We consider the quantum Heisenberg antiferromagnet in a magnetic field on two one-dimensional lattices containing equilateral triangles (a chain of corner-sharing double tetrahedra and a frustrated three-leg ladder) which support localized-magnon states. By mapping of the localized-magnon degrees of freedom on a classical lattice gas we obtain high-field thermodynamic quantities of the models at low temperatures.
We consider the one-orbital N-site repulsive Hubbard model on two kagome like chains, both of which yield a completely dispersionless (flat) one-electron band. Using exact many-electron ground states in the subspaces with n ≤ n_{max} (n_{max} ∝ N) electrons, we calculate the square of the total spin in the ground state to discuss magnetic properties of the models. We have found that although for n < n_{max} the ground states contain fully polarized states, there is no finite region of electron densities n/{\cal N} < 1 ({\cal N} = N/3 or {\cal N} = N/5) where ground-state ferromagnetism survives for {\cal N} → ∞.
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