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Extensions of Single Site DMFT and its Applications to Correlated Materials. On the road towards understanding superconductivity in strongly correlated materials. Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University. Workshop on Quantum Materials

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Extensions of Single Site DMFT and its Applications to Correlated Materials


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    1. Extensions of Single Site DMFT and its Applications to Correlated Materials On the road towards understanding superconductivity in strongly correlated materials Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University • Workshop on Quantum Materials • Heron Island Resort • New Queensland Australia • 1-4 June 2005

    2. Mott Transition in the Actinide Series Lashley et.al.

    3. Mott transition in open (right) and closed (left) shell systems. Superconductivity is an unavoidable consequence to the approach to the Mott transition with a singlet closed shell state. S S g T Tc Log[2J+1] ??? Uc J=0 U U g ~1/(Uc-U)

    4. Cuprate superconductors and the Hubbard Model . PW Anderson 1987 . Connect superconductivity to an RVB Mott insulator. Science 235, 1196 (1987). Hubbard , t-J model . • Baskaran Zhou and Anderson (1987). slave boson approach, S-wave • Pairing. Connection to an insulator with a Fermi surface. • .

    5. RVB phase diagram of the Cuprate Superconductors and Superexchange. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) k, D singlet formation order parameters • The approach to the Mott insulator renormalizes the kinetic energy . Kinetic energy renormalizes to zero. • Attraction in the d wave channel of order J Not renormalized. Trvb increases. • The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero. • Superconducting dome. Pseudogap evolves continously into the superconducting state.

    6. Problems with the approach. • Neel order. How to continue a Neel insulating state ? • Stability of the pseudogap state at finite temperature. [Ubbens and Lee] • Missing incoherent spectra . [ fluctuations of slave bosons ] • Temperature dependence of the penetration depth [Wen and Lee , Ioffe and Millis ] . Theory:r[T]=x-Ta x2 , Exp: r[T]= x-T a. • Mean field is too uniform on the Fermi surface, in contradiction with ARPES. The development of DMFT solves many of these problems.!!

    7. Also, one would like to be able to evaluate from the theory itself when the approximation is reliable!!

    8. Cluster Extensions of Single Site DMFT

    9. Medium of free electrons : impurity model.Solve for the medium usingSelf Consistency. Extraction of lattice quantities. G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

    10. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension 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) ] U/t=4.

    11. Finite T Mott transtion in CDMFT Parcollet Biroli and GK PRL, 92, 226402. (2004))

    12. Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

    13. Evolution of the k resolved Spectral Function at zero frequency. (QMC study Parcollet Biroli and GK PRL, 92, 226402. (2004)) ) U/D=2.25 U/D=2 Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W

    14. Physical Interpretation • Momentum space differentiation. The Fermi liquid –Bad Metal, and the Bad Insulator - Mott Insulator regime are realized in two different regions of momentum space. • Cluster of impurities can have different characteristic temperatures. Coherence along the diagonal incoherence along x and y directions.

    15. Cuprate superconductors and the Hubbard Model . PW Anderson 1987

    16. CDMFT study of cuprates . • Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition. • Earlier studies (Katsnelson and Lichtenstein, M. Jarrell, M Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett 85, 1524 (2000) ) used QMC as an impurity solver and DCA as cluster scheme. • We use exact diag ( Krauth Caffarel 1995 with effective temperature 32/t=124/D ) as a solver and Cellular DMFT as the mean field scheme.

    17. Superconducting State t’=0 • Does the Hubbard model superconduct ? • Is there a superconducting dome ? • Does the superconductivity scale with J ?

    18. Superconductivity in the Hubbard model role of the Mott transition and influence of the super-exchange. ( work with M. Capone V. Kancharla. CDMFT+ED, 4+ 8 sites t’=0) .

    19. Superconducting State t’=0 • Does it superconduct ? • Yes. Unless there is a competing phase. • Is there a superconducting dome ? • Yes. Provided U /W is above the Mott transition . • Does the superconductivity scale with J ? • Yes. Provided U /W is above the Mott transition .

    20. Competition of AF and SC or SC AF SC AF AF+SC d d

    21. D wave Superconductivity and Antiferromagnetism t’=0 M. Capone V. Kancharla (see also VCPT Senechal and Tremblay ). Antiferromagnetic (left) and d wave superconductor (right) Order Parameters

    22. Competition of AF and SC U /t << 8 or SC AF AF SC AF+SC d d

    23. Can we connect the superconducting state with the “underlying “normal” state “ ? What does the underlying “normal” state look like ?

