Emergent Magnetic Monopoles in Quantum Matter

Abstract

Magnetic monopoles, though elusive as elementary particles, emerge as quantum excitations in granular quantum materials. Under certain conditions, they can undergo Bose condensation, leading to the formation of a novel state of matter known as the superinsulator. In this state, charge carriers, Cooper pairs and anti-Cooper pairs, are bound together by an electric flux string, forming neutral electric pions. This confinement mechanism results in an infinite resistance that persists even at finite temperatures. Superinsulators behave, thus, as dual superconductors.

1. Introduction

In a vacuum, Maxwell’s equations exhibit a fundamental self-duality under the exchange of electric and magnetic fields, expressed as follows: $\mathbf{E}\to \mathbf{B},\mathbf{B}\to -\mathbf{E}$. This symmetry suggests a deep connection between electric and magnetic phenomena. However, when matter is introduced into the system, maintaining this duality requires the inclusion of magnetic monopoles, hypothetical particles that serve as magnetic counterparts to electric charges [1]. The existence of such monopoles would extend the conventional formulation of electrodynamics and restore the fundamental symmetry between electric and magnetic fields. In classical electrodynamics, the fundamental dynamical variables are the components of the field strength tensor $F^{\mu \nu }$, which encode the electric and magnetic field components. The standard Maxwell equations describe the behavior of electric and magnetic fields in the presence of electric charges and currents, but they do not naturally accommodate the existence of magnetic charges. To modify Maxwell’s equations in a way that incorporates magnetic monopoles, we introduce a magnetic current $m^{\mu }$ which corresponds to a source term associated with magnetic charges g (in natural units $c=1$, $\hslash =1$, and ${\epsilon }_0=1$, we use Greek letters for spacetime indices):

$$\begin{array}{c}\hfill {\partial }_{\mu }{F}^{\mu \nu }=qe{j}^{\nu },\\ \hfill {\partial }_{\mu }{\tilde{F}}^{\mu \nu }=g{m}^{\nu },\end{array}$$

(1)

where ${\tilde{F}}^{\mu \nu }=(1/2){ϵ}^{\mu \nu \alpha \beta }{F}_{\alpha \beta }$ is the dual electromagnetic field tensor. Despite the theoretical appeal of magnetic monopoles, no elementary particles carrying a net magnetic charge have ever been observed in nature. As a result, they were not incorporated into Maxwell’s equations, which describe classical electrodynamics based solely on electric charges and currents. The absence of experimental evidence for monopoles has kept them as a speculative extension rather than an established component of fundamental physics.

In the quantum formulation of electrodynamics, however, the canonical variables are no longer the components of the field strength tensor $F_{\mu \nu }$ as in the classical case. Instead, they are given by the components of the vector potential $A_{\mu }$. The relationship between these quantities is expressed as follows:

$$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.$$

From this definition, it follows that the field strength tensor must satisfy the equation: ${\partial }_{\mu }{\tilde{F}}^{\mu \nu }=0$, provided that the vector potential $A_{\mu }$ is well behaved everywhere in spacetime. However, this condition is violated if $A_{\mu }$ possesses a singularity, such as the so-called Dirac string, a theoretical construct representing an infinitely thin and infinitely long solenoid that extends to infinity, carrying magnetic flux inward. The Dirac string provides a way to model magnetic monopoles within quantum mechanics, but its physical observability is a crucial question [1]. Remarkably, if the Dirac quantization condition is satisfied,

$$qeg=2\pi n,n\in \mathbb{Z},$$

(2)

then the singularity associated with the Dirac string can be completely removed by a gauge transformation. This means that the string itself becomes unobservable, even in non-local quantum interference experiments such as those based on the Aharonov–Bohm effect. The significance of this result lies in the fact that magnetic monopoles, if they exist, must have charges quantized in a manner dictated by this condition. This theoretical argument, first proposed by Dirac, provides a compelling explanation for why electric charge is observed to be quantized in nature. Despite its elegance, however, no experimental confirmation of magnetic monopoles has been found, leaving their existence an open question in fundamental physics.

