ARTICLE

DOI: 10.1038/s42005-018-0073-9 OPEN Confinement and asymptotic freedom with Cooper pairs

M.C. Diamantini1, C.A. Trugenberger2 & V.M. Vinokur 3 1234567890():,; One of the most profound aspects of the standard model of particle physics, the mechanism of confinement binding into , is not sufficiently understood. The only known semiclassical mechanism of confinement, mediated by chromo-electric strings in a con- densate of magnetic monopoles still lacks experimental evidence. Here we show that the infinite resistance superinsulating state, which emerges on the insulating side of the superconductor-insulator transition in superconducting films offers a realization of confine- ment that allows for a direct experimental access. We find that superinsulators realize a single-color version of quantum chromodynamics and establish the mapping of quarks onto Cooper pairs. We reveal that the mechanism of superinsulation is the linear binding of Cooper pairs into neutral “mesons” by electric strings. Our findings offer a powerful laboratory for exploring and testing the fundamental implications of confinement, asymptotic freedom, and related quantum chromodynamics phenomena via desktop experiments on superconductors.

1 NiPS Laboratory, INFN and Dipartimento di Fisica e Geologia, University of Perugia, via A. Pascoli, I-06100 Perugia, Italy. 2 SwissScientific Technologies SA, rue du Rhone 59, CH-1204 Geneva, Switzerland. 3 Materials Science Division, Argonne National Laboratory, 9700S. Cass Avenue, Lemont, IL 60637, USA. Correspondence and requests for materials should be addressed to V.M.V. (email: [email protected])

