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**One-dimensional approach to frustrated magnets**

Akira Furusaki (RIKEN) Collaborators: Oleg Starykh (Utah) Leon Balents (UCSB) PRB 72, (2005) Oct. 28, 2005 KIAS

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**Phases of antiferromagnets**

Neel ordered phase Valence Bond Solid Spin liquid (RVB) Tomonaga-Luttinger liquid (d=1) staggered magnetization gapless magnons, S=1 dimer LRO broken translational symmetry gapped magnons, S=1 No local order parameter deconfined massive spinons, S=1/2 algebraic correlations deconfined massless spinons, S=1/2 Oct. 28, 2005 KIAS

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**Frustrated antiferromagnets: a route to exotic spin liquids ?**

(classical) antiferromagnets on frustrated lattices have extensive ground-state degeneracy Lifting degeneracy by quantum fluctuations exotic phases ? fractionalized spin liquids? cf. fractional quantum Hall liquids Landau levels with extensive degeneracy + interactions Laughlin states, anyons,.. Oct. 28, 2005 KIAS

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**Geometric frustration**

Lattices with triangles geometric frustration spins on a triangle interacting via antiferromagnetic exchange 2d Triangular lattice -(ET)2X, Cs2CuCl4…. Kagome lattice 3d Pyrochlore lattice spinel oxides,…. ? 2-fold degeneracy Oct. 28, 2005 KIAS

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**Pyrochlore: most frustrated 3d lattice**

Corner-sharing 3d lattice of tetrahedra (4 spins in each) On every tetrahedron: Extensive degeneracy = # tetrahedra Oct. 28, 2005 KIAS

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**2-dimensional pyrochlore = checkerboard lattice**

X J1 J2 corner-sharing geometry, but and are not equivalent on and S=1/2 antiferromagnetic Heisenberg model Oct. 28, 2005 KIAS

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**related 2d lattice models**

J1-J2 model Shastry-Sutherland model J1: N.N. AF exchange J2: N.N.N. AF exchange Oct. 28, 2005 KIAS

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**J1-J2 model 2d J1-J2 model J2/J (p,0) Columnar Dimer phase (p,p) J1/J**

LRO LRO finite-temperature Ising transition Maximal frustration 2d J1-J2 model Anisotropic model Columnar Dimer phase Classical phase boundary J2=J1/2 J2/J J1/J (p,0) (p,p) Starykh & Balents, PRL (2004) Oct. 28, 2005 KIAS

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**Shastry-Sutherland model [Physica 108B, 1069 (1981)]**

dimer plaquette singlet Neel Koga and Kawakami, PRL 84, 4461 (2000) Oct. 28, 2005 KIAS

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**Ground-state phase diagram of the checkerboard model (2d pyrochlore)**

earlier results J1/J2 ? Neel (square lattice of J1 bonds) Plaquette phase Chains J2>>J1 1 (classically decoupled) = 4-spin singlet ? Oct. 28, 2005 KIAS

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**J１>>J2 B. Canals [PRB 65, 184408 (2002)] spin wave analysis J(q)**

Eigenvector of lowest eigenstate Oct. 28, 2005 KIAS

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**J１～J2 Exact diagonalization [Fouet et al., PRB 67, 054411 (2003)]**

J1=J2, up to N=36 (N=40) ground state S=0: 2-fold degeneracy (N>>1) valence bond crystal (plaquette singlet) many low-lying excited states in the S=0 sector Oct. 28, 2005 KIAS

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**Series expansion [Brenig and Grzeschik, PRB 69, 064420 (2004)]**

triplet dispersion Oct. 28, 2005 KIAS

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**Quadrumer boson approximation**

hard-core bosons Oct. 28, 2005 KIAS

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**rewrite the Hamiltonian in terms of the T bosons **

discard quartic and higher-order terms quadratic approximation diagonalization by Bogoliubov transformation dispersion of triplet bosons (1) unstable when bosons condense at Antiferromagnetic LRO: Neel state (2) unstable when bosons condense at Magnetic LRO Oct. 28, 2005 KIAS

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**Comparison with 1/S (a) (b) (c) (d) (e)**

Tchernyshyov, Starykh, Abanov & Moessner, PRB 68, (2003) Oct. 28, 2005 KIAS

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**J１<<J2 decoupled spin chains sliding Luttinger liquids ?**

Starykh, Singh & Levin, PRL 88, (2002) AF coupling is frustrated. Oct. 28, 2005 KIAS

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**A single Heisenberg chain (S=1/2)**

staggered dimerization Low-energy theory SU(2)1 WZW theory Oct. 28, 2005 KIAS

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**Quasi LRO (algebraic correlations)**

dominant correlations staggered magnetization Neel order Dimer order (VBS) staggered dimerization uniform magnetization is subdominant Oct. 28, 2005 KIAS

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**Weak perturbation: J1 Inter-chain interaction**

If we keep only the term, V is not relevant sliding Luttinger liquid However, the and terms are dangerously irrelevant! Oct. 28, 2005 KIAS

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**2nd order perturbation in V**

Operator Product Expansion uniform magnetization staggered magnetization = staggered dimerization relevant operator allowed by symmetry Mean-field analysis crossed-dimer state Oct. 28, 2005 KIAS

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**comparison with exact diagonalization (36 spins)**

Sindzingre, Fouet, Lhuillier, PRB 66, (2002) dimer-dimer correlations Oct. 28, 2005 KIAS

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Global phase diagram scenario I: direct transition between crossed-dimer and plaquette VBS Neel Plaquette Crossed Dimer ? ? = 1st order transition or intermediate coexistence phase Continuous transitions are forbidden by Landau-Ginzburg-Wilson theory. Oct. 28, 2005 KIAS

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**scenario II: crossed-dimer and plaquette VBS via an intermediate ordered phase**

? Crossed Dimer Neel* Plaquette Neel O(3) ? = 1st order transition or intermediate coexistence phase Oct. 28, 2005 KIAS

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**application to 3d pyrochlore**

3d crossed-dimer phase for Oct. 28, 2005 KIAS

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Summary Global phase diagram of the AF Heisenberg model on the 2d-pyrochlore (checkerboard lattice) 1d-approach can give some useful hints for understanding 2d and 3d frustrated magnets. ? ? Crossed Dimer Plaquette Neel PRB 72, (2005) Oct. 28, 2005 KIAS

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appendix uniform part of magnetization Oct. 28, 2005 KIAS

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