Theoretical Condensed Matter Physics

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Master Concepts Fondamentaux de la Physique 2015-2016 Theoretical Condensed Matter

, Xavier Leyronas, Christophe Mora, Olivier Parcollet, Michel Ferrero Lecture 1 (AG) -

1 Master Concepts Fondamentaux de la Physique Theoretical Condensed Matter Physics A.G., Xavier Leyronas, Christophe Mora, Olivier Parcollet, Michel Ferrero Lecture 1 (AG) - 07/09/2015 : 1. Organisation of the lectures 2. Introduction to the field seminar style 3. The Bose-Hubbard model Antoine Georges antoine.georges@college-de-france.fr

2 ORGANISATION OF THE LECTURES

3 «Y-SHAPED» STRUCTURE: Common part (6 lectures) UPMC-INSP 317 QUANTUM MANY-BODY THEORY: BASICS Lectures: A.Georges (1-4), X.Leyronas (5-6) Exercises: C.Mora, X.Leyronas Branch 1 ENS Magnetism, etc C.Mora/X.Leyronas Branch 2 - Orsay (LPS) [ Parcours MC] Response functions, Spectroscopies, Transport Cours: O.Parcollet TD: M.Ferrero

4 Common Part (6 1 st lectures): Lectures 1-2 (AG): The Bose-Hubbard model and the superfluid-insulator transition Lectures 3-6 (AG, then XL): Introduction to some aspects of interacting fermions: Introduction; Reminders on free fermions The `Hubbard atom ; coupling to a bath: Anderson impurity model and quantum dots The Hubbard model. Weak-coupling properties; guessing some properties at strong coupling Quasiparticles, Spectral function, etc.

5 Second semester (March à ) A number of optional lectures, in particular: ``Introduction to Strongly Correlated Systems (O. Parcollet, C. Mora)

6 Recommended Reading Strongly Correlated Bosons and Fermions in Optical Lattices T.Giamarchi & A.G. arxiv: [Lectures notes at Les Houches, quite close to these lectures] Many-Body Quantum Theory in Condensed Matter Physics H.Bruus, K.Flensberg Oxford University Press Many-Particle Physics G.D. Mahan, Kluwer Some lecture notes will be made available. Don t expect that they follow the course of the lectures, word by word, though

7 Website, Slides, Lecture Notes teaching/master2_icfp_ageorges.htm Now on website: 2014 documents (no major differences) will be gradually uploaded

8 Condensed Matter Physics: Understand the structure, physical properties and collective phenomena in ``organised forms of matter A very broad field! In these lectures: Quantum Systems of Interacting Fermions (and some bosons)

9 A few exemples

10 Organised forms of CARBON D=2 Diamond Graphite Lonsdaléite ß «Buckminsterfullerènes» à D=0 Amorphous (~ glass) Carbon Nanotubes D=1 Image : wikipedia

11 Deux formes cristallines du carbone : Diamant Graphite

face centred» lattices interpenetrate; tetrahedral coordination of each atom) o o

12 Diamond Transparent The hardest material Very good thermic conductors Very bad electric conductor (insulator (isolant)) Cristal formed by Carbon atoms (2 «cubic face centred» lattices interpenetrate; tetrahedral coordination of each atom) o o Other elements of the IVth column of the periodic table with Diamond structure: Si, Ge, α-sn o o

13 Graphite Black Cleavable (friable) (clivage selon plans) Conductor Cristalline layers of graphene «Honeycomb lattice» («nid d abeille») Lien Java

14 GRAPHENE: 2D carbon sheet The Nobel Prize in Physics 2010 was awarded jointly to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene"

15 Oxides ou Sr Li LiCoO 2 : an intercalated compound, Crucial for «Lithium ion» batteries La 2-x Sr x CuO 4 : a SUPERCONDUCTOR with «high» critical temperature ß CuO 2 plane

16 K.A. Müller A revolution (1986) : superconductors with «high» critical temperature J.G. Bednorz RFeAsO (2008) Température de L Azote liquide (77 degrés K) Température de L Hélium liquide (4 degrés K)

