# Protected Weyl semimetals within 2D chiral classes

Faruk Abdulla1,2 1Harish-Chandra Research Institute, A CI of Homi Bhabha
National Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211019, India
2Physics Department, Technion - Israel Institute of Technology, Haifa 32000,
Israel

###### Abstract

Weyl semimetals in three dimensions can exist independently of any symmetry
apart from translations. Conversely, in two dimensions, Weyl semimetals are
believed to require additional symmetries including crystalline symmetries to
exist. In this study, we demonstrate that a 2D Weyl semimetal phase can exist
in systems with Hamiltonians possessing internal symmetries, such as time
reversal ${\cal T}$, charge conjugation ${\cal C}$ , and their combined chiral
symmetry ${\cal S}={\cal CT}$, only. Starting from a minimal-dimension Dirac
Hamiltonian, we establish the presence of a stable Weyl semimetal phase in
each of the five chiral classes: AIII, BDI, CII, DIII, and CI in two
dimension. Similar to 3D Weyl semimetals where Weyl points possess nontrivial
topological charges (Chern number), the Weyl points in WSMs within the 2D
chiral classes are characterized by the $Z$ winding numbers. In accordance
with the bulk-boundary correspondence, protected Fermi arc edge states emerge,
connecting the projections of Weyl points that carry opposite topological
charges. Unlike the surface states in 3D WSMs, the edge states of WSMs within
2D chiral classes are completely dispersionless and are always at the zero
energy due to the protecting chiral symmetry.

##  I Introduction

In the recent decades, three dimensional topological Weyl semimetals (WSMs)
[1, 2, 3, 4, 5, 6, 7, 8, 9] have gone through an intensive studies owing to
their many exotic properties, such as chiral anomaly [10, 11, 12, 13],
negative magneto resistance [14, 15, 16, 17, 18, 19, 20], quantum oscillation
and topologically protected Fermi arc surface states [21, 22, 23, 24]. In 3D
WSMs, non degenerate valence and conduction bands touch at a finite number of
isolated points in 3D Brillouin zone (BZ), in such a way that the low energy
dispersion around each of the touching points is linear in every direction.
Thus, when the chemical potential is pinned at the Weyl point (WP), the low
energy physics in 3D WSMs are governed by the relativistic massless Weyl
fermions. The WSM state is the most robust topological gapless phase in three
dimensions because it does not require any symmetry (except lattice
translation) for the protection of the Weyl points.

Considering the novelty of the WSM state and the exotic phenomena it exhibits
in three dimensions, it is natural to ask whether WPs can occur in two
dimensional materials also. Let us first clarify the usage of the terms WPs
and WSMs state in two-dimensional systems. Just as in three-dimensional WSMs,
two-dimensional systems can exhibit instances where the nondegenerate valence
and conduction bands come into contact at isolated points within the two-
dimensional BZ. Around these specific points of contact, the energy dispersion
is linear in all directions. These touching points within two-dimensional
systems, akin to the WPs found in three-dimensional WSMs, exhibit analogous
energy dispersion characteristics and possess nontrivial topological charges
that safeguard their existence. Consequently, these touching points are also
termed WPs, and materials hosting them are referred to as 2D WSMs [25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. A Dirac semimetal in 2D, where doubly
degenerate valence and conduction bands touch linearly at isolated points
(called Dirac points) in 2D BZ, is distinguished from a 2D WSM by the
degeneracy of the touching point.

The presence of nontrivial topology in the band structure of gapless
semimetals and metals can be understood by enclosing the band touching
point(s) in a lower dimensional manifold within the BZ. By construction the
Hamiltonian, which is restricted on the enclosing manifold is always gapped,
describes a lower dimensional insulators. Thus, the topological categorization
of band touching phenomena in metals and semimetals simplifies to the
classification of topological insulators found on the lower-dimensional
enclosed manifold [38, 39, 40, 41, 42]. For instance, in the context of 3D
Weyl semimetals, a WP is surrounded by a two-dimensional surface, and the
Hamiltonian confined to this surface characterizes Chern insulator. Therefore,
the WPs in 3D WSMs carry a nonzero Chern numbers which protect them from gap
opening when subjected to an arbitrary but small perturbation.

Unlike 3D WSM, the reduced dimension believed to require additional
symmetries, including crystalline symmetries, to protect a WP in two
dimensional materials. Previous theoretical investigations find existence of
2D WSM state in a square lattice with non zero phase in the hopping where WPs
are protected by an antiunitary composite symmetry which is a combination of a
translation and sublattice symmetry [25, 26], WPs protected by space-time
inversion [30, 43, 36], WPs protected by time-reversal, two fold rotation and
nonsymmorphic symmetries [27], mirror symmetry [31], 2D WPs induced by non-
Abelian gauge field [32], WPs protected by two fold as well as three fold
crystalline rotational symmetry [35, 37]. Recently, a set of 2D materials,
namely PtCl3, MnNBr, Cr2C, NiCS3, CrN, Mn2NT2, and NpAs, VI3 [44, 45, 35, 46,
47, 48, 49, 50], have been proposed as candidates for WSMs in two dimensions.
Since these WSMs in 2D require some form of crystalline symmetry, the edge
states at the boundary are in general not protected. Thus identifying 2D WSMs
by employing ARPES measurement like in 3D WSM [6, 7, 8], poses a challenge to
the experimentalist.

As we discuss in Sec. II, chiral symmetry $\cal{S}$, which anticommutes with
the Bloch Hamiltonian ${\cal S}H({\bf k}){\cal S}^{-1}=-H({\bf k})$ for all
${\bf k}=(k_{x},k_{y})$, is crucial to have WPs protected by internal
symmetries in two dimensions. By internal symmetries, we mean time-reversal
${\cal T}$, particle-hole ${\cal C}$ and their product, the chirality
$\cal{S}=\cal{TC}$ [38] only. In what follows, other symmetries such as
crystalline symmetries involving space, spin-rotation symmetry are irrelevant
in our classification. Since our classification involves only internal
symmetries, we focus on the Altland-Zirnbauer (AZ) symmetry class. There are
five such AZ symmetry classes : AIII (${\cal T}^{2}={\cal C}^{2}=0$), BDI
(${\cal T}^{2}={\cal C}^{2}=1$), CII (${\cal T}^{2}={\cal C}^{2}=-1$), CI
(${\cal T}^{2}=-{\cal C}^{2}=1$) and DIII (${\cal T}^{2}=-{\cal C}^{2}=-1$)
which have chiral symmetry.

It is known for decades that the transition from a 3D topological insulator to
a trivial insulator, within non chiral classes, passes through an intermediate
gapless phase which is a topological WSM in three dimensions. Murakami [1] and
Burkov-Hook-Balents [4] demonstrated that the generic Hamiltonian in class AII
in 3D possesses a stable gapless phase (as an intermediate phase between
topological and trivial insulating states) which is generically a topological
WSM. Recently, the authors in the Refs. [51, 52] have shown that the generic
Hamiltonian within the 3D chiral classes AIII, DIII, CI and CI also possesses
a stable gapless phase which is generically a topological nodal line semimetal
in three dimensions. Interestingly, the nodal line semimetal state which
appears as an intermediate gapless phase between topological and trivial
insulating states does not require any crystal symmetry for protection.

Previous studies assumed the necessity for the class to have a nontrivial
topological insulator phase in three dimensions, in order to have a stable
gapless phase as an intermediate state in the transition from topological to
trivial insulating states. However, this is not a necessary condition. A
general Hamiltonian within a given class in a given dimension, which has
trivial topology in that dimension, can still possesses a stable gapless phase
with nontrivial topology. Here we focus on the chiral classes in two
dimensions.

In two dimensions, Hamiltonian in the chiral classes (excluding DIII) has
trivial topology [38]. In this work, first, we show that the generic
Hamiltonian in any of the five AZ chiral class in two dimensions possesses a
robust semimetal phase. By generic Hamiltonian, we mean that all coupling
terms allowed within a given symmetry class are present in the Hamiltonian.
Second, as a consequence of the first, we show that this gapless state, which
is a WSM in two dimensions, does not require any crystalline symmetry or spin
rotation symmetry for the protection of the Weyl points. In fact, we find that
the WSM is the only stable semimetal phase within the 2D chiral classes.
Third, we show that the WSMs, which we have uncovered within the chiral
classes, are topologically nontrivial because the WPs carry a nontrivial $Z$
topological charge which is the topological winding number

$\displaystyle W=\frac{1}{2\pi i}\int_{C}d{\bf
k}\cdot\mathrm{Tr}\left(Q^{-1}\nabla_{\bf k}Q\right),$ (1)

where $C$ is an enclosing loop surrounding a WP in the 2D BZ. Hamiltonian in a
chiral class can always be brought to block off-diagonal form and the matrix
$Q({\bf k})$ in Eq. 1 is the upper right block. Because the WPs carry a
nontrivial topological charge (an 1D invariant), they cannot be removed
immediately by adding perturbations to the Hamiltonian in a given chiral
symmetry class.

We verify our claims by writing down the generic, minimal dimension
Hamiltonian (Dirac Hamiltonian with all the possible terms allowed in the
symmetry class) in every chiral class. We solve the Hamiltonian for gapless
solutions to show explicitly that there always exist a stable semimetal phase
which is a topological WSM in two-dimension. We verify its topological
protection by computing the topological charge $W$ of the WPs using Eq. 1. The
WPs always come in pair with opposite topological charges $W=\pm 1$. When
translation and the chiral symmetry preserved, the only way to destroy a WP is
to bring it close to another WP of opposite topological charge to gap them out
pairwise.

By the bulk boundary correspondence, there exists topologically protected zero
energy edge states at the boundary. Like the Fermi arc surface states in 3D
WSMs, the zero energy edge states (Fermi arcs edge states) in 2D WSMs always
connect a pair of WPs which carry opposite topological charges (an example in
Fig. 2). However there is an important distinction between the surface states
of a 3D WSMs and the edge states of a 2D WSMs: For WSMs in 3D, the surface
states are dispersionless along one direction but disperse linearly along the
transverse direction, whereas the edge states of WSMs in 2D are completely
dispersionless due to the reduced dimensionality.

The fact that the Hamiltonian in classes BDI, CII, CI and DIII has antiunitary
time-reversal and particle-hole symmetry, puts a restriction on the minimum
number of WPs which can appear in the theory. Similar to WSMs in 3D, WPs of 2D
WSMs always come in pair so that net topological charge remains zero. The
number of WPs in a WSM which belongs to BDI and CII (${\cal T}^{2}={\cal
C}^{2}$) is always a multiple of two, whereas in class CI and DIII (${\cal
T}^{2}=-{\cal C}^{2}$) the number of WPs must be a multiple of four.

The plan of the paper is as follows: In Sec. II, we briefly review band
touching in a generic two bands model to illustrate how WPs can occur in two
dimensions when chiral symmetry is imposed. Then in Sec. III, we set up and
fully explore a minimal model with a four by four Hamiltonian in every chiral
classes and show how a generic gapless phase in a given chiral class can be
obtained. We write down lattice models in Sec. IV to demonstrate the Fermi arc
edge states associated with the WPs of WSM in two dimensions. We discuss our
findings and conclude in Sec. V. Technical details regarding the
transformation of the winding invariant $W$ under time-reversal can be found
in the Appendix A.

