Thermodynamical stability of odd-frequency superconducting state

Odd-frequency pairing mechanism of superconductivity has been investigated for several decades. Nevertheless, its properties, including the thermodynamic stability, have remained unclear. In particular, it has been argued that the odd-frequency state is thermodynamically unstable, has an unphysical (anti-)Meissner effect, and thus cannot exist as a homogeneous equilibrium phase. We argue that this conclusion is incorrect because it implicitly relies on the inappropriate assumption that the odd-frequency superconductor can be described by an effective Hamiltonian that breaks the particle conservation symmetry. We demonstrate that the odd-frequency state can be properly described within the functional-integral approach using nonlocal-in-time effective action. Within the saddle-point approximation, we find that this phase is thermodynamically stable, exhibits ordinary Meissner effect, and therefore can be realized as an equilibrium homogenous state of matter.