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BCS to BEC Crossover and the Unitary Fermi Gas

1. BCS to BEC Crossover and the Unitary Fermi Gas. Mohit Randeria The Ohio State University Columbus, OH 43210. Pedagogical Lecture at RPMBT Columbus, OH, July 2009. 2. Outline: Introduction to BCS-BEC crossover Two-body scattering: a s Ground state & Excitations

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BCS to BEC Crossover and the Unitary Fermi Gas

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  1. 1 BCS to BEC Crossover and the Unitary Fermi Gas Mohit Randeria The Ohio State University Columbus, OH 43210 Pedagogical Lecture at RPMBT Columbus, OH, July 2009

  2. 2 • Outline: • Introduction to BCS-BEC crossover • Two-body scattering: as • Ground state & Excitations • Transition Temperature Tc & pairing T* • Pairing Pseudogap • Vortices • Critical Velocity • Connections to solid state, • high energy and nuclear physics

  3. 3 • Routes to Strongly Interacting Fermions • in Cold Atom Systems: • Feshbach resonance enhance interactions • 3D BCS-BEC crossover • Optical lattice suppress “kinetic energy” • repulsion >> bandwidth • 2D Hubbard model • “high Tc superconductivity” • Rotation  Quantum Hall physics goal

  4. 4 23 6 Na: BEC; Li: BEC  BCS Experimental Observation of Condensation and Superfluidity in Strongly Interacting Fermi Gases 40 K: BCS  BEC D. Jin group (JILA) W. Ketterle Group (MIT)

  5. 6 6 Fermi Gas Li Experiments K 40 “up” & “down” species: two different hyperfine states e.g. Li  Pairing of “spin up” and “down” fermions interacting via a tunable 2-body interaction: Feshbach Resonance Wide resonance  single channel model with s-wave scattering length as 6 Typical Numbers: Ef ~ 100 nK -1 mK T ~ 0.05 - 0.1 Ef 1/kF ~ 0.3 mm TF radius ~ 100 mm Trap freq. ~ 20 - 100 Hz Experiments: Jin (JILA) Ketterle (MIT) Salomon (ENS) Grimm (Innsbruck) Hulet (Rice) Thomas (Duke)

  6. 7 • Recent Reviews • I. Bloch, J. Dalibard and W. Zwerger, • Rev. Mod. Phys. 80, 885 (2008). • S. Giorgini, L. P. Pitaevskii and S. Stringari, • Rev. Mod. Phys. 80, 1215 (2008). • Ultracold Fermi Gases, Proceedings of the • Varenna ‘Enrico Fermi’ Summer School 2007, • W. Ketterle, M. Inguscio and C. Salomon (editors).

  7. 8 • Outline: • Introduction to BCS-BEC crossover • Two-body scattering: Scattering length as • Ground state & Excitations • Transition Temperature Tc & pairing T* • Pairing Pseudogap • Vortices • Critical Velocity • Connections to condensed matter, • high energy and nuclear physics

  8. Closed channel Open channel 9 Feshbach Resonance: external B field  tune bound state in closed channel & modify the effective interaction in open channel “Wide” resonance: Linewidth a single-channel effective model is sufficient = two-body potential with a variable depth = low-energy description as

  9. 10 Attractive Fermi Gas Hamiltonian: * * Two-body interaction in Dilute gas “g”: Range of V(r) << interparticle distance Low-energy effective interaction: s-wave scattering length Dimensionless Coupling constant

  10. 11 Two-body interaction in Dilute gas  s-wave scattering length as

  11. 12 Two-body problem: Low-energy effective interaction: s-wave scattering length Scattering amplitude T 2-body bound state in vacuum size

  12. 13 Two-body interactions Many-body state: ? Unitarity BEC limit BCS limit

  13. BEC • tightly bound • molecules • pair size • BCS • cooperative • Cooper pairing • pair size pair size Unitarity BCS-BEC Crossover • D. M. Eagles, PR 186, 456 (1969) T=0 variational BCS gap eqn. • A.J. Leggett, Karpacz Lectures (1980) plusm renormalization • Ph. Nozieres & S. Schmitt-Rink, JLTP 59, 195 (1985) diagrams: Tc • C. sa deMelo, MR, J. Engelbrecht, Functional Integral:T*,Tc, TDGL • PRL71, 3202 (1993), PRB 55, 15153 (1997) T=0; gap & collective modes

