## Odd-frequency Superconductivity

In general, the symmetry of a superconductor can be characterized by studying the anomalous Green’s function:

\[ F_{1;2}=-i\langle T \psi_{1}\psi_{2}\rangle \]

where \(\psi_i\) annihilates an electron with indices labeling spin, \(\sigma_i\), position, \(\textbf{r}_i\), time, \(t_i\), and orbital/band degrees of freedom, \(\alpha_i\), and T is the time-ordering operator. Using the fermionic properties of electrons it is straightforward to show that: \(F_{1;2}=-F_{2;1}\). This relation tells us that the wavefunction describing the Cooper pairs, \(\Psi\), must obey \(\mathcal{S}\mathcal{P}\mathcal{O}\mathcal{T}\Psi=-\Psi\) where: \(\mathcal{S}\) acts on spin \((\sigma_1\leftrightarrow\sigma_2)\); \(\mathcal{P}\) is the spatial parity operator \((\textbf{r}_1\leftrightarrow\textbf{r}_2)\); \(\mathcal{O}\) interchanges orbital degrees of freedom \((\alpha_1\leftrightarrow\alpha_2)\); and \(\mathcal{T}\) reverses the time coordinates \((t_1\leftrightarrow t_2)\). Using this property of \(\Psi\) 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 \(\mathcal{S}\), \(\mathcal{P}\), \(\mathcal{O}\), and \(\mathcal{T}\):

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.