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Confinement and Asymptotic Freedom with Cooper Pairs 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 quarks into hadrons, 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]) COMMUNICATIONS PHYSICS | (2018) 1:77 | DOI: 10.1038/s42005-018-0073-9 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | DOI: 10.1038/s42005-018-0073-9 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) Cooper pair-vortex topological interaction, analogy in a Gedanken experiment to explain quark 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 superconductivity, 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.
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