Reply to “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””

We reply to the Comment by D. Belitz and T. R. Kirkpatrick on our article “Instability of the ferromagnetic quantum critical point and symmetry of the ferromagnetic ground state in 2D and 3D electron gases with arbitrary spin-orbit splitting.” They correctly point out that nonanalyticities in the thermodynamic potential Ω of 2D and 3D Fermi liquids that originate from the backscattering processes are suppressed by the square of the Cooper logarithm. This suppression is not essential in 2D Fermi liquids where the nonanalyticity is cubic and, therefore, is much more relevant than the quartic Ginzburg-Landau term. However, in 3D Fermi liquids, the Cooper renormalization renders the marginally relevant quartic nonanalyticity marginally irrelevant. We argue that the leading Ω a B⁴ ln(E_F/B) form of the 3D nonanalyticity in the thermodynamic potential Ω as a function of the Zeeman field B survives due to a contribution coming from the arbitrary-angle scattering that is not subject to the Cooper renormalization. In perturbation theory, such a correction first emerges in third order in the interaction, as was identified by Maslov and Chubukov [Phys. Rev. B 79, 075112 (2009)]. However, the sign of this third-order correction is positive, which contradicts the physical argument on the sign of nonanalytic corrections presented by D. Belitz and T. R. Kirkpatrick in their Comment. Maslov and Chubukov argued that the overall sign of the nonanalytic correction is not universal and strongly depends on the angular structure of the Landau scattering amplitude in the regime where the interaction is no longer weak; this is also the regime where the ferromagnetic quantum phase transition is expected. They also showed that the nonanalyticity remains marginally relevant in the vicinity of the 3D ferromagnetic quantum critical point, Ω a — B⁴ ln ln(E_F/B). Though this nonanalyticity can formally destabilize the T — 0 ferromagnetic quantum critical point, it becomes irrelevant at any experimentally achievable temperature.