# Multiplicative topological semimetals

Adipta Pal  Max Planck Institute for Chemical Physics of Solids, Nöthnitzer
Strasse 40, 01187 Dresden, Germany  Max Planck Institute for the Physics of
Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany  Joe H. Winter
Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
01187 Dresden, Germany  Max Planck Institute for the Physics of Complex
Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany  SUPA, School of
Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews
KY16 9SS, UK  Ashley M. Cook  Max Planck Institute for Chemical Physics of
Solids, Nöthnitzer Strasse 40, 01187 Dresden, Germany  Max Planck Institute
for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden,
Germany

###### Abstract

Exhaustive study of topological semimetal phases of matter in equilibriated
electonic systems and myriad extensions has built upon the foundations laid by
earlier introduction and study of the Weyl semimetal, with broad applications
in topologically-protected quantum computing, spintronics, and optical
devices. We extend recent introduction of multiplicative topological phases to
find previously-overlooked topological semimetal phases of electronic systems
in equilibrium, with minimal symmetry-protection. We show these multiplicative
topological semimetal phases exhibit rich and distinctive bulk-boundary
correspondence and response signatures that greatly expand understanding of
consequences of topology in condensed matter settings, such as the limits on
Fermi arc connectivity and structure, and transport signatures such as the
chiral anomaly. Our work therefore lays the foundation for extensive future
study of multiplicative topological semimetal phases.

## I Introduction

Topological semimetals are a vast family Soluyanov  _et al._ (2015a); Bradlyn
_et al._ (2016) of topological phases of matter studied in great depth
experimentally Huang  _et al._ (2015a); Lv  _et al._ (2015a, b); Xu  _et al._
(2015a, b); Borisenko  _et al._ (2014); Liu  _et al._ (2014a, b); Neupane  _et
al._ (2014) in the search for table-top, quasiparticle realizations of high-
energy physics Wan  _et al._ (2011a). At the simplest level, the topological
degeneracies of band structures in these topological semimetal phases are
realized quite generically if either time-reversal symmetry Burkov and Balents
(2011) or inversion symmetry Halász and Balents (2012) are broken. This is the
requirement for two-fold topological degeneracies characteristic of the Weyl
semimetal phase, although it is desireable to realize such degeneracies in the
vicinity of the Fermi level Armitage  _et al._ (2018); Sun  _et al._ (2015),
with minimal contributions to the Fermi surface from other electronic states.
In such cases, the key signatures of Weyl semimetals are especially prominent,
including the distinguishing Fermi arc surface states Baum  _et al._ (2015);
Potter  _et al._ (2014a); Gyenis  _et al._ (2016); Batabyal  _et al._ (2016),
and transport signatures associated with the chiral anomaly Nielsen and
Ninomiya (1983); Son and Spivak (2013); Parameswaran  _et al._ (2014); Huang
_et al._ (2015b); Zhang  _et al._ (2016); Shekhar  _et al._ (2015); Arnold
_et al._ (2016). Such isolation of Weyl nodes in the vicinity of the Fermi
level is also facilitated—and the physics of topological semimetals
enriched—by systematic study of these topological phases in compounds with
wide-ranging phenomena, including superconductivity, strong spin-orbit
coupling, and strong correlations Graf  _et al._ (2011); Butch  _et al._
(2011); Tafti  _et al._ (2013); Pan  _et al._ (2013); Nakajima  _et al._
(2015); Fisk  _et al._ (1991); Kawasaki  _et al._ (2018). Much progress has
also been made in identifying other topological semimetals with more complex
topological degeneracies Murakami (2007); Wang  _et al._ (2012a); Young  _et
al._ (2012a); Bradlyn  _et al._ (2016) in electronic band structures,
protected by a large set of crystalline point group symmetries in combination
with additional anti-unitary symmetries such as time-reversal.

The present work returns to the foundations of topological semimetal studies
by introducing previously-unidentified topological semimetal phases of matter
of electronic systems in equilibrium, which may then be generalized in the
same manner as outlined above. We do so by studying the first topological
semimetal realizations of multiplicative topological phases, a recently-
identified set of topological phases of matter described by Bloch Hamiltonians
in an infinitely-large, periodic bulk, which are symmetry-protected tensor
products of “parent” Bloch Hamiltonians. These multiplicative topological
semimetal (MTSM) phases are therefore straightforward constructions described
by tensor products of Weyl semimetal Bloch Hamiltonians, yet exhibit rich
phenomena distinct from all other known topological semimetals.

We first review the Weyl semimetal phase and its canonical models. We then
construct the first examples of multiplicative topological semimetal phases
using these past results. The multiplicative topological semimetals are then
first characterized in the bulk, and their bulk-boundary correspondence
established.

## II Review of the Weyl semimetal phase and suitable models for constructing
multiplicative phases

The Weyl semimetal is a topologically non-trivial phase of matter
characterized by topologically-protected, doubly-degenerate and linearly-
dispersing band crossings in the Brillouin zone Wan  _et al._ (2011b). That
is, these band-crossings, known as Weyl points or nodes, cannot be removed
from the electronic structure through smooth deformations of the Hamiltonian,
but rather only through mutual annihilation of the Weyl nodes, by bringing two
nodes of opposite topological charge to the same point in the Brillouin zone
to gap out these band-touchings. When the Fermi level intersects only the Weyl
nodes of this semimetal phase, their low-energy physics dominates, yielding a
variety of intensely-studied exotic phenomena of interest for applications. At
the simplest level, the Weyl nodes serve as quasiparticle, table-top
realizations of Weyl fermions predicted in high-energy physics. However, they
are also a starting point in going well beyond high-energy physics, by tilting
the Weyl cone to realize a type-II Weyl semimetal phase Soluyanov  _et al._
(2015b), in which the low-energy physics of the Weyl nodes is not Lorentz-
invariant.

Weyl semimetal phases can be realized in effectively non-interacting systems
where certain discrete symmetries are broken rather than respected, in
contrast to many other effectively non-interacting topological phases. They
may be derived through symmetry-breaking starting from the Dirac semimetal
state Wang  _et al._ (2012b); Young  _et al._ (2012b), for instance, (which
could be topologically-robust or fine-tuned) by breaking either time-reversal
symmetry $\mathcal{T}$ or inversion symmetry $\mathcal{I}$, which pulls the
two Weyl nodes comprising the Dirac node away from one another in momentum-
space Zyuzin  _et al._ (2012). This phase, characterized by Weyl nodes in the
Brillouin zone, is topologically stable so long as Weyl nodes of opposite
topological charge do not annihilate one another Morimoto and Furusaki (2014).

$\mathcal{I}$-breaking Weyl semimetal phases are of tremendous experimental
interest, but are described by Bloch Hamiltonian models with four bands at
minimum. A more natural starting point in deriving multiplicative topological
semimetal phases is instead to use the minimal Weyl semimetal Bloch
Hamiltonian achieved by breaking $\mathcal{T}$, which possesses only two
bands. Such two-band models for the Weyl semimetal correspond to the non-
trivial homotopy group $\pi_{3}(S^{2})$ and, similarly to the two-band Chern
and Hopf insulators Moore  _et al._ (2008) and the two-band Kitaev chain model
Kitaev (2001), may be combined using known constructions to form a
multiplicative counterpart of the Weyl semimetal phase, the multiplicative
Weyl semimetal phase (MWSM).

We therefore consider a well-established two-band Bloch Hamiltonian previously
used in study of Weyl nodes, with various instances of this model serving as
the parents of the MWSM.

$\begin{split}\mathcal{H}_{WSM}(\bm{k})=&t_{1}\sin k_{x}\tau^{x}+t_{2}\sin
k_{y}\tau^{y}\\\ &+t_{3}(2+\gamma-\cos k_{x}-\cos k_{y}-\cos
k_{z})\tau^{z}.\end{split}$ (1)

where the $\tau^{j}$ ($j=x,y,z$) are the Pauli matrices in the orbital basis.
The two band spectrum,

$\begin{split}&E(\bm{k})=\pm\sqrt{t_{1}^{2}\sin^{2}k_{x}+t_{2}^{2}\sin^{2}k_{y}+\epsilon(\bm{k})^{2}},\\\
&\epsilon(\bm{k})=t_{3}(2+\gamma-\cos k_{x}-\cos k_{y}-\cos
k_{z}),\end{split}$ (2)

has two gapless nodes at $\bm{k}=(0,0,\pm k_{0})$, for $\cos k_{0}=\gamma$. We
refer to these as the Weyl nodes. The equation of motion for Bloch electrons
in the $\bm{k}$-space in the presence of Berry curvature is represented by
$\dot{\mathbf{r}}=\mathbf{v}_{\bm{k}}+\dot{\bm{k}}\times\mathbf{F}(\bm{k})$.
For the equation of motion to remain invariant under $\mathcal{T}$-symmetry,
one must have the equality, $\mathbf{F}(\bm{k})=-\mathbf{F}(-\bm{k})$. The
breaking of $\mathcal{T}$-symmetry, then involves a minimum of two Weyl nodes
with opposite Berry curvature at opposite momenta. Therefore, close to the
Weyl nodes, we have,

$\mathcal{H}_{\pm}(\bm{k})=\pm t_{1}k_{x}\tau^{x}+t_{2}k_{y}\tau^{y}\pm
t_{3}\sin k_{0}k_{z}\tau^{z},$ (3)

which in turn corresponds to the Berry curvatures,

$\mathbf{F}^{\pm}(\bm{k})|_{0,0,\pm k_{0}}=\pm\frac{t_{1}t_{2}t_{3}\sin
k_{0}}{2[t_{1}k_{x}^{2}+t_{2}k_{y}^{2}+(t_{3}\sin
k_{0})^{2}k_{z}^{2}]^{3/2}}(k_{x},k_{y},k_{z}).$ (4)

The Chern number of the lower-energy band for the range, $k_{x}=0$, $k_{y}=0$
and $k_{z}\in(-k_{0},k_{0})$ is $C=\pm 1$ depending on the direction of the
magnetic field corresponding to the monopoles at the two Weyl points. The Weyl
nodes are involved with exotic boundary states at surfaces perpendicular to
the $z$-axis, called the Fermi Arc surface states. For the case where the
surfaces are open in the $x$-direction, the surface dispersion is given by,

$E(k_{y})=\pm t_{2}\sin k_{y},$ (5)

and the arc-states,

$\Psi(x,k_{y},k_{z})=e^{+ik_{y}y+ik_{z}z}(e^{-\lambda_{1}x}-e^{-\lambda_{2}x})\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\pm i\end{pmatrix}.$

In the $k$-space, this includes all contours $\cos k_{y}+\cos k_{z}>1+\cos
k_{0}$.

