Slowdown of the Electronic Relaxation Close to the Mott Transition

We investigate the time-dependent reformation of the quasiparticle peak in a correlated metal near the Mott transition, after the system is quenched into a hot electron state and equilibrates with an environment which is colder than the Fermi-liquid crossover temperature. Close to the transition, we identify a purely electronic bottleneck time scale, which depends on the spectral weight around the Fermi energy in the bad metallic phase in a nonlinear way. This time scale can be orders of magnitude larger than the bare and renormalized electronic hopping time, so that a separation of electronic and lattice time scales may break down. The results are obtained using nonequilibrium dynamical mean-field theory and a slave-rotor representation of the Anderson impurity model.

Figure 1

(a) Equilibrium density of state $A(\omega )$ for three different temperatures throughout the metal-insulator crossover at $U=4$. (b),(c) $A(\omega )$ in equilibrium (dashed line) compared to the time-dependent spectral function $A(t,\omega )$ for $\lambda =0.5$ at times $t=20$ (blue), $t=50$ (red), and $t=80$ (green). (d) The height of the quasiparticle peak $A(t,\omega =0)$ as a function of time for $U=4$, bath temperatures $\beta =5$, 6.5, 7.5, 8, 10 (from bottom to top) and $\lambda =0.5$. Symbols on the right vertical axis correspond to the equilibrium value $A(\omega =0)$ at the same temperatures. (e) The height of the quasiparticle peak $A(t,\omega =0)$ as a function of time for $U=4$, 4.1, 4.25, bath temperature $\beta =10$, and $\lambda =0.5$. Arrows indicate the time $t_{\max }$ [Fig. 3]. The inset of (e) plots the time-dependent spectral function $A_{\mathrm{sc}}(t_{\mathrm{sc}})=A(t,\omega =0)$ as a function of rescaled time $t_{\mathrm{sc}}=t/{\tau }^{*}$ with arbitrary rescaling of the vertical axis (see main text).

Figure 2

(a)–(c) Spinon ($f$) and rotor ($X$) spectral functions in equilibrium for $U=4$ and temperatures in the bad metal regime [$\beta =1$, (a)], the crossover [$\beta =5$, (b)] and the metallic phase ([$\beta =7$, (c)]. (d) The spinon lifetime (inverse width of the peak) as a function of $\beta$ for different values of $U$. Triangular points are calculated using the approximate expression Eq. (4). Dashed lines indicate the maximum spinon lifetime ${\tau }_{\max }$ during the relaxation process (see main text and Fig. 3).

Figure 3

(a) Retarded spinon Green’s function $G_f^{\text{ret}}(t,t-s)$ as a function of relative time $s$ for various different times $t$ ($U=4$, $\beta =10$, $\lambda =0.5$), and in equilibrium (dashed line). The slope decreases for $t\lesssim 32$ and increases for $t\gtrsim 32$. (b) Inverse of the slope [Eq. (3)] as a function of $t$ for $\lambda =0.5$, $\beta =10$, $s_0=16$ and various values $U$ below the metal-insulator transition; $s_0=16$ is chosen large enough so that $G_f(t,t-s_0)$ reflects the low-energy part of the Green’s function (the Lorentzian peak). (c) Crossover time $t_{\max }$ plotted against ${\tau }_{\max }$, where $(t_{\max },{\tau }_{\max })$ corresponds to the maximum of the curves ${\tau }_{\mathrm{ne}}(t)$ in panel (b).

Figure 4

(a) Test of the quasiequilibrium relation $G_X^{>}(\omega ,t)/G_X^{<}(\omega ,t)=e^{{\beta }_{\text{eff}}\omega }$ at $t=80$ for $U=4$, 4.2, 4.3 ($\lambda =0.75$, $\beta =10$). (b) Integrated spectral weight $I(t)={\int }_0^{0.5}d\omega A_X(t,\omega )$ of the rotor at low energy. Square half-filled points at $t=80$ correspond to the equilibrium values. Arrows indicate the $t_{\max }$ [Fig. 3].