Dynamical quantum phase transitions in Weyl semimetals

The quench dynamics in type-I inversion-symmetric Weyl semimetals (WSM) is explored in this work which, due to the form of the Hamiltonian, may be readily extended to two-dimensional Chern insulators. We analyze the role of equilibrium topological properties characterized by the Chern number of the prequench ground state in dictating the nonequilibrium dynamics of the system, specifically, the emergence of dynamical quantum phase transitions (DQPT). By investigating the ground-state fidelity, it is found that a change in the signed Chern number constitutes a sufficient but not necessary condition for the occurrence of DQPTs. Depending on the ratio of the transverse and longitudinal hopping parameters, DQPTs may also be observed for quenches lying entirely within the initial Chern phase. Additionally, we analyze the zeros of the boundary partition function discovering that while the zeros generally form two-dimensional structures resulting in one-dimensional critical times, infinitesimal quenches may lead to one-dimensional zeros with zero-dimensional critical times, provided the quench distance scales appropriately with the system size. This is strikingly manifested in the nature of nonanalyticities of the dynamical free energy, revealing a logarithmic singularity. Moreover, we show that these zero-dimensional critical times may be detected only by a direct observation of the dynamical free energy. Finally, following recent advances in observing the dynamical zeros of the Loschmidt overlap through azimuthal Bloch phase vortices, we rigorously investigate the same in WSMs, demonstrating that only one-dimensional critical times arising from two-dimensional manifolds of zeros of the boundary partition function lead to dynamical vortices.

Figure 1

Chern-number phase diagram for the type-I inversion-symmetric single WSM lattice model (2). The equations of the boundaries are obtained from the degeneracies beginning with the uppermost one are $\cos \left(k_z\right)=\frac{t^{\prime }}{2t}(2-b),\cos \left(k_z\right)=\frac{t^{\prime }}{2t}(-b)$, and $\cos \left(k_z\right)=\frac{t^{\prime }}{2t}(-2-b)$.

Figure 2

The domains in parameter space for the Hamiltonian (2) with $t=-t^{\prime }=1$ which yield DQPTs for prequench (a) $b_i=6.0$, (b) $b_i=4.5$, and (c) $b_i=2.5$. Regions shown in white denote the occurrence of a DQPT. Red lines show the Chern phase boundaries (Fig. 1), while the yellow line shows the $b_i$. A change in the Chern number guarantees a DQPT. Note that quenches from the upper $C=0$ phase to the lower $C=0$ phase in general do not trigger a DQPT except in a small region near the $C=0/-1$ phase boundary. (d) Same as before, but for $t=-\frac{t^{\prime }}{3}=1$ which shows DQPTs even for quenches lying entirely within the initial Chern phase.

Figure 3

${\left|F\right|}^2$ for the quenches described in the captions, at $k_z=0$. In each subfigure, the panels (1) show quenches with the same initial and final Chern phases while the panels (2) show Chern phase boundary crossing. Only in panels (2), at the $(k_x,k_y)$ values sufficiently close to the corresponding Chern phase boundary degeneracies, ${\left|F\right|}^2$ crosses half and triggers DQPTs. In (d), for the quench from the upper to the lower Chern zero phases, $(k_x,k_y)$ corresponding to all the boundaries lying in the quench path, which in fact includes all the Chern boundaries, contribute.

Figure 4

The domain in parameter space yielding DQPTs (white), near the Chern phase boundary between the $C=-1$ and $C=0$ (lower) phases, for a system described by $t=1$, and (a) $t^{\prime }=-2.2$, (b) $t^{\prime }=-2.5$, and (c) $t^{\prime }=-2.8$. The quench is described by $b_i=-\frac{2t}{t^{\prime }}-\delta$ with $\delta =0.5$, which equals (a) 0.4091, (b) 0.3, and (c) 0.2143, respectively, for the two panels. With decreasing $t^{\prime }$, the region yielding DQPTs within the initial $C=-1$ phase enlarges.

