Abstract
We show that vertex corrections to Migdal's theorem in general induce an odd-frequency spin-triplet superconducting order parameter, which coexists with its more commonly known even-frequency spin-singlet counterpart. Fully self-consistent vertex-corrected Eliashberg theory calculations for a two dimensional cuprate model, isotropically coupled to an Einstein phonon, confirm that both superconducting gaps are finite over a wide range of temperatures. The subordinate $d$-wave odd-frequency superconducting gap is found to be one order of magnitude smaller than the primary even-frequency $d$-wave gap. Our study provides a direct proof of concept for a previously unknown generation mechanism of odd-frequency superconductivity as well as for the generic coexistence of both superconducting states in bulk materials.
Highlights
Odd-frequency superconductivity describes a Cooper-pair condensate in which the quantum mechanical wave function is odd under exchange of relative time between two electrons of a Cooper pair
As first proposed by Berezinskii [1], this ordered state is compatible with the Pauli principle as long as either the momentum-space parity is even and the electrons are of spin-triplet type, or the parity is odd and one considers spin-singlet electron pairs
By deriving self-consistent and vertex-corrected Eliashberg equations, we prove that a finite odd-frequency spin-triplet order parameter can theoretically always coexist along with the “standard” even-frequency spin-singlet superconducting gap
Summary
Odd-frequency superconductivity describes a Cooper-pair condensate in which the quantum mechanical wave function is odd under exchange of relative time between two electrons of a Cooper pair. It has been argued that electron-phonon coupling can theoretically lead to oddfrequency superconductivity, provided that the interaction has a sufficiently strong odd momentum component as a result of coupling between electrons and acoustic phonons [4]. Such a scenario was considered unlikely if no spin dependence enters into the interaction [5]. By deriving self-consistent and vertex-corrected Eliashberg equations, we prove that a finite odd-frequency spin-triplet order parameter can theoretically always coexist along with the “standard” even-frequency spin-singlet superconducting gap.
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