1. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Spontaneous Hexagon Organization in Pyrochlore Lattice Jung Hoon Han & Chenglong Jia (Sung Kyun Kwan University)

2. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Examples of Frustrated Lattice - Triangular Ground state of Heisenberg model Mean-field theory predicts long-range spin ordering with for nearest neighbors

3. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU More Examples - Kagome Mean-field theory predicts but this is insufficient to define a unique ground state Macroscopic degeneracy of (N=number of triangles) Absence of LRO, despite local spin ordering

4. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU 3D Frustrated Lattice - Pyrochlore Tetrahedron as a building block Ground state condition for each tetrahedron Canals & Lacroix, PRB00 Lee et al. Nature 02

5. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU 3D Frustrated Lattice - Pyrochlore indeterminate!! -> No local rigidity of spins Continuous manifold of ground states highly susceptible to perturbation!!

6. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Experiments on Pyrochlore – ZnCr 2 O 4 Below Tc S.H.Lee et al. PRL, 2000 : Spins on Cr 3+ (S=3/2) order antiferromagnetically at as first-order transition, acccompanied by cubic-to-tetragonal distortion.

7. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Experiments on Pyrochlore – ZnCr 2 O 4 Above Tc S.H.Lee et al. Nature, 2002 : Neutron scattering of paramagnetic state at Structure factor consistent with hexagon spin cluster (spin-loop director)

8. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU 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. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Theory of Spin-Lattice Coupling Experimental fact: Short-range spin correlation persists at temperatures above transition (“Hexagon protectorate”) A “glue” is needed to protect correlation within a hexagon Perhaps it comes from spin-lattice coupling Exchange integral depends only on relative distance of ions: i j uiui ujuj R ij

10. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU 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

11. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Antiferomagnetic Spins on a Hexagon Ring Each hexagon can “Shrink” to minimize exchange energy Spins are collinear antiferromagnet (Holstein-Primakoff boson analysis of spin-lattice model) Spin Lattice

12. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU “Hidden geometry” of Pyrochlore ( S.H.Lee et al. Nature 2002 ) Pyrochlore can be decomposed in terms of hexagons, instead of tetrahedra Each site belongs to one and only one hexagon Four hexagon types of different orientations Non-overlapping hexagons form a lattice

13. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Neutrons scatter off hexagons (S.H.Lee, nature 02) Extrapolated correlation length remains finite

14. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Interpreting Experiments as spin-lattice coupling Invoking spin-lattice coupling, each independent hexagon shrinks, taking advantage of finite lattice stiffness and lowering exchange energy Nearby spin-loop directors are orthogonal, to be consistent with uniform contraction of hexagons

15. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Spins within a hexagon are collinear Spins of nearby hexagons are orthogonal Director-Director Interaction

16. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Emergence of 3-states Potts Model R G B

17. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Projecting onto a Kagome plane …

18. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU A picture of paramagnetic state in ZnCr 2 O 4 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 ZnCr 2 O 4

19. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Spin-Lattice Coupling in Other Frustrated Lattices Local rigidity of spin-spin correlation leads to absence of lattice distortion within our model

20. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU YMnO 3 There is direct evidence of coupling between spin and lattice Distortion of triangular lattice of Mn ions takes place Reason remains unclear yet

21. The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Outlook Spin-lattice coupling is the likely reason for the formation of a “super-structure” in frustrated lattices Different types of super-structures (hexagon vs. tetrahedron) may compete in a given lattice