Metal-insulator transitions of
correlated lattice fermions
with disorder
Disorder in Condensed Matter and Cold Atoms
Lorentz Center, Leiden; September 26, 2007
Dieter Vollhardt
Supported by Deutsche Forschungsgemeinschaft through SFB 484
• Metal-Insulator transitions: Examples
• Lattice models
• Dynamical Mean-Field Theory (DMFT)
of correlated fermions & bosons
• Mott-Hubbard transition
• Disorder and averaging
• Binary alloy disorder:
Mott-Hubbard transition at fractional filling
• Mott-Hubbard transition vs. Anderson localization
In collaboration with:
Krzysztof Byczuk
Walter Hofstetter
Band insulator
qualitatively
different?
Mott insulator
Need
Needbetter
betterunderstanding
understandingof
ofinsulators
insulators
Two-band Hubbard model (analytic): Smooth crossover
Rosch (2006)
Metal-Insulator Transitions: Examples
1. Squeezable nanocrystal film switching between
metal and insulator
Compressed film: metal
(metallic sheen)
Uncompressed film: insulator
(shininess is gone)
Discontinuous transition
Collier, Saykally, Shiang, Henrichs, Heath (1997)
2. Mott transition in (TMTSF)2PF6
Dressel et al. (1996)
d=1
Giamarchi (1997)
3. Metal-insulator transition in a dilute,
low-disordered Si MOSFET
d=2
Kravchenko, Mason, Bowker,
Furneaux, Pudalov,
D’Iorio (1995)
Anissimova, Kravchenko,
Punnoose, Finkel’stein,
Klapwijk (2007)
4. Metal-Insulator transitions in La1-xSrxMnO3
d=3
5. Anderson metal-insulator transition:
(disorder induced)
d=3
6. Mott metal-insulator transition in V2O3
(interaction/correlation induced)
d=3
7. Superfluid–Mott transition of cold bosons in an optical lattice
d=3
Superfluid
(coherent)
V0
Mott insulator
(incoherent)
Greiner, Mandel, Esslinger, Hänsch, Bloch (2002)
Correlated Lattice Fermions: Models
e-, 6Li,
40K, 171Yb,…
1. Hubbard model:
t (=J)
Simplest lattice
fermion model
U
Local (“Hubbard“) physics:
H = −t
∑σ
i,j ,
ci†σ cjσ + U ∑ ni↑ni↓
i
ni↑ ni↓ ≠ ni↑ ni↓
Correlation phenomena:
•Metal-insulator transition
•Ferromagnetisms,...
2. Falicov-Kimball model
t (=J)
x
U
mobile fermion (“d”)
x
x
immobile fermion (“f”)
x
171Yb
annealed disorder
H FK = −t ∑ ci†↑cj↑ + U ∑ ni↑ni↓
i,j
n=1
i
= −t ∑ d i†d j + U ∑ nid nif
i,j
6Li
checker
board
i
d=2,3: unsolvable many-body problems
Reliable approximations ?
Dynamical Mean-Field Theory (DMFT)
of Correlated Lattice Fermions
Dynamical mean-field theory of correlated electrons
d=3; coord. # Z
fcc lattice: Z=12
Metzner, DV (1989)
i
d →∞
Z →∞
⎯⎯⎯
→
Scaling of t = t *
hopping
Z
Σ(ω)
i
G(ω)
"single-impurity Anderson model"
+ self-consistency
Hubbard model
Georges and Kotliar (1992)
Σ( k , ω )
d →∞
Z →∞
⎯⎯⎯
→
Σ(ω )
Dynamical mean-field theory of correlated electrons
Proper time resolved treatment of local electronic interactions:
Σ( k , ω )
Σ(ω )
Kotliar, DV (2004)
DMFT: “local” theory with full many-body dynamics
Reliable approximation in d=3
DMFT self-consistency equations
(i) Effective single impurity problem
(ii) k-integrated Dyson equation (lattice enters)
Solve with an „impurity solver“, e.g., QMC, NRG, ED,...
Σ(ω)
i
G(ω)
DMFT helped to advance our understanding of, e.g.:
•
Correlation phenomena at intermediate couplings
• Mott-Hubbard metal-insulator transition
Georges, Kotliar, Krauth, Rozenberg (1996)
Kotliar, DV (2004)
Mott-Hubbard metal-insulator transition
at integer filling
Application of DMFT: Mott-Hubbard metal-insulator transition
Hubbard model, n=1
lower
Hubbard band
LHS
upper
Hubbard band
UHS
Atomic levels
t=0
Hubbard bands
t/U<<0
Application of DMFT: Mott-Hubbard metal-insulator transition
Hubbard model, n=1
Density of states
U
U
quasiparticles
lower
Hubbard band
LHS
upper
Hubbard band
UHS
U
EF
Quasiparticle
renormalization
m*
→∞
m
E
Hubbard model (n=1)
Strongly correlated
electron materials:
V2O3
NiSe2-xSx
κ-organics, ...
