center for electronic correlations and magnetism university of … · 2019-11-12 · • dynamical...
TRANSCRIPT
Dieter Vollhardt
The art of modeling in solid state physics
Johannes Kepler Universität Linz; November 6, 2019
Center for Electronic Correlations and MagnetismUniversity of Augsburg
• Modeling in the natural sciences
• Step 1: From materials to models
• Electronic correlations
• Dynamical mean-field theory
• Step 2: From models back to materials
Outline
Modeling in the natural sciences
Observation Interpretation Modeling→ ↔
Micro-cosmos Macro-cosmos
Observation and interpretation
• What is observed?• How to interpret it?• How can it be explained?
Fundamental research
Requires idealization and abstraction (“reductionism“) models↔
The art of modeling
Weighing scale
Model (Archimedes)
Every-day use
©ge
ttyi
mag
es
Spherical lodestone: model of Earth
⇒ Earth is a magnet
1600 William GILBERT of Colchester
De Magnete
(“Father of magnetism“)
1269 Pierre PELERIN de Maricourt: spherical lodestone (=magnet) has poles
The art of modeling
Earth‘s magnetic field
©Sh
utte
rsto
ck
Known since antiquity
Papin (1680)
blac
inc.
com
The art of modeling
Steam engine
Every-day use
Simple models
Mechanical models: very successful
Electromagnetism
The art of modeling
vortices
Allowed to discuss the displacement current, but… Maxwell equations: 1864
J. C. Maxwell (1857):"I have been grinding at … a vortical theory of magnetism and electricity which is very crude but has some merits."
electrical current of particles rotation of vortices gh opposite motion of particles above gh
(induced opposite electric current) anti-rotation transmitted to vortices kl, …
layers of particles(“rolling contact”)
“It seems to me that the test of ‘Do we or do we not understand a particular point in physics ?’ is, ‘Can we make a mechanical model of it ?’ “ Lord Kelvin (1884)
Most models are not successful
Maxwell’s mechanical model of electromagnetism: Vortices in a molecular medium
The art of modeling
Bohr model of the atom Bohr (1913)
Circumference de Broglienu nλ=
+ Postulates:• discrete energies• frequency of emitted light • quantization condition for orbits
/n mE E h= −
• Correspondence principle• Semiclassical limit of quantum mechanics
Example: Iron (Fe)
Microscopic view: O(1023) interacting electrons + ions
The art of modeling
Properties of solids
Macroscopic view:
How to investigate/model?
M(T)
T TC
How to understand the• origin of ferromagnetism in Fe• T-dependence of the magnetization M ?
Simplest quantum model for interacting electrons in a solid:
modeling
Þ
H
eld, 2004
Theoretical investigation of materials
Fe: Interacting many-particle system Maximal simplification needed(… more art than science)
Hubbard model (1963)to explain ferromagnetism in transition metals
Step 1: From materials to models
How to explain ferromagnetism in transition metals?
skul
lsin
thes
tars
.com
Weiss model of magnetic domains (1906)
Alignment of “elementary magnets” in each domain due to a “molecular field“
(Weiss mean field)
Microscopic origin?
How to explain ferromagnetism in transition metals?
Z. Physik 31, 253 (1925)
web
.sta
nfor
d.ed
u
Exact solution in d=1: no order (also in d=3?!)
Ising(-Lenz) model (1925)
But: Magnetism is a quantum effect Bohr (1911), van Leeuwen (1919)
How to explain ferromagnetism in transition metals?
Classical spin model
Z. Physik 49, 619 (1928)
But: Electrons are mobile Bloch (1929)
Heisenberg model (1928)
Weiss molecular field originates fromquantum-mechanical exchange processes
com
plex
ity-
cove
ntry
.org
must include their kinetic energy + Coulomb interaction
How to explain ferromagnetism in transition metals?
Quantum spin model
Next year: Hubbard model ??
Not for another 34 years!
1929
Development of many-body theory for condensed matter:
• Field-theoretic/diagrammatic methods (Feynman 1949)
• Superconductivity (Bardeen, Cooper, Schrieffer 1957)
• Single-impurity model (Anderson 1961)
• Elementary excitations/quasiparticles, Fermi liquid theory (“Standard model of condensed matter physics“) (Landau 1956)
U
Non-interactingconduction (s-) electrons
+Immobile d-electrons with
interaction U on a single site ("impurity")
Single-impurity Anderson model Anderson (1961)
t
Us,d-hybridization V
+
• Characteristic 3-peak structure• non-perturbative energy scale("Kondo physics")
Bulla, Hewson, Pruschke (1998)
Single-impurity Anderson model Anderson (1961)
Non-interactingconduction (s-) electrons
+Immobile d-electrons with
interaction U on a single site ("impurity")
t
Two fundamental unsolved problems
Beginning of the 1960s
Ni,V 3d electrons form narrow bands (“correlated”): How to model?