## III Multiplicative Weyl Semimetal (MWSM) in the bulk

A protocol for constructing the child Hamiltonian for the MWSM,
$\mathcal{H}_{c}$ derived from $\mathcal{H}_{p1}$ and $\mathcal{H}_{p2}$ as
first reported in Cook and Moore Cook and Moore (2022), is given as follows.
Given two two-band Bloch Hamiltonians $\mathcal{H}_{p1}$ and
$\mathcal{H}_{p2}$ written in a general form, with momentum-dependence
suppressed, as

$\begin{split}\mathcal{H}_{p1}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix};\quad\mathcal{H}_{p2}=\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix},\\\ \end{split}$ (6)

the multiplicative child Bloch Hamiltonian constructed from these two parents
can be written as $\mathcal{H}^{c}_{12}$, where

$\mathcal{H}^{c}_{12}=\begin{pmatrix}a\delta&-a\gamma&b\delta&-b\gamma\\\
-a\beta&a\alpha&-b\beta&b\alpha\\\ c\delta&-c\gamma&d\delta&-d\gamma\\\
-c\beta&c\alpha&-d\beta&d\alpha\end{pmatrix}.$ (7)

Expressing the two-band parent Bloch Hamiltonians $\mathcal{H}_{p1}(\bm{k})$
and $\mathcal{H}_{p2}(\bm{k})$ more compactly as the following,

$\begin{split}&\mathcal{H}_{p1}(\bm{k})=\mathbf{d}_{1}(\bm{k})\cdot\bm{\tau};\quad\mathcal{H}_{p2}(\bm{k})=\mathbf{d}_{2}(\bm{k})\cdot\bm{\sigma},\end{split}$
(8)

where $\mathbf{d}_{1}(\bm{k})$ and $\mathbf{d}_{2}(\bm{k})$ are momentum-
dependent, three-component vectors of scalar functions, and each of
$\bm{\sigma}$ and $\bm{\tau}$ is the vector of Pauli matrices, the
multiplicative child Hamiltonian may more compactly be written as,

$\begin{split}&\mathcal{H}^{c}_{12}(\bm{k})=(d_{11},d_{21},d_{31})\cdot\bm{\tau}\otimes(-d_{12},d_{22},-d_{32})\cdot\bm{\sigma},\end{split}$
(9)

to highlight the tensor product structure of the child Hamiltonian, which can
be symmetry-protected as discussed in earlier work by Cook and Moore on
multiplicative topological phases, and therefore can describe phases of
matter, even in the presence of additional bands Cook and Moore (2022).

The tensor-product structure guarantees that the energy spectrum of the child
Hamiltonian is a product of the energy spectrum of $\mathcal{H}_{p1}(\bm{k})$,
$E_{p1}(\bm{k})$, and of $\mathcal{H}_{p2}(\bm{k})$, $E_{p2}(\bm{k})$,
respectively,

$E^{c}_{12}(\bm{k})=\pm E_{p1}(\bm{k})E_{p2}(\bm{k}).$ (10)

This implies that bands of the child Hamiltonian dispersion are at least
doubly degenerate everywhere in the bulk Brillouin zone.

We will consider two cases in this work: (1) the Weyl node separation of each
parent is along one axis in the Brillouin zone, and (2) the axis along which
Weyl nodes are separated in one parent is perpendicular to the axis along
which Weyl nodes are separated in the other parent. Spectral and magneto-
transport properties differ significantly between these two cases, as we will
show, demonstrating the richness of MTSM phases of matter.

### III.1 Multiplicative Weyl Semimetal - parallel axis parents

The construction of the MWSM for both parents along the same axis is derived
from two parent WSMs. As an example, we consider the following parents and the
resulting child:

$\begin{split}\mathcal{H}_{p1}(\bm{k})=&t_{11}\sin k_{x}\tau^{x}+t_{21}\sin
k_{y}\tau^{y}\\\ &+t_{31}(2+\gamma_{1}-\cos k_{x}-\cos k_{y}-\cos
k_{z})\tau^{z},\\\ \end{split}$ (11a)
$\begin{split}\mathcal{H}_{p2}(\bm{k})=&t_{12}\sin k_{x}\sigma^{x}+t_{22}\sin
k_{y}\sigma^{y}\\\ &+t_{32}(2+\gamma_{2}-\cos k_{x}-\cos k_{y}-\cos
k_{z})\sigma^{z},\end{split}$ (11b)
$\begin{split}\mathcal{H}_{c}(\bm{k})=&[t_{11}\sin k_{x}\tau^{x}+t_{21}\sin
k_{y}\tau^{y}\\\ &+t_{31}(2+\gamma_{1}-\cos k_{x}-\cos k_{y}-\cos
k_{z})\tau^{z}]\\\ &\otimes[-t_{12}\sin k_{x}\sigma^{x}+t_{22}\sin
k_{y}\sigma^{y}\\\ &-t_{32}(2+\gamma_{2}-\cos k_{x}-\cos k_{y}-\cos
k_{z})\sigma^{z}].\end{split}$ (11c)

Each parent Hamiltonian realizes Weyl nodes at
$\bm{k}=\left(0,0,\cos^{-1}\gamma_{i}\right)$ when $-1<\gamma_{i}<1$,
$(i=1,2)$. Examples of such topologically non-trivial dispersion are shown in
Fig. 1 (a) and (b), respectively.

Figure 1: Dispersion $E(\bm{k})$ for (a) WSM Parent Hamiltonian with
$\gamma_{1}=0.5$ along $k_{z}$ and $t_{11}=t_{21}=t_{31}=1$, (b) WSM Parent
Hamiltonian with $\gamma_{2}=-0.5$ along $k_{z}$ and $t_{12}=t_{22}=t_{32}=1$,
and (c) the resulting MWSM parallel Child Hamiltonian along $k_{z}$.

From these parent Hamiltonian dispersions, we can find the dispersion of the
child. As given in Eq. 10, the bulk spectrum is doubly degenerate and
determined by the spectra of the parent $1$, $E_{p1}(\bm{k})$, and parent $2$,
$E_{p2}(\bm{k})$, respectively, which take the following forms:

$\begin{split}&E_{p1}(\bm{k})=[t_{11}^{2}\sin^{2}k_{x}+t_{21}^{2}\sin^{2}k_{y}+\epsilon_{1}(\bm{k})^{2}]^{1/2},\\\
&E_{p2}(\bm{k})=[t_{12}^{2}\sin^{2}k_{x}+t_{22}^{2}\sin^{2}k_{y}+\epsilon_{2}(\bm{k})^{2}]^{1/2},\end{split}$
(12)

where $\epsilon_{1/2}(\bm{k})=t_{31/2}(2+\gamma_{1/2}-\cos k_{x}-\cos
k_{y}-\cos k_{z})$.
For the sake of convenience, we refer to the MWSM with Weyl node separation
for each parent along the same axis in the Brillouin zone (as in the case of
parents given by Eq. 22a and Eq. 22b) as MWSM$||$. For the MWSM$||$ bulk
spectrum given by Eq. 10 and Eq. 12, gapless points occur at the positions in
the Brillouin zone where gapless points are present for the parent systems. As
$\gamma_{1}$ and $\gamma_{2}$ control separation of the Weyl nodes in the
Brillouin zone for the parents, they play a major role in determining the
number of nodes, the location of the nodes, and the polynomial order of the
nodes in the Brillouin zone for the child. When $\gamma_{1}=\gamma_{2}$, for
instance, we have two gapless points but the dispersion near the nodes is
quadratic. In contrast, for $\gamma_{1}\neq\gamma_{2}$ as for parents depicted
in Fig. 1 (a) and (b), the child MWSM$||$ has four nodes, and bands disperse
linearly in the vicinity of the nodes, as shown in Fig. 1 (c). Each node is
four-fold degenerate.

While such degeneracy naively suggests Dirac nodes or Weyl nodes of higher
charge, the multiplicative nodes are distinct in a number of ways. To examine
this difference, we look at the child Hamiltonian in the vicinity of each
multiplicative node for the case $-1<\gamma_{1}\neq\gamma_{2}<1$. From the
tensor product structure, it easy to check that $\frac{\partial
E_{\pm}}{\partial k_{i}}=const.$ which implies that the dispersion is linear
at each of the gapless nodes of the MWSM. Therefore the possibility of a
higher order Weyl node is nullified. The position of each of the
multiplicative nodes are determined by the nodes in the respective parents. We
refer to $(0,0,\pm k_{01})$ as the Weyl node positions derived from the first
parent, and $(0,0,\pm k_{02})$ as the Weyl node positions derived from the
second parent. Here $\gamma_{i}=\cos k_{0i}$, $(i=1,2)$. If the gapless point
is $(0,0,k_{02})$, then we define MWSM$||$ in the vicinity as
$\mathcal{H}^{c}_{||,2}$, and,

$\mathcal{H}^{c}_{||,2}=t_{31}(\gamma_{1}-\gamma_{2})\tau^{z}(-t_{12}k_{x}\sigma^{x}+t_{22}k_{y}\sigma^{y}-t_{32}\sin
k_{02}\bar{k}_{z,2}\sigma^{z}),$ (13)

where $\bar{k}_{z,2}=(k_{z}-k_{02})$. Surprisingly, this looks like a Dirac
semimetal Hamiltonian, whose Dirac node has been shifted in $\bm{k}$-space.
Since it is no longer at the origin, the time-reversal symmetry is broken. For
the other node, $\gamma_{1}=\cos k_{01}$ for $(0,0,k_{01})$, we define the
multiplicative Hamiltonian in the vicinity as $\mathcal{H}^{c}_{||,1}$, so
that,

$\mathcal{H}^{c}_{||,1}=(t_{11}k_{x}\tau^{x}+t_{21}k_{y}\tau^{y}+t_{31}\sin
k_{01}\bar{k}_{z,1}\tau^{z})t_{32}(\gamma_{1}-\gamma_{2})\sigma^{z},$ (14)

where $\bar{k}_{z,1}d=(k_{z}-k_{01})$ and contains off-diagonal terms for the
block Hamiltonian. But again, it is possible to perform a similarity
transformation on this Hamiltonian, in the form
$U=R^{-1}_{\tau}(\theta,\phi)\otimes R_{\sigma}(\theta,\phi)$, so that we get
another ‘shifted’ Dirac semimetal type Hamiltonian,

$\bar{H}^{c}_{||,1}=t_{32}(\gamma_{1}-\gamma_{2})\tau^{z}(t_{11}k_{x}\sigma^{x}+t_{21}k_{y}\sigma^{y}+t_{31}\sin{k_{01}}\bar{k}_{z,1}\sigma^{z}).$
(15)

Again, the shift from the origin breaks the time-reversal symmetry of the
original Dirac semimetal. It is therefore appropriate to refer to the MWSM$||$
as possessing degeneracies consisting of Weyl nodes, rather than possessing
Dirac nodes, and exhibit strikingly different physics as a result.

