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https://doi.org/10.1038/s42005-020-00410-5 OPEN Direct probe of the interior of an electric pion in a Cooper pair superinsulator
M. C. Diamantini1, S. V. Postolova2,3, A. Yu. Mironov 2,4, L. Gammaitoni 1, C. Strunk5, C. A. Trugenberger6 & ✉ V. M. Vinokur 7,8 1234567890():,;
The nature of hadrons is one of the most fundamental mysteries of physics. It is generally agreed that they are made of “colored” quarks, which move nearly free at short scales but are confined inside hadrons by strong interactions at large distances. Because of confinement, quarks are never directly observable and, experimentally, their properties can be tested only indirectly, via high energy collisions. Here we show that superinsulating films realize a complete, one-color model system of hadron physics with Cooper pairs playing the role of quarks. We report measurements on highly controlled NbTiN films that provide a window into the interior of "Cooper pair mesons" and present the first direct evidence of asymptotic freedom, ‘t Hooft’s dual superconductivity confinement mechanism, and magnetic monopoles.
1 NiPS Laboratory, INFN and Dipartimento di Fisica e Geologia, University of Perugia, via A. Pascoli, I-06100 Perugia, Italy. 2 A. V. Rzhanov Institute of Semiconductor Physics SB RAS, 13 Lavrentjev Avenue, Novosibirsk, Russia 630090. 3 Institute for Physics of Microstructures RAS, GSP-105, Nizhny Novgorod, Russia 603950. 4 Novosibirsk State University, Pirogova str. 2, Novosibirsk, Russia 630090. 5 Institute of Experimental and Applied Physics, University of Regensburg, D-93025 Regensburg, Germany. 6 SwissScientific Technologies SA, rue du Rhone 59, CH-1204 Geneva, Switzerland. 7 Materials Science Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60637, USA. 8 Consortium for Advanced Science and Engineering (CASE), ✉ University of Chicago, 5801 S Ellis Ave, Chicago, IL 60637, USA. email: [email protected]
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uark interactions are described by quantum chromody- decreasing the distance between electrodes, realizing the obser- namics (QCD), a non-Abelian gauge theory. A salient vation spatial scale. Using the compact QED mapping we Q 1,2 feature of QCD is asymptotic freedom , the weakening demonstrate the electric Meissner effect and calculate the I–V- of the interaction coupling strength at short distances (ultraviolet characteristics in the confinement regime. Comparing our (UV) limit). At large distances (infrared (IR) limit), the quarks are experimental results with theoretical predictions yields the string thought to be confined within hadrons, which are physical tension and constitutes the first ever “look inside a meson”, observable excitations, by strings. Quarks themselves cannot be directly confirming asymptotic freedom and ’t Hoofts confine- extracted from hadrons and be seen in isolation. The mechanism ment mechanism by magnetic monopoles. for the transition from weak quark interactions in the UV regime to confinement and strings in the IR regime remains an open Results issue. Strings and the Meissner and mixed states of a superinsulator. ‘t Hooft3 put forth an appealing confinement mechanism, the A quantitative theory of the response of a superinsulator to a dc dual superconductivity where the condensate of magnetic electric field rests on the fact that in a superinsulator the fun- monopoles4 constricts the chromoelectric field into thin flux damental excitations are electric strings with linear tension σ11. tubes binding quarks into mesons. He coined the term “super- The electric Meissner state is obtained from the dual analog of the insulator”, as opposite to superconductor, for the confined quark London equations7,8,21, describing the electrodynamics of strings matter with infinite (chromo)-electric resistance. Polyakov connecting charged particles. Deferring the technical derivation showed that this confinement mechanism occurs also in Abelian of the Meissner state and its vanishing static electric permittivity, ε = gauge theories, provided they are compact and hence support