KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

Valence bond solid orderthrough

spin-lattice coupling

Jung Hoon Han & Chenglong JiaSung Kyun Kwan U.

成均館大

Ref: Jia(Kim)2Han, PRB (2005); Jia & Han, cond-mat/0505573



Lattice-Coupled Antiferromagnetic Spin Model

Spin and lattice are coupled through magneto-strictioneffect (Pytte, PRB 1974)

20 1 ( )

2ij i j i j i

ij i

KH J J e u u S S u

1

iji i j

j i

Ju e S S

K

i

j

ui

ujRij

ˆ jie

Lattice displacement related tolocal spin-spin correlation by



Application to Frustrated Lattices – Triangular, Kagome

NO LATTICE DEFORMATION!



Triangular/Kagome lattices – meanfield theory

Ground state of Heisenberg model

i jH J S S

gives 120 degrees for neighboring spins for triangular/Kagome2

2i j

SS S

According to linearized spin-lattice theory local displacement is zero.

Classical degeneracy of Kagome is not lifted.

ˆ 0i ij i j ijj i j i

u e S S e



3D Frustrated Lattice - Pyrochlore

Tetrahedron as a building block

Ground state condition

for each tetrahedron

4

1

0ii

S

Macroscopic classical ground state degeneracy in pristine

Heisenberg model



Lifting of degeneracy through spin-lattice coupling

For pyrochlore lattice, there are infinitely many ways to deform the lattice through linearized spin-lattice coupling

Some clues to the deformation pattern must come from experiment



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



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



Experiments on Pyrochlore – ZnCr2O4

Above TcS.H.Lee et al. Nature, 2002

: Neutron scattering of paramagnetic state at T=15K Structure factor consistent with hexagon spin cluster (spin-loop director)

Conclude: Neutrons scatter off a hexagon cluster of spins, rather individual, fluctuating spins (very unusual paramagnetic state!)



Lifting of degeneracy through spin-lattice coupling

TetrahedronUnit (Tschernyshyov

et al. PRL 2002)

HexagonUnit

(Jia et al. PRB 2005)



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

Directors of nearby hexagons

interact via



Spins within a hexagon are collinearSpins of nearby hexagons are orthogonal

Director-Director Interaction



Hexagon Organization of Spins in Pyrochlore

Hexagons form a super-lattice with each color representing a director orientation

All nearby directors are orthogonal

(3-states Potts)



Energy Comparison

In comparing energies for both distortion modes,

2

2

( ) 1 2

1( ) 1

2

E tetrahedron

E hexagon

Tetrahedron-based distortion has a lower energy, buthexagon shrinkage is a viable, metastable state



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



Spin-Lattice Coupling in YMnO3

Mn3+ (S=2) spins form triangular network in MnO2

plane

Katsufuji et al. PRB (2001)



Experiments on triangular S=2 AFM – YMnO3

Expansion of Mn triangles (red) at the onset of magnetic orderBut simple spin-lattice theory won’t work….

S Lee et al. PRB (2005)



New order parameter ?

Possible order parameters of spin:

(1) spin density wave – magnetism

(2) valence-bond – VBS or VBL(RVB)

1( )2VB i j i ji j



1D Chain (S=1)

2D square (S=2) 2D hexagonal (S=3/2)

Examples of VBS (AKLT states)



IDEA: Spin-Lattice Coupling-induced VBS ?

Magnetic order doesn’t work, but valence-bond order might.

PARTIAL VBS ORDER

At most 4 VB’s can formfor coordination = 6

Without SLC, VB’s will resonante, but with SLC, VBS patt

erns can be formed.

1( )2VB i j i ji j



Calculations

Take spin-lattice Hamiltonian and use Schwinger boson mean-field theory

(Jia & Han, cond-mat/0505573)

20 1 ( )

2ij i j i j i

ij i

KH J J e u u S S u

ij i j i jQ a b ba

Bond order

0i ia b Bose condensation

Spin singlet

LRO



Results

Energy corresponding to a particular VBS pattern below is lower than

that of uniform state for sufficiently large spin-lattice coupling strength

(Not spin Jahn-Teller type)

2 23 3( 3 0.1161)

2 2uniformE Q S

2 2 2 2 2 22 1 2 1

1

2

1 3( )

2 8

3 0.2128

3 0.1823

non uniformE Q Q Q Q

Q S

Q S

0.4946c ( 2)S



Results

'

| | |

| ( ) | 0i j i jij

GS LRO PVBS

PVBS a b b a

This state is also Bose-condensed, so LRO co-exists with VBS

Elementary excitations:

(1) Spin waves for |LRO>

(2) Triplet wave for |PVBS>



A New Excitation

We calculate excitation spectra for the VBS within SMA

VBS Spin wave

Comparison

3

1( )

,

1 (1/ 3) cos[ ][ [ , ]( )

2 [ i j

k kik r r

k k i ji j

k eS H Sk

S S S S e



Interpretation

(1) A new spectrum is gapped, has a minimum at (4pi/3a,0), same as spin wave minimum

(2) Dispersion is extremely flat! Reminiscent of the flat band in kagome AFM.



Relation to Experiments? – YMnO3

Tc=70K

Quasi-elastic peak at high T, into Neel region (Takagi: fluctuating XY spins but 40K seems too far from Tc)

Any sign of flat band? (must extract spin waves first)

Park et al. PRB (2003)



Summary

A naïve inclusion of magneto-striction leads to a variety of interesting phenomena

(1) Classical spin degeneracy of pyrochlore lattice can be lifted through magneto-striction in a number of interesting ways; tetrahedron-based or hexagon-based

(2) Formation of valence-bond-solid is facilitated by the spin-lattice coupling and may lead to an interesting excitation spectrum

Documents

KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun