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
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
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 (200
3)
Our approach
Metallic or SC region in between SDW and CDW proposed
in simplified pictures
Two parameters:α=g/ω: # of phonons / site,
λ=2g2/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
DMRG + pseudo-site method
Pseudo-site method for Einstein phononsE. Jeckelmann and S.R. White, PRB 57, 6376 (1998)
Phonon system
Electron system
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 eigenv
alues
Transfer operators and Hamiltonian using the original
U, g
A bare U (i.e., not the phonon-renormalized Ueff) added at in
termediate stages : does not give a good density matrix for the new basis modify U
A difficulty whenphonon-mediated attraction ≒ Hubbard we propose a new (compensation)
method
Improved ground state
-3.98
-3.97
-3.96
-3.95
-3.94
-3.93
-3.92
1001020100
compensationno compensation
number of sites in the left block
(U, g, ω)=(0, 3, 5)L=20, 4 pseudo-sites/site,m=200
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 distancedistance
Cor
rela
tion
func
tion
Exponents versus
On-site SC correlation does not dominateunlike the previous proposal
U
Exp
onen
t
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
SDW
Y. Nagaoka, Prog. Theor. Phys. 52, 1716 (1974).
The model coupled to phonons
Degraded electron-hole symmetry
On-site SC indeed dominates !
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-/a -/2a 0 /2a /a
k
E/t
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
Cor
rela
tion
func
tion
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.
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