SI 0, to the end of the paper, we note here that the strings can topological excitations, magnetic monopoles, which are instan- be either closed, describing pure gauge excitations (the analogs of tons in 2D and solitons in 3D5,6. The fact that the compact QED glueballs in QCD)22 or open, representing Cooper pairs–anti- in the confinement regime maps onto a confining string theory7,8 Cooper pair dipoles (the analogs of mesons in QCD). The pre- makes it a perfect model for ’t Hooft’s dual superconductivity sence of such Cooper pair dipoles follows from either self- mechanism. induced or imposed electronic granular structure of a system Remarkably, Abelian confinement emerged in a condensed supporting superinsulation. When a Cooper pair tunnels from matter realization of superinsulators, predicted first for Josephson one granule to another it leaves behind a +2e charge excess junction arrays (JJA) in ref. 9 and rediscovered in ref. 10 in films representing a Cooper pair “hole”. This picture is fully supported experiencing the superconductor-insulator transition (SIT). These by the experimental observation of the charge BKT transition in electric superinsulators constitute a new state of matter with such materials13. The energy to create pure gauge excitations is fi fi Δ = 2 fi = εμ 1/2 in nite resistance at nite temperatures due to electric strings G mv , with m being the gauge- eld mass and v c/( ) “ ”11 binding Cooper pairs into electric mesons . Transport mea- (μ ≈ 1 is the magnetic permeability) being the light velocitypffiffiffiffiffiffiffiffiffiffi in the surements revealed superinsulation in titanium nitride (TiN) medium. Open strings have the typical length d ¼Oð _v=σÞ fi 10,12 fi 13 pffiffiffiffiffiffiffiffi s lms , niobium titanium nitride (NbTiN) lms , and, albeit Oð _ σÞ fi 14,15 and a gap of the order v . The width of the strings is under a different name, InO lms . The long-distance elec- fi λ = ℏ 22 Δ ’ de ned by the screening length el /mv . Both G and the tromagnetic response of superinsulators is exactly Polyakov s σ Λ = 11 string tension are expressed via the ultraviolet (UV) cutoff 0 compact QED with the effective coupling constant ℏ ≃ ξ fi v/r0, where r0 in lms and is of order of the plaquette size in 2 ¼ α ðκÞ=ε : ð Þ 23 eeff 4 f g 1 JJA, as functions of the effective coupling (Eq. (1)) and v , Here α = e2/(ℏc) is the fine structure constant, κ = λ/ξ is the _ 1 π e2 v2 _v π m / exp À ; σ ¼ eff m / exp À : Ginzburg-Landau parameter of the superconducting material, vr e2 4e2 4r r2 4e2 with λ its London penetration depth and ξ its coherence length 0 eff eff 0 0 eff and ε is the dielectric permittivity of the normal insulating state. ð2Þ Finally, g is the tuning parameter driving the system across the ≫ fi ≃ κ For samples with linear dimension L ds, small electric elds SIT, so that g 1 near the transition. The function f( ) is smooth fi and is Oð1=κÞ for κ ≫ 111. Pure gauge compact QED in 2D is not below a critical value Ec1 are suf cient only to excite isolated strings of typical length ds much smaller than the sample size. In renormalizable. However, coupling the action to dynamical fi matter results in a non-trivial fixed point16,17. The same occurs in this regime, the applied electric eld does not penetrate the sample, only neutral “pion-like” dipole excitations made of a our case: compact QED is induced by an underlying matter “ ” dynamics from which it inherits the corresponding fixed point Cooper pair and a Cooper hole can be created. This is the 18,19 Meissner state of the superinsulator. When the applied electric structure encoded in g. One can show that g, and hence also fi 2 – – eld reaches the critical value Ec1, enough electric pions can be the effective coupling eeff , have a Berezinskii Kosterlitz Thouless (BKT) (see ref. 20 for a review) infrared (IR) fixed point at the created such that a chain of them reaches from one end of the sample to the other. At this point, a single electric flux tube can critical value gc, at which the string tension diverges. The tension, fl traverse the sample end-to-end and the Meissner state is and thus