1. 48 Sanibel Symposium 2008 Cluster Dynamical Mean Field Approach to Strongly Correlated Materials K Haule Rutgers University

2. Sanibel 2008 References and Collaborators  Strongly Correlated Superconductivity: a plaquette Dynamical mean field theory study, K. H. and G. Kotliar, Phys. Rev. B 76, 104509 (2007). Phys. Rev. B 76, 104509 (2007).  Nodal/Antinodal Dichotomy and the Energy-Gaps of a doped Mott Insulator, M. Civelli, M. Capone, A. Georges, K. H., O. Parcollet, T. D. Stanescu, G. Kotliar, Phys. Rev. Lett. 100, 046402 (2008).Phys. Rev. Lett. 100, 046402 (2008).  Modelling the Localized to Itinerant Electronic Transition in the Heavy Fermion System CeIrIn5, J.H. Shim, K. Haule and G. Kotliar, Science 318, 1615 (2007),Science 318, 1615 (2007)  Quantum Monte Carlo Impurity Solver for Cluster DMFT and Electronic Structure Calculations in Adjustable Base, K. H., Phys. Rev. B 75, 155113 (2007).Phys. Rev. B 75, 155113 (2007).  Optical conductivity and kinetic energy of the superconducting state: a cluster dynamical mean field study, K. H., and G. Kotliar, Europhys Lett. 77, 27007 (2007).Europhys Lett. 77, 27007 (2007)  Doping dependence of the redistribution of optical spectral weight in Bi2Sr2CaCu2O8+delta, F. Carbone, A. B. Kuzmenko, H. J. A. Molegraaf, E. van Heumen, V. Lukovac, F. Marsiglio, D. van der Marel, K. H., G. Kotliar, H. Berger, S. Courjault, P. H. Kes, and M. Li, Phys. Rev. B 74, 064510 (2006).Phys. Rev. B 74, 064510 (2006)  Avoided Quantum Criticality near Optimally Doped High Temperature Superconductors, K.H. and G. Kotliar, Phys. Rev. B 76, 092503 (2007).Phys. Rev. B 76, 092503 (2007). Thanks to Ali Yazdani for unpublished data!

3. Sanibel 2008 Standard theory of solids Band Theory: electrons as waves: Rigid band picture: En(k) versus k Landau Fermi Liquid Theory applicable Very powerful quantitative tools: LDA,LSDA,GW Predictions: total energies, stability of crystal phases optical transitions M. Van Schilfgarde

4. Sanibel 2008 Fermi Liquid Theory does NOT work. Need new concepts to replace rigid bands picture! Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). Non perturbative problem. Strong correlation – Standard theory fails

5. Sanibel 2008 Non perturbative methods On site correlations usually the strongest -> Mott phenomena at integer fillings Successful theory which can describe Mott transition: Dynamical Mean Field Theory

6. Sanibel 2008 Mott phenomena at half filling 1B HB model (plaquette): Bad insulator Bad metal Single site DMFT Georges, Kotliar, Krauth, Rozenber, Rev. Mod. Phys. 1996

7. Sanibel 2008 Dynamical Mean Field Theory  For a given lattice site, DMFT envisions the neighboring sites on the lattice as a Weiss field of conduction electrons, exchanging electrons with that site. Maps lattice model to an effective quantum impurity model More rigorously: DMFT sumps up all local diagrams (to all orders in perturbation theory)

8. Sanibel 2008 DMFT in single site approximation Successfully describes spectra and response functions of numerous correlated materials: Mott transition in V 2 O 3 LaTiO 3 actinides (Pu,…) Lanthanides (Ce,…) and … far to many to mention all, KH et al. 2003, KH et al. 2007 Recent review: (G. Kotliar S. Savrasov K.H., V. Oudovenko O. Parcollet and C. Marianetti, RMP 2006).

9. Sanibel 2008 Non-f spectra at 10K Ce In Band structure and optics of heavy fermion CeIrIn 5 300K eV J.H. Shim, KH, and G. Kotliar, Science 318, 1618 (2007).

10. Sanibel 2008 Later verified by Yang & Pines Remarkable agreement with Y. Yang & D. Pines cond-mat/0711.0789! Anomalous Hall coefficient

11. Sanibel 2008 High Tc: Need non-local self-energy d-wave pairing: 2x2 cluster-DMFT necessary to capture the order parameter Fermi surface evolution with doping can not be understood within single site DMFT.

