Interaction-induced quantum spin Hall insulator in the organic Dirac electron system $\alpha \text{−}{\text{(BEDT-TSeF)}}_2{\mathrm{I}}_3$

Focusing on the recently-discovered candidate topological insulator $\alpha \text{−}{\text{(BEDT-TSeF)}}_2{\mathrm{I}}_3$—having two-dimensional charge-neutral Dirac cones in a low symmetry lattice—we combine ab initio and extended-Hubbard model calculations to deal with spin-orbit and nonlocal repulsive interactions, and find a realization of an interaction-induced quantum spin Hall (QSH) insulator, similar to the one proposed in the honeycomb lattice under next-nearest-neighbor repulsions. In the absence of repulsive interactions, a topological insulator appears by the spin-orbit coupling and is characterized by a nonzero spin Chern number. By considering up to next-nearest-neighbor repulsions at Hartree-Fock level, the intrinsic spin-orbit gap is found to grow by orders of magnitude and a QSH insulating phase appears that has both a finite spin Chern number and order parameter. Transport coefficients and spin susceptibility are calculated and found to consistently account for most of the experimental findings, including the metal-to-insulator crossover occurring at $\sim 50\mathrm{K}$ as well as the Berry phase change from 0 to $\pi$ under hydrostatic pressure. We argue that such a QSH insulating phase does not necessitate a sizable spin-orbit interaction to generate a large insulating gap, which is highly advantageous for the search of novel topological phases in generic materials having low symmetry lattice and/or small spin-orbit coupling.

Figure 1

(a) Crystal structure and real-space distribution of the MLWFs for $\alpha$-(${\mathrm{BETS})}_2{\mathrm{I}}_3$ under ambient pressure at 30 K in a unit cell drawn by vesta [74]. The box drawn by the black line represents the unit cell, and there are four BETS molecules (sites) labeled A, A′, B, and C in the unit cell. The A and A′ sites are crystallographically equivalent but the A, B and C sites are nonequivalent. (b) Energy band structure with SOC effect obtained by first-principles calculation.

Figure 2

Schematic of the lattice structure of $\alpha$-(${\mathrm{BETS})}_2{\mathrm{I}}_3$ and definition of the transfer integrals and the repulsive interactions for the (a) nearest-neighbor and (b) next-nearest-neighbor sites. The shaded pink parallelogram denotes a the unit cell.

Figure 3

$\xi$ dependence of the (a) energy gap $\mathrm{\Delta }$ and (b) the absolute values of order parameter in the interaction-induced QSH insulating phase which modulate transfer integrals $t_{b1}^{\prime }$ and $t_{b4}$: $|{\chi }^{\mathrm{QSH}}{|}_{b{1}^{\prime }}$ and $|{\chi }^{\mathrm{QSH}}{|}_{b4}$, respectively.

Figure 4

The energy band structure $E_{\nu ,\sigma }\left(\mathit{k}\right)$ in the (a) TI ($\xi =0$) and (b) interaction-induced QSH insulator ($\xi =0.5$) states near the Fermi energy ($\nu =1,2$) for $\sigma =\uparrow$. $E_{\nu ,\uparrow }\left(\mathit{k}\right)$ and $E_{\nu ,\downarrow }\left(\mathit{k}\right)$ are degenerated. The inset in (a) is the magnified view of $E_{\nu ,\sigma }\left(\mathit{k}\right)$ in the TI state in the $-0.02<E_{\nu ,\sigma }\left(\mathit{k}\right)<0.02$ energy range. (c) The Berry curvature $B_{\nu ,\sigma }^z\left(\mathit{k}\right)$ in the interaction-induced QSH insulating phase for $\nu =1$.

Figure 5

(a) $V\text{−}{V}^{\prime }$ phase diagram with SOC for $(U,T)=(0.5,1\times {10}^{-4})$. For $V^{\prime }<V^{\prime \mathrm{CO}}$, the inversion symmetry is broken ($\langle n_{\mathrm{A}}\rangle \ne \langle n_{{\mathrm{A}}^{\prime }}\rangle$) and the horizontal stripe charge ordered insulating phase appears. (b) and (c) show the schematic of the unit cell and definition of the phases around the (b) three nearest-neighbor bonds and (c) three nearest-neighbor bonds and one next-nearest-neighbor bond. (d) and (e) represent the $V^{\prime }$ dependence of the phases on the loops shown in (b) and (c): ${\phi }_n$ ($n=1,\cdots ,10$) at $(U,V,T)=(0.5,0.1,1\times {10}^{-4})$ as an example. (f) $V^{\prime }$ dependence of spin Chern number $N^{\mathrm{sCh}}$ at $(U,V,T)=(0.5,0.1,1\times {10}^{-4})$. Dotted line of integer $-1$ is drawn for eye guide.

