Wave function for odd-frequency superconductors

We revisit the question of the nature of odd-frequency superconductors, first proposed by Berezinskii in 1974 (JETP Lett. 20 287). We start with the notion that the order parameter of odd-frequency superconductors can be thought of as a time derivative of the odd-time pairing operator. This leads to the notion of the composite boson condensate (Abrahams et al 1995 Phys. Rev. B 52 1271; Balatsky and Bonca 1993 Phys. Rev. B 48 7445). To elucidate the nature of broken symmetry states in odd-frequency superconductors, we consider a wave function that properly captures the coherent condensate of composite charge 2e bosons in an odd-frequency superconductor. We consider the Hamiltonian that describes the equal-time composite boson condensation as proposed earlier by Abrahams et al (1995 Phys. Rev. B 52 1271). We propose a Bardeen-Cooper-Schrieffer (BCS)-like wave function that describes a composite condensate comprised of a spin-0 Cooper pair and a spin-1 magnon excitation. We derive the quasi-particle dispersion, the self-consistent equation for the order parameter and the density of states. We show that the coherent wave function approach recovers all the known proprietaries of odd-frequency superconductors: the quasi-particle excitations are gapless and the superconducting transition requires a critical coupling.