Dynamical Topological Quantum Phase Transitions in Nonintegrable Models

We consider sudden quenches across quantum phase transitions in the $S=1$ $XXZ$ model starting from the Haldane phase. We demonstrate that dynamical phase transitions may occur during these quenches that are identified by nonanalyticities in the rate function for the return probability. In addition, we show that the temporal behavior of the string order parameter is intimately related to the subsequent dynamical phase transitions. We furthermore find that the dynamical quantum phase transitions can be accompanied by enhanced two-site entanglement.

Figure 1

(a) The main panel shows the rate function for $L=80$, $\mathrm{\Delta }=1$ and various values of $D$ as indicated by the legend in panel (b). The inset figure of panel (a) shows the finite-size effects around the first critical time for $D=4$; the dotted, dashed, and solid lines denote $L=30$, 60, and 120, respectively. (b) The $x$ component of the SOP for various values of $D$, $\mathrm{\Delta }=1$ and for chain length $L=80$. (c) The $z$ component of the SOP for the same parameters as in panels (a) and (b).

Figure 2

The rate function as a function of time and single-ion anisotropy using a color code in the sidebar, for $\mathrm{\Delta }=1$ and $L=80$.

Figure 3

(a) The rate function for various values of $\mathrm{\Delta }$ and $D=0$. Panels (b) and (c) show the $x$ and $z$ components of the SOP for the same values of $\mathrm{\Delta }$ and $D=0$ as in panel (a). The chain length is $L=80$ in all cases.

Figure 4

(a) The entanglement distance as a function of time calculated with different exponents for the quench with $\mathrm{\Delta }=4$, $D=0$. (b) The entanglement patterns in the chain at time $t=0.53$ for the same quench as in panel (a). The lines encode the magnitude of the mutual information between different sites using the gray scale sidebar. Note that 20 sites are discarded at both ends. (c) Similar to panel (b) but for $t=0.89$. The chain length is $L=80$ in all cases.