From Materials to Models and Back. Dieter Vollhardt

Transcription

1 From Materials to Models and Back Dieter Vollhardt 28 th Edgar Lüscher Seminar, Klosters; February 8, 2017

2 From Materials to Models and Back - The Need for Models in Condensed Matter Physics - Outline: The art of modeling From materials to models and back: From models to materials Universality of models

3 The need for models Natural phenomena: often complicated (or even complex) What do we observe? How can we explain it? Can we predict new phenomena? Idealization, abstraction, reduction necessary Need models!

4 The need for models Natural phenomena: often complicated (or even complex) Models must be simple, yet sufficiently specific Micro-cosmos Macro-cosmos Construction of models: Somewhere between art and science

5 The art of modeling Weighing scale Every-day use Simple model for theoretical investigation (Archimedes)

6 The art of modeling Steam engines Every-day use blacinc.com Simple model Papin (1680) Extreme idealization

7 The art of modeling Earth s magnetic field 1269 Pierre PELERIN de Maricourt: spherical lodestone has poles 1600 William GILBERT of Colchester ( Father of magnetism ) De Magnete Spherical lodestone = model of the earth The earth is a magnet

8 The art of modeling Electromagnetism It seems to me that the test of Do we or do we not understand a particular point in physics? is, Can we make a mechanical model of it? (Lord Kelvin) J. C. Maxwell (1857): "I have been grinding at many things and lately during this letter at a vortical theory of magnetism & electricity which is very crude but has some merits." Maxwell s mechanical model of electromagnetism: Vortices in a molecular medium provided basis for the displacement current Maxwell equations: 1864

9 The art of modeling Electromagnetism It seems to me that the test of Do we or do we not understand a particular point in physics? is, Can we make a mechanical model of it? (Lord Kelvin) Mechanical model for the coupling between two electrical circuits Boltzmann (1891)

10 The art of modeling Properties of solids Example: Magnetite (Fe 3 O 4 ) Macroscopic view Explanation of the origin T-dependence of the magnetization M? Microscopic view: O(10 23 ) interacting electrons + ions Much more difficult to model

11 How to explain ferromagnetism? From materials to models

12 How to explain ferromagnetism? Weiss model of magnetic domains (1906) skullsinthestars.com Alignment of elementary magnets due to a molecular field ( Weiss mean field ) Microscopic explanation?

13 How to explain ferromagnetism? Ising model as proposed by Lenz (1920) Z. Physik 31, 253 (1925) web.stanford.edu Exact solution in d=1 Ising (1925) d=2 Onsager (1944) d= Weiss mean-field theory Single-site approximation molecular ( mean ) field

14 How to explain ferromagnetism? Ising model as proposed by Lenz (1920) Z. Physik 31, 253 (1925) web.stanford.edu Exact solution in d=1 Ising (1925) d=2 Onsager (1944) d= Weiss mean-field theory But: Magnetism is a quantum effect Bohr (1911), van Leeuwen (1919)

15 How to explain ferromagnetism? Heisenberg model Z. Physik 49, 619 (1928) complexity-coventry.org Exact solution in d=1 by Bethe ansatz Bethe (1931) d= : Weiss mean-field theory But: Electrons are mobile Bloch (1929)

16 Birth of many-body theory for condensed matter: Feynman (1949): Diagrams Landau (1956): Anderson impurity model (1961) Hubbard model (1963) quasiparticles, Fermi liquid theory ( Standard model of condensed matter physics )

17 Hubbard model A model to explain metallic ferromagnetism?

18 Hubbard model Simplest model for interacting electrons in solids Gutzwiller, 1963 Hubbard, 1963 Kanamori, 1963 time H = t c c + U n n ij,, σ iσ jσ i i i n n n n i i i i Exact solution in d=1 by Bethe ansatz Lieb, Wu (1968) d= : mean-field solution? Hartree mean-field theory (static!) generally insufficient

19 Hubbard model Simplest model for interacting electrons in solids Gutzwiller, 1963 Hubbard, 1963 Kanamori, 1963 time H = t c c + U n n ij,, σ iσ jσ i i i Purely numerical approaches (d=2,3): hopeless Theoretical challenge: Construct reliable, comprehensive, non-perturbative approximation schemes n n n n i i i i Hartree mean-field theory (static!) generally insufficient

20 Early non-perturbative approximation schemes: Hubbard III = coherent potential approximation (CPA) Hubbard (1964) Gutzwiller approximation /Gutzwiller variational wave function (1963/65) Ferromagnetism?! Gutzwiller approximation describes Mott transition Brinkman, Rice (1970) Anderson, Brinkman (1978) DV (1984) V 2 O 3 McWhan et al. (1973)

21 Mean-field theory in the limit d of the Hubbard model? H = t c c + U n n ij,, σ iσ jσ i i i Face-centered cubic lattice (d=3) time Metzner, DV (1989) dz, dynamical mean-field Self-consistent single-impurity Anderson model Z=12 Georges, Kotliar (1992)