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he standard model of particle physics is extraordinarily correspond to fixed φ, hence indefinite N. Inversely, fixed N and successful at explaining many facets of the physical realm. indefinite φ characterizes the superinsulating state. As a Cooper T π Yet, one of its profound aspects, the mechanism of con- pair is a charge quantum, while a vortex carries the 2 phase finement binding quarks into hadrons, is not sufficiently under- quantum, the SIT is driven by the competition between charge stood. The only known semiclassical mechanism of confinement (Cooper pairs) and vortex degrees of freedom, in accordance with is mediated by chromo-electric strings in a condensate of mag- early ideas11. netic monopoles1–3 but its relevance for quantum chromody- We turn now to the construction of the action of the Cooper namics still lacks experimental evidence. This suggests a quest for pair-vortex system near the SIT, where both degrees of freedom systems that could allow for direct experimental tests of the string are to be included on an equal footing. The key contribution is confinement mechanism. To identify such a system we follow a the infinite-range (i.e. non-decaying with distance) Aharonov- brilliant insight of ‘t Hooft4, who appealed to a solid state physics Bohm-Casher (ABC) -vortex topological interaction, analogy in a Gedanken experiment to explain confinement. embodying the quantum phase acquired either by a charge He demonstrated that it is realized in a phase which is a dual twin encircling a vortex or by a vortex encircling a charge. To ensure a to , in a sense that it has zero particle mobility, local formulation of the action, we must introduce two emergent and called hence this phase a “superinsulator”. The infinite- gauge fields, aμ and bμ mediating these ABC interactions. Then resistance superinsulating state was indeed first predicted to the topological part of the action assumes the form Z emerge in Josephson junction arrays (JJA)5 and then in dis- hipffiffiffi pffiffiffi fi 6,7 CS ¼ 3 n ϵ ∂ þ þ ; ð Þ ordered superconducting lms at the insulating side of the S d xi π aμ μαν αbν i naμQμ i nbμMμ 1 superconductor-insulator transition (SIT)8–12. Experimentally, 2 fi 7,13 superinsulators were observed in titanium nitride (TiN) lms ϵ and, albeit under a different name, InO films14 and have become where μαν is the completely antisymmetric tensor, and –  15 17 P R ðiÞðτÞ ever since a subject of an intense study, see and references dxqμ 3 ðiÞ Qμ ¼ ðiÞ dτ τ δ x À xq ðτÞ ; therein. xq d 5,7 i ð2Þ Originally, the idea of superinsulation grew from the sup- P R ðiÞ ðτÞ dxmμ 3 ðiÞ posed 2D logarithmic Coulomb interactions between Cooper ¼ ðiÞ τ δ À ðτÞ ; Mμ x d dτ x xm pairs in the critical vicinity of the SIT realized in lateral Josephson i m junction arrays5,12. Here we show that, starting with the uncer- tainty principle for Cooper pairs7 and building solely on the most are the world-lines of elementary charges and vortices labeled by ðiÞ ðiÞ general locality and gauge invariance principles, one constructs the index i, parametrized by the coordinates xq and xm , the effective action for superinsulators, which is exactly Poly- respectively, n is the dimensionless charge, and Greek subscripts akov’s compact quantum electrodynamic (QED) action3,18. run over the Euclidean three-dimensional space encompassing Accordingly, superinsulation emerges as an explicit realization of the 2D space coordinates and the Wick rotated time coordinate. the Mandelstam–‘t Hooft S-duality1,2 in materials that harbor Equation (1)defines the mixed Chern-Simons (CS) action21 and Cooper pairs and constitutes a single-color version of the quan- represents the local formulation of the topological interactions tum chromodynamic (QCD) vacuum, in which Cooper pairs play between charges and vortices, whereno the ABC phasesno are encoded fi ð Þ ð Þ the role of quarks. We thus nd that the Cooper pair binding in the Gauss linking number of the x i and x i world-lines. mechanism in a superinsulator, leading to the infinite resistance q m → at finite temperatures, is the linear, rather than logarithmic, The CS action is invariant under the gauge transformations aμ + ∂ λ → + ∂ χ fl confinement of charges into neutral “mesons” due to Polyakov’s aμ μ and bμ bμ μ ,re ecting the conservation of the electric strings3,18, arising in the vortex condensate. The Abelian charge and vortex numbers, and is the dominant contribution to fi character of the compact QED, albeit a strong coupling gauge the action at long distances, since itp containsffiffiffi only one eld = ðÞ= π ϵ ∂ ϕ = theory, allows for an analytical derivation of the linear confine- derivative.pffiffiffi In this representation jμ n 2 μαν αbν and μ ment by electric strings, at variance to the QCD whose com- ðÞn=2π ϵμαν∂αaν are the continuous charge and vortex number plexity requires heavy numerical computations. current fluctuations, while Qμ and Mμ stand for integer point Since linear confinement by strings is not restricted to 2D, we charges and vortices. We use natural units c = 1, ħ = 1 but establish that superinsulation is a distinct genuine state of matter restore physical units when necessary. Also, from now on we set that appears in both 2D or 3D realizations and calculate the the charge unit n = 2 for Cooper pairs. deconfinement temperature that marks the phase transition of The next-order terms in the effective action of the SIT contain superinsulators into conventional insulators and which, in 2D, two field derivatives. Gauge invariance requires that they be coincides with the Berezinskii-Kosterlitz-Thouless (BKT) transi- constructed in terms of the “electric” and “magnetic” fields tion temperature. Finally we also unearth a Cooper pair analogue corresponding to the two gauge fields. Introducing the dual field 19 of the asymptotic freedom effect , which suggests that systems strengths fμ ¼ ϵμαν∂αbν and gμ ¼ ϵμαν∂αaν one identifies the smaller than the string scale appear in a quantum metallic fi fi magnetic elds as f0 and g0 and the electric elds as fi and gi, state. Our findings offer thus an easy access tool for testing “ ” fi where 0 denotes the Wick rotated time and Latin indices denote fundamental implications of con nement, asymptotic freedom, purely spatial components. We thus arrive at the full action and related QCD phenomena via desktop experiments on R ¼ 3 1 ϵ ∂ þ 1 2 superconductors. S d xiπ aμ μαν αbν 2μ f 2D 2ev P 0 ε þ P 2 þ 1 2 2 f 2 μ g ð Þ 2ev i 2eq P 0 3 ε pffiffiffi pffiffiffi Results þ P 2 þ þ : 2 gi i 2aμQμ i 2bμMμ Action in two-dimensional systems. We start by showing how 2eq dual superconducting and superinsulating states can be under- μ ε Δ Δφ ⩾ Here P is the magnetic permeability and P is thepffiffiffiffiffiffiffiffiffi electric stood from the uncertainty principle, N 1 between the 20 fi ¼ = μ ε number of charges, N ¼ 2jjΨ 2, and the phase φ of the Cooper permittivity , which de ne the speed of light vc 1 P P in 2 ¼ 2= 2 ¼ pairs quantum field Ψ = N exp(iφ), bound by the commutation the material. The two coupling constants, eq e d and ev 15,20 2 2 relation [N, φ] = i . At zero temperature, superconductors π =ðÞe λ? are the characteristic energies of a charge and a vortex

