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