22 Dynamical mean-field theory (DMFT) of correlated electrons Σ( k, ω) Σ( ω) U Kotliar, DV (2004) Dynamics of local electronic interaction described exactly Spectral function Definition of Correlations: Transfer of spectral weight Experimentally detectable by PES

23 DMFT: Mott-Hubbard metal-insulator phase diagram Strongly correlated electron materials V 2 O 3 NiSe 2-x S x κ-organics,... Kotliar, DV (2004)

24 DMFT: Ferromagnetism ferromagnetic metal Ulmke (1998) Generalized fcc lattice ( d ) Ferromagnetic order of itinerant local moments

25 From materials to models and back: From models to materials

26 Need: Realistic Models Electronic lattice (Hubbard) model Gutzwiller/Hubbard/Kanamori (1963) Density functional theory (DFT) Hohenberg, Kohn (1964) Kohn, Sham (1965) 1965 ca Implementation of DFT + study of materials with DFT Exploration of the properties of many-body models Solid State Physics vs. Statistical Physics

27 Theoretical approximation schemes for real materials, GGA How to combine? time Held (2004)

28 Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, GGA,...) or GW + Local electronic correlations (Many-body theory: DMFT) = LDA+DMFT Anisimov et al. (1997) Lichtenstein, Katsnelson (1998) Held et al. (2003) Kotliar et al. (2006)

29 Computational scheme for correlated electron materials: Material specific electronic structure (Density functional theory: LDA, GGA,...) or GW + Local electronic correlations (Many-body theory: DMFT) = X+DMFT X=LDA, GGA; GW, Solve self-consistently with an impurity solver, e.g., QMC, NRG, ED,...

30 Goal: Dynamical mean-field approach with predictive power for strongly correlated materials Research Unit FOR

31 LDA+DMFT: Application Fe Most abundant element by mass on Earth Ferromagnetism: Exceptionally high Curie temperature (T C = 1043 K) Still most widely used metal in modern day industry ( iron age )

32 Ferromagnetism in DMFT and LDA+DMFT ferromagnetic metal DMFT for one-band Hubbard model Ulmke (1998) Generalized fcc lattice ( d ) Ferromagnetic order of itinerant local moments LDA+DMFT Lichtenstein, Katsnelson, Kotliar (2001)

33 Until recently: LDA+DMFT investigations of correlated materials for given lattice structure Electrons + ions interact with each other Which lattice structure is (de)stabilized?

34 Investigation of the structural stability of Fe Fe T struct austenite ferrite hexaferrum DFT/GGA: Paramagnetic α-phase unstable (i) Why is the paramagnetic α-phase stable? (ii) How to compute T struct?

35 Investigation of the structural stability of Fe Fe T struct austenite ferrite hexaferrum GGA+DMFT: Electronic correlations (local repulsion) - increase unit cell volume correct density + compressibility - stabilize paramagnetic α-phase T struct > T C - cause the bcc-fcc structural phase transition Leonov, Poteryaev, Anisimov, DV (2011)

36 Lattice dynamics of paramagnetic α-fe Non-magnetic GGA phonon dispersion Pressure, GPa 1. Brillouin zone Dynamically unstable + elastically unstable (C 11, C < 0) Experiment: Neuhaus, Petry, Krimmel (1997)

37 Lattice dynamics of paramagnetic α-fe phonon frequencies calculated with frozen-phonon method harmonic approximation Pressure, GPa GGA+DMFT phonon dispersion at 1.2 T C Calculated: equilibrium lattice constant a~2.883 Å (a exp ~2.897 Å) Debye temperature Θ~458 K Theory: Leonov, Poteryaev, Anisimov, DV (2012) Experiment: Neuhaus, Petry, Krimmel (1997)

38 Universality of models Need for modeling in physics + Universality of physical laws Surprising universal insights from simple models

39 Universality of models From low to high energies: Kondo model

40 Magnetic impurity coupled (J) to non-interacting (mobile) electrons J J T>T K (high energies) Asymptotically free local moment T<T K (low energies) Screening of moment (confinement) Kondo effect (1964) K F 1/ F T Ee Λ J ( ) N( E ) Prototypical interaction problem with running coupling constant J(Λ) QED, QCD

41 Universality of models From high to low energies: Kibble-Zurek model ( mechanism ) of defect formation

42 Universality in continuous phase transitions BANG! short-range order T>T c Spins: paramagnetic Helium: normal liquid Universe: Unified forces and fields T=T c Phase transition ferromagnetic superfluid elementary particles, fundamental interactions Defects: domain walls vortices, etc. cosmic strings, etc. Kibble (1976) long-range order T<T c nucleation of galaxies?

43 Rapid thermal quench through 2 nd order phase transition Bäuerle et al. (1996) 1. Local temperature T T C Expansion + rapid cooling 3. Defects overlap 2. Nucleation of independently ordered regions Clustering of ordered regions Defects 4. T < T C : Vortex tangle Estimate of density of defects: Zurek (1985) Kibble-Zurek mechanism: General model for defect formation Now only used in condensed matter physics