1. Extended t-J model – the variational approach T. K. Lee Institute of Physics, Academia Sinica, Taipei, Taiwan July 10, 2007, KITPC, Beijing

2. Outline Background and motivation Variational Monte Carlo method -- basic approach -- improve trial wave functions increase variational parameters power Lanczos method Hole-doped cases, antiferromagnetic (AFM) and superconducting states. -- ground state -- excitations Electron-doped cases Summary

3. Damscelli, Hussain, and Shen, Rev. Mod. Phys. 2003 Phase diagram holeelectron

4. Phase diagram Only AFM insulator (AFMI)? How about metal (AFMM), if no disorder? Coexistence of AF and SC? Related to the mechanism of SC?

5. H Mukuda et al., PRL (’06) Phase diagram for multi-layer systems UD Hg-1245 ( T c =72K, T N ~290K ) AFM (0.69μ B ) AFM (0.67μ B ) AFM (0.69μ B ) AFM (0.67μ B ) SC (72K) +AFM (0.1μ B )

6. “ideal” Phase diagram for hole-doped cupartes?

7. H. Mukuda et al., PRL 96, 087001 (2006).

8. Kitaoka, Mukuda et al. PRL (‘06) Zero-field NMR spectrum at 1.4K Coexisting state of SC and AFM is realized on a single CuO 2 plane (OP) !! N h (IP)~0.06, N h (OP)=0.22 N h (IP)~0, N h (OP)~0.08? N h (IP)~0.08?, N h (OP)=0.24 AFMM

9. OP IP+ IP* Linewidth  50 Oe (IP) Flatness of CuO 2 plane 150 Oe 50 Oe Narrowest among high-T c cuprates ! ・ Ideal flatness ・ Hole is homogeneously doped Slide From H. Mukuda

10. Basic info from experiments 5 possible phases: AFMI, AFMM, d-wave SC, and AFM+d- SC, normal metal.  e-doped system is different from hole- doped.  Broken symmetries: particle and hole; AFM, SC… Theoretical challenge: the simplest model to account for all these properties? To start, only consider the homogeneous solution.

11. Minimal theoretical models 2D Hubbard model – U and t differ by almost an order of magnitude, reliable numerical approaches: exact diagonalization (ED) for 18 sites with 2 holes (Becca et al, PRB 2000) finite temp. quantum Monte Carlo (QMC), fermion sign problem ( Bulut, Adv. in Phys. 2002) 2D t-J type models –

12. Three species: an up spin, a down spin or an empty site or a “hole” Model proposed by P.W. Anderson in 1987: t-J model on a two-dimensional square lattice, generalized to include long range hopping Constraint: For hole-doped systems two electrons are not allowed on the same lattice site t ij = t for nearest neighbor charge hopping, t’ for 2nd neighbors, t’’ for 3rd, t’ and t’’ breaks the particle-hole symmetry J is for n.n. AF spin-spin interaction

13. Minimal theoretical models 2D Hubbard model – U and t differ by almost an order of magnitude, reliable numerical approaches exact diagonalization (ED) for 18 sites with 2 holes (Becca et al, PRB 2000) finite temp. quantum Monte Carlo (QMC), fermion sign problem ( Bulut, Adv. in Phys. 2002) 2D t-J type models – no finite temp. QMC – sign problem and strong constraint ED for 32 sites with 1,2 and 4 holes (Leung, PRB)

14. Variational Monte Carlo method Expectation values of an operator O in |  >: Bases: -- the probability of config. α Metropolis algorithm Slater determinant

15. After a series of configurations (  1,  2,…,  M ) is generated, expectation values of the operator O is given by with error To estimate the accuracy of, Do M G independent Monte- Carlo runs with different initial configurations. with error Here, M  total number of configurations! Is it accurate?