the interaction strength, ow to smaller values in the UV fi limit and, as a result, the induced compact QED becomes destroyed. For E > Ec1 electric elds penetrate the superinsulator 2 fl λ asymptotically free, albeit this applies near the confining IR fixed in form of ux tubes of typical width el and the mixed state of point, instead of near the UV-free fixed point, as in QCD. the superinsulator sets in. This is the dual state of the Abrikosov fi Confinement by strong interactions prevents a direct view on lattice in superconductors. Finally, at the critical eld Ec2, quarks despite that they move nearly free at small scales. Since superinsulation breaks down. electric Cooper pair mesons are generated by much weaker Coulomb interactions, they have a macroscopic dimension and Current-voltage characteristics. We start the derivation of the I are accessible to direct experimental study. Here we investigate (V) response by writing down the compact QED potential for the superinsulators, which allow for a direct observation of the interaction between Cooper pairs interior of electric mesons made of Cooper pairs by standard c_π λ r transport measurements. We reveal the transition from the ð Þ¼σð Þ À þ el À ; ð Þ U r T r a ln K0 λ 3 confined to the asymptotic free Cooper pair motion upon 24r r0 el
2 COMMUNICATIONS PHYSICS | (2020) 3:142 | https://doi.org/10.1038/s42005-020-00410-5 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-00410-5 ARTICLE where the second term is the so-called Lüscher term24. The third the activated current, is r* = a/F, so that the energy barrier is term is the screened 2D Coulomb potential (K is the MacDonald U* ≡ U ðrÃÞ = a½ln ða=Fr ÞÀ1 .Inequilibrium,theionization 0 Eext 0 function) that reduces to a ln ðr=r Þ at r ≪ λ and decays expo- R ∝ ðÀ Ã= Þ R 0 el rate exp U kBT and the recombination rate, r,ofthe± nentially at r ≫ λ .Atr > r , the Lüscher term is negligible, so R ∝ = 2 el 0 charges, are equal.pffiffiffiffiffi Since r n+n− nf , then with logarithmic that U(r ) ≃ 0. Near the SIT, the strength of the Coulomb à a=ð2k TÞ 0 accuracy n = R ∝ expðÀU =2k TÞ ∝ ðFr =aÞ B ,andEq. potential becomes11 f B 0 (6)yields ¼ð 2= πε ε Þð ðκÞ= Þ : ð Þ a 4e 2 0 d f g 4 þ = L k T λ I /ðV À σL=2eÞ1 Tdec 16T ; V
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a b c 200 100M L=0.2 mm 1µ 1µ 0 150 /r c ) ) λ ) AlN/Si −1 −1 100 L=2.4 mm Ω Ω Ω NbTiN 100n 100n 50 0.1 10M 012 L(mm) 0.2 2.4 10n 10n Resistance (
1M Conductance ( Conductance ( 2.40 mm λ 0.55 mm c
1n 0.2 mm 1n 8
1234 0.4 0.8 1.2 123 -1/2 1/T(1/K) Temperature (K) (T/Tdec-1)
Fig. 1 Insulating NbTiN film’s sheet resistance evolution with bridge length. a The Arrhenius plot of sheet resistance R□ vs. reversed temperature 1/T for fridges of various length L. Dashed line shows R / expð1=TÞ. Inset: Experimental setup. The Si substrate with AlN buffer layer is shown with light gray and Hall bridge of NbTiN is dark gray. The square gold contacts are given in yellow. All lateral sizes are given in millimeters. The bridges lengths (i.e., distances between measuring electrodes) shown in the legend in panel b. b Same data as in a but replotted as conductance G = 1/R□ vs. T in log-line scale. A few curves are omitted to avoid overcrowding. Dotted lines are fits using a two dimensional Coulomb gas model20 that generalizes the Berezinskii–Kosterlitz–Thouless (BKT) G / ½ÀðT=T À Þ1=2 13 formula for conductance exp dec 1 by incorporating a self-consistent solution to the effects of electrostatic screening , where the screening λ T fi fi T ≈ ðT=T À Þ1=2 length c and dec enter as tting parameters. For all bridges we nd dec 400 mK. c Same data as in b but for temperature renormalized as dec 1 . Solid line corresponds to the case of an infinite electrostatic screening length λc → ∞. Inset: Screening length as function of the bridge length L. Symbols correspondtoBKT-fits of G(T) (dotted lines in b and c) and dashed line is the eye guide. The error bars are obtained from variations of parameters leaving self- consistent solutions of electrostatic equations of ref. 13 unchanged and are much less than the size of symbols.
