Super Efimov effect. Sergej Moroz University of Washington. together with Yusuke Nishida and Dam Thanh Son. Tuesday, April 1, 14

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1 Super Efimov effect together with Yusuke Nishida and Dam Thanh Son Sergej Moroz University of Washington

2 Few-body problems They are challenging but useful: Newton gravity Quantum atoms Quantum molecules Born-Oppenheimer perturbation theory, chaos variational Hartree-Fock Efimov effect is new entry

3 Few-body universality Low energies, short-range interactions in 3d: scattering length a,... Universal regime: a other length scales Two-body bound state near resonance E = 2 ma 2 a>0

4 Efimov problem Three bosons near resonance: Universality At resonance as n E (n+1) T E (n) T e 2 /s 0 40 years later

5 Basics intuition How can short-range forces create infinite number of bounds states? Born-Oppenheimer approximation: Window of universality Due to separation of scales long-range effective potential! Scale-invariant potential

6 Energy spectrum V (R) = 1/4 +s2 0 R 2 V R Landau&Lifshitz: Fall to center for strong attraction s 0 > 0 Semiclassical solution E n = s2 R 2 0 exp 2 n s + Efimov geometric spectrum

7 Experimental signatures Three-body loss: ṅ = L 3 n 3 Florence 39 K enhanced when trimers merge with atom threshold Bar Ilan 7 Li First experiment Innsbruck 2006 Rice 7 Li

8 Beyond standard model? d =3 d =2 d =1 No scale invariant two-body attraction away from 3d s-wave! Efimov effect was liberated from 3d Nishida, Tan

9 Beyond standard model? d =3 d =2 d =1 No Efmov effect, but...

10 p-wave in 2d Square well solution dj l (kr)/dr J l (kr) = dk l( r)/dr K l ( r) In p-wave critical attraction needed 0 r 2 0 =5.784 Normalized wave-function (r) = 2 K 1 ( r) No scale invariance Point-like boson as r 0 0 ln( r 0 )

11 This talk p-wave Few-body quantum physics of resonantly interacting fermions in flatland

12 Superfluids T=0 state of a neutral many-body system No dissipation, quantum vortices,... Old: 4 He and 3 He New: Bose and Fermi ultracold atoms

13 P-wave superfluids From mean-field: Volovik, Read, Green,... Chiral condensate p =(p x ± ip y ) ˆ preferred Topological phase transition at µ =0 Chiral Majorana modes on boundaries Toy model for a film of 3 Sometimes mean-field is not good enough near resonance!

14 Super Efimov effect At resonance near threshold: Infinite tower of l = ±1 E (n) 3 exp 2e 3 n/4+ trimer bound states Infinite set of l = ±2 tetramer resonances E (n) exp 2e 3 n/ Super exponential scaling!

15 P-wave model in d=2 L = i t a i t a +g a ( i a ) + g ( i a ) a +v 3 a a + v 4 a a a a + v 4 a a a a spinless composite fermion l = ±1 boson P-wave resonance zero energy bound state All dimensionless couplings are included

16 Efimov effect from RG Flow of atom-dimer vertex: RG=one-loop diagrams bosons vs fermions in 3d Β Λ 3 R Tetramers can be found from RG T 1/s 0 Λ 3 R limit cycle t

17 Super Efimov from RG Two-body: Three-body: Perturbative counting is reliable! s = ln /k irrelevant in IR like QED g 2 (s) = v 3 (s) 2 s 1 cot 1 s + 1 g 2 (0) Double log periodic solution: 4 3 (ln s ) Divergences= trimer bound states

18 T-matrix solution One-channel model: H = dk k 2 (2 ) 2 2 a(p) a(q) Separable interaction k k v 0 a=± k dkdpdq (2 ) 6 2 +p k 2 p k 2 q k 2 +q a(p) =p a e p2 /(2 2 )

