1. Stability of RVB state with respect to charge modulations Rastko Sknepnek Iowa State University and DOE Ames Lab In collaboration with: Jun Liu and Joerg Schmalian

2. Is RVB state stable against charge order? Very strong pairing interaction  suggests local pairing. Local pairing may be strongly affected by: charge order charge order disorder (impurities) disorder (impurities) We assume a resonating valence bond (RVB) ground state. Correlation length

3. Mean-field theory predicts very strong spatial variations in the gap!  1+p    1-p   p  = 0.2 How important are electronic correlations? Perform a toy Bogoliubov-de Gennes (mean-filed) calulation.

4. Model – t-J Hamiltonian Study the t-J Hamiltonian using variational Monte Carlo method. Minimize E {  } as function variational parameters {  }. We use J = 0.3t

5. RVB wave function Variational parameters: t        1+p    1-p   Minimize with respect to: t,        but fixed p  ! (P.W. Anderson, Science,1987)

6. Superconducting order parameter  measures local d-wave pairing. (Paramekanti, et al. PRB 70, 054504 (2004)) Superconductivity is characterized by the existence of Off Diagonal Long Range Order in the reduced density matrix. (Penrose&Onsager, Phys. Rev. 104, 576 (1956); Yang, Rev. Mod. Phys. 34, 694 (1962) Nieh, et al. PRB 51, 3760 (1995)) Nieh, et al. PRB 51, 3760 (1995)) Reduced density matrix:   is largest eigenvalue of   and is O(N) and  corresponding eigenfunction.

7. Superconducting order parameter: Note: In a mean-field theory gap and SC order parameter are the same!

8. Minimization is performed on 8x8 system with 60 electrons – = 0.9375. We have minimized energy with respect to the 5 variational parameters. Difficulty: local minima with very small energy differences. local minima with very small energy differences. Requires very high precision energy evaluation.

9. System exhibits long range superconducting order.

10. Superconducting order parameter is insensitive to charge modulation! For:

11. Summary We have use variational Monte Carlo to study stability of the RVB We have use variational Monte Carlo to study stability of the RVB state to charge modulation. state to charge modulation. This approach allows us to independently probe paring amplitude and This approach allows us to independently probe paring amplitude and the superconducting order parameter. We find that the superconducting order parameter is insensitive even relatively We find that the superconducting order parameter is insensitive even relatively large charge modulation.