a 10n b 10n
1n 1n
100p 100p Vc1
Vc1 10p 10p Vc1 Current (A) Current (A)
1p 1p
100f 100f 10m 100m 1 10m 100m 1 10 Voltage (V) Voltage (V) cd6 3 2.0 e
1.5 0.78⋅L 2.1⋅L 4 2 1.02⋅L (V) (V) 1.0 (V) c1 c2 c2 V V V 2 1 0.5
0.0 0 0 01230123 0123 L(mm) L(mm) L(mm)
Fig. 2 Threshold voltage of bridges of different length. The I–V curves of two different NbTiN samples with (a) R□(T = 2K)= 0.2 MΩ and (b) R□(T = 2 K) = 2MΩ taken at the same temperature T = 50 mK. Different colors correspond to different bridge’s length L (distance between electrodes). c
Dependence of the threshold voltage Vc2 on the bridge’s length for 0.2 MΩ sample. The dashed line depicts Vc2 = 0.78 ⋅ L dependence. The error bars are defined as the width of the jump and are less than the symbol sizes. d Threshold voltage Vc2 for the 2 MΩ sample; Vc2 = 2.1 ⋅ L (dashed line). Error bars are given by the width of the jump. e The dependence of the kink voltage marked by arrows in main panel, which we associate with the critical voltage Vc1 upon the bridge length. The error bars are determined from the errors in the slopes of the I–V curves and are much less than the symbol sizes.
K. The current jumps span a range of a few orders of magnitude shown in Fig. 2a, the string tension can be estimated from ds ≃ and become less sharp upon shortening the distance between 0.13 mm as σ ≃ 4.15 × 10−21 J/m = 26 meV/m, which leads to a −5 electrodes. The threshold voltage Vc2 exhibits a linear dependence contribution σ/2e = 1.3 × 10 V/mm to the slope. This is neg- upon L, see in Fig. 2c, d, implying that the threshold electric field ligible with respect to the Coulomb contribution (second term in Ec2 = Vc2/L is independent on the distance between electrodes in Eq. (10)), which amounts to a predicted slope of 0.63 V/mm, full accordance with Eq. (10). For the low-resistance sample, again in remarkable quantitative agreement with the measured
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slope of 0.78 V/mm. The I–V characteristics below the threshold, ÀÁμν & þð 2Þ2 F μν ¼ Λ 2 2J~ ; & ∂ ∂ À 2∇2 ; ð Þ measured at 50 mK, are linear, which agrees reasonably well with mv 2 v 0 0 v 13 the theoretically predicted power-law exponent 1 + (T /16T) = dec ~ 0i ~ ij 1 + (400 mK/16 × 50 mK) = 1.5. where J ¼ vJ 0i, J ¼ð1=vÞJ ij, and time derivatives and F 0i ∂ The I–V characteristics of the high-resistance sample are factors in Eq. (12) must be substituted by (1/v) 0 and shown in Fig. 2b. The three largest bridges exhibit two-kink I–V ð1=vÞF0i, respectively. Equation (13) implies the electric Meiss- curves. The low-voltage kinks are identified as Vc1, and the upper ner effect for applied voltages below a critical value Vc1 where ones represent Vc2. The two smallest bridges do not resolve Vc1. strings do not penetrate the sample. In the static situation with no This is because their size is so small that the strings penetrate the strings, Eq. (13) reduces to entire sample in the whole experimentally accessible voltage range ÀÁ ∇2 Àð Þ2 F 0i ¼ : ð Þ and, thus, the Meissner state is not visible. The linear dependence mv 0 14 ∝ Vc2 L of the bridge length L has the slope 2.1 V/mm, see Fig. 2d. fi For a superinsulator occupying the z > 0 half-space and a uniform Three available values of Vc1 are apparently not suf cient for fi electric eld Eext applied in the x direction with boundary con- conclusive evidence of linearity, see Fig. 2e. Let us however F 01ð ¼ Þ¼ fi dition z 0 Eext one nds assume that Vc1 ∝ L in accordance with (7). Then the mean F 01ð Þ¼ ðÀ =λ Þ; ; ð Þ square deviation estimate for the approximating straight line z Eext exp z el z>0 15 (shown in red) would give 1.02 V/mm for the slope. Making use λ = Eq. (10), one would obtain the slope 2.1 − 1.02 ≈ 1.1 character- where el 1/(vm) is the electric analog of the London penetra- izing the pure Coulomb contribution. From Eq. (10), we then tion length. Hence, in a bulk superinsulator in the electric ε = estimate the deconfinement temperature 692 mK for this Meissner state, the static dielectric permittivity SI 0. + = + sample, yielding the exponent 1 (Tdec/16T) 1 (692 mK/16 × To describe the electric Meissner effect in terms of magnetic 50 mK) = 1.86 for the I–V characteristics in the mixed state, monopoles, let us recall that the electrodynamics of the dual Vc1 < V < Vc2. Comparing this value with the measured ones for charge-monopole ensemble is governed by the Maxwell-Dirac the three largest bridges, 1.6, 1.8 and 2, respectively, we conclude equations4 ∝ μν that the assumption Vc1 L results in a fair quantitative agree- ∂ μν ¼ ν ; ∂ ~ ¼ ν ; ð Þ ment between the