  14. 15 “Universality”: Unitary Fermi gas Only energy scales Bertsch (2003) Ho (2004) Quantum critical point Sachdev & Nikolic (2007) More generally: Even more generally, away from unitarity:

  15. Outline: • Introduction to BCS-BEC crossover • Two-body scattering: as • Ground state; Excitations; Quantum Fluctns. • Transition Temperature Tc & pairing T* • Pairing Pseudogap • Vortices • Critical Velocity • Connections to condensed matter, • high energy and nuclear physics

  16. Fermions with Attractive interaction: 17 Hubbard-Stratonovich transformation: Mean Field Theory: Uniform, static Saddle-point Gap Equation Number Equation

  17. 18 Leggett-BCS Mean Field Theory at T=0: SadeMelo, MR, Engelbrecht, PRL (93), PRB(97) • MFT Qualitatively correct at T=0: • all the way from Cooper pairs to composite bosons! • will address Quantitative limitations later • Note crossover region:

  18. BCS-BEC crossover: n(k) & length scales healing  length BCS BEC Engelbrecht, MR & Sa de Melo, PRL(1993),PRB(1997)

  19. 20 Energy Gap for Fermionic Excitations: “weak pairing” (BCS regime) “strong pairing” (BEC regime) * Leads to a Phase transition (not a crossover) for non-s-wave pairing!

  20. Bosonic collective excitations: Sound Goldstone mode of broken U(1) in superfluid Gaussian fluctuations about the saddle point BCS limit: (Anderson, Bogoliubov) BEC limit: (Bogoliubov-Beliaev)

  21. 22 Quantitative Limitations of T=0 mean field theory MF Ground state energy too large by ~ 35% compared to QMC & expts • MFT: • Incorrect dimer • scattering length • misses Lee-Yang • correction to g.s. • energy –- quantum • depletion • MFT misses: • Fermi-liquid • corrections to • g.s. energy • (power law in as) • Gorkov, Melik-Barkhudarov • pre-exponential in gap

  22. 23 MFT + Quantum Fluctuations Ground state energy reduced by ~ 35% compared to MFT • improved dimer • scattering length • obtain 94% of • Lee-Yang correction • -- qtm. depletion • Fermi-liquid • corrections to • g.s. energy Diener, Sensarma & MR Phys. Rev A 77, 023626 (2008)

  23. 24 • Equation of state of Fermi gas in • BCS-BEC crossover: • * Include quantum corrections to • Thermodynamic potential • -- zero point motion of collective modes • -- virtual scattering of quasiparticle excitations • * satisfy Goldstone’s theorem • * tame ultraviolet divergences Ground state energy density  “cosmological constant” in field theory

  24. Fluctuations about MF saddle-point Gaussian Approximation Inverse fluctuation propagator: Nambu-Gorkov Green’s function 25

  25. 26 * “Improved” estimate of thermodynamic potential:  mean field plus  Fluctuation contributions Convergence factors * Solve Saddle Point and (new) Number Equation

  26. Saddle Point + Gaussian Fluctuations: Thermodynamic potential MFT quantum corrections • Zero point • motion of • collective modes • Virtual scattering • of quasiparticles regularization p-p channel excitations 27

  27. 28 Equation of State through BCS-BEC Crossover Ground state energy Look at limits in detail …

  28. BCS Limit: continuum dominates collective mode poles Functional integral result equivalent to: Lee-Yang, Galitskii For as > 0 BCS Normal Dilute Fermi Gas energy BCS Condensation Energy ground state energy - = 29

  29. 30 Ground State Energy Density at Unitarity Reduction Due to Quantum Fluctuations

  30. 31 BEC Limit: Zero point Motion of Collective modes Condensate Depletion due to Quantum fluctuations Lee,Yang & Huang (58) 0.94  1 Compare with: Exact 4-body result Petrov, Shlyapnikov & Salomon (04) ; Expt: Innsbruck (04) MFT: Sa deMelo, MR, Engelbrecht (93,97)

  31. 32 MFT Fermi-Fermi Mixture: Unequal mass pairing R. Diener & MR; arXiv (2009) g.s. energy (unitarity) Dimer scattering (BEC limit) MFT exact 1/ N QMC: Gezerlis et al, arXiv (2009) Exact 4-body: Petrov et al(2006) MFT: Iskin & Sade Melo (2007)

  32. 33 Summary: • Beyond Mean Field Theory at T=0: • Include quantum fluctuations • zero point motion of collective modes • virtual scattering of quasiparticles • Results: •  reduction of the ground state energy • at unitarity •  Lee-Yang & Galitskii theory of the • interaction corrections to • g.s. energy in BCS limit • improved estimate of dimer-dimer • scattering and quantum depletion • in BEC limit • These are observable effects in experiments!

