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

Faruk Abdulla<EMAIL_ADDRESS>Harish-Chandra Research Institute, A CI of
Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad)
211019, India

###### 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 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.

##  I Introduction

Weyl semimetals (WSMs) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] are examples of three
dimensional topological semimetals where nondegenerate valence and conduction
bands touch at an even number of isolated points in the 3D Brillouin zone (BZ)
called Weyl nodes (WNs). Each WN carries a topological charge and has a
definite chirality. The fact that WNs carry nontrivial topological charges
leads to existence of special kind of surface states called surface Fermi arc
which joins the projections of WNs of opposite chiralities onto the surface
Brillouin zone (SBZ).

Weyl semimetal is a robust topological state of quantum matter. When spatial
translation symmetry is preserved, the only way to destroy the state is to
bring two WNs of opposite chiralities (or topological charges) close to each
other to annihilate them pairwise [1]. Weyl semimetals which are known for
many exotic properties such as chiral anomaly [11, 12, 13], negative
magnetoresistance [14, 15, 16, 17, 18, 19, 20], planar Hall effect [21, 22,
23, 24, 25], and Fermi arc mediated quantum oscillations and 3D quantum Hall
effect [26, 27, 28, 29, 30, 31, 32, 33, 34, 35], require a presence of
external magnetic fields to exhibit the above mentioned properties. However an
external magnetic field, if strong enough, can couple a pair of WNs of
opposite chiralities and can potentially annihilate them to destroy the WSM
state. The Refs. 36 and 37 found that pairwise annihilation of WNs by external
magnetic field can happen when the inverse magnetic length
$l^{-1}_{B}=\sqrt{eB/\hbar}$ becomes close to or larger than the momentum
space separation $2k_{0}$ between the two WNs of opposite chirality.

To investigate the pairwise annihilation of WNs by external magnetic fields,
the Refs. 36, 37 considered a simple model of Weyl semimetal, with two WNs
only, in the continuum approximation. A Hamiltonian with two WNs located at
${\bf k}_{w}=(k_{0},0,0)$ and $-{\bf k}_{w}$ may be approximated in the
continuum as

$\displaystyle H_{con}({\bf k})=(k_{0}^{2}-{\bf
k}^{2})\sigma_{x}+k_{y}\sigma_{y}+k_{z}\sigma_{z}.$ (1)

The WNs are separated along the $k_{x}$ axis and the distance is $2k_{0}$ in
momentum space. Working with such a Hamiltonian of WSMs, Refs. 36, 37 found
that a pair of Weyl nodes get annihilated when the strength of the magnetic
field is such that $l_{B}$ becomes close to $1/2k_{0}$ or smaller than this
value. Such pairwise annihilation of Weyl nodes which causes a transition from
gapless semimetal to an insulator, has been also observed in the experiments
by measuring the resistivity of Weyl materials TaP [38] and TaAs [39] at a
high applied magnetic fields.

Working with the low-energy continuum Hamiltonian outlined in Eq. 1 for a Weyl
semimetal presents several limitations. Firstly, the applicability of the
continuum Hamiltonian in Eq. 1, derived from the full lattice model of a Weyl
semimetal, is constrained to situations where the separation ($2k_{0}$)
between the two Weyl nodes is relatively small. Secondly, when dealing with
strong magnetic fields in the regime where the magnetic length $l_{B}$ is
comparable to the lattice constant (Hofstadter regime, $l_{B}\sim a$), any
continuum approximation of the complete lattice model exceeds its range of
relevance. The efficacy of the low energy continuum Hamiltonian in Eq. 1 for a
Weyl semimetal with two Weyl nodes is confined to the conditions: $\ell_{B}\gg
a$ and $1/2k_{0}\gg a$.

The preceding discussion underscores the increasing complexity associated with
investigating pairwise annihilation of WNs within a continuum model of WSMs
featuring multiple WNs (e.g. time reversal preserved WSMs), especially when
multiple node separations are involved. Another crucial constraint is that a
continuum model cannot anticipate the subsequent state following the pairwise
annihilation of Weyl nodes. This process may yield not only normal insulating
states but also states with nontrivial topological characteristics.

Addressing the aforementioned challenges can be achieved by exploring a
lattice model of a WSM to investigate the pairwise annihilation of WNs induced
by external magnetic fields. The authors of the Ref. 40 did some insightful
investigation in this direction. In their work, they specifically examined a
complicated lattice model of a time-reversal broken WSM which involves many
parameters, with a primary emphasis on constructing phase diagrams in the
presence of commensurate magnetic fields. Their findings revealed new phases
which include layered Chern insulator (LCI), insulator which is trivial in the
bulk but has counter propagating surface states on certain open surface (I′),
and a coexistent phase (W2′) where Chern bands and WPs coexist with their own
Fermi arc surface states. The emergence of these diverse phases from a given
WSM state was not immediately apparent.

In examining the pairwise annihilation WNs, it is essential to recognize that
an external magnetic field, aligned with the direction of separation between
two WNs of opposite chiralities, cannot couple these two nodes. The
possibility of pairwise annihilation by an external field arises only when the
field is not parallel to the direction of separation between two WNs of
opposite chirality. To streamline the computation without sacrificing the
essence of the problem, we will presume that the external magnetic field’s
direction is perpendicular to the separation between two Weyl nodes of
opposite chirality.

In this work, first, we consider a simple model of time-reversal broken WSM
with only one parameter $k_{0}$ ($2k_{0}$ is the separation between two WNs in
the momentum space) to understand how the pairwise annihilation of WNs by
external magnetic fields can lead to different states. We show that the
pairwise annihilation in a WSM with two WNs leads to either a normal insulator
(no surface states) or a layered Chern insulator. Then based on the concept
that a pair of WNs (separated by $2k_{0}$) gets annihilated when $l_{B}\sim
1/2k_{0}$, we develop a model independent intuitive representation of pairwise
annihilation process (an example in Fig. 2) by external magnetic fields.
Importantly, this intuitive picture of pairwise annihilation only requires
information about the Weyl nodes’ locations and the connectivities of Fermi
arcs in the surface BZ to accurately predict the phases which can appear after
pairwise annihilation of Weyl nodes. We apply the intuitive picture of
pairwise annihilation to demonstrate how the states like LCI, I′ and W2′ can
be straightforwardly obtained from a simpler WSM state, without resorting to
any complicated model as was considered by the Ref. 40.

Second, we consider a minimal model of time-reversal preserved WSM with four
Weyl nodes. The minimal model with four WNs has two free parameters $k_{1}$
and $k_{2}$ (see Fig. 4) which provide momentum space separation between Weyl
nodes of opposite chirality. In a WSM with four WNs, there are three distinct
perpendicular directions in which a magnetic field can be applied to cause
pairwise annihilation of Weyl nodes. We meticulously construct phase diagrams
for each of the three cases by solving the model with thorough effort.
Subsequently, we assert that these phase diagrams can be easily derived from
the intuitive representation of pairwise annihilation of WNs, requiring only
minimal information about the WNs’ locations and the connectivities of Fermi
arcs on the surface BZ.

We also touched upon pairwise annihilation of WNs by magnetic fields in a WSM
with six Weyl nodes. We analyze a simple case where all the WNs are located in
a single plane. The intuitive picture of pairwise annihilation of WNs
immediately predicts emergence of two new coexistence phases denoted as W2′′
and W4′ in Fig. 9.

The plan of the paper is as follows: In Sec. II, we conduct a thorough
examination of pairwise annihilation of WNs by external magnetic fields in a
simple model of WSM with two Weyl nodes (time-reversal broken case). Then in
Sec. III, we study the pairwise annihilation in a minimal model of time-
reversal preserved WSM with four WNs, which has two free parameters -the
separations between WNs of opposite chirality. We discuss our findings in Sec.
IV and summarize them in Sec. V. In the Appendix A, we discuss about nature of
the insulators which appear after pairwise annihilation of WNs in the time-
reversal preserved model.

Figure 1: ($a$) Phase diagram of the time-reversal broken WSM (Eq. 2) 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. II.1. 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. 1($b$). We notice a similarity
between the two phase diagrams for the small and large $q$ values (details in
the text).

##  II Time-reversal broken WSM

We consider the following lattice model of time-reversal broken Weyl semimetal

$\displaystyle H({\bf k})=$
$\displaystyle(2+\cos{k_{0}}-\cos{k_{x}}-\cos{k_{y}}-\cos{k_{z}})\sigma_{x}$
(2) $\displaystyle+\sin{k_{y}}\sigma_{y}+\sin{k_{z}}\sigma_{z}.$

with minimal two WNs at ${\bf k}_{w}=(k_{0},0,0)$ and $-{\bf k}_{w}$ carrying
monopole charges $C=1$ and $-1$ respectively. The two WNs are separated along
the $k_{x}$ axis by $2k_{0}$ and the parameter $k_{0}$ lies in the range
$0\leq k_{0}\leq\pi$. It is easily checked that the Hamiltonian respects
neither time-reversal or particle-hole symmetry. However, the Hamiltonian is
symmetric under space inversion ${\cal P}H({\bf k}){\cal P}^{-1}=H(-{\bf k})$,
with ${\cal P}=\sigma_{x}$. Because of the inversion symmetry, the Fermi arc
which exist on $k_{x}$-$k_{y}$ and $k_{x}$-$k_{z}$ surface BZs is a straight
arc joining the projections of the two Weyl points (see Fig. 2$a$).

