BCSL 1

Application of BCS-like Ideas to Superfluid 3-He

Tony Leggett

University of Illinois at Urbana-Champaign BCSL 2

Electrons in Metals (BCS):

4 Fermions of spin ½, TKTKFc∼∼10 , 10 ⇒ strongly degenerate at onset of superconductivity Normal state: in principle described by Landau Fermi-liquid theory, but “Fermi-liquid” effects often small and generally very difficult to see. BCS: model normal state as Weakly interacting gas with weak “fixed” attractive interaction Liquid 3He: −3 also fermions of spin ½ TKTKFc∼∼1, 10 ⇒ again, strongly degenerate at onset of superfluidity Normal state: must be described by Landau Fermi-liquid theory, Fermi-Liquids effects very strong. (e.g. Wilson ratio ∼4) ⇒ low-lying states (inc. effects of pairing) must be described in terms of Landau quasiparticles.

What is Common: 2-particle density matrix has single macroscopic (~N) eigenvalue, with associated eigenfunction

FF()(:)rr1212σ σσσ≡ R r 12 “wave function of Cooper pairs”

(for r ,, σ 12 σ fixed: GL “macroscopic wave function Ψ(R)) BCSL 3

STRUCTURE OF COOPER-PAIR WAVE FUNCTION

(in original BCS theory of superconductivity, for fixed R, σ1, σ2)

⎛⎞ −1 FF||()rr==Δ⎜⎟ ∑(2Eik ) exp kr⋅ ⎝⎠ k

2 12/ ⎛⎞2 Energy gap ⎜⎟εk +Δ|| ⎝⎠

⎛⎞12/ sin krF ≅Δconst.N⎜⎟ /EF Ω exp −r / ξ ⎝⎠ kFr ξυ= "pair radius"~ /~AΔ 104 F ( )

“Number of Cooper pairs” (N ) = normn of F(r) −1 o a ~k B F ~ξ

n “Number of Cooper pairs” (No) = norm of F(r) N 22Δ 1 ≡ |F()r |2 d~r~ ξ N/Δ E ~10−4N ∫ 22 ()F Ω EkFF cf: N/N~10% in 4He ()0 ______In original BCS theory of superconductivity, 1 F:()r σσ1 2 = ↑↓ −↓↑ F ()r 2 ( 12 12) spin singlet orbital s-wave ⇒PAIRS HAVE NO “ORIENTATIONAL” DEGREES OF FREEDOM (⇒stability of supercurrents, etc.) BCSL 4 THE FIRST ANISOTROPIC COOPER-PAIRED SYSTEM: SUPERFLUID 3HE solid 2-PARTICLE DENSITY MATRIX ρ 2 A still has one and only one macroscopic eigenvalue ↑ B ⇒ can still define “pair wave A N function” F(R,r:σ1σ2 ) However, even when FF ≠ ( R ) , T→ 3mK

F ()rσ 22σ HAS ORIENTATIONAL DEGREES OF FREEDOM!

(i.e. depends nontrivially on ˆr, σ 12σ )

All three superfluid phases have =S =1 ______A phase (“ABM”) Spin triplet

1 F ()rr:fσσ12=↑↓↓↑112x + × () 2 ( )dˆ char. “spin axis”

ff()rr=×⋅sin θ exp iϕ o ()()ˆ char. “orbital axis”

Properties anisotropic in orbital and spin space separately, e.g. ˆˆk k ˆ ˆ! Δ=KoΔ=Δ×( ) ⇐ nodes at ±

2 ⎛⎞⎛⎞⎛⎞ WHAT IS TOTAL ANG. MOMENTUM? ⎜⎟N/N⎜⎟⎜⎟ΔΔ /EFF /N /E ? ⎝⎠⎝⎠⎝⎠ BCSL 5

B phase (“BW”) For any particular direction nˆ (in real or k-space) can always choose spin axis s.t. 1 F:nˆ ~ ()σσ12 ↑↓12 +↓↑ 12 2 ( )dˆ i.e. ddnˆˆ = () ˆ . Alternative description:

3 o BW phase is Po state “spin-orbit rotated” by 104 .

-1 L=S=J=O because of dipole force cos (-1/4)=θo Note: rotation (around axis ωˆ ) breaks P but not T

ext field Ho inversion time reversal Orbital and spin behavior individually isotropic, but: properties involving spin-orbit correlations anisotropic! Example: NMR dipole energy dES δ =×SHo + D Ho,ωˆ dt δθ ↑Hrf (long) ∠of rotation about rf field Hrf (transverse) ˆ direction Hrf ˆ In transverse resonance, rotation around H rf equiv. rotation of ωˆ with θo unchanged ⇒ No dipole torque.

In longitudinal resonance, rotation changes θo ⇒ finite-frequency resonance! BCSL 6 BCSL 7

“EXOTIC” PROPERTIES OF SUPERFLUID 3HE

A. Orientation const. in space, varying in time: — spin dynamics (NMR) — orbital dynamics (“normal locking”) (A phase) — effect of macroscopic ang. momentum? (A phase) ______B. Orientation const. in time, varying in space — spin textures (3He-A) ()dlˆˆ= ± in equation ˆ ↑↑↑↑↑↑↑↑↑ ˆ d ↑↑ ↓↓↓ (carries spin current)

— orbital textures — topological singularities (boojums, “half-quantum” vortices. . . .) — instability of supercurrents in 3He-A. ______C. Orientation varying in both space and time — spin waves — orbital waves —”flapping” and “clapping” modes D. Amplification of ultra-weak effects BCSL 8 BCSL 9 BCSL 10