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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,

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

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

2 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

3 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?

4 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,….

5 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

6 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

7 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

8 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  

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

10 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 ?

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

12 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)

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

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

15 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

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

17 Oct. 28, 2005 KIAS 17 J 1 <

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

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

20 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!

21 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

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

23 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.

24 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

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

26 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)

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


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