Critical quench dynamics of random quantum spin chains: ultra-slow relaxation from initial order and delayed ordering from initial disorder

By means of free fermionic techniques combined with multiple precision arithmetic we study the time evolution of the average magnetization, $\overline{m}(t)$, of the random transverse-field Ising chain after global quenches. We observe different relaxation behaviors for quenches starting from different initial states to the critical point. Starting from a fully ordered initial state, the relaxation is logarithmically slow described by $\overline{m}(t) \sim \ln^a t$, and in a finite sample of length $L$ the average magnetization saturates at a size-dependent plateau $\overline{m}_p(L) \sim L^{-b}$; here the two exponents satisfy the relation $b/a=\psi=1/2$. Starting from a fully disordered initial state, the magnetization stays at zero for a period of time until $t=t_d$ with $\ln t_d \sim L^{\psi}$ and then starts to increase until it saturates to an asymptotic value $\overline{m}_p(L) \sim L^{-b'}$, with $b'\approx 1.5$. For both quenching protocols, finite-size scaling is satisfied in terms of the scaled variable $\ln t/L^{\psi}$. Furthermore, the distribution of long-time limiting values of the magnetization shows that the typical and the average values scale differently and the average is governed by rare events. The non-equilibrium dynamical behavior of the magnetization is explained through semi-classical theory.