Fig. 9: The loop of winding number and additional phase diagrams.

a The loop of \(\det (q(k))\) winds around the origin (black dot) in the topological region for a₁ = 2, m_z = 2.4 and m_o = 1.8 as the wavenumber k runs through the reduced Brillouin zone. b The loop of \(\det (q(k))\) passes though the origin (black dot) at at the critical point for a₁ = 2, m_z = 2.4 and \({m}_{o}=\sqrt{1.76}\). Momentum-space winding number ν in the m_o-m_z plane for (c) a₁ = 2, t  = 1, and t_s = 0.95; d t = t_s = 1 with (a₁, a₂) = (3, 7); (e) t = t_s = 1 with (a₁, a₂) = (2, 5); and (f) t = t_s = 1 with (a₁, a₂) = (3, 5). Red dashed curves in (c–f) are phase boundaries solved from the renormalized Zeeman strength.