### III.2 MWSM - perpendicular axis parents

Before characterizing bulk-boundary correspondence and transport signatures of
MTSMs, we explore further richness of multiplicative constructions by
considering cases where parent Weyl nodes are separated along orthogonal axes
in $\bm{k}$-space. As a specific case, we choose parent Hamiltonians such that
the first parent has Weyl node separation along the y-axis, while the second
one has Weyl node separation along the z-axis,

$\begin{split}\mathcal{H}_{p1}(\bm{k})=&t_{11}\sin k_{x}\tau^{x}+t_{21}\sin
k_{z}\tau^{y}\\\ &+t_{31}(2+\gamma_{1}-\sum_{i}\cos
k_{i})\tau^{z},\end{split}$ (16a)
$\begin{split}\mathcal{H}_{p2}(\bm{k})=&t_{12}\sin k_{x}\sigma^{x}+t_{22}\sin
k_{y}\sigma^{y}\\\ &+t_{32}(2+\gamma_{2}-\sum_{i}\cos
k_{i})\sigma^{z}.\end{split}$ (16b)

Again the bulk spectrum is derived from the tensor product structure,

$\begin{split}&E_{p1}(\bm{k})=[t_{11}^{2}\sin^{2}k_{x}+t_{21}^{2}\sin^{2}k_{z}+\epsilon_{1}^{2}(\bm{k})]^{1/2},\\\
&E_{p2}(\bm{k})=[t_{12}^{2}\sin^{2}k_{x}+t_{22}^{2}\sin^{2}k_{y}+\epsilon^{2}_{2}(\bm{k})]^{1/2},\\\
&E^{c}_{\perp}{\bm{k}}=\pm E_{p1}(\bm{k})E_{p2}(\bm{k}),\end{split}$ (17)

where $\epsilon_{1/2}(\bm{k})=t_{31/32}(2+\gamma_{1/2}-\cos k_{x}-\cos
k_{y}-\cos k_{z})$. Examples of parent and child dispersion in this case are
shown in Fig. 2 for the values, $\gamma_{1}=0.5$ and $\gamma_{2}=-0.5$.

Figure 2: Dispersion $E(k)$ ($t_{11}=t_{12}=1$, $t_{21}=t_{22}=1$,
$t_{31}=t_{32}=1$) for (a) WSM Parent Hamiltonian with $\gamma_{1}=0.5$ along
$k_{y}$, (b) WSM Parent Hamiltonian with $\gamma_{2}=-0.5$ along $k_{z}$ and
the resulting MWSM perpendicular Child Hamiltonian along (c) $k_{z}$ and (d)
$k_{z}$. The energy dispersion plotted along both $k_{y}$ and $k_{z}$ is shown
in (e) and the dispersion along a high-symmetry path in the first quadrant of
the two-dimensional (2d) BZ is shown in (f). Inversion symmetry relates the
nodes in the first quadrant to those in the other quadrants, giving rise to
four gapless nodes in the 2d BZ.

We gain greater understanding of the multiplicative structure in this case by
examining the low-energy expansion of the Child Hamiltonian in the vicinity of
its nodes. Taylor expanding up to linear order around the point,
$(0,k_{0,1},0)$ for $\gamma_{1}=\cos k_{0,1}$, one gets,

$\begin{split}\mathcal{H}^{c}_{\perp,1}(\bm{k})=&(t_{11}k_{x}\tau^{x}+t_{21}k_{z}\tau^{y}+t_{31}\sin
k_{0,1}\bar{k}_{y,1}\tau^{z})\\\ &\otimes(t_{22}\sin
k_{0,1}\sigma^{y}-t_{32}(\gamma_{2}-\gamma_{1})\sigma^{z}).\end{split}$ (18)

Similarly, expanding around $(0,0,k_{0,2})$ for $\gamma_{2}=\cos k_{0,2}$, we
get,

$\begin{split}\mathcal{H}^{c}_{\perp,2}(\bm{k})=&(t_{21}\sin
k_{0,2}\tau^{y}+t_{31}(\gamma_{1}-\gamma_{2})\tau^{z})\\\
&\otimes(-t_{12}k_{x}\sigma^{x}+t_{22}k_{y}\sigma^{y}-t_{32}\sin
k_{0,2}\bar{k}_{z,2}\sigma^{z}).\end{split}$ (19)

One notices that $\mathcal{H}^{c}_{\perp,2}(\bm{k})$ is equivalent to a DSM
when $\gamma_{1}=\gamma_{2}$.

### III.3 Discrete Symmetries of the MWSM

The discrete symmetries satisfied by the parent WSMs include invariance under
particle-hole conjugation given by $\mathcal{P}=\sigma^{x}\kappa$, such that
the Hamiltonian satisfies,

$\sigma^{x}\mathcal{H}^{*}_{1/2}(\bm{k})\sigma^{x}=-\mathcal{H}_{1/2}(-\bm{k}),$

and invariance under spatial inversion given by $\mathcal{I}=\sigma^{z}$, such
that the Hamiltonian satisfies,

$\sigma^{z}\mathcal{H}_{1/2}(\bm{k})\sigma^{z}=\mathcal{H}_{1/2}(-\bm{k}).$

The MWSM$||$ or $\perp$ child systems are instead invariant under time
reversal given by $\mathcal{T}=i\tau^{x}\sigma^{x}\kappa$ corresponding to the
transformation,

$\tau^{x}\sigma^{x}\mathcal{H}^{*}_{c}(\bm{k})\tau^{x}\sigma^{x}=\mathcal{H}_{c}(-\bm{k}).$

They are also invariant under spatial inversion given by
$\mathcal{I}=\tau^{z}\sigma^{z}$, corresponding to the transformation,

$\tau^{z}\sigma^{z}\mathcal{H}_{c}(\bm{k})\tau^{z}\sigma^{z}=\mathcal{H}_{c}(-\bm{k}).$

The MWSM should then satisfy the symmetry, $\mathcal{TI}$, which may also
protect the Dirac semi-metal phase. Indeed, in some cases, the Dirac
Hamiltonian for the MWSM near the nodes is reminiscent of the corresponding
low-energy Hamiltonian for a Dirac semi-metal. This invariance of the
multiplicative bulk Hamiltonian under products of transformations, which leave
each parent Hamiltonian invariant, is expected given the multiplicative
dependence of the child on the parents.

### III.4 Bulk characterization of topology with Wilson loops

As calculated in Supplementary section S1, the Berry connection for the MWSM
is given as

$\mathbf{A}=(A_{1,k_{x}}-A_{2,k_{x}},A_{1,k_{y}}-A_{2,k_{y}},A_{1,k_{z}}-A_{2,k_{z}}),$
(20)

where
$A_{j,l}=(i\left<+_{j}\right|\partial_{l}\left|+_{j}\right>,i\left<-_{j}\right|\partial_{l}\left|-_{j}\right>)$.
Using this expression for the Berry connection, we compute Wilson loops and
associated Wannier spectra by integrating over $k_{x}$ for a given $k_{y}$, as
detailed in Alexandradinata  _et al_ Alexandradinata  _et al._ (2014). In the
parallel case illustrated in Fig. 3(a), the Wannier spectra derived from
Wilson loop calculations show that only in regions where only one of the
parent phases is non-trivial do we get non-trivial Wannier spectra
distinguished by $\pi$ values for Wannier charge centers. However, the Wannier
spectra in the region where each parent is topological appears trivial, given
the dependence of child Wannier spectra on parent Wannier spectra distinctive
of multiplicative topological phases. We have referred to a pair of Weyl nodes
of equal and opposite topological charge as a ‘dipole’. We observe, that the
orientation of this dipole due to the two constituent parents is important, as
anti-parallel dipoles, as depicted in Fig. 3(b), show non-trivial Wilson loop
eigenvalues in a region in the 2d BZ where neither of the parent systems have
non-trivial topological character. Analogous results for the MWSM$\perp$ are
shown in Fig. 4, although the Wannier spectrum structure is far richer than in
the parallel case.

(a) Each parent has Weyl node ‘dipole’ oriented in $+\hat{z}$ direction.

(b) Parent 1 with dipole oriented along $+\hat{z}$ direction and parent 2 with
dipole oriented in $-\hat{z}$ direction.

Figure 3: Wannier spectra in MWSM parallel for two filled bands derived from
Wilson loop around $k_{x}$ for parent 1 with $\gamma_{1}=-0.5$ and parent 2
with $\gamma_{2}=0.5$. The upper row (a) and the lower row (b) have opposite
orientation of the Weyl node ’dipole’ for parent 2. Corresponding Wannier
spectra of the MWSM for the lowest-energy and second-lowest in energy occupied
bands is shown in (c) and (d), respectively.

(a) Parent 1 has Weyl node ‘dipole’ oriented along $+\hat{y}$ direction and
parent 2 along $-\hat{z}$ direction.

(b) Parent 1 has Weyl node ’dipole’ oriented in $+hat{y}$ direction and parent
2 Weyl node dipole oriented in $-\hat{z}$ direction.

Figure 4: Wannier spectra in MWSM perpendicular for two filled bands derived
from Wilson loop around $k_{x}$ for parent 1 with $\gamma_{1}=-0.5$ and parent
2 with $\gamma_{2}=0.5$. The upper row (a) and the lower row (b) have opposite
orientation of the Weyl node ’dipole’ for parent 2. Corresponding Wannier
spectra of the MWSM for the lowest-energy and second-lowest in energy occupied
bands is shown in (c) and (d), respectively.

## IV MWSM with open-boundary conditions–

#### IV.0.1 Slab spectra of MWSM:

An important aspect of WSM physics is its distinctive bulk-boundary
correspondence: Weyl nodes in the three-dimensional bulk Brillouin zone serve
as termination points of topologically-protected boundary states known as
Fermi arcs when projected to a slab Brillouin zone corresponding to open
boundary conditions in one direction. We expect analogous topologically-
protected surface states in MTSMs and explore possible realizations of these
Fermi arc states in this section.

