Comment on "Instability of the ferromagnetic quantum critical point and symmetry of the ferromagnetic ground state in two-dimensional and three-dimensional electron gases with arbitrary spin-orbit splitting"

Metallic quantum ferromagnets in the absence of quenched disorder are known to generically undergo a first-order quantum phase transition, avoiding the quantum critical point that had originally been expected. This is due to soft modes in the underlying Fermi liquid that lead to long-ranged correlations. These correlations in turn yield a nonanalytic dependence of the free energy on the magnetization even at a mean-field level that results in a fluctuation-induced first-order transition. Kirkpatrick and Belitz [Phys. Rev. Lett. {\bf 124}, 147201 (2020)] have pointed out that one notable exception are non-centrosymmetric metals with a strong spin-orbit interaction. In such materials the spin-orbit interaction gives the relevant soft modes a mass, which inhibits the mechanism leading to a first-order transition. Miserev, Loss, and Klinovaja [Phys. Rev. B {\bf 106}, 134417 (2022)] have claimed that this conclusion does not hold if electron-electron interactions in the particle-particle channel, or 2$\kF$ scattering processes, are considered. They concluded that this interaction channel leads to soft modes that are not rendered massive by the spin-orbit interaction and again lead to a first-order quantum phase transition. In this Comment we show that this conclusion is not correct in three-dimensional magnets if the screening of the interaction is properly taken into account.

💡 Research Summary

The paper is a comment on the recent work by Miserev, Loss, and Klinovaja (Phys. Rev. B 106, 134417, 2022), which claimed that in non‑centrosymmetric metals with strong spin‑orbit coupling (SOC) the particle‑particle (Cooper) and 2k_F interaction channels generate soft modes that remain massless despite SOC. According to their argument, these modes would re‑introduce a non‑analytic term in the free energy, thereby restoring the fluctuation‑induced first‑order ferromagnetic quantum phase transition (QPT) that is otherwise avoided in clean itinerant ferromagnets.

The authors of the comment first recall the well‑established mechanism for a first‑order transition in clean Fermi liquids: soft particle‑hole excitations produce long‑range correlations, leading to a non‑analytic contribution ∝ M⁴ ln M (or equivalently a term ∝ h² ln h in the spin susceptibility). In non‑centrosymmetric metals, strong SOC gives these soft modes a mass, cutting off the singularity and allowing a genuine quantum critical point (QCP) to exist, as shown in Kirkpatrick and Belitz, Phys. Rev. Lett. 124, 147201 (2020).

Miserev et al. argued that the particle‑particle (Cooper) channel and the 2k_F particle‑hole channel are not affected by SOC, because the SOC does not open a gap in the corresponding propagators. They therefore concluded that the non‑analyticity survives and the QCP is destroyed even in the presence of strong SOC.

The comment refutes this conclusion for three‑dimensional (3D) systems by emphasizing the role of Cooper‑pair screening (often called “Cooper screening”). The bare interaction in the particle‑particle singlet channel is taken as a constant γ_c. Summing the ladder diagrams of Fig. 2 yields the screened amplitude

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