Interaction-induced phase transitions of type-II Weyl semimetals

The study of Weyl semimetals (WSMs) lies at the forefront of the nontrivial topological phenomena in condensed-matter physics. In this work, we study the effect of on-site repulsive Hubbard interaction on the WSM system with a nonzero tilt at half filling. Within the Hartree-Fock mean-field approximation, we treat the Hubbard interaction self-consistently and find that the Fock exchange field vanishes, while the Hartree field can renormalize the topological mass, the tilt, and the Fermi velocity of the Weyl cones. When the renormalized tilt is larger than the renormalized Fermi velocity, the Hubbard interaction will induce the quantum phase transition from a type-I WSM to a type-II WSM. We then provide the interaction-induced phase diagrams of WSMs in different parametric spaces, in which the antiferromagnetic order at strong interaction is also considered. In addition, we analyze another model hosting two pairs of Weyl nodes, and similar results are obtained. The implications of these results are discussed.

Figure 1

Schematic plot of the tilting Weyl cones in the $k_x\text{−}{k}_z$ plane of (a) WSM1 with ${\gamma }_z=0.4$ and (b) WSM2 with ${\gamma }_z=1.2$. The Chern numbers are shown in different layers. As shown in (a), when the Hubbard interaction induces the renormalized topological mass $m_1^{\prime }$ increases, the Weyl nodes will move to the edge of the BZ, and the Weyl cones will become more tilted. Note the energies of Weyl nodes are unequal, leading to the existence of electron and hole Fermi surfaces at half filling.

Figure 2

Phase diagram of the WSM in the noninteracting case with $m_0=2$, where the different phases are shown with different colors. The phase boundary of solid lines characterizes the metal-insulator topological phase transition, while the dotted lines describes the WSM1-WSM2 continuous transitions. Stars a–c are the initial phases for interaction-induced phase transitions, as denoted by the arrows in Fig. 3.

Figure 3

The magnetization ${\mathcal{M}}_z$ vs (a) the topological mass $m_1$ and (b) the tilting parameter $a_t$ at different Hubbard interaction strengths and Wilson masses $(U,m_0)$. We have fixed $a_t=1$ in (a) and $m_1=-0.8$ in (b). The legend is the same for both plots.

Figure 4

Interaction-induced phase diagrams of WSM (a) in the parametric space of $(U,m_1)$ with $a_t=1$ and (b) in $(U,a_t)$ with $m_1=-0.8$. The different phases are shown in different colors. Note the linear phase boundaries in (a) and the nonlinear phase boundaries in (b).

Figure 5

Interaction-induced phase diagram of WSM at finite temperature, which is shown in the parametric space of $(U,\ln \beta )$ with $\beta =\frac{1}{k_{B}T}$. The different phases are shown in different colors.

Figure 6

(a) Schematic plot of the lattice structure in 3D space. The yellow cell represents nearest-neighbor four-site cells in the $({\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{b}}_3)$ basis. (b) The first BZ for the one- (four-) site cell is shown by the black (green) lines. (c) and (d) Band structures and DOS per unit cell obtained from the tight-binding model, where (c) is calculated along the high-symmetry lines in the BZ, as shown by the purple lines in (b). The other parameters are set as $a_t=1,m_0=2$, and $m_1=0.5$.

Figure 7

Interaction-induced phase diagrams in WSM at the large-$U$ limit. The red solid (blue dotted) lines in both plots are the separations between the QAHI and AFM-$xy$ (AFM-$z)$ phase, but actually, these phases cannot exist in the system due to their higher ground-state energies compared with AFM-$xyz$ order. The parameters are the same as in Fig. 4.

Figure 8

(a) Schematic plot of the lattice structure in the $x\text{−}y$ plane including one-site and two-site cells. The four yellow cells represent nearest-neighbor cells in the $({\mathbf{b}}_1,{\mathbf{b}}_2)$ basis. (b) The first $k_x\text{−}{k}_y$ BZ is plotted for the one- (two-) site cell in the solid black (dotted green) square. (c) and (d) Band structures and DOS per unit cell obtained from the tight-binding model, where (c) is calculated with $k_z=3$ and along the high-symmetry lines in the BZ, as shown by the purple lines in (b). The other parameters are set as $a_t=1,m_0=2$, and $m_1=0.5$.

Figure 9

(a) Schematic plot of the lattice structure in the 3D space, including one-site and two-site cells in the $z$ direction. The yellow cells represent nearest-neighbor cells. (b) The first $k_y\text{−}{k}_z$ BZ is plotted for the one- (two-) site unit cell in the solid black (dotted green) square. (c) and (d) Band structures and DOS per unit cell obtained from the tight-binding model, where (c) is calculated when $k_x=0$ and along the high-symmetry lines in the BZ, as shown by the purple lines in (b). The other parameters are the same as in Fig. 7.

Figure 10

Plot of the magnetization ${\mathcal{M}}_{\alpha z}$ vs the Hubbard interaction $U$ for the enlarged unit cell in (a) the $x\text{−}y$ plane, (b) the $z$ direction, and (c) all three directions. The parameters are chosen as $m_0=2,m_1=-1.2$, and $a_t=1$. (d) The total energy of the ground state $E/N_e$ for different AFM orders.