Some Applications of Dynamical Mean Field Theory (DMFT). Density Functional Theory Meets Strong Correlation. Montauk September 5-9 (2006). G.Kotliar Physics.

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Some Applications of Dynamical Mean Field Theory (DMFT). Density Functional Theory Meets Strong Correlation. Montauk September 5-9 (2006). G.Kotliar Physics Department Center for Materials Theory Rutgers University.

Outline Sketch of some DMFT ideas. Application : Localization-Delocalization Transition in actinides.a) Pu b) Am DFT and DMFT formalism. Acknowledgment: Collaborators K. Haule (Rutgers) S. Savrasov (UCDavis) and N. Zein (Kurchatov) Discussions : M. Fluss J. C Griveaux G Lander A. Lawson A. Migliori J.Singleton J.Smith J Thompson J. Tobin $upport: NSF DOE- BES

n=1 Order in Perturbation Theory Order in PT Range of the clusters Basis set size. DMFT GW r site CDMFT l=1 l=2 l=lmax r=1 r=2 n=2 GW+ first vertex correction

DMFT Cavity Construction. Happy marriage of atomic and band physics. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti Rev. Mod. Phys. 78, 865 (2006). G. Kotliar and D. Vollhardt Physics 53 Today (2004) Extremize a functional of the local spectra. Local self energy.

T/W Phase diagram of a Hubbard model with partial frustration at integer filling. [Rozenberg et. al. PRL 1995] Evolution of the Local Spectra as a function of U,and T. Mott transition driven by transfer of spectral weight Zhang Rozenberg Kotliar PRL (1993).. Mott transition in one band model. Review Georges et.al. RMP 96

Pu issues All the spin density functional studies of fcc Pu, using either LDA or GGA, predict magnetic long range order with a large moment. Experimentally  Pu is not magnetic. If one treats the f electrons as part of the core LDA overestimates the volume by 30% Valence of Pu controversy. LDA+U Schick, Havela Lichtenstein et.al. Anisimov et.al.(5f)^6. Erickson and Wills (5f)^4.

Pu phases: A. Lawson Los Alamos Science 26, (2000) Experimentally Pu is not magnetic. No trace of ordered or low frequency fluctuating local moments. [Lashley et. al. cond-matt ] PRB (2005). Approach the Mott transition from the left. (delocalized side).

Approach the Mott point from the right Am under pressure 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 ~6  B (Soderlind et.al., PRB 2000).  Experimentally, Am has non magnetic f 6 ground state with J=0 ( 7 F 0 ) Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard”

Resistivity of Am under pressure. J. C. Griveau J.Rebizant G. Lander and G. Kotliar PRL 94, (2005).

LS coupling L=0 S=7  jj coupling J=7/2   =2S+L Expt monent. is closer to L S coupling Curium is magnetic Hurray et.al. Physica. B (1980) 217

Applications of DMFT to actinides S. Savrasov G.Kotliar and E. Abrahams [neglect multiplets] energy and spectra IPT imp. Solver Phonons [Dai et.al.] [Hubbard I imp solver] Could not address the existence of magnetically ordered states or the interplay of magnetism multiplets and Kondo physics. Recent review see G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti Rev. Mod. Phys. 78, 865 (2006) Recent progress : K. Haule SUNCA imp. solver, full multiplet structure and Kondo physics compete on equal footing. Can consider DMFT eqs. in different ordered phases.

Minimum in melting curve and divergence of the compressibility at the Mott endpoint

Total Energy as a function of volume for Pu Total Energy as a function of volume for Pu  (ev) vs (a.u ev) (Savrasov, Kotliar, Abrahams, Nature ( 2001) Non magnetic correlated state of fcc Pu. Zein (2005) Following Aryasetiwan Imada Georges Kotliar Bierman and Lichtenstein. PRB (2004) Pu

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

DMFT Phonons in fcc  -Pu C 11 (GPa) C 44 (GPa) C 12 (GPa) C'(GPa) Theory Experiment ( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

Double well structure and  Pu Qualitative explanation of negative thermal expansion[ Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Phys vol.126, (2002)] 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. F(T,V)=Fphonons +Finvar

Curie-Weiss Tc Photoemission in Actinides alpa->delta volume collapse transition Curium has large magnetic moment and orders antif Pu does is non magnetic. F0=4,F2=6.1 F0=4.5,F2=7.15 F0=4.5,F2=8.11

The “DMFT- valence” in the late actinides

Approach the Mott point from the right Am under pressure 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 ~6  B (Soderlind et.al., PRB 2000).  Experimentally, Am has non magnetic f 6 ground state with J=0 Experimental Equation of State (after Heathman et.al, PRL 2000) Mott Transition? “Soft” “Hard”

Am equation of state. LDA+DMFT.New acceleration technique for solving DMFT equations S. Savrasov K. Haule G. Kotliar cond-mat (2005)

Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, (2005).

Photoemission spectra using Hubbard I solver [Lichtenstein and Katsnelson, PRB 57, 6884,(1998 ), Svane cond-mat ] and Sunca. [Savrasov Haule and Kotliar cond-mat ] Hubbard bands width is determined by multiplet splittings.

