1. The Ohio State University
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. ~ Hz
N ~ 10
Ef ~ 100 nK -1 mK
T ~ 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. external B field  tune bound state in closed channel
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. effective interaction: s-wave scattering length
Two-body problem:
Low-energy
effective interaction:
s-wave scattering length
2-body bound state
in vacuum
size
as B field increases
decreases

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

8. BCS-BEC Crossover BEC BCS tightly bound cooperative molecules
Cooper pairing
pair size
BEC
tightly bound
molecules
pair size
 B
D. M. Eagles, PR 186, 456 (1969) T=0 variational BCS gap eqn.
A.J. Leggett, Karpacz Lectures (1980) plus m 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. Trivedi pseudogap;
2-dimensions

9. BCS to BEC crossover at T=0 “gap” D chemical potential m
momentum distribution n(k)
collective modes
Crossover:
Engelbrecht, MR & Sa de Melo, PRB 55, (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, (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, (2004)
Li: M. Zwierlein, et al., PRL 92, (2004) 40 6
analysis of projection: R. Diener and T. L. Ho, cond-mat/
Theoretical Tc: C. Sa deMelo, MR, J. Engelbrecht, PRL 71, 3202 (1993)

13. Only scales in the problem: Energy & Length
“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
d-wave Tc
s-wave
Superfluid
0.2
Carrier (hole)
concentration
BEC
BCS
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. Repulsive interactions d-wave pairing near Mott transition
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 normal states
- 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. brief review of BCS-BEC crossover pseudogap vortex structure
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, (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. brief review of BCS-BEC crossover pseudogap vortex structure
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, (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. Energy Gap v/s. D in BCS-BEC crossover:
Recall:
Energy Gap v/s. D in 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. brief review of BCS-BEC crossover pseudogap vortex structure
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, (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
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. c ~ N(0) both strongly T-dep dn/dm very weakly T-dep
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)