One might expect that the tensor product structure of the multiplicative
phases is visible in the surface spectrum of the MWSM. Numerical simulations
show that this is the case. For the parent WSMs, the surface spectra is given
as, $E(k_{y})\sim\sin(k_{y})$(Fig. 5(a) and (b) and Fig. 6 2nd row) for nodes
along the z-axis and open boundaries along the x-direction and
$E(k_{z})\sim\sin(k_{z})$(Fig. 6 1st row) for nodes along the y-axis and open
boundaries along the x-direction. Indeed, corresponding surface spectra of
child Hamiltonians depend on these surface spectra in a multiplicative way.
Numerical simulation from Fig. 5 (c) shows that, for MWSM$||$, the slab
spectra disperses as $E(k_{y})\sim\sin^{2}(k_{y})$ for two parents each with
surface spectrum $E(k_{y})\sim\sin(k_{y})$. In contrast, Fig. 6 (c) shows that
the surface spectrum instead disperses as
$E(k_{y},k_{z})\sim\sin(k_{y})\sin(k_{z})$ for MWSM$\perp$ when one parent has
the former surface spectrum and the other has the latter. We also show, for
the case of each parent surface spectrum along $k_{z}$, which exhibits flat
bands between the two Weyl nodes (Fig. 5(a) and (b)) corresponds to flat bands
between all four gapless points in the MWSM parallel system (Fig. 5(c)).
However, fitting $\sin^{2}(k_{y})$ curves to each of the parallel and
perpendicular MWSM spectra reveals that, except in special cases when
$\gamma_{1}=\gamma_{2}$ where the fit is exact, the slab spectra does not
disperse as $\sin^{2}(k_{y})$ and instead exhibits $k_{z}$-dependence. One can
check this by comparing $E$ vs. $k_{y}$ slab spectra in the range
$-\text{min}(k_{0,1},k_{0,2})<k_{z}<\text{min}(k_{0,1},k_{0,2})$ and
$\text{min}(k_{0,1},k_{0,2})<k_{x}<\text{max}(k_{0,1},k_{0,2})$. The spectra
appears linear near zero in the latter case.

Figure 5: Finite slab spectra(in x-direction, $L_{x}=80$) along $k_{y}$
($k_{z}=0$) and $k_{z}$($k_{y}=0$) respectively for (a,b) WSM with
$\gamma_{1}=-0.5$, (c,d) WSM with $\gamma_{2}=0.5$. In (e,f) the slab
spectra($L_{x}=80$) $E$ vs. $k_{y}$ for the MWSM$||$ child created from the
above two parents for $k_{z}=0$ and $k_{z}=\frac{\pi}{2}$ respectively. (g)
shows the slab spectra $E$ vs. $k_{z}$ at $k_{y}=0$ for the same MWSM$||$
child system. Figure 6: Finite slab spectra (in x direction, $N_{x}=80$) with
the constituent parent Hamiltonians - WSM parent Hamiltonian 1 with
$\gamma_{1}=0.5$ and Weyl nodes along the $k_{y}$-direction is shown along
column (a), WSM parent Hamiltonian 2 with $\gamma=-0.5$ with Weyl nodes along
the $k_{z}$-direction along column (b) and the MWSM perpendicular child
Hamiltonian along column (c). It is apparent how the surface spectra along
$k_{z}$(for $k_{y}=0$) and $k_{y}$(for $k_{z}=0$) combine multiplicatively to
create the surface spectra for the MWSM perpendicular system. The lowest
diagram along column (c) especially shows the spectra along the diagonal
$k_{z}+k_{y}$ direction where the component spectra $\sin(k_{z})$ and
$\sin(k_{y})$ have combined to produce $\sin(k_{z})\sin(k_{y})$ as the leading
term.

#### IV.0.2 Stability of surface states of MWSM

For the MWSM$||$ system, the low-energy expansion about a node is reminiscent
of a Dirac node, and it is therefore possible to break apart the four-fold
degeneracy at each of the nodes by introducing an external magnetic field. We
introduce minimal coupling, $k_{y}\rightarrow k_{y}-eBx$ for the MWSM$||$ to
simulate the effect of applied magnetic field on the spectral density of the
Fermi arc surface states. We observe that the Fermi arcs split but are not
destroyed by the applied field as in the case of the DSM.

#### IV.0.3 Fermi Arcs for the MWSM as a stack of MCIs:

WSMs can be interpreted as a set of Chern insulators(CIs), each defined in a
2d submanifold of the 3d BZ of the WSM (e.g., each $k_{x}$-$k_{y}$ plane) for
a given value of $k_{z}$, yielding a stack of CIs in the $k_{z}$-direction.
The Weyl nodes then correspond to topological phase transitions—corresponding
to gap-closings—in the stack between intervals in $k_{z}$ with topologically-
distinct CIs. Specifically, we use the SCZ model Qi  _et al._ (2006),
$\mathcal{H}_{CI}=B(2+M-\cos k_{x}-\cos k_{y})\sigma^{z}+\sin
k_{x}\sigma^{x}+\sin k_{y}\sigma^{y}$ in particle-hole space. In the WSM, the
mass term is given as $M=\gamma-\cos k_{z}$. Here, for the range,
$-1<\gamma<1$, $k_{z}\in[-\cos^{-1}\gamma,\cos^{-1}\gamma]$. The Fermi arcs we
observe in the 2d BZ defined in the $k_{y}-k_{z}$ for open boundary conditions
in the x-direction are projections of the chiral edge states of the slices of
the corresponding CIs in the stack.

The multiplicative counterpart of a Chern insulator was introduced recently by
Cook and Moore Cook and Moore (2022) as Multiplicative Chern Insulators(MCIs).
Here, one must notice that the MCI has two mass terms derived from each of the
parent systems, one from each of the parent systems. Hence, there exists more
than one way to stack the MCIs in the $k_{z}$ direction. Either parent mass
term can be $k_{z}$-dependent, for instance, or both can be. Here, we have
attached the momentum dependence to both the mass terms, so that the
difference in parent mass parameters remains constant. We then characterize
the multiplicative Fermi arc states by opening boundary conditions in the x-
and y-directions, and plotting the probability density for the sum of 40
eigenstates nearest in energy to zero in Fig. 7 for $k_{z}=0$ (a 2D
submanifold of the BZ realizing an MCI) and $k_{z}=\frac{\pi}{2}$ (a 2D
submanifold of the BZ that is topologically trivial). For the former case
shown in Fig. 7(a) and (b), the probability density in the corresponding child
is localized at sites at the boundary, but also at the sites adjacent to these
sites. For the latter case, parent 1 has edge states and parent 2 does not as
shown in Fig. 7(d) and (e). The resultant child probability density shows low-
energy states localize only at the boundary sites as shown in Fig. 7(f). This
localization behavior is similar to that of the multiplicative Kitaev chain
presented in a second work by the present authors, where, if each parent is
topological, edge states are localized at lattice sites right at the edge, but
also at sites adjacent to these sites. We expect such localization to protect
the edge states from backscattering to some extent, which we will explore in
future work.

Figure 7: Probability densities of superposition of 40 edge state eigenvectors
in a $30\times 30$($L_{x}\times L_{y}$) square lattice at $k_{z}=0$ and
$k_{z}=\frac{\pi}{2}$ for (a, d) Parent WSM 1 ($\gamma_{1}=-0.5$), (b, e)
Parent WSM 2($\gamma_{2}=0.5$) and (c, f) MWSM $||$ child ($\gamma_{1}=-0.5$
and $\gamma_{2}=0.5$) respectively. At $k_{z}=0$, both the parent systems are
topological as seen from a visible edge state which results in localization at
both the edge and second last edge sites in the MWSM $||$ child system. When
$k_{z}=\frac{\pi}{2}$, the parent 1 is still topological but the parent 2 is
trivial as seen from the absence of edge states which results in localization
only at the edge sites of the MWSM $||$ child system.

#### IV.0.4 Boundary states disconnected from bulk states—

The MCI introduced by Cook and Moore Cook and Moore (2022) can exhibit
topologically robust yet floating edge states, which are separated from the
bulk by a finite energy gap. MTSMs constructed from MCIs can inherit this
exotic boundary state connectivity, displaying boundary states disconnected
from the bulk band structure.

To realize such a MWSM, we first note when edge states are disconnected from
bulk states for the case of the MCI:

$\begin{split}\mathcal{H}_{CI,p1}(\bm{k})=&B_{1}(2+M_{1}-\cos k_{x}-\cos
k_{y})\tau^{z}\\\ &+\sin k_{x}\tau^{x}+\sin k_{y}\tau^{y},\end{split}$ (21a)
$\begin{split}\mathcal{H}_{CI,p2}(\bm{k})=&B_{2}(2+M_{2}-\cos k_{x}-\cos
k_{y})\sigma^{z}\\\ &+\sin k_{x}\sigma^{x}+\sin k_{y}\sigma^{y},\end{split}$
(21b) $\begin{split}\mathcal{H}_{MCI,c}(\bm{k})=&[B_{1}(2+M_{1}-\cos
k_{x}-\cos k_{y})\tau^{z}\\\ &+\sin k_{x}\tau^{x}+\sin k_{y}\tau^{y}]\\\
&\otimes[-B_{2}(2+M_{2}-\cos k_{x}-\cos k_{y})\sigma^{z}\\\ &-\sin
k_{x}\sigma^{x}+\sin k_{y}\sigma^{y}],\end{split}$ (21c)

the range of parameters over which this is possible is $M_{1}\in[-4,-2]$ and
$M_{2}\in[-2,0]$ which corresponds to Chern numbers $C=+1$ and $C=-1$
respectively. We therefore construct a MWSM for which the Weyl nodes of one
parent WSM are separated in k-space by a stack of Chern insulators, each with
total Chern number $C=+1$, and the Weyl nodes of the other parent are
separated by a stack of Chern insulators, each with total Chern number $C=-1$.
Comparing Eqn. (22a) with (21a) and Eqn. (22b) with (21b), it is clear that,
for each Chern insulator in the stack, the following mapping holds,
$M_{i}=\gamma_{i}-\cos k_{z}$, $i\in\\{1,2\\}$, and $i$ labeling the parent.
From this mapping, it is not possible to have $M_{2}\in(-4,-2)$ while
$\gamma_{i}\in(-1,1)$, $i\in\\{1,2\\}$. We therefore generalize the mapping to
the following form, $M_{i}=\gamma_{i}-r_{i}\cos k_{z}$, $i\in\\{1,2\\}$, so
that the parents and the child Hamiltonian for the MWSM parallel are,

$\begin{split}\mathcal{H}_{p1}(\bm{k})=&t_{11}\sin k_{x}\tau^{x}+t_{21}\sin
k_{y}\tau^{y}\\\ &+t_{31}(2+\gamma_{1}-\cos k_{x}-\cos k_{y}-r_{1}\cos
k_{z})\tau^{z},\\\ \end{split}$ (22a)
$\begin{split}\mathcal{H}_{p2}(\bm{k})=&t_{12}\sin k_{x}\sigma^{x}+t_{22}\sin
k_{y}\sigma^{y}\\\ &+t_{32}(2+\gamma_{2}-\cos k_{x}-\cos k_{y}-r_{2}\cos
k_{z})\sigma^{z},\end{split}$ (22b)
$\begin{split}\mathcal{H}_{c}(\bm{k})=&[t_{11}\sin k_{x}\tau^{x}+t_{21}\sin
k_{y}\tau^{y}\\\ &+t_{31}(2+\gamma_{1}-\cos k_{x}-\cos k_{y}-r_{1}\cos
k_{z})\tau^{z}]\\\ &\otimes[-t_{12}\sin k_{x}\sigma^{x}+t_{22}\sin
k_{y}\sigma^{y}\\\ &-t_{32}(2+\gamma_{2}-\cos k_{x}-\cos k_{y}-r_{2}\cos
k_{z})\sigma^{z}].\end{split}$ (22c)

