Comparative study of nonequilibrium insulator-to-metal transitions in electron-phonon systems

We study equilibrium and nonequilibrium properties of electron-phonon systems described by the Hubbard-Holstein model using dynamical mean-field theory. In equilibrium, we benchmark the results for impurity solvers based on the one-crossing approximation and slave-rotor approximation against non-perturbative numerical renormalization group reference data. We also examine how well the low-energy properties of the electron-boson coupled systems can be reproduced by an effective static electron-electron interaction. The one-crossing and slave-rotor approximations are then used to simulate insulator-to-metal transitions induced by a sudden switch-on of the electron-phonon interaction. The slave-rotor results suggest the existence of a critical electron-phonon coupling above which the system is transiently trapped in a nonthermal metallic state with coherent quasiparticles. The same quench protocol in the one-crossing approximation results in a bad metallic state.

Figure 1

DMFT phase diagram of the Hubbard-Holstein model obtained for ${\omega }_0=0.2$ using SR-WC (purple circles points), OCA-WC (brown square points), NRG (orange triangle points), and QMC [10] (blue diamond points with error bars) impurity solvers. The arrows mark the el-ph couplings for which the analysis of the spectral functions is presented in Fig. 2. The additional green line represents the SR-WC phase boundary at $\beta =20$. The NRG phase boundary was determined for a larger discretization $\mathrm{\Lambda }=2.5$ and broadening $\alpha =0.3$ parameter than in the rest of the manuscript.

Figure 2

Equilibrium spectral function $A\left(\omega \right)$ obtained from OCA-WC (solid red lines), NRG (dashed blue lines), and SR-WC (dashed-dotted green lines) for $U\in \{4.2,4.6,5.5\}$, $g\in \{0.134,0.268,0.4,0.44\}$, and fixed phonon frequency ${\omega }_0=0.2$. Panels on the same row show results for the same Hubbard interaction, while the vertically aligned panels correspond to a fixed el-ph coupling. For $U=4.2$ and $\lambda =0.968$, the OCA-WC calculation fails to converge.

Figure 3

Equilibrium spectral function $A\left(\omega \right)$ obtained from OCA-WC (solid red lines), NRG (dashed blue lines), and SR-WC (dashed-dotted green lines) for $U\in \{4.2,4.6,5.5\}$, $g\in \{0.3,0.6,0.9\}$, and fixed phonon frequency ${\omega }_0=1.0.$ Missing data for OCA-WC indicate that a converged solution could not be obtained.

Figure 4

Effect of the phonon frequency on the electronic spectral function for ${\omega }_0\in \{0.2,1.0\}$ at $\beta =30$ ($\beta =20$ for SR-WC). (a), (c), and (e) present spectral densities for $U=4.6$ while (b), (d), and (f) show the corresponding results for $U=5.5$. The el-ph coupling strength is fixed at $\lambda =0.81$.

Figure 5

(a) SR-WC results for the integral over the low-energy PES $I={\int }_{-0.2}^{0.2}I\left(\omega \right)d\omega$ obtained from the purely electronic model (red line) and the electron-boson coupled system (horizontal lines) for $U\in \{4.2$ (solid bars with circles), 4.6 (dashed bars with triangles)$\}$, and different el-ph couplings $g\in \{0.134,0.268,0.4,0.44\}$, whose values are given in the color bar. Comparison of the spectral function for the Hubbard-Holstein model (blue full line) at $g=0.4$ and $U=4.6$ (b) and $U=4.2$ (c) and the Hubbard model with the effective interactions $U_{\text{eff}}\approx 4.49$ (b) and $4.11$ (c). The phonon frequency is ${\omega }_0=0.2$.

Figure 6

Similar analysis as in Fig. 5 for the OCA-WC method. The effective interaction for $U=4.6,g=0.4$ is $U_{\text{eff}}\approx 4.4$ (b), while for $U=4.2,g=0.4$, it is given by $U_{\text{eff}}=4.04$ (c). The phonon frequency is ${\omega }_0=0.2$.

Figure 7

Time-dependent double occupancy (a) and kinetic energy (b) within OCA-WC (solid line) and SR-WC (dashed line) at $U\in \{4.4,4.6\}$, ${\omega }_0=0.2$, $g=0.44$, and $\beta =\{20$ (SR-WC), $30\left(\mathrm{OCA}\text{−}\mathrm{WC}\right)\}$.

Figure 8

Time evolution of the double occupancy (colored bar) within SR-WC (a) and OCA-WC (b) at $\beta =\{20$ (SR-WC), $30\left(\mathrm{OCA}\text{−}\mathrm{WC}\right)\}$. The color bar represents time. Blue lines indicate the phase transition of the Hubbard model and red lines show the corresponding results for the Hubbard-Holstein model at ${\omega }_0=0.2$ and $g=0.44$.

Figure 9

(a)–(c) SR-WC results for the time-dependent spectral functions ($A(\omega ,t)$) at $U\in \{4.4,4.6,5.5\}$, ${\omega }_0=0.2$ and final el-ph coupling $g=0.44$. (d)–(f) Analogous OCA-WC results as a function of time at $U\in \{4.4,4.6,5.5\}$, ${\omega }_0=0.2$ and final el-ph coupling $g=0.44$. The color bar represents time. (g)–(i) Difference between the initial ($t=2$) and final ($t=24$) spectral functions plotted in (a)–(f) at $U\in \{4.4,4.6,5.5\}$, ${\omega }_0=0.2$ from left to right for SR-WC (dashed black line) and OCA-WC (orange solid line).

