Strong-coupling superconductivity with mixed even- and odd-frequency pairing

We investigate the general structure of Landau free energy for stabilizing a novel superconducting state with both even-frequency and odd-frequency components in the gap function. On the basis of the Luttinger-Ward functional, we elucidate an emergent mixing between different ``parities'' in time. The simplest case of the conventional s-wave singlet mixed with the odd-frequency triplet state under broken time-reversal symmetry is examined to demonstrate the anomalous structure of the free-energy functional. The induced odd-frequency component alters behaviors of physical quantities from those obtained by neglecting the odd-frequency component. The novel mixed state may also be relevant to strong-coupling superconductivity coexisting with ferromagnetism.