2 COMMUNICATIONS PHYSICS | (2018) 1:77 | DOI: 10.1038/s42005-018-0073-9 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | DOI: 10.1038/s42005-018-0073-9 ARTICLE in the film, respectively20. Here d is the thickness of the film, confining Cooper pairs in superinsulators into “mesons” (Fig. 1). λ ¼ λ2= λ γ fi ? L d is the Pearl length, and L is the London length of the Indeed, as seen from the action (4), at large , the dynamical elds bulk. The effective action in this order of the expansion with get squeezed into the vicinity of the paths minimizing the action, 2 respect to derivatives is perfectly dual under the mutual exchange to form quantized fluxes ‘ Fμ. The quantized electric flux tubes of charge and vortex degrees of freedom and the corresponding are the analogues of the strings mediating linear confinement of coupling constants. The charge-vortex duality is expressed by the quarks into hadrons. Like Abrikosov vortices, for which the ≡ ↔ action symmetry with respect to the transformation g ev/eq London penetration depth, the inverse of the Anderson-Higgs 1/g. Thus, g is the tuning parameter driving the system across the photon mass, sets the spatial scale of the decay of encircling SIT, and the SIT itself corresponds to g = g = 1. The possible supercurrents and magnetic field associated with the vortex, the c fi duality breaking is a higher order effect. In field theory, this characteristic lateral scale wstring for the decay of electric elds 22 = duality goes under the name of S-duality (strong-weak coupling around the string is the inverse of the photon mass mγ , wstring “ ” duality). Note that the addition of kinetic terms generates the 1/(vcmγ). The typical meson size instead, is given by the string fi σ 23 topological Chern-Simons mass mT for both gauge elds. In the tension . In the 2D relativistic model these are given by μ = ε = = γ2 2 relativistic case, P P 1, and the CS mass becomes mT ¼ pffiffi Àγ =2π; π21 fi mγ π ‘ e eqev/ . In the non-relativistic case the CS mass is modi ed to vc ð Þ 2 2 = 5 = μ π ¼ π mγv π3 2 Àγ2= π mpTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPeqev/ and the dispersion relation becomes E σ ¼ c ¼ vc 2 : 2D ‘γ2 ‘2 e 4 4 þ 2 2 4 4 mTvc vc p (see Methods, Lattice Chern-Simons operator). We stress here that we derived the action (3) describing the system of interacting Cooper pairs and vortices using solely Unlike vortices, however, long strings are unstable: it is symmetry and gauge invariance considerations. Importantly, the 5,12 energetically favorable to break a string into a sequence of action describing Josephson junction arrays is a special case of segments via the creation of charge-anticharge pairs, see Fig. 2. the same action with ε = 1, μ → ∞, e → 4E , e → 2π2E , P P q C v J This process corresponds top theffiffiffiffiffiffiffiffiffi creation of neutral “mesons” with where EC and EJ are the charging energy and the Josephson the typical size d ¼ v =σ. From the dependence of mγ energy of a single junction, respectively (see Supplementary string c on γ2, one finds, for the non-relativistic case Note 1). This provides a crosscheck for our general result.  