BCS - BEC Crossover: Pseudogap, Vortices & Critical Current

1. BCS - BEC Crossover: Pseudogap, Vortices &Critical Current Mohit Randeria The Ohio State University Columbus, OH 43210, USA Nordita, June 2006

2. Outline: • review of BCS-BEC crossover theory • pseudogap • vortex structure • fermionic bound states in vortex core • critical current  unitary gas is the most robust superfluid

3. Two routes to Strongly Interacting Fermions • in Cold Atom Systems: • Feshbach resonance enhance interactions • attraction > Ef • 3D BCS-BEC crossover • Optical lattice suppress “kinetic energy” • repulsion >> bandwidth • 2D Hubbard model • high Tc “superconductivity” • Feshbach Resonance + Optical lattice goal

4. 6 Fermi Atoms: Li 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 6 Typical Numbers: Trap freq. ~ 20 - 100 Hz N ~ 10 Ef ~ 100 nK -1 mK T ~ 0.05 - 0.1 Ef 1/kF ~ 0.3 mm TF radius ~ 100 mm Experiments: Jin (JILA) Ketterle (MIT) Grimm (Innsbruck) Hulet (Rice) Thomas (Duke) Salamon (ENS) 6

5. Feshbach Resonance: external B field  tune bound state in closed channel & modify the effective interaction in open channel Closed channel Open channel adapted from Ketterle group (MIT) “Wide” resonance: Linewidth a single-channel effective model is sufficient

6. Two-body problem: Low-energy effective interaction: s-wave scattering length 2-body bound state in vacuum size asB field increases decreases

7. Many-body Problem: Dilute gas: range << interparticle distance  Low-energy effective interaction: BCS limit Unitarity BEC limit Strongly Interacting regime Dimensionless Coupling constant

8. BCS-BEC Crossover • BEC • tightly bound • molecules • pair size • BCS • cooperative • Cooper pairing • pair size  B • 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) diagrammatic • theory of Tc • M. Randeria, in “Bose Einstein Condensation” (1995) T*,Tc, T=0; • with C. sa deMelo, J. Engelbrecht; and N. Trivedipseudogap; • 2-dimensions

9. BCS to BEC crossover at T=0 • “gap”D • chemical potentialm • momentumdistribution n(k) • collective modes Crossover: Engelbrecht, MR & Sa de Melo, PRB 55, 15153 (1997)

10. Functional Integral Approach: 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)

11. How reliable is “saddle-point + Gaussian fluctuations”? Effect of (static) 4th order terms  Ginzburg criterion Sa de Melo, MR & Engelbrecht, PRL 71, 3202 (1993) PRB 55, 15153 (1997)

12. Comparison between Theory & Experiment: “Condensate fraction” measured on molecular (BEC) side after rapid sweep from initial state  `Projection’ Experimental data: K: C. A. Regal, M. Greiner, and D. S. Jin, PRL 92, 040403 (2004) Li: M. Zwierlein, et al., PRL 92, 120403 (2004) 40 6 analysis of projection: R. Diener and T. L. Ho, cond-mat/ 0401517 Theoretical Tc: C. Sa deMelo, MR, J. Engelbrecht, PRL 71, 3202 (1993)

13. “Universality” for Only scales in the problem: Energy& Length Bertsch - Baker (2001); K. O’Hara et al., Science (2002); T. L. Ho, PRL (2004). Mean field theory* + fluctuations At unitarity: Monte Carlo** BEC limit: exact 4-body result! *C. Sa deMelo, MR, J. Engelbrecht, PRL (1993) & PRB (1997) Petrov, Shlyapnikov & Salamon, PRL (2003) ** T= 0 QMC: J. Carlson et al. PRL (2003); G. Astrakharchik et al. PRL(2004) T>0 QMC: A. Bulgac et al., (2005); E. Burovski et al., (2006); V. Akkineni, D.M. Ceperley & N. Trivedi (2006).

14. Outline: • brief review of BCS-BEC crossover • pseudogap • vortex structure • fermionic bound states in vortex core • critical current

15. Landau’s Fermi-Liquid Theory: • Strongly Interacting  Weakly-interacting • Normal Fermi systems Quasiparticle gas • e.g., He3; electrons in metals; heavy fermions • BCS theory: pairing instability in a normal Fermi-liquid • Qualitatively new physics in Strongly Interacting Fermions: • * Breakdown of Landau’s Fermi-liquid Theory • e.g., • Normal states of High Tc cuprate superconductors • pseudogap in BCS-BEC crossover • * Superconductivity/fluidity is not • a pairing instability in a normal Fermi liquid.

16. Breakdown of Fermi-liquid theory: Crossover from to Normal Fermi Gas Normal Bose Gas Pseudogap:Tc < T < T* Pairing Correlations in a degenerate Fermi system M. Randeria et al., PRL (1992) N. Trivedi & MR, PRL (1995) • pairing gap in above Tc • strong T-dep. suppression of • spin susceptibility above Tc • no anomalous features in Pseudo -gap

17. Strongly correlated non-Fermi-liquid superconductors normal states High Tc Cuprates Cold Fermi Gases T* T normal Bose gas Pseudo -gap Fermi Liquid Tc d-wave s-wave Superfluid 0 0.2 0 Carrier (hole) concentration BCS BEC • low-energy pseudogap • high-energy pseudogap • strange metal: w/T scaling • Spin-Charge separartion? M. Randeria in “Bose Einstein Condensation” (1995) & Varenna Lectures (1997).

