Emergence of Dirac electron pair in the charge-ordered state of the organic conductor $\alpha$-(BEDT-TTF)${}_2$I${}_3$

We reexamine the band structure of the stripe charge ordered state of $\alpha$-(BEDT-TTF)${}_2$I${}_3$ under pressure by using an extended Hubbard model within the Hartree mean-field theory. By increasing pressure, we find a topological transition from a conventional insulator with a single-minimum in the dispersion relation at the $M$ point in the Brillouin zone, toward a new phase which exhibits a double minimum. This transition is characterized by the appearance of a pair of Dirac electrons with a finite mass. Using the Luttinger-Kohn representation at the $M$ point, it is shown that such a variation of the band structure can be described by an effective $2\times 2$ low-energy Hamiltonian with a single driving parameter. The topological nature of this transition is confirmed by the calculation of the Berry curvature which vanishes in the conventional phase and has a double peak structure with opposite signs in the new phase. We compare the structure of this transition with a simpler situation which occurs in two-component systems, like boron-nitride.

Figure 1

The model describing the electronic system of $\alpha$-(BEDT-TTF)${}_2$I${}_3$ (Refs. 13, 25, and 26). The unit cell consists of four BEDT-TTF molecules $\mathrm{A}$, ${\mathrm{A}}^{\prime }$, $\mathrm{B}$, and $\mathrm{C}$ with seven transfer energies. The nearest-neighbor repulsive interactions are given by $V_a$ and $V_b$. The $a$ and $b$ axis in the conventional notation correspond to the $y$ and $x$ axis in the present paper.

Figure 2

The phase diagram on the uniaxial pressure along the $a$ axis ($P_a$ [kbar]) and the repulsive interaction between nearest-neighbor sites along the $a$ axis ($V_a$ [eV]), where $U$ $=$ 0.4 eV and $V_b$ $=$ 0.05 eV. The CO and COM denote insulating and metallic states, respectively. In addition to the phase (I) of the previous work,[9] there exists a new phase (II), which is characterized by a double minimum in the up-spin band.

Figure 3

Schematic behavior of the energy spectrum close to the Fermi energy (horizontal line) in the different subphases of the CO phase: (a) CO(I), (b) CO(II), (c) COM(I), (d) COM(II). The red and blue bands correspond, respectively, to $\uparrow$ and $\downarrow$ spins. The center of the horizontal line corresponds to the $M$ point [${\mathbf{k}}_M=(\pi ,\pm \pi )$].

Figure 4

The $\mathbf{q}$ dependence of the conduction and valence bands for $\uparrow$ spin where $\mathbf{q}=\mathbf{k}-{\mathbf{k}}_M$, and the center denotes the $M$ point [${\mathbf{k}}_M=(\pi ,\pm \pi )$]. (a) Spectrum of the CO(I) phase with a single minimum at the $M$ point. (b) Spectrum of the CO(II) phase with a double minimum around the $M$ point. Here the parameters are $V_a=0.18$ eV and $V_b=0.05$ eV, and the pressures are, respectively, $P_a=4.5$ kbar and $P_a=5.4$ kbar, for the CO(I) and CO(II) phases.

Figure 5

(a) Contours of the gap function ${\Delta }_{\mathbf{k}}=[{\xi }_{+}\left(\mathbf{k}\right)-{\xi }_{-}\left(\mathbf{k}\right)]/2$ for $P_a=5.4$ kbar in CO (II) phase, where $0<k_y<2\pi$ and $0<k_x<2\pi$. The two Dirac points emerge from the $M$ point ${\mathbf{k}}_M=(\pi ,\pm \pi )$. Points O, X, and Y denote (0,0), ($\pi$,0), and (0,$\pi$), respectively. (b) The same contour ${\Delta }_{\mathbf{q}}$ in the vicinity of the $M$ point, which is taken as the origin ($\mathbf{q}=\mathbf{k}-{\mathbf{k}}_M$). The red curve represents the energy contour ${\Delta }_{\mathbf{q}}=\Delta$. The two red lines denote the angles ${\theta }_{\min }$ and ${\theta }_{\max }$ defined in the text (29). Note that on Fig. (b), the scales are different along the two axes.

Figure 6

Pressure dependence of (a) $\Delta$, (b) $v_j$, (c) $w_3^{ij}$, (d) ${\overline{w}}_3^{ij}$, (e) det$S_M$, and (f) ${\theta }_{\max }$, ${\theta }_{\min }$ for $V_a=0.18$ eV and $V_b=0.05$ eV. The pressure $P_a$ is in kbar.

Figure 7

Contours of $\Delta \left(\mathbf{q}\right)$ for the BN model. Here we have taken the same parameters as for the CO(I)-CO(II) transition in $\alpha$(BEDT-TTF)${}_2$I${}_3$, $\Delta$, $w_1=w_1^{yy}$, $w_3=w_3^{yy}$, $v_x$, other $w_1^{ij}$, $w_3^{ij}$, $v_y$ being zero. In this simple case, since the parameter ${\overline{w}}_3^{xy}=0$, the Dirac points stay along the $y$ axis. The red curve corresponds to $\Delta \left(\mathbf{q}\right)=\Delta$. The two lines indicate the directions ${\theta }_{\max }$ and ${\theta }_{\min }$.

Figure 8

(a) Berry curvature $B_{1\uparrow }\left(\mathbf{k}\right)$ in CO(I) with $P_a=4.4$ kbar; (b) Berry curvature $B_{1\uparrow }\left(\mathbf{k}\right)$ in CO(II) with $P_a=5.4$ kbar, where $V_a=0.18$ eV and $V_b=0.05$ eV. The center denotes the $M$ point.

Figure 9

$P_a$ dependence of the Berry phase $\left|\Gamma \right({\mathbf{k}}_{\pm }\left)\right|$ defined in Eq. (54) (solid line) compared with the expression ${\Gamma }_{\pm }$ given in Eq. (55) (dashed line). The inset denotes ${q_{+}}_y$, the $y$ component of one Dirac point ${\mathbf{k}}_{+}$, where $\mathbf{q}=\mathbf{k}-{\mathbf{k}}_M$.

Figure 10

The energy bands of ${\xi }_{1\sigma }$ and ${\xi }_{2\sigma }$ for $\uparrow$ spin band (a) and the $\downarrow$ spin band (b) in the CO(II) for $P_a=5.4$ kbar, $V_a=0.18$ eV, and $V_b=0.05$ eV. The center denotes the $M$ point.