Figure 5

System has $t=-t^{\prime }=1$. (a)–(c) Each set of zeros has three tails, with each tail corresponding to a Chern phase boundary: thin lower tail $C=0\left(\mathrm{upper}\right)/1$, thick middle tail $C=1/-1$ and thin upper tail $C=-1/0\left(\mathrm{lower}\right)$. The tails corresponding to the Chern phase boundaries not crossed in the quench move toward the left extending up to $-\infty$, thereby preventing real-time zeros. (d) On increasing the quench distance across the Chern phase boundary (here $C=0/1$ boundary), the wave vectors ${\mathit{k}}^{*}:p_{{\mathit{k}}^{*}}=\frac{1}{2}$ move farther away from the degeneracies and thus the real-time intersections occur at smaller times. (e) Zeros of the boundary partition function for the quench $b=4.1\to 3.9$ at the $C=0$ and 1 phase boundaries with the zoomed inset showing the true 2D nature of the seemingly 1D tail which leads to 1D critical times. (f) Zero-dimensional critical time (marked in red) in $z_0$ for the quench described by $b=4.1\to 4.1-\mathrm{\Delta }b=3.9999$ with $\mathrm{\Delta }b$ given by (19) in a system of size $N=6001$. Dashed line in inset shows the zero on the lower tail closest to the imaginary axis with $\text{Re}{z}_0=-2.45$.

Figure 6

Isocircles of critical wave vectors $\left\{(k_x,k_y):p_{\mathit{k}}=\frac{1}{2}\right\}$ at the $C=0/1$ Chern phase boundary. The concentric circles centered at $(k_x,k_y)=(0,0)$, which is the point of degeneracy, with increasing radius and lighter colors correspond to ${\mathrm{\Delta }}_i=0.1,0.2,0.3$, and 0.4, respectively. Note that $\kappa =\frac{{\mathrm{\Delta }}_i}{2\sqrt{2}}$ corresponds to $(k_x,k_y)=(\kappa ,\kappa )$, implying that the corresponding isocircle has a radius $\kappa \sqrt{2}=\frac{{\mathrm{\Delta }}_i}{2}$ as seen from the figure.

Figure 7

Histogram plots of $\mathrm{Re}\left(z_0\right)$ evaluated at all the in-plane wave vectors ${\mathit{k}}_{\parallel }\in 2\text{D}$ BZ with $k_z=0$ in a system described by $t=-t^{\prime }=1$ with system size (a) $N=101$ and (b) $N=6001$, for the quench beginning at $b_1=4.1$ and $b_2=b_1-\mathrm{\Delta }b$ as described by (19). The 0D critical time with $\mathrm{Re}\left(z_0\right)=0$ is well isolated from the remaining zeros in the thermodynamic limit, the nearest one of which is located at $\text{Re}{z}_0\approx -2.45$ (blue dashed line). (c) The real parts of the zeros for $k_{x}a=k_{y}a\in [-\pi ,\pi ]$, showing the wave vector corresponding to the zero immediately neighboring the 0D critical time on the lower tail. [See accompanying text and Fig. 5 for the structure of the zeros.]

Figure 8

(a) $\mathrm{Re}f(z=it)$ (black line) and $\mathrm{Re}{f}^{\prime }(z=it)$ (red line) for the same quench as in Fig. 5. The colors of the shaded regions and the dashed lines match the colors of the zeros they depict in Fig. 5. $\mathrm{Re}f(z=it)$ shows nonanalyticities only at the imaginary-axis intersections of the zeros shown in Fig. 5. (b) $\mathrm{Re}f(z=it)$ (black line) and $\mathrm{Re}{f}^{\prime }(z=it)$ (red line) at the seemingly 1D tail at $t\approx 0.767$, denoted by the dashed green line in Fig. 5. There are two closely spaced sharp nondifferentiable points in $\mathrm{Re}{f}^{\prime }(z=it)$, bringing out the true 1D nature of the critical time, establishing that the thin tails are in fact 2D tails. (c) $\mathrm{Re}f(z=it)$ (black line) and $\mathrm{Re}{f}^{\prime }(z=it)$ (red line) for the quench $b=4.1\to 3.9999$ for a system with $N=6001\left(\mathrm{\Delta }k=\frac{2\pi }{Na}\right)$. A true 1D tail with a 0D critical time is found as shown in Fig. 5, showing a logarithmic singularity at the 0D critical time.

Figure 9

(a)–(f) Azimuthal phase map for the quench described by $t=-t^{\prime }=1,k_z=0$, and $b=\infty \to 1.9$. Static vortices are marked in black while the dynamical vortices with positive (negative) vorticity are marked in red (blue). The static vortices appearing at $(k_x,k_y)=(0,\pm \pi )$ and $(\pm \pi ,0)$ continue over the extended BZ and thus represent the same static vortices. The vortices develop and move along the $p_{\mathit{k}}=\frac{1}{2}$ contours in the 2D BZ, appearing initially in the middle of each $p_{\mathit{k}}=\frac{1}{2}$ contour and moving outward toward the edge of the BZ with increasing time. (g) $p_{\mathit{k}}=\frac{1}{2}$ contours shown in white. (h) The zeros $z_0$ and $z_1$, with the boundaries of the corresponding critical times marked.