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}:

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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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Key papers:

  1. Review article:
    Odd-frequency superconductivity
    Jacob Linder, Alexander V. Balatsky
  2. General conditions for proximity-induced odd-frequency superconductivity in two-dimensional electronic systems
    Christopher Triola, Driss M. Badiane, Alexander V. Balatsky, E. Rossi
    Phys. Rev. Lett. 116, 257001 (2016)
  3. Proximity-induced unconventional superconductivity in topological insulators
    Annica M. Black-Schaffer, Alexander V. Balatsky
    Phys. Rev. B 87, 220506(R) (2013)