Geometric frustration and magnetization plateaus in quantum spin and Bose-Hubbard models on tubes

We study $\mathrm{XXZ}$ Heisenberg models on frustrated triangular lattices wrapped around a cylinder. In addition to having interesting magnetic phases, these models are equivalent to Bose-Hubbard models that describe the physical problem of adsorption of noble gases on the surface of carbon nanotubes. We find analytical results for the possible magnetization plateau values as a function of the wrapping vectors of the cylinder, which in general introduce extra geometric frustration besides the one due to the underlying triangular lattice. We show that for particular wrapping vectors $(N,0),$ which correspond to the zigzag nanotubes, there is a macroscopically degenerate ground state in the classical Ising limit. The Hilbert space for the degenerate states can be enumerated by a mapping first into a path in a square lattice wrapped around a cylinder (a Bratteli diagram), and then to free fermions interacting with a single ${\mathbf{Z}}_{N}$ degree of freedom. From this model we obtain the spectrum in the anisotropic Heisenberg limit, showing that it is gapless. The continuum limit is a $c=1$ conformal field theory with compactification radius $R=N$ set by the physical tube radius. This result cannot be checked against a Lieb-Schultz-Mattis argument, for the argument is inconclusive when applied to this problem. We show that the compactification radius quantization is exact in the projective ${J}_{\mathrm{xy}}{/J}_{z}\ensuremath{\ll}1$ limit, and that higher-order corrections reduce the value of R. The particular case of a $(N=2,0)$ tube, which corresponds to a two-leg ladder with cross links, is studied separately and shown to be gapped because the fermion-mapped problem contains superconducting pairing terms.