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Kagome Spin Liquid Assa Auerbach Ranny Budnik Erez Berg
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Classical Heisenberg AFM Macroscopic degeneracy Kagome O(3)xO(2)/O(2) -> O(4) critical pt Three sublattice N’eel state Huse, Singh Triangular a cb a b c a b b
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Experiments Strong quantum spin fluctuations (spin gap?) S=3/2 layered Kagome ‘90 However: Large low T specific heat
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S=1/2 Kagome: Numerical Results 1. Short range spin correlations : Zheng & Elser ’90; Chalker & Eastmond ‘92 Spin gap 0.06J 2. Finite spin gap E(S min +1)-E(S min )=
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Lots of Low Energy Singlets Mambrini & Mila Finite T=0 entropy? energy Log (# states) Number of sites Misguich&Lhuillier Log (# states) Massless nonmagnetic modes? S=0 S=1 E
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RVB on the Kagome Mambrini & Mila, EPJB 2000 Weak bonds strong bonds 6-site singlet “dimer” Perturbation theory in weak/strong bonds. 1. Number of dimer coverings is 2. Dimers (10 -5 of all singlets N=36) exhaust low energy spectrum.
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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. Truncate small longer range interactions 2. Interactions range N From exact diagonalization of clusters 2. Effective Hamiltonian (exact)
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Kagome CORE step 1 Triangles on a triangular superlattice States of
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Dominant range 2 interactions 2 triangles Heisenberg Dimerization field TEST Supertriangle has 4-fold degeneracy For Heisenberg, and CORE range 2 supertriangle
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Range 3 corrections
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Effective Bond Interactions Large Dimerization fields. Contributions will cancel for uniform ! 0.953 0.2111 0.053 0.1079 0.2805 0.0598 0.038
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Variational theory Columnar dimers win! Barrier between ground states is 0.66/site Spin Order E = -0.134/site Columnar Dimers. E=-0.2035/site
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Energies of dimer configurations Defect in Columnar state: Flipping dimers using 0.038 Quantum Dimer Model (Rokhsar, Kivelson) H = -t + V 0.038 -0.0272
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Quantum Dimer Model Quantum Dimer Model (Rokhsar, Kivelson) H = -t + V 0.038 -0.0272 Moessner& Sondhi: For t/V=1: an exponentially disordered dimer liquid phase! Here t/V<0.
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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 ~ T 2
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The triangular Heisenberg Antiferromagnet Comparison to the Kagome: 1.Je, and h are smaller. 2.Jyy is negative! 3.Variationally: Triangular Heisenberg also prefers Columnar Dimers.
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Kagome Triangular Iterated Core Transformations
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Second Renormalization 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 Kagome triangular Dominant “ferromagnetic” interaction. Leads to > 0 in the ground state Pseudospins align ferromagnetically in xz plane
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Proposed RG flow 3 sublattice Neel spinwaves O(2)-spin liquid Massless singlets triangular Kagome 0 Spin gap, 6 sites 18 sites 54 sites
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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.
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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. block excitations are the ''atoms'' (composite particles) Truncate: M lowest states per block Reduced Hilbert space: ( dim= M N )
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CORE Step II: The Effective Hamiltonian on a particular cluster 1.Diagonalize H on the connected cluster. Old perturbative RG 2.Project on reduced Hilbert space3. Orthonormalize from ground state up. (Gramm-Schmidt)
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CORE Step III: The Cluster Expansion Effective Interactions: 2. CORE Exact Identity: + + ++ d>1: only rectangular shapes! E. Altman's thesis. 3. If long range interactions are sufficiently small, truncate H eff at finite range. 4. is the size ("coherence length") of the renormalized degrees of freedom. Note: H eff is not perturbative in h i j, and not a variational approximation. All the error is in the discarded longer range interactions.
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pseudospin S=1/2 Tetrahedra Psedospins 2 J S=1 S=2 S=0 E tetrahedron = super-tetrahedron pseudospin S=1/2 E. Berg, E. Altman and A.A, cond-mat/0206384, PRL (03)
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10 -2 Cubic 16-site singlets 2 CORE Steps to Ground State pyrochlore 1 E/J Heisenberg antiferromagnet Fcc 10 -1 CORE step 1 Anisotropic spin half model: frustrated CORE step 2 Ising like model: not frustrated
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Variational comparison (S=1/2) Hexagons Versus Supertetrahedra What do experiments say?
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Ground state Moessner, Tshernyshyov, Sondhi Domain wall singlet excitations The Checkerboard Palmer and Chalker (2001)
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Geometrical Frustration on Pyrochlores 2D Checkerboard 3D Pyrochlore Non dispersive zero energy modes. Spinwave theory is poorly controlled Villain (79); Moessner and Chalker (98); free hexagons Free plaquettes
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Insufficient Renormalization! Remaining Mean-Field zero energy modes Perturbative Expansions+spinwave theory Harris, Berlinsky,Bruder (92), Tsunetsugu (02) Pseudospins defined on a FCC lattice Range 3 CORE +0.4 J ( 0.1 J Interactions between pseudospins
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10 -1 Fcc pyrochlore 1 No order! Macroscopic degeneracy! Spin-½ Pyrochlore Antiferromagnet E/J Mean Field OrderEffective model 4 sublattice “order”: Harris, Berlinsky,Bruder (92) Pseudospins Macroscopic degeneracy! 10 -2 Cubic Ising-like AFM: not frustrated
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CORE: Correlations: Theory vs Experiment Ansatz: Theory: S=3/2 S=1/2 E. Berg AA.,, to be published Tchernyshyov et.al. S.H. Lee et. al. magnon gap fixed q 1 meV
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