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Strongly correlatedCooper pair insulators and
superfluids
Predrag Nikolić
George Mason University
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Acknowledgments
Affiliations and sponsors
W.M.Keck Program inQuantum Materials
Collaborators
Subir Sachdev Anton Burkov Arun ParamekantiEun-Gook Moon
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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
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Unitarity: two-body picture
Universality: irrelevant microscopic details Two-body resonant scattering Bound state at zero energy
,
,
,
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Unitarity: many-body picture
Universality Quantum critical point
P.N., S.Sachdev; Phys.Rev. A 75, 033608 (2007)
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Theoretical Approaches Mean-field approximation Perturbation theory Renormalization group
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Perturbation theory
Action: SP(2N ), imaginary time
Feynman diagrams
physical atom(fermion)
Cooper pair, molecule(boson)
vertex
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Full bosonic propagator (Dyson equation)
Perturbation theory: 1/N expansion
No natural small parameter Semi-classical expansion: N=∞ is mean-field approximation Physical: N=1
Seff
=
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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
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1st order superfluid-metal transitions: 2nd 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
hc=0.807μ+O(1/N)
P.N., S.Sachdev; Phys.Rev. A 75, 033608 (2007)
T=0 phase diagram with population imbalance
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Time-reversal symmetry violations
Orbital effects Superfluid → vortex lattice Fermi liquid → fermionic quantum Hall state Correlated insulators → many possibilities
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?
Zeeman effects FFLO states, magnetized correlated insulators
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Vortices in superconductors
Conventional BCS superconductivity s-wave → vortex core states Large cores Heavy vortex, large friction
“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
P.N., S.Sachdev; Phys.Rev.B 73, 134511 (2006)
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Quantum vortex liquid
Vortices in the normal phase of cuprates, even at T=0
Y.Wang, et.al.; Phys.Rev.B 73, 024510 (2006)
T.Hanaguri, et.al.; Nature 430, 1001 (2004)
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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
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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
Pairing instability
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Pairing instability
Quantum Hall → superfluid 2nd order (saddle-point)
P.N, Phys.Rev.B 79, 144507 (2009)
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Superfluids & Vortex lattice FFLO states
Competing forces Pairing, orbital, Zeeman
FFLO-”metals” and FFLO-”insulators”P.N, Phys.Rev.A 81, 023601 (2010)
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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)
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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)
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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
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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
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Quantum vortex lattice melting
Vortex mass Compression of the stiff superfluid
Neutral:
Vortex lattice potential energy
Π is degenerate → Epot
~ Φ0
4
Vortex localization energy
Ekin
~ p2/2mv ... p2 ~ B
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Vortex liquid
strong (BEC) weak (BCS)pairing
Genuine phases at T=0
Vortex lattice potential energy: Δ0
4
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)
P.N, Phys.Rev.B 79, 144507 (2009)
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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)
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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, 230403 (2007)
M.P.A.Fisher, P.B.Weichman, G.Grinstein, D.S.Fisher;Phys.Rev.B 40, 546 (1989)
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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 E
r
MIT experiment: VC~ 6 E
r
Fluctuation effects?
Critical lattice depth
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Pair density wave
Pair density wave Supersolid without the uniform component Pairing instability in a band-insulator generally occurs at a finite crystal momentum
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PDW evolution
Incommensurate PDW Vertex q-dependence Weak coupling (BCS limit)
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Commensurate PDW Energy q-dependence Strong inter-band coupling Halperin-Rice in p-p
P.N., A. Burkov, A.Paramekanti, Phys.Rev.B 81, 012504 (2010)
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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?
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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
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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”)
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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
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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
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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...)
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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
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