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 quench, in which transverse field $g$ suddenly changes from zero to nonzero value. The long-timescale reduced single spin and two nearest-neighbor spins, involves evaluation expectation values odd operators that break fermion parity, is exactly obtained for finite-size but large rings through use recently developed Pfaffian method [N. Wu, Phys. Rev. E 101, 042108 (2020)]. Time dependence longitudinal magnetizations (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}\ensuremath{\rangle}}_{t}$ ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x,y}\ensuremath{\rangle}}_{t}$), single-spin purity, value string operator ${X}_{j}={\ensuremath{\prod}}_{l=1}^{j\ensuremath{-}1}{\ensuremath{\sigma}}_{l}^{z}{\ensuremath{\sigma}}_{j}^{x}$ (${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$), several equal-time two-site correlators (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x,z}{\ensuremath{\sigma}}_{j+1}^{x,z}\ensuremath{\rangle}}_{t}$, ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{y}\ensuremath{\rangle}}_{t}$, ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{z}\ensuremath{\rangle}}_{t}$), pairwise concurrence quenches different phases are numerically studied. Our main findings (i) generic approaches long-time limit; (ii) ${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$ exhibits $j$-independent exponential decay quench $g=1$ time at reaches its first maximum scales linearly with $j$; (iii) purity mainly controlled by ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}\ensuremath{\rangle}}_{t}$ (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}\ensuremath{\rangle}}_{t}$) $g<1$ ($g\ensuremath{\ge}1$). For disordered phase $g\ensuremath{\gg}1$, tends be maximally mixed state ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}{\ensuremath{\sigma}}_{j+1}^{z}\ensuremath{\rangle}}_{t}$ ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{x}\ensuremath{\rangle}}_{t}$, respectively, $\ensuremath{-}0.25$ $0.5$ thermodynamic (iv) entanglement acquires finite plateau increases increasing $g$, saturated $\ensuremath{\sim}0.125$ $g\ensuremath{\gg}1$.