1. Kagome Spin Liquid Assa Auerbach Ranny Budnik Erez Berg

2. 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

3. Experiments Strong quantum spin fluctuations (spin gap?) S=3/2 layered Kagome ‘90 However: Large low T specific heat

4. 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 )=

5. 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

6. 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.

7. 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)

8. Kagome CORE step 1 Triangles on a triangular superlattice States of

9. Dominant range 2 interactions 2 triangles Heisenberg Dimerization field TEST Supertriangle has 4-fold degeneracy For Heisenberg, and CORE range 2 supertriangle

10. Range 3 corrections

11. 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

12. 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 

13. 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

14. 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.

15. 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

16. 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.

17. Kagome Triangular Iterated Core Transformations

18. 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

19. Proposed RG flow 3 sublattice Neel spinwaves O(2)-spin liquid Massless singlets triangular Kagome 0 Spin gap, 6 sites 18 sites 54 sites

20. 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.

21. 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 )

22. 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)

23. 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.

24. 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)

25. 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

26. Variational comparison (S=1/2) Hexagons Versus Supertetrahedra What do experiments say?

27. Ground state Moessner, Tshernyshyov, Sondhi Domain wall singlet excitations The Checkerboard Palmer and Chalker (2001)

28. 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

29. 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

30. 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

31. 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