We want to investigate pairwise annihilation of WNs by external magnetic
fields for field’s strength which ranges all the way from small ($\ell_{B}\gg
a$) to a very large value ($\ell_{B}\sim a$) in the Hofstadter regime. Our
goal is to identify the states that emerge following the pairwise annihilation
of Weyl nodes. We know for sure that the pairwise annihilation in a WSM with
two WNs always leads to insulating states. The question we are asking is what
is the nature of these insulating states. Specifically, we seek to determine
whether the insulator exhibits surface states and whether it possesses
topological nontriviality.

As outlined in the previous section, the applied magnetic field which is not
aligned along the direction of separation of two WNs of opposite chirality can
couple the nodes and hence can potentially annihilate them. In our model Eq.
2, the Weyl nodes are separated along the $k_{x}$ axis. To simplify the
analysis, we assume the magnetic field is aligned perpendicular to the
$x$-axis, specifically aligned with the $z$-axis. For a WSM with two WNs
separated along the $k_{x}$-direction, pairwise annihilation by magnetic field
applied along either the $z$ or $y$-direction would result in identical sets
of phases.

###  II.1 Hofstadter Hamiltonian and gapless solutions

An external magnetic field can be easily coupled to the Hamiltonian in Eq. 2
by taking it to the real space,

$\displaystyle\begin{aligned} H=\sum_{{\bf n},j}c^{\dagger}({\bf
n})2M\sigma_{x}c({\bf n})-\left(c^{\dagger}({\bf
n}+a\hat{e}_{j})~{}T_{j}~{}c({\bf n})+H.c.\right)\end{aligned}$ (3)

where ${\bf n}=a(n_{x},n_{y},n_{z})$, $n_{i}$ being integers, denote the
lattice sites, $\hat{e}_{j}$ is the unit vector along $j^{th}$ direction, and
$M=2+\cos{k_{0}}$. The hopping matrices $T_{j}$, $j=(x,y,z)$, are given by:
$T_{x}=\sigma_{x}$, $T_{y}=\sigma_{x}+i\sigma_{y}$ and
$T_{z}=\sigma_{x}+i\sigma_{z}$. The lattice constant $a$ is set to be unity
for the rest of the paper.

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 $2k^{\prime}_{0}=2\pi-2k_{0}$ 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 $k_{0}<k^{\prime}_{0}$, 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 $k_{0}>k^{\prime}_{0}$, pairwise annihilation occurs at the
boundary of the BZ by leaving the Fermi arc as depicted in figure ($b$). Hence
a LCI state emerges.

In presence of an external magnetic fields, the hopping terms in the
Hamiltonian Eq. 3 pick up a nontrivial phase factor under Peierls
substitution[41]. We choose to work in the Landau gauge ${\bf A}=(-y,0,0)B$,
where only the hopping in the $x$-direction picks up a nontrivial phase so
that the Hamiltonian in a magnetic field is obtained from Eq. 3 by the
replacement $T_{x}\to T_{x}\exp(-i2\pi y\phi/\phi_{0})$. We restrict ourselves
to the case where the flux $\phi$ (in units of the quantum flux
$\phi_{0}=h/e$) per unit cell is commensurate i.e.
$\phi/\phi_{0}=Ba^{2}/\phi_{0}=p/q$, where $p$ and $q$ are relatively prime,
so that translation symmetry along the $y$-direction is restored with a larger
unit cell [42]. In order to diagonalize the Hamiltonian, we introduce a
magnetic unit cell that expands $q$ times in comparison to the original unit
cell, elongating along the y-direction. Employing Fourier transformation in
relation to the Bravais lattice positions within the magnetic unit cell yields
following Bloch Hamiltonian

$\displaystyle h_{\phi}({\bf k})=\sum_{\alpha=0}^{q-1}$ $\displaystyle
c_{\alpha}^{\dagger}({\bf k})\left[f_{1}^{\alpha}({\bf
k})\sigma_{x}+f_{3}^{\alpha}({\bf k})\sigma_{z}\right]c_{\alpha}({\bf k})$
$\displaystyle-\left(c_{[\alpha+1]}^{\dagger}({\bf
k})e^{iqk_{y}\delta_{(\alpha,q-1)}}~{}T_{y}~{}c_{\alpha}({\bf
k})+H.c.\right).$ (4)

We will refer to $h_{\phi}({\bf k})$ as the Bloch-Hofstadter Hamiltonian,
where $\alpha=0,1,...,q-1$ are the sublattice indices in the magnetic unit
cell and $\mathbf{k}$ lies in the reduced (magnetic) Brillouin zone (MBZ),
$i.e.$, ${\bf k}$: $k_{x}\in\left(0,2\pi\right)$,
$k_{y}\in\left(0,2\pi/q\right)$, $k_{z}\in\left(0,2\pi\right)$. The square
bracket notation in $c_{[\alpha+1]}^{\dagger}({\bf k})$ implies that the
values of $\alpha$ are taken modulo $q$ \- $i.e.$, $[\alpha]=\alpha\mod q$.
The functions $f^{\alpha}_{1}$ and $f^{\alpha}_{3}$ are

$\displaystyle f^{\alpha}_{1}({\bf k})=$ $\displaystyle
2\left(M-\cos{\left(k_{x}+\frac{2\pi p}{q}\alpha\right)}-\cos{k_{z}}\right)$
(5a) $\displaystyle f^{\alpha}_{3}({\bf k})=$ $\displaystyle f_{3}({\bf
k})=2\sin{k_{z}}.$ (5b)

Note that there is only one free parameter $k_{0}$ (enters through
$M=2+\cos{k_{0}}$) which determines the separation between the two WNs of
opposite chirality in our zero field model. Our goal is to determine the phase
diagram for different values of $p/q$. The phase diagrams can be constructed
if we can find all the gapless points (band touching points) of the Bloch-
Hofstadter Hamiltonian $h_{\phi}({\bf k})$.

It is typically not possible to analytically determine the full energy
spectrum of a Bloch-Hofstadter Hamiltonian for all combinations of $p$ and $q$
values. However in some special cases, all the zeros (gapless solutions) of a
Bloch-Hofstadter Hamiltonian can be found for different values of $p$ and $q$.
Following the Ref. [40], we find the energy spectrum of $h_{\phi}({\bf k})$

$\displaystyle E_{n}({\bf k})=\pm\sqrt{\gamma_{n}({\bf
k},p,q)+\left(f_{3}({\bf k})\right)^{2}},$ (6)

which is symmetric about the zero energy, $n=1,2,....,q$, is the Landau level
index and $\gamma_{n}({\bf k},p,q)\geq 0$ for all ${\bf k}$, $p$ and $q$.
Clearly the zero energy solutions are given by $f_{3}({\bf k})=\sin{k_{z}}=0$
and $\gamma_{1}({\bf k},p,q)=0$. The first condition tells that band touching
along the $k_{z}$ direction can occur only at $k_{z}=0$ and/or $\pi$. Though
the quantity $\gamma_{1}({\bf k},p,q)$ is not known explicitly (as a function
of ${\bf k},p$ and $q$), $\gamma_{1}({\bf k},p,q)=0$ can be solved exactly for
all ${\bf k},p$ and $q$ values. Solving $\gamma_{1}({\bf k},p,q)=0$, we find
band touching along the $k_{y}$ and $k_{z}$ directions can happen only at
$k_{y}=0$ and $k_{z}=0$ respectively. We notice that the external magnetic
field (aligned along the $z$-direction) did not alter the band touching points
along the $k_{y}$ and $k_{z}$ directions i.e. the touching point remains at
$k_{y}=k_{z}=0$. The corresponding $k_{x}$ values are given by

$\displaystyle\cos{qk_{x}}=(-1)^{p}\left[-T_{q}(g)+2^{q-1}~{}\right],$ (7)

where $g=1+\cos{k_{0}}$ and $T_{q}(g)$ is a Chebyshev polynomial of degree $q$
of first kind. The gapless solutions (exists only when the R.H.S of Eq. 7 lies
in the range $[-1,1]$) describes isolated point touchings which are the Weyl
nodes in the theory. The WNs remain separated along the $k_{x}$-direction.
Note that though the integer $p$ of flux $p/q$ can change the sign of the
R.H.S of Eq. 7, it does not affect the region of gapless solutions and hence
the phase diagrams. In what follows we will assume $p=1$, unless it is stated.