Excursion:
Dynamical Mean-Field Theory of
Correlated Lattice Bosons (B-DMFT)
1. Bose-Hubbard model
2. Bose-Falicov-Kimball model
mobile
immobile
7Li
87Rb
annealed disorder
Scaling of hopping to derive B-DMFT
Byczuk, DV (arXiv:0706.0839)
N0=const
Scaling not possible on Hamiltonian level,
but in the action (cumulant expansion)
Bose-Hubbard model: B-DMFT
Byczuk, DV (arXiv:0706.0839)
Condensate wave fct.
New: Coupling to generalized Gross-Pitaevskii eq. for condensate wave fct.
Generalizes static MFT of Fisher, Weichman, Grinstein, Fisher (1989)
DMFT for fermions
B-DMFT
Æ Poster
K. Byczuk, DV
B-DMFT for Bose-Falicov-Kimball model
Hardcore f-bosons: nf=0,1
Byczuk, DV (arXiv:0706.0839)
d=3, simple cubic lattice
• Correlation induced band splitting Æ TBEC increases
• Narrowing of lower subband Æ TBEC decreases
Æ Enhancement of TBEC due to interactions
Disorder
(quenched)
Anderson disorder model on the lattice
H=
∑σ
i,j ,
tij ci†σ cjσ + ∑ ε i niσ
Random hopping
iσ
Random local potential
Anderson disorder model on the lattice
H = −t
∑σ
i,j ,
ci†σ cjσ + ∑ ε i niσ
iσ
εi
Random local potential
Disorder Æ Scattering of quantum particle
Scattering time τ Æ
Weak scattering (d=3)
Σ(ω = 0) ∝
1
τ
Anderson disorder model on the lattice
H = −t
∑σ
i,j ,
ci†σ cjσ + ∑ ε i niσ
iσ
εi
Random local potential
Disorder distributions, e.g.,:
Δ: disorder strength
Box disorder
1/ Δ
Oi
arith
= ∫ d ε i P(ε i )Oi
e.g., local DOS
ρ (ε i )
Binary alloy disorder (alloys A1-xBx, e.g., .Fe1-xCox)
x
1-x
arith
iφ ( r )
ψ
(
r
)
=
ψ
(
r
)
e
Disorder affects wave fct.
Localization σ (0) = 0 due to, e.g.,
Anderson localization
Alloy band splitting
Binary alloy disorder, bounded Hamiltonian
Δ > Δ c max( t ,U ), for d ≥ 1
Upper alloy
Subband (UAB)
Lower alloy
Subband (LAB):
DMFT for disordered systems
Anderson disorder model on the lattice
H = −t
∑σ
i,j ,
DMFT with
ci†σ cjσ + ∑ ε i niσ
iσ
ρ (ε i )
arith
⇔
CPA
Vlaming, DV (1992)
Janis, DV (1992)
CPA: Coherent potential approximation
(“best single-site approx.”)
Soven (1967)
Taylor (1967)
• robust results for ρ (ε i ) arith
• cannot describe Anderson localization
Example: CPA results for phonon DOS for disordered cubic crystal
Elliot, Krumhansl, Leath (RMP, 1974)
Interactions + binary disorder:
Mott-Hubbard metal-insulator transition
at fractional filling
Anderson-Hubbard Hamiltonian
H = −t
∑σ
i,j ,
ci†σ cjσ + U ∑ ni↑ni↓ + ∑ ε i niσ
iσ
i
• Binary alloy disorder
x
1-x
Δ: disorder strength
# A,B-atoms: NA,B
# lattice sites: NL
Æ concentration x=NA/NL, 1-x=NB/NL
# electrons: N Æ filling n=N/NL
• Arithmetic averaging:
• DMFT(NRG)
Oi
arith
= ∫ d ε i P(ε i )Oi
Mott-Hubbard transition at fractional filling (binary alloy disorder)
0
No disorder
Byczuk, Hofstetter, DV (2004)
Mott-Hubbard transition at fractional filling (binary alloy disorder)
x=NA/NL, 1-x=NB/NL
0
Δc
No interaction
Byczuk, Hofstetter, DV (2004)
Mott-Hubbard transition at fractional filling (binary alloy disorder)
x=NA/NL, 1-x=NB/NL
0
Δc
Filling n=x: half filled LAB
Mott-Hubbard
transition at
fractional filling
Byczuk, Hofstetter, DV (2004)
Mott-Hubbard transition at fractional filling (binary alloy disorder)
Filling n=x=0.5
x=NA/NL, 1-x=NB/NL
Ground-state phase diagram
Hysteresis
Effective theory for
LHB for Δ U , t
← Uc = 6 x
x =0.5
= 3/ 2
Byczuk, Hofstetter, DV (2004)
Mott-Hubbard Transition
vs.
