We study a topological phase transition between a normal insulator and a quantum spin Hall insulator in two-dimensional (2D) systems with time-reversal and twofold rotation symmetries. Unconventional Topological Phase Transition in Two-Dimensional Systems with Space-Time Inversion Symmetry This allows us to calculate the corresponding momentum- space entanglement entropy that surprisingly carries information about the real- space conformal field theory describing the defect lines that can be created on the gapped boundary. The Witten black-hole metric is a solution of this gauge theory and coincides with the Bures metric. The gauge connection for this model is associated to the Maxwell algebra, which takes into account the Lorentz symmetries related to the Dirac theory and the momentum- space magnetic translations connected to the magnetic perturbation. We first derive the Chern number from the cigar geometry and we then show that the quantum metric can be seen as a solution of two-dimensional non-Abelian BF theory in momentum space. The quantum Bures metric acquires a central role in our discussion and identifies a cigar geometry. The gap is induced by introducing a magnetic perturbation, such as an external Zeeman field or a ferromagnet on the surface. We focus, for simplicity, on the gapped boundary of three-dimensional topological insulators in class AII, which are described by a massive Dirac Hamiltonian and characterized by an half-integer Chern number. In this paper, we stress the importance of momentum- space geometry in the understanding of two-dimensional topological phases of matter. Momentum- space cigar geometry in topological phases
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