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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 · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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Page 1: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 2: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

• 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

Page 3: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 4: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

U

3U

Ut

Correlated lattice bosons: Bosonic Hubbard modelCorrelated lattice bosons: Bosonic Hubbard modelCorrelated lattice bosons: Bosonic Hubbard model

Page 5: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 6: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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)]

Page 7: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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.

Page 8: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 9: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 10: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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)

Page 11: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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)

Page 12: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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)

Page 13: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

[ , ][ , ] 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)

Page 14: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

[ , ][ , ] 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)

Page 15: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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)

Page 16: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 17: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 18: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 19: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 20: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 21: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 22: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

numerical

Efficient numerical computation of all correlation functions on the Bethe lattice

Page 23: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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

Page 24: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

• 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

Page 25: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

• 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

Page 26: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

• 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

Page 27: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

• 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

Page 28: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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)

Page 29: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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)

Page 30: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

,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)

Page 31: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

.. .

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)

Page 32: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

.. .

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)

Page 33: DMFT for correlated bosons and boson-fermion mixtures · DMFT for correlated bosons and boson-fermion mixtures Supported by Deutsche Forschungsgemeinschaft through SFB 484 Workshop

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