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Bosonic Topological Insulator Intermediate State in the Superconductor-Insulator Transition

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Bosonic Topological Insulator Intermediate State in the Superconductor-Insulator Transition Physics Letters A 384 (2020) 126570 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Bosonic topological insulator intermediate state in the superconductor-insulator transition M.C. Diamantini a, A.Yu. Mironov b,c, S.M. Postolova b,d, X. Liu e, Z. Hao e, D.M. Silevitch f, ∗ Ya. Kopelevich g, P. Kim e, C.A. Trugenberger h, V.M. Vinokur i,j, a NiPS Laboratory, INFN and Dipartimento di Fisica e Geologia, University of Perugia, via A. Pascoli, I-06100 Perugia, Italy b A.V. Rzhanov Institute of Semiconductor Physics SB RAS, 13 Lavrentjev Avenue, Novosibirsk 630090, Russia c Novosibirsk State University, Pirogova str. 2, Novosibirsk 630090, Russia d Institute for Physics of Microstructures RAS, GSP-105, Nizhny Novgorod 603950, Russia e Department of Physics, Harvard University, Cambridge, MA 02138, USA f Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA g Universidade Estadual de Campinas-UNICAMP, Instituto de Física “Gleb Wataghin”/DFA, Rua Sergio Buarque de Holanda, 777, Brazil h SwissScientific Technologies SA, rue du Rhone 59, CH-1204 Geneva, Switzerland i Materials Science Division, Argonne National Laboratory, 9700 S. Cass Avenue, Lemont, IL 60637, USA j Consortium for Advanced Science and Engineering (CASE), University of Chicago, 5801 S Ellis Ave, Chicago, IL 60637, USA a r t i c l e i n f o a b s t r a c t Article history: A low-temperature intervening metallic regime arising in the two-dimensional superconductor-insulator Received 12 April 2020 transition challenges our understanding of electronic fluids. Here we develop a gauge theory revealing Accepted 12 May 2020 that this emergent anomalous metal is a bosonic topological insulator where bulk transport is Available online 19 May 2020 suppressed by mutual statistics interactions between out-of-condensate Cooper pairs and vortices and Communicated by L. Ghivelder the longitudinal conductivity is mediated by symmetry-protected gapless edge modes. We explore Keywords: the magnetic-field-driven superconductor-insulator transition in a niobium titanium nitride device and Berezinskii-Kosterlitz-Thouless transition find marked signatures of a bosonic topological insulator behavior of the intervening regime with the Superconductor-insulator transition saturating resistance. The observed superconductor-anomalous metal and insulator-anomalous metal dual Bose metal phase transitions exhibit quantum Berezinskii-Kosterlitz-Thouless criticality in accord with the gauge Bosonic topological insulator theory. Josephson junction arrays © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND Quantum Berezinskii-Kosterlitz-Thouless license (http://creativecommons.org/licenses/by-nc-nd/4.0/). transition Introduction SIT [20] and unravel the fundamental role of the infrared-dominant Aharonov-Bohm-Casher (ABC) mutual statistics interactions that An anomalous metallic regime intervening between the su- determine the SIT phase structure. Here utilizing the technique perconductor and insulator has been reported in a wide va- of [20], we construct a gauge theory of the Bose metal (BM) in riety of two-dimensional electronic systems experiencing the which strong quantum fluctuations prevent Bose condensation of superconductor-to-insulator transition (SIT) [1–15], and is often re- both vortices and Cooper pairs (CP) and their mutual statistics in- ferred to as “Bose metal” [16]. Despite decades of dedicated studies teractions, see Fig. 1, induce a gap in their fluctuation spectrum, [17,18], its nature remains unclear and poses a challenge to our quantified by the Chern-Simons mass, mCS [21], preventing bulk understanding of electron fluids. The very existence of a 2D metal transport. The longitudinal conductance is mediated by symmetry- is at odds with the 2D orthodoxy, as conventional theories expect protected U(1)ZT edge modes, where ZT denotes time-reversal a direct quantum SIT with no intermediate metallic phase. 