  The Meissner effect is one of the defining properties of superconductivity,
with a conventional superconductor completely repelling an external magnetic
field. In contrast to this diamagnetic behavior, odd-frequency superconducting
pairing has often been seen to produce a paramagnetic Meissner effect, which
instead makes the superconductor unstable due to the attraction of magnetic
field. In this work we study how both even- and odd-frequency superconducting
pairing contributes to the Meissner effect in a generic two-orbital
superconductor with a tunable odd-frequency pairing component. By dividing the
contributions to the Meissner effect into intra- and inter-band processes, we
find that the odd-frequency pairing actually generates both dia- and
paramagnetic Meissner responses, determined by the normal-state band structure.
More specifically, for materials with two electron-like (hole-like) low-energy
bands we find that the odd-frequency inter-band contribution is paramagnetic
but nearly canceled by a diamagnetic odd-frequency intra-band contribution.
Combined with a diamagnetic even-frequency contribution, such superconductors
thus always display a large diamagnetic Meissner response to an external
magnetic field, even in the presence of large odd-frequency pairing. For
materials with an inverted, or topological, band structure, we find the
odd-frequency inter-band contribution to instead be diamagnetic and even the
dominating contribution to the Meissner effect in the near-metallic regime.
Taken together, our results show that odd-frequency pairing in multi-orbital
superconductors does not generate a destabilizing paramagnetic Meissner effect
and can even generate a diamagnetic response in topological materials.

Superconductivity