KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through...

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

Application to Frustrated Lattices – Triangular, Kagome

NO LATTICE DEFORMATION!

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group 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

KIAS Emergence Workshop 2005

Manybody Physics Group 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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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!)

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

Lifting of degeneracy through spin-lattice coupling

TetrahedronUnit (Tschernyshyov

et al. PRL 2002)

HexagonUnit

(Jia et al. PRB 2005)

KIAS Emergence Workshop 2005

Manybody Physics Group 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

Directors of nearby hexagons

interact via

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

Spins within a hexagon are collinearSpins of nearby hexagons are orthogonal

Director-Director Interaction

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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)

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group 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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

Spin-Lattice Coupling in YMnO3

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

plane

Katsufuji et al. PRB (2001)

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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)

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

1D Chain (S=1)

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

Examples of VBS (AKLT states)

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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>

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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.

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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)

KIAS Emergence Workshop 2005

Manybody Physics Group SKKU

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