from gutzwiller wave functions to dynamical mean-field theory
TRANSCRIPT
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From Gutzwiller Wave Functions to Dynamical Mean-Field Theory
Dieter Vollhardt
Autumn School on Correlated Electrons DMFT at 25: Infinite Dimensions
Forschungszentrum Jülich, September 15, 2014
Supported by FOR 1346
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Outline:
• Electronic correlations • Approximation schemes: Mean-field theories, variational wave functions
• Gutzwiller wave functions, Gutzwiller approximation
• Derivations of the Gutzwiller approximation
• From one to infinite dimensions
• Simplification of many-body perturbation theory in
• Dynamical mean-field theory (DMFT)
• Applications: LDA+DMFT for correlated electron materials
d = ∞
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Correlations in fermionic matter
- Mott insulators/metal-insulator transitions - Ferromagnetism in 3d transition metals Liquid Helium-3 Kondo effect (1964)
Heavy fermion systems
- Fractional quantum Hall effect - High-Tc superconductivity Colossal magnetoresistance Cold fermionic atoms in optical lattices etc.
< 1950s
1950s
1960s
1970s
1980s
1990s
2000s
Hubbard model (1963)
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material realistic model
Þ modeling
maximal simplification ? Þ
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Hubbard model Gutzwiller, 1963 Hubbard, 1963 Kanamori, 1963
time
ˆ ˆ ˆ ˆn n n n↑ ↓ ↑ ↓≠i i i i
Static (Hartree-Fock-type) mean-field theories generally insufficient
†
, ,
ˆ ˆ ˆ ˆ ˆH c c nt U nσ σσ
↑ ↓−= +∑ ∑i j i ii j i
Dimension of Hilbert space L: # lattice sites
(4 )LO
Computational time for N2 molecule: ca. 1 year with 50.000 compute nodes
Purely numerical approaches (d=2,3): hopeless!
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Reliable, widely applicable, non-perturbative
approximation scheme for
correlated electron models
AND
PROFESSORSHIP
Best-known approaches: • Mean-field theories • Variational wave functions
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1) Construction by decoupling/factorization
AB A B→
i j i jS S S S→
e.g., spins:
Weiss molecular field theory (1907)
. .
. .
. .
. mean field
Mean-field theory (MFT)
Prototypical “single site” MFT
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2) Construction by continuum/infinity limit Spin S Degeneracy N Dimension d /coordination number Z
} ∞
Weiss MFT for spin models
Mean-field theory (MFT)
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Variational parameters
Expectation value of some operator
Energy expectation value exactE≥
*
*var 0i
ii
E
λ
λλ
∂= ⇒
∂Minimization determines variational parameters
Applications, e.g.: - Quantum liquids 3He, 4He - Nuclear physics - Correlated electrons - Heavy fermions - FQHE - High-Tc superconductivity
Variational Wave Functions
One-particle wave function
Correlation operator reduces energetically unfavorable configurations in
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Gutzwiller Wave Functions
Martin Gutzwiller (1925-2014)
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Gutzwiller (1963) Gutzwiller variational wave functions
Gutzwiller wave functions
intˆ
ˆ
HG
C
e FGλψ −=
One-particle (product) wave function
kin int†
, , ˆ
ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ i
i j i ii j i
ij
D
D
H H Hc c n nt Uσ σ
σ↑ ↓
= +
= +∑ ∑
Op. of local double occupation
Op. of total double occupation
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kin int†
, , ˆ
ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ i
i j i ii j i
ij
D
D
H H Hc c n nt Uσ σ
σ↑ ↓
= +
= +∑ ∑
Op. of local double occupation
Op. of total double occupation
, 0 1g≤ ≤intˆ ˆ ˆ
U
H UD De g
e e gλ
λ λ− ≡
− −= =1 00
g Ug U
=⇔== ⇔ = ∞
i
i
Global reduction of configurations with too many doubly occupied sites
“Gutzwiller projection“
Gutzwiller wave function
Gutzwiller (1963) Gutzwiller variational wave functions
intˆ
ˆ
HG
C
e FGλψ −=
Fermi gas
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Electronic configuration with D doubly occupied sites
Set of all different electron configurations with given D
⇒
Probability of the electron configuration iD
How to calculate?
