Odd-frequency superfluidity from a particle-number-conserving perspective

We investigate odd-in-time (or odd-frequency) pairing [1] of fermions in equilibrium systems within the particle-number-conserving framework of Penrose, Onsager and Yang [2,3], where superfluid order is defined by macrosocopic eigenvalues of reduced density matrices [4]. We show that odd-frequency pair correlations are synonymous with even fermion-exchange symmetry in a time-dependent correlation function that generalises the two-body reduced density matrix [5]. Macroscopic even-under-fermion-exchange pairing is found to emerge from conventional Penrose-Onsager-Yang condensation in two-body or higher-order reduced density matrices through the symmetry-mixing properties of the Hamiltonian. We identify and characterise a transformer matrix responsible for producing macroscopic even fermion-exchange correlations that coexist with a conventional Cooper-pair condensate, while a generator matrix is shown to be responsible for creating macroscopic even fermion-exchange correlations from hidden orders such as a multi-particle condensate. The transformer scenario is illustrated using the spin-imbalanced Fermi superfluid [6] as an example. The generator scenario is demonstrated by the composite-boson condensate arising for itinerant electrons coupled to magnetic excitations [7]. Structural analysis of the transformer and generator matrices is shown to provide general conditions for odd-frequency pairing order to arise in a given system.

References
[1] J. Linder and A. V. Balatsky, Rev. Mod. Phys. 91, 045005 (2019).
[2] O. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).
[3] C. N. Yang, Rev. Mod. Phys. 34, 694 (1962).
[4] A. J. Leggett, Quantum Liquids, Oxford UniversityPress, Oxford, UK, 2006.
[5] K. Thompson, U. Zülicke, J. Schmalian, M. Governale, and J. Brand, Phys. Rev. Res. 6, 033165 (2024).
[6] D. Chakraborty and A. M. Black-Schaffer, Phys. Rev. B 106, 024511 (2022).
[7] E. Abrahams, A. Balatsky, D. J. Scalapino, J. R. Schrieffer, Phys. Rev. B 52, 1271 (1995).