predicted and measured exponents. Thus, the μF jq μF jϕ 16 fi μν observation of Vc1 is the rst ever direct measurement of Poly- ~ ¼ð = Þϵμναβ fi ’ where F 1 2 Fαβ is the dual electromagnetic eld ten- akov s string tension. ϵμναβ μ sor ( is the totally antisymmetric tensor), jq is the charge Finally, let us discuss why the kink associated with Vc1 is not μ seen in the IV-curves of the low-resistance sample. Due to its current and jϕ is the magnetic monopole current. Consider an lower resistance, this sample is closer to the SIT, where the infinite superinsulating slab harboring an ensemble of vortices deconfinement temperature decreases towards zero. Since mea- aligned along the z axis perpendicular to the slab surfaces, with surements of both samples are made at the same temperature, this their magnetic monopole endpoints. The applied dc electric field ≡ means that the low-resistance sample is closer to the deconfine- E (Ex, 0, 0) generates a magnetic monopole current, according ment temperature, where the string tension vanishes and strings to the dual Ampère law ∇ × E = − jϕ, circulating around the slab become infinitely long. As a consequence, strings in the lower- in the ±y-direction, which, in turn, induces a shielding electric resistance sample are longer than in the high-resistance sample field opposite to the applied one. Since, in the condensate, and penetrate the sample end-to-end for all accessible voltages, so monopole motion is dissipationless, the entire external applied fi that the lower kink Vc1 is not observable. In order to resolve the electric eld is screened. Hence the electric Meissner effect. lower kink and access the Meissner state one needs larger sam- The above consideration captures the physics of bulk super- ples, which are sufficiently far from the the SIT and have mea- insulators, but thep caseffiffiffiffiffiffiffiffi of thin films is more subtle. The screening λ ≲ ’ =σ11 surements carried out at low enough temperatures so that the length el ds v , increases with temperature and diver- fi 26 ≃ μ 11 kink was not fogged by the noise. ges at the decon nement transition . For TiN, ds 60 m .In typical experiments on TiN and NbTiN films, where the film = − ≪ λ A theory of the electric Meissner state thickness d 3 20 nm el, the concept of circular monopole Now we discuss the electric Meissner effect and formation of the currents does not apply literally, since the physical size of the “ ” fi electric Meissner state at electric fields E < E which is another bulk monopoles exceeds the lm thickness. In this 2D case, c1 6 spectacular manifestation of Cooper pair confinement. The action monopoles become instantons , describing tunneling events at “ ” of the Abelian confining string, in its local formulation, is induced which vortex particles appear and disappear in the condensate. by the antisymmetric tensor gauge field of the second kind7,8. The vector current of point-like particles is replaced by the tensor Discussion μν current J of strings (Greek letters denote space-time indices, The novel material presented here, a superinsulator based on an Latin letters stand for spatial indices, and we use natural units atomic layer deposition grown NbTiN film, opens a new route for c = ℏ = 1). Accordingly, the vector potential Aμ, related to the further exploring the superinsulating state. Mapping the super- electromagnetic field via Fμν = ∂μAν − ∂νAμ, and coupling to the insulator onto compact QED enabled us to construct an electric particle vector current, is replaced by the fundamental field tensor analog of the superconducting London theory and to describe the μν – F μν coupled directly to the string tensor current J . The electric response and the I V characteristics of a superinsulator. equations for this field, see “Methods” section The device geometry, a Hall bar pattern, allowed us to investigate ÀÁ the details of the electric behavior of a superinsulator as function & þ 2 F μν ¼ Λ2J μν ; ð Þ m 2 11 of the system size and to reveal the linear dimension dependen- cies of both critical voltages, Vc1, at which the electric mixed state μν μα ∂α ∂μF À ∂ν ∂μF ¼ 0 ; ð12Þ forms, and Vc2, at which superinsulation breaks down. These pffiffiffi observations open the door for a direct observation of asymptotic = Λ Λ ¼ Λ = fi with m /e, 0 z 4, can be viewed as the dual London freedom phenomena and allow rst-ever measurements of the equations. In a superinsulator with the light speed v they linear tension of Polyakov’s strings and investigations of the become, see “Methods” section interior of an electric meson via desktop experiments.
