Parallel magnetic field driven quantum phase transition in a thin topological insulator film

It is well known that helical surface states of a three-dimensional topological insulator (TI) do not respond to a static in-plane magnetic field. Formally this occurs because the in-plane magnetic field appears as a vector potential in the Dirac Hamiltonian of the surface states and can thus be removed by a gauge transformation of the surface electron wave functions. Here we show that when the top and bottom surfaces of a thin film of TI are hybridized and the Fermi level is in the hybridization gap, a nonzero diamagnetic response appears. Moreover, a quantum phase transition occurs at a finite critical value of the parallel field from an insulator with a diamagnetic response to a semimetal with a vanishing response to the parallel field.

Figure 1

$k_x=0$ section of the surface band dispersion, plotted at different values of the tunneling matrix element $\Delta$ and fixed nonzero magnetic field. (a) $\Delta =0$ band structure: two Dirac cones shifted by $2{\kappa }_B$ with respect to each other along the $y$ axis. (b) Band structure for $\Delta <{\epsilon }_B$, i.e., in the semimetal phase. Hybridization gaps open only at the two $k_y=0$ crossing points. (c) Critical point $\Delta ={\epsilon }_B$. The dispersion of the two bands that touch is quadratic along the $y$ direction, but remains linear in the $x$ direction. (d) Band structure in the insulating phase, $\Delta >{\epsilon }_B$.