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Oct. 28, 2005KIAS1 One-dimensional approach to frustrated magnets Akira Furusaki (RIKEN) Collaborators: Oleg Starykh (Utah) Leon Balents (UCSB) PRB 72, (2005)

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

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Oct. 28, 2005 KIAS 3 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 ? cf. fractional quantum Hall liquids Landau levels with extensive degeneracy + interactions Laughlin states, anyons,.. fractionalized spin liquids?

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Oct. 28, 2005 KIAS 4 Geometric frustration ? spins on a triangle interacting via antiferromagnetic exchange 2-fold degeneracy Lattices with triangles geometric frustration 2d Triangular lattice -(ET) 2 X, Cs 2 CuCl 4 …. Kagome lattice 3d Pyrochlore lattice spinel oxides,….

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Oct. 28, 2005 KIAS 5 Pyrochlore: most frustrated 3d lattice Corner-sharing 3d lattice of tetrahedra (4 spins in each) On every tetrahedron: Extensive degeneracy = # tetrahedra

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Oct. 28, 2005 KIAS 6 2-dimensional pyrochlore = checkerboard lattice corner-sharing geometry, but and are not equivalent Y X J1J1 J 2 S=1/2 antiferromagnetic Heisenberg model on and

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Oct. 28, 2005 KIAS 7 related 2d lattice models J 1 -J 2 model Shastry-Sutherland model J 1 : N.N. AF exchange J 2 : N.N.N. AF exchange

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Oct. 28, 2005 KIAS 8 J 1 -J 2 model 2d J 1 -J 2 model LRO finite-temperature Ising transition Maximal frustration Anisotropic model Starykh & Balents, PRL (2004) Columnar Dimer phase Classical phase boundary J 2 =J 1 /2 J 2 /J J 1 /J

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Oct. 28, 2005 KIAS 9 Shastry-Sutherland model [Physica 108B, 1069 (1981)] dimer plaquette singlet Neel Koga and Kawakami, PRL 84, 4461 (2000)

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Oct. 28, 2005 KIAS 10 Ground-state phase diagram of the checkerboard model (2d pyrochlore) J 1 /J 2 ? 0 Neel (square lattice of J 1 bonds)Plaquette phase Chains J 2 >>J 1 1 (classically decoupled) = 4-spin singlet earlier results ?

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Oct. 28, 2005 KIAS 11 J １ >> J 2 B. Canals [PRB 65, (2002)] spin wave analysis J(q) Eigenvector of lowest eigenstate

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Oct. 28, 2005 KIAS 12 J１～J2J１～J2 Exact diagonalization [Fouet et al., PRB 67, (2003)] J 1 =J 2, up to N=36 (N=40) ground state S=0: 2-fold degeneracy (N>>1) many low-lying excited states in the S=0 sector valence bond crystal (plaquette singlet)

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Oct. 28, 2005 KIAS 13 Series expansion [Brenig and Grzeschik, PRB 69, (2004)] triplet dispersion

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Oct. 28, 2005 KIAS 14 Quadrumer boson approximation hard-core bosons

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Oct. 28, 2005 KIAS 15 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

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Oct. 28, 2005 KIAS 16 Comparison with 1/S Tchernyshyov, Starykh, Abanov & Moessner, PRB 68, (2003) (a) (b)(c)(d)(e)

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Oct. 28, 2005 KIAS 17 J １ <

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Oct. 28, 2005 KIAS 18 A single Heisenberg chain (S=1/2) Low-energy theory SU(2) 1 WZW theory staggered dimerization

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Oct. 28, 2005 KIAS 19 Quasi LRO (algebraic correlations) dominant correlations staggered magnetization staggered dimerization Neel order Dimer order (VBS) uniform magnetization is subdominant

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Oct. 28, 2005 KIAS 20 Weak perturbation: J 1 Inter-chain interaction If we keep only the term, V is not relevantsliding Luttinger liquid However, the and terms are dangerously irrelevant!

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Oct. 28, 2005 KIAS 21 2 nd order perturbation in V Operator Product Expansion uniform magnetization staggered magnetization = staggered dimerization relevant operator allowed by symmetry Mean-field analysis crossed-dimer state

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Oct. 28, 2005 KIAS 22 comparison with exact diagonalization (36 spins) Sindzingre, Fouet, Lhuillier, PRB 66, (2002) dimer-dimer correlations

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Oct. 28, 2005 KIAS 23 Global phase diagram scenario I: direct transition between crossed-dimer and plaquette VBS NeelPlaquetteCrossed Dimer ?? ? = 1 st order transition or intermediate coexistence phase Continuous transitions are forbidden by Landau-Ginzburg-Wilson theory.

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Oct. 28, 2005 KIAS 24 scenario II: crossed-dimer and plaquette VBS via an intermediate ordered phase ? ? Crossed DimerNeel*PlaquetteNeelO(3) ? = 1 st order transition or intermediate coexistence phase

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Oct. 28, 2005 KIAS 25 application to 3d pyrochlore 3d crossed-dimer phase for

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Oct. 28, 2005 KIAS 26 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. NeelPlaquetteCrossed Dimer ?? PRB 72, (2005)

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Oct. 28, 2005 KIAS 27 appendix uniform part of magnetization

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