Entanglement in quenched extended Su-Schrieffer-Heeger model with anomalous dynamical quantum phase transitions

The formal establishment of the theory of dynamical quantum phase transitions (DQPTs) in 2013 [1], followed by the studies of its fundamental properties [2, 3, 4, 5, 6] and successful experimental realization [7, 8, 9, 10], leads to an alternative approach in studying non-equilibrium physics in quantum many-body systems. DQPTs have been witnessed and substantially studied in spin models [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 98], topological models [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 99] and high energy physics [58, 59, 60, 61, 62, 100, 101], via various protocols such as a quench or a periodical drive [63, 64, 65, 66, 67, 68, 69, 70] to bring the system out of equilibrium. There are three types of DQPTs. The type-I DQPTs (DPTs-I) considers the kinks at the transition point in the long-time average of the order parameter of a system [5, 6, 71, 72, 73]. On the other hand, the type-II DQPTs (DPTs-II), which will be the main focus of this work, concerns the transient time dynamics, and are defined by the zeros of the Loschmidt amplitude (LA), the overlap of the time-evolved state driven by the unitary evolution of the final Hamiltonian $H_{f}$ onto the initial state $\ket{\psi_{0}^{i}}$

and in turn the nonanalyticities in the Loschmidt rate (also known as the dynamical free energy) $\lambda_{0}(t)=-\lim_{N\rightarrow\infty}N^{-1}\ln|\mathcal{G}_{0}(t)|^{2}$ with $N$ the system size [1, 74, 75]. Recently, a third type of DQPT (DPTs-III) has also been introduced as the change in the scaling behavior of a physical observable upon approaching the steady state in the long-time limit [76, 77, 78, 79].

The type-II DQPTs are commonly found in models being quenched across the underlying equilibrium phase boundary, however, exceptions exist [12, 18, 20, 39, 40, 49]. On the other hand, DQPTs can also occur for quench within the same phase [12, 19, 19, 18, 20, 38, 40, 53, 55, 57]. The connection, if any, between the equilibrium and the dynamical phase diagrams remain to be addressed.

In topological models, it is found that there can be more than one critical momenta (and correspondingly critical time scales) [43, 53, 54, 55, 56, 57]. These critical momenta originate from the different topological structures between the initial and final phases, in which their number must be at least the difference in the winding number between the two phases for 1D models. The cases with excessive critical momenta are referred to as accidental cases in Ref. [53]. For 2D topological models, robust DQPTs are present if there are at least one node in the $k$-space wavefunction overlap, and such nodes are guaranteed when the pre- and postquenched phases have a different Chern number [55, 56]. The 2D case in general has areas of zeros of the Loschmidt amplitude distributed on the complex time plane and a continuum of solutions of the critical times, but it is found in a number of cases that have line-like distribution of zeros whose critical time usually agrees well with the Loschmidt rate non-analytic peak [80].

In this work, we show that the previously referred to accidental DQPTs in topological models in fact contains three notions, distinguished by quenches (1) within the same phase, (2) across the phase boundary but without a change in the winding numbers, and (3) across the boundary with a change in the winding numbers. We demonstrate our findings via an extended version of the Su-Schrieffer-Heeger (SSH) model, a commonly studied topological model in condensed matter physics, due to its simplicity and accessibility to experiments [81]. The extended SSH model possesses a rich equilibrium phase diagram, allowing us to probe the quantum dynamics of quenches involving different topological and the trivial phases. Specifically, we realize for quenches between trivial and the topological phases, and between phases with the same winding number, there exists regions in the dynamical phase diagram, where the number of critical momenta $n_{k^{*}}>|\Delta W|$, the absolute difference of winding number between the two phases. Distinctive features are found between the cases of $n_{k^{*}}=|\Delta W|$ and $n_{k^{*}}>|\Delta W|$. For quench between phases with the same winding number, DQPTs are not just found to be possible for quench within the same phase, but also across the equilibrium phase boundary. We reckon the observations may be a generic nonequilibrium feature for topological models beyond nearest-neighbor hoppings, and further formulate and term these kinds of accidental DQPTs the type-B anomalous DQPTs. Note that the type-B anomalous DQPTs here is different from the anomalous DQPTs (we call them type-A anomalous DQPTs for unambiguity) described in Ref. [5, 72, 83, 84, 82, 6, 16, 85, 86, 100], which emphasizes the minimum in rate functions before the first critical time.