##  II Weyl touching in a generic two bands model

Since Weyl touching, either in 2D or 3D, involves only two non degenerate
bands, one can consider a generic two bands model,

$\displaystyle H({\bf k})=f_{1}({\bf k})\sigma_{x}+f_{2}({\bf
k})\sigma_{y}+f_{3}({\bf k})\sigma_{z},$ (2)

to see whether a robust Weyl touching is possible between the two bands. We
have ignored the identity term $f_{0}({\bf k})\sigma_{0}$ because it is not
relevant to our analysis below. The $\sigma$’s in Eq. 2 are the two by two
Pauli matrices. For touching between the two bands, all the three function
$f_{i}({\bf k})$, $i=(1,2,3)$, must vanish simultaneously for some ${\bf
k}={\bf k}_{0}$. This is always possible in 3D because one has three variables
${\bf k}=(k_{x},k_{y},k_{z})$ to tune to make the three function $f_{i}({\bf
k})$ vanish simultaneouesly. If there is no additional constraint on
$f_{i}({\bf k})$ due to some crystalline symmetry, the touching at ${\bf
k}_{0}$ will be in general linear in every direction. Therefore Weyl points in
3D appear without any symmetry requirement except translation. Clearly, any
small perturbation $\delta f_{i}({\bf k})\sigma_{i}$ cannot remove the
touching point immediately, but can shift the touching point in the k-space.
Topological nature of the WP at ${\bf k}_{0}$ is revealed by considering a 2D
enclosing surface surrounding the WP in the 3D BZ. For a generic enclosing 2D
surface, the Hamiltonian restricted on it belongs to class A and a 2D
Hamiltonian in this class has nontrivial topology which is characterized by
the Chern number [53]

$\displaystyle C=\frac{1}{2\pi}\int_{S}d{\bf S}\cdot{\bf F},$ (3)

where $S$ is a 2D enclosing surface within the 3D BZ and ${\bf
F}=\nabla\times{\bf A}$ is the Berry curvature. So WPs in three dimensions
carry nonzero Chern numbers which provide them topological protection against
gap opening.

Coming to the band touching in 2D, we see that it is in general not possible
to make three function vanish by tuning only two variables ${\bf
k}=(k_{x},k_{y})$. However, if one imposes appropriate symmetries on $H({\bf
k})$, then one can obtain robust touching protected by those symmetries. As it
was shown by the earlier works, some form of crystalline symmetry (sometimes
more than one) may stabilize a Weyl point in 2D. Here comes our crucial
observation: if chiral symmetry is imposed on $H({\bf k})$, then one of the
$f_{i}({\bf k})$ must vanish identically for all ${\bf k}=(k_{x},k_{y})$.
Suppose, we represent the chiral symmetry operation by ${\cal S}=\sigma_{z}$,
then the generic two bands model can have only two terms in the Bloch-
Hamiltonian

$\displaystyle H({\bf k})=f_{1}({\bf k})\sigma_{x}+f_{2}({\bf k})\sigma_{2}.$
(4)

For band touching, the two function $f_{1}$, $f_{2}$ must vanish
simultaneously for some ${\bf k}={\bf k}_{0}$. And this is always possible in
2D because one has two variables ${\bf k}=(k_{x},k_{y})$ to tune to make the
two function $f_{1}$, $f_{2}$ vanish. Clearly, any small perturbation $\delta
h=\delta f_{1}({\bf k})\sigma_{x}+\delta f_{2}({\bf k})\sigma_{y}$ cannot
remove the touching point immediately but shifts it in the 2D k-space. So in
2D, WPs are generically obtained in a chiral symmetric two bands model without
requirement of any crystalline symmetry. Topological nature of the WP at ${\bf
k}_{0}$ is revealed by considering a 1D enclosing loop surrounding the WP in
the 2D BZ. As we have discussed in the previous section, the WPs in 2D systems
(which belongs to a chiral class) are characterised by the 1D AIII topological
insulator invariant $W$ given in Eq. 1.

Note that the chiral symmetry may originate from sublattice, orbital or is
simply obtained from the combined symmetry of time-reversal and particle-hole
[38]. It may also be an emergent symmetry in a theory [54]. The above example
illustrates the possibility of a 2D WSM with chiral symmetry (but no
crystalline symmetries) but is too simple to accommodate the full complexity
of a generic 2D Hamiltonian with spin and orbital degrees of freedom. As we
have stated earlier, we claim that the generic Hamiltonian in all the five AZ
chiral classes namely AIII, BDI, CII, CI and DIII has a robust gapless phase
which is a WSM in two-dimension. In what follows, we will verify our claim by
writing down a Dirac Hamiltonian for each of the five chiral classes, which we
will then perturb by all the symmetry-allowed terms and solve to explicitly
show that the gapless phase is indeed a topologically protected WSM in two-
dimension.

##  III Weyl semimetals in the 2D chiral classes

It is straightforward to argue that if the generic Hamiltonian within the
chiral classes possesses a stable gapless phase, then it cannot be a Dirac
semimetal. Since our Hamiltonian has all the terms in the symmetry class, the
inversion and the other crystalline symmetry are in general broken. Therefore
the gapless phase cannot be a Dirac semimetal which necessarily requires
crystalline symmetry for protection [27]. It cannot be a protected nodal loop
semimetal either: Suppose it is a topologically protected nodal loop
semimetal. Then the Hamiltonian restricted on an enclosing manifold must be
topologically nontrivial. For a nodal line in 2D BZ, the enclosing surface is
a zero dimensional manifold which consists of only two points: one inside and
the other one outside the nodal loop. In general the Hamiltonian, which is
restricted on this manifold, has chiral symmetry only. Therefore, the
Hamiltonian restricted on the enclosing manifold belongs to class AIII. Since
0D Hamiltonian in class AIII has trivial topology, the gapless phase cannot be
a stable nodal loop semimetal. Therefore we assert that the gapless phase (if
exists) must be a WSM in two dimensions and the WPs within the 2D chiral
classes are classified by the winding numbers $W$ which characterize
topological insulators belonging to class AIII in one dimension.

Reason for the WPs in any of the five chiral classes can be classified by a
single topological invariant $W$ may be understood through the following
argument: Suppose there is a WP in the theory. Its topology can be revealed by
investigating the lower dimensional insulator (1D insulator) which is living
on an enclosing manifold surrounding the Weyl point. For a WP in a 2D BZ, the
enclosing manifold is a closed loop surrounding the Weyl point. For a generic
enclosing loop, the ${\bf k}$ and $-{\bf k}$ points of the 2D BZ are not
included. This implies that the Hamiltonian which is restricted on a generic
enclosing loop will not respect the anti-unitary time-reversal and the
particle-hole symmetry of the full Hamiltonian in the class. So the
Hamiltonian which is restricted on a generic enclosing loop belongs to class
AIII. We know that AIII has nontrivial topology in one dimension: The 1D
insulators in class AIII are classified by a $Z$ topological invariant which
is the winding number $W$ defined in Eq. 1. Therefore, the WPs (if exist) in
the chiral classes can always be classified by a universal topological
invariant $W$.

Our calculation scheme is very similar to those presented in the Refs. [51,
52]. The goal is to initially write down the most general, spinful, $4\times
4$ Hamiltonian within each chiral classes. Subsequently, we proceed to
calculate the energy spectrum, with a particular focus on solving for zero
energy, corresponding to the occurrence of band touching. In all the chiral
classes, we explicitly show that the condition for zero energy always consists
of two independent equations which involve momenta and the parameters in the
Hamiltonian. Then we apply our arguments given in the previous section that we
have two momenta $k_{x}$, $k_{y}$ in 2D, which we can tune to satisfy two
equations whose solution space generically describes a WSM in two-dimension.
After establishing the existence of 2D WSM gapless phase, we verify
topological protection of the WPs by computing their topological charges $W$
using Eq. 1.

###  III.1 Class AIII

To write down a generic spinful Hamiltonian in class AIII, let us begin with a
simple Dirac Hamiltonian

$\displaystyle H_{0}({\bf
k})=\eta_{x}\otimes\left(k_{x}\sigma_{x}+k_{y}\sigma_{y}\right)+m\eta_{y}.$
(5)

Here $\sigma$’s and $\eta$’s, are two by two Pauli matrices, act on spin and
pseudo spin degrees of freedom, respectively. In addition to the chirality
${\cal S}=\eta_{z}$, this Hamiltonian is also symmetric under the time-
reversal ${\cal T}=i\eta_{x}\sigma_{y}{\cal K}$ (${\cal K}$ complex conjugates
anything to its right). We must break this time-reversal symmetry to obtain
Hamiltonian in class AIII. The Hamiltonian in Eq. 5 has other crystalline
symmetries as well. For instance, it has inversion symmetry realized by ${\cal
I}=\eta_{y}$, and mirror symmetries in the axes directions realized by ${\cal
M}_{i}=\eta_{y}\sigma_{i}$, $i=x,y$. Below, in the process of constructing the
generic Hamiltonian in class AIII, we will break the crystalline symmetries.

The Hamiltonian in class AIII preserves only chiral symmetry. There are five
more terms which can be added to $H_{0}({\bf k})$ without breaking the chiral
symmetry: $\eta_{x}$, $\eta_{x}\sigma_{z}$ and $\eta_{y}\sigma_{i}$
($i=1,2,3$). Therefore the most general Hamiltonian in class AIII is

$\displaystyle H_{\textrm{AIII}}({\bf k})=H_{0}({\bf
k})+\tilde{m}\eta_{x}\sigma_{z}+V\eta_{x}+\eta_{y}\otimes{\bf
A}\cdot{\boldsymbol{\sigma}},$ (6)

where $\tilde{m}$, $V$ are real scalar and ${\bf A}=(A_{x},A_{y},A_{z})$ is a
real vector so that the Hamiltonian remain hermitian. Note that the terms
$\tilde{m}\eta_{x}\sigma_{z}$ and $\eta_{y}\otimes{\bf
A}\cdot{\boldsymbol{\sigma}}$ make sure that time reversal is broken and the
Hamiltonian in Eq. 6 belongs to class AIII. Further the terms
$\tilde{m}\eta_{x}\sigma_{z}$ and $V\eta_{x}$ break inversion as well as
mirror symmetries. Since crystalline symmetries are irrelevant in our
discussion, the three scalars $m,\tilde{m}$, $V$ and the vector ${\bf A}$ can
be any function of momenta $k_{x},k_{y}$.

