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 ϕ in the Cooper pair condensate, ing numbers do not contribute to the partition function. [N, ϕ] = 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 ∆ϕ∆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- rent√ 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 √ √ 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 {Qi} and {Mi} 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). ~ 2 Integrating out the gauge fields (one needs an intermedi- where ∇3 = ∂0∂0 + v ∇2 and v is the speed of light in ate regulator for this) gives the medium. In Coulomb gauge A0 = 0, ∂iAi = 0, the effective action reduces to Z 3 ∂α Slinking = 2πi d x Qµµαν Mν . (3) Z −∇2 SC 3 Seff ∝ d x AiAi , (6) For charge-anticharge and vortex-antivortex fluctuations, represented by closed loops {Qi} and {Mi}, this is the and the induced current ji = δSeff /δAi satisfies the Lon- 3 don equations, As we have have mentioned above, the next order terms in the derivative expansion of the effective action (1) con- ∂ j ∝ E , 0 tain two derivatives: gauge invariance requires then that rot j ∝ B. (7) they must be built from the “electric” (fi and gi) and This implies that the electric condensation phase is a “magnetic” (f0 and g0) fields of the two gauge poten- superconductor. tials. The most general possible gauge invariant action If only vortices condense, the effective action is derived up to two field derivatives is then given by by setting Qµ = 0 and coupling, as before, a probe elec- Z 3 κ tromagnetic gauge field to the charge current, S = d x i aµµαν ∂αbν 2π √ Z 1 ε 1 ε 3 κ κ P P S = d x i aµµαν ∂αbν + i bµ (κeFµ + 2πMµ) . + 2 f0f0 + 2 fifi + 2 g0g0 + 2 gigi 2π 2π 2evµP 2ev 2eqµP 2eq (8) √ √ + i κaµQµ + i κbµMµ , (10) As before, we need gauge invariant regulators to obtain the effective action, but for both the gauge fields aµ and with the magnetic permeability µP and the electric per- bµ this time. Since the Chern-Simons term survives in mittivity ε [24], which determine the speed of light √ P this case, both fictitious gauge fields are massive. Af- v = 1/ µPεP in the material. The two coupling con- ter removing the regulator we obtain the local effective 2 2 stants eq and ev are phenomenological parameters having action the dimensionality of [mass] and comprising the remain- Z ing material characteristics relevant to this order. It can SI 3 2 1 2 S ∝ d x v (κeF0 + 2πM0) + (κeFi + 2πMi) . 2 2 2 2 2 eff v be shown [24] that eq = e /d, ev = π /(e λ⊥), with d be- 2 (9) ing the thickness of the 2D film and λ⊥ = λL/d the Pearl which is nothing else but the non-relativistic version of length and λL being the London length of the bulk. This the Polyakov compact QED action [23] (a completely rig- identification, however, is not relevant for the the struc- orous formulation requires, of course, the introduction tures of phases emerging in the critical vicinity of the SIT. 2 2 of a lattice regularization). An important point here is One can simply consider eq and ev as phenomenological that, in the vortex condensate, the vortex number is not parameters embodying material parameters and effects of conserved. This is reflected by the presence of instan- disorder. Note that to this order in the derivative expan- tons M = ∂0M0 + ∂iMi at the end of the vortex world sion, the effective action is perfectly dual with respect to lines Mµ [23]. These instantons force the electric flux the mutual exchange of charge and vortex degrees of free- into strings, dual images of Abrikosov fluxes, that cause dom and the corresponding coupling constants. Possible linear confinement of charges [23], leading to an infinite duality breaking is a higher order effect. In field theory, resistance state, the superinsulator [9–12, 18]. this duality under the transformation g = ev/eq → 1/g Finally, the state where none of the condensates