One-dimensional approach to frustrated magnets Akira Furusaki (RIKEN) Collaborators: Oleg Starykh (Utah) Leon Balents (UCSB) PRB 72, 094416 (2005) Oct. 28, 2005 KIAS

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

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

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

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

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

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

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

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

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

J１>>J2 B. Canals [PRB 65, 184408 (2002)] spin wave analysis J(q) Eigenvector of lowest eigenstate Oct. 28, 2005 KIAS

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

Series expansion [Brenig and Grzeschik, PRB 69, 064420 (2004)] triplet dispersion Oct. 28, 2005 KIAS

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

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

Comparison with 1/S (a) (b) (c) (d) (e) Tchernyshyov, Starykh, Abanov & Moessner, PRB 68, 144422 (2003) Oct. 28, 2005 KIAS

J１<<J2 decoupled spin chains sliding Luttinger liquids ? Starykh, Singh & Levin, PRL 88, 167203 (2002) AF coupling is frustrated. Oct. 28, 2005 KIAS

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

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

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

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

comparison with exact diagonalization (36 spins) Sindzingre, Fouet, Lhuillier, PRB 66, 174424 (2002) dimer-dimer correlations Oct. 28, 2005 KIAS

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

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

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

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) Oct. 28, 2005 KIAS

appendix uniform part of magnetization Oct. 28, 2005 KIAS