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One-dimensional approach to frustrated magnets

One-dimensional approach to frustrated magnets. Akira Furusaki (RIKEN) Collaborators: Oleg Starykh (Utah) Leon Balents (UCSB). PRB 72, 094416 (2005). Phases of antiferromagnets. staggered magnetization. gapless magnons , S=1. Neel ordered phase Valence Bond Solid

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One-dimensional approach to frustrated magnets

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  1. One-dimensional approach tofrustrated magnets Akira Furusaki (RIKEN) Collaborators: Oleg Starykh (Utah) Leon Balents (UCSB) PRB 72, 094416 (2005) KIAS

  2. Phases of antiferromagnets staggered magnetization gapless magnons, S=1 • Neel ordered phase • Valence Bond Solid • Spin liquid (RVB) • Tomonaga-Luttinger liquid (d=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 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 ? fractionalized spin liquids? cf. fractional quantum Hall liquids Landau levels with extensive degeneracy + interactions Laughlin states, anyons,.. KIAS

  4. ? 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 KIAS

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

  6. Y J1 on and X J2 2-dimensional pyrochlore = checkerboard lattice corner-sharing geometry, but and are not equivalent S=1/2 antiferromagnetic Heisenberg model KIAS

  7. related 2d lattice models • J1-J2 model • Shastry-Sutherland model J1: N.N. AF exchange J2: N.N.N. AF exchange KIAS

  8. J2/J Classical phase boundary J2=J1/2 (p,0) Columnar Dimer phase (p,p) J1/J LRO LRO • J1-J2 model finite-temperature Ising transition Maximal frustration 2dJ1-J2 model Anisotropic model Starykh & Balents, PRL (2004) KIAS

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

  10. ? J1/J2 0 1 Chains J2>>J1 Plaquette phase Neel (square lattice of J1 bonds) = 4-spin singlet (classically decoupled) Ground-state phase diagram ofthe checkerboard model (2d pyrochlore) earlier results ? KIAS

  11. J１>>J2 • B. Canals[PRB 65, 184408 (2002)] spin wave analysis J(q) Eigenvector of lowest eigenstate KIAS

  12. 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 KIAS

  13. Series expansion [Brenig and Grzeschik, PRB 69, 064420 (2004)] triplet dispersion KIAS

  14. Quadrumer boson approximation hard-core bosons 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 KIAS

  16. Comparison with 1/S (a) (b) (c) (d) (e) Tchernyshyov, Starykh, Abanov & Moessner, PRB 68, 144422 (2003) KIAS

  17. J１<<J2 decoupled spin chains sliding Luttinger liquids ? Starykh, Singh & Levin, PRL 88, 167203 (2002) AF coupling is frustrated. KIAS

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

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

  20. If we keep only the term, V is not relevant sliding Luttinger liquid Weak perturbation: J1 Inter-chain interaction However, the and terms are dangerously irrelevant! KIAS

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

  22. comparison with exact diagonalization (36 spins) Sindzingre, Fouet, Lhuillier, PRB 66, 174424 (2002) dimer-dimer correlations KIAS

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

  24. ? Crossed Dimer Neel* O(3) Plaquette ? Neel • scenario II: crossed-dimer and plaquette VBS via an intermediate ordered phase ? = 1st order transition or intermediate coexistence phase KIAS

  25. 3d crossed-dimer phase for application to 3d pyrochlore 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. ? ? Crossed Dimer Plaquette Neel PRB 72, 094416 (2005) KIAS

  27. uniform part of magnetization appendix KIAS

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