Nonequilibrium steady-state theory of photodoped Mott insulators

Photodoped states are widely observed in laser-excited Mott insulators, in which charge excitations are quickly created and can exist beyond the duration of the external driving. Despite the fruitful experimental explorations, theoretical studies on the microscopic models face the challenge to simultaneously deal with exponentially separated time scales, especially in multiband systems, where the longtime behaviors are often well beyond the reach of state-of-the-art numerical tools. Here, we address this difficulty by introducing a steady-state description of photodoped Mott insulators using an open-system setup, where the photodoped system is stabilized as a nonequilibrium steady state by a weak external driving. Taking advantage of the stationarity, we implement and discuss the details of an efficient numerical tool using the steady-state dynamical mean-field theory combined with the noncrossing approximation. We demonstrate that these stationary photodoped states exhibit the same properties of their transient counterparts, while being solvable with reasonable computational efforts. Furthermore, they can be parametrized by just a few physical quantities, including the effective temperature and the density of charge excitations, which confirms the universal nature of photodoped states indeed independent of the excitation protocols. As a first application, we consider the stationary photodoped states in a two-band Hubbard model with intertwined spin-and-orbital ordering and find a family of hidden phases unknown from the previous studies, implying an apparently unexplored time regime of the relaxation of the intertwined orders.

Figure 1

Illustration of real laser-induced photodoping and the auxiliary bath doping protocol: (a) In a Mott insulator, photoexcitation creates charge excitations. This effectively dopes both doublons and holons into the system. (b) Coupling a Mott insulator to suitably chosen fermion baths can lead to nonequilibrium doping effects similar to the photoexcitation. The spectral function is calculated for the Hubbard model with $U=8,\beta =12.5$ in the antiferromagnetic ground state and averaged over spins. The blue shades schematically indicate the charge occupation.

Figure 2

Distribution function $f\left(\omega \right)$ of bath-doped systems in the AFM phase. The minority spin is shown. The dark-red line is the spectral function $A\left(\omega \right)$. The blue-shaded region indicates the fermion-bath occupation. The bath coupling is changed in the range $\mathrm{\Gamma }=0.2,0.4,0.6...,1.8\times {10}^{-4}$, shown as solid lines from blue to red colors, respectively.

Figure 3

Distribution function of bath-doped systems in the PM phase. The dark-red line is the spectral function $A\left(\omega \right)$. The chosen $\mathrm{\Gamma }$'s and the color scheme are identical to Fig. 2.

Figure 4

Charge excitations induced in the one-band Hubbard model through fermion-bath coupling. Lines with colors from blue to red show occupied density of states $-i{G}^{<}\left(\omega \right)$ in the upper Hubbard band (for minority spin). The blue areas indicate the shape and position of the baths. Different panels show the distribution for baths with different shapes (square or semielliptic) and offsets $V$. The colors from blue to red correspond to increasing bath couplings. The dark-red line $A\left(\omega \right)$ indicates the spectrum function of the upper Hubbard band of the minority spin. The panel of transient states shows the distribution created by an electric pulse in the real-time DMFT simulation. The $E_0$ is varied from 0.1, 0.2, ..., 0.6. For the steady states, the damping $\mathrm{\Gamma }=$ (a) $0.2,0.4,...,1.8\times {10}^{-4}$, (b) $0.2,0.4,...,1.2\times {10}^{-5}$, (c) $0.2,0.4,...,1.8\times {10}^{-4}$, (d) $0.2,0.4,...,1.4\times {10}^{-5}$, (e) $0.4,0.6,...,2\times {10}^{-6}$.

Figure 5

The antiferromagnetic order parameter $S_z(T_{\mathrm{eff}},n_{\mathrm{ex}})$ in equilibrium and nonequilibrium. In this plot, we have chosen $U=8.0$ and $\beta =12.5$ for the equilibrium reference system in an antiferromagnetic phase. The points show $S_z$ in the nonequilibrium steady state with $\mathrm{\Gamma }=0.5,1.0,...,5.0\times {10}^{-7}$, plotted against $n_{\text{ex}}$ and $T_{\mathrm{eff}}$ as defined in the text. The surface shows equilibrium $S_z$ for different temperatures and chemical potentials. In the equilibrium case, we define $n_{\mathrm{ex}}=\frac{1}{2}|n_{\uparrow }+n_{\downarrow }-1|$. The factor $1/2$ is introduced to fairly compare with the bath-doped system where both doublon and holons are present.

Figure 6

(a) Spectral function $A\left(\omega \right)$ and (b) occupied spectrum $A\left(\omega \right)f\left(\omega \right)$ under various bath coupling $\mathrm{\Gamma }/{10}^{-5}=0.5,1.0,...,4.5$ from blue to red. The dashed line is the result of real-time simulation. It roughly fits the red curve with $\mathrm{\Gamma }=4.5\times {10}^{-5}$. The inset of (b) compares the real-time solution, the steady state with square bath coupling, and the equilibrium state of $\beta =11.2$ at half filling, which all have rather close AFM order parameters. $W=1.0$ and $V=1.8$ are used for the bath-coupled steady-state system.

Figure 7

Intertwined spin and orbital order in a photodoped two-band Hubbard model. (a) A-type AFM order parameter $S_z$ versus orbital order parameter $X_3$. The inset shows a sketch of the intertwined spin and orbital order in equilibrium, indicating the orbital wave function and the spin (green arrows) of a hole per site. The yellow solid line shows orders of transient photodoped systems, i.e., after excitation with an electric field pulse. The yellow dashed line shows equilibrium order for different temperatures ($T$ increases from right to left along the curve, indicated by the yellow arrow). The red line and dots show orders in the presence of semielliptic baths (the steady-state photodoped states), with $\mathrm{\Gamma }=0.3,0.4,...,1.5\times {10}^{-5}$. A bath bandwidth $W=0.3$ and asymmetric $V_U=1.2,V_L=-1.27$ are chosen to maintain the filling $n=3$. The blue hollow squares correspond to a system coupled to a square bath, with a variety of different $\mathrm{\Gamma }$'s. $W=0.05$ and $V_U=1.3,V_L=-1.4$. The point corresponding to $\mathrm{\Gamma }=0$ (equilibrium) is marked as a black triangle. (b) Distribution functions obtained for a coupling to square baths. The curves from blue to red, with increasing excitation density, correspond to the hollow squares in panel (a).

Figure 8

Nonequilibrium steady states pumped up by the fermion baths. In the left panel, the dashed line indicates an equilibrium initial guess, while the solid lines correspond to a “polarized” initial guess. In the right panel, the red dashed line shows the distribution function of the polarized initial state. The blue dashed line shows the fermion-bath spectrum (in the upper Hubbard band) and the blue shaded area shows its occupation.