The emergence of monopole singularities, which correspond to the open ends of Dirac strings, is a direct consequence of the gauge structure of the theory. Specifically, such singularities appear when the gauge group is the compact group $U\left(1\right)$, rather than the non-compact group $\mathbb{R}$. The distinction between these two cases is crucial, as compactness imposes topological constraints that allow for the existence of magnetic monopoles. A compact $U\left(1\right)$ gauge group can be obtained as the lowest-energy surviving symmetry after spontaneous symmetry breaking of a larger compact group, such as the grand-unified groups (GUTs) $SU\left(5\right)$ or $SO\left(10\right)$. Within this framework, the existence of magnetic monopoles was first theoretically predicted by ’t Hooft and Polyakov [1]. These monopoles are not point-like elementary particles but rather solitonic solutions—stable, finite-energy field configurations—arising due to the topology of the spontaneously broken gauge field. Their theoretical mass is extraordinarily large, on the order of O(${10}^{16}$) GeV. This energy scale is far beyond anything achievable in current or foreseeable particle accelerators, implying that if such monopoles exist, they could only have been produced under the extreme conditions of the early universe, particularly during the high-energy epochs following the Big Bang. Despite their theoretical appeal and their natural emergence in GUT frameworks, no experimental evidence for magnetic monopoles has ever been found. Various searches, including direct detection experiments and indirect astrophysical observations, have yielded no conclusive results. This absence remains one of the significant puzzles in high-energy physics and cosmology, leading to ongoing investigations into whether monopoles were somehow diluted, confined, or otherwise rendered undetectable in the present universe.

Polyakov, in his seminal works [2,3], introduced a lattice regularization framework to describe monopole singularities. This approach has provided a robust theoretical foundation for studying the role of magnetic monopoles in various physical systems. By employing this regularization technique, we have demonstrated that much lighter monopoles can emerge as collective excitations in quantum granular materials [4,5,6,7]. Their presence has profound consequences, as they can drastically alter the underlying behavior of the theory, leading to the confinement of electric charges [2,3]. Quantum matter describes systems that exist at extremely low temperatures where quantum mechanical effects play a fundamental role in determining their properties and collective quantum phenomena emerge. At these low temperatures, classical descriptions of matter break down, and quantum coherence, entanglement, and fluctuations become dominant. Such materials exhibit exotic phases that cannot be understood within the framework of conventional condensed matter physics. Among these, superinsulators stand out as a particularly intriguing example, where magnetic monopoles manifest as emergent quantum excitations in granular systems, leading to unique charge confinement effects.

A particularly well-studied example of such systems is planar Josephson junction arrays (JJAs) [4,6], which are easily accessible in laboratory experiments. These systems serve as practical models for superconducting thin films [8], where emergent granularity has been observed [9]. The presence of magnetic monopoles in these materials leads to novel quantum phenomena, which are crucial for understanding unconventional phases of matter. This phenomenon is not restricted to planar systems. Recent experimental studies have revealed the presence of emergent granularity in bulk superconductors [10]. Furthermore, emergent granularity has been detected in high-temperature superconductors, particularly in the underdoped regime [11].

In planar Josephson junction arrays and superconducting films, magnetic monopoles manifest themselves as instantons, non-perturbative tunneling events that play a crucial role in quantum dynamics. In higher-dimensional settings, specifically in (3 + 1)-dimensional spacetime, monopoles instead take the form of topological solitons. These solitons are dual counterparts to Cooper pairs, which are associated with Noether charges. Within this framework, we have previously shown [5,7] that these emergent monopole (instanton) excitations in condensed matter systems can form a quantum Bose condensate, effectively behaving as a plasma of instantons. This realization opens the door to understanding novel phases of matter driven by monopole condensation.

One of the most striking implications of this phenomenon is the interplay between electric and magnetic quantum orders, which lies at the heart of the superconductor–insulator transition (SIT) [12,13,14]. At the SIT, the condensation of monopoles forming a plasma of instantons in (2 + 1) dimensions gives rise to an entirely new phase of matter, known as the superinsulator [5]. This phase is fundamentally distinct from conventional insulators and superconductors. In a superinsulator, Cooper pairs experience linear confinement due to the presence of Polyakov’s electric confining strings [15], which act as the dual analog of superconducting vortices. This mechanism is strikingly similar to the way quarks are confined within hadrons in quantum chromodynamics (QCD).

In essence, superinsulators provide an Abelian realization of the confinement mechanism originally proposed by t Hooft [16] in the context of QCD, where confinement arises due to the dual Meissner effect. This analogy suggests that the physics of condensed matter systems, particularly granular superconductors, can serve as a powerful platform for exploring fundamental aspects of non-Abelian gauge theories and confinement. Understanding the role of emergent monopoles in these systems not only sheds light on novel quantum phases but also deepens our comprehension of confinement mechanisms that play a crucial role in high-energy physics.

