1. Van Roekeghem et al., EPL (2014) Electronic structure calculations in strongly correlated materials A short overview Pascal Delange - Journée scientifique de l'EDPIF1

2. The many-body problem Pascal Delange - Journée scientifique de l'EDPIF2 Typical google results for « 3-body problem » Chaotic behaviour, numerically intractable

3. The many-body problem Pascal Delange - Journée scientifique de l'EDPIF3 How about 3.6x10 24 bodies instead ? (number of electrons in 1g of C) Is the electron-electron repulsion important ?  Strongly material dependent ! Fe - correlated Al – free electron like Why the difference ?

4. Band structure calculations What is a band ? Pascal Delange - Journée scientifique de l'EDPIF4 Principles of general Chemistry, Averill & Elderedge

5. Band structure calculations What is a band ? The band is to the molecular bond what the crystal is to the molecule Pascal Delange - Journée scientifique de l'EDPIF 5 molécule … … Infinite 1D atom chain a « wave vector »

6. Band structure calculations Realistic calculations : Pascal Delange - Journée scientifique de l'EDPIF6 Ec électrons Ec ions répulsion ions répulsion ion-électron répulsion électron-électron

7. Band structure calculations Realistic calculations : Pascal Delange - Journée scientifique de l'EDPIF7 Ec électrons Ec ions répulsion ions répulsion ion-électron répulsion électron-électron Born-Oppenheimer approximation : immobile ions

8. Band structure calculations Realistic calculations : Born-Oppenheimer for ions Still, what about e-e interaction ? Modern methods circumvent this annoying problem  Treat an effective problem, “equivalent” to the original one (in some sense…) Pascal Delange - Journée scientifique de l'EDPIF8

9. Band structure calculations Pascal Delange - Journée scientifique de l'EDPIF9 Hohenberg-Kohn theorem: ”The ground state density uniquely determines the potential and consequently all properties of the system, including the many-body wave-function”  Work on the density instead of individual wave-functions  Work on an effective potential instead of explicit electron-electron repulsion

10. Band structure calculations Pascal Delange - Journée scientifique de l'EDPIF10 Aluminium [Ne]3s 2 3p 1 Free electrons

11. Band structure calculations Pascal Delange - Journée scientifique de l'EDPIF11 Flat bands -> localisation of electrons -> strong interactions up down 3d, 4f : strong correlations = problems

12. Band structure calculations Pascal Delange - Journée scientifique de l'EDPIF12 P. Seth wide orbital = strong hopping = wide band & screened interaction Tight orbital = small hopping = narrow band & strong interaction

13. Can we see “real” bands ? Pascal Delange - Journée scientifique de l'EDPIF13 Photoemission spectroscopy : Cu

14. Can we see “real” bands ? Pascal Delange - Journée scientifique de l'EDPIF14 More complicated in reality : we do not see “pure” bands

15. From bands to localized electrons Pascal Delange - Journée scientifique de l'EDPIF15 The Mott transition : metal to insulator U : local Coulomb interaction parameter A(w) : spectral function (~measured in photoemission) States at Fermi level : metal Band gap : insulator Increasing local Coulomb interaction

16. From bands to localized electrons Pascal Delange - Journée scientifique de l'EDPIF16 The Mott transition : metal to insulator The Hubbard model

17. Conclusion A large family of phenomena due to e-e interaction Magnetism (Ferro, Para, Anti-ferro…) Magnetoresistance (high-Tc) superconductivity Luminescence of phosphors Metal-insulator transition … Do you have questions ? Pascal Delange - Journée scientifique de l'EDPIF17