12. Sanibel 2008 Cluster DMFT approaches In the Baym Kadanoff functional, the interacting part  is restrictied to the degrees of freedom (G) that live on the cluster.  R=(0,0)  R=(1,0)  R=(1,1)  [G plaquette ] periodization Maps the many body problem onto a self consistent impurity model Momentum space approach-Dynamical cluster approximation (Hettler, Maier, Jarrel) Real space approach – Cellular DMFT (Kotliar,Savrasov,Palson) Impurity solvers: Continuous time QMC Hirsh-Fye QMC NCA ED

13. Sanibel 2008 General impurity problem Diagrammatic expansion in terms of hybridization  +Metropolis sampling over the diagrams Exact method: samples all diagrams! No severe sign problem K.H. Phys. Rev. B 75, 155113 (2007) ; P Werner, PRL (2007); N. Rubtsov PRB 72, 35122 (2005). An exact impurity solver, continuous time QMC - expansion in terms of hybridization

14. Sanibel 2008 Approach  Understand the physics resulting from the proximity to a Mott insulator in the context of the simplest models.  Construct mean-field type of theory and follow different “states” as a function of parameters – superconducting & normal state. [Second step compare free energies which will depend more on the detailed modeling and long range terms in Hamiltonian…..]  Leave out disorder, electronic structure, phonons … [CDMFT+LDA second step, under way]  Approach the problem from high temperatures where physics is more local. Address issues of finite frequency– a nd finite temperature crossovers.

15. Sanibel 2008  (i  ) with CTQMC Hubbard model, T=0.005t on-site largest nearest neighbor smaller next nearest neighbor important in underdoped regime

16. Sanibel 2008 Momentum space differentiation t-J model, T=0.005t Normal state T>Tc SC state T<<Tc with large anomalous self- energy …gets replaced by coherent SC state Normal state T>Tc: Very large scattering rate at optimal doping

17. Sanibel 2008 SC Tunneling DOS Large asymmetry at low doping Gap decreases with doping DOS becomes more symmetric Asymmetry is due to normal state DOS -> Mottness Computed by the NCA for the t-J model SC  =0.08 SC  =0.20 NM  =0.08 NM  =0.20 K. H. and G. Kotliar, Phys. Rev. B 76, 104509 (2007).

18. Sanibel 2008 Ratio A S /A N Ratio more universal, more symmetric With decreasing doping gap increases, coherence peaks less sharp->Non BCS McElroy,.. JC Davis, PRL 94, 197005 (2005) Exp:Bi2212 with STM

19. Sanibel 2008 Yazdani’s experiment on Bi2212, 30K, slightly overdoped A.N. Pasupathy (1), A. Pushp (1,2), K.K. Gomes (1,2), C.V. Parker (1), J. Wen (3), Z. Xu (3), G. Gu (3), S. Ono (4), Y. Ando (5), and Ali Yazdani (1), (1)Princeton University, (2)Urbana-Champaign (3)Brookhaven N.L., (4)CRIEPI, Tokyo, Japan, (5)ISIR, Osaka, Japan. Unpublished, shown with permission Ratio A S /A N CDMFT calculation ratio almost symmetric Pronounced dip-hump feature can not be fitted with BCS

20. Sanibel 2008 Non-BCS DOS in normal state decreases Gap increases

21. Sanibel 2008 Normal state DOS a nd SC gap Not using realistic band structure (t’) CDMFT A.N. Pasupathy (1), A. Pushp (1,2), K.K. Gomes (1,2), C.V. Parker (1), J. Wen (3), Z. Xu (3), G. Gu (3), S. Ono (4), Y. Ando (5), and Ali Yazdani (1), (1)Princeton University, (2)Urbana-Champaign (3)Brookhaven N.L., (4)CRIEPI, Tokyo, Japan, (5)ISIR, Osaka, Japan. Unpublished, shown with permission Yazdani’s experiment on Bi221,30K, slightly overdoped

22. Sanibel 2008 Gap changes, mode does not J.C. Davis, Nature 442, 546 (2006)

23. Sanibel 2008 Where does the dip-hump structure come from? J.C. Davis, Nature 442, 546 (2006)

24. Sanibel 2008  Eliashberg theory A(  )   phonon frequency Gap up to    Real part constant No scattering up to   

25. Sanibel 2008 A S /A N Dip-hump structure Kink in normal self-energy Sharp rise of scattering rate in SC state Most important: Dip and peak in anomalous self-energy Normal self-energy in SC stateNormal self-energy in NM state anomalous self-energy

26. Sanibel 2008 Phenomenology Similar frequency dependence of gap recently introduced by W.Sacks and B. Doucot PRB 74,174517 (2006) to fit experiments.

