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Kagome Spin Liquid Assa Auerbach Ranny Budnik Erez Berg
Triangular Kagome a a b c b O(3)xO(2)/O(2) -> O(4) critical pt b a c b Three sublattice N’eel state Huse, Singh Macroscopic degeneracy Classical Heisenberg AFM
Experiments S=3/2 layered Kagome ‘90 Strong quantum spin fluctuations (spin gap?) ‘90 However: Large low T specific heat
Spin gap 2. Finite spin gap 0.06J E(Smin+1)-E(Smin)= S=1/2 Kagome: Numerical Results 1. Short range spin correlations : Zheng & Elser ’90; Chalker & Eastmond ‘92
Lots of Low Energy Singlets Misguich&Lhuillier Mambrini & Mila E S=0 S=1 Log (# states) Log (# states) energy Number of sites Finite T=0 entropy? Massless nonmagnetic modes?
Weak bonds strong bonds 6-site singlet “dimer” 1. Number of dimer coverings is 2. Dimers (10-5 of all singlets N=36) exhaust low energy spectrum. RVB on the KagomeMambrini & Mila, EPJB 2000 Perturbation theory in weak/strong bonds.
2. Interactions range N From exact diagonalization of clusters 2. Effective Hamiltonian (exact) Truncate small longer range interactions Contractor Renormalization (CORE) C.Morningstar, M.Weinstein, PRD 54, 4131 (1996). E. Altman and A. A, PRB 65, 104508, (2002). Details: Ehud Altman's Ph.D. Thesis.
States of Kagome CORE step 1Triangles on a triangular superlattice
2 triangles Heisenberg Dimerization field TEST Supertriangle has 4-fold degeneracy For Heisenberg, and CORE range 2 supertriangle Dominant range 2 interactions
0.953 0.2111 0.053 0.1079 0.2805 0.0598 0.038 Effective Bond Interactions Large Dimerization fields. Contributions will cancel for uniform <SS>!
Spin Order E = -0.134/site Columnar Dimers. E=-0.2035/site × Columnar dimers win! Barrier between ground states is 0.66/site Variational theory
Defect in Columnar state: 0.038 Flipping dimers using Quantum Dimer Model (Rokhsar, Kivelson) -0.0272 0.038 + V H = -t Energies of dimer configurations
Quantum Dimer Model Quantum Dimer Model (Rokhsar, Kivelson) -0.0272 0.038 + V H = -t Moessner& Sondhi: For t/V=1: an exponentially disordered dimer liquid phase! Here t/V<0.
Long Wavelength GL Theory 2+1 dimensional N=6 Clock model, Exponentially suppressed mas gap. Extremely close to the 2+1 D O(2) model Cv ~ T2
The triangular Heisenberg Antiferromagnet • Comparison to the Kagome: • Je, and h are smaller. • Jyy is negative! • Variationally: Triangular Heisenberg also prefers Columnar Dimers.
Kagome Triangular Iterated Core Transformations
0.081 0.005 0.039 - 0.112 0.1 -0.018 0.004 0.039- 0.005 0.037 - 0.038 0.05 -0.03 -0.05 Second Renormalization Kagome triangular Pseudospins align ferromagnetically in xz plane Dominant “ferromagnetic” interaction. Leads to <ly> > 0 in the ground state
Proposed RG flow Spin gap, 6 sites 18 sites 54 sites triangular Kagome 0 3 sublattice Neel spinwaves O(2)-spin liquid Massless singlets
Conclusions • Using CORE, we derived effective low energy models for the Kagome and Triangular AFM. • The Kagome model, describes local singlet formation, and a spin gap. • We derive the Quantum Dimer Model parameters and find the Kagome to reside in the columnar dimer phase. • Low excitations are described by a Quantum O(2) field theory, with a 6-fold Clock model mass term. This leads to an exponentially small mass gap in the spinwaves. • The triangular lattice flows to chiral symmetry breaking, probably the 3 sublattice Neel phase. • Future: Investigations of the quantum phase transition in the effective Hamiltonian by following the RG flow.
Truncate: M lowest states per block Reduced Hilbert space: ( dim= MN ) block excitations are the ''atoms'' (composite particles) Contractor Renormalization (CORE) C.Morningstar, M.Weinstein, PRD 54, 4131 (1996). E. Altman and A. A, PRB 65, 104508, (2002). Details: Ehud Altman's Ph.D. Thesis. Step I: Divide lattice to disjoint blocks. Diagonalize H on each Block.
Old perturbative RG CORE Step II: The Effective Hamiltonian on a particular cluster 2. Project on reduced Hilbert space 1. Diagonalize H on the connected cluster. 3. Orthonormalize from ground state up. (Gramm-Schmidt)
Effective Interactions: 2. CORE Exact Identity: + + + + d>1: only rectangular shapes! E. Altman's thesis. 3. If long range interactions are sufficiently small, truncate Heff at finite range. Note: Heff is not perturbative in hi j, and not a variational approximation. All the error is in the discarded longer range interactions. 4. x is the size ("coherence length") of the renormalized degrees of freedom. ! CORE Step III: The Cluster Expansion
Tetrahedra Psedospins E. Berg, E. Altman and A.A, cond-mat/0206384, PRL (03) E S=2 = tetrahedron S=1 S=1 S=1 2 J S=0 S=0 pseudospin S=1/2 super-tetrahedron pseudospin S=1/2
Heisenberg antiferromagnet E/J pyrochlore 1 CORE step 1 Anisotropic spin half model: frustrated Fcc 10-1 CORE step 2 Ising like model: not frustrated Cubic 10-2 16-site singlets 2 CORE Steps to Ground State
Variational comparison (S=1/2) Hexagons Versus Supertetrahedra What do experiments say?
Palmer and Chalker (2001) Ground state Moessner, Tshernyshyov, Sondhi Domain wall singlet excitations The Checkerboard
3D Pyrochlore 2D Checkerboard Free plaquettes free hexagons Geometrical Frustration on Pyrochlores Villain (79); Moessner and Chalker (98); Non dispersive zero energy modes. Spinwave theory is poorly controlled
Pseudospins defined on a FCC lattice Range 3 CORE 0.1 J +0.4 J ( Perturbative Expansions+spinwave theory Harris, Berlinsky,Bruder (92), Tsunetsugu (02) Remaining Mean-Field zero energy modes Interactions between pseudospins Insufficient Renormalization!
pyrochlore No order! 1 Macroscopic degeneracy! 4 sublattice “order”: Harris, Berlinsky,Bruder (92) 10-1 Fcc Cubic Ising-like AFM: not frustrated 10-2 Spin-½ Pyrochlore Antiferromagnet Effective model Mean Field Order E/J Macroscopic degeneracy! Pseudospins
Correlations: Theory vs Experiment S=3/2 fixed q S.H. Lee et. al. 1 meV Ansatz: Tchernyshyov et.al. CORE: Theory: E. Berg AA.,, to be published magnon gap S=1/2