Doping effects in high-harmonic generation from correlated systems

Using the one-dimensional Hubbard model, which is commonly used for describing, e.g., high-$T_c$ superconducting cuprates, we study high-harmonic generation (HHG) from doped, correlated materials. Doping is modeled by changing the number of electrons in the lattice from the conventional half-filling case. For relatively small Hubbard $U$, i.e., small electron-electron correlation, we find little to no effect of doping on the dynamics and the HHG spectra. For increasing $U$ the degree of doping has a marked effect on the dynamics and spectra. We explain these findings through the quasiparticle-based doublon-holon picture. The dynamics are separated into two types, first, doublon and holon movement, and, second, doublon-holon pair creation and annihilation. Doping results in all configurations containing doublons or holons. Those quasiparticles can move at no extra cost in energy regardless of the correlation level. This motion at no energy cost increases the high-harmonic gain for low- and medium-harmonic orders. We discuss that in the high-$U$ limit, antiferromagnetic ordering becomes increasingly unlikely with increasing doping rates and explain an associated drop in the high-order harmonics relative to the case of half-filling.

Figure 1

Illustration of the separation of the dynamics into currents $j_{\text{ca}}\left(t\right)$ associated with the (a) annihilation and (b) creation of doublon-holon pairs, and into currents $j_{\text{hop}}\left(t\right)$ associated with the hopping of a (c) doublon or a (d) holon. Sites are illustrated by black circles. In each panel the upper sites are the initial configuration, and the bottom the final configuration. The transition is created by the operator written in the middle of each panel. A red arrow indicates that a spin-down electron occupies the given site, similarly, a blue arrow indicates that a spin-up electron occupies the given site.

Figure 2

HHG spectra [Eq. (3)] for different degrees of band filling and Hubbard $U$. (a) $U=0.1t_0$, (b) $U=2t_0$, (c) $U=5t_0$, and (d) $U=10t_0$. The dashed, black vertical lines indicate the $U$ value in terms of harmonic orders, the full black vertical lines on the left indicate the Mott gap [see Eq. (10)], and the full black vertical lines on the right indicate the Mott gap plus twice the energy width of the underlying Bloch band, i.e., ${\mathrm{\Delta }}_{\text{Mott}}\left(U\right)+2{\mathrm{\Delta }}_{\text{width}}$.

Figure 3

The current generated via Eq. (4), for different degrees of band filling and Hubbard $U$. (a) $U=0.1t_0$, (b) $U=2t_0$, (c) $U=5t_0$, and (d) $U=10t_0$. In all panels the vector potential is shown by the dashed, black line and the Bloch current from the half-filled lattice $j_0\left(t\right)$ [Eq. (15)] is shown via the thin, gray line.

Figure 4

The $D$ measure of Eq. (8) for different degrees of band filling and Hubbard $U$. (a) $U=0.1t_0$, (b) $U=2t_0$, (c) $U=5t_0$, and (d) $U=10t_0$.

Figure 5

The $P_{\text{af}}$ measure of Eq. (9), for different degrees of band filling and Hubbard $U$. (a) $U=0.1t_0$, (b) $U=2t_0$, (c) $U=5t_0$, and (d) $U=10t_0$.

Figure 6

The creation and annihilation current $j_{\text{ca}}\left(t\right)$ from Eq. (5), and the hopping current $j_{\text{hop}}\left(t\right)$ from Eq. (6), for the $U=10t_0$ system for lattices with $n$ electrons. Half-filling corresponds to $n=L=12$. The total current is shown by the full, blue curve and the vector potential is shown by the black, dashed line. Finally, the scaled down Bloch current is shown by the thin, gray line. The scaling factors are given by $max\left[\right|j\left(t\right)\left|\right]/|2a{E}_{\text{GS}}|$, (a) $\simeq 1.9\times {10}^{-3}$, (b) $\simeq 0.31$, (c) $\simeq 0.52$. Note that in (a), $j_{\text{ca}}\left(t\right)$ is identical to the total current $j\left(t\right)$ on the scale of the figure.

Figure 7

The creation and annihilation $S_{\text{ca}}\left(\omega \right)$ and hopping $S_{\text{hop}}\left(\omega \right)$ spectra [Eq. (7)] from lattices with various values of $U$: (a) $U=5t_0$, (b) $U=10t_0$, (c)–(e) $U=10t_0$, and degrees of band filling (a)–(c) $n=12$, (d) $n=14$, and (e) $n=16$. The dashed, black vertical lines indicate the $U$ value in terms of harmonic orders, the full black lines on the left indicate the Mott gap in terms of harmonic orders, and the full black lines on the right indicate the Mott gap plus two times the energy width of the Bloch band.