Gutzwiller wave function
with
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Gutzwiller Approximation
Approximates quantum mechanical expectation values by counting classical spin configurations
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with
Gutzwiller approximation
Simplest approximation:
?
....
Probability for a configuration of electrons with spin σ
ND : # electron configurations with given D
⇒
const=
Neglect correlations between electrons: count classical configurations of electrons all probabilities equal
,↑ ↓
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Thermodynamic limit: Sum over D determined by the largest term:
2( ) 0DD
D
d g NdD
= /D L d≡→ ( )d g⇒
Physical meaning?
Gutzwiller approximation
L: # lattice sites
Nσ: # electrons with spin σ D: # doubly occupied sites
: # singly occupied sites with spin σ : # empty sites
: # spin configurations with given D
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Law of mass action for the “chemical reaction”
2DDg N
Quasi-chemical equilibrium: Typical mean-field result!
[ ][ ].... .....[ ][... . ]...
=
+ + 2g
Gutzwiller approximation
Norm , ( )D D g=
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Gutzwiller approximation
Similarly:
U int
ˆˆ G G
GG G
UDH E UD
ψ ψψ ψ
= ≡ =
01 ( ( ), )ˆ ( , )G
G
E d g nH q d n UdL L
ε≡ = − +
*
0 ( *)G
d
E g dd
∂= ⇒
∂
Interaction energy
Ground state energy
*d d≡In the following:
Study origin of band ferromagnetism Gutzwiller (1965)
2nn n↑ ↓= =
2
2
2(1 2 )( ) , =1-1
d d dq nδ δ δδ
− − + +=
−Interactions multiplicative renormalization of kinetic energy
Kinetic energy
0
0kinkin kin kin( , )
ˆˆ G G
GG G L
HH q d EnE
ε
ψ ψψ ψ
−
= ≡ =
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Gutzwiller-Brinkman-Rice theory Brinkman, Rice (1970)
kin int: 0 and 0cU U E E→ → → Electrons localize no charge current
1 14 c
UdU
= −
2
1c
UqU
= −
08cU ε=1n =
Gutzwiller approximation describes a correlation induced ("Mott") metal-insulator transition as in V2O3
Applicable to normal liquid 3He? Anderson, Brinkman (1975)
2
0 1G
c
E UL U
ε
= − −
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Properties of normal liquid 3He
Why does χs become so large?
Spin susceptibility
Fermi liquid theory:
Anderson, Brinkman (1975)
Paramagnon theory: 1,
1
os
s FF
U NUNχχ −= →
− 3He close to ferromagnetic instability
Effective mass m*/m
Wilson ratio
3He close to Mott (localization) transition?
Strongly renormalized Fermi gas
Greywall (1983) DV (1984)
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Gutzwiller approximation and Fermi liquid theory
Change of the ground state energy
Correspondence with Landau Fermi liquid theory:
FLEδ
Brinkman, Rice (1970) DV (1984)
change of: particle energy
distribution of particles
0 0 0 *( )G G
d
E Eq Ud q qL L
δε δε δ ε= + ⇒ = +
⇓ 1l <
0 0
bare* (0) (0), *
*p pm mm m
N N σ σε ε= =
.........................................
*mq
m⇒ =
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Gutzwiller approximation
Gutzwiller approximation and Fermi liquid theory
DV (1984)
Fermi liquid relations Spin susceptibility
Compressibility
c
UUU
= 0 0( ) ( )a sF U F U = −
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Gutzwiller approximation
Gutzwiller approximation and Fermi liquid theory
DV (1984)
Fermi liquid relations Spin susceptibility
Compressibility
c
UUU
=
034
3 const4
ca U UF p→⇒ − =→−
Wilson ratio 4 constcU U→→ =
No divergence!