We further investigate the origin of the critical momenta and the time evolution of the entanglement entropy (EE) and the correlation matrix spectrum in type-B anomalous DQPTs. The main findings are summarized in Table 1. It is found that the extra $k^{*}$’s in the anomalous regions are in fact complex conjugate pair of the roots from the orthogonality of the initial and final Bloch vectors. Motivated by the distinctive behavior of the EE in the equilibrium case, we categorize the four phases in the extended SSH model into two classes according to the extent of non-locality of the ground state to extract the generic features in the EE. We considered a variety of quenches within a class and across two classes, and the EE extremum properties near DQPTs in accordance with the regular ($n_{k^{*}}=|\Delta W|$) and type-B anomalous ($n_{k^{*}}>|\Delta W|$) regions are presented in Table 2. Furthermore, distinguishing features in the correlation matrix spectrum are also found between the regular and type-B anomalous dynamical phases. In the latter case, a significantly larger number of eigen-levels are involved in the evolution in the long-time regime.

This paper is organized as follows: Sec. II demonstrates the dynamical phase diagram of some selected initial points in each equilibrium phase of extended SSH model, for which we provide analytical characteristics and reasoning as to how the type-B anomalous regions emerge and their distinction to the regular case. We further discuss the differences between type-B anomalous and regular regions regarding the entanglement entropy in the vicinity of DQPTs in different quench scenarios in Sec. III, and the time evolution of the correlation matrix spectrum in Sec. IV. Finally, we give a conclusion and outlook for future study in Sec. V.

The Hamiltonian of the extended SSH model reads [87]

where L is the number of unit cells, A and B denote the two sublattices, $t_{a}=-t(1+\eta)$ and $t_{b}=-t(1-\eta)$ are the intra- and inter-cell hoppings in the original SSH model. The $t_{c}$ and $t_{d}$ terms are added to include chiral-symmetry-preserving next-nearest-neigbhor hoppings (see Fig. 1 for an illustration of the model). The model possesses topological phases with winding numbers $W=1,-1$, and 2 respectively, and a trivial phase with winding number $W=0$, in which we denote them as $W_{0}$, $W_{1}$, $W_{2}$ and $W_{-1}$ in the remaining text. Without the loss of generality, we will consider $\eta=0,t=1$, and adopt periodic boundary condition in the following, unless otherwise specified.

The model can be solved exactly via Fourier transformation as

which brings the Hamiltonian of the system into the form of

where $\Psi_{k}=(a_{k},b_{k})^{T}$ and $H(k)$ is the Bloch Hamiltonian $H(k)=\vec{d}(k)\cdot\vec{\sigma}$ where the associated Bloch vector has the components

and $\vec{\sigma}$ is the Pauli matrix vector.

The system is first prepared in the ground state $\ket{\psi_{0}^{i}}$ of the Hamiltonian $H_{i}=H(t_{c}^{i},t_{d}^{i})$, then is brought out of equilibrium by a sudden quench and allowed to evolve with the Hamiltonian $H_{f}=H(t_{c}^{f},t_{d}^{f})$. Throughout the paper, we use the simplified notation $(t_{c}^{i},t_{d}^{i})\rightarrow(t_{c}^{f},t_{d}^{f})$ to indicate a quench. The Loschmidt amplitude in two-band models can be analytically calculated as [43]

with $|g_{k}|^{2}=(1+\hat{d}_{i}(k)\cdot\hat{d}_{f}(k))/2$ and $|e_{k}|^{2}=(1-\hat{d}_{i}(k)\cdot\hat{d}_{f}(k))/2$, $\hat{d}(k)=\vec{d}(k)/|\vec{d}(k)|$ is the unit Bloch vector and the quasiparticle energy $\varepsilon_{k}^{f}=\pm|\vec{d}_{f}(k)|$. The orthogonality between the two Bloch vectors, i.e. $\vec{d}_{i}(k^{*})\cdot\vec{d}_{f}(k^{*})=0$ results in a cubic equation of $\cos k$ that defines the critical momenta $k^{*}$ associated with the critical times

where $n=1,2,\cdots$, when DQPTs take place. The number of real critical momenta $n_{k^{*}}$ ranges from 0 to 3, depending on the quench parameters. In the following discussion, the critical times are obtained from Eq. (8) in the thermodynamic limit.