    24. Follow the “normal state” with doping. Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

    25. : Spectral Function A(k,ω→0)= -1/π G(k, ω→0) vs k U=16 t hole doped K.M. Shen et.al. 2004 2X2 CDMFT

    26. Approaching the Mott transition: CDMFT Picture • Fermi Surface Breakup. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! • D wave gapping of the single particle spectra as the Mott transition is approached. • Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, 226402. (2004) .

    27. What about the electron doped semiconductors ?

    28. Spectral Function A(k,ω→0)= -1/π G(k, ω→0) vs k electron doped P. Armitage et.al. 2001 Momentum space differentiation a we approach the Mott transition is a generic phenomena. Location of cold and hot regions depend on parameters. Civelli et.al. 2004

    29. Qualitative Difference between the hole doped and the electron doped phase diagram is due to the underlying normal state.” In the hole doped, it has nodal quasiparticles near (p/2,p/2) which are ready “to become the superconducting quasiparticles”. Therefore the superconducing state can evolve continuously to the normal state. The superconductivity can appear at very small doping. • Electron doped case, has in the underlying normal state quasiparticles leave in the (p, 0) region, there is no direct road to the superconducting state (or at least the road is tortuous) since the latter has QP at (p/2, p/2).

    30. Can we connect the superconducting state with the “underlying “normal” state “ ? • Yes, within our resolution in the hole doped case. • No in the electron doped case. • What does the underlying “normal state “ look like ? • Unusual distribution of spectra (Fermi arcs) in the normal state.

    31. Mott transition into a low entropy insulator. Is superconuctivity realized in Am ? “Soft” Mott Transition? “Hard” 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~6mB (Soderlind et.al., PRB 2000). • Experimentally, Am hasnon magnetic f6ground state with J=0(7F0)

    32. Am under pressure: J.C. GriveauJ. Rebizant G. Lander G. Kotliar PRL (2005)

    33. Mott transition into a low entropy insulator. Is it realized in Am ? • Yes! But there are additional suprises, which are specific to Am, such as the second maximum in Tc vs pressure. Additional system specific properties.

    34. Conclusions • Correlated Electron materials, as a second frontier in materials science research, the “in between “ regime between itinerant and localizedis very interesting. • Getting the general picture, and the material specific details are both important.. • Mott transition : open shell (finite T Mott endpoint in V2O3, NiSeS, K-organics, Pu ) closed shell case (Am, cuprates…….)connection to superconductivity. • The challenge of material design using correlated materials.

    35. Conclusions • DMFT is a useful mean field tool to study correlated electrons. Provide a zeroth order picture of a physical phenomena. • Provide a link between a simple system (“mean field reference frame”) and the physical system of interest. [Sites, Links, and Plaquettes] • Formulate the problem in terms of local quantities (which we can usually compute better). • Allows to perform quantitative studies and predictions . Focus on the discrepancies between experiments and mean field predictions. • Generate useful language and concepts. Follow mean field states as a function of parameters. • Controlled approach!

    36. Conjecture, Mott transition with Zcold finite ? Continuity with the insulator at one point in the zone.

    37. Conjecture, Mott transition with Zcold finite ? Continuity with the insulator at one point in the zone.

    38. Conjecture, Mott transition with Zcold finite ? Continuity with the insulator at one point in the zone.

    39. Is the formation of the hot and cold regions is the result of the proximity to Antiferromagnetism ?Study various values of t’/t, U=16.

    40. Introduce much larger frustration: t’=.9t U=16tn=.69 .92 .96

    41. Is the momentum space differentiation a result of proximity to an ordered state , e.g. antiferromagnetism? • Fermi Surface Breakup or Momentum space differentiation takes place irrespectively of the value of t’. The gross features are the result of the proximity to a Mott insulating state irrespective of whether there is magnetic long range order.

    42. How is the Mott insulatorapproached from the superconducting state ? Work in collaboration with M. Capone

    43. Evolution of the low energy tunneling density of state with doping. Decrease of spectral weight as the insulator is approached. Low energy particle hole symmetry.

    44. Alternative view