In three dimensions, quantum artificial superlattices are promising platforms for testing the effects of quantum magnetic monopoles. In [17], an artificial supersolid was realized in a lanthanum-based cuprate superlattice composed of artificial quantum wells [18,19,20,21,22], opening the possibility of experimentally investigating the effects of monopoles in quantum artificial materials.

In [23], magnetic monopoles as emergent excitations in exotic magnets known as spin ice have been proposed. It is stated that, in spin ice, the dipole moments of localized spins fractionalize into emergent magnetic monopoles. These monopoles behave as sources and sinks of an emergent magnetic field, forming a classical Coulomb gas. However, unlike the emergent monopole excitations we describe here, monopoles in spin ice do not satisfy the Dirac quantization condition. In our case, monopoles (instantons) arise in a phase governed by compact QED as an effective field theory. In QED, Dirac monopoles require the presence of fundamental electric charges to satisfy the quantization condition. In contrast, spin ice features only emergent magnetic excitations, with no corresponding free electric charges. As a result, the Dirac quantization condition does not apply.

Superinsulators are materials that can transition sharply to a highly resistive state under external influence. This property makes them promising for developing innovative devices, such as resistive switching memory elements, where the high-resistance state represents binary “1” and the low-resistance state represents binary “0”. Additionally, superinsulators could serve as the basis for superinsulating qubits, which are conceptually dual to superconducting qubits, offering an alternative way to encode quantum information.

Moreover, while superconductors enable perfect current storage by allowing dissipationless electrical flow, superinsulators provide perfect charge storage by preventing any charge transport. This unique property suggests their potential use in “perfect” batteries, revolutionizing energy storage in the same way superconductors have transformed energy transport.

However, a major challenge is that superinsulators currently exist only at ultra-low temperatures, limiting their immediate practical applications. Just as superconductivity was initially confined to cryogenic conditions but later extended to higher temperatures through material discoveries, the next major step in superinsulation research will be to identify materials and mechanisms that sustain superinsulating behavior at more accessible temperatures. Achieving this goal could open the door to a wide range of technological applications, from energy storage to next-generation quantum devices. Furthermore, we plan to extend this research to explore superinsulation in three-dimensional granular materials, expanding the potential impact of this phenomenon.

This review is organized as follows: In Section 2, we show how compact electrodynamics arises as an effective gauge theory for granular quantum materials in two spatial dimensions (2D). We will then analyze, in Section 3, how confinement arises by the formation of electric flux tubes (confining strings) in a plasma of instantons, which bind Cooper pairs and anti-Cooper pairs, causing infinite resistance. The last section is devoted to the extension to three spatial dimensions (3D) and to the study of the deconfinement criticality which allows us to distinguish between superinsulation in 2D and 3D.

2. Compact Effective Electromagnetic Action for Quantum Granular Materials

As a primary example, we will examine superconducting films where superinsulation was first experimentally observed. These materials, near the superconductor–insulator transition (SIT), exhibit emergent granularity characterized by the spontaneous formation of superconducting droplets [24]. Due to this granular nature, as we explained, they can be effectively modeled using JJA, which provide a theoretical framework for understanding their collective quantum behavior.

In what follows, we will consider an Euclidean lattice with lattice spacing $v{\mathsf{\ell }}_0=\mathsf{\ell }=1$, where ${\mathsf{\ell }}_0$ is the lattice spacing in the “time” direction, and the velocity of light $v=1/\sqrt{\epsilon \mu }<1$. Correspondingly, we will also rescale the gauge potentials $A_0\to A_0/v$.

As illustrated in Figure 1, when the phases of the local order parameter exhibit a non-trivial circulation around neighboring droplets, winding by an integer multiple of $2\pi$, a vortex is formed. Unlike conventional vortices, these vortices lack a normal-state core and resemble those found in the XY model [25], where phase coherence persists between the droplets. We have thus the formation of XY vortices, for which dissipation is negligible.

