Transcript A DMRG study of correlation functions in the Holstein

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
Holstein-Hubbard model
Superconductivity
Electronphonon
coupling
Electronelectron
interaction
Electron-electron repulsion
Electron-phonon coupling
What
happens
when
they coexist?
phonons
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, JPSJ 65, 1544 (1996)
Y. Takada and Chatterjee, PRB 67, 081102 (2003)
Metallic or SC region in between
SDW and CDW proposed
in simplified pictures
Charge
Our approach
Treat the HH model on a long chain
with DMRG to determine phases by
calculating correlation functions.
Spin
on-site SC
n.n. singlet SC
n.n. triplet SC
DMRG + pseudo-site method
Pseudo-site method for Einstein phonons
E. Jeckelmann and S.R. White, PRB 57, 6376 (1998)
Phonon system
Electron system
A difficulty when
phonon-mediated attraction ≒ Hubbard
 we propose a new (compensation) method
When we add the first few pseudo-sites,
A bare U (i.e., not the phononrenormalized Ueff) added at
intermediate stages : does not
give a good density matrix for
the new basis
 modify U
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
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
number of sites in the left block
10
Correlation function
Result for correlation functions
distance
distance
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
Exponent
Exponents versus
U
On-site SC correlation does not dominate
unlike the previous proposal
Correlation functions when an
electron-hole symmetry exists
CDW
on-site pair
SDW
SDW
Y. Nagaoka, Prog. Theor. Phys. 52, 1716 (1974).
• 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?
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
Correlation function
Degraded electron-hole symmetry
0.8
0.6
0.4
0.2
0
E/t -0.2
-0.4
-0.6
-1.023±0.004
-0.8
-1
-1.2
-/a
-/2a
0
/2a
/a
k
distance
On-site SC indeed dominates !
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