Kagome Spin Liquid

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Kagome Spin Liquid

• Assa Auerbach
• Ranny Budnik
• Erez Berg

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Classical Heisenberg AFM
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Experiments
S3/2 layered Kagome
90
Strong quantum spin fluctuations (spin gap?)
90
However Large low T specific heat
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S1/2 Kagome Numerical Results
1. Short range spin correlations Zheng Elser
90 Chalker Eastmond 92
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Lots of Low Energy Singlets
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RVB on the Kagome Mambrini Mila, EPJB 2000
Perturbation theory in weak/strong bonds.
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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.
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Kagome CORE step 1 Triangles on a triangular
superlattice
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Dominant range 2 interactions
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Range 3 corrections
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Effective Bond Interactions
Large Dimerization fields. Contributions will
cancel for uniform ltSSgt!
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Variational theory
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Energies of dimer configurations
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Quantum Dimer Model
Quantum Dimer Model (Rokhsar, Kivelson)
-0.0272
0.038
V
H -t
Moessner Sondhi For t/V1 an exponentially
disordered dimer liquid phase! Here t/Vlt0.
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Long Wavelength GL Theory
21 dimensional N6 Clock model,
Exponentially suppressed mas gap. Extremely close
to the 21 D O(2) model Cv T2
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The triangular Heisenberg Antiferromagnet

• Comparison to the Kagome
• Je, and h are smaller.
• Jyy is negative!
• Variationally Triangular Heisenberg also prefers
  Columnar Dimers.

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Iterated Core Transformations
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Second Renormalization
Kagome
triangular
Pseudospins align ferromagnetically in xz plane
Dominant ferromagnetic interaction. Leads to
ltlygt gt 0 in the ground state
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Proposed RG flow
Spin gap, 6 sites
18 sites
54 sites
triangular
Kagome
0
3 sublattice Neel spinwaves
O(2)-spin liquid Massless singlets
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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.
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CORE Step II The Effective Hamiltonian on a
particular cluster
1. Diagonalize H on the connected cluster.
2. Project on reduced Hilbert space
3. Orthonormalize from ground state up.
(Gramm-Schmidt)
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CORE Step III The Cluster Expansion
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Tetrahedra Psedospins
E. Berg, E. Altman and A.A, cond-mat/0206384,
PRL (03)
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2 CORE Steps to Ground State
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Hexagons Versus Supertetrahedra
What do experiments say?
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The Checkerboard
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Geometrical Frustration on Pyrochlores
Villain (79) Moessner and Chalker (98)
Non dispersive zero energy modes. Spinwave
theory is poorly controlled
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Interactions between pseudospins
Insufficient Renormalization!
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Spin-½ Pyrochlore Antiferromagnet
Mean Field Order
Effective model
E/J
Macroscopic degeneracy!
Pseudospins
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Correlations Theory vs Experiment
S3/2
fixed q
S.H. Lee et. al.
1 meV
Ansatz
Tchernyshyov et.al.
CORE
Theory
E. Berg AA.,, to be published
magnon gap
S1/2