Correlation functions in the Holstein-Hubbard model calculated with an improved algorithm for DMRG

1. Correlation functions in the Holstein-Hubbard model calculated with an improved algorithm for DMRG Masaki Tezuka, Ryotaro Arita and Hideo Aoki Dept. of Physics, Univ. of Tokyo

2. Motivation and model Holstein-Hubbard model Superconductivity Electron-phonon coupling Electron-electron interaction Electron-electron repulsion Electron-phonon coupling What happens when they coexist? phonons

3. Y. Takada, JPSJ 65, 1544 (1996) What to expect ? Two parameters: α=g/ω: # of phonons / site, λ=2g2/ω: measure of the phonon-mediated attraction ↓ Phase diagram vs α and λ ? Y. Takada and Chatterjee, PRB 67, 081102 (2003) Metallic or SC region in between SDW and CDW proposed in simplified pictures Charge Our approach Spin Treat the HH model on a long chain with DMRG to determine phases by calculating correlation functions. on-site SC n.n. singlet SC n.n. triplet SC

4. DMRG + pseudo-site method Pseudo-site method for Einstein phonons E. Jeckelmann and S.R. White, PRB 57, 6376 (1998) Phonon system Electron system

5. A difficulty whenphonon-mediated attraction ≒ Hubbard  we propose a new (compensation) method When we add the first few pseudo-sites, A bare U (i.e., not the phonon-renormalized Ueff) added at intermediate stages : does not give a good density matrix for the new basis  modifyU Add a new term to the Hamiltonian, which effectively changes the values of U and/or g so that the # of electrons = band filling (unity here) Diagonalize ρ and choose eigenstates that have large eigenvalues Transfer operators and Hamiltonian using the original U, g

6. Improved ground state -3.92 compensation no compensation -3.93 -3.94 -3.95 -3.96 (U, g, ω)=(0, 3, 5) L=20, 4 pseudo-sites/site, m=200 -3.97 -3.98 0 10 20 10 0 10 number of sites in the left block

7. Result for correlation functions t=1, (g, ω)=(3, 5), 40-site chain, 4 phonon pseudo-sites/site, m=600 • U≪λ: (CDW～on-site SC) • U～λ: all power-law • U≫λ: SDW • Surprising for an electron-phonon coupled system • Consistent with the calculated charge- and spin- gaps [H. Fehske, G. Wellein, G. Hager, A. Weiße and A. R. Bishop, PRB 69 , 165115 (2004)] Correlation function distance distance distance

8. Exponent U Exponents versus On-site SC correlation does not dominate unlike the previous proposal

9. Correlation functions when an electron-hole symmetry exists SDW CDW • For electron-hole symmetric models, CDW and on-site pair have the same exponent. • The exponents are still about the same for the HH model with finite ω, where the electron-phonon interaction is not exactly e-h symmetric. • What happens if we destroy the electron-hole symmetry of the electron system? SDW on-site pair Y. Nagaoka, Prog. Theor. Phys. 52, 1716 (1974).

10. The model coupled to phonons t=1, t’=0.2, (U, g, ω)=(1, 4, 10), 40-site chain, 4 phonon pseudo-sites/site, m=600 -1.118±0.009 Degraded electron-hole symmetry -1.023±0.004 Correlation function distance On-site SC indeed dominates !

11. Conclusion • Correlation functions calculated for the first time for the 1D Holstein-Hubbard model with DMRG + pseudo-site method. • A new algorithm to deal with the difficulty that arises when the phonon-mediated attraction ≒ Hubbard U. • For the electron-hole symmetric chain, superconducting phases do not dominate even around λ=U for the case of half-filling. • In a system( model here) with broken electron-hole symmetryon-site pair correlation can dominate.

12. Future problems • Analysis of the (s-wave) SC observed in A3C60 (A=K, Rb). • Further evaluation of the compensation method • Other applications, e.g. molecules and chains with many branches