1. Dynamical Mean Field Theory of the Mott Transition Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University UBC September 2003

2. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Abstract: The Mott metal to insulator transition is realized in systems as diverse as the kappa organics, vanadium oxide and nickel selenide nickel sulfide mixtures. We will review the single site DMFT studies as well as a recent cluster DMFT study of this problem (O. Parcollet G. Biroli and G. Kotliar, cond-mat/0308577). We will also review critically recent experiments that test various DMFT predictions in these materials.

3. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Outline Relevance of the Mott transition problem. Dynamical Mean Field Theory Single Site and Extensions [I] Predictions of single site DMFT and observations. Prediction of cluster DMFT. Conclusions. Dynamical Mean Field Theory Single Site and Extensions EDMFT+GW and life without U [II]

4. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS The Mott transition problem Universal and non universal aspects. Frustration. t vs U the fundamental competition.

5. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V 2 O 3 under pressure or

6. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS NiSe 2-x S x

7. 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)

8. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Pressure Driven Mott transition

9. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS A more general perspective on DMFT. DMFT as an exact theory. (Chitra and Kotliar PRB 2001 Savrasov and GK cond- matt 2003) DMFT as an approximation (Chitra and Kotliar PRB2002) DMFT as a new starting point for perturbative expansions. ( P. Sun and G.K PRB 2002)

10. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT as an exact theory, analogy with DFT Start with TOE

11. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT: effective action construction(Fukuda et.al. ) Chitra and GK

12. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DFT: Kohn Sham formulation =

13. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Exchange and correlation energy Exact formal expressions can be given in terms of a coupling constant integration.[Harris-Jones, adiabatic connection] DFT is useful because practical accurate expressions for Exc, exist. LDA, GGA, hybrids,

14. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kohn Sham reference system

15. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Kohn Sham Greens function is an good point to compute spectra in perturbation theory in screenedCoulomb interaction GW,G0W0 Practical implementations, introduce a finite basis set. Division into valence (active ) degrees of freedom and core.

16. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Strongly correlated systems are usually treated with model Hamiltonians Conceptually one wants to restrict the number of degrees of freedom by eliminating high energy degrees of freedom. In practice other methods (eg constrained LDA are used)

17. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Functional derivation.

18. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Model Hamiltonian. Exact functional of the local Greens function A +

19. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT for model Hamiltonians. Kohn Sham formulation. Introduce auxiliary field Exact “local self energy”

20. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS About XC functional. One can derive a coupling constant integration formulae (Harris Jones formula) for Generate approximations. The exact formalism generates the local Greens function and  ii is NOT the self energy. However one can use the approach as starting point for computing other quantities.

21. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mapping onto impurity models. The local Greens function A, and the auxilliary quantity  can be computed from the solution of an impurity model that describes the effects of the rest of the lattice on the on a selected central site.

22. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on functional construction Atoms as a reference point. Expansion in t. Locality does not necessarily mean a single point. Extension to clusters. Jii ---  Jii Ji i+  Aii ---  Ai i+   ii ---   i i+  Exact functional  Aii,Ai i+   he lattice self energy and other non local quantities extending beyond the cluster are OUTSIDE the formalism and need to be inferred.

23. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Comments on funct. construction. Construction of approximations in the cluster case requires care to maintain causality. One good approximate construction is the cellular DMFT: a) take a supercell of the desired range,b) c) obtain estimate of the lattice self energy by restoring translational symmetry. Many other cluster approximations (eg. DCA, the use of lattice self energy in self consitency condition, restrictions of BK functional, etc. exist). Causality and classical limit of these methods has recently been clarified [ G Biroli O Parcollet and GK]

24. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice and cluster self energies

25. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT: Cavity Construction

26. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS DMFT Impurity cavity construction

27. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Solving the DMFT equations Wide variety of methods of solution, and they are straighforwardly extended to the cluster setting. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

28. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Single site DMFT and expt. Single site DMFT study of the Mott transition, based on a study of the Hubbard model on frustrated lattices made several interesting qualitative predictions. New experiments and reexamination of old ones give some credence to that the local picture is quite good.

29. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS V 2 O 3 under pressure or

30. 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)

31. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS NiSe 2-x S x

32. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Qualitative single site DMFT predictions. Spectra of the strongly correlated metallic regime contains both quasiparticle-like and Hubbard band-like features. Mott transition is drive by transfer of spectral weight.

33. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic Phase Diagram of the Frustrated Hubbard model

34. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

35. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Searching for a quasiparticle peak

36. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Schematic DMFT phase diagram Hubbard model (partial frustration). Evidence for QP peak in V2O3 from optics. M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

37. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS. ARPES measurements on NiS 2-x Se x Matsuura et. Al Phys. Rev B 58 (1998) 3690. Doniaach and Watanabe Phys. Rev. B 57, 3829 (1998)

38. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS QP in V2O3 was recently found Mo et.al

39. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Different transport regimes.

40. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Anomalous Resistivity and Mott transition Ni Se 2-x S x Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator.

41. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

42. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Phase Diagram k Organics

43. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

44. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Transport in k organics

45. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising endpoint finally found

46. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

47. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS

48. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Ising critical endpoint! In V 2 O 3 P. Limelette et.al.

49. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conclusion. An electronic model accounts for all the qualitative features of the finite temperature of a frustrated system at integer occupancy. The observation of the spinodal lines and the wide classical critical region where mean field holds indicate the coupling to the lattice is quantitatively very important.

50. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cluster studies of the Mott transition.

51. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Cluster extensions of single site DMFT M. Jarrell Dynamical Cluster Approximation [Phys. Rev. B 7475 1998] Continuous version [periodic cluster] M. Katsenelson and A. Lichtenstein PRB 62, 9283 (2000). Cellular DMFT [ G. Kotliar et.al. PRL87, 186401 2001] PCMDFT Biroli Parcollet and GK cond-matt 0307587

52. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Connection between cluster schemes. where tˆ(K) is the hopping expressed in the superlattice notations, with K in the Reduced Brillouin Zone

53. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Relation between CDMFT and exact studies of small systems and DMRG CDMFT improves on finite size studies by considering a system in an enviroment (described by the hybridization function) instead of an isolated system. This idea is similar the DMRG method of White, the key difference that it avoids diagonalizing a larger system to infer the nature of the bath, and instead it uses the periodicity which is contained in the CMDFT self consistency condition.

54. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Issue: best choice of G0 CDMFT, DCA, periodic cluster schemes, Nested cluster schemes Biroli Parcollet and GK cond-matt 0307587

55. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS C-DMFT: test in one dimension. (Bolech, Kancharla GK cond-mat 2002) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1

56. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS N vs mu in one dimension. Comparaison of 2+8 vs exact Bethe Anzats, [M. Capone and M.Civelli]

57. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS How good is the local approximation ? Study of the Mott transition within CDMFT. Are the single site DMFT results robust ? How are they modified by short range magnetic correlations? Study a frustrated model.

58. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS  organics ET = BEDT-TTF=Bisethylene dithio tetrathiafulvalene  (ET)2 X Increasing pressure -----  increasing t’  ------------ X0 X1 X2 X3 (Cu)2CN)3 Cu(NCN)2 Cl Cu(NCN2)2Br Cu(NCS)2 Spin liquid Mott transition

59. 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)

60. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Controversy on the unfrustrated case. Comment on "Absence of a Slater Transition in the Two- Dimensional Hubbard Model" B. KyungB. Kyung, J.S. Landry, D. Poulin, A.- M.S. Tremblay Phys. Rev. Lett. 90, 099702-1 (2003)J.S. LandryD. PoulinA.- M.S. Tremblay

61. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Double Occupancy vs U CDMFT Parcollet, Biroli GK

62. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Compare with single site results Rozenberg Chitra Kotliar PRL 2002

63. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition in cluster

64. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral FunctionU/D=2, U/D=2.25 (Parcollet et.al.) Uc=2.35+-.05, Tc/D=1/44

65. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the distribution in k space of the low energy spectral intensity as the Mott transition is approached. Spectral FunctionU/D=2, U/D=2.25 (Parcollet et.al.) Uc=2.35+-.05, Tc/D=1/44

66. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral FunctionU/D=2, U/D=2.25 (Parcollet et.al.) Uc=2.35+-.05, Tc/D=1/44

67. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Conjecture Formation of hot regions is a more general phenomena due to the proximity to the Mott point.

68. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Deviations from single site DMFT

69. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Lattice and cluster self energies

70. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mechanism for hot spot formation

71. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Evolution of the Spectral FunctionU/D=2, U/D=2.25 (Parcollet et.al.) Uc=2.35+-.05, Tc/D=1/44

72. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Mott transition and d wave symmetry

73. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions

74. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Start with the TOE

75. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Rewrite the TOE as an electron boson problem.

76. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Build effective action for the local greens functions of the fermion and Bose field r=R+  R unit cell vector  position within the unit cell. Ir>=|R,  Couple sources to

77. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Legendre transfor the sources, eliminating the field  Build exact functional of the correlation functionsW(r R,r’ R) and G (r R,r’ R)

78. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS “Kohn Sham “ decomposition.

79. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS (E)DMFT pproximation to Sum over all LOCAL 2PI graphs (integrations are restricted over the unit cell ) built with W and G Map into impurity model to generate G and W Go beyond this approximation by returning to many body theory and adding the first non local correction.

80. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Test on extended Hubbard model V/U=.25, P Sun and GK

81. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS EDMFT functional.

82. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Returning to many body physics.

83. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Take the solution of the EDMFT equations as an approximation for the TRUE local self energy, and add the leading NON LOCAL corrections to the self energy G_NL W_NL, as a correction. Do it self consistently and as a one shot iteration G0_NL W0_NL and compare the results.

84. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Average Z vs U (P. Sun 2003)

85. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Z1: K dependent part of QP residue.

86. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS Local DOS beta=44

87. THE STATE UNIVERSITY OF NEW JERSEY RUTGERS What was he correct solution within DMFT, was not accepted until recently. Phys. Rev. Lett. 82, 4890 (1999)