ARTICLE

https://doi.org/10.1038/s42005-021-00531-5 OPEN Quantum condensate ✉ M. C. Diamantini1, C. A. Trugenberger2 & V. M. Vinokur 3

Despite decades-long efforts, magnetic monopoles were never found as elementary particles. Monopoles and associated currents were directly measured in experiments and identified as topological quasiparticle excitations in emergent condensed matter systems. These mono- poles and the related electric-magnetic symmetry were restricted to classical electro- dynamics, with monopoles behaving as classical particles. Here we show that the electric- magnetic symmetry is most fundamental and extends to full quantum behavior. We demonstrate that at low temperatures magnetic monopoles can form a quantum Bose condensate dual to the charge condensate in superconductors. The monopole

1234567890():,; Bose condensate manifests as a superinsulating state with infinite resistance, dual to . The monopole supercurrents result in the electric analog of the Meissner effect and lead to linear confinement of the Cooper pairs by Polyakov electric strings in analogy to in .

1 NiPS Laboratory, INFN and Dipartimento di Fisica e Geologia, University of Perugia, Perugia, Italy. 2 SwissScientific Technologies SA, Geneva, Switzerland. ✉ 3 Terra Quantum AG, Rorschach, Switzerland. email: [email protected]

COMMUNICATIONS PHYSICS | (2021) 4:25 | https://doi.org/10.1038/s42005-021-00531-5 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00531-5