can goes under the name of S duality (strong-weak coupling form, Qµ = 0, Mµ = 0, is characterized by the purely duality). topological mixed Chern-Simons long distance effective Upon addition of the usual electromagnetic action action. This intermediate state, that can appear between both gauge fields acquire a topological Chern-Simons the superconductor and the superinsulator, was origi- mass. In the relativistic case µP = εP = 1 this is nally called a quantum or Bose metal [25]. Our approach m = κeqev/2π [22]. In the non-relativistic case it is shows that this intermediate Bose metal is a topological modified to m = µP κeqev/2π if the dispersion relation insulator [26], with the quantum resistance arising exclu- E = pm2v4 + v2p2 is used [24]. Note that both the 2 2 sively from the conductance along the edges [24]. Note, coupling constants eq and ev have dimension [mass]. As however that, while the flux of this topological insulator a consequence, the quadratic terms are power-counting is π, the charge is 2e instead of e since it is a Cooper pair infrared irrelevant: this is the reason why they do not state. This bosonic topological insulator is thus a level enter in the determination of the nature of the various 1 topological insulator, with no ground state degeneracy phases. Chern-Simons theories, however, are plagued by on the torus. an anomaly: the ground state wave function differs if Note that no material or disorder parameter has en- the model is considered as a purely topological model tered the above derivations, which are entirely predicated or as the limit m → ∞ of a topologically massive the- on the topological interactions alone. The material pa- ory [27]. The former is not normalizable, only the lat- rameters do determine the London penetration depth of ter makes physical sense. This has the consequence that the superconductor and the string tension of the superin- the quadratic term are actually non-perturbatively rele- sulator and, thus, as we now show, the conditions for the vant, since they can drive the system toward fixed points particular scenario of the transition between the phases, different from the bosonic topological insulator. These but they are totally irrelevant as far as the nature of the correspond to continuous phase transitions [28], whose possible phases is concerned. location is determined by an energy-entropy balance for 4

threshold voltage. Finally we conclude by commenting on the sugges- tions [30, 31] that many-body localization (MBL) [32] may be an alternative to the string confinement mech- anism for superinsulation. This is not so. Indeed, it has been recently pointed out [33–35] that MBL can arise without exogenous disorder due to strong, confining inter- actions alone, by mechanisms such as the mixing of the charge superselection sectors implied by a gauge symme- try. In the example discussed in [33] this mixing arises in the course of the temporal evolution of quantum states, the mixing mechanism playing effectively the role of a FIG. 1. Phase diagram of the vicinity of the SIT. Tuning the 2 p disorder average. This process was identified exactly as parameter g = (π/e ) d/λ⊥, one drives the system across the SIT. The quantity η characterizes the strength of quantum a transport-inhibiting mechanism due to confinement in fluctuations in a given material. the Schwinger model in 1D. In the present setting, it is the Polyakov monopole instantons that play the role of endogenous, spontaneous disorder. Accordingly, the the topological excitations Qµ and Mµ [9, 24]. The tran- summation over the instanton gas configurations acts as sition can also be a direct first-order transition between a averaging over disorder as pointed out already in the orig- superconductor and a superinsulator. The proper deriva- inal literature [23]. Importantly, the instanton formula- tion of these results requires