1. Hideo AOKI Dept Phys, Univ Tokyo Hideo AOKI Dept Phys, Univ Tokyo Strongly correlated electrons COE symposium “ Physics of strongly correlated systems --- from neutron stars to cold atoms ”, Tokyo, 19 Jan 2007

2. Kazuhiko Kuroki Univ Electro-Commun Ryotaro Arita MPI Stuttgart, now at RIKEN Seiichiro Onari Nagoya Univ  spin kxkx kyky Shiro Sakai Univ. Tokyo Ryotaro Arita RIKEN Karsten Held MPI Stuttgart Masaki Tezuka Univ. Tokyo Ryotaro Arita RIKEN SC Ryotaro Arita MPI Stuttgart  RIKEN Shiro Sakai Univ. Tokyo ferromagnetism Electron mechanism for SC Orbital degrees of freedom El-el + el-phonon

3. HTC FQHE Tatsuya Nakajima Tohoku Univ Masaru Onoda Electron Correlation Lab Takahiro Mizusaki Senshu Univ Takaharu Otsuka Univ Tokyo Tatsuya Nakajima Tohoku Univ Masaru Onoda Electron Correlation Lab Takahiro Mizusaki Senshu Univ Takaharu Otsuka Univ Tokyo

4. Condensed matter Electron correlation SC F F d-electrons s,p-electrons

5. attraction phonon el-el repulsion (spin /charge) Tc ～ 0.1ω D Tc ～ 0.01t anisotropic pairing 10000K  100K 100K  10K isotropic pairing

6. room T liq N Tc (K) liq 4 He year organics conventional oxides © H Aoki

7. ● Weak / strong correlation ● Spin and/or charge fluctuations （ Int ’ action range short  long ） ● Internal degrees of freedom （ orbitals ） ● Electron and/or phonon mechanisms ● Spin-off to FQHE physics Factors governing pairing

8. Hubbard Gutzwiller Kanamori Moriya HTC apres HTC Studies for correlated electron systems 1960 1970 1980 1990 2000 FLEX (Bickers et al, 1989) SCR (Moriya) RVB (Anderson, …) FRG (Metzner,..) QMC (Kuroki & Aoki, 1997) VMC (Yokoyama, Yamaji,..) DCA (Maier et al, 2005)

9. 1. Heavy fermion superconductors  spin fluctuation mediated 2. Superfluid 3 He  hard core interaction 3. SC in the Coulomb gas (GIC, STO?) Non-phonon mechanism SC Kohn & Luttinger 1965; Chubukov 1993: Repulsively interacting fermions  Attractive pairing channel exists for T  0 (weak, dilute case)

10. High-Tc cuprate (La 2 CuO 4 )

11. Hubbard model (a generic model) U t U ～ 5 eV t ～ 0.4 eV

13. FLEX Dyson’s eq effective interaction self energy self-consistent loop

14. Attraction  SC V(k,k ’) Repulsion  SC:  nothing strange =-V X X Repulsion  anisotropic SC + + - - attraction if  has nodes

15. ● Weak / strong correlation ● Spin and/or charge fluctuations （ Int ’ action range short  long ） ● Internal degrees of freedom （ orbitals ） ● Electron and/or phonon mechanisms Factors governing pairing

16. (Maier et al, Rev Mod Phys 2005) DCA hole concentration d-SC

17. Uemura plot T=T F TFTF TCTC Tc upperbounded by T C < 0.03t (FLEX, DCA) Tc ～ T F /100 is VERY low ! for cuprates : T F ～ 10000 K  T C < 100K

18. (Uemura 2004) Tc (1)Pairing int ’ action from el-el repulsion = weak Cf. Laser-cooled Fermi gas goes superfulid (Science, 2004) Tc  0.1 E F but attractive int ’ action ↑Feschbach resonance (3) Pairing from el-el repulsion = anisotropic (i.e., nodes in  BCS ) Good reasons why Tc is so low (2) Self-energy correction  quasi-particles short-lived - - + +

19. 2D or 3D ? (Arita et al, JPSJ; PRB1999; Monthoux-Lonzarich PRB 1999) kxkx kyky kxkx kyky kzkz > Hf Aoki, J Phys Condens. Matter(2004)

20. spin fluctuation ー＋ ＋＋ V singlet : V triplet : d x2-y2 Q spin + p, f + + - + + 2D 3D singlet   triplet Pairing interaction

