such a connection between the currents reflects the dual nature of the
corresponding theories.

In the zero mass limit the Thirring model in 1+1 dimensions is equivalent to a
free scalar field, since in this case $\alpha=0$. Thus, as elaborated in
Appendix A, we obtain that the current correlation function
$C_{\mu\nu}(x)=\langle j_{\mu}(x)j_{\nu}(0)\rangle$ is given by,

$C_{\mu\nu}(x)=-\frac{1}{2\pi(\pi+\lambda)x^{2}}\left(\delta_{\mu\nu}-\frac{2x_{\mu}x_{\nu}}{x^{2}}\right).$
(8)

This result highlights the future importance of the prefactor arising in the
current correlator, which is dependent on the interaction coupling $\lambda$.

###  II.2 Fermions in the presence of a background gauge field

There is yet another instance where a current having the form (6) plays a
role. Let us consider the action of a massless Dirac fermion in the presence
of a background gauge field,

$S=\int d^{2}x\bar{\psi}(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-i\hbox to0.0pt{\hbox
to7.50002pt{\hfil/\hfil}\hss}A)\psi.$ (9)

This action features besides the local $U(1)$ gauge symmetry also the local
chiral symmetry $\psi(x)\to e^{i\gamma_{5}\varphi(x)}$, since this
transformation is compensated by $A_{\mu}(x)\to
A_{\mu}(x)-\epsilon_{\mu\nu}\partial_{\nu}\varphi$. This is true because in
$d=2$ the property that $\gamma_{\mu}\gamma_{5}=\epsilon_{\mu\nu}\gamma_{\nu}$
holds. Hence, the gauge field is entirely parameterized in terms of a scalar
field. At the classical level a gauge transformation in the fermionic fields
removes henceforth the gauge field completely from the action, leading to a
theory of a free massless scalar fermion. As is well known, such a result is
not correct at the quantum level, as the theory of Eq. (9) features a chiral
anomaly. Recall that the effective action,

$S_{\rm eff}=-\ln\det(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-i\hbox to0.0pt{\hbox
to7.50002pt{\hfil/\hfil}\hss}A),$ (10)

needs to be regularized. Using the standard Pauli-Villars regularization, one
obtains that the scalar field parametrization of the gauge field cannot be
gauged away any longer. The final result is a theory consisting of a free
massless Dirac fermion decoupled from a free massless scalar field Zinn-Justin
(2002),

$S=\int d^{2}x\left[\bar{\psi}\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial\psi+\frac{1}{2\pi}(\partial_{\mu}\varphi)^{2}\right].$
(11)

The scalar field term in the above action corresponds to the $O(2)$ nonlinear
sigma model considered in detail in Appendix A (with $g=\pi$). As the vector
gauge field $A_{\mu}$ is eliminated, instead of the fermionic current
$j_{\mu}=i\bar{\psi}\gamma_{\mu}\psi$ there is a corresponding topological
current,
$\tilde{J}_{\mu}=\frac{1}{\pi}\epsilon_{\mu\nu}\partial_{\nu}\varphi$,
obtained from Eq. (11). Analogously, the current
$J_{\mu}=\partial_{\mu}\varphi/\pi$ is identified to the fermionic chiral
current $j_{\mu}^{5}=i\bar{\psi}\gamma_{5}\gamma_{\mu}\psi$. Due to the chiral
anomaly, the current correlation function for $J_{\mu}$ is not transverse in
momentum space, as shown in Appendix A. Therefore, through a proper Pauli-
Villars regularization one obtains a bosonized version of the
(1+1)-dimensional massless fermionic theory with a background gauge field.

##  III Dynamically generated Chern-Simons term and Bosonization

Bosonization in 2+1 dimensions works very differently from the
(1+1)-dimensional case. One distinctive feature is the absence of a
$\gamma_{5}$ matrix and, as a consequence, a chiral symmetry in the case of
massless fermions 111Regarding this point, the following remark is in order.
In some versions of QED in 2+1 dimensions a different version of the chiral
symmetry can be introduced Appelquist  _et al._ (1986a). In this case one
considers four component Dirac fermions and a $\gamma_{5}$-like matrix, anti-
commuting with all other Dirac $\gamma$ matrices can be defined. However, such
a $\gamma_{5}$-like matrix is not given as a product of the other Dirac
matrices. Such a version of QED in 2+1 dimensions has numerous applications in
condensed matter physics, see for example Refs. Rantner and Wen (2001); Franz
_et al._ (2002); Herbut (2006, 2002); Kleinert and Nogueira (2002); Kaul and
Sachdev (2008); Nogueira (2008); Gusynin  _et al._ (2003). However, in this
case there is no parity anomaly, since neither time-reversal nor spatial
inversion are broken by adding a mass term. The latter actually leads to a
breaking of the chiral symmetry (defined in the mentioned sense) instead.
Although in 2+1 dimensions the product of Dirac matrices does not yield a
$\gamma_{5}$ matrix, the trace of such a product produces ${\rm
tr}(\gamma_{\mu}\gamma_{\nu}\gamma_{\lambda})=2i\epsilon_{\mu\nu\lambda}$,
which is of paramount importance in the dynamical generation of the CS term.
The possibility of a CS term is another distinctive feature of systems at an
odd number of spacetime dimensions. The simplest occurrence of a CS term is in
0+1 dimensions, where most calculations can be performed exactly. This also
provides a toy model to illustrate some key features of the bosonization
duality in 2+1 dimensions. Concerning the latter, an early example amounts to
extend to 2+1 dimensions the (1+1)-dimensional bosonization of the Thirring
model Fradkin and Schaposnik (1994). We briefly review this result in Sect.
III.2 below, since some elements of it will be useful latter on. Also, for
completeness, since the Thirring model is being discussed in both 1+1 and 2+1
dimensions, we connect the exact bosonization duality in 0+1 dimensions to a
(0+1)-dimensional version of the Thirring model, which just turns out to
correspond to the Hubbard model in the atomic limit.

###  III.1 Exact bosonization in 0+1 dimensions

####  III.1.1 Bosonization duality and parity anomaly in 0+1 dimensions

The bosonization duality takes different forms depending on the dimensionality
of the theories. In pursuit of a clear narrative behind the (2+1)$d$
bosonization that we are set to obtain, let us start by demonstrating an exact
bosonization duality in (0+1)$d$ with only the time component present. In the
imaginary time formalism, a fermionic theory has the following Lagrangian,

$\mathcal{L}_{f}=\psi^{\dagger}(\partial_{\tau}+m-iA(\tau))\psi,$ (12)

where $\psi(\tau)$ is a fermionic field coupled to the gauge field $A$ and
satisfying an anti-periodic boundary condition, $\psi(0)=-\psi(\beta)$, with
$\beta=1/T$ being an inverse temperature. While the Lagrangian is gauge
invariant, the partition function fails to have this property when
topologically nontrivial gauge transformations are considered. By the latter
we mean a gauge transformation $\psi\to e^{i\lambda}\psi$, $A\to
A+\partial_{\tau}\lambda$, where $\lambda\in S^{1}$. We will see how this
works within this simple setting below.

Once the fermionic field is integrated out, the partition yields,

$Z_{f}=\frac{\det(\partial_{\tau}+m-iA(\tau))}{\det(\partial_{\tau}+m)},$ (13)

where for normalization purposes there is a division by the free theory. The
fermion determinant is given by the product of the eigenvalues of the operator
$(\partial_{\tau}+m+A(\tau))$. The boundary conditions on the fermionic field
allow to obtain a discretized set of eigenvalues, and thus, the partition
function becomes,

$\displaystyle Z_{f}$ $\displaystyle=$
$\displaystyle\prod_{n=-\infty}^{\infty}\left[1+\frac{\beta
m-i\phi}{i(2n+1)\pi}\right]$ $\displaystyle\times$
$\displaystyle\prod_{n=-\infty}^{\infty}\left[1+\frac{\beta
m}{i(2n+1)\pi}\right]^{-1},$

where we used the notation,

$\phi=\int_{0}^{\beta}d\tau A(\tau).$ (15)

The infinite products arising in the partition function (III.1.1) can be
evaluated by manipulating the well known infinite product representation of
the cosine hyperbolic function Arfken and Weber (2005). Eventually, one
obtains following Refs. Deser  _et al._ (1997); Dunne  _et al._ (1997),

$Z_{f}=\frac{\cosh\left(\frac{\beta m-i\phi}{2}\right)}{\cosh\left(\frac{\beta
m}{2}\right)}=\cos\left(\frac{\phi}{2}\right)-i\tanh\left(\frac{\beta
m}{2}\right)\sin\left(\frac{\phi}{2}\right).$ (16)

Assuming the mass, $m$, to be positive and taking the limit $\beta
m\to\infty$, we obtain the zero temperature result,

$Z_{f}=e^{-\frac{i\phi}{2}}.$ (17)

Note that the zero temperature limit resembles a Pauli-Villars type of
regularization, if the limit is interpreted as a large mass one. Importantly,
if the “flux” of Eq. (15) is $2\pi$, in analogy with the single unit flux in
3+1 dimensions, we obtain that $Z_{f}=-1$, so the partition function is
negative. This result is a toy example for a more general scenario provided by
the Atiyah-Patodi-Singer (APS) theorem Atiyah  _et al._ (1975); see also Ref.
Witten (2016) for a discussion in the context of partition functions of
fermionic systems in manifolds with and without boundaries.

A topologically nontrivial gauge transformation, $A\to
A+\partial_{\tau}\lambda$ yields,

$Z_{f}^{\prime}=(-1)^{n}Z_{f},$ (18)

where $n$ is the winding number,

$n=\frac{1}{2\pi}\int_{0}^{\beta}d\tau\partial_{\tau}\lambda,~{}~{}~{}~{}~{}n\in\mathbb{Z}.$
(19)

The above sign flip in the partition function that is caused by the gauge
transformation is the (0+1)-dimensional version of the well known parity
anomaly in 2+1 dimensions Redlich (1984a, b). By means of this toy example we
can see that the anomaly is canceled at zero temperature if we add a level
-1/2 CS term to the action, which in (0+1)$d$ is simply a gauge field itself
with an imaginary prefactor. The resulting theory without the anomaly is then
given by,

$Z_{f}^{\prime}(A)=\int\mathcal{D}\psi^{\dagger}\mathcal{D}\psi
e^{-S_{f}+\frac{i}{2}\int_{0}^{\beta}d\tau A(\tau)}=1.$ (20)

A similar argument like the one above plays an important role in (2+1)$d$, as
we will see a couple of times later on.

Although we have considered a massive theory, we note that the anomaly
persists in the massless case, since for $m=0$ we have,

$Z_{f}|_{m=0}=\cos(\phi/2),$ (21)

which again gets a $(-1)^{n}$ factor after performing a topologically
nontrivial gauge transformation.

Now we consider the bosonic Lagrangian in (0+1) dimensions containing a level
1 CS term,

$\mathcal{L}_{b}=\frac{1}{2g}(\partial_{\tau}\theta-A)^{2}+iA,$ (22)

where $\theta\in S^{1}$. This Lagrangian can be regarded as a topological
Abelian Higgs model in 0+1 dimensions.

Performing the shift $A\to A+\partial_{\tau}\theta$ we obtain,

$\mathcal{L}_{b}^{\prime}=\frac{1}{2g}A^{2}+i(A+\partial_{\tau}\theta),$ (23)

and thus in view of Eq. (15),

$Z_{b}=e^{-i\phi},$ (24)

after using a proper normalization. The important result for us is the phase
factor. From the discussion for the fermion case, we can therefore write,

$\displaystyle\int\mathcal{D}\psi^{\dagger}\mathcal{D}\psi
e^{-S_{f}-\frac{i}{2}\int_{0}^{\beta}d\tau A(\tau)}$
$\displaystyle=\int\mathcal{D}\theta
e^{-\int_{0}^{\beta}d\tau\left[\frac{1}{2g}(\partial_{\tau}\theta-A)^{2}+iA\right]}.$
(25)

The above is the simplest example of the bosonization duality involving CS
terms. We note in particular a feature that will appear later in our
discussion of bosonization in 2+1 dimensions, namely, that the fermionic side
features a level 1/2 CS term, while a level 1 CS occurs in the bosonic side.

####  III.1.2 The Thirring model in 0+1 dimensions

Given the results of the previous subsection, we see that if we promote $A$ to
be a dynamical gauge field $a$ over which we integrate over in the path
integral, the Lagrangian of Eq. (12) actually imposes a constraint
$\psi^{\dagger}\psi=0$. Thus, inspired also by Eq. (23), we add a term
$a^{2}/(2g)$ to the Lagrangian of Eq. (12). The result obtained upon
integrating out $a$ is an interacting fermionic theory,

$\mathcal{L}^{\prime}_{f}=\psi^{\dagger}(\partial_{\tau}+m)\psi+\frac{g}{2}(\psi^{\dagger}\psi)^{2},$
(26)

where in the present case we must require that there are at least two Fermi
field components, otherwise $(\psi^{\dagger}\psi)^{2}$ vanishes, since $\psi$
is a Grassmann field. The Lagrangian (26) corresponds to the Thirring model in
0+1 dimensions. In the two-component case [for instance,
$\psi=(\psi_{\uparrow},\psi_{\downarrow})$], the Lagrangian (26) is equivalent
to the atomic limit of the Hubbard model. In this two-component example where
$a$ arises as an auxiliary field (via a so called Hubbard-Stratonovich
transformation), integrating out the fermions yields in the zero temperature
limit,

$Z=\int\mathcal{D}ae^{-i\phi-\frac{1}{2g}\int d\tau a^{2}(\tau)}.$ (27)

The partition function above presents a bosonized version of the Thirring
model.

To build up the solid groundwork for the discussion of the (2+1)-dimensional
case, we also need general current correlation functions of massless scalar
and fermionic theories. The expressions are known, but for the sake of
completeness of the narrative these results are derived in Appendix B. With
this and bosonization in lower dimensions in mind, we transition to the
bosonization occurring in (2+1)$d$. As before, the conversation is opened by
the fermionic case. While there are similarities when approaching the (1+1)$d$
and (2+1)$d$ cases, as was hinted above they are not fully analogues. For
instance, the same action of Eq. (9) taken in 2+1 dimensions does not have a
chiral symmetry, since $\gamma_{5}$ matrices can only be defined for even
dimensions of spacetime. However, just as in 0+1 dimensions, the theory
features a parity anomaly, since the effective action (10) in 2+1 dimensions
receives a factor $(-1)^{n}$ under a topologically nontrivial gauge
transformation, with $n$ being the winding number introduced in Eq. (19) and
associated to the nontrivial topology Redlich (1984b).

###  III.2 The Thirring model in 2+1 dimensions

A bosonization procedure for the Thirring model in 2+1 dimensions consists in
performing a straightforward Hubbard-Stratonovich transformation to introduce
an auxiliary vector field $a_{\mu}$,

${\cal L}^{\prime}_{\rm Th}=\bar{\psi}(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial+i\hbox to0.0pt{\hbox
to5.28589pt{\hfil/\hfil}\hss}a+M)\psi+\frac{1}{2\lambda}a_{\mu}^{2}.$ (28)

Integrating out fermions generates the following effective Lagrangian Fradkin
and Schaposnik (1994) in the regime where the mass $M$ is large,

${\cal L}_{\rm eff}=\frac{1}{2\lambda}a_{\mu}^{2}+\frac{i{\rm
sgn}(M)}{8\pi}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}+\frac{1}{48\pi|M|}(\epsilon_{\mu\nu\lambda}\partial_{\nu}a_{\lambda})^{2}.$
(29)

This effective Lagrangian follows directly from the large mass expansion of
the current correlation function for free fermions of Eq. (B.2) in Appendix
B.2, since it also corresponds to the one-loop vacuum polarization.

