DMFT for correlated bosons and boson-fermion mixtures

Transcription

1 DMFT for correlated bosons and boson-fermion mixtures Workshop on Recent developments in dynamical mean-field theory ETH ürich, September 29, 2009 Dieter Vollhardt Supported by Deutsche Forschungsgemeinschaft through SFB 484

2 Contents Bosonic Hubbard model Construction of a DMFT for lattice bosons ( B-DMFT ) Properties of the B-DMFT B-DMFT solution of the bosonic Falicov-Kimball model Boson-Fermion mixtures In collaboration with Krzysztof Byczuk

3 Superfluid Mott transition of of cold bosons in in an an optical lattice Superfluid (coherent) V 0 Mott insulator (incoherent) Greiner, Mandel, Esslinger, Hänsch, Bloch (2002)

4 Correlated lattice bosons: Bosonic Hubbard model t U U 3U

5 Correlated lattice bosons: Bosonic Hubbard model t U U 3U Bose-Einstein condensation: T T BEC N ( ) BEC T N L BEC (distance independent) + normal bosons N BEC : # condensed bosons N L : # lattice sites H t N BEC ( T ) t kin ij T T BEC ij : coordination number

6 Theory of of correlated lattice bosons Liquid 4 He Matsubara, Matsuda (1956, 1957) Morita (1957) 4 He in porous media Superfluid-insulator transition Fisher et al. (1989) Batrouni, Scalettar, imanyi (1990) Roksar, Kotliar (1991) Sheshadri et al. (1993) Freericks, Monien (1994, 1996) Granular superconductors/ Josephson junctions ( small Cooper pairs) Bosonic atoms in optical lattices ( 7 Li, 87 Rb, ) Kampf, imanyi (1993) Bruder, Fazio, Schön (2005) Jaksch et al. (1998) Bloch, Dalibard, werger [RMP (2008)] BEC of magnons in TlCuCl 3 Giamarchi, Rüegg, Tchernyshov (2008)

7 Exact limits and standard approximation schemes U=0: free bosons t ij =0: immobile bosons ( atomic limit ) Weak coupling theories 1 st order in U: Bogoliubov approx. (no normal diagrams) Hartree-Fock-Bogoliubov approx. static mean-field theories 2 nd order in U: Beliaev-Popov approx.

8 Exact limits and standard approximation schemes U=0: free bosons t ij =0: immobile bosons ( atomic limit ) Weak coupling theories 1 st order in U: Bogoliubov approx. (no normal diagrams) Hartree-Fock-Bogoliubov approx. static mean-field theories 2 nd order in U: Beliaev-Popov approx. Fisher et al. mean-field theory (1989) t ij t N L = const infinite range hopping Gutzwiller approx. for variational wave fct. Roksar, Kotliar (1991) Properties: Immobile normal bosons No dynamic coupling between condensed/normal bosons Static mean-field theory

9 Goal Comprehensive scheme for correlated lattice bosons Valid for all parameter values t, U, n, T, Thermodynamically consistent Concerving Small (control) parameter 1/d d Dynamical mean-field theory for lattice bosons (B-DMFT)? Problem: How to rescale E kin with d? H kin t N BEC ( T ) t ij T T BEC ij

10 d mean-field theory: Bosonic Hubbard model Condensed bosons (T<T BEC ) H kin t bi bj 1 i j( NN i) BEC distance independent Classical rescaling J J * Brout (1960) Normal bosons H kin t 1 i j( NN i) bib j 1 Quantum rescaling t t * Metzner, DV (1989)

11 d mean-field theory: Bosonic Hubbard model Condensed bosons (T<T BEC ) H kin t bi bj 1 i j( NN i) BEC distance independent Classical rescaling J J * Brout (1960) Normal bosons H kin t 1 i j( NN i) bib j 1 Quantum rescaling t * Metzner, DV (1989) t Hkin t N ( T) t b b BEC i j 1 1 ij T TBEC normal Rescaling of normal and condensed bosons not possible on the Hamiltonian level

12 0 Construction of of B-DMFT by by rescaling in in the action Cavity method (Georges et al., 1996) D[ b, b] e Sb [, b] Scaling of hopping in cumulant expansion w.r.t. e.g., 4 th order term: Byczuk, DV; Phys. Rev. B 77, (2008) d1d2d3d4 t i tj0tk0tl0 bi bjbkbi l bbbb i jkl0 1 3 BEC (distance 2 independent) Action 0 S S S S S0, i 0 i0 0, i0 i, j0 site i 0 G( )

13 0 Construction of of B-DMFT by by rescaling in in the action Cavity method (Georges et al., 1996) D[ b, b] e Sb [, b] Scaling of hopping in cumulant expansion w.r.t. e.g., 4 th order term: Byczuk, DV; Phys. Rev. B 77, (2008) d1d2d3d4 t i tj0tk0tl0 bi bjbkbi l bbbb i jkl0 1 3 BEC (distance 2 independent) Action 0 S S S S S0, i 0 i0 0, i0 i, j0 site i 0 G( )

14 0 Construction of of B-DMFT by by rescaling in in the action Cavity method (Georges et al., 1996) D[ b, b] e Sb [, b] Scaling of hopping in cumulant expansion w.r.t. e.g., 4 th order term: Byczuk, DV; Phys. Rev. B 77, (2008) d1d2d3d4 t i tj0tk0tl0 bi bjbkbi l bbbb i jkl0 1 3 BEC (distance 2 independent) Action 0 S S S S S0, i 0 site i 0 G( ) d, only 1 st and 2 nd order terms remain Linked cluster theorem local action S loc i0 0, i0 i, j0