16. Linear optimization method Variational parameters set { } Search for the optimized parameters opt which give the minimum energy, Employ a simple linear optimization of each parameter in 1D with the other fixed. Linear optimization may not work well if correlated trial wave functions require the use of many parameters. Then, what should we do?

17. Stochastic reconfiguration (SR) method Optimization for variational energy: By Taylor expansion of for, p i : variational parameters, i=1,2,…,v p.  : a given configuration {R 1,…,R N }, where the O i operator is Up to O(  p i 2 ) First, Casula, et. al., J. Chem. Phys. ’04 Sorella, PRB ’05 Yunoki and Sorella, PRB ‘06

18. Similar to steepest descent (SD) method, we define a “force”: The energy improvement is approximately written as Energy will converge to the minimum when all F i =0. We can tune  t(>0) to control the convergence of iteration. We can obtain the next parameters p i ’ by iterating Similar to steepest descent (SD) method, we define a “force”:

19. However, SD method overlooks a possibility that a small  p i may lead to a large change of the wave function… Therefore, we also need to minimize a “distance”: we ignore O(  p i 3 ) terms A functional is defined as is a Lagrange multiplier

20. Minimization of f(  p i ) with respect to  p i, we have a “new” iterated formula:  t=1/2 Then, the energy improvement for SR method becomes S ij matrix remains positive definite. Sometimes S ij has no inverse matrix for some unstable iterations. If so, we use the SD method instead.

21. How do we choose  t in SR method? 2D Lattice size=64 t-t’-t’’-J model with (t’,t’’,J)=(-0.3,0.2,0.3) Trial wave function: d-wave RVB wave function n is the number of iterations linear SR

22. For a given trial wave function,, we approach the ground state in two steps: 1. Lanczos iteration C 1 and C 2 are taken as variational parameters 2. Power Method Beyond VMC approach – the Power-Lanczos method We denote this state as |PL1> |PL2>

23. J=0.3E Exact-0.583813 RVB(PL0)-0.5431 RVB(PL1)-0.5654 RVB(PL2)-0.5709 PL Energy After PL, nk, HH(R), and energy get closer to exact results! For the t-J model, Leung PRB 2006, 4 holes in 32 sites

24. HH(R) – the hole-hole correlation function

25. Trial wave functions for hole-doped systems Must account for : Mott insulator AFM, SC, AFM+SC?

26. This provides the pairing mechanism! It can be easily shown that near half-filling this term only favors d-wave pairing for 2D Fermi surface! In 1987, Anderson pointed out the superexchange term Spin or charge pairing? AFM is natural! D-wave SC?

27. The resonating-valence-bond (RVB) variational wave function proposed by Anderson ( originally for s-wave and no t’, t’’), d-RVB = A projected d-wave BCS state! The Gutzwiller operator P d enforces no doubly occupied sites for hole-doped systems AFM was not considered. four variational parameters, t v ’ t v ’’ ∆ v, and μ v

28. The simplest way to include AFM: Lee and Feng, PRB 1988, for t-J

29. Assume AFM order parameters: staggered magnetization And uniform bond order Use mean field theory to include AFM, Two sublattices and two bands – upper and lower spin-density-wave (SDW) bands Lee and Feng, PRB38, 1988; Chen, et al., PRB42, 1990; Giamarchi and Lhuillier, PRB43, 1991; Lee and Shih, PRB55, 1997; Himeda and Ogata, PRB60, 1999

30. RVB + AFM for the half-filled ground state (no t, t’ and t’’) PdPd & Ne = # of sites Variational results staggered moment m = 0.367 “best” results -0.3344 0.375 ~ 0.3 Liang, Doucot And Anderson

31. The wave function for adding holes or removing electrons from the half-filled RVB+AFM ground state A down spin with momentum –k ( & – k + (π, π ) ) is removed from the half-filled ground state. --- This is different from all previous wave functions studied. Lee and Shih, PRB55, 5983(1997); Lee et al., PRL 90 (2003); Lee et al. PRL 91 (2003). The state with one hole (two parameters: m v and Δ v ) Creating charge excitations to the Mott Insulator “vacuum”.