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Considering the crudeness of our long-wavelength model, the of H2, CO, and CO2 to less than 1 ppb and O2 and H2O to less than 100 ppt. The thermal ALD growth of the AlN/NbTiN multilayer was performed using alter- observed Vc2(L) dependencies demonstrate an amazingly good agreement with the theoretical predictions. At the same time, nating exposures to the following gaseous reactants with the corresponding timing sequence (exposure-purge) in seconds: AlCl3 (anhydrous, 99.999%, Sigma-Aldrich) while the results for V are in an excellent concert with the – – c1 (1 10), NbCl5 (anhydrous, 99.995%, Sigma-Aldrich) (1 10), TiCl4 (99.995%, – – theoretical estimates, provided the linear dependence Vc1 is Sigma-Aldrich) (0.5 10) and NH3 (anhydrous, 99.9995%, Sigma-Aldrich) (1.5 10). The intrinsic silicon substrates were initially cleaned insitu using a 60 s exposure to assumed, more experimental research is required to establish a ∘ calibrated tool for measurements of the string tension. 7.5 ± 0.5 nm was deposited at 450 C with 200 ALD cycles. The chamber tem- perature was then lowered to 350 ∘C to synthesize the NbTiN layers. The growth + + cycle of the NbTiN is 2× (TiCl4 NH3) and 1× (NbCl5 NH3) that was repeated Methods 80 times with the corresponding total ALD cycles 240 to produce the film thick- Electric London equations. The tensor current describing the motion of strings nesses 10 nm as measured ex-istu by X-ray reflectivity (XRR). The chemical replaces the vector current encoding the motion of point particles. Correspond- composition measured by X-ray Photoemission Spectroscopy (XPS) and Ruther- ingly, the vector potential Aμ related to the original electromagnetic field strength ford Backscattering Spectroscopy (RBS) show consistently for the AlN layer 5.5 ± fi via Fμν = ∂μAν − ∂νAμ is replaced by a fundamental tensor field F μν coupled to the 0.3% of Cl impurities and a Al/N ratio of 1 ± 0.05, whereas for the NbTiN lms 3 ± μν + tensor current J . The associated field strength is given by the three-tensor 0.3% of Cl impurities, a Nb/Ti ratio of 2.3 ± 0.03 and a (Nb Ti)/N ratio of 1 ± 0.03. 3 Hμνρ ¼ ∂μF νρ þ ∂ν F ρμ þ ∂ρF μν . The Euclidean action for electric and magnetic The material densities measured by RBS and XRR are 2.5 ± 0.01 g/cm in AlN and 3 fi fields in the superinsulating vortex condensate8 acquires the form (we use natural 6 ± 0.05 g/cm in the NbTiN. After deposition lms were stored at room conditions units c = 1, ℏ = 1) in ten years. Z 4 1 1 1 S ¼ d x HμνρHμνρ þ F μνF μν þ i F μνJ μν ; ð17Þ 12Λ2 4e2 2 Samples parameters and measurements. The sample parameters are as follows: pffiffiffi 2 −1 ξ Λ ¼ Λ = Λ = the diffusion constant D ~0.25cm s , the superconducting coherence length (0) where 0 z 4, with 0 1/r0 the UV cutoff and z the magnetic monopole ⋅ 21 −3 27 fi F ~ 5 nm, the density of states as n ~4 10 cm . NbTiN lm was patterned quantum fugacity. Varying with respect μν yields the equations of motion using photolithography and plasma etching into resistivity bar 50 μm wide and μαβ 2 αβ 2 αβ with 100, 250 and 100 μm separation between additional perpendicular bars. On ∂μH þ m F ¼ 2Λ J ; ð18Þ this additional bars were making gold contacts. The chosen design allows for two- which, in turn, reduce to Eqs. (11) and (12) of the main text. probe resistivity measurements of regions with different length (2.4, 1.34, 0.55, For the sake of transparency, we now show how they reduce to the usual 0.44, 0.32, 0.2 μm). Measurements of the temperature dependencies of the resis- Maxwell equations in the limit of vanishing monopole fugacity, z → 0, when the tance were carried