Density wave instabilities and surface state evolution in interacting Weyl semimetals

We investigate the interplay of many-body and band-structure effects of interacting Weyl semimetals (WSMs). Attractive and repulsive Hubbard interactions are studied within a model for a time-reversal-breaking WSM with tetragonal symmetry, where we can approach the limit of weakly coupled planes and coupled chains by varying the hopping amplitudes. Using a slab geometry, we employ the variational cluster approach to describe the evolution of WSM Fermi arc surface states as a function of interaction strength. We find spin and charge density wave instabilities which can gap out Weyl nodes. We identify scenarios where the bulk Weyl nodes are gapped while the Fermi arcs still persist, hence realizing a quantum anomalous Hall state.

Figure 1

Tight-binding band structure of the noninteracting ${\mathrm{WSM}}_6$ phase $\left(m_x=-1,m_0=2,t_x=2\right)$ in the surface BZ, which is shown in transparent gray with $(k_x,k_y)\in [-\pi ,\pi ]\times [-\pi ,\pi ]$. A surface state (light red) including Fermi arcs (red), and the valence (gold) and conduction bulk bands (blue) with the Weyl nodes are shown. The surface state containing the three Fermi arcs evolves through the entire surface BZ.

Figure 2

“Paramagnetic” interacting phasediagram (i.e., ignoring many-body instabilities) containing a QAH insulating state, various WSM phases which are distinguished by the number of Weyl nodes [39], and a trivial Mott insulator. $W$ denotes the bandwidth.

Figure 3

Interacting phase diagram including SDW and CDW orders for (a) $t_x=1$ and (b) $t_x=2$. Besides the normal semimetallic phases, we also find them accompanied by SDW/CDW orders. Moreover, trivial insulating SDW and CDW phases appear as well as an antiferromagnetic QAH phase. For discussion, see the main text. (c) Movement of two Weyl nodes towards each other influenced by the repulsive interaction leads to simultaneous annihilation of the Weyl nodes at $k_x=0$ and the Fermi arc. (d) Backfolding of the BZ can lead to doubling of the Fermi arc, realizing a QAH state after the Weyl nodes are annihilated at $k_x=\pi /2$.

Figure 4

(a)–(c) Evolution of the interacting surface states at $t_x=1.0$ for repulsive interactions. We show the high-symmetry path $(k_x=0,k_y=0)\to (\pi ,0)$. Note that the path $(0,0)\to (-\pi ,0)$ is symmetric and therefore omitted. We only select the spectral weight which is localized at the top surface, guaranteeing the best resolution of the surface states. Note that the backfolded Fermi arc is plotted on a logarithmic scale to increase its visibility. (b) Spectral function of the magnetic QAH phase with the gapped Weyl nodes. (a), (c) Spectral functions of the magnetic ${\mathrm{WSM}}_2$ phase. (d) Position $k_x/\pi$ of the Weyl nodes and the single-particle gap $\mathrm{\Delta }$, identifying the QAH phase. (e) Schematic picture of the slab geometry with only two clusters (i.e., four sites) along $z$.

Figure 5

A representative cut through the surface states along $k_y$ (for fixed $k_x=3\pi /4)$ is shown at the transition from ${\mathrm{WSM}}_4$ into the CDWI phase $\left(t_x=1\right)$. (a) Surface state with negative Fermi velocity and the backfolded surface state (possessing much weaker intensity) with positive velocity: They hybridize and acquire a small gap. (b) A further increase of attractive interactions opens the gap further. More details are delegated to the Supplemental Material [53].