Pairwise annihilation of Weyl nodes induced by magnetic fields in the Hofstadter regime

Weyl semimetal, which does not require any symmetry except translation for protection, is a robust gapless state of quantum matter in three dimensions. When translation symmetry is preserved, the only way to destroy a Weyl semimetal state is to bring two Weyl nodes of opposite chirality close to each other to annihilate pairwise. An external magnetic field can destroy a pair of Weyl nodes (which are separated by a momentum space distance $2k_0$) of opposite chirality, when the magnetic length $l_B$ becomes close to or smaller than the inverse separation $1/2k_0$. In this work, we investigate pairwise annihilation of Weyl nodes induced by an external magnetic field which ranges from a small to a very large value in the Hofstadter regime. We show that this pairwise annihilation in a Weyl semimetal featuring two Weyl nodes leads to the emergence of either a normal insulator or a layered Chern insulator. In the case of a Weyl semimetal with multiple Weyl nodes, the potential for generating a variety of states through external magnetic fields emerges. Our study introduces a straightforward and intuitive representation of the pairwise annihilation process induced by magnetic fields, enabling accurate predictions of the phases that may appear after pairwise annihilation of Weyl nodes.

Figure 1

An intuitive picture of how and when a normal insulator (NI) and a LCI state appear after pairwise annihilation of two WNs separated by a momentum space distance $2k_0$. (a) shows projections of the WNs (black dots) and the Fermi arc in the $k_x\text{−}{k}_y$ surface BZ. The parameter $2k_0^{\prime }=2\pi -2k_0$ measure the inter-BZ separation between the two WNs of opposite chirality. Two Weyl nodes get pairwise annihilated by magnetic field when the inverse magnetic length $l_B^{-1}$ becomes close to or larger then momentum space separation between them. There are two scenarios prevail. When $k_0<k_0^{\prime }$, the inverse magnetic length $l_B^{-1}$ first reaches the intra-BZ separation $2k_0$. In this case, when the magnetic field is increased, two WNs approach each other along the Fermi arc to meet at a point inside the BZ. This leads annihilation of the two nodes without leaving the Fermi arc. Hence a normal insulator emerges. On the other hand, if $k_0>k_0^{\prime }$, pairwise annihilation occurs at the boundary of the BZ by leaving the Fermi arc as depicted in (c). Hence a LCI state emerges.

Figure 2

(a) Phase diagram of the time-reversal broken WSM [Eq. (2)] with two WNs in presence of commensurate flux $\varphi /{\varphi }_0=1/q$ (magnetic flux quantum ${\varphi }_0=h/e$) per unit cell, for small $q$ values (large fluxes) in the Hofstadter regime $l_B\sim a$. This phase diagram is obtained from the gapless solution of the Bloch-Hofstadter Hamiltonian Eq. (4). Here $k_0$ (in units of inverse lattice constant $a$) represents half the distance separating the two Weyl nodes of opposite chirality. The regions with grey, blue and orange color represent a normal insulator (NI), WSM and a LCI state respectively. (b) Energy gap $\mathrm{\Delta }$ (in arbitrary unit) is plotted as a function of the separation parameter $k_0$ for large $q$ values (small fluxes). (c) This phase diagram is derived with inputs from the Fig. 2. We notice a similarity between the two phase diagrams for the small and large $q$ values (details in the text).

Figure 3

An intuitive representation of pairwise annihilation process of WNs of opposite chirality by an external magnetic field. Figures (a) and (d) show the projections of the WNs (black dots) and the Fermi arcs on the $k_x\text{−}{k}_y$ surface BZ. For a magnetic field aligned in the $y$-direction, separations of WNs along the $k_x$ direction are relevant for pairwise annihilation. $2k_{01}$ and $2k_{02}$ are the intra-BZ separations and $2k_{01}^{\prime }=2\pi -2k_{01}$ and $2k_{02}^{\prime }=2\pi -2k_{02}$ are the corresponding inter-BZ separations. If $k_{02}^{\prime }<k_{01}$, the pair of WNs separated by $2k_{02}$ will be annihilated at the boundary of BZ by leaving the Fermi arc states. Thus a coexistence phase ${\mathrm{W}2}^{\prime }$ emerges (see figure (b)). If $k_{02}^{\prime }>k_{01}$, then the pair of WNs separated by $2k_{01}$ will be annihilated at some point inside the BZ without leaving the Fermi arcs. This results in a WSM (labelled W2) with two Weyl nodes (see (c)). Suppose $k_{01}=k_{02}=k_0$ as shown in (d). Now, it is clear that a normal insulator (NI) emerges when $k_0<k_0^{\prime }$, and an insulator (${\mathrm{I}}^{\prime }$) with counter propagating surface states appears when $k_0>k_0^{\prime }$. Note that we would get the same set of phases if the magnetic field was aligned in the $z$ direction, provided the separations of the WNs along the $k_y$ direction is kept maximum.

Figure 4

Projections of the WNs and the Fermi arcs on the $k_x\text{−}{k}_y$ surface BZ of the model in Eq. (8). The arrows indicate that the states on the two Fermi arcs are counter propagating. Clearly, there are two separations $2k_1$ and $2k_2$ between Weyl nodes of opposite chirality.