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
$2k^{\prime}_{01}=2\pi-2k_{01}$ and $2k^{\prime}_{02}=2\pi-2k_{02}$ are the
corresponding inter-BZ separations. If $k^{\prime}_{02}<k_{01}$, 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 $k^{\prime}_{02}>k_{01}$, 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 $k_{0}<k^{\prime}_{0}$, and an insulator
with counter propagating surface states appear (I′) when
$k_{0}>k^{\prime}_{0}$. 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.

###  II.2 Phase diagrams

The phase diagram, for a given $q$, can be obtained from the Eq. 7 by solving
it for allowed $k_{0}$ values such that the R.H.S remains in the range
$[-1,1]$. The phase diagram is shown in Fig. 1a for small $q$ values. There
are two insulating regions in the phase diagram. We note that the gapless
condition Eq. 7 is not enough to determine the nature of the two insulators in
the phase diagram. We use numerics (to compute Chern numbers and surface
states) to find the nature of the two insulators in the phase diagram. We will
shortly see that a simple intuitive representation of pairwise annihilation of
WNs by external fields can accurately predict the entire phase diagram
including the nature of the insulators.

The WSM state, which existed for $k_{0}$ in the range $0<k_{0}<\pi$, now in an
applied magnetic field exists in a smaller region (see Fig. 1$a$). The WSM
states with either small or large separation of WNs get gapped out first by
the applied magnetic field and transform to insulators. The nature of the
resulting insulators depends on the separation and the Fermi arc connectivity
between the two WNs in the zero field model. In our model, the Fermi arc is an
intra-BZ straight arc (along $k_{x}$) joining the projections of WNs of
opposite chirality as shown in Fig. 2a. We observe that a WSM state with small
separation between the WNs produces a normal insulator, while a WSM state
characterized by a large separation between the WNs gives rise to a layered
Chern insulator (LCI) [40] after pairwise annihilation. The LCI state carries
nonzero Chern numbers $C(k_{x})=1$ for all $k_{x}$ values.

Let us take a closer look at the phase diagram Fig. 1$a$ for small $q$ values.
We notice that as $q$ is decreased, the region of the gapless WSM state
expands. From physical point of view, this may seem counterintuitive because
the magnetic length $l_{B}=\sqrt{\hbar/eB}=\sqrt{q}a$ ($a$ lattice constant)
decreases with decreasing $q$ and hence we expect the region of the gapless
WSM state in the phase diagram Fig. 1a to contract with decreasing $q$ (recall
pairwise annihilation occurs when $l_{B}\lesssim 1/k_{0}$). Actually, this
behavior of the system for small $q$ values in the Hofstadter regime
$l_{B}\sim a$ is not contradictory but is consistent with what we expect in a
lattice: The system should go towards the zero field limit as we decrease $q$
because in the limit $q\to 1$, the phase factor $\exp(-i2\pi y_{j}p/q)$
($y_{j}$ is an integer) in the hopping term also approaches $1$.

Now let us focus on higher $q$ values for which the magnetic length is much
larger than the lattice constant. From the phase diagram Fig. 1$a$, we see
that the gapless region shrinks as $q$ is increased. From the gapless
condition Eq. 7, we find that the gapless region actually shrinks almost to a
point for $q$ value as small as $q\sim 10$. This implies that for $q\gtrsim
10$ the transition from the normal insulator to the LCI state goes through a
point instead of a region in the $k_{0}$-space. So the system remains gapless
(WSM) only at the point $k_{0}=k_{0c}\approx 1.318$ for $q\gtrsim 10$.

This apparently means that an applied magnetic field with very large values of
$q$ i.e. an arbitrarily small field can destroy a WSM state with two WNs of
arbitrary separation. From a physical point of view, an arbitrary small field
cannot destroy a WSM state. There must be some additional information which is
missing when we construct phase diagram from the gapless condition Eq. 7 only.
A crucial information which is missing is that the energy gap ($\Delta$) falls
exponentially [40] with increasing $q$. Therefore to find the correct phase
diagram for large values of $q$ in the regime $l_{B}\gg a$, we need to compute
the energy gap $\Delta$ as a function of $k_{0}$ and $q$. We have computed the
energy gap $\Delta$ numerically as a function of $k_{0}$ and plotted it for a
series of values of $q$ in Fig. 1$b$. Now we find that the gapless WSM state
exists in a finite region for a large value of $q=200$. We also see that the
gapless region contracts as we decrease the value of $q$ from 200 to 100, 80,
60,…,10, which is according to our expectation: With decreasing $q$, the
magnetic length $l_{B}=\sqrt{q}a$ decreases and hence the applied field
annihilates a pair of WNs of higher and higher separation.

In summary, we find that the analytically obtained gapless condition produces
correct phase diagram for small $q$ values in the regime $l_{B}\sim a$. Since
energy gap decreases exponentially with increasing $q$, the gapless condition
is not enough to obtain correct phase diagram for large values of $q$ in the
regime $l_{B}\gg a$. For large values of $q$, we obtain phase diagram by
computing the energy gap numerically.

###  II.3 An intuitive representation of pairwise annihilation process

We have obeservd that pairwise annihilation of WNs by external fields in a WSM
with two WNs results in a normal insulator when WNs are closed spaced.
However, a LCI state emerges when the separation between the two WNs is large.
The above result can be understood through a simple picture based on the
argument that a pair of WNs get annihilated when the inverse magnetic length
$l_{B}^{-1}$ becomes close to or larger than the separation $2k_{0}$ between
the two Weyl nodes of opposite chirality. Note that in a periodic BZ, there
are two separations between two WNs of opposite chirality located at
$(k_{0},0,0)$ and $(-k_{0},0,0)$: (i) intra-BZ separation $2k_{0}$ and (ii)
inter-BZ separation $2k^{\prime}_{0}=2\pi-2k_{0}$. Clearly it is the shorter
separation which determines how the pair will be annihilated by the applied
field. Now consider a WSM with $k_{0}<k^{\prime}_{0}$. In this case, the
inverse magnetic length $l_{B}^{-1}$ will first reach $2k_{0}$. As magnetic
field is increased, two WNs approach each other along the Fermi arc to meet at
a point inside the BZ and get annihilated without leaving the Fermi arc
(demonstrated in Fig. 2). This results in the formation of a normal insulator
which possesses no surface states. On the other hand if
$k_{0}>k^{\prime}_{0}$, the inverse magnetic length $l_{B}^{-1}$ will first
hit $2k^{\prime}_{0}$ and consequently the pair of WNs is expected to get
annihilated at the boundary of BZ by leaving the surface Fermi arc states.
This results in a LCI state. The process is demonstrated in the Fig. 2. Note
that the maximum separation occurs when $k_{0}=\pi/2$ or
$k^{\prime}_{0}=\pi/2$. This implies that a very strong field is needed to
destroy a WSM in which WNs are separated by a distance $2k_{0}=\pi$. This is
the reason why the WSM state survives in the central region of the phase
diagram Fig. 1 in presence of an external magnetic field.

Here we want to point out that the region, in which the WSM state survives,
shrinks with the increase in the strength of the field. At certain field
values (around $q\sim 10$), this region contracts to a singular point. Based
on the reasoning presented in the preceding paragraph, it is anticipated that
this point is positioned at $k_{0}=\pi/2$. On the contrary, in the model we
have considered, the gapless region shrinks to the point $k_{0c}\approx
1.318$, as mentioned earlier. It is important to note that we do not consider
this value of $k_{0c}$ to be universally applicable to all WSMs; rather, it
may be contingent on specific yet unknown details of the considered model.

###  II.4 Pairwise annihilation in a WSM with multiple Weyl nodes

Examining pairwise annihilation becomes more challenging as the count of WNs
rises because of a corresponding increase in the model’s free parameters.
Despite this complexity, the intuitive insights gained from studying pairwise
annihilation in a WSM with two nodes can be readily extended to predict
potential new states which can result in after pairwise annihilation in a WSM
with multiple nodes. To illustrate let us consider a WSM with four Weyl nodes
placed in a magnetic field which is aligned along the $z$-direction. For
simplicity, let us assume all the four Weyl nodes are located in the
$k_{x}$-$k_{y}$ plane at a constant $k_{z}=0$, and they are at a maximum
separation of $\pi$ along the $k_{y}$ direction. Clearly, maximum information
about the location of the WNs are retained when they are projected on the
$k_{x}$-$k_{y}$ surface BZ. Projections of the WNs with a possible Fermi arc
connectivity on the $k_{x}$-$k_{y}$ surface BZ are depicted in Figs. 3$a$ and
3$d$. Since the WNs are located at the maximum separation along the $k_{y}$
direction, the relevant separation parameters are $k_{01}$ and $k_{02}$ as
shown in Fig. 3$a$. Suppose $k_{02}>\pi/2$ and also $k_{02}\gg k_{01}$. Now if
$k^{\prime}_{02}<k_{01}$, then the magnetic length will first hit
$k^{\prime}_{02}$. In this situation as magnetic field is increased, the two
WNs (separated by $k_{02}$) will approach each other across the BZ to meet at
the boundary of the BZ. This results in pairwise annihilation of the two WNs
(separated by $k_{02}$) by leaving the Fermi arc states. Thus we get a state
with two WNs but with an additional surface Fermi arc (see Fig. 3$b$). This
state is a coexistent phase called W2′ which Ref. 40 found in a complicated
model with many parameters. Now consider $k^{\prime}_{02}>k_{01}$. In this
case, the pairwise annihilation of the two WNs separated by $k_{01}$ leads to
a WSM state with only two Weyl nodes as demonstrated in Fig. 3$c$.