To show that the Hamiltonian $H_{\textrm{AIII}}$ in Eq. 6 possesses a robust
gapless phase, first we have to find its spectrum and then solve for band
touching in the six dimensional parameter space ${\bf X}=(m,\tilde{m},V,{\bf
A})$. Since the Hamiltonian has chiral symmetry, the band touching can occur
at the zero energy only. To find the spectrum of $H_{\textrm{AIII}}$, we
square it, collect the terms and repeat this until the resulting Hamiltonian
becomes proportional to identity. We find the following energy spectrum,

$\displaystyle E^{2}({\bf k})=\tilde{\bf k}^{2}+{\bf A}^{2}+V^{2}+m^{2}\pm
2\sqrt{(V\tilde{\bf k}+m{\bf A})^{2}+(\tilde{\bf k}\times{\bf A})^{2}},$ (7)

where $\tilde{\bf k}=(k_{x},k_{y},\tilde{m})$ and $\tilde{\bf
k}^{2}=k_{x}^{2}+k_{y}^{2}+\tilde{m}^{2}$. For band touching (zero energy
solution), we must have

$\displaystyle\left(\tilde{\bf k}^{2}+{\bf
A}^{2}+V^{2}+m^{2}\right)^{2}=4\left((V\tilde{\bf k}+m{\bf A})^{2}+(\tilde{\bf
k}\times{\bf A})^{2}\right),$ (8)

which can be rewritten as

$\displaystyle\left(\tilde{\bf k}^{2}+m^{2}-{\bf
A}^{2}-V^{2}\right)^{2}+4\left(\tilde{\bf k}\cdot{\bf A}-mV\right)^{2}=0.$ (9)

Clearly the zero energy solutions, which are given by the the intersection of
two curves $k_{x}^{2}+k_{y}^{2}+\tilde{m}^{2}+m^{2}={\bf A}^{2}+V^{2}$ and
$A_{1}k_{x}+A_{2}k_{y}=mV-\tilde{m}A_{3}$ in the $k_{x}$-$k_{y}$ space,
generically describe point touchings (two fold degenerate). These touching
points are the WPs in the theory. To see the zero energy solutions describe a
stable WSM phase, suppose there is a WP at ${\bf k}={\bf k}_{0}$ for a given
choice of the parameters ${\bf X}={\bf X}_{0}$. Given the smoothness of the
curves in parameters (only polynomial dependence), the solution for WPs should
exist for any arbitrary but small change in parameters $\delta{\bf X}_{0}$. A
small change in parameters ${\bf X}_{0}+\delta{\bf X}_{0}$ cannot gap out a WP
immediately but only can shift its location to ${\bf k}_{0}+{\bf\delta
k}_{0}$. Therefore we have established our primary claim that the generic
Hamiltonian in class AIII possesses a robust gapless semimetal phase which is
a WSM in two dimensions.

As previously discussed, the WSM phase we have identified, has topological
protection which is revealed by computing the topological charge $W$ of the
WPs using Eq. 1. When the parameters ${\bf X}$ are taken momentum independent,
there exists only two Weyl points. Numerical computations of the invariant $W$
reveal that these two WPs bear opposite topological charges $W=\pm 1$. Given
their nontrivial topological charge, perturbations within the symmetry class
cannot eliminate them unless two WPs with opposing topological charges
approach each other and undergo pairwise annihilation.

###  III.2 Class BDI

A generic spinful Hamiltonian in 2D belonging to class BDI has time-reversal
${\cal T}$, particle-hole ${\cal C}$ and their product ${\cal S}={\cal T}{\cal
C}$ preserved with ${\cal T}^{2}={\cal C}^{2}=1$. To write down a generic
Hamiltonian in this class, let us begin with a simple Dirac Hamiltonian (as we
did for the class AIII )

$\displaystyle H_{0}({\bf
k})=\eta_{y}\otimes\left(k_{x}\sigma_{x}+k_{y}\sigma_{z}\right)+m\eta_{x}.$
(10)

Here $\sigma$’s and $\eta$’s, are two by two Pauli matrices, act on spin and
pseudo spin degrees of freedom respectively. The time-reversal and the
particle-hole operators are realized by ${\cal T}=\cal{K}$ and ${\cal
C}=i\eta_{z}{\cal K}$ respectively, obeying ${\cal T}^{2}=1$, ${\cal
C}^{2}=1$. Note the Hamiltonian in Eq. 10 is not the most general Hamiltonian
in class BDI. There are three additional terms which can be added without
breaking time-reversal and particle-hole symmetry: a scalar
$\delta=\eta_{y}\sigma_{y}$, and a vector
${\boldsymbol{\alpha}}=(\eta_{x}\sigma_{x},\eta_{x}\sigma_{3})$. Now we can
write down the most general (minimal dimension) Hamiltonian in class BDI,

$\displaystyle H_{\textrm{BDI }}({\bf
k})=\eta_{y}\otimes\left(k_{x}\sigma_{x}+k_{y}\sigma_{z}\right)+m\eta_{x}+\tilde{m}\delta+{\bf
A}\cdot{\boldsymbol{\alpha}},$ (11)

where ${\bf A}=(A_{1},A_{2})$ is a two component real vector and $\tilde{m}$
is a real scalar. The set of parameters ${\bf X}=(m,\tilde{m},{\bf A})$ can
only be an even function of momenta to keep the time-reversal and particle-
hole symmetry preserved. Without loss of generality, we can consider the
following quadratic dependence on momenta: $X_{\alpha}({\bf k})=X_{\alpha
0}-\sum_{ij}t^{(X_{\alpha})}_{ij}k_{i}k_{j}$, where $X_{\alpha}$,
$\alpha=1,2,..4$, is a component of the four-dimensional vector ${\bf X}$, and
$X_{\alpha 0}$, $t^{(X_{\alpha})}_{ij}$ are all real numbers. It is easily
checked that the crystalline symmetries, including inversion and mirrors, are
broken.

Now by computing the energy spectrum and solving it for zero energy, we can
explicitly show whether the Hamiltonian $H_{\textrm{BDI}}({\bf k})$ in Eq. 11
possesses a stable gapless phase. Repeating the same calculation as we did for
class AIII Hamiltonian, we find the following energy spectrum of
$H_{\textrm{BDI}}({\bf k})$,

$\displaystyle E^{2}({\bf k})={\bf k}^{2}+\mu^{2}+{\bf A}^{2}\pm
2\sqrt{\mu^{2}{\bf A}^{2}+({\bf k}\times{\bf A})^{2}},$ (12)

where $\mu^{2}=m^{2}+\tilde{m}^{2}$. The condition for band touching (or zero
energy) is easily obtained:

$\displaystyle\left({\bf k}^{2}+\mu^{2}-{\bf A}^{2}\right)^{2}+4\left({\bf
k}\cdot{\bf A}\right)^{2}=0.$ (13)

Clearly the zero energy solutions, which are given by the the intersection of
two smooth curves (i) ${\bf k}^{2}+\mu^{2}={\bf A}^{2}$ and (ii) ${\bf
k}\cdot{\bf A}=0$ in the k-space, generically describe point touchings i.e.
Weyl points. Further from the smoothness of the two curves in parameters, it
follows that for a given choice of the parameters ${\bf X}={\bf X}_{0}$, if
${\bf k}={\bf k}_{0}$ is a solution, then a small change in the parameters
${\bf X}_{0}+\delta{\bf X}_{0}$ cannot remove the touching point immediately
but can shift it in the k-space to ${\bf k}_{0}+\delta{\bf k}_{0}$. This
implies that there exists a finite region for solution of WPs in the parameter
space. Therefore (like in class AIII) a generic Hamiltonian in class BDI also
possesses a stable gapless WSM phase in two-dimension.

The WSM phase has topological protection which can be verified by computing
the topological charge $W$ of the WPs by surrounding them in a closed loop.
When the parameters ${\bf X}$ are independent of momenta, the location of the
WPs can be easily obtained by finding the intersection of the circle ${\bf
k}^{2}+\mu^{2}={\bf A}^{2}$ with the straight line $A_{1}k_{x}+A_{2}k_{y}=0$.
Clearly there are only two WPs which are time-reversal partner of each other.
We have computed the topological charge $W$ of these WPs, and we found they
carry opposite topological charges $W=\pm 1$. Non zero topological charges
make the WPs topologically robust. The only way to gap out a WP is to approach
another WP of opposite topological charge, causing them to mutually
annihilate.

###  III.3 Class CII

A Hamiltonian belonging to the symmetry class CII possesses time-reversal
(${\cal T}$), particle-hole (${\cal C}$) and the chiral (${\cal S}$) symmetry
with ${\cal T}^{2}=-1,{\cal C}^{2}=-1$. Following the similar steps as we did
for class AIII and BDI, the most general (minimal dimension) spinful
Hamiltonian in class CII is

$\displaystyle H_{\textrm{CII}}({\bf
k})=\eta_{x}\otimes\left(k_{x}\sigma_{x}+k_{y}\sigma_{z}\right)+V_{1}\eta_{x}+V_{2}\eta_{y}\sigma_{y}+{\bf
A}\cdot{\boldsymbol{\alpha}},$ (14)

where ${\boldsymbol{\alpha}}=(\eta_{y}\sigma_{1},\eta_{y}\sigma_{3})$, ${\bf
A}=(A_{1},A_{2})$ is a two component real vector, and $V_{1}$, $V_{2}$ are
real scalars. Like in class BDI, the set of parameters ${\bf
X}=(V_{1},V_{2},{\bf A})$ can be any even function of momentum without
breaking the time-reversal and particle-hole symmetry and we can choose them
to be quadratic in momenta: $X_{\alpha}({\bf k})=X_{\alpha
0}-\sum_{ij}t^{(X_{\alpha})}_{ij}k_{i}k_{j}$, where $X_{\alpha}$,
$\alpha=1,2,..4$, is a component of the four-dimensional vector ${\bf X}$, and
$X_{\alpha 0}$, $t^{(X_{\alpha})}_{ij}$ are all real numbers. For a generic
choice of $t^{(X_{\alpha})}_{ij}$, it is easily checked that crystalline
symmetries, including inversion and mirrors, are broken. The energy spectrum
is given by

$\displaystyle E^{2}({\bf k})={\bf k}^{2}+{\bf A}^{2}+V^{2}\pm
2\sqrt{V^{2}{\bf k}^{2}+({\bf k}\times{\bf A})^{2}},$ (15)

where $V^{2}=V_{1}^{2}+V_{2}^{2}$. We note that this energy spectrum is
identical to the energy spectrum of $H_{\textrm{BDI}}({\bf k})$ given in Eq.
12 with the replacement $\mu^{2}\to V^{2}$. So, the arguments given after Eq.
13 for the BDI class, apply to this case as well. Therefore, we arrive at the
conclusion that a generic Hamiltonian in class CII possesses a robust gapless
phase which is WSM in two dimensions.

We have confirmed the topological robustness of the WPs by calculating their
topological charges $W$. Our results indicate that the WPs carry non-zero
topological charges. Notably, pairs of Weyl points serving as time-reversal
partners exhibit opposite topological charges $W=\pm 1$.