15 B-DMFT self-consistency equations Byczuk, DV; PRB 77, (2008) Nambu notation, etc. (i) Effective single impurity problem site i 0 G( ) Lower band edge k0 i0 t i0 R i (ii) Condensate wave function Hybridization with bath (iii) k-integrated Dyson equ. (lattice)

16 B-DMFT self-consistency equations Byczuk, DV; PRB 77, (2008) Nambu notation, etc. (i) Effective single impurity problem site i 0 G( ) Fisher et al. MFT/ Gutzwiller approx. Fermionic DMFT B-DMFT (ii) Condensate wave function (iii) k-integrated Dyson equ. (lattice)

17 Condensate wave function (order parameter) S b 0 loc eq. of motion * for b( ) b( ) Ub 0 ( ) b( ) b( ) d b b ' ( ') ( ') ( ') ( ') b( ) b( ) S ( ) (approximation?) loc Classical eq. of motion of homogeneous condensate for d ( ) 2 U ( ) ( ) d ' ( ') ( ') ( ') ( ') 0 Retardation effect due to coupling to normal bosons Generalized time-dependent Gross-Pitaevskii eq. for order parameter

18 Boson mean-field theory of Fisher et al. = Gutzwiller approx.

19 How to to solve the B-DMFT eqs.? B-DMFT New kind of self-consistent bosonic quantum impurity problem NRG Lee, Bulla (2007) CT-QMC Winter, Rieger, Vojta, Bulla (2009) Werner et al.

20 Numerical solution of the B-DMFT equations ED-solution of B-DMFT eq. with Bethe DOS arxiv: v1

21 Numerical solution of the B-DMFT equations arxiv: v2 Expansion around Gutzwiller approximation ( with t t/ scaling) to O(1/) For actions agree No scaling in final equ. terms O(1) and O(1/) treated equally at large (?) ED-solution of B-DMFT equ. with Bethe DOS =4

22 numerical Efficient numerical computation of all correlation functions on the Bethe lattice

23 Application of of B-DMFT to to bosonic Falicov-Kimball model b i 2 : b, f bosons bi b f f f ti j ibj f i j bf n i i ff i i ij i i i H b f f U n U n n mobile 7 Li immobile 87 Rb n f i, H 0 4U bf U bf annealed disorder 3U bf

24 Application of of B-DMFT to to bosonic Falicov-Kimball model Byczuk, DV; Phys. Rev. B 77, (2008) f Hard-core f-bosons U ff ni simple cubic lattice n b =0.65, n f =0.8 0,1 U bf >U c (n f ) Correlation gap U bf lower upper Hubbard bands

25 Application of of B-DMFT to to bosonic Falicov-Kimball model Byczuk, DV; Phys. Rev. B 77, (2008) f Hard-core f-bosons U ff ni simple cubic lattice n b =0.65, n f =0.8 0,1 = const Increasin gu: condensate fraction increases mmmmmmmmm U T U / T Oe ( )

26 Application of of B-DMFT to to bosonic Falicov-Kimball model Byczuk, DV; Phys. Rev. B 77, (2008) f Hard-core f-bosons U ff ni simple cubic lattice n b =0.65 0,1 n f =0.8 Correlation effect! TBEC increases Increasin gu: condensate fraction increases mmmmmmmmm UT U/ T Oe ( ) = const

27 Application of of B-DMFT to to bosonic Falicov-Kimball model Byczuk, DV; Phys. Rev. B 77, (2008) f Hard-core f-bosons U ff ni 0,1 simple cubic lattice n b =0.65 n f =0.8 Correlation effect = const TBEC increases Increasin gu: condensate fraction increases mmmmmmmmm UT U/ T Oe ( )

28 Boson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009) Model I: Spinless particles/atoms e.g., 87 Rb (boson) + 40 K (fermion) in only one hyperfine state)

29 Boson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009) Model I: Spinless particles/atoms e.g., 87 Rb (boson) + 40 K (fermion) in only one hyperfine state) Cavity method Sbf [, ] D[ b] D[ f] e Action lim S S b 0 f 0 bf i Si Si0 U bf complicated effective dynamics of bosons Even for U b =0 effective bosonic action not bilinear non-trivial effective interaction between bosons

30 Boson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009) Model I: Spinless particles/atoms e.g., 87 Rb (boson) + 40 K (fermion) in only one hyperfine state) Expansion of bosonic action in U bf b, S RPA i0 Effective static interaction between bosons fermionic bubble retarded interaction between bosons due to fermions Attraction if Phase separation/bosonic molecules

31 Boson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009) Model II: Spinless bosons + S=1/2 fermions e.g., 87 Rb (boson) + 40 K (fermion) with two hyperfine states...

32 Boson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009) Model II: Spinless bosons + S=1/2 fermions e.g., 87 Rb (boson) + 40 K (fermion) with two hyperfine states... Expansion of fermionic action in U bf, S f RPA i0 bosonic bubble Effective static interaction between fermions retarded interaction between fermions due to bosons Attraction if Cooper pair formation

33 Bosonic dynamical mean-field theory for correlated lattice bosons (B-DMFT) Construction via d limit in cumulant expansion Generalizes static MFT of Fisher et al. (1989) Bosonic Falicov-Kimball model: Increase of n BEC (T), T BEC for increasing U bf Prediction for 7 Li, 87 Rb in optical lattices To do: Develop efficient B-DMFT impurity solvers For bosons and boson-fermion mixtures calculate - phase diagrams -n BEC (T), T BEC - compressibility / other susceptibilities - dynamic quantities -disorder effects