The Unitary Fermi Gas

Transcription

1 Local Density unctional Theoy fo Supefluid emionic Systems The Unitay emi Gas A. Bulgac, Univesity of Washington axiv:cond-mat/070356, PRA, R, in pess (007

2 Unitay emi gas in a hamonic tap Chang and Betsch, Phys. Rev. A 76, 01603(R (007

3 Outline: What is a unitay emi gas Vey bief/sewed summay of DT Bogoliubov-de Gennes equations, enomalization Supefluid Local Density Appoximation (SLDA fo a unitay emi gas emions at unitaity in a hamonic tap within SLDA and compaison with ab intio esults

4 What is a unitay emi gas

5 Betsch Many-Body X challenge, Seattle, 1999 What ae the gound state popeties of the many-body system composed of spin ½ femions inteacting via a zeo-ange, ange, infinite scatteing-length contact inteaction. In 1999 it was not yet clea, eithe theoetically o expeimentally, whethe such femion matte is stable o not. - systems of bosons ae unstable (Efimov effect - systems of thee o moe femion species ae unstable (Efimov effect Bae (winne of the MBX challenge concluded that the system is stable. See also Heiselbeg (enty to the same competition Chang et al (003 ixed-node Geen unction Monte Calo and Astahachi et al. (004 N-DMC povided best the theoetical estimates fo the gound state enegy of such systems. Thomas Due goup (00 demonstated expeimentally that such systems ae (metastable.

6 Conside Betsch s MBX challenge (1999: ind the gound state of infinite homogeneous neuton matte inteacting with an infinite scatteing length. 0 << λ << 0 a Calson, Moales, Pandhaipande and Ravenhall, PRC 68, 0580 (003, with Geen unction Monte Calo (GMC EN N 3 ε 5 α N, α N 0.54 nomal state Calson, Chang, Pandhaipande and Schmidt, PRL 91, (003, with GMC E S N 3 ε 5 α S, α S 0.44 supefluid state This state is half the way fom BCS BEC cossove, the paiing coelations ae in the stong coupling limit and HB invalid again.

7 BEC side BCS side Solid line with open cicles Chang et al. PRA, 70, (004 Dashed line with squaes - Astahachi et al. PRL 93, (004

8 E G 3 5 m Geen unction Monte Calo with ixed Nodes Chang, Calson, Pandhaipande and Schmidt, PRL 91, (003

9 a m e a m e BCS Goov exp 8 exp 7/3 π π ixed node GMC esults, S. ixed node GMC esults, S.-Y. Chang Y. Chang et al. et al. PRA 70, (004 PRA 70, (004

10 BCS BEC cossove Leggett (1980, Noziees and Schmitt-Rin (1985, Randeia et al. (1993, If a<0 at T0 a emi system is a BCS supefluid e 7 /3 m π exp a << ε, iff a << 1 and ξ 1 ε >> 1 If a and n 03 1 a emi system is stongly coupled and its popeties ae univesal. Calson et al. PRL 91, (003 E N nomal 3 Esupefluid ε, 0.44 ε and ξ O 5 N 5 ( λ, O( ε If a>0 (a 0 and na 3 1 the system is a dilute BEC of tightly bound dimes n ε << > ma 3 f and na 1, whee and b nb abb 0.6a 0

11 Calson s s tal at Pac oest, WA, August, 007

12 emi gas nea unitaity has a vey complex phase diagam (T0 Bulgac, obes, Schwen, PRL 97, 0040 (007

13 Vey bief/sewed summay of DT

14 Density unctional Theoy (DT Hohenbeg and Kohn, 1964 Egs Ε[ n( ] Local Density Appoximation (LDA Kohn and Sham, 1965 paticle density only! The enegy density is typically detemined in ab initio calculations of infinite homogeneous matte. 3 Egs d τ( + ε[ n( ] n( m n ( ( ( ( N N ψi τ ψi i 1 i 1 ψi U ψi εψ i i ( + ( ( ( m Kohn-Sham equations

