Relaxation dynamics of an exactly solvable electron-phonon model

We address the question whether observables of an exactly solvable model of electrons coupled to (optical) phonons relax into large time stationary state values and investigate if the asymptotic expectation values can be computed using a stationary density matrix. Two initial nonequilibrium situations are considered. A sudden quench of the electron-phonon coupling, starting from the noninteracting canonical equilibrium at temperature $T$ in the electron as well as in the phonon subsystems, leads to a rather simple dynamics. A richer time evolution emerges if the initial state is taken as the product of the phonon vacuum and the filled Fermi sea supplemented by a highly excited additional electron. Our model has a natural set of constants of motion, with as many elements as degrees of freedom. In accordance with earlier studies of such type of models, we find that expectation values which become stationary can be described by the density matrix of a generalized Gibbs ensemble which differs from that of a canonical ensemble. For the model at hand, it appears to be evident that the eigenmode occupancy operators should be used in the construction of the stationary density matrix.

Figure 1

(Color online) Eigenmodes ${\lambda }_{\pm }\left(q\right)$ and coefficients $c_q^2$ and $s_q^2$ of the eigenvectors for $\Gamma =0.01$ and $\Omega =0.1$. Note the sharp transition (discontinuity) to the noninteracting values at $q_c$.

Figure 2

(Color online) The GGE and canonical momentum distribution function of the fermions. The system parameters are $\Gamma =0.01$, $\Omega =0.1$, $\nu =2\pi /\left(L{q}_{\text{c}}\right)={10}^{-3}$ and the dimensionless initial temperature is $\tau =T/\left(v_{\text{F}}{q}_{\text{c}}\right)=0.1$. The canonical distribution is the best fit to the GGE one with the fitted temperature ${\tilde{\tau }}_{\text{b}}=0.10275$. The inset shows the absolute value of the difference of the two functions.

Figure 3

(Color online) Time evolution of the (excess) energies in the fermion (solid line, top) and phonon (solid line, middle) subsystems as well as of the energy in the interacting part of the Hamiltonian (solid line, bottom). The dashed lines for $t∊\left[1000,1500\right]$ are the corresponding GGE expectation values. The parameters are $\Gamma =0.01$, $\Omega =0.1$ and the initial temperature is given by $\tau =0.1$. The symbols in the inset show data for the energy in the fermion subsystem obtained at a finite system size $\nu =2\times {10}^{-3}$. The recurrence time is then given by $\left(v_{\text{F}}{q}_{\text{c}}\right)t_r=v_{\text{F}}2\pi /\left(L\nu \right)L/v_{\text{F}}=2\pi /\nu \approx 3142$.

Figure 4

(Color online) The time-dependent part of the phonon momentum distribution function $\delta N_{\text{q}}\left(q,t\right)$ as a function of $q$ and $t$. The parameters are $\Gamma =0.01$, $\Omega =0.2$, and the initial temperature is given by $\tau =0.1$.

Figure 5

(Color online) Momentum distribution function of the fermions as a function of $k$ for different $t$. The GGE distribution obtained for the same system size is shown for comparison. The parameters are $\Gamma =0.03$, $\Omega =0.1$, $\nu ={10}^{-3}$, and the initial temperature is given by $\tau =0.1$.

Figure 6

(Color online) The same as in Fig. 5 but as a function of $t$ for different $k$. For the momenta $k<0$, $1-n_{\text{q}}\left(k,t\right)$ is shown. The GGE expectation values are indicated by the arrows.

Figure 7

(Color online) The momentum distribution function of the fermions $n_{k_0}\left(k,t\right)$. The parameters are $\Gamma =0.001$, $\Omega =0.1$, $\nu ={10}^{-3}$, and $k_0/q_c=0.5$. The dark feature at $k=k_0$ is the occupation of the initially filled level which strongly exceeds the scale of the $z$ axis. Note the scale of the $z$ axis.

Figure 8

(Color online) The same as in Fig. 7 but for $\Gamma =0.01$.

Figure 9

(Color online) The momentum distribution function of the fermions $n_{k_0}\left(k,t\right)$ as a function of $k>0$ for fixed $\left(v_{\text{F}}{q}_{\text{c}}\right)t={10}^3$ and different $\nu =4\times {10}^{-3}$, $2\times {10}^{-3}$, ${10}^{-3}$. The other parameters are $\Gamma =0.01$, $\Omega =0.1$, and $k_0/q_c=0.5$. The inset shows $n_{k_0}\left(k,t\right)/\nu$.

Figure 10

(Color online) Momentum distribution function of the fermions $n_{k_0}\left(k,t\right)$ as a function of $k$ for different $t$. The GGE distribution obtained for the same system size is shown for comparison. The parameters are $\Gamma =0.03$, $\Omega =0.1$, $\nu ={10}^{-3}$, and $k_0/q_c=0.5$.

Figure 11

(Color online) The same as in Fig. 10 but as a function of $t$ for different $k$. For the momenta $k<0$, $1-n_{k_0}\left(k,t\right)$ is shown. The GGE expectation values are indicated by the arrows.