To construct one parent with Chern number of this stack non-trivial and
opposite in sign to the Chern number of the stack in the other parent, we
first introduce some terminology. We refer to the region between Weyl nodes
including $k_{z}=0$ as regular Weyl region (RWR) and the region including
$k_{z}=\pm\pi$ as the irregular Weyl region (IWR). The existence of Weyl nodes
requires $|r_{1,2}|\geq 1$ for $|\gamma_{1,2}|<1$. It is then possible to
realize a RWR with negative Chern number by varying $r_{1,2}$, so that
$\gamma_{1,2}-r_{1,2}\cos k_{z}\in(-4,-2)$. These RWRs—one of each parent
system—must then occur over the same interval in $k_{z}$, however, to realize
topological floating surface states. We set $\gamma_{2}=0$ and $r_{2}=3$,
which means we have $C=-1$ for the range
$[-\cos^{-1}(\frac{2}{3}),\cos^{-1}(\frac{2}{3})]$ when
$M_{2}=\gamma_{2}-r_{2}\cos k_{z}\in[-3,-2]$. Then we must have
$\gamma_{1}=\cos\frac{\pi}{3}=0.5$ and $r_{1}=1$ so that in the region
$k_{z}\in[-\cos^{-1}(\frac{2}{3}),\cos^{-1}(\frac{2}{3})]$, we have the same
kind of MCI with edge states gapped from the bulk as described in Cook and
MooreCook and Moore (2022). These results are shown in Fig. 8.

Figure 8: Slab spectra along $k_{y}$ (subfigure a) and $k_{z}$ (subfigure b)
for WSM parent 1 with $\gamma_{1}=0$, $r_{1}=3$, and slab spectra along
$k_{y}$ (subfigure c) and $k_{z}$ (subfigure d) for WSM parent 2 with
$\gamma_{2}=2/3$, $r_{2}=1$, respectively. Corresponding slab spectra for the
MWSM$||$ with $t_{11}=t_{12}=1$, $t_{21}=t_{22}=1$, $t_{31}=t_{32}=1$ along
(e) $k_{y}$ and (f) $k_{z}$, respectively, with edges separate from the bulk
slab spectra along $k_{y}$.

The MWSM$\perp$ case of topologically robust yet floating Fermi arc surface
states is constructed similarly, and we defer thorough investigation of this
case to later work.

## V Effect of Magnetic field on MWSM and Chiral anomaly

We now investigate response signatures of MTSMs. As we consider MWSMs here,
which may be constructed from WSM parent systems, we focus in particular on
the question of whether there is a multiplicative generalization of the chiral
anomaly, one of the most important signatures of Weyl semimetals: application
of non-orthogonal electric and magnetic fields can pump electrons between Weyl
nodes of opposite chirality Jia  _et al._ (2016). More specifically, applying
an external magnetic field parallel to the axis along which Weyl nodes are
separated in k-space yields a single chiral Landau level near each of the Weyl
nodes. In Weyl semimetals, this suppresses backscattering of electrons with
opposite chirality, manifesting as a negative magnetoresistance (MR). Weyl
semimetals therefore serve as condensed matter platforms for study of the
chiral anomaly, also known as Adler-Bell-Jackiw anomaly, associated with the
Standard Model of particle physics Yan and Felser (2017). When the external
magnetic field is instead oriented perpendicular to the k-space axis along
which Weyl nodes are separated, semi-classical calculations indicate the
presence of quantum oscillations in the density of states Potter  _et al._
(2014b), observable in magnetization, magnetic torque, and MR measurements Yan
and Felser (2017).

To study the effects of external fields on the MWSM, we first derive the
Landau level structure for the the Weyl semimetal in the cases of external
magnetic fields applied parallel and perpendicular to the Weyl node axis. We
can then draw parallels between these results and their generalizations in the
case of the MWSM.

### V.1 Chiral anomaly in WSM

To study the chiral anomaly in a WSM, we consider a particular Bloch
Hamiltonian $H_{WSM}(\bm{k})$ characterizing a Weyl semimetal phase and its
expansion around the $k_{z}$-axis, i.e. $\bm{k}\rightarrow(0,0,k_{z})$ (up to
2nd order in $k_{x}$ and $k_{y}$),

$\begin{split}H_{WSM}(\bm{k})=&t(2+\gamma-\cos k_{x}-\cos k_{y}-\cos
k_{z})\sigma^{z}\\\ &+t^{\prime}\sin k_{y}\sigma^{y}+t^{\prime}\sin
k_{x}\sigma^{x},\\\
\approx&t(Q+\frac{1}{2}(k_{x}^{2}+k_{y}^{2}))\sigma^{z}+t^{\prime}k_{y}\sigma^{y}+t^{\prime}k_{x}\sigma^{x},\end{split}$
(23)

where $Q=\gamma-\cos k_{z}$. Applying the magnetic field,
$\mathbf{B}=B\hat{\mathbf{z}}$ along the Weyl node axis, Peierls substitution
changes the momenta in the following way, $k_{x}\rightarrow
k_{x}^{\prime}=k_{x}$, $k_{y}\rightarrow k_{y}^{\prime}=k_{y}+eBx$, and
$k_{z}\rightarrow k_{z}^{\prime}=k_{z}$. The position-momentum commutator,
implies, $[k_{y}^{\prime},k_{x}^{\prime}]=ieB$, so that, it is possible to
define bosonic ladder operators,

$\begin{split}a=\frac{k_{x}^{\prime}-ik_{y}^{\prime}}{\sqrt{2eB}};\quad
a^{\dagger}=\frac{k_{x}^{\prime}+ik_{y}^{\prime}}{\sqrt{2eB}};\quad[a,a^{\dagger}]=1.\end{split}$
(24)

Applying Eqn.S18, after substituting $\bm{k}\rightarrow\bm{k}^{\prime}$, we
get the following system which looks similar to the polariton conserving
Jaynes-Cummings Hamiltonian,

$H_{WSM}(\bm{k}^{\prime})\approx
t(Q+eB(a^{\dagger}a+\frac{1}{2}))\sigma^{z}+t^{\prime}\sqrt{2eB}(a\sigma^{+}+a^{\dagger}\sigma^{-}),$
(25)

where $\sigma^{\pm}=\frac{1}{2}(\sigma^{x}\pm i\sigma^{y})$ are the spin
ladder operators in the basis $\\{\left|+\right>,\left|-\right>\\}$ of
$\sigma^{z}$ ($\sigma^{z}\left|\pm\right>=\pm\left|\pm\right>$). The ground
state from the above Hamiltonian is given by the eigenvector,
$\left|\psi_{LLL}\right>=\left|0;-\right>$ (states denoted as
$\left|n;s\right>$ where n is the bosonic number and $s$ is the spin
direction), which leads to the lowest Landau level energy,

$E_{LLL}=-t(Q+\frac{1}{2}eB).$ (26)

Near each of the Weyl nodes, it is easy to observe that
$\left|\psi_{LLL}\right>$ is chiral as shown in Fig. 9. The other Landau
levels can be derived by restricting to the two dimensional disjoint spaces,
$\\{\left|n,-\right>,\left|n-1,+\right>\\}$, parametrized by the bosonic
number, $n$ so that in each such basis, the Hamiltonian is,

$H(k_{z},n)=-\frac{teB}{2}\sigma^{0}-t(Q+eBn)\sigma^{z}+t^{\prime}\sqrt{2eBn}\sigma^{x}.$
(27)

The energy for the other Landau levels parametrized by $n=1,2,...$ is given by
the eigenvalues of Eqn. 27,

$E_{nLL}=-\frac{teB}{2}\pm\sqrt{t^{2}(Q+eBn)^{2}+2{t^{\prime}}^{2}eBn}.$ (28)

We have illustrated the analytically calculated Landau levels in Fig. 9 and
compared them to numerical calculations of Landau levels. The numerical
computation involves plotting the bands for the Peierls substituted Weyl
semimetal with periodic boundary conditions, say in the x-direction, and
subjected to magnetic field in integer multiples of $\frac{2\pi}{L}$, where L
is the size of the lattice in the x-direction . We observe that the chiral
Landau level from both analytical and numerical methods overlap, with an
approximate overlap of the other Landau levels since we only considered till
second order in $k_{x}$ and $k_{y}$.

Figure 9: Landau Levels for the two-band Weyl Semimetal calculated
analytically from Eqn. 25 and numerically, with $t=1=t^{\prime}$, $\gamma=0$
and $\mathbf{B}=\frac{2\pi}{51}\hat{\mathbf{z}}$(upper) and
$\mathbf{B}=\frac{2\pi}{51}\hat{\mathbf{y}}$(lower). The (black) bands
indicate the numerically calculated Landau levels and the (red) bands for the
analytically calculated Landau levels for $n=1,2,...,19$. The (blue) band and
the dotted (magenta) band is the Lowest Landau Level(LLL) calculated
numerically and analytically, and is responsible for the Chiral anomaly in the
upper figure and Weyl orbits in the lower figure.

Next we consider the case when the magnetic field is directed perpendicular to
the Weyl node axis, say $\mathbf{B}=B\hat{\mathbf{y}}$. Expanding the first
line of Eqn. 23 around the Weyl node, $\bm{k}=(0,0,k_{0}=\cos^{-1}\gamma)$ of
positive chirality, and setting $t=t^{\prime}=1$, we get,

$\begin{split}H_{WSM}(\bm{k})\approx&\sin
k_{0}(k_{z}-k_{0})\sigma^{z}+k_{y}\sigma^{y}+k_{x}\sigma^{x},\\\ \implies
H^{\prime}_{WSM}(\bm{k})\approx&-k_{y}\sigma^{z}+k_{x}\sigma^{x}+\sin
k_{0}(k_{z}-k_{0})\sigma^{y},\end{split}$ (29)

where in the second line we have rotated the Hamiltonian to a new basis via,
$\sigma^{x}\rightarrow\sigma^{z}$ and $\sigma^{x}\rightarrow-\sigma^{x}$. In
the presence of mentioned magnetic field perpendicular to the Weyl node axis,
the Peierls substitution is applied as $k_{x}\rightarrow
k_{x}^{\prime}=k_{x}$, $k_{y}\rightarrow k_{y}^{\prime}=k_{y}$ and
$k_{z}\rightarrow k_{z}^{\prime}=k_{z}-eBx$. The commutation relation,
$[k_{x},\sin k_{0}(k_{z}-k_{0}-eB_{x})]=ieB\sin k_{0}$, then constructs the
bosonic ladder operators,

$b=\frac{k_{z}-k_{0}-eBx-ik_{x}}{\sqrt{2eB\sin k_{0}}};\quad
b^{\dagger}=\frac{k_{z}-k_{0}-eBx+ik_{x}}{\sqrt{2eB\sin k_{0}}}.$ (30)

The system in Eqn. 29 then changes to,

$H_{WSM}(\bm{k}^{\prime})\approx-k_{y}\sigma^{z}+\sqrt{2eB\sin
k_{0}}(b\sigma^{+}+b^{\dagger}\sigma^{-}).$ (31)

Similar to the previous case, it is possible to resolve the Hamiltonian into
the subspaces spanned by $\\{\left|n,-\right>,\left|n-1,+\right>\\}$, where
$n$ is the eigenvalue of the number operator, $b^{\dagger}b$. We get two
chiral lowest Landau levels with energies, $E=\pm k_{y}$ in the bulk, which
are responsible for the chiral anomaly Yan and Felser (2017).