###  III.4 Class CI

Hamiltonians belonging to class CI must have time-reversal (${\cal T}$),
particle-hole (${\cal C}$), and the chiral (${\cal S}$) symmetry with ${\cal
T}^{2}=1,{\cal C}^{2}=-1$. To write down a generic Hamiltonian in this class,
let us begin with a minimal dimension Dirac Hamiltonian (as we did for other
classes)

$\displaystyle H_{0}({\bf
k})=\left(k_{x}\eta_{x}+k_{y}\eta_{y}\right)\otimes\sigma_{y}$ (16)

Here $\sigma$’s and $\eta$’s, are two by two Pauli matrices, act on spin and
pseudo spin degrees of freedom, respectively as it was for the other classes.
The time-reversal and the particle-hole operators are realized by ${\cal
T}=\eta_{x}\cal{K}$ and ${\cal C}=-i\eta_{y}{\cal K}$ respectively, obeying
${\cal T}^{2}=1$, ${\cal C}^{2}=-1$. The chirality operator is obtained from
the product of time-reversal and particle-hole: ${\cal S}={\cal TC}=\eta_{z}$.
Note that there is no mass term (which anticommutes with the kinetic terms)
for the $4\times 4$ Dirac Hamiltonian in class CI. This was also the case for
class CII. The four fold degenerate Dirac point located at $k_{x}=k_{y}=0$ is
a fine tuned point. We will shortly see when the remaining terms allowed by
the symmetry class are added, the Dirac point immediately goes to Weyl points.

To write down the most general Hamiltonian in the class, we have to find all
the terms allowed by the symmetry class and add them to $H_{0}({\bf k})$ in
Eq. 16. There are six such terms which can be arranged as two scalars and two
(two component) vectors: $\eta_{x}$, $\eta_{y}$,
${\boldsymbol{\alpha}}=(\eta_{x}\sigma_{x},\eta_{x}\sigma_{z})$ and
${\boldsymbol{\beta}}=(\eta_{y}\sigma_{x},\eta_{y}\sigma_{z})$. The most
general four by four Hamiltonian in class CI is then

$\displaystyle H_{\textrm{CI}}({\bf k})=$
$\displaystyle\left(k_{x}\eta_{x}+k_{y}\eta_{y}\right)\otimes\sigma_{y}+V_{1}\eta_{x}+V_{2}\eta_{y}$
(17) $\displaystyle+{\bf A}\cdot{\boldsymbol{\alpha}}+{\bf
B}\cdot{\boldsymbol{\beta}},$

where $V_{1}$, $V_{2}$ are scalars and ${\bf A}=(A_{1},A_{2})$, ${\bf
B}=(B_{1},B_{2})$ are two components vectors. It is easily checked that the
Hamiltonian $H_{\textrm{CI}}({\bf k})$ respects both time-reversal ${\cal
T}H_{\textrm{CI}}({\bf k}){\cal T}^{-1}=H_{\textrm{CI}}(-{\bf k})$ and
particle-hole ${\cal C}H_{\textrm{CI}}({\bf k}){\cal
C}^{-1}=-H_{\textrm{CI}}(-{\bf k})$ symmetry implemented by ${\cal
T}=\eta_{x}\cal{K}$ and ${\cal C}=-i\eta_{y}{\cal K}$ respectively, obeying
${\cal T}^{2}=1$, ${\cal C}^{2}=-1$). The parameters ${\bf
X}=\left(V_{1},V_{2},{\bf A},{\bf B}\right)$ can be any even function of
momenta without altering the time-reversal and the particle-hole symmetry.
Without loss of generality, we can choose the following quadratic dependence
on momenta: $X_{\alpha}({\bf k})=X_{\alpha
0}-\sum_{ij}t^{(X_{\alpha})}_{ij}k_{i}k_{j}$, where $X_{\alpha}$,
$\alpha=1,2,..6$, is a component of the six-dimensional vector ${\bf X}$, and
$X_{\alpha 0}$, $t^{(X_{\alpha})}_{ij}$ are all real.

In the following we will show the Hamiltonian $H_{\textrm{CI}}({\bf k})$
possesses a robust gapless phase by computing its energy spectrum and solving
it for zero energy. Following the routine steps as before, we get the
following condition for zero energy (band touching)

$\displaystyle\left(\tilde{k}^{2}-\tilde{V}^{2}+{\bf A}^{2}-{\bf
B}^{2}\right)^{2}+\left(k_{x}k_{y}+{\bf A}\cdot{\bf
B}-V_{1}V_{2}\right)^{2}=0$ (18)

where $\tilde{k}^{2}=k_{x}^{2}-k_{y}^{2}$ and
$\tilde{V}^{2}=V_{1}^{2}-V_{2}^{2}$. Since the sum of two squares is zero,
they must vanish separately

$\displaystyle k_{x}^{2}-k_{y}^{2}$ $\displaystyle={\bf B}^{2}-{\bf
A}^{2}-\tilde{V}^{2}$ (19a) $\displaystyle k_{x}k_{y}$
$\displaystyle=V_{1}V_{2}-{\bf A}\cdot{\bf B}.$ (19b)

Similar to the preceding instances within class AIII, BDI, and CII, we observe
that the solution space, which is given by the intersection of two smooth
curves in the 2D k-space, generically describes Weyl touchings. Note that for
any small deformations in the parameters ${\bf X}=\left(V_{1},V_{2},{\bf
A},{\bf B}\right)$ in the Hamiltonian of Eq. 17 will slightly deform the two
intersecting curves, and thus slightly shift the location of the WPs in
k-space. This implies that there exists a finite region in the parameter space
for solution of WPs. So, the WSM phase is a stable semimetal phase. When the
parameters ${\bf X}$ are taken independent of momenta, the locations of WPs
are given by the intersection of two hyperbolas. Note that the hyperbola
represented by the first condition is symmetric about both the $k_{x}$ and
$k_{y}$ axes, where the second hyperbola is symmetric about the straight line
$k_{y}=\pm k_{x}$. Intersections can occur only at two points (WPs). The WPs
are time-reversal partner of each other. We have computed their topological
charges $W$ and we find that they carry identical charges $W=-1$. Since the
WPs carry nontrivial topological charges, they cannot be gapped out by small
perturbations unless two WPs of opposite charges come close and annihilate
each other pairwise.

###  III.5 Class DIII

Hamiltonian that belongs to AZ symmetry class DIII must have time-reversal
(${\cal T}$), particle-hole (${\cal C}$) and the chiral (${\cal S}$) symmetry
and they should obey ${\cal T}^{2}=-1,{\cal C}^{2}=1$. Following the similar
steps as we did for the other classes, we obtain the most general (minimal
dimension) spinful Hamiltonian in class DIII:

$\displaystyle H_{\textrm{DIII}}({\bf
k})=\eta_{x}\otimes\left(k_{x}\sigma_{x}+k_{y}\sigma_{y}\right)+m\eta_{y}+V\eta_{x},$
(20)

where $m$, $V$ are two real scalars. The time-reversal and the particle-hole
operators are implemented by ${\cal T}=-i\eta_{x}\sigma_{y}\cal{K}$ and ${\cal
C}=\eta_{y}\sigma_{y}{\cal K}$ respectively, obeying ${\cal T}^{2}=-1$, ${\cal
C}^{2}=1$. The chirality operator is obtained from the product of time-
reversal and particle-hole: ${\cal S}={\cal TC}=\eta_{z}$. The inversion and
mirrors symmetries are broken due to the term $V\eta_{x}$. Any left over
crystalline symmetry can be removed by considering $V$ and $m$ an arbitrary
but even function of momenta. The evenness of $V({\bf k})$ and $m({\bf k})$ is
required to keep the antiunitary time-reversal and particle-hole symmetry
preserved. Without loss of generality, we can consider the following quadratic
dependence on momenta: $m({\bf k})=m_{0}-\sum_{ij}t^{(m)}_{ij}k_{i}k_{j}$,
$V({\bf k})=V_{0}-\sum_{ij}t^{(V)}_{ij}k_{i}k_{j}$, where $m_{0}$, $V_{0}$,
$t^{()}_{ij}$ are all real.

The energy spectrum is found to be,

$\displaystyle E^{2}({\bf k})={\bf k}^{2}+m^{2}({\bf k})+V^{2}({\bf k})\pm
2\sqrt{V^{2}({\bf k})m^{2}({\bf k})}.$ (21)

The condition for band touching (or zero energy) is readily obtained:

$\displaystyle k_{x}^{2}+k_{y}^{2}=V^{2}({\bf
k}),~{}~{}~{}~{}\sum_{ij}t^{(m)}_{ij}k_{i}k_{j}=m_{0}.$ (22)

Clearly, the solution space, which is given by the intersection of two smooth
curves in the 2D k-space, generically describes Weyl touchings. From the
smoothness of the two curves, it follows that the solution for WPs must exist
in a finite region in the parameter space. Therefore we have established our
claim that the generic Hamiltonian within the class DIII possesses a stable
gapless phase which is a WSM in two-dimension.

This WSM is a topologically robust gapless phase. We have confirmed the
topological robustness of the WPs by calculating their topological charges
$W$. To find the location of the WPs, let us assume $V$ is independent of
momenta. Then the solution for WPs is given by the intersection of the circle
$k_{x}^{2}+k_{y}^{2}=V^{2}$ with the conic section
$\sum_{ij}t^{(m)}_{ij}k_{i}k_{j}=m_{0}$. Since there is no linear term in
momenta in the conic section, the curve has to be symmetric about the origin
$k_{x}=k_{y}=0$. So, the intersections between the circle and the conic
section, should occur at four points (WPs). We have numerically computed the
topological charge $W$ of these four WPs. We find that the pair of WPs, which
are time-reversal partners, carries same charges, which is opposite to the
charges carried by the other pair. The topological charges of the two pairs
are $W=\pm 1$, thus establishing the WPs topologically nontrivial.

###  III.6 Discussion

In summary of the foregoing analysis, we have demonstrated that the generic
Hamiltonian within each chiral class exhibits a robust semimetal phase which
is WSM in two dimensions. Notably, within the realm of 2D chiral classes, WSM
stands as the sole stable gapless semimetal phase. Importantly, the WSMs
within these 2D chiral classes do not necessitate any crystalline symmetry
(except for lattice translations) for the protection of Weyl points. The
topological stability of this 2D WSM state stems from the nontrivial
topological charge carried by the Weyl points. This topological charge,
denoted as the winding number $W$, serves as the invariant for 1D topological
insulators in class AIII. Weyl points always appear in pairs. Similar to WSMs
in three dimensions, the sole means of annihilating a WP in a symmetry class
is by bringing two WPs of opposite topological charges into proximity and
pairwise annihilation.

We have observed that in classes BDI and CII (CI and DIII), a WP carries a
topological charge which is opposite (identical) to its time-reversal partner.
This occurrence is a consequence of the class-specific constraints: ${\cal
T}^{2}={\cal C}^{2}$ in BDI and CII, compelling the topological charge $W$ of
a WP to be opposite to its time-reversed partner’s charge. Conversely, the
constraint ${\cal T}^{2}=-{\cal C}^{2}$ in CI and DIII results in WPs and
their time-reversed partners possessing identical topological charges (see
Appendix A for a derivation). We will shortly see a consequence of this in a
lattice model of WSMs in two dimensions.

