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The Superconductor-Superinsulator Transition: S-Duality and QCD on The The Superconductor-Superinsulator Transition: S-duality and the QCD on the desktop M. Cristina Diamantini,1 Luca Gammaitoni,1 Carlo A. Trugenberger,2 and Valerii M. Vinokur3 1NiPS Laboratory, INFN and Dipartimento di Fisica, University of Perugia, via A. Pascoli, I-06100 Perugia, Italy 2SwissScientific, rue due Rhone 59, CH-1204 Geneva, Switzerland 3Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA We show that the nature of quantum phases around the superconductor-insulator transition (SIT) is controlled by charge-vortex topological interactions, and does not depend on the details of ma- terial parameters and disorder. We find three distinct phases, superconductor, superinsulator and bosonic topological insulator. The superinsulator is a state of matter with infinite resistance in a finite temperatures range, which is the S-dual of the superconductor and in which charge transport is prevented by electric strings binding charges of opposite sign. The electric strings ensuring linear confinement of charges are generated by instantons and are dual to superconducting Abrikosov vor- tices. Material parameters and disorder enter the London penetration depth of the superconductor, the string tension of the superinsulator and the quantum fluctuation parameter driving the transi- tion between them. They are entirely encoded in four phenomenological parameters of a topological gauge theory of the SIT. Finally, we point out that, in the context of strong coupling gauge theories, the many-body localization phenomenon that is often referred to as an underlying mechanism for superinsulation is a mere transcription of the well-known phenomenon of confinement into solid state physics language and is entirely driven by endogenous disorder embodied by instantons with no need of exogenous disorder. The superconductor-insulator transition (SIT) [1{7] perimental observations [13, 14]. Superinsulators where is a paradigmatic quantum phase transition found in identified in [11] as a low-temperature charge Berezinskii- Josephson junction arrays (JJA) [1,6] and in 2D disor- Kosterlitz-Thouless (BKT)[16, 17] phase emerging at the dered superconducting films at low temperatures T [2{5]. temperature of the charge BKT transition and was de- The tuning parameter driving the SIT is the ratio of the rived from the wave function { phase-amplitude duality single junction Coulomb energy to the Josephson cou- of the uncertainty principle. pling. In 2D films this ratio is effectively controlled by Motivated by 't Hooft's beautiful idea [8] and building varying the film thickness d which regulates the strength on the framework proposed in [9], a comprehensive theory of disorder and hence of Coulomb screening or by apply- of the SIT and superinsulators was developed in [18]. It ing a magnetic field that suppresses the Josephson cou- was shown that in duality to the Meissner effect in super- pling. This can cause a dramatic change in the ground conductors, which constricts the magnetic field lines pen- state so that superconductivity is lost in favour of insu- etrating a type II superconductor into Abrikosov vortices, lating behaviour. in superinsulators, electric flux tubes that linearly bind Cooper pairs into neutral \mesons" form. These electric In 1978, G. 't Hooft [8] appealed to a solid state physics strings confine fluctuating ±Cooper pair charges com- analogy in a Gedankenexperiment to explain quark con- pletely, thereby impeding electric conductance: Cooper finement and demonstrated that this is realized in a pairs are confined exactly as quarks in hadrons [18], the phase which is in many respects similar to the super- finite-temperature deconfinement transition for linear po- conducting phase, but is in a sense a zero particle mo- tentials coinciding with the 2D BKT transition [19, 20]. bility phase, the extreme opposite of a superconductor This established superinsulators as a novel, distinct state and called hence this phase a "superinsulator." In 1996, of matter. two of the present authors (mcd and cat) [9] developed a comprehensive field theory framework for the description The interpretation of the infinite-resistance state in of the SIT in JJA. They predicted that, on the insulating terms of the superinsulating state dual to superconduc- arXiv:1810.06862v1 [cond-mat.supr-con] 16 Oct 2018 side of the SIT, a new ground state forms, correspond- tivity was fiercely criticized in [21] by questioning the cor- ing to a novel phase with infinite resistance. This novel rectness of the microscopic modeling of the superconduct- phase is dual to the superconductor, characterized by ing films as a Josephson junction array (JJA) and attack- zero resistance, and they thus independently also called ing the treatment of this array by [11]. Here we demon- this phase a superinsulator. Independently, superinsula- strate by a straightforward