Interband tunneling near the merging transition of Dirac cones

Motivated by a recent experiment in a tunable graphene analog [Tarruell et al., Nature 483, 302 (2012)], we consider a generalization of the Landau-Zener problem to the case of a quadratic crossing between two bands in the vicinity of the merging transition of Dirac cones. The latter is described by the so-called universal Hamiltonian. In this framework, the interband tunneling problem depends on two dimensionless parameters: one measures the proximity to the merging transition and the other the adiabaticity of the motion. Under the influence of a constant force, the probability for a particle to tunnel from the lower to the upper band is computed numerically in the whole range of these two parameters and analytically in different limits using (i) the Stückelberg theory for two successive linear band crossings, (ii) diabatic perturbation theory, (iii) adiabatic perturbation theory, and (iv) a modified Stückelberg formula. We obtain a complete phase diagram and explain the presence of probability oscillations in terms of interferences between two poles in the complex time plane. We also compare our results to the above-mentioned experiment.

Figure 1

Energy spectrum in the gapless phase (${\Delta }_{*}<0$): energy $E=\pm \sqrt{{(p_x^2/\left(2m^{*}\right)+{\Delta }_{*})}^2+c_y^2{p}_y^2}$ as a function of momentum $p_x\sim t$ and $p_y\sim k$. The distance between the two Dirac cones is controlled by the merging gap ${\Delta }_{*}\propto d$. The perpendicular gap $c_y{p}_y\propto k$ controls how far the particle is from hitting the Dirac cones, which are located at $(t=\pm \sqrt{\left|d\right|}$, $k=0)$, directly at their tip. Black lines are lines of constant $k$.

Figure 2

Energy $E$ as a function of time $t$ when $d=-2<0$. (a) Diabats $E_{1,2}=\pm [t^2+d]$ are parabolas that intercept in real time. (b) Adiabats $E_{+,-}=\pm \sqrt{{[t^2+d]}^2+k^2}$ (with $k=0.5$) do not intercept in real time but do intercept in complex time. Solid (blue) line is for a plus sign; dashed (red) line is for a minus sign.

Figure 3

Contour plot of the transition probability $P$ computed with the Stückelberg approach as a function of the merging gap $d$ and the perpendicular gap $k$. The white region corresponds to $d\ge 0$, where the Stückelberg approach is not defined. (a) In the coherent case, interferences as a function of both $d$ and $k$ are clearly visible as well as the vanishing of $P$ in the $k\to 0$ and $k\to \infty$ limits. The probability varies between 0 and 1 as given by the color code (color steps corresponds to 0.1). (b) In the incoherent case, the oscillations are washed out and the maximum probability is $1/2$ instead of 1 in the coherent case (note the change of color scale for $P$).

Figure 4

Contour plot of the transition probability $P$ as a function of the merging gap $d$ and the perpendicular gap $k$. For small $k$, it is computed with the diabatic perturbation theory, Eq. (17); while for large $k$, it is computed using adiabatic perturbation theory, Eq. (30) (see Sec. 5). Interferences as a function of $d$ are clearly visible in the gapless phase ($d<0$). In the gapped phase ($d>0$), the probability vanishes exponentially. White regions corresponds to a probability exceeding 1. Indeed, close to $k\sim 1$, the two perturbative approaches break down. The color code is the same as in Fig. 3a.

Figure 5

Transition probability $P$ as a function of $d$ for fixed $k=0.1$ as computed in diabatic perturbation theory [see Eq. (17)]. For a negative argument (gapless phase, $d<0$) it shows Stückelberg oscillations and then decays exponentially for a positive argument (gapped phase, $d>0$). There is an inflexion point right at the merging (when the argument vanishes, $d=0$).

Figure 6

Poles $t_1$ and $t_4=-t_1^{*}$ in the complex time plane. Poles are represented at the merging transition ($d=0$ such that $\arg t_1=\pi /4=-\arg t_4$). The short horizontal (green) arrows indicate their motion when $d$ increases (gapped phase, $d>0$); the long vertical (red) arrows, that when it decreases (gapless phase, $d<0$). When the poles have a finite real part, there are oscillations in the probability, whereas a finite imaginary part implies an exponential decay of the probability. Note that there is a branch cut relating the two poles.