Alternatively, one also detects DQPTs by the dynamical topological order parameter (DTOP) defined as [43]

where $\phi_{k}^{G}(t)=\phi_{k}^{\mathcal{G}_{0}}(t)-\phi_{k}^{\text{dyn}}(t)$ is the Pancharatnam geometric phase (PGP) [88, 89]. $\phi_{k}^{\mathcal{G}_{0}}(t)$ is the argument of $\mathcal{G}_{k}(t)$ and the dynamical phase takes the form

DTOP usually jumps by an integer at critical times while stays constant for $t\neq t_{n}^{*}$. In some quench scenarios in some models, the jumps can be of half-integers [25, 34]. The jump in DTOP can be positive or negative depending on the critical momenta. The dynamics of the $\mathcal{G}_{k}(t)$, combined with the PGP, can be visualized by the dynamical vector [25]

It was shown that the trajectory of $\vec{r}_{k}$ in the first Brillouin zone at the first critical time reveals distinct features for quenches from the equilibrium phase boundary [25]. Here, we find the trajectories of $\vec{r}_{k}$ exhibit different characteristics for the quenches involving the regular and anomalous regions in the dynamical phase diagram, and the results will be presented below.

In the original SSH model, there are only two phases, the trivial $W_{0}$ phase and the topological $W_{1}$ phase, with $\eta$ as the driving parameter. The condition for a DQPT is determined by the equation $\cos k^{*}=-(1+\eta_{i}\eta_{f})/(1-\eta_{i}\eta_{f})$. Given the quantum critical point at $\eta_{c}=0$, quenching the original SSH model guarantees at most one critical momentum. On the other hand, the addition of $t_{c},t_{d}$ hoppings in the extended model introduces two additional topological phases – the $W_{2}$ and $W_{-1}$ phases, and one can have DQPTs with at most three critical momenta.

Figure 2 shows the dynamical phase diagram defined by the number of critical momenta $n_{k^{*}}$ for an initial state taken from each of the four equilibrium phases. Interestingly, for the quench between $W_{0}$ and $W_{1}$ phases, a dynamical phase boundary separating the $1k^{*}$ (the regular) and $3k^{*}$ (the anomalous) regions within the same equilibrium phase is observed (Figs. 2(a) and 2(b)). We also find that DQPTs can occur for quenches without crossing an equilibrium phase boundary. Especially for quenches within the $W_{2}$ or $W_{-1}$ phases (Figs. 2(c) and 2(d)), there exists a finite region in the dynamical phase diagram that DQPTs take place within the same phase. These observations motivate us to define the type-B anomalous DQPT as the regions where $n_{k^{*}}$ exceeds $|\Delta W|$. In later parts we will elucidate theoretically that all these type-B anomalous DQPTs originate from the same cause.

The Loschmidt rate (LR) and DTOP for some selected quench cases are shown in Fig. 3. It clearly shows the difference for quenches to the anomalous regions and the regular regions. Namely, there is one non-analytical peak in LR associated with the upward integer jump on DTOP in Fig. 3(a), whereas in Fig. 3(b) two extra critical times result when quenched to the $3k^{*}$ region for $W_{1}\rightarrow W_{0}$. On the other hand, Fig. 3(c) presents the quench with the three $k^{*}$’s, the highest possible number of critical momenta, from $W_{2}$ to $W_{-1}$ phase, where three kinks at distinct times can be found. Figure 3(d) shows the DQPT for quench within the lower $W_{2}$ phase and we clearly see two finite critical times with $\nu_{D}(t)$ jumped up at $t_{1}^{*}$ and down at $t_{2}^{*}$.