axwell’s equations in vacuum are symmetric under the depending on the ratio of the strength of Josephson coupling and duality transformation E → B and B → − E (we use the Coulomb energy of the elemental excessive charge on a single M = ℏ = ε = natural units c 1, 1, 0 0). Duality is preserved, granule, the system can be either an insulator, superconductor, or provided that both electric and magnetic sources (magnetic topological insulator12,23. If the system is at the insulating side of monopoles and magnetic currents) are included1. Magnetic the SIT, the global phase coherence is absent and each condensate monopoles, while elusive as elementary particles2, exist in many bubble is characterized by an independent phase. The relevant materials in the form of emergent quasiparticle excitations3. degrees of freedom in such systems are single Cooper pairs that Magnetic monopoles and associated currents were directly mea- can tunnel from one island to the next one, leaving behind a sured in experiments4, confirming the predicted symmetry Cooper hole, and vortices, resulting from non-trivial phase between electricity and magnetism. The existence of monopoles circulations over adjacent granules. In a 2D system such a vortex requires that gauge fields are compact, implying, in turn, the would be a usual Josephson vortex, and in 3D such an elemental quantization of charge and Dirac strings5 or a core with addi- ‘minimal’ vortex is similar to a pancake vortex in a layered tional degrees of freedom to regularize the singularities of the cuprate24. In the usual configuration, these pancake vortices vector potential6,7. aggregate on top of each other to make stacks, or chains. When The importance of electric-magnetic duality was first realized the pancake stack forms, the magnetic monopoles and antimono- by Nambu8, Mandelstam9, and ’t Hooft10, who proposed that poles at the “bottom” and “top” of each of such a pancake color confinement in quantum chromodynamics (QCD) can be “annihilate” and one long vortex forms. Only the monopoles and understood as a dual Meissner effect. In the present context, anti-monopoles at the very end of this configuration survive and, electric-magnetic duality manifestly realizes the symmetry correspondingly, one has such monopoles and anti-monopoles between Cooper pairs, which are Noether charges, and magnetic only at the surfaces of the sample24. When pancake vortices are monopoles, which are topological solitons, and forms the foun- ballistic and the layers are only weakly coupled, however, the dation for the superconductor-insulator transition11 and the vertical stacks can break in the middle, resulting in monopole and appearance of the superinsulating state12–14. Magnetic monopoles anti-monopole pairs joined by a shorter vortex in the interior of arise also as instantons in Josephson junction arrays (JJA)15, the sample. As always, when there are both dynamic charges and which are easily accessible experimental systems themselves and vortices in the spectrum, the latter acquires a topological gap25. provide an exemplary model for superconducting films16. The inverse of the gap sets the spatial scale for the width of the So far, monopoles in emergent condensed matter systems were vortices and radius of monopoles. treated as classical excitations. Here we show that the electric- On distances larger than the vortex width, long vortices appear magnetic symmetry is most fundamental and extends to the full as quantized flux tube singularities that become unobservable quantum realm. We demonstrate that at low temperatures, because of the compactness of the U(1) gauge fields1, and a magnetic monopoles form a quantum Bose condensate dual to neutral condensate of point magnetic monopoles satisfying the the charge condensate in superconductors and generate a Dirac quantization1 condition forms. Remarkably, as we show superinsulating state with infinite resistance, dual to below, the monopole condensate strongly suppresses Cooper pair superconductivity12,13. We show that magnetic monopole tunneling providing thus a mechanism that stabilizes the granular supercurrents result in the direct electric analog of the Meissner structure. effect and lead to linear confinement of the Cooper pairs by Polyakov’s electric strings14,17,18 (dual to superconducting vor- Long-distance effective field theory. Having established that in tices) in analogy to quarks within hadrons19. The monopole the vicinity of the SIT an ensemble of Cooper pairs acquires a condensate realizes a 3D version of superinsulators, that have granular structure, we construct a Ginzburg-Landau-type effec- been previously observed in 2D superconducting films, and result fi – tive eld theory of such granular Cooper pair condensates. We from quantum tunneling events, or instantons12 14,17,18. focus on the London limit, i.e., on long distances, much larger than the topological width scale so that vortices and monopoles appear as point-like singularities. As a first step, following general Results principles by Wilczek25, we identify that the dominant interac- Magnetic monopoles in granular superconductors.Togain tions are the topological mutual statistics interaction setting the insight into the nature of a system that can harbor Cooper pairs and Aharonov-Bohm phases between charges and vortices. A stan- magnetic monopoles simultaneously, we first reiterate that mono- dard description of the interaction part of the system’s Lagran- poles naturally emerge in 2D JJA15 as instantons and provide the gian is achieved by introducing two fictitious gauge fields, a vector underlying mechanism of 2D superinsulation as quantum tunneling field aμ and an antisymmetric pseudotensor gauge field bμν26: events. To establish that JJA indeed offers a universal model describing superinsulation in films,onerecallstheearlyhypothesis 1 μναβ μ 1 μν L ¼ bμνϵ ∂αaβ þ aμq þ bμνm ; ð1Þ that, in the vicinity of the superconductor-insulator transition (SIT), BI 4π 2 fi the lms acquire self-induced electronic granular texture and can be μ μν viewed as a set of superconducting granules immersed into an where q and m are the charge and vortex currents, respectively. insulating matrix20. Employing the model of the 2D JJA21 to treat This so-called BF model26 is topological, since it is metric- the experimental data of refs. 17,22, perfectly confirmed this picture, independent. It is invariant under the usual gauge transforma- and settled that granules have the typical size of order of the tions aμ → aμ + ∂μξ and under the gauge transformation of the superconducting coherence length, ξ, and are coupled by Josephson second kind26, under which the antisymmetric tensor transforms links. On the theory side, this granular structure was derived in the as bμν → bμν + ∂μλν − ∂νλμ, a vector field λμ becoming itself the framework of the gauge theory of the SIT in ref. 23, conclusively gauge function. The BF action for a model defined on a compact establishing the JJA-like texture in 2D systems experiencing the SIT. space endowed with the non-trivial topology, yields a ground As a next step, we generalize the reasoning of ref. 23 onto 3D state with the degeneracy reflecting, one-to-one, this topology and systems. This enables us to adopt the original gauge theory of is referred to as topological order27. The coefficient 1/4π of the JJA12 for 3D superconductors and consider a general inhomo- first term in Eq. (1) ensures that the system does not have such a geneous system of superconducting granules, i.e., bubbles of topological order28. In turn, the topological coupling between a Cooper pair condensate, coupled by tunneling links. Then, vector and a pseudotensor in 3D, ensures the parity (ZP) and