32. Dispersion for a single hole. t’/t= - 0.3, t”/t= 0.2 The same wave function is used for both e-doped and hole-doped cases. There is no information about t’ and t” in the wave function used. Energy dispersion after one electron is doped. The minimum is at (π, 0). t’/t= 0.3, t”/t= - 0.2 J/t=0.3 1st e - 1st h + □ Kim et. al., PRL80, 4245 (1998); ○ Wells et. al.. PRL74, 964(1995); ∆ LaRosa et. al. PRB56, R525(1997). ● SCBA for t-t’-t’’-J model, Tohyama and Maekawa, SC Sci. Tech. 13, R17 (2000)

33. ARPES for Ca 2 CuO 2 Cl 2 The lowest energy at Ronning, Kim and Shen, PRB67 (2003) Nd 2-x Ce x CuO 4 -- with 4% extra electrons Fermi surface around (π,0) and (0, π)! Armitage et al., PRL (2002)

34. Momentum distribution for a single hole calc. by the quasi-particle wave function Exact results for the single- hole ground state for 32 sites. Chernyshev et al. PRB58, 13594(98’) 64 sites

35. Wave Functions for one hole system Quasi-particle state: Hole momentum (k h )=unpaired spin momentum (k s ) = k Spin-bag state:

36. a QP state k h =( ,0) = k s a SB state k s =( ,0), k h =(  /2,  /2) Exact 32 site result from P. W. Leung for the lowest energy (π,0) state t-J t-t’-t’’-J

37. Takami TOHYAMA et al., J. Phys. Soc. Jpn. Vol 69, No1, pp. 9-12 (0,0) QP (0,0) SB (  /2,  /2) QP ( ,0) QP ( ,0) SB a 0.188-0.0288-0.00440.123-0.0313 b 0.188-0.0254-0.00520.159-0.0085 c 0.202-0.0302-0.00050.071-0.002 d -0.273-0.203-0.2241-0.353-0.1921 e -0.264-0.195-0.2154-0.279-0.2115 Our Result: (π, 0) is QP at t-J model, but SB for t-t’-t”-J. (0, 0) is QP for both QP:Quasi-Particle state SB: Spin-Bag state J/t=0.4, t’/t  -α/3 and t”/t’=2/3

38. Ground state is k h =(π/2, π/2) for two 0e holes; k h =(π,0) for two 2e holes. The state with two holes Similar construction for more holes and more electrons. The Mott insulator at half-filling is considered as the vacuum state. Thus hole- and electron-doped states are considered as the negative and positive charge excitations. Fermi surface becomes pockets in the k-space! Lee et al., PRL 90 (2003); Lee et al. PRL 91 (2003).

39. The new wave function has AFM but negligible pairing. It could be used to represent an AFM metallic (AFMM) phase. d-wave pairing correlation function 2 holes in 144 sites

40. 2 holes in 144 sites

41. Increase doping, pockets are connected to form a Fermi surface: Cooper pairs formed by SDW quasiparticles three new variational parameters: μ v, t’ v and t” v

42. AFMM shows stronger hole- hole repulsive correlation than AFMM+SC. The pair-pair correlation of AFMM is much smaller than AFMM+SC.

43. μ v, t ’v and t” v m v and Δ v Δ v, μ v, t ’v and t” v

44. All trial wave functions: RVB(SC): AFMM+SC: AFMM:

45. Variational Energy Phase diagram

46. SDW: The energy difference among these states

47. CT Shih et al., LTP (’05) and PRB (‘04) t’/t=0 t’’/t=0 t’/t=-0.3, t’’/t=0.2 AFMM+SC AFMM x 0.60.50.40.30.2 0.1 Possible Phase Diagrams for the t-J model t’/t=-0.2 t’’/t=0.1

48. Phase diagram for hole-doped systems H Mukuda et al., PRL (’06) The “ideal” Cu-O plane Extended t-J model, t’/t=-0.2, t’’/t=0.1

49. Summary – I Variational approach provides a quantitative way to understand the t-J or extended t-J model by taking into account the projection rigorously. Based on the RVB concept, trial wave functions for the doped system have been constructed to represent the AFMM, AFMM+SC and SC phases observed in multilayer cuprates. With the values of t, t’ and J in the range of experiments, the phase diagram obtained agree with cuprates below optimal doping.

