Topological quantum phase transitions: The vacua at b0 ≠ 0 and b > M have Fermi surfaces. At b2 > b20 + M2, these Fermi surfaces have nonzero global topological charges N3 = +1 and N3 = −1. At the quantum phase transition occurring on the line b0 = 0, b > M (thick horizontal line) the Fermi surfaces shrink to the Fermi points with nonzero N3. At M2 < b2 < b20 + M2 the global topology of the Fermi surfaces is trivial, N3 = 0. At the quantum phase transition occurring on the line b = M (thick vertical line), the Fermi surfaces shrink to the points; and since their global topology is trivial the zeroes disappear at b < M where the vacuum is fully gapped. The quantum phase transition between the Fermi surfaces with and without topological charge N3 occurs at b2 = b20 + M2 (dashed line). At this transition, the Fermi surfaces touch each other, and their topological charges annihilate each other.
What we have assumed here is that the Fermi point in the Standard Model above the electroweak energy scale is marginal, i.e. its total topological charge is N3 = 0. Since the topology does not protect such a point, everything depends on symmetry, which is more subtle. In principle, one may expect that the vacuum is always fully gapped. This is supported by the Monte-Carlo simulations which suggest that in the Standard Model there is no second-order phase transition at finite temperature, instead one has either the first-order electroweak transition or crossover depending on the ratio of masses of the Higgs and gauge bosons. This would actually mean that the fermions are always massive.
Such scenario does not contradict to the momentum-space topology, only if the total topological charge N3 is zero. However, from the point of view of the momentum-space topology there is another scheme of the description of the Standard Model. Let us assume that the Standard Model follows from the GUT with SO(10) group. Here, the 16 Standard Model fermions form at high energy the 16-plet of the SO(10) group. All the particles of this multiplet are left-handed fermions. These are: four left-handed SU(2) doublets (neutrino-electron and 3 doublets of quarks) + eight left SU(2) singlets of anti-particles (antineutrino, positron and 6 anti-quarks). The total topological charge of the Fermi point at p = 0 is N3 = −16, and thus such a vacuum is topologically stable and is protected against the mass of fermions. This topological protection works even if the SU (2) × U (1) symmetry is violated perturbatively, say, due to the mixing of different species of the 16-plet. Mixing of left leptonic doublet with left singlets (antineutrino and positron) violates SU(2) × U(1) symmetry, but this does not lead to annihilation of Fermi points and mass formation since the topological charge N3 is conserved.
What this means in a nutshell is that if the total topological charge of the Fermi surfaces is non-zero, the gap cannot appear perturbatively. It can only arise due to the crucial reconstruction of the fermionic spectrum with effective doubling of fermions. In the same manner, in the SO(10) GUT model the mass generation can only occur non-perturbatively. The mixing of the left and right fermions requires the introduction of the right fermions, and thus the effective doubling of the number of fermions. The corresponding Gor’kov’s Green’s function in this case will be the (16 × 2) × (16 × 2) matrix. The nullification of the topological charge N3 = −16 occurs exactly in the same manner, as in superconductors. In the extended (Gor’kov) Green’s function formalism appropriate below the transition, the topological charge of the original Fermi point is annihilated by the opposite charge N3 = +16 of the Fermi point of “holes” (right-handed particles).
This demonstrates that the mechanism of generation of mass of fermions essentially depends on the momentum space topology. If the Standard Model originates from the SO(10) group, the vacuum belongs to the universality class with the topologically non-trivial chiral Fermi point (i.e. with N3 ≠ 0), and the smooth crossover to the fully-gapped vacuum is impossible. On the other hand, if the Standard Model originates from the left-right symmetric Pati–Salam group such as SU(2)L × SU(2)R × SU(4), and its vacuum has the topologically trivial (marginal) Fermi point with N3 = 0, the smooth crossover to the fully-gapped vacuum is possible.