15 Kohn-Sham theoem Injective map (one-to to-one one H T ( i + U( ij + U( ij V ( i HΨ (1,,... N E Ψ (1,,... N 0 N N N N i i< j i< j< i N n ( Ψ δ ( Ψ 0 i 0 i Ψ (1,,... N V ( n( ext E0 d m + n + V n n ( (, 3 min τ( ε[ ( ] ext( ( n( N N ϕi τ( ϕi( i i ext Univesal functional of density independent of extenal potential

16 How to constuct and validate an ab initio ED? Given a many body Hamiltonian detemine the popeties of the infinite homogeneous system as a function of density Extact the enegy density functional (ED Add gadient coections, if needed o nown how (? Detemine in an ab initio calculation the popeties of a select numbe of wisely selected finite systems Apply the enegy density functional to inhomogeneous systems and compae with the ab initio calculation, and if lucy declae Victoy!

17 One can constuct howeve an ED which depends both on paticle density and inetic enegy density and use it in a extended Kohn-Sham appoach (petubative esult Notice that dependence on inetic enegy density and on the gadient of the paticle density emeges because of finite ange effects. Bhattachayya and unstahl, Nucl.. Phys. A 747, 68 (005

18 The single-paticle spectum of usual Kohn-Sham appoach is unphysical, with the exception of the emi level. The single-paticle spectum of extended Kohn-Sham appoach has physical meaning.

19 Extended Kohn-Sham equations Position dependent mass 3 Egs d ( [ ( ] ( * τ + ε n n m [ n( ] n ( ( ( ( N N ψi τ ψi i 1 i 1 ( ( ( ( * ψi + U ψi εψ i i m [ n( ] Nomal emi systems only!

20 Howeve, not eveyone is nomal!

21 Supeconductivity and supefluidity in emi systems Dilute atomic emi gases T c ev Liquid 3 He T c 10-7 ev Metals, composite mateials T c ev Nuclei, neuton stas T c ev QCD colo supeconductivity T c ev units (1 ev 10 4 K

22 Bogoliubov-de Gennes equations and enomalization

23 SLDA - Extension of Kohn-Sham appoach to supefluid emi systems 3 Egs d ε( n(, τ(, ν( n ( v (, τ ( v ( ν ( u ( v ( * T + U( µ ( u( u( E * ( ( T U( µ + v( v( Mean-field and paiing field ae both local fields! (fo sae of simplicity spin degees of feedom ae not shown Thee is a little poblem! The paiing field diveges.

24 Why would one conside a local paiing field? Because it maes sense physically! The teatment is so much simple! Ou intuition is so much bette also. 0 p 1 adius of inteaction inte-paticle sepaation 1 ω << V N DExp ε ξ 1 ε >> 0 coheence length size of the Coope pai

25 Natue of the poblem Natue of the poblem * (, v ( u ( (, (, (, E V ν ν > at small sepaations It is easie to show how this singulaity appeas in infinite homogeneous matte., ( sin( (, 1, v u, ( 1 1 v exp( u ( u, exp( v ( v d m m U m i i δ π ν δ ε µ ε µ ε

26 Pseudo Pseudo-potential appoach potential appoach (appopiate fo vey slow paticles, vey tanspaent, (appopiate fo vey slow paticles, vey tanspaent, but somewhat difficult to impove but somewhat difficult to impove Lenz (197, emi (1931, Lenz (197, emi (1931, Blatt Blatt and and Weisopf Weisopf (195 (195 Lee, Huang and Yang (1957 Lee, Huang and Yang (1957 [ ] [ ] B B A g V ia m a g i a f O a f i f i R V E V m ( ( (... ( : Example ( ( ( ( then 1 if... (1 4, 1 1 ( exp( exp( ( 0 if (, ( ( ( ( δ ψ δ ψ ψ δ ψ π ψ ψ ψ ψ + + << > +

27 The SLDA (enomalized equations { ε (, τ ( ε (, ν ( } 3 E gs d N n + S n def S n c geff c (, ( ( ( ( ( ε ν ν ν [ h( µ ]u( i + ( v( i Eiu( i * ( u i( [ h( µ ]v i( Eiv i( h( + U ( m ( ( geff ( ν c ( 1 1 m( c( ( c( + ( 1 ln geff ( g[ n( ] π c( c( ( E c ρ ( v (, ν ( v ( u ( c E 0 i i E c * c i E 0 c ( ( Ec + µ + U (, µ + U ( m( m( Position and momentum dependent unning coupling constant Obsevables ae (obviously independent of cut-off enegy (when chosen popely. i i

