Lindbladian route towards thermalization of a Luttinger liquid

We study the nonequilibrium dynamics of a Luttinger liquid after a simultaneous quantum quench of the interaction and dissipative quench to the environment within the realm of the Lindblad equation. When the couplings to the environment satisfy detailed balance, the system is destined to thermalize, which we follow using bosonization. The thermodynamic entropy of the system, which also encodes information about the entanglement with the environment, either exhibits a maximum and minimum or grows monotonically before reaching its thermal value in a universal fashion. The single-particle density matrix reveals interesting features in the spatiotemporal “phase diagram,” including thermal, prethermal, and dissipation enhanced sudden quench Luttinger liquid behavior, as well as dissipation, enhanced Fermi liquid response.

Figure 1

The time evolution of the entropy at several temperatures and $K=0.7$. The black dotted line indicates the steady, thermal value. The gray, dashed lines exhibit Eq. (7) describing the late time behavior of the entropy. (Inset) $K$ dependence of the temperature $T_s$ separating the two characteristics of the entropy evolution.

Figure 2

Spatial decay of the single-particle density matrix at several time instants in the early stage of time evolution for $\gamma t\ll {\ell }_T$ with $K=0.5$, $c/\gamma ={10}^3$, and $\alpha /{\ell }_T={10}^{-5}$. The three characteristic behaviors, namely, thermal LL, dissipation enhanced sudden quench LL and dissipation enhanced Fermi liquid behaviors are clearly visible with increasing $x$.

Figure 3

Spatial decay of the single-particle density matrix at several time instants for late times, $\gamma t\gg {\ell }_T$ with $K=0.5$, $c/\gamma ={10}^3$, and $\alpha /{\ell }_T={10}^{-5}$. The three main characteristics, namely, power-law decay within the thermal correlation length, thermal exponential decay above the thermal correlation length and prethermal LL behavior are identified with increasing $x$.

Figure 4

The spatiotemporal map or “phase diagram” of the single-particle density matrix, displaying the various behaviors such as dissipation enhanced sudden quench LL, dissipation enhanced Fermi liquid, prethermal, and thermal LL. The dominant timescales are the same as for the time evolution of the entropy.

Figure 5

Time dependence of $n_q\left(t\right)$ at different wave numbers. The function starts at $n_0\left(K\right)={(1-K)}^2/\left(4K\right)$ in each wave-number channel and then monotonously tends to the thermal value, $n(T,q)$.

Figure 6

The time evolution of total energy at several temperature values. The black dotted lines indicate the steady value of the energy, (A4). The gray dashed lines show the long time behavior, (A3). For the plot, $K=0.7$ is set.

Figure 7

Time dependence of $N_q\left(t\right)$ at different wave numbers. At high wave numbers, $N_q\left(t\right)$ exhibits a maximum and decreases back to its thermal value, $n(T,q)$. At low wave numbers, the function increases monotonously. Furthermore, it is remarkable that high momentum modes thermalize faster than low energy modes.

Figure 8

The time evolution of the Rényi entropy at several temperature values. The black dotted lines indicate the steady state thermal value, $S_{2,T}$. The gray dashed lines show the long time behavior which is analytically obtained as $S_2\left(t\right)=S_{2,T}-L/\left(\pi \gamma t\right)$. For the plot, $K=0.7$ is set.