Odd-frequency superconducting states with different types of Meissner response: Problem of coexistence

We consider physical properties of a superconductor with a recently proposed type of odd-frequency pairing that exhibits diamagnetic Meissner response (``odd-dia state''). Such a state was suggested in order to address stability issues arising in an odd-frequency superconducting state with paramagnetic Meissner response (``odd-para state''). Assuming the existence of an odd-dia state (due to a proper retarded interaction), we study its coexistence with an odd-para state. The latter is known to be generated as an induced superconducting component in, e.g., singlet superconductor/ferromagnet proximity structures or triplet superconductor/normal metal systems. Calculating the superfluid density of the mixed odd-para/odd-dia state and the Josephson current between the odd-para and odd-dia states, we find that the expressions for the currents in both cases have nonvanishing imaginary contributions and are therefore unphysical. We show that a realization of the odd-dia state implies the absence of a Hamiltonian description of the system, and suggest that there exists no physically realizable perturbation that could give rise to the spontaneous symmetry breaking necessary for an actual realization of the odd-dia superconducting state.