  We theoretically investigate Josephson junctions with a phase shift of $\pi$
in various proximity induced one-dimensional superconductor models. One of the
salient experimental signatures of topological superconductors, namely the
fractionalized $4 \pi$ periodic Josephson effect, is closely related to the
occurrence of a characteristic zero energy bound state in such junctions. We
make a detailed analysis of a more general type of $\pi$-junctions coined
"phase winding" junctions where the phase of the order parameter rotates by an
angle $\pi$ while its absolute value is kept finite. Such junctions have
different properties, also from a topological viewpoint, and there are no
protected zero energy modes. We compare the phenomenology of such junctions in
topological ($p$-wave) and trivial ($s$-wave) superconducting wires, and
briefly discuss possible experimental probes. Furthermore, we propose a
topological field theory that gives a minimal description of a wire with
defects corresponding to $\pi$-junctions. This effective theory is a
one-dimensional version of similar theories describing Majorana bound states in
half-vortices of two-dimensional topological superconductors.

Mesoscale and Nanoscale Physics;Superconductivity