Here, we define the regions with $n_{k^{*}}$ exceeding $|\Delta W|$ as the type-B anomalous regions and present the possible cause and ways to distinguish them from regular DQPTs. Note that in both anomalous cases presented in Table 1, two additional critical momenta are found (comparing column 1 and 3, and column 2 and 4). The two extra $k^{*}$’s are originated from a complex conjugate pair of $k^{*}$ in $\vec{d}_{i}(k^{*})\cdot\vec{d}_{f}(k^{*})=0$, where they share the same real part. As the postquenched parameters enter the anomalous region (the $3k^{*}$ region for quench between $W_{0}$ and $W_{1}$, and the $2k^{*}$ region for quench within the $W_{-1}$ or $W_{2}$ phases), the imaginary part of the two critical momenta vanishes and they transform to the real critical momenta, giving rise to extra real critical times. This ensures that the parity of $n_{k^{*}}$ is conserved in both regions. For example, a $W_{2}\rightarrow W_{2}$ quench gives either zero $k^{*}$ or 2 $k^{*}$’s, while a $W_{1}\rightarrow W_{0}$ quench gives either 1 $k^{*}$ or 3 $k^{*}$’s.

The emergence of the complex-conjugate pair is visualized in Fig. 4, in which it illustrates the locations of $k^{*}$’s on the complex plane around the dynamical phase boundary. The complex conjugate pairs zone in the real axis when the final quench parameters approach the dynamical phase boundaries. The two critical momenta overlap right at the dynamical phase boundary, implying only one additional critical time resulted. Notice, however, that this is to be differentiated from the degenerate critical modes observed in quenched clustered Ising model, which is related to the structure of its energy spectrum [38]. When quenched further into the anomalous regions, the two critical momenta separate farther apart from each other on the real axis and two distinct critical times emerge.

Apart from the location of $k^{*}$’s, the vector $\vec{r}_{k}$ also reveals distinct features between regular and type-B anomalous regions, as summarized in Table 1. Figure 5 shows the trajectories of $\vec{r}_{k}$ for the respective quench cases in Fig. 3 at the first critical time. For the regular case where $n_{k^{*}}=|\Delta W|$, each $k^{*}$ is “assigned” a separate loop in the trajectory of $\vec{r}_{k}$. It is clearly seen in Fig. 5(a), quenching from $W_{1}$ phase to $1k^{*}$ region in $W_{0}$ phase where we have one loop for one critical momentum, and Fig. 5(c), quenching from the bottom $W_{2}$ phase to the left $W_{-1}$ phase in the phase diagram shown where we have three loops, each containing one $k^{*}$. On the other hand, for the anomalous case with quench to $3k^{*}$ in Fig. 5(b) and the quench within the bottom $W_{2}$ phase in Fig. 5(d), the critical momentum pair gathers in the same loop instead of staying in two separate loops, which further suggests the same origin of the two extra critical momenta. This feature still holds true for the SSH model with even longer range hoppings as discussed in Appendix A. One thing to notice is the extra loop for quench $W_{1}\rightarrow W_{0}$ to $1k^{*}$ region in Fig. 5(a) that has no critical momentum contained in it. This extra loop corresponds to the imaginary critical momentum pair that are not yet landed onto the real axis and cosine range. For the postquenched Hamiltonian approaching the dynamical boundary, the loop will eventually touch the origin, marking the emergence of the $k^{*}$ pair.

In the above discussions and those that follow, we focus on the type-B anomalous DQPTs. It is worth noting that type-A anomalous DQPTs are also observed in this system. The results and the corresponding discussions are presented in Appendix B.

In this section, we consider the time evolution of entanglement in different quench cases. We only focus on the short-time behavior of the entanglement entropies (EE) near DQPT. Our results suggest the EE signals the DQPT by its extremum, which is usually subsystem dependent except for quenches deep inside the phase where the dynamics is quite coherent for all the subsystems. The entanglement is quantified by the von-Neumann entropy defined as

where $\rho_{\Omega}$ is the reduced density matrix of the subsystem $\Omega$.

In non-interacting systems, the entanglement entropy can be calculated by

where $0\leq\xi_{j}\leq 1$ are the eigenvalues of the correlation matrix $C_{\mu\nu}=\braket{\hat{c}_{\mu}^{\dagger}\hat{c}_{\nu}}$ for $\mu,\nu\in\Omega$ [48, 90, 91, 92]. The eigenvalues of the correlation matrix measure the probabilities that the states with energies $\ln[(1-\xi_{j})/\xi_{j}]$ in the single-particle entanglement spectrum are occupied [93].

