  The association of broken symmetries with phase transitions is ubiquitous in
condensed matter physics: crystals break translational symmetry, magnets break
rotational symmetry, and superconductors break gauge symmetry. However, despite
the frequency with which it is made, this last statement is a paradox. A gauge
symmetry, in this case the U(1) gauge symmetry of electromagnetism, is a
redundancy in our description of nature, so the notion of breaking such a
"symmetry" is unphysical. Here, we will discuss how gauge symmetry breaks, and
doesn't, inside a superconductor, and explore the fundamental relationship
between gauge invariance and the striking phenomena observed in
superconductors. The majority of this article is intended to be accessible to
readers with only a basic knowledge of quantum mechanics.
