DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop on Recent developments in dynamical mean-field theory ETH Zürich, September 29, 2009 Dieter Vollhardt
DMFT for correlated bosons and boson-fermion mixtures
Supported by Deutsche Forschungsgemeinschaft through SFB 484
Workshop on Recent developments in dynamical mean-field theory
ETH Zürich, September 29, 2009
Dieter Vollhardt
• 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
ContentsContents
Superfluid(coherent)
Mott insulator(incoherent)
Greiner, Mandel, Esslinger, Hänsch, Bloch (2002)
Superfluid–Mott transition of cold bosons in an optical lattice Superfluid–Mott transition of cold bosons in an optical lattice
V0
U
3U
Ut
Correlated lattice bosons: Bosonic Hubbard modelCorrelated lattice bosons: Bosonic Hubbard modelCorrelated lattice bosons: Bosonic Hubbard model
U
3U
Ut
Correlated lattice bosons: Bosonic Hubbard modelCorrelated lattice bosons: Bosonic Hubbard modelCorrelated lattice bosons: Bosonic Hubbard model
Bose-Einstein condensation:
NBEC : # condensed bosonsNL : # lattice sites
Z: coordination number
BEC(distance
independent)
+normal bosons
( )BEC
L
N TN
BECT T
( )BEC
BEC ijjT iT
tZ N T t
kinH
Theory of correlated lattice bosonsTheory of correlated lattice bosons
• Liquid 4He Matsubara, Matsuda (1956, 1957)Morita (1957)
• 4He in porous media• Superfluid-insulator transition
Fisher et al. (1989)
Batrouni, Scalettar, Zimanyi (1990)Roksar, Kotliar (1991)Sheshadri et al. (1993)Freericks, Monien (1994, 1996)
• Granular superconductors/Josephson junctions (“small” Cooper pairs)
Kampf, Zimanyi (1993)Bruder, Fazio, Schön (2005)
• BEC of magnons in TlCuCl3 Giamarchi, Rüegg, Tchernyshov (2008)
• Bosonic atoms in optical lattices(7Li, 87Rb, …)
Jaksch et al. (1998)Bloch, Dalibard, Zwerger [RMP (2008)]
Exact limits and standard approximation schemesExact limits and standard approximation schemes
• U=0: free bosons
• tij=0: immobile bosons (“atomic limit”)
• Weak coupling theories
1st order in U: Bogoliubov approx. (no normal diagrams)Hartree-Fock-Bogoliubov approx.
static mean-field theories2nd order in U: Beliaev-Popov approx.
Exact limits and standard approximation schemesExact limits and standard approximation schemes
• U=0: free bosons
• tij=0: immobile bosons (“atomic limit”)
• Weak coupling theories
1st order in U: Bogoliubov approx. (no normal diagrams)Hartree-Fock-Bogoliubov approx.
static mean-field theories2nd order in U: Beliaev-Popov approx.
• Fisher et al. mean-field theory (1989) ijL
ttN
= 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
GoalGoal
• Valid for all parameter values t, U, n, T, …• Thermodynamically consistent• Concerving• Small (control) parameter
Comprehensive scheme for correlated lattice bosons
Dynamical mean-field theory for lattice bosons (B-DMFT) ? d
1/d
Problem: How to rescale Ekin with d ?
