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 $\left(\lambda \right)$ while the latter is quenched either slowly [in a linear fashion in time $\left(t\right)$ 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 nonanalyticities [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 $\left(\gamma \right)$ and $\tau$, thereby establishing the existence of boundaries in the $\left(\gamma \text{−}\tau \right)$ 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.

Figure 1

Phase diagram of the Hamiltonian Eq. (7) in the $\alpha \text{−}\lambda$ plane for the isotropic case $\gamma =1$ and an anisotropic case with $\gamma =0.5$. For a fixed $\alpha$ and a large magnitude of the rescaled field $\lambda$, the spin chain is antiferromagnetic (AF). On the other hand, when $\lambda$ is reduced, the system undergoes quantum phase transitions from Gapless-I (GPI) phase characterized by two Fermi points to the Gapless-II (GPII) phase characterized by four Fermi points. The vertical line shows the direction of quenching. After Ref. [59].

Figure 2

The rate function $I\left(t\right)$ as obtained by numerically integrating Eq. (3) is plotted as a function of time following a slow quench from a large negative to a large positive value of $\lambda$. The upper panel corresponds to $\gamma =1$, where there are periodic occurrences of DPTs for all values of $\tau$'s subject to the condition that $p_k{|}_{k=0}<0.5$, i.e., $\tau >{\tau }_1{\left(\gamma \right)|}_{\gamma \to 1}$. The lower panel corresponds to the anisotropic situation with $\gamma =0.5$ for which the critical value of $\tau$ as obtained from Eq. (9) is given by ${\tau }_2\left(\gamma \right)=1.76$. As discussed in the text, the figure shows the presence of DPTs for ${\tau }_1\left(\gamma \right)<\tau <{\tau }_2\left(\gamma \right)$, while they disappear for $\tau >{\tau }_2\left(\gamma \right)$. The red colored solid line shows sharp peaks, whereas the black colored dashed lines shows peaks that are rounded off.

Figure 3

The phase diagram in the $\gamma \text{−}\tau$ plane showing the DPT and the no-DPT regions following a slow quench as discussed in Fig. 2. The upper curve corresponds to the condition presented in Eq. (9); i.e., ${\tau }_2\left(\gamma \right)=2log2/\left[\pi {(1-\gamma )}^2\right]$, which diverges in the isotropic case $\left(\gamma =1\right)$. The lower curve denoted by ${\tau }_1\left(\gamma \right)=2log2/\left[\pi {(1+\gamma )}^2\right]$ is obtained from the requirement $p_{k=0}=1/2$ and ${\tau }_1{\left(\gamma \right)|}_{\gamma =1}=log2/2\pi$. It is to be noted that the values of both ${\tau }_1\left(\gamma \right)$ and ${\tau }_2\left(\gamma \right)$ are zoomed by a factor of 10 for better visibility. The DPTs exist for the $\gamma$ and $\tau$ values lying in the region bounded by ${\tau }_2\left(\gamma \right)$ and ${\tau }_1\left(\gamma \right)$.

Figure 4

The presence and the absence of DPTs following a sudden quenching of $\lambda$ from $-\lambda$ to $+\lambda$ for $\alpha =1$. The upper panel corresponds to $\gamma =1$ where there are periodic occurrences of DPTs when $\lambda =2$, while DPTs get rounded off when $\lambda =2.5$. The lower panel corresponds to the anisotropic situation with $\gamma =0.5$, where the presence of DPTs are wiped out when $\lambda =1.8$ $[>(1+\gamma \left)\right]$ or $\lambda =0.4$ $[<(1-\gamma \left)\right]$ while these are prominently present when for $\lambda =1.5[=(1+\gamma \left)\right]$ and $\lambda =0.5[=(1-\gamma \left)\right]$ as discussed in the text. It should also be noted that the position of the maxima (or nonanalyticities) depend on the magnitude $\lambda$.

Figure 5

The phase diagram in the ${\lambda }_f\text{−}\gamma$ plane showing the regions where DPTs will occur following sudden quenches with ${\lambda }_i=-1$ (red triangles) and $=-2$ (blue circles) to a final value ${\lambda }_f$. In both cases, DPTs occur when ${\lambda }_f$ lies within the range ${(1-\gamma )}^2/\left|{\lambda }_i\right|$ and ${(1+\gamma )}^2/\left|{\lambda }_i\right|$ as elaborated in the text. In the isotropic situation $\left(\gamma =1\right)$, the condition gets simplified to $0\le {\lambda }_f\le 4/\left|{\lambda }_i\right|$.