Strongly correlated Cooper pair insulators and superfluids

Transcription

1 Strongly correlated Cooper pair insulators and superfluids Predrag Nikolić George Mason University

2 Acknowledgments Collaborators Subir Sachdev Eun-Gook Moon Anton Burkov Arun Paramekanti Affiliations and sponsors W.M.Keck Program in Quantum Materials 2/33

3 Overview Unitarity: the most correlated pairing BCS-BEC crossover in uniform systems Vortex lattices and liquids near unitarity (re-entrant superfluids, FFLO states, quantum Hall and paired insulators) Unitarity in periodic potentials (band-mott crossover, pair density waves and Bose-insulators) Conclusions 3/33

4 Unitarity: two-body picture Universality: irrelevant microscopic details Two-body resonant scattering Bound state at zero energy,,, 4/33

5 Unitarity: many-body picture Universality Quantum critical point Theoretical Approaches Mean-field approximation Perturbation theory Renormalization group P.N., S.Sachdev; Phys.Rev. A 75, (2007) 5/33

6 Perturbation theory Action: SP(2N ), imaginary time Feynman diagrams physical atom (fermion) Cooper pair, molecule (boson) vertex 6/33

7 Perturbation theory: 1/N expansion Full bosonic propagator (Dyson equation) Seff = No natural small parameter Semi-classical expansion: N= is mean-field approximation Physical: N=1 7/33

8 BCS-BEC crossover in uniform systems Attractive interactions & pairing correlations Weak => many-body bound state, BCS superconductor Strong => two-body bound state, BEC condensate of molecules Unitarity Feshbach resonance The strongest pairing correlations and quantum entanglement Novel state uniquely accessible in atomic physics Fundamental questions The evolution of states between BCS and BEC limits New quantum phases 8/33

9 T=0 phase diagram with population imbalance 1st order superfluid-metal transitions: h =0.807μ+O(1/N) c nd 2 order superfluid-insulator (vacuum) transition Smooth BEC-BCS crossover Uniform magnetized BEC superfluid phase for μ<0 Normal metallic phases with one or two Fermi seas P.N., S.Sachdev; Phys.Rev. A 75, (2007) 9/33

10 Time-reversal symmetry violations Orbital effects Superfluid vortex lattice Fermi liquid fermionic quantum Hall state Correlated insulators many possibilities Zeeman effects FFLO states, magnetized correlated insulators Strongly correlated quantum insulators Quantum Hall liquid or density wave of Cooper pairs? Questions Phase transitions or crossovers between different normal states? The nature of paired insulators? Topological order? 10/33

11 Vortices in superconductors Fluctuating d-wave superconductivity Massless Dirac fermions Lattice + Coulomb repulsion + pairing no vortex core states Small cores Light and friction-free vortices Quantum vortex dynamics Conventional BCS superconductivity s-wave vortex core states Large cores Heavy vortex, large friction P.N., S.Sachdev; Phys.Rev.B 73, (2006) 11/33

12 Quantum vortex liquid Vortices in the normal phase of cuprates, even at T=0 T.Hanaguri, et.al.; Nature 430, 1001 (2004) Y.Wang, et.al.; Phys.Rev.B 73, (2006) 12/33

13 Superfluids in the quantum Hall regime Normal state quantum Hall insulator Localized particles (cyclotron orbitals) Discrete Landau levels Macroscopic degeneracy: two particles per flux quantum Superfluid 13/33

14 Pairing instability No px dependence to all orders of 1/N charged bosonic excitations live on degenerate Landau levels Macroscopically many modes turn soft simultaneously The nature of condensate is determined by interactions 14/33

15 Pairing instability Quantum Hall superfluid 2nd order (saddle-point) P.N, Phys.Rev.B 79, (2009) 15/33

16 Superfluids & Vortex lattice FFLO states Competing forces Pairing, orbital, Zeeman FFLO- metals and FFLO- insulators P.N, Phys.Rev.A 81, (2010) 16/33

17 Superfluids & Vortex lattice FFLO states Re-entrant pairing (superfluidity) In arbitrarily large magnetic fields With arbitrarily weak attractive interactions P.N, Phys.Rev.A 81, (2010) 17/33

18 FFLO states FFLO states Condensates in higher (n>0) bosonic Landau levels Vortex lattice in the level n: n extra vortex-antivortex pairs per unit-cell Driven by Zeeman effect (more order parameter supression) 18/33

19 Hypothetical experimental signatures Trapped gasses Sharp shell boundaries FFLO: ρs 0 & p 0 FFLO-insulator: quantized p FFLO-metal: variable p Features Polarized outer shells FFLO rings, abrupt appearance 19/33

