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