Equation (13) can also be viewed as the Shannon entropy of a Bernoulli distribution among the eigen-levels of the correlation matrix [91]. The largest contribution to the entanglement entropy comes from an eigenvalue of $\xi_{j}=0.5$, meaning the eigen-level is simultaneously occupied and unoccupied. In the original SSH model under the periodic boundary condition, the correlation matrix spectrum is doubly degenerated at $\xi_{j}=0.5$ in the topological phase, which corresponds to the boundary modes at two ends of the subsystem. In the quasiparticle picture, this induces a position uncertainty in the occupation which further translates to the momentum uncertainty, leading to an increase of entropy [93]. In the following, the time evolution of EE will be revealed in the vicinity of DQPTs, followed by further investigations on the correlation matrix spectrum in Sec. IV.

In this section, we further label the upper/lower $W_{2}$ phases as $W_{2}^{+/-}$ phase and the left/right $W_{-1}$ phases as $W_{-1}^{-/+}$ phase for clearer distinction in analysis (if not specified, we consider both). Following the analysis on EE, we study the evolution of the correlation matrix spectrum mentioned in the previous section under different quench scenarios.

In the original SSH model, it has been found that the eigen-levels of the correlation matrix cross at the Fermi level ($\xi_{j}=0.5$) if one quenches from the trivial $W_{0}$ phase to the topological $W_{1}$ phase and no crossings occur for quenches within the trivial phase [96]. Similar phenomenon happens for the trivial to topological phase quench in a model of the same Altland-Zirnbauer BDI class as the SSH model [48]. Moreover, persistent double degeneracy at the Fermi level is observed for quenches from topological to trivial phase [48]. The literature further proposed a dynamical quantity, termed entanglement echo, to indicate the crossings via a sudden jump in its rate function at the corresponding times. We found that the entanglement echo can behave very differently for different subsystem sizes and it does not always resemble the evolution of the Loschmidt rate in large subsystems, unlike the cases shown in Ref. [48] (see Appendix C).

The above phenomena in the correlation matrix spectrum are not limited to the mentioned quench scenarios in the extended SSH model. Figure 11 shows the correlation matrix spectrum for quenches from each phase in the extended model. In addition to the consistent features found in the literature [96, 48] for $W_{1}\leftrightarrow W_{0}$ phase quenches (not presented here), we also observe similar features in $W_{-1}\leftrightarrow W_{0}$ phase quenches, where there are level crossings in the quench from the trivial to the topological phase (Fig. 11(a)), and a persistent double degeneracy at $\xi_{j}(t)=0.5$, i.e. the Fermi level for the opposite quench (Fig. 11(d)). The latter can be deduced by noting that the Fermi level is doubly degenerated initially in the $W_{-1}$ phase and its evolution is a constant over time. Such a persistent degeneracy can also be observed for quenches between topological phases as shown in Fig. 11(b). Furthermore, we notice that level crossings occur when the system is quenched from the topological $W_{2}$ phase to the trivial phase, as shown in Fig. 11(c). Note that the Fermi level is four-fold degenerate initially in the $W_{2}$ phase and evolve into a pair of doubly degenerated levels until they cross at the Fermi level again. The crossing results in an instantaneous four-fold degeneracy of the Fermi level, retaining the number of edge modes in $W_{2}$ phases at the crossing times.

We further investigate the correlation matrix spectrum’s evolution of different subsystem sizes for quenches starting from the $W_{1}$ and $W_{2}$ phases regarding the anomalous regions found in Fig. 2. Figures 12(a) and 12(b) present the correlation matrix spectrum for $W_{1}\rightarrow W_{0}$ quench in the regular and anomalous region respectively. Level crossing in the half-block’s correlation matrix spectrum is observed in $3k^{*}$ phase but not in $1k^{*}$ phase (bottom panels of Figs. 12(a) and 12(b)). The cases for $L_{A}=2$ behave the same but in a more drastic manner – level crossings barely occur when quenched to $1k^{*}$ region (top panel of Fig. 12(a)), whereas the near-Fermi levels frequently cross each other for quenches to $3k^{*}$ region, as shown in the top panel of Fig. 12(b). Subsystems $L_{A}=4$ seem to be insensitive to this kind of dynamical change, where there is no significant distinct feature in their correlation matrix spectra, as observed from the middle panels of Figs. 12(a) and 12(b). Overall, the quench to the anomalous regions triggers more frequent correlation redistributions for both very small and very large subsystems.