A couple of remarks are in order. First, we note that the Lagrangian (29) is
not gauge invariant because of the term $\sim a_{\mu}^{2}$. However, even in
the absence of such a term (for instance, in the limit $\lambda\to\infty$),
gauge invariance is lost if one allows for topologically nontrivial gauge
transformations, since the CS term is half-quantized in the above calculation
Redlich (1984a). Strictly speaking, such large gauge transformations are not
required in the case of the Thirring model, since $a_{\mu}$ here is a non-
compact auxiliary field. Nevertheless, let us take the opportunity to consider
this strong coupling regime and the possibility of allowing for large gauge
transformations, since these are of paramount importance in most contexts
considered in this paper. In fact, this leads us to consider Pauli-Villars
regularization in the form,

$S_{\text{eff}}=-\operatorname{Tr}\ln(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial+M-i\hbox to0.0pt{\hbox
to5.28589pt{\hfil/\hfil}\hss}a)+\operatorname{Tr}\ln(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-M-i\hbox to0.0pt{\hbox
to5.28589pt{\hfil/\hfil}\hss}a),$ (30)

assuming $M>0$ infinitely large such as to suppress the Maxwell term and turn
the generated CS term integer-quantized. In this way, we obtain,

${\cal L}^{\prime}_{\rm
eff}=\frac{1}{2\lambda}a_{\mu}^{2}+\frac{i}{4\pi}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda},$
(31)

which is invariant under topologically nontrivial gauge transformations when
$\lambda\to\infty$.

The field equation for the effective Lagrangian (31) expresses that the theory
is self-dual (in the sense of field equations) Townsend  _et al._ (1984);
Deser and Jackiw (1984),

$a_{\mu}=-i\frac{\lambda}{2\pi}\epsilon_{\mu\nu\lambda}\partial_{\nu}a_{\lambda}.$
(32)

The above self-duality may be explored further by introducing an auxiliary
field $b_{\mu}$ to rewrite (31) as,

$\mathcal{L}^{\prime\prime}_{\rm
eff}=\frac{\lambda}{8\pi^{2}}(\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{\lambda})^{2}+\frac{i}{4\pi}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}\left(a_{\lambda}-b_{\lambda}\right).$
(33)

Integrating out $a_{\mu}$ yields then the equivalent dual effective
Lagrangian,

$\widetilde{L}_{\rm
eff}=\frac{\lambda}{8\pi^{2}}(\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{\lambda})^{2}-\frac{i}{4\pi}\epsilon_{\mu\nu\lambda}b_{\mu}\partial_{\nu}b_{\lambda},$
(34)

which is gauge invariant. Hence, similarly to the (1+1)-dimensional case, the
massive Thirring model offers a path to bosonization in 2+1 dimensions as
well. However, the result is not exact in the latter case. Nevertheless, we
see already here that the level 1 CS term appears naturally in the
(2+1)-dimensional bosonization to preserve the gauge invariance. This
subsection highlights the interplay between gauge noninvariance, parity
anomaly and the role CS term plays in both of them.

##  IV The critical point and particle-vortex duality

By this point, we touched on the correspondence between the current
correlation functions occurring in (1+1)-dimensional dualities in Section II.
We now generalize this discussion as well as examine the (2+1)-dimensional
case in detail. Current correlators describe the behavior of a theory in the
near-critical regime, and so as dual theories are expected to behave similarly
the notions of criticality and duality are intertwined. This section will
demonstrate this connection through the bosonic particle-vortex duality as
well as consider some of the more intricate procedures involved.

###  IV.1 Scale invariance and current conservation

It is well known that scale invariance and current conservation imply that the
scaling behavior of the current correlation function does not have an
anomalous dimension Collins (1985); Kovtun and Ritz (2008). This means that if
a theory features a critical point, the power-law behavior of the current
correlation function is determined by dimensional analysis. Thus, the power
law obtained for the free theory is also valid for the interacting case at an
RG fixed point, therefore, only the amplitude is changed. To account for this
possibility, we generalize the current correlation functions of a free
massless bosonic theory in momentum space calculated in Eq. (147) of Appendix
B to be,

$C_{\mu\nu}(q)=-vc_{d}|q|^{d-2}\left(\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right),$
(35)

where $v$ is a universal amplitude equal to unity for a free scalar theory and
$c_{d}$ is a coefficient defined by the dimensionality of the theory and given
explicitly in Eq. (140).

Interestingly, as is shown in Appendix B, the case of $v=1$ also corresponds
to the one-loop vacuum polarization of the critical Higgs theory that we will
encounter further in Eq. (41). This one-loop value changes for a
multicomponent Higgs theory. For instance, in the case of a $O(2N)$-invariant
theory, where $N$ is the number of complex fields, we have, $v=N$, since one
has to sum over all $N$ complex scalar loops. This is the well-known leading
order large $N$ result.

As was discussed in the previous section, fermionic currents are intrinsically
dependent on the structure of $\gamma_{\mu}$ matrices, therefore, the
generalization to $d$ dimensions when calculating the current correlation
functions is tiresome. Nevertheless, in 2+1 dimensions the current correlation
function of free massless Dirac fermions coincides with the bosonic one
presented in Eq. (35) for $v=1$ (see Appendix B). The crucial feature of the
$d=3$ case lies in the fact that in the most general situation one has to
account for a possibility of the parity symmetry breaking in the form of a CS
term Witten ,

$C_{\mu\nu}(p)=-\frac{|p|}{16}\left[v\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+w\epsilon_{\mu\nu\lambda}\frac{p_{\lambda}}{|p|}\right],$
(36)

where $w$ is another universal amplitude. This equation will be shown to be
consistent with our original calculations in 2+1 dimensions in Sections V and
VI.

###  IV.2 Particle-vortex duality and the existence of a critical point

Current correlation functions at criticality were shown to yield universal
results reflecting the duality as fermionic and bosonic correlators possess
the same form. Let us demonstrated now how duality can be used as a tool to
claim the existence of a critical point. Particle-vortex duality in 2+1
dimensions is performed exactly on the lattice using a Villain form of the
$XY$ model Peskin (1978); Thomas and Stone (1978). In the version discussed
originally by Peskin Peskin (1978), it is shown that the $XY$ model is
equivalent by duality to a lattice Abelian Higgs model with infinite phase
stiffness in Euclidean spacetime. In a classical statistical mechanics sense
this was treated as a zero temperature limit in Ref. Peskin (1978), which was
dubbed ”frozen superconductor”. On the other hand, the temperature of the
original $XY$ model becomes the inverse of the gauge coupling constant $e^{2}$
in the dual model. This line of reasoning ultimately led to the conclusion in
a later work by Dasgupta and Halperin Dasgupta and Halperin (1981) that the
Abelian Higgs model (in the form of a lattice Ginzburg-Landau superconductor
Kleinert (1989)) features a critical point, thus undergoing a second-order
phase transition. This result is very hard to obtain using RG methods (see,
for instance, Ref. Ihrig  _et al._ (2019) for a recent account and the closing
paragraph of this subsection). More sophisticated numerical studies Hove and
Sudbø (2000) have confirmed the duality scenario of earlier papers Peskin
(1978); Thomas and Stone (1978); Dasgupta and Halperin (1981); Kleinert
(1982), including an analysis of the scaling dimensions. This convincingly
confirms by means of lattice simulations the validity of the scale invariant
form of the current correlation function as given in Eq. (36) with $w=0$
(i.e., corresponding to the situation where parity symmetry breaking is
absent). Furthermore, since the renormalization of the gauge coupling is well
known to be given in terms of the vacuum polarization by,

$e^{2}(p)=\frac{e^{2}}{1+e^{2}\Pi(p)},$ (37)

Eq. (36) implies, $e^{2}(p)=e^{2}/[1+e^{2}v/(16|p|)]$. We obtain that the
dimensionless coupling $\hat{e}^{2}(p)=e^{2}(p)/|p|$ reaches the IR stable
fixed point, $\hat{e}^{2}_{*}=16/v$, as $|p|\to 0$. We conclude that this
theory is strongly coupled as $e^{2}\to\infty$ causes the dimensionless
coupling to flow to the same fixed point. In this case, the bare coupling
constant $e^{2}$ serves as a UV scale. The same line of argument holds for
fermionic QED in 2+1 dimensions Appelquist  _et al._ (1986a).

The discussion above allows us to establish the field theory version of the
bosonic particle-vortex duality on the lattice as the equality of the
partition functions in the IR regime, i.e., the one in which $e^{2}\to\infty$,
so the Maxwell term can be suppressed 222However, the Maxwell term is
frequently needed for obtaining more rigorous results, since it provides a UV
regulator, also within the bosonization duality context involving Chern-Simons
actions. See, for instance, Ref. Shyta  _et al._ (2021a). In order to account
for the current response, a background field has to be included in the
corresponding partition functions. The way to proceed is already thoroughly
discussed in Peskin’s seminal work Peskin (1978) and leads to the inclusion of
what is now known as a BF term,

$S_{BF}[a;A]=\frac{i}{2\pi}\int
d^{3}xa_{\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}A_{\lambda},$ (38)

where $A_{\mu}$ denotes a background gauge field and $a_{\mu}$ represents a
dynamical gauge field. As the field theory action associated to the $XY$ model
is simply a $|\phi|^{4}$ field theory Zinn-Justin (2002), this bosonic field
theory in the presence of background field $A_{\mu}$ is defined via,

$\displaystyle Z_{bQED}[A]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\phi^{*}\mathcal{D}\phi e^{-S_{bQED}[A]},$ (39)
$\displaystyle S_{bQED}[A]$ $\displaystyle=$ $\displaystyle\int
d^{3}x\left[|(\partial_{\mu}-iA_{\mu})\phi|^{2}+m^{2}|\phi|^{2}+\frac{\lambda}{2}|\phi|^{4}\right].$

When the background field vanishes, $A_{\mu}=0$, the partition function
$Z_{\rm bQED}$ is just the field theory version of the $XY$ model, and we can
write $Z_{bQED}[A=0]=Z_{XY}$. Following the prescription above regarding an
additional BF term, we have Seiberg  _et al._ (2016); Karch and Tong (2016),

$\displaystyle Z_{bQED}[A]=\int\mathcal{D}\phi^{*}\mathcal{D}\phi
e^{-S_{bQED}[A]}$ (40) $\displaystyle=$
$\displaystyle\lim_{e^{2}\to\infty}\int\mathcal{D}\varphi^{*}\mathcal{D}\varphi\mathcal{D}a_{\mu}e^{-\int
d^{3}x\mathcal{L}_{\rm sQED}[a]+S_{BF}[a;A]}$ $\displaystyle=$ $\displaystyle
Z_{sQED}[A]$

where

$\mathcal{L}_{\rm
sQED}[a]=\frac{1}{4e^{2}}F_{\mu\nu}^{2}+|(\partial_{\mu}-ia_{\mu})\varphi|^{2}+m^{2}|\varphi|^{2}+\frac{u}{2}|\varphi|^{4}$
(41)

is the usual scalar QED Lagrangian. Note the difference in the two QED models
yielding partition functions denoted $Z_{bQED}[A]$ and $Z_{sQED}[A]$. In the
former the fluctuating scalar field is coupled minimally to a (classical)
background field $A_{\mu}$, while in the latter both the scalar and the
minimally coupled gauge field are being integrated over. In $Z_{sQED}[A]$ the
background field couples to the fluctuating gauge field via a BF term.

Equation (40) is interesting, since it establishes the particle-vortex duality
as something that mimics dual variables in ordinary quantum mechanics, the
most prosaic example being given by the duality between the position and
momentum operators via the commutator, $[x,p]=i$, which is also reflected in
terms of a plane wave basis via Fourier transforms of wave functions. In the
functional integral of Eq. (40) the phase factor contains the BF term, which
plays a role similar to the phase factor in an ordinary Fourier transform
underlying the particle-wave duality in quantum mechanics. In this sense, the
partition function of Eq. (39) acquires the Fourier representation in terms of
fluxes as represented in Eq. (40). Note that by promoting $A_{\mu}$ to a
dynamical gauge field $b_{\mu}$ on the RHS of Eq. (40), adding a BF term
coupling $b_{\mu}$ to a new background field $B_{\mu}$, trivially yields the
LHS upon subsequently integrating out $b_{\mu}$, which enforces
$a_{\mu}=B_{\mu}$ (note that this simple calculation can also be done assuming
$e^{2}$ finite).

Along a similar vein, a fermionic analog of the bosonic particle-vortex
duality expressed in Eq. (40) has been proposed Son (2015); Metlitski and
Vishwanath (2016). In the absence of parity symmetry breaking this amounts to
a replacement of the scalar QED by a fermionic QED in Eq. (40), along with the
switching to corresponding path integrals over spinor (Grassmann) fields.

From Eq. (40) it follows that the current correlation function of the $XY$
model field theory,

$C_{\mu\nu}(x-x^{\prime})=\left.\frac{\delta\ln Z_{bQED}[A]}{\delta
A_{\mu}(x)\delta A_{\nu}(x^{\prime})}\right|_{A=0},$ (42)

is connected to the correlation function
$\mathcal{K}_{\mu\nu}(x-x^{\prime})=\langle
J_{\mu}(x)J_{\nu}(x^{\prime})\rangle$ of the topological current,

$J_{\mu}=\frac{i}{2\pi}\varepsilon_{\mu\nu\lambda}\partial_{\nu}a_{\lambda}.$
(43)

These correlation functions will be explored in a more quantitative fashion in
the next section.

Let us discuss the limit $e^{2}\to\infty$ further. Integrating out $a_{\mu}$
exactly on the RHS of Eq. (40) yields the effective Lagrangian,

$\displaystyle\mathcal{L}_{\rm eff}$ $\displaystyle=$
$\displaystyle|\partial_{\mu}\varphi|^{2}+m^{2}|\varphi|^{2}+\frac{u}{2}|\varphi|^{4}$
(44) $\displaystyle+$
$\displaystyle\frac{\left(\varphi^{*}\partial_{\mu}\varphi-\varphi\partial_{\mu}\varphi^{*}-\frac{\epsilon_{\mu\nu\lambda}\partial_{\nu}A_{\lambda}}{2\pi}\right)^{2}}{4|\varphi|^{2}}.$

When the background field is set to zero, the last term of Eq. (44) represents
a (perturbatively) non-renormalizable interaction in 2+1 dimensions, as near a
$U(1)$ broken symmetry state the denominator $|\varphi|^{2}\sim{\rm const}$ at
leading order. This is an irrelevant operator in the RG sense, so essentially
upon integrating $a_{\mu}$ out in the RHS of Eq. (40) we indeed obtain
(asymptotically) its LHS.

The discussion of the previous paragraph glosses over the effect of the
functional measure arising in the path integral after integrating out
$a_{\mu}$ in the limit $e^{2}\to\infty$. In fact, after integrating out
$a_{\mu}$ the measure becomes formally,
$\mathcal{D}\varphi^{*}\mathcal{D}\varphi/|\varphi|$ (implicit in this
expression is a formal infinite product of the form,
$\prod_{x}[d\varphi^{*}(x)d\varphi(x)/|\varphi(x)|]$). Although this has no
major consequences perturbatively, it will be important in the following
manipulations of the path integral.

