De latome au 1 supraconducteur haute temprature critique O. - - PowerPoint PPT Presentation

De l’atome au supraconducteur à haute température critique

• O. Parcollet

Institut de Physique Théorique CEA-Saclay, France

1

Quantum liquids

2

Ultra-cold atoms in optical lattices “Artificial solids”

• f atoms & light

+5'

Correlated metal/superconductors at interface of oxides

SrTiO3/LaTiO3

Ohtomo et al, Nature 2002

Quantum many-body systems, fermions (or bosons), with interactions, at low temperature

Materials High Temperature superconductors Transition metal oxides,

Cuprate (1986) Fe-Based (2008)

Quantum liquids : motivations (I)

• Fundamental issues : physical phenomena, new states of matter
• Fermi liquids
• Metal-insulator transition
• Superconductors
• Quantum magnetism (Ferro/Antiferromagnetism, ...)
• Quantum phase transitions.
• Quantum Hall effect
• ...
• Quantum, collective phenomena
• Require a specific theoretical description : we can not “simply” solve

the Schrödinger equation for the N-bodies. “More is different”

• Experimentally driven field.

3

Quantum liquids : motivations (II)

• Material science : quantitative questions & applications
• Electrons in a solids = “electronic structure”
• Determines a lot of physical properties of a material :
• Electric conductivity
• Interaction with light : optical conductivity, ...
• Magnetism
• Volume, thermal coefficient.
• Thermoelectric effect (Seebeck, Peltier).
• Challenge : Material design

Make this a (more) predictive science Crucial for many applications.

4

+5' Fe-Based high temperature

superconductors (2008)

• Bloch’s theorem : Independent electrons in periodic potential

form bands = eigenstates of H labelled by band index n and wave vector k

Fermions in a periodic potential

5

 ~2 2mr2 + V

• |Ψkni = ✏kn|Ψkni
• Quantum many-body state = Slater determinant of these states.
• Fermi sea, Fermi surface, ...

U"

'/

7

flo*;uJ

;"k*-hJ

I

U"

'/

7

flo*;uJ

;"k*-hJ

I

Metal Band Insulator

T=0

✏F D(✏) D(✏) Gap

Fermi Level

D(ε) =

• k

δ(ε − εk)

Density of states

Interactions ?

• Kinetic energy

Characteristic energy scale = D = 1-10 eV

• Interaction energy :

Coulomb interaction matrix element : U ~ 1-10 eV

• U and D of the same order of magnitude.
• Why is this independent electrons picture of any use ???
• Short answer :
• Fermi liquid theory (Landau 50’s)
• Density functional theory (Kohn, 60’s).
• ... and in some (many) systems, it does not work :

⟶ strongly correlated (quantum) systems

6

Fermi Liquid

• At low energy, a regular fermionic liquid (metal) ressembles

a gas of quasi-particles = renormalized weakly interacting fermions, with a weight Z, an effective mass m*, lifetime

7

ARPES Quasi-particle peak A(k, ω) = 1 π Im

• dxdtei(kx−ωt)iθ(t)[c(x, t), c†(0, 0)]⇥
• L. Landau, 50’s

Fermi Liquid (II)

• Fixed point of Renormalization Group.
• Determines low temperature behavior of various quantities, e.g. :
• magnetic susceptibility
• specific heat
• Not a weak coupling theory.

In some material m*/m > 100 !

• A low energy theory, below coherence energy E_coh.

For a good metal (Cu), E_coh is very large (~ eV).

• A fundamental notion for many theories, e.g. BCS.

8

χ(T) ∝ cte Cv ∝ T

• pper

Density functional theory

• Effective way to compute electronic

structure in weakly correlated systems.

• Functional of the electronic density.

(Kohn-Hohenberg theorem).

• Solve using an auxiliary problem :

independent electrons in an “effective” Kohn-Sham potential VKS, determined self- consistently.

