Strongly correlated Cooper pair insulators and superfluids Predrag Nikolić George Mason University Acknowledgments Collaborators Subir Sachdev Eun-Gook Moon Anton Burkov Arun Paramekanti Affiliations and sponsors W.M.Keck Program in Quantum Materials Strongly correlated Cooper pair insulators and superfluids 2/33 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 Strongly correlated Cooper pair insulators and superfluids 3/33 Unitarity: two-body picture Universality: irrelevant microscopic details Two-body resonant scattering Bound state at zero energy , , , Strongly correlated Cooper pair insulators and superfluids 4/33 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, 033608 (2007) Strongly correlated Cooper pair insulators and superfluids 5/33 Perturbation theory Action: SP(2N ), imaginary time Feynman diagrams physical atom Cooper pair, molecule vertex (fermion) (boson) Strongly correlated Cooper pair insulators and superfluids 6/33 Perturbation theory: 1/N expansion Full bosonic propagator (Dyson equation) S = eff No natural small parameter Semi-classical expansion: N=∞ is mean-field approximation Physical: N=1 Strongly correlated Cooper pair insulators and superfluids 7/33 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 limit @ 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 T=0 phase diagram with population imbalance 1st order superfluid-metal transitions: h =0.807μ+ (1/N) c O 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, 033608 (2007) Strongly correlated Cooper pair insulators and superfluids 9/33 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? Strongly correlated Cooper pair insulators and superfluids 10/33 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, 134511 (2006) Strongly correlated Cooper pair insulators and superfluids 11/33 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, 024510 (2006) Strongly correlated Cooper pair insulators and superfluids 12/33 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 Strongly correlated Cooper pair insulators and superfluids 13/33 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 Strongly correlated Cooper pair insulators and superfluids 14/33 Pairing instability Quantum Hall → superfluid nd 2 order (saddle-point) P.N, Phys.Rev.B 79, 144507 (2009) Strongly correlated Cooper pair insulators and superfluids 15/33 Superfluids & Vortex lattice FFLO states Competing forces Pairing, orbital, Zeeman FFLO-”metals” and FFLO-”insulators” P.N, Phys.Rev.A 81, 023601 (2010) Strongly correlated Cooper pair insulators and superfluids 16/33 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, 023601 (2010) Strongly correlated Cooper pair insulators and superfluids 17/33 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) Strongly correlated Cooper pair insulators and superfluids 18/33 Hypothetical experimental signatures Trapped gasses Sharp shell boundaries FFLO: ρ ≠0 & p≠0 s Features FFLO-insulator: quantized p Polarized outer shells FFLO-metal: variable p FFLO rings, abrupt appearance Strongly correlated Cooper pair insulators and superfluids 19/33 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 Strongly correlated Cooper pair insulators and superfluids 20/33 Quantum vortex lattice melting Vortex mass Compression of the stiff superfluid Neutral: Vortex localization energy 2 2 E ~ p /2m ... p ~ B kin v Vortex lattice potential energy 4 Π is degenerate → E ~ Φ pot 0 Strongly correlated Cooper pair insulators and superfluids 21/33 Vortex liquid Genuine phases at T=0 4 Vortex lattice potential energy: Δ 0 -1 Melting kinetic energy gain: log (Δ ) 0 st 1 order vortex lattice melting as Δ →0 0 Low energy spectrum inconsistent with fermionic quantum Hall states Δ 0 Non-universal properties (by RG) strong (BEC) pairing weak (BCS) P.N, Phys.Rev.B 79, 144507 (2009) Strongly correlated Cooper pair insulators and superfluids 22/33 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, 144507 (2009) Strongly correlated Cooper pair insulators and superfluids 23/33 BCS-BEC crossover in lattice potentials nd 2 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; M.P.A.Fisher, P.B.Weichman, G.Grinstein, D.S.Fisher; Phys.Rev.Lett. 99, 230403 (2007) Phys.Rev.B 40, 546 (1989) Strongly correlated Cooper pair insulators and superfluids 24/33 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? Strongly correlated Cooper pair insulators and superfluids 25/33 Pair density wave Pair density wave Supersolid without the uniform component Pairing instability in a band-insulator generally occurs at a finite crystal momentum Strongly correlated Cooper pair insulators and superfluids 26/33 PDW evolution Incommensurate PDW Commensurate PDW Vertex q-dependence Energy q-dependence Weak coupling (BCS limit) Strong inter-band coupling Halperin-Rice in p-p P.N., A. Burkov, A.Paramekanti, Phys.Rev.B 81, 012504 (2010) Strongly correlated Cooper pair insulators and superfluids 27/33 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 Strongly correlated Cooper pair insulators and superfluids 28/33 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”) Strongly correlated Cooper pair insulators and superfluids 29/33 Renormalization of fermion spectra Pairing fluctuations near the superfluid-insulator transition Worst-case scenario: Goldstone-like bosons (c→0) -1/2 2D: small real self-energy ~ bandgap 3D: large cutoff-dependent self-energy Bose-insulator is protected only in 2D Strongly correlated Cooper pair insulators and superfluids 30/33 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