Title: One-Dimensional Symmetry Protected Topological Phases and their Transitions,
Abstract: We present a unified perspective on symmetry protected topological (SPT)
phases in one dimension and address the open question of what characterizes
their phase transitions. In the first part of this work we use symmetry as a
guide to map various well-known fermionic and spin SPTs to a Kitaev chain with
coupling of range $\alpha \in \mathbb Z$. This unified picture uncovers new
properties of old models --such as how the cluster state is the fixed point
limit of the Affleck-Kennedy-Lieb-Tasaki state in disguise-- and elucidates the
connection between fermionic and bosonic phases --with the Hubbard chain
interpolating between four Kitaev chains and a spin chain in the Haldane phase.
In the second part, we study the topological phase transitions between these
models in the presence of interactions. This leads us to conjecture that the
critical point between any SPT with $d$-dimensional edge modes and the trivial
phase has a central charge $c \geq \log_2 d$. We analytically verify this for
many known transitions. This agrees with the intuitive notion that the phase
transition is described by a delocalized edge mode, and that the central charge
of a conformal field theory is a measure of the gapless degrees of freedom.