out in 3He/4He dilution refrigerator Triton400. Two-probe vortex condensate disappears.R In this limit we have Λ → 0 and finiteness of the technique was used with the 100 μV and 1 Hz. The chosen values of applied voltage partition function Z ¼ DF μν expðÀSÞ requires that the 3-tensor field strength guaranteed that the resistance was obtained in the linear response regime by the vanishes, Hμνρ ¼ 0. This is generically the case for “pure gauge configurations” direct measurements of current-voltage characteristics. Ac measurements were performed using SR830 lock-in amplifier, dc measurements were performed using F μν ¼ ∂μAν À ∂νAμ ¼ Fμν so that, in this limit, the partition function of classical nanovoltmeter Agilent 34420, current was transformed into voltage with using Maxwell electrodynamics is recovered, Z Z SR570 low-noise current preamplifier with filtration system. 4 1 Z ! DAμ exp À d x Fμν Fμν þ iAμjμ ; ð19Þ 4e2 μ ¼ ∂ J μν Data availability with the current j ν describing the now free charges at the end of the The authors declare that all relevant data supporting the findings of this study are original strings. This shows what is the relation between electrodynamics within available within the article. the vortex condensate and the familiar Maxwell theory. The latter is recovered as a “pure gauge” version of the former. Received: 16 April 2020; Accepted: 24 July 2020; pffiffiffiffiffi Electric field equations in materials with v = c/ εμ. The above “electric London equations” hold if the speed of light is c (=1 in our natural units), which is reflected in the Lorentz invariance of the action (Eq. 17) and the Lorentz covariance of the equations of motion (Eq. 11) of the main text. For real materials, however, this is surely not the case. The (Euclidean) action for electromagnetic fields in a linear material is given by References Z 1 1 1. Gross, D. J. & Wilczek, F. Ultraviolet behaviour on non-Abelain gauge S ¼ dt d3x E2 þ vB2 ; ð20Þ theories. Phys. Rev. Lett. 30, 1343–1346 (1973). linear 2e2 v pffiffiffiffiffi 2. Poolitzer, H. D. Reliable perturbative results for strong interactions. Phys. Rev. where, v ¼ 1= με is the light velocity in the material, expressed in terms of the Lett. 30, 1346–1349 (1973). dielectric permittivity ε and the magnetic permeability μ. Due to v < 1, the sym- 3. t Hooft, G. On the phase transition towards permanent quark confinement. metry of this action is now restricted to non-relativistic Galilean invariance. Cor- Nucl. Phys. B 138,1–25 (1978). respondingly, the Galilean-invariant action for a superinsulating thin film of this 4. Goddard, P. & Olive, D. I. Magnetic monopoles in gauge fields theories. Rep. material when vortices condense, is Progr. Phys. 41, 1357–1437 (1978). Z 5. Polyakov, A. M. Compact gauge fields and the infrared catastrophe. Phys. Lett. 1 3 S ¼ dt d3x vH H þ H H 59,82–84 (1975). 12 v2Λ2 ijk ijk v 0ij 0ij ð21Þ 6. Polyakov, A. M. Gauge Fields and Strings. (Harwood Academic Publisher, 1 1 1 Chur, Switzerland, 1987). þ vF F þ 2 F F þ i F μν J μν ; 4e2 ij ij v 0i 0i 2 7. Polyakov, A. M. Confining strings. Nucl. Phys. B 486,23–33 (1997). 8. Diamantini, M. C., Quevedo, F. & Trugenberger, C. A. Confining strings with where, v is the light velocity expressed in terms of material parameters of the – normal insulating state. 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6 COMMUNICATIONS PHYSICS | (2020) 3:142 | https://doi.org/10.1038/s42005-020-00410-5 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-00410-5 ARTICLE