19 T-matrix solution Two-fermion scattering T-matrix: T (E; p, q) = 16 p q cos( p q) e (p2 +q 2 )/(2 2 ) 2 v 0 2 Ee E/ 2 E 1 ( E/ 2 ) = +

20 T-matrix solution Three-fermion scattering T-matrix: Near binding energy T ab (E; p, q) Z a ( p)z b ( q)/(e + 2 ) Z a (p) = dq 2 (p+2q) a e (5p2 +5q 2 +8p q)/(8 2 ) p 2 +q 2 +p q+ 2 P b=± (2p+q) b Z b (q) ( 3 4 q2 + 2 ) e ( 3 4 q2 + 2 )/ 2 E 1 (( 3 4 q2 + 2 )/ 2 ) = +

21 T-matrix solution Partial wave decomposition: Z a (p) =e i p z a (p) s and d waves are coupled! Similar to deuteron due to tensor one-pion force

22 T-matrix solution = + n ln ln / n Analytic solution leading log approximation Numeric solution n n n n

23 T-matrix solution Near binding energy T ab (E; p, q) Z a ( p)z b ( q)/(e + 2 ) = + E (n) 3 exp 2e 3 n/4+ Agreement with RG result

24 Effective potential V (R) = 1 1/4 +r 2 4R 2 (R ln R R 0 )2 Semiclassical solution with double Langer correction V E n = R r2 R 2 0 exp e 2 2e r n+ r n+ super exponential scaling

25 Hyperspherical calculation Volosniev et al.; Gao&Yu Adiabatic approximation l=1 = R 3/2 f l=1 (R) l=1 ( ; R) s-d wave mixing is well captured Diagonal corrections important for super Efimov effect Is adiabatic approximation reliable?

26 Tetramer states in 3d Universal tetramer resonances Hammer&Platter von Stecher et al,... Innsbruck Universality and # tetramers not settled Hadizadeh et al

27 Tetramers from RG l=2 sector: s v 4 ' 30 Similar to Efimov tetramers: dv 4 ds = 4g4 + 2g2 v 3 2g 2 v 4 + 2v 2 4 Numerical solution necessary Singularities understood analytically E (n) exp 2e 3 n/ Tetramer ground state Tetramer resonances ln s

28 Super Efimov in 3d? Recent RG calculation includes trimer degrees of freedom Jaramillo Avila&Birse Suggests super Efimov tower of tetramers for every Efimov trimer in 3d! k (n) 4 = k 3 exp( e n ) Hand-waving RG argument: appears due to logarithmic trimer divergences that feed into the four-body solution

29 Experiments Great success in three dimensions Quasi 2d fermions near p-wave resonance ETH 2005 Trimers sizes: many-body physics µ but quasi 2d! No tuning possible in this theory!

30 Open questions Im s v 4 ' 50 peak width Decay of tetramers from RG threshold behavior ln s Tetramers and higher-body from T-matrix? Superfluid near resonance

31 Conclusion Super Efimov -- double exponential scaling Many question to be asked and answered...

32 Ultracold atoms Ensembles of neutral alkali atoms Low densities n gases Laser cooling T 10 9 quantum Tunable interactions and geometry Harmonic trap keeps atoms together

33 Quantum simulator Let nature do the calculation Quantum High degree of tuning Table-top size Lattice models high Tc superconductors Artificial gauge fields topological states of matter Precision measurements equation of state of neutrons Few-body physics quantum chemistry Single atom manipulation quantum computer

34 Experimental achievements Mott MPI&Harvard

35 Quantum simulator Let Nature do the calculation Quantum High degree of tuning Table-top size Lattice models high Tc superconductors Artificial gauge fields topological states of matter Precision measurements equation of state of neutrons Few-body physics quantum chemistry Single atom manipulation quantum computer

36 Feshbach resonance Tunable interactions in ultracold gases Innsbruck Interaction strength tuned by magnetic field B

37 Feshbach resonance Tunable interactions in ultracold gases MIT Resonance phenomenon: a(b) a 1+ B B B 0