kvantni materiali : od pasovne teorije do koreliranih...
TRANSCRIPT
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Kvantni materiali : od pasovne teorije do koreliranih
elektronov
Jernej Mravlje
Gradivo za predavanja pri specialističnem seminarju: "Izbrane teme iz
teoretične fizike"
V Ljubljani, Feb 2018.
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Redneck perspective on strong correlations
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● a working-class white person from the southern US, especially a politically reactionary one.
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Left-wing radicals vs. conservative
● Left-wing:
– emergence (Phil Anderson, Nobel ‘77)
–
● Right-wing
– atomistic details (Walter Kohn, Nobel prize 98)
From : http://condensedconcepts.blogspot.si
. .. we have the conservatives who believe that the key ingredients are atomistic detail, good density functionals, perturbation theory, mean-field theory, and the random-phase approximation.
New concepts and methods are not really needed. They have a good system [just like capitalism].
In particular, we don't need a revolution, just bigger computers!
Completely new conceptual structures and techniques are needed. We must go beyond Landau's mean-field theory and Fermi liquid theory: topological order, quasi-particles with fractional quantum numbers, AdS-CFT, quantum criticality, ...
To the barricades!
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To balance Rok’s part I will discuss more the redneck part
● Density functional theory
● Dynamical mean-field theory(a centrist thread - a bridge between left and right)
● The idea was also to bring the discussion in contact to recent research work ruthenates and on TaS
2 (next week)
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Some (10?) facts about SCES
(Initially, I wanted to make 10 – like 10 commandments from the bible. Then it was 7, like 7 things one should absolutely avoid on the first date. I ended up at only a few (tend to forget ...))
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Fact 1: there is a thing called a DFT and it describes the bandstructure of
many compounds well
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Fact 1: there is a thing called DFT and it describes the bandstructure of
many compounds well
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Bandstructure
DOS
E
● k-resolved spectrum A(k,E)=Σknδ[E-E
n(k)];
DOS=Σkδ[E-E(k)]. DOS determines many
properties of solid state
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Well: Fact 2: Several experimental techniques exist that allow
quantitative comparison with theory
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Measuring spectra PES,ARPES
● Probes occupied one-electron states
I(k,E)=f(E) A(k,E)
● PES : k-integrated, probes DOS
● ARPES : k-resolved, probes A(k,E)
http://www.stanford.edu/group/photontheory/ARPES.html
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ARPES
● Probes occupiedpart of spectral function
● Energy resolution
10meV (sinhrotron)1meV(laser)
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STS (measures ~DOS)
● STM :
DOS (sample)
E
DOS (metal)
E
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Optical conductivity
● Also gives information about DOS (more precisely, joint DOS)
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● Schroedinger equation for
there is a bijective relation between V(r), n(r). V(r) defines the ground state density. The relation can be inverted. There is a unique functional
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Fact 1: there is a thing called DFT and it describes the bandstructure of
many compounds well
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ARPES on Cu
● Sharp excitations over
wide energy range
● Agrees with band theorycf. H. Alloul, Springer-Verlag 2011:
Symbols: peak of I(k,E)
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This DFT looks quite neat. What is it? (blackboard)
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● Comments: information about exact energy of homogeneous electron gas is built in
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● LDA is ab-initio technique: you put in coordinate of atoms and you get E (can investigate which structures are stable)
● It describes state in terms of auxiliary Kohn-Sham states
● Energies of these eigenstates for some reason are close to the experiments for several cases (noone really knows why)
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● The energies also have meaning: take SrVO3
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Transition metal oxides; structure and band-structure
Oxygen
3d (t
2g )
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An observation
● Bands quite narrow (2eV) in contrast to Cu (~10eV)
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Part 2
When LDA breaks down:
electronic correlations
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Redneck perspective on strong correlations (part 2)
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From past week
Fact 1. there is a thing called DFT(LDA) and it describes the bandstructure of many compounds well
Fact 2. Several experimental techniques exist that allow quantitative comparison with theory
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● LDA breaks down when interactions get strong
● Electronic correlations and Mott transition
● DMFT
● Sr2RuO
4: Correlated metal with low coherence
scale due to Hund's rule coupling
● Spin-liquid behavior in TaS2
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Resistivity, magnetic susceptibility, and specific heat : band-insulators
vs. metals
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Resistivity, magnetic susceptibility, and specific heat : band-insulators
vs. metals
Note: band insulator needs even # of els. / band.