Anderson Localization
Anderson-Hubbard Hamiltonian
H = −t
∑σ
i,j ,
ci†σ cjσ + U ∑ ni↑ni↓ + ∑ ε i niσ
i
n=1
iσ
1/ Δ
Box disorder
Δ: disorder strength
Δ=0: Mott-Hubbard metal-insulator transition for U>Uc
U=0: Anderson localization for Δ > Δ c > 0 in d>2
d=2: QMC study
Chakraborty, Denteneer, Scalettar (2007)
1.
1.Can
Canboth
bothtransitions
transitionsbe
becharacterized
characterizedby
bythe
the(average
(averagelocal)
local)DOS?
DOS?
2.
2.Further
Furtherdestabilization
destabilizationof
ofcorrelated
correlatedmetallic
metallicphase
phaseby
bydisorder?
disorder?
3.
3.Are
Arethe
theMott
Mottinsulator
insulatorand
andAnderson
Andersoninsulator
insulatorseparated
separatedby
by
another
another(metallic)
(metallic)phase?
phase?
Anderson (1958)
Usually unknown
Approximation of PDF: calculate averages + moments
Which average?
ρi ( E )
arith
> 0 does not detect localization Why?
PDF of disordered systems: very broad with long tails
Æ system not self-averaging
Anderson localization:
ρ i ( E ) typical = ρ i ( E )
geometric
=e
Anderson (1958)
ln ρi ( E )
DMFT for Anderson-Hubbard model
lattice Green function
Dobrosavljevic, Pastor, Nikolic (2003)
Better “typical” values of the local DOS by the Hölder mean?
x ∈ \+ , q ∈ \
qÆ0 geometric mean
q=1 arithmetic mean
q < 0.5 : ρ q (ω = 0) = 0 at Δ cq (U )
Souza, Maionchi, Herrmann; cond-mat/0702355
Mott-Hubbard Transition vs. Anderson Localization
Anderson
insulator
Δc
metal
?
U c Mott-Hubbard
insulator
Anderson-Hubbard Hamiltonian
H = −t
∑σ
i,j ,
1/ Δ
ci†σ cjσ + U ∑ ni↑ni↓ + ∑ ε i niσ
i
iσ
Δ: disorder strength
Solution by DMFT(NRG)
Non-magnetic phase diagram; n=1,T=0
1
Local DOS
1
2
3
2
3
Critical behavior at
localization transition
Byczuk, Hofstetter, DV (2005)
Anderson-Hubbard Hamiltonian
H = −t
∑σ
i,j ,
ci†σ cjσ + U ∑ ni↑ni↓ + ∑ ε i niσ
i
1/ Δ
iσ
Δ: disorder strength
Non-magnetic phase diagram; n=1,T=0
• Disorder increases U c
• Interaction in/decreases Δ cA
Hubbard
Andersonand
Mottsubbands
insulator
neighboring
vanish
Æ Interactions may increase
metallicity
d=2:
Denteneer, Scalettar, Trivedi (1999)
Byczuk, Hofstetter, DV (2005)
Anderson-Hubbard Hamiltonian
H = −t
∑σ
i,j ,
ci†σ cjσ + U ∑ ni↑ni↓ + ∑ ε i niσ
i
1/ Δ
iσ
Δ: disorder strength
DMRG for disordered bosons in d=1 Non-magnetic phase diagram; n=1,T=0
Hubbard
Andersonand
Mottsubbands
insulator
neighboring
vanish
Rapsch, Schollwöck, Zwerger (1999)
Byczuk, Hofstetter, DV (2005)
Antiferromagnetism
vs.
Anderson Localization
Anderson-Hubbard Hamiltonian
H = −t
∑σ
i,j ,
1/ Δ
ci†σ cjσ + U ∑ ni↑ni↓ + ∑ ε i niσ
i
iσ
Δ: disorder strength
NN hopping, bipartite lattice, n=1:
Take into account antiferromagnetic order
Unrestricted Hartree-Fock, d=3
Tusch, Logan (1993)
Anderson-Hubbard Hamiltonian
H = −t
∑σ
i,j ,
1/ Δ
ci†σ cjσ + U ∑ ni↑ni↓ + ∑ ε i niσ
iσ
i
Δ: disorder strength
NN hopping, bipartite lattice, n=1:
Take into account antiferromagnetic order order
DMFT: Non-magnetic phase diagram
Magnetic phase diagram
No boundary between Slater- and Heisenberg AFI
Poster Å
Byczuk, Hofstetter, DV (2007), unpubl.