Abstract
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=Π^j-1_l=1σ^z_lσ^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) X_j_t exhibits j-independent exponential decay for a quench to g=1 and the time at which X_j_t reaches its first maximum scales linearly with j; (iii) The single-spin purity dynamics is mainly controlled by σ^x_j_t (σ^z_j_t) for a quench to g<1 (g≥ 1). For quenches to the disordered phase with g≫1, the single-spin tends to be in the maximally mixed state and the transverse and longitudinal correlators σ^z_jσ^z_j+1_t and σ^x_jσ^x_j+1_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 ∼0.125 for g≫1.