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

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

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

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

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

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

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

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.

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??

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

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!

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

“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

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

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

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

Emergence of 3-states Potts Model in Pyrochlore X Y Z

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)

Emergence of 3-states Potts Model in Pyrochlore X Y Z

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

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