21. Better the nesting, the better for SC ? (Onari et al, PRB 2003) Im   k max   Peak position/width in  (k,  )  band dispersion  (k ) ()()

22. Why cuprates ? (1) small  dp  large t eff ～ t dp 2 /  dp (2) single-band Hubbard (Wilson, 1988) Ag Au Ir p d

23. ● Weak / strong correlation ● Spin and/or charge fluctuations （ Int ’ action range short  long ） ● Internal degrees of freedom （ orbitals ） ● Electron and/or phonon mechanisms Factors governing pairing

24. V singlet : V triplet : charge fluctuationspin fluctuation ー＋ ＋＋  ↑↓    ↑↑ Spin- and charge-fluctuation mediated pairing more effective for longer-ranged repulsion

25. U=4 t’=0 same peak positions different positions n =0.7 n =0.6 n Phase diagram for the extended Hubbard (DCA+QMC: Arita et al, PRL 2004; FLEX: Onari et al, PRB 2004) V U triplet pairing

26. kxkx  charge 0  kyky General physical picture: Peak position/height/width  pairing symmetry General physical picture: Peak position/height/width  pairing symmetry 0  spin  kxkx kyky (Onari et al, 2004) Crossover to electron gas

27. (B)lattice (half-filled meaningful) (Takada, PRB ’ 93) (A)on-site U  extended Hubbard  1/r electron gas (spin fl dominated) (charge fl dominated) rsrs 8.63.3 sp dx 2 -y 2 12 th neighbour extended H (Onari et al, cond-mat/05) p s dxy (Takada, 1993) p s (A)1/r  Hubbard U dxydx 2 -y 2 p         spin

28. (Waber & Cromer, 1965) 4d and 5d orbital radius (A) atomic # p d Nb 4d 4 Ag Au Ir

29. Sr 2 RuO 4 Sr Ru(4d)  Cu(3d) O q2D Sr 2 RuO 4 (Maeno et al ) Pairing symmetry: triplet p+ip (Sigrist & Rice) p x+y + i = p x-y (Arita et al, PRL 2004) Time-reversal broken triplet p

30. When T-broken pairing can occur? When the space group of the pair has two-dimensional rep: as in ● p + ip in tetragonal systems ● d + id in hexagonal systems (  6 + ) (Onari et al, PRB 2002) d1d1 d2d2 + i More recent candidate --- Skutterdite RET 4 X 12

31. Spin: T-reversal Sr 2 RuO 4 Orbit: T-reversal Spin: T-reversal  non-unitary states Orbit: T-reversal (ie, broken SU(2)) Non-unitary SC ● Magnetic-field induced triplet pairing (Arita, Kuroki & Aoki, JPSJ 2004) 3 He ● A 1 phase of superfluid 3 He ● Ferromagnetic SC(UGe 2, etc) p T A1A1 Solid Super- fluid A B Liquid B

32. FFLO: Cooper pair with a momentum a q Fulde-Ferrell-Larkin-Ovchinikov k spacereal space CeCoIn 5 Ultrasonic NMR Watanabe et al Kakuyanagi et al

33. T μBμB Quark-gluon plasma ~ 1 GeV ~ 170 MeV Colour super- conductor Hadronic fluid Vacuum Outlook (5) Colour superconductivity ● neutron star FFLO

34. ● Weak / strong correlation ● Spin and/or charge fluctuations （ Int ’ action range short  long ） ● Internal degrees of freedom （ orbitals ） ● Electron and/or phonon mechanisms Factors governing pairing

35. Multi-orbital systems J: Hund ’ s coupling Superconductivity U - 3J U Intraorbital Coulomb U - 2J Interorbital Coulomb Hund’s coupling Magnetism

36. (a)(b) (c)(d) (a) Singlet Triplet (b) Triplet Singlet Cooper pair = (real space)x(spin) x(orbital) = antisymmetric Hund ’ s coupling  pairing symm DMFT+QMC result ( Sakai et al, PRB 2004 )

37. ● Weak / strong correlation ● Spin and/or charge fluctuations （ Int ’ action range short  long ） ● Internal degrees of freedom （ orbitals ） ● Electron and/or phonon mechanisms Factors governing pairing

38. attraction phonon el-el repulsion isotropic pairing Tc ～ 0.1ω D Tc ～ 0.01t anisotropic pairing Which is better ? or, what if they coexist ?

