1. Simulating High Tc Cuprates T. K. Lee Institute of Physics, Academia Sinica, Taipei, Taiwan December 19, 2006, HK Forum, UHK

2. Collaborators 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 Acknowledgement Kitaoka and Mukuda, Osaka Univ.

3. Introduction To start simulation, need Hamiltonians! minimal models: 2D Hubbard model – 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) dynamic mean field theory and density matrix renormalization group,.. 2D t-J type models –

4. 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 J is for n.n. AF spin-spin interaction

5. Introduction To start simulation, need Hamiltonians! minimal models: 2D Hubbard model – 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). dynamic mean field theory and density matrix renormalization group,.. 2D t-J type models – no finite temp. QMC – sign problem and strong constraint ED for 32 sites with 4 holes (Leung, PRB 2006) Variational Approach!

6. Trial wave function:  is configuration index a  is Slater determinant To calc. a quantity Variational approach : 1.Construct the trial wave function 2.Treat the constraint exactly by using Monte Carlo method

7. Damscelli, Hussain, and Shen, Rev. Mod. Phys. 2003 Phase diagram hole electron Expt. Info. electron – hole asymmetry!

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

9. Important 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: electron and hole symmetry Afmm, afm+sc and sc all have different broken symmetries

10. t’ and t’’ break the symmetry between doping electrons and holes! Without the long range hopping terms, t’ and t’’, the model Hamiltonian, t-J model, has the particle- hole symmetry: From the theory side: From hole-doped to electron-doped, just change t’/t → – t’/t and t”/t → – t”/t Different Hamiltonians – different phase diagram.

11. 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?

12. 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

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

14. 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 Chen, et al., PRB42, 1990; Giamarchi and Lhuillier, PRB43, 1991; Lee and Shih, PRB55, 1997; Himeda and Ogata, PRB60, 1999

15. 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

16. The wave function for adding holes to 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”.

17. 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)

18. 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)

19. 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’)

20. 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

21. 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

22. 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). The rotation symmetry of the wave function alternates between s and d symmetry for 2h,4h, 6h,…!

23. The new wave function has AFM but negligible pairing. An AFM metallic (AFMM) phase but no SC! d-wave pairing correlation function 2 holes in 144 sites

24. 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

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

26. Difference between AFMM and AFMM+SC The doping dependence of AFMM is quite different from AFMM+SC.

27. AFMM shows more hole-hole repulsive correlation than AFMM+SC. The pair-pair correlation of AFMM is much lower than AFMM+SC. 12X12 16X16

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

29. Variational Energy Phase diagram

30. 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

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

32. 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

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

34. spectral weight: STM Conductance is related to the spectral weight

35. 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

36. 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>

37. Controversies about pairing Only t-J (without t’) is sufficient to explain high Tc? No! --- Shih, Chen, Lin and Lee, PRL 81, 1294 (1997) 64 sites, J=0.4, PL0=VMC, PL1-1st order Lanczos, PL2-2nd order P. W. Leung, PRB 2006

38. Summary Variational approach has provided a number of semi-quantitative successes in understanding high Tc cuprates : ground state phase diagram and excitaiytons, spectral weight, STM conductance asymmetry, etc.. Detailed questions about pseudogap, effects of disorder and electron-phonon interactions, etc.. are still to be resolved. Finite temp? Thank you for your attention!b