Fig. 1: Schematic of separation of a time-reversal invariant system with mirror symmetry into two symmetry broken subsystems with mirror parity.
(Left panel) The time-reversal invariant system is represented by the cyan color, with the wave functions of the Dirac states on the top and bottom surfaces denoted by \(\left\vert {\Psi }_{t/b}\right\rangle\), respectively. The presence of mirror symmetry connects these two states: \(i{U}_{{M}_{z}}\left\vert {\Psi }_{t}(z)\right\rangle=\left\vert {\Psi }_{b}(-z)\right\rangle\). The cyan-colored circles with arrows indicate the helical mirror current centered at the mirror plane. (Middle panel) The system characterized by definite mirror parity, features a single Dirac cone within the Brillouin zone. The wavefunction of surface states exhibits either even (χ = + 1, red color) or odd (χ = − 1, yellow color) parity with respect to the z-axis, which is represented by \(\left\vert {\Psi }_{e/o}(z)\right\rangle=\frac{1}{\sqrt{2}}(\left\vert {\Psi }_{t}(z)\right\rangle \pm \left\vert {\Psi }_{b}(z)\right\rangle )\), respectively. (Right panel) After transitioning to the eigenbasis of mirror operator \({\hat{\phi }}_{z,\chi }={\hat{\psi }}_{z}\mp i\chi {U}_{{M}_{z}}{\hat{\psi }}_{-z}\), each mirror eigensector is equivalent to a topological insulator thin film with a single surface subjected to a symmetry-breaking term. The wavefunction of the surface state, denoted by \(\left\vert {\Phi }_{\chi }(z)\right\rangle\), is distributed on a single surface within the new coordinate framework. The red and yellow circles with arrows represent the counter-propagating mirror current Jχ=± associated with the half-quantum mirror Hall effect. The arrows pointing along the z-direction indicate the effective magnetization on the surface in the mirror eigenbasis.
