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Towards a first Principles Electronic Structure Method Based on Dynamical Mean Field Theory Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University Montauk Long Island September 13-17 2009

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Outline Dynamical Mean Field Theory: Basic Ideas Dynamical Mean Field Theory and Electronic Structure, LDA+ DMFT Illustrative Applications Reviews: G. Kotliar et. al. Reviews of Modern Physics 78, 865-951, (2006). K. Held Advances in Physics 56, 829 (2007)

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Correlated Electron Systems Pose Basic Questions in CMT FROM ATOMS TO SOLIDS How to describe electron from localized to itinerant ? How do the physical properties evolve ? Spectra and Total Energies

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D MFT Local Physics of a solid as atom in a medium10

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Early Review: Georges Kotliar Krauth Rozenberg RMP Early Review: Georges Kotliar Krauth Rozenberg RMP 68, 13 (1996) 12 Spectra=- Im G(k, ) Self consistency for V and Simple extensions to phases with LRO Locality: simple extensions to cluster of sites. Rapid advances in impurity solvers

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But how accurate is it ? Important tests in Cold Atom Traps

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Cluster DMFT Reviews: T. Maier et. al. Rev. Mod. Phys. 77, 1027, (2005). G. Kotliar et. al. Rev. of Mod. Phys. 78, 865, (2006). A.M Tremblay B. Kyung D. Senechal JLT Phys. 32, 424-451 (2006).

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Cluster DMFT Difficulties 2x2 cluster DMFT equations are considerably harder to solve and to interpret than single site DMFT. Uniqueness: No unique formulation of cluster DMFT. Reconstruction of k dependence of quantities. Multiplicity of Solutions.

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CDMFT vs BA in the 1D Hubbard Model density n vs chemical potential Gap vs U at half filling V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]

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Outline Dynamical Mean Field Theory: Basic Idea Dynamical Mean Field Theory and Electronic Structure and LDA+ DMFT Applications to 3d Materials Applications to 4f Materials Applications to 5f Materials Outlook

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THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional formulation. Chitra and Kotliar Phys. Rev. B 63, 115110 (2001) Ambladah et. al Int. Jour Mod. Phys. B 13, 535 (1999). Ir>=|R, > Double loop in Gloc and Wloc

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Full implementation in the context of a a one orbital lattice model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). After finishing the loop one can treat the graphs involving Gnonloc Wnonloc in perturbation theory.. Phys. Rev. Lett. 92, 196402 (2004) Limiting case (perturbation theory as solvers) Zeyn and Antropov. N. E. Zein and V. P. Antropov, J. Appl. Phys. 89, 7314 (2001), Phys. Rev. Lett. 89, 126402 (2002) Application to semiconductors N. Zeyn S. Savrasov and G. Kotliar PRL 96, 226403, 2006 EDMFT loop Chitra and Kotliar Phys. Rev. B 63, 115110 (2001). G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301. cond-mat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)

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Determine energy and and self consistently from extremizing a functional : the spectral density functional. Chitra and Kotliar (2001). Determine energy and and self consistently from extremizing a functional : the spectral density functional. Chitra and Kotliar (2001). R. Chitra and G. Kotliar, Phys. Rev. B 63, 115110 Savrasov and Kotliar (2001) Full self consistent implementation. Review: Kotliar et.al. RMP (2006) (2001). Savrasov and Kotliar (2001) Full self consistent implementation. Review: Kotliar et.al. RMP (2006) 12 LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). U is parametrized in terms of Slater integrals F0 F2 F4 .

