Study of Collective Modes in Stripesby Means of RPA

E. Kaneshita, M. Ichioka, K. Machida

1. Introduction

3. Collective excitations in stripes

Stripes in High Tc cuprates

5. Summary

4. Phonon anomaly due to stripes

Random phase approximation

Results of RPA

2. Mean-field approach to stripes

Self-consistent calculation

Phonon anomalies in High Tc cuprates

High Tc cuprates

La2-xAxCuO4 (A=Ba, Sr, Ca) Tc ~ 40K

Bi2Sr2CaCu2O8 Tc ~ 80K

Bi2Sr2Ca2Cu3O10 Tc ~ 110K

YBa2Cu3O6+x Tc ~ 95K

Tl2Ba2Ca2Cu3O10 Tc ~ 120K

Hg2Ba2Ca2Cu3O8 Tc ~ 135K

Nd2-xCexCuO4 Tc ~ 25K

CuO2 plane

structure

La2-xSrxCuO4 YBa2Cu3O6+x

CuO2 plane

Before doping After doping

Anti Ferro

holehole doping

Hole doping

2-D square lattice(Cu site only)

CuO2 plane

Phase diagram

B. Keimer, et al,Phys. Rev. B 46, 14034 (1992)

Spin and charge ordered structure

La2-xSrxCuO4

Considering the underdoped region

Underdoped

AF

SC T

Hole concentration

Stripe

Spin ordering vector Q

Charge ordering vector ２ Q

filled ： up spinopen ： down spin

AF domain

hole

spin-charge ordering

Vertical stripe Diagonal stripe

Superconductor Insulator

In the neutron scattering experiments for LSCO,these stripe structures are observed .

Elastic neutron scattering experiment

Incommensurate peak

H. Yamada, et al.,Phys. Rev. B 59 (1998) 6165

S. Wakimoto, et al.,Phys. Rev. B 61 (2000) 3699

Diagonal stripe Vertical stripe

Incommensurability

M. Matsuda, et al.,Phys. Rev. B 62 (2000) 9148

diagonal stripe vertical stripe

Anomaly appears in phonon spectrum observed by neutron inelastic scattering.

R. J. McQueeney, et al.,Phys. Rev. Lett. 82, 628 (1999)

H. A. Mook, and F. Dogan,Nature 401, 145 (1999)

La1.85Sr0.15CuO4 YBa2Cu3O7-x

Phonon spectrum

Formulation

stripe order

self-consistent mean-field calculation

collective mode

random phase approximation

①

②

phonon spectrum

renormalize the collective stripe mode to the phonon spectrum

③

Hubbard model Assumingthe spin order

Mean field approximation

Hubbard model

Mean field approximation

t : nearest neighbor hoppingt’ : next nearest neighbor hoppingU: on-site coulomb

Fourier transformation

Periodicity

(k0 : within a reduced zone)

Assuming the spin order : (N-site periodicity)

diagonal stripe case vertical stripe case

path path

reduced zone

(N: periodicity)

diagonalization

self-consistent condition

diagonal stripe (insulator)

vertical stripe (insulator)

vertical stripe (metal)

charge density

spin density

charge density

spin density

spin density

charge density

Collective excitations in stripes

Stripe stateself-consistent mean-field calculation

RPA

single-particle Green function

HF susceptibilities

single-particle Green function

HF susceptibilities

= k2 k1

=

Dynamicalsusceptibilities

: charge excitation (phason mode)

: spin longitudinal excitation (phason mode)

: spin transverse excitation (spin wave mode)

We calculate these values by means of RPA:

RPA

i for spin flip

Spin wave excitation

direction (A): effective J is smaller

Comparing the spin velocities

direction A direction B

path anisotropy of spin wave

A

B

exchange coupling

path

period ：８ site

spin density excitation (phason)

sliding mode

meandering mode compression mode

Meandering mode has lower energy.

Anisotropy of phason mode

sliding mode of stripe(Just Q point)

sliding mode of stripe

Charge collective mode at 2Q

sliding mode of stripe

Charge excitation

phonon green function

k-k’ = l ×(l ： integer)

Umklapp process

2Q

Spectral function

phonon self energy

electron-phononinteraction

Of Frohlich type

Results 1

unperturbedphonon H. A. Mook, and F. Dogan,

Nature 401, 145 (1999)

log-plot

charge order

（　　　： energy of free phonon ）

Effect of stripe

sliding mode

charge order band foldingsliding mode gap

oscillation mode

below the gap （ in-phase)

above the gap （ out of phase ）

difference of oscillation mode above and below the gap

Results 2

R. J. McQueeney, et al.,Phys. Rev. Lett. 82, 628 (1999)

unperturbedphonon

dynamical susceptibility by RPA

collective modes in the stripe

Summary

Phonon anomaly

•Anisotropy of collective excitations•sliding mode of the stripe

Coupling with the sliding mode

referencesE. Kaneshita, et al., J. Phys. Soc. Jpn. 70 (2001) 866E. Kaneshita, et al., Phys. Rev. Lett. 88 (2002) 115501