In general, the symmetry of a superconductor can be characterized by studying the anomalous Green’s function:
where annihilates an electron with indices labeling spin, , position, , time, , and orbital/band degrees of freedom, , and is the time-ordering operator. Using the fermionic properties of electrons it is straightforward to show that: . This relation tells us that the wavefunction describing the Cooper pairs, , must obey where: acts on spin (); is the spatial parity operator (); interchanges orbital degrees of freedom (); and reverses the time coordinates (). Using this property of together with the fact that all four transformations square to the identity, the possible symmetries of the Cooper pair wavefunction may be divided into 8 different classes based on how they transform under , , , and :
While no examples of bulk odd-frequency superconductors have yet been identified, there are a growing number of proposals for engineering these exotic amplitudes in heterostructures and driven systems, magnetic states and systems with Majorana modes.
Pair Symmetry Conversion in Driven Multiband Superconductors
By subjecting a multiband superconductor to a time-dependent drive, even-frequency pair amplitudes can be converted to odd-frequency pair amplitudes and vice versa. In the movie below we evaluate, as a function of time, both the even-frequency and odd-frequency pairing amplitudes of a multiband superconductor driven by a time-periodic chemical potential. At generic times during the period, contributions to the odd-frequency and even-frequency pair amplitudes are non-zero. The corrections to the odd-frequency amplitudes are largest exactly when the drive vanishes and smallest exactly when the drive reaches its maximum amplitude; the corrections to the even-ω amplitudes behave in the opposite manner.
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