The Ohio State University BCS - BEC Crossover: Pseudogap, Vortices & Critical Current Mohit Randeria The Ohio State University Columbus, 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 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

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

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

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

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

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

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, 15153 (1997)

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)

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)

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)

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).

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

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.

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

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).

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

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

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: At Unitarity: the two scales merge BCS limit (cf. GL theory) Two length scales! initial rise: (analytical result) approach to on scale:

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

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, 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). STM: Davis group (Cornell) D0 D(r) “Andreev” bound states in the core: “minigap” & spacing r Very low-energy excitations in vortex core

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

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)

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

Bound state wavefunctions

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, 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?

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

Current Flow around a vortex: Critical current:

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:

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

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)

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

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)