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Liquid 3He: "almost localized Fermi liquid" (close to Mott transition at Uc)
Gutzwiller approximation and Fermi liquid theory
Spin susceptibility
34
← −
Conclusion:
DV (1984)
• no magnetic instability
• χs becomes large because m*/m becomes large!
Gutzwiller approximation
Gutzwiller approximation
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Gutzwiller approximation and Fermi liquid theory
Spin susceptibility
34
← −
Conclusion:
DV (1984)
• no magnetic instability
• χs becomes large because m*/m becomes large!
Gutzwiller approximation
Gutzwiller approximation
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Gutzwiller approximation: 1) Quasi-classical, non-perturbative approximation scheme for correlated fermions 2) Describes a: - correlated Fermi liquid with a - Mott metal-insulator transition for n=1 at a critical Uc 3) Shows mean-field behavior Open question: Systematic derivation by more conventional methods of quantum many-body theory?
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Introduce local Bose fields
ie ip ↓ idip ↑
Hubbard model
Local constraints
Saddle point approximation results of Gutzwiller approximation
iz σ : renormalization factors (operators)
Lecture G. Kotliar Slave-boson mean-field theory
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Metzner, DV (1987, 1988)
Diagrammatic evaluation of ˆ
G GG
G G
HE
ψ ψψ ψ
=Gutzwiller wave function
int
2 2 1
1
ˆ ˆ
( )
( , )
( )1
G G
mm
m
H U D LU
LUg
g n
ng
d
c∞
−
=
= =
= −∑
1)
( )mc nDiagrammatic representation of
i∏
, 0 1g≤ ≤
0 † 0, 0
FT
ij i jg c c nσ σ σ σ= ↔ k
(probability amplitude/ momentum distribution) Diagrams for the Hubbard interaction
(φ4 theory)
( )mc n
Lines correspond to
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Metzner, DV (1987, 1988)
2) kinˆ ˆ( )
n
GGH n
σ
σσ
ε= ∑k
kk
k
( )mf σ k
2
2
( , ) ... ... ( ( )1)mm
mn g n g f σσ
∞
=
= + −∑k k
Diagrammatic representation of
Diagrammatic evaluation of ˆ
G GG
G G
HE
ψ ψψ ψ
=Gutzwiller wave function
i∏
, 0 1g≤ ≤
Diagrams for the kinetic energy
( )mf σ k
0 † 0, 0
FT
ij i jg c c nσ σ σ σ= ↔ k
(probability amplitude/ momentum distribution)
Lines correspond to
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Metzner, DV (1987, 1988)
Diagrams for the kinetic energy
( )mf σ k
Diagrams for the Hubbard interaction
( )mc n
d=1: analytic calculation of all diagrams possible
Diagrammatic evaluation of ˆ
G GG
G G
HE
ψ ψψ ψ
=Gutzwiller wave function
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( , )d g n
density of doubly occupied sites int )ˆ ˆ ( ,G G
H U LUdD g n= =1)
momentum distribution 2) kinˆ ˆ( )
n
GGH n
σ
σσ
ε= ∑k
kk
k
Minimization of EG(g,d(g),n,U):
kin intˆ ˆ
GG GH H E⇒ + ≡
0
2 2
014
l:
n( / )GtU EU U
επ ε = −
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Gebhard, DV (1987, 1988) Correlation functions
Spin, density, empty sites, doubly occupied sites, local Cooper pairs
E. g., spin-spin correlations:
U = ∞
d=1: analytic calculation of all diagrams possible
1n =
Excellent agreement with exact result for antiferromagnetic Heisenberg chain (j=1,2)
Gutzwiller wave function = Anderson‘s RVB state in 1D
Exact result for S=1/2 antiferromagnetic Heisenberg chain with exchange coupling Haldane(1988), Shastry (1988)
Very interesting wave function! d=2,3?
