  We give a classification of gapped quantum phases of one-dimensional systems
in the framework of Matrix Product States (MPS) and their associated parent
Hamiltonians, for systems with unique as well as degenerate ground states, and
both in the absence and presence of symmetries. We find that without
symmetries, all systems are in the same phase, up to accidental ground state
degeneracies. If symmetries are imposed, phases without symmetry breaking
(i.e., with unique ground states) are classified by the cohomology classes of
the symmetry group, this is, the equivalence classes of its projective
representations, a result first derived in [X. Chen, Z.-C. Gu, and X.-G. Wen,
Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. For phases with symmetry
breaking (i.e., degenerate ground states), we find that the symmetry consists
of two parts, one of which acts by permuting the ground states, while the other
acts on individual ground states, and phases are labelled by both the
permutation action of the former and the cohomology class of the latter. Using
Projected Entangled Pair States (PEPS), we subsequently extend our framework to
the classification of two-dimensional phases in the neighborhood of a number of
important cases, in particular systems with unique ground states, degenerate
ground states with a local order parameter, and topological order. We also show
that in two dimensions, imposing symmetries does not constrain the phase
diagram in the same way it does in one dimension. As a central tool, we
introduce the isometric form, a normal form for MPS and PEPS which is a
renormalization fixed point. Transforming a state to its isometric form does
not change the phase, and thus, we can focus on to the classification of
isometric forms.
