Disorder-induced dynamical Griffiths singularities after certain quantum quenches

We demonstrate that in a class of disordered quantum systems the dynamical partition function is not an analytical function in a time window after certain quantum quenches. We related this behavior to rare and large regions with atypical inhomogeneity configurations. We also quantify the strength of the associated singularities and their signatures in experiments and numerical studies.

Figure 1

Schematics of (a) the equilibrium phase diagram and the class of quantum quenches studied: from a point far from the quantum phase transition to a point in the nearby Griffiths phase. (b) The rare-region-induced zeros of the dynamical partition function $Z$, and (c) the associated dynamical free energy $f$ (solid line). Each set of zeros (stars of a given color) and the corresponding dynamical free energy $f_{\text{RR}}$ (dashed line) are due to a single rare region. The singular part of $f$ is a simple superposition of all $f_{\text{RR}}$. The time window $t_A<t<t_B$ is the dynamical quantum Griffiths phase.

Figure 2

(a) The dynamical free energy $f$ as a function of the real time $t$ for three different chains after the quantum quench from $h_0=\infty$ to finite $h$. The first chain (black dotted line) is homogeneous, $L=30$ sites long with periodic boundary conditions, and has couplings $J_{\text{B}}=h/5$. The second chain (red solid line) is identical to the first one except that it contains a RR of size $L_{\text{RR}}=8$ inside which the couplings are $J_{\text{RR}}=3h$. The third chain (blue dashed line) is homogeneous, $L_{\text{RR}}$ sites long with open boundary conditions, and has couplings $J_{\text{RR}}$. (b) The corresponding Yang-Lee-Fisher zeros of $Z\left(z\right)$ for these three chains: open symbols, solid symbols, and $\times$ symbols, respectively. The zeros' trajectories of the second chain (when changing $J_{\text{RR}}$ from $h/5$ to $3h$) are given by the gray dots (see text).

Figure 3

(a) The mean energy density above the ground state $\delta {\mathcal{E}}_i$ and (b) the corresponding density current ${\mathcal{J}}_i$ as a function of the real time $t$ for each lattice site $i$ (see text).