Tem
pera
ture
(K)
Pressure (4 kbar/division)
Electronic Correlations
AB A B≠
quantifies correlations
AB A B−
Correlations are present when
Wigner (1934)
Definition of electronic correlations (I): Effects beyond factorization of the interaction (Hartree-Fock)
e.g., 2( ) ( ') ( ) ( ')n n n n n≠ =r r r r
Narrow band systems :
Interaction correlations magnetismkin intW E E<
Pauli principle
Unusual properties of correlated electron materials
• huge resistivity changes• gigantic volume anomalies• colossal magnetoresistance
• high-Tc superconductivity
• correlated metallic behavior at interfaces of insulators, …
• sensors, switches• thermoelectrics• high-Tc superconductor devices• functional materials: oxide heterostructures, …
Interesting for technological applications:
• How to explain these properties?• What is the minimal model?
Diagonal in momentum space(waves)
Diagonal in position space(particles)
,
, UH n D D n nσσ
ε ↑ ↓= + =∑ ∑k i i i ik i
k
Gutzwiller (1963)Hubbard (1963)Kanamori (1963)• tight binding
• extreme screening: local interaction allowedby Pauli principle
no classical analogue
Hubbard model
Minimal lattice model for correlated electrons
• How to solve?• No fully numerical solutions possible• Find good approximation (mean-field?)
Single-band model:
1. Construction by factorization
- Spin models (finite range interaction), e.g., IsingFactorization Weiss MFT
Mean-Field Theory (MFT)
- Spin S- Degeneracy N- Range of interaction R - Spatial dimension d or coordination number Z
}2. Construction by exaggeration
→∞
Hypercubic lattices: Z=2d
- Electronic models (local interaction), e.g., HubbardFactorization Hartree MFT
Ising model: all yield Weiss MFT
n n n n↑ ↓ ↑ ↓→i i i i
S S S S→i j i j
Mean-field theory of the Hubbard model
( )ωΣ
Metzner, DV (1989)
†
, ,σ σ
σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i
i j i
Purely local interaction:independent of d,Z
,d Z→∞→ • what simplifications arise ?• what type of MFT ?
n n n n↑ ↓ ↑ ↓≠i i i i
Quantum fluctuations neglected static (no correlations)
Local quantum fluctuations always present dynamic
No factorization: very different from Hartree-Fock
Mean-field theory of the Hubbard model
( )ωΣMüller-Hartmann (1989)
Janiš (1991)
Metzner, DV (1989)
†
, ,σ σ
σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i
i j i
Strong diagrammaticsimplifications
Purely local interaction:independent of d,Z
mutuallydependent
Self-consistent mean-field theory
Georges, Kotliar (1992), Jarrell (1992)
( )ωΣ
Mean-field theory of the Hubbard model
Janiš (1991)Self-consistent mean-field theory
†
, ,σ σ
σ↑ ↓= +− ∑ ∑H c c n nt Ui j i i
i j i
Metzner, DV (1989)
Purely local interaction:independent of d,Z
Dynamical mean-field theory (DMFT)
Hubbard model single-impurity Anderson model+ self-consistency condition
→∞→,d Z
Strong diagrammaticsimplifications
DMFT:
• Fully dynamical, but mean-field in position space
• New type of MFT for quantum particles
DMFT of the Hubbard model: Summary
,d Z →∞• Exact in
( , )ωΣ k ( )ωΣKotliar, DV (2004)
Characteristic features of DMFT
U
EEd+U/2Ed-U/2
U
Ed
Atomic limit (no kinetic energy)
( , )ωΣ k ( )ωΣ
Density of states/Spectral function
Kotliar, DV (2004)
EEd+U/2Ed-U/2
U
Ed
lowerHubbard bandincoherent
upperHubbard bandincoherent
Excitations at Fermi level
(quasiparticles)
fermions (with kinetic energy)
U( , )ωΣ k ( )ωΣ
Kotliar, DV (2004)
Characteristic features of DMFT
Density of states/Spectral function
coherent
U
Experimentallydetectable
(PES, ARPES, …)
Definition of electronic correlations (II):• transfer of spectral weight• finite lifetime of excitations
U( , )ωΣ k ( )ωΣ
Kotliar, DV (2004)
Characteristic features of DMFT
Density of states/Spectral function
Application of DMFTMott metal-insulator transition
Intermediate-coupling problemin
sula
tor
met
al
lowerHubbard band
upperHubbard band
quasiparticles
Mott metal-insulator transition
→
ParamagneticMott insulator
P←Kotliar, DV (2004)
Mott metal-insulator transition: phase diagram
V2O3κ-organics, ...
