\section{Introduction}
\noindent Two-dimensional superconductors, that come into being as superconducting thin films, differ crucially in their properties from bulk superconductors. First of all, global superconductivity, where the whole system becomes phase-coherent, sets in at temperatures below the Berezinskii-Kosterlitz-Thoules (BKT) transition, see\,\cite{Goldman2013,tinkham,minnhagen} for a review. In other words, `two-dimensional", or as we refer hereafter, ``planar superconductivity", arises due to a topological phase transition. Next, planar superconductors harbor the celebrated superconductor-insulator transition (SIT), see\,\cite{Goldman2010} for a review. The SIT has been demonstrated to be a topological phase transition, and the phases that emerge in the vicinity of the SIT were identified as topological phases\,\cite{dtv2019}.


The macroscopic physics of superconductors is commonly described by the Ginzburg-Landau (GL) theory\,\cite{tinkham} in which the Coulomb interaction is usually neglected.
Planar superconductivity, in films with the thickness $d$ comparable or smaller than the coherence length $\xi$, establishing at $T\leq T_{\rm\scriptscriptstyle BKT}$,  is still discussed within the GL framework with the order parameter having a fixed amplitude and a phase\,\cite{feigelman}. The bosonic model of the SIT typically treats fluctuations of the phase in the XY model\,\cite{minnhagen} and the Coulomb interaction is still neglected, since it is modelled by the same 1/r potential as in the 3D case. Here we demonstrate that the common treatment of planar superconductivity neglects crucial aspects distinguishing films from their bulk counterparts and outline the topological gauge theory replacing the standard approaches\,\cite{topsc, higgsless}. 

In the vicinity of the SIT, superconducting films develop a huge normal-state dielectric constant $\varepsilon$ and the Coulomb interaction maintains the 2D Coulomb logarithmic form in the spatial scale range $d$$\ll$$r$$\ll$$\Lambda_{\mathrm c}$, where the 2D screening length $\Lambda_{\mathrm c}\simeq\varepsilon d$\,\cite{BV2013} reaches macroscopic scales (for TiN films with $d=5$\,nm, $\Lambda_{\mathrm c}\simeq 200\,\mu$m) (this is analogous to the behaviour of the finite-temperature Coulomb potential in 3D, with $\beta = 1/T$ playing the same role as $d$, see, e.g.,\,\cite{zarembo}).
This implies that, on the relevant spatial scales, one can consider electric interactions as genuinely 2D. As the strong logarithmic interactions cannot be neglected, the Ginzburg-Landau model has to be coupled to 2D electromagnetism. In the common approach, having absorbed the phase with the usual Anderson-Higgs mechanism, see e.g.,\,\cite{tinkham}, the gauge fields become massive and remain coupled to charge fluctuations of the condensate. Importantly, in thin films, the scale on which gauge fields are screened, i.e., the inverse of the Anderson-Higgs mass, is given by the Pearl length $\lambda_{\perp} = \lambda_{\rm\scriptscriptstyle L}^2/d$, which is also of a macroscopic scale ( for $d\simeq\xi$, the Pearl length in, for example, TiN films is $\approx$$100\,\mu$m, comparable to the characteristic sizes of the experimental systems).  This means that, on the relevant spatial scales, the gauge fields remain massless within the film. 

Let us consider the perturbative expansion around the trivial gauge vacuum corresponding to vanishing fields. The resulting model is non-compact QED in (2+1) dimensions, which is famously infrared divergent\,\cite{polyakov, jackiw1}, since the perturbation parameter is $\propto {\rm log}\ R/a$ where $a$ is the ultraviolet (UV) cutoff, in our case the coherence length $\xi$, and $R$ the infrared (IR) cutoff, the system size. As a consequence the model becomes non-perturbative (and thus ill-defined) for system sizes $\xi \ll R \le \lambda_{\perp} $. Such superconductors epytomize what we call planar superconductors: in the formal limit $d\to 0$ every superconductor is in this class. This implies that for planar superconductors the Anderson-Higgs mechanism is ineffective. In the following we will describe how these planar superconductors cure their infrared divergences. 