Josephson vortices are dual to Cooper pairs, representing the charge degrees of freedom in superconducting films. Their mutual statistics interaction, the Aharonov–Bohm–Casher (ABC) interaction, is the dominant one at large distances. Wilczek [26] demonstrated that this interaction can be locally described by introducing two emergent gauge fields, $a_{\mu }$ and $b_{\mu }$, which couple through a mixed topological Chern–Simons action [27]:

$$S_{\mathrm{matter}}=\sum _{x}i{\displaystyle \frac{1}{2\pi }}{a}_{\mu }{ϵ}^{\mu \alpha \nu }{\partial }_{\alpha }{b}_{\nu }+i{a}_{\mu }{Q}_{\mu }+i{b}_{\mu }{M}_{\mu },$$

(3)

where $Q_{\mu }$ and $M_{\mu }$ are the charge and vortex currents, respectively. When the dynamical terms admitted by symmetry for the two gauge fields $a_{\mu }$ and $b_{\mu }$ are included in Equation (3), the matter action becomes the following:

$$S_{\mathrm{matter}}=\sum _{x}i{\displaystyle \frac{1}{2\pi }}{a}_{\mu }{ϵ}^{\mu \alpha \nu }{\partial }_{\alpha }{b}_{\nu }+{\displaystyle \frac{1}{4e_v^2}}{f}_{\mu \nu }{f}_{\mu \nu }+{\displaystyle \frac{1}{4e_q^2}}{g}_{\mu \nu }{g}_{\mu \nu }+i{a}_{\mu }{Q}_{\mu }+i{b}_{\mu }{M}_{\mu },$$

(4)

with $f_{\mu \nu }$ and $g_{\mu \nu }$ field strengths for the $a_{\mu }$ and $b_{\mu }$ gauge fields, respectively. The duality between Cooper pairs and vortices arises because the electric field lines are confined in the plane up to a screening length of $O\left(d\epsilon \right)$ with $\epsilon$ being the relative dielectric permittivity of the insulating normal state [28]. The Coulomb interaction is, thus, logarithmic. The two coupling constants $e_q^2$ and $e_v^2$, which represent the typical electric and magnetic energy scales, are related to the film and JJA parameters by the following relations:

$$\begin{array}{ccc}\hfill e_q^2& & =O\left({\displaystyle \frac{e^2}{2\pi d}}\right)=8E_C,\hfill \\ \hfill e_v^2& & =O\left({\displaystyle \frac{\pi d}{e^2{\lambda }_L^2}}\right)=4{\pi }^2{E}_J.\hfill \end{array}$$

(5)

The phase diagram of superconducting films is governed by the interplay between the Coulomb and magnetic energy scales, characterized by the ratio $g=e_v/e_q$. While the universality class of the transition may be influenced by specific lattice properties, exploring these effects lies beyond the scope of this review. There are two dual phases: a global Higgsless superconducting phase [29], when charges form a condensate, and a superinsulating phase, when vortices proliferate [4,30,31]. If the film parameters satisfy certain conditions [4], an intermediate Bose metal phase [4] can form. This Bose metal phase is a phase in which statistical interactions cause both charges and vortices to be suppressed in the bulk. This phase is a bosonic topological insulator and it is characterized by the presence of protected edge modes [32]. These 1D conducting edge channels, although protected by symmetries, are not superconducting, since a resistance is caused by quantum phase slips [33], resulting in a metallic behavior. A parameter $\eta$, which naturally arises in our theory [4], distinguishes between the direct transition and the transition through the intermediate Bose metal phase.

To gain insight into phase slips, consider a one-dimensional granular material modeled as a Josephson junction chain [33]. This system consists of superconducting islands arranged along the vertices of a 1D lattice, each characterized by a phase $\varphi$. The key parameters governing its behavior include the Josephson coupling energy $E_J$, relative capacitance C, ground capacitance $C_0$, and charging energy $E_C=2e/C$. In the regime where $C_0>>C$, charge dynamics can be effectively described as a point interaction with strength $E_C=2e/C$. Quantum tunneling between islands occurs when the phases are sufficiently aligned. These tunneling events correspond to instances where one of the phases undergoes a discrete jump of $2\pi$, a process known as a quantum phase slip [34]. In a phase where these phase slips proliferate (for a comprehensive review, see [25]), the application of a current to the quantum wire disrupts the balance between phase slips of opposite chiralities. This imbalance leads to a fundamental change in the system’s transport properties, effectively transforming the originally superconducting wire into a metallic state by introducing a finite resistance [33]. The phase diagram is shown in Figure 2.

