Mass and Chirality Inversion of a Dirac Cone Pair in Stückelberg Interferometry

We show that a Stückelberg interferometer made of two massive Dirac cones can reveal information on band eigenstates such as the chirality and mass sign of the cones. For a given spectrum with two gapped cones, we propose several low-energy Hamiltonians differing by their eigenstates properties. The corresponding interband transition probability is affected by such differences in its interference fringes being shifted by a new phase of geometrical origin. This phase can be a useful bulk probe for topological band structures realized with artificial crystals.

Figure 1

Stückelberg interferometer made of a pair of one-dimensional Dirac cones described by Hamiltonian (1) $H(p_x,p_y=0)$ with mass function $M_z(p_x)=M[\cos \beta -(p_x/\sqrt{2m{\mathrm{\Delta }}_{*}})\sin \beta ]$ tunable by the flux parameter $\beta \in [0,\pi /2]$. (a) $\beta =0$, $M_z(D)=M_z(D^{\prime })$ (equal mass); $\beta =\pi /2$, $M_z(D)=-M_z(D^{\prime })$ (opposite mass); (b) $\beta =\pi /4$, $M_z(D)\ne M_z(D^{\prime })=0$ (single massless cone); (c) interband transition probability $P_f$, as a function of $\beta$ and the intercone distance $d=2\sqrt{2m{\mathrm{\Delta }}_{*}}$, obtained with the replacement $p_x\to F_{x}t$ (see text) and $M=0.8(\hslash F_x{)}^{2/3}/(2m{)}^{1/3}$.

Figure 2

Momentum space trajectories driven by the force $\overrightarrow{F}=(F_x,F_y)$ in the vicinity of two Dirac points $D$, $D^{\prime }$: (a) diagonal trajectory [$\overrightarrow{p}\to (F_{x}t,F_{y}t)$], (b) parallel trajectory [$\overrightarrow{p}\to (F_{x}t,p_y=\text{const})$].

Figure 3

Hamiltonian (1) with $M_z(\overrightarrow{p})=M$ is written as $H(\overrightarrow{p})=\overrightarrow{B}(\overrightarrow{p})·\overrightarrow{\widehat{\sigma }}$ where the effective magnetic field $\overrightarrow{B}=E_{+}(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$ defines $\varphi$ and $\theta$. (a) Vector plot of $\varphi (\overrightarrow{p})$ in the vicinity of a Dirac cone pair seen as a vortex and an antivortex indicated by dots. Parallel (black) and diagonal (red) trajectories are represented. (b) Berry connection $\dot{\varphi }(t)\mathbf{(}1-\cos \theta (t)\mathbf{)}$ (in the south pole gauge [39]) along the parallel (black) and diagonal trajectories (red) as a function of time $t/t_0$ in the Stückelberg limit $t_0\gg t_{\mathrm{LZ}}$. Two values of the mass $M=0$ (higher peak) and $M\ne 0$ are considered. The geometrical phase ${\phi }_g$ is obtained as a line integral of the Berry connection between $-t_0$ and $+t_0$ [thick lines in (a)].

Figure 4

Transition probability $P_f$ as a function of the time interval $2t_0=2\sqrt{{\mathrm{\Delta }}_{*}}$ between the two cones. Comparison between the Stückelberg theory [Eq. (5), full curve] and numerical solution (dots) for the parameters: $c_y{F}_y=0.01$, $\mathrm{\Delta }=0.3$.