• Interactions taken into account in

VKS

• Effective one-body physics.

9

Band structure of Cu For simple metals, we have the concepts and predictive computational tools. What about more complicated cases ?

• W. Kohn, 60’s

Computation Experiments

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Mott transition

• Take a textbook metal : electrons on a lattice, one per site on

average (half-filled band)... ... add strong on-site Coulomb repulsion : charge motion frozen.

Mott insulator

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Mott insulator

Large Coulomb repulsion U ∼ eV ∼104 K

Electron, spin +1/2 Electron, spin -1/2

H = −

• ⇤ij⌅,σ=,⇥

tijc†

iσcjσ + Unini⇥,

niσ ≡ c†

iσciσ

δ = 1 ⇥n↑ + n↓⇤

Hubbard model

• N. Mott, 50’s

Interaction Driven Mott Transition

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Mc Whan et al, 1973

V2O3

First order transition

100 200 300 400 500 600 700 800 10 20 30 40 50 60 70 80

A.F insulator

Semiconductor Bad metal Fermi liquid Mott insulator P (bar)

T (K)

P

c 1 (T)

P

c 2 (T)

T

* Met

!(")max T

* Ins

(d#/dP)max

• P. Limelette, et al. PRL 91, 016401 (2003)

κ-(BEDT-TTF)2Cu[N(CN)2]Cl

• Vary pressure P ⇔ 1/U

Critical Point

2-d organics : resistivity versus T and P

(but has a simple hubbard modelization)

Mott Metal

Example of “application” : Intelligent windows

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Morin et al.1959

1000/T Conductivity (1/(ohm.cm))

TiO2 VO2 SiO2 dTiO2 dVO2 dSiO2=3mm Air Air

• Use Mott transition in

VO2 (film).

• High temperature : metal.
• Lower temperature : insulator.
• Reliable computation of material

specific electronic properties ?

340 K

• Figs. from S. Biermann

Strongly correlated (Mott) metals

• Adding holes in a Mott insulator

(chemical doping)

• Holes form a liquid : a

(correlated) metal.

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Holes = charge carriers

• Mott metals are fragile :
• Many instabilities.

Rich equilibrium phase diagrams.

• Low coherence energy scale, e.g. can vanish at the transition
• Ideal candidates to go far from equilibrium, beyond linear response.

Mott transition: an intermediate coupling problem

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U/t

metal metal Mott insulator

?

δ

Interaction driven Doping driven How is the metal destroyed close to a Mott transition ?

δ = 1 ⇥n↑ + n↓⇤

A simple correlated (Mott) metal

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Hubbard band Quasi-particle peak

Theoretical expectation

Photoemission experiments (not k resolved)

Competition : QP peak (delocalized) Hubbard bands (atomic like excitations)

Sekiyama et al. PRL 93, 2004

• 3
• 2
• 1
• 3

Intensity (arb. units) (a)

SrVO3 Bulk V 3d

PES FLAPW FLAPW (x 0.6 in energy)

Narrowing of quasiparticle bands due to correlations (Brinkman-Rice)

• High-Tc cuprate superconductors

(Bednorz, Muller1986)

Cuprate high-Tc superconductors

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La2−xSrxCuO4, Bi2Sr2CaCu2O8+δ

No FL FL SC PG AF

Tc T* x 0.15 0.0

under doped

• ver doped

T

Doping/ Number of charge carriers

Generic phase diagram

T

Cuprate high-Tc superconductors

18

No FL FL SC PG AF

Tc T* x 0.15 0.0

under doped

• ver doped

T

Doping / % of holes

“Strange metal” Non-BCS superconductivity

Schematic, generic phase diagram

Doped Mott insulators

Pseudogap Antiferromagnetism ~ Fermi liquid

No Quasi-Particle Quasi-Particle

Mott insulator

1/4 Brillouin zone

(0, 0)

(π, π)

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A bit of theory ...