Let us now express the complex scalar field in terms of an amplitude and a
phase as, $\varphi=\rho e^{i\theta}/\sqrt{2}$, with the result,

$\displaystyle\mathcal{L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(\partial_{\mu}\rho)^{2}+m^{2}\rho^{2}+\frac{u}{4}\rho^{4}\right]$
(45) $\displaystyle+$
$\displaystyle\frac{1}{8\pi^{2}\rho^{2}}(\epsilon_{\mu\nu\lambda}\partial_{\nu}A_{\lambda})^{2}-\frac{i}{2\pi}\partial_{\mu}\theta\epsilon_{\mu\nu\lambda}\partial_{\nu}A_{\lambda},$

while the path integral measure becomes
$\mathcal{D}\varphi^{*}\mathcal{D}\varphi/|\varphi|=\prod_{x}[d\rho(x)d\theta(x)]$.
Partial integration of the last term of the effective Lagrangian (45) in the
action leads to a direct coupling of $A_{\mu}$ to loops of vortex lines, since
periodicity of $\theta$ implies that the spacetime curl of
$\partial_{\mu}\theta$ does not vanish along vortex lines, yielding instead
Kleinert (1989),

$\epsilon_{\mu\nu\lambda}\partial_{\nu}\partial_{\lambda}\theta=2\pi\sum_{a}n_{a}\oint_{L_{a}}dy_{\mu}^{(a)}\delta^{3}(x-y^{(a)}),$
(46)

where $n_{a}\in\mathbb{Z}$ and $L_{a}$ denote vortex loop paths. Thus, the
Wilson loop factors arise in the partition function,

$\prod_{a}e^{i2\pi n_{a}\oint_{L_{a}}dy_{\mu}^{(a)}A_{\mu}(y^{(a)})},$ (47)

with the functional integral over the phase $\theta(x)$ becoming a path
integral over all possible particle trajectories along the loops $L_{a}$.
Therefore, the current correlation function can be expressed in terms of a
vortex loop correlation function, i.e., a correlation function between
topological defects. The lattice version of this statement can already be
encountered in Ref. Peskin (1978). Here we are rephrasing it in terms of a
continuum field theory approach, which also highlights the role played by
amplitude fluctuations with respect to the background field: from Eq. (45) we
note that $\rho^{2}$ plays the role of a gauge coupling in the Maxwell term
for the background field.

The correctness of the effective Lagrangian (45) can be cross checked by
showing that it indeed yields the LHS of Eq. (40) after promoting the
background field $A_{\mu}$ to a dynamical gauge field $b_{\mu}$, adding a BF
term $S_{BF}[b;B]$ in the process, with $B_{\mu}$ being a new background
field. We have already seen that this is a trivial calculation if done before
integrating out $a_{\mu}$, since $b_{\mu}$ in this case simply acts as a
Lagrange multiplier field. It should not matter whether this is done before or
after integrating out $a_{\mu}$. To see this, we integrate out $b_{\mu}$
exactly, but the presence of the coefficient $1/\rho^{2}(x)$ multiplying the
Maxwell term makes it necessary to proceed with some care. First, we insert
the identity,

$\int\mathcal{D}s_{\mu}\delta(s_{\mu}-\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{\lambda})=1,$
(48)

and use the integral representation of the delta function above, which makes a
Lagrange multiplier field $\lambda_{\mu}$ appear in the action,

$\displaystyle\mathcal{L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[(\partial_{\mu}\rho)^{2}+m^{2}\rho^{2}+\frac{u}{4}\rho^{4}\right]$
(49) $\displaystyle+$
$\displaystyle\frac{1}{8\pi^{2}\rho^{2}}s_{\mu}^{2}+is_{\mu}\left(\frac{B_{\mu}}{2\pi}-\frac{\partial_{\mu}\theta}{2\pi}-\lambda_{\mu}\right)$
$\displaystyle+$ $\displaystyle
i\lambda_{\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{\lambda}.$

Integrating out $b_{\mu}$ leads to the constraint
$\epsilon_{\mu\nu\lambda}\partial_{\nu}\lambda_{\mu}=0$, which is immediately
solved as $\lambda_{\mu}=\partial_{\mu}\omega$ and trivially gauged away via
$\theta\to\theta-2\pi\omega$. Finally, integrating over $s_{\mu}$ yields,

$\mathcal{L}_{XY}=\frac{1}{2}\left[(\partial_{\mu}\rho)^{2}+m^{2}\rho^{2}+\frac{u}{4}\rho^{4}\right]+\frac{\rho^{2}}{2}\left(\partial_{\mu}\theta-
B_{\mu}\right)^{2},$ (50)

while restoring the Jacobian factor back to
$\prod_{x}[d\rho(x)\rho(x)d\theta(x)]$. The result can obviously be rewritten
back in the form (39) by reintroducing the complex fields via $\phi=\rho
e^{i\theta}/\sqrt{2}$.

We note here the power of the duality in establishing the existence of a
critical point, something otherwise difficult to obtain by purely perturbative
means Kolnberger and Folk (1990); Folk and Holovatch (1996); Ihrig  _et al._
(2019) (for a non-perturbative analysis of the problem using the functional
RG, see Refs. Bergerrhoff  _et al._ (1996); Bergerhoff  _et al._ (1996)).
Although the existence of an IR fixed point for a properly defined (i.e.,in
terms of some chosen scaling variable) dimensionless gauge coupling is easy to
obtain at one-loop order, the same does not hold with the $|\phi|^{4}$
coupling, since the one-loop calculation requires a number $N$ of complex
field components to be larger than 183 Halperin  _et al._ (1974); Hikami
(1979). This critical number of fields gets significantly reduced in a four-
loop calculation, but an IR stable fixed point is still not found even in this
case Ihrig  _et al._ (2019). Extensive numerical simulations were made in a
recent work Bonati  _et al._ (2021) for several values of $N$ (specifically,
$N=2,4,10,15,25$), starting with $N=2$, conclusively finding a second-order
phase transition for $N\geq 10$. Importantly, a first-order phase transition
is found for the $N=2$ case Bonati  _et al._ (2021), leading to an agreement
with earlier RG results Halperin  _et al._ (1974); Hikami (1979). Thus, it is
quite remarkable that the field theory duality can establish the existence of
a critical point for the single complex field case in the strong-coupling
regime $e^{2}\to\infty$. As a side remark, let us point out that
perturbatively even the RG flow for the dimensionless gauge coupling
introduces difficulties at higher loop order. For instance, while at one-loop
a an IR fixed point is found for any number of Higgs scalars, the same fails
to be true at two-loops: the $\epsilon$-expansion for the dimensionless gauge
coupling yields a fixed point provided more than 18 Higgs fields are available
Kolnberger and Folk (1990); Folk and Holovatch (1996). However, the duality
result implies that such an IR fixed point must exist non-perturbatively for a
single Higgs scalar. This in turn implies the scaling behavior of the current
correlation function. Most importantly, no detailed information is needed
about the RG flow of the $|\phi|^{4}$ coupling, which underlies most of the
difficulties involving the establishment of Wilson-Fisher critical behavior
Halperin  _et al._ (1974); Hikami (1979); Herbut and Tešanović (1996); Ihrig
_et al._ (2019). Importantly, the vanishing of the RG $\beta$ function for the
gauge coupling is well known to exactly determine the anomalous dimension of
the gauge field, which is in this case given by $4-d$ Herbut and Tešanović
(1996); Bergerhoff  _et al._ (1996); Calan and Nogueira (1999); Hove and Sudbø
(2000). Thus, since the gauge field couples linearly to the conserved current,
the absence of an anomalous dimension in the latter follows immediately,
leading in this way to a current correlator given by Eq. (35).

###  IV.3 From background to dynamical field via fluctuating vortex loops

The discussion of the particle-vortex duality in the previous subsection
involved the promotion of the background gauge field to a dynamical one. Let
us explore the meaning of this procedure and physical motivation behind it in
terms of vortex loops.

In a regime where $\rho$ is approximately uniform, we can adopt a different
point of view about promoting $A_{\mu}$ to a dynamical field that emphasizes
the role of vortex loops. In this sense, promoting $A_{\mu}$ to a dynamical
gauge field corresponds to solving the field equation for $A_{\mu}$ as,

$A_{\mu}(x)=B_{\mu}(x)+i\frac{\rho^{2}}{2}\sum_{a}n_{a}\oint_{L_{a}}\frac{dy_{\mu}^{(a)}}{|x-y^{(a)}|},$
(51)

where $B_{\mu}(x)$ is a non-singular classical background contribution that
does not depended on vortex loops. At first sight, the fact that this solution
is complex might seem problematic, since $A_{\mu}$ is supposed to be real.
However, this is actually not a problem at all, since the imaginary
contribution is interpreted in the sense of quantum fluctuations in an
ensemble of vortex loops. After all, performing a Gaussian integral in this
case amounts to simply solving the field equations and plugging the solution
(51) back into the action. In this way the Wilson loop term generates a
contribution that yields the interaction between vortex loops in the effective
action,

$\displaystyle S_{\rm Vortex-loop-int}=\pi\rho^{2}\sum_{a,b}n_{a}n_{b}$
$\displaystyle\times\oint_{L_{a}}dy_{\mu}^{(a)}\oint_{L_{b}}dy_{\mu}^{(b)}\frac{1}{|y^{(a)}-y^{(b)}|}.$
(52)

Such a vortex interaction term naturally appears in lattice Villain models
Savit (1980). In the particular case of Eq. (IV.3), it just corresponds to the
continuum version of the vortex loop representation of the Villain form of the
$XY$ model in 2+1 dimensions Dasgupta and Halperin (1981). The analysis above
emphasizes at the end the precise meaning of the particle-vortex duality,
since we ultimately have to perform a path integral over vortex loops that are
direct worldlines of particle trajectories. Also, this subsection, in
particular Eq. (51), motivates the notion of promoting a background gauge
field to be dynamical.

##  V Bosonization duality and critical current correlators in the
$U(1)$-symmetric case

In the previous section we demonstrated how the existence of a critical regime
can follow from the particle-vortex duality. As the end-goal of this work is
to examine the bosonization duality between the CS $U(1)\times U(1)$ Higgs
theory and interacting massless fermions, this section sets up the necessary
methods on a simpler model. Here we will obtain the bosonization duality for
the CS $U(1)$-symmetric Higgs theory employing the flux attachment approach.
Moving further, we will show the way in which the duality is reflected in the
relation between current correlation functions corresponding to different
sides of the duality.

###  V.1 Bosonization via flux attachment

The flux attachment technique is very powerful in its simplicity as it uses
the two basic bosonization equations as building blocks of the duality web
Karch and Tong (2016); Chen  _et al._ (2018),

$Z_{fQED}[A]e^{-\frac{1}{2}S_{CS}[A]}=Z_{bQED+flux}[A],$ (53)
$Z_{fQED+flux}[A]=Z_{bQED}[A]e^{S_{CS}[A]},$ (54)

where $S_{CS}[A]$ is the action for a level 1 CS term,

$S_{CS}=\frac{i}{4\pi}\epsilon_{\mu\nu\lambda}A_{\mu}\partial_{\nu}A_{\lambda},$
(55)

and $A_{\mu}$ is the background field. Let us describe the elements arising in
Eqs. (53) and (54) in detail. While the bosonic partition function was
introduced in Eq. (39), the fermionic side is given by,

$\displaystyle Z_{fQED}[A]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi e^{-S_{fQED}[A]},$
$\displaystyle S_{fQED}[A]$ $\displaystyle=$ $\displaystyle\int
d^{3}x\bar{\psi}(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-i\hbox to0.0pt{\hbox
to7.50002pt{\hfil/\hfil}\hss}A)\psi.$ (56)

The fermionic flux attachment is defined as,

$\displaystyle Z_{fQED+flux}[A]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}a_{\mu}Z_{fQED}[a]e^{-\frac{1}{2}S_{CS}[a]-S_{BF}[a;A]},$

while for bosons one has,

$\displaystyle Z_{bQED+flux}[A]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}a_{\mu}Z_{bQED}[a]e^{S_{CS}[a]+S_{BF}[a;A]},$

with the BF term defined in Eq. (38).

Equation (54) can easily be shown to follow from Eq. (53) Tong (2018), so
actually the latter can be regarded as the fundamental result out of which all
other dualities belonging to the duality web are derived Seiberg  _et al._
(2016). Conversely, a similar reasoning would allow us to derive Eq. (53)
starting from (54), so it would have been equally possible to assume the
latter as a seed to all other dualities as well. This way one can obtain the
exact particle-vortex duality discussed in Sect. IV.2 from one of the flux
attachment conjectures and its time-reversed version. Such an elegant approach
is also consistent with other well established dualities and can be used to
provide new reliable results.

Note that the flux attachment for fermions operates slightly differently from
the bosonic case. Indeed, while the bosons get a level 1 CS term, the
fermionic flux attachment features a level 1/2 CS term. The reason for this is
the parity anomaly Redlich (1984a, b). We have encountered this problem before
in our discussion of the Thirring model in 2+1 dimensions. It is now the
moment of being more thorough on this point. It turns out that massless Dirac
fermions are invariant under TR and inversion of one of the two spatial
coordinates, but not both, since in two spatial dimensions this would
correspond to a rotation by $\pi$. In Euclidean three-dimensional spacetime
both symmetries translate into parity symmetry, corresponding to an inversion
of any of the three coordinates. It turns out that quantum fluctuations (i.e.,
integrating out the fermions in a gauge field background) would either break
gauge invariance or the parity Redlich (1984b). The solution to this problem
is well known. Just like in the (1+1)-dimensional case, we have to employ
Pauli-Villars regularization in the fermionic theory.

The effective action for free massless fermions can be obtained assuming
dimensional regularization for the parity even component of current
correlation function (as shown in Eq. (B.2) in Appendix B). Nevertheless, this
type of regularization is not suitable to obtain a CS term in the massless
case. In order to do so, let us consider the theory with an additional level
1/2 CS gauge term and integrate out fermionic fields. In this case, the
following effective action is obtained

$S_{\text{eff}}=-\operatorname{Tr}\ln(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-i\hbox to0.0pt{\hbox
to7.50002pt{\hfil/\hfil}\hss}A)+\operatorname{Tr}\ln(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-M-i\hbox to0.0pt{\hbox
to7.50002pt{\hfil/\hfil}\hss}A)+\frac{1}{2}S_{CS}(A),$ (59)

where $M>0$. The second tracelog in Eq. (59) arises from the Pauli-Villars
regulator fields. When the limit $M\to\infty$ is taken the regularized theory
yields an additional level 1/2 CS contribution that results in an overall
level 1 CS term. Were we add a level -1/2 CS term, no CS term at all would be
present in the effective action. We have noticed a similar behavior within the
simple setting in 0+1 dimensions, see, for instance, Eq. (20). Note that
nothing like this happens in the bosonic theory, since scalar field
fluctuations do not generate any anti-symmetric contribution in the vacuum
polarization.

Curiously, from Eqs. (40) and (54) we can write,

$\displaystyle Z_{fQED+flux}[A]e^{-S_{CS}[A]}=Z_{bQED}[A]$ (60)
$\displaystyle=$
$\displaystyle\int\mathcal{D}a_{\mu}Z_{bQED}[a]e^{S_{BF}[a;A]},$

so we note that for $A_{\mu}=0$ there must be no parity symmetry breaking
occurring in $Z_{fQED+flux}[A=0]$, in spite of the (dynamical) CS term in Eq.
(V.1), since $Z_{bQED}[A=0]$ does not break parity symmetry. This is
consistent with the discussion above. Although the classical fermionic action
seem to break parity, the latter is actually not broken at the quantum level
Seiberg  _et al._ (2016).

