  Motivated by the presence of Ising transitions that take place entirely in
the singlet sector of frustrated spin-1/2 ladders and spin-1 chains, we study
two types of effective dimer models on ladders, a quantum dimer model and a
quantum loop model. Building on the constraints imposed on the dimers, we
develop a Density Matrix Renormalization Group algorithm that takes full
advantage of the relatively small Hilbert space that only grows as Fibonacci
number. We further show that both models can be mapped rigorously onto a
hard-boson model first studied by Fendley, Sengupta and Sachdev [Phys. Rev. B
69, 075106 (2004)], and combining early results with recent results obtained
with the present algorithm on this hard-boson model, we discuss the full phase
diagram of these quantum dimer and quantum loop models, with special emphasis
on the phase transitions. In particular, using conformal field theory, we fully
characterize the Ising transition and the tricritical Ising end point, with a
complete analysis of the boundary-field correspondence for the tricritical
Ising point including partially polarized edges. Finally, we show that the
Fibonacci anyon chain is exactly equivalent to special critical points of these
models.
