Correlated Mott insulators in strong electric fields: Role of phonons in heat dissipation

We study the spectral and transport properties of a Mott insulator driven by a static electric field into a nonequilibrium steady state. For the dissipation, we consider two mechanisms: wide band fermion reservoirs and phonons included within the Migdal approximation. The electron correlations are treated via nonequilibrium dynamical mean field theory with an impurity solver suitable for strong correlations. We find that dissipation via phonons is limited to restricted ranges of field values around Wannier-Stark resonances. To cover the full range of field strengths, we allow for a small coupling with fermionic baths, which stabilizes the solution. When considering both dissipation mechanisms, we find that phonons enhance the current for field strengths close to half of the gap while lowering it at the gap resonance as compared to the purely electronic dissipation used by Murakami and Werner [Murakami and Werner, Phys. Rev. B 98, 075102 (2018)]. Once phonons are the only dissipation mechanism, the current in the metallic phase is almost one order of magnitude smaller than the typical values obtained by coupling to a fermionic bath. In this case, the transport regime is characterized by an accumulation of charge in the upper Hubbard band described by two effective chemical potentials.

Figure 1

(a) Spectral function $A\left(\omega \right)/t^{*-1}$ at zero electric field $F=0$ for different values of the ratio $\gamma$. The inset (b) shows the magnified quasiparticle peak at $\omega =0$. Default parameters are specified at the beginning of Sec. 4.

Figure 2

(a) Spectral function $A\left(\omega \right)/t^{*-1}$ and (b) nonequilibrium distribution function $F_{\text{neq}}\left(\omega \right)$ for various field strengths at $\gamma =0.065$. Default parameters are specified at the beginning of Sec. 4.

Figure 3

Spectral function $A\left(\omega \right)/t^{*-1}$ (solid) and spectral occupation $N_{\text{e}}\left(\omega \right)/t^{*-1}$ (shaded area) at (a) $F=3t^{*}$, (b) $F=4t^{*}$, (c) $F=5.6t^{*}$, and (d) $F=8t^{*}$. Here $\gamma =0.065$. Default parameters are specified at the beginning of Sec. 4.

Figure 4

(a) WS in-gap states at ${\omega }_{\text{WS},\pm ,\mp }=\pm (U/2)\mp F$ in the spectral function $A\left(\omega \right)/t^{*-1}$, for various fields $F$ at $\gamma =0.065$. Black arrows point to the position of ${\omega }_{\text{WS2},\pm ,\mp 2}=\pm (3U/2)\mp 2F$, while inset (b) shows the magnified region $\omega /t^{*}\in [-1.5,1.5]$ for selected field strengths. (c), (d) Same for $\gamma =0.108$. (e), (f) WS in-gap states ${\omega }_{\text{WS},\pm ,\mp }$ for $\gamma =0.162$ and $\gamma =0.324$. Default parameters are specified at the beginning of Sec. 4.

Figure 5

(a)–(c) Electric field dependence of the steady-state current $j$, double occupancy $d$, and kinetic energy $E_{\text{kin}}$ for various $\gamma ={\mathrm{\Gamma }}_{\text{e}}/{\mathrm{\Gamma }}_{\text{ph}}$. Dashed vertical lines denote the resonant fields $F=U/3, U/2$, and $U$, while the black arrow highlights the resonance at $F\approx 5.6t^{*}$. Default parameters are specified at the beginning of Sec. 4.

Figure 6

(a) Frequency resolved current $j\left(\omega \right)$ (26) at $F=3t^{*}$ and $4t^{*}$. (b) Same quantity shown for $F=5.6t^{*}$ and $8t^{*}$. Dashed vertical lines mark the maxima discussed in the text. Default parameters are specified at the beginning of Sec. 4.

Figure 7

(a) Electron and hole spectral occupation functions $N_{\text{e}}\left(\omega \right)/t^{*-1}$ and $N_{\text{h}}(\omega +F)/t^{*-1}$ and their product at $F=3t^{*}$. Black arrows point at the subpeaks discussed in the text [see also Fig. 6]. (b) Same quantities for $F=5.6t^{*}$. Here $\gamma =0.065$ while default parameters are specified at the beginning of Sec. 4.

Figure 8

Electron spectral function $A\left(\omega \right)/t^{*-1}$ and e-ph SE $-\text{Im}\left[{\mathrm{\Sigma }}_{\text{e-ph}}^{\text{R}}\left(\omega \right)\right]/t^{*}$ at $F=0$ for $\gamma =0$. Dashed vertical lines mark the peaks of $A$ and $-\text{Im}\left({\mathrm{\Sigma }}_{\text{e-ph}}^{\text{R}}\right)$, while arrows highlight the shifts between them. Default parameters are specified at the beginning of Sec. 4.

Figure 9

Color map of the spectral function $A\left(\omega \right)/t^{*-1}$ in the accessible regions (a) $F/t^{*}\in \{3.80,3.82,\cdots ,4.60\}$ and (b) $F/t^{*}\in \{7.80,7.82,\cdots ,8.60\}$. Default parameters are specified at the beginning of Sec. 4.

Figure 10

(a), (c) Spectral function $A\left(\omega \right)/t^{*-1}$, spectral occupation $N_{\text{e}}\left(\omega \right)/t^{*-1}$ (shaded area), nonequilibrium distribution function $F_{\text{neq}}\left(\omega \right)$, and e-ph SE $-\text{Im}\left[{\mathrm{\Sigma }}_{\text{e-ph}}^{\text{R}}\left(\omega \right)\right]/t^{*}$ at $F/t^{*}=4.20$. (b), (d) Same quantities for $F/t^{*}=8.42$. The arrow highlights the energy splitting $\mathrm{\Delta }=F$ that introduces effective chemical potentials for the Hubbard bands. Default parameters are specified at the beginning of Sec. 4.

Figure 11

Electric field dependence of the current (a), (b), double occupation (c), (d), and kinetic energy (e), (f) near the resonances $F/t^{*}\approx U/2$ and $U$, for ${\omega }_{\text{ph}}/t^{*}=\{0.025,0.050,0.075,0.100\}$. The default value of ${\mathrm{\Gamma }}_{\text{ph}}/t^{*}$ is specified at the beginning of Sec. 4.

Figure 12

Frequency resolved current $j\left(\omega \right)$ near the resonances at (a) $F/t^{*}\approx U/2$ and (b) $F/t^{*}\approx U$. Default parameters are specified at the beginning of Sec. 4.

Figure 13

Forward (blue) and backward (red) flow contributions to the frequency-dependent tunneling current (30). (a), (b) For $F=7.8t^{*}$, the backflow overcomes the forwards flow and leads to a negative current. (c) At $F=8.42t^{*}$, the forwards flow dominates. Default parameters are specified at the beginning of Sec. 4.

Figure 14

(a), (d) Spectral function $A\left(\omega \right)/t^{*-1}$ and e-ph SE $-\text{Im}\left[{\mathrm{\Sigma }}_{\text{e-ph}}^{\text{R}}\left(\omega \right)\right]/t^{*}$ displayed for ${\omega }_{\text{ph}}/t^{*}=0.1$ at field strengths which correspond to the current maxima in Figs. 11 and 11. The same quantities are shown for (b), (e) ${\omega }_{\text{ph}}/t^{*}=0.05$ and (c), (f) ${\omega }_{\text{ph}}/t^{*}=0.025$. Default parameters are specified at the beginning of Sec. 4.

Figure 15

Keldysh contour $C_{\kappa }={\gamma }_{-}\cup {\gamma }_{+}$ for real-time arguments.