Tuning the presence of dynamical phase transitions in a generalized $XY$ spin chain

We study an integrable spin chain with three spin interactions and the staggered field ($\lambda$) while the latter is quenched either slowly (in a linear fashion in time ($t$) as $t/\tau$ where $t$ goes from a large negative value to a large positive value and $\tau$ is the inverse rate of quenching) or suddenly. In the process, the system crosses quantum critical points and gapless phases. We address the question whether there exist non-analyticities (known as dynamical phase transitions (DPTs)) in the subsequent real time evolution of the state (reached following the quench) governed by the final time-independent Hamiltonian. In the case of sufficiently slow quenching (when $\tau$ exceeds a critical value $\tau_1$), we show that DPTs, of the form similar to those occurring for quenching across an isolated critical point, can occur even when the system is slowly driven across more than one critical point and gapless phases. More interestingly, in the anisotropic situation we show that DPTs can completely disappear for some values of the anisotropy term ($\gamma$) and $\tau$, thereby establishing the existence of boundaries in the $(\gamma-\tau)$ plane between the DPT and no-DPT regions in both isotropic and anisotropic cases. Our study therefore leads to a unique situation when DPTs may not occur even when an integrable model is slowly ramped across a QCP. On the other hand, considering sudden quenches from an initial value $\lambda_i$ to a final value $\lambda_f$, we show that the condition for the presence of DPTs is governed by relations involving $\lambda_i$, $\lambda_f$ and $\gamma$ and the spin chain must be swept across $\lambda=0$ for DPTs to occur.