Nonlinear doublon production in a Mott insulator: Landau-Dykhne method applied to an integrable model

Doublon-hole pair production, which takes place during dielectric breakdown in a Mott insulator subject to a strong laser or a static electric field, is studied in the one-dimensional Hubbard model. Two nonlinear effects cause the excitation, i.e., multiphoton absorption and quantum tunneling. Keldysh crossover between the two mechanisms occurs as the field strength and photon energy are changed. The calculation is done analytically by the Landau-Dykhne method in combination with the Bethe ansatz solution, and the results are compared with those of the time-dependent density matrix renormalization group. Using this method, we calculate the distribution function of the generated doublon-hole pairs and show that it drastically changes as we cross the Keldysh crossover line. After calculating the tunneling threshold for several representative one-dimensional Mott insulators, possible experimental tests of the theory are proposed, such as angle-resolved photoemission spectroscopy of the upper Hubbard band in the quantum tunneling regime. We also discuss the relationship of the present theory with a many-body extension of electron-positron pair production in nonlinear quantum electrodynamics, known as the Schwinger mechanism.

Figure 1

(a) Schematic plot of the nonlinear optical absorption spectrum of a Mott insulator in a strong ac electric field. $F_0(>0)$ is the field strength and $\Omega$ is the photon energy of the applied laser. In addition to the contribution from linear response theory (Kubo formula), subgap excitations occur due to nonlinear processes. Mechanisms are quantum tunneling and multiphoton absorption. They are governed by the Mott gap ${\Delta }_{\mathrm{Mott}}$ and correlation length $\xi$. (b) Correlation length $\xi$ is the typical size of doublon-hole pairs in the Mott insulating ground state. Pair production and annihilation occur during the virtual process.

Figure 2

(a) The excitation energy $\Delta E$ of the doublon-hole pair as a function of the hole momentum $p$ in the $U=8$ Hubbard model. ${\mathcal{P}}_p$ is the tunneling probability to create a state ${|p\rangle }_{dh}$ from $|0\rangle$. (b) The real part of the excitation energy plotted for complex $p$. The $\mathrm{Im}p=0$ slice is equivalent to the left half of (a). The gap closes at the level crossing point $p=i{p}_c$, where a gapless line starts. Paths ${\Gamma }_1$ and ${\Gamma }_2$ are used in the integral in Eq. (25).

Figure 3

(a) Correlation length $\xi$ (Ref. 44) and (b) Mott gap ${\Delta }_{\mathrm{Mott}}$ (Lieb-Wu solution) of the 1D Hubbard model at half-filling.

Figure 4

Schematic rigid band description of the dielectric breakdown of Mott insulators in dc electric fields. The upper and lower “Hubbard bands” are tilted with static potential $V\left(x\right)=F_{0}x$ and quantum tunneling starts to occur when the energy drop $F_0\xi$ between the doublon pairs separated by $\xi$ becomes comparable to the excitation gap ${\Delta }_{\mathrm{Mott}}$.

Figure 5

Schwinger limit (i.e., the tunneling threshold)[15] of the 1D Hubbard model at half-filling. In (c), the solid line is the Landau-Dykhne result given by Eq. (35), whereas the dashed line is its approximate form, Eq. (37). The Landau-Zener result in Eq. (39) with $v_{\mathrm{eff}}=2$ is plotted as a dotted line. The Landau-Zener result is only accurate for small $U$.

Figure 6

Tunneling probability of the $dh$ pair obtained for the $U=8$ Hubbard model in an ac field. Parts (a) and (c) correspond to $F_0=1.0$ and $\Omega =4.5$, respectively, which are in the multiphoton regime, while (b) and (d) are for $F_0=1.0$ and $\Omega =0.1$, respectively, which are in the quantum tunneling regime. In (a) and (b), the tunneling probability is indicated by the size of the circle plotted on top of the $dh$-pair spectrum.

Figure 7

(a) Total production rate of the $U=8$ Hubbard model in an ac field. The dashed line corresponds to Keldysh crossover $\gamma =1$, where the excitation mechanism changes from multiphoton absorption to quantum tunneling. The Schwinger limit is $F_{\mathrm{th}}=1.668$. (b), (c) Schematic pictures of the nonlinear excitation in the two regimes. In the quantum tunneling regime (b), the doublon-hole distribution becomes momentum-independent, which means that the “upper Hubbard band” becomes populated by photocarriers.

Figure 8

(a) Time evolution of doublon density in ac fields calculated by td-DMRG for the $U=8$ Hubbard model ($L=30,F_0=4,\Omega =2$). (b) Time evolution of the averaged doublon density $d$. Results for $L=30$ and 20 are plotted. Upper panel is the electric field $F\left(t\right)=F_0\cos \Omega t$. Doublon density shows initial increase and then saturation occurs. (c) Resonant oscillation between the ground state and the excited state with a neighboring doublon and hole pair.

Figure 9

Total production rate for the $U=8,7$ Hubbard model with photon energy $\Omega =1$ plotted against field strength $F_0$ [(b) is the logarithmic plot]. The td-DMRG result is obtained using the fitting of Eq. (48), which is compared with the Landau-Dykhne result. Photon energy is $\Omega =1$ and system size is $L=30$.

Figure 10

(a) Shape of pulse field with duration $\sigma =2$. (b) Time evolution of the doublon density $d\left(t\right)$ for the $U=10$ Hubbard model obtained by td-DMRG. (c) Increase in the doublon density $\Delta d$ obtained by td-DMRG and the total production $\Gamma$ obtained by the Landau-Dykhne method. See the main text for the error bar. Inset: $p$-resolved distribution function of doublon-hole pairs obtained by the Landau-Dykhne method.

Figure 11

Typical $IV$ characteristics of strongly correlated insulators. We have a threshold behavior, negative differential resistance, and a transition to a metallic state. The present theory is only applicable to explain the threshold behavior in the small current regime shown in the dashed box.