Let us revisit our decision to utilize a four-by-four Dirac Hamiltonian across
all five chiral classes in two dimensions. While it is established that the
minimal dimension of the Dirac Hamiltonian in 2D chiral classes is four, a
notable constraint emerges when employing a four-by-four Dirac Hamiltonian
specifically within the two classes, CII and CI. We observe that the Dirac
Hamiltonian $H_{0}({\bf k})$ in class CII (Eq. 14) and CI (Eq. 16) is
inherently massless. In fact, it is not possible to add a mass term to a four
by four Dirac Hamiltonian belonging to chiral classes CII and CI in two
dimensions. The minimum dimension required for a massive Dirac Hamiltonian
within these classes is eight in two dimensions. This prompts an inquiry into
whether the generic Hamiltonian in the chiral classes CII and CI, constructed
from an eight-by-eight massive Dirac Hamiltonian, still exhibits the WSM state
as a robust, gapless phase. The authors in the Refs. [51, 52] have constructed
the generic Hamiltonian from a eight-by-eight Dirac Hamiltonian by adding all
the terms allowed in the symmetry classes CII and CI. Although, the focus was
in three spatial dimensions, their calculations extend to two dimensions with
an appropriate modification of the kinetic term—specifically, the $k_{z}$ term
is excluded. Subsequently, upon analyzing the solutions, we observe the
emergence of a WSM state as a robust gapless phase for the generic (eight-by-
eight) Hamiltonian in both the CII and CI classes.

We seek to juxtapose our findings with those of Refs. [1, 3, 51, 52]. In their
works [1, 3], it was established that the generic Hamiltonian in the non
chiral class AII in three dimensions possesses a stable semimetal phase which
exists as an intermediate phase between the topological and trivial insulating
states. This semimetal phase is a WSM in three dimensions. Recently, the Refs.
[51, 52] demonstrated that the generic Hamiltonian in the chiral classes AIII,
DIII, CII and CI in three dimensions, possesses a stable nodal line semimetal
phase which exists as an intermediate phase between the topological and
trivial insulating states. The presence of a stable semimetal phase within
classes AII, AIII, DIII, CII, and CI in 3D can be intuitively grasped as
follows. In three dimensions, each of the classes AII, AIII, DIII, CII, and CI
accommodates a topologically nontrivial insulator. The transition from a
topologically nontrivial to a trivial insulating state must necessarily
traverse through an intermediate gapless state. Consequently, the assurance of
an intermediate gapless state’s existence in a given dimension is inherent to
an AZ class exhibiting nontrivial topology in that dimension.

In view of the above argument, our finding may not appear immediately
apparent. With the exception of DIII, all other chiral classes exhibit trivial
topology in two dimensions. However, we have demonstrated the existence of a
stable gapless phase within all five chiral classes in two dimensions.
Remarkably, across all chiral classes, this gapless phase manifests as a WSM
in the two-dimensional realm. This WSM state is topologically robust and does
not require any crystalline symmetry for protection of the Weyl points.

##  IV Weyl semimetals and their boundary states in a lattice

Having established the existence of 2D topological WSM as a robust gapless
phase in the continuum models within the chiral classes, now we want to write
down a few illustrative lattice models of 2D WSMs to study their topologically
protected edge states. Before we move on to the discussion of edge states, we
must understand how the topological charge $W$ transforms under time-reversal
in the different chiral classes. In 3D WSM, the presence of additional time-
reversal symmetry forces a WP to have monopole charge which is opposite to its
time-reversal partner. Similarly, the presence of time-reversal and particle-
hole symmetry in the BDI, CII, CI and DIII classes forces a WP to have a
topological charge $W$ which is fixed by the charge of its time-reversal
partner.

As we have shown in the Appendix A, a pair of WPs, which are related by time-
reversal symmetry, carries opposite charges in class BDI, CII and identical
charges in class CI and DIII. It is the relation ${\cal T}^{2}={\cal C}^{2}$
in class BDI and CII, which forces a WP to have charge $W$ which is opposite
to its time-reversal partner. Similarly it is ${\cal T}^{2}=-{\cal C}^{2}$ in
class CI and DIII, which puts condition on a WP to have charge $W$ which is
identical to the charge carried by its time-reversal partner.

Figure 1: ($a$) Two Weyl points (black dots) in a 2D BZ, with enclosing loops
$C_{1}$, $C_{2}$. ($b$) A pair of Weyl points with opposite topological
charges $W=\pm 1$. The vertical lines $L_{1}$ and $L_{2}$ are two closed lines
at a constant $k_{x}$ values.

Now we can easily establish from the periodicity of the BZ in a lattice that
the minimal number of WPs is two (four) in class BDI, CII (CI, DIII). Consider
a loop $C_{1}$ which encloses only one WP, then its topological charge is
given by the integration (in Eq. 1) over the closed loop $C_{1}$. Now consider
another closed loop $C_{2}$ which encloses all the WPs in the 2D BZ (as in
Fig. 1$a$), then the total charge is given by

$\displaystyle\sum_{i}W_{i}=\frac{1}{2\pi i}\int_{C_{2}}d{\bf
k}\cdot\mathrm{Tr}\left(Q^{-1}\nabla_{\bf k}Q\right),$ (23)

where $W_{i}$’s are the charges of the individual WPs. Since there are no WPs
(or gapless points) outside the loop $C_{2}$, we can expand the loop $C_{2}$
continuously to the BZ boundary without altering the total charge
$\sum_{i}W_{i}$. By periodicity of the BZ, the boundary of the BZ is
equivalent to a point. This implies that the line integral in Eq. 23 must
vanish. So the sum of the topological charges of all the WPs must be zero in a
lattice. We note that the above result for WPs in 2D is same as the Nielsen-
Ninomiya theorem [11] for Weyl fermions in 3D in a lattice. Now it is clear
why the minimal number of WPs has to be two (four) in class BDI, CII (CI,
DIII). Since, there is no time-reversal or particle-hole symmetry in class
AIII, the total number of WPs in AIII models of WSMs will be a multiple of
two.

###  IV.1 Fermi arc edge states

In 3D WSMs, the existence of surface states is attributed to the nontrivial
topological charge (Chern number) carried by the Weyl points. Consider a WSM
with two WPs at ${\bf k}=(k_{0},0,0)$ and $-{\bf k}=(-k_{0},0,0)$. The Fermi
arc surface states of the WSM are understood as a collection of the Chern
insulator’s (which exist for fixed $k_{x}$ values in between $k_{0}$ and
$-k_{0}$) chiral edge states [2, 3, 4].

Figure 2: Energy spectrum (about zero energy) of the Hamiltonian in Eq. 24 in
a cylinder geometry with finite length $L_{y}=60$. The edge states
(highlighted in blue), which exist only at the zero energy, forms a Fermi arc
by joining the projections of WPs of opposite charges located at
$k_{x}/\pi=-1/3$ and $1/3$.

The above argument about the existence of Fermi arc surface states in 3D WSM
also applies to 2D WSMs which we have uncovered in this work. The difference
is that now we have to consider 1D closed lines (instead of planes) and the
insulators living on the closed lines describe 1D AIII insulators (instead of
Chern insulators) for the case of 2D WSMs in the chiral classes. Let us
briefly go through the argument to establish that the 2D WSMs must have Fermi
arc edge states, similar to the Fermi arc surface states in 3D Weyl
semimetals. Consider two closed lines $L_{1}$ and $L_{2}$ as shown in Fig.
1$b$. Suppose the Hamiltonian, which is restricted on the line $L_{1}$,
describes a trivial insulator. Then the Hamiltonian restricted on the line
$L_{2}$ must describes a 1D AIII topological insulator, because the line
$L_{2}$ has crossed a WP of nonzero topological charge. Since AIII topological
insulator in 1D has protected edge states, the 2D WSM must have topologically
protected edge states which are given by the collection of the 1D topological
insulator’s edge states for all the $k_{x}$ values in between the two WPs of
opposite topological charges. The zero energy edge states must lie on an arc
(Fermi arc edge states) joining the WP projections of opposite charges on the
1D edge BZ.

An important point to note about the edge states of WSMs in 2D is that the
edge states are completely dispersionless. For WSMs in 3D, the surface states
is dispersionless along one direction forming Fermi arc states, but they
disperse linearly along the transverse direction. For WSMs in 2D, there is no
transverse direction because of the reduced dimensionality of the system.
Therefore all the edge states of WSMs within the 2D chiral classes live solely
at the zero energy, forming Fermi arcs joining the projections of WPs on the
1D edge BZ.

Below we demonstrate the Fermi arc edge states of a 2D WSM, belonging to BDI
class, though a simple model. Let us consider a two bands model with minimal
two WPs,

$\displaystyle H({\bf
k})=\left(1+\cos{k_{0}}-\cos{k_{x}}-\cos{k_{y}}\right)\eta_{x}+\sin{k_{y}}\eta_{y},$
(24)

where $\eta$’s are the Pauli matrices which represent a pseudo spin. The
Hamiltonian preserve time-reversal with ${\cal T}={\cal K}$ and particle-hole
with ${\cal C}=\eta_{z}{\cal K}$. The chiral symmetry is given by the product
of time-reversal and particle-hole ${\cal S}={\cal TC}=\eta_{z}$. Since ${\cal
T}^{2}={\cal C}^{2}=1$, the Hamiltonian in Eq. 24 belongs to class BDI. There
are only two WPs located at ${\bf k}_{w}=(\pm k_{0},0)$. Their topological
charges $W$ of the two WPs are computed using Eq. 1, we find they carry $W=\pm
1$ respectively. To see the edge states associated with the WPs, we have to
take the system finite along an appropriate direction. Since the two WPs has
non overlapping projections only on the 1D $k_{x}$ BZ, the system has to be
taken finite along the $y$-direction to see its topological Fermi arc edge
states. We have computed the energy spectrum by exact diagonalization, in a
cylinder geometry with finite length $L_{y}=60$. In Fig. 2, we have shown the
relevant energy spectrum about the zero energy to focus on the edge states. We
can clearly see that the zero energy edge states (all the edge states are at
zero energy) form a Fermi arc joining the projections of WPs of opposite
charges.

Figure 3: ($a$) Location of the WPs of the Hamiltonian in Eq. IV.1.1. ($b$)
Energy spectrum (about zero energy) in a cylinder geometry with finite length
$L_{y}=80$. The edge states (highlighted in blue), which exist only at the
zero energy, forms a Fermi arc by joining the projections of WPs of opposite
charges $W=\pm 1$.

