  We study lattice superconductors such as attractive Hubbard models. As is
well known, Bloch's theorem asserts absence of persistent current in ground
states and equilibrium states for general fermion systems. While the statement
of the theorem is true, we can show that the theorem cannot exclude possibility
of a surface persistent current. Such a current can be stabilized by boundary
magnetic fields which do not penetrate into the bulk region of a
superconductor, provided emergence of massive photons, i.e., Meissner effect.
Therefore, we can expect that a surface persistent current is realized for a
ground/equilibrium state in the sense of stability against local perturbations.
We also apply Elitzur's theorem to superconductors at finite temperatures. As a
result, we prove absence of symmetry breaking of the global $U(1)$ phase of
electrons for almost all gauge fields. These observations suggest that the
nature of superconductivity is the emergence of massive photons rather than the
symmetry breaking of the $U(1)$ phase of electrons.
