Signatures of bosonic excitations in high-harmonic spectra of Mott insulators

The high-harmonic spectrum of the Mott insulating Hubbard model has recently been shown to exhibit plateau structures with cutoff energies determined by $n\mathrm{th}$-nearest-neighbor doublon-holon recombination processes. The spectrum thus allows one to extract the on-site repulsion $U$. Here, we consider generalizations of the single-band Hubbard model and discuss the signatures of bosonic excitations in high-harmonic spectra. Specifically, we study an electron-plasmon model which captures the essential aspects of the dynamically screened Coulomb interaction in solids and a multiorbital Hubbard model with Hund coupling which allows one to analyze the effect of local spin excitations. For the electron-plasmon model, we show that the high-harmonic spectrum can reveal information about the screened and bare on-site interaction, the boson frequency, as well as the relation between boson coupling strength and boson frequency. In the multiorbital case, string states formed by local spin excitations result in an increase of the radiation intensity and cutoff energy associated with higher-order recombination processes.

Figure 1

Real and imaginary parts of the frequency-dependent interaction, $U\left(\omega \right)$, for ${\omega }_0=17$, $U_{\text{scr}}=5$, and $\lambda =20$. The horizontal dashed lines are the asymptotic values for $\mathrm{Re}U$ in the limit of $\omega \to \pm \infty$ and $\omega \to 0$. Besides the single-boson case ($\sigma =0$), we also show the results for a distribution of bosons with a width defined by $\sigma$ [see Eq. (8)].

Figure 2

Spectral functions for different boson distributions with $\sigma =0,1,2$. The parameters are ${\omega }_0=17$, $U_{\mathrm{scr}}=5$, $\lambda =20$, and the inverse temperature is $\beta =5$. For $\sigma =1$ (2), we use 31 (61) bosonic modes. The vertical lines show $U_{\text{scr}}/2$ (solid black) and $U_{\text{bare}}/2$ (dashed black) and the line $U_{\text{scr}}/2+{\omega }_0$ (gray). For the main Hubbard band, all the curves overlap.

Figure 3

Spectral functions for ${\omega }_0=2$, $U_{\mathrm{scr}}=5$, $\lambda =10$, $\beta =5$, and different boson distributions parametrized by $\sigma$. The vertical lines indicate $U_{\text{scr}}/2$ (solid black), $U_{\text{bare}}/2$ (dashed black), and $U_{\text{scr}}/2+{\omega }_0$ (gray).

Figure 4

Equilibrium spectral functions of the two-orbital model for $U=10$, inverse temperature $\beta =5$, and indicated values of $J$. The left panel is for the half-filled system and the right panel for the quarter-filled system.

Figure 5

HHG spectrum of the electron-plasmon model with ${\omega }_0=17$, $U_{\text{scr}}=5$, $\lambda =20$, and $\sigma =0$. The black solid line indicates $U_{\text{scr}}+E_0$, the black dashed line $U_{\text{bare}}+E_0$, while the gray line indicates $U_{\text{scr}}+{\omega }_0+E_0$. The gray vertical line is at $U_{\text{scr}}+{\omega }_0$. Inset: Shape of the 10-cycle excitation pulse with central frequency $\mathrm{\Omega }=1$ and a ${cos}^2$-type envelope.

Figure 6

Top panel: Harmonic spectra for the Hubbard-Holstein model with ${\omega }_0=2$, $U_{\mathrm{scr}}=5$, and $\lambda =10$. Solid black lines indicate $U_{\mathrm{scr}}+n{E}_0$ and dashed black lines $U_{\mathrm{bare}}\pm n{E}_0$ ($n=1,2,3$). Black dots mark the cutoff energies of the first plateau and gray dots those of the second plateau. Bottom panel: HHG spectra at $E_0=6$ and 40. Here, the solid (dashed) vertical lines indicate $U_{\mathrm{scr}}+E_0$ ($U_{\mathrm{bare}}+E_0$) as well as $U_{\mathrm{scr}}+3E_0$ ($U_{\mathrm{bare}}+3E_0$). The arrows mark the edges of the plateaus in the $E_0=40$ case.

Figure 7

Left panel: Ratio between the HHG spectra of the Holstein-Hubbard model with ${\omega }_0=17$, $U_{\text{scr}}=5$, $\lambda =20$, and the Hubbard model with $U=U_{\text{scr}}$ and renormalized bandwidth. Right panel: Analogous comparison between the Holstein-Hubbard model with ${\omega }_0=2$, $U_{\text{scr}}=5$, $\lambda =10$, with the Hubbard model with $U=U_{\mathrm{scr}}$ and renormalized bandwidth. Black solid lines indicate $\omega =U_{\text{scr}}+n{E}_0$ and black dashed lines $\omega =U_{\text{bare}}+n{E}_0$. The vertical gray lines indicate ${\omega }_0+U_{\mathrm{scr}}$ in both figures.

Figure 8

Upper panels: Electron-plasmon model with ${\omega }_0=17$, $U_{\mathrm{scr}}=5$, and $\lambda =20$. Upper left: $\mathrm{\Omega }=1$ with $\sigma =0.25$. Upper right: $\mathrm{\Omega }=1$ with $\sigma =1$. Lower panels: ${\omega }_0=2$, $U_{\mathrm{scr}}=5$, and $\lambda =10$. Lower left: $\mathrm{\Omega }=1$ with $\sigma =0.06$. Lower right: $\mathrm{\Omega }=1$ with $\sigma =0.24$. The arrows point out some features not seen in the left panel.

Figure 9

Left panel: Shape of the few-cycle excitation pulse with frequency $\mathrm{\Omega }=0.5$ and amplitude $E_0=1$ used in the two-orbital simulations. Right panel: Current for indicated values of $E_0$ ($U=10,J=1$).

Figure 10

High-harmonic spectra ${\left|\omega j\left(\omega \right)\right|}^2$ as a function of $E_0$ for $J=0$ (left) and $J=3$ (right). The gray dashed lines represent the cutoffs $U+J\pm n{E}_0$ ($n=1,2,3,4$).

Figure 11

High-harmonic spectra for $E_0=8$ and $J=3$. The vertical lines show $U+J+n{E}_0$ (gray dashed), $U+J+n{E}_0+(n-1)J$ (solid black), and $2(U+J)+(n-1)E_0$ (dark-red dashed), with $n\ge 1$.

Figure 12

The left panel shows a logarithmic plot of the ratio of the HHG spectra for the half-filled two-orbital models with $U=13,J=0$ and $U=10,J=3$ (original bandwidth), while the right panel shows the ratio between the HHG spectra for $U=13,J=0$ and $U=12.6,J=0$ with a bandwidth multiplied by 0.53. The gray dashed lines represent the cutoffs $U+J\pm n{E}_0$ and the white lines the maximum cutoffs including spin-flip processes, $U+J+n{E}_0+(n-1)J$. The black dashed lines are the cutoffs associated with second order in $U$ processes.

Figure 13

HHG spectra of the quarter-filled two-orbital model. The left panel shows the result for $U=14$, $J=2$ and the right panel the difference of this spectrum to the result for $U=9$, $J=0$. Light gray dashed lines show the cutoffs $U-3J+n{E}_0$, white dashed lines $U-2J+n{E}_0$, and dark-gray dashed lines $U+n{E}_0$ (for $U=14$, $J=2$).