####  IV.1.1 Fermi arc edge states in CI and CII models

A straightforward way to obtain lattice model is to take the continuum model
of 2D WSMs (as given in Eqs. 6, 11, 14, 17 and 20) and replace i) the linear
terms in “$k_{i}$” by “$\sin{k_{i}}$” and ii) the quadratic terms by
“$\cos{k_{i}}$”. Following this prescription, we can write down a general
lattice (hopping) model

$\displaystyle H=\sum_{{\bf n},j}c^{\dagger}({\bf n})~{}2{\cal M}~{}c({\bf
n})-\left(c^{\dagger}({\bf n}+a\hat{e}_{j})~{}{\cal T}_{j}~{}c({\bf
n})+H.c\right),$ (25)

which is defined on a square lattice. Here $n=a(n_{1},n_{2})$, $n_{i}$ being
integers, denotes the lattice sites, $\hat{e}_{j}$ is the unit vector along
$j^{th}$ direction. We will set the lattice constant $a=1$. The matrix
dimension of the onsite term ${\cal M}$ and the hopping term ${\cal T}_{j}$
(should not be confused with the time-reversal operator ${\cal T}$ which does
not have subscript) depends on the symmetry class, and it is given by the
matrix dimension of the Dirac Hamiltonian in the symmetry classes. Note that
the onsite term ${\cal M}$ includes all the momentum-independent terms (as
given in the continuum Dirac models) allowed by the symmetry class. The
hopping matrices ${\cal T}_{j}$, $j=1,2$, may be expressed in a generic form
${\cal T}_{j}=t_{j}\tilde{\gamma}+i\gamma_{j}$, where $i$ is the imaginary
unit and $t_{j}$’s are real numbers. The $\gamma_{j}$ matrices, $j=1,2$,
anticommutes with each other and $\tilde{\gamma}$ need not to anticommute with
$\gamma_{j}$ (as is the case for four bands models in class CI and CII). Now
we can explicitly specify $\gamma_{j},\tilde{\gamma},{\cal M}$ and ${\cal
T}_{j}$ to obtain a lattice model of 2D WSM in each of the five chiral classes
and investigate their Fermi arc edge states.

We have seen earlier that the Hamiltonian in class BDI, CII obey ${\cal
T}^{2}={\cal C}^{2}$ and those in class CI, DIII obey ${\cal T}^{2}=-{\cal
C}^{2}$, and a WP carries a topological charge which is opposite (equal) to
the charge carried by its time-reversal partner in class BDI, CII (CI, DIII)
in two dimensions. It is needless to go through all the chiral classes to
demostrate the Fermi arc edge states, here we choose one chiral class with
${\cal T}^{2}={\cal C}^{2}$ and another chiral class with ${\cal T}^{2}=-{\cal
C}^{2}$ to demonstrate the existence of Fermi arc edge states associated with
the Weyl points.

In class CII, the Dirac Hamiltonian in Eq. 14 is four by four, therefore
$c^{\dagger}({\bf n})$ and $c({\bf n})$ are the four components Dirac fermion
creation and annihilation operators respectively. From the Dirac Hamiltonian
in Eq. 14, we read the two anticommuting Gamma matrices
$\gamma_{1}=\eta_{x}\sigma_{x}$ and $\gamma_{2}=\eta_{x}\sigma_{z}$. We choose
$\tilde{\gamma}=\eta_{x}\sigma_{0}$, which is the $V_{1}$ term in Eq. 14. Then
the matrix ${\cal M}$ which contains the momentum independent terms of
$H_{\textrm{CII}}$ in Eq. 14 is

$\displaystyle{\cal M}=V_{1}\eta_{x}\sigma_{0}+V_{2}\eta_{y}\sigma_{y}+{\bf
A}\cdot{\boldsymbol{\alpha}}.$

Therefore, a CII lattice model (Fourier transforming $H$ in Eq. 25) of WSM in
two dimensions is given by

$\displaystyle H_{\textrm{CII, lat}}({\bf k})=$
$\displaystyle\sin{k_{x}}\cdot\eta_{x}\sigma_{x}+\sin{k_{y}}\cdot\eta_{x}\sigma_{z}$
$\displaystyle+\left({\cal
M}-(t_{1}\cos{k_{x}}+t_{2}\cos{k_{y}})\eta_{x}\right).$ (26)

For the Femi arc edge states associated with the WPs, we take a representative
choice of the parameters: $t_{1}=0.7,t_{2}=0.8$, $V_{1}=0.5,V_{2}=0.2$ and
${\bf A}=(0.1,0.4)$. For this choice of parameters, there exist four WPs as
shown in Fig. 3$a$. Clearly, the projections of the WPs on the $k_{x}$ axis
are non-overlapping and they are well separated. Therefore the Fermi arc edge
states should be clearly seen when the system is taken finite along the
$y$-direction so that $k_{x}$ remains a good quantum number. To see the Fermi
arc edge states, we have numerically computed the spectrum of the lattice
model in Eq. IV.1.1, in a cylinder geometry with finite length $L_{y}=80$
along the $y$-direction and plotted it in Fig. 3$b$. We clearly see that the
zero energy edge states, which lie in between the bulk valence and conduction
bands, join the projections of WPs of opposite charges to form an arc.

Figure 4: (a) Location of the WPs of the Hamiltonian in Eq. IV.1.1. Note that
the pair of WPs, which are related by time-reversal, carries identical charge
in a CI model of WSM. (b) Energy spectrum (about zero energy) in a cylinder
geometry with finite length $L_{y}=80$. The edge states (highlighted in blue),
which exist only at the zero energy, forms a Fermi arc by joining the
projections of WPs of opposite charges $W=\pm 1$.

Like class CII, the $c^{\dagger}({\bf n})$ and $c({\bf n})$ are the four
components Dirac fermion creation and annihilation operators respectively in
class CI. The two anticommuting Gamma matrices can be easily read from the
Dirac Hamiltonian in Eq. 17: $\gamma_{1}=\eta_{x}\sigma_{y}$ and
$\gamma_{2}=\eta_{y}\sigma_{y}$. We can choose
$\tilde{\gamma}=\eta_{x}\sigma_{0}$, which is the $V_{1}$ term in Eq. 17. Then
the matrix ${\cal M}$ which contains the momentum independent terms of
$H_{\textrm{CI}}$ in Eq. 17 is

$\displaystyle{\cal M}=V_{1}\eta_{x}+V_{2}\eta_{y}+{\bf
A}\cdot{\boldsymbol{\alpha}}+{\bf B}\cdot{\boldsymbol{\beta}},$

where ${\boldsymbol{\alpha}}$, ${\boldsymbol{\beta}}$ have been defined in Eq.
17. Therefore, a CI lattice model (Fourier transforming $H$ in Eq. 25) of WSM
is given by

$\displaystyle H_{\textrm{CI, lat}}({\bf k})=$
$\displaystyle\sin{k_{x}}\cdot\eta_{x}\sigma_{y}+\sin{k_{y}}\cdot\eta_{y}\sigma_{y}$
$\displaystyle+\left({\cal
M}-(t_{1}\cos{k_{x}}+t_{2}\cos{k_{y}})\eta_{x}\right).$ (27)

We take a representative choice of parameters: $t_{1}=0.7,t_{2}=0.9$,
$V_{1}=0.4,V_{2}=0.3$, ${\bf A}=(0.3,0.2)$ and ${\bf B}=(0.5,0.3)$, so that
the Hamiltonian in Eq. IV.1.1 describes a WSM state. For this choice of
parameters, the WSM state hosts four WPs as depicted in Fig. 4$a$. Since the
WPs are well separated along the $k_{x}$-direction, their projections on the
$k_{x}$ axis are also well separated. Therefore the Fermi arc edge states
should be clearly seen when the system is taken finite along the $y$-direction
so that $k_{x}$ remains a good quantum number. For the Fermi arc edge states,
we take the system finite along the $y$-direction and compute the spectrum
numerically. The spectrum is plotted in Fig. 4$b$. Similar to the previous
cases of edge states of WSM models in class BDI and CII, here also in class
CI, the edge states, which exist only at zero energy, form an arc by joining
the projections of WPs of opposite topological charges.

In the preceding discussion, we derived lattice models from the continuous
Dirac models (given in Sec. III) in a straightforward way by substituting the
linear terms in “$k_{i}$” with “$\sin{k_{i}}$” and the quadratic terms with
“$\cos{k_{i}}$”, without delving into the specifics of the lattice structure
and the sources of these terms. Now, as we move forward, we turn our attention
to more realistic systems, namely Graphene and a 2D SSH model, to demonstrate
that when chiral symmetry is maintained, Weyl points consistently emerge
within their gapless phases.

###  IV.2 Graphene and 2D SSH model

Graphene is a topological semimetal with Dirac cones at the Fermi energy. The
Bloch Hamiltonian [55, 56] (for a nearest neighbour hopping model) is

$\displaystyle H({\bf k})=\begin{pmatrix}0&f({\bf k})\\\ f^{*}({\bf
k})&0\end{pmatrix}=f_{1}({\bf k})\sigma_{x}+f_{2}({\bf k})\sigma_{y},$ (28)

where $f({\bf k})=1+e^{i{\bf k}\cdot{\bf R}_{2}}+e^{-i{\bf k}\cdot{\bf
R}_{3}}$ and $f_{1}({\bf k})=1+\cos{{\bf k}\cdot{\bf R}_{2}}+\cos{{\bf
k}\cdot{\bf R}_{3}}$, $f_{2}({\bf k})=-\sin{{\bf k}\cdot{\bf R}_{2}}+\sin{{\bf
k}\cdot{\bf R}_{3}}$. The lattice vectors ${\bf R}_{n}$ ($n=1,2,3$) are given
by ${\bf R}_{n}=a\left(\cos{2\pi(n-1)/3},~{}\sin{2\pi(n-1)/3}\right)$. Here
$\sigma$’s represent the two sublattices A and B of the graphene honeycomb
lattice. The spinless graphene Hamiltonian in Eq. 28 obey time-reversal with
${\cal T}={\cal K}$ and particle-hole with ${\cal C}=\sigma_{z}{\cal K}$. The
Hamiltonian has chiral symmetry ${\cal S}=\sigma_{z}$ which originates due to
the sublattice symmetry. Clearly the time-reversal and particle-hole symmetry
obey ${\cal T}^{2}={\cal C}^{2}=1$. Therefore the spinless graphene
Hamiltonian in Eq. 28 belongs to the chiral class BDI. There are two gapless
points in the energy spectrum at $\pm{\bf K}=\pm\frac{4\pi}{3a}(1,0)$ and the
low energy spectrums around $\pm{\bf K}$ form Dirac cone. For spinless
graphene, the Dirac cones are nondegenerate. Therefore, according to the
definition of WPs in 2D (discussed in the introduction), the touching points
$\pm{\bf K}$ of spinless graphene are called Weyl points. Recall that a WSM
belonging to the BDI class has a minimum of two Weyl points carrying opposite
topological charges, and Graphene satisfies this requirement. To verify
topological protection of the two WPs, we compute the topological charge $W$
(using Eq. 1) in a close loop surrounding the WPs. We find that the two WPs at
$\pm{\bf K}$ carry opposite topological charges $W=\pm 1$. Thus, the spinless
graphene (described by Eq. 28) is a two dimensional topological WSM in class
BDI.

