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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)
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
Examples of Frustrated Lattice - Triangular
Ground state of Heisenberg modeli jH J S S
Mean-field theory predicts long-range spin ordering
with for nearest neighbors
2
2i j
SS S
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
More Examples - Kagome
Mean-field theory predictsbut this is insufficient to define a unique ground state Macroscopic degeneracy of (N=number of triangles)
2N
Absence of LRO, despite local spin ordering
2
2i j
SS S
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
4
1
0ii
S
Cana
ls &
Lac
roix
, PRB
00
Lee e
t al.
N
atu
re 0
2
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
3D Frustrated Lattice - Pyrochlore
i jS S indeterminate!! -> No local rigidity of spins
Continuous manifold of ground states
highly susceptible to perturbation!!
4
1
0ii
S
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
Experiments on Pyrochlore – ZnCr2O4
Below TcS.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.
cT 12.5K
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
Experiments on Pyrochlore – ZnCr2O4
Above TcS.H.Lee et al. Nature, 2002
: Neutron scattering of paramagnetic state at Structure factor consistent with hexagon spin cluster (spin-loop director)
T =15K
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, 2002TMS, PRB, 2002
Elongation (contraction) of a tetrahedron along an axis
And collinear antiferromagnetic spins is the ground state
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:
0 1
( )
ˆ- ( - )
ij ij
ji j i
J J R
J J e u u
i
j
ui
ujRij
ˆ jie
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
20 1 ( )
2ij i j i j i
ij i
KH J J e u u S S u
iji i jj i
u e S S
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
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 ofdifferent orientations
Non-overlapping hexagons form a lattice
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
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
iji i jj i
u e S S
Nearby spin-loop directors areorthogonal, to be consistent withuniform contraction of hexagons
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
Spins within a hexagon are collinearSpins of nearby hexagons are orthogonal
Director-Director Interaction
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
Emergence of 3-states Potts Model
R
G
B
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
Projecting onto a Kagome plane …
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
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 ofspin-loop directors, but it appears that a tetragonal distortionpre-empts this possibility in ZnCr2O4
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
0iu
Spin-Lattice Coupling in Other Frustrated Lattices
Local rigidity of spin-spin correlation leads to
absence of lattice distortion within our model
2
2i j
SS S
The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System
Manybody Lab, SKKU
YMnO3
There is direct evidence of coupling between spin and lattice
Distortion of triangular lattice of Mn ions takes place
Reason remains unclear yet
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
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