50. Acknowledgement Y. C. Chen, Tung Hai University, Taichung, Taiwan R. Eder, Forschungszentrum, Karlsruhe, Germany C. M. Ho, Tamkang University, Taipei, Taiwan C. Y. Mou, National Tsinghua University, Taiwan Naoto Nagaosa, University of Tokyo, Japan C. T. Shih, Tung Hai University, Taichung, Taiwan Students: Chung Ping Chou, National Tsinghua University, Taiwan Wei Cheng Lee, UT Austin Hsing Ming Huang, National Tsinghua University, Taiwan

51. Outline Background and motivation Variational Monte Carlo method -- basic approach -- improve trial wave functions increasing variational parameters power Lanczos method Hole-doped cases, antiferromagnetic (AFM) and superconducting states. -- ground state -- excitations Electron-doped cases Summary

52. Excitations in the SC state With a fixed-number of holes, the d-RVB state becomes The ground state Quasi-particle excitation Excitation energies are fitted with

53. Example: QP excitation t’/t, t’’/t, and  /t are renormalized and “Fermi surface” is determined by Setting  =0 in the excitation energy.

54. J/t=0.3, t=0.3eV t’/t=-0.3 & t’’/t=0.2 for YBCO and BSCO but t’/t=-0.1 & t’’/t=0.05 for LSCO Our VMC overestimates the gap by a factor of 2, PRB 2006

55. Anomalies in the spectral weight for the low-lying excitations. For Gutzwiller-projected wave functions?? For ideal Fermi gas : For BCS theory :

56. Exact Identity : S. Yunoki, cond-mat/0508015, C.P. Nave et al., cond-mat/0510001 No exact relation is known about Using the identities,

57. Is this relation between pairing amplitude and spectral weight also true for projected wave functions? BCS theory predicts So what is ? YES!!

58. Pairing amplitude by VMC (II): C.T. Shih et al., PRL Long-range pair-pair correlation

59. Anomalies in STS conductance S.H. Pan et al., unpublished data OPD T. Hanaguri et al., Nature Ca 2-x Na x CuO 2 Cl 2 MgB 2

60. McElroy et al. Science (05)

61. Could the asymmetry be DOS effects ? d-BCS with 2 bands B. W. Hoogenboom et al. PRB (‘03) as hole doped d-wave gap opened ! DOS singularity Hoffman M. Cheng et al. PRB (‘05)

62. Particle-hole asymmetry for STS conductance the tunneling conductance at negative bias: For positive bias: Conductance is related to the spectral weight

63. Quasi-particle contribution to the conductance ratio d-RVB (t’=-0.3, t’’=0.2) d-BCS  E=0.3t for 12X12  E=0.2t for 20X20

64. Quasi-particle contribution to the conductance ratio : d-RVB (t’=-0.3, t’’=0.2) d-BCS

65. Spectral weight PEAK HEIGHTS and GAP SIZES

66. Why is d-wave SC so robust? McElroy et al. Science (05)

67. Consider the impurity elastic scattering matrix element For BCS theory, this is just u k u k’ – v k v k’ But for projected states, there is a strong renormalization

68. g t =2x/(1+x) Renormalized Mean-field theory, Garg et al, Cond-mat/0609666 15 holes in 12*12

69. Summary - II In addition to studying ground states, variational approach could also study excitation spectra, STM conductance asymmetry, effects of disorder (why d- wave is so robust) etc..