28 Supefluid Local Density Appoximation (SLDA fo a unitay emi gas

29 The naïve SLDA enegy density functional suggested by dimensional aguments τ ε( α + β + γ /3 5/3 ( 3(3 π n ( ν( 1/3 5 n ( n ( v( τ( v( ν( u(v( *

30 The enomalized SLDA enegy density functional /3 5/3 τc( 3(3 π n ( ε( α + β + geff ( νc( 5 τ ( v(, ( u(v( c * νc E< E E< E c 1/3 1 n ( c( 0( c( + 0( 1 ln eff ( γ π α c( c( 0( c ( 0( c + µ α + U (, µ α + U ( g E U ( β + V ( + small coection /3 /3 (3 π n ( ( /3 3 γn ( ( g ( ν ( eff c c ext

31 How to detemine the dimensionless paametes α, β and γ? 3 3 α / + β / µ d n π ( π ( α + β µ / / + 3 d ε 1 3 ( π E d ε ε ξ ε β + n n α 1 S ( π E E 1/3 3 n d γ ( π α E

32 One thus obtains: 5E ξ 0.4( s 3 N ε α 1.14 η 0.504(4 β ε µ 1 ς 0.4( ε γ

33 Bonus! Quasipaticle spectum in homogeneous matte solid/dotted blue line - SLDA, homogeneous GMC due to Calson et al ed cicles - GMC due to Calson and Reddy dashed blue line - SLDA, homogeneous MC due to Juillet blac dashed-dotted dotted line meanfield at unitaity Two moe univesal paamete chaacteizing the unitay emi gas and its excitation spectum: effective mass, meanfield potential

34 Exta Bonus! The nomal state has been also detemined in GMC ξ N 5E 3N ε 0.55( SLDA functional pedicts ξ α + β N 0.59

35 emions at unitaity in a hamonic tap within SLDA and compaison with ab intio esults GMC - Chang and Betsch, Phys. Rev. A 76, 01603(R (007 N-DMC - von Steche,, Geene and Blume,, axiv: axiv:

36 emions at unitaity in a hamonic tap GMC - Chang and Betsch, Phys. Rev. A 76, 01603(R (007 N-DMC - von Steche,, Geene and Blume,, axiv:

37 NB Paticle pojection neithe equied no needed in SLDA!!!

38 SLDA - Extension of Kohn-Sham appoach to supefluid emi systems 3 E { gs d ε( n(, τ(, ν( + V n n ( v (, τ ( v ( ν ( u ( v ( * * ext( ( + ext( ν( + ext( ν ( * T + U( µ ( u( u( * ( ( T + U( µ E v( v ( univesal functional (independent of extenal potential }

39 1 δ EN ( EN ( EN ( EN ( 1 [ ]

40 Densities fo N8 (solid, N14 (dashed and N0 (dot-dashed dashed GMC (ed, SLDA (blue

41 New extended N-DMC esults D. Blume,, J. von Steche,, and C.H. Geene, axiv:

42 New extended N-DMC esults D. Blume,, J. von Steche,, and C.H. Geene, axiv:

43 Ageement between GMC/N-DMC and SLDA extemely good, a few pecent (at most accuacy Why not bette? A bette ageement would have eally signaled big toubles! Enegy density functional is not unique, in spite of the stong estictions imposed by unitaity Self-inteaction coection neglected smallest systems affected the most Absence of polaization effects spheical symmety imposed, odd systems mostly affected Spin numbe densities not included extension fom SLDA to SLSD(A needed ab initio esults fo asymmetic system needed Gadient coections not included

44 Outloo Extension away fom unitaity - tivial Extension to (some excited states - easy Extension to time dependent poblems appeas easy Extension to finite tempeatues - easy, but one moe paamete is needed, the paiing gap dependence as a function of T Extension to asymmetic systems staightfowad (at unitaity quite a bit is aleady now about the equation of state