gη d ’ ‘exp K ; ð6Þ Superinsulator. We are now equipped to discuss the nature of string v2 the superinsulating state. To that end, we couple the charge c current jμ to the physical electromagnetic gauge field Aμ by ≈ η where Kpisffiffiffiffiffiffiffiffiffi a numerical constant. Near the SIT, where g 1/ and adding to the action the minimal coupling term 2eAμjμ. Setting ¼ = μ ε  vc 1 P P c due to the divergence of the electric Qμ = 0, since charges are dilute, integrating out the gauge fields aμ ε 7,15  ‘ permittivity P , dstring , and the electric string is a well- and bμ, and summing over the condensed vortices Mμ, we arrive defined object. This establishes superinsulators as a single-color at the effective action Seff (Aμ) describing the electromagnetic response of an ensemble of charges in a superinsulator. On a realization of QCD. Cooper pairs assume the role of quarks that ‘ are bound by electric strings into neutral mesons and this linear discretized lattice with spacing (see Methods, Lattice Chern- fi fi Simons action), the effective action takes a form in which con nement is the origin of the in nite resistance of super- one immediately recognizes a non-relativistic version of the insulators. As quarks cannot be observed outside hadrons, 3,18 Cooper pairs do not exists outside neutral bound states, and Polyakov action for the compact QED model (see Supple- fi mentary Note 2): the absence of free charge carriers causes the in nite resistance.  2D Action and superinsulator in three-dimensional systems. The S Aμ ¼ S eff compact string confinement mechanism of superinsulation allows to P ¼ γ2 ½ŠÀ ðÞ‘2 generalize the concept of a superinsulator to higher dimensions, 2π2 vc 1 cos 2e F0 ð Þ fi fi x 4 since linear con nement by electric strings is not speci c to the ) 2D realm. Hence, superinsulators can exist in 3D exactly as QCD P þ 1 ½ŠÀ ðÞ‘2 : exists in 3D. The 3D analogue of the topological action (3) v 1 cos 2e Fi 24 x;i c involves the so called BF term , combining the standard gauge field aμ with the Kalb-Ramond antisymmetric gauge field of the second kind25 bμν, μ = μν ν Here the summation runs over the lattice grid {x}, F k A is R ε fi ¼ 4 1 ϵ ∂ þ 1 2 þ P 2 the dual eld strength, kμν is the lattice Chern-Simons operator S3D d xiπ aμ μναβ νbαβ 2e2μ f0 2e2fi 2 v P vpffiffi ϵμαν∂α (see Methods, Lattice Chern-Simons operator), and γ = ε pffiffiffi ð7Þ þ 1 2 þ P 2 þ þ 2 : η η = α 2 μ b 2 e i 2aμQμ i bμνMμν C g/vc with C being a numerical constant. The quantity (1/ ) 2eq i 2eq i 2 κ fl P ל ( , vc) characterizes the strength of quantum uctuations (see Supplementary Note 3). Here κ = λ⊥/ξ is the Ginzburg-Landau Here e = ∂ a − ∂ a and b ¼ ϵ ∂ a are the usual electric and fi ξ i 0 i i 0 i ijk j k parameter of the lm, is the superconducting coherence length, magnetic fields associated with the gauge field aμ, while fμ ¼ taking on the role of the ultraviolet cutoff ‘, and, finally, α = e2/ ð1=2Þϵμναβ∂νbαβ is the dual field strength associated with the (ħc) ≈ 1/137 is the fine structure constant. antisymmetric gauge field bμν. In addition to the gauge symmetry The physics of a superinsulator is governed by the spontaneous → + λ 18 = ∂ under transformations aμ aμ , this action is invariant under proliferation of instantons M μMμ, corresponding to → + ∂ χ − ∂ χ magnetic monopoles, so that the vortex number is not gauge symmetries of the second rank, bμν bμν μ ν ν μ,in conserved in the vortex condensate. Then, in a mirror analogue which the gauge function itself is a vector. In 3D, vortices are one- to the formation of Abrikosov vortices in superconductors due to dimensional extended objects and their world-surfaces are the Meissner effect mediated by the Cooper pair condensate, the described by the two-index antisymmetric tensor Mμν. Cooper fl ¼ condensate constricts electric field lines pairs,ÀÁpffiffiffi Qμ, and the related uctuation number current jμ 3,18 = π connecting the charge-anticharge pair into electric strings 2 2 fμ retain their point charge character. In 3D, eq is a