2 COMMUNICATIONS PHYSICS | (2021) 4:25 | https://doi.org/10.1038/s42005-021-00531-5 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00531-5 ARTICLE time-reversal (ZT) symmetries of the model. The field strength Phase transitions and phase diagram associated with aμ is fμν = ∂μaν − ∂νaμ, whereas bμν has the Let us consider a granular system (irrespective to whether the 13 associated 3-tensor field strength hμνα = ∂μbνα + ∂νbαμ + ∂αbμν.It granularity is the self-induced electronic granularity or is of the can be easily checked that this expression is invariant under the structural origin, such as, e.g., granular diamond30), characterized ℓ gauge transformations of the second kind; it plays for bμν the by the length scale playing the role of the granule size, and same role that the field strength fμν plays for the usual gauge field examine the various phases that can emerge. We focus on cubic μ μ μναβ aμ. The dual field strengths j = (1/2π)h = (1/4π)ϵ ∂νbαβ lattices since, as in 2D, the paradigmatic system for monopoles μν ~μν μναβ and the transitions they induce is a Josephson junction array31. and m ¼ð1=2πÞf ¼ð1=4πÞϵ f αβ represent the conserved Of course, the universality class of the transition can depend on charge and vortex currents, respectively. For Cooper pairs, the the lattice details but the discussion of such effects is beyond the charges are measured in integer units of the elemental charge of a scope of this work. We thus formulate the action (Eq. 1)ona Cooper pair, 2e. Accordingly, the magnetic charge of monopoles cubic lattice of spacing ℓ and we add all possible local gauge and vortices is measured in integer units of 2π/2e = π/e. If one fi invariant terms. In the presence of magnetic monopoles the de nes a fundamental charge as a charge of a single , e, tensor current mμν does not conserve any more, and the gauge then our magnetic monopoles of strength π/e should be viewed, invariance of the second kind of the tensor field bμν is effectively strictly speaking, as half-monopoles. However, we always deal broken at the vortex endpoints. Longitudinal components of the with the phases of matter where charge unit is 2e rather than e. tensor gauge field bμν become the usual vector gauge fields for the Thus we shall call our objects the unit monopoles. This complies magnetic monopoles and induce for the monopoles the same type with condensed matter notations, where a unit vortex carries fl π of Coulomb interaction which experience electric charges. How- magnetic ux quantum /e, since it always appears in a Cooper ever, this Coulomb interaction is subdominant to the linear pair condensate, but differs from the field theory notation where π tension created by vortices connecting monopole and anti- /e is half a vortex. monopole pair, and one can neglect it when determining the The developed model describes what is known today as a fi 29 phase structure. More speci cally, when inverting the Kalb- (simple, as opposed to strong) bosonic topological insulator Ramond kernel, we will consider only the transverse components and is an intensely investigated state of matter. Open vortices of the vortices and neglect their endpoints. emerging in this model carry magnetic monopole-antimonopole ν μν Rotating to Euclidean space-time we arrive at the action pairs with current m = ∂μm at their ends, and the gauge symmetry of the second kind is broken. The state of the X ‘4 ‘4 ‘4 S ¼ f μνf μν À i aμkμαβbαβ þ hμναhμνα monopole ensemble is self-consistently harnessed with vortex 4f 2 4π 12g2 properties. Existence of the appreciable vortex tension implies x ð2Þ fi 2 1 that vortices are short and taut and linearly con ne the þ i‘aμqμ þ i‘ bμνmμν; monopoles into ‘small’ dipoles, as illustrated in Fig. 1a. The 2 vanishing vortex tension allows monopoles to break loose, while where kμνα is the lattice BF term12,see“Methods” section, f is a vortices themselves grow infinitely long and loose and turn into dimensionless coupling, and g has the canonical dimension of unobservable Dirac strings, as shown in Fig. 1b. Accordingly, the mass ([mass]). To describe materials with the relative electric world-lines of monopoles become infinitely long as well, which ε μ permittivity and relative magneticpffiffiffiffiffi permeability , we incorporate means that they Bose condense. Determining the condensation the velocity of light v ¼ 1= εμ < 1bydefining the Euclidean time ℓ = ℓ point is a dynamical problem. One can say that the condensation lattice spacing as 0 /v and by rescaling all time derivatives, point is set by the moment when quantum corrections to the currents, and zero-components of gauge fields by the factor 1/v. vortex tension are large enough to turn it negative so that As ap consequence,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi both gauge fields acquire a dispersion relation vortices become Dirac strings. The rest of this paper is devoted to E ¼ m2v4 þ v2p2 with the topological mass32 m = fg/2πv. The answering this question and to deriving the nature of the ensuing dimensionless parameter f ¼OðeÞ encodes the effective Coulomb new state of matter. interaction strength in the material, g is the magnetic scale To conclude here, we generalize the 2D consideration of refs. 12,23 =λ λ g ¼Oð1 LÞ, where L is the London penetration depth of the onto 3D systems and predict that in three dimensions super- superconducting phase. conductors also acquire self-induced emergent granularity in the To analyze how the additional interactions can drive quantum vicinity of the SIT. Magnetic monopoles play a crucial role in phase transitions taking the system out of the bosonic insulator, the formation and properties of this new superconducting state. we integrate over the fictitious gauge fields to obtain a Euclidean action Scv for point charges and line vortices alone. As we show in Methods, this is proportional to the length of the charge world- lines and to the area of vortex world-surfaces, exactly as their configurational entropy. Charges and vortices can thus be assigned an effective action (equivalent to a quantum "free energy" in this Euclidean field theory context) ; Fcv ¼ ðÞsc À hc N þ ðÞsv À hv A ð3Þ where N and A are the length of word-lines in number of lattice links and the area of world-surfaces in number of lattice pla- quettes, respectively, and sc,v and hc,v denote the action and entropy contributions (per length and area) of charges and vor- tices, respectively. The possible phases that are realized are determined by the relation between the values of the Coulomb Fig. 1 Magnetic monopole states at low temperatures. a Monopoles are and magnetic scales and by materials parameters determining confined into dipoles state by tension of vortices connecting them. b As the whether the coefficients of N and A are positive or negative. For vortex tension vanishes, the monopoles at their endpoints become loose positive coefficients, long world-lines and large wold-surfaces are free particles that can condense. suppressed. If either of the parenthesis becomes negative, then

