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

3. Treat the HH model on a long chain with DMRG to determine phases by calculating correlation functions. What to expect ? Y. Takada, JPSJ 65, 1544 (1996) Y. Takada and Chatterjee, PRB 67, 081102 (2003) Our approach Metallic or SC region in between SDW and CDW proposed in simplified pictures Two parameters: α=g/ω: # of phonons / site, λ=2g 2 /ω: measure of the phonon-mediated attraction ↓ Phase diagram vs α and λ ? Charge Spin 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. 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) When we add the first few pseudo-sites, Diagonalize ρ and choose eigenstates that have large eigenvalues Transfer operators and Hamiltonian using the original U, g A bare U (i.e., not the phonon- renormalized U eff ) added at intermediate stages : does not give a good density matrix for the new basis  modify U A difficulty when phonon-mediated attraction ≒ Hubbard  we propose a new (compensation) method

6. Improved ground state -3.98 -3.97 -3.96 -3.95 -3.94 -3.93 -3.92 100 20100 compensation no compensation number of sites in the left block (U, g, ω)=(0, 3, 5) L=20, 4 pseudo-sites/site, m=200

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)] distance Correlation function

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

9. Correlation functions when an electron-hole symmetry exists 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? CDW on-site pair SDW Y. Nagaoka, Prog. Theor. Phys. 52, 1716 (1974).

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

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 symmetry on-site pair correlation can dominate.

12. Future problems Analysis of the (s-wave) SC observed in A 3 C 60 (A=K, Rb). Further evaluation of the compensation method Other applications, e.g. molecules and chains with many branches