As we have discussed earlier, the WPs, in spinless graphene which belongs to
class BDI, cannot be removed by small perturbations unless the two WPs come
close and annihilate each other pairwise. However, if the chiral symmetry
broken perturbations are added, the WPs are not topologically protected and
can get gapped out immediately. One illustrative instance of such perturbation
is the inclusion of a spin-orbit coupling term [56]

$\displaystyle h_{\lambda}({\bf k})=\lambda g({\bf k})\sigma_{z}\otimes
s_{z},$ (29)

where $g({\bf k})=-\frac{2}{3\sqrt{3}}\sum_{n=1,2,3}\sin{{\bf k}\cdot{\bf
R}_{n}}$, and the Pauli matrix $s_{z}$ represents electron’s spin. This term
breaks the chiral symmetry because of the $\sigma_{z}$ dependence in
$h_{\lambda}({\bf k})$, immediately creates a gap in the spectrum.

Let us now examine another example of WSM protected by internal symmetry in
two-dimension: In a 2D SSH (hopping) model [57, 36] featuring dimerized bonds
along both the $x$ and $y$ directions, as depicted in Fig. 5, we can represent
it as follows

$\displaystyle H=\sum_{{\bf R}}$ $\displaystyle\left(t_{x}d^{\dagger}_{1}({\bf
R})d_{2}({\bf R})+td^{\dagger}_{4}({\bf R})d_{3}({\bf R})+H.c\right)$
$\displaystyle+\left(t_{y}d^{\dagger}_{2}({\bf R})d_{3}({\bf
R})+td^{\dagger}_{1}({\bf R})d_{4}({\bf R})+H.c\right)$
$\displaystyle+\left(td^{\dagger}_{2}({\bf R})d_{1}({\bf
R}+\hat{x})+t_{x}d^{\dagger}_{3}({\bf R})d_{4}({\bf R}+\hat{x})+H.c\right)$
$\displaystyle+\left(t_{y}d^{\dagger}_{4}({\bf R})d_{1}({\bf
R}+\hat{y})+td^{\dagger}_{3}({\bf R})d_{2}({\bf R}+\hat{y})+H.c\right).$

There are four lattice sites, labelled $\alpha=1,2,3,4$, in the unit cell. To
diagonolize $H$, we Fourier transform $d^{\dagger}_{\alpha}({\bf R})$ and
$d_{\alpha}({\bf R})$ to get the following Bloch Hamiltonian in the basis
$d({\bf k})=\left(d_{1}({\bf k})~{}d_{3}({\bf k})~{}d_{2}({\bf
k})~{}d_{4}({\bf k})\right)^{T}$

$\displaystyle H({\bf k})=\begin{pmatrix}0&Q({\bf k})\\\ Q^{\dagger}({\bf
k})&0\\\ \end{pmatrix},$ (31)

where $Q({\bf k})$ is a two by two matrix

$\displaystyle Q({\bf
k})=\begin{pmatrix}t_{x}+te^{ik_{x}}&t+t_{y}e^{ik_{y}}\\\
t_{y}+te^{-ik_{y}}&t+t_{x}e^{-ik_{x}}\\\ \end{pmatrix}.$ (32)

For symmetry analysis, it is useful to write the Bloch Hamiltonian in Eq. 31
in the following compact form

$\displaystyle H({\bf k})=$ $\displaystyle
f_{x1}\cdot\eta_{x}\tau_{0}+f_{x2}\cdot\eta_{y}\tau_{0}+f_{x3}\cdot\eta_{x}\tau_{z}+f_{x4}\cdot\eta_{y}\tau_{z}$
$\displaystyle+f_{y1}\cdot\eta_{x}\tau_{x}+f_{y2}\cdot\eta_{y}\tau_{x}+f_{y3}\cdot\eta_{x}\tau_{y}+f_{y4}\cdot\eta_{y}\tau_{y},$
(33)

where $f_{x1}({\bf k})=(t+t_{x})(1+\cos{k_{x}})/2$, $f_{x2}({\bf
k})=(t_{x}-t)\sin{k_{x}}/2$, $f_{x3}({\bf k})=(t_{x}-t)(1-\cos{k_{x}})/2$,
$f_{x4}({\bf k})=-(t_{x}+t)\sin{k_{x}}/2$, $f_{y1}({\bf
k})=(t_{y}+t)(1+\cos{k_{y}})/2$, $f_{y2}({\bf k})=(t-t_{y})\sin{k_{y}}/2$,
$f_{y3}({\bf k})=(t_{y}+t)\sin{k_{y}}/2$, $f_{y4}({\bf
k})=(t_{y}-t)(1-\cos{k_{y}})/2$. We notice that the function $f_{xi/yi}({\bf
k})$ multiplied to $\eta_{\mu}\tau_{\nu}$ is odd (even), when
$\eta_{\mu}\tau_{\nu}$ is complex (real). Therefore the Hamiltonian $H({\bf
k})$ in Eq. 31 is symmetric under time-reversal with ${\cal T}={\cal K}$. It
also obeys particle-hole symmetry with ${\cal C}=\eta_{z}{\cal K}$. The chiral
symmetry ${\cal S}=\eta_{z}$ is obvious from the block-offdiagonal form of
$H({\bf k})$ in Eq. 31. Since ${\cal T}^{2}={\cal C}^{2}=1$, the 2D SSH
Hamiltonian $H$ in Eq. IV.2 belongs to the chiral class BDI.

Figure 5: Schematic of 2D lattice where SSH model is defined. The dimerized
hopping along $x$-direction is denoted by $t$, $t_{x}$ and those along
$y$-direction are $t$, $t_{y}$.

In the following, we will see that the 2D SSH model in Eq. IV.2 has a stable
gapless phase (in its phase diagram) which is a topological WSM in two-
dimension. To find gapless phases, we need to compute the zeros of the energy
spectrum $E({\bf k})$. The energy spectrum is given by

$\displaystyle E^{2}_{\xi}({\bf k})=A_{\xi}^{2}({\bf k})+B_{\xi}^{2}({\bf
k}),$ $\displaystyle A_{\xi}({\bf
k})=(t+t_{x})\cos{\frac{k_{x}}{2}}+\xi(t+t_{y})\cos{\frac{k_{y}}{2}},$
$\displaystyle B_{\xi}({\bf
k})=(t-t_{x})\sin{\frac{k_{x}}{2}}-\xi(t-t_{y})\sin{\frac{k_{y}}{2}},$

where $\xi=\pm 1$. The gapless points, which are given by the intersection of
the two smooth curves $A_{\xi}({\bf k})=0$ and $B_{\xi}({\bf k})=0$, are
generically Weyl points. The Weyl points are located at $\pm{\bf
K}=\pm(K_{x},-K_{y})$, where $K_{x/y}$ is given by

$\displaystyle
K_{x/y}=2\cos^{-1}\left(\sqrt{\frac{(t+t_{y/x})^{2}(2t-t_{x}-t_{y})}{4t(t^{2}-t_{x}t_{y})}}\right).$
(35)

As long as, the quantity inside the square root lies in the range $[0,1)$,
small change in the parameters $t,t_{x},t_{y}$ can not remove these Weyl
points. There is a finite region in the parameter space for solution of Weyl
points. The pair of WPs are topologically protected because they carry nonzero
and opposite topological charges $W=\pm 1$. Any small perturbation, within the
class BDI, cannot gap out the WPs unless they meet at some point in the BZ and
annihilate each other.

##  V Discussion and Conclusion

Previously, the WSMs in two dimension have been believed to require some form
of crystalline symmetry (e.g. inversion, mirror, rotation, nonsymmorphic
symmetries), in addition to the internal symmetries, to exist as a robust
phase. Depending on which crystalline symmetries are present, the
classification of the WPs gets modified and so the topological charges they
carry. Furthermore, the boundary states which are considered as hallmark of
nontrivial bulk topology, may not be protected because the protecting
crystalline symmetries could be in general broken for a finite size systems.

In this work, we have demonstrated the existence of a robust WSM state within
2D chiral classes by formulating a comprehensive Hamiltonian for each chiral
class. This Hamiltonian is crafted by incorporating all permissible terms, in
accordance with the symmetry class, into the minimal Dirac Hamiltonian. We
have unambiguously shown the generic Hamiltonian within the 2D chiral classes
possesses a robust semimetal phase which is a topological Weyl semimetal.
Notably, within the realm of 2D chiral classes, WSM stands as the sole stable
gapless semimetal phase. Most importantly, this WSM state within the chiral
classes does not require any crystalline symmetry for its protection.
Independent to which chiral class, the WSM state belongs to, the state is
characterized by $Z$ type invariant which is the winding number $W$ defined in
Eq. 1. The topological invariant is calculated on a close loop surrounding a
WP. The enclosing manifold can be a generic loop which may not include ${\bf
k}$ and its time reversed momenta $-{\bf k}$. Therefore, the Hamiltonian
restricted on this close loop in general belongs to class AIII which has
nontrivial topology in one dimension. The topological invariant is the winding
number ($Z$) defined in Eq. 1.

Like 3D WSMs, the WPs in 2D WSM always come in pairs with opposite charges so
that the net topological charge is zero. The only way to eliminate a WP within
the 2D chiral classes is by bringing two WPs of opposite charges into close
proximity for mutual annihilation. A Weyl semimetal state belonging to 2D
chiral classes AIII, BDI, and CII possesses at least two WPs with opposite
charges. However, within the DIII and CI classes, it is impossible for a WSM
state to exist with just two WPs; the minimum requirement is four, due to the
constraints imposed by ${\cal T}^{2}=-{\cal C}^{2}$ in these two classes.

Similar to the Fermi arc surface states of WSM in 3D, the 2D WSMs in the
chiral class also host topological Fermi arc edge states which always join the
projections of WPs of opposite topological charges on the edge BZ. However
there is an important distinction between the Fermi arc surface states in 3D
WSMs and Fermi arc edge states in 2D WSMs: In the case of three-dimensional
WSMs, the surface states exhibit no dispersion in one direction, but they
disperse linearly in the perpendicular direction. Conversely, in two-
dimensional WSMs, edge states are completely dispersionless due to the reduced
dimensionality.

Upon incorporating our findings with those in the Refs. [1, 3, 51, 52], a
noteworthy relationship becomes evident—linking an AZ symmetry class to the
topological semimetal (protected by internal symmetry) it harbors as a robust
gapless phase: The generic Hamiltonian belonging to the non chiral classes
(e.g. AII, A, AI) possesses WSM as a robust semi-metallic phase in three
dimensions. Nodal line semimetals in 3D protected by the internal symmetries
exist as a robust gapless phase within the 3D chiral classes. However within
the 2D chiral classes, WSMs emerge as a robust topological semimetal protected
by internal symmetries. Given this interesting observation, it is natural to
ask whether there exists a robust topological semimetal protected by internal
symmetry within the non chiral classes in two dimensions. We hope to address
this question in the near future.