### V.2 Chiral anomaly in the MWSM

We now study the response of the MWSM to external fields for comparison with
the signatures of the chiral anomaly in the WSM reviewed in the previous
section. We treat the MWSM parallel and perpendicular cases separately, given
the expected sensitivity of the response to orientation of the axes of node
separation relative to the orientation of the external fields.

#### V.2.1 Landau levels in the MWSM parallel system:

In Sec. III.1 we have derived the Dirac Hamiltonian for the MWSM$||$ in the
vicinity of each of its two nodes, $(0,0,k_{01})$ and $(0,0,k_{02})$ derived
respectively from each of its two parents.

$\displaystyle H^{c}_{||,1}(\bm{k})=$
$\displaystyle(t_{1}^{\prime}k_{x}\tau^{x}+t_{1}^{\prime}k_{y}\tau^{y}$
$\displaystyle+t_{1}\sin
k_{01}\bar{k}_{z,1}\tau^{z})t_{2}(\gamma_{1}-\gamma_{2})\sigma^{z},$
$\displaystyle H^{c}_{||,2}(\bm{k})=$ $\displaystyle
t_{1}(\gamma_{1}-\gamma_{2})\tau^{z}(-t_{2}^{\prime}k_{x}\sigma^{x}+t_{2}^{\prime}k_{y}\sigma^{y}$
$\displaystyle-t_{2}\sin k_{02}\bar{k}_{z,2}\sigma^{z})$

In this section, we will only consider cases where $\gamma_{1}\neq\gamma_{2}$.
To investigate the response to external fields for the MWSM$||$, we consider
the effect of magnetic field along the Weyl node axis, i.e.,
$\mathbf{B}=B\hat{\mathbf{z}}$. We use the exact Peierls substitution in Eqn.
S18, so that the two expressions above transform as follows,

$\begin{split}H^{c}_{||,1}(\bm{k}^{\prime})=&t_{2}(\gamma_{1}-\gamma_{2})(t_{1}\sin
k_{01}\bar{k}_{z,1}\tau^{z}\\\
&+t_{1}^{\prime}\sqrt{2eB}(a\tau^{+}+a^{\dagger}\tau^{-}))\sigma^{z},\\\
H^{c}_{||,2}(\bm{k}^{\prime})=&t_{1}(\gamma_{1}-\gamma_{2})\tau^{z}(-t_{2}\sin
k_{01}\bar{k}_{z,2}\sigma^{z}\\\
&-t_{2}^{\prime}\sqrt{2eB}(a\sigma^{-}+a^{\dagger}\sigma^{+})).\end{split}$
(32)

Here $\tau^{\pm}=\frac{1}{2}(\tau^{x}\pm i\tau^{y})$ and
$\sigma^{\pm}=\frac{1}{2}(\sigma^{x}\pm i\sigma^{y})$ are the pseudo-spin
ladder operators in the $\tau$ and $\sigma$ spaces. The lowest Landau levels
from the above two expressions are given below,

$\begin{split}H^{c}_{||,1}\rightarrow&E_{1,LLL}=\pm(\gamma_{1}-\gamma_{2})t_{1}t_{2}\sin
k_{01}\bar{k}_{z,1},\\\ &\left|\psi_{1,LLL}\right>=\left|0;-,\pm\right>,\\\
H^{c}_{||,2}\rightarrow&E_{2,LLL}=\mp(\gamma_{1}-\gamma_{2})t_{2}t_{2}\sin
k_{02}\bar{k}_{z,2},\\\
&\left|\psi_{2,LLL}\right>=\left|0;\pm,+\right>.\end{split}$ (33)

One may notice that the eigenvector $\left|0;-,+\right>$ occurs in the
vicinity of each node. Therefore, we calculate its energy eigenvalue if one
expands the MWSM parallel system in the vicinity of the $k_{z}$ axis. The
details of the calculation can be found in the Supplementary Materials S3. We
find the energy is given as,

$E_{\left|0;-,+\right>}=(Q_{1}Q_{2}+\frac{1}{2}eB(Q_{1}+Q_{2})).$ (34)

We show that this expression is consistent with the numerically calculated
Landau levels in Fig. 10. The other chiral Landau level consistent with the
other two eigenvectors, $\left|0;-,-\right>$ and $\left|0;+,+\right>$ near
their respective Weyl nodes appears distinct from $\left|0;-,+\right>$ away
from the Weyl nodes.

Figure 10: The Landau Levels for the MWSM parallel Hamiltonian with
$\gamma_{1}=-0.5$, $\gamma_{2}=0.5$,
$t_{1}=t_{1}^{\prime}=t_{2}=t_{2}^{\prime}=1$ and $B=\frac{2\pi}{80}$. (a) and
(b) show the Landau levels for the magnetic field along the Weyl axis and
perpendicular to the Weyl axis (at Weyl node $(0,0,\frac{\pi}{3})$
respectively. The (red) bands refer to the lowest Landau levels and the
(black) bands form the bulk Landau levels.

In Fig. 10, for certain values of $\gamma_{1}$ and $\gamma_{2}$, it appears,
at first glance, as if there are two separate, chiral Landau levels
corresponding to $\left|0;-,-\right>$ and $\left|1;,-,-\right>$ respectively.
All four Weyl nodes are connected by each of these LLLs, however, and the two
LLLs in combination furthermore account for each chirality at each node.
Although this is reminiscent of the Dirac semimetal, there is potentially a
distinction in character between the chiralities at each node. If each parent
corresponds to a particular degree of freedom, for instance, and these dofs
are physically distinct from one another in some sense, such as one parent
corresponding to a two-fold valley dof, and the other corresponding to a two-
fold layer dof, the chiral anomalies are inequivalent and do not compensate
one another as they would for a Dirac semimetal.

The two apparently ’separated’ LLLs seem to only scatter between the Weyl
nodes derived from their respective parents, i.e. intra-parent scattering.
Upon closer inspection, however, we see the intersection point between two
apparently separated Landau levels is actually a very small gap. We have
verified in Supplementary Sec. S3, that the gap is finite in analytical
calculations performed to second order in momenta. The gap is an emergent
feature of the multiplicative chiral anomaly, with the single LLL reducing to
$\left|0;-,-\right>$ and $\left|0;+,+\right>$ at nodes associated with a
particular parent. We therefore interpret the multiplicative chiral anomaly as
exhibiting parent-graded features as well as emergent features not associated
with either individual parent. This is reminiscent of the topologically robust
floating bands of the multiplicative Chern insulator Cook and Moore (2022).

#### V.2.2 Landau levels in the MWSM perpendicular system:

In Sec. III.2, we had shown the linear expansion of the MWSM$\perp$ Bloch
Hamiltonian near each of the nodes corresponding to one parent with Weyl nodes
separated along the $k_{y}$ axis and the other parent with Weyl nodes
separated along the $k_{z}$ axis in Eqn. 18 and 19. Without loss of
generality, we consider, $t_{31}=t_{32}=t_{21}=t_{22}=1=t_{11}=t_{12}$. There
exists three separate cases one needs to check - (i) magnetic field along the
Weyl axis of the first parent, $\mathbf{B}=B\hat{y}$, (ii) magnetic field
along the Weyl axis of the second parent, $\mathbf{B}=B\hat{\mathbf{z}}$, and
(iii) magnetic field perpendicular to the Weyl axis of both parents,
$\mathbf{B}=B\hat{\mathbf{x}}$.

  * •

Case 1 ($\mathbf{B}=B\hat{\mathbf{y}}$) : Substituting, $k_{x}\rightarrow
k_{x}^{\prime}=k_{x}+eBz$, and using the bosonic ladder operators,
$a_{\perp,y}=\frac{k_{z}-ik_{x}^{\prime}}{\sqrt{2eB}}$,
$a^{\dagger}_{\perp,y}=\frac{k_{z}+ik_{x}^{\prime}}{\sqrt{2eB}}$, we have,
from Eqn. 18,

$\begin{split}H_{\perp,1}(\bm{k}^{\prime})=&(\sin
k_{0,1}(k_{y}-k_{0,1})\tau^{z}+\sqrt{2eB}(a_{\perp,y}\tau^{+}+a_{\perp,y}^{\dagger}\tau^{-})\\\
&\otimes(\sin
k_{0,1}\sigma^{y}+(\gamma_{1}-\gamma_{2})\sigma^{z}).\end{split}$ (35)

For the expression from Eqn. 19, we instead consider the following bosonic
ladder operators,
$\tilde{a}_{\perp,y}=\frac{\tilde{k_{z}}-ik_{x}^{\prime}}{\sqrt{2eB\sin
k_{0,2}}}$ and
$\tilde{a}_{\perp,y}^{\dagger}=\frac{\tilde{k_{z}}+ik_{x}^{\prime}}{\sqrt{2eB\sin
k_{0,2}}}$, which gives us,

$\begin{split}H_{\perp,2}(\bm{k}^{\prime})=&(\sin
k_{0,2}\tau^{y}+(\gamma_{1}-\gamma_{2})\tau^{z})\\\
&\otimes(k_{y}\sigma^{y}-\sqrt{2eB\sin
k_{0,2}}(\tilde{a}_{\perp,y}\sigma^{+}_{y}+\tilde{a}_{\perp,y}^{\dagger}\sigma^{-}_{y})).\end{split}$
(36)

It is easy to find the lowest Landau level energies in the vicinity of each
node. From Eqn. 35 and 36, we respectively have the LLL energies,

$\begin{split}E_{y,1,LLL}=&\pm\sqrt{\sin^{2}k_{0,1}+(\gamma_{1}-\gamma_{2})^{2}}\sin
k_{0,1}(k_{y}-k_{0,1}),\\\
E_{y,2,LLL}=&\pm\sqrt{\sin^{2}k_{0,2}+(\gamma_{1}-\gamma_{2})^{2}}k_{y}.\end{split}$
(37)

We then find two $k_{y}$-dependent chiral LLLs connecting the nodes of the
first parent, while we have two chiral LLLs at $k_{y}=0$ due to the second
parent, as shown in Fig. 11 (a). The following result was expected if one
considers the Landau levels for the parents for different directions of the
magnetic field discussed in the previous subsection. For the MWSM
perpendicular case, the incident magnetic field in this case is both parallel
to the Weyl axis of parent 1 and perpendicular to the Weyl axis of parent 2,
so that we get both kinds of Landau levels simultaneously.