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Breakdown of band picture: correlated metals and Mott
insulators
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Fact 3: strong Coulomb repulsion leads to a breakdown of band theory
and occurrence of Mott insulator
● Partial breakdown: quasiparticles with large renormalizations compared to band-theory
● Complete breakdown: Mott insulator
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Partial breakdown (Sr2RuO
4)
● Completely differen
picture.
● Coherent excitatio
only up to a 0.1e
● Heavy quasipartic
(E=k2/2m, dE/dk
Red: band-theory Strong renormalizations
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Transition metal oxides; structure and band-structure
Oxygen
3d (t
2g )
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TMOs: structure and el. structure
● PES: satellites (not present in LDA)
● ARPES: coherent excitations
only up to a 0.5eV
● Renormalization of slope (mass) by 2Yoshida, …, ZX Shen (2008)
c
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TMOs: photoemission
● PES:atomic satellites (not presen
● QP peak (narrowed by 2 from LD
● ARPES: coherent excitations
only up to a 0.5eV
● Renormalization of slope (mass) b
Yoshida, …, ZX Shen (2008)
c
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La2CuO
4
● 9 els in Cu 3d orbitals: 1 hole in x2-y2
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Mott transition: La2CuO
4 is (Mott)
insulator● PES shows a 2eV gap instead of a metal
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Optical conductivity also shows gap
Uchida'91
Notice overall LDA describes result well, only low energy spectral weight ismissing
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On doping, metallicity is restored; filling controlled Mott transition
● (Actually, more than that : doped LSCO is a high Tc superconductor!)
Ando'04
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Band-width controlled driven Mott transition
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Fact 4: orbitals matter!Why 3d special?
● 3d orbitals don't have nodes, reach further into core
● -> charge is screened less
● -> kinetic energy smaller
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Values of interaction can also be calculated
● Atomic Uat(3d orbitals) ~e2/4πε
0r ~ 15 eV
● In oxides screened to U~5eV
● 4d orbitals ~2.5eV ; 5d orbitals 1.5eV
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● One has an insulating state in a half-filled band
● Mott insulator
● Hubbard model: takeband of x2-y2 orbitalsand supplement it by Hubbard repulsion U
● LDA bands as a starting point for many body calculations
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Hubbard model
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Mott picture of the Mott transition
● (atomic : blackboard)
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Fact 5 : Mott insulators are magnets
● localized electrons/site = spin
● Spins are coupled by exchange interaction -> Heisenberg model and magnetism
● Magnetic susceptibility rises as one cools down and magnetism arises
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● Mott picture: adiabatic continuity with an insulator, but not with a metal:
● Does not described correctly doped Mott insulator
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Brinkman-Rice picture
● Solving Hubbard model with a variational ansatz
Doping driven: Z ~ δ
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● Mott transition: on metallic side adiabatic continuity with a noninteracting gas (with reduced bandwidth by Z), on
insulating with an atom.
Fact 6: DOS of correlated metal consists of quasiparticle peak and
Hubbard bands
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DMFT
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Kondo effect
Resistivity minimum: associated to presence of magnetic impurities
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RG picture of Kondo: infrared slavery (like quarks)
● Anderson model -> Kondo model -> Kondo effect
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NRG results
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Spectral function of Anderson model
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● Good numerics to calculate, tests in quantum dots and transmission through molecules
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Back to bulk
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Dynamical mean-field theory (DMFT)
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Dynamical mean-field theory (DMFT)
In limit of large d, self energy becomes local (independent of k)
Georges et al. RMP'96
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DMFT results
● single orbital Hubbard model
● Metal-insulator transition
Georges, Kotliar,Krauth,Rozenberg,RMP'96
INT
ER
AC
TIO
N (U
)
ATOMIC-like
BAND-like
Quasiparticle band (weight Z)
Atomic satellites (weight 1-Z)
Uc
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ARPES
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LDA+DMFT
● Take LDA bands
● Describe interactions with DMFT (substracting mean-free part included in LDA).
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Sr2RuO
4
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Sr2RuO
4 properties
p-wave supercond.
Tc ~2K
Correlated metal: Fermi liquid, (m*/m~4)
4 el. in Ru t2g
orbitals
Maeno et al., Nature'94
Rice and Sigrist , J.Phys.CM'95
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Fermi liquid behavior below 20KFact 7: In Fermi liquids ρ~T2
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Fermi surfaces
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Sr2RuO
4: el. structure
LDA and experiments give very similar Fermi surfaces. Shape can be understood (blackboard)
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Strong renormalization. ~ 4
Band-width 3eV, U~2.5eV.
