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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun Kwan U. 成均館大 Ref: Jia(Kim) 2 Han, 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-striction effect (Pytte, PRB 1974) i j uiui ujuj R ij Lattice displacement related to local spin-spin correlation by

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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 gives 120 degrees for neighboring spins for triangular/Kagome According to linearized spin-lattice theory local displacement is zero. Classical degeneracy of Kagome is not lifted.

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU 3D Frustrated Lattice - Pyrochlore Tetrahedron as a building block Ground state condition for each tetrahedron Macroscopic classical ground state degeneracy in pristine Heisenberg model

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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

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Experiments on Pyrochlore – ZnCr2O4 Below Tc S.H.Lee et al. PRL, 2000 : Spins on Cr 3+ (S=3/2) order antiferromagnetically at as first-order transition, acccompanied by cubic-to-tetragonal distortion.

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Theory of spin-Peierls transition by Tschernyshyov, Moessner, and Sondhi TMS, PRL, 2002 TMS, PRB, 2002 Elongation (contraction) of a tetrahedron along an axis And collinear antiferromagnetic spins is the ground state

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Experiments on Pyrochlore – ZnCr2O4 Above Tc S.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!)

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Lifting of degeneracy through spin-lattice coupling Tetrahedron Unit (Tschernyshyov et al. PRL 2002) Hexagon Unit (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 Directors of nearby hexagons interact via

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Spins within a hexagon are collinear Spins 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, Tetrahedron-based distortion has a lower energy, but hexagon shrinkage is a viable, metastable state

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU A picture of paramagnetic state in ZnCr 2 O 4 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 of spin-loop directors, but it appears that a tetragonal distortion pre-empts this possibility in ZnCr 2 O 4

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Spin-Lattice Coupling in YMnO 3 Mn 3+ (S=2) spins form triangular network in MnO 2 plane Katsufuji et al. PRB (2001)

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Experiments on triangular S=2 AFM – YMnO 3 Expansion of Mn triangles (red) at the onset of magnetic order But 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)

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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 form for coordination = 6 Without SLC, VB’s will resonante, but with SLC, VBS patterns can be formed.

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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) Bond order 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)

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Results This state is also Bose-condensed, so LRO co-exists with VBS Elementary excitations: (1) Spin waves for |LRO> (2) Triplet wave for |PVBS>

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU A New Excitation We calculate excitation spectra for the VBS within SMA VBS Spin waveComparison

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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.

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KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Relation to Experiments? – YMnO 3 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)

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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

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