17 Copper-oxide superconductors: A headache for theorists - Rich phase diagram with mysterious electronic phases Temperature (Kelvin) Magnetic State (AF) «Strange» Metal (not a conventional Fermi liquid) Superconducting state Electron injection Mott Insulator Hole injection

18 «Artificial Materials» Epitaxie par jet moléculaire: élaboration de matériaux «plan atomique par plan atomique» Multicouches Fer/Chrome: Les débuts de la «magnétorésistance géante» (A.Fert, P.Grünberg, prix Nobel 2007) ``Point Quantique dans un gaz 2D d electrons

19 A new frontier: ultra-cold atoms and condensed-matter physics Optical lattices: Crystals of light and atoms

20 Experimental observation of the Mott transition (bosons) (Greiner et al, 2002) Phase coherence between wells in superfluid phase >interference pattern

22 From the micro- to the macro- scale: orders of magnitude

23 Seven orders of magnitude! Distances: (mètres) Å nm µm mm Atomic ß Mesoscopic à Macroscopicà Energy / Temperature: (électronvolts) =10mK (Degrés Kelvin)

24 Different systems, Very different energy scales Yet, common physical issues! NOTE: 1 electron-volt (ev) = K 1 µk = 20.8 khz

25 Two classes of theoretical approaches, broadly defined I. Directly at the macro-scale: Low-energy effective theories of collective emergent phenomena (i.e: identify phases, broken symmetries, effective theory of low-energy excitations in each phase) II. Start from the atomic scale and attempt at deducing quantitatively the large-scale properties No need to choose your camp: LEARN TO DO BOTH!

26 Low-energy effective theory of elementary excitations in graphene: Dirac fermions! à Study relativistic fermions in the solid-state lab!

27 All our natural world (copper, or DNA or viruses) contained in one big universal equation!? Wave-function: positions of electrons {r i }, of nuclei {R p } (Schrödinger, 1926) 1 st line: kinetic energy of electrons and nuclei 2 nd line: interactions: nuclei-nuclei, nuclei-electrons and between electrons (this is the difficult term!) Coulomb electrostatic interaction (1785)

28 Paul Dirac, 1929 ``Quantum Mechanics of Many-Electron Systems ``The general theory of quantum mechanics is now almost complete ( ). The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.'' P. A. M. Dirac, "Quantum Mechanics of Many-Electron Systems, Proceedings of the Royal Society of London, Series A, Vol.123, April 1929, pp 714.

29 Dirac (suite ) `` It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. précisément l objet de ce cours!

30 ``Weakly correlated systems One-particle hamiltonian in which each particle sees an ``effective potential provides a satisfactory description, allowing for a basic understanding of all physical properties Simplest approximation: Hartree

31 Examples: CAN/CANNOT be explained from a 1-particle effective potential approach : CAN: Why is diamond an insulator and graphene a (semi-) metal? CANNOT: Why are NiO or La 2 CuO 4 insulators?

32 Strongly correlated materials: the suspects are the localized orbitals! * d- or f- orbitals are quite close to ions nuclei (particularly 3d and 4f, for orthogonality reasons) They do not behave as regular band-forming orbitals (e.g sp-bonding) and retain atomic-like aspects à Electrons hesitate between localized and itinerant behaviour! Materials: transition-metals and their oxides, rare-earth/actinides and their compounds, but also some organic materials

33 Transition Metals Rare earths and Actinides

34 Materials with «strong electron correlations» still resist full understanding A «Mott insulator»: LaTiO 3

35 The Mott phenomenon: when repulsive intercations block motion Animation: par permission de Hidetoshi Fukuyama

Matériaux et dispositifs à fortes corrélations électroniques

Chaire de Physique de la Matière Condensée Matériaux et dispositifs à fortes corrélations électroniques Ingénierie quantique = vers le contrôle de leurs fonctionnalités Antoine Georges Cycle 2014-2015