an ultraviolet regularization tion describes not only 1D, but the 2D and 3D physical of the model on a scale corresponding to the coherence dimensions as well. This spontaneous disordering mecha- length ξ, for example a lattice of spacing ` = ξ. The nism by instantons has the same effect, that of mixing, in resulting phase structure [9, 24] is shown in Fig. 1. this case, the flux superselection sectors, leading to the The two parameters driving the quantum phase struc- survival of only the neutral charge sector as the physi- 2 p ture are the conductance g = (π/e ) d/λ⊥ and η = cal state, while all other, charged states are localized on (1/α)f(K, v) with α the fine structure constant and K the string scale. Hence inhibition of the charge transport the Landau parameter of the material. The appearance and the infinite resistance. In the present context, MBL of α in the denominator of η shows that the intermediate is a different name for the 50-year old phenomenon of bosonic topological insulator (Bose metal) phase opens confinement and again it is endogenous, external disor- up only if quantum fluctuations are strong enough in a der plays no role. The same confinement mechanism that particular material. prevents the observation of quarks is thus responsible for The same approach applies also to describe the finite- the absence of charged states and the infinite resistance temperature behaviour of the phases near the SIT [18]. in superinsulators. Specifically, linearly confined charges in the superinsu- Acknowledgments– M. C. D. thanks CERN, where she lator are liberated at the deconfinement phase transi- completed this work, for kind hospitality. The work at tion, where the string tension vanishes. In 2D, this de- Argonne (V.M.V.) was supported by the U.S. Depart- confinement transition is of the Berezinskii-Kosterlitz- ment of Energy, Office of Science, Basic Energy Sciences, Thoulsess (BKT) [16, 17] type [19] and has been recently Materials Sciences and Engineering Division. observed experimentally [15]. In 3D superinsulators [18], instead, the resistance is predicted to have Vogel-Fulcher- Tamman criticality [20]. This critical behaviour has also been recently experimentally observed in InO films [29], which have a thickness d  ξ, making them good candi- [1] Efetov, K. B. Phase Transition in Granulated Supercon- dates for 3D superinsulators. This is confirmed also by ductors. Sov. Phys. JETP 51, 1015 – 1022 (1980). [2] Haviland, D., Liu, Y., & Goldman, A. Onset of Super- the apparent violation of charge-vortex duality in InO conductivity in the Two-Dimensional Limit. Phys. Rev. films [30]: in 3D, duality is between electric and mag- Lett. 62, 2180 – 2183 (1989). netic fields, not between charges and vortices. A further [3] Hebard, A. & Paalanen, M. A. Magnetic-Field- confirmation of the string confinement picture of superin- Tuned Superconductor-Insulator Transition in Two- sulation comes from the recent measurement of strong Dimensional Films. Phys. Rev. Lett. 65, 927 – 930 (1990). noise near the threshold voltage [30]. Indeed, the thresh- [4] Fisher, M. P. A., Grinstein, G., & Girvin, S. M. Pres- old voltage corresponds to the critical strength when an ence of Quantum Diffusion in Two Dimensions: Universal Resistance at the Superconductor-Insulator Transition. applied voltage starts creating strips of normal insula- Phys. Rev. Lett. 64, 587 – 590 (1990). tor, carrying the current, mixed with the superinsulat- [5] Fisher, M. P. A. Quantum Phase Transitions in Dis- ing matrix. The proximity to this dynamic phase transi- ordered Two-Dimensional Superconductors. Phys. Rev. tion implies exactly strong current fluctuations near the Lett. 65, 923 – 926 (1990). 5

