Quantum quenches in the thermodynamic limit. II. Initial ground states

A numerical linked-cluster algorithm was recently introduced to study quantum quenches in the thermodynamic limit starting from thermal initial states [M. Rigol, Phys. Rev. Lett. 112, 170601 (2014)]. Here, we tailor that algorithm to quenches starting from ground states. In particular, we study quenches from the ground state of the antiferromagnetic Ising model to the $XXZ$ chain. Our results for spin correlations are shown to be in excellent agreement with recent analytical calculations based on the quench action method. We also show that they are different from the correlations in thermal equilibrium, which confirms the expectation that thermalization does not occur in general in integrable models even if they cannot be mapped to noninteracting ones.

Figure 1

Nearest [(a)–(d)] and next-nearest [(e)–(h)] neighbor ${\sigma }^z{\sigma }^z$ correlations for quenches with $\Delta =1$ [(a),(e)], $\Delta =4$ [(b),(f)], $\Delta =7$ [(c),(g)], and $\Delta =10$ [(d),(h)]. We show results for the last nine orders of the NLCE for the diagonal ensemble (DE) and for the grand-canonical ensemble (GE). Lines joining the symbols are provided to guide the eye. For all $\Delta$'s, the GE results are essentially converged for all orders shown here. The horizontal continuous lines are the results from the quantum action method (QA) [28, 29], and the horizontal dashed lines are the NLCE results after resummations (see text) using Wynn's algorithm (Resum). The results of the resummations, and of the NLCE bare sums when converged, are virtually indistinguishable from those of the quantum action method. The GE results, on the other hand, are clearly different except for $\langle {\sigma }_1^z{\sigma }_2^z\rangle$ when $\Delta =1$ (see text).

Figure 2

Relative differences ${\delta }_{{\sigma }_1^z{\sigma }_2^z}$ and ${\delta }_{{\sigma }_1^z{\sigma }_3^z}$ between the GE and the quantum action method vs. $\Delta$. The differences are seen to go to zero as $\Delta \to 1$ and $\Delta \to \infty$ and peak for $\Delta \approx 2.0$. (Inset) Effective temperature of the GE vs. $\Delta$.