Interconnections between equilibrium topology and dynamical quantum phase transitions in a linearly ramped Haldane model

We study the behavior of Fisher zeros (FZs) and dynamical quantum phase transitions (DQPTs) for a linearly ramped Haldane model occurring in the subsequent temporal evolution of the same, and we probe the intimate connection with the equilibrium topology of the model. Here, we investigate the temporal evolution of the final state of the Haldane Hamiltonian (evolving with the time-independent final Hamiltonian) reached following a linear ramping of the staggered (Semenoff) mass term from an initial to a final value, first selecting a specific protocol, so chosen that the system is ramped from one nontopological phase to the other through a topological phase. We establish the existence of three possible behaviors of areas of FZs corresponding to a given sector: (i) no-DQPT, (ii) one-DQPT (intermediate), and (iii) two-DQPTs (reentrant), depending on the inverse quenching rate $\tau$. Our study also reveals that the appearance of the areas of FZs is an artefact of the nonzero (quasi-momentum-dependent) Haldane mass ($M_H$), whose absence leads to an emergent one-dimensional behavior indicated by the shrinking of the area's FZs to lines and the nonanalyticity in the dynamical “free energy” itself. Moreover, the characteristic rates of crossover between the three behaviors of FZs are determined by the time-reversal-invariant quasimomentum points of the Brillouin zone where $M_H$ vanishes. Thus, we observe that through the presence or absence of $M_H$, there exists an intimate relation to the topological properties of the equilibrium model even when the ramp drives the system far away from equilibrium.

Figure 1

(a) The phase diagram of the Haldane model in the $M\text{−}\varphi$ plane; topological phases and the nontopological phase correspond to Chern numbers $C=\pm 1$ and 0, respectively. We employ a quenching scheme $M\left(t\right)=-t/\tau$ with $M_i=3$ (green dot) and $M_f=-3$ (red dot) with $\varphi =1$, so that the system is quenched across both of the critical lines (i.e., gapless DPs) in the process of quenching. [One may also consider a slow ramping from $M_i=3$ (green dot) to $M_f=1$ (yellow dot).] (b) The rhomboidal Brillouin zone (BZ) that we have chosen in this paper is illustrated for a $10\times 10$ system. The Dirac points are shown in black (biggest dots) while other small dots represent the points of the BZ. The four corner points and the highest symmetry point at the center [with coordinates $(2\pi /3,0)$] marked by dots of medium size are time-reversal-invariant momentum (trim) points; while the left corner point $(0,0)$ (in green) and the central point (in cyan) are included in the BZ, other trim points (in red) are not. At all trim points the Haldane mass is zero.

Figure 2

The variation of the Haldane mass $M_H\left(\mathbf{k}\right)$ in the Brillouin zone (BZ) as a function of $k_x$ and $k_y$. It is zero along the $k_x$ axis when $k_y=0$ while it is positive (negative) in the lower (upper) half-plane. As a consequence of the time-reversal invariance of this mode, $M_H$ vanishes at all the trim points. Following any contour of fixed $M_H\left(\mathbf{k}\right)$ in the BZ, one can immediately conclude that the Haldane mass must explicitly depend on $k_x$ and $k_y$ and not $\left|k\right|$ only.

Figure 3

(a) The variation of $p_{\mathbf{k}}$ with $h\left(\mathbf{k}\right)=\sqrt{h_1^2\left(\mathbf{k}\right)+h_2^2\left(\mathbf{k}\right)+M_f^2}$ has been plotted for a quench from $M_i=3$ to $M_f=-3$ with a $\tau =0.1$ for a small $\varphi =0.1$. The $p_{{\mathbf{k}}^{*}}=1/2$ (in red) line intersects with the curve at only one point, indicating that all ${\mathbf{k}}^{*}$'s lead to the same value of $h\left({\mathbf{k}}^{*}\right)$. (b) The multiple intersections of the $p_{{\mathbf{k}}^{*}}=1/2$ line in the plot of $p_{\mathbf{k}}$ vs $M_H$ show that different ${\mathbf{k}}^{*}$s yield different contributions to $e_f^2\left({\mathbf{k}}^{*}\right)$ rendering it anisotropic, thereby leading to a continuum of critical $t_c$'s.