70. Hole-dopedElectron-doped T’ phaseT phase Electron-doped systems

71. Damscelli, Shen and Hussain, Rev. Mod. Phys. 2003 Phase diagram:

72. t for n.n., t’ for 2 nd n.n., and t” for 3 rd n.n. Consider t-t’-t”-J model: For electron-doped, no empty sites!! “Lower Hubbard band is projected out”

73. Experiments M. Ikeda, thesis ‘06 T. Kubo, Physica C ‘02

74. VMC (0.1,-0.05) (t’,t’’) (0.15,-0.1) (0.3,-0.2)

75. Recent ARPES for Nd 2-x Ce x CuO 4 : 16.5 eV --- Armitage et al., PRL 2002 55 eV --- Armitage et al., PRL 2001 (x=0.15) 400 eV --- Claesson et al., PRL 2004 (x=0.15) 22 eV --- H. Matsui et al., 0410388 (x=0.13)

76. NCCO SCCO Non-uniform gap! (x=0.13, T N =110K, T=30K) x=0.14, Tc=13K,T~18K, T N =110K

77. A gap near (π,0) region, but not near (π/2,π/2) The gap increases as it moves away from (π,0) The “lower” band approaches Fermi energy near (π /2, π /2) and the peaks are broad Could these be lower and upper SDW bands?? Near (π,0)Near (π/2,π/2)

78. t-J model only has the upper Hubbard band + “AF LRO” k and k + Q are coupled!! Q= (π,π) In weak coupling analogy, 2 “SDW” bands

79. staggered magnetization: uniform bond order: d-wave RVB pairing: 2 sublattices and 2 bands – upper and lower SDW bands G.J. Chen at al. 1990, Giamarchi and Lhuiller, 1991 Same asa Same asa same as the hole-doped case

80. Mean-field Hamiltonian for lower and upper SDW bands: with

81. A down spin with –k (–k+Q) is removed from the half-filled ground state. Dope one hole Create charge excitations in the Mott Insulator “vacuum”, The AFMM state: Dope one electron T. K. Lee, et al., PRB 1997; T. K. Lee, et al., PRL 2003; W. C. Lee, et al., PRL 2003. all quantities could be calculated with hole wf except t’/t → – t’/t and t”/t → – t”/t

82. 1 electron wave function: Another similar wave function: These 2 wave functions have a finite overlap!

83. (0,0)(π,0)(π,π)(π,π) (0,0) Fermi level 2 occupied states for the same k Consider 20 doped electrons in 144 sites:

84. or Outside (π,0) region (0,0)(π,0)(π,π)(π,π)(0,0) Fermi level Remove an electron from this (π,0) region (0,0)(π,0)(π,π)(π,π)(0,0) Fermi level (1) (2)

85. Consider 19 doped electrons in 144 sites: another similar wave function: QP state (1) (2)

86. Spectral Weight (π,0) (5π/6,π/6)(2π/3,π/3)(π/2,π/2) USDW (19e) 3.23E-012.99E-012.08E-068.61E-08 USDW (27e) 3.08E-013.00E-019.76E-068.31E-08 LSDW (19e) 5.63E-027.51E-021.41E-021.29E-02 LSDW (27e) 1.08E-011.12E-012.07E-021.90E-02 12 by 12, x=013 & 0.19 16 by 16, x=0.17 Gap exists only near (π,0)! Gap is K-dependent!

89. Conclusions: The e-doped t-t’-t’’-J model has been studied in the low energy states. At low doping, small FS pockets appear around anti-nodal points. Upon increased doping, AF will become weaker. Based on the AFMM states and coupling between k and k+ Q= (π,π), t he evolution of the FS with doping and the calculated spectral weight are in good qualitative agreement with the ARPES data. Questions remained:  The doping dependence of phase diagram?  Other excited states to be considered?  SC+AFM coexistent state ? Thank you for your attention!