Now it is clear that if $k_{01}=k_{02}$, then their pairwise annihilation by
external magnetic fields would result either a normal insulator or an
insulator (called I′) with counter propagating surface states as shown in Fig.
3$f$.

For a magnetic field along the $y$-direction, the separation parameters
$k_{01}$ and $k_{02}$ are relevant only. In this case, the separation of the
WNs along the $k_{y}$ direction is completely irrelevant for pairwise
annihilation of WNs. Hence, pairwise annihilation by magnetic fields aligned
in the $y$-direction would result in an identical set of phases as the
previous case. We study this case in the Sec. III.2 in details.

The authors in Ref. 40 explored an intricate model of a WSM featuring eight
Weyl nodes. They successfully addressed the complexities of the multi-
parameter model and identified phases such as W2′ and I′ in the presence of a
magnetic field. In our work, we have demonstrated how these phases could be
derived from a simpler WSM model with only four Weyl nodes. Crucially, our
approach does not rely on a particular model; instead, all that is necessary
is knowledge of the WNs’ positions and the Fermi arc connectivity in the
surface BZ. This enables us to precisely predict the potential phases that may
emerge in the presence of an external field.

###  II.5 Effect of Zeeman energy on the phase diagram

So far, we have completely ignored effect of Zeeman energy on the phase
diagram. For a magnetic field along the $z$-direction, the Zeeman term
($H_{Z}$) would be proportional to the $\sigma_{z}$ i.e.
$H_{Z}=E_{Z}\sigma_{z}$, where $E_{Z}\propto 1/q$. Addition of this term to
the Bloch-Hofstadter Hamiltonian Eq. II.1, will modify only the quantity
$f^{\alpha}_{3}({\bf k})\to f^{\alpha}_{3}({\bf k})+E_{Z}$. Therefore the band
touching along the $k_{z}$ direction will move from the point $k_{z0}=0$ to
$k_{z0}=\sin^{-1}(E_{Z}/t_{z})$ (hopping along the $z$-direction is
parametrized by $t_{z}$). This change in band touching along the $k_{z}$
direction flows to the quantity $f^{\alpha}_{1}({\bf k})$ to cause a shift in
the parameter $k_{0}$. The final result is that the critical point
$k_{0}=k^{*}_{0}$, at which a transition from WSM to insulator occurs, moves
with the change in the Zeeman energy. Of course, this would not give any new
phase, but can move the phase boundary.

Figure 4: Projections of the WNs and the Fermi arcs on the $k_{x}$-$k_{y}$
surface BZ of the model in Eq. 8. The arrows says 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.

##  III Time-reversal preserved WSM

In the previous section, we have explored pairwise annihilation in a WSM with
two Weyl nodes. A WSM with two WNs necessarily breaks time-reversal symmetry.
In this section, we want to examine the pairwise annihilation of WNs by
external magnetic fields in a time-reversal preserved Weyl semimetals. A
minimal model of time-reversal preserved WSM has four Weyl nodes. Now there
will be two independent separation parameters (see Fig. 4). In Sec. II.4, We
have briefly looked at pairwise annihilation in a WSM with four WNs through
the intuitive picture of pairwise annihilation. We restricted ourselves to a
case where separation of the WNs along the $k_{y}$ direction was fixed to
simplify the analysis. We predicted emergence of an insulator, labelled I′,
with counter propagating surface states (see Figs. 3$d$-$f$). Here we will
verify this prediction. Below we explore pairwise annihilation in a WSM with
four WNs in full details. The separations along both the $k_{x}$ and $k_{y}$
directions will be considered as free parameters in the theory.

To study pairwise annihilation of WNs in a time-reversal preserved WSM by
external magnetic field whose strength can range all the way from small
($l_{B}\ll a$) to a very large value in the Hofstadter regime ($l_{B}\sim a$),
we consider the following lattice model of WSM

$\displaystyle H({\bf k})=$
$\displaystyle(\cos{k_{2}}-\cos{k_{y}})\sigma_{y}+\sin{k_{z}}\sigma_{z}$ (8)
$\displaystyle+(1+\cos{k_{1}}-\cos{k_{x}}-\cos{k_{z}})\sigma_{x},$

with a minimal of four WNs located at ${\bf k}_{w_{1}}=(k_{1},k_{2},0)$,
$-{\bf k}_{w_{1}}$, ${\bf k}_{w_{2}}=(k_{1},-k_{2},0)$, and $-{\bf
k}_{w_{2}}$. They all lie in the same $k_{x}$-$k_{y}$ plane at $k_{z}=0$. The
two Weyl nodes at ${\bf k}_{w_{1}}$ and $-{\bf k}_{w_{1}}$ are time-reversal
partner of each other and they carry identical chiral charge $\chi=1$. On the
other hand the pair ${\bf k}_{w_{2}}$ and $-{\bf k}_{w_{2}}$ carries opposite
chiral charge $\chi=-1$. The projections of the WNs with the Fermi arcs on the
$k_{x}$-$k_{y}$ surface BZ are depicted in the Fig. 4.

Here $\sigma$’s, which are the two by two Pauli matrices, represent pseudo-
spin degree of freedom. Time-reversal symmetry ${\cal T}H({\bf k}){\cal
T}^{-1}=H({\bf-k})$ is realised by ${\cal T}=i\sigma_{x}{\cal K}$, where
${\cal K}$ acts by taking complex conjugation of any quantities appearing on
the right of it.

Now we are ready to couple magnetic field to the Hamiltonian in Eq. 8 to study
pairwise annihilation of WNs. Unlike the time-reversal broken model with two
WNs, here in the time-reversal preserved model with four WNs, the external
magnetic field applied along any of the three axis direction can couple the
WNs and can potentially annihilate them. The minimal model Eq. 8 has two free
parameters $k_{1}$ and $k_{2}$ which provide separations of WNs of opposite
chirality as shown in the Fig. 4. For magnetic field applied along the
$y$-direction ($x$-direction), the separation parameter $k_{1}$ ($k_{2}$) is
relevant only. This case is similar to the two WNs’ problem where we had only
one separation parameter. For magnetic field along the $y$-direction, the
intuitive picture of pairwise annihilation immediately tells that the new
state which appears after pairwise annihilation is either a normal insulator
or the insulator I′ with counter propagating surface states (see Fig. 3$f$).
We will verify our prediction by solving the model for phase diagrams in
presence of an external commensurate magnetic field.

For magnetic field along the $z$-direction, both the separation parameter
plays significant role in pairwise annihilation of Weyl nodes. First, we solve
this model for phase diagrams in presence of an external commensurate magnetic
field. Then, we argue that the phase diagrams can be derived, based on the
intuitive picture of pairwise annihilation of WNs by external magnetic field.

Figure 5: Phase diagrams in Figs. 5($a$)-($d$) 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, 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. 5 ($i$-$l$) 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
$k_{1}\sim\pi/2$, $|k_{2}-\pi/2|\sim\pi/2$.