COMMUNICATIONS PHYSICS | (2018) 1:77 | DOI: 10.1038/s42005-018-0073-9 | www.nature.com/commsphys 3 ARTICLE COMMUNICATIONS PHYSICS | DOI: 10.1038/s42005-018-0073-9

a Quark confinement

q q

b Superconductor N S

Solenoid Solenoid

c Superinsulator

2e –2e

‘Anti-Cooper pair Cooper pair

Fig. 1 Dual Mandelstam–‘t Hooft–Polyakov confinement. a Quark confinement by chromo-electric strings. b Magnetic tube (Abrikosov vortex) that forms in a superconductor between two magnetic monopoles. c Electric string that forms in a superinsulator between the Cooper pair and anti-Cooper pair. The lines are the force lines for magnetic and electric fields respectively. In all cases the energy of the string (the binding energy) is proportional to the distance between either the monopoles or the charges

2e –2e 2e –2e ... 2e –2e

dstring

Fig. 2 Splitting electric strings into neutral mesons. The formation of a long string is energetically unfavorable, and small size charge-anticharge pairs emerge, splitting the string into a sequence of segments, each constituting a neutral meson

= dimensionless parameter, eq O(e), while ev has the dimension exact mapping between QCD and the physics of superinsulators, = λ λ of mass, ev O(1/ ), with being the bulk London length of the both in 2D and 3D. material. The topological mass arising from the BF coupling26 Finally, let us mention that, unlike in 2D, in 3D, the minimal = μ π maintains the same form as in 2D, mT Peqev/ . coupling of charges to electromagnetismR pffiffiffi can be complemented by The derivation of the effective action for a superinsulator in 3D 4 ðθ= π Þϕ a topological couplingÀÁpffiffiffi d xi 8 2 μνFμν of the vortex follows exactly the same steps as in 2D (Supplementary Note 2, current ϕ = 2=2π ϵμναβ∂αaβ to the electromagentic field Effective action for the superinsulator), with the result μν ( strength Fμν. This leads to an axion term28 S ¼  R axion 2 P ÂÃÀÁ 4 ðÞθ= π2 ~ SI ¼ 3D ¼ γ À ‘2~ d xi 16 FμνFμν in the electromagnetic effective action. Seff Aμ Scompact π2 2vc 1 cos 2e F0i 2 ; This is a surface term, since the partition function exp(−S )is x i ) ð Þ axion hi 8 invariant under shifts θ → θ + 2π. Time reversal, T , maps θ → P −θ θ T θ = þ 1 1 À cos 2e‘2F~ . So the only values of compatible with -invariance are vc ij θ = π π θ = π x;i;j 0 and , modulo 2 . For the string becomes fermionic18, acquiring a topological contribution (−1)ν in the ~ partition function, where ν is the signed self-intersection number where Fμν ¼ kμναAα is the 3D dual field strength (kμνα being the of the world-sheet in four-dimensional Euclidean space-time. The 3D lattice BF term- see Methods, Lattice BF term). This is again a 27 non-relativistic version of Polyakov’s compact QED model, this string tension changes to 3,18 27 pffiffiffi time in 3D , with the relativistic string tension given by v z pffiffiffi σ ¼ c K ð10Þ v z 3D 64π‘2 0 16γ σ ¼ c K γ ; ð9Þ 3D 64π‘2 0 4π Because the factor γ is now in the denominator, the fermionic where K0 is the McDonald function and z is the monopole Cooper pair mesons are large also in the deep superinsulating fugacity. Equations (4) and (8) are our key results, establishing an region, where ηg  1 and v = O(1).

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Finite temperatures. Now we turn to the finite temperature a M behavior and the deconfinement transition at which string con- finement of Cooper pairs ceases to exist and the superinsulator Superinsulator ‘ ’ 1 Deconfinement transforms to a conventional insulator. This happens at the transition critical temperature Tdc where the linear tension of the string ≡ 29 turns to zero. While it is known that, in 2D, Tdc TBKT , we can calculate Tdc straightforwardly as the temperature of dis- appearance of the vortex condensate. This is done in methods Q (finite temperature deconfinement transition), with the result that the superinsulator experiences a direct deconfinement transition –1 0 1 to an insulating state at the critical deconfinement temperature η = determined by relation /1/(g ) S(T ) where the function S(T) ) dc 1.5 T is derived by a geometric condition for the two competing con- ( b