###  V.2 Current correlation functions

With the duality (53) in mind, let us calculate the current correlators for
bosonic and fermionic sides. We will start with the former by first
integrating over the scalar field $\phi$ at one-loop order in Eq. (V.1). Then,
the CS $U(1)$ Higgs action can be rewritten as a quadratic form,

$S_{B}=\frac{1}{2}\int_{p}a_{\mu}(p)M_{\mu\nu}(p)a_{\nu}(-p),$ (61)

with

$M_{\mu\nu}(p)=\frac{|p|}{16}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)-\frac{1}{2\pi}\varepsilon_{\mu\nu\lambda}p_{\lambda}+\frac{|p|}{\alpha}\frac{p_{\mu}p_{\nu}}{p^{2}},$
(62)

where the last term contains a gauge fixing constant $\alpha$. Inverting the
tensor we obtain a propagator for $a_{\mu}$,

$D_{\mu\nu}(p)=\frac{\pi^{2}}{4\left(\frac{\pi^{2}}{64}+1\right)|p|}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}+\frac{8}{\pi|p|}\varepsilon_{\mu\nu\lambda}p_{\lambda}\right),$
(63)

written here in the Landau gauge $\alpha=0$.

The theory features the topological current of Eq. (43), and one can thus see
that the current correlation function in this case is directly derived from
the propagator $D_{\mu\nu}$ and yields in momentum space,

$\displaystyle\mathcal{K}_{\mu\nu}^{B}(p)=\left\langle
J_{\mu}(p)J_{\nu}(-p)\right\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{\frac{\pi^{2}}{64}+1}\left[\frac{|p|}{16}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\right.$
(64) $\displaystyle+$
$\displaystyle\left.\frac{1}{2\pi}\varepsilon_{\mu\nu\lambda}p_{\lambda}\right].$

The obtained topological current correlator has to be compared to the one on
the fermionic side of the bosonization duality in Eq. (53). To do so, we
recall that the current correlation function of a free massless fermion is
given by Eq. (36) with $v=1$ and $w=0$ (see Eq. (156) in Appendix B.2). Note
that an anti-symmetric contribution to the current correlator is generated by
the proper Paulli-Villars regularization as shown in Eq. (59). Finally, the
fermionic side of the duality at hand contains an additional level 1/2 CS
term. Therefore, the fermionic current correlation function reads,

$\mathcal{C}^{F}_{\mu\nu}(p)=-\frac{|p|}{16}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)-\frac{1}{2\pi}\epsilon_{\mu\nu\lambda}p_{\lambda}.$
(65)

Comparing the above expression to the general scale invariant form in Eq.
(36), we see that $v=1$ and $w=8/\pi$. Hence, the topological current
correlation function is proportional to $\mathcal{C}^{F}_{\mu\nu}(p)$,

$\mathcal{K}_{\mu\nu}^{B}(p)=\frac{1}{\frac{\pi^{2}}{64}+1}\mathcal{C}^{F}_{\mu\nu}(p).$
(66)

As was conveyed in Section IV, such a relation between the current correlators
not only supports the validity of the bosonization duality but also points
towards the existence of a critical regime for both theories in the duality.

The interpretation of the coefficient relating topological and Noether current
correlators is a well known manifestation of S-duality in 2+1 dimensions
Witten : the coefficients of the transverse and parity odd components of
$\mathcal{K}_{\mu\nu}^{B}(p)$ are the real and imaginary parts of
$1/(4\pi^{2}\tau)$, where $\tau=-1/16+i/(2\pi)$ is viewed as a complex-valued
coupling constant that gets inverted by the duality. The real and imaginary
parts of $\tau^{*}$ are the transverse and parity odd coefficients of
$\mathcal{C}^{F}_{\mu\nu}(p)$.

##  VI Bosonization duality in $U(1)\times U(1)$ topological Abelian Higgs
model

We have seen by now that bosonization duality for the one-component CS Higgs
model results in current correlation functions on both sides of the relation
coinciding up to a universal prefactor. We showed in the previous section how
the duality between bosonic and fermionic theories in 2+1 dimensions can be
obtained via flux attachment. Now, the discussion above will finally result in
the derivation of the bosonization duality involving the $U(1)\times U(1)$
topological Abelian Higgs model. We will also calculate corresponding current
correlation functions and conclude whether their form implies the existence of
a critical point in the duality. To elaborate on the motivation to study such
a duality, let us first put this model in the condensed matter physics
context.

Some quantum antiferromagnets in two spatial dimensions are describable by
means of a CP1 model with an additional Maxwell term Motrunich and Vishwanath
(2004); Senthil  _et al._ (2004a); Sachdev (2011). In this case the direction
of the magnetization $\bm{n}$ is given in the terms of the CP1 fields $z_{a}$
($a=1,2$) by $\bm{n}=z_{a}^{*}{\mbox{\boldmath$\sigma$}}_{ab}z_{b}$. Such a
model is known to be self-dual when an additional easy-plane anisotropy term,
$Kn_{z}^{2}/2=K(|z_{1}|^{2}-|z_{2}|^{2})^{2}/2$, $K>0$ is included in the
Lagrangian, thus making the theory $U(1)\times U(1)$-symmetric Motrunich and
Vishwanath (2004). Originally derived in the lattice Motrunich and Vishwanath
(2004), this self-duality was rederived in Ref. Karch and Tong (2016) by means
of the bosonization duality technique described in Sec. V.

Recently we have considered a version of such a quantum antiferromagnet with
an additional CS term Shyta  _et al._ (2021a). One of the goals of that study
was to show that the CS term causes the model to undergo a second-order phase
transition, unlike the case where the CS term is absent Kuklov  _et al._
(2006); Kragset  _et al._ (2006). Specifically, the theory has a Lagrangian,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{a=1,2}|(\partial_{\mu}-ia_{\mu})z_{a}|^{2}+\frac{i}{4\pi}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}$
(67) $\displaystyle+$ $\displaystyle
m_{0}^{2}(|z_{1}|^{2}+|z_{2}|^{2})+\frac{u}{2}(|z_{1}|^{4}+|z_{2}|^{4}),$

where $a_{\mu}$ is an emergent gauge field. The Lagrangian above may also be
viewed as a generalization of the theory of Eq. (V.1) to include two complex
fields rather than a single one. This observation will be useful in the
discussion that will follow later on.

The theory is dual to another bosonic theory, which was established through
the standard particle-vortex duality Shyta  _et al._ (2021a), which closely
parallels the dualities performed in the lattice, though it was carried out
directly in the continuum. The main reason for using a continuum approach is
related to the technical difficulties one faces while trying to write a CS
term in the lattice Eliezer and Semenoff (1992); Berruto  _et al._ (2000);
DeMarco and Wen (2021). The dual bosonic field theory obtained in Ref. Shyta
_et al._ (2021a) has the following form yielding a second-quantized
representation for the ensemble of vortex loops Kleinert (1989),

$\displaystyle\mathcal{L}_{dual}$ $\displaystyle=$
$\displaystyle\sum_{I=1,2}\left[|(\partial_{\mu}-ib_{I\mu})\phi_{I}|^{2}+m^{2}|\phi_{I}|^{2}+\frac{\lambda}{2}|\phi_{I}|^{4}\right]$
(68) $\displaystyle-$
$\displaystyle\frac{i}{4\pi}\epsilon_{\mu\nu\lambda}(b_{1\mu}+b_{2\mu})\partial_{\nu}(b_{1\lambda}+b_{2\lambda}).$

We can establish the bosonization duality leading to Eq. (68) using the flux
attachment technique discussed in Sec. V. We use the basic flux attachment
dualities (53) and (54) to derive the bosonization duality for the Lagrangian
of Eq. (67) by simply multiplying both these relations together,

$\displaystyle Z$ ${}_{fQED}[A]e^{-\frac{1}{2}S_{CS}[A]}Z_{fQED+flux}[A]$
$\displaystyle=$ $\displaystyle Z$
${}_{bQED+flux}[A]Z_{bQED}[A]e^{S_{CS}[A]},$ (69)

After we promote the background field $A_{\mu}$ to a dynamical field
$b_{\mu}$, the left-hand side of the Eq. (VI) takes the form,

$\displaystyle\int\mathcal{D}b_{\mu}Z_{fQED}[b]e^{-\frac{1}{2}S_{CS}[b]}Z_{fQED+flux}[b]$
(70) $\displaystyle=$
$\displaystyle\int\mathcal{D}b_{\mu}Z_{bQED+flux}[b]Z_{bQED}[b]e^{S_{CS}[b]},$

which upon renaming the gauge fields, $a_{\mu}\to b_{1\mu}$ [remember that
there is an integral over $a_{\mu}$ in $Z_{fQED+flux}[A]$, recall Eq. (V.1)],
$b_{\mu}\to b_{2\mu}$, yields the fermionic action,

$\displaystyle S_{f}$ $\displaystyle=$ $\displaystyle\int
d^{3}x\left\\{\sum_{I=1,2}\left[\bar{\psi}_{I}(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-i\hbox to0.0pt{\hbox
to5.00002pt{\hfil$b$\hfil}\hss}/_{I})\psi_{I}+\frac{i}{8\pi}b_{I\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{I\lambda}\right]\right.$
$\displaystyle+$
$\displaystyle\left.\frac{i}{2\pi}b_{1\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{2\lambda}\right\\}.$

Proceeding similarly for the right-hand side of the Eq. (VI), we obtain that a
bosonic action with the Lagrangian (68). Therefore, we have obtained that the
derivation given via flux attachments to fermions and bosons reproduces the
result (68) derived in Ref. Shyta  _et al._ (2021a) using particle-vortex
duality in terms of vortex loops.

While promoting the background field to a dynamical gauge field we have
intentionally not introduced a new background field. The reason for this is
that what has been done above would not have given the current response of the
original model of Eq. (67) directly, since both Eqs. (68) and (VI) are models
dual to (67). What we need is to derive the dual fermionic model (VI) directly
from the original action (67). In this way introducing background fields in an
appropriate fashion leads ultimately to a relationship between the conserved
current response of the model (67) to the topological current response of the
fermionic dual model (VI). Hence, we will now show that the fermionic dual
theory follows by starting directly from the Lagrangian (67). This is achieved
by first considering a time-reversed version of the Eq. (54),

$\bar{Z}_{fQED+flux}[A]=Z_{bQED}[A]e^{-S_{CS}[A]},$ (72)

where time reversal changes the signs in the flux attachment as follows,

$\bar{Z}_{fQED+flux}[A]=\int\mathcal{D}a_{\mu}Z_{fQED}[a]e^{\frac{1}{2}S_{CS}[a]+S_{BF}[a;A]}.$
(73)

Then, the expression in Eq. (72) is used twice in the following way,

$\displaystyle\bar{Z}_{fQED+flux}[A-C]\bar{Z}_{fQED+flux}[A+C]e^{S_{CS}[A-C]}$
(74) $\displaystyle=$ $\displaystyle
Z_{bQED}[A-C]Z_{bQED}[A+C]e^{-S_{CS}[A+C]}.$

Both sides of the equation are then multiplied by a BF term introducing a new
background gauge field $B$,

$\displaystyle\int\mathcal{D}a_{\mu}Z_{\rm
bQED}[a-C]Z_{bQED}[a+C]e^{-S_{CS}[a+C]+S_{BF}[a;B]{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-S_{CS}[B]}}$
$\displaystyle=$
$\displaystyle\int\mathcal{D}a_{\mu}\mathcal{D}b_{1\mu}\mathcal{D}b_{2\mu}Z_{\rm
fQED}[b_{1}]Z_{\rm
fQED}[b_{2}]e^{\frac{1}{2}\left(S_{CS}[b_{1}]+S_{CS}[b_{2}]\right)+S_{BF}[b_{1};a+C]+S_{BF}[b_{2};a-C]+S_{CS}[a-C]+S_{BF}[a;B]{-\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}S_{CS}[B]}},$

where the background field $A$ was promoted to be dynamical $A\to a$ and we
have included a CS term for the background field $B$ on both sides of Eq.
(VI). After $a$ is integrated out in the RHS leading to $a=C-(b_{1}+b_{2}+B)$,
we obtain,

$\displaystyle\int\mathcal{D}a_{\mu}Z_{\rm
bQED}[a-C]Z_{bQED}[a+C]e^{-S_{CS}[a+C]+S_{BF}[a;B]{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-S_{CS}[B]}}$
$\displaystyle=$
$\displaystyle\int\mathcal{D}b_{1\mu}\mathcal{D}b_{2\mu}Z_{\rm
fQED}[b_{1}]Z_{\rm
fQED}[b_{2}]e^{-\frac{1}{2}\left(S_{CS}[b_{1}]+S_{CS}[b_{2}]\right)-S_{BF}[b_{1};b_{2}]+2S_{BF}[b_{1};C]-S_{BF}[b_{1}+b_{2};B]-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}2}S_{CS}[B]+S_{BF}[B;C]}.$

Setting $B=C$, the duality takes the form,

$\displaystyle Z_{B}[B]$ $\displaystyle\equiv$
$\displaystyle\int\mathcal{D}a_{\mu}Z_{\rm
bQED}[a-B]Z_{bQED}[a+B]e^{-S_{CS}[a]-2S_{CS}[B]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}b_{1\mu}\mathcal{D}b_{2\mu}\left(\prod_{I=1,2}\mathcal{D}\bar{\psi}_{I}\mathcal{D}\psi_{I}\right)e^{-S_{f}+S_{BF}[b_{1}-b_{2};B]}\equiv
Z_{F}[B].$

The first line of Eq. (VI) reproduces precisely the path integral for the
Lagrangian of Eq. (67) with bosons additionally coupled to the background
field $B_{\mu}$. The second line features the fermionic action $S_{f}$ of Eq.
(VI). Both sides of the expression contain additional terms coupling dynamical
fields to the background field $B_{\mu}$. The notations $Z_{B}[B]$ and
$Z_{F}[B]$ are introduced to refer to the bosonic and fermionic sides of the
expression. It is clear, therefore, that Eq. (VI) expresses an explicit
bosonization duality for the theory in Eq. (67).