calculation that the nature tors where also soon proposed in [10]. Finally, the name of the phases in the critical vicinity of the SIT is deter- and phenomenon of superinsulation was rediscovered and mined solely by fundamental topological interactions and experimentally detected by one of the authors (vmv) by gauge invariance. Disorder plays only the role of the and his collaborators in [11, 12] based on the earlier ex- tuning mechanism cranking up and down the effective 2 strength of the Coulomb interaction that drives the sys- sum of the integer Gauss linking numbers between closed tem across the SIT. Our finding relates the topological loops of the two kinds. These linking numbers represent nature of the superinsulator with the earlier finding [12] the Aharonov-Bohm/Aharonov-Casher phases accumu- that superinsulators, dual to superconductors, emerge as lated when one charge completely encircles a vortex and a consequence of quantum conjugation of the number of viceversa. Because of the factor (2πi) such integer link- particles N and phase ' in the Cooper pair condensate, ing numbers do not contribute to the partition function. [N; '] = i, and the related competition of the uncertain- They do, however for generic, infinitely extended world- ties of these variables due to the Heisenberg principle, lines of charges and vortices. The action (1) is the local ∆'∆N > 1. This completely invalidates the critique [21] representation of these topological interactions. since microscopic details of the model are irrelevant. The charge and vortex number currents Qµ and Mµ The T = 0 partition function of a quantum many body are conserved. Correspondingly, the gauge fields are system in D spatial dimension is determined by the Eu- invariant under the U(1) gauge transformations aµ ! clidean action of a classical system in (D+1) dimensions. aµ + @µλ and bµ ! bµ + @µχ. The full effective action for The effective action governing this partition function can the SIT must then respect these two gauge invariances. then be expanded in a series of derivatives. The universal The Chern-Simons term is the only marginal gauge in- properties of the system, including the phase structure variant term in 2D since it is the unique gauge invari- and the nature of the possible phases are determined by ant term involving only one field derivative. Topological the relevant and marginal terms in this effective action. interactions thus dominate near the SIT. From a purely The more terms one includes, the more microscopic de- field-theoretic point of view, the charge and vortex world- tails are modeled and taken into account. lines represent the singularities in the dual field strengths It has been recognized since the very early days of fµ = µαν @αbµ and gµ = µαν @αaµ arising from the com- the SIT studies, that the relevant degrees of freedom pactness of the two U(1) gauge groups [23]. A proper for the SIT are charges (Cooper pairs) and vortices and formulation of a compact U(1) gauge theory, however, that the possible phases are determined by the compe- requires the introduction of an ultraviolet lattice regular- tition between these two types of degrees of freedom [4{ ization [23]. This was done for the mixed Chern-Simons 6]. Charges and vortices are subject to topological in- model in [9]. teractions, embodied by the Aharonov-Bohm/Aharonov- It is now easy to derive what the possible phases Casher (ABC) phases they acquire when encircling one near the SIT are. If only charges condense, the cur- another. A local formulation of such topological inter- rentp Qµ becomes a field that can be expressed as Qµ = actions requires the introduction of two emergent gauge ( κ/2π)µαν @αcν , while Mµ vanishes. The field cµ can fields aµ and bµ coupled to the conserved charge and then be reabsorbed by a shift of bµ. If we couple the vortex currents, respectively. The Euclidean topological charge current to a probe electromagnetic gauge field Aµ action for these gauge fields is the mixed Chern-Simons we obtain the action action [22] Z 3 κ κ Z κ p p S = d x i aµµαν @αdν + i bµFµ ; (4) S = d3x i a @ b +i κa Q +i κb M ; (1) 2π 2π 2π µ µαν α ν µ µ µ µ where dµ = bµ + cµ and Fµ = µαν @αAν is the dual where κ is the dimensionless charge (κ = 2 for a Cooper field strength. This shows that the Chern-Simons term, pair) and that must be integrated over aµ and dµ, decouples. The Z (i) integration over bµ, instead, yields the electromagnetic X dqµ (τ) Q = dτ δ3 x − q(i)(τ) ; effective action. To do it, we need a gauge invariant µ dτ i Qi regulator, which, to dominant order, must be constructed Z (i) from the \electric" and \magnetic" fields of the b gauge X dmµ (τ) µ M = dτ δ3 x − m(i)(τ) ; (2) µ dτ potential. This gives i Mi Z 1 1 1 with fQig and fMig representing the world-lines, SC 3 Seff / d x vF0 2 F0 + Fi 2 Fi ; (5) parametrized by q(i) and m(i), of elementary charges and −∇3 v −∇3 vortices, respectively (we use natural units c = 1, = 1).
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