S 1.4 densations (see Supplementary Note 3, Quantum Phase transi- Topological –1 tions) and is shown in Fig. 3. This equation uniquely determines insulator 1.3 the deconfinement temperature as a function of material 1.2 parameters. 1.1 1.0 Scale factor Scale factor 0.0 0.2 0.4 0.6 0.8 1.0 υ Experimental implications. To explore the far reaching experi- Tl / (h c) mental implications of the confining string theory of super- insulation we note first that the deconfinement criticality depends Fig. 3 Deconfinement transition. a Finite temperature deconfinement on the space dimension30. In 2D it coincides with that of the BKT transition from a superinsulator (magnetic numbers M = ±1 fall into the ÀÁpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q = transition29, and the resistance R / exp b= jjT=T À 1 .In interior of the ellipse, while electric numbers ±1 remain outside) to an 2D BKT insulator (no non-trivial quantum numbers fall within the ellipse). b The 3D, instead, the resistance exhibits the so-called Vogel-Fulcher- finite-temperature scaling factor that determines the critical temperature Tamman (VFT) criticality, R ∝ exp[b′/|T/T − 1|]30. Juxta- 3D dc for the superinsulator deconfinement transition, v is the light velocity in the posing the critical behaviors of the NbTiN film, having a super- c material conducting coherence length ξ ≳ d17 and that of the InO film, where ξ  d16, one sees that the NbTiN film shows the BKT- while the InO film exhibits the VFT divergence, in compliance with our predictions about 3D superinsulation. superinsulating had their size exceeded the typical dimension of fi ≲ ħ The deconfinement transition can be realized as a quantum the con ning string, estimated as dstring vc/kBTBKT. Using the fi 7,15 ≲ μ dynamical phase transition driven by an applied electric field E TiN lms parameters one obtains dstring 60 m. Remarkably, that would tear the electric strings. The threshold voltage, Vt ∝ the study of the size dependence of superinsulating properties in σL, corresponding to the pair-breaking critical current in TiN films33 revealed that in films with lateral sizes, of 20 μm and superconductors, breaks down the neutral meson chains, and a less, the insulating, thermally activated behavior saturates to the strip of ‘normal’ insulator forms along the former string path, metallic one upon cooling to ‘superinsulating temperatures’. This carrying the current. This pretty much resembles the conven- complies with the expected asymptotic freedom behavior. tional dielectric breakdown where the electric field burns a However, it would be premature to take it as a conclusive conducting channel in otherwise insulating environment and evidence for the asymptotic freedom in superinsulators, and triggers avalanche-like current jumps. The dielectric breakdown is further experimental research is needed. usually accompanied by current noise. Such a noise has indeed been recently observed in InO films31. Experiments demonstrat- ing the linear dependence of the threshold voltage on the sample Discussion size in films are still to come. Yet the evidence for linear We conclude by pointing out a close connection of the string confinement was provided by the analysis of the superinsulating confinement mechanism to concepts of many-body-localization behavior in the ultrathin TiN films32, which revealed that the (MBL)34. It was recently shown that MBL-like behavior may arise fi 35 magnetic eld dependence of Vt is exactly that of the 1D without exogenous disorder, due to strong interactions alone , Josephson ladder. and that, in gauge theories, this is due to the endogenous disorder In QCD, the flip side of the string confinement mechanism is embodied by the mixing of superselection sectors36, this process asymptotic freedom, i.e. the unconstrained dynamics of quarks at being identified as a transport-inhibiting mechanism due to spatial scales smaller than the string size19. While, strictly confinement in the Schwinger model in 1D. In our setting, it is speaking, asymptotic freedom refers to the running of the the Polyakov monopole instantons that play the role of endo- dimensionless gauge coupling to zero in the ultraviolet limit, it genous spontaneous disorder. Accordingly, our summation over can be viewed, from the string point of view, as the “slackening” the instanton gas configurations acts as averaging over endo- of the string so that quarks feel only weak short-range potentials genous disorder3,18. Importantly, the instanton formulation at small scales. One would thus expect that, in superinsulators, describes not only 1D, but the 2D and 3D physical dimensions as asymptotic freedom, in this string sense, should map onto the well. This spontaneous disordering mechanism has the unconstrained motion of the Cooper pairs at scales smaller than same effect, that of mixing, in this case, the flux superselection dstring. The ratio of the string width to the string length is wstring/ sectors, leading to the survival of only the neutral charge sector ∝ γ2 γ2 dstring (vc/ )exp(K /vc) with K being a numerical constant. as the physical state, while all other, charged states are localized For systems with small K and large γ2 this ratio is small. At scales on the string scale. Hence inhibition of the charge transport and fi fi wstring < r < dstring, Cooper pairs do not feel the string tension the in nite resistance. The same con nement mechanism that anymore but neither do they feel Coulomb interactions screened prevents the observation of quarks is thus responsible for the by the photon mass. Hence, one can expect a metallic-like low- absence of charged states and the infinite resistance in temperature behavior of small samples that should have turned superinsulators.