(SrVO3 had
renormalization 2, but U~5eV)
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Going multi-orbital:new ingredient:Hund's rule coupling● “Bus seat rule”
● Maximize 1st spin, 2nd orbital momentum, 3rd spin-orbit
● Coulomb interaction → direct repulsion and exchange
3-orbital: t case
Vc=1/(r-r')
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DMFT results: 3orbital Kanamori model
● H=Hband
+Hatomic
● Large and filling-dependent effects of J!
● 1. shift of Uc 2.additional effect on correlations
● At N=2, correlated state (a “Hund's metal”) far from Mott insulator (for physical values of J)
Quasiparticle weight Z ~ (mLDA
/m):
L.de Medici, JM, and A. Georges,PRL'11
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Why? : Two effects of J 1st modified atomic charge gap
● Effective interaction
Ueff
=E(N+1)+E(N-1) – 2 E(N)
- U-3J away from half-filling (Ueff
diminished by J)
- U+(M-1)J at half filling (Ueff
increased by J)
● Slater all d-states Hamiltonian (# of orbs. M=5)
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2nd effect: J suppresses tunneling
● J lowers atomic degeneracy, blocks orbital fluctuations, hence prohibits some of the hopping
● Example: Create a charge excitation in half-filled two orbital problem. J allows it to move only in one of two possible ways.
● Suppression of tunneling → Kondo
temperature known for a
long time in Kondo problem
(Schrieffer 1967, Okada, Yosida 1973, Jayaprakash et al, 1981, Nevidomskiy-Coleman 2009).
Yanase et al, J.Phys.Soc.Jpn. '97
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A localized perspective
single occ. half-filled other(“Janus”)
atomic deg.
E ofchargeexcit
drives
6 (unchanged by J) 20 (supp → 4 by J) 15 (supp → 9 by J)
diminished increased diminished
metallic insulating incoherent metallic
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Quasiparticle weight Z ~ (mLDA
/m):
L.de Medici, JM, and A. Georges,PRL'11
Correlated state, far from Mott, ruthenates and pnictides.
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Coherence-incoherence crossover in Sr
2RuO
4
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ARPES
Shen et al.,PRL'07 Wang et al.,PRL'04 Ingle et al.,PRB'05
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Magnetic response
Pauli susceptibilty1/T
1 ~ T
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Fact 8: Many correlated electron systems are well described by
LDA+DMFT
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TaS2
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TaS2
● TaS2 one electron in Ta d-bands (local t
2g env.)
● Layers of triangular lattices
● 5-d electrons (small U)
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Mott transition
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● STS shows insulating behavior (also optics)
● Small gap (0.1eV)
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● Actually, a set of structural transitions. Mott state is realized in CDW at low-T.
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Narrow band close to FL following structural transition
● 1 band per cluster of 13 Ta atoms relevant at low energies
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● If it is a Mott insulator, what do spins do ?
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Mott insulator, but Pauli susceptibility?
● Pauli magnetic susceptibility
J. Phys. C: Solid State Phys. 18 (1985)
3175.
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● Finite linear in T specific heat. Like in metals.
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Collaboration with F5/F7
● Klanjsek et al, Nphys'17
● No magnetic order (muon spectroscopy)
● 1/T1 shows unusual T2 dependence
● Spin liquid!
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Summary
● Atomistic/ band-structure point of view on correlated electron systems
● LDA describes band-structure of many compounds well, but breaks down when U is strong
● Mott transition ; DMFT
● Sr2RuO
4 : low coherence scale & Hund's
coupling
● TaS2 Mott insulator at small U; spin-liquid
behavior
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Fact 8: Many correlated electron systems are well described by DMFT (LDA+DMFT)
Fact 4: orbitals matter!
Fact 3: strong Coulomb repulsion leads to a breakdown of band theory and occurrence of Mott insulator
Fact 1: there is a thing called DFT and it describes the bandstructure of many compounds well
Fact 2: Several experimental techniques exist that allow quantitative comparison with theory
Fact 6: DOS of correlated metals consists of quasiparticle peak and Hubbard bands
Fact 5: Mott insulators are magnets
Fact 7: in Fermi liquids ρ~T2
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Thank you!