  We use the multiscale entanglement renormalisation ansatz (MERA) to
numerically investigate three critical quantum spin chains with Z_2 x Z_2
on-site symmetry: a staggered XXZ model, a transverse field cluster model, and
the quantum Ashkin-Teller model. All three models possess a continuous
one-parameter family of critical points. Along this critical line, the
thermodynamic limit of these models is expected to be described by classes of
c=1 conformal field theories (CFTs) of two possible types: the S^1 free boson
and its Z_2-orbifold. Our numerics using MERA with explicitly enforced Z_2 x
Z_2 symmetry allow us to extract conformal data for each model, with strong
evidence supporting the identification of the staggered XXZ model and critical
transverse field cluster model with the S^1 boson CFT, and the Ashkin-Teller
model with the Z_2-orbifold boson CFT. Our first two models describe the phase
transitions between symmetry protected topologically ordered phases and trivial
phases, which lie outside the usual Landau-Ginsburg-Wilson paradigm of symmetry
breaking. Our results show that a range of critical theories can arise at the
boundary of a single symmetry protected phase.