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Effective interaction among electrons. Constrained RPA (cRPA) Ferdi Ariasetiwan,A, M Imada, A Georges, G Kotliar, S Biermann, AI Lichtenstein, PRB 70, 195104 (2004) energy-dependent effective interaction between the 3d electrons Can be used to extract a screened U Identity:

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THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT functional Sum of local 2PI graphs with local U matrix and local G

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LDA+DMFT Self-Consistency loop DMFT U E dc

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Practical Matters Choice of the projector, in the simplest case choice of orbital. (i.e. Projective LMTOs ) Basis in which to truncate the Kohn Sham Hamiltonian. Implementation of charge self consistency Impurity Solvers: slave bosons, NCA, OCA, CTQMC, Hubbard I, etc. tradeoff between speed and accuracy. Choice of U and double counting.

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Total Energy as a function of volume for Pu. Wrest( ) Total Energy as a function of volume for Pu. Wrest( ) (ev) vs (a.u. 27.2 ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. N, Zein, Following Aryasetiwan Imada Georges Kotliar Bierman and Lichtenstein. PRB 70 195104. (2004) Pu

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DMFT Phonons in fcc -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Experiment 36.28 33.59 26.73 4.78 ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

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Main DMFT Concepts Valence Histograms. Describes the history of the atom in the solid, multiplets! Weiss Weiss field, collective hybridizationfunction, quantifies the degree of localization Functionals of density and spectra give total energies Local Self Energies and Correlated Bands

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Qualitative Phase diagram :frustrated Hubbard model, integer filling M. Rozenberg et.al. 75, 105 (1995) T/W 10 CONCEPT: (orbitally resolved) spectral function. Transfer of spectral weight. CONCEPT: Mott transition. DMFT view of Pu: adding orbitals and coupling to strucure to this bare bones phase diagram

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What is the valence in the late actinides ? Plutonium has an unusual form of MIXED VALENCE

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LDA results Finding the f occupancyTobin et. al. PRB 72, 085109 2005 K. Moore and G. VanDerLaan RMP (2009). Shim et. al. Europhysics Lett (2009)

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Looking for moments. Pu under (negative ) pressure. C Marianetti, K Haule GK and M. Fluss Phys. Rev. Lett. 101, 056403 (2008)

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Application to Electron and Hole Doped Cuprates : Review: Armitage Fournier Green (arXiv:0906.2931 )

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Single Site vs 2 site CDMFT Phase Diagram

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Doping NCCO.2 ev.03 ev N. L. Wang, G. Li, D. Wu, X. H. Chen, C. H. Wang, and H. Ding, Phys. Rev. B 73, 184502 (2006). Y. Onose et al., Phys. Rev. B, 69, 024504 (2004)

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Optical Spectral Weights. C. Weber et. al. Not a very sensitive probe of the strength of correlations around the intermediate correlation regime. Expt points : Y. Onose et al., Phys. Rev. B, 69, 024504 (2004). S. Uchida et al., Phys. Rev. B 43, 7942 (1991).

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Underdoped vs Overdoped T=0 M. Civelli PRB 79,195113 (2009)arXiv:cond-mat/0508302arXiv:cond-mat/0508302T. Stanescu and G. Kotliar Phys. Rev. B 74, 125110 (2006) F. F. Balakirev et. al. arXiv.org:0710.4612 (2007). Phys. Rev. B 74, 125110 (2006)

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Avoided Quantum Criticality : QCP under the dome. arXiv:cond-mat/0605149K. Haule and GK Phys. Rev. B 76, 092503 (2007)Avoided Quantum Criticality : QCP under the dome. arXiv:cond-mat/0605149K. Haule and GK Phys. Rev. B 76, 092503 (2007). Coherence vanishes underdoped overdoped optimally scattering at Tc

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Real Space Picture Momentum Space Picture: High T Singlet formation. S,T N=2 singlet, triplet E N=0 1+ states with 1 electron in + orb Underdoped region: arcs shrink as T is reduced. Overdoped region FS sharpens as T is reduced.

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Conclusion Dynamical Mean Field Theory: Locality as a Basic Idea Dynamical Mean Field Theory and Electronic Structure. Some Interesting Applications Many others taking place, many groups working in this area all over the world.

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Thanks for your Attention!!

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Kinetic energy

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Comparison of 2 and 4 sites