Differentiating $\ln Z_{B}[B]=\ln Z_{F}[B]$ with respect to the background
field, $B_{\mu}$, and setting $B_{\mu}=0$ at the end, one obtains the equality
between the expectation values of the currents from both sides of the duality.
The fermionic current is a topological one,

$j^{F}_{\mu}=\frac{i}{2\pi}\epsilon_{\mu\nu\lambda}\partial_{\nu}(b_{2\lambda}-b_{1\lambda}),$
(78)

while the bosonic side yields the conserved current associated to the
$U(1)\times U(1)$ symmetry,

$j^{B}_{\mu}=\sum_{a=1,2}(-1)^{a}[i(z_{a}^{*}\partial_{\mu}z_{a}-z_{a}\partial_{\mu}z_{a}^{*})+2|z_{a}|^{2}a_{\mu}].$
(79)

For current correlation functions for both bosonic and fermionic sides of the
duality (VI) one can write,

$\displaystyle\mathcal{C}^{B}_{\mu\nu}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\left.\frac{\delta^{2}\ln Z_{B}[B]}{\delta B_{\mu}(x)\delta
B_{\nu}(x^{\prime})}\right|_{B=0},$
$\displaystyle\mathcal{K}^{F}_{\mu\nu}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\left.\frac{\delta^{2}\ln Z_{F}[B]}{\delta B_{\mu}(x)\delta
B_{\nu}(x^{\prime})}\right|_{B=0}.$ (80)

It is anticipated that the current correlation functions are equal up to an
overall universal amplitude prefactor. Note that for the fermionic correlation
function non-diagonal gauge field propagators like $\langle
b_{1\mu}(p)b_{2\nu}(-p)\rangle$ are needed. To obtain these we have to
calculate the effective gauge action by integrating the fermions out. There is
a simpler way, however. We can integrate the fermions out before we integrate
$a_{\mu}$ out, i.e., using Eq. (VI) instead. Following this procedure generate
an additional $e^{\frac{1}{2}\left(S_{CS}[b_{1}]+S_{CS}[b_{2}]\right)}$ factor
in the integrand. The integrals over $b_{1\mu}$ and $b_{2\mu}$ have the same
(Gaussian) form, and thus we obtain upon integrating over both $b_{1\mu}$ and
$b_{2\mu}$ the effective action,

$\displaystyle S_{\rm
eff}^{\prime}=\frac{1}{4\pi^{2}}\int_{x}\int_{x^{\prime}}\bar{D}_{\mu\nu}(x-x^{\prime})\epsilon_{\mu\alpha\beta}\epsilon_{\nu\sigma\rho}\partial_{\alpha}\partial^{\prime}_{\sigma}\left[a_{\beta}(x)a_{\rho}(x^{\prime})+B_{\beta}(x)B_{\rho}(x^{\prime})\right]-S_{CS}[a],$
(81)

where in momentum space $\bar{D}_{\mu\nu}(p)=D_{\mu\nu}(-p)$, with
$D_{\mu\nu}(p)$ being given by Eq. (63), and we have set $B=C$ as before. By
applying the prescription given in Eq. (VI), we obtain,

$\mathcal{K}_{\mu\nu}^{F}(x-x^{\prime})=-\left.\frac{\delta^{2}S_{\rm
eff}^{\prime}}{\delta B_{\mu}(x)\delta B_{\nu}(x^{\prime})}\right|_{B=0},$
(82)

and therefore,

$\mathcal{K}_{\mu\nu}^{F}(p)=-\frac{8|p|}{\pi^{2}+64}\left[\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}-\frac{8}{\pi}\frac{\epsilon_{\mu\nu\lambda}p_{\lambda}}{|p|}\right].$
(83)

Similarly, upon integrating out the scalar fields $z_{a}$ (at one-loop), Eq.
(VI) yields,

$\mathcal{C}_{\mu\nu}^{B}(p)=-\frac{|p|}{8}\left[\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}-\frac{8}{\pi}\frac{\epsilon_{\mu\nu\lambda}p_{\lambda}}{|p|}\right].$
(84)

Henceforth we verify the relation,

$\mathcal{K}^{F}_{\mu\nu}(p)=\frac{1}{\frac{\pi^{2}}{64}+1}\mathcal{C}_{\mu\nu}^{B}(p).$
(85)

The universal amplitude relating $\mathcal{C}_{\mu\nu}^{B}(p)$ to
$\mathcal{K}^{F}_{\mu\nu}(p)$ in Eq. (85) is the same as the one appearing in
Eq. (66), with the difference that there the topological current correlator
refers to the bosonic side of the duality and the $U(1)$ conserved current
correlator is associated to the fermionic side of the duality. Actually what
determines the nature of the current correlators (i.e., either a conserved
current associated to global symmetry or a topological current) is how each
side of the duality is coupled to the background field. In Eq. (60) the
background field $A$ is coupled to a $U(1)$ current on the fermionic side of
the duality while it couples to a topological current on the bosonic side. As
a consequence, Eq. (66) ultimately follows from this fact. In the duality of
Eq. (VI), on the other hand, the background field is coupled to the
$U(1)\times U(1)$ conserved current in the bosonic side of the duality, while
the fermionic side couples the background field to the topological current.
Another interesting aspect in this comparison is the presence of a CS term for
the background field on the side where the latter is coupled to the conserved
current following from a global (Noether) symmetry. In this case the
coefficient of the CS term reflects the effective $U(1)$ charge of the theory.
We note that for the Lagrangian (67) we can define an operator
$z_{1}^{*}z_{2}$ that creates a monopole excitation of charge $+1-(-1)=2$,
which reflects the level 2 of the CS term in Eq. (VI).

Working directly with the fermionic side of Eq. (VI) where $a_{\mu}$ has
already been integrated out is more cumbersome, since we have to deal with a
matrix gauge field propagator with matrix elements, $\langle
b_{I\mu}(p)b_{J\nu}(-p)\rangle$. While this is not strictly necessary for the
calculation of the current correlators, it is important for obtaining other
dynamical physical properties (we will encounter a situation in the next
section where this happens, namely, in the study of the fermion self-energy as
obtained from the Schwinger-Dyson equation). Let us therefore discuss this
matrix propagator below.

From the one-loop fermionic vacuum polarization, Eq. (VI) leads to the
effective gauge field action in the form,

$\displaystyle S_{\rm eff}^{\rm gauge}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{p}\left[\begin{array}[]{cc}b_{1\mu}(p)&b_{2\mu}(p)\end{array}\right]\left[\begin{array}[]{cc}\frac{|p|}{16}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)&-\frac{1}{2\pi}\epsilon_{\mu\nu\lambda}p_{\lambda}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr-\frac{1}{2\pi}\epsilon_{\mu\nu\lambda}p_{\lambda}&\frac{|p|}{16}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\end{array}\right]\left[\begin{array}[]{c}b_{1\nu}(-p)\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr b_{2\nu}(-p)\end{array}\right]$ (91)
$\displaystyle+$
$\displaystyle\frac{1}{2\pi}\int_{p}\epsilon_{\mu\nu\lambda}p_{\lambda}[b_{2\mu}(p)-b_{1\mu}(p)]B_{\nu}(-p),$
(92)

where the CS terms cancel due to the level 1/2 CS terms with opposite signs
generated by integrating out fermions.

From Eq. (91) one derives the matrix gauge field propagator in the Landau
gauge,

$\mathbb{D}_{\mu\nu}(p)=\left[\begin{array}[]{cc}D_{\mu\nu}(p)&E_{\mu\nu}(p)\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
E_{\mu\nu}(p)&D_{\mu\nu}(p)\end{array}\right],$ (93)

where diagonal elements of the matrix are,

$\displaystyle D_{\mu\nu}(p)=\langle b_{1\mu}(p)b_{1\nu}(-p)\rangle=\langle
b_{2\mu}(p)b_{2\nu}(-p)\rangle$ (94) $\displaystyle=$
$\displaystyle\frac{16\pi^{2}}{|p|(\pi^{2}+64)}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right),$

and the off-diagonal elements can be calculated to be,

$E_{\mu\nu}(p)=\langle
b_{1\mu}(p)b_{2\nu}(-p)\rangle=\frac{128\pi}{(\pi^{2}+64)p^{2}}\epsilon_{\mu\nu\lambda}p_{\lambda}.$
(95)

The topological fermionic current correlation function corresponding to Eq.
(78) written in terms of the matrix propagator elements is,

$\mathcal{K}_{\mu\nu}^{F}(p)=-\frac{1}{2\pi^{2}}\epsilon_{\mu\alpha\beta}\epsilon_{\nu\sigma\rho}p_{\alpha}p_{\sigma}\left[D_{\beta\rho}(p)-E_{\beta\rho}(p)\right].$
(96)

After a lengthy but straightforward calculation, one obtains once more the
result of Eq. (83).

##  VII Discussion: bosonization and criticality in $U(1)\times U(1)$
symmetric theories

As the $U(1)\times U(1)$ Abelian Higgs model that is most often considered in
the literature does not contain a CS term in its Lagrangian Motrunich and
Vishwanath (2004); Senthil  _et al._ (2004a, b), both its bosonic Motrunich
and Vishwanath (2004) and fermionized Karch and Tong (2016) forms of particle-
vortex duality behave very differently relative to the model discussed in the
previous section. Thus, we will focus the discussion on the critical behavior
of the topological two-component Higgs model and how it compares to the non-
topological version. First, from the standard Villain lattice form of
particle-vortex duality Motrunich and Vishwanath (2004) we easily infer that
the effective continuum dual theory has the form 333We have dropped the
Maxwell terms associated to the gauge fields $b_{I\mu}$, since they do not
essentially contribute in the IR.,

$\displaystyle\widetilde{\mathcal{L}}$ $\displaystyle=$
$\displaystyle\sum_{I=1,2}\left[|(\partial_{\mu}-ib_{I\mu})\phi_{I}|^{2}+m^{2}|\phi_{I}|^{2}+\frac{\lambda}{2}|\phi_{I}|^{4}\right]$
(97) $\displaystyle+$ $\displaystyle\frac{M^{2}}{2}(b_{1\mu}+b_{2\mu})^{2}.$

The gauge invariance of the above dual theory is more restrictive than the one
of Eq. (68). Indeed, while for the latter independent gauge transformations,
$b_{I\mu}\to b_{I\mu}+\partial_{\mu}\xi_{I}$, $\phi_{I}\to
e^{i\xi_{I}}\phi_{I}$, leave the action invariant, we have to use $b_{1\mu}\to
b_{1\mu}+\partial_{\mu}\xi$, $b_{2\mu}\to b_{2\mu}-\partial_{\mu}\xi$,
$\phi_{1}\to e^{i\xi}\phi_{1}$, $\phi_{2}\to e^{-i\xi}\phi_{2}$ in Eq. (97).
Closely related to this is the fact that effectively $M^{2}$ is very large,
since it is proportional to $e^{2}$; recall the closely related discussion in
this context in Sect. IV. This leads effectively to only one gauge field,
ultimately implying that the theory is self-dual Motrunich and Vishwanath
(2004); Senthil  _et al._ (2004b). On the contrary, there is nothing similarly
happening to the mixed CS term arising in Eq. (68). The two gauge fields
appearing there are a robust feature of the theory, which is also reflected in
its broader gauge freedom. There is no need to restrict $\xi_{2}=-\xi_{1}$ in
this case.

The effective reduction to a single gauge field also occurs in the
bosonization duality of the theory leading to the bosonic dual Lagrangian
(97). This contrasts to the fermionic dual theory of Eq. (VI). This fact has
far reaching consequences in the study of possible critical points Wang  _et
al._ (2017); Shyta  _et al._ (2021a). One major difficulty in this case is
whether or not the system becomes critical and this has been a matter of some
controversy in the field of deconfined quantum critical points. Numerical
results typically yield a first-order phase transition for the standard (i.e.,
without a CS term) $U(1)\times U(1)$ Abelian Higgs model Kuklov  _et al._
(2006); Kragset  _et al._ (2006), so quantum criticality would be absent.
Nevertheless, as pointed out recently in Ref. Wang  _et al._ (2017), the
bosonization duality rather suggests that the theory becomes critical. The
dual fermion model in this case is given by the Lagrangian,

$\mathcal{L}_{f}=\sum_{I=1,2}\bar{\psi}_{I}(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-i\hbox to0.0pt{\hbox
to5.28589pt{\hfil/\hfil}\hss}a)\psi_{I},$ (98)

where we have omitted the BF terms. At first sight it seems obvious that this
theory is critical, with a current correlator being given by twice the
expression of Eq. (156) of Appendix B.2, since the above theory features two
massless fermions. This result is the same as the one-loop vacuum polarization
at the critical point (meaning $e^{2}\to\infty$). This corresponds to an IR
stable fixed point in terms of a properly defined dimensionless gauge
coupling. Higher loops do not seem to change this picture. However, non-
perturbative behavior may spoil the criticality, causing a dynamical
generation of a fermion mass Appelquist  _et al._ (1988); Rantner and Wen
(2001); Franz  _et al._ (2002); Herbut (2006, 2002); Gusynin  _et al._ (2003).
This is best seen by generalizing the Lagrangian of Eq. (98) to include $2N$
fermion components rather than two. Grouping the $2N$ two-component Dirac
spinors $\psi_{I}$ into $N$ four-component Dirac spinors $\Psi_{a}$, we can
introduce $4\times 4$ gamma matrices that also satisfy a Clifford algebra,
along with a $\gamma_{5}$-like matrix Appelquist  _et al._ (1986a),
introducing a chiral symmetry reminiscent of the actual chiral symmetry for
massless fermions in even dimensions of spacetime 444Putting the fermions in
the lattice, this chiral symmetry actually manifests itself as a sublattice
symmetry. Graphene constitutes a realization of this type of chiral symmetry;
see, for instance, Ref. Herbut (2006).. A fermion QED in 2+1 dimensions
defined in this way does not break parity if a mass term of the form
$m\sum_{a=1}^{N}\bar{\Psi}_{a}\Psi_{a}$ is added, since in terms of the
original two-component fermions it corresponds to having half of the fermions
with mass $m$ and the other half with mass $-m$. Hence, there is no anti-
symmetric contribution to the vacuum polarization in this case and no CS term
is generated. Since $e^{2}\to\infty$, the gauge field propagator is obtained
by inverting the vacuum polarization. The Schwinger-Dyson equation for the
fermion propagator then is known to imply a dynamically generated mass that
vanishes for $N\geq 32/\pi^{2}\approx 3.24$, a result following from an
approximate self-consistent solution Appelquist  _et al._ (1988). Thus, if one
assumes the validity of this non-perturbative result 555Strictly speaking,
this result is derived by first assuming a large $N$ limit and neglecting
vertex corrections in the Schwinger-Dyson equation. We could in principle
question a critical value of $N$ that is not even large. However, numerical
analyses using staggered fermions in the lattice seem to corroborate the
existence of a critical value of $N$ that is not large Strouthos and Kogut
(2009); see Ref. Braun  _et al._ (2014) for a recent account of this problem
and references therein., we obtain that the theory of Eq. (98) becomes gapped
once non-perturbative quantum fluctuation effects are accounted for. However,
if this is the case, it would be difficult to even claim the existence of a
phase transition at all for $N=2$. On the other hand, there is some evidence
favoring a critical value of $N$ lying in the interval $1<N<2$ Braun  _et al._
(2014); Appelquist  _et al._ (1999); Gusynin and Pyatkovskiy (2016). In this
case, we can safely argue that the bosonization duality rather implies a
quantum critical point for the $U(1)\times U(1)$ symmetric Abelian Higgs
model. However, recent conformal bootstrap calculations Li (2018, 2021);
Albayrak  _et al._ (2022) show that the theory is gapped for $N=1$ (or two
two-component fermions), so the theory is not critical.

It is unclear at the moment whether such an IR form (98) of the dual theory
can actually capture the actual phase structure of the lattice model, which in
this case corresponds to a quantum antiferromagnet with easy-plane anisotropy
Motrunich and Vishwanath (2004, 2008); Shyta  _et al._ (2021b). Furthermore,
so far simulations address rather the so called deep easy-plane limit, so the
phase stiffnesses are the same. On the other hand, there is numerical evidence
for a second-order phase transition when the phase stiffnesses are different
Herland  _et al._ (2010).