COMMUNICATIONS PHYSICS | (2021) 4:25 | https://doi.org/10.1038/s42005-021-00531-5 | www.nature.com/commsphys 3 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00531-5 either a charge or a monopole condensate forms. In the case of is a dual superinsulating phase governed by the magnetic charges, the proliferation of long-world lines is the geometric monopole condensate, which is the main subject of this paper. picture of Bose condensation first put forth by Onsager33 and To conclude this section, we would like to point out that the elaborated by Feynman34, see ref. 35 for a recent discussion. In the direct transition between superinsulator and superconductor for case of vortices, the string between magnetic monopole endpoints η < 1 shown in Fig. 2 matches perfectly the low-temperature, becomes loose and assumes the role of an unobservable Dirac quantum phase structure of the QCD as a function of the density, string. In this case, the monopoles are characterized by long with a transition between confined hadronic matter and color world-lines describing their Bose condensate phase, see Fig. 1. superconductivity39. The details of this vortex transition have been discussed in 36,37 refs. . The resulting phase diagramffiffiffiffiffiffiffiffiffiffiffi is determined by the value Electromagnetic response and the electric string tension η π ‘ =pμ μ of the parameter ¼ 2 ðmv ÞG N A encoding the strength of To reveal the nature of the superinsulating phase, we examine its quantum fluctuations and the materialpffiffiffiffiffiffiffiffiffiffiffiffiffi properties of the system18 electromagnetic response. To that end we again minimally couple γ =‘ μ =μ fi and tuning parameter, ¼ðf gÞ N A, taking the system the electric current jμ to the electromagnetic gauge eld Aμ and across superconductor-insulator transition (SIT). Here compute its effective action, see “Methods” section. Taking the G ¼OðGðmv‘ÞÞ, where G(mvℓ) is the diagonal element of the limit mvℓ ≫ 1, we find − P latticeÀÁ kernel G(x y) representing the inverse of the operator X 1 2 À Fμν À2πmμν ‘2 2 ∇2 μ μ 8f 2 x;μ;ν ðÞ; ðmvÞ À , and N and A are the entropy per unit length of expðÞ¼ÀSe:m: e ð5Þ the world line and per unit area of the world surface, respectively. mμν The phase structure at T = 0 and the domains of different phases fi where mμν are integer numbers. The monopole condensation for in the critical vicinity of the SIT are de ned by the relations, see fi “ ” strong f renders the real electromagnetic eld a compact variable, Methods section for details: fi −π +π  de ned on the interval [ , ] and the electromagnetic γ < 1; charge Bose condensate ; response is given by Polyakov’s compact QED action40,41. This η ! < 1 γ ; ; changes drastically the Coulomb interaction. To see that, let us > 1 monopole Bose condensate fi take two external probe charges ±qext and nd the expectation 8 value for the corresponding Wilson loop operator W(C), where C γ 1 ; ; <> < η charge Bose condensate is the closed loop in 4D Euclidean space-time (the factor ℓ is η 1 γ η ; ; absorbed into the gauge field Aμ to make it dimensionless) >1 ! > η < < bosonic insulator ð4Þ Z P : X þπ 2 P 1 π γ η ; : 1 À 2 ;μν ðÞFμν À2 mμν iq lμAμ > monopole Bose condensate hWðCÞi¼ DAμ e 8f x e ext C ; ZAμ;mμν Àπ and are shown in Fig. 2. The finite-temperature decay of the fmμν g condensates, corresponding to the deconfinement transitions into ð6Þ the bosonic insulator, is described by the same approach38. The effect of charge condensation is immediately revealed by where lμ takes the value 1 on the links forming the Wilson loop C minimally coupling the current jμ = (1/2π)hμ to the electro- and 0 otherwise. When the loop C is restricted to the plane fi formed by the Euclidean time and one of the space coordinates, magnetic gauge potential Aμ and integrating out all matter elds 〈 〉 to obtain the electromagnetic response. The latter acquires a W(C) measures the interaction energy between charges ±qext.A μ perimeter law indicates a short-range potential, while an area-law photon mass term ∝ AμA so that the induced charge current jμ ∝ is tantamount to a linear interaction between them41. For Cooper Aμ. This is nothing but London equation i.e., the major mani- = “ ” h ð Þi ¼ ðÀσ Þ festation of superconductivity16. Not dwelling on this standard pairs, qext 1, see Methods section, W C exp A derivation, but just stressing that our approach reproduces that where A is the area of the surface S enclosed by the loop C. This yields a linear interaction between probe Cooper pairs, which charge condensate phase is a superconductor, we now focus on fi the new phase induced by the condensation of magnetic mono- therefore can be viewed as con ned by the elastic string with the string tension poles. At f =‘

