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

Abstract

Weyl semimetal, which does not require any symmetry except translation for protection, is a robust gapless state of quantum matters 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 external magnetic field which ranges all the way from small to a very large value in the Hofstadter regime $l_B \sim a$. We show that this pairwise annihilation in a WSM 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.

Figures (12)

• Figure 1: ($a$) Phase diagram of the time-reversal broken WSM (Eq. \ref{['eq:H0TRB']}) with two WNs in presence of commensurate flux $1/q$ per unit cell, for small $q$ values in the Hofstadter regime $l_B \sim a$. This phase diagram is obtained from the gapless solution of the Bloch-Hofstadter Hamiltonian Eq. \ref{['eq:blochH2']}. The regions with grey, blue and orange color represent a normal insulator (NI), WSM and a LCI state respectively. ($b$) Energy gap $\Delta$ is plotted as a function of the separation parameter $k_0$ for large $q$ values. ($c$) This phase diagram is derived with inputs from the Fig. \ref{['fig:PhaseTRB1']}($b$). We notice a similarity between the two phase diagrams for the small and large $q$ values (details in the text).
• Figure 2: 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 $2k_0$. Figure ($a$) shows projections of the WNs (black dots) and the Fermi arc in the $k_x$-$k_y$ surface BZ. The parameter $2k'_0 = 2\pi - 2k_0$ 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 $l^{-1}_B$ becomes close to or larger then momentum space separation between them. There are two scenarios prevail. When $k_0 < k'_0$, the inverse magnetic length $l^{-1}_B$ 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 $k_0> k'_0$, pairwise annihilation occurs at the boundary of the BZ by leaving the Fermi arc as depicted in figure ($c$). Hence a LCI state emerges.
• 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 $k_x$-$k_y$ 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 $2k'_{01} = 2\pi - 2k_{01}$ and $2k'_{02}= 2\pi - 2k_{02}$ are the corresponding inter-BZ separations. If $k'_{02}<k_{01}$, 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 W2$'$ emerges (see figure($b$)). If $k'_{02}>k_{01}$, 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 $k_0<k'_0$, and an insulator (I$'$) with counter propagating surface states appears when $k_0>k'_0$. 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 $k_x$-$k_y$ surface BZ of the model in Eq. \ref{['eq:HKTRP']}. 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. \ref{['fig:Phaseqz_TRP']}($a$)-($d$) for small $q=2, 3, 4$, and 5 are obtained from the gapless (analytical) solutions of the Bloch-Hofstadter Hamiltonian Eq. \ref{['eq:HKTRPB']}. 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, 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 is computed numerically and plotted in Figs. \ref{['fig:Phaseqz_TRP']} ($i$-$l$) for different values of $q$. 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 $k_1 \sim \pi/2$, $|k_2 - \pi/2| \sim \pi/2$.