1. Computational Studies of Strongly Correlated Materials Using Dynamical Mean Field Theory Gabriel Kotliar Center for Materials Theory Rutgers University Research Support: NSF DMR 0096462, DOE DE-FG02-99ER45761, ONR N00014-21-0766

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline, Collaborators, References Introduction to extensions of DMFT for applications to electronic structure. [ S. Savrasov and GK cond- matt0308053]  C-DMFTstudy of the Mott transition. [O. Parcollet G. Biroli and GK cond-mat 0308577 ] Applications to materials: MIT in Ti2O3[S. Poteryaev S. Lichtenstein and GK cond-mat 0311319 ] Delta –Epsilon transition in Plutonium [Xi Dai S. Savrasov GK A Migliori H. Ledbetter E. Abrahams Science 300, 953 (2003)] Outlook.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Cavity Construction: A. Georges and G. Kotliar PRB 45, 6479 (1992). Figure adapted from : Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004) http://www.physics.rutgers.edu/~kotliar/RI_gen.html http://www.physics.rutgers.edu/~kotliar/RI_gen.html

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS EDMFT [H. Kajueter Rutgers Ph.D Thesis 1995 Si and Smith PRL77, 3391(1996) R. Chitra and G. Kotliar PRL84,3678 (2000)]

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Site  Cell. Cellular DMFT. C-DMFT. G. Kotliar,S.. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) hopping expressed in the superlattice notations. Other cluster extensions (DCA Jarrell Krishnamurthy, Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes Schiller Ingersent ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two paths for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

7. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988). The light, SP (or SPD) electrons are extended, well described by LDA.The heavy, D (or F) electrons are localized treat by DMFT. LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term)  Kinetic energy is provided by the Kohn Sham Hamiltonian (sometimes after downfolding ). The U matrix can be estimated from first principles of viewed as parameters. Solve resulting model using DMFT.

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Functional formulation. Chitra and Kotliar (2001), Ambladah et. al. (1999) Savrasov and Kotliarcond- matt0308053 (2003). Ir>=|R,  > Double loop in Gloc and Wloc

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Next Step: GW+EDMFT S. Savrasov and GK.(2001). P.Sun and GK. (2002). S. Biermann F. Aersetiwan and A.Georges. (2002). P Sun and G.K (2003) Implementation in the context of a model Hamiltonian with short range interactions.P Sun and G. Kotliar cond-matt 0312303 or with a static U on heavy electrons, without self consistency. Biermann et.al. PRL 90,086402 (2003) W W

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Self-Consistency loop. S. Savrasov and G. Kotliar (2001) and cond-matt 0308053 DMFT U E

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How good is this approximation ? It becomes exact as the coordination number increases or in the limit of infinite dimensions introduced by Metzner and Vollhardt. PRL 62,34, (1989). How good is it in low dimensions ? Promising recent developments from theory and experiments.

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Impurity Solvers. Hubbard I. Quantum Montecarlo. Rational Approximations to the self energy, constructed with slave bosons. cond-mat/0401539 V. Oudovenko, K. Haule, S. Savrasov D. Villani and G. Kotliar. Extensions of NCA. Th. Pruschke and N. Grewe, Z. Phys. B: Condens. Matter 74, 439, 1989. SUNCA K. Haule, S. Kirchner, J. Kroha, and P. W¨olfle, Phys. Rev. B 64, 155111, (2001). K. Haule et. al. (2004)

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS One dimensional Hubbard model. Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats, [V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][ [M. Capone M.Civelli V Kancharla C.Castellani and GK cond-mat 0401060] See presentation S20.012 at 16:42. U/t=4.

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram and DOS of a partially frustrated integer filled Hubbard model and pressure driven Mott transition. S Lefebvre et al. PRL (2000)

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Recent Experiments support qualitative single site DMFT predictions Mo et al., Phys. Rev.Lett. 90, 186403 (2003). Limelette et. al.(2003) Ito et. al. (1995)

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Theoretical issue: is there a Mott transition in the integer filled Hubbard model, and is it well described by the single site DMFT ? Study frustrated t t’ model t’/t=.9 YES! Parcollet Biroli Kotliar cond-matt 0308577

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the k resolved Spectral Function at zero frequency. Uc=2.35+-.05, Tc/D=1/44 U/D=2 U/D=2.25 Qualitative effect, formation of hot regions! D wave gapping of the single particle spectra as the Mott transition is approached. Very strong k dependece near the trasition.

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3 : Coulomb or Pauling C.E.Rice et all, Acta Cryst B33, 1342 (1977) LTS 250 K, HTS 750 K.

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ti2O3. Isostructural to V2-xCrxO3. Al lot of the qualitative physics of the high temperature part of the phase diagram of V2O3 can be understood within single site DMFT. Is this true in Ti2O3? Band Structure Calculations good metal. L.F. Mattheiss, J. Phys.: Condens. Matter 8, 5987 (1996).Unrestricted Hartree Fock calculations produce large antiferromagnetic gap. M. Cati, et. al. Phys. Rev. B. f55, 16122 (1997).