15. Ovadia, M. et al. Evidence for a finite-temperature insulator. Scientific Reports Author contributions 5, 13503 (2015). C.M.D., C.A.T. and V.M.V. conceived the work and performed calculations; S.V.P. and 16. Nogueira, F. S. & Kleinert, H. Compact quantum electrodynamics in 2.1 A.Yu.M. carried out experiment and analyzed the data; L.G. and C.S. discussed an dimensions and spinon confinement: a renormalization group analysis. Phys. experiment; all authors discussed the results and wrote the manuscript. Rev. B 77, 045107 (2008). 17. Kleinert, H., Nogueira, F. S. & Sudbo, A. Kosterlitz-Thouless-like deconfinement mechanism in the (2.1)-dimensional Abelian Higgs model. Competing interests – Nucl. Phys. B 666, 361 395 (2003). The authors declare no competing interests. 18. Diamantini, M. C. et al. Bosonic topological intermediate state in the superconductor-insulator transition. Phys. Lett A 384, 126570 (2020). 19. Diamantini, M. C. & Trugenberger, C. A. Superinsulators, a toy realization of Additional information QCD in condensed matter. Invited Contribution to R. Jackiwʼs80ʼs Birthday Supplementary information is available for this paper at https://doi.org/10.1038/s42005- Festschrift, pp. 275–286 (World Scientific, Singapore, 2020). 020-00410-5. 20. Minnhagen, P. The two-dimensional Coulomb gas, vortex unbinding, and superfluid-superconducting films. Rev. Mod. Phys. 59, 1001 (1987). Correspondence and requests for materials should be addressed to V.M.V. 21. Quevedo, F. & Trugenberger, C. A. Phases of antisymmetric tensor field theories. Nucl. Phys. B 501, 143–172 (1997). Reprints and permission information is available at http://www.nature.com/reprints 22. Caselle, M., Panero, M. & Vadacchino, D. Width of the flux tube in compact U (1) gauge theory in three dimensions. JHEP 02, 180 (2016). Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in 23. Kogan, I. I. & Kovner, A. Compact QED3: a simple example of a variational published maps and institutional affiliations. calculation in a gauge theory. Phys. Rev. D 51, 1948–1955 (1995). 24. Lüscher, M. Symmetry-breaking aspects of the roughening transition in gauge theories. Nucl. Phys. B 180, 317–329 (1981). 25. Agasyan, N. O. & Zarembo, K. Phase structure and nonperturbative states in Open Access This article is licensed under a Creative Commons three-dimensional adjoint Higgs model. Phys. Rev. D 57, 2475 (1998). Attribution 4.0 International License, which permits use, sharing, 26. Diamantini,M.C.,Gammaitoni,L.,Trugenberger,C.A.&Vinokur,V.M.Vogel- adaptation, distribution and reproduction in any medium or format, as long as you give Fulcher-Tamman criticality of 3D superinsulators. Phys. Rep. 8, 15718 (2018). appropriate credit to the original author(s) and the source, provide a link to the Creative 27. Burdastyh, M. V. et al. Dimension effects in insulating NbTiN disordered films Commons license, and indicate if changes were made. The images or other third party and asymptotic freedom of Cooper pairs. Pis'ma v ZhETF 109, 833–838 (2019). material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory Acknowledgements regulation or exceeds the permitted use, you will need to obtain permission directly from This work was supported by the U.S. Department of Energy, Office of Science, Basic the copyright holder. To view a copy of this license, visit http://creativecommons.org/ Energy Sciences, Materials Sciences and Engineering Division (V.M.V.). Experimental licenses/by/4.0/. work (S.V.P. and A.Yu.M.) were supported by Russian Science Foundation project No. 18-72-10056. M.C.D. thanks CERN for hospitality during completion the work. We are grateful to Dr. T. I. Baturina for valuable contribution at the initial stage of research of © The Author(s) 2020 superinsulator in NbTiN.
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