**The Mott Transition and the Challenge of Strongly Correlated** Electron Systems. G. Kotliar Physics Department and Center for Materials Theory Rutgers PIPT Showcase Conference UBC Vancouver May 12th 2005

**The Standard Model of Solid State Physics. ** • Itinerant limit. Band Theory. Wave picture of the electron in momentum space. . Pauli susceptibility. • Localized model. Real space picture of electrons bound to atoms. Curie susceptibility at high temperatures, spin-orbital ordering at low temperatures.

**Correlated Electron Materials** • Are not well described by either the itinerant or the localized framework . • Compounds with partially filled f and d shells. Need new starting point for their description. Non perturbative problem. New reference frame for computing their physical properties. • Have consistently produce spectacular “big” effects thru the years. High temperature superconductivity, colossal magneto-resistance, huge volume collapses……………..

**Large Metallic Resistivities**

**Transfer of optical spectral weight non local in frequency** Schlesinger et. al. (1994), Vander Marel (2005) Takagi (2003 ) Neff depends on T

**Breakdown of the standard model of solids. ** • Qualitative Issues : i.e. Large metallic resistivities. Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics…………… • The quantitative tools of the standard model fail (i.e. density functional + GW )

**Two paths for calculation of electronic structure of** strongly correlated materials Crystal structure +Atomic positions Model Hamiltonian Correlation Functions Total Energies etc. DMFT ideas can be used in both cases.

**MODEL HAMILTONIAN AND OBSERVABLES** Parameters: U/t , T, carrier concentration, frustration : Local Spectral Function Limiting case itinerant electrons Limiting case localized electrons Hubbard bands

**Limit of large lattice coordination** Metzner Vollhardt, 89 Muller-Hartmann 89

**Dynamical Mean-Field Theory** A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992) Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

**Mott transition in V2O3 under pressure or chemical** substitution on V-site. How does the electron go from localized to itinerant.

**The Mott transition and Universality** Same behavior at high tempeartures, completely different at low T

**COHERENCE INCOHERENCE CROSSOVER** T/W Phase diagram of a Hubbard model with partial frustration at integer filling. M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995). .

**Transfer of optical spectral weight** M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

**Transfer of optical spectral weight** M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

**Anomalous transfer of optical spectral weight, NiSeS.** [Miyasaka and Takagi 2000]

**Anomalous Resistivity and Mott transition Ni Se2-x Sx** Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator. M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

**Single-site DMFT and expts**

**Conclusions: lessons from the application of DMFT to toy** model. • Three peak structure, quasiparticles and Hubbard bands. • The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase. • The method can describe coherent and incoherent phenomena and their crossover. Access to non perturbative regime . Real and momentum space. • Theory and experiments begin to agree on a broad picture.

**Realistic Descriptions of Materials and a First Principles** Approach to Strongly Correlated Electron Systems. • Incorporate realistic band structure and orbital degeneracy. • Incorporate the coupling of the lattice degrees of freedom to the electronic degrees of freedom. • Predict properties of matter without empirical information.

**LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin** and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). • Realistic band structure and orbital degeneracy. Describes the excitation spectra of many strongly correlated solids. . Spectral Density Functionals. Chitra and Kotliar PRB 2001 Savrasov et. al. Nature (2001) Savrasov and Kotliar PRB (2005) • Determine the self energy , the density and the structure of the solid by extremizing a functional of these quantities. Coupling of electronic degrees of freedom to structural degrees of freedom.

**Mott Transition in the Actinide Series**

**Pu phases: A. Lawson Los Alamos Science 26, (2000) ** LDA underestimates the volume of fcc Pu by 30%. Within LDA fcc Pu has a negative shear modulus. LSDA predicts d Pu to be magnetic with a 5 ub moment. Experimentally it is not. Treating f electrons as core overestimates the volume by 30 %

**Total Energy as a function of volume for PU** (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu.

**Double well structure and d Pu** Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ] Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the volume expands the insulator and contract the metal.

**Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science** vol 300, 953, 2003

**Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.** E = Ei - Ef Q =ki - kf

** C11 (GPa)** C44 (GPa) C12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 DMFT Phonons in fcc d-Pu ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

**J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003**

**First Principles DMFT Studies of Pu** • Pu strongly correlated element, at the brink of a Mott instability, which could not be described within the standard model. • Quantitative computations : total energy, photoemission spectra and phonon dispersions of delta Pu. • Qualitative Insights and quantitative studies. Double well. Alpha and Delta Pu.Other Pu anomalies.

**Approach the Mott point from the right Am under pressure** Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard” Density functional based electronic structure calculations: • Non magnetic LDA/GGA predicts volume 50% off. • Magnetic GGA corrects most of error in volume but gives m~6mB (Soderlind et.al., PRB 2000). • Experimentally, Am hasnon magnetic f6ground state with J=0(7F0)

**Mott transition in open (right) and closed (left) shell** systems. Realization in Am ?? S S g T Tc Log[2J+1] ??? Uc J=0 U U g ~1/(Uc-U)

**Cluster Extensions of Single Site DMFTCaptures new physics** beyond single site DMFT , i.e. d wave superconductivity, and other novel aspects of the Mott transition in two dimensional systems.

**Conclusions Future Directions** • DMFT: Method under development, but it already gives new insights into materials……. • Exciting development: cluster extensions. Allows us to see to check the accuracy of the single site DMFT corrections, and obtain new physics at lower temperatures and closer to the Mott transition where the single site DMFT breaks down. • Development of DMFT allow us to focus on deviations of experiments from mean field theory. • DMFT and RG developments

**Some References** • Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996). • Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005). • Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)