One of 4 sets of diagrams
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Analytic evaluation of all diagrams not possible in d=2,3
Metzner, DV (1988)
Diagrams for the kinetic energy
( )mf σ k
Diagrams for the Hubbard interaction
( )mc n
Diagrammatic evaluation of ˆ
G GG
G G
HE
ψ ψψ ψ
=Gutzwiller wave function
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i j
dimension Diagrams for the Hubbard interaction
( )mc n
Diagrammatic evaluation of ˆ
G GG
G G
HE
ψ ψψ ψ
=Gutzwiller wave function
Calculation of individual diagrams:
Numerical check by Monte-Carlo integration :
3 4
d
n
→∞
Each diagram takes simple value # lines
2
n n n↑ ↓= =
3
4
2 , 13 2
, 2
d
n d
n = = ∞
=
Example:
Great simplifications for : internal momenta become independent
d →∞
Metzner, DV (1988)
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i j
Numerical check by Monte-Carlo integration
dimension
Conclusion: Gutzwiller approximation gives the exact result for expectation values calculated with the Gutzwiller wave function in d →∞
Diagrammatic evaluation of ˆ
G GG
G G
HE
ψ ψψ ψ
=Gutzwiller wave function
:d →∞Calculation of individual diagrams:
Diagrammatic derivation of Gutzwiller approximation
Metzner, DV (1988)
:3 4
d
n
→∞
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Optimal form of Gutzwiller wave functions in d = ∞
arbitrary one-particlewave f
ˆ0
unction
DG gΨ = Ψ
Gebhard (1990) Re-write by gauge transformation ˆ
0 0
chose local chemical potentialssuch that all Hartree bubbles disappear in
i ii
n
d
gσ
σσµ−
=∞
∑Ψ = Ψ
In spite of diagrammatic collapse: Remaining diagrams have to be calculated with difficult 0Ψ
0,
,
ˆi j ij iG ij i
dH t q q g U dσ σ σ
σ
=∞
= − +∑ ∑
0 0ˆ( , ), iq d n n n nσ σ σ σ→ = Ψ Ψ
For arbitrary : 0Ψ
Diagrammatic derivation in of the Kotliar-Ruckenstein slave boson saddle point solution
d = ∞
renormalization factors
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limit for lattice fermions d →∞
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= Probability for hopping j NN i Amplitude for hopping j NN i 2 1Z
∝
Amplitude for hopping j NN i 0,
1 1or ijgZ dσ= ∝⇒
In general:
0,
†kin 00
, ( )1
ˆ ˆ ˆ
ijZZ
gNN
H c ct
σ
σ σσ
∝
−= ∑ ∑ i ji j i
Probability amplitude for hopping j NN i
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or Z d→∞→
1Z1
Z1Z1
Z
Example:
Great simplification of many-body perturbation theory, e.g., self-energy diagram purely local
Collapse of all connected, irreducible diagrams in position space
or Z d→ ∞→
0,
†kin 00
, ( )1
ˆ ˆ ˆ
ijZ gZ
NNH c ct
σ
σ σσ
∝
−= ∑ ∑ i ji j i
ijδ
Metzner (1989)
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0 † 0, 0
FT
ij i jg c c nσ σ σ σ= ↔ kValid for probability amplitude/momentum distribution and the full, time-dependent propagator, since
or Z d→∞→
1Z1
Z1Z1
Z
Example:
Collapse of all connected, irreducible diagrams in position space
or Z d→ ∞→
0,
†kin 00
, ( )1
ˆ ˆ ˆ
ijZ gZ
NNH c ct
σ
σ σσ
∝
−= ∑ ∑ i ji j i
ijδ
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Collapse of all connected, irreducible diagrams in position space
Excellent approximation for d=3
or Z d→∞→
1Z1
Z1Z1
Z
Example: correlation energy of Hubbard model
22E /U ∝
or Z d→ ∞→
0,
†kin 00
, ( )1
ˆ ˆ ˆ
ijZ gZ
NNH c ct
σ
σ σσ
∝
−= ∑ ∑ i ji j i
ijδ
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Excellent approximation for d=3
or Z d→∞→
1Z1
Z1Z1
Z
Example: correlation energy of Hubbard model
22E /U ∝
1Z∝
*t
Zt
= Scaling †kin 00
, ( 1
)
ˆ ˆ ˆ
ZZ
NNtH c cσ σ
σ
∝
= − ∑ ∑ i ji j i