V2O3
P →McWhan, Menth, Remeika, Brinkman, Rice (1973)
ParamagneticMott insulator
Fermi liquid
Critical point
P →Kotliar, DV (2004)
V2O3κ-organics, ...
U←
Mott metal-insulator transition: phase diagram
Pustogow, Rösslhuber, Tan, Uykur, Böhme, Löhle, Hübner, Schlueter, Dressel, Dobrosavljević(arXiv:1907.04437)
U
ParamagneticMott insulator
Fermi liquid
Critical point
P →Kotliar, DV (2004)
V2O3κ-organics, ...
←
Mott metal-insulator transition: phase diagram
Step 2: From models back to materialsApplication of DMFT to correlated materials
Combination ?
, GGA
Non-perturbative approaches for real materials
UHel
d (2
004)
Hel
d (2
004)
+Local electronic correlations
Double counting correction
(Many-body theory: DMFT)
LDA+DMFTAnisimov, Poteryaev, Korotin, Anokhin, Kotliar (1997)Lichtenstein, Katsnelson (1998)
Material specific electronic structure(Density functional theory: LDA)
Computational scheme for correlated electron materials
=
-
Simplest approach:
Early application of DFT+DMFT
(Sr,Ca)VO3: 3d1 system
TheoryElectronic structure
180V VO∠ − − = °
162V VO∠ − − ≈ °
No correlation effects/spectral transfer
LDA+DMFT results
Constrained LDA:U=5.55 eV, J=1.0 eV
k-in
tegr
ated
spe
ctra
l fun
ctio
n A
(ω) , CaVO3
Osaka – Augsburg – Ekaterinburg collaboration:Sekiyama et al. (2004)
Pavarini et al. (2004)
Spectral function ( , )A ωkSrVO3:
Byczuk et al. (2007)
LDA
LDA+DMFT
Electronic correlations • band narrowing• quasiparticle damping
occupied states unoccupied states
LDA+DMFT for (Sr,Ca)VO3: Comparison with experiment
Correlation-induced3-peak structure
confirmed
Sekiyama et al. (2004, 2005) [Osaka – Augsburg – Ekaterinburg collaboration]
One-band Hubbard model on a generalized fcc lattice ( )Z →∞
Paradox “spins vs. electrons” reconciled
Curie-WeissBrillouin-function-typeferromagnetism of itinerantelectrons with non-integer magneton #
Intermediate-coupling problem
TC
Application of DMFTMetallic ferromagnetism
mean-field critical exponents
( ) , ( )c F cM T T T Tβ γχ −∝ − ∝ −
1 , 12
β γ= =
Ulmke (1998)
Ulmke (1998)
Curie-Weiss
mean-field critical exponents
( ) , ( )c F cM T T T Tβ γχ −∝ − ∝ −
LDA+DMFT
Microscopic origin of exchange interactions in Fe
Fe,Ni
TC
Lichtenstein, Katsnelson, Kotliar (2001)
Kvashnin et al. (2016)
One-band Hubbard model on a generalized fcc lattice ( )Z →∞
Brillouin-function-typeferromagnetism of itinerantelectrons with non-integer magneton #
1 , 12
β γ= =
0.35, 1.33β γ≈ ≈Exp.:
Application of DMFTMetallic ferromagnetism
Ulmke (1998)
Curie-Weiss
mean-field critical exponents
( ) , ( )c F cM T T T Tβ γχ −∝ − ∝ −
LDA+DMFT Fe at Earth’s core conditions
TC
Hausoel et al. (2017)Pourovskii (2019)
One-band Hubbard model on a generalized fcc lattice ( )Z →∞
Brillouin-function-typeferromagnetism of itinerantelectrons with non-integer magneton #
1 , 12
β γ= =
astr
obob
.are
avoi
ces.
com
Application of DMFTMetallic ferromagnetism
- transition-metal oxides- f-electron systems- cuprates, iron pnictides, …
Jacob, Haule, Kotliar (2009,2010)al-Badri et al. (arXiv:1811.05739)
Molecular electronics, quantum chemistry, ligand binding
Bulk materials
Heterostructures, interlayers, surfaces
Assm
ann
et a
l.(2
013) Okamoto, Millis (2004)
Janson, Held (2018)
Ni dimer
Cu nanowire Cu nanowire
Lechermann (2018)
DMFT: Applications from bulk matter to heterostructures
Lech
erm
ann
(201
8)
ca. 150 lecture notes freely available on:https://www.cond-mat.de/events/correl.html
DMFT: Generic mean-field theory of correlated electrons Wide field of applications
http://gamediv1.weebly.comHemberger et al. (2002)
Bruker: EPR in Life SciencePauli, Willmott (2008)
±∞
Perfetti et al. (2006)
Pustogow et al, arXiv:1907.04437