In the superinsulating phase, electric topological excitations $Q_{\mu }$ are suppressed, allowing us to set $Q_{\mu }=0$. The effective electromagnetic action can be derived by coupling the matter action in Equation (4) to the electromagnetic gauge field $A_{\mu }$ and subsequently integrating out the fields $a_{\mu }$ and $b_{\mu }$. This results in the expression:

$$S_{\mathrm{eff}}={\displaystyle \frac{1}{2e_{\mathrm{eff}}^2}}\sum _{x,i}{\left(F_i+2\pi M_i\right)}^2={\displaystyle \frac{1}{2e_{\mathrm{eff}}^2}}\sum _{x,i}{E}_i^2+{\displaystyle \frac{2{\pi }^2}{e_{\mathrm{eff}}^2}}\sum _{x,i}{m}_x{\displaystyle \frac{1}{-{\nabla }_2^2}}{m}_x,$$

(6)

with $F_{\mu }=(1/2){ϵ}_{\mu \alpha \beta }{F}_{\alpha \beta }$. The electric field components are represented as $E_i$, and the quantity $m=d_i{M}_i$ corresponds to instantons that arise due to the model’s compact structure. The effective coupling constant $e_{\mathrm{eff}}$ is given by the following:

$$e_{\mathrm{eff}}^2\approx {\displaystyle \frac{1}{\kappa }},$$

(7)

with $\kappa =e_v/e_q$. We note that Equation (6) is the non-relativistic limit of Polyakov compact QED l [2,3]

$$S_{\mathrm{eff}}={\displaystyle \frac{1}{2e_{\mathrm{eff}}^2}}\sum _{x,\mu \nu }\left(1-\cos \left(F_{\mu \nu }\right)\right).$$

(8)

Using the Villain approximation, Equation (8) can be rewritten as follows:

$$S_{\mathrm{eff}}={\displaystyle \frac{1}{4e_{\mathrm{eff}}^2}}\sum _x{\left(F_{\mu \nu }-2\pi M_{\mu \nu }\right)}^2,$$

(9)

where we introduce integer plaquette variables $M_{\mu \nu }\in \mathbb{Z}$ which implement periodicity and over which we have to sum in the partition function. Although the variables $F_{\mu \nu }$ satisfy the Bianchi identity $d_i{\tilde{F}}_{i0}=\mathrm{div}{e}_{\mathrm{eff}}\mathbf{B}=0$ (with $d_i$ lattice derivatives), in the presence of the integer fields $M_{\mu \nu }$, it can be readily demonstrated that the overall compact magnetic field naturally allows for the existence of magnetic monopoles [2,3],

$${\displaystyle \frac{1}{e_{\mathrm{eff}}}}{d}_i\left({\tilde{F}}_{i0}-2\pi {\tilde{M}}_{i0}\right)={\displaystyle \frac{2\pi }{e_{\mathrm{eff}}}}{\delta }_{\mathbf{x},{\mathbf{x}}_0}.$$

(10)

The non-relativistic limit arises due to the diverging dielectric constant $\epsilon \to \infty$, which corresponds to the limit where the velocity of light $v\to 0$. In this limit, only static electric configurations matter. In the gauge theory description of JJA, which models quantum films, the purely electric action is obtained directly [6]. The gauge invariance of the emergent $b_{\mu }$ field in (3) forces the condition: $d_{\mu }{M}_{\mu }=0$, which, in Minkowski spacetime coordinates, gives the following:

$$m=d_i{M}_i=-d_t{M}_0.$$

(11)

Since, in 3D Euclidean space, $F_{\mu \nu }$ in Equation (23) represents the magnetic field, with $F_0$ as its component in the Euclidean time direction, $M_0$ represents the vortices. In Euclidean spacetime, instantons can be interpreted as non-relativistic magnetic monopoles that mediate transitions between the one-vortex and zero-vortex sectors, as illustrated in Figure 3.

Instantons proliferate for large effective coupling, i.e., for $\kappa <1$, when Coulomb interactions dominate magnetic ones.

3. Confinement and Superinsulation

The partition function in the superinsulating phase is given by the following:

$$Z=\sum _{\left\{M_i\right\}}{\int }_{-\pi }^{+\pi }\mathcal{D}{A}_{\mu }{\mathrm{e}}^{S_{\mathrm{eff}}\left(A_{\mu },M_i\right)}.$$

(12)

With $S_{\mathrm{eff}}$ given by Equation (6), it can be written as follows:

$$Z=Z_0{Z}_{\mathrm{inst}}=Z_0\sum _{\left\{m\right\}}{e}^{-{\displaystyle \frac{2{\pi }^2}{e_{\mathrm{eff}}^2}}{\sum }_x{m}_x{\displaystyle \frac{1}{-{\nabla }_2^2}}{m}_x}$$

