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), with interactions, at low temperature 2 Materials High Temperature superconductors Transition metal oxides, Correlated metal/superconductors at interface of oxides SrTiO3/LaTiO3 Ohtomo et al, Nature 2002 Ultra-cold atoms in optical lattices Fe-Based (2008) Artificial solids of atoms & light Cuprate (1986)
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) 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 Make this a (more) predictive science Crucial for many applications. Fe-Based high temperature superconductors (2008)
Fermions in a periodic potential 5 Bloch s theorem : Independent electrons in periodic potential form bands = eigenstates of H labelled by band index n and wave vector k apple ~ 2 2m r2 + V kni = kn kni Quantum many-body state = Slater determinant of these states. Fermi sea, Fermi surface,... Metal Band Insulator D( ) D( ) T=0 Density of states D( )= ( k) flo*;uj ;"k*-hj I F Fermi Level Gap 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 L. Landau, 50 s 7 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 ARPES Quasi-particle peak A(k, ) = 1 Im dxdtei(kx t) i (t) [c(x, t), c (0, 0)]
Fermi Liquid (II) 8 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! (T ) / cte C v / T 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 Computation Experiments 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 VKS, determined selfconsistently. Interactions taken into account in VKS Effective one-body physics. Band structure of Cu For simple metals, we have the concepts and predictive computational tools. What about more complicated cases?
Mott transition 10
Mott insulator 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. 11 H = Hubbard model ij, =, t ij c i c j + Un i n i, n i c i c i Electron, spin +1/2 Electron, spin -1/2 = 1 n + n Mott insulator Large Coulomb repulsion U ev 10 4 K
Interaction Driven Mott Transition Vary pressure P 1/U V 2 O 3 80 2-d organics : resistivity versus T and P -(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl (but has a simple hubbard modelization) 12 Mc Whan et al, 1973 70 60 Semiconductor P c 1 (T) T * Met T * Ins Pc 2 (T)!(") max (d#/dp) max 50 Mott Metal Bad metal T (K) 40 30 Mott insulator Critical Point 20 10 A.F insulator Fermi liquid 0 100 200 300 400 500 600 700 800 P (bar) P. Limelette, et al. PRL 91, 016401 (2003) First order transition
Metal-insulator transition accompanied by dimerization of V atoms: 2 Example of application : Intelligent windows 13 Use Mott transition in VO2 (film). High temperature : metal. Lower temperature : insulator. Reliable computation of material specific electronic properties? Intelligent Window? TiO 2 on VO 2 on SiO 2 Conductivity (1/(ohm.cm)) p.11 340 K Air TiO 2 d TiO2 VO 2 d VO2 Figs. from S. Biermann SiO 2 Air d SiO2 =3mm 1000/T Morin et al.1959
Strongly correlated (Mott) metals 14 Adding holes in a Mott insulator (chemical doping) Holes = charge carriers 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.
Interaction driven Mott transition: an intermediate coupling problem 15 U/t Mott insulator? Doping driven metal metal δ = 1 n + n How is the metal destroyed close to a Mott transition?
A simple correlated (Mott) metal Photoemission experiments (not k resolved) 16 Theoretical expectation Quasi-particle peak SrVO 3 Bulk V 3d PES FLAPW FLAPW (x 0.6 in energy) (a) Hubbard band Competition : QP peak (delocalized) Hubbard bands (atomic like excitations) (arb. units) -3-2 -1 0 Sekiyama et al. PRL 93, 2004 Narrowing of quasiparticle bands due to correlations (Brinkman-Rice)
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+ T T Generic phase No FL diagram T* PG Tc FL AF SC 0.0 under doped 0.15 over doped x Doping/ Number of charge carriers
Mott insulator Cuprate high-tc superconductors Schematic, generic phase diagram Pseudogap Strange metal Quasi-Particle (, ) T No FL 18 T* 0, 0) No Quasi-Particle 1/4 Brillouin zone AF Antiferromagnetism PG 0.0 under doped 0.15 over doped Tc Non-BCS SC superconductivity FL Doping / % of holes Doped Mott insulators x ~ Fermi liquid
A bit of theory... 19
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... H = ij, =, t ij c i c j + Un i n i, n i c i c i Hubbard model
Dynamical Mean Field Theory 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 21 H = 0 =, c c + Un n Local site + k k k V k k c + h.c. + k, =, k, =, 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. G0 Still a quantum many-body problem, but simpler (local).