  33. Outline: • Introduction to BCS-BEC crossover • Two-body scattering: as • Ground state; Excitations & Qtm. Fluctns. • Transition Temperature Tc & pairing T* • Pairing Pseudogap • Vortices • Critical Velocity • Connections to condensed matter, • high energy and nuclear physics

  34. 35 BCS-BEC crossover Saha ionization T*: Pairing temperature saddle-point Tc: Phase Coherence saddle-point + Gaussian fluctuations BEC BCS Sa de Melo, MR & Engelbrecht, PRL 71, 3202 (1993)

  35. Transition Temperature: 36 “Tc” at which trivial saddle pt. D=0 becomes unstable Number Equation Pairing T* MFT: Including Gaussian Fluctns. Phase Transition Tc

  36. Tc, correlation length, Ginzburg region Sa de Melo, MR & Engelbrecht, PRL (1993) & PRB (1997)

  37. 38 BCS-BEC Experiments Quantum Monte Carlo Experimental data: K: Regal, Greiner & Jin, PRL (‘04) Li: Zwierlein, et al., PRL (‘04) analysis: Diener & Ho, cond-mat (‘04) Theoretical Tc: Sa deMelo, MR, Engelbrecht, PRL (‘93) Burovski et al, PRL (2008)

  38. 39 * Pairing pseudogap: MR, Trivedi, Moreo, Scalettar PRL (92) Trivedi & MR, PRL (95) * Based on Sa de Melo, MR & Engelbrecht, PRL (1993) from: Sa de Melo, Phys.Today (Oct. 2008)

  39. Outline: • Introduction to BCS-BEC crossover • Two-body scattering: as • Ground state & Excitations • Transition Temperature Tc & pairing T* • Pairing Pseudogap • Vortices • Critical Velocity • Connections to solid state, • high energy and nuclear physics

  40. T*/ t 41 BCS-BEC in Attractive Hubbard Model (2D) • Differences with continuum: • CDW + SC at half-filling; only SC away from it • BEC limit: boson hopping Kosterlitz-Thouless Tc Tc/ t r = 0.7 |U|/t QMC: Paiva, Scalettar, MR & Trivedi, arXiv (2009)

  41. T/ t Normal Bose Liquid ? Normal Fermi Liquid |U|/t 42 “Normal” State Crossover Answer: Pairing Pseudogap in Single-particle spectrum & Spin Correlations in a highly degenerate Fermi system

  42. 43 Pairing Pseudogap in 2D Attractive Hubbard Model QMC: MR, Trivedi, Moreo & Scalettar, PRL 69, 2001 (1992) Trivedi & MR, PRL 75, 381 (1995)

  43. Outline: • Introduction to BCS-BEC crossover • Two-body scattering: as • Ground state & Excitations • Transition Temperature Tc & pairing T* • Pairing Pseudogap • Vortices • Critical Velocity • Connections to solid state, • high energy and nuclear physics

  44. 45 Quantized Vortices in Rotating Superfluid Fermi Gases 6 Li Fermi gas through a Feshbach Resonance M.W. Zwierlein et al., Nature, 435, 1047, (2005)

  45. 46 Unitary Fermi gas: Most strongly interacting superfluid Qs: Does this lead to any “striking” behavior in a physical observable at unitarity? • Structure of a Vortex through BCS-BEC crossover • Max Critical Current at Unitarity R. Sensarma, MR & T. L. Ho, PRL 96, 090403 (2006)

  46. 47 How does a vortex evolve through the crossover? Motivation: healing length & pair size BCS BEC unitarity Sa de Melo, MR & Engelbrecht, PRL (1993) & PRB (1997)

  47. 48 Bogoliubov-DeGennes (BdG) Theory: Mean-field th’y: spatially varying order parameter & density T=0 Self-consistency conditions Gap and Number equations for crossover vortex

  48. BCS limit (cf. GL theory) • Two length scales! • initial rise: • approach • on scale: At Unitarity: single length scale 49 Self-consistent solution of BdG Eqns. Order Parameter at T=0

  49. 50 Density Profiles: BCS limit: Core density ~ n Unitarity: Core density depleted BEC limit: “Empty” core o.p. ~ density Vortices much easier to Image in BEC regime

  50. Current Flow around a vortex: “vortex size” & Critical current: 51

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