Photoinduced electron-electron pairing in the extended Falicov-Kimball model

By employing the time-dependent exact diagonalization method, we investigate the photoexcited states of the excitonic insulator in the extended Falicov-Kimball model (EFKM). We here show that the pulse irradiation can induce the interband electron-electron pair correlation in the photoexcited states, while the excitonic electron-hole pair correlation in the initial ground state is strongly suppressed. We also show that the photoexcited states contain the eigenstates of the EFKM with a finite number of interband electron-electron pairs, which are responsible for the enhancement of the electron-electron pair correlation. The mechanism found here is due to the presence of the internal SU(2) pairing structure in the EFKM and thus it is essentially the same as that for the photoinduced $\eta$ pairing in the repulsive Hubbard model reported recently [T. Kaneko et al., Phys. Rev. Lett. 122, 077002 (2019)]. This also explains why the nonlinear optical response is effective to induce the electron-electron pairs in the photoexcited states of the EFKM. Furthermore, we show that, unlike the $\eta$ pairing in the Hubbard model, the internal SU(2) structure is preserved even for a nonbipartite lattice when the EFKM has the direct-type band structure, in which the pulse irradiation can induce the electron-electron pair correlation with momentum $\mathit{q}=\mathit{0}$ in the photoexcited states. We also discuss briefly the effect of a perturbation that breaks the internal SU(2) structure.

Figure 1

Schematic band structures of (a) a semimetal ($U=0$) and (b) an excitonic insulator in the EFKM with $t_h^{\left(1\right)}·t_h^{\left(2\right)}<0$.

Figure 2

(a) Excitonic (i.e., electron-hole pair) structure factor $N_X\left(q\right)$ and (b) single-particle excitation spectrum $A(k,\omega )$ for the ground state of the 1D EFKM with $t_h^{\left(2\right)}=-t_h^{\left(1\right)}=t_h,U=8t_h$, and $D=0.75t_h$ in $L=16$, where $N_1=12$ and $N_2=4$. The Fermi energy is set at $\omega =0$ in (b). Here, the single-particle excitation spectrum is defined as $A(k,\omega )={\sum }_{\alpha }\langle {\psi }_0|{\widehat{c}}_{k,\alpha }^{†}{\delta }_{\gamma }(\omega +\widehat{\mathcal{H}}-E_0){\widehat{c}}_{k,\alpha }|{\psi }_0\rangle +{\sum }_{\alpha }\langle {\psi }_0|{\widehat{c}}_{k,\alpha }{\delta }_{\gamma }(\omega -\widehat{\mathcal{H}}+E_0){\widehat{c}}_{k,\alpha }^{†}|{\psi }_0\rangle$, where ${\widehat{c}}_{k,\alpha }^{†}$ is the Fourier transform of ${\widehat{c}}_{j,\alpha }^{†}$ [also see Eq. (36)]. The broadening factor $\gamma$ in $A(k,\omega )$ is $0.1t_h$.

Figure 3

(a) Time evolution of the onsite electron-electron pair correlation function $P(j,t)$. (b) $P(j,t)$ at $t=0$ (blue circles) and $t=30/t_h$ (orange squares). (c) Time evolution of the electron-electron pair structure factor $P(q,t)$ and the excitonic (i.e., electron-hole pair) structure factor $N_X(q,t)$ at $q=0$. (d) $P(q,t)$ at $t=0$ (blue circles) and $t=30/t_h$ (orange squares). The results are for the 1D EFKM with $t_h^{\left(2\right)}=-t_h^{\left(1\right)}=t_h,U=8t_h$, and $D=0.75t_h$ in $L=16$. In this case, the initial state before the pulse irradiation (i.e., the ground state of the 1D EFKM) has $N_1=12$ and $N_2=4$. We set $A_0=0.4,{\omega }_p=7t_h,{\sigma }_p=2/t_h$, and $t_0=10/t_h$ for $A\left(t\right)$. The vertical dashed lines in (a) and (c) indicate $t_0$.

Figure 4

Contour plots of (a) the excitonic structure factor $N_X(q=0,t)$ averaged from $t=20/t_h$ to $40/t_h$ and (b) the electron-electron pair structure factor $P(q=0,t)$ at $t=30/t_h$ in the parameter space of ${\omega }_p$ and $A_0$. (c) Optical spectrum ${\chi }_{JJ}\left(\omega \right)$ calculated for the ground state of the EFKM, which is compared with $P(q=0,t=30/t_h)$ as a function of ${\omega }_p$ for different values of $A_0$. The results are for the 1D EFKM with $t_h^{\left(2\right)}=-t_h^{\left(1\right)}=t_h,U=8t_h$, and $D=0.75t_h$ in $L=16$. In this case, the initial state before the pulse irradiation (i.e., the ground state of the 1D EFKM) has $N_1=12$ and $N_2=4$. We set ${\sigma }_p=2/t_h$ and $t_0=10/t_h$ for $A\left(t\right)$. The broadening factor $\gamma$ in ${\chi }_{JJ}\left(\omega \right)$ is $0.2t_h$ in (c).