( )BEC
BEC ijjT iT
tZ N T t
kinH
mean-field theory: Bosonic Hubbard modelmean-field theory: Bosonic Hubbard modeld
1
†kin
( )
Z
i ji j NN i
Z
H bt b BEC distance independent
Condensed bosons (T<TBEC)
*JZJ
Classicalrescaling
Brout (1960)
1
†kin
( )1
i ji j NN i
Z ZZ
btH b
Normal bosons
*t
Zt
Quantumrescaling
Metzner, DV (1989)
Rescaling of normal and condensed bosonsnot possible on the Hamiltonian level
†
kin normal1 1( )
BEC
BEC i jijT
Z ZT
H Z N T bt t b
mean-field theory: Bosonic Hubbard modelmean-field theory: Bosonic Hubbard modeld
1
†kin
( )
Z
i ji j NN i
Z
H bt b BEC distance independent
Condensed bosons (T<TBEC)
*JZJ
Classicalrescaling
Brout (1960)
1
†kin
( )1
i ji j NN i
Z ZZ
btH b
Normal bosons
*t
Zt
Quantumrescaling
Metzner, DV (1989)
3
2
1 2 3 4 000
0 0
11
0 00
1
0 0
4
0
1
j k l j k i
ZZ
ii
j ki
l
Z
Z
ZZ
t bt t t b bd d d d b b b bb
BEC (distance independent)
G( )
isite 0
l
Construction of B-DMFT by rescaling in the actionConstruction of B-DMFT by rescaling in the action
[ , ][ , ] S b bZ D b b e
Byczuk, DV; Phys. Rev. B 77, 235106 (2008)
Scaling of hopping in cumulant expansion w.r.t. e.g., 4th order term:
0, 0iS
0 0, 0 , 0i i i jS S S S Action
Cavity method (Georges et al., 1996)
[ , ][ , ] S b bZ D b b e
Byczuk, DV; Phys. Rev. B 77, 235106 (2008)
3
2
1 2 3 4 000
0 0
11
0 00
1
0 0
4
0
1
j k l j k i
ZZ
ii
j ki
l
Z
Z
ZZ
t bt t t b bd d d d b b b bb
BEC (distance independent)
G( )
isite 0
Construction of B-DMFT by rescaling in the actionConstruction of B-DMFT by rescaling in the action
l
Scaling of hopping in cumulant expansion w.r.t. e.g., 4th order term:
0, 0iS
0 0, 0 , 0i i i jS S S S Action
Cavity method (Georges et al., 1996)
[ , ][ , ] S b bZ D b b e
Byczuk, DV; Phys. Rev. B 77, 235106 (2008)
3
2
1 2 3 4 000
0 0
11
0 00
1
0 0
4
0
1
j k l j k i
ZZ
ii
j ki
l
Z
Z
ZZ
t bt t t b bd d d d b b b bb
BEC (distance independent)
G( )
isite 0
,d Z
only 1st and 2nd order terms remain
Linked cluster theorem local action Sloc
Construction of B-DMFT by rescaling in the actionConstruction of B-DMFT by rescaling in the action
l
Scaling of hopping in cumulant expansion w.r.t. e.g., 4th order term:
0, 0iS
0 0, 0 , 0i i i jS S S S Action
Cavity method (Georges et al., 1996)
G( )
isite 0
Nambu notation , etc.
(i) Effective single impurity problem
(iii) k-integrated Dyson equ. (lattice)
(ii) Condensate wave function
Lower band edge 00
0i
iR
i
tZ
k
Hybridizationwith bath
B-DMFT self-consistency equationsB-DMFT self-consistency equations Byczuk, DV; PRB 77, 235106 (2008)
G( )
isite 0
Nambu notation , etc.
(i) Effective single impurity problem
B-DMFT
(ii) Condensate wave function
Fisher et al. MFT/Gutzwiller approx.
B-DMFT self-consistency equationsB-DMFT self-consistency equations Byczuk, DV; PRB 77, 235106 (2008)
(iii) k-integrated Dyson equ. (lattice)
Fermionic DMFT
Generalized time-dependent Gross-Pitaevskii eq. for order parameter
Condensate wave function (order parameter)Condensate wave function (order parameter)
loc* ( )0 b
Sb
eq. of motionfor
11 1
2
2
0
( ) ( ) ( )
( ') ( ')' ( ') ( ')
U
d
Retardation effect due to coupling to normal bosons
Classical eq. of motion of homogeneous condensate for d loc
() )( ( )S
b b (approximation ?)
11 12
0
( ) ( ) ( ) ( )
( ') (' ) )' ( ' '( )
b b
b b
U b
d
b
Boson mean-field theory of Fisher et al. = Gutzwiller approx.
New kind of self-consistent bosonic quantum impurity problem
How to solve the B-DMFT eqs.?How to solve the B-DMFT eqs.?
NRG Lee, Bulla (2007)CT-QMC Winter, Rieger, Vojta, Bulla (2009)
Werner et al.
B-DMFT
arXiv:0907.2928v1
Numerical solution of the B-DMFT equationsNumerical solution of the B-DMFT equations
ED-solution of B-DMFT eq. with Bethe DOS
arXiv:0902.2212v2
• Expansion around Gutzwiller approximation ( with t t/Z scaling) to O(1/Z)• For actions agree• No scaling in final equ. terms O(1) and O(1/Z) treated equally at large Z (?)