The difficulties above are not shared by the $U(1)\times U(1)$ symmetric
Abelian Higgs model with a CS term considered recently in Ref. Shyta  _et al._
(2021a) and analyzed in Sec. VI. Importantly, the fermionic dual theory given
in Eq. (VI) features two gauge fields and a reduction to a single gauge field
does not occur. This theory features also dynamical CS terms, including mixed
ones 666If background fields are included, CS terms involving only background
fields also appear in the theory of Eq. (98) Wang  _et al._ (2017). This does
not affect the critical behavior, however. A one-loop RG analysis of the
original model (67) shows that a critical point indeed exists Shyta  _et al._
(2021a). Most importantly, the fermionic dual action does not lead to
dynamical mass generation. To see this, we consider the Schwinger-Dyson
equation for one of the fermions $\psi_{I}$,

$G^{-1}(p)=i\gamma\cdot
p+\int\frac{d^{3}q}{(2\pi)^{3}}\gamma_{\mu}G(q)\gamma_{\nu}D_{\mu\nu}(p-q),$
(99)

where $D_{\mu\nu}(p)$ is the propagator of Eq. (94) and
$G(p)=[Z(p)i\gamma\cdot p+\Sigma(p)]^{-1}$ represents the full fermionic
propagator. In the above equation vertex corrections are being neglected. In
order to gain better control of the calculation, we introduce $M$ copies of
fermions $\psi_{1}$ and $\psi_{2}$, and thus we have to multiply the diagonal
matrix elements in Eq. (91) by $M$. This modifies $D_{\mu\nu}(p)$ to,

$D_{\mu\nu}(p)=\frac{16\pi^{2}M}{(64+M^{2}\pi^{2})|p|}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right).$
(100)

Equation (99) leads then to two self-consistent integral equations of the
form,

$\Sigma(p)=\frac{32\pi^{2}M}{64+M^{2}\pi^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\Sigma(k)}{[Z^{2}(k)k^{2}+\Sigma^{2}(k)]|k+p|},$
(101)
$Z(p)=1-\frac{16\pi^{2}M}{(64+M^{2}\pi^{2})p^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{[k^{2}-p^{2}+(k+p)^{2}](k+p)\cdot
pZ(k)}{[Z^{2}(k)k^{2}+\Sigma^{2}(k)]|k+p|^{3}}.$ (102)

Using a well-known procedure Appelquist  _et al._ (1986a, 1988); Nash (1989),
we easily convert Eqs. (101) and (102) to nonlinear differential equations,

$\frac{d}{ds}\left[s^{2}\frac{d\Sigma(s)}{ds}\right]=-\frac{16M}{64+M^{2}\pi^{2}}\frac{s^{2}\Sigma(s)}{Z^{2}(s)s^{2}+\Sigma^{2}(s)},$
(103)
$\frac{d}{ds}\left[s^{4}\frac{dZ(s)}{ds}\right]=\frac{16M}{64+M^{2}\pi^{2}}\frac{s^{4}Z(s)}{Z^{2}(s)s^{2}+\Sigma^{2}(s)},$
(104)

where $s\equiv|p|$. These equations are solved with the boundary conditions,
$\lim_{s\to 0}s\Sigma(s)=0$, $sd\Sigma(s)/ds|_{s=\Lambda}=-\Sigma(\Lambda)$,
$sdZ(s)/ds|_{s=\Lambda}=3[1-Z(\Lambda)]$, where $\Lambda$ is a UV cutoff. In
addition, the spectral representation implies that $0<Z(0)\leq 1$ must hold.
Typically a first approximation to solve Eqs. (103) and (104) is by
linearizing them. Using the leading order result where $Z(0)=1$, we obtain a
nonzero gap of the form,

$\Sigma_{\pm}(0)=\pm\Lambda e^{-2\pi/\gamma},$ (105)

where,

$\gamma=\sqrt{\frac{64M}{64+\pi^{2}M^{2}}-1}.$ (106)

We have included two possible signs for the generated gap, since we are
dealing with two-component spinors here, so strictly speaking a chiral
symmetry breaking is only possible for $M=2N$, with half of the fermions
getting a mass $\Sigma_{+}(0)$ and the other half $\Sigma_{-}(0)$. For
$M=2N+1$ it is the parity that is being spontaneously broken.

We obtain that $\Sigma(0)/\Lambda$ is only nonzero for $M_{-}<M<M_{+}$, where
$M_{\pm}=(64\pm\sqrt{4096-256\pi^{2}})/(2\pi^{2})$ are the critical values of
$N$ for which the gap vanishes. We note that $M_{-}\approx 1.235$, so this
solution does not yield any gap generation for $M=1$. Even for the case where
$M$ lies in the interval $(M_{-},M_{+})$, the corresponding value of
$\Sigma(0)/\Lambda$ is tiny. Thus, for all practical purposes we can assume
that there is no gap generation at all. This situation is in sharp contrast
with the one corresponding to the Lagrangian of Eq. (98) when it is
generalized to include $2N$ fermion fields. Of course, one important
difference is that in the case just discussed $M$ fermions are coupled to the
gauge field $b_{1\mu}$ while the other $M$ fermions couple to the gauge field
$b_{2\mu}$, but this is actually not the main point. The fact is that applying
the same Schwinger-Dyson method to Eq. (98), would amount to replacing
$64M/(64+\pi^{2}M^{2})$ in Eq. (106) by $32/(\pi^{2}N)$, leading in this way
to the already quoted result that the gap vanishes only for $N\geq
32/\pi^{2}\approx 3.24$. Note that the number $f=32/(\pi^{2}N)$ is
proportional to the dimensionless gauge coupling at the IR stable fixed point
(to see this we have to include a Maxwell term as a UV regulator, similarly to
the discussion in Sect. IV.2) Appelquist  _et al._ (1986a), and thus
$64M/(64+\pi^{2}M^{2})$ with $M=2N$ would coincide with $f$ only in the large
$N$ limit.

We have thus found that only values of $M=2N>1$ would allow for chiral
symmetry breaking. We note however that the interval $(M_{-},M_{+})$ includes
the values 2, 3, and 4. Among those values, $M=3$ is special, since it is odd
and this would mean that parity gets dynamically broken in this case. Such a
result is not expected in ordinary QED in 2+1 dimensions Appelquist  _et al._
(1986b) and seems to violate a theorem by Vafa and Witten Vafa and Witten
(1984), which states that in massless QED in (2+1)$d$ it is not possible to
spontaneously break parity for any value of $M$. However, in our theory the
dependence on $M$ arises in a somewhat different way, since the photon
propagator is obtained from inverting the matrix appearing in Eq. (91). Thus,
the conditions underlying the Vafa-Witten result Vafa and Witten (1984) and
the early Schwinger-Dyson calculation performed in Ref. Appelquist  _et al._
(1986b) do not apply in this case. Furthermore, for the $M=1$ case, which is
our main interest here, no dynamical mass generation occurs. Hence, the
critical current correlator obtained in Sect. VI should correctly account for
the scale invariance of the theory.

##  VIII Conclusion

Let us summarize the main results of our paper. The first important result we
have obtained is the bosonization duality for the topological $U(1)\times
U(1)$ Abelian Higgs model, which in terms of Lagrangians can be schematically
represented in the form,

$\displaystyle\mathcal{L}_{b}$ $\displaystyle=$
$\displaystyle\sum_{a=1,2}|(\partial_{\mu}-ia_{\mu}-i(-1)^{a}B_{\mu})z_{a}|^{2}$
$\displaystyle+$
$\displaystyle\frac{i}{4\pi}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}+\dots$
$\displaystyle\hskip 56.9055pt\big{\Updownarrow}$
$\displaystyle\mathcal{L}_{f}$ $\displaystyle=$
$\displaystyle\sum_{I=1,2}\left[\bar{\psi}_{I}(\hbox to0.0pt{\hbox
to5.30904pt{\hfil/\hfil}\hss}\partial-i\hbox to0.0pt{\hbox
to5.00002pt{\hfil$b$\hfil}\hss}/_{I})\psi_{I}+\frac{i}{8\pi}b_{I\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{I\lambda}\right]$
$\displaystyle+$
$\displaystyle\frac{i}{2\pi}b_{1\mu}\epsilon_{\mu\nu\lambda}\partial_{\nu}b_{2\lambda}+\frac{i}{2\pi}\epsilon_{\mu\nu\lambda}B_{\mu}\partial_{\nu}(b_{2\lambda}-b_{1\lambda}),$

where $B_{\mu}$ is a background field. While paving the way to derive the main
duality (VIII), we have revisited some of the known results of the duality web
with focus on inferring critical properties and current correlators. Note that
in contrast with the duality of Eq. (53) for the $U(1)$-invariant case (i.e.,
featuring a single scalar), the fermionic dual model is not a free theory in
the case of Eq. (VIII). We have demonstrated for the duality (VIII) that the
topological and Noether current correlators only differ by a universal
prefactor we have calculated, giving the result of Eq. (85). The agreement of
bosonic and fermionic current correlation functions in the bosonization
duality also highlights that theories at criticality flow to the same fixed
points and, therefore, exhibit similar critical behavior. Thus, an important
feature of the topological $U(1)\times U(1)$ Abelian Higgs model is that the
system undergoes a second-order phase transition, in contrast to the usual
non-topological case, where the existence of the critical point is still under
debate. Indeed, the Schwinger-Dyson analysis of the fermionic dual model
appearing in Eq. (VIII) complements and strengthens our RG analysis of the
bosonic Lagrangian (67) [represented in a more schematic form in Eq. (VIII)]
in Ref. Shyta  _et al._ (2021a), since it shows that no dynamical mass
generation occurs. As was extensively discussed in the text, the bosonization
duality for the non-topological $U(1)\times U(1)$ theory, despite having a
much simpler looking form, yields far less conclusive results regarding its
phase structure.

###### Acknowledgements.

We thank the Deutsche Forschungsgemeinschaft (DFG) for support through the
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter – ct.qmat (EXC 2147, project-id 39085490) and the Collaborative
Research Center SFB 1143 (project-id 247310070).

##  Appendix A Current correlator in the $O(2)$ nonlinear sigma model

If we consider a scalar massless theory with an additional constraint,
$|\varphi|^{2}=1$, the model turns into an $O(2)$ nonlinear $\sigma$ model. In
this case the constraint is easily solved by $\varphi=e^{i\theta}$, and the
Lagrangian reduces to,

$\mathcal{L}=\frac{1}{2g}(\partial_{\mu}\theta)^{2},$ (108)

where we have included now a coupling constant $g$. This is the model for a
free scalar field theory. However, since $\theta$ is periodic, such a model is
nontrivial. For instance, when supplemented by a coupling constant, it is
known to lead to a Berezinskii-Kosterlitz-Thouless (BKT) phase transition at
$d=2$.

The conserved current associated to the Lagrangian (108) is given by,

$J_{\mu}=\frac{1}{g}\partial_{\mu}\theta,$ (109)

and makes its appearance in Sec. II.2 in the context of a bosonized version of
massless Dirac fermions. The corresponding current correlation function is
given by,

$K_{\mu\nu}(x-x^{\prime})=\frac{1}{g^{2}}\langle\partial_{\mu}\theta(x)\partial_{\nu}^{\prime}\theta(x^{\prime})\rangle.$
(110)

This can be rewritten as,

$K_{\mu\nu}(x)=-\frac{1}{g^{2}}\partial_{\mu}\partial_{\nu}\langle\theta(x)\theta(0)\rangle.$
(111)

Since,

$\langle\theta(x)\theta(0)\rangle=\frac{g}{S_{d}(d-2)|x|^{d-2}},$ (112)

we easily obtain that,

$K_{\mu\nu}(x)=\frac{1}{gS_{d}|x|^{d}}\left(\delta_{\mu\nu}-d\frac{x_{\mu}x_{\nu}}{x^{2}}\right),$
(113)

and we also verify that this result is transverse,

$\partial_{\mu}K_{\mu\nu}=0.$ (114)

However, the current correlation function is not transverse in momentum space,

$K_{\mu\nu}(p)=\frac{p_{\mu}p_{\nu}}{gp^{2}}.$ (115)

This is reminiscent of the axial anomaly occurring naturally in even spacetime
dimensions. This behavior has another consequence: the trace $K_{\mu\mu}(x)$
vanishes, but the same does not happen for $K_{\mu\mu}(p)$, which yields the
unity. In fact, we actually have,

$K_{\mu\mu}(x)=\frac{1}{g}\delta^{d}(x),$ (116)

i.e., $K_{\mu\mu}(x)$ indeed vanishes everywhere, except at $x=0$ where a
delta function singularity arises.

On the other hand, we can recall that there is another conserved current, an
axial one, namely Zinn-Justin (2002),

$\tilde{J}_{\mu}=\frac{1}{g}\epsilon_{\mu\nu}\partial_{\nu}\theta,$ (117)

which is dual to $J_{\mu}$ of Eq. (109). The current correlation function
$\widetilde{K}_{\mu\nu}(x)=\langle\tilde{J}_{\mu}(x)\tilde{J_{\nu}}(0)\rangle$
is simply given by,

$\widetilde{K}_{\mu\nu}(x)=-K_{\mu\nu}(x),$ (118)

implying they are both transverse in real space. However, as we have already
seen, $K_{\mu\nu}(p)$ is longitudinal. The interesting result following from
the simple calculation above is that, $\widetilde{K}_{\mu\nu}(p)$, by
contrast, is transverse,

$\widetilde{K}_{\mu\nu}(p)=\frac{1}{g}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right).$
(119)

##  Appendix B Current correlation function of free theories

###  B.1 Free scalar bosons in $d$ dimensions

A general expression for the bosonic current correlation function of the
massless theory in $d$ dimensions can be obtained explicitly. The vacuum
polarization in scalar QED in the massless limit is easily shown to precisely
correspond to the exact current correlation function of the free theory,

$\mathcal{L}=|\partial_{\mu}\varphi|^{2}+m^{2}|\varphi|^{2},$ (120)

when $m=0$. The theory possesses a conserved $U(1)$ current given by,

$j_{\mu}=-i(\varphi^{*}\partial_{\mu}\varphi-\varphi\partial_{\mu}\varphi^{*}).$
(121)

The corresponding current correlation function for a free complex scalar boson
can be calculated explicitly and initially is written as,

$\displaystyle C_{\mu\nu}(x-x^{\prime})$ $\displaystyle=$
$\displaystyle\langle
j_{\mu}(x)j_{\nu}(x^{\prime})\rangle=-\langle\varphi^{*}(x)\partial_{\mu}\varphi(x)\varphi^{*}(x^{\prime})\partial_{\nu}^{\prime}\varphi(x^{\prime})\rangle-\langle\varphi(x)\partial_{\mu}\varphi^{*}(x)\varphi(x^{\prime})\partial_{\nu}^{\prime}\varphi^{*}(x^{\prime})\rangle$
(122) $\displaystyle+$
$\displaystyle\langle\varphi^{*}(x)\partial_{\mu}\varphi(x)\varphi(x^{\prime})\partial_{\nu}^{\prime}\varphi^{*}(x^{\prime})\rangle+\langle\varphi(x)\partial_{\mu}\varphi^{*}(x)\varphi^{*}(x^{\prime})\partial_{\nu}^{\prime}\varphi(x^{\prime})\rangle.$

Since the theory is free, we can use Wick’s theorem to decouple the
correlations above. The result is,

