Odd-frequency superfluidity from a particle-number-conserving perspective

We investigate odd-in-time—or odd-frequency—pairing of fermions in equilibrium systems within the
particle-number-conserving framework of Penrose, Onsager, and Yang, where superfluid order is defined
by macroscopic eigenvalues of reduced density matrices. We show that odd-frequency pair correlations are
synonymous with even fermion-exchange symmetry in a time-dependent correlation function that generalises
the two-body reduced density matrix. Macroscopic even-under-fermion-exchange pairing is found to emerge
from conventional Penrose-Onsager-Yang condensation in two-body or higher-order reduced density matrices
through the symmetry-mixing properties of the Hamiltonian. We identify and characterize a transformer matrix
responsible for producing macroscopic even fermion-exchange correlations that coexist with a conventional
Cooper-pair condensate, while a generator matrix is shown to be responsible for creating macroscopic even
fermion-exchange correlations from hidden orders such as a multiparticle condensate. The transformer scenario
is illustrated using the spin-balanced s-wave superfluid with Zeeman splitting as an example. ... mehrThe generator
scenario is demonstrated by the composite-boson condensate arising for itinerant electrons coupled to magnetic
excitations. Structural analysis of the transformer and generator matrices is shown to provide general conditions
for odd-frequency pairing order to arise in a given system. Our formalism facilitates a fully general derivation
of the Meissner effect for odd-frequency superconductors that holds also beyond the regime of validity for
mean-field theory.