Odd-frequency pairing of Bogoliubov quasiparticles in superconductor junction

TL;DR

The paper develops a McMillan Green's-function framework to study Josephson junctions formed by Bogoliubov Fermi surfaces, where bogolons exhibit bulk odd-frequency pairing due to self-energy effects in a non-Hermitian effective Hamiltonian. At interfaces, translational symmetry breaking induces an even-frequency p-wave component, while the bulk remains odd-frequency, leading to a π-junction–like current-phase relation for bogolons and distinctive LDOS behavior. A detailed comparison with conventional s-wave junctions reveals opposite frequency dependencies of the interface-induced versus bulk pairing, different ABS structures, and barrier-strength–dependent spectral and transport responses. The authors also formulate a quasiclassical Green's function to address slowly varying spatial components, linking full Green's-function results to experimentally accessible quantities and outlining implications for BFS materials such as Fe(Se,S). Overall, the work provides a foundational framework for understanding surface and interface Cooper pairing in BFS-hosting superconductors.

Abstract

We study a superconductor Josephson junction with a Bogoliubov Fermi surface, employing McMillan's Green's function technique. The low-energy degrees of freedom are described by spinless fermions (bogolons), where the characteristic feature appears as an odd-frequency pair potential. The differential equation of the Green's function is reduced to the eigenvalue problem of the non-Hermitian effective Hamiltonian. The physical quantities such as the density of states and pair amplitude are then extracted from the obtained Green's function. We find that the zero energy local density of states at the interface decreases as the relative phase of the Josephson junction increases. This decrease is accompanied by the generation of an even-frequency pair amplitude near the interface. We also clarify that the $π$-junction-like current phase relation is realized in terms of bogolons. In contrast to conventional $s$-wave superconductor junctions, where even-frequency pairs dominate in the bulk and odd-frequency pairs are generated near the interface, our findings illuminate the distinct behaviors of junctions with Bogoliubov Fermi surfaces. We further explore spatial dependences of these physical quantities systematically using quasiclassical Green's functions.

Figures (7)

• Figure 1: Schematic figure of one-dimensional SC junction. We consider two systems: (a) Bogolon junction and (b) $s$-wave SC junction.
• Figure 2: Four types of eigenfunctions. The red ($+$) arrows and blue ($-$) arrows indicate the particle- and the hole-like plane waves, respectively. The dashed lines represent the incident waves. The directions of arrows indicate the group velocity. The sign in the superscript of $\Psi$, $a$, $b$, $c$, and $d$ indicates particle- $(+)$ or hole-like particle $(-)$ incident.
• Figure 3: Physical quantities at $x = 0$ for $s$-wave SC junction without barrier potential. (a) The $s$-wave component of the pair amplitude, (b) the $p$-wave component of the pair amplitude, (c) the LDOS normalized by its value in normal state ($\Delta_{\mathrm{L}} = 0$), (d) the LDOS in the complex plane for $\theta = 2\pi/3$, which is defined by the extension of $\omega$ to the complex energy plane $z$, i.e., $D(\omega, x=0) \to D(z, x=0)$, (e) $j(\mathrm{i}\omega_{n})=j(\mathrm{i}\omega_{n},x=0)$, and (f) the Josephson current $J(x=0)$. Figure legend of (c) is the same as that of (a), and the legend of (e) is the same as that of (b).
• Figure 4: Frequency-dependence of physical quantities of $s$-wave SC at $x = 0$ for several values of $Z$. (a) The $s$-wave component of the pair amplitude (even-frequency), (b) the $p$-wave component of the pair amplitude (odd-frequency), (c) the LDOS normalized by its value in normal state ($\Delta_{\mathrm{L}} = 0$), and (d) $j(\mathrm{i}\omega_{n})=j(\mathrm{i}\omega_{n},x=0)$. The relative phase is chosen as $\theta = 2\pi/3$.
• Figure 5: Physical quantities at $x = 0$ for bogolon junction without barrier potential. (a) The $s$-wave component of the pair amplitude, (b) the $p$-wave component of the pair amplitude, (c) the LDOS of bogolons normalized by its value of clean limit ($\Gamma_{1\mathrm{L}} = \Gamma_{2\mathrm{L}} = 0$), (d) the LDOS of bogolons in the complex energy plane for $\theta = 2\pi/3$, which is defined by the extension of $\omega$ to the complex plane $z$, i.e., $D(\omega, x=0) \to D(z, x=0)$, (e) $j(\mathrm{i}\omega_{n}) = j(\mathrm{i}\omega_{n}, x = 0)$, and (f) $J(x=0) = (1/\beta)\sum_n j(\mathrm{i}\omega_n)$. Figure legends of (b)-(d) are the same as that of (a). We set $|\Gamma_{2\mathrm{L}}|/\Gamma_{1\mathrm{L}}=0.9$ in (a)--(e).