Many other open questions remain. An important one is to pursue whether a WSM
state in the 2D chiral classes survive in presence of disorder. In case of WSM
in 3D, it is known that the WSM state is perturbatively stable to disorder
[58, 59, 60, 61, 62, 63, 64, 65], where the WSM undergoes a transition to a
diffusive metal at a finite critical disorder strength. There is another line
of argument where the rare region effects have been believed to destroy the
WSM at arbitrarily weak disorder [66, 67, 68, 69]. However in case of WSM
state in the 2D chiral classes, a generic disorder which may break the
protecting chiral symmetry is expected to destroy the WSM state by gapping out
the WPs. An interesting future direction would involve investigating the fate
of the WSM state within 2D chiral classes, in presence of chiral symmetric
disorder (off-diagonal disorders).

Another interesting direction would be to investigate pairwise annihilation of
WPs induced by the orbital magnetic fields. Recent research, as highlighted in
Refs. [70, 71], has showcased the profound implications of pairwise
annihilation in a three dimensional WSMs, unveiling a myriad of novel phases
in the presence of an external orbital magnetic fields. We foresee the
emergence of similarly compelling physics in the context of WSMs in two
dimensions. We hope to investigate these and other intriguing questions in the
near future.

###### Acknowledgements.

The author wants to thank Ganpathy Murthy and Ankur Das for insightful
discussion. I would like to thank Sumathi Rao for her valuable comments on the
manuscript. The author acknowledges the financial support provided by the
Infosys Foundation. Additionally, 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.

##  Appendix A Transformation of $W$ under time-reversal in different chiral
classes

In the main text, we have seen that a WP carries a topological charge $W$
(defined in Eq. 1) which is opposite (equal) to the charge carried by its
time-reversed partner in class BDI, CII (DIII, CI). In this appendix, we
provide a proof of this. For convenience, let us work with a basis in which
the chirality operator ${\cal S}=\eta_{z}\otimes{\mathbb{I}}_{n}$
(${\mathbb{I}}_{n}$ is a $n\times n$ identity matrix, $\eta$’s are Pauli
matrices) is diagonal. The Bloch-Hamiltonian is a $2n\times 2n$ dimensional
matrix. Given the chirality operator ${\cal
S}=\eta_{z}\otimes{\mathbb{I}}_{n}$ , we can choose the time-reversal operator
to be (i) ${\cal T}=i\eta_{0}U{\cal K}$ or (ii) ${\cal
T}=i\eta_{x}\tilde{U}{\cal K}$, where $U$, $\tilde{U}$ are $n\times n$ unitary
matrices satisfying $UU^{*}=\alpha$, $\tilde{U}\tilde{U}^{*}=\alpha$, with
$\alpha=\pm 1$. Note that we have an alternative choice (i) ${\cal
T}=i\eta_{z}U{\cal K}$ or (ii) ${\cal T}=i\eta_{y}\tilde{U}{\cal K}$, but they
are identical to the former choice upto a unitary transformation in the $\eta$
space. Without loss of generality, we will work with the former choice. Now
from ${\cal S}={\cal TC}$, it follows that (i) ${\cal C}=i\eta_{z}U{\cal K}$
or (ii) ${\cal C}=i\eta_{y}{\tilde{U}}{\cal K}$. Therefore, in the basis in
which ${\cal S}=\eta_{z}{\mathbb{I}}_{n}$, the time-reversal and particle-hole
operators are given by either

$\displaystyle\textrm{(i)}~{}~{}{\cal T}=i\eta_{0}U{\cal K},~{}~{}{\cal
C}=i\eta_{z}U{\cal K},~{}~{}U^{*}U=\alpha,$ (36)

or

$\displaystyle\textrm{(ii)}~{}~{}{\cal T}=i\eta_{x}\tilde{U}{\cal
K},~{}~{}{\cal C}=i\eta_{y}\tilde{U}{\cal
K},~{}~{}\tilde{U}^{*}\tilde{U}=\alpha.$ (37)

The case (i), in which ${\cal T}^{2}={\cal C}^{2}=\alpha$, represents the
symmetry class BDI, if $\alpha=1$, and CII, if $\alpha=-1$. And the case (ii),
in which ${\cal T}^{2}=-{\cal C}^{2}=\alpha$, represent the DIII (for
$\alpha=-1$) or CI (for $\alpha=1$) symmetry class. Now we are ready to show
how the topological charge of a WP transform under time-reversal within the
BDI, CII, DIII, CI classes. To begin, we will concentrate on the BDI and CII
classes. Suppose we have a pair of WPs at ${\bf k_{0}}$ and at time-reversed
momentum $-{\bf k_{0}}$ which carry charges $W$ and $W^{\prime}$ respectively.
We want to find a relation between $W$ and $W^{\prime}$. The charge $W$
carried by the WP at ${\bf k}_{0}$ is given by the line-integral in Eq. 1 over
the close loop $C$ (which is traversed counterclockwise) as shown in Fig. 6.
Under time-reversal, the topological charge $W$ goes to

$\displaystyle W$ $\displaystyle\to\frac{1}{2\pi i}\int_{C^{\prime}}d{\bf
k}\cdot\mathrm{Tr}\left({Q^{\prime}}^{-1}\nabla_{\bf k}Q^{\prime}\right)$ (38)
$\displaystyle=\frac{1}{2\pi}\int_{C^{\prime}}d{\bf
k}\cdot\textrm{Im}\left[\mathrm{Tr}\left({Q^{\prime}}^{-1}\nabla_{\bf
k}Q^{\prime}\right)\right]$

which must be equal to the topological charge $W^{\prime}$ of the time-
reversed WP at $-{\bf k}_{0}$. In going from first to second line, we have
used the fact that $W$ is real to rewrite the integral in a form which will be
useful later. To relate $W^{\prime}$ with $W$, we have to find how $Q({\bf
k})$ transforms under time-reversal i.e. we have to relate $Q^{\prime}({\bf
k})$ with $Q({\bf k})$.

Figure 6: Two WPs, which are time-reversal partner of each other at ${\bf
k}_{0}$ and $-{\bf k}_{0}$, carry topological charges $W$ and $W^{\prime}$
respectively. Under time-reversal, the integration loop $C$ goes to
$C^{\prime}$ without altering the orientation i.e. it remains
counterclockwise.

The $Q^{\prime}({\bf k})$ can be obtained by evaluating ${\cal T}H({\bf
k}){\cal T}^{-1}$. In BDI and CII class

$\displaystyle{\cal T}\begin{pmatrix}0&Q({\bf k})\\\ Q^{\dagger}({\bf
k})&0\end{pmatrix}{\cal T}^{-1}$ $\displaystyle=i\eta_{0}U{\cal
K}\begin{pmatrix}0&Q({\bf k})\\\ Q^{\dagger}({\bf
k})&0\end{pmatrix}i\alpha\eta_{0}U{\cal K}$
$\displaystyle=\begin{pmatrix}0&\alpha UQ^{*}({\bf k})U^{*}\\\ \alpha
UQ^{T}({\bf k})U^{*}&0\end{pmatrix},$ (39)

where we have used ${\cal T}^{2}=\alpha$ and ${\cal T}^{-1}=\alpha{\cal T}$.
From Eq. A, we read the transformed $Q^{\prime}({\bf k})$ and
${Q^{\prime}}^{\dagger}({\bf k})$ to be

$\displaystyle Q({\bf k})\to Q^{\prime}({\bf k})=\alpha UQ^{*}({\bf k})U^{*}$
(40) $\displaystyle Q^{\dagger}({\bf k})\to{Q^{\prime}}^{\dagger}({\bf
k})=\alpha UQ^{T}({\bf k})U^{*}.$

From the above equation, the transformation of $Q^{\prime-1}({\bf k})$ can be
obtained

$\displaystyle Q^{\prime-1}({\bf k})=\alpha^{-1}U^{T}\left(Q^{-1}({\bf
k})\right)^{*}U^{\dagger}.$ (41)

Now we can express the integral in Eq. 38 in terms of $Q({\bf k})$ by
inserting the above expressions of $Q^{\prime}({\bf k})$ and
${Q^{\prime}}^{-1}({\bf k})$. We simplify the integrand as follows

$\displaystyle\mathrm{Tr}\left({Q^{\prime}}^{-1}\nabla_{\bf
k}Q^{\prime}\right)$ (42)
$\displaystyle=\mathrm{Tr}\left(\alpha^{-1}U^{T}\left(Q^{-1}({\bf
k}))\right)^{*}U^{\dagger}\nabla_{\bf k}\alpha UQ^{*}({\bf k})U^{*}\right)$
$\displaystyle=\alpha^{-1}\alpha~{}\mathrm{Tr}\left(U^{T}\left(Q^{-1}({\bf
k}))\right)^{*}\nabla_{\bf k}Q^{*}({\bf k})U^{*}\right)$
$\displaystyle=\mathrm{Tr}\left(\left(Q^{-1}({\bf k})\right)^{*}\nabla_{\bf
k}Q^{*}({\bf k})U^{*}U^{T}\right)$
$\displaystyle=\mathrm{Tr}\left(\left(Q^{-1}({\bf k})\right)^{*}\nabla_{\bf
k}Q^{*}({\bf k})\right)$ $\displaystyle=\mathrm{Tr}\left(Q^{-1}({\bf
k})\nabla_{\bf k}Q({\bf k})\right)^{*}.$

In the above manipulation, we have used
$U^{*}U^{T}=\left(U^{\dagger}U\right)^{T}=1$, cyclic property of trace, and
assumed the unitary matrix $U$ to be momentum independent. The above
simplification implies the following important relation

$\displaystyle\textrm{Im}\left[\mathrm{Tr}\left({Q^{\prime}}^{-1}\nabla_{\bf
k}Q^{\prime}\right)\right]=-\textrm{Im}\left[\mathrm{Tr}\left({Q}^{-1}\nabla_{\bf
k}Q\right)\right].$ (43)

Therefore, we find that

$\displaystyle W^{\prime}=$ $\displaystyle\frac{1}{2\pi}\int d{\bf
k}\cdot\textrm{Im}\left[\mathrm{Tr}\left({Q^{\prime}}^{\dagger}\nabla_{\bf
k}Q^{\prime}\right)\right]$ (44a) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int d{\bf
k}\cdot\textrm{Im}\left[\mathrm{Tr}\left({Q}^{\dagger}\nabla_{\bf
k}Q\right)\right]$ (44b) $\displaystyle=$ $\displaystyle-W$ (44c)

i.e. the topological charge $W$ of a WP is opposite to the charge carried by
its time-reversed partner in class BDI and CII. A similar relation between the
topological charges of two WPs which are related by the time-reversal symmetry
can be established in class DIII and CI. Going through a similar exercise as
above, we find that

$\displaystyle\textrm{Im}\left[\mathrm{Tr}\left({Q^{\prime}}^{-1}\nabla_{\bf
k}Q^{\prime}\right)\right]=\textrm{Im}\left[\mathrm{Tr}\left({Q}^{-1}\nabla_{\bf
k}Q\right)\right].$ (45)

in class DIII and CI. Therefore the topological charge $W$ of a WP has to be
equal to the charge carried by its time-reversed partner in class DIII and CI.

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