Exact relations for few-body and many-body problems with short-range interactions. Félix Werner UMass (Amherst)

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1 Exact relations for few-body and many-body problems with short-range interactions Félix Werner UMass (Amherst Yvan Castin Ecole Normale Supérieure (Paris [arxiv 00]

2 -body scattering state: f k = k 0 Φ k (r = r a + ik k r e +... e ik r + f k e ikr r +... a Φ k 0(r r a + r A. FERMIONS Spin / N + N = N Zero-Range Model: ψ(r,..., r, r }{{ N } N +,..., r N }{{} spin ψ(r,..., r N = rij 0 N i= [ m r i + U(r i ( ] r ij a ψ = E ψ spin A ij (R ij, (r k k i,j + O(r ij r i + r j

3 Exact relations [Tan 008]: for any eigenstate: de d( /a = 4π m ( (A (, A ( i<j k i,j (A, A where d d r k d d R ij (A ( ij A( ij (R ij, (r k k i,j Lemma: ψ : a, A ( { ψ : a, A ( ψ, Hψ Hψ, ψ = 4π m ( a a (A (, A ( [ Ostrogradsky s Thm.]

4 C lim k k4 n σ (k = (4π (A, A d 3 k (π 3 n σ(k = N σ. Derivation à la Olshanii-Dunjko [003, in D]: n (k = N i= ( l i dr l dr i e ik r i ψ(r,..., r N }{{} k j,j i r ij A ij (r j, (r l l i,j.

5 g ( (r, r ˆn (r ˆn (r 3 d 3 R g ( ( R + r, R r r 0 (A, A r 4 E E trap = C 4πma + σ d 3 k k (π 3 m [n σ (k Ck 4 ] N i= U(r i Numerical verification of exact relations for 4 fermions [Daily&Blume 009]

6 Contact measurements for balanced Fermi gas at low T in a trap 0 C / (N k F 0 - FNMC [Trento] g ( [Hu et al.] n(k [Stewart et al.] #closed-channel molecules [Partridge et al. + Werner et al.] - 0 /(k a F

7 C also appears in radiofrequency spectra Exp: [Stewart, Gaebler, Drake & Jin 00] Theo: [Punk&Zwerger 007] [Baym, Pethick, Yu&Zwierlein 007] [Pieri, Perali&Strinati 009] [Schneider, Shenoy&Randeria 009] [Haussmann, Punk&Zwerger 009] [Braaten, Kang&Platter 00] [Schneider&Randeria 00]

8 Some new relations d E n d( /a = ( 4π m (A(n, A (n E n E n n,e n E n C (/a > 0 at constant T or S

9 3 ( dˆr d d R ˆψ 4π σ R r ( ˆψσ R + r = N C r 0 4π r m ( 3 E E trap C r πma σ=, 4 ( E r e a = π i<j d 3 R ( k i,j E + 4m R + m d 3 r k A ij (R, (r k k i,j k i,j rk N l= U(r l A ij (R, (r k k i,j generalises [Efimov 993] [ Ostrogradsky s Thm. + effective range model]

10 generalisations to D and to finite-range interactions: see arxiv

11 B. BOSONS Spinless Efimov effect additional boundary condition in Zero-Range Model: ψ(r,..., r N R 0 R sin [ s 0 ln RRt ] Φ(Ω B(C, r 4,..., r N R ( r + r 3 + r 3 /3 R t = 3-body parameter (directly related to binding energy of Efimov trimers s 0 = i is imaginary solution of ( s cos s π + 8 ( sin s π = Φ(Ω = hyperangular part of wavefunction of Efimov trimer

12 ( de d( /a d 3 R g ( Rt = 4π m (A, A ( R + r, R r r 0 (A, A r 3 ( E ln(r t a = m 3 3 s 0 N(N (N dc dr 4... dr N B(C, r 4,..., r N [ Ostrogradsky s Thm.] 4 Virial Thm. [Werner 008]: (E E trap = N i= + a r i U( r i ( E (/a R t ( E ln(r t a

13 For Efimov trimer: C n(k = k k 4 + D [ ] k 5 cos s 0 ln(k 3/κ 0 + ϕ +... E trimer = κ0 m d 3 k k (π 3 m [n(k Ck 4 ] diverges conjecture : D ( E ln(r t a

14 s cos ( s π + η 4 3 sin 3 particles ( s π 6 = 0 a = where η = for bosons, for fermions Efimov trimers: wavefunction known [Efimov 970] ( E ( /a In isotropic harmonic trap: wavefunctions known [Jonsell, Heiselberg&Pethick 00; Tan 004; Werner&Castin 005] (universal states; l = 0; q = 0 E ( /a ( E r e a = = R t = 3 ω 8m 3 ω m E m Γ(s + cos ( s π π tan(sπ sin ( s π s π sin ( s π 4π 3 cos ( s π 3 6 Γ(s + s sin ( s π [ Γ(s + cos ( s π s π sin ( s π η π 3 cos ( ] s π 3 6 Γ(s s(s sin ( s π [ cos ( s π + s π sin ( s π + η π 3 cos ( s π 3 6 Agreement with numerical results of: Braaten & Hammer; von Stecher, Greene & Blume; Werner & Castin ]

15 Critical Temperature of Unitary Gas T c / T F Continuous space model 0.05 Hubbard model [Burovski et al., 006] k F r e T c T F k F r e % for 6 Li 6% for 40 K (TBC