  * •

Case 2 ($\mathbf{B}=B\hat{z}$): This produces results similar to Case 1, as
shown in Fig. 11 (b). A similar calculation gives us the lowest Landau level
energies,

$\begin{split}E_{z,1,LLL}=&\pm\sqrt{\sin^{2}k_{0,1}+(\gamma_{1}-\gamma_{2})^{2}}k_{z},\\\
E_{z,2,LLL}=&\pm\sqrt{\sin^{2}k_{0,2}+(\gamma_{1}-\gamma_{2})^{2}}\sin
k_{0,2}(k_{z}-k_{0,2}).\end{split}$ (38)

Figure 11: Landau levels for the MWSM perpendicular system with
$\gamma_{1}=-0.5$ and $\gamma_{2}=0.5$ representing separation of Weyl nodes
along the $k_{y}$ and $k_{z}$ direction respectively. We show two cases, (a)
when magnetic field is along the y-direction and (b) when magnetic field is
along the z-direction. Red lines indicate the chiral Landau levels. since the
magnetic field is paralle to one Weyl node separation and perpendicular to
another Weyl node separation, the above behaviour is expected.

## VI Discussion and Conclusion

In this work, we have introduced the previously-unidentified multiplicative
topological semimetal phases of matter, distinguished by Bloch Hamiltonians
with a symmetry-protected tensor product structure. Parent Bloch Hamiltonians,
with either one or both of the parents being topologically non-trivial, may
then be combined in the tensor product to realize multiplicative topological
semimetal phases inheriting topology from the parent states.

We consider foundational examples of multiplicative topological semimetals,
with Bloch Hamiltonians constructed as tensor products of two-band Bloch
Hamiltonians, each characterizing a Weyl semimetal phase. These multiplicative
topological semimetal phases are protected by a combination of symmetries of
class DIII at the level of the child, and each parent Bloch Hamiltonian in
class D. Given the great variety of exotic crystalline point group symmetries
considered to protect most recently-identified topological semimetal phases,
it is remarkable that the symmetry-protection of these multiplicative
semimetal phases is relatively simple, and suggests many additional
multiplicative semimetal phases may be identified by enforcing these many
other symmetries on parent Bloch Hamiltonians.

We first characterize multiplicative topological semimetal phases in the bulk,
showing the bulk spectrum of the child Bloch Hamiltonian depends in a
multiplicative way on the spectra of the parent Bloch Hamiltonians: each
eigenvalue of the child, at a given point in k-space, is a product of
eigenvalues, one from each parent. We furthermore consider two different
constructions of the multiplicative Weyl semimetal, either for the case of
each parent having a pair of Weyl nodes separated along the same axis in
$\bm{k}$-space (parallel construction), or along perpendicular axes in
$\bm{k}$-space (perpendicular construction). For either construction, the
multiplicative symmetry-protected structure can then naturally yield nodal
degeneracies reminiscent of Dirac nodes or higher-charge Weyl nodes. However,
the multiplicative degeneracies are distinguished from these more familiar
quasiparticles by distinctive Wannier spectra signatures in the bulk, and
exotic bulk-boundary correspondence. Importantly, bulk characterization by
Wannier spectra reveals a complex dependence of Berry connection in the child
Bloch Hamiltonian on Berry connection of each parent Bloch Hamiltonian,
depending on whether the parents are constructed with Weyl nodes separated
along the same axis in momentum-space (parallel) or not (perpendicular).
Additionally, the connectivity of Fermi arc surface states for the
multiplicative Weyl semimetal is far more complex than in standard Dirac or
Weyl semimetals, reflecting the underlying dependence of the child topology on
the topology of the parents. An especially interesting example is the
realization of topologically-protected—yet floating—boundary states.

Response signatures of the multiplicative Weyl semimetal also inherit response
signatures of the parents, with the potential for emergent phenomena beyond
that of either parent individually. Here, we consider the multiplicative
analog of one of the defining response signatures of the Weyl semimetal, the
chiral anomaly, finding instead multiple co-existing chiral anomalies graded
by the parent degrees of freedom, as well as emergent features in the Landau
level structure not inherited from a particular parent. In the case of parents
corresponding to effectively the same degree of freedom, the response reduces
to a signature reminiscent of a Dirac semimetal. This brings up the
possibility of controlled manipulation of particular properties of an
electronic system similar to spintronics.

Future work will characterize other signatures of multiplicative topological
semimetals anticipated given the extensive characterization of Weyl and Dirac
semimetals, particularly optical and non-linear responses given the tremendous
interest in the bulk photovoltaic effect in Weyl semimetals, as well as
symmetry-protection of more exotic topological quasiparticles, such as
multiplicative generalizations of multifold fermions or nodal lines. Given the
immense body of work on topological semimetals and the surprising consequences
of multiplicative topology for bulk-boundary correspondence, nodal band
structure, and Berry phase structure, our introduction of previously-
unidentified multiplicative topological semimetals into the literature lays
the foundation for considerable future theoretical and experimental study,
which will greatly expand and deepen our understanding of topological
semimetal phases.

Acknowledgements \- We gratefully acknowledge helpful discussions with J. E.
Moore, I. A. Day, D. Varjas and R. Calderon.

Correspondence \- Correspondence and requests for materials should be
addressed to A.M.C. (email: cooka@pks.mpg.de).

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Supplemental material for “Multiplicative topological semimetals”
Adipta Pal1,2, Joe H. Winter1,2,3, and Ashley M. Cook1,2,∗

1Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Strasse 40,
01187 Dresden, Germany
2Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse
38, 01187 Dresden, Germany
3SUPA, School of Physics and Astronomy, University of St. Andrews, North
Haugh, St. Andrews KY16 9SS, UK

∗Electronic address<EMAIL_ADDRESS>
## S1 Wilson loops for multiplicative Weyl semi-metal:

Labelling the parent Hamiltonians as
$\mathcal{H}_{p1}=(d_{1x},d_{1y},d_{1z})\cdot\bm{\tau}$ and
$\mathcal{H}_{p2}=(d_{2x},d_{2y},d_{2z})\cdot\bm{\sigma}$, with eigenvectors,
$\\{\left|+_{1}\right>\left|-_{1}\right>\\}$ and
$\\{\left|+_{2}\right>,\left|-_{2}\right>\\}$ respectively, the child
Hamiltonian is given by
$\mathcal{H}_{c}=\mathcal{H}_{p1}\otimes\mathcal{H}_{p2}^{\prime}$, where
$\mathcal{H}_{p2}^{\prime}=(-d_{2x},d_{2y},-d_{2z})\cdot\bm{\sigma}$. The
ground state subspace of the child Hamiltonian is then spanned by,
$\\{\left|+_{1}\right>\left|-_{2}\right>^{\prime},\left|-_{1}\right>\left|+_{2}\right>^{\prime}\\}$
$=\\{\left|+_{1}\right>\overline{\left|+_{2}\right>},\left|-_{1}\right>\overline{\left|-_{2}\right>}\\}$,
where $\overline{\left|\psi\right>}$ denotes complex conjugation and ’ denotes
an eigenstate of $\mathcal{H}_{p2}^{\prime}$. The non-abelian Berry connection
is then given as follows:

$\begin{split}A_{\mu}=&i\begin{pmatrix}\left<+_{1},\overline{+}_{2}\right|\partial_{\mu}\left|+_{1},\overline{+}_{2}\right>&\\\
&\left<-_{1},\overline{-}_{2}\right|\partial_{\mu}\left|-_{1},\overline{-}_{2}\right>\\\
\end{pmatrix}=i\begin{pmatrix}\left<+_{1}\right|\partial_{\mu}\left|+_{1}\right>&\\\
&\left<-_{1}\right|\partial_{\mu}\left|-_{1}\right>\\\
\end{pmatrix}+i\begin{pmatrix}\overline{\left<+_{2}\right|}\partial_{\mu}\overline{\left|+_{2}\right>}&\\\
&\overline{\left<-_{2}\right|}\partial_{\mu}\overline{\left|-_{2}\right>}\\\
\end{pmatrix},\\\ =&\begin{pmatrix}A_{1,\mu}^{+}-A_{2,\mu}^{+}&\\\
&A_{1,\mu}^{-}-A_{2,\mu}^{-}\\\ \end{pmatrix},\end{split}$ (S1)

where $A_{j,\mu}^{l}=i\left<l_{j}\right|\partial_{\mu}\left|l_{j}\right>$. For
Berry connection around a loop in the Brillouin zone, the values of $\mu$ are
$\\{k_{x},k_{y},k_{z}\\}$ for a 3d Brillouin Zone. This clearly shows the
difference between the parallel multiplicative phases and the perpendicular
multiplicative phases. For a 1d BZ, as shown in past work Cook and Moore
(2022), the connection for parallel MKC is
$\mathbf{A}=(A_{1,k_{x}}-A_{2,k_{x}},0,0)$, while for the perpendicular MKC it
is, $\mathbf{A}=(A_{1,k_{x}},-A_{2,k_{y}},0)$. For 2d or 3d parent systems, it
then becomes very straightforward to extrapolate this trend such that the
Berry connection looks qualitatively like the combination of the parallel and
perpendicular MKC connections based on which directions the parents have in
common. This is particularly interesting for the case of parallel and
perpendicular Multiplicative Chern Insulators(MCIs), where parent CIs are each
defined over a 2d BZ, and the parents can share one or two axes. We illustrate
the MCI parallel with two parent CIs on the x-y plane. The resultant Berry
connection is then
$\mathbf{A}=(A_{1,k_{x}}-A_{2,k_{x}},A_{1,k_{y}}-A_{2,k_{y}},0)$. The MCI
perpendicular on the other hand is constructed with one parent in the x-y
plane and another in the x-z plane. The resulting Berry connection is then,
$\mathbf{A}=(A_{1,k_{x}}-A_{2,k_{x}},A_{1,k_{y}},-A_{2,k_{z}})$. The MWSM, on
the other hand, is a 3d system, so we instead consider parent Weyl nodes
separated along parallel or perpendicular axes in k-space. As explained in the
main text, the parallel MWSM has parent Weyl nodes separated along the same
axis in k-space (the $k_{z}$ axis) while the perpendicular MWSM has parent $1$
and parent $2$ Weyl nodes separated along the $k_{y}$-axis and $k_{z}$-axis,
respectively. The resultant Berry connection is then,
$\mathbf{A}=(A_{1,k_{x}}-A_{2,k_{x}},A_{1,k_{y}}-A_{2,k_{y}},A_{1,k_{z}}-A_{2,k_{z}})$.