Z Z
ED-solution of B-DMFT equ. with Bethe DOS
Z=4
Numerical solution of the B-DMFT equationsNumerical solution of the B-DMFT equations
numerical
Efficient numerical computation of all correlation functions on the Bethe lattice
Application of B-DMFT to bosonic Falicov-Kimball modelApplication of B-DMFT to bosonic Falicov-Kimball model
i ib b : ,2 fb bosons
mobile7Li
immobile87Rb
annealed disorder
† † f f fi j bf i ff
bi j f i i
i i ij
ji i
i
H b f f U n U n nb nt
, 0fin H
Ubf
4Ubf
3Ubf
• Hard-core f-bosons • simple cubic lattice• nb=0.65, nf=0.8
Byczuk, DV; Phys. Rev. B 77, 235106 (2008)
Ubf>Uc(nf) Correlation gap lower upper
Hubbard bands
Ubf
Application of B-DMFT to bosonic Falicov-Kimball modelApplication of B-DMFT to bosonic Falicov-Kimball model
0,1fff iU n
• Hard-core f-bosons • simple cubic lattice• nb=0.65, nf=0.8
/( )U T U TO e
mmmmmmmmm
= const
/( )U T U TO e
mmmmmmmmm
= const
increases
condensate fraction
Increasin
g U:
Byczuk, DV; Phys. Rev. B 77, 235106 (2008)
Application of B-DMFT to bosonic Falicov-Kimball modelApplication of B-DMFT to bosonic Falicov-Kimball model
0,1fff iU n
• Hard-core f-bosons • simple cubic lattice• nb=0.65
BECT increases
Correlation effect !
/( )U T U TO e
mmmmmmmmm
= const
increases
condensate fraction
Increasin
g U:
nf=0.8
Byczuk, DV; Phys. Rev. B 77, 235106 (2008)
Application of B-DMFT to bosonic Falicov-Kimball modelApplication of B-DMFT to bosonic Falicov-Kimball model
0,1fff iU n
• Hard-core f-bosons • simple cubic lattice• nb=0.65 nf=0.8
/( )U T U TO e
mmmmmmmmm
= const
increases
condensate fraction
Increasin
g U:
Byczuk, DV; Phys. Rev. B 77, 235106 (2008)
Application of B-DMFT to bosonic Falicov-Kimball modelApplication of B-DMFT to bosonic Falicov-Kimball model
BECT increases
Correlation effect
0,1fff iU n
Boson-Fermion mixturesBoson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009)
Model I: Spinless particles/atomse.g., 87Rb (boson) + 40K (fermion) in only one hyperfine state)
Model I: Spinless particles/atomse.g., 87Rb (boson) + 40K (fermion) in only one hyperfine state)
[ , ][ ] [ ] S b fZ D b D f e 0 0 0lim b f bf
i i iZS S S SAction
Cavity method
• Ubf complicated effective dynamics of bosons• Even for Ub=0 effective bosonic action not bilinear
non-trivial effective interaction between bosons
Boson-Fermion mixturesBoson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009)
,0
biS RPA
fermionicbubble
retarded interaction between bosons due to fermions Effective static interaction between bosons
Attraction if Phase separation/bosonic molecules
Expansion of bosonic action in Ubf
Model I: Spinless particles/atomse.g., 87Rb (boson) + 40K (fermion) in only one hyperfine state)
Boson-Fermion mixturesBoson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009)
.. .
Model II: Spinless bosons + S=1/2 fermionse.g., 87Rb (boson) + 40K (fermion) with two hyperfine states
Boson-Fermion mixturesBoson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009)
.. .
Model II: Spinless bosons + S=1/2 fermionse.g., 87Rb (boson) + 40K (fermion) with two hyperfine states
Expansion of fermionic action in Ubf
,0
fiS RPA
retarded interaction between fermions due to bosons
bosonicbubble
Attraction if Cooper pair formation
Effective static interaction between fermions
Boson-Fermion mixturesBoson-Fermion mixtures K. Byczuk, DV; Ann. Phys. (Berlin) 18, 622 (2009)
Bosonic Falicov-Kimball model:Increase of nBEC(T), TBEC for increasing Ubf
Prediction for 7Li, 87Rb in optical lattices
Bosonic dynamical mean-field theoryfor correlated lattice bosons (B-DMFT)Bosonic dynamical mean-field theoryfor correlated lattice bosons (B-DMFT)
• Construction via limit in cumulant expansion• Generalizes static MFT of Fisher et al. (1989)
d
To do: • Develop efficient B-DMFT impurity solvers
• For bosons and boson-fermion mixtures calculate- phase diagrams- nBEC(T), TBEC
- compressibility / other susceptibilities- dynamic quantities- disorder effects