(13)

where $Z_0$ is the Gaussian integral over $A_{\mu }$. From Equation (26), it follows that instantons interact through the inverse of the spatial Laplacian, resulting in a logarithmic potential of the form $(e_{\mathrm{eff}}^2/2\pi )\log \left|\mathbf{x}\right|$. This behavior contrasts with the relativistic case, where the interaction potential scales as $\propto 1/x$ and is, thus, always in a confining plasma phase. In superconducting films, compact QED arises as an induced effective action in the magnetic condensation phase in a theory in which there is an ABC interaction between electric and magnetic degrees of freedom. This fact dramatically changes the behavior of the theory and leads to the presence of a quantum BKT transition, where the role of temperature is played by the coupling constant The brackets are not matched correctly; please check if they can be removed or if a right bracket is missing. $e_{\mathrm{eff}}$: for high values of $e_{\mathrm{eff}}$, instantons are free and confine probe charges. At the SIT, which corresponds to low values of $e_{\mathrm{eff}}$, instantons are confined, and probe charges are liberated.

To analyze the confinement properties of the model, we introduce the Wilson loop operator $W\left(C\right)$, where C represents a closed contour in three-dimensional Euclidean spacetime. When confined to a plane defined by Euclidean time and one spatial coordinate, the expectation value of the Wilson loop provides insight into the interaction potential between two external probe charges $\pm q_{\mathrm{ext}}$:

$$\langle W\left(C\right)\rangle ={\displaystyle \frac{1}{Z_{A_{\mu },M_i}}}\sum _{\left\{M_i\right\}}{\int }_{-\pi }^{+\pi }\mathcal{D}{A}_{\mu }{e}^{-{\displaystyle \frac{2{\pi }^2}{e_{\mathrm{eff}}^2}}{\sum }_x{\left(F_i-2\pi M_i\right)}^2}{\mathrm{e}}^{i{q}_{\mathrm{ext}}{\sum }_C{A}_{\mu }},$$

(14)

where we absorbed a factor ℓ in $A_{\mu }$. A linear potential between the probe charges will result in an area law in the expectation value of the Wilson loop [3]:

$$\langle W\left(C\right)\rangle =e^{-\sigma A},$$

(15)

where A is the area of the surface S enclosed by the loop C, which amounts to confinement by formation of a string with string tension $\sigma$ [3]. A perimeter law is, instead, the hallmark of a short-range potential between probe charges.

The expectation value of the Wilson loop Equation (14) gives an area law $\langle W\left(C\right)\rangle =e^{-\sigma A}$ with string tension $\sigma$ (we will not repeat here the details of the calculation that can be found in [5,6,7]):

$$\sigma ={\displaystyle \frac{\sqrt{8}}{{\mathsf{\ell }}_0\mathsf{\ell }}}{\displaystyle \frac{e_{\mathrm{eff}}}{\pi }}{\mathrm{e}}^{-{\displaystyle \frac{{\pi }^2}{e_{\mathrm{eff}}^2}}{G}_2\left(0\right)}.$$

(16)

Here, $G_2\left(0\right)$ represents the inverse of the infrared-regularized two-dimensional lattice Laplacian evaluated at coinciding points. If we instead consider a single-electron probe with charge $q_{\mathrm{ext}}=1/2$, subject to the same boundary conditions, it experiences confinement by a string with tension given by the following:

$${\sigma }_{\mathrm{electrons}}={\displaystyle \frac{1}{2}}\sigma .$$

(17)

As a result, charge transport, which relies on thermally excited normal quasiparticles, becomes strongly suppressed in superinsulators. Instantons generate an endogenous disorder in the system and give a mass for the photon [2,3]:

$$m_{\gamma }={\displaystyle \frac{8{\pi }^2}{e_{\mathrm{eff}}^2}}z,$$

(18)

where the parameter z:

$$z=e^{{\displaystyle \frac{-2{\pi }^{2}G\left(0\right)}{e_{eff}^2}}},$$

(19)

is the instanton fugacity. The effect of instantons is, thus, to screen the Coulomb potential with a a screening length ${\lambda }_{\mathrm{el}}=1/m_{\gamma }$. We have, thus, the formation of an electric string of typical width ${\lambda }_{\mathrm{el}}$ [35] and typical length $d_s=1/\sqrt{\sigma }$, which binds together Cooper pairs and Cooper holes in an “electric pion” on films (or JJAs) of a sufficient size. If we apply an external voltage, Cooper pairs and Cooper holes will move apart stretching the string, until it becomes energetically favorable for the system to pop out a Cooper pair–Cooper hole pair, forming two shorter strings. In the superinsulating phase, there are, thus, only neutral states, and we have an infinite resistance since free charges do not exist.

