strongly correlated cooper pair insulators and...

Strongly correlatedCooper pair insulators and

superfluids

Predrag Nikolić

George Mason University

Strongly correlated Cooper pair insulators and superfluids 2/33

Acknowledgments

Affiliations and sponsors

W.M.Keck Program inQuantum Materials

Collaborators

Subir Sachdev Anton Burkov Arun ParamekantiEun-Gook Moon

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


Unitarity: two-body picture

Universality: irrelevant microscopic details Two-body resonant scattering Bound state at zero energy

,

,

,


Unitarity: many-body picture

Universality Quantum critical point

P.N., S.Sachdev; Phys.Rev. A 75, 033608 (2007)


Theoretical Approaches Mean-field approximation Perturbation theory Renormalization group

Perturbation theory

Action: SP(2N ), imaginary time

Feynman diagrams

physical atom(fermion)

Cooper pair, molecule(boson)

vertex


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

=


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

Strongly correlated Cooper pair insulators and superfluids

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)


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)


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?


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


strongly correlated cooper pair insulators and...

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