Discussion Superinsulation has been observed in 2D, where magnetic monopoles are instantons rather than particles13,17,18 and the structure of the phase diagram is conclusively established. The charge Berezinskii-Kosterlitz-Thouless (BKT) transition into the low-temperature confined superinsulating phase was measured. Fig. 2 Cooper pairs–quantum magnetic monopoles phase diagram at T = 0. 18 pffiffiffiffiffiffiffiffiffiffiffi Reference presented measurements of the electromagnetic η ¼ πðmv‘ÞG= μ μ The parameter 2 N A encodes the strength of quantum response of the superinsulating phase and of properties of elec- fluctuations and the material properties of the system. The inter-phases tric strings. In particular, the observed double kinks in the I-V lines are η = 1/γ(γ <1,η > 1), γ = 1(η < 1), and η = γ (η >1, γ > 1). characteristics indicate the predicted electric Meissner effect

4 COMMUNICATIONS PHYSICS | (2021) 4:25 | https://doi.org/10.1038/s42005-021-00531-5 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00531-5 ARTICLE characteristic to superinsulators. The experimental results exhibit Phases of monopoles.Tofind the topological action for monopoles, we start from a fair agreement with the theoretical predictions. The existence of Eq. (2) and integrate out fictitious gauge fields aμ and bμν X 2 a bosonic insulator in various materials and, in particular, in the f δμν g2 δμαδνβ À δμβδνα fi – Stop ¼ 2 qμ 2 qν þ mμν 2 mαβ same NbTiN lms where the kinked I V curves were measured, 2‘ ðmvÞ À ∇2 8 ðmvÞ À ∇2 42 x fi 2 ð12Þ was unambiguously established in ref. .ThatTiN lms with π kμαβ ðmvÞ ÀÁ: smaller η < 1 exhibit the direct SIT, while NbTiN films endowed þ i qμ mαβ ‘ ∇2 ðmvÞ2 À ∇2 with η > 1 show the SIT across the intermediate topological The last term represents the Aharonov-Bohm phases of charged particles around insulator phase, complies with the theoretical expectations. λ λ vortices of width L. On scales much larger than L, where the denominator However, more research on various systems is required to spot reduces to (mv)2∇2, this term becomes i2π integer, as can be easily recognized by the exact location of the tricritical point. To establish the Cooper expressing qμ = (1/2)ℓkμαβyαβ. This reflects the absence of Aharonov–Bohm phases pair-based nature of the bosonic topological insulator, shot noise between charges ne and magnetic fluxes 2π/ne. Accordingly, we shall henceforth measurements similar to those carried out in ref. 43 are desirable. neglect this term. The signature of 3D superinsulation44 has been detected in InO The important consequence of the topological interactions is that they induce fi 45,46 self-energies in form of the mass of Cooper pairs and tension for vortices between lms . However, more studies are necessary for drawing magnetic monopoles. These self-energies are encoded in the short-range kernels in reliable conclusions that InO can be considered as a material the action (Eq. 12), which we approximate by a constant. World-lines and world- hosting a magnetic monopole condensate. Strongly type II surfaces are thus assigned “energies” (formally Euclidean actions in the present fi superconductors with a fine, inherent or self-induced granular statistical eld theory setting and thus dimensionless in our units) proportional to their length N and area A (measured in numbers of links and plaquettes), electronic texture, are other most plausible candidates to house 3D superinsulators. Finally, another class of candidates are π ‘ f 2 ; SN ¼ 2 ðmv ÞG ‘ Q N layered materials. Vortex lines in such materials can be regarded g ‘ ð13Þ 24 π ‘ g 2 ; as stacks of pancake vortices . If the layers are weakly coupled SA ¼ 2 ðmv ÞG M A and vortices are ballistic, the pancakes can split and form mag- f netic monopoles at the inner layers’ intersections, albeit very where G = O(G(mvℓ)), with G(mvℓ) the diagonalÀÁ element of the lattice kernel G(x 2 2 2 anisotropic ones. These can then condense into