Figure 5

(a) All the eigenenergies ${\varepsilon }_m$ and $P(q=0)$ for the eigenstates $|{\psi }_m\rangle$ of the half-filled 1D EFKM $\widehat{\mathcal{H}}$ with $t_h^{\left(2\right)}=-t_h^{\left(1\right)}=t_h$ for $L=10$ under PBC at $U=8t_h$ and $D=0.4t_h$, where $N_1=6$ and $N_2=4$. The color of each point (diamond) indicates the weight ${\left|\langle {\psi }_m|\mathrm{\Psi }\left(t\right)\rangle \right|}^2$ of the eigenstate $|{\psi }_m\rangle$ in the photoinduced state $|\mathrm{\Psi }\left(t\right)\rangle$ at $t=30/t_h$ for $A\left(t\right)$ with $A_0=0.3,{\omega }_p=7t_h,{\sigma }_p=2/t_h$, and $t_0=10/t_h$. When the eigenstates are degenerate, the color indicates the sum of ${\left|\langle {\psi }_m|\mathrm{\Psi }\left(t\right)\rangle \right|}^2$ over these degenerate states. The inset shows the time evolution of $P(q=0,t)$ for $|\mathrm{\Psi }\left(t\right)\rangle$. (b) The total weight $w\left(\mathrm{\Delta }\right)$ of ${\left|\langle {\psi }_m|\mathrm{\Psi }\left(t\right)\rangle \right|}^2$ over the eigenstates $|{\psi }_m\rangle$ with the same value of $\mathrm{\Delta }$ in (a). Note that ${\sum }_{\mathrm{\Delta }}w\left(\mathrm{\Delta }\right)=1$.

Figure 6

(a) Onsite electron-electron pair correlation function $P\left(j\right)$ and (b) the corresponding structure factor $P\left(q\right)$ for the half-filled eigenstate $|{\psi }_{N_{\mathrm{\Delta }}}\rangle$ with the different number $N_{\mathrm{\Delta }}$ of $\mathrm{\Delta }$ pairs. The eigenstate $|{\psi }_{N_{\mathrm{\Delta }}}\rangle$ is constructed from the exact ground state of the 1D EFKM with $N_1-N_{\mathrm{\Delta }}$ and $N_2-N_{\mathrm{\Delta }}$ electrons for orbitals 1 and 2, respectively, calculated by the ED method, for $U=8t_h$ and $D=0.4t_h$ in $L=10$ under PBC, where $N_1=6$ and $N_2=4$.

Figure 7

Time evolution of the electron-electron pair structure factor $P(\mathit{q},t)$ and the excitonic (i.e., electron-hole pair) structure factor $N_X(\mathit{q},t)$ at $\mathit{q}=\mathit{0}=(0,0)$ for the 2D EFKM with $t_h^{\left(2\right)}=-t_h^{\left(1\right)}=t_h,U=8t_h$, and $D=t_h$ in a $4\times 4$ square lattice under PBC. In this case, the initial state before the pulse irradiation (i.e., the ground state of the 2D EFKM) has $N_1=12$ and $N_2=4$. The time-dependent vector potential $\mathit{A}\left(t\right)=A\left(t\right)({\mathit{e}}_x+{\mathit{e}}_y)$ is applied along the diagonal direction (indicated in the figure). We set $A_0=0.4,{\omega }_p=8t_h,{\sigma }_p=2/t_h$, and $t_0=10/t_h$ for $A\left(t\right)$. The vertical dashed line indicates $t_0$.

Figure 8

Time evolution of the electron-electron pair structure factor $P(\mathit{q},t)$ and the excitonic (i.e., electron-hole pair) structure factor $N_X(\mathit{q},t)$ at $\mathit{q}=\mathit{0}=(0,0)$ for the 2D EFKM with $t_h^{\left(2\right)}=-t_h^{\left(1\right)}=t_h,U=8t_h$, and $D=1.3t_h$ in a $4\times 4$ triangular lattice under PBC. In this case, the initial state before the pulse irradiation (i.e., the ground state of the 2D EFKM) has $N_1=12$ and $N_2=4$. The time-dependent vector potential $\mathit{A}\left(t\right)=A\left(t\right)(\frac{1}{2}{\mathit{e}}_x+\frac{\sqrt{3}}{2}{\mathit{e}}_y)$ is applied along the direction indicated in the figure. We set $A_0=0.6,{\omega }_p=8t_h,{\sigma }_p=2/t_h$, and $t_0=10/t_h$ for $A\left(t\right)$. The vertical dashed line indicates $t_0$.

