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The odd-frequency superconducting state appears generally in hybridstructures consisting of conventional superconductors and other materials, andfeatures electrons that form temporally non-local Cooper pairs. Thequasiclassical theory of superconductivity has been extensively used to modelsuch systems, finding in many cases excellent agreement with experimentalmeasurements. Therefore, it is of interest to study effective models ofodd-frequency superconductivity to predict new phenomena associated with thisform of unconventional pairing. We establish necessary criteria that thequasiclassical Green functions in odd-frequency superconducting systems in thedirty limit must satisfy in order to be physically reasonable, includingconservation of spectral weight. We show that it is possible to write downeffective models which satisfy all the abovementioned criteria, but which stillpredict different behavior when it comes to the density of states and themagnetic response of the superconductor. For instance, an effective model forthe odd-frequency anomalous Green function that gives a conserved spectralweight can yield either a peaked or gapped density of states at the Fermienergy, and exhibit conventional, zero, or unconventional Meissner response.This finding demonstrates the importance of carefully considering theproperties of effective models describing odd-frequency superconductivity inorder to obtain physically reasonable results.

Effective quasiclassical models for odd-frequency superconductivity: energy-symmetry, preserved spectral weight, and Meissner response