The features discussed above can also be spotted in quenches within the $W_{2}$ phase with anomalous DQPTs. Eigen-levels in the half-block’s correlation matrix cross at the Fermi level, as shown in the bottom panel of Fig. 12(d). In addition, more eigen-levels away from the Fermi level approach the Fermi level as compared to its regular counterpart (bottom panel of Fig. 12(c)). This suggests the quench dynamics inside the anomalous regions involves more eigen-levels, leading to a more dramatic phenomenon with the addition of the two extra critical momenta. On the other hand, there are a couple of differences between the $W_{2}^{-}\rightarrow W_{2}^{-}$ and $W_{1}\rightarrow W_{0}$ quenches. In particular, level crossings in half-block subsystem for $W_{2}^{-}\rightarrow W_{2}^{-}$ occur even without the emergence of DQPTs (bottom panel of Fig. 12(c)). In Figs. 12(c) and 12(d), we also found a relatively steady evolution of the eigen-levels away from the fermi level for all subsystems shown. For the quench within $W_{2}$ phase, the evolutions are less entangled and crossings rarely occur among the eigen-levels away from the two around the Fermi level, unlike the situation in $W_{1}\rightarrow W_{0}$ 3$k^{*}$. Note that these features also apply to quenches between 2 distinct $W_{2}$ phases ($W_{2}^{-}\leftrightarrow W_{2}^{+}$).

In the previous section, we argued that the $W_{2}$ and $W_{-1}$ phases have higher non-locality where the quench dynamics, especially in the vicinity of DQPTs, has their own characteristics compared to that for quenches from the topological phase $W_{1}$ (See Table 2). In Fig. 11(d), we found that the two eigen-levels (each doubly degenerated) around the Fermi level are locally the furthest away from the Fermi level near the first critical time. These levels have the highest contribution to the time evolution of the entanglement entropy since most of the other levels evolve relatively steadily. As a result, the entanglement entropy exhibits a local minimum, implying a temporary gain of information in the subsystem around the first critical time, as presented for the case of the anomalous DQPT involving class 2 $\rightarrow$ class 2 quench in the previous section.

On the contrary, one witnesses the rapid reorganization of correlation in a regular DQPT involving the class 2 $\rightarrow$ class 2 deep quench. Figure 13 shows the time evolution of the correlation matrix for a deep quench from $W_{2}^{-}$ to $W_{-1}^{-}$ phase. Note that for such kind of deep quenches, the originally three distinct critical times become close to each other due to the increasingly large energy each k mode possesses for $t_{c}^{f}\gg 1$ or $t_{d}^{f}\gg 1$. One observes level crossings around the critical times for all the subsystems concerned. Moreover, especially for $L_{A}=2,4$, the upper and lower levels also have their closest approach to the Fermi level around the first critical time. Same phenomenon can be observed for quenches to $W_{-1}^{+}$ phase. This is consistent with the observation that the entanglement entropy attains a local maximum around the first critical times as shown in Fig. 10(a). In the vicinity of DQPT, the system experiences a temporary redistribution of correlation in the subsystem, where a short instant of momentum uncertainty is introduced by the formation of quasiparticle modes at the Fermi level resulting in a restoration of the initial topological feature. This uncertainty in turn causes the loss of information to correctly describe the whole system and thus an increase in the entanglement entropy. The features regarding the correlation matrix spectrum presented in this section can also be observed in the extended SSH model with further neighbor hoppings, as shown in Appendix A.2.