a − y) representing the inverse of the operator ‘ ðmvÞ À ∇ , and Q and M are the superinsulating phase. integer quantum numbers carried by the two kinds of topological defects. However, also the entropy of link strings and plaquette surfaces is proportional to their length Finally, while the monopole condensate existence is strongly 47 μ μ fi μ μ and area , NN and AA. Both coef cients are non-universal: N ’ lnð7Þ since supported by the observation of the superinsulating state and the at each step the non-backtracking string can choose among 7 possible directions on μ corresponding experimental implications are reliably established how to continue, while A does not have such a simple interpretation but can be “ by recent transport measurements18, the conclusive evidence for estimated numerically. This gives for both types of topological defects a free energy” proportional to their dimension and with coefficients that can be positive monopoles should come from their direct observations. One of or negative depending on the parameters of the theory. The total free energy is the ways to implement such an observation may be extending the "# ! ‘ SQUID-on-tip device method of ref. 4 to lower temperatures. F f 2 1 g 2 1 ; π ‘ ¼ ‘ Q À η N þ M À η A 2 ðmv ÞG g Q f M Methods where we have defined Lattice BF term. To formulate the gauge-invariant lattice BF-term, we follow12 and 2πðmv‘ÞG 2πðmv‘ÞG introduce the lattice BF operators η ; η : Q ¼ μ M ¼ μ ð14Þ N A fi kμνρ  Sμϵμανρdα ; If the coef cients are positive, the self-energy dominates and large string/surface fi ^ ^ ^ ð8Þ con gurations are suppressed in the partition function. In this regime Cooper pairs kμνρ  ϵμναρdαSρ; and/or vortices are gapped excitations, suppressed by their large action. If the coefficients, instead are negative, the entropy dominates and large configurations “ ” where are favored in the free energy (effective action). The phase in which long world- lines of Cooper pairs dominate the Euclidean partition function is a charge Bose condensate, as discussed originally by Onsager33 and Feynmann34 (for a recent f ðx þ ‘μ^ÞÀf ðxÞ discussion see ref. 35). This phase is the Bose condensate of magnetic monopoles. dμf ðxÞ ; Sμf ðxÞf ðx þ ‘μ^Þ ; ‘ For vortices, proliferation of large world-surfaces means that the strings binding ð9Þ ‘μ^ monopoles and antimonopoles into neutral pairs become loose. We will show ^ f ðxÞÀf ðx À Þ ; ^ ‘μ^ ; dμf ðxÞ ‘ Sμf ðxÞf ðx À Þ below that in this case the long real monopole world-lines dominate the electromagnetic response. The combined energy–entropy balance equations are best viewed as defining are the forward and backward lattice derivative and shift operators, respectively. the interior of an ellipse on a 2D integer lattice of electric and magnetic quantum Summation by parts on the lattice interchanges both the two derivatives (with a numbers, minus sign) and the two shift operators; gauge transformations are defined using the forward lattice derivative. The two lattice BF operators are interchanged (no Q2 M2 þ < 1 ; ð15Þ minus sign) upon summation by parts on the lattice and are gauge invariant so r2 r2 that: Q M where the semiaxes are given by rffiffiffiffiffiffi ^ ‘ ‘ μ kμνρdν ¼ kμνρdρ ¼ dμkμνρ ¼ 0; 2 g 1 g N 1 ; ð10Þ rQ ¼ η ¼ μ η ^ ^ ^ ^ ^ f Q f A kμνρdρ ¼ dμkμνρ ¼ dν kμνρ ¼ 0 rffiffiffiffiffiffi ð16Þ f 1 f μ 1 r2 ¼ ¼ A ; M ‘g η ‘g μ η And satisfy the equations M N with  pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ^ 2 η η η π ‘ = μ μ : kμνρkρλω ¼À δμλδνω À δμωδνλ ∇ ¼ Q M ¼ 2 ðmv ÞG N A ð17Þ  ^ ^ ^ ^ Of course, configurations with Q ≠ 0 and M ≠ 0 must be excluded since the two þ δμλdν dω À δνλdμdω þ δνωdμdλ À δμωdνdλ ; ð11Þ  types of excitations are different, only pairs {0, M}or{Q, 0} have to be considered. ^ ^ 2 ^ The phase diagram is found by establishing which integer charges lie within the kμνρkρνω ¼ kμνρkρνω ¼ 2 δμω∇ À dμdω ; ellipse when the semi-axes are varied. This yields Eq. (4) in the main text.

