Correlations in a band insulator

We study a model of a covalent band insulator with on-site Coulomb repulsion at half filling using dynamical mean-field theory. Upon increasing the interaction strength the system undergoes a discontinuous transition from a correlated band insulator to a Mott insulator with hysteretic behavior at low temperatures. Increasing the temperature in the band insulator close to the insulator-insulator transition we find a crossover to a Mott insulator at elevated temperatures. Remarkably, correlations decrease the energy gap in the correlated band insulator. The gap renormalization can be traced to the low-frequency behavior of the self-energy, analogously to the quasiparticle renormalization in a Fermi liquid. While the uncorrelated band insulator is characterized by a single gap for both charge and spin excitations, the spin gap is smaller than the charge gap in the correlated system.

Figure 1

$T\text{−}U$ phase diagram at fixed $V=0.5$. The critical values of the interaction strength $U$ are determined from the double occupancy (see inset). Below a critical temperature both band and Mott insulating solutions of the DMFT equations are found depending on the initial guess for the self-energy. For temperatures above the critical end point of the coexistence region, there is a regime where the spectral function has a single peak at the Fermi energy accompanied by broad Hubbard bands (see Fig. 3). Inset: the double occupancy $D=⟨n_{\uparrow }{n}_{\downarrow }⟩$ as a function of $U$ for $T=1/30$ (circles) and $T=1/60$ (squares) indicates the phase transition from the correlated BI (larger $D$) to the MI (smaller $D$). Both values of the double occupancy are shown in the region where two solutions are found in the DMFT self-consistency cycle. Note that all energies are given in units of the hopping integral $t=1$.

Figure 2

Local spectral function $A\left(\omega \right)$ at fixed interlayer coupling $V=0.5$ and temperature $T=1/30$ for various values of the interaction strength $U$. The noninteracting density of states $\left(U=0\right)$ has a charge gap ${\Delta }_c=2V=1$. The gap in the correlated band insulator shrinks with increasing $U$ until a discontinuous transition to a Mott insulator occurs with a hysteresis region $5.35<U<5.82$. For $U=5.79$ both the band (solid line) and Mott insulating (dashed line) solutions are displayed. All energies are given in units of the hopping integral $t=1$.

Figure 3

Local spectral function $A\left(\omega \right)$ at fixed hybridization $V=0.5$ and interaction strength $U=5$ for various values of the temperature $T$. Upon increasing $T$ the correlated band insulator (split peak around $\omega =0$ plus Hubbard bands) shows a crossover to a Mott insulator (broad Hubbard bands and reduced spectral weight near $\omega =0$ at $T=1/5$). At intermediate temperatures ($T=1/8$ and $T=1/10$) there is a single peak at the Fermi energy accompanied by broad Hubbard bands. Note that the curves are shifted by a vertical offset for clarity.

Figure 4

The self-energy within second-order weak-coupling perturbation theory at zero temperature [see Eq. (11)] for $V=0.5$. The real part (solid line) is linear around $\omega =0$; the slope (indicated by the dot-dashed line) determines the $Z$ factor. The imaginary part (dashed line) is gapped for $\left|\omega \right|<3V$. The dotted vertical lines indicate the bare gap ${\Delta }_c^0=1$. The scale of the vertical axis is $U^2/t$.

Figure 5

Imaginary part of the self-energy as a function of Matsubara frequencies ${\omega }_n$ for three values of the interaction strength $U$. In the correlated BI the self-energy has a negative slope at small ${\omega }_n$. In the coexistence region $\left(U=5.79\right)$ both solutions are displayed.

Figure 6

Local dynamical susceptibilities in the correlated BI. At $V=0.5$ and temperature $T=1/30$ the imaginary parts of the spin susceptibility ${\chi }_s$ (solid lines) and the bubble diagram ${\chi }_B$ (dashed lines) are shown for $U∊\left\{2,3,3.5\right\}$. The susceptibilities are obtained from the QMC data by analytic continuation. The bubble diagram is calculated from the convolution of the fully dressed Green’s functions [see Eq. (16)]. Thus the gap in $\text{Im}{\chi }_B\left(\omega \right)$ corresponds to the single-particle energy gap in $A\left(\omega \right)$, which is apparently larger than the spin gap observed in $\text{Im}{\chi }_s\left(\omega \right)$ in the correlated insulator.

Figure 7

Spin and charge gaps in the correlated BI as a function of $U$ for $V=0.5$ as determined from the spectral function and the spin susceptibility at $T=1/30$, respectively.[29] The thick solid line shows the charge gap obtained from second-order perturbation theory [Eq. (9)]. The squares represent the $Z$ factor [Eq. (14)]. Inset: discontinuous change in spin and charge gaps at the BI to MI transition. In the Mott insulator ${\Delta }_s=0$.