27. Sanibel 2008 Fermi surface  =0.09 Arcs FS in underdoped regime pockets+lines of zeros of G == arcs Cumulant is short in ranged: Single site DMFT PD

28. Sanibel 2008 Nodal quasiparticles

29. Sanibel 2008 Nodal quasiparticles the slope=v nod almost constant V nod almost constant up to 20% v  dome like shape Superconducting gap tracks Tc! M. Civelli, cond-mat 0704.1486

30. Sanibel 2008 Energy scale of peak in antinodal (nodal) region increases (decreases) with decreasing doping in underdoped cuprates. Two energy scales in Raman Spectrum in the SC State of Underdoped Cuprates Le Tacon et al, Nat. Phys. 2, 537 (2006) doping

31. Sanibel 2008 Evolution of Nodal and Antinodal energy scales with doping Le Tacon et al, Nat. Phys. 2, 537 (2006)

32. Sanibel 2008 Antinodal gap – two gaps M. Civelli, using ED, PRL. 100, 046402 (2008). “true” superconducting gap has a dome like shape (like v  ) Normal state “pseudogap” monotonically increasing with underdoping

33. Sanibel 2008 Optical conductivity Basov et.al.,PRB 72,54529 (2005)  Low doping: two components Drude peak + MIR peak at 2J  For x>0.12 the two components merge  In SC state, the partial gap opens – causes redistribution of spectral weight up to 1eV

34. Sanibel 2008 Optical spectral weight - Hubbard versus t-J model t-J model J Drude no-U Experiments intraband interband transitions ~1eV Excitations into upper Hubbard band Kinetic energy in Hubbard model: Moving of holes Excitations between Hubbard bands Hubbard model Drude U t 2 /U Kinetic energy in t-J model Only moving of holes f-sumrule

35. Sanibel 2008 Optical spectral weight & Optical mass F. Carbone,et.al, PRB 74,64510 (2006) Bi2212 Weight increases because the arcs increase and Zn increases (more nodal quasiparticles) mass does not diverge approaches ~1/J Basov et.al., PRB 72,60511R (2005)

36. Sanibel 2008 Temperature/doping dependence o f the optical spectral weight Single site DMFT gives correct order of magni tude (Toshi&Capone) At low doping, single site DMFT has a small cohere nce scale -> big change Cluser DMF for t-J: Carriers become more coherent In the overdoped regime -> bigger change in kinetic energy for large 

37. Sanibel 2008 Bi2212 ~1eV Weight bigger in SC, K decreases (non-BCS) Weight smaller in SC, K increases (BCS-like) Optical weight, plasma frequency F. Carbone,et.al, PRB 74,64510 (2006) A.F. Santander-Syro et.al, Phys. Rev. B 70, 134504 (2004)

38. Sanibel 2008 Phys Rev. B 72, 092504 (2005) cluster-DMFT, Eu. Lett. 77, 27007 (2007). Kinetic energy change Kinetic energy decreases Kinetic energy increases Exchange energy decreases and gives largest contribution to condensation energy same as RVB (see P.W. Anderson Physica C, 341, 9 (2000)

39. Sanibel 2008 Origin of the condensation energy Resonance at 0.16t~5Tc (most pronounced at optimal doping) Second peak ~0.38t~120meV (at opt.d) substantially contributes to condensation energy Scalapino&White, PRB 58, (1998) Main origin of the condensation energy

40. Sanibel 2008 Conclusions Plaquette DMFT provides a simple mean field picture of the underdoped, optimally doped and overdoped regime One can consider mean field phases and track them even in the region where they are not stable (normal state below Tc) Many similarities with high-Tc’s can be found in the plaquette DMFT: Strong momentum space differentiation with appearance of arcs in UR Superconducting gap tracks Tc while the PG increases with underdoping Nodal fermi velocity is almost constant Tunneling DOS As/An has a dip hump dip structure -> comes from the structure in the anomalous self-energy Optical conductivity shows a two component behavior at low doping Optical mass ~1/J at low doping and optical weigh increases linearly with  In the underdoped system -> kinetic energy saving mechanism overdoped system -> kinetic energy loss mechanism exchange energy is always optimized in SC state

41. Sanibel 2008 Issues The mean field phase diagram and finite temperature crossover between underdoped and over doped regime Study only plaquette (2x2) cluster DMFT in the strong coupling limit (at large U=12t) Can not conclude if SC phase is stable in the exact solution of the model. If the mean field SC phase is not stable, other interacting term in H could stabilize the mean-field phase (long range U, J)