39. Anti-adiabatic limit Adiabatic limit = U/tU/t ω/t λ/tλ/t (Tezuka et al, 2005) Parameter space in Holstein-Hubbard model Parameter space in Holstein-Hubbard model U -t-t

40. (U,t’)=(2.0,-0.5) http://buckminster.physics.sunysb.edu/ trestle lattice cf. A 3 C 60 : fcc SC dominates! Break the el-hole symmetry  sc can dominate (Tezuka, Arita & Aoki, PRL 2005) In 1D Hubbard model: degeneracy  CDW =  SC  lifted when el-hole asymm

41. Ferromagnetism

42. E = p 2 /2m s d transition metals p EFEF Ferromagnetism  very difficult to realise

43. Repulsively interacting electrons (Hubbard model)  Ferromagnetism ? Stoner (UD(E F ) > 1) too crude a criterion (1) Any rigorous F ? (Nagaoka; Lieb, Mielke, Tasaki) (2) Why are real metallic magnets (Ni, Co, Fe) F ? Band (itinerant) ferromagnetism

44. U t U t 15 puzzle U >> t

45. He Multiple-exchange in He Ceperley

46. Design of a ferromagnetic polymer polyaminotriazole N C ・・・

47. Kanamori theory (T-matrix approx.) χ pp = + +… negligible Particle-particle scattering Particle-hole scattering J can be important! χ ph = + +… 1-orb. FLEX: Arita et al.’00. Reliable for low electron densities Ni: 2 holes in 5 orbitals General band filling Insulating FM with an antiferro-orbital order Metallic FM (Ni, Fe, Co) --- how does Hund’s coupling work? ? Itinerant FM in multiorbital systems (Sakai, PhD thesis 2006)

48. 3D fcc lattice with t=4t’=0.28 (W eff =4). Cf.) 1 orb: Ulmke’98 Lattice structure is important. U’ suppresses FM. J enhances FM. U’ & J are crucial both for n=1.5 and n=0.75 (~Ni). n=1.5 U=4 (i)U’=J=0 (1 orb.) (ii) U’=4, J=0 (iii) U’=3.5, J=0.25 (iv) U’=3, J=0.5 U’=U-2J for (ii)-(iv). Itinerant FM in multiorbital systems - 1st numerical result (Sakai, PhD thesis 2006)

49. Correlated electron systems FQHE systems

50. ρ xy B Fractional quantum Hall effect 1/ν ( ∝ B ) Pan et al 2002 ρ xx 0 1 2 3 4 2DE G ρ xy ρ xx B

51. Where does the quantum zero point come from ? Liquid He  [x, p] = ih FQHE  [x, y] = ih H = (1/2m)  2,  = p +eA R = (X,Y), [X, Y] = il 2  = -(1/eB) e z x , [  x,  y ] = -il 2 non-commutative space ！ l = (h/eB) 1/2 : magnetic length (  80A for B=10T) FQHE system ＝ Many-fermion system with Coulomb repulsion accompanied by uncertainty in (X, Y)

52. N S Composite fermion picture -- flux attachment Composite fermion picture -- flux attachment

53. 00.10.20.30.40.5  N =0 N =2 N =1 Compressible liquid Stripe Laughlin state Wigner crystal Bubble DMRG result: Shibata & Yoshioka 2003 1

54. Triplet p-ip (Pfaffian state  3 He A1  Sr 2 RuO 4 ) Trial wf: Moore-Read, Greiter-Wen-Wilczek 1991 Numerical: Morf 1998, Rezayi-Haldane 2000; Onoda-Mizusaki-Aoki, 2003 Experiment: Willett-West-Pfeiffer 1998, 2002 CF liquid  BCS paring at = 5/2 ? CF B p T A1 Solid Superfluid B A 3 He Sr Ru(4d)  Cu(3d) O  Reminds us of Kohn-Luttinger 1965 every metal superconducting with p,d,f, … pairing T  0

55. Interaction form Coulomb gauge field range Landau level  spin/charge  CF interaction HTC FQHE Band structure anisotropic isotropic Summary

56. © Aoki 2005 composite-boson picture for the Bose-Einstein condensate (©Kasamatsu et al, 2003) (Nakajima & Ueda, 2001; 2003;2004)

57. Magnetar (4 papers in Nature 31 August 2006) B > 10 11 T ?

58. Summary: correlated electrons Outlook ● Relation with neutron star physics Colour SC Ferromagnetism,... ● Relation with cold atom physics SC ferromagnetism ● Lattice models (Hubbard, etc) ● Coulomb gas ● Hard-core system ● FQHE