1. Spontaneous Hexagon Organization in Pyrochlore Lattice
韓 政勳, 價 成龍
(成均館大)

2. Examples of Frustrated Lattice - Triangular
Ground state of Heisenberg model
Mean-field theory predicts long-range spin ordering
with for nearest neighbors

3. Absence of LRO, despite local spin ordering
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. 3D Frustrated Lattice - Pyrochlore
Tetrahedron as a building block
Ground state condition
for each tetrahedron
Lee et al. Nature 02
Canals & Lacroix, PRB00

5. 3D Frustrated Lattice - Pyrochlore
indeterminate!! -> No local rigidity of spins
Continuous manifold of ground states
highly susceptible to perturbation!!

6. Theory of Spin-Lattice Coupling
Spin ordering mediated by superexchange
Assume superexchange integral depends only on relative distance of ions: i j ui uj
Rij

7. 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

8. 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.

9. Experiments on Pyrochlore – ZnCr2O4
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) ->
Neutrons scatter off a hexagon cluster of spins,
rather individual, fluctuating spins
Theory??

10. 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

11. Theory of TMS based on behavior of a single tetrahedron
Assume all tetrahedra behave the same way (q=0)
Different approach required to understand hexagons!

12. 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

13. “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

14. Neutrons scatter off hexagons (S.H.Lee, nature 02)
Extrapolated correlation length remains finite

15. 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

16. Director-Director Interaction
Spins within a hexagon are collinear Spins of nearby hexagons are orthogonal

17. Emergence of 3-states Potts Model in PyrochloreX Y Z

18. Setting up Hexagon Coordinates….
A-type hexagons located at
(B;3m,2n+1,p) (C;3m-1,n,2p+1) (D;3m-1,n,2p)

19. Emergence of 3-states Potts Model in PyrochloreX Y Z

20. 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

21. 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