###  III.1 Field along $z$-direction

For a constant magnetic field ${\bf B}=B\hat{z}$, we can choose the Landau
gauge ${\bf A}=(-y,0,0)B$ to work with. After going through same exercise as
in Sec. II, we arrive at the following Bloch-Hofstadter Hamiltonian

$\displaystyle h^{(z)}_{\phi}({\bf k})=\sum_{\alpha=0}^{q-1}$ $\displaystyle
c_{\alpha}^{\dagger}({\bf k})\left[f_{1}^{\alpha}({\bf
k})\sigma_{x}+f_{2}^{\alpha}({\bf k})\sigma_{y}+f_{3}^{\alpha}({\bf
k})\sigma_{z}\right]c_{\alpha}({\bf k})$ (9)
$\displaystyle-\left(c_{[\alpha+1]}^{\dagger}({\bf
k})e^{iqk_{y}\delta_{(\alpha,q-1)}}~{}T_{y}~{}c_{\alpha}({\bf k})+H.c.\right)$

for commensurate flux $\phi/\phi_{0}=1/q$ per unit cell. The functions
$f^{\alpha}_{i}({\bf k})$, $i=1,2,3$ are given by

$\displaystyle f^{\alpha}_{1}({\bf k})=$ $\displaystyle
2\left(M-\cos{\left(k_{x}+\frac{2\pi p}{q}\alpha\right)}-\cos{k_{z}}\right),$
(10a) $\displaystyle f^{\alpha}_{2}({\bf k})\equiv$ $\displaystyle f_{2}({\bf
k})=2\cos{k_{2}},$ (10b) $\displaystyle f^{\alpha}_{3}({\bf k})\equiv$
$\displaystyle f_{3}({\bf k})=2\sin{k_{z}},$ (10c)

where $M=1+\cos{k_{1}}$ and the hopping matrix $T_{y}$ in the second term of
$h^{(z)}_{\phi}({\bf k})$ is $T_{y}=\sigma_{y}$. The Bloch-Hofstadter
Hamiltonian $h^{(z)}_{\phi}({\bf k})$ is to be diagonalized in the magnetic
BZ: $k_{x}\in\left(0,2\pi\right)$, $k_{y}\in\left(0,2\pi/q\right)$,
$k_{z}\in\left(0,2\pi\right)$. We want to find all the gapless points in
energy spectrum (zeros of the Bloch-Hofstadter Hamiltonian
$h^{(z)}_{\phi}({\bf k})$) to construct the phase diagrams for different
values of $q$. We choose a basis
$\Psi=(\psi_{\uparrow},\psi_{\downarrow})^{T}$, where
$\psi_{s}=\left(c_{0,s}({\bf k}),c_{1,s}({\bf k}),...,c_{q-1,s}({\bf
k})\right)^{T}$ and $s\equiv\left(\uparrow,\downarrow\right)$, so that
$h^{(z)}_{\phi}({\bf k})$ can be expressed as a matrix,

$\displaystyle\tilde{h}^{(z)}_{\phi}({\bf k})=\begin{pmatrix}{\bf A}&{\bf
B}\\\ {\bf C}&{\bf D}\end{pmatrix}$ (11)

where the diagonal blocks ${\bf A}=-{\bf D}=2\sin{k_{z}}{\bf I}_{q}$ are
proportional to identity ${\bf I}_{q}$ of dimension $q\times q$, and

$\displaystyle{\bf B}=\begin{bmatrix}m_{0}&u&0&0&...&ue^{ik_{y}q}\\\
u&m_{1}&u&0&...&0\\\ 0&u&m_{2}&u&...&...\\\ ..&..&..&..&..&..\\\
0&0&...&u&m_{q-2}&u\\\ ue^{-ik_{y}q}&0&...&...&u&m_{q-1}~{}\\\
\end{bmatrix}=~{}{\bf C}^{\dagger}.$ (12)

Here $m_{\alpha}=f^{\alpha}_{1}({\bf k})-if^{\alpha}_{2}({\bf k})$,
$\alpha\in[0,q-1]$, and $u=-i$. The eigenvalues $E({\bf k})$ (energy spectrum
of the Bloch-Hofstadter Hamiltonian $h^{(z)}_{\phi}({\bf k})$) are given by

$\displaystyle\textrm{det}\begin{bmatrix}{\bf A}-E({\bf k}){\bf I}_{q}&{\bf
B}\\\ {\bf C}&{\bf D}-E({\bf k}){\bf I}_{q}\\\ \end{bmatrix}=0.$ (13)

Since the diagonal blocks commutes with the off-diagonal blocks, the above
condition reduces to

$\displaystyle\textrm{det}\left(\gamma{\bf I}_{q}-{\bf
BB}^{\dagger}\right)=0,$ (14)

where we have used ${\bf\tilde{A}\tilde{D}}=\gamma{\bf I}_{q}$,
$\gamma=E^{2}({\bf k})-\left(f_{3}({\bf k})\right)^{2}$. Note that $\gamma$ is
the eigenvalue of the positive definite matrix ${\bf BB}^{\dagger}$, so
$\gamma\geq 0$. The energy spectrum is

$\displaystyle E_{n}({\bf k})=\pm\sqrt{\gamma_{n}(q,{\bf k})+\left(f_{3}({\bf
k})\right)^{2}},$ (15)

where $n=1,2,3,...q$, are the Landau level indices. Clearly the spectrum is
symmetric about the zero energy. Therefore the gapless points between the
highest occupied and lowest unoccupied bands are given by $E_{1}({\bf k})=0$,
which leads to two separate conditions $f_{3}({\bf k})=2\sin{k_{z}}=0$ and
$\gamma_{1}(q,{\bf k})=0$. The first condition tells that band touching along
the $k_{z}$ direction can occur only at $k_{z}=0$ and/or $\pi$. Band touchings
along the $k_{x}$ and $k_{y}$ directions can be found from Eq. 14 by setting
$\gamma=0$. Then the Eq. 14 reduces to

$\displaystyle\textrm{det}({\bf B})=0,$ (16)

which is to be solved for a fixed $q$ to find the $k_{x}$ and $k_{y}$ values
at which band touching can occur. For the case of Bloch-Hofstadter Hamiltonian
in Eq. II.1, a special form of the matrix $T_{y}=\sigma_{x}+i\sigma_{y}$
brought ${\bf B}$ in (almost) triangular form which made us possible to solve
the above equation for arbitrary values of $q$. This is not the situation for
the present case. Nevertheless we can make a progress for small $q$ values,
where the Eq. 16 can be solved by brute force. We have learned in Sec. II that
the exact solution of the zeros of the Bloch-Hofstadter Hamiltonian produces
correct phase diagram in the Hofstadter regime $l_{B}\sim a$ (i.e. small $q$
values) only. For large $q$ values in the regime $l_{B}\gg a$, we construct
the phase diagrams numerically by computing the energy gap. Below we
analitically compute the zeros of the Bloch-Hofstadter Hamiltonian to
construct phase diagrams for small $q$ values $q=2,3,4$ and $5$ only.

####  III.1.1 $q=2$

Since the magnetic field is aligned along the $z$-direction, band touching
points along the $k_{z}$ direction remain at $k_{z0}=0$ and/or $\pi$ for all
$q$ values. The corresponding $k_{x}$ and $k_{y}$ values for $q=2$ are given
by the condition

$\displaystyle\textrm{det}\begin{bmatrix}m_{0}&ue^{-ik_{y}q}\\\
ue^{ik_{y}q}&m_{1}\\\ \end{bmatrix}$ $\displaystyle=0$ (17a) $\displaystyle
m_{0}m_{1}-u^{2}$ $\displaystyle=0,$ (17b)

which can be simplified to a set of two conditions

$\displaystyle 2\cos{qk_{x}}$ $\displaystyle=4g^{2}-4\cos^{2}{k_{2}}-1,$ (18a)
$\displaystyle g\cos{k_{2}}$ $\displaystyle=0,$ (18b)

where $g=1+\cos{k_{1}}-\cos{k_{z0}}$. We notice that the momentum $k_{y}$ does
not appear in the above conditions. Therefore the gapless solution (if exists
for some $k_{1}$ and $k_{2}$) describes a nodal line semimetal. It is easily
checked that $k_{z0}=\pi$ does not allow simultaneous solution of the two
conditions above. The nodal line is located at a fixed $k_{z}=0$ and
$k_{x}=k_{x0}$ (determined by Eq. 18a), but it spreads along the entire
$k_{y}$ axis. The full gapless solution is shown as a shaded region in Fig.
5$a$. The phase diagram is very simple which consists of only two phases: a
topologically trivial insulator and a gapless state which is a nodal line
semimetal.

####  III.1.2 $q=3$

Solving $\textrm{det}({\bf B})=0$ for $q=3$, we get the following two
conditions

$\displaystyle\cos{qk_{x}}$ $\displaystyle=F_{3}(g,\cos{k_{2}}),$ (19a)
$\displaystyle\cos{qk_{y}}$ $\displaystyle=F_{3}(\cos{k_{2}},g),$ (19b)

which $k_{x}$ and $k_{y}$ must satisfy in order to have gapless solution. The
function $F_{3}(u,v)=12uv^{2}-4u^{3}$, and $g=1+\cos{k_{1}}-\cos{k_{z0}}$.
Recall that bands touching along the $k_{z}$ direction can occur only at
$k_{z0}=0$ and/or $\pi$. It is easily checked that $k_{z0}=\pi$ does not admit
simultaneous solution of the two equations above. Therefore bands touching
happens only at $k_{z0}=0$ and the corresponding $k_{x}$, $k_{y}$ values are
determined by Eqs. 19a and 19b. Clearly the solution space describes
generically point touchings which are the Weyl points in the theory. A gapless
solution exists in a finite region in the $k_{1}$-$k_{2}$ space as shown in
Fig. 5$b$. The full phase diagram consists of only two phases: a topologically
trivial insulator and a gapless phase which is a WSM.