There are two ways in which gauge fields in (2+1) dimensions can get rid of their divergences. One is to generate a mass via magnetic monopole instantons, describing tunnelling between non-trivial topological gauge vacua \cite{polyakov}. The other is to generate a topological Chern-Simons (CS) mass \cite{jackiw1, jackiw2}. This second way, however entails breaking the discrete symmetries ${\cal P} $ and ${\cal T}$ of parity and time-reversal invariance. This undesired property can be circumvented, however, when the CS mass arises in a model with two gauge fields and a mixed topological term\,\cite{dst}. Depending on the value of $d$, the system chooses one of these possibilities. The former leads to superinsulators\,\cite{dtv1,dtv2,dtv3}, the latter results in planar superconductors that are the focus of present communication. Actually, there is a third possibility, which is eliminating completely any condensate, leading to Bose metal\,\cite{dst,das1,bm,dtv2021}. The topological CS mass comprises the product of two characteristic magnetic and electric length scales, so that the film thickness $d$ falls out. The resulting gauge screening length is not the Pearl length but the original bulk London penetration depth
\begin{equation}
\lambda_{\rm top} = {\cal O}\left( \lambda_{\rm\scriptscriptstyle L}\right)  \ .
\label{renlon}
\end{equation}
The topological gauge theory of superconductivity replaces the Ginzburg-Landau model for planar superconductors\,\cite{topsc, higgsless}. In this model there is no Higgs field and no Abrikosov vortices, only Josephson vortices. 


\iffalse
where $\alpha $ is the fine structure constant, $g$ is the quantum conductivity, representing the ratio of magnetic to electric energies, and $\ell$ is a dynamically generated size, as we now discuss. Deep in the superconducting region, both $g$ and $\ell$ become large and the  screening length decreases. When $g$ approaches 1, the dynamical scale $\ell \approx \xi/\alpha$ and the screening length approaches the Pearl length: at this point the system switches to the monopole instanton solution to the infrared catastrophe and becomes a superinsulator. 

\fi



Let us now describe in detail how does the topological gauge theory work. If we freeze instantons, the system has no other choice to regularize itself but breaking up the condensate into ``perturbative" droplets of the size ${\cal O}(\xi)$. These droplets are indeed observed in thin superconducting films \cite{sacepe}. Each of these droplets is characterized by an independent phase of the condensate and, thus, there exist topological configurations in which the circulation of these phases on neighbouring droplets is a multiple of $2\pi$. These constitute Josephson vortices see, e.g.,\,\cite{tinkham}, having a nontrivial phase structure but no normal-state core. Indeed, within the droplets, Abrikosov vortices cannot exist if the droplet size compares to $\xi$\,\cite{Likharev1979} (so-called Likharev vortex explosion phenomenon). The system forms global superconductivity as the Cooper pair tunneling between the droplets establish the global phase coherence.

The importance of droplet structure and vortices has already been noticed in very early studies of thin superconducting films. It was posited  that these vortices can condense\,\cite{fisher1, fisher2, fisher3, fazio, girvin} at the insulating side of the SIT, but that the vortex condensation is possible only because the vortices are Josephson ones, without a dissipative core, was never stressed. Two other crucial aspects of these vortices have been remaining underexplored. First, that near enough to the SIT the parameter window in which they become ballistic, thus moving without dissipation, gets really wide and diverges near the transition\,\cite{ballistic1}. Second, charges and vortices are subject to topological interactions leading to Aharonov-Bohm \cite{aharonovbohm} and Aharonov-Casher \cite{casher} phases. These topological interactions are the infrared-dominant ones and crucially influence the long-range properties of the system. This is ultimately why the treatment of thin superconducting films only in terms of a phase governed by the XY model is not correct since the most important interaction has been neglected. 

