The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Spontaneous Hexagon Organization in Pyrochlore Lattice Jung

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)


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


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


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


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


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


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


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


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


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


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


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


Manybody Lab, SKKU

Neutrons scatter off hexagons (S.H.Lee, nature 02)

Extrapolated correlation length remains finite


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


Manybody Lab, SKKU

Spins within a hexagon are collinearSpins of nearby hexagons are orthogonal

Director-Director Interaction


Manybody Lab, SKKU

Emergence of 3-states Potts Model

R

G

B


Manybody Lab, SKKU

Projecting onto a Kagome plane …


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


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


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


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

Documents

The 5th Korea-Japan-Taiwan Symposium on Strongly Correlated Electron System Manybody Lab, SKKU Spontaneous Hexagon Organization in Pyrochlore Lattice Jung