Nonequilibrium self-energy functional approach to the dynamical Mott transition

The real-time dynamics of the Fermi-Hubbard model, driven out of equilibrium by quenching or ramping the interaction parameter, is studied within the framework of the nonequilibrium self-energy functional theory. A dynamical impurity approximation with a single auxiliary bath site is considered as a reference system, and the time-dependent hybridization is optimized as prescribed by the variational principle. The dynamical two-site approximation turns out to be useful to study the real-time dynamics on short and intermediate time scales. Depending on the strength of the interaction in the final state, two qualitatively different response regimes are observed. For both weak and strong couplings, qualitative agreement with previous results of nonequilibrium dynamical mean-field theory is found. The two regimes are sharply separated by a critical point at which the low-energy bath degree of freedom decouples in the course of time. We trace the dependence of the critical interaction of the dynamical Mott transition on the duration of the interaction ramp from sudden quenches to adiabatic dynamics and therewith link the dynamical to the equilibrium Mott transition.

Figure 1

Illustration of the original (left) and the reference system (right). Black solid lines indicate nearest-neighbor hopping between correlated sites (red filled dots). The original system is the Hubbard model with additional uncorrelated sites (blue filled dots) which, however, are decoupled and merely ensure that both Hamiltonians operate in the same Hilbert space. In the reference system (right) these “bath sites” are locally coupled to the otherwise disconnected correlated sites via a hybridization parameter $V^{\prime }\left(t\right)$, indicated by blue solid lines.

Figure 2

(a) Optimal variational parameter $V_{\mathrm{opt}}^{\prime }$ as a function of the interaction for different inverse temperatures increasing from $\beta =10$ (red curve) to $\beta =200$ (blue curve), and (b) the respective double occupancies. The inset in (b) shows the Maxwell construction for $\beta =100$: The mid arrow indicates the value for $U_{\mathrm{c}}$, the outer arrows point at the spinodal points, which define the region where metallic and insulating solutions coexist.

Figure 3

Phase diagram of the Mott transition in the half-filled Hubbard model on a one-dimensional lattice as obtained from the DIA with a two-site reference system. Below the critical temperature $T_c$ metallic solutions exist up to interactions $U\le U_{\mathrm{c}2}$, insulating solutions exist down to $U\ge U_{\mathrm{c}1}$, and in between both coexist. Red line: first-order phase boundary $U_{\mathrm{c}}\left(T\right)$.

Figure 4

Time dependencies of the optimal hybridization $V_{\mathrm{opt}}^{\prime }$, the double occupancy $〈n_{\uparrow }{n}_{\downarrow }〉$, and the total energy $E_{\mathrm{tot}}$ for interaction quenches starting from $U_{\mathrm{ini}}=0.01$ to different $U_{\mathrm{fin}}$ (see color labels). Left: weak-coupling regime, $U_{\mathrm{fin}}<U_{\mathrm{c}}^{\mathrm{dyn}}$. Right: strong-coupling regime, $U_{\mathrm{fin}}>U_{\mathrm{c}}^{\mathrm{dyn}}$.

Figure 5

Long-time averages (points) and fluctuations (shaded areas) of the optimal hybridization ${V^{\prime }}_{\mathrm{opt}}^2$, the double occupancy $〈n_{\uparrow }{n}_{\downarrow }〉$, and the total energy $E_{\mathrm{tot}}$. Red lines: critical interaction $U_{\mathrm{c}}^{\mathrm{dyn}}$ separating weak- and strong-coupling regime. Blue line: equilibrium values of the double occupancy at zero temperature. Green lines: thermal values of the double occupancy, which in the weak-coupling regime almost coincide with the long-time average, but in the strong-coupling regime match the minima of the double-occupancy oscillations (squares). Orange line: total energy right after the quench.

Figure 6

Left: Fourier transform ($\mathcal{F}$) of the double occupancy. Plots have been shifted for better visibility. Right: linear dependence of the dominant frequency on the final interaction $U_{\mathrm{fin}}$ in the strong coupling regime $U_{\mathrm{fin}}>U_{\mathrm{c}}^{\mathrm{dyn}}$. Note the interaction-independent small beating frequencies (left).

Figure 7

Dynamical decoupling of the bath site at the critical point ($U_{\mathrm{fin}}=U_{\mathrm{c}}^{\mathrm{dyn}}$) (top) and the corresponding dynamics of the double occupancy (bottom). Additional curves: dynamics for final interactions differing by less than $0.3\%$ from $U_{\mathrm{c}}^{\mathrm{dyn}}$. Note the strong impact upon tiny changes of $U_{\mathrm{fin}}$ indicating a sharp transition between the two regimes.

Figure 8

Check of adiabaticity for ramps of the interaction from $U_{\mathrm{ini}}=0.01$ to $U_{\mathrm{fin}}=12$ with different ramp times ($\mathrm{\Delta }{t}_{\mathrm{ramp}}\in \{0.5,1,2,4,6,8,10,12,15\}$, colored from purple to cyan). The total energy during the ramp is plotted as a function of the instantaneous interaction (top left), i.e., $E_{\mathrm{tot}}\left(U\right)\equiv E_{\mathrm{tot}}\left(t\left(U\right)\right)$, as obtained from the inverse of (cosine) ramp profile $U\left(t\right)$, Eq. (6), (bottom) and from the time-dependent total energy $E_{\mathrm{tot}}\left(t\right)$ (right). The results are contrasted with the total energy after a quench (straight red line) and the equilibrium energy at the same final interaction (dashed red line).

Figure 9

Time dependencies of the optimal hybridization $V_{\mathrm{opt}}^{\prime }$, the double occupancy $〈n_{\uparrow }{n}_{\downarrow }〉$, and the total energy $E_{\mathrm{tot}}$ (from top to bottom) for ramps of the interaction with $\mathrm{\Delta }{t}_{\mathrm{ramp}}=7$, starting from $U_{\mathrm{ini}}=0.01$ to different $U_{\mathrm{fin}}$ (as indicated in the top panel) in the weak- and the strong-coupling regime as well as right at the dynamical critical point.

Figure 10

Time dependent momentum distribution for a ramp with $\mathrm{\Delta }{t}_{\mathrm{ramp}}=7$ ending at different interactions (a) below, (b) above, and (c) right at the critical point. Precise numbers are given in the top left corner of each plot. For the latter the long-time average of the momentum distribution is fitted with an equilibrium distribution within the Hubbard-I approximation (red line). Relative errors of fits for the momentum distribution, the double occupancy, and the total energy (from top to bottom) at different temperatures are shown in (d), an error estimate for the effective temperature $T_{\mathrm{eff}}\approx 0.5$ is indicated by a red shaded area.

Figure 11

Dynamical critical interaction $U_{\mathrm{c}}^{\mathrm{dyn}}$ as a function of the ramp time $\mathrm{\Delta }{t}_{\mathrm{ramp}}$ determined for cosine-shaped ramps starting from $U_{\mathrm{ini}}=0.01$. The equilibrium value $U_{\mathrm{c}2}\approx 8.59$ for the critical interaction at zero temperature is indicated by a gray dashed line.