\iffalse
 near the SIT were completely neglected moreover. First, for the same reason that they can condense, i.e. the absence of dissipation, the Josephson vortices are ballistic \cite{ballistic1, ballistic2} and their currents must be taken into account. Secondly, and even more importantly, charges and vortices are subject to topological interactions, which encode the Aharonov-Bohm \cite{aharonovbohm} and Aharonov-Casher \cite{casher} phases acquired by their respective wave functions when one excitation goes around the other. These mutual statistics phases have to be taken into account in the partition function of the system. This is particularly important, given that these topological interactions are the infrared-dominant ones and crucially influence the long-range properties of the system. This is ultimately why the treatment of thin superconducting films only in terms of a phase governed by the XY model is not correct: the most important interaction has been neglected. 
\fi

Following \cite{dst, bm} we treat the condensate droplets of Cooper pairs and vortices in the continuum space-time with coordinates $x = (x_0, {\bf x})$  as the point-like objects by introducing the fields 
\begin{eqnarray}
Q^{\mu} &&= \sum_I Q_I \int ds \ {dx_I^{\mu} \over ds} \ \delta^3 \left( x -x_I (s) \right) \ ,
\nonumber \\
M^{\mu} &&= \sum_J M_J \int dt \ {dx_J^{\mu} \over dt} \ \delta^3 \left( x -x_J(t) \right) \ ,
\label{pointsources}
\end{eqnarray}
with $x_I (s)$ and $x_J (t)$ parametrizing closed (for fluctuations) or infinitely long space-time trajectories so that $\partial_\mu Q^{\mu} = 0$, $\partial_\mu M^{\mu} = 0$. We will then introduce the droplet size as the necessary ultraviolet cutoff. The mutual statistics interactions are encoded in the Euclidean partition function by phases (we use natural units $c=1$, $\hbar =1$, $\varepsilon_0=1$) 
\begin{equation}
S_{\rm top}  =  i \ 2\pi   \int d^3x \  Q_{\mu} \epsilon_{\mu \alpha \nu} {\partial_{\alpha} \over \nabla^2} M_{\nu} \ ,
\label{effec3}
\end{equation}
where $\epsilon^{\mu \alpha \nu}$ denotes the totally antisymmetric tensor. 
Using the 3D Green function
\begin{equation}
{1\over -\nabla^2} \delta^3 (x) = {1\over 4\pi} {1\over |x|} \ ,
\label{3dG}
\end{equation} 
one obtains
\begin{eqnarray}
S_{\rm top} \left( Q_I, M_J \right) &&=  i 2\pi  \sum_{IJ} Q_I M_J \Phi \left( C_I, C_J \right) \ ,
\nonumber \\
\Phi \left( C_I, C_J \right) &&= {1\over 4\pi} \int_0^1 ds \int_0^1 dt \ {dx_I^{\mu} \over ds} \epsilon_{\mu \alpha \nu}
{(x_I -x_J)^{\alpha} \over |x_I-x_J|^3} {dx_J^{\nu} \over dt} \ .
\label{gauss}
\end{eqnarray} 
For closed Euclidean trajectories $C_I$ and $C_J$, the quantity $\Phi \left( C_I, C_J \right)$ represents the integer Gauss linking number, the simplest topological knot invariant (for a review see\cite{kaufmann}). If the excitations satisfy the quantization condition $Q_I M_J = {\rm integer} $ for all $I, J$, the above phase becomes trivial for closed trajectories, corresponding, e.g., to a Minkowski space-time fluctuation creating a charge and a hole that annihilate after having encircled a vortex. In general, however, Aharonov-Bohm-Casher (ABC) phases lead to non-trivial quantum interference effects that cannot be neglected. 


Superficially, it looks like the topological interactions are non-local, see (\ref{effec3}), and thus difficult to treat. As was pointed out by Wilczek \cite{wilczek}, however, there exists a local formulation by coupling the charge and vortex trajectories to two gauge fields $a_{\mu}$ and $b_{\mu}$ with a mixed Chern-Simons interaction \cite{dst}, leading to the Euclidean partition function
\begin{eqnarray}
Z &&= \sum_{ \{ Q^{\mu}, M^{\mu} \} } \int {\cal D} a_{\mu} {\cal D} b_{\mu} {\rm e}^{-S} \ ,
\nonumber \\
S &&= \int d^3 x {i\over 2\pi} a_{\mu} \epsilon^{\mu \alpha \nu} \partial_{\alpha} b_{\nu} + i a_{\mu} Q^{\mu} + i b_{\mu} M^{\mu} \ .
\label{wilc}
\end{eqnarray}
This is the reason why the effective field theory of planar superconductors ${\it must}$ be formulated in terms of gauge fields. 

