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

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 $X_j={\prod }_{l=1}^{j-1}{\sigma }_l^z{\sigma }_j^x$, 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 $⟨{\sigma }_j^x{⟩}_t$ ($⟨{\sigma }_j^z{⟩}_t$) for a quench to $g<1$ ($g\ge 1$). For quenches to the disordered phase with $g\gg 1$, the single-spin tends to be in the maximally mixed state and the transverse and longitudinal correlators $⟨{\sigma }_j^z{\sigma }_{j+1}^z{⟩}_t$ and $⟨{\sigma }_j^x{\sigma }_{j+1}^x{⟩}_t$ 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 $\sim 0.125$ for $g\gg 1$.

arXiv comment13 pages, 10 figures

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