Figure 6

(a) $V\text{−}{V}^{\prime }$ phase diagram without SOC for $(U,T)=(0.5,1\times {10}^{-4})$. $V^{\prime \mathrm{C}}$ indicates the value of $V^{\prime }$ that opens the energy gap. (b) and (c) represent the $V^{\prime }$ dependence of the phases on the loops shown in (b) and (c): ${\phi }_n$ ($n=1,\cdots ,10$) at $(U,V,T)=(0.5,0.1,1\times {10}^{-4})$ as an example. (d) $V^{\prime }$ dependence of spin Chern number $N^{\mathrm{sCh}}$ at $(U,V,T)=(0.5,0.1,1\times {10}^{-4})$. Dotted lines of integers zero and $-1$ are drawn for eye guide.

Figure 7

(a) $V^{\prime }$ dependence of the absolute value of the Onsager phase factor $\left|\gamma \right|$ at $(U,V,T)=(0.5,0.1,1\times {10}^{-4})$. [(b) and (c)] Illustration of the Berry curvature $B_{\nu ,\sigma }^z\left(\mathit{k}\right)$ and the energy band $E_{\nu ,\sigma }\left(\mathit{k}\right)$ near the Dirac point for two $\mathrm{\Delta }$ values: $\mathrm{\Delta }\ll |E_{\mathrm{F}}|$ and $\mathrm{\Delta }>|E_{\mathrm{F}}|$.

Figure 8

$T$ dependence of (a) the energy gap $\mathrm{\Delta }$ at $(U,V,V^{\prime })=(0.5,0.1,0.16)$ and (b) the DC resistivity along the $a$-axis direction ${\rho }_a\left(T\right)/{\rho }_0$ in units of the universal conductivity reciprocal ${\rho }_0\equiv {\sigma }_0^{-1}={(4e^2/\pi h)}^{-1}$ at $(U,V,V^{\prime })=(0.5,0.1,0.16)$.

Figure 9

Illustration of the $T$ dependence of the electronic state obtained by calculation with the Hartree-Fock approximation, with onsite $U$, nearest-neighbor sites $V$, and next-nearest-neighbor sites $V^{\prime }$.

Figure 10

$T$ dependence of the site-resolved Knight shift $K_{\alpha }\equiv {\sum }_{\beta }\mathrm{Re}\left[{\chi }_{\alpha ,\beta }^S(\mathit{0},0)\right]$ at $U=0.13$ (thick line) and $U=0$ (thin line) for (a) $\alpha =\mathrm{A}$, (b) C, and (c) B. The inset in (c) is the $T$ dependence of $K_{\mathrm{B}}$ divided into two components: intraband (dotted line) and interband (one-dot chain line).

Figure 11

Momentum $\mathit{q}$ dependence of the real part of the site-resolved spin susceptibility of the A site $\mathrm{Re}\left[{\chi }_{\mathrm{A},\beta }^S(\mathit{q},0)\right]$ at $(U,T)=(0.46,0.007)$ for (a) $\beta =\mathrm{A}$ and (b) ${\mathrm{A}}^{\prime }$. (c) $T$ dependence of $\mathrm{Re}\left[{\chi }_{\mathrm{A},\mathrm{A}}^S(\mathit{0},0)\right]$ and $\mathrm{Re}\left[{\chi }_{\mathrm{A},{\mathrm{A}}^{\prime }}^S(\mathit{0},0)\right]$.

Figure 12

$T$ dependence of $1/T_{1}T$ at $U=0$, 0.13, and 0.46.

Figure 13

Range $\left|R\right|$ dependence of the bare direct integral $V_{\alpha }\left(R\right)$ and the static effective direct integral $W_{\alpha }\left(R\right)$ at the A site ($\alpha$ = A) for (a) $\alpha$-(${\mathrm{BETS})}_2{\mathrm{I}}_3$ at 30 K, and (b) result of aluminum (Al) as an example.

Figure 14

$U\text{−}{\lambda }_{\mathrm{SOC}}$ phase diagram for $(V,V^{\prime },T)=(0,0,1\times {10}^{-4})$. In $U<U^{\mathrm{SMD}}$, the topological insulator (TI) phase appears owing to the SOC contribution. In $U>U^{\mathrm{SMD}}$, the spin ordered massive Dirac electron phase suggested in our previous study appears [67]. In the spin ordered massive Dirac electron phase, the time-reversal symmetry is broken owing to antiferromagnetism between the A and ${\mathrm{A}}^{\prime }$ sites in the unit cell.

Figure 15

$U\text{−}{V}^{\prime }$ phase diagram for $({\lambda }_{\mathrm{SOC}},V,T)=(1,0.1,1\times {10}^{-4})$. In $U<U^{\mathrm{CO}}$, the horizontal strip charge ordered insulating phase phase appears because $V$ is fixed at $V=0.1$. In $U>U^{\mathrm{SMD}}$, the spin ordered massive Dirac electron phase appears owing to the contribution of $U$ [67]. The interaction-induced QSH insulating phase appears during the transition between the spin ordered massive Dirac electron and horizontal strip charge ordered insulating phases, $U^{\mathrm{CO}}<U<U^{\mathrm{SMD}}$.