Quantum Quenches in the Thermodynamic Limit

We introduce a linked-cluster based computational approach that allows one to study quantum quenches in lattice systems in the thermodynamic limit. This approach is used to study quenches in one-dimensional lattices. We provide evidence that, in the thermodynamic limit, thermalization occurs in the nonintegrable regime but fails at integrability. A phase transitionlike behavior separates the two regimes.

Figure 1

Relative difference $\mathrm{\Delta }(E{)}_{17}$ between the last 2 orders in the NLCE calculation of $E^{\mathrm{DE}}$ vs $T_I$ for different values of $t^{\prime }=V^{\prime }$ in the final Hamiltonian. For $T_I\gtrsim 0.7$, $\mathrm{\Delta }(E^{\mathrm{DE}}{)}_{17}$ is zero within machine precision. (Insets) $\mathrm{\Delta }(E{)}_l$ vs $l$ for two values of $T_I$ and the same quenches as in the main panel. The plots show that $E_l^{\mathrm{DE}}$ approaches $E_{18}^{\mathrm{DE}}$ exponentially fast with $l$.

Figure 2

Last order ($l=18$) of the NLCE of $\mathrm{\Delta }{E}^2$ in the DE (open symbols) and the GE (filled symbols) vs $T$, for quenches with $T_I\ge 1$. The quenches with a different initial state ($t^{\prime }=V^{\prime }=0.5$) have $t_I=0.5$, $V_I=1.5$, $t_I^{\prime }=V_I^{\prime }=0.5$, while $t=V=1$ as in the other quenches. Lines joining the data points for the quenches with $t^{\prime }=V^{\prime }=0.5$ are meant to guide the eye and show that $\mathrm{\Delta }{E}^2$ in the DE depends on the initial state. The inset shows $\delta (\mathrm{\Delta }{E}^2{)}_l$ vs $l$ for $T_I=1$ for the same quenches as in the main panel.

Figure 3

Momentum distribution and kinetic energy $K$ after a quench. (a), Last order ($l=18$) of the NLCE for the momentum distribution in the initial state ($T_I=2$), and in the DE and the GE after quenches with $t^{\prime }=V^{\prime }=0$ and 0.5. (b)–(d), Relative differences between $(m_k{)}_l^{\mathrm{DE}}$ and $(m_k{)}_{18}^{\mathrm{GE}}$ [(b),(c)] and between $K_l^{\mathrm{DE}}$ and $K_{18}^{\mathrm{GE}}$ (d) vs $l$ in four sets of quenches with $T_I=2$ [(b),(d)] and $T_I=10$ (c). These results provide strong evidence that thermalization occurs in nonintegrable systems but fails at integrability. Inset in (a), open (filled) symbols report the relative difference between $(m_k{)}_l^{\mathrm{DE}}$ [$(m_k{)}_l^{\mathrm{GE}}$] in the last 2 orders of the NLCE vs $T_I$, for the same quenches as in (b)–(d). For ease of display, there is a four decade gap in the $y$ axis of the inset.

Figure 4

Entropy after a quench. Relative difference $\delta (S{)}_l$ between $S_l^{\mathrm{DE}}$ and $S_{18}^{\mathrm{GE}}$ vs $l$ in quenches with $T_I=2$ (top four plots) and $T_I=10$ (bottom four plots). (Inset) Open (filled) symbols report the relative difference $\mathrm{\Delta }(S{)}_{17}$ between $S_l^{\mathrm{DE}}$ ($S_l^{\mathrm{GE}}$) in the last 2 orders of the NLCE expansion vs $T_I$, for the same quenches as in the main panel. The results in this figure are qualitatively similar to those in Fig. 3. For ease of display, there is a four decade gap in the $y$ axis of the inset.