Figure 4

This set of figures presents the variation of $p_{\mathbf{k}}$ in the $k_x\text{−}{k}_y$ plane and no-DQPT, reentrant, and intermediate behaviors of FZs for the sector $n=0$ with $M_i=3$, $M_f=-3$, $\varphi =1$, and $L=100$. In (a) and (b), $\tau$ is small ($<{\tau }_c^1$) so that the minimum value of $p_{\mathbf{k}}\ge 1/2$ for all values of $\mathbf{k}$ and the area never crosses the real axis. The inset of (b) validates the observation that there is no DQPT by showing that the area formed by $e^2\left(\mathbf{k}\right)$ does not touch the line $p_{\mathbf{k}}=p_{{\mathbf{k}}^{*}}=1/2$ for any value of the $\mathbf{k}$. On the contrary, (c) shows that in the extreme slow limit, i.e., $\tau >{\tau }_c^2$, $p_{\mathbf{k}}$ becomes less than 1/2 along the dotted line connecting two DPs, and there is a reentrant behavior of FZs evident in (d): the inset shows four boundary points of $e^2\left(\mathbf{k}\right)$ on the line $p_{{\mathbf{k}}^{*}}=1/2$ leading to four instants of time: $t_b^1$, $t_e^1$, $t_b^2$, and $t_e^2$ [as obtained from Eq. (8)], where $\mathrm{Re}\left[f^{\prime }\left(t^{\prime }\right)\right]$ is nonanalytic [see Fig. 5]. For ${\tau }_1^c<\tau <{\tau }_c^2$, (e) shows that $p_{\mathbf{k}}$ never becomes less that $1/2$ along the dotted line, and consequently the area of FZs crosses the real-time axis only once (f): the inset shows the boundary points that lead to $t_b$ and $t_e$, where $\mathrm{Re}\left[f^{\prime }\left(t^{\prime }\right)\right]$ is nonanalytic, as shown in Fig. 5.

Figure 5

The nonanalyticities (cusp singularities) in $\mathrm{Re}\left[f^{\prime }\left(t^{\prime }\right)\right]$ in the reentrant and the intermediate situations [as predicted in Figs. 4–4] are shown in Figs. 5 and 5, respectively, for $n=0$. Here, although we have used the same $\tau$, $\varphi$, $M_i$, and $M_f$ values as in Fig. 4, $L$ is chosen to be 1000 to accentuate the instants of real time at which sharp nonanalyticities appear.

Figure 6

This set of figures presents the scenario occurring at the critical values of $\tau$, namely ${\tau }_c^1$ and ${\tau }_c^2$, with $M_i=3$, $M_f=-3$, $\varphi =1$, and $L=100$. As shown in (a), at $\tau ={\tau }_c^1$, the excitation probability $p_{\mathbf{k}}$ becomes $1/2$ for the mode $k_x=k_y=0$ [the trim point at the corner of BZ in Fig. 1] for which $M_H=0$. Part (b) shows that the area of FZs touches the imaginary (real-time) axis for this mode at the instant $t_m$; the inset shows that $e^2\left(\mathbf{k}\right)$ touches the $p_{{\mathbf{k}}^{*}}=1/2$ line for this momentum mode. On the contrary, at $\tau ={\tau }_c^2$ (c), $p_{\mathbf{k}}$ just becomes 1/2 for the trim point at the center of the BZ. Consequently, as shown in (d), the internal boundary of the area of FZs touches the real-time axis for this mode; this is more transparently depicted in the inset, where there is an additional boundary point for the energy $e_m^2\left(\mathbf{k}\right)$ (that corresponds to the instant $t_m$) when compared to the inset of Fig. 4.

Figure 7

These figures demonstrate the effective 1D behavior that emerges when the parameter $\varphi$ and consequently $M_H$ vanishes; we choose $\tau =0.3333$, so that for $\varphi \ne 0$ there is a reentrant behavior as presented Figs. 4 and 4. Part (a) shows that the excitation probability $p_{\mathbf{k}}\sim 1$, around both the DPs. Nevertheless, as shown in (b), the FZs constitute a line rather than an area; furthermore, the reentrant behavior completely disappears. As elaborated in the text, in (c), we show that there exist sharp nonanalyticities in $\mathrm{Re}\left[f\left(t^{\prime }\right)\right]$ itself at different instants of time, and we choose $n=0$, 1, and 2. In (d), it is illustrated that even for very small $\varphi$, there are four boundary points of $e^2\left(\mathbf{k}\right)$ on the line $p_{{\mathbf{k}}^{*}}=1/2$, which shrink to one point as soon as $\varphi$ vanishes (inset).

Figure 8

(a) The FZs when $M_H\left(\mathbf{k}\right)$ assumes two constant values (with different signs) at two DPs; we obtain two disjoint lines of FZs both crossing the real-time axis at two different instants. (b) This refers to a hypothetical situation in which the Haldane mass assumes three constant values in three regions of the BZ as elaborated in the text, and consequently one obtains three disjoint lines.

Figure 9

A hexagonal lattice on which the topological Haldane model resides with ${\mathbf{a}}_{\mathbf{1}}$ and ${\mathbf{a}}_{\mathbf{2}}$ as the lattice vectors.