Non-Abelian anomalies in multi-Weyl semimetals

We construct the effective field theory for time-reversal symmetry-breaking multi-Weyl semimetals (MWSMs), composed of a single pair of Weyl nodes of (anti)monopole charge $n$, with $n=1,2,3$ in crystalline environment. From both the continuum and lattice models, we show that a MWSM with $n>1$ can be constructed by placing $n$ flavors of linearly dispersing simple Weyl fermions (with $n=1$) in a bath of an $\text{SU}\left(2\right)$ non-Abelian static background gauge field. Such an $\text{SU}\left(2\right)$ field preserves certain crystalline symmetry (fourfold rotational or $C_4$ in our construction), but breaks the Lorentz symmetry, resulting in nonlinear band spectra, namely, $E\sim {(p_x^2+p_y^2)}^{n/2}$, but $E\sim |p_z|$, for example, where momenta $\mathbf{p}$ is measured from the Weyl nodes. Consequently, the effective field theory displays $\text{U}\left(1\right)\times \text{SU}\left(2\right)$ non-Abelian anomalies, yielding the anomalous Hall effect, its non-Abelian generalization, and various chiral conductivities. The anomalous violation of conservation laws is determined by the monopole charge $n$ and a specific algebraic property of the $\text{SU}\left(2\right)$ Lie group, which we further substantiate by numerically computing the regular and isospin densities from the lattice models of MWSMs. These predictions are also supported from a strongly coupled (holographic) description of MWSMs. Altogether our findings unify the field-theoretic descriptions of MWSMs of arbitrary monopole charge $n$ (featuring $n$ copies of the Fermi arc surface states), predict signatures of non-Abelian anomaly in table-top experiments, and pave the way to explore the structure of anomalies for multifold fermions, transforming under arbitrary half-integer or integer spin representations.

Figure 1

Energy dispersion ${\epsilon }_{\mathbf{p}}$ for multi-Weyl fermions in the $(p_x,p_y)$ plane for (a) $n=1$, (b) $n=2$, and (c) $n=3$ around a Weyl node. For $n=1$, 2, and 3 the valence and conduction bands display, respectively, linear, quadratic, and cubic touching in this plane. However, the dispersion always scales linearly with $p_z$ irrespective of $n$, as shown in (d). The momentum $\mathbf{p}$ is measured from the Weyl nodes, placed at $\mathbf{p}=0$.

Figure 2

Band structure of (a) simple ($n=1$), (b) double ($n=2$), and (c) triple ($n=3$) Weyl semimetals from the corresponding effective two band models [see Eqs. (15, 16, 17, 18)]. We set $t=t_0=t_z=1$ in the tight-binding model, and the lattice constant $a=1$. We here take the path $K\to \mathrm{\Gamma }\to W\to R\to K$, where $K$, $\mathrm{\Gamma }$, $W$, and $R$ stand for $(\pi ,\pi ,\pi /2)$, $(0,0,0)$, $(0,0,\pi /2)$, and $(\pi ,0,\pi /2)$, respectively. The energy $E$ is measured in units of $t$. Note that Weyl nodes are located at $(0,0,\pm \pi /2)$, around which dispersion always scales linearly in the $z$ direction. The insets display linear, quadratic, and cubic dispersion in the $(k_x,k_y)$ plane, respectively. Due to the absence of particle-hole asymmetry, the Weyl nodes are always pinned at zero energy. Realizations of double- and triple-Weyl systems from four- and six-band models, respectively, are shown in Fig. 3.

Figure 3

Band structure of (a) double- and (b) triple-Weyl semimetals from four- and six-band models, respectively [see Eqs. (19) and (20)]. For numerical diagonalization we set $t=t_0=t_z=\mathrm{\Delta }=1$ in the tight-binding models, and the lattice spacing $a=1$. Note that these models also display quadratic [see (a)] and cubic [see (b)] dispersion in the $(k_x,k_y)$ plane around the Weyl nodes located at $W=(0,0,\pm \pi /2)$ [compare with Figs. 2 and 2]. The dispersion always scales linearly with $k_z$.