Figure 9

Contour plots of (a) the excitonic structure factor $N_X(\mathit{q}=\mathit{0},t)$ averaged from $t=20/t_h$ to $40/t_h$ and (b) the electron-electron pair structure factor $P(\mathit{q}=\mathit{0},t)$ at $t=30/t_h$ in the parameter space of ${\omega }_p$ and $A_0$. (c) Optical spectrum ${\chi }_{JJ}\left(\omega \right)$ calculated for the ground state of the EFKM, which is compared with $P(\mathit{q}=\mathit{0},t=30/t_h)$ as a function of ${\omega }_p$ for different values of $A_0$. The results are for the 2D EFKM with $t_h^{\left(2\right)}=-t_h^{\left(1\right)}=t_h,U=8t_h$, and $D=1.3t_h$ in a $4\times 4$ triangular lattice under PBC. In this case, the initial state before the pulse irradiation (i.e., the ground state of the 2D EFKM) has $N_1=12$ and $N_2=4$. We set ${\sigma }_p=2/t_h$ and $t_0=10/t_h$ for $A\left(t\right)$. The broadening factor $\gamma$ in ${\chi }_{JJ}\left(\omega \right)$ is $0.2t_h$ in (c).

Figure 10

(a) All the eigenenergies ${\varepsilon }_m$ and $P(\mathit{q}=\mathit{0})$ for the eigenstates $|{\psi }_m\rangle$ of the half-filled 2D EFKM $\widehat{\mathcal{H}}$ with $t_h^{\left(2\right)}=-t_h^{\left(1\right)}=t_h$ in a $4\times 3$ triangular cluster under PBC at $U=8t_h$ and $D=0.8t_h$, where $N_1=9$ and $N_2=3$. The color of each point (diamond) indicates the weight ${\left|\langle {\psi }_m|\mathrm{\Psi }\left(t\right)\rangle \right|}^2$ of the eigenstate $|{\psi }_m\rangle$ in the photoinduced state $|\mathrm{\Psi }\left(t\right)\rangle$ at $t=30/t_h$ for $A\left(t\right)$ with $A_0=0.4,{\omega }_p=9t_h,{\sigma }_p=2/t_h$, and $t_0=10/t_h$. When the eigenstates are degenerate, the color indicates the sum of ${\left|\langle {\psi }_m|\mathrm{\Psi }\left(t\right)\rangle \right|}^2$ over these degenerate states. The inset shows the time evolution of $P(\mathit{q}=\mathit{0},t)$ for $|\mathrm{\Psi }\left(t\right)\rangle$. (b) The total weight $w\left(\mathrm{\Delta }\right)$ of ${\left|\langle {\psi }_m|\mathrm{\Psi }\left(t\right)\rangle \right|}^2$ over the eigenstates $|{\psi }_m\rangle$ with the same value of $\mathrm{\Delta }$ in (a). Note that $\mathrm{\Delta }$ corresponds to the number $N_{\mathrm{\Delta }}$ of $\mathrm{\Delta }$ pairs at half filling and ${\sum }_{\mathrm{\Delta }}w\left(\mathrm{\Delta }\right)=1$.

Figure 11

Time evolution of (a) the electron-electron pair structure factor $P(q=0,t)$ and (b) the excitonic (i.e., electron-hole pair) structure factor $N_X(q=0,t)$ for the 1D EFKM with different bandwidths ($t_h=|t_h^{\left(1\right)}|>|t_h^{\left(2\right)}|$) at half filling. The results are calculated by the ED method for $L=16$ at $U=8t_h$ under PBC. We set $D=0.75t_h,0.65t_h,0.6t_h$, and $0.55t_h$ for $t_h^{\left(2\right)}/t_h^{\left(1\right)}=-1.0,-0.9,-0.8$, and $-0.7$, respectively, in which $N_1=12$ and $N_2=4$. The vector potential $A\left(t\right)$ is adopted with $A_0=0.4,{\omega }_p=7t_h,{\sigma }_p=2/t_h$, and $t_0=10/t_h$. The vertical dashed lines indicate $t_0$.

Figure 12

The onsite electron-electron pair correlation function $P(j,t)$ in the photoexcited state at (a) $t=12.5/t_h$ and (b) $t=30/t_h$ indicated by arrows in Fig. 11. The results are for the 1D EFKM with $U=8t_h$ in $L=16$ at half filling. We set $D=0.75t_h,0.65t_h,0.6t_h$, and $0.55t_h$ for $t_h^{\left(2\right)}/t_h^{\left(1\right)}=-1.0,-0.9,-0.8$, and $-0.7$, respectively, in which $N_1=12$ and $N_2=4$. The vector potential $A\left(t\right)$ is adopted with $A_0=0.4,{\omega }_p=7t_h,{\sigma }_p=2/t_h$, and $t_0=10/t_h$.

Figure 13

(a) Time evolution of the onsite electron-electron pair correlation function $P(j,t)$ and (b) $P(j,t)$ at $t=0$ (blue circles) and $t=30/t_h$ (orange squares). The results are for the 1D EFKM under PBC with $t_h^{\left(1\right)}=t_h^{\left(2\right)}=t_h,U=8t_h$, and $D=0.75t_h$ in $L=16$, for which $N_1=12$ and $N_2=4$. We set $A_0=0.4,{\omega }_p=7t_h,{\sigma }_p=2/t_h$, and $t_0=10/t_h$ for the vector potential $A\left(t\right)$ defined in Eq. (19).