Odd-frequency Superconductivity

Due to the fermionic nature of electrons, the spatial symmetry (s-wave, p-wave, d-wave, etc.) of a superconducting gap is intimately related to the spin state (singlet or triplet) of the Cooper pairs making up the condensate. In the limit of equal-time pairing this relationship is quite simple, even-parity gaps (like s-wave, or d-wave) correspond to spin singlet states while odd-parity gaps (like p-wave or f -wave) correspond to spin triplet states. However, if the electrons are paired at unequal times the superconducting gap could be odd in time or, equivalently, odd in frequency [1], in which case the condensate could be even in spatial parity and spin triplet or odd in spatial parity and spin singlet. This possibility, is intriguing both because of the unconventional symmetries which it permits and for the fact that it represents a class of hidden order, due to the vanishing of equal time correlations.

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

(1)   \begin{equation*} F_{1;2}=-i\langle T \psi_{1}\psi_{2}\rangle \end{equation*}

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}, see Table 1.

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The first two columns in Table 1 encapsulate the standard pair symmetries: s-wave singlet and p-wave triplet. However, the other columns represent generalizations allowing for both odd-orbital and odd-time (or, equivalently, odd-frequency) pairings. 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 [2] and driven systems [3]. It has also been argued that these amplitudes should be ubiquitous in multiband superconductors [4,5]. Furthermore, similar odd-frequency orders have been predicted in magnetic states [6] and in systems hosting Majorana modes [7].

Pair Symmetry Conversion in Driven Multiband Superconductors

Building on previous work demonstrating the emergence of odd-frequency pairing in driven systems [3] and in multiband superconductors [4,5], it was recently shown that 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 [8]. 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, see Eq (1) of [8]. The movie shows essentially the same plots as in Figs 1 and 2 of [8] time-resolved over a full period of the drive (plotted in the bottom panel of the video).

From this movie we observe that, at generic times during the period, contributions to the odd-frequency and even-frequency pair amplitudes are non-zero. Furthermore, we can see that the corrections to the odd-frequency amplitudes are largest exactly when the drive vanishes and smallest exactly when the drive reaches its maximum amplitude. On the other hand the corrections to the even-ω amplitudes behave in the opposite manner, obtaining their largest contribution exactly when the drive is at its maximum amplitude and smallest contribution when the drive vanishes.

  1. Alexander Balatsky and Elihu Abrahams, Phys. Rev. B 45, 13125(R) (1992)
  2. Christopher Triola, Driss M. Badiane, Alexander V. Balatsky, and E. Rossi, Phys. Rev. Lett. 116, 257001 (2016)
  3. Christopher Triola and Alexander V. Balatsky, Phys. Rev. B 94, 094518 (2016)
  4. Annica M. Black-Schaffer and Alexander V. Balatsky, Phys. Rev. B 88, 104514 (2013)
  5. L. Komendová, A. V. Balatsky, and A. M. Black-Schaffer, Phys. Rev. B 92, 094517 (2015)
  6. A. V. Balatsky and Elihu Abrahams, Phys. Rev. Lett. 74, 1004 (1995)
  7. Zhoushen Huang, P. Wölfle, and A. V. Balatsky, Phys. Rev. B 92, 121404(R) (2015)
  8. Christopher Triola and Alexander V. Balatsky, arXiv:1704.04170

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