Robustness of the Kohn-Luttinger mechanism against symmetry breaking

TL;DR

This work probes the robustness of the Kohn–Luttinger mechanism against strong spatial-symmetry breaking in two-dimensional, spin–orbit-coupled systems. By introducing a symmetry-breaking field $\gamma$ in models with Ising and Rashba SOC, the authors show that $T_c$ remains finite for all $\gamma$, typically displaying a nonmonotonic dependence with a maximum near the Fermi energy and exponential suppression at large $\gamma$. The analysis highlights that ISing SOC can enhance or reduce $T_c$ depending on the DOS and channel mixing, while Rashba SOC introduces more pronounced interchannel mixing, yet KL-type superconductivity persists. A non-generic case demonstrates invariance of the KL mechanism under Fermi-surface shifts via a gauge transformation, underscoring the generality of the mechanism beyond perfectly rotationally symmetric settings. Overall, KL-type superconductivity can survive in a broad class of spin–orbit-coupled materials, suggesting experimental relevance in heterostructures with strong SOC and broken point-group symmetries.

Abstract

We investigate how strongly broken spatial symmetries affect the Kohn--Luttinger (KL) mechanism, in which superconductivity emerges purely from repulsive interactions. While the original KL argument assumes continuous rotational symmetry, real materials possess only discrete point-group symmetries, raising a central question: can sufficiently strong symmetry breaking suppress or eliminate KL superconductivity? Using controlled perturbation theory and explicit two-dimensional models with Ising and Rashba spin--orbit coupling (SOC), we find that KL superconductivity is broadly robust and exhibits qualitatively universal behavior across models: the transition temperature $T_c$ is nonmonotonic in the symmetry-breaking field, shows a pronounced maximum at scales of the order of the Fermi energy, and decays exponentially toward zero at asymptotically large fields. However, the physical mechanisms determining this suppression may differ between models. Overall, these results demonstrate that KL-type superconductivity can persist across a wide class of spin--orbit-coupled systems.

Figures (5)

• Figure 1: The four diagrams contributing to the pairing vertex at second order in the bare interaction.
• Figure 2: (a) Fermi surfaces split by Ising spin-orbit coupling Eq. \ref{['eq:H_SO_quad']} in a quadratic system or equivalently of the Hamiltonian Eq. \ref{['Eq:H0quartic']}. The spin-orbit coupling shifts the Fermi surfaces without distorting them. (b) The polarization bubble of a parabolic and quartic dispersion relation with full rotational symmetry normalized by the density of states, Eq. \ref{['eq:Pi2']} and Eq. \ref{['eq:Pi4']}, respectively. (c) The Fermi surfaces of the dispersion relation Eq. \ref{['eq:xi_ising_gen']} with $\gamma = {\epsilon}_0$ and $\kappa = 0.4 {\epsilon}_0$. (d) The polarization bubble Eq. \ref{['eq:Pi']} for the Fermi surfaces in panel (c). As can be seen it develops anisotropy and stronger momentum dependence as compared with $\Pi_4$ in panel (b).
• Figure 3: Results for the KL instability for the model with Ising SOC, Eq. \ref{['eq:xi_ising_gen']}, computed with fixed density. (a) $\log_{10}T_c/\Lambda$ normalized by the coupling constant $(\nu_0 U_0)^2$ vs. the symmetry breaking field ${\gamma}/{\epsilon}_0$ for three different values of $\kappa$ indicated in the legend and calculated for the dispersion relation in Eq. \ref{['eq:xi_ising_gen']}. Here $\nu_0$ is the density of states per spin at the Fermi level for ${\gamma}=0$. Note that in this calculation the density of electrons $n$ is held fixed and therefore the chemical potential is changing as a function of ${\gamma}$ and ${\kappa}$. (b) The normalized $T_c/T_c^0$ vs. ${\gamma}/{\epsilon}_0$. Here $T_c^0$ is the transition temperature at ${\gamma}=0$ given by Eq. \ref{['eq:Tc_sym']}. (c) The density of states $\nu$ normalized by $\nu_0$ vs. $\gamma/{\epsilon}_0$. (d) The difference $\Delta V$ from Eq. \ref{['eq:DeltaV']} normalized by its value at ${\gamma}=0$ vs. ${\gamma}/{\epsilon}_0$. (e) The same as panel (a), but here we tune $U_0$ such that the product of $U_0 \nu = 0.25$ is fixed for every value of ${\gamma}$ and ${\kappa}$. This last panel shows that the origin of the decay of $T_c$ at large ${\gamma}$ is the reduction in the density of states at the Fermi level.
• Figure 4: The solution of the gap equation \ref{['eq:gapeqtoytoy']} overlaid on top of the spin-up Fermi surface, for various values of ${\gamma}$ and ${\kappa}$ indicated in the panels. $\kappa$ and ${\gamma}$ are measured in units of ${\epsilon}_0 = \pi^2\beta n^2$, $k_x$ and $k_y$ are measured in units of $k_0 = \sqrt{\pi n}$, which are omitted for brevity.
• Figure 5: Results for the model with Rashba spin–orbit coupling, Eq. \ref{['eq:H_rashba']}, computed at fixed chemical potential. (a) Heat map of $\log_{10}(T_c/\Lambda)$, obtained from Eq. \ref{['eq:gap_eq_rashba']}, shown as a function of the Fermi energy $\epsilon_F/\epsilon_0$ and the symmetry–breaking field $\gamma/\epsilon_0$, for $U_0 m/2\pi = 0.15$. (b) The same quantity, $\log_{10}(T_c/\Lambda)$, computed instead at fixed $U_0 \nu = 0.15$, where $\nu$ is the density of states at Fermi level for a given ${\gamma}$ and ${\epsilon}_F$. This panel therefore isolates the influence of $\Delta V$ and channel mixing by removing variations due to changes in the density of states. (c) $T_c / T_c^{max}$ for three slices of panel (a), ${\epsilon}_F/{\epsilon}_0 = -0.9,\,-0.7$ and $-0.4$, as a function of ${\gamma}/{\epsilon}_0$. $T_c^{max}$ is the maximum value of $T_c$ in the plotted range. (d) The density of states averaged over the Fermi surfaces normalized by $\nu_0$ as a function of ${\gamma}/{\epsilon}_0$. $\nu_0$ is the average density of states per spin at the symmetric point ${\gamma} = 0$. (e) The measure of attraction strength ${\Delta} V / {\Delta} V_0$ vs. ${\gamma} / {\epsilon}_0$.