Time-domain pumping a quantum-critical charge density wave ordered material

We determine the exact time-resolved photoemission spectroscopy for a nesting driven charge density wave (described by the spinless Falicov-Kimball model within dynamical mean-field theory). The pump-probe experiment involves two light pulses: the first is an ultrashort intense pump pulse that excites the system into nonequilibrium, and the second is a lower amplitude, higher frequency probe pulse that photoexcites electrons. We examine three different cases: the strongly correlated metal, the quantum-critical charge density wave, and the critical Mott insulator. Our results show that the quantum critical charge density wave has an ultraefficient relaxation channel that allows electrons to be de-excited during the pump pulse, resulting in little net excitation. In contrast, the metal and the Mott insulator show excitations that are closer to what one expects from these systems. In addition, the pump field produces spectral band narrowing, peak sharpening, and a spectral gap reduction, all of which rapidly return to their field free values after the pump is over.

Figure 1

Kadanoff-Baym-Keldysh time contour, which runs from a minimum time to a maximum time along the real time axis, then backwards to the minimum time, and then parallel to the imaginary time axis for a length given by the inverse of the initial equilibrium temperature.

Figure 2

Equilibrium phase diagram of the Falicov-Kimball model as functions of temperature and the Coulomb interaction. There are five different phases: disordered metal (strongly correlated metal) with no gap in the DOS (red); Mott insulator with a Mott gap in the DOS (blue); weakly correlated CDW insulator with a CDW gap (green); strongly correlated CDW insulator with CDW gap in the DOS (purple); correlated CDW metal with a CDW gap in the DOS but nonzero DOS at the chemical potential level (gold).

Figure 3

Equilibrium DOS for $U=0.5$ (correlated metal) at different temperatures: $T=0.0178$ corresponds to $\mathrm{\Delta }{n}_f=0.49$ (blue); $T=0.0278$ corresponds to $\mathrm{\Delta }{n}_f=0.4$ (green); $T=0.0326$ corresponds to $\mathrm{\Delta }{n}_f=0.2$ (red); $T=0.04$ corresponds to $\mathrm{\Delta }{n}_f=0$ (black, dashed). Insets show the temperature dependence of the corresponding order parameters of the localized (left) and itinerant (right) electrons. Note how the subgap DOS rapidly forms and fills in the gap region, while the signature of the spectral gap in the DOS remains fixed at $U$ for all $T$ in the ordered phase.

Figure 4

Equilibrium DOS for $U=0.86$ (material at the CDW quantum critical point) at different temperatures: $T=0.03$ corresponds to $\mathrm{\Delta }{n}_f=0.495$ (blue); $T=0.047$ corresponds to $\mathrm{\Delta }{n}_f=0.4$ (green); $T=0.053$ corresponds to $\mathrm{\Delta }{n}_f=0.2$ (red); $T=0.06$ corresponds to $\mathrm{\Delta }{n}_f=0$ (black, dashed). Insets show the temperature dependence of the order parameters of localized (left) and itinerant (right) electrons. Note how the subgap DOS rapidly forms at the chemical potential producing a metal and then fills in the gap region, while the signature of the gap in the DOS remains fixed at $U$ for all $T$ in the ordered phase.

Figure 5

Equilibrium DOS for $U=1.4$ (critical point for the Mott transition) at different temperatures: $T=0.033$ corresponds to $\mathrm{\Delta }{n}_f=0.495$ (blue); $T=0.0596$ corresponds to $\mathrm{\Delta }{n}_f=0.4$ (green); $T=0.07$ corresponds to $\mathrm{\Delta }{n}_f=0.2$ (red); $T=0.08$ corresponds to $\mathrm{\Delta }{n}_f=0$ (black, dashed). Insets show the temperature dependence of the order parameters of the localized (left) and itinerant (right) electrons. Note how the subgap DOS rapidly forms but does not fill the entire gap region due to Mott physics suppressing the DOS at the chemical potential; the signature of the gap in the DOS still remains fixed at $U$ for all $T$ in the ordered phase.

Figure 6

False color plot of the PES response function for $U=0.5$ at different temperatures with a logarithmic color scale: (a) $T=0.0178$ corresponds to $\mathrm{\Delta }{n}_f=0.49$; (b) $T=0.0278$ corresponds to $\mathrm{\Delta }{n}_f=0.4$; (c) $T=0.0326$ corresponds to $\mathrm{\Delta }{n}_f=0.2$; (d) $T=0.04$ corresponds to $\mathrm{\Delta }{n}_f=0$. The pump field with $E_0=30$ is plotted above, and the probe pulse width is ${\sigma }_b=7$. Note how the pump pulse does excite a substantial number of electrons to the upper band, but it also de-excites electrons (similar to what happens in the simplified CDW model) so that the net excitation at the end of the pulse is small.