Superinsulation was predicted in 1996 [4], in the study of a gauge theory description of Josephson junction arrays. It was then, independently, found in 2008 in [30]. Experimental evidence of superinsulation has been found in TiN, NbTiN, InO, and NbSi films. Experimental evidence of the linear confinement due to the formation of a confining string (electric flux tube) has been found in [36,37].

4. Deconfinement Criticality

We have seen in the previous section that monopoloes (instantons) appear as emergent excitations in granular 2D systems. Ganularity, however, is not confined to two dimensions. In fact, in [10], granularity was detected in bulk superconductors. The two-dimensional core-less vortices become, in 3D, 1D extended objects which can end in magnetic monopole–antimonopole pairs.

In 3D, the ABC interactions between charges and vortices, in their local formulation, are described by the action:

$$\mathcal{L}={\displaystyle \frac{i}{4\pi }}{b}_{\mu \nu }{ϵ}_{\mu \nu \alpha \beta }{\partial }_{\alpha }{a}_{\beta }+i{a}_{\mu }{Q}_{\mu }+{\displaystyle \frac{i}{2}}{b}_{\mu \nu }{M}_{\mu \nu },$$

(20)

where $Q_{\mu }$ and $M_{\mu \nu }$ describe the “Euclidean world-lines” of point charges and “Euclidean world-surfaces” of vortices on the lattice. Equation (20), representing the BF action [38], is the generalization to 3D of Wilczek’s Chern–Simons construction, where point-like charges couple to the vector gauge fields $a_{\mu }$, while vortices couple to Kalb–Ramond antisymmetric pseudotensor fields $b_{\mu \nu }$ [39] so that the theory is invariant under parity. The BF model is topological, since it is metric-independent. It is invariant under the usual gauge transformations $a_{\mu }\to a_{\mu }+{\partial }_{\mu }\xi$ but also under gauge transformations of the second kind,

$$b_{\mu \nu }\to b_{\mu \nu }+{\partial }_{\mu }{\lambda }_{\nu }-{\partial }_{\nu }{\lambda }_{\mu }.$$

(21)

The 3D, Euclidean-space action corresponding to Equation (4) is as follows:

$$S=\sum _x{\displaystyle \frac{{\mathsf{\ell }}^4}{4f^2}}{f}_{\mu \nu }{f}_{\mu \nu }+i{\displaystyle \frac{{\mathsf{\ell }}^4}{4\pi }}{a}_{\mu }{k}_{\mu \alpha \beta }{b}_{\alpha \beta }+{\displaystyle \frac{{\mathsf{\ell }}^4}{12{\mathsf{\Lambda }}^2}}{h}_{\mu \nu \alpha }{h}_{\mu \nu \alpha }+i\mathsf{\ell }{a}_{\mu }{Q}_{\mu }+i{\mathsf{\ell }}^2{\displaystyle \frac{1}{2}}{b}_{\mu \nu }{M}_{\mu \nu },$$

(22)

where all zero-components of vectors are rescaled by the light velocity $v=1/\sqrt{\epsilon \mu }$ in the medium so that $x^0=vt$. The two parameters, $f=\mathrm{O}\left(e\right)$, dimensionless, and $\mathsf{\Lambda }=\mathrm{O}(1/{\lambda }_{\mathrm{L}})$, with ${\lambda }_{\mathrm{L}}$ being the London penetration depth of the superconducting granules, encode the effective Coulomb interaction strength in the material and the magnetic scale, respectively.

The computation of the electromagnetic response $S_{\mathrm{eff}}\left(A_{\mu }\right)$ in the magnetic monopole condensate can be performed as in 2D. This is given by the following:

$$\begin{array}{c}\hfill {\mathrm{e}}^{-S_{\mathrm{eff}}\left(A_{\mu }\right)}=\sum _{M_{\mu \nu }}{\mathrm{e}}^{-{\displaystyle \frac{1}{8f^2}}{\sum }_{x,\mu ,\nu }{\left({\tilde{\mathcal{F}}}_{\mu \nu }-2\pi M_{\mu \nu }\right)}^2}.\end{array}$$

(23)

For sufficiently strong f, monopole condensation effectively transforms the physical electromagnetic field into a compact variable, constrained within the interval $[-\pi ,+\pi ]$. Consequently, the electromagnetic response is given by Polyakov’s compact QED once again.

