Superinsulator: a new topological state of matter

Transcription

1 Superinsulator: a new topological state of matter M. Cristina Diamantini Nips laboratory, INFN and Department of Physics and Geology University of Perugia Coll: Igor Lukyanchuk, University of Picardie Carlo A. Trugenberger, SwissScientific Valerii Vinokur, Argonne National Laboratory

2 Superconductor: zero electrical resistance, conductance electric current can flow indefinitely in a superconducting wire without power source Superinsulator: dual of a superconductor, resistance indefinitely blocks electric charges Superinsulator: in 1996 (P. Sodano, C.A. Trugenberger, MCD, Nucl. Phys. B474 (1996) 641) quantum phase transition (T=0) between a (topological ) superconductor and a superinsulator ( resistivity) was theoretically predicted; experimentally observed in disordered titanium nitride (TiN) films in 2008 (Vinokur et al, Nature 452 (2008) 613). This superconducting films cooled towards zero temperature have resistence

3 Results for homogeneously disordered TiN film (2D) T 0 exhibit superconducting/insulator transition near the transition a Cooper pair insulator forms, that undergoes a transition to a superinsulator state with resistance at low temperature (also direct superconductor/superinsulator observed) transition driven by tuning disorder (thickness of the film) putting the system in a external magnetic field (P, T breaking) and varying it (Vinokur et al. Nature)

4 theoretical understanding modeled on results derived for a regular array of Josephson junctions: superinsulating state dual to the superconducting state experiments carried out on homogeneously disordered films emergent granularity (Fazio, Nature 452 (2008) 542)

5 Outline Topological states of matter Gauge theory of Jja in 2d Phase diagram Superinsulating phase as confinement phase Topologically insulating phase Granularity 3d?

6 Topological states of matter Different states of matter are distinguished by their internal structure orders orders are associated with symmetries (lack of) 1982 FQE states (2d): more rigid than a solid, incompressible no symmetry breaking new quantum order, topological order (Wen) quantum degrees of freedom organize themselves into robust emergent macroscopic states When we talk about orders in FQH liquids, we are really talking about the internal structure of FQH liquids at zero temperature. In other words, we are talking about the internal structure of the quantum ground state of FQH systems. So the topological order is a property of ground state wave function. (Wen)

7 QUANTUM HALL EFFECT: 2 dim electron gas at low temperature T ~ 10 mk and high magnetic field B ~ 10 Tesla conductance tensor: J i = σ ij E j Plateaux: σ xx = 0 no longitudinal conductance gap σ xy = ν e 2 / h ; ν filling fraction, e 2 /h conductance quantum ν integer (10-8 ) = 1/m ; m = 2k +1, k integer (10-6 ) (m=2k+1/p Jain, p integer) quantization independent of microscopic details: symmetry universality topology

8 FQE states: gapped in the bulk; gapless edge excitations; low energy effective field theory, Chern-Simons: background independent; ground state degeneracy; quasiparticles have fractional charge and statistics; breaks P and T symmetries Recently new topological phases of matter with P and T symmetries have been discovered in 2 and 3d (Kane, Mele, Fu, Moore, Balents, Zhang, König): topological insulators: insulating in the bulk support conducting edge excitations topological BF theory, mixed Chern-Simons in 2d ground state degeneracy BF action was first proposed as a Field theory description of topological phases of condensed matter systems in 1996 (Sodano, Trugenberger, MCD, recently reintroduced by Moore) : topological superconductors; topological insulators; superinsulator/topological confinement.

9 Wen's idea: excitations over top. ground states described by conserved matter currents (2d): topological field theory at low energy: conserved matter current, b μ U(1) pseudovector gauge field CS, P and T breaking can we have P and T symmetries? Key feature: existence of a gap conserved vortex current mixed CS, U(1) x U(1), PT invariant if a μ is a vector How can we get a mass for a U(1) gauge theory (2d)?

