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

We study the nonequilibrium 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)2470-004510.1103/PhysRevE.101.042108]. Time dependence of the transverse and longitudinal magnetizations ((σjz)t and (σjx,y)t), single-spin purity, expectation value of the string operator Xj=l=1j-1σlzσjx ((Xj)t), several equal-Time two-site correlators ((σjx,zσj+1x,z)t, (σjxσj+1y)t, and (σjxσj+1z)t), 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) (Xj)t exhibits j-independent exponential decay for a quench to g=1 and the time at which (Xj)t reaches its first maximum scales linearly with j; (iii) The single-spin purity dynamics is mainly controlled by (σjx)t ((σjz)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 (σjzσj+1z)t and (σjxσj+1x)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.