Exact one- and two-site reduced dynamics in a finite-size quantum Ising ring after a quench: A semi-analytical approach

We study the non-equilibrium dynamics of a homogeneous quantum Ising ring after a quench, in which the transverse field $g$ suddenly changes from zero to a nonzero value. The long-timescale reduced dynamics of a single spin and of two nearest-neighbor spins, which involves the evaluation of expectation values of odd operators that break the fermion parity, is exactly obtained for finite-size but large rings through the use of a recently developed Pfaffian method [N. Wu, Phys. Rev. E 101, 042108 (2020)]. Time dependence of the transverse and longitudinal magnetizations, single-spin purity, expectation value of the string operator $X_j=\prod^{j-1}_{l=1}\sigma^z_l\sigma^x_j$, several equal-time two-site correlators, and pairwise concurrence after quenches to different phases are numerically studied. Our main findings are that (i) The expectation value of a generic odd operator approaches zero in the long-time limit; (ii) $\langle X_j\rangle_t$ exhibits $j$-independent exponential decay for a quench to $g=1$ and the time at which $\langle X_j\rangle_t$ reaches its first maximum scales linearly with $j$; (iii) The single-spin purity dynamics is mainly controlled by $\langle\sigma^x_j\rangle_t$ ($\langle\sigma^z_j\rangle_t$) for a quench to $g<1$ ($g\geq 1$). For quenches to the disordered phase with $g\gg1$, the single-spin tends to be in the maximally mixed state and the transverse and longitudinal correlators $\langle\sigma^z_j\sigma^z_{j+1}\rangle_t$ and $\langle\sigma^x_j\sigma^x_{j+1}\rangle_t$ respectively approaches $-0.25$ and $0.5$ in the thermodynamic limit; (iv) The nearest-neighbor entanglement acquires a finite plateau value that increases with increasing $g$, and approaches a saturated value $\sim0.125$ for $g\gg1$.