De l’atome au 1 supraconducteur à haute température critique O. Parcollet Institut de Physique Théorique CEA-Saclay, France
Quantum liquids Quantum many-body systems, fermions (or bosons), 2 with interactions, at low temperature Correlated metal/superconductors Materials at interface of oxides High Temperature superconductors Transition metal oxides, SrTiO 3 /LaTiO 3 Ohtomo et al, Nature 2002 Ultra-cold atoms in optical lattices +5' Fe-Based (2008) “Artificial solids” of atoms & light Cuprate (1986)
Quantum liquids : motivations (I) 3 • 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.
Quantum liquids : motivations (II) 4 • 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 +5' Fe-Based high temperature Make this a (more) predictive science superconductors (2008) Crucial for many applications.
7 U" Fermions in a periodic potential 5 7 U" • Bloch’s theorem : Independent electrons in periodic potential '/ '/ form bands = eigenstates of H labelled by band index n and wave vector k � ~ 2 � 2 m r 2 + V | Ψ kn i = ✏ kn | Ψ kn i • Quantum many-body state = Slater determinant of these states. • Fermi sea, Fermi surface, ... Band Insulator Metal D ( ✏ ) D ( ✏ ) T=0 Density of states flo*;uJ flo*;uJ ;"k*-hJ ;"k*-hJ I ✏ F Gap I Fermi Level � D ( ε ) = δ ( ε − ε k ) k
Interactions ? 6 • 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
Fermi Liquid 7 L. Landau, 50’s • 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 Quasi-particle ARPES peak A ( k, ω ) = 1 � dxdte i ( kx − ω t ) i θ ( t ) � [ c ( x, t ) , c † (0 , 0)] ⇥ π Im
Fermi Liquid (II) 8 • Fixed point of Renormalization Group. • Determines low temperature behavior of various quantities, e.g. : • magnetic susceptibility χ ( T ) ∝ cte • specific heat C v ∝ T • 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.
Density functional theory 9 W. Kohn, 60’s • 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 V KS , determined self- consistently. Computation • Interactions taken into account in Experiments V KS • Effective one-body physics. opper Band structure of Cu For simple metals, we have the concepts and predictive computational tools. What about more complicated cases ?
10 Mott transition
Mott insulator 11 N. Mott, 50’s • 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. � t ij c † n i σ ≡ c † H = − i σ c j σ + Un i � n i ⇥ , i σ c i σ ⇤ ij ⌅ , σ = � , ⇥ δ = 1 � ⇥ n ↑ + n ↓ ⇤ Electron, spin +1/2 Hubbard model Electron, spin -1/2 Large Coulomb repulsion U ∼ eV ∼ 10 4 K Mott insulator
Interaction Driven Mott Transition 12 • Vary pressure P ⇔ 1/U 2-d organics : resistivity versus T and P κ -(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl V 2 O 3 (but has a simple hubbard modelization) 80 c c P 1 (T) P 2 (T) 70 * ! ( " ) max Mc Whan et al, 1973 T Met * (d # /dP) max T Ins 60 Semiconductor Mott Metal Bad metal 50 T (K) Critical Point 40 Mott insulator 30 20 Fermi liquid A.F insulator 10 0 100 200 300 400 500 600 700 800 P (bar) P. Limelette, et al. PRL 91, 016401 (2003) First order transition
Example of “application” : Intelligent windows 13 Conductivity (1/(ohm.cm)) • Use Mott transition in VO 2 (film). • High temperature : metal. • Lower temperature : insulator. • Reliable computation of material 340 K specific electronic properties ? Air d TiO 2 TiO 2 d VO 2 VO 2 SiO 2 d SiO 2 =3mm 1000/T Air Morin et al.1959 Figs. from S. Biermann
Strongly correlated (Mott) metals 14 • Adding holes in a Mott insulator Holes = charge carriers (chemical doping) • Holes form a liquid : a (correlated) metal. • 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 15 Mott insulator U/t Doping driven Interaction driven ? metal δ = 1 � ⇥ n ↑ + n ↓ ⇤ metal δ How is the metal destroyed close to a Mott transition ?
A simple correlated (Mott) metal 16 Photoemission experiments (not k resolved) (a) SrVO 3 Bulk V 3d Theoretical expectation PES FLAPW Quasi-particle FLAPW peak (x 0.6 in energy) Intensity (arb. units) Hubbard band -3 -2 -1 0 -3 Sekiyama et al. PRL 93, 2004 Competition : Narrowing of quasiparticle bands QP peak (delocalized) due to correlations (Brinkman-Rice) Hubbard bands (atomic like excitations)
Cuprate high-Tc superconductors 17 • High-Tc cuprate superconductors (Bednorz, Muller1986) La 2 − x Sr x CuO 4 , Bi 2 Sr 2 CaCu 2 O 8+ δ T T Generic phase diagram No FL T* PG Tc FL AF SC x 0.0 0.15 under doped over doped Doping/ Number of charge carriers
Cuprate high-Tc superconductors 18 Schematic, generic phase diagram Pseudogap “Strange metal” T Quasi-Particle ( π , π ) No FL Mott insulator T* PG (0 , 0) Tc No Quasi-Particle FL 1/4 Brillouin zone Non-BCS AF SC superconductivity ~ Fermi liquid x 0.0 0.15 under doped over doped Doping / % of holes Antiferromagnetism Doped Mott insulators
19 A bit of theory ...
Theory for strongly correlated systems 20 • 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 ... � t ij c † n i σ ≡ c † H = − i σ c j σ + Un i � n i ⇥ , i σ c i σ ⇤ ij ⌅ , σ = � , ⇥ Hubbard model
Dynamical Mean Field Theory 21 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) • Reference system : An atom coupled to an effective bath self-consistently determined • Example : fermionic Hubbard model ⟶ Anderson impurity model � � � V k σ ⇥ † � k σ ⇥ † c † H = � 0 σ c σ + Un ↑ n ↓ k σ c σ + h.c. + k σ ⇥ k σ + σ = ↑ , ↓ k, σ = ↑ , ↓ k, σ = , ↓ ⌅ ⇤⇥ ⇧ ⌅ ⇤⇥ ⇧ Local site Coupled to an effective electronic bath • 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). G0
Comparing lattice and impurity 22 Lattice Anderson impurity Quasi-particle-peak A ( k, ω ) − 1 π Im G c ( ω ) ω ω Free electrons Fermi liquid Hubbard bands • Abrikosov-Suhl resonance Mott physics : Hubbard band (localized) • Local Fermi liquid with vs coherence temperature T K Q.P . peak (delocalized) Nozières, 1974 DMFT transform the analogy into a formalism
DMFT is a diagrammatic method 23 Γ BK [ G ij ] = Tr ln G ij − Tr ( g − 1 0 ij G ij ) + Φ BKLW [ G ij ] De Dominicis, Σ ij = δ Φ BKLW Martin (1964) � � Tc i ( t ) c † G ij ( t ) ≡ − j (0) δ G ij X Φ Hubbard [ G ij ] = 2 particle-irreducible (2PI) diagrams X X X = φ 1 ( G ii ) + φ 2 ( G i,j ) + φ 3 ( G i,j , G i,k , G j,k ) + . . . i h i,j i h i,j,k i | {z } | {z } Local = DMFT Non local = clusters φ 1 ( G ii ) = Φ Anderson ( G ii ) • Local approximation : Exact in non-interacting and atomic limit • Impurity model resums all local diagrams. • DMFT : a systematic expansion.
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