Self-consistent tight-binding description of Dirac points moving and merging in two-dimensional optical lattices

We present an accurate ab initio tight-binding model, capable of describing the dynamics of Dirac points in tunable honeycomb optical lattices following a recent experimental realization [Tarruell et al., Nature (London) 483, 302 (2012)]. Our scheme is based on first-principle maximally localized Wannier functions for composite bands. The tunneling coefficients are calculated for different lattice configurations, and the spectrum properties are well reproduced with high accuracy. In particular, we show which tight-binding description is needed in order to accurately reproduce the position of Dirac points and the dispersion law close to their merging, for different laser intensities.

Figure 1

Bravais lattice associated to the potential in Eq. (8) for the stretched-honeycomb configuration. Black and white circles refers to minima of type $A$ and $B$, respectively. The elementary cell is highlighted in gray. The various diagonal and off-diagonal tunneling coefficients of our tight-binding expansion are indicated for the site of type $A$ in the central cell.

Figure 2

(a) Stretched-honeycomb potential (8) for $V_{\overline{X}}=5$, $V_X=0.28$, and $V_Y=1.8$. Hot and cold colors denote high and low values of the potential, respectively. The unit cell is indicated by solid (black) lines. (b) Structure of the calculated MLWF for sublattice $A$ for the same potential setup as in (a) (see text).

Figure 3

Behavior of the various tunneling coefficients as function of $V_{\overline{X}}$. Panel (a) covers the experimental regime, $V_X=0.28$, $V_Y=1.8$, while in (b) we consider a proper tight-binding regime, $V_X=0.56$, $V_Y=3.6$.

Figure 4

Cut of the exact energy bands (black solid line) compared to the two tight-binding approximations with just $t_0$, $t_1$, and $t_2$ (red dotted line), and with all the coefficients in Fig. 1 (green dotted-dashed line). Panels (a) and (b) respectively show cuts along $k_x$ ($k_y=0$) and $k_y$ ($k_x=0$), for $V_X=0.28$, $V_Y=1.8$, and $V_{\overline{X}}=5$; (c) and (d) refer to $V_X=0.56$, $V_Y=3.6$, and $V_{\overline{X}}=8.5$.

Figure 5

Position of the Dirac points along the $k_y$ axis as a function of $V_{\overline{X}}$ for (a) the parameter regime of Tarruell et al. [8], and (b) the tight-binding regime discussed in the text. The exact positions (circles) extracted from the Bloch spectrum are compared with the predictions of Eqs. (28) and (29).

Figure 6

Unit cell in quasimomentum space, with the location of the possible merging of the Dirac points. Equivalent points (connected by a reciprocal space vector $\mathit{G}$) are depicted with the same color. Given the actual values of the tunneling coefficients, only the points at $k_x=0,k_y=\pm 1$ can be realized (larger red dots).

Figure 7

Cuts of the energy bands around the merging point ${\mathit{k}}_M=(0,1)$, for the tight-binding regime discussed in the text ($V_X=0.56$, $V_Y=3.6$). The exact Bloch bands (red solid lines) are compared to the approximate expressions in Eq. (38), as a function of $k_y$ (at $k_x=0$) [(a),(b),(c)], and of $k_x$ (at $k_y=1$) [(d),(e),(f)]. Each column corresponds to a different value of $V_{\overline{X}}$: (a),(d) $V_{\overline{X}}=8$, (b),(e) $V_{\overline{X}}=6.94$ (merging point), and (c),(f) $V_{\overline{X}}=6.54$. Note that the cut along $k_x$ in (d) does not cross the Dirac point, as the latter is located at $k_y\simeq 0.68$; this is the reason why Eq. (38) provides a poorer approximation in this case.

Figure 8

Cuts of the energy bands around the merging point ${\mathit{k}}_M=(0,1)$ for different values of the parity breaking angle $\theta$. The exact Bloch bands (dots) are compared to the full tight-binding model (solid line), as a function of $k_y$ (at $k_x=0$) (a), and of $k_x$ (at $k_y=1$) (b). The picture refers to the tight-binding regime $V_X=0.56$, $V_Y=3.6$, and $V_{\overline{X}}=6.54$.

Figure 9

Same as Fig. 8, but for a case with two Dirac points located at $\mathit{k}=(0,0.75)$ ($V_X=0.56$, $V_Y=3.6$, and $V_{\overline{X}}=8.05$).

Figure 10

Dimer ($V_{\overline{X}}=1$) and 1D-chain ($V_{\overline{X}}=8$) limits for fixed $V_X=0.28$ and $V_Y=1.8$. Panels (a) and (b) show the respective potential structures, and (c) and (d) show associated MLWFs. Color code as in Fig. 2.

Figure 11

Spread of the MLWFs as a function of $V_{\overline{X}}$, in the regime of the experiment [8] ($V_X=0.28$ and $V_Y=1.8$).

Figure 12

Evolution of various tunneling coefficients as a function of $V_{\overline{X}}$, covering the whole range from the dimer ($V_{\overline{X}}\approx 1$) to the 1D-chain ($V_{\overline{X}}\approx 1$) limits in the regime of the experiment [8] ($V_X=0.28$ and $V_Y=1.8$).

Figure 13

Cut of the exact energy bands along $k_y$ ($k_x=0$) compared to the two tight-binding approximations discussed in the text. Panels (a) and (c) represent the dimer limits in the experimental and tight-binding regimes, respectively. Panels (b) and (d) are analogous in the 1D-chain limit.

Figure 14

Calculated energy mismatch $\delta E_n$ for the two bands including the two tight-binding approximations discussed in the text. Panels (a) and (b) respectively show the results for the experimental and tight-binding regimes.

Figure 15

Asymmetric structure corresponding to the angle $\theta =\pi +0.1$, with potential parameters $V_X=0.56$, $V_Y=3.6$, and $V_{\overline{X}}=6.94$ (merging point). Panel (a) illustrates the structure of the potential for this configuration, showing a deeper minimum at sublattice $B$ than in $A$ (color code as in Fig. 2). Panels (b) and (c) show one-dimensional profiles of $|w_{0A}{(x,y=0)|}^2$ (solid, blue) and $|w_{0B}{(x,y=0)|}^2$ (dashed, red) in the central unit cell. Note the different distributions of the two MLWFs, as a consequence of parity breaking. This is evident also from the two-dimensional plots of $|w_{0A}{\left(\mathbf{r}\right)|}^2$ and $|w_{0B}{\left(\mathbf{r}\right)|}^2$, in (d) and (e), respectively.

Figure 16

Splitting of the diagonal coefficients as a function of the angle $\theta$ (at the merging point: $V_X=0.56$, $V_Y=3.6$, and $V_{\overline{X}}=6.94$; cf. Fig. 3). Note that $|E_A-E_B|=2|\epsilon |$; see Eq. (19).