![Page 1: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/1.jpg)
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
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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
![Page 3: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/3.jpg)
KIAS Emergence Workshop 2005
Manybody Physics Group SKKU
Application to Frustrated Lattices – Triangular, Kagome
NO LATTICE DEFORMATION!
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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
![Page 5: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/5.jpg)
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
![Page 6: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/6.jpg)
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
![Page 7: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/7.jpg)
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
![Page 8: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/8.jpg)
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
![Page 9: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/9.jpg)
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!)
![Page 10: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/10.jpg)
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)
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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
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KIAS Emergence Workshop 2005
Manybody Physics Group SKKU
Spins within a hexagon are collinearSpins of nearby hexagons are orthogonal
Director-Director Interaction
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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)
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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
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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
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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)
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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)
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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
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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)
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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
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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
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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
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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>
![Page 24: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/24.jpg)
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
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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.
![Page 26: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/26.jpg)
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)
![Page 27: KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun](https://reader035.vdocument.in/reader035/viewer/2022081518/5516e260550346fe558b4592/html5/thumbnails/27.jpg)
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