The confinement properties of the theory are studied by computing the expectation value of the Wilson loop in this phase. The result is, as in 2D, an area law:

$$\langle W\left(C\right)\rangle ={\mathrm{e}}^{-\sigma A}$$

(24)

where A is the area of the surface S enclosed by the loop C, and the string tension is given by

$$\sigma ={\displaystyle \frac{32f}{\pi \sqrt{\epsilon \mu }}}{\displaystyle \frac{1}{{\mathsf{\ell }}^2}}\exp \left(-{\displaystyle \frac{\pi G\left(0\right)}{8f^2}}\right),$$

(25)

where $G\left(0\right)$ is the 4D lattice Coulomb potential at coinciding points. Similar to the two-dimensional case, the monopole condensate in three dimensions generates a string that binds charges to their corresponding anti-charges, effectively inhibiting charge transport in sufficiently large systems. In this context, a magnetic monopole condensate behaves as a three-dimensional superinsulator. The transition to the superinsulating phase occurs when the effective Coulomb interaction strength reaches the critical value: $f_{\mathrm{crit}}=O(\mathsf{\ell }/{\lambda }_L)$ [4,5,7].

Which is the experimental hallmark of superinsulation that can discriminate between 2D and 3D superinsulators, exposing the linear nature of the underlying confinement caused by monopole (instanton) condensation? As demonstrated, in the superinsulating phase, Cooper pair dipoles are tightly bound into neutral “meson-like” states by Polyakov’s confining strings [15]. The action of these strings arises from their interaction with a massive Kalb–Ramond tensor gauge field, which couples to the world-sheet elements. This action can be explicitly derived for compact QED in both two and three dimensions [40], as well as for Abelian-projected $SU\left(2\right)$ [41,42]. The world-sheet formulation of the confining string action is given in terms of a non-local, long-range interaction between surface elements [43]. In [44], the finite-temperature behavior of these confining string has been studied.

The confining string action describes the superinsulating phase and can be used to study all its physical properties. In the finite-temperature deconfinement transition of superinsulators, the critical behavior corresponds to the regime where the confining strings extend indefinitely relative to the cutoff scale, leading to the liberation of the particles at their endpoints. The deconfinement criticality for the string has been derived in [45] both in 2D and 3D. This critical behavior of the string tension can be connected to the critical behavior of the conductance G using the critical scaling of the correlation length, interpreted as the maximum size of the bound charge–anticharge pair. In 2D, the criticality, as shown in [45], is the BKT criticality for charges, which turns out to be the same for both logarithmic and linear confinement, confirming the Svetitsky–Yaffe conjecture [46]. In this case, the correlation length scales as follows [47,48,49,50]:

$${\xi }_{\pm }\propto exp\left[{\displaystyle \frac{b_{\pm }}{\sqrt{|T/T_{\mathrm{c}}-1|}}}\right],$$

(26)

where the ± subscript labels the $T>T_{\mathrm{c}}$ and $T<T_{\mathrm{c}}$ regions, respectively. The conductance becomes the following:

$$G\propto exp[-b/\sqrt{|T/T_{\mathrm{CBKT}}-1|}],$$

(27)

where $T_{\mathrm{CBKT}}$ is the temperature of the charge BKT transition, and b is a constant of order unity. This behavior has been experimentally observed in NbTiN films [51], as shown in Figure 4.

In [45], we demonstrated how the deconfinement scaling described by Equation (26) extends to three-dimensional systems. In 3D, the critical behavior of superinsulators deviates from the BKT form and instead follows the Vogel–Fulcher–Tammann (VFT) scaling:

$${\xi }_{\pm }\propto exp\left({\displaystyle \frac{b_{\pm }^{\prime }}{|T/T_{\mathrm{c}}-1|}}\right).$$

(28)

This behavior is a defining feature of one-dimensional confining strings in three-dimensional systems, where the interactions between world-surface elements follow a logarithmic scaling, similar to how particles interact in two dimensions. The VFT scaling behavior is considered a hallmark of glassy behavior, derived [52] for the 3D XY model with quenched disorder. Here, the glassy behavior arises due to the presence of topological defects endowed with long-range interactions without any quenched disorder.

The scaling behavior described by Equation (28) has been experimentally observed in [53] for the finite-temperature insulating phase in InO disordered films, in which the thickness is much larger than the superconducting coherence length. The VFT scaling is shown in Figure 5. This experimental result can be considered as the first experimental evidence, although not conclusive of linear confinement in 3D superinsulators.