Figure 10

[(a) and (b)] SR-WC results for $A^{<}(\omega ,t)/A^{>}(\omega ,t)$ at $U\in \{4.4,4.6\}$, ${\omega }_0=0.2$ and final el-ph coupling $g=0.44$. [(c) and (d)] Analogous OCA-WC results as a function of time at $U\in \{4.4,4.6\}$, ${\omega }_0=0.2$ and final el-ph coupling $g=0.44$. The color bar represents time. Dashed black and brown lines are low-energy linear fits of $exp(-{\beta }_{\mathrm{eff}}\omega )$ to $A^{<}(\omega ,t)/A^{>}(\omega ,t)$ at $t=1.5$ and $t=24$, respectively.

Figure 11

(a) SR-WC results for the time-dependent integral over the low-energy PES $I\left(t\right)={\int }_{-0.2}^{0.2}I(\omega ,t)d\omega$ over the associated value at $t=15$ as a function of time at $U\in \{4.2,4.4,4.6\}$. (b) SR-WC results for the integral over the low-energy PES obtained from the purely electronic model (red line) and the electron-boson coupled system (colored bars) for $U\in \{4.4,4.6\}$, ${\omega }_0=0.2$, $g=0.44$, and $\beta =20$.

Figure 12

(a) OCA-WC results for the time-dependent integral over the low-energy PES $I\left(t\right)={\int }_{-0.2}^{0.2}I(\omega ,t)d\omega$ over the associated value at $t=15$ as a function of time at $U\in \{4.2,4.4,4.6,4.8\}$, ${\omega }_0=0.2$, $g=0.44$ and $\beta =30$. (b) OCA-WC results for the integral over the low-energy PES obtained from the purely electronic model (red line) and the electron-boson coupled system (colored bars) for $U\in \{4.4,4.6\}$ and $\beta =30$.

Figure 13

Equilibrium spectral function $A\left(\omega \right)$ obtained from OCA-WC, NRG, and SR-WC for $\lambda \in \left\{0.09\text{(red}\text{lines),}0.36\text{(blue}\text{lines),}0.81\text{(green}\text{lines)}\right\}$, ${\omega }_0=0.2$ and $U\in \{4.2,4.6,5.5\}$. Panels on the same row are computed using the indicated approximation. The vertically aligned panels describe systems at a fixed Hubbard interaction.

Figure 14

Equilibrium spectral function $A\left(\omega \right)$ obtained from OCA-WC, NRG, and SR-WC for $\lambda \in \left\{0.09\text{(red}\text{lines),}0.36\text{(blue}\text{lines),}0.81\text{(green}\text{lines)}\right\}$, ${\omega }_0=1.0$ and $U\in \{4.2,4.6,5.5\}$. Panels on the same row are computed within the mentioned approximation. The vertically aligned panels describe systems at a fixed electron-phonon coupling. Missing OCA-WC data indicate that the solutions cannot be converged.

Figure 15

Renormalized phonon frequency obtained from various approximations. The dashed (dot-dashed) blue (dark green) line is the linear fit of the SR-WC (OCA-WC) results for ${\omega }_0=\{0.2,0.5\}$ and $U<U_{\mathrm{c}}$. SR-WC results are computed at $\beta =20$.

Figure 16

Metal to Mott insulator crossover in the noncrossing approximation with the Lang-Firsov transformation (NCA-LF) for $\beta =30$ and ${\omega }_0\in \{0.2,1,5\}$. (Inset) Metal to Mott insulator phase boundaries in the one crossing approximation with Lang-Firsov transformation (OCA-LF) at $\beta =30$ and ${\omega }_0=0.2$. The phase boundary in the large phonon-frequency limit, $U_c=U_c(g=0)+\frac{2g^2}{{\omega }_0}$, is also shown (dotted black lines).

Figure 17

Spectral functions from OCA-LF and NRG averaged over six different discretization parameters, namely $\mathrm{\Lambda }\in \{1.8,1.9,2.0,2.1,2.2,2.3\}$ and reduced broadening parameter $\alpha =0.05$, at $U=10$, $\beta =30$, and ${\omega }_0=1$ for varying electron phonon coupling strength $g^2/{\omega }_0\in \{0.81,1,1.5,2\}$.

Figure 18

Spectral functions from OCA-LF and NRG averaged over six different discretization parameters, namely $\mathrm{\Lambda }\in \{1.8,1.9,2.0,2.1,2.2,2.3\}$ and reduced broadening parameter $\alpha =0.05$, at $U=10$, $\beta =30$, and ${\omega }_0=0.2$ for varying electron phonon coupling strength $g^2/{\omega }_0\in \{0.81,1,1.5,2\}$.

Figure 19

Spectral functions from NRG at four different discretization parameters, namely $\mathrm{\Lambda }\in \{1.8,2.0,2.1,2.3\}$ and reduced broadening parameter $\alpha =0.05$, at $U=10$, $\beta =30$, and ${\omega }_0=0.2$ for varying electron phonon coupling strength $g^2/{\omega }_0\in \{0.81,1,1.5,2\}$.