Post on 12-Jan-2016
description
BCS - BEC Crossover:
Pseudogap, Vortices
& Critical CurrentMohit Randeria
The Ohio State UniversityColumbus, OH 43210, USA
Nordita, June 2006
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
Two routes to Strongly Interacting Fermions in Cold Atom Systems:
• Feshbach resonance enhance interactionsattraction > Ef
3D BCS-BEC crossover
• Optical lattice suppress “kinetic energy” repulsion >> bandwidth 2D Hubbard model high Tc “superconductivity”
• Feshbach Resonance + Optical lattice
goal
Fermi Atoms: Li K
6
40
Experiments:Jin (JILA)Ketterle (MIT)Grimm (Innsbruck)Hulet (Rice)Thomas (Duke)Salamon (ENS)
Typical Numbers:Trap freq. ~ 20 - 100 HzN ~ 10Ef ~ 100 nK -1 KT ~ 0.05 - 0.1 Ef1/kF ~ 0.3 mTF radius ~ 100 m
“up” & “down” species: two different hyperfine statese.g. Li
Pairing of “spin up” and “down” fermions interactingvia a tunable 2-body interaction: Feshbach Resonance
6
6
Feshbach Resonance:external B field tune bound state in closed channel & modify the effective interaction in open channel
Openchannel
Closedchannel
adapted from Ketterle group (MIT)
“Wide” resonance: Linewidth a single-channel effective model is sufficient
2-body bound state in vacuum size
Two-body problem:
Low-energyeffective interaction:s-wave scattering length
as B field increases decreases
BCS limit
BEC limit
Unitarity
Many-body Problem:
Dilute gas: range << interparticle distance
Low-energy effective interaction:
Dimensionless Coupling constant
Strongly Interacting regime
BCS• cooperative Cooper pairing• pair size
BEC• tightly bound molecules• pair size
• D. M. Eagles, PR 186, 456 (1969) T=0 variational BCS gap eqn.
• A.J. Leggett, Karpacz Lectures (1980) plus 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
BCS-BEC Crossover
B
BCS to BEC crossover at T=0• “gap” • chemical potential • momentum distribution n(k)• collective modes
Engelbrecht, MR & Sa de Melo, PRB 55, 15153 (1997)
Crossover:
T*: Pairing temperature saddle-point
BCS
BEC
Saha ionization
Sa de Melo, MR & Engelbrecht, PRL 71, 3202 (1993)
Tc:Phase Coherence saddle-point + Gaussian fluctuations
Functional Integral Approach:
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)
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)
406
Theoretical Tc: C. Sa deMelo, MR, J. Engelbrecht, PRL 71, 3202 (1993)
Comparison between Theory & Experiment:“Condensate fraction” measured on molecular (BEC) side after rapid sweep from initial state `Projection’
analysis of projection: R. Diener and T. L. Ho, cond-mat/ 0401517
*C. Sa deMelo, MR, J. Engelbrecht, PRL (1993) & PRB (1997)
** 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).
“Universality” forOnly scales in the problem: Energy & Length
Bertsch - Baker (2001); K. O’Hara et al., Science (2002); T. L. Ho, PRL (2004).
At unitarity: Monte Carlo**
BEC limit:
Petrov, Shlyapnikov & Salamon, PRL (2003)
exact4-bodyresult!
Mean field theory* + fluctuations
Outline:
• brief review of BCS-BEC crossover
• pseudogap
• vortex structure
• fermionic bound states in vortex core
• critical current
Qualitatively new physics in Strongly Interacting Fermions: * Breakdown of Landau’s Fermi-liquid Theorye.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.
• Landau’s Fermi-Liquid Theory:
Strongly Interacting Weakly-interactingNormal Fermi systems Quasiparticle gas
e.g., He3; electrons in metals; heavy fermions
• BCS theory: pairing instability in a normal Fermi-liquid
Breakdown of Fermi-liquid theory:
Crossover from toNormal Fermi Gas Normal Bose Gas
Pseudogap: Tc < T < T* Pairing Correlations in a degenerate Fermi system
Pseudo -gap
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
Carrier (hole)concentration
d-wave
T*
BEC BCS
Tc Fermi Liquid s-wave
Superfluid
Pseudo -gap
High Tc Cuprates Cold Fermi Gases
0 0 0.2
M. Randeria in “Bose Einstein Condensation” (1995) & Varenna Lectures (1997).
normal Bose gas
Strongly correlated non-Fermi-liquid superconductors normal states
• low-energy pseudogap• high-energy pseudogap• strange metal: scaling Spin-Charge separartion?
T
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 • 10 A• Tc ~ s << • Mean-field theory fails• anomalous normal states - strange metal & pseudogap Breakdown of Fermi-liquid theorySpin-charge separation?
BCS-BEC crossover• Highest known Tc/Ef ~ 0.2 * cold Fermi atoms
• Attractive interactions• s-wave pairing• only pairing instability
• attraction > Ef• kf• Tc ~ s << • Mean-field theory fails• pairing pseudogap
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)
Vortices in Rotating Fermi Gases
M.W. Zwierlein et al., Nature, 435, 1047, (2005)
Li Fermi gas through a Feshbach Resonance6
Quantized vortices unambiguous signature of superfluidity
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
Order Parameter Profile at T=0:
BCS limit (cf. GL theory)
Two length scales!• initial rise: (analytical result)
• approach to on scale:
At Unitarity:
the two scales merge
Density Profiles:
BCS limit: Core density ~ n
Unitarity:Core densitydepleted
BEC limit:“Empty” coreorder parameter ~ density
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)
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).
0
(r)
“Andreev” bound states in the core: “minigap” & spacing
r
STM: Davis group (Cornell)
Very low-energy excitations in vortex core
Spectrum of Fermionic Excitations
at unitarity
continuum
Bound states:Core states“edge” states
Minigap followsC-dG-Mpredictions Through unitarity!
Leggett (1980)MR, Duan, Shieh (1990)
Energy Gap v/s. in BCS-BEC crossover:
Recall:
Fermionic Excitations in BEC regime
E
continuum
Bound state!
Fermion bound state in Vortex core persists into molecular BEC regime!
probe bound states viaRF spectroscopy
Bound state wavefunctions
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
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
Current Flow around a vortex:
dependence?
fromEngelbrecht, MR & Sa de Melo,PRB (1997)
Vortex Core Size from Current flow
BEC BCS
Current Flowaround a vortex:
Critical current:
• 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:
The unitary gas is the most robust Superfluid
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
The end
Randeria, Trivedi, Moreo & Scalettar, PRL 69, 2001 (1992)Trivedi & Randeria, PRL 75, 381 (1995)
(T,U) + Un/2 + 4 > T
Degenerate “normal” Fermi system
Tc ~ 0.05t < T < t for |U| = 4t
Pseudogap in 2D Attractive Hubbard Model
Randeria, Trivedi, Moreo & Scalettar, PRL 69, 2001 (1992)
• d/dT > 0
• 1/(T1T) T-dep
• 1/(T1T) ~ (T)
Pseudogap AnomalousSpin Corelations
Trivedi & Randeria, PRL 75, 381 (1995)
• N(0) both strongly T-dep
• dn/d very weakly T-dep
Pseudoagap: Compressibility looks ordinary Spin susceptibility reflects one-particle Energy gap