Constructing a Weyl semimetal by stacking one-dimensional topological phases

Topological semimetals in three-dimensions (e.g., a Weyl semimetal) can be built by stacking two-dimensional topological phases. The interesting aspect of such a construction is that even though the topological building blocks in the low dimension may be gapped, the higher dimensional semimetallic phase emerges as a gapless critical point of a topological phase transition between two distinct insulating phases. In this work, we extend this idea by constructing three-dimensional topological semimetallic phases akin to Weyl systems by stacking one-dimensional Aubry-Andre-Harper (AAH) lattice tight-binding models with nontrivial topology. The generalized AAH model is a family of one-dimensional tight-binding models with cosine modulations in both hopping and on-site energy terms. In this paper, we present a two-parameter generalization of the AAH model that can access topological phases in three dimensions within a unified framework. We show that the $\pi$-flux state of this two-parameter AAH model manifests three-dimensional topological semimetallic phases where the topological features are embedded in one dimension. The topological nature of the band touching points of the semimetallic phase in 3D is explicitly established both analytically and numerically from the 1D perspective. This dimensional reduction provides a simple protocol to experimentally construct the three-dimensional Brillouin zone of the topological semimetallic phases using “legos” of simple 1D double well optical lattices. We also propose Zak phase imaging of optical lattices as a tool to capture the topological nature of the band touching points. Our work provides a theoretical connection between the commensurate AAH model in 1D and Weyl semimetals in 3D, and points toward practical methods for the laboratory realization of such three-dimensional topological systems in atomic optical lattices.

Figure 1

Numerical energy spectrum for tight-binding chain of 100 sites with open boundary conditions and parameter value ${\lambda }_{xy}={\lambda }_{xz}=0.3,{\lambda }_y={\lambda }_z=0,t=1$. (a)–(c) shows the evolution of the spectrum as a function of ${\varphi }_y$ for ${\varphi }_z=0,\pi /2,\pi$, respectively. (d) Region plot for the inequality condition $|{\Delta }_{+}({\varphi }_y,{\varphi }_z)|>|{\Delta }_{-}({\varphi }_y,{\varphi }_z)|$ as a function of for existence of zero-energy modes. The shaded region corresponds to the zero-energy states and the boundary of the region corresponds to the topological phase transition where the zero-energy states terminate as seen in (a)–(c).

Figure 2

Solution of $({\varphi }_y,{\varphi }_z)$ satisfying Eqs. (9) and (10) for ${\lambda }_y={\lambda }_{xz}=0.6,{\lambda }_z={\lambda }_{xy}=0.4$, and $t=1$. (a) The red line denotes the contour satisfying the first equality condition of Eq. (9) for which the on-site modulation vanishes. Shaded region denotes the Brillouin zone patch satisfying the second inequality condition in Eq. (10) for the existence of the zero modes. (b) The red line corresponds to the zero-energy (Weyl) arcs, which is the region satisfying the intersection of both constraints.

Figure 3

Numerical energy spectrum for tight-binding chain of 100 sites with open boundary conditions and parameter value ${\lambda }_z={\lambda }_{xy}=0.3,{\lambda }_y={\lambda }_{xz}=0,t=1$. (a)–(c) show the evolution of the spectrum as a function of ${\varphi }_y$ for ${\varphi }_z=0,\pi /4,\pi /2$, respectively. (d) Region plot for the inequality condition $|{\Delta }_{+}|>|{\Delta }_{-}|$ as a function of $({\varphi }_y,{\varphi }_z)$ for existence of zero-energy modes. The red lines corresponds to the zero-energy states and the ends of the lines corresponds to the topological phase transition point where the zero-energy states terminate as seen in (a)–(c).

Figure 4

Numerical energy spectrum for tight-binding chain of 100 sites with open boundary conditions and parameter value ${\lambda }_z={\lambda }_{xy}=0.4,{\lambda }_{xz}=0.6,{\lambda }_y=0.4,t=1$. (a) and (b) show the evolution of the spectrum as a function of ${\varphi }_y$ for ${\varphi }_z=-\pi ,-3\pi /4$, respectively. (c)–(e) show the evolution of the spectrum as a function of ${\varphi }_y$ for ${\varphi }_z=-\pi /2,-\pi /4,0$ respectively. (f) Region plot tracing the locus of zero-energy Fermi surface or the Weyl arc states for the condition given in Eqs. (9) and (10) as a function of $({\varphi }_y,{\varphi }_z)$. The red lines corresponds to the zero-energy states and end of the line corresponds to the topological phase transition point where the zero-energy states terminate or the Weyl nodes as seen in the spectrum plot (c).

Figure 5

Average hybrid Wannier charge center for tight-binding chain of 100 sites with open boundary conditions and parameter value ${\lambda }_z={\lambda }_{xy}=0.4,{\lambda }_{xz}=0.6,{\lambda }_y=0.4,t=1$. (b) and (d) show $\overline{n}({\varphi }_y,{\varphi }_z)\text{vs}{\varphi }_y$ for trivial (insulator) Brillouin zone slices ${\varphi }_z=-\pi ,-3\pi /4$. (h) and (j) show $\overline{n}({\varphi }_y,{\varphi }_z)\text{vs}{\varphi }_y$ for topologically nontrivial (two mirror copies of the Chern insulator) BZ slices ${\varphi }_z=-\pi /4,\text{and}0$. (f) show the topological transition at ${\varphi }_z=-\pi /2$, which is a semimetallic 2D plane containing the band touching points or Weyl nodes. We plot the bands corresponding to the polarization plot for each 2D plane in (a), (c), (e), (g), and (i).

Figure 6

Schematic of a double well potential with $\Delta ({\varphi }_y,{\varphi }_z)$ modulating the relative hopping amplitude of the dimerized state. $\epsilon ({\varphi }_y,{\varphi }_z)$ modulates the relative on-site energy of the different realizations of the 1D lattice.

Figure 7

Cubic lattice unit cell with tilted magnetic flux in $y\text{−}z$ plane. The solid black lines denote the nearest-neighbor hopping and the dashed black lines denote the off diagonal or next nearest-neighbor hopping. B is the magnetic field confined to the $y\text{−}z$ plane tilted by an angle $\alpha$ with respect to the $z$ axis.