####  III.1.3 $q=4$

Solving $\textrm{det}({\bf B})=0$ for $q=4$, we get the following two
conditions for bands touching

$\displaystyle\cos{qk_{x}}+\cos{qk_{y}}=8(g^{4}-6g^{2}\cos^{2}{k_{2}}+\cos^{4}{k_{2}})+2,$
(20a) $\displaystyle g\cos{k_{2}}(g^{2}-\cos^{2}{k_{2}})=0.$ (20b)

Like the previous cases, it is checked that only $k_{z0}=0$ admits
simultaneous solution of the two conditions above. We notice that the
corresponding $k_{x}$, $k_{y}$ values, at which bands touching can occur, are
solely determined by the first condition Eq. 20a (provided the second
condition is satisfied). The second condition, which involves only the two
parameters $k_{1}$ and $k_{2}$ but no momenta, forces the gap closing to occur
only on a contour (not a region) in the $k_{1}$, $k_{2}$ parameters space.
Since the $k_{x}$, $k_{y}$ values for bands touching are determined by only an
one condition, the gapless solution describes only nodal line. This is similar
to what we have seen for the case of $q=2$. The full phase diagram (depicted
in Fig. 5$c$) consists of only two phases: a topologically trivial insulator
and a gapless state which is a nodal line semimetal.

####  III.1.4 $q=5$

Solving $\textrm{det}({\bf B})=0$ for $q=5$, we get the following two
conditions for a gapless solution

$\displaystyle\cos{qk_{x}}$ $\displaystyle=F_{5}(g,\cos{k_{2}}),$ (21a)
$\displaystyle\cos{qk_{y}}$ $\displaystyle=F_{5}(-\cos{k_{2}},g),$ (21b)

where $F_{5}(u,v)=16u(u^{4}-10u^{2}v^{2}+5v^{4})+5u(1+\sqrt{5})/2$, and
$g=1+\cos{k_{1}}-\cos{k_{z0}}$. Similar to the case of $q=3$, bands touching
for $q=5$ occurs at isolated points in the BZ: $k_{z}=0$, and $k_{x}$, $k_{y}$
values are given by the simultaneous solution of the Eqs. 21a and 21b. The
band touching points are the Weyl points in the theory. A gapless solution
exists in a finite region in the $k_{1}$-$k_{2}$ space as shown in Fig. 5$d$.
Like the phase diagram for $q=3$, the phase diagram for $q=5$ (see Fig. 5$d$)
also consists of a WSM phase and a topologically trivial insulating phase
only.

Figure 6: Bulk energy dispersion of the Bloch-Hofstadter Hamiltonian
$h^{(z)}_{\phi}$, 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.

Finding gapless solution analytically becomes challenging as $q$ increases.
For $q>5$, the phase digram can be understood by computing the energy gap as a
function of $k_{1}$ and $k_{2}$. The result is shown in the second and third
row of Fig. 5. We find that all the insulating regions are adiabatically
connected, and all the gapless regions describe a same phase. For any odd
values of $q=1,3,5,7,9,...$, the gapless regions describe a Weyl semimetal
state. Though for small even values of $q=2,4,6,...$, the gapless regions
describe a nodal line semimetal state, for large values of $q$, the system, in
the gapless regions, behaves like a WSM in terms of the low energy dispersion.
As $q$ increases, the energy bands along the $k_{x}$, $k_{y}$ directions
becomes flatter [40]. For large $q$ values in the regime $l_{B}\gg a$
(semiclassical regime), the bands along the $k_{x}$, $k_{y}$ directions become
almost flat to form dispersionless Landau levels as expected from the
continuum approximation in the semiclassical regime [43, 44, 45, 17]. For an
illustration, energy dispersion, for $q=40$, along all the three $k_{x}$,
$k_{y}$, $k_{z}$ directions are depicted in Fig. 6.

From the phase diagrams for large $q$ values, we observe that the WSM state
gets gapped out in some specific regions in the $k_{1}$-$k_{2}$ parameter
space and the area of the insulating region increases with the strength of the
applied magnetic field $|{\bf B}|\propto 1/q$. Let us closely examine the
phase diagram for $q=100$ (Fig. 5$l$), and focus on how the phase diagrams
evolve as the value of $q$ decreases.. We notice that regions, where
$|k_{1}-\pi/2|\sim\pi/2$ and $k_{2}\sim\pi/2$ are gapped. Similarly, the
regions where $k_{1}\sim\pi/2$ and $|k_{2}-\pi/2|\sim\pi/2$ are also gapped.
However, the region in which $k_{1}\sim\pi/2$ and $k_{2}\sim\pi/2$ remains
gapless (WSM). As $q$ decreases (see Figs. 5$i$-$j$), the area of the gapped
insulating regions increases and simultaneously the area of the gapless region
decreases. All these can be understood from the very fundamental concept that
a pair of WNs of opposite chirality, which are separated by a momentum space
distance $2k_{0}$, annihilates each other when the magnetic length
$l_{B}=\sqrt{q}a$ hits the inverse separation $1/2k_{0}$. Clearly the regions
with either small $k_{1}$ or small $k_{2}$ values will be gapped out first
after pairwise annihilation of Weyl nodes. Recall when $k_{1}>\pi/2$ or
$k_{2}>\pi/2$, one should compare the momentum space distances
$k^{\prime}_{1}=\pi-k_{1}$ and $k^{\prime}_{2}=\pi-k_{2}$ with the inverse
magnetic length. Therefore the regions with either $|k_{1}-\pi/2|\sim\pi/2$ or
$|k_{2}-\pi/2|\sim\pi/2$ will be gapped out first. The separations between WNs
of opposite chirality are maximum in the central region $k_{1}\sim\pi/2$ and
$k_{2}\sim\pi/2$. This is the reason why it requires a very strong fields to
gap out the central region.

Figure 7: Phase diagrams of the time-reversal preserved WSM with four WNs in
presence of $1/q$ commensurate flux per unit cell, along the $y$-direction.
The regions shaded in grey, blue and orange represent a normal insulator, Weyl
semimetal and an insulator (I′) which has counter-propagating surface states
on the open surface along the $z$-direction.

An interesting tension occurs when $k_{2}\approx k_{1}$ or $k_{2}\approx\pi-
k_{1}$. In this situation a single WN of chirality $\chi$ gets simultaneously
coupled with two WNs of chirality $-\chi$. This leads to an effective coupling
between two WNs of same chirality. Since a WN cannot annihilate another WN of
same chirality, we get a partial annihilation of WNs along the line
$k_{2}\approx k_{1}$ or $k_{2}\approx\pi-k_{1}$ in the phase diagrams.

What is common among all the phase diagrams is that there are only two phases:
an insulator and a gapless state. Let us focus on a phase diagram for a
particular value of $q=40$. The entire insulating region in the phase diagram
may be split into four subregions: left, right, top and bottom insulating
regions. From the intuitive picture of pairwise annihilation of WNs, we expect
the insulators which are living in the left, top and bottom regions will not
have any surface states. The reason is the following. In the left region where
$k_{1}\ll\pi/2$ and $k_{2}\sim\pi/2$, the separation parameter $k_{1}$ is
relevant for pairwise annihilation. In this case, WNs which are separated by
$2k_{1}$ get pairwise annihilated at some point inside the BZ. Hence no Fermi
arc states are left. In the bottom region ($k_{2}\ll\pi/2$) and top region
($k_{2}\sim\pi$), the separation parameter $k_{2}$ is relevant for pairwise
annihilation. Since the Fermi arcs are counter propagating, pairwise
annihilation of WNs either at a point inside the BZ or at the boundary of the
BZ cannot leave the Fermi arc states. We have verified this numerically.
However the insulator, which is living in the right insulating region where
$k^{\prime}_{1}\ll k_{1}$ and $k_{2}\sim\pi/2$, can have surface states in
accordance with our intuitive picture of pairwise annihilation of Weyl nodes
(see Appendix A for details). The bulk of this insulating state is of course
trivial and the state is adiabatically connected to the adjacent insulating
states.

###  III.2 Field along $y$-direction

An external magnetic field oriented in the y-direction cannot couple WNs which
are separated along $k_{y}$ direction. In this case, the crucial separation to
consider for pairwise annihilation of WNs is $k_{1}$. The current problem can
be thought of as a two copies of a two WNs’ problem, similar to the time-
reversal broken case studied in the Sec. II. Here, the separation parameter
$k_{1}$ plays the role of the parameter $k_{0}$ of the time-reversal broken
case (see Eq. 2) with two Weyl nodes. The intuitive picture of pairwise
annihilation of WNs (see Figs . 8 and 3$d$-$f$) immediately tells that the
phase which appear after annihilation is either a normal insulator or an
insulator (I′) with counter propagating surface states on the $k_{x}$-$k_{y}$
surface BZ. In the following, we verify this prediction by solving the model
for phase diagram in presence of commensurate magnetic fields.

