Local Density Functional Theory for Superfluid Fermionic Systems. The Unitary Fermi Gas

Transcription

1 Local Density unctional Theoy fo Supefluid emionic Systems The Unitay emi Gas

2 Unitay emi gas in a hamonic tap Chang and Betsch, physics/070390

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

4 What is a unitay emi gas

5 Betsch Many-Body X challenge, Seattle, 999 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 999 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 999: 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, , 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 9, , 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. physics/04045 Dashed line with squaes - Astahachi et al. cond-mat/04063

8 E G 3 5 m Geen unction Monte Calo with ixed Nodes S.-Y. Chang, J. Calson, V. Pandhaipande and K. Schmidt physics/040304

9 a m e a m e BCS Goov exp 8 exp 7/3 π π ixed node GMC esults, S.-Y. Chang et al. 003

10 BCS BEC cossove Leggett 980, Noziees and Schmitt-Rin 985, Randeia et al. 993, If a<0 at T0 a emi system is a BCS supefluid e 7 /3 m π exp a << ε, iff a << and ξ ε >> If a and n 03 a emi system is stongly coupled and its popeties ae univesal. Calson et al. PRL 9, 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 the system is a dilute BEC of tightly bound dimes n ε << > ma 3 f and na, whee and b nb abb 0.6a 0

11 Vey bief/sewed summay of DT

12 Density unctional Theoy DT Hohenbeg and Kohn, 964 E gs Ε[ ρ ] paticle density only! Local Density Appoximation LDA Kohn and Sham, 965 The enegy density is typically detemined in ab initio calculations of infinite homogeneous matte. 3 Egs d τ + ε[ ρ ] ρ m ρ ψ τ ψ N N i i i i ψi U ψi εψ i i + m Kohn-Sham equations

13 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/040804

14 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.

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

16 Howeve, not eveyone is nomal!

17 Supeconductivity and supefluidity in emi systems Dilute atomic emi gases T c ev Liquid 3 He T c 0-7 ev Metals, composite mateials T c ev Nuclei, neuton stas T c ev QCD colo supeconductivity T c ev units ev 0 4 K

18 Bogoliubov-de Gennes equations and enomalization

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

20 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 adius of inteaction inte-paticle sepaation ω << V N DExp ε ξ ε >> 0 coheence length size of the Coope pai

21 Natue of the poblem Natue of the poblem * 0, v u,,, E V ν ν > at small sepaations It is easie to show how this singulaity appeas in infinite homogeneous matte., sin,, v u, v exp u u, exp v v 0 d m m U m i i δ π ν δ ε µ ε µ ε

22 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 97, emi 93, Lenz 97, emi 93, Blatt Blatt and and Weisopf Weisopf Lee, Huang and Yang 957 Lee, Huang and Yang 957 [ ] [ ] 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 if... 4,... exp exp 0 if, 0 0 δ ψ δ ψ ψ δ ψ π ψ ψ ψ ψ + + << > +

23 The SLDA enomalized equations { ε ρ, τ ε ρ, ν } 3 Egs d N + S def S c eff c, g ε ρ ν ν ν [ h µ ]u i + v i Eiu i * u i [ h µ ]v i Ei v i h + U m geff ν c m c c + ln geff g[ ρ ] π c c Ec ρ v, ν v u c E 0 i i Ec * 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

24 Supefluid Local Density Appoximation SLDA fo a unitay emi gas

25 The naïve SLDA enegy density functional suggested by dimensional aguments τ ε α + β + γ /3 5/3 33 π n ν /3 5 n n v τ v ν uv *

26 The enomalized SLDA enegy density functional /3 5/3 τc 33 π n ε α + β + geff νc 5 τ v, uv c * νc E< E E< E c + g E /3 n c 0 c 0 ln eff γ π α c c 0 c 0 c + µ α + U, µ α + U U β + V + small coection /3 /3 3 π n /3 3 γn g ν eff c c ext

27 How to detemine the dimensionless paametes α, β and γ? n d α / + β / µ 3 π π α / + β / µ d 3 π 3 3 d ε εnξs εnβ + α 5 5 π /3 3 n d 3 γ π α E 3 3 E E ε E

28 One thus obtains: 5E ξ 0.4 s 3 N ε α.4 η β ε µ ς ε γ

29 Bonus! Quasipaticle spectum in homogeneous matte solid line - SLDA cicles - GMC due to Calson and Reddy

30 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

31 emions at unitaity in a hamonic tap GMC calculations of Chang and Betsch

32 GMC - Chang and Betsch

33

34 δ EN EN EN + + EN [ ]

35 Densities fo N8 solid, N4 dashed and N0 dot-dashed dashed GMC ed, SLDA blue

36 Ageement between GMC and SLDA vey good a few pecent 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 SLSDA needed ab initio esults fo asymmetic system needed Gadient coections not included