Quantum Quenches and Relaxation Dynamics in the Thermodynamic Limit

We implement numerical linked cluster expansions (NLCEs) to study dynamics of lattice systems following quantum quenches, and focus on a hard-core boson model in one-dimensional lattices. We find that, in the nonintegrable regime and within the accessible times, local observables exhibit exponential relaxation. We determine the relaxation rate as one departs from the integrable point and show that it scales quadratically with the strength of the integrability breaking perturbation. We compare the NLCE results with those from exact diagonalization calculations on finite chains with periodic boundary conditions, and show that NLCEs are far more accurate.

Figure 1

Dynamics of the kinetic energy per site $k(\tau )$ (main panels) and of the nearest neighbor interaction energy per site $u_1(\tau )$ (insets) following quantum quenches in which (a) $t^{\prime }=V^{\prime }=0$, (b) $t^{\prime }=V^{\prime }=0.1$, (c) $t^{\prime }=V^{\prime }=0.4$, and (d) $t^{\prime }=V^{\prime }=0.8$, after the quench. Results are shown for the 18th and 19th orders of the NLCE, and for periodic chains with 19 and 20 sites from exact diagonalization. The horizontal lines depict the 19th order result of the NLCE for the diagonal (DE) and grand-canonical (GE) ensembles, and for the DE results from exact diagonalization in chains with 20 sites.

Figure 2

Normalized difference ${\mathrm{\Delta }}_l[k(\tau )]$, see Eq. (4), as a function of $l$ at four times ($\tau =2$, 10, 50, and 100) for (a) $t^{\prime }=V^{\prime }=0$, (b) $t^{\prime }=V^{\prime }=0.1$, (c) $t^{\prime }=V^{\prime }=0.4$, and (d) $t^{\prime }=V^{\prime }=0.8$.

Figure 3

(Main panels) ${\delta }_l^{\mathrm{DE}}[k(\tau )]$ (a) and ${\delta }_l^{\mathrm{DE}}[u_1(\tau )]$ (b) [see Eq. (5)] as a function of $\tau$ for $l=18$ (open symbols) and $l=19$ (filled symbols), and for $t^{\prime }=V^{\prime }=0.4$, 0.6, and 0.8. The straight lines depict exponential fits to the $l=19$ results, from which we extract the relaxation rate for every value of $t^{\prime }=V^{\prime }$. (Insets) Relaxation rates, obtained from fits as those depicted in the main panels, for $l=18$ and $l=19$. The dashed lines depict fits to $a{t^{\prime }}^{\alpha }$ for $0.4\le t^{\prime }=V^{\prime }\le 0.7$, with $a$ and $\alpha$ reported in the labels of the curves.