2 2 symmetry. Hence the BM realizes the long sought [22] bosonic Yet a gauge theory of Josephson junction arrays (JJA) at T = 0 topological insulator. We demonstrate that at T = 0the transitions [19]predicted a metallic phase intervening between the supercon- between the superconductor and BM and between the BM and ductor and superinsulator at T = 0. Extending the approach of [19] superinsulator are the respective quantum Berezinskii-Kosterlitz- to finite temperatures, enabled us to develop a field theory of the Thouless (BKT) transitions. We report transport measurements on niobium titanium nitride (NbTiN) and van der Waals heterostruc- * Corresponding author. tures of twisted double bilayer graphene (TDBG) films offering E-mail address: [email protected] (V.M. Vinokur). strong support to the proposed picture. https://doi.org/10.1016/j.physleta.2020.126570 0375-9601/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2 M.C. Diamantini et al. / Physics Letters A 384 (2020) 126570 parameter η describes the strength of quantum fluctuations. Near the SIT, where eq ≈ ev, η acquires the form (in physical units) 1 π 2 π η = G , (1) α μeλ⊥ αλ⊥ Fig. 1. Mutual statistics interactions. a: A charge (blue ball) evolves along the 2 Euclidean time axis and is encircled by a vortex (red). The two trajectories are where α = e /(hc¯ ) ≈ 1/137 is the fine structure constant, μe is the topologically linked since one cannot decouple the charge trajectory from the cir- positional entropy (see SI), and G is the diagonal element of the 3D cle without breaking it. This linking encodes the Aharonov-Casher effect. b: A dual Green function describing electromagnetic interactions screened by situation in which a vortex (red ball) evolves along the Euclidean time direction the CS mass, see SI. Identifying the UV cutoff with the supercon- and is encircled by a charge. This linking represent the Aharonov-Bohm effect. c: The linked charge-anticharge and vortex-antivortex fluctuations representing cou- ducting coherence length ξ , yields the geometric factor as d/(κλL) pled Aharonov-Bohm-Casher interactions which are encoded in the Chern-Simons where κ = λL/ξ is the Landau-Ginzburg parameter. term in the action. (For interpretation of the colors in the figure(s), the reader is Bose condensation of charges and/or vortices means prolifera- referred to the web version of this article.) tion of strings of an arbitrary size and occurs if F is negative, i.e. if Phase structure of the SIT Q 2/g + gM2 < 1/η . (2) We consider a two-dimensional superconducting film in the The phase emerging at particular values of g and η is determined vicinity of the SIT at temperatures T Tc0, where Tc0 is the mean- field temperature of formation of Cooper pairs with the infinite by the geometric condition that the nods on a square lattice of in- lifetime. Near the SIT, a film acquires self-induced electronic gran- teger electric and magnetic charges, {Q , M}, fall within the interior = 1/2 = 1/2 ularity which, conjectured in [23,24], became a paradigmatic at- of an ellipse with semi-axes rQ (g/η) and rM 1/(gη) , see tribute of the SIT [25,26]. This implies that in the critical vicinity Fig. 2(a–d). The Bose metal emerges if none of the condensates can of the SIT the 2D superconducting film can be perfectly modeled form i.e. if simultaneously as the granular superconductor or JJA. The latter harbors interact- ing elementary excitations, Cooper pairs which in this case can g/η > 1andgη > 1 . (3) be considered as charged bosons [27,28] and vortices with char- This relation resolves the enigma of why some materials exhibit acteristic energies 4E and 2π 2 E , respectively, E and E being C J C J a direct SIT while others go through the intermediate Bose metal the charging energy and Josephson coupling energy of a single phase. The direct SIT at g = 1 corresponds to η < 1; tuning g, one junction. The mapping on the real film is achieved by replace- crosses the region η < g < 1/η, where both vortex and Cooper ment 4E → e2 = 4e2/ and 2π 2 E → e2 = 2/λ⊥ where e is the C q J v 0 pair condensates coexist, i.e. the direct SIT is a first-order quan- electron charge (we use natural units, c = 1, h¯ = 1, restoring phys- tum transition. The Bose metal phase opens up for η > 1 and is ical units when necessary), = π/e is the flux quantum, the 0 favored by thicker films and for materials with low κ. Its domain ultraviolet (UV) cutoff of the theory is defined as min{d,ξ}, is delimited by the lines g = η and g = 1/η. As we show below, with d and ξ being the thickness of the film and the supercon- these lines represent quantum BKT transitions [32,33], and g = 1, = 2 ducting coherence length, respectively, and λ⊥ λL /d is the Pearl η = 1is a quantum tri-critical point. length of the film, with λL being the London penetration depth of To describe the magnetic-field-driven SIT in systems with g ≈ 1 the bulk material. Note that in a superconducting film the mean i.e. which are already on the brink of the SIT, we introduce the distance between Cooper pairs is much less than the size of a sin- frustration factor f = B/B, where B is the magnetic field corre- gle Cooper pair, hence, strictly speaking, the latter are not bosons. sponding to one flux quantum 0 = π/e per unit cell. Then the ex- However, since in a superconducting film the Cooper pair conden- ternal magnetic field shifts M → M + f and modifies the conden- sate maintains phase coherence, one can still describe the ensem- sation conditions, see Fig.
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