Photomission Spectra of Am under pressure. Sunca. Onset of mixed valence. Savrasov Haule Kotliar PRL (2005)

The “DMFT- valence” in the late actinides

in the late actinides [DMFT results: K. Haule and J. Shim ]

Conclusions 5f systems at the Mott boundary. Pu and Am. Single site DMFT describes well, the total energy of phases, the phonon spectra, the photoemission spectra, of Am and Pu. Am Pu are non magnetic Pu is magnetic in agreement with expts. Qualitative explanation of mysterious phenomena, such as the negative thermal expansion in delta Pu, the volume contraction in the delta-epsilon transition, the anomalous raise in resistivity as one applies pressure to Am metal, etc…..

Density functional and Kohn Sham reference system Kohn Sham spectra, proved to be an excellent starting point for doing perturbation theory in screened Coulomb interactions GW.

= W = [ - ] -1 = G - [ - ] GW approximation (Hedin )

Weak correlations Intermediate -Strong Correlations One electron spectral function

Strongly Correlated Materials Can we construct a conceptual framework and computational tools, for studying strongly correlated materials, which will be as successful as the Fermi Liquid –LDA-GW program ? Is a local perspective reasonable ? accurate ? Dynamical Mean Field Theory. Unify band theory and atomic physics. Use an impurity model, (local degrees of freedom + free electron enviroment ) to describe the local spectra of a correlated system.

Spectral density functional. Effective action construction.e.g Fukuda et.al

In practice we need good approximations to the exchange correlation, in DFT LDA. In spectral density functional theory, DMFT. Review: Kotliar et.al. Rev. Mod. Phys. 78, 865 (2006) Kohn Sham equations

Different methods differ by the choice of variable a used. DFT Spin and Density FT Spectral Density Functional R. Chitra and G.K Phys. Rev. B 62, (2000). S. Savrasov and G.K PRB (2005) Phys. Rev. B 62, (2000).

C DMFT extend the notion of “locality”to several unit cells U (and form of dc) are input parameters. EDMFT a=“ Gloc Wloc” Cluster Greens Function and Screened interaction, No input parameters. Recently impelemented and tested for sp systems. Si C …. N. Zein et.al.PRL 96, (2006) Zein and Antropov PRL 89, Review: Kotliar et.al. Rev. Mod. Phys. 78, 865 (2006) DFT+DMFT

Conclusion DMFT, method under very active development. But there is now a clear formulation (and to large extent implementation) as a fully self consistent, controlled many body approach to solids. It gives good quantitative results for total energies, phonon and photoemission spectra, and transport of materials. Many examples…all over the periodic table. Helpful in developing intuition and qualitative insights in correlated electron materials. With advances in implementation, we will be able to focus on deviations from (cluster) dynamical mean field theory.

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, (2000) and Phys. Rev.B (2001). Phys. Rev. B 62, (2000) Ex. Ir>=|R,  > Gloc=G(R , R  ’)  R,R’ ’ Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc. Sum of 2PI graphs One can also view as an approximation to an exact Spetral Density Functional of Gloc and Wloc.

n=1 Order in Perturbation Theory Order in PT Range of the clusters Basis set size. DMFT GW r site CDMFT l=1 l=2 l=lmax r=1 r=2 n=2 GW+ first vertex correction

Classical case Quantum case A. Georges, G. Kotliar (1992) Mean-Field : Classical vs Quantum Easy!!! Hard!!! QMC: J. Hirsch R. Fye (1986) NCA : T. Pruschke and N. Grewe (1989) PT : Yoshida and Yamada (1970) NRG: Wilson (1980) Pruschke et. al Adv. Phys. (1995) Georges et. al RMP (1996) IPT: Georges Kotliar (1992).. QMC: M. Jarrell, (1992), NCA T.Pruschke D. Cox and M. Jarrell (1993), ED:Caffarel Krauth and Rozenberg (1994) Projective method: G Moeller (1995). NRG: R. Bulla et. al. PRL 83, 136 (1999),……………………………………...

How good is the local approximation ??? Exact in infinite dimensions, very good also in one dimension! Cellular DMFT [Kotliar et. al. PRL (2001) ] Test in 1d Hubbard model Capone Civelli Sarma Castellani and Kotliar PRB 69, (2004) ]

CDMFT: removes limitations of single site DMFT No k dependence of the self energy. No d-wave superconductivity. No Peierls dimerization. No (R)valence bonds. Reviews: Georges et.al. RMP(1996). Th. Maier, M. Jarrell, Th.Pruschke, M.H. Hettler RMP (2005); G. Kotliar S. Savrasov K. Haule O. Parcollet V. Udovenko and C. Marianetti RMP (2006) Various cluster approaches, DCA momentum spcace. Cellular DMFT G. Kotliar et.al. PRL (2004). O Parcollet G. Biroli and G. Kotliar B 69, (2004) T. D. Stanescu and G. Kotliar cond-mat/

 is local by construction and plays auxiliary role exactly like Kohn-Sham potential in DFT  Energy dependent Kohn-Sham (Dyson) equations give rise to energy-dependent band structure  have physical meaning in contrast to Kohn-Sham spectra. are designed to reproduce local spectral density Local Self-Energy of Spectral Density Functional

LDA+GW: semiconducting gaps