Fig. 2: Symmetry-breaking gapless Dirac cones and half-quantum mirror Hall effect.
a Schematic diagram of a film exhibiting mirror symmetry. \({H}_{2D}(z)={H}_{0}+{\sum }_{{{\boldsymbol{\delta }}}=\hat{{{\bf{x}}}},\hat{{{\bf{y}}}}}({H}_{{{\boldsymbol{\delta }}}}{e}^{i{{\boldsymbol{\delta }}}\cdot {{\bf{k}}}}+{{\rm{h.c.}}})\) represents the Hamiltonian of each layer, while Tz,z+1 and Tz+1,z represent the hopping terms between adjacent layers in the z-direction. The gray plane represents the mirror plane. b The band structure of topological insulator thin film (gray lines) is based on a 3D tight-binding model calculation with open boundary conditions parallel to the mirror plane and periodic boundary conditions along the remaining two directions. The red circles represent the energy spectrum by diagonalizing the Hamiltonian describing half thickness of the film (\({\hat{{{\mathcal{H}}}}}_{\chi }^{0}\)) with a time-reversal symmetry breaking term \({\hat{{{\mathcal{V}}}}}_{\chi }\) on the bottom layer. The blue dashed lines are from the effective Hamiltonian [Eq. (7)]. c The mirror Hall conductance as a function of chemical potential μ calculated by using Eq. (8) based on the 3D tight-binding model for TI and the model for the surface states in Eq. (7). The blue line marked with triangles represents the contributions from the lowest-energy four gapless bands as determined by the tight-binding model. d Schematic diagram of two separated classes of the gapless Dirac cones with even and odd parity. Black arrows represent the pseudo-spin texture (i.e., \(\hat{{{\bf{d}}}}=\langle \widetilde{{{\boldsymbol{\sigma }}}}\rangle\)) of two gapless Dirac cones with opposing mirror eigenvalues iχ. The time-reversal symmetry \(\hat{{{\mathcal{T}}}}\) transforms one Dirac cone into another with the opposite mirror eigenvalue. For each mirror sector, we can define a parity operator \(\widetilde{M}\), which transforms (kx, ky) → (−kx, ky) and is represented by \(i{\widetilde{\sigma }}_{y}\) in the 2 × 2 subspace. This parity symmetry is disrupted in high-energy states (indicated by light colors) but is preserved in low-energy states (indicated by dark colors).