[6] Fazio, R. & Sch¨on, G. Charge and Vortex Dynamics in [21] Efetov K. B. Feigel’man M. V. & Wiegmann P. B. Com- Arrays of Tunnel Junctions. Physical Review B 43, 5307 ment on “Superinsulator and Quantum Synchroniza- – 5320 (1991). tion,” Phys. Rev. Lett. 102, 049701 (2009). [7] Goldman, A. M. Superconductor-Insulator Transitions. [22] Deser, S., Jackiw, R. & Templeton, S. Three-Dimensional Int. J. Mod. Phys. B24, 4081 – 4101 (2010). Massive Gauge Theories. Phys. Rev. Lett. 48 975 (1982). [8] ’t Hooft, G. On the phase transition towards permanent [23] Polyakov, A. M. Gauge Fields and Strings, Harwood Aca- quark confinement. Nucl. Phys. B138 1-25 (1978). demic Publisher, Chur (Switzerland) (1987). [9] Diamantini, M. C., Sodano, P. & Trugenberger, C. A. [24] Diamantini M. C. Trugenberger C. A. Lukyanchuk Gauge Theories of Josephson Junction Arrays. Nuclear I. & Vinokur V. M. Superconductor-insulator tran- Physics B474, 641 – 677 (1996). sition and topological nature of the Bose metal. [10] Kr¨amer, A. & Doniach, S. Superinsulator phase of two- arXiv:1710.10575. dimensional superconductors. Physical Review Letters [25] Das, D., and Doniach, S. Bose metal: Gauge-field Fluctu- 81, 3523 – 3527 (1998). ations and Scaling for Field-Tuned Quantum Phase Tran- [11] Vinokur, V. M. et al. Superinsulator and quantum syn- sitions. Phys. Rev. B 64 134511 (2001). chronization. Nature 452, 613 – 615 (2008). [26] Cho, G. Y. & Moore, J. E. Topological BF Field Theory [12] Baturina T. I. & Vinokur, V. M. Superinsulator– Description of Topological Insulators. Ann. Phys. 326 superconductor duality in two dimensions. Ann. 1515-1535 (2011). Phys. 331 236 – 257 (2013). [27] Dunne, G., Jackiw, R. & Trugenberger, C. A. Topological [13] Sambandamurthy, G., Engel, L. M., Johansson A., (Chern-Simons) quantum mechanics. Phy. Rev. D41 661- Peled, E. & Shahar, D. Experimental evidence for a 666 (1990). collective insulating state in two-dimensional supercon- [28] Su-Peng, K., Jing, Y., & Wen, X.G. Mutual Chern- ductors. Phys. Rev. Lett. 94, 017003 (2005). Simons Landau-Ginzburg Theory for Continuous Quan- [14] Baturina, T. I., Mironov, A. Yu., Vinokur, V. M., Bak- tum Phase Transitions of Z2 Topological Order, Phys. lanov, M. R., Strunk, C. Localized superconductivity Rev. B 80, 125101 (2009). in the quantum-critical region of the disorder-driven [29] Ovadia, M., et al. Evidence for a finite-temperature insu- superconductor-insulator transition in TiN thin films. lator. Scientific Reports 5, 13503 (2015). Phys. Rev. Lett. 99, 257003 (2007). [30] Tamir, I., Levinson, T., Gorniaczyk, F., Doron, A., Lieb, [15] Mironov A. Yu. et. al. Charge Berezinskii-Kosterlitz- J. & Shahar, D. Excessive Noise as Test fo Many-Body Thouless transition in superconducting NbTiN films. Sci. Localization. arXiv:1806.09492. Rep 8 (2018) 4082. [31] Feigel’man, M. V., Ioffe, L. B. & M´ezard, M. [16] V. L. Berezinski V. L. Sov. Phys. JETP 32 493 (1971), Superconductor-insulator transitions and energy local- Sov. Phys. JETP 34 610 (1972). ization. Phys. Rev. B82 184534 (2010). [17] Kosterlitz J. M. & Thouless D. J. Phys. C5 L124 (1972), [32] Basko, D. M., Aleiner, I. L. & Altshuler, B. L. Metal- J. Phys.C6 1181 (1973). insulator transition in a weakly interacting many-electron [18] Diamantini, M. C., Trugenberger, C. A. & Vinokur, V., system wihth localized single-particle states. Ann. Phys. M. Confinement and Asymptotic Freedom with Cooper 321 1126 (2006). Pairs. arXiv:1807.01984. [33] Brenes, M., Dalmonte, M., Heyl, M., & Sardicchio, A. [19] Svetitsky B. & Yaffe L. Critical Behavior at Finite Tem- Many-body localization dynamics from gauge invariance. perature Confinement Transitions. Nucl. Phys. B210 Phys. Rev. Lett. 120 030601 (2018). [FS6] 423-447 (1982). [34] Kormos, M., Collura, M., Takacs, G. & Calabrese, P. [20] Diamantini, C. M., Trugenberger, C. A., Vinokur, V. M. Nat. Phys. 13 246-249 (2017). Vogel-Fulcher-Tamman criticality of 3D superinsulators. [35] Mazza, P., Perfetto, G., Lerose, A., Collura, M. & Gam- arXiv:1710.10575. bassi, A. arXiv1806.09674.