Theory for strongly correlated systems

• Answer fundamental & qualitative questions e.g. :
• What is the glue for Cooper pairs in superconductors ?
• Develop quantitative, controlled computation methods.
• How to change the material to increase Tc ? or make a good

thermoelectric, or etc...

• Make strongly correlated physics predictive !
• Intermediate/strong coupling problems : all methods used nowadays

are non-perturbative.

• Today : one route of research ...

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H = −

• ⇤ij⌅,σ=,⇥

tijc†

iσcjσ + Unini⇥,

niσ ≡ c†

iσciσ

Hubbard model

Dynamical Mean Field Theory

• Reference system :

An atom coupled to an effective bath self-consistently determined

• Example : fermionic Hubbard model ⟶ Anderson impurity model

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• A. Georges, G. Kotliar,
• W. Krauth and M. Rozenberg, Rev. Mod. Phys. 68, 13, (1996)
• G. Kotliar, S.
• Y. Savrasov, K. Haule,
• V. S. Oudovenko, OP, C. Marianetti, Rev. Mod. Phys. 78, 865 (2006)

Local site Coupled to an effective electronic bath

H = 0

• σ=↑,↓

c†

σcσ + Un↑n↓

⌅ ⇤⇥ ⇧ +

• k,σ=↑,↓

Vkσ⇥†

kσcσ + h.c. +

• k,σ=,↓

kσ⇥†

kσ⇥kσ

⌅ ⇤⇥ ⇧

G0

• DFT : an electron in an effective Kohn-Sham potential
• DMFT : a “dual” method which puts the atom back at

the center of the description of the solid.

• Still a quantum many-body problem, but simpler (local).

Comparing lattice and impurity

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Lattice Anderson impurity

• Abrikosov-Suhl resonance
• Local Fermi liquid with

coherence temperature TK Nozières, 1974 Hubbard bands Quasi-particle-peak Mott physics : Hubbard band (localized) vs Q.P . peak (delocalized) DMFT transform the analogy into a formalism

− 1 π ImGc(ω)

A(k, ω)

ω ω

Free electrons Fermi liquid

• Local approximation : Exact in non-interacting and atomic limit
• Impurity model resums all local diagrams.
• DMFT : a systematic expansion.

DMFT is a diagrammatic method

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De Dominicis, Martin (1964)

φ1(Gii) = ΦAnderson(Gii)

ΓBK[Gij] = Tr ln Gij − Tr(g−1

0ijGij) + ΦBKLW [Gij]

Gij(t) ≡ −

• Tci(t)c†

j(0)

• Σij = δΦBKLW

δGij

ΦHubbard[Gij] = X 2 particle-irreducible (2PI) diagrams = X

i

φ1(Gii) | {z }

Local = DMFT

+ X

hi,ji

φ2(Gi,j) + X

hi,j,ki

φ3(Gi,j, Gi,k, Gj,k) + . . . | {z }

Non local = clusters

No FL FL SC PG AF

Tc T* x 0.15 0.0

under doped

• ver doped

T

DMFT : cluster methods

• Towards a controlled solution of e.g. Hubbard model.
• Non local terms : larger impurity models.

24 G0

G0

short range quantum fluctuations local quantum fluctuations

D A B C

Reciprocal space picture Brillouin zone patching

Size of cluster = momentum resolution

• f self-energy

= Control parameter

Real space picture

DMFT weakly corrected

C l u s t e r c

• r

r e c t i

• n

s i n c r e a s e

25

Cluster DMFT : what can we do today ?

• Thanks to modern algorithms, solve “easily” up to 16 sites, T ≈200K

(π, π) (0, 0) (0, π) (π, 0) (0, 0) (π, π) (0, π) (π, 0) (π, π) (0, 0) (0, 0) (π, π) (π/2, π/2) (π, 0) (0, 0) (π, π)

• Reproduce qualitatively various aspects of cuprate phase diagram.
• Example : our recent 8, 16 sites computations in the SC phase.