18. High Tc SC in cuprates • Highest known Tc (in K) • * electrons • Repulsive interactions • d-wave pairing • near Mott transition • competing orders: AFM,CDW • repulsion U >> bandwidth • x ~ 10 A • Tc ~ rs << D • Mean-field theory fails • anomalous normalstates • - strange metal & pseudogap • Breakdown of Fermi-liquid theory • Spin-charge separation? • BCS-BEC crossover • Highest known Tc/Ef ~ 0.2 • * cold Fermi atoms • Attractive interactions • s-wave pairing • only pairing instability • attraction > Ef • x ~ 1/kf • Tc ~ rs << D • Mean-field theory fails • pairing pseudogap

19. Outline: • brief review of BCS-BEC crossover • pseudogap • vortex structure • fermionic bound states in vortex core • critical current R. Sensarma, MR & T. L. Ho, PRL 96, 090403 (2006) See also: N. Nygaard et al., PRL (2003); Bulgac & Y. Yu, PRL(2003). M. Machida & T. Koyama, PRL (2005); K. Levin et al, cond-mat (2005)

20. Vortices in Rotating Fermi Gases Quantized vortices  unambiguous signature of superfluidity 6 Li Fermi gas through a Feshbach Resonance M.W. Zwierlein et al., Nature, 435, 1047, (2005)

21. Bogoliubov-DeGennes Theory: mean field theory with a spatially-varying order parameter (can also include external trapping potential; not included here) T=0 Self-consistency: vortex

22. Order Parameter Profile at T=0: At Unitarity: the two scales merge • BCS limit (cf. GL theory) • Two length scales! • initial rise: • (analytical result) • approach to • on scale:

23. Density Profiles: BCS limit: Core density ~ n Unitarity: Core density depleted BEC limit: “Empty” core order parameter ~ density

24. Outline: • brief review of BCS-BEC crossover • pseudogap • vortex structure • fermionic bound states in vortex core • critical current R. Sensarma, MR & T. L. Ho, PRL 96, 090403 (2006)

25. Fermionic Bound States in the Vortex Core: Theoretical prediction (BCS limit): C. Caroli, P. deGennes, J. Matricon, Phys. Lett. 9, 307 (1964) STM Expts. NbSe2: H. Hess et al., PRL (1989). STM: Davis group (Cornell) D0 D(r) “Andreev” bound states in the core: “minigap” & spacing r Very low-energy excitations in vortex core

26. Spectrum of Fermionic Excitations at unitarity continuum Bound states: Core states “edge” states Minigap follows C-dG-M predictions Through unitarity!

27. Recall: Energy Gapv/s.Din BCS-BEC crossover: Leggett (1980) MR, Duan, Shieh (1990)

28. Fermionic Excitations in BEC regime Fermion bound state in Vortex core persists into molecular BEC regime! continuum E Bound state! probe bound states via RF spectroscopy

29. Bound state wavefunctions

30. Outline: • brief review of BCS-BEC crossover • pseudogap • vortex structure • fermionic bound states in vortex core • critical current  unitary gas is the most robust superfluid R. Sensarma, MR & T. L. Ho, PRL 96, 090403 (2006) and unpublished

31. Qs: Is there anything “special” about the unitary superfluid? • max • but similar for all • superfluid density • (Gallilean invaraince) for all • (analog of ) • hard to define – centrifugal effects • critical velocity Vc: non-linear response to flow

32. Current Flow around a vortex: dependence?

33. Vortex Core Size from Current flow from Engelbrecht, MR & Sa de Melo, PRB (1997)  BCS BEC 

34. Current Flow around a vortex: Critical current:

35. The unitary gas is the most robust Superfluid • max Tc ~ 0.2Ef(but similar for all 1/kfas > 0) • max critical velocity: BCS limit: Vc  Pair breaking BEC limit: Vc  Collective modes Landau Criterion:

36. Conclusions: • single-channel model (interaction  as) • sufficient for wide resonances in Fermi gases • “mean-field theory + fluctuations” is qualitatively • correct for BCS-BEC crossover, • but no small parameter near unitarity • pairing pseudogap: breakdown of Fermi-liquid theory • Vortices evolve smoothly through crossover • Order Parameter, density & current profiles, Fermion bound states • Fermionic bound states exist even on BEC side • Critical velocity is nonmonotonic across resonance • Unitary gas is the most robust superfluid

37. The end

38. Pseudogap in 2D Attractive Hubbard Model Degenerate “normal” Fermi system Tc ~ 0.05t < T < t for |U| = 4t m(T,U) + Un/2 + 4 > T Randeria, Trivedi, Moreo & Scalettar, PRL 69, 2001 (1992) Trivedi & Randeria, PRL 75, 381 (1995)

39. Pseudogap  Anomalous Spin Corelations • dc/dT > 0 • 1/(T1T) T-dep • 1/(T1T) ~ c(T) Randeria, Trivedi, Moreo & Scalettar, PRL 69, 2001 (1992)

40. Pseudoagap: Compressibility looks ordinary Spin susceptibility reflects one-particle Energy gap • c ~ N(0) both strongly T-dep • dn/dm very weakly T-dep Trivedi & Randeria, PRL 75, 381 (1995)