While $Q_{\mu}$ represents the quantized charges on the droplets and their currents, the gauge invariant quantity
\begin{equation}
j^{\mu} = {1\over 2\pi} \epsilon^{\mu \alpha \nu} \partial_{\alpha} b_{\nu} \ ,
\label{chargefluc}
\end{equation} 
describes the fluctuations of the droplet condensates and the tunnelling currents between them. $M_{\mu}$ and 
\begin{equation} 
\phi^{\mu} = {1\over 2\pi} \epsilon^{\mu \alpha \nu} \partial_{\alpha} a_{\nu} \ ,
\label{vortexfluc}
\end{equation}
are the dual quantities for vortices. 

Once we have recognized that the effective field theory must be formulated in terms of gauge fields we can easily add the next-order terms in a derivative expansion, since they are completely dictated by gauge invariance,
\begin{equation}
S=\int d^3 x \  {i\over2 \pi} a_{\mu} \epsilon^{\mu \alpha \nu} \partial_{\alpha} b_{\nu} 
+{1\over 2e^2_v} f_{\mu}f_{\mu} 
+{1 \over 2e^2_q} g_{\mu} g_{\mu} 
+i a_{\mu}  Q_{\mu} +i b_{\mu}  M_{\mu} \ , 
\label{nonrelac3}
\end{equation}
where 
\begin{eqnarray}
f^{\mu} = {1\over 2} \epsilon^{\mu \alpha \nu}  f_{\mu \nu} = \epsilon^{\mu \alpha \nu} \partial_{\alpha} b_{\nu} \ ,
\nonumber \\
g^{\mu} = {1\over 2} \epsilon^{\mu \alpha \nu}  g_{\mu \nu} = \epsilon^{\mu \alpha \nu} \partial_{\alpha} a_{\nu} \ ,
\label{fields}
\end{eqnarray}
are the dual Maxwell field strength of the two gauge fields. The two parameters $e_{\mathrm q}$ and $e_{\mathrm v}$ encode the electric and magnetic scales of a droplet,
\begin{eqnarray}
e_{\mathrm q}^2 &&= {\cal O} \left( {e^2\over 2\pi d} \right) \ ,
\nonumber \\
e_{\mathrm v}^2 &&= {\cal O} \left( {\pi \over e^2 \lambda_{\perp} } \right) = {\cal O} \left( {\pi d \over e^2 \lambda_{\rm\scriptscriptstyle L}^2}\right)  \ .
\label{couplings}
\end{eqnarray}
Non-relativistic effects can be taken into account by considering the ``space component" $x_0 = vt$ where $v$ is the velocity of propagation in the medium and, correspondingly, all time derivatives with respect to $x_0$ and gauge field components with index ``0" as time derivatives with a pre-factor $1/v$. 

The two parameters $e_{\mathrm q}^2$ and $e_{\mathrm v}^2$ can be traded for the topological CS mass \cite{jackiw1, jackiw2} $m=e_{\mathrm q}e_{\mathrm v}/2\pi v$, appearing in the dispersion relation of both charges and vortices,
\begin{equation}
E = \sqrt{m^2 v^4 + v^2 p^2} \ ,
\label{dis}
\end{equation}
and one dimensionless parameter $g=e_{\mathrm v}/e_{\mathrm q} = O(d/(\alpha \lambda_L))$, where $\alpha = e^2/4\pi$ is the fine structure constant. This parameter plays the role of a dimensionless conductivity and is the quantum parameter which ``selects" between the two infrared-catastrophe-avoiding models \cite{dst, bm}. 