No FL FL SC PG AF

Tc T* x 0.15 0.0

under doped

• ver doped

T

0.1 0.2 4 4.5 5 5.5 6 6.5 7

U/t

= 4 = 8 =16

E Gull, A. Millis, OP, arxiv:1207.2490

X U/t

See also work by Haule, Tremblay, Civelli, Capone, Kotliar, Maier, Jarrell, Lichtenstein, Werner et al..

No Quasi-Particle Pseudogap

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Quasi-Particle “Good” Fermi liquid

No FL FL SC PG AF

Tc T* x 0.15 0.0

under doped

• ver doped

T

Cluster DMFT : what can we do today ?

• Another example : reproduce (with limited resolution) the node-

antinode dichotomy observed in experiments.

• Interpretation as a selective Mott transition (in momentum space)

(π, π) (0, 0) (0, π) (π, 0) (0, 0) (π, π) (0, π) (π, 0) (π, π) (0, 0) (0, 0) (π, π) (π/2, π/2) (π, 0) (0, 0) (π, π)

• OP. et al. 2004,
• M. Ferrero, O.P. 2009-2010

Experimental spectral intensity map at Fermi level (ARPES)

G.$!#$%$H$I$.? #)$" J$" K'3--,L2.($ )*&*.)$D!&F

• Ab-initio method for spectra & structure of correlated materials
• Use real atoms instead of one site of Hubbard model
• Select correlated orbitals. Downfolding.
• One-body part with DFT
• LDA (LDA+DMFT)
• How do you compute U ??

Example of DMFT as a realistic computation method

27

Dynamical mean-field theory within an augmented plane-wave framework: Assessing electronic correlations in the iron pnictide LaFeAsO

Markus Aichhorn,1 Leonid Pourovskii,1 Veronica Vildosola,1,2,3 Michel Ferrero,1,4 Olivier Parcollet,4 Takashi Miyake,3,5,6 Antoine Georges,1,3,7 and Silke Biermann1,3

1Centre de Physique Théorique, École Polytechnique, CNRS, 91128 Palaiseau Cedex, France

PHYSICAL REVIEW B 80, 085101 2009

+5'

Fe-Based (2008) See also. K. Haule (PRL 2008), Anisimov (2008)

Algorithms are the key ...

• Our reference system is a local-many body problem, not a well

known mathematical problem like linear algebra, PDE, ... New methods/algorithms are needed.

• The continuous time QMC revolution

A.N. Rubtsov et al., (2005), P. Werner et al(2006), E.Gull et al. (2008)

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10 20 30 40 50 60 70

!t

50 100 150

Matrix Size Weak Coupling Algorithm Hybridization Expansion Hirsch Fye

• E. Gull et al, Phys. Rev. B 76, 235123 (2007)
• Complexity ≈ Matrix Size^3
• Today :
• Solving simple models is a lot

simpler, faster, more reliable than a few years ago.

• Still work in progress for realistic,

large atoms (d, f shells), large clusters. Old New

1/T

Summary

• Strongly correlated quantum systems raises fundamental issues.
• “Materials Design” require better, more reliable, controlled

computation methods, without adjustable parameters (ab-initio).

• DMFT : a family of methods with atoms in effective baths as a

reference system, allows such ab-initio computations.

• Cluster DMFT methods reproduce many aspects of the high

temperature superconductors.

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Current directions of research

• Unify DMFT with methods treating long range e.g. AF fluctuations,

GW+ DMFT.

• Out of equilibrium physics.
• Still better impurity solvers (many open algorithmic problems) ...

30

Transition Metals Rare earth and actinides

Strong correlations : role of d- or f- orbitals

31

• Relatively close to nuclei, in particular 3d, 4f (orthogonality)
• Often transition metal (oxides), rare-earth and actinides compounds.