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

Abstract

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)]. Time dependence of the transverse and longitudinal magnetizations (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}\ensuremath{\rangle}}_{t}$ and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x,y}\ensuremath{\rangle}}_{t}$), single-spin purity, expectation value of the 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 (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x,z}{\ensuremath{\sigma}}_{j+1}^{x,z}\ensuremath{\rangle}}_{t}$, ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{y}\ensuremath{\rangle}}_{t}$, and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{z}\ensuremath{\rangle}}_{t}$), 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) ${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$ exhibits $j$-independent exponential decay for a quench to $g=1$ and the time at which ${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$ reaches its first maximum scales linearly with $j$; (iii) The single-spin purity dynamics is mainly controlled by ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}\ensuremath{\rangle}}_{t}$ (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}\ensuremath{\rangle}}_{t}$) for a quench to $g<1$ ($g\ensuremath{\ge}1$). For quenches to the disordered phase with $g\ensuremath{\gg}1$, the single-spin tends to be in the maximally mixed state and the transverse and longitudinal correlators ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}{\ensuremath{\sigma}}_{j+1}^{z}\ensuremath{\rangle}}_{t}$ and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{x}\ensuremath{\rangle}}_{t}$, respectively, approaches $\ensuremath{-}0.25$ 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 $\ensuremath{\sim}0.125$ for $g\ensuremath{\gg}1$.