Nodal chain semimetal in geometrically frustrated systems

Geometrically frustrated systems and topological semimetals have both attracted much interest and been studied in various systems in recent years. Here we study the interplay between these two systems. We show that a Weyl point can be extended to a chain of degeneracy (a nodal chain) with nonzero charge of Berry flux through geometric frustration. We propose to realize such a charged nodal chain in an acoustic metamaterial, based on both tight-binding and full-wave numerical simulations. Moreover, we observe a fanlike surface state spectrum, the dispersion of which is controlled by the bulk band properties. Our paper points to a class of band degeneracy that carries nonzero Berry flux. The resulting topological metamaterial may be useful for controlling the flow of sound and light.

Figure 1

(a) Berry flux density distribution of a charged nodal line, where the arrows represent the direction and amplitude of the local Berry flux density and the red cylinder represents the charged nodal line. (b) Two nodal lines in the reciprocal space crossing each other can be regarded as two nodal rings chained together (c) as the reciprocal space is periodic.

Figure 2

(a) A sketch of the tight-binding model. The spheres represent lattice sites. Each unit cell consists of four sites with different heights. The heights of the red, blue, magenta, and cyan spheres are $h/4,2h/4,3h/4$, and $h$, respectively, where $h$ is the unit length along the $z$ direction. Black and yellow cylinders represent hopping between nearest neighbor and next-nearest neighbor, respectively. The hopping strengths represented by the black bonds are the same as denoted by $t_0$, while those of the yellow bonds are slightly different. The transparent light blue plane is drawn only for better illustration. (b) The projection of the tight-binding model on the $xy$ plane. The projection forms a square lattice with lattice constant $a$. The number on each sphere represents the height of that sphere. The black and yellow arrows represent, respectively, the projection of the black and yellow cylinders in (a), where the arrows point in the upward direction. The hopping strengths of the yellow bonds are either $t_1$ or $t_2$, as marked in (b). We set $t_1=\left(1+\delta \right)t_c$ and $t_2=\left(1-\delta \right)t_c$, where $\delta$ is a small number. (c) The reciprocal space with positions of relevant high-symmetric points labeled and nodal lines highlighted in red.

Figure 3

(a), (b) The band structures of the tight-binding model in Fig. 2, where $t_0=-2\sqrt{2},t_c=-1$, and $\delta =0.1$ for (a) and $\delta =0$ for (b). The inset in (b) shows the amplitude distributions of the eigenmodes on the flat bands, where red, cyan, and gray represent +1, $-1$, and zero, respectively. Here all the eigenmodes on the flat bands are the same except for a global gauge freedom. The Weyl points at $S$ and $Z$ in (a) get flattened by geometric frustration and form charged nodal lines in (b). The positions of the charged nodal lines are highlighted in Fig. 2, which hence form nodal chains in the reciprocal space similar to Fig. 1. (c), (d) Logarithm of the absolute value of the Berry flux density of the highest band at $k_{z}h=0.999\pi$, where (c) and (d) correspond to the band structures in (a) and (b), respectively. Here (c) and (d) share the same color bar, and red (blue) represents the maximum (minimum).

Figure 4

(a) and (b) show the side view and top view of a unit cell of an acoustic metamaterial which possesses nodal chains with nonzero Berry charges. Here blue and yellow represent the surfaces where hard boundary and periodic boundary conditions are applied, respectively. The system is filled with air (density $1.29\mathrm{kg}/{\mathrm{m}}^3$ and speed of sound 343 m/s). (c) The band dispersion of the system along several high-symmetry directions, which are quite similar to the tight-binding model. The bands form a nodal chain as highlighted in Fig. 2. The parameters used are $a=50\mathrm{cm},r=5\mathrm{cm},u=10\mathrm{cm},h=24\mathrm{cm},w_0=2.4\mathrm{cm}$, and $w_c=1.6\mathrm{cm}$.

Figure 5

(a)–(d) show, respectively, the charge distributions (left panel) in the reciprocal space and Chern numbers for fixed $k_z$ (right panel) of the first (lowest) band to the fourth (highest) band, where the red and orange spheres represent Weyl points with charge +2 and +1, respectively, cyan marks chain degeneracy with Berry charge $-2$, and gray marks chain degeneracy without Berry charge. The parameters used are the same as Fig. 4.

Figure 6

(a) The gray area represents the projected band structure of a strip of the system [shown in (b)], which is periodic along the $y$ and $z$ directions and finite along the $x$ direction. The number of unit cells along the $x$ direction is chosen to be large enough that the dispersions of the surface states (represented by gray, cyan, and red curves) are stable. Here the cyan and red curves represent, respectively, one-way edge states localized on the left and right boundary, and gray represents the trivial edge state. The color code in (b) represents the absolute value of the eigenpressure distribution of one edge state at $k_{y}a=0.1\pi$ and 200.9 Hz, where red and blue correspond to maximum value and zero, respectively. The parameters of the system used are the same as Fig. 4. (c) Fermi arcs on the $k_y-k_z$ surface Brillouin zone with different truncations calculated with the tight-binding model. We consider a strip of unit cells which is periodic along the $y$ and $z$ directions, while finite along the $x$ direction. Here black and magenta represent the cases when sublattice 1 (2) is the outmost sublattice (the inset on the lower right) and sublattice 3 (4) is the outmost sublattice (the inset on the upper left), respectively. The red dot and cyan line represent the projection of the charged +2 Weyl point and the charged $-2$ nodal chain, respectively. In this calculation, we set $t_0=-2\sqrt{2}$ and $t_c=-1$ and the energy is fixed at 4, which is also the energy of the nodal lines and Weyl points between band 3 and band 4.

Figure 7

(a) Spectrum of the surface state between band 3 and band 4 and localized on the upper surface of a strip of the system. The semi-infinite system is semifinite along the $z$ direction with sublattice 4 on the outmost and periodic along the $x$ and $y$ directions. (b) The Zak phase of band 4 along the black circle as shown in the inset, which is a circle with $a\sqrt{k_x^2+k_y^2}=0.1\pi$ and centers at $k_x=k_y=0$. The cyan lines and red sphere of the inset represent the projection of the nodal chain and the Weyl point, respectively. (c) The Zak phase of band 4 as a function of $k_x$ and $k_y$. (d) The Fermi arc at the energy of the nodal chain on the upper surface of a semi-infinite system as in (a) but with sublattice 3 on the outmost. Here we plot the density of state of the unit cell on the surface (surface local density of states). In this calculation, we set $t_0=-2\sqrt{2}$ and $t_c=-1$ and the energy is fixed at 4.

Figure 8

(a)–(d) show, respectively, the charge distributions when $\delta =0.1$ (left panel) and $\delta =0$ (middle panel) in the reciprocal space and Chern numbers for fixed $k_z$ (right panel) of band 1 to band 4. Here the Chern numbers as a function of $k_z$ for the left panel and middle panel are the same. On the left and middle panel, the red, orange, and blue spheres represent Weyl points with Berry charges +2, +1, and $-2$, respectively, the cyan lines denote line degeneracy with charge $-2$, and the gray sphere and lines denote point and line degeneracy without Berry charge. Here the tight-binding model in Eq. (A1) with $t_0=-2\sqrt{2}$ and $t_c=-1$ is assumed.

Figure 9

The Zak phase of each band as a function of $k_x$ for fixed $k_{z}h=0.5\pi$ (a) and $k_{z}h=0.1\pi$ (b), where black, red, blue, and magenta represent the Zak phases of band 1, 2, 3, and 4, respectively. The geometric parameters of the system under consideration are the same as Fig. 4.