Figure 4

Topologically protected Fermi arc surface states for (a) simple ($n=1$), (b) double- ($n=2$), and (c) triple- ($n=3$) Weyl semimetals, obtained from their two-band models [see Eqs. (15, 16, 17, 18)]. For numerical diagonalization we implement a mixed momentum (along $k_y$ and $k_z$) and real-space or Wannier (along $x$) representation. The linear dimensionality of the system along $x$ is $L=60$ (see Sec. 3b for details). Consequently, the Fermi arc surface states are localized on the top and bottom surfaces. We set $t=t_z=t_0=1$. We display here the square of the amplitude of the low-energy states within an energy window $\mathrm{\Delta }E=0.1$ for $n=1$, $\mathrm{\Delta }E=0.08$ for $n=2$, and $\mathrm{\Delta }E=0.05$ for $n=3$ around zero energy. The number of Fermi arcs is equal to $n$, anchoring the bulk-boundary correspondence for multi-Weyl semimetals. In addition, the Fermi arcs from opposite surfaces (top and bottom) get connected through the bulk Weyl points (acting as defects, namely, monopole and antimonopole, in the momentum space), where the band gap vanishes.

Figure 5

Fermi arc surface states for (a) double- ($n=2$) and (b) triple- ($n=3$) Weyl semimetals, obtained, respectively, from four- and six-band models [see Eqs. (19) and (20)]. For numerical diagonalization we follow the same approach, mentioned in the caption of Fig. 4, and set $t=t_z=t_0=\mathrm{\Delta }=1$ and $a=1$. We display here the square of the amplitude of the low-energy states within an energy window $\mathrm{\Delta }E=0.05$ for $n=2$ and $\mathrm{\Delta }E=0.04$ for $n=3$. Note that double- and triple-Weyl semimetals support, respectively, two and three copies of Fermi arc surface states, as we obtained from their two-band representations [see Figs. 4 and 4]. This observation establishes the topological equivalence among these models and the bulk-boundary correspondence in these gapless topological systems. The color scale is the same as in Fig. 4.

Figure 6

Charge density $\delta Q$, measured in units of $e/\pi$, in the vicinity of the flux tubes placed at $x=L/4$ (yielding $\delta Q>0$) and $x=-L/4$ (yielding $\delta Q<0$) [see Eq. (65)] for simple Weyl (SW), double Weyl (DW), and triple Weyl (TW) semimetals. Results for simple Weyl semimetal are always obtained from the two-band model. By contrast, for the double- (triple-) Weyl semimetals, we compute $\delta Q$ from (a) four- (six-) and (b) two-band models. Here $\mathrm{\Phi }$ is the magnetic flux, measured in units of the flux quanta ${\mathrm{\Phi }}_0=2\pi /e$ (in natural units $\hslash =c=1$). For numerical analysis, we always set $t=t_z=t_0=1$ and $L_z=200$, where $L_z$ is the number of points in the momentum space along the $k_z$ direction. Specifically, for (a) we set $\mathrm{\Delta }=0.2$ and $L=24$, while for (b) $L=48$, where $L$ is the linear dimension of the system in the $x$ and $y$ directions. The dots correspond to numerically computed values of $\delta Q$, whereas the straight lines follow the relation $\delta Q=n\frac{e}{2\pi }(\mathbf{b}·\mathbf{B})\equiv nb(\mathrm{\Phi }/{\mathrm{\Phi }}_0)$, where $n$ is the monopole charge of the Weyl nodes, located at $\mathbf{b}=(0,0,\pm b)$, and $b=\pi /2$. This analysis establishes excellent agreement between the field-theoretic predictions and the scaling of the corresponding observable computed from the lattice models.

Figure 7

Non-Abelian or isospin density ${\rho }_3$ as a function of the non-Abelian magnetic field ${\mathrm{\Delta }}^2$, computed numerically from the four- and six-band models for the double-Weyl and triple-Weyl semimetals, respectively. For numerical analysis we set $t=t_z=t_0=1$ and discretize each momentum direction into 41 points in the Brillouin zone. In our construction, $|{\mathbf{B}}_3|={\mathrm{\Delta }}^2$ and $b=\pi /2$.

Figure 8

Renormalization of the background non-Abelian gauge field for different chemical potentials (see the inset) as a function of $\overline{\mathrm{\Delta }}$. The IR values correspond to $\overline{\mathrm{\Delta }}\to \infty$. The dashed line corresponds to a linear fit $Z=\overline{\mathrm{\Delta }}$.

Figure 9

Isospin chiral magnetic conductivity as a function of $\overline{\mathrm{\Delta }}$ for vanishing chemical potential $\mu$.

Figure 10

Isospin current as a function of $\overline{\mathrm{\Delta }}$ for vanishing isospin chemical potential ${\mu }_3$. The dashed line corresponds to a quadratic fitting $Z_{\mathrm{\Delta }}={\overline{\mathrm{\Delta }}}^2$.