Figure 7

PES response function for $U=0.5$ at temperature $T=0.0178$ $\left(\mathrm{\Delta }{n}_f=0.49\right)$; this corresponds to vertical cuts through the false color image in the previous figure. Different curves correspond to different time delays $t_0{}^{\prime }$ for the probe pulse and have been offset in the vertical for clarity. Thin red lines are a guide to the eye. The upper inset shows results for the total spectral weight of the PES response for different $t_0{}^{\prime }$. The loss of weight at the edges signifies that there is not enough information in the calculated Green's function to properly construct the PES. Data from those red points are not included in the main figure. The lower inset shows the time evolution of the conduction electron order parameter, which goes from high order to low order after the pulse has ended.

Figure 8

False color plot of the PES response function for $U=0.86$ (quantum critical material) at different temperatures on a logarithmic color scale: (a) $T=0.03$ corresponds to $\mathrm{\Delta }{n}_f=0.495$; (b) $T=0.047$ corresponds to $\mathrm{\Delta }{n}_f=0.4$; (c) $T=0.053$ corresponds to $\mathrm{\Delta }{n}_f=0.2$; (d) $T=0.06$ corresponds to $\mathrm{\Delta }{n}_f=0$. The pump field with $E_0=30$ is plotted above, and the probe pulse width is ${\sigma }_b=7$.

Figure 9

PES response function for $U=0.86$ at temperature $T=0.03$ $\left(\mathrm{\Delta }{n}_f=0.495\right)$. Different curves correspond to different times $t_0{}^{\prime }$ of the probe pulse. Thin red lines are a guide to the eye. The upper inset shows results of integration of the PES response for different $t_0{}^{\prime }$. The lower inset shows the time evolution of the conduction electron order parameter, which transiently is reduced, but then is restored as the pulse ends.

Figure 10

False color image of the PES response function for $U=1.4$ (critical Mott insulator) at different temperatures on a logarithmic color scale: (a) $T=0.033$ corresponds to $\mathrm{\Delta }{n}_f=0.495$; (b) $T=0.0596$ corresponds to $\mathrm{\Delta }{n}_f=0.4$; (c) $T=0.07$ corresponds to $\mathrm{\Delta }{n}_f=0.2$; (d) $T=0.08$ corresponds to $\mathrm{\Delta }{n}_f=0$. The pump field with $E_0=30$ is plotted above, and the probe pulse width is ${\sigma }_b=7$.

Figure 11

PES response function for $U=1.4$ at temperature $T=0.033$ $\left(\mathrm{\Delta }{n}_f=0.495\right)$. Different curves correspond to different times $t_0{}^{\prime }$ of the probe pulse. Thin red lines are a guide to the eye. The upper inset shows the results of the integration of the PES response for different $t_0{}^{\prime }$. The lower inset shows the time evolution of the conduction electron order parameter, which goes from high order to low order after the pulse has ended.

Figure 12

DOS (black) and convolved DOS (red) for $U=0.5$ at different temperatures: (a) $T=0.0178$ corresponds to $\mathrm{\Delta }{n}_f=0.49$; (b) $T=0.0278$ corresponds to $\mathrm{\Delta }{n}_f=0.4$; (c) $T=0.0326$ corresponds to $\mathrm{\Delta }{n}_f=0.2$; (d) $T=0.04$ corresponds to $\mathrm{\Delta }{n}_f=0$.

Figure 13

DOS (black) vs convolved DOS (red) for $U=0.86$ at different temperatures: (a) $T=0.03$ corresponds to $\mathrm{\Delta }{n}_f=0.495$; (b) $T=0.047$ corresponds to $\mathrm{\Delta }{n}_f=0.4$; (c) $T=0.053$ corresponds to $\mathrm{\Delta }{n}_f=0.2$; (d) $T=0.06$ corresponds to $\mathrm{\Delta }{n}_f=0$.

Figure 14

DOS (black) vs convolved (red) for $U=1.4$ at different temperatures: (a) $T=0.033$ corresponds to $\mathrm{\Delta }{n}_f=0.495$; (b) $T=0.0596$ corresponds to $\mathrm{\Delta }{n}_f=0.4$; (c) $T=0.07$ corresponds to $\mathrm{\Delta }{n}_f=0.2$; (d) $T=0.08$ corresponds to $\mathrm{\Delta }{n}_f=0$.