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS U = 2, J = 0.5, W = 0.5 β = 20 eV -1, LT structure U = 2, J = 0.5, W = 0.5 β = 10 eV -1, HT structure 2site-Cluster DMFT with intersite Coulomb A. Poteryaev

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pauling and Coulomb Ti2O3[S. Poteryaev S. Lichtenstein and GK cond-mat 0311319 ] Dynamical Goodenough- Honing picture

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Ti2O3 2 site cluster DMFT describes the MIT in Ti2O3. Different from V2O3 where single site DMFT works well, and cluster corrections are small [A. Poteryaev] It requires the Coulomb interactions, and a frequency dependent enhancement of the a1g-a1g hopping, induced by the Coulomb interactions. [Haldane Ph.D thesis, Q Si and GK 1993 ].Dynamical Pauling- Goodenough mechanism is able to trigger the MIT at low enough temperatures. Coulomb and Pauling synergistically cooperate.

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pu phases: A. Lawson Los Alamos Science 26, (2000) LDA underestimates the volume of fcc Pu by 30% Predicts magnetism in d Pu and gives negative shear Core-like f electrons overestimates the volume by 30 %

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The delta –epsilon transition The high temperature phase, (epsilon) is body centered cubic, and has a smaller volume than the (fcc) delta phase. What drives this phase transition? Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)

25. Total Energy as a function of volume for Pu (after Savrasov, Kotliar, Abrahams, 2001,410,793, 2001)

26. 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 (after Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003) (experiments from Wong et.al, Science, 22 August 2003)

27. DMFT Phonons in bcc  -Pu

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phonon entropy drives the epsilon delta phase transition Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta. Different from Cerium see Jeong et. al. cond- mat/0308416 At the phase transition the volume shrinks but the phonon entropy increases. Estimates of the phase transition following Drumont and Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outlook Dynamical mean field theory,” local reference “ for correlated electron systems. Analogy to FLT, DFT. The need of simpler reference frames for thinking about complex problems. Future directions: downfolding and RG, algorithmic speedups. While a general method is under construction, the extensions described in this talk, already allow to perform quantitative calculations and obtain quantitative insights.

30. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion Introduction to DMFT and its extensions. Flexibility of a local approach. DMFT describes well the Mott transitions. Formation of hot and cold regions near FS. MIT in Ti2O3 cluster DMFT.Dynamical Pauling-Coulomb mechanism. Delta-Epsilon Plutonium. Correlations and phonon entropy.

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Alpha Gamma Cerium B. Johansson, Philos. Mag. 30, 469 (1974). Mott transition of the f electrons as a function of pressure. Ce alpha gamma transition. spd electrons are spectators..  J.W. Allen and R.M. Martin, Phys. Rev. Lett. 49, 1106 (1982); Kondo volume collapse picture. The dominant effect is the spd-f hybridization. J.W. Allen and L.Z. Liu, Phys. Rev. B 46, 5047 (1992).

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS LDA+DMFT:Ce M..Z¨olfl,I.A.NekrasovTh.Pruschke,V.I.Anisimov J. Keller,Phys.Rev. Lett 87, 276403 (2001). K. Held, A.K. McMahan, and R.T. Scalettar, Phys. Rev.Lett. 87, 276404 (2001) A.K.McMahan,K.Held,andR.T.Scalettar,Phys Rev. B 67, 075108 (2003). 1.Successful calculations of thermodynamics. 2.Mott transition and Kondo collapse give rise to similar spectra and phase diagram.

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS To resolve the conflict between the Mott transition and the volume collapse picture : Turn to Optics! Haule et.al. Qualitative idea. The spd electrons have much larger velocities, so optics will be much more sensitive to their behavior. See if they are simple spectators (Mott transition picture ) or wether a Kondo binding unbinding takes pace (Kondo collapse picture). General method, bulk probe.

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Optics formula double pole single pole One divergence integrated out!

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The volume of alpha is 28.06°A and the temperature 580K. The volume of the gamma phase is 34.37°A and T = 1160K. Experiments : alpha at 5 K and gamma phase at 300 K. Theory: Haule et. al. cond-matt 04 Expt: J.W. vanderEb PRL 886,3407 (2001)

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Temperature dependence of the optical conductivity.

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Origin of the features.

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion: Cerium Qualitatively good agreement with existing experiment. Some quantitative disagreement. Experiments should study the temperature dependence of the optics. Optics + Theory can provide a simple resolution of the Mott vs K-Collapse conundrum.

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Benchmarking SUNCA, V. Udovenko and K. Haule

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Interpolative scheme with slave bosons.

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Photoemission&experiment A. Mc Mahan K Held and R. Scalettar (2002) Zoffl et. al (2002) K. Haule V. Udovenko S. Savrasov and GK. (2004)

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Application to Materials Ti2O3-V2O3 Cerium: Alpha to Gamma Transition. Plutonium : Alpha-Delta-Epsilon.

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)