[Paper Review] Obstructions to odd-frequency superconductivity in Eliashberg theory

Edwin Langmann, Christian Hainzl+2 more|arXiv (Cornell University)|Jul 5, 2022

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper establishes a no-go theorem in Eliashberg theory proving that standard one-band superconductors with electron-phonon or spin-fluctuation-mediated pairing cannot host odd-frequency superconductivity due to strong self-energy renormalizations that suppress the gap equation. The results explain the rarity of odd-f SC in conventional materials and identify specific interaction symmetries required to enable such states.

ABSTRACT

We present a necessary condition for odd-frequency (odd-f) superconductivity (SC) to occur in a large class of materials described by Eliashberg theory. We use this condition to prove a no-go theorem ruling out the occurrence of odd-f SC in standard one-band superconductors with pairing interactions mediated by phonon exchange. We also present a corresponding no-go theorem for superconductors with interactions mediated by spin-fluctuations. Our results explain why odd-f SC is rare in conventional materials, and they open up the possibility for a search for materials with interactions designed so as to allow for odd-f SC.

Motivation & Objective

• To identify fundamental obstructions preventing odd-frequency superconductivity in conventional superconductors described by Eliashberg theory.
• To explain why odd-frequency pairing is rare in nature despite being theoretically allowed.
• To derive a no-go theorem for electron-phonon and spin-fluctuation-mediated pairing in one-band models.
• To establish conditions under which odd-frequency superconductivity could be stabilized in materials with engineered interactions.
• To provide a design principle for materials with interaction potentials that allow non-trivial odd-f superconducting solutions.

Proposed method

• Formulates the linearized Eliashberg equations for the superconducting critical temperature $T_c$ in the presence of frequency- and momentum-dependent interactions.
• Derives a necessary condition for odd-frequency pairing by analyzing the gap equation and self-energy corrections in the Matsubara frequency representation.
• Applies the SP∗T∗-rule to classify pairing symmetries and identify constraints on interaction kernels $V(i\omega_n, i\omega_m)$ that allow odd-f solutions.
• Uses numerical diagonalization of infinite matrices to compute the effective function $g(X)$ that determines $T_c$, with convergence checks across different $N$.
• Analyzes model potentials such as the inverted $\gamma$-model ($f(x) = 1/x^\gamma$) and screened Coulomb-like interactions to derive analytic and numeric solutions.
• Extends results from simplified to general Eliashberg equations, confirming the no-go theorems hold without local approximations.

Experimental results

Research questions

• RQ1 Why is odd-frequency superconductivity so rare in conventional superconductors despite being theoretically possible?
• RQ2 What fundamental obstruction prevents odd-frequency pairing in electron-phonon coupled one-band systems within Eliashberg theory?
• RQ3 Can a no-go theorem be rigorously derived for both phonon- and spin-fluctuation-mediated pairing in standard superconductors?
• RQ4 What specific symmetry or structure in the interaction kernel is required to allow odd-frequency superconducting solutions?
• RQ5 How does the critical temperature $T_c$ scale with coupling strength $\lambda$ in odd-frequency pairing for different interaction models?

Key findings

• A no-go theorem is proven for odd-frequency superconductivity in one-band superconductors with electron-phonon coupling, showing that strong self-energy renormalization prevents non-trivial solutions.
• For the inverted $\gamma$-model with $f(x) = 1/x^\gamma$, the critical temperature scales as $T_c \approx K\Omega\lambda^{1/\gamma}$, with $K \approx 0.1822$ for $\gamma = 0.5$, $0.1911$ for $\gamma = 1$, and $0.1829$ for $\gamma = 2$.
• Numerical results suggest $\lambda_c = 0$ for the inverted $\gamma$-model, implying odd-f SC is possible even for arbitrarily small coupling, while $\lambda_c > 0$ for finite $V$ at $|\omega_n - \omega_m| \to 0$.
• The $T_c$-curve for odd-f SC with repulsive interactions shows monotonic growth with $\lambda$, similar to conventional even-f SC, but with a model-dependent $g(X)$ function instead of $g(X) = -\log X$.
• For large $\lambda$, the $T_c$ curves for inverted phonon and screened Coulomb interactions are well approximated by $T_c \approx 0.18\Omega\lambda^{1/2}$ and $T_c \approx 0.19\Omega\lambda$, respectively.
• The study confirms that the no-go theorems hold in the general Eliashberg formalism without local approximations, validating the results across full frequency and momentum dependence.