Two-particle correlation effects on nonlinear optical responses in the one-dimensional interacting Rice-Mele model

Nonlinear responses in crystalline solids are attracting a great deal of attention because of exciting phenomena, such as the bulk photovoltaic effect in noncentrosymmetric crystals and the third-harmonic generation related to Higgs modes in superconductors, and their potential applicability to electronic devices. Recently, nonlinear responses have also been studied in strongly correlated electron systems. Experimental evidence has revealed that correlations play a significant role in nonlinear responses. However, most theoretical calculations only consider excitonic effects or involve numerically demanding approaches, making interpreting the results challenging. In this paper, we adopt another approach, which is based on real-time evolution using the correlation expansion method. We focus in particular on the one-dimensional interacting Rice-Mele model. We analyze the impact of the density-density interaction on the linear and nonlinear conductivities, and we demonstrate that two-particle correlations beyond the mean-field level enhance second-order nonlinear responses, especially the second-harmonic generation, while the linear response is not strongly affected. Furthermore, by decomposing the current into a one-particle contribution and six two-particle contributions, we show that the “biexciton transition” term and its nonlinear oscillations are the most dominant two-particle contribution to the nonlinear response. In addition, we also show that the intercell charge-charge correlation is strongly enhanced when the system is driven with the frequency corresponding to the excitonic peak, and it can even exceed the intracell correlation. This implies the possibility of manipulating two-particle correlations with external fields.

Figure 1

Band dispersion of the Rice-Mele model for $Q_x=0.25,Q_y=0.3,\mathrm{and}{Q}_{\text{on}}=0.25$. The red curve denotes the conduction band, and the blue curve denotes the valence band. The vertical axis is in units of $\hslash {\omega }_0$, and the horizontal axis is in units of $a^{-1}$.

Figure 2

Comparison between conductivities calculated by the Kubo formula (Kubo) and tdMF (tdMF) for $V=0$. (a) Linear conductivity ${\sigma }_{\mathrm{Linear}}$. (b) Photovoltaic conductivity ${\sigma }_{\mathrm{PV}}$. (c) SHG conductivity ${\sigma }_{\mathrm{SHG}}$. ${\sigma }_{\mathrm{Linear}}, {\sigma }_{\mathrm{PV}}, {\sigma }_{\mathrm{SHG}}$, and $\mathrm{\Omega }$ are in units of $\frac{e^2{a}^2}{\hslash },\frac{e^3{a}^3}{{\hslash }^2{\omega }_0},\frac{e^3{a}^3}{{\hslash }^2{\omega }_0},\mathrm{and}{\omega }_0$, respectively.

Figure 3

(a) Linear conductivity calculated by tdMF for $V=0,0.05,0.15$ and linear conductivity calculated by IPA for $V=0.15$. (b) Magnification of (a), showing the linear conductivity around $\mathrm{\Omega }=0.7\text{−}0.8$ calculated by tdMF for $V=0.0,0.15$. The dashed line in (b) is a Lorentzian fit of the peak of the conductivity for $V=0.15$. ${\sigma }_{\mathrm{Linear}}, \mathrm{\Omega }$, and $V$ are in units of $\frac{e^2{a}^2}{\hslash },{\omega }_0,\mathrm{and}\hslash {\omega }_0$, respectively.

Figure 4

Linear conductivity calculated by the correlation expansion including two-particle correlations (2P) and by tdMF for $V=0.03$ (a) and $V=0.15$ (b). ${\sigma }_{\mathrm{Linear}}, \mathrm{\Omega }$, and $V$ are in units of $\frac{e^2{a}^2}{\hslash },{\omega }_0,\mathrm{and}\hslash {\omega }_0$, respectively.

Figure 5

Photovoltaic conductivity calculated by the correlation expansion (2P) and by tdMF for $V=0.03$ (a) and $V=0.15$ (b). ${\sigma }_{\mathrm{PV}}, \mathrm{\Omega }$, and $V$ are in units of $\frac{e^3{a}^3}{{\hslash }^2{\omega }_0},{\omega }_0,\mathrm{and}\hslash {\omega }_0$, respectively.

Figure 6

SHG conductivity calculated by the correlation expansion (2P) and tdMF for $V=0.03$ (a) and $V=0.15$ (b). ${\sigma }_{\mathrm{SHG}}, \mathrm{\Omega }$, and $V$ are in units of $\frac{e^3{a}^3}{{\hslash }^2{\omega }_0},{\omega }_0,\mathrm{and}\hslash {\omega }_0$, respectively.

Figure 7

Interaction strength dependence of the peak intensity in the SHG conductivity calculated by the correlation expansion (2P) and by tdMF for the two-photon peak (a) and the one-photon peak (b). The peak intensity of the one-photon peak corresponds to the negative peak in Fig. 6. The peak value of SHG and $V$ are in units of $\frac{e^3{a}^3}{{\hslash }^2{\omega }_0}\mathrm{and}\hslash {\omega }_0$, respectively.

Figure 8

(a) Comparison between the one-particle contribution and $S_{ccvv}$ contribution to the SHG conductivity for $V=0.15$. (b) A magnification of the upper panel around $\mathrm{\Omega }\sim 0.7$–0.9. ${\sigma }_{\mathrm{SHG}}, \mathrm{\Omega }$, and $V$ are in units of $\frac{e^3{a}^3}{{\hslash }^2{\omega }_0},{\omega }_0,\mathrm{and}\hslash {\omega }_0$, respectively.

Figure 9

Contributions of different harmonics ($n_{\max }=0,1,2,\infty$) to the one-photon peak in the SHG conductivity for $V=0.15$. ${\sigma }_{\mathrm{SHG}}$ and $\mathrm{\Omega }$ are in units of $\frac{e^3{a}^3}{{\hslash }^2{\omega }_0}\mathrm{and}{\omega }_0$, respectively.

Figure 10

(a) Time dependence of the averaged intercell and intracell density-density correlation for $\mathrm{\Omega }=0.78$ and 0.85. (b) Comparison of intercell and intracell correlations in the steady state for different frequencies. Around the excitonic peak, the intercell correlations are strongly enhanced and exceed the intracell correlations. $t$ and $\mathrm{\Omega }$ are in units of ${\omega }_0^{-1}\mathrm{and}{\omega }_0$, respectively.

Figure 11

One-particle expectation values $f_c\left(k\right), y\left(k\right)$, and two-particle correlation $S_{ccvv}$ during the adiabatic switching on of the interaction strength for $V=0.15, T=50$, and different momenta. All correlations reach a plateau at $t=T(=50)$ without large oscillations. $t$ and $k$ are in units of ${\omega }_0^{-1}\mathrm{and}{a}^{-1}$, respectively.

Figure 12

Comparison of the conduction-electron–valence-hole correlations calculated in real space for different distances between electron and hole. The upper panel (a) shows the time-resolved correlations at the excitonic peak, demonstrating a substantial enhancement of the local correlations. The lower panel (b) shows the time-resolved correlations away from the excitonic peak, where these correlations are still comparably strong for large distances. $t$ is in units of ${\omega }_0^{-1}$.