We choose to work with the Landau gauge ${\bf A}=(z,0,0)B$. In this choice of
gauge, the Bloch-Hofstadter Hamiltonian takes the following form (after a
unitary rotation in $\sigma$’s space about the $x$-direction)

$\displaystyle h^{(y)}_{\phi}({\bf k})=\sum_{\alpha=0}^{q-1}$ $\displaystyle
c_{\alpha}^{\dagger}({\bf k})\left[f_{1}^{\alpha}({\bf
k})\sigma_{x}+f_{3}^{\alpha}({\bf k})\sigma_{z}\right]c_{\alpha}({\bf k})$
$\displaystyle-\left(c_{[\alpha+1]}^{\dagger}({\bf
k})e^{iqk_{z}\delta_{(\alpha,q-1)}}~{}T_{z}~{}c_{\alpha}({\bf
k})+H.c.\right),$ (22)

for commensurate flux $\phi/\phi_{0}=1/q$ per unit cell. The functions
$f^{\alpha}_{1}({\bf k})$, $f^{\alpha}_{3}({\bf k})$ are given by

$\displaystyle f^{\alpha}_{1}({\bf k})=$ $\displaystyle
2\left(M-\cos{\left(k_{x}+\frac{2\pi}{q}\alpha\right)}\right),$ (23a)
$\displaystyle f^{\alpha}_{3}({\bf k})\equiv$ $\displaystyle f_{3}({\bf
k})=2\left(\cos{k_{2}}-\cos{k_{y}}\right),$ (23b)

where $M=1+\cos{k_{1}}$ and the hopping matrix $T_{z}=\sigma_{x}-i\sigma_{y}$.
Note that the Bloch-Hofstadter Hamiltonian $h^{(y)}_{\phi}({\bf k})$ is
defined in the magnetic BZ: $k_{x}\in\left(0,2\pi\right)$,
$k_{y}\in\left(0,2\pi\right)$, $k_{z}\in\left(0,2\pi/q\right)$. We can obtain
the phase diagrams by solving the spectrum for gapless points. Writing
$h^{(y)}_{\phi}({\bf k})$ in a matrix form (see Eq. 11), we obtain ${\bf
A}=-{\bf D}=2\left(\cos{k_{2}}-\cos{k_{y}}\right){\bf I}_{q}$ and the block
matrix ${\bf B}$ is

$\displaystyle{\bf B}=\begin{bmatrix}m_{0}&-2&0&0&...&0\\\
0&m_{1}&-2&0&...&0\\\ 0&0&m_{2}&-2&...&...\\\ ..&..&..&..&..&..\\\
0&0&...&0&m_{q-2}&-2\\\ 2e^{-iqk_{z}}&0&...&...&0&m_{q-1}~{}\\\
\end{bmatrix},$ (24)

is almost an upper triangular matrix except the element $2e^{-ik_{z}q}$.

Figure 8: ($a$) The Fermi arcs and the projections of the WNs on the
$k_{x}$-$k_{y}$ 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. If $k_{1}<k^{\prime}_{1}$, the pairwise annihilation, which
occurs at a point inside the BZ, does not leave behind the Fermi arc states.
Hence a normal insulator results in. If $k_{1}>k^{\prime}_{1}$, pairwise
annihilation occurs at the boundary of the BZ by leaving the Fermi arc states.
Hence, the insulator I′ emerges.

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
$k_{x}$-$k_{y}$ 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 W2′′ emerges when
$k_{1}<k^{\prime}_{2}$ (see ($b$)) and an insulator I′ with counter
propagating surface states appear when $k_{1}>k^{\prime}_{2}$ (see ($c$))
after pairwise annihilation by magnetic fields. In the second scenario ($d$),
pairwise annihilation results in either a coexistence phase W4′ or a WSM with
two WNs, depending on the relative values of $k^{\prime}_{1}$ and $k_{2}$.

The quantity $m_{\alpha}$, $\alpha=0,1,2,...(q-1)$, is equal to
$f_{1}^{\alpha}({\bf k})$ i.e. $m_{\alpha}=f_{1}^{\alpha}({\bf k})$. The
Landau level energy spectrum is

$\displaystyle E_{n}({\bf k})=\pm\sqrt{\gamma_{n}(q,{\bf k})+\left(f_{3}({\bf
k})\right)^{2}},$ (25)

where $\gamma_{n}(q,{\bf k})\geq 0$ are the eigenvalues of the positive
definite matrix ${\bf B}{\bf B}^{\dagger}$, and $n=1,2,3,...$ are the Landau
level indices. Since the spectrum is symmetric about the zero energy, band
touching points are given by the zero energy solutions. Clearly, for zero
energy, we must have (i) $f_{3}({\bf k})=2(\cos{k_{2}}-\cos{k_{y}})=0$ and
(ii) $\gamma_{1}(q,{\bf k})=0$. We see from the (i) condition that the band
touching along the $k_{y}$ direction remains at $k_{y0}=\pm k_{2}$ as we
expected. The corresponding $k_{x}$ and $k_{z}$ values at which band touching
can occur are determined by the (ii) condition. The condition (ii) tells that
the determinant of the matrix ${\bf B}$ must vanish. Since ${\bf B}$ is almost
an upper triangular, the determinant can be easily evaluated to be

$\displaystyle\textrm{det}({\bf B})=$
$\displaystyle\prod_{\alpha}m_{\alpha}-2^{q}e^{-iqk_{z}}$ (26)
$\displaystyle=$ $\displaystyle
2\left(T_{q}(g)-\cos{qk_{x}}\right)-2^{q}e^{-iqk_{z}}$

Setting $\textrm{det}({\bf B})=0$ and comparing its real and imaginary parts,
we arrive at the following two conditions

$\displaystyle\sin{qk_{z}}=0$ (27a)
$\displaystyle\cos{qk_{x}}=T_{q}(g)-2^{q-1}\cos{qk_{z}},$ (27b)

where $T_{q}(g)$ is a Chebyshev polynomial of degree $q$ of first kind and
$g=M=1+\cos{k_{1}}$. The condition $\sin{qk_{z}}=0$ gives two values of
$k_{z0}=0$ and $\pi/q$ at which gap closing can happen. However, the solution
$k_{z0}=\pi/q$ does not satisfy the condition Eq. (27b) because the right hand
side of Eq. (27b) is always greater than the unity for all $q$’s. Therefore
band touching along the $k_{z}$ direction remains at $k_{z}=0$ and the
corresponding $k_{x}$ values are given by

$\displaystyle\cos{qk_{x}}=T_{q}(g)-2^{q-1}.$ (28)

We notice that this condition is identical to the condition in Eq. (7) for the
time-reversal broken case with two WNs, provided, we have made the replacement
$k_{0}\to k_{1}$. The above condition describe a region in the $k_{1}$
parameter’s space for gapless solutions. The gapless phase is the Weyl
semimetal phase. The full phase diagram is shown in Fig. 7 for multiple values
of $q$. We notice that the phase diagrams are very similar to the phase
diagrams of time-reversal broken case with two WNs. Now the WSM state has four
WNs and the LCI state is to be replaced by the insulator I′ which has a pair
of counter propagating Fermi arc surface states which are separated by a
distance $k_{2}$ along the $k_{y}$-direction in the $k_{x}$-$k_{y}$ surface
BZ. This confirms our prediction derived from the intuitive picture of
pairwise annihilation: The phase which results in after pairwise annihilation
by magnetic field aligned along the $y$-direction is either a normal insulator
or an insulator (I′) with counter propagating surface states on the
$k_{x}$-$k_{y}$ surface BZ.

###  III.3 Field along $x$-direction

For magnetic field aligned along the $x$-direction, the separation parameter
$k_{2}$ is relevant for pairwise annihilation of Weyl nodes. We do not need to
go through the whole calculation to find what would be the possible phases. We
can easily guess the phase diagram from the intuitive picture of pairwise
annihilation of Weyl nodes. In the zero field model, the Fermi arcs join
projections of WPs which are separated along the $k_{x}$ direction. Since the
two Fermi arcs are counter propagating, the insulator which results in after
pairwise annihilation of WNs either at a point inside the BZ or at the
boundary of the BZ will be devoid of surface states. Therefore, the phase
diagram should consist of of two insulating regions (representing normal
insulators) which are separated by a WSM phase in the central region.