2 ^ ^ where ∇ ¼ dμdμ is the lattice Laplacian. We use the notation Δμ and Δμ for the Electromagnetic response in the magnetic monopole condensate. To establish forward and backwards finite difference operators. the electromagnetic response of the monopole condensate we add the minimal

COMMUNICATIONS PHYSICS | (2021) 4:25 | https://doi.org/10.1038/s42005-021-00531-5 | www.nature.com/commsphys 5 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00531-5 coupling of the charge current jμ to the electromagnetic field, S, X X 1 Δ^ Δ χcl π Δ^ 2 χcl ; ‘4 ‘4 ; 1 1 2 ¼À2 qext 1S12 þ M sin 2 S ! S À i Aμjμ ¼ S À i π Aμkμαβbαβ ð18Þ ð28Þ x x 4 Δ^ Δ χcl π Δ^ 2 χcl ; 2 2 1 ¼À2 qext 2S21 þ M sin 1 and we compute its effective action by integrating over the fictitious gauge fields aμ Following41, we solve these equations in the continuum limit, and bμν. This requires no new computation since, by a summation by parts, the ∂ ∂ χcl π δ0 2 χcl ; above coupling amounts only to a shift 1 1 2 ¼ 2 qext ðx1ÞþM sin 2 ð29Þ ∂ ∂ χcl ¼ 2πq δ0ðx ÞþM2 sin χcl : 1 2^ 2 2 1 ext 2 1 mμν ! mμν À ‘ kμναAα ; ð19Þ π = 2 For qext 1 (corresponding to Cooper pairs in our case), the classical solutions χcl ∣ ∣ → ∞ in Eq. (12). Setting qμ = 0 for the phase with gapped Cooper pairs gives Eq. (5)in with the boundary conditions 1;2 ! 0 for x1,2 are the main text. χcl ¼ signðx Þ 4 arctan eÀMjx2j ; 1 2 ð30Þ χcl ¼ signðx Þ 4 arctan eÀMjx1j : Computation of the string tension. The starting point is Eq. (6) in the main text. 2 1 For large values of the coupling f, the action is peaked around the values Fμν = Inserting this back in (Eq. 26) we get formula (Eq. 7) in the main text. 2πmμν, allowing for the saddle-point approximation to compute the Wilson loop. Using the lattice Stoke’s theorem, one rewrites Eq. (6)as Data availability Z π P P X þ 1 ~ 2 q 1 À Fμν À2πmμν ext ~ π Data sharing not applicable to this article as no datasets were generated or analyzed 8f 2 x ðÞi 2 Sμν ðFμν À2 mμν Þ ; hWðCÞi ¼ DAμ e e S ð20Þ Z ; π during this study. Aμ mμν fmμν g À where the quantities Sμν are unit surface elements perpendicular (in 4D) to the Received: 16 June 2020; Accepted: 15 January 2021; plaquettes forming the surface S encircled by the loop C and vanish on all other plaquettes. WeP have also multiplied the Wilson loop operator by 1 in the form π 41 expðÀi qext xSμν mμν Þ. Following Polyakov , we decompose mμν into transverse and longitudinal components,

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