20 Caveats in the superfluid phases Effects of quantum fluctuations Shear vortex motion restores U(1) symmetry in the superfluid No long-range phase coherence of the order parameter Algebraic correlations Vortex lattice order Space-group symmetry breaking (vortex lattice) survives at T=0 All symmetries restored at T>0 Algebraic correlations between vortex positions at low T Order parameter description is approximate True free energy density is: OK at energy scales above 20/33

21 Quantum vortex lattice melting Vortex mass Compression of the stiff superfluid Neutral: Vortex localization energy Ekin ~ p2/2mv... p2 ~ B Vortex lattice potential energy Π is degenerate Epot ~ Φ04 21/33

22 Vortex liquid Genuine phases at T=0 Vortex lattice potential energy: Δ04 Melting kinetic energy gain: log-1 (Δ0) 1st order vortex lattice melting as Δ0 0 Low energy spectrum inconsistent with fermionic quantum Hall states Δ0 Non-universal properties (by RG) strong (BEC) pairing weak (BCS) P.N, Phys.Rev.B 79, (2009) 22/33

23 The nature of vortex liquids Non-universal properties At Gaussian and unitarity fixed points of RG All interactions are relevant in d=2 Dimensional reduction Many stable interacting fixed points? P.N, Phys.Rev.B 79, (2009) 23/33

24 BCS-BEC crossover in lattice potentials 2nd order superfluid-insulator phase transition at T=0, h=0 Band-Mott insulator crossover at unitarity (s-wave) E.G.Moon, P.Nikolić, S.Sachdev; Phys.Rev.Lett. 99, (2007) M.P.A.Fisher, P.B.Weichman, G.Grinstein, D.S.Fisher; Phys.Rev.B 40, 546 (1989) 24/33

25 Critical lattice depth Saddle-point approximation Diagonalize in continuum space near unitarity Single-band Hubbard models: only deep in BCS or BEC limits... Fix density - completely filled bands At unitarity: Our result: VC~ 70 Er MIT experiment: VC~ 6 Er Fluctuation effects? 25/33

26 Pair density wave Pair density wave Supersolid without the uniform component Pairing instability in a band-insulator generally occurs at a finite crystal momentum 26/33

27 PDW evolution Incommensurate PDW Vertex q-dependence Weak coupling (BCS limit) Commensurate PDW Energy q-dependence Strong inter-band coupling Halperin-Rice in p-p P.N., A. Burkov, A.Paramekanti, Phys.Rev.B 81, (2010) 27/33

28 Fluctuation effects Incommensurate supersolid? Pairing bubble has linear q-dependence at small q Inconsistent with q=0 pairing ( Goldstone modes) Robust finite-q pairing against fluctuations But, frustrated on the lattice! Fluctuation effects Stabilize a commensurate supersolid order Looks like Mott physics! Are there non-trivial paired insulators? Near the superfluid-insulator transition Fermions have a large (band) gap Collective bosonic modes are low energy excitations Charge conservation => infinite lifetime for gapped bosons 28/33

29 Bose insulator Preformed Cooper pairs Not a new thermodynamic phase Singularities in the excited state spectrum Non-equilibrium phase transitions Sharp signature: driven condensate (Cooper pair laser ) 29/33

30 Renormalization of fermion spectra Pairing fluctuations near the superfluid-insulator transition Worst-case scenario: Goldstone-like bosons (c 0) 2D: small real self-energy ~ bandgap-1/2 3D: large cutoff-dependent self-energy Bose-insulator is protected only in 2D 30/33

31 Renormalization group analysis Pairing in a band-insulator near a band-edge One type of active fermions (electrons or holes) Vacuum ground-state => exact RG Unstable fixed-point at unitarity Run-away flow to superfluid or Mott-insulator (Which one? Decided at cutoff scales) Pairing in the middle of the bandgap Particles and holes => perturbative RG 8 fixed points ( Bragg images of unitarity ) Analogous run-away flows No natural particle-hole instabilities 31/33

32 Two gaps scenario for cuprates Very low doping AF correlations (short range) Large charge gap, small spin gap Frustrated motion of a single hole => fermion gap ( pseudogap ) A pair of holes could gain kinetic energy (Anderson) (adapts to AF domain walls) Attractive interactions develop between holes Larger doping (still underdoped) Shorter AF correlations => smaller fermion gap (T*) Attractive interactions win at large scales Large spin gap, small charge gap... eventually SC Dual vortex picture is valid (Tesanovic, Balents, Sachdev...) 32/33

33 Conclusions Unitarity The most correlated pairing Zero-density quantum critical point Pairing with violated time-reversal symmetry Re-entrant superfluidity FFLO states Non-universal vortex liquids Pairing in lattice potentials PDW instability in band-insulators Cooper-pair insulators 33/33