42. Sanibel 2008 Doping dependence of the spectral weight F. Carbone,et.al, PRB 74,64510 (2006) Comparison between CDMFT&Bi2212

43. Sanibel 2008 RVB phase diagram of the t-J m. Problems with the RVB slave bosons:  Mean field is too uniform on the Fermi surfa ce, in contradiction with ARPES.  Fails to describe the incoherent finite temperature regime and pseudogap regime.  Temperature dependence of the penetration depth. Theory:  [T]=x-Ta x 2, Exp:  [T]= x-T a.  Can not describe two distinctive gaps: normal state pseudogap and superconducting gap

44. Sanibel 2008 Similarity with experiments Louis Taillefer, Nature 447, 565 (2007). A. Kanigel et.al., Nature Physics 2, 447 (2006) Arcs FS in underdoped regime pockets+lines of zeros of G == arcs de Haas van Alphen small Fermi surface Shrinking arcs On qualitative level consistent with

45. Sanibel 2008 Fermi surface  =0.09 Arcs FS in underdoped regime pockets+lines of zeros of G == arcs Arcs shrink with T! Cumulant is short in ranged:

46. Sanibel 2008 Insights into superconducting state (BCS/non-BCS)? J. E. Hirsch, Science, 295, 5563 (2002) BCS: upon pairing potential energy of electrons decreases, kinetic energy increases (cooper pairs propagate slower) Condensation energy is the difference non-BCS: kinetic energy decreases upon pairing (holes propagate easier in superconductor)

47. Sanibel 2008 Pengcheng et.al., Science 284, (1999) YBa 2 Cu 3 O 6.6 (Tc=62.7K) Origin of the condensation energy local susceptibility Resonance at 0.16t~5Tc (most pronounced at optimal doping) Second peak ~0.38t~120meV (at opt.d) substantially contributes to condensation energy Scalapino&White, PRB 58, (1998) Main origin of the condensation energy

48. Sanibel 2008 Similarity with experiments Louis Taillefer, Nature 447, 565 (2007). Arcs FS in underdoped regime pockets+lines of zeros of G == arcs de Haas van Alphen small Fermi surface On qualitative level consistent with

49. Sanibel 2008 Anomalous self-energy t-J model: Hubbard: 0

50. Sanibel 2008 Momentum versus real space In plaquette CDMFT cluster quantities are diagonal matrices in cluster momentum base In analogy with multiorbital Hubbard model exist well defined orbitals But the inter-orbital Coulomb repulsion is nontrivial and tight-binding Hamiltonian in this base is off-diagonal

51. Sanibel 2008 J.H. Shim, K.H., G. Kotliar, Nature 446, 513 (2007). Mixed valence nature of  -Pu and magnetism of Cm, generalized valence of a solid Magnetization of Cm:

52. Sanibel 2008 Normal state T>Tc  (0,0) orbital reasonable coherent Fermi liquid t-J model, T=0.01t Momentum space differentiation  ( ,0) very incoherent around optimal doping (   ~0.16 for t-J and   ~0.1 for Hubbard U=12t)  (  ) most incoherent and diverging at another doping (  1 ~0.1 for t-J and  1 ~0 for Hubbard U=12t)

53. Sanibel 2008 Momentum space differentiation t-J model, T=0.005t Normal state T>Tc SC state T<<Tc with large anomalous self- energy …gets replaced by coherent SC state Normal state T>Tc: Very large scattering rate at optimal doping  ( ,0) orbital T

54. Sanibel 2008 Anomalous self-energy and order parameter Anomalous self-energy: Monotonically decreasing with i  Non-monotonic function of doping (largest at optimal doping) Of the order of t at optimal doping at T=0,  =0 Order parameter has a dome like shape and is small (of the order of 2Tc) Hubbard model, CTQMC

55. Sanibel 2008 Anomalous self-energy on real axis t-J model: It does not change sign at certain frequency  D ->attractive for any  Many scales can be identified J,t,W Although it is peaked around J, it remains large even for  ~W

56. Sanibel 2008 Superfluid density at low T Low T expansion using imaginary axis QMC data. Current vertex corrections are neglected In RVB the coefficient b~  2 at low  [Wen&Lee, Ioffe&Millis]

57. Sanibel 2008 Superfluid density close to T c Computed by NCA, current vertex corrections neglected underdoped