Figure 5

Phase diagrams in Figs. 5 for small $q=2,3,4$, and 5 are obtained from the gapless (analytical) solutions of the Bloch-Hofstadter Hamiltonian Eq. (9). The dark-blue region(s) represent a gapless phase which is a WSM for $q=3,5$ (odd) and nodal line semimetal for $q=2,4$ (even). The white region represent a normal insulator. For larger $q$ values (smaller fluxes), the phase diagrams can be derived by computing the energy gap as a function of the two separation parameters $k_1$ and $k_2$. The bulk energy gap (in arbitrary units) is computed numerically and plotted in Figs. 5 for different values of flux $1/q$. The dark-blue regions represent a gapless phase. All the insulating regions (in yellow) are adiabatically connected. For large $q$ value, say $q=81$, we notice that the insulating regions appear where $|k_1-\pi /2|\sim \pi /2$, $k_2\sim \pi /2$ or $k_1\sim \pi /2$, $|k_2-\pi /2|\sim \pi /2$.

Figure 6

Bulk energy dispersion $E$ (in arbitrary units) of the Hofstadter Hamiltonian $H_{\varphi }^{\left(z\right)}$, Eq. (9), for $q=40$. The separation parameters are $k_1=2.0$ and $k_2=1.4$. For each of the plots, one of the momenta is allowed to vary, and the other ones are fixed at $k_x=0.3\pi /q$, $k_y=0.2\pi /q$ and $k_z=0.1\pi$ appropriately. Energy bands along the $k_x$ and $k_y$ directions form flat Landau levels.

Figure 7

(a) The Fermi arcs and the projections of the WNs on the $k_x\text{−}{k}_y$ surface BZ of the WSM defined in Eq. (8). For magnetic field along $y$ direction, the separation parameter $k_1$ is relevant for pairwise annihilation of Weyl nodes. If $k_1<k_1^{\prime }$, pairwise annihilation of the WMs, which occurs at a point inside the BZ, does not leave the Fermi arc states. Hence a normal insulator results in. If $k_1>k_1^{\prime }$, pairwise annihilation occurs at the boundary of the BZ by leaving the Fermi arc states. Hence, the insulator ${\mathrm{I}}^{\prime }$ emerges.

Figure 8

Phase diagrams of the time-reversal preserved WSM [Eq. (8)] with four WNs in presence of $1/q$ commensurate flux per unit cell, along the $y$ direction. $k_1$ and $k_2$ are the separation parameters (in units of the inverse lattice constant $a$) between Weyl nodes of opposite chirality. The shaded areas in grey, blue, and orange signify a normal insulator, Weyl semimetal, and an insulator (${\mathrm{I}}^{\prime }$), respectively. The latter exhibits counterpropagating surface states along the open surface in the $z$ direction.

Figure 9

An intuitive representation of pairwise annihilation of WNs by external magnetic field in a WSM with six Weyl nodes. Figures (a) and (d) depict the projections of the WNs (black dots) and the Fermi arcs in the $k_x\text{−}{k}_y$ surface BZ. For a magnetic field aligned in the $y$-direction, only the two separation parameters $k_1$ and $k_2$ are relevant. In the first scenario (a), a coexistence phase ${\mathrm{W}2}^{\prime \prime }$ emerges when $k_1<k_2^{\prime }$ [see (b)] and an insulator ${\mathrm{I}}^{\prime }$ with counter propagating surface states appear when $k_1>k_2^{\prime }$ [see (c)] after pairwise annihilation by magnetic fields. In the second scenario (d), pairwise annihilation results in either a coexistence phase ${\mathrm{W}4}^{\prime }$ or a WSM with two WNs, depending on the relative values of $k_1^{\prime }$ and $k_2$.

Figure 10

(a) Longitudinal conductance $G_{zz}$ as a function of separation parameter $k_0$ (in units of the inverse lattice constant $a$) between two WNs of opposite chirality for three different values of flux $1/q=1/3,1/5,1/7$. Conductance is computed for a WSM slab of length $L_z=100$ and width $L_x=L_y=25$ [the model is defined in Eq. (2)]. The chemical potential is fixed at $\mu =0.1$. The conductance $G_{zz}$ is maximum for intermediate separation but vanishes for small separation and drops to finite value for large separation. (b)–(d) show the energy spectrum $E$ (in arbitrary units) of the slab (taken periodic along the transport direction $z$) for three different values of $k_0=1.0$ (normal insulator), $k_0=1.3$ (WSM), and $k_0=2.0$ (LCI) for a fixed $q=5$.

Figure 11

Energy gap (in arbitrary units) of the system [Eq. (8)] in a slab geometry (finite along the $z$ direction) is plotted as a function of the separation parameters $k_1$ and $k_2$. The slab has zero energy surface states in the dark-blue regions. Comparing with the phase diagrams Figs. 5 and 5, we see that the insulator which is living on the “right insulating region” has zero energy surface states. We take a representative point $k_1=2.6$, $k_2=2.2$ from the right insulating region for $q=3$ to show the surface states in the Fig. 12.

Figure 12

(a) The zero energy surface states in the $k_x\text{−}{k}_y$ surface BZ and (b) the dispersion along the $k_y$ direction for $q=3$ of the system [Eq. (8)] in a slab geometry (finite along the $z$ direction). Values of the separation parameters are $k_1=2.6$, $k_2=2.2$, which represent a model for insulator [see Fig. 5]. (b), the surface states, which lie in the bulk gap of the insulator, are highlighted.