Given that $e_{\mathrm q}^2$ and $e_{\mathrm v}^2$ have canonical dimension [1/length] and, correspondingly, the two kinetic terms in (\ref{nonrelac3}) are infrared-irrelevant, one might be tempted to leave them out and consider anyway only the infrared-dominant mixed Chern-Simons theory. Unfortunately, this leads to wrong results and the origin of the problem lies exactly in the role of the droplet charges $Q_{\mu}$ and the Josephson vortices $M_{\mu}$. Indeed, the limit $m \to \infty$ does not commute with quantization since it involves a phase-space reduction\cite{csqm}. For physical applications (as opposed to purely mathematical ones concerning knot theory), the topological theory has {\it always} to be considered as the $m \to \infty$ limit of a topologically massive \cite{jackiw2} one since, otherwise, physical states are not normalizable. However, when two gauge fields are present, the ``topological limit" $e_{\mathrm q}^2 \to \infty$ and $e_{\mathrm v}^2 \to \infty$ is not well defined without specifying the value of $g$ in this limit. The behaviour of the charges and vortices depends crucially on this value and, as a consequence one obtains very different ground states when $g$ is varied \cite{dst, bm}. 

We now minimally couple the charge current $j^{\mu}$ to the real electromagnetic field $A_{\mu}$ by adding to the action the term
\begin{equation}
S \to S + i\int d^3x \ A_{\mu} j^{\mu} \ ,
\label{mincoup}
\end{equation}
and we compute the effective action $S_{\rm eff}\left( A_{\mu}, Q_{\mu}, M_{\mu} \right)$ by integrating over the fictitious gauge fields $a_{\mu}$ and $b_{\mu}$. This gives $S_{\rm eff} = S_{\rm eff}\left( A_{\mu}, Q_{\mu} \right)$ as
\begin{eqnarray}
S_{\rm eff} &&= \int d^3x \left[ 
{e_{\mathrm q}^2 \over 2} Q_\mu { \delta_{\mu\nu}  \over -\Delta + m^2 v^2}  Q_\nu \right.\nonumber \\&& \left.+i 2 \pi  m^2v^2 Q_\mu { \epsilon_{\mu \alpha \nu} \partial_\alpha \over  \Delta (-\Delta + m^2 v^2 )} \left( M_\nu + {1\over 2\pi}  F_{\nu}\right) 
+  \right. \nonumber  \\ && \left. +  {e_{\mathrm v}^2 \over 2} \left( M_\mu + {1\over 2\pi}  F_{\mu }\right) { \delta_{\mu\nu}  \over -\Delta + m^2 v^2}  \left( M_\nu + {1\over 2\pi}  F_{\nu} \right) 
 \right]   \ ,
 \label{topact}
 \end{eqnarray}
with $ F_\mu$$=$$\epsilon_{\mu\nu\alpha} \partial_\nu A_\alpha$ being the real dual electromagnetic field strength. 
Following the typical approach in lattice gauge theories\,\cite{kogut} we retain only the self-interaction terms in\,Eq.\,(\ref{topact}), replacing  the interaction kernel with its diagonal component 
\begin{equation}
G(x-y) =  {1\over m^2v^2 - \Delta} \delta^3(x-y)  \to \ell^2 G(mv\ell) \ \delta^3(x-y) \ , 
\label{green}
\end{equation}
where the necessary UV cutoff $\ell $ represents, as anticipated, the droplet size. We also introduce the numerical parameter $\eta = (mv\ell) G(mv\ell)$. This is a numerical constant of ${\cal O}(1)$ depending on the dimensionless quantity $mv\ell$ and is the second parameter determining the quantum phase structure \cite{dst, bm}.
Eq. (\ref{topact}) become thus
\begin{eqnarray}
S_{\rm eff} &&= \int d^3x \left[
 {1\over 2} {2\pi \ell \eta  \over g} Q^2_\mu  +  {1 \over 2}  2\pi \ell \eta g \left( M_\mu + {1\over 2 \pi}  F_\mu \right)^2  + \right.  \nonumber \\  && \left.  +
 i 2 \pi  (mv\ell)  \eta Q_\mu { \epsilon_{\mu\nu\alpha} \partial_\nu \over  \Delta }  \left(M_\alpha + {1\over 2 \pi}  F_\alpha \right)   \right]   \ .
 \label{topactdiag}
 \end{eqnarray}