10 Spontaneous Symmetry breaking Add unbroken vacuum: complex scalar field with two real massive degrees of freedom and the gauge field with one massless excitation; broken vacuum: massive gauge boson, two degrees of freedom; one massive chargeless Higgs boson PT invariant Topologically massive gauge theory (Deser, Jackiw, Templeton) Add equation of motion which describe the propagation of a single degree of freedom with mass m=ke 2, gauge symmetry is not broken; Gauss law: magnetic field acts as source for electric field; P and T breaking. generalize to BF theory (PT invariant)

11 Josephson junction arrays Planar arrays of spacing l V x, φ x superconducting island C 0 = ground capacitance Josephson coupling E J ; C nearest neighbours capacitance, E C = e 2 /2C charging energy V= electric potential of the island; φ = phase of its order parameter ; C C 0 (C 0 =0) two relevant parameter: (8 E J E C ) 1/2 Josephson plasma frequency; E J / E C

12 Phase φ x canonically conjugate to the charges Q x on the island: Q = 2 e p 0 p 0 Z Poisson s equation: -C Δ V x = Q x degrees of freedom: Cooper pairs and vortices Action: two Coulomb gases for charges and vortices + kinetic term for the charges +Aharonov-Bohm topological interaction between charges and vortices (Fazio, Schön) add kinetic term for vortices (induced by integration over the charge dynamics) perfect duality between charges and vortices (Sodano Trugenberger MCD)

13 T=0 exact mapping (Sodano Trugenberger MCD) between the action of Jja and (Euclidean lattice, with lattice spacing l) BF theory mixed Chern-Simons, a μ vector, b μ pseudovector, PT invariant conserved charge current conserved vortex current mixed Chern-Simons term is periodic presence of topological defects (instantons) phase structure determined by the Q μ electric topological defects condensation (lack of) of M μ magnetic topological defects topological defects

14 T=0, B=0 Phase structure η dimensionless parameter that depends on the lattice: with G(ml) diagonal part of the lattice kernel μ = ln5 entropy of a string

15 electric condensation superconductor magnetic condensation superinsulator electric condensation superconductor no condensation top. Insulator/quantum metal magnetic condensation superinsulator η < 1: η < < 1/η coexistence region for electric and magnetic charges first-order direct transition superinsulator /superconductor with metastable coexistence of phases granularity

16 B 0 f = number of elementary fluxes π/e per plaquette piercing the array; 0 f <1 a) η > 1; transition from a superconductor to a topological insulator b) η <1 direct transition from a superconductor to a superinsulator (exotic quantum Hall phases may appear)

17 Electromagnetic response: compute the induced effective action S eff (A μ ) for the electromagnetic gauge potential A μ, induced currents: 1. Superconducting phase: closed Q μ condense, M μ diluted effective electromagnetic action for energy << gap m=(8 E C E J ) 1/2 non-local form of the London equation Coulomb gauge London penetration depth

18 2. Superinsulating phase, M μ condense, Q μ diluted >>1 Villain approximation compact QED in 2d S QED = 1/2e 2 x ( 1 - cos F μ ) (Polyakov) quantization of electric fluxes in integer multiples 2e dual Meissner effect: electric fields are expelled and can penetrate only as quantized electric flux tubes

19 QCD confining string like picture: linear confinement of Cooper pairs into neutral "U(1) mesons string tension (Polyakov) typical size determined by the scale: L (1/T) 1/2 L O(l) near the direct superconductor to superinsulator transition for η < 1 (but not too small) new spatial scale spontaneously emerging near these transitions, describing U(1) mesons extending over several lengths of the UV cutoff emergent granularity

20 3. Topological insulating phase: M μ and Q μ diluted Insulator?? infrared dominant action: effective electromagnetic action for energy << gap m=(8 E C E J ) 1/2 gap presence of boundary currents ρ = boundary charge density s = coordinate along the boundary electric conductivity 0 at T=0, realized by the edge modes, conductance =conductance quantum

21 Vortex current: Hall current for vortices vortex Hall conductance: σ V = 2e/2π "integer quantum Hall effect" of charges quantized in Cooper pairs

22 Emergent granularity L η< 1, droplet of true superconducting vacuum in the false superinsulating vacuum a true false b a= energy associated with the true vacuum b= energy of escaping fluctuations, boundary effect E(L) = const a L 2 + b L would like to expand, no false vacuum would like to contract, escaping fluct. cost energy for η < < 1/η b dominates, formation of small droplet favored

23 3d charge current, b μν pseudotensor vortex current, a μ vector + kinetic for a μ and b μν compact gauge theory (or periodic BF coupling) topological defects, condensation (lack of) determines the phase diagram electromagnetic response: phases: topological superconductor topological insulator superinsulator BiSe-BiTe-BiSb (Sodano, Trugenberger, MCD)

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