Study of Collective Modes in Stripes by 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 T c cuprates La 2-x A x CuO 4 (A=Ba, Sr, Ca) T c ~ 40K Bi 2 Sr 2 CaCu 2 O 8 T c ~ 80K Bi 2 Sr 2 Ca 2 Cu 3 O 10 T c ~ 110K YBa 2 Cu 3 O 6+x T c ~ 95K Tl 2 Ba 2 Ca 2 Cu 3 O 10 T c ~ 120K Hg 2 Ba 2 Ca 2 Cu 3 O 8 T c ~ 135K Nd 2-x Ce x CuO 4 T c ~ 25K CuO 2 plane

structure La 2-x Sr x CuO 4 YBa 2 Cu 3 O 6+x CuO 2 plane

Before dopingAfter doping Anti Ferro hole hole doping Hole doping 2-D square lattice (Cu site only) CuO 2 plane

Phase diagram B. Keimer, et al, Phys. Rev. B 46, (1992) Spin and charge ordered structure La 2-x Sr x CuO 4 Considering the underdoped region Underdoped AF SC T Hole concentration

Stripe Spin ordering vector Q Charge ordering vector ２Q２Q filled ： up spin open ： down spin AF domain hole spin-charge ordering

Vertical stripe Diagonal stripe SuperconductorInsulator 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 stripeVertical stripe

Incommensurability M. Matsuda, et al., Phys. Rev. B 62 (2000) 9148 diagonal stripevertical 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) La 1.85 Sr 0.15 CuO 4 YBa 2 Cu 3 O 7-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 Assuming the spin order

Mean field approximation Hubbard model Mean field approximation t : nearest neighbor hopping t’ : next nearest neighbor hopping U: on-site coulomb

Fourier transformation Periodicity (k 0 : within a reduced zone) Assuming the spin order : (N-site periodicity)

diagonal stripe case vertical stripe case 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 charge density

Collective excitations in stripes Stripe state self-consistent mean-field calculation RPA single-particle Green function HF susceptibilities

single-particle Green function HF susceptibilities = k2k2 k1k1 =

Dynamical susceptibilities : 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 Adirection B pathanisotropy 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 2Q2Q Spectral function phonon self energy electron-phonon interaction Of Frohlich type

Results 1 unperturbed phonon 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 folding sliding modegap

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) unperturbed phonon

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