Let us now show how superconductivity emerges from the global condensation of the droplet charges in a phase with no vortices, $M_{\mu}=0$. This means that droplets get connected by quantum tunnelling, forming global phase coherence (therefore no vortices), i.e. a global condensate. To this end we integrate over the conserved currents $Q_{\mu}$ in (\ref{topactdiag}) by solving first the constraint $\partial_{\mu}Q_{\mu}=0$ with the introduction of a new gauge field $n_{\mu} $ defined as $Q_{\mu } = \epsilon_{\mu \alpha \nu} \partial_\alpha n_{\nu}$. This is a gauge field since shifts of $n_{\mu}$ by derivatives leave the conserved charge currents invariant. The effective action for the electromagnetic field $A_\mu$ becomes
\begin{equation}
S_{\rm eff} = \int d^3x  \ {1 \over 2} {2\pi \ell \eta \over g} H^2_\mu  + {1\over 2} { \ell \eta g \over 2\pi} F^2_\mu
- i  (mv\ell) \eta \ n_\mu \epsilon_{\mu\alpha\nu} \partial_\alpha  A_\nu  \ ,
\label{scinter}
\end{equation}
where we have introduced the dual field strength $H_\mu = \epsilon_{\mu\alpha \nu} \partial_\alpha  n_\nu$. As anticipated, this effective action is not a Ginzburg-Landau/Higgs model but, rather, a mixed Chern-Simons topological theory. The electromagnetic gauge field does not acquire its mass by the Anderson-Higgs mechanism but by a topological mass generation mechanism \cite{jackiw1, jackiw2} with gauge field mass $m = e_q e_v/2\pi $. The corresponding screening length $\lambda_{\rm top}=1/mv$ is the bulk London penetration depth (\ref{renlon}). Note also that the effective coupling of the Maxwell term in the action is $O(d\ell/e^2 \lambda_{\rm top})$, as can be seen using (\ref{couplings}). This shows that the the Coulomb coupling constant is renormalized to an effective coupling $e^2 \lambda_{\rm top}/\ell$: when $\ell \approx \lambda_{\rm top}$ we have the usual electron charge, when $\ell$ becomes small the effective Coulomb interaction increases. 

\iffalse


By integrating also over the gauge field $n_{\mu}$ representing the global condensate fluctuations we obtain an effective action for $A_{\mu}$ alone,
\begin{equation}
S_{\rm eff} \left( A_{\mu} \right) = \int d^3 x \ {1\over 2\lambda_{\rm top}} A_{\mu} \left( \delta_{\mu \nu} -{\partial_{\mu} \partial_{\nu} \over \Delta}\right) A_{\nu} + \dots \ ,
\label{lond}
\end{equation}
with the dots denoting higher-order terms in the derivative expansion and $\lambda_{\rm top}$ given by (\ref{renlon}). This is the quantity entering the London equations and representing thus the effective London penetration depth of planar superconductivity. 

\fi

In the dual phase with independent droplet phases and a vortex condensate we can set $Q_{\mu}=0$ while we keep the integration over the vortex degrees of freedom $M_{\mu}$. This phase is particularly interesting for materials characterized by very high dielectric constants, so that $v\ll1$ and we can neglect the magnetic components with respect to the electric ones in the effective action, which becomes
\begin{equation}
S_{\rm eff} \left( M_{\mu}, A_{\mu} \right) = \int d^3x \ {e_v^2  \ell^2 G \over v^2 8\pi^ 2 } \left(F_i + 2 \pi M_i  \right)^2 \ .
\label{polac}
\end{equation}
This is a non-relativistic version \cite{monjja} of Polyakov's compact QED\,\cite{polyakov} in which magnetic monopole instantons create a confining linear potential between probe charges and generate a photon mass. This dual phase is a superinsulator\,\cite{dst, dtv1}.

To conclude, planar superconductors cure their infrared divergences by realizing one of two possible phases with effective actions corresponding to the two known mechanisms of gauge field mass generation different from the Anderson-Higgs mechanism. This Higgsless, topological superconductivty \cite{topsc, higgsless} is the only possibility in 2D, but may be realized also in 3D bulk materials \cite{moncon, pseudo}. Indeed, 3D bulk materials with the emergent granularity typical of planar superconductors have been recently found\,\cite{parra}. 

\smallskip

\textit{Acknowledgments--} The work of V.M.V. was supported by the Terra Quantum AG.





 