$\displaystyle C_{\mu\nu}(x-x^{\prime})$ $\displaystyle=$
$\displaystyle-\langle\varphi^{*}(x)\partial_{\nu}^{\prime}\varphi(x^{\prime})\rangle\langle\partial_{\mu}\varphi(x)\varphi^{*}(x^{\prime})\rangle-\langle\varphi(x)\partial_{\nu}^{\prime}\varphi^{*}(x^{\prime})\rangle\langle\partial_{\mu}\varphi^{*}(x)\varphi(x^{\prime})\rangle$
(123) $\displaystyle+$
$\displaystyle\langle\varphi^{*}(x)\varphi(x^{\prime})\rangle\langle\partial_{\mu}\varphi(x)\partial_{\nu}^{\prime}\varphi^{*}(x^{\prime})\rangle+\langle\varphi(x)\varphi^{*}(x^{\prime})\rangle\langle\partial_{\mu}\varphi^{*}(x)\partial_{\nu}^{\prime}\varphi(x^{\prime})\rangle.$

We note the Fourier representations of the correlation functions,

$\langle\varphi(x)\varphi^{*}(x^{\prime})\rangle=\int_{p}\frac{e^{ip\cdot(x-x^{\prime})}}{p^{2}+m^{2}},$
(124)
$\langle\partial_{\mu}\varphi(x)\varphi^{*}(x^{\prime})\rangle=i\int_{p}\frac{p_{\mu}e^{ip\cdot(x-x^{\prime})}}{p^{2}+m^{2}},$
(125)
$\langle\partial_{\mu}\varphi(x)\partial_{\nu}^{\prime}\varphi^{*}(x^{\prime})\rangle=\int_{p}\frac{p_{\mu}p_{\nu}e^{ip\cdot(x-x^{\prime})}}{p^{2}+m^{2}},$
(126)

while the other correlation functions entering Eq. (123) follow by complex
conjugation. In the expressions above, we used the shorthand notation,

$\int_{k}=\int\frac{d^{d}k}{(2\pi)^{3}}.$ (127)

After inserting the Fourier representations into Eq. (123), we obtain the
Fourier transform of the current correlation function as,

$C_{\mu\nu}(k)=\int d^{3}xe^{-ik\cdot x}C_{\mu\nu}(x).$ (128)

Finally, some straightforward algebraic manipulations and simplifications
yield,

$C_{\mu\nu}(k)=I_{\mu\nu}(k),$ (129)

where $I_{\mu\nu}(k)$ is given in momentum space by the integral,

$I_{\mu\nu}(q)=\int_{k}\frac{(2k-q)_{\mu}(2k-q)_{\nu}}{[(k-q)^{2}+m^{2}](k^{2}+m^{2})}.$
(130)

Since we can write,

$I_{\mu\nu}(q)=A(q)\delta_{\mu\nu}+B(q)q_{\mu}q_{\nu},$ (131)

where,

$A(q)=\frac{1}{d-1}\left[I_{\mu\mu}(q)-\frac{q_{\mu}q_{\nu}}{q^{2}}I_{\mu\nu}(q)\right],$
(132)
$B(q)=\frac{1}{(1-d)q^{2}}\left[I_{\mu\mu}(q)-d\frac{q_{\mu}q_{\nu}}{q^{2}}I_{\mu\nu}(q)\right],$
(133)

we obtain after straightforward simplifications,

$\displaystyle A(q)$ $\displaystyle=$
$\displaystyle-\frac{1}{d-1}\left\\{\frac{4m^{2}}{d-2}\int_{k}\frac{1}{(k^{2}+m^{2})^{2}}\right.$
$\displaystyle+$
$\displaystyle\left.(q^{2}+4m^{2})\int_{k}\frac{1}{[(k-q)^{2}+m^{2}](k^{2}+m^{2})}\right\\}.$
$\displaystyle B(q)$ $\displaystyle=$
$\displaystyle\frac{1}{(d-1)q^{2}}\left\\{-4m^{2}\int_{k}\frac{1}{(k^{2}+m^{2})^{2}}\right.$
$\displaystyle+$
$\displaystyle\left.(q^{2}+4m^{2})\int_{k}\frac{1}{[(k-q)^{2}+m^{2}](k^{2}+m^{2})}\right\\}.$

We now decompose $I_{\mu\nu}(q)$ into transverse and longitudinal components,

$I_{\mu\nu}(q)=A(q)\left(\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right)+\left[B(q)q^{2}+A(q)\right]\frac{q_{\mu}q_{\nu}}{q^{2}}.$
(136)

The longitudinal component is given simply by,

$B(q)q^{2}+A(q)=-\frac{4m^{2}}{d-2}\int_{k}\frac{1}{(k^{2}+m^{2})^{2}},$ (137)

and we see that for $m^{2}=0$ the result for $I_{\mu\nu}(q)$ is purely
transverse,

$I_{\mu\nu}(q)|_{m^{2}=0}=-\Pi(q)(q^{2}\delta_{\mu\nu}-q_{\mu}q_{\nu}),$ (138)

with

$\Pi(q)=\int_{k}\frac{1}{(k-q)^{2}k^{2}}=c_{d}|q|^{d-4},$ (139)

where $c_{d}$ is given by,

$c_{d}=\frac{d-2}{(4\pi)^{d/2}}\Gamma\left(2-\frac{d}{2}\right)\frac{\Gamma^{2}(d/2-1)}{\Gamma(d)}.$
(140)

Let us now specialize to $d=3$. In this case we have,

$\int_{k}\frac{1}{(k^{2}+m^{2})^{2}}=\frac{1}{8\pi|m|},$ (141)
$\displaystyle\int_{k}\frac{1}{[(k-q)^{2}+m^{2}](k^{2}+m^{2})}=\frac{1}{4\pi|q|}\arctan\left(\frac{|q|}{2|m|}\right).$

Hence, the transverse and longitudinal components of $I_{\mu\nu}(q)$ read,

$I_{t}(q)=-\frac{1}{4\pi}\left[|m|+\frac{(q^{2}+4m^{2})}{2|q|}\arctan\left(\frac{|q|}{2|m|}\right)\right],$
(143) $I_{l}(q)=-\frac{|m|}{2\pi}.$ (144)

In the expression for $A(q)$ we have used dimensional regularization to write,

$\int_{k}\frac{1}{k^{2}+m^{2}}=-\frac{2m^{2}}{d-2}\int_{k}\frac{1}{(k^{2}+m^{2})^{2}},$
(145)

which removes a power of the UV cutoff $\Lambda$ of the form
$\sim\Lambda^{d-2}$ spoiling gauge invariance.

Therefore, the current correlation function of a massless bosonic theory is
purely transverse and we obtain in real space that,

$C_{\mu\nu}(x)=\frac{2}{S_{d}^{2}(d-2)|x|^{2(d-1)}}\left(\delta_{\mu\nu}-2\frac{x_{\mu}x_{\nu}}{x^{2}}\right),$
(146)

where $S_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the solid angle in $d$ dimensions.
Indeed, we verify that $\partial_{\mu}C_{\mu\nu}(x)=0$, and in momentum space
we have,

$C_{\mu\nu}(q)=-c_{d}|q|^{d-2}\left(\delta_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}}\right).$
(147)

###  B.2 Free fermions in 2+1 dimensions

It is a straightforward exercise to exactly compute the current correlation
function for free fermions. In momentum space we have,

$C_{\mu\nu}^{F}(p)=\langle j_{\mu}(p)j_{\nu}(-p)\rangle=-\int_{q}{\rm
tr}[\gamma_{\mu}G(q)\gamma_{\nu}G(p-q)],$ (148)

where,

$G(p)=\frac{M-i\hbox to0.0pt{\hbox
to5.03125pt{\hfil/\hfil}\hss}p}{p^{2}+M^{2}}.$ (149)

Thus,

$C_{\mu\nu}^{F}(p)=\int_{q}\frac{{\rm tr}[\gamma_{\mu}(M-i\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/)\gamma_{\nu}(M-i\hbox to0.0pt{\hbox
to5.03125pt{\hfil/\hfil}\hss}p+i\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/)]}{(q^{2}+M^{2})[(p-q)^{2}+M^{2}]}.$ (150)

The trace in the numerator of the above expression is rewritten as,

$\displaystyle{\rm tr}[\gamma_{\mu}(M-i\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/)\gamma_{\nu}(M-i\hbox to0.0pt{\hbox
to5.03125pt{\hfil/\hfil}\hss}p+i\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/)]=2(M^{2}+q^{2}$ (151) $\displaystyle-$
$\displaystyle p\cdot
q)\delta_{\mu\nu}-2(p_{\mu}q_{\nu}+p_{\nu}q_{\mu})+4q_{\mu}q_{\nu}$
$\displaystyle+$ $\displaystyle 2M\epsilon_{\mu\nu\lambda}p_{\lambda},$

where we have made use of the identities,

${\rm tr}(\gamma_{\mu}\gamma_{\nu})=2\delta_{\mu\nu},$ (152) ${\rm
tr}(\gamma_{\mu}\gamma_{\nu}\gamma_{\lambda})=2i\epsilon_{\mu\nu\lambda},$
(153) ${\rm
tr}(\gamma_{\mu}\gamma_{\lambda}\gamma_{\nu}\gamma_{\rho})=2(\delta_{\mu\lambda}\delta_{\nu\rho}+\delta_{\mu\rho}\delta_{\lambda\nu}-\delta_{\mu\nu}\delta_{\lambda\rho}).$
(154)

Performing explicitly the several integrals yields,

$\displaystyle-
C_{\mu\nu}^{F}(p)=\frac{1}{4\pi}\left[|M|+\frac{p^{2}-4M^{2}}{2|p|}\arctan\left(\frac{|p|}{2|M|}\right)\right]$
$\displaystyle\times$
$\displaystyle\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\frac{M}{2\pi|p|}\arctan\left(\frac{|p|}{2|M|}\right)\epsilon_{\mu\nu\lambda}p_{\lambda},$

where UV finiteness of the parity-even transverse term has been obtained via
dimensional regularization. We will see later that a different gauge-invariant
regularization is more appropriate in this case.

For $M=0$ the above result becomes simply,

$C_{\mu\nu}^{F}(p)|_{M=0}=-\frac{|p|}{16}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right).$
(156)

Up to a minus sign, Eq. (156) coincides with the bosonic current correlation
function for free massless bosons of Eq. (147) when $d=3$ is set,

$C_{\mu\nu}^{F}(p)|_{M=0}=C_{\mu\nu}(p)|_{m=0}.$ (157)