Figure 7

Integral $J\equiv \mathrm{Im}{\int }_0^{u_1}du\sqrt{1+{(u^2+D)}^2}$, giving $\mathrm{Im}{\varphi }_1=2k^{3/2}J(d/k)$, plotted as a function of $D$. The numerical calculation is shown by the solid (red) line and is compared to different analytical results: $\pi /\left(8\sqrt{\left|D\right|}\right)$, dashed (green) line; $\frac{\Gamma {(1/4)}^2}{12\sqrt{\pi }}+\frac{{\pi }^{3/2}}{\Gamma {(1/4)}^2}D$, dotted (magenta) line; and $2D^{3/2}/3+\ln D/\left(4\sqrt{D}\right)$, dot-dashed (blue) line. Interpolation formula (for positive $d$) $\frac{\Gamma {(1/4)}^2}{12\sqrt{\pi }}+\frac{2}{3}{D}^{3/2}$ is shown by the thin black line.

Figure 8

Transition probability $P$ as a function of $d$ (at fixed $k=2.5$) computed in adiabatic perturbation theory [solid (red) line]; see Eq. (30). Also shown, by the dashed (black) line, is the incoherent probability $P_{\mathrm{incoh}}$; see Eq. (34).

Figure 9

Contour plot of the modified Stückelberg transition probability $P$ as a function of the merging gap $d$ and the perpendicular gap $k$. (a) Coherent case [see Eq. (35)]: the probability is between 0 and 1 [the color code is the same as in Fig. 3a]. Note that the modified Stückelberg formula does not work in the ($d\ge 0,k<1$) region, as the nonadiabatic phase is not properly given by the Stokes phase. (b) Incoherent case [see Eq. (36)]: the probability is between 0 and 0.5 [the color code is the same as in Fig. 3b].

Figure 10

Contour plot of the numerically computed transition probability $P$ as a function of the merging gap $d$ and the perpendicular gap $k$. Oscillations are clearly visible in the gapless phase, whereas the probability is vanishingly small in the gapped phase. The vanishing of the probability in both the diabatic $k\ll 1$ and the adiabatic $k\gg 1$ limits is also visible. The color code is the same as in Fig. 3a.

Figure 11

Transition probability $P$ at the merging $d=0$ as a function of $k$. The numerically exact result is shown by the solid (blue) line. The diabatic perturbative result, Eq. (17), is shown by the dashed (red) line; the adiabatic perturbative result, Eq. (30), by the dotted (black) line. (a) $k$ between 0 and 2. (b) $k$ between 1.5 and 3 (note the change in the vertical scale by a factor of ${10}^3$): there is a tiny oscillation due to the interference between two poles.

Figure 12

Transition probability $P$ at fixed $d=-1$ as a function of $k$. The numerically exact result is shown by the solid (blue) line; the diabatic perturbative result, Eq. (17), by the short-dashed (red) line; the Stückelberg result, Eq. (12), by the dotted (black) line; the adiabatic perturbative result, Eq. (30), by the dot-dashed (green) line; and the modified Stückelberg formula, Eq. (35), by the long-dashed (magenta) line.

Figure 13

Transition probability $P$ at fixed $k$ as a function of $d$. The numerically exact result is shown by the solid (blue) line; the diabatic perturbative result, Eq. (17), by the short-dashed (red) line; the Stückelberg result, Eq. (12), by the dotted (black) line; the adiabatic perturbative result, Eq. (30), by the dot-dashed (green); and the modified Stückelberg formula, Eq. (35), by the long-dashed (magenta) line. (a) $k=0.1$ (the modified Stückelberg formula coïncides with the Stückelberg probability when $d<0$—it is not shown for clarity—and is not applicable when $d\ge 0$); (b) $k=1$; (c) $k=2.5$. Note the different probability scales in the three graphs.