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2 ^ Methods where ∇ ¼ dμdμ is the lattice Laplacian. The Euclidean lattice BF model in 3D is Lattice Chern-Simons operator. The formulation of a gauge-invariant lattice then given by the action Chern-Simons term requires particular care. Following5 we introduce first the P ‘4 ‘4 ‘4ε ‘4 ¼ þ 2 þ P 2 þ μ 2 forward and backward derivatives and shift operators on a three-dimensional S i π aμkμαβbαβ 2e2 μ bi 2e2 ei 2e2 Pf0 x q P q v Euclidean lattice with sites denoted by {x}, directions indicated by Greek letters and pffiffiffi pffiffi ð22Þ ‘4ϵ ‘ þ P 2 þ ‘ þ ‘2 2 ; lattice spacing , 2 f i 2aμQμ i bμν Mμν 2ev i 2 f ðxþ‘μ^ÞÀf ðxÞ ^ dμf ðxÞ¼ ‘ ; Sμf ðxÞ¼fxðÞþ ‘μ ; ð11Þ where the dual field strengths are now defined by ^ f ðxÞÀfxðÞþ‘μ^ ^ ^ dμf ðxÞ¼ ‘ ; Sμf ðxÞ¼fxðÞÀ ‘μ : 1 ~ ^ fμ ¼ kμνρbνρ; fμν ¼ kμνρaρ; ð23Þ Summation by parts on the lattice interchanges both the two derivatives (with a 2 fi minus sign) and the two shift operators. Gauge transformations are de ned by = − ¼ ~ fi fi and ei d0ai dia0 and bi f0i are the usual electric and magnetic elds asso- using the forward lattice derivative. In terms of these operators one can then de ne fi two lattice Chern-Simons terms ciated with the gauge eld aμ. The dispersion relation and mass remain identical to the 2D formulas. In this case they are the non-relativistic generalizations of the BF ^ ^ ^ 26 kμν ¼ Sμϵμανdα; kμν ¼ ϵμαν dαSν ; ð12Þ mass . where no summation is implied over equal indices. Summation by parts on the Finite temperature deconfinement transition. In the field theory, the finite lattice interchanges also these two operators (without any minus sign). Gauge temperature T is introduced by formulating the action on a Euclidean time of finite invariance is then guaranteed by the relations length β = 1/T, with periodic boundary conditions (we have reabsorbed the fi ¼ ^ ¼ ; ^ ¼ ^ ^ ¼ : ð Þ Boltzmann constant into the temperature). If the original eld theory model is kμαdν dμkαν 0 kμνdν dμkμν 0 13 defined on a Euclidean lattice of spacing ‘, then β is quantized in integer multiples ‘= fi fi of vc. This representation of the nite-temperature eld theory holds as long as Note that the product of the two Chern-Simons terms gives the lattice Maxwell β  ‘ vc , or, equivalently, if the temperature is much lower than the UV cutoff, operator  =‘ fi T vc , as expected. Because of the lattice structure, energies are de ned only π=‘ ^ ^ 2 ^ within a Brillouin zone of length 2vc , due to the periodic boundary condition in kμαkαν ¼ kμαkαν ¼Àδμν ∇ þ dμdν ; ð14Þ the Euclidean time direction, however the energy k0 must be also quantized in the integer multiples of 2π/β. This gives 2 ^ where ∇ ¼ dμdμ is the 3D Laplace operator. The discrete version of the mixed Z 2πv  c Xn¼b Chern-Simons gauge theory can thus be formulated as ‘ ÀÁ 2π 2πv n P 0 0 ! c ; ð24Þ ‘3 ‘3 ‘3ε ‘3 dk fk f ¼ þ 2 þ P 2 þ 2 β b‘ S i π aμkμνbν 2 μ f 2 f 2 μ g 0 n¼0 2ev P 0 2ev i 2eq P 0 x pffiffiffi pffiffiffi ð15Þ ‘3ε þ P 2 þ ‘ þ ‘ ; 2 g i 2aμQμ i 2bμMμ where β ¼ b‘=v and the factor within the sum represents the density of states. The 2eq i c integers n in the summation are known as Matsubara frequencies. Typically, fi however momenta integral are defined over the fundamental Brillouin zone where the discrete dual eld strengths are given by ½Àπ =‘; π =‘Š ½ ; π =‘Š fi vc vc , rather then 0 2 vc . The corresponding nite temperature ¼ ; ¼ : ð Þ 0 ! 