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"

Abstract

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.

Figures (3)

• Figure 1: Interaction amplitude $\gamma_c$ in the particle-particle spin-singlet channel, denoted by a dotted line. The vertices shown as filled circles carry a Pauli matrix $\sigma_1$.
• Figure 2: The Cooper-screened particle-particle interaction amplitude $\Gamma_c(q)$,screening_footnote denoted by a double dotted line.
• Figure 3: One-loop contributions in the particle-particle channel to the spin susceptibility. The external vertex denoted by an open triangle carries a Pauli matrix $\sigma_3$.