We attempted to integrate several types of DQPTs into the same framework, namely, the type-B anomalous DQPTs, by comparing it with regular DQPTs in the extended SSH model with two chiral-symmetry-preserving next-nearest-neighbor hopping terms added to the original SSH model. The introduction of these hopping terms induces two additional phases on top of $W_{0}$ and $W_{1}$, namely $W_{2}$ and $W_{-1}$, in equilibrium. We investigated the dynamical phase diagram by counting the number of critical momenta $n_{k^{*}}$ of the corresponding quench, and defined the type-B anomalous DQPTs as $n_{k^{*}}$ bigger than the absolute winding number difference between the pre- and postquenched phases. Under this formulation, type-B anomalous DQPTs incorporate the possibility of DQPTs occuring for quenches within the same phase (both with and without crossing equilibrium phase boundary), and quenches that involve multiple $n_{k^{*}}$ and thus critical time scales. On top of that, we elucidated the extra critical momenta always come in a pair due to the fact that they are complex conjugate pair of each other. The trajectories of the dynamical vector $\vec{r}_{k}$ helps to distinguish the nature of the regular critical momenta and the extra critical momenta, where the latter pair stays in the same loop of trajectory instead of two separate loops.

Furthermore, we classified the quench scenarios by the equilibrium entanglement entropy phase diagram, where $W_{2}$ and $W_{-1}$ are the more non-local (overall higher EE) phases forming class 2, while $W_{0}$ and $W_{1}$ are the less non-local (lower EE) phases forming class 1. As such, there are six scenarios of entropy dynamics quench, namely, from class 1 to class 2, and vice versa, from class 2 to itself with/without changing winding number, and from class 1 to itself with $1k^{*}$/ $3k^{*}$. We discussed the time evolution of the EE near the first DQPT regarding these six different situations. On one hand, if we quench deep inside the phase, DQPT occurs near the local maximum of EE for class 1 to class 2 and from class 2 to itself with $n_{k^{*}}=|\Delta W|$, regardless of the size of the subsystem. In these quench cases, the deeper the quench to the phase, the more alignment between the time EE is maximized and the first critical time. The cause of this maximum comes from the level crossings occurred among the correlation matrix spectrum, giving the highest momentum uncertainty during the reorganization of correlation during DQPTs.

On the other hand, with an appropriate subsystem choice, the DQPT occurs near the local minimum of the entanglement entropy for quenches from class 2 to class 1 and class 2 to itself with $n_{k^{*}}>|\Delta W|$. The variation in time of the correlation matrix spectrum in this case is the least and the effective eigen-levels are farthest away from the Fermi level near the critical time. In addition, one might be able to realize a quench that attains lower entropy with a small subsystem near DQPT compared to the pre-quenched phase. A possible application is that one can utilize such property to perform measurement on just a small subsystem, yet still capable of capturing significant amount of information of the whole system comparing to its pre-quenched phase. For the quenches within class-1 phases with $3k^{*}$, it is consistent with the original SSH model, where $W_{1}\rightarrow W_{0}$ and $W_{0}\rightarrow W_{1}$ attain minimum and maximum for subsystem $L_{A}=1$ in the vicinity of DQPT respectively. Our results show DQPT occurs around the second and the first minimum of the EE time evolution in the $1k^{*}$ and $3k^{*}$ regions, respectively. We interpret the first minimum in EE in the $1k^{*}$ region as a preparation for the DQPT that emerges when entering the $3k^{*}$ region.

Our work may serve as an attempt to extend the study of DQPTs in topological models with more than one topological phases with distinct topological characters. The question of any physical quantities like the order parameter, whether equilibrium or dynamical, can detect the anomalous regions studied in this work remains to be explored. Topological models that happen to display type-B anomalous dynamical regions in the dynamical phase diagram are of particular interest in the future research. A natural extension would be to generalize the definition of type-B anomalous DQPTs to other models of different topological invariants such as Chern number [55, 97] and models in higher dimensions. For example, for models without chiral symmetry where the winding number is not a good topological invariant [54]. It is also intriguing to explore the applicability of our results to generalization of unit cells instead of hopping range, which also exhibits type-B anomalous DQPTs [55]. Besides, investigating the type-B anomalous DQPTs under the influence of non-Hermitian couplings in the SSH model [102] also presents an interesting research direction. On the other hand, the Loschmidt amplitude spectrum [33, 41] may also serve as a probe to study the physical effect of the extra critical momenta in topological models in the future work.