*JZJ
=Classical rescaling
*t
Zt
= Quantum scaling
or Z d → ∞
ijδ
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• Fermi liquid • Only Hubbard interaction remains dynamical
ωΣ ⇒( , )k
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Solution of the Hubbard model in : Towards dynamical mean-field theory
d →∞
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Self-consistent DMFT equations:
Generalization of coherent potential approximation (CPA) to interacting lattice fermions
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Dynamical (single-site) mean-field theory
free electrons in a dynamic potential Σ(ω)
0( ( ))G ω ω= −Σ
Generalization of coherent potential approximation (CPA) to interacting lattice fermions
Numerical solution?
( )ω ε µ ω− + − Σ
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Lecture A. Georges
Solution of the Hubbard model in : = Mapping to a single-impurity Anderson model + self-consistency condition
d →∞
Jarrell (1992)
Preprint May 6, 1991
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(i) Effective impurity problem: “local propagator“
(ii) k-integrated Dyson equation (“lattice Green function“: lattice enters)
, ...............................single-site ("impurity") action A
“Impurity solver“ QMC Hirsch-Fye (1986) Jarrell (1992) ED Caffarel, Krauth (1994), Si et al. (1994) NRG Bulla (1999) …
Self-consistent DMFT equations
( )ω ε µ ω− + − Σ
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( , )ωΣ k ( )ωΣKotliar, DV (2004)
DMFT: local theory with full many-body dynamics
Mott-Hubbard metal-insulator transition
Spec
tral
fun
ctio
n
U
Review: Georges, Kotliar, Krauth, Rozenberg (RMP, 1996)
The solution of the Hubbard model in provides a dynamical mean-field theory (DMFT) of correlated electrons
d →∞
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LDA+DMFT for Correlated Electron Materials
Lecture A. Lichtenstein
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How to combine?
Held (2004)
Hel
d (2
004)
time
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Material specific electronic structure (Density functional theory: LDA, GGA, ...) or GW
Computational scheme for correlated electron materials:
+ Local electronic correlations (Many-body theory: DMFT)
X+DMFT X=LDA, GGA; GW, …
Anisimov, Poteryaev, Korotin, Anokhin, Kotliar(1997) Lichtenstein, Katsnelson (1998)
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Material specific electronic structure (Density functional theory: LDA, GGA, ...) or GW
Computational scheme for correlated electron materials:
+ Local electronic correlations (Many-body theory: DMFT)
“LDA+DMFT”
Anisimov, Poteryaev, Korotin, Anokhin, Kotliar(1997) Lichtenstein, Katsnelson (1998)
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Material specific electronic structure (Density functional theory: LDA, GGA, ...) or GW
Computational scheme for correlated electron materials:
+ Local electronic correlations (Many-body theory: DMFT)
LDA+DMFT
Held, Nekrasov, Keller, Eyert, Blümer, McMahan, Scalettar, Pruschke, Anisimov, DV (Psi-k 2003)
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Conclusions and Outlook
• Quantum many-body perturbation theory for lattice fermions greatly simplifies in • Calculations with Gutzwiller wave function in = Gutzwiller approximation • DMFT: canonical mean-field theory for correlated electrons • Goal: Development of “LDA+DMFT“ into a computational scheme for correlated materials with predictive power
d →∞
d →∞