Comparing lattice and impurity 22 A(k, ) Lattice Quasi-particle-peak Anderson impurity 1 ImGc ( ) Free electrons Fermi liquid Mott physics : Hubbard band (localized) vs Q.P. peak (delocalized) Hubbard bands DMFT transform the analogy into a formalism Abrikosov-Suhl resonance Local Fermi liquid with coherence temperature TK Nozières, 1974
DMFT is a diagrammatic method 23 Γ BK [G ij ] = Tr ln G ij Tr(g 0ij 1 G ij) + Φ BKLW [G ij ] G ij (t) T c i (t)c j (0) Σ ij = δφ BKLW δg ij 1(G ii )= Anderson (G ii ) De Dominicis, Martin (1964) Hubbard[G ij ]= X 2 particle-irreducible (2PI) diagrams = X 1(G ii ) + X 2(G i,j )+ X 3(G i,j,g i,k,g j,k )+... i hi,ji hi,j,ki {z } {z } Local = DMFT Non local = clusters Local approximation : Exact in non-interacting and atomic limit Impurity model resums all local diagrams. DMFT : a systematic expansion.
DMFT : cluster methods Towards a controlled solution of e.g. Hubbard model. Non local terms : larger impurity models. local quantum fluctuations Real space picture short range quantum fluctuations Reciprocal space picture Brillouin zone patching 24 T G0 AF T* PG No FL Cluster corrections increase Tc SC 0.0 under doped 0.15 over doped FL x G0 DMFT weakly corrected A B D C Size of cluster = momentum resolution of self-energy = Control parameter
Cluster DMFT : what can we do today? Thanks to modern algorithms, solve easily up to 16 sites, T 200K (, ) (0, ) (, ) (0, ) (, ) (, 0) (, ) (, ) ( /2, /2) 25 (0, 0) (0, 0) (, 0) (0, 0) (, 0) (0, 0) (0, 0) Reproduce qualitatively various aspects of cuprate phase diagram. Example : our recent 8, 16 sites computations in the SC phase. E Gull, A. Millis, OP, arxiv:1207.2490 7 T 6.5 AF T* PG No FL Tc SC 0.0 under doped 0.15 over doped FL x U/t 0 0.1 0.2 See also work by Haule, Tremblay, Civelli, Capone, Kotliar, Maier, Jarrell, Lichtenstein, Werner et al.. 6 5.5 5 4.5 4 U/t X
Cluster DMFT : what can we do today? 26 (, ) (0, ) (, ) (0, ) (, ) (, 0) (, ) (, ) ( /2, /2) (0, 0) (0, 0) (, 0) (0, 0) (, 0) (0, 0) (0, 0) Another example : reproduce (with limited resolution) the nodeantinode dichotomy observed in experiments. Interpretation as a selective Mott transition (in momentum space) Quasi-Particle T T* No FL OP. et al. 2004, M. Ferrero, O.P. 2009-2010 Experimental spectral intensity map at Fermi level (ARPES) PG No Quasi-Particle Pseudogap AF Tc SC 0.0 under doped 0.15 over doped FL x Good Fermi liquid
Example of DMFT as a realistic computation method 27 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?? K'3--,L2.($ )*&*.)$D!&F G.$!#$%$H$I$.? See also. K. Haule (PRL 2008), Anisimov (2008) #)$" J$" Fe-Based (2008) PHYSICAL REVIEW B 80, 085101 2009 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 Biermann 1,3 1 Centre de Physique Théorique, École Polytechnique, CNRS, 91128 Palaiseau Cedex, France
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. 28 The continuous time QMC revolution A.N. Rubtsov et al., (2005), P. Werner et al(2006), E.Gull et al. (2008) 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. Matrix Size 150 100 50 E. Gull et al, Phys. Rev. B 76, 235123 (2007) Old Weak Coupling Algorithm Hybridization Expansion Hirsch Fye New 0 0 10 20 30 40 50 60 70!t 1/T
Summary Strongly correlated quantum systems raises fundamental issues. Materials Design require better, more reliable, controlled computation methods, without adjustable parameters (ab-initio). 29 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. 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
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. Transition Metals Rare earth and actinides