0 À π =‘ fμ kμν bν gμ kμν aν 16 expression can be readily obtained from (24) by the shift k k vc , Z πv  c ÀÁ Xk¼b As we show below, this action describes two massive modes with dispersion ‘ π πv k dk0fk0 ! f c ; ð25Þ relation and mass given Àπvc β b‘ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ‘ k¼Àb μ e e E ¼ m2 v4 þ v2k2; m ¼ P q v : ð17Þ T c c T π where k = 2n − b and thus correspondingly, the density of states must be divided pffiffiffiffiffiffiffiffiffi by a factor 2. ¼ = μ ε fi η where vc 1 P P is the light velocity in the medium. This is the non-relativistic The nite temperature T > 0 affects primarily the parameter (see 21 fi ðÞ‘ version of the celebrated Chern-Simons mass . Supplementary Note 3, Quantum phase structure) via the coef cient Gmvc .At the zero temperature this is given by 24 Z π Lattice BF operator. The formulation of a discrete 3D lattice BF model can be 1 1 5 GmðÞ‘v ¼ d4k : achieved along the same lines as in 2D. Following we introduce the lattice BF c ð πÞ4 P3 ÀÁ ð Þ 2 Àπ ðÞ‘ 2þ ki 2 26 operators m vc 4sin ¼ 2 ^ ^ ^ i 0 kμνρ  Sμϵμανρdα kμνρ  ϵμναρdαSρ; ð18Þ At finite temperatures it has to be modified according to (25), where Z kX¼þb π 1 π 1 2 3 fxðÞÀþ‘μ^ f ðxÞ ðÞ¼‘ ; dk dk dk ; dμf ðxÞ ; Sμf ðxÞfxðÞþ ‘μ^ ; Gmvc T 4 P3 ð Þ ‘ ð πÞ 2 π 2 i 2 27 ð Þ 2 ¼À b Àπ ðÞm‘v þ4 sinðÞk þ 4 sinðÞk ð ÞÀ ðÞÀ‘μ^ 19 k b c 2b 2 ^ f x fx ^ ^ i¼1 dμf ðxÞ ‘ ; Sμf ðxÞfxðÞÀ ‘μ ; where T ¼ v =b‘. As we have verified over three orders of magnitude (m‘v ¼ are the forward and backward lattice derivative and shift operators, respectively. c c 0:001 to m‘v ¼ 1) the ratio SðTÞ = GmðÞ‘v ; T =GmðÞ‘v does not depend on the Summation by parts on the lattice interchanges both the two derivatives (with a c c c parameter m‘v but is rather a function of the temperature alone. As a minus sign) and the two shift operators; gauge transformations are defined using c consequence, η and the semiaxes of the ellipse determining the phase structure, see the forward lattice derivative. Also the two lattice BF operators are interchanged Supplementary Note 3, Supplementary Eq. (33), scale with the inverse of the (no minus sign) upon summation. Moreover they are gauge invariant, in the sense function S(T). This means that with the increasing temperature the whole ellipse that they obey the following equations: shrinks by the scale factor S(T). Magnetic quantum numbers M = ±1 that are ^ = fi kμνρdν ¼ kμνρdρ ¼ dμkμνρ ¼ 0; within the ellipse at T 0, will exit its interior at some critical temperature de ned ð Þ ^ ^ ^ ^ ^ 20 by the condition kμνρdρ ¼ dμkμνρ ¼ dνkμνρ ¼ 0: 1 ¼ ðÞ; ð Þ η STc 28 Finally, they satisfy also the equations g  ^ 2 kμνρkρλω ¼À δμλδνω À δμωδνλ ∇ assuming that the quantity on the left-hand side is larger than one (i.e. there is a  superinsulator at T = 0). Since the magnetic semiaxis is always longer and thus no þ δ ^ À δ ^ μλdνdω νλdμdω electric quantum numbers may appear within the ellipse interior when the magnetic  fi ^ ^ ð Þ ones have fallen outside, the superinsulator experiences a direct decon nement þ δνωdμdλ À δμωdν dλ ; 21 = transition into a topological insulator at T Tc. Correspondingly, superconductors ~ fi ^ ^ undergo a phase transition to topological insulators at Tc de ned by kμνρkρνω ¼ kμνρkρνω  ÀÁ g ~ 2 ^ ¼ S T : ð Þ ¼ 2 δμω∇ À dμdω ; η c 29

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