## References

  * Lüscher (1989) M. Lüscher, “Bosonization in 2 + 1 dimensions,” Nuclear Physics B 326, 557–582 (1989).
  * Polyakov (1988) A. M. Polyakov, “Fermi-bose transmutations induced by gauge fields,” Modern Physics Letters A 03, 325–328 (1988).
  * Fradkin and Schaposnik (1994) Eduardo Fradkin and Fidel A. Schaposnik, “The fermion-boson mapping in three-dimensional quantum field theory,” Physics Letters B 338, 253–258 (1994), arXiv:hep-th/9407182 .
  * Giamarchi (2003) Thierry Giamarchi,  _Quantum physics in one dimension_ , Vol. 121 (Clarendon press, 2003).
  * Zinn-Justin (2002) Jean Zinn-Justin,  _Quantum field theory and critical phenomena_ , 4th ed. (Clarendon Press, 2002).
  * Seiberg  _et al._ (2016) Nathan Seiberg, T. Senthil, Chong Wang, and Edward Witten, “A duality web in 2+1 dimensions and condensed matter physics,” Annals of Physics 374, 395 – 433 (2016), arXiv:1606.01989 .
  * Aharony (2016) Ofer Aharony, “Baryons, monopoles and dualities in chern-simons-matter theories,” Journal of High Energy Physics 2016, 93 (2016), arXiv:1512.00161 .
  * Karch and Tong (2016) Andreas Karch and David Tong, “Particle-vortex duality from 3d bosonization,” Phys. Rev. X 6, 031043 (2016), arXiv:1606.01893 .
  * Wang  _et al._ (2017) Chong Wang, Adam Nahum, Max A. Metlitski, Cenke Xu, and T. Senthil, “Deconfined quantum critical points: Symmetries and dualities,” Phys. Rev. X 7, 031051 (2017), arXiv:1703.02426 .
  * Metlitski (2015) Max A. Metlitski, “$s$-duality of $u(1)$ gauge theory with $\theta=\pi$ on non-orientable manifolds: Applications to topological insulators and superconductors,” (2015), arXiv:1510.05663 [hep-th] .
  * Nastase and Núñez (2018) Horatiu Nastase and Carlos Núñez, “Deriving three-dimensional bosonization and the duality web,” Physics Letters B 776, 145 – 149 (2018), arXiv:1703.08182 .
  * Chen  _et al._ (2018) Jing-Yuan Chen, Jun Ho Son, Chao Wang, and S. Raghu, “Exact boson-fermion duality on a 3d euclidean lattice,” Phys. Rev. Lett. 120, 016602 (2018), arXiv:1705.05841 .
  * Türker  _et al._ (2020) Oguz Türker, Jeroen van den Brink, Tobias Meng, and Flavio S. Nogueira, “Bosonization in $2+1$ dimensions via chern-simons bosonic particle-vortex duality,” Phys. Rev. D 102, 034506 (2020), arXiv:2004.10789 .
  * Shyta  _et al._ (2021a) Vira Shyta, Jeroen van den Brink, and Flavio S. Nogueira, “Deconfined criticality and bosonization duality in easy-plane chern-simons two-dimensional antiferromagnets,” Phys. Rev. Lett. 127, 045701 (2021a), arXiv:2008.13446 .
  * Coleman (1975) Sidney Coleman, “Quantum sine-gordon equation as the massive thirring model,” Phys. Rev. D 11, 2088–2097 (1975).
  * Witten (1984) Edward Witten, “Non-abelian bosonization in two dimensions,” Communications in Mathematical Physics 92, 455–472 (1984).
  * Peskin (1978) Michael E. Peskin, “Mandelstam-’t Hooft duality in abelian lattice models,” Ann. Phys. (N. Y). 113, 122–152 (1978).
  * Thomas and Stone (1978) Paul R. Thomas and Michael Stone, “Nature of the phase transition in a non-linear o(2)3 model,” Nuclear Physics B 144, 513 – 524 (1978).
  * Dasgupta and Halperin (1981) C. Dasgupta and B. I. Halperin, “Phase transition in a lattice model of superconductivity,” Phys. Rev. Lett. 47, 1556–1560 (1981).
  * Kleinert (1982) H. Kleinert, “Disorder version of the abelian higgs model and the order of the superconductive phase transition,” Lettere al Nuovo Cimento (1971-1985) 35, 405–412 (1982).
  * Banks  _et al._ (1977) T. Banks, R. Myerson, and J. Kogut, “Phase transitions in abelian lattice gauge theories,” Nuclear Physics B 129, 493–510 (1977).
  * José  _et al._ (1977) Jorge V. José, Leo P. Kadanoff, Scott Kirkpatrick, and David R. Nelson, “Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model,” Phys. Rev. B 16, 1217–1241 (1977).
  * Savit (1980) Robert Savit, “Duality in field theory and statistical systems,” Rev. Mod. Phys. 52, 453–487 (1980).
  * Cardy (1982) John L. Cardy, “Duality and the $\theta$ parameter in abelian lattice models,” Nuclear Physics B 205, 17–26 (1982).
  * Cardy (1980) John L. Cardy, “Universal properties of u(1) gauge theories,” Nuclear Physics B 170, 369–387 (1980).
  * Chen  _et al._ (1993) Wei Chen, Matthew P. A. Fisher, and Yong-Shi Wu, “Mott transition in an anyon gas,” Phys. Rev. B 48, 13749–13761 (1993), arXiv:cond-mat/9301037 .
  * Ezawa (2013) Zyun Francis Ezawa,  _Quantum Hall effects: Field theoretical approach and related topics_ , 3rd ed. (World Scientific Publishing Company, Singapore, 2013).
  * Nayak  _et al._ (2008) Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008), arXiv:0707.1889 .
  * Arovas  _et al._ (1984) Daniel Arovas, J. R. Schrieffer, and Frank Wilczek, “Fractional statistics and the quantum hall effect,” Phys. Rev. Lett. 53, 722–723 (1984).
  * Stern (2008) Ady Stern, “Anyons and the quantum hall effect—a pedagogical review,” Annals of Physics 323, 204–249 (2008), january Special Issue 2008, arXiv:0711.4697 .
  * Jackiw (1990) R Jackiw, “Topics in planar physics,” in  _Fields and Particles_ (Springer, 1990) pp. 117–169.
  * Forte (1992) Stefano Forte, “Quantum mechanics and field theory with fractional spin and statistics,” Rev. Mod. Phys. 64, 193–236 (1992).
  * Wen (2004) Xiao-Gang Wen,  _Quantum field theory of many-body systems: from the origin of sound to an origin of light and electrons_ (Oxford University Press on Demand, 2004).
  * Fradkin (2013) Eduardo Fradkin,  _Field theories of condensed matter physics_ (Cambridge University Press, 2013).
  * Seiberg and Witten (2016) Nathan Seiberg and Edward Witten, “Gapped boundary phases of topological insulators via weak coupling,” Progress of Theoretical and Experimental Physics 2016 (2016), arXiv:1602.04251 .
  * Kleinert (1989) Hagen Kleinert,  _Gauge Fields in Condensed Matter: Vol. 1: Superflow and Vortex Lines (Disorder Fields, Phase Transitions) Vol. 2: Stresses and Defects (Differential Geometry, Crystal Melting)_ (World Scientific, 1989).
  * Goldman and Fradkin (2018) Hart Goldman and Eduardo Fradkin, “Loop models, modular invariance, and three-dimensional bosonization,” Phys. Rev. B 97, 195112 (2018), arXiv:1801.04936 .
  * Halperin  _et al._ (1974) B. I. Halperin, T. C. Lubensky, and Shang-keng Ma, “First-order phase transitions in superconductors and smectic-$a$ liquid crystals,” Phys. Rev. Lett. 32, 292–295 (1974).
  * Hikami (1979) Shinobu Hikami, “Renormalization Group Functions of CP(N-1) Non-Linear sigma-Model and N-Component Scalar QED Model,” Progress of Theoretical Physics 62, 226–233 (1979).
  * Ihrig  _et al._ (2019) Bernhard Ihrig, Nikolai Zerf, Peter Marquard, Igor F. Herbut, and Michael M. Scherer, “Abelian higgs model at four loops, fixed-point collision, and deconfined criticality,” Phys. Rev. B 100, 134507 (2019), arXiv:1907.08140 .
  * Hove and Sudbø (2000) J. Hove and A. Sudbø, “Anomalous scaling dimensions and stable charged fixed point of type-ii superconductors,” Phys. Rev. Lett. 84, 3426–3429 (2000), arXiv:cond-mat/0002197 .
  * Kuklov  _et al._ (2006) A.B. Kuklov, N.V. Prokof’ev, B.V. Svistunov, and M. Troyer, “Deconfined criticality, runaway flow in the two-component scalar electrodynamics and weak first-order superfluid-solid transitions,” Annals of Physics 321, 1602 – 1621 (2006), july 2006 Special Issue, arXiv:cond-mat/0602466 .
  * Kragset  _et al._ (2006) S. Kragset, E. Smørgrav, J. Hove, F. S. Nogueira, and A. Sudbø, “First-order phase transition in easy-plane quantum antiferromagnets,” Phys. Rev. Lett. 97, 247201 (2006), arXiv:cond-mat/0609336 .
  * Hove and Sudbø (2007) J. Hove and A. Sudbø, “Deconfined quantum criticality in the two-dimensional antiferromagnetic heisenberg model with next-nearest-neighbor ising exchange,” Phys. Rev. B 75, 104405 (2007), arXiv:cond-mat/0605118 .
  * Herland  _et al._ (2013) Egil V. Herland, Troels A. Bojesen, Egor Babaev, and Asle Sudbø, “Phase structure and phase transitions in a three-dimensional $su(2)$ superconductor,” Phys. Rev. B 87, 134503 (2013), arXiv:1301.3142 .
  * Zhang  _et al._ (2018) Xue-Feng Zhang, Yin-Chen He, Sebastian Eggert, Roderich Moessner, and Frank Pollmann, “Continuous easy-plane deconfined phase transition on the kagome lattice,” Phys. Rev. Lett. 120, 115702 (2018), arXiv:1706.05414 .
  * Călugăreanu (1959) George Călugăreanu, “L’intégrale de gauss et l’analyse des nœuds tridimensionnels,” Rev. Math. pures appl 4, 5–20 (1959).
  * Călugăreanu (1961) G Călugăreanu, “Sur les classes d’isotopie des noeuds tridimensionnels et leurs invariants,” Czechoslovak Mathematical Journal 11, 588–625 (1961).
  * Moffatt and Ricca (1992) Henry Keith Moffatt and Renzo L. Ricca, “Helicity and the călugăreanu invariant,” Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439, 411–429 (1992).
  * Deser  _et al._ (1997) S. Deser, L. Griguolo, and D. Seminara, “Gauge invariance, finite temperature, and parity anomaly in d=3,” Phys. Rev. Lett. 79, 1976–1979 (1997), arXiv:hep-th/9705052 .
  * Dunne  _et al._ (1997) Gerald Dunne, Kimyeong Lee, and Changhai Lu, “Finite temperature chern-simons coefficient,” Phys. Rev. Lett. 78, 3434–3437 (1997), arXiv:hep-th/9612194 .
  * Note (1) Regarding this point, the following remark is in order. In some versions of QED in 2+1 dimensions a different version of the chiral symmetry can be introduced Appelquist  _et al._ (1986a). In this case one considers four component Dirac fermions and a $\gamma_{5}$-like matrix, anti-commuting with all other Dirac $\gamma$ matrices can be defined. However, such a $\gamma_{5}$-like matrix is not given as a product of the other Dirac matrices. Such a version of QED in 2+1 dimensions has numerous applications in condensed matter physics, see for example Refs. Rantner and Wen (2001); Franz  _et al._ (2002); Herbut (2006, 2002); Kleinert and Nogueira (2002); Kaul and Sachdev (2008); Nogueira (2008); Gusynin  _et al._ (2003). However, in this case there is no parity anomaly, since neither time-reversal nor spatial inversion are broken by adding a mass term. The latter actually leads to a breaking of the chiral symmetry (defined in the mentioned sense) instead.
  * Arfken and Weber (2005) George B Arfken and Hans J Weber,  _Mathematical methods for physicists_ (Elsevier, 2005).
  * Atiyah  _et al._ (1975) M. F. Atiyah, V. K. Patodi, and I. M. Singer, “Spectral asymmetry and riemannian geometry. i,” Mathematical Proceedings of the Cambridge Philosophical Society 77, 43–69 (1975).
  * Witten (2016) Edward Witten, “Fermion path integrals and topological phases,” Rev. Mod. Phys. 88, 035001 (2016), arXiv:1508.04715 .
  * Redlich (1984a) A. N. Redlich, “Gauge noninvariance and parity nonconservation of three-dimensional fermions,” Phys. Rev. Lett. 52, 18–21 (1984a).
  * Redlich (1984b) A. N. Redlich, “Parity violation and gauge noninvariance of the effective gauge field action in three dimensions,” Phys. Rev. D 29, 2366–2374 (1984b).
  * Townsend  _et al._ (1984) P.K. Townsend, K. Pilch, and P. Van Nieuwenhuizen, “Self-duality in odd dimensions,” Physics Letters B 136, 38 – 42 (1984).
  * Deser and Jackiw (1984) S. Deser and R. Jackiw, ““self-duality” of topologically massive gauge theories,” Physics Letters B 139, 371 – 373 (1984).
  * Collins (1985) John C Collins,  _Renormalization: an introduction to renormalization, the renormalization group and the operator-product expansion_ (Cambridge university press, 1985).
  * Kovtun and Ritz (2008) Pavel Kovtun and Adam Ritz, “Universal conductivity and central charges,” Phys. Rev. D 78, 066009 (2008), arXiv:0806.0110 .
  * (62) Edward Witten, “$sl(2,z)$ action on three-dimensional conformal field theories with abelian symmetry,” in  _From Fields to Strings: Circumnavigating Theoretical Physics_, pp. 1173–1200, arXiv:hep-th/0307041 .
  * Appelquist  _et al._ (1986a) Thomas W. Appelquist, Mark Bowick, Dimitra Karabali, and L. C. R. Wijewardhana, “Spontaneous chiral-symmetry breaking in three-dimensional qed,” Phys. Rev. D 33, 3704–3713 (1986a).
  * Note (2) However, the Maxwell term is frequently needed for obtaining more rigorous results, since it provides a UV regulator, also within the bosonization duality context involving Chern-Simons actions. See, for instance, Ref. Shyta  _et al._ (2021a).
  * Son (2015) Dam Thanh Son, “Is the composite fermion a dirac particle?” Phys. Rev. X 5, 031027 (2015), arXiv:1502.03446 .
  * Metlitski and Vishwanath (2016) Max A. Metlitski and Ashvin Vishwanath, “Particle-vortex duality of two-dimensional dirac fermion from electric-magnetic duality of three-dimensional topological insulators,” Phys. Rev. B 93, 245151 (2016), arXiv:1505.05142 .
  * Kolnberger and Folk (1990) S. Kolnberger and R. Folk, “Critical fluctuations in superconductors,” Phys. Rev. B 41, 4083–4088 (1990).
  * Folk and Holovatch (1996) R Folk and Yu Holovatch, “On the critical fluctuations in superconductors,” Journal of Physics A: Mathematical and General 29, 3409–3425 (1996).
  * Bergerrhoff  _et al._ (1996) B. Bergerrhoff, D.F. Litim, S. Lola, and C. Wetterich, “Phase transitions of n-component superconductors,” International Journal of Modern Physics A 11, 4273–4306 (1996), arXiv:cond-mat/9502039 .
  * Bergerhoff  _et al._ (1996) B. Bergerhoff, F. Freire, D. F. Litim, S. Lola, and C. Wetterich, “Phase diagram of superconductors from nonperturbative flow equations,” Phys. Rev. B 53, 5734–5757 (1996), arXiv:hep-ph/9503334 .
  * Bonati  _et al._ (2021) Claudio Bonati, Andrea Pelissetto, and Ettore Vicari, “Lattice abelian-higgs model with noncompact gauge fields,” Phys. Rev. B 103, 085104 (2021), arXiv:2010.06311 .
  * Herbut and Tešanović (1996) Igor F. Herbut and Zlatko Tešanović, “Critical fluctuations in superconductors and the magnetic field penetration depth,” Phys. Rev. Lett. 76, 4588–4591 (1996), arXiv:cond-mat/9605185 .
  * Calan and Nogueira (1999) C. de Calan and F. S. Nogueira, “Scaling critical behavior of superconductors at zero magnetic field,” Phys. Rev. B 60, 4255–4262 (1999), arXiv:cond-mat/9903247 .
  * Tong (2018) David Tong, “Gauge theory,”  (2018), see section 8.6.3.
  * Motrunich and Vishwanath (2004) Olexei I. Motrunich and Ashvin Vishwanath, “Emergent photons and transitions in the $\mathrm{O}(3)$ sigma model with hedgehog suppression,” Phys. Rev. B 70, 075104 (2004), arXiv:cond-mat/0311222 .
  * Senthil  _et al._ (2004a) T. Senthil, Ashvin Vishwanath, Leon Balents, Subir Sachdev, and Matthew P. A. Fisher, “Deconfined quantum critical points,” Science 303, 1490–1494 (2004a), arXiv:cond-mat/0311326 .
  * Sachdev (2011) S. Sachdev,  _Quantum Phase Transitions_ , 2nd ed. (Cambridge University Press, 2011).
  * Eliezer and Semenoff (1992) D. Eliezer and G.W. Semenoff, “Intersection forms and the geometry of lattice chern-simons theory,” Physics Letters B 286, 118 – 124 (1992), arXiv:hep-th/9204048 .
  * Berruto  _et al._ (2000) F. Berruto, M.C. Diamantini, and P. Sodano, “On pure lattice chern–simons gauge theories,” Physics Letters B 487, 366 – 370 (2000), arXiv:hep-th/0004203 .
  * DeMarco and Wen (2021) Michael DeMarco and Xiao-Gang Wen, “Compact ${\mathrm{u}}^{k}(1)$ chern-simons theory as a local bosonic lattice model with exact discrete 1-symmetries,” Phys. Rev. Lett. 126, 021603 (2021), arXiv:1906.08270 .
  * Senthil  _et al._ (2004b) T. Senthil, Leon Balents, Subir Sachdev, Ashvin Vishwanath, and Matthew P. A. Fisher, “Quantum criticality beyond the landau-ginzburg-wilson paradigm,” Phys. Rev. B 70, 144407 (2004b), arXiv:cond-mat/0312617 .
  * Note (3) We have dropped the Maxwell terms associated to the gauge fields $b_{I\mu}$, since they do not essentially contribute in the IR.
  * Appelquist  _et al._ (1988) Thomas Appelquist, Daniel Nash, and L. C. R. Wijewardhana, “Critical behavior in (2+1)-dimensional qed,” Phys. Rev. Lett. 60, 2575–2578 (1988).
  * Rantner and Wen (2001) Walter Rantner and Xiao-Gang Wen, “Electron spectral function and algebraic spin liquid for the normal state of underdoped high ${T}_{c}$ superconductors,” Phys. Rev. Lett. 86, 3871–3874 (2001), arXiv:cond-mat/0010378 .
  * Franz  _et al._ (2002) M. Franz, Z. Tešanović, and O. Vafek, “${\mathrm{qed}}_{3}$ theory of pairing pseudogap in cuprates: From $d$-wave superconductor to antiferromagnet via an algebraic fermi liquid,” Phys. Rev. B 66, 054535 (2002), arXiv:cond-mat/0203333 .
  * Herbut (2006) Igor F. Herbut, “Interactions and phase transitions on graphene’s honeycomb lattice,” Phys. Rev. Lett. 97, 146401 (2006), arXiv:cond-mat/0606195 .
  * Herbut (2002) Igor F. Herbut, “${\mathrm{qed}}_{3}$ theory of underdoped high-temperature superconductors,” Phys. Rev. B 66, 094504 (2002), arXiv:cond-mat/0202491 .
  * Gusynin  _et al._ (2003) V. P. Gusynin, D. V. Khveshchenko, and M. Reenders, “Anomalous dimensions of gauge-invariant amplitudes in massless effective gauge theories of strongly correlated systems,” Phys. Rev. B 67, 115201 (2003), arXiv:cond-mat/0207372 .
  * Note (4) Putting the fermions in the lattice, this chiral symmetry actually manifests itself as a sublattice symmetry. Graphene constitutes a realization of this type of chiral symmetry; see, for instance, Ref. Herbut (2006).
  * Note (5) Strictly speaking, this result is derived by first assuming a large $N$ limit and neglecting vertex corrections in the Schwinger-Dyson equation. We could in principle question a critical value of $N$ that is not even large. However, numerical analyses using staggered fermions in the lattice seem to corroborate the existence of a critical value of $N$ that is not large Strouthos and Kogut (2009); see Ref. Braun  _et al._ (2014) for a recent account of this problem and references therein.