Spin Waves in Stripe Ordered Systems E. W. Carlson D. X. Yao D. K. Campbell

Strong Correlations  nickelates  manganites  cuprate superconductors  organic superconductors All show some evidence of real space order

Strong Correlation Fermi Liquid Interaction energy is important Real space structure –spin –charge Kinetic energy is minimized k-space structure Real space homogeneity

Organic Superconductors  -(ET) 2 X (TMTST) 2 PF 6 From E. Dagotto, cond-mat/ From S. Lefebvre et al., Physica B 312 : (2002)

Organic superconductors CDW, SDW Bond Order  BCSDW (Campbell) (Mazumdar, Clay, and Campbell, Synth. Met. 137, 1317 (2003) D. Chow et al., Phys Rev Lett (2000)

Cuprates and Nickelates Layered structure  quasi-2D system Cu-O or Ni-O Planes Other Layers Layered structure  quasi-2D system

Cuprates and Nickelates Dope with holes (remove spins) Topological Doping Ni: S=1 Cu: S=1/2 OxygenCu or Ni

Cuprates Dope with holes Superconducts at certain dopings OxygenCu or Ni T x AF SC

Neutron Scattering in Cuprates and Nickelates Disappearance of (π,π) peak with doping Appearance of satellite peaks AFM signal averages to zero antiphase domain walls π π δ=0 π π

Issues: nature – static vs. dynamic orientation – vertical vs. diagonal spacing – commensurate vs. incommensurate width – one atom vs. two... location of holes – site-centered vs. bond-centered

Cuprates from Almason and Maple (1991) stripes: interleaved charge and spin density (Kivelson, Emery) (Zaanen) (Castro Neto, Morais-Smith) bond-ordered charge density (Sachdev) 2D magnetic/current textures: DDW (Marsten, Chakravarty, Morr); Staggered flux (Lee); Loops (Varma)

Scattering Probes Energy, Momentum Phase Information?  Yes in certain cases Goals:  Phase-sensitive information from diffraction probe  Guidance for microscopic theories of superconductivity in cuprates, organics

Site or Bond-Centered J a > 0 (AFM) J b > 0 (AFM) JbJb JaJa JbJb J a > 0 (AFM) J b < 0 (FM) JaJa Site-centered p=3Bond-centered p=3 π π Both produce weight at (π+ π/p, π)

Model and Method JbJb Bond-centered, p=3 J a > 0 (AFM) J b < 0 (FM) JaJa Heisenberg model

Elastic Response

Magnetic Reciprocal Lattice Vectors Site-centered p=3 Bond-centered p=3 π π Spacing p=3

Magnetic Reciprocal Lattice Vectors Bond-centered p=4 π π Spacing p=4 Site-centered p=4

Elastic Neutron Scattering g(m) f(n)

Elastic Neutron Scattering p=3 Site-centered π π g(m) f(n)

Elastic Neutron Scattering p=3 Site-centered g(m) f(n) π π

Elastic Neutron Scattering p=3 Bond-centered g(m) f(n) π π

Elastic Neutron Scattering p=3 Bond-centered g(m) f(n) π π

Site vs. Bond-Centered p=3 Bond-centered p=3 g(m) f(n) π π Site-centered p=3 g(m) f(n) π π

Site vs. Bond-Centered p=4 Bond-centered p=4 g(m) f(n) Site-centered p=4 g(m) f(n) π π π π

Elastic Peaks 2D Antiphase Domain Walls Site-centered: never weight at Bond-centered: no weight at for p=EVEN generic weight at for p=ODD The presence of weight at with incommensurate peaks at is positive evidence of a bond- centered configuration

Elastic Peaks 3D Antiphase Domain Walls Site-centeredBond-centered p=EVEN Vertical/Vertical -- Diagonal/Vertical -- Diagonal/Diagonal -- Bond-centered p=ODD (0, ,  ) (0, ,0) (0,0,0)

Inelastic Response: Spin Waves

Model and Method JbJb Bond-centered, p=3 J a > 0 (AFM) J b < 0 (FM) JaJa Heisenberg model

Model and Method Heisenberg model Up Spins:Down Spins: Holstein-Primakoff Bosons

Model and Method Heisenberg model Fourier transformation + symplectic transformation yield spectrum and eigenstates

Spin Structure Factor

Number of Bands Bond-centered p=4 Site-centered p=4 p-1 spins per unit cell Spin up/Spin down degeneracy ) (p-1)/2 bands 3 bands for p=4 p spins per unit cell Spin up/Spin down degeneracy ) p/2 bands 4 bands for p=4

Site-Centered: S(k, w) J b =0.4 J a J b =1.0 J a J b =2.5 J a kxkx p=3 p=4 π π N.B. Site-centered consistent with F.Kruger and S. Scheidl, PRB 67, (2003)

J b = J a J b =-0.56 J a J b =-1.0 J a Note the elastic weight for p=3 π π Bond-Centered: S(k, w ) p=3 p=4 p=2 kxkx

S3 k=(0,0) k=(π, π) S4 Energy dependence on λ=

k=(0,0) k=(π, π)B2 B3 B4 Energy dependence on λ=

Site-centered velocities v velocity along the stripe direction v velocity perpendicular to the stripe direction v velocity of pure 2D antiferromagnet || AF

Bond-centered velocities v velocity along the stripe direction v velocity perpendicular to the stripe direction v velocity of pure 2D antiferromagnet || AF

Conclusions Elastic: For both 2D and 3D antiphase domain walls, bond-centered p=ODD stripes show new peaks, forbidden for site-centered Inelastic: –Number of bands distinguishes site- or bond-centered Site: (p-1) bandsBond: (p) bands –Qualitatively different spin wave spectra Site: all bands increase with J_b Bond: lower bands independent of J_b top band ~ 2 J_b –Velocity anisotropy Bond-centered is rather isotropic over a large range of parameters Extensions: –Diagonal spin waves –Other spin textures