1. Magnetic Interactions and Order-out-of-disorder in Insulating Oxides Ora Entin-Wohlman, A. Brooks Harris, Taner Yildirim Robert J. Birgeneau, Marc A. Kastner, Koichi Katsumata R. Ramirez, C. Broholm, J. W. Lynn TAU, BGU, U Penn, NIST, MIT, RIKEN, Lucent, JHU Les Houches summer school on Quantum Magnetism, June 2006 Amnon Aharony

2. 2 Lecture 2: Generalized superexchange: add spin-orbit and involve all 10 d orbitals Tetragonal lattice: bond dependent anisotropies, Remove frustration by spin wave zero-point energy; Order out of disorder Orthorhombic lattice: Dzyaloshinskii-Moriya, dependence on Cu — O — Cu bond angle Interplanar coupling: Shender mechanism, Pseudo-dipolar anisotropies

3. 3 What determines the easy axes for the spins (in plane and between planes)? LCO, 214 YBCO, 123

4. 4 Simple theory: Super-exchange 1 2 Heisenberg Hamiltonian: ordered moment: Hubbard Hamiltonian: (the manifold, each site has only a single electron) Perturb in t, keep low energy states

5. 5 Superexchange: No phase transition in the 2D isotropic Heisenberg model?? Order arises due to small anisotropies plus weak interplane coupling Plus quantum fluctuations!

6. 6 Theory On Cu Cu---O---Cu On Cu

7. 7 cuprates: full Hamiltonian spin-orbit interaction; full Coulomb interaction; hopping among all 5 d-orbitals; further hopping processes;

8. 8 Tetragonal symmetry: x y z 1 2 3 Spin-orbit anisotropy

9. 9 Orthorhombic symmetry: Also Dzyaloshinskii-Moriya antisymmetric exchange Cu O Oxygen tilted along z D along y, AFM along x FM along z

10. 10 Moriya's anisotropic superexchange interaction, frustration, and Dzyaloshinsky's weak ferromagnetism, L. Shekhtman, O. Entin-Wohlman and AA, Phys. Rev. Lett. 69, 000836 (1992)

11. 11 For a 90 degrees Cu — O — Cu bond, electrons must hop from each Cu ion to two orthogonal orbitals on the oxygen, and then Hund ’ s rule turns the superexchange ferromagnetic. Superexchange then turns antiferromagnetic as the bond angle increases towards 180 degrees.

12. 12 NNN NN

13. 13 x y z 1 2 3 f Frustration! Tetragonal symmetry: Mean field theory:

14. 14 x z 1 2 3 f Order out of disorder!

15. 15 The “Shender” mechanism: 1 2 3 12 3 2342 2122

16. 16 xy I1 I2 II xy I1 I2 II Pseudodipolar interactions

17. 17 Interplane interactions 12 Also explain: need Nd-Cu and Pr-Cu interactions

18. 18 Order determined by delicate competition With quartic spin anisotropies on the Pr or Nd

19. 19 2342 Fits to magnetic moment versus Direction of magnetic field yield Many of the interesting parameters

20. 20 Spin wave theory for 2342 Shender 4-fold

21. 21 ESR: in plane anisotropy gap in 2122 and 2342, In and out of plane gaps for CuII at low T

22. 22 Spin chains, ladders: same interactions as in 2342?

23. 23 Conclusions: Cuprates: taylored 2D and 1D S=1/2 AFMs Hubbard + spin-orbit effective magnetic anisotropies Pseudodipolar, Dzyaloshinskii-Moriya Fluctuations: Shender, in-plane 4-fold Explain structures of many cuprates 2342: measure many anisotropies, useful for chains and ladders Relevance to ladders, chains, nickelates, cobaltates, titanates…

24. 24 THE END (More tomorrow)