Valence bond solid order through spin-lattice coupling

1. Valence bond solid orderthroughspin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun Kwan U. 成均館大 Ref: Jia(Kim)2Han, PRB (2005); Jia & Han, cond-mat/0505573

2. uj Rij j ui i Lattice-Coupled Antiferromagnetic Spin Model Spin and lattice are coupled through magneto-striction effect (Pytte, PRB 1974) Lattice displacement related to local spin-spin correlation by

3. Application to Frustrated Lattices – Triangular, Kagome NO LATTICE DEFORMATION!

4. Triangular/Kagome lattices – meanfield theory Ground state of Heisenberg model gives 120 degrees for neighboring spins for triangular/Kagome According to linearized spin-lattice theory local displacement is zero. Classical degeneracy of Kagome is not lifted.

5. 3D Frustrated Lattice - Pyrochlore Tetrahedron as a building block Ground state condition for each tetrahedron Macroscopic classical ground state degeneracy in pristine Heisenberg model

6. Lifting of degeneracy through spin-lattice coupling For pyrochlore lattice, there are infinitely many ways to deform the lattice through linearized spin-lattice coupling Some clues to the deformation pattern must come from experiment

7. Experiments on Pyrochlore – ZnCr2O4 Below Tc S.H.Lee et al. PRL, 2000 : Spins on Cr3+(S=3/2) order antiferromagnetically at as first-order transition, acccompanied by cubic-to-tetragonal distortion.

8. Theory of spin-Peierls transition by Tschernyshyov, Moessner, and Sondhi TMS, PRL, 2002 TMS, PRB, 2002 Elongation (contraction) of a tetrahedron along an axis And collinear antiferromagnetic spins is the ground state

9. Experiments on Pyrochlore – ZnCr2O4 Above Tc S.H.Lee et al. Nature, 2002 : Neutron scattering of paramagnetic state at T=15K Structure factor consistent with hexagon spin cluster (spin-loop director) Conclude: Neutrons scatter off a hexagon cluster of spins, rather individual, fluctuating spins (very unusual paramagnetic state!)

10. Lifting of degeneracy through spin-lattice coupling Tetrahedron Unit (Tschernyshyov et al. PRL 2002) Hexagon Unit (Jia et al. PRB 2005)

11. Interpreting Experiments as Spin-Lattice Coupling Invoking spin-lattice coupling, each independent hexagon shrinks, taking advantage of finite lattice stiffness and lowering exchange energy Directors of nearby hexagons interact via

12. Director-Director Interaction Spins within a hexagon are collinearSpins of nearby hexagons are orthogonal

13. Hexagon Organization of Spins in Pyrochlore Hexagons form a super-lattice with each color representing a director orientation All nearby directors are orthogonal (3-states Potts)

14. Energy Comparison In comparing energies for both distortion modes, Tetrahedron-based distortion has a lower energy, but hexagon shrinkage is a viable, metastable state

15. A picture of paramagnetic state in ZnCr2O4 Spin-lattice interaction leads to enhanced singlet (collinear antiferromagnet) tendency within a hexagon Residual spin-lattice interaction leads to orthogonality of nearby directors (3-states Potts model) At finite temperature, thermal fluctuations smear out the inter-hexagon interaction, spin-spin correlation remains confined to a single hexagon Further lowering temperature might lead to condensation of spin-loop directors, but it appears that a tetragonal distortion pre-empts this possibility in ZnCr2O4

16. Spin-Lattice Coupling in YMnO3 Katsufuji et al. PRB (2001) Mn3+ (S=2) spins form triangular network in MnO2 plane

17. Experiments ontriangular S=2 AFM – YMnO3 S Lee et al. PRB (2005) Expansion of Mn triangles (red) at the onset of magnetic order But simple spin-lattice theory won’t work….

18. New order parameter ? • Possible order parameters of spin: • spin density wave – magnetism • valence-bond – VBS or VBL(RVB)

19. Examples of VBS (AKLT states) 1D Chain (S=1) 2D square (S=2) 2D hexagonal (S=3/2)

20. IDEA: Spin-Lattice Coupling-induced VBS ? Magnetic order doesn’t work, but valence-bond order might. PARTIAL VBS ORDER At most 4 VB’s can form for coordination = 6 Without SLC, VB’s will resonante, but with SLC, VBS patterns can be formed.

21. Calculations Take spin-lattice Hamiltonian and use Schwinger boson mean-field theory (Jia & Han, cond-mat/0505573) Bose condensation Bond order LRO Spin singlet

22. Results Energy corresponding to a particular VBS pattern below is lower than that of uniform state for sufficiently large spin-lattice coupling strength (Not spin Jahn-Teller type)

23. Results This state is also Bose-condensed, so LRO co-exists with VBS • Elementary excitations: • Spin waves for |LRO> • Triplet wave for |PVBS>

24. A New Excitation We calculate excitation spectra for the VBS within SMA VBS Spin wave Comparison

25. Interpretation (1) A new spectrum is gapped, has a minimum at (4pi/3a,0), same as spin wave minimum (2) Dispersion is extremely flat! Reminiscent of the flat band in kagome AFM.

26. Relation to Experiments? – YMnO3 • Tc=70K • Quasi-elastic peak at high T, into Neel region • (Takagi: fluctuating XY spins but 40K seems too far from Tc) • Any sign of flat band? (must extract spin waves first) Park et al. PRB (2003)

27. Summary • A naïve inclusion of magneto-striction leads to a variety of interesting phenomena • Classical spin degeneracy of pyrochlore lattice can be lifted through magneto-striction in a number of interesting ways; tetrahedron-based or hexagon-based • Formation of valence-bond-solid is facilitated by the spin-lattice coupling and may lead to an interesting excitation spectrum