Figure 10: ($a$) Longitudinal conductance $G_{zz}$ as a function of separation
parameter $k_{0}$ between two WNs of opposite chirality for three different
values of $q=3,5,7$. 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. The figures (b)-(d) show the spectrum 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 $q=5$.

##  IV Discussion

We have explored the minimal model of time-reversal broken and time-reversal
preserved WSM with two and four WNs respectively to demonstrate how phase
diagrams in presence of an external magnetic fields can be derived from an
intuitive picture of pairwise annihilation of Weyl nodes. As the number of WNs
increases, the complexity of solving the model to determine the phase diagram
grows due to the escalating number of free parameters. The true strength of
the intuitive representation of the pairwise annihilation process lies in its
independence from intricate model details. It only necessitates information
about the locations of WNs and Fermi arc connectivities in the surface BZ to
predict the potential phases that may emerge after pairwise annihilation
induced by magnetic fields.

Let us consider a WSM with six WNs and see if there is any new phase which was
not there in the previous models with two and four Weyl nodes. Imagine all the
WNs are located at the $k_{x}$-$k_{y}$ plane at $k_{z}=0$. Suppose the Fermi
arcs connect the projection of WNs which are separated along the $k_{x}$
direction as depicted in Fig. 9. Assume the magnetic field is applied along
the $y$ direction so that the separation $k_{1}$ and $k_{2}$ (as in Fig. 9)
are relevant for pairwise annihilation. We have considered two scenarios. In
the first scenario, we have $k_{1}\ll\pi/2$ and $k_{2}\sim\pi\to
k^{\prime}_{2}\ll\pi/2$. Now depending on the relative values of $k_{1}$ and
$k^{\prime}_{2}$, the pairwise annihilation by magnetic field results in
either an insulator (I′) or a new coexistent phase W2′′. The second scenario,
where we have $k_{1}\sim\pi\to k^{\prime}_{1}\ll\pi/2$ and $k_{2}\ll\pi/2$,
results in either a WSM state with two WNs or a new coexistence phase W4′. So
we find that, in a WSM with six WNs, pairwise annihilation of WNs by external
fields can lead to at least two new phases which were not possible in a WSM
with two and four Weyl nodes.

We have seen that the pairwise annihilation of WNs by external field in a WSM
results in a state which can be an insulator (e.g. NI, LCI, I′), a coexistence
phase (e.g. W2′, W2′′, W4′) or a WSM with reduced number of Weyl nodes. A
pertinent question arises: are there any experimental signatures of these
transitions? One potential quantity to investigate is the magnetoconductanc.
For example, the transition from a WSM state to a normal insulator can be
distinguished from the transition of a WSM to a LCI state by measuring the
magnetoconductance. Though, both the normal insulator and the LCI state are
gapped in bulk, the LCI state has protected zero energy surface states.
Suppose the WNs are at zero energy in the model (as we have in our case).
There are no states available near zero energy in the normal insulating state
to carry current. Therefore, we expect the conductance, at the transition from
the WSM state to the normal insulating state, to drop to zero. However, the
conductance at the transition from the WSM to the LCI state should be finite
because there are finite number of states near zero energy due to the zero
energy surface states in the LCI state.

We have computed the magnetoconductance for the WSM model (Eq. 2) with two
Weyl nodes. The magnetic field is aligned along the $z$-direction. The
quantity of interest is $G_{zz}$ which measures the longitudinal conductance
along the $z$-direction i.e. along the direction of the applied magnetic
field. We employ KWANT [46] simulation to compute the longitudinal conductance
$G_{zz}$. The conductance $G_{zz}$ is plotted in Fig. 10 for three different
$q$ values. Because of the computational limitation arising due to the finite
size of the system along the transverse directions ($L_{x}$ and $L_{y}$), we
restrict ourselves to only small $q$ values. The chemical potential is fixed
at $\mu=0.1$. We clearly see that the conductance vanishes for small WNs
separation (normal insulator) and it drops but remains finite for large WNs
separation (LCI state). This demonstrates that the transition from a WSM state
to a normal insulator can be distinguished from the transition of a WSM to a
LCI state by measuring the longitudinal conductance in the experiment.

##  V Summary and Conclusion

An external magnetic field, when aligned in the appropriate direction, can
couple a pair of WNs of opposite chirality and can potentially annihilate the
pair. Pairwise annihilation of WNs occurs when the inverse magnetic length
$l^{-1}_{B}$ becomes close to or larger than the momentum space separation
$2k_{0}$ between the two WNs of opposite chirality. In this work, we have
investigated pairwise annihilation of WNs by external magnetic field which
ranges all the way from small ($l_{B}\ll a$) to a very large value in the
Hofstadter regime ($l_{B}\sim a$). We have shown that pairwise annihilation of
WNs by external magnetic field in a WSM with two WNs results in either a
normal insulator or a layered Chern insulator. For a WSM with more than two
WNs which are not collinear, the magnetic field which is applied along any of
the three perpendicular directions can cause pairwise annihilation of WNs. The
set of phases which appear for fields along, say, $x$-direction is not
identical to the set of phases for fields aligned in the $z$-direction.

We conducted a comprehensive investigation into pairwise annihilation
phenomena within both the time-reversal broken and time-reversal preserved
models of WSMs. Our findings reveal that the pairwise annihilation of WNs
induced by external magnetic fields leads to an emergence of a new state which
can be an insulating state (e.g., NI, LCI, I′), a coexistence phase (e.g.,
W2′, W2′′, W4′), or a WSM with a reduced number of Weyl nodes.

We have developed a model independent intuitive representation of pairwise
annihilation process of WNs by external magnetic fields. This conceptual
framework relies solely on information pertaining to the locations of the WNs
and the connectivities of Fermi arcs on the surface BZ. With these essential
inputs, our intuitive model accurately predicts the resulting phases following
the pairwise annihilation of WNs by external magnetic fields.

This conceptual framework is versatile and can extend its applicability to
elucidate the pairwise annihilation processes induced by external magnetic
fields in other point node semimetals, such as three-dimensional Dirac
semimetals, as well as two-dimensional point node semimetals such as Weyl
semimetals and Dirac semimetals [47, 48, 49, 50, 51, 52, 53]. We anticipate
further exploration of these systems in the future research.

###### Acknowledgements.

The author acknowledges the financial support provided by the Infosys
Foundation. I wish to express my sincere thanks to ICTS for their warm
hospitality during my visit, where a significant portion of this work was
undertaken.

Figure 11: Energy gap 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$b$ and 5$d$, 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 $k_{x}$-$k_{y}$ 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$b$). In figure ($b$), the surface states,
which lie in the bulk gap of the insulator, are highlighted.

##  Appendix A Insulating states of the phase diagram in Fig. 5

In Sec. III, we have studied pairwise annihilation of WNs induced by external
fields in a WSM with four Weyl nodes. All four WNs are located at the
$k_{x}$-$k_{y}$ plane at $k_{z}=0$. Pairwise annihilation of WNs by magnetic
field aligned in the $z$ direction, results in a simple phase diagram as shown
in Fig. 5). The phase diagram consists of only two phases: a gapless phase
(WSM) and an insulator. Let us focus on a phase diagram for a particular value
of $q=100$ (Fig. 5$l$). The entire insulating region in the phase diagram may
be split into four subregions: left, right, top and bottom insulating regions.
As we have argued in main text, the insulators which are living in the left,
top and bottom regions will not have any surface states. However the
insulator, which is living in the right insulating region where
$k^{\prime}_{1}\ll k_{1}$ and $k_{2}\sim\pi/2$ , can have surface states (in
the $k_{x}$-$k_{y}$ surface BZ) in accordance with our intuitive picture of
pairwise annihilation process of Weyl nodes. The bulk of this insulating state
is of course trivial and the state is adiabatically connected to the adjacent
insulating states. We can numerically confirm whether the insulator living on
the right insulating region has any zero energy surface states. Because of
computational limitation, we do this for small values of $q=3,5$ (large $q$
values require more computational resource). Note that even for small values
of $q$, we can split the entire insulating regions into for subregions. The
previous argument about existence of surface states for $q=100$ also applies
to the small values of $q$. Therefore, we expect the insulator living on the
‘right insulating region’ in the phase diagram for small values of $q$ should
have zero energy surface states. We have numerically computed energy gap of
the system in a slab geometry (finite in the $z$-direction) to look for the
zero energy surface states. Since the spectrum of the Bloch-Hofstadter
Hamiltonian is symmetric about the zero energy, we know for sure that the
surface states (if exist) will be at the zero energy. The energy gap of the
system in a slab geometry is plotted in Fig. 11. We can clearly see the
insulator which is living on the ‘right insulating region’ has zero energy
surface states. The plots in the Fig. 12 show the zero energy surface states
in the $k_{x}$-$k_{y}$ surface BZ and the dispersion along the $k_{y}$
direction.

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