## S2 Calculation for the surface state spectrum of MWSM:

We write down here the derivation for the surface state energy for the MWSM
parallel and MWSM perpendicular Hamiltonians, for the case of open boundary
conditions in the $\hat{x}$ direction and periodic boundary conditions in the
$\hat{y}$ and $\hat{z}$ directions. First, we briefly specify how such a
calculation should be done for the two band Weyl semi-metal.

### S2.1 Slab spectra for WSM:

We start by writing down the WSM Hamiltonian used,

$\begin{split}\mathcal{H}_{WSM}(\bm{k})=&t_{3}(2+\gamma-\cos k_{x}-\cos
k_{y}-\cos k_{z})\sigma^{z}+t_{2}\sin k_{y}\sigma^{y}+t_{1}\sin
k_{x}\sigma^{x},\\\ =&t_{3}(f-\cos k_{x})\sigma^{z}+t_{2}\sin
k_{y}\sigma^{y}+t_{1}\sin k_{x}\sigma^{x},\end{split}$ (S2)

where $f=2+\gamma-\cos k_{y}-\cos k_{z}$. Surface states decay into the bulk,
so for open boundaries in the x-direction, we carry out the transformation,
$k_{x}\rightarrow iq$ for edge states on the left side $(x=0)$, so that,

$\mathcal{H}_{WSM}(iq,k_{y},k_{z})=t_{3}(f-\cosh q)\sigma^{z}+t_{2}\sin
k_{y}\sigma^{y}+it_{1}\sinh q\sigma^{x}.$ (S3)

We claim that the determinant derived from the matrix due to the following
limit must be zero,

$\lim_{q_{1}\rightarrow
q_{2}}\frac{\mathcal{H}(iq_{1})-\mathcal{H}(iq_{2})}{2\sinh q_{-}}=-t_{3}\sinh
q_{+}\sigma^{z}+it_{1}\cosh q_{+}\sigma^{x},$ (S4)

where $q_{\pm}=\frac{1}{2}(q_{1}\pm q_{2})$. Carrying out the determinant, we
get the following two conditions,

$t_{3}\sinh q_{+}=\pm t_{1}\cosh q_{+}.$ (S5)

Choosing the $+$ sign, the RHS in Eqn. S4 becomes, $-t_{1}\cosh
q_{+}(\sigma^{z}-i\sigma^{x})$, so that the null eigenvector derived from it
is one of the eigenvectors for the surface spectra,

$\left|\psi_{+}\right>=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ i\end{pmatrix}.$
(S6)

The energy corresponding to this eigenvector can be found by solving the
eigenvalue for the RHS in Eqn. S3 with the above eigenvector. This gives us
the eigen-energy, $E$, and the equation to determine the eigen-function, for
the left boundary

$E=t_{2}\sin k_{y},$ (S7a)
$\begin{split}&(t_{3}+t_{1})e^{-2q}+2fe^{-q}+(t_{3}-t_{1})=0,\\\
\implies&e^{-q_{\pm}}=\frac{-f\pm\sqrt{f^{2}-(t_{3}^{2}-t_{1}^{2})}}{(t_{3}+t_{1})},\\\
&\Psi_{+}(x,y,z)\sim(e^{-q_{+}x}-e^{-q_{-}x})e^{ik_{y}y+ik_{z}z}\left|\psi_{+}\right>.\end{split}$
(S7b)

The eigen-function in the last line has the following form based on the
boundary condition on the left edge, $\Psi(x=0)=0$. The other edge can be
derived similarly by shifting $x\rightarrow L+1-x$ where L is the length of
the system along the x-direction.

### S2.2 Slab spectra for MWSM parallel:

We use the same method as in section S2A of the supplementary materials to
derive surface states and spectra for the MWSM parallel system. The
Hamiltonian is given as follows,

$\begin{split}\mathcal{H}_{MWSM||}(\bm{k})=[t_{31}(f_{1}-\cos
k_{x})\tau^{z}+t_{21}\sin k_{y}\tau^{y}+t_{11}\sin
k_{x}\tau^{x}]\otimes[-t_{32}(f_{2}-\cos k_{x})\sigma^{z}+t_{22}\sin
k_{y}\sigma^{y}-t_{12}\sin k_{x}\sigma^{x}],\end{split}$ (S8)

where $f_{1/2}=2+\gamma_{1/2}-\cos k_{y}-\cos k_{z}$. To ease our
calculations, we carry out the following rotation on the four band basis,
$\tau^{z}\rightarrow\tau^{y}$, $\tau^{y}\rightarrow-\tau^{z}$ and
$\sigma^{z}\rightarrow-\sigma^{y}$, $\sigma^{y}\rightarrow-\sigma^{z}$. The
Hamiltonian then becomes,

$\begin{split}\mathcal{H}_{MWSM||}(\bm{k})=&[t_{31}(f_{1}-\cos
k_{x})\tau^{y}-t_{21}\sin k_{y}\tau^{z}+t_{11}\sin
k_{x}\tau^{x}]\otimes[-t_{32}(f_{2}-\cos k_{x})\sigma^{y}-t_{22}\sin
k_{y}\sigma^{z}-t_{12}\sin k_{x}\sigma^{x}],\\\ =&[t_{31}(f_{1}-\cos
k_{x})\tau^{y}+t_{11}\sin k_{x}\tau^{x}][-t_{32}(f_{2}-\cos
k_{x})\sigma^{y}-t_{12}\sin k_{x}\sigma^{x}]\\\ &-t_{21}\sin
k_{y}\tau^{z}[-t_{32}(f_{2}-\cos k_{x})\sigma^{y}-t_{12}\sin
k_{x}\sigma^{x}]-t_{22}\sin k_{y}[t_{31}(f_{1}-\cos k_{x})\tau^{y}+t_{11}\sin
k_{x}\tau^{x}]\sigma^{z}\\\
&+t_{21}t_{22}\sin^{2}k_{y}\tau^{z}\sigma^{z}.\end{split}$ (S9)

Again, without loss of generality, we set
$t_{11}=t_{21}=t_{31}=1=t_{32}=t_{22}=t_{12}$. The edge modes on the left edge
$(x=0)$, require we carry out the substitution, $k_{x}\rightarrow iq$, and the
Hamiltonian is now,

$\begin{split}\mathcal{H}_{MWSM||}(iq,k_{y},k_{z})=&[(f_{1}-\cosh
q)\tau^{y}+i\sinh q\tau^{x}][-(f_{2}-\cosh q)\sigma^{y}-i\sinh q\sigma^{x}]\\\
&-\sin k_{y}\tau^{z}[-(f_{2}-\cosh q)\sigma^{y}-i\sinh q\sigma^{x}]-\sin
k_{y}[(f_{1}-\cosh q)\tau^{y}+i\sinh q\tau^{x}]\sigma^{z}\\\
&+\sin^{2}k_{y}\tau^{z}\sigma^{z}.\end{split}$ (S10)

Carrying out our previous limit on the rotated Hamiltonian above, we get the
following matrix,

$\begin{split}\lim_{q_{1}\rightarrow
q_{2}}\frac{\mathcal{H}_{MWSM||}(iq_{1})-\mathcal{H}_{MWSM||}(iq_{2})}{2\sinh
q_{-}}=\begin{pmatrix}0&i\sin k_{y}S_{+}&-i\sin
k_{y}S_{+}&S_{+}(-(f_{1}+f_{2})+2S_{+})\\\ i\sin
k_{y}S_{-}&0&-f_{1}S_{-}+f_{2}S_{+}&i\sin k_{y}S_{+}\\\ -i\sin
k_{y}S_{-}&f_{1}S_{+}-f_{2}S_{-}&0&-i\sin k_{y}S_{+}\\\
S_{-}((f_{1}+f_{2})-2S_{-})&i\sin k_{y}S_{-}&-i\sin k_{y}S_{-}&0\\\
\end{pmatrix},\end{split}$ (S11)

where $S_{\pm}=\cosh q_{+}\pm\sinh q_{+}$. The determinant of the RHS of Eqn.
S11 must be zero, i.e., we have the condition,

$\begin{split}&S_{-}S_{+}[\sin^{2}k_{y}S_{-}(f_{1}+f_{2}-2S_{+})(f_{1}-f_{2})(S_{-}+S_{+})-\sin^{2}k_{y}S_{+}(f_{1}+f_{2}-2S_{-})(f_{1}+f_{2})(S_{+}-S_{-})\\\
&-(f_{1}+f_{2}-2S_{+})(f_{1}+f_{2}-2S_{-})(f_{1}S_{-}-f_{2}S_{+})(-f_{2}S_{-}+f_{1}S_{+})]=0\end{split}$
(S12)

Let us start with the first condition, $S_{-}=0$. The RHS of Eqn. S11 then
becomes,

$\begin{split}\lim_{q_{1}\rightarrow
q_{2}}\frac{\mathcal{H}_{MWSM||}(iq_{1})-\mathcal{H}_{MWSM||}(iq_{2})}{2\sinh
q_{-}}=&S_{+}(-(f_{1}+f_{2})+2S_{+})\tau^{+}\sigma^{+}+f_{2}S_{+}\tau^{+}\sigma^{-}+f_{1}S_{+}\tau^{-}\sigma^{+}\\\
&+i\sin k_{y}S_{+}\tau^{z}\sigma^{+}-i\sin
k_{y}S_{+}\tau^{+}\sigma^{z},\end{split}$ (S13)

where $\tau^{\pm}=\frac{1}{2}(\tau^{x}\pm i\tau^{y})$ and
$\sigma^{\pm}=\frac{1}{2}(\sigma^{x}\pm i\sigma^{y})$ are the two level ladder
operators. Here, if $\\{\left|+\right>,\left|-\right>\\}$ are eigen-vectors of
$\tau^{z}$, then $\tau^{+}\left|-\right>=\left|+\right>$,
$\tau^{+}\left|+\right>=0$, $\tau^{-}\left|-\right>=0$ and
$\tau^{-}\left|+\right>=\left|-\right>$. Similar relations exist for the
$\sigma$ counterpart.
$\left|\psi_{1}\right>=\left|+\right>\otimes\left|+\right>$ is a null eigen-
vector to the above expression on the RHS. We solve for the energy eigenvalue
first for the special case $\gamma_{1}=\gamma_{2}$.Then from
$\mathcal{H}_{MWSM||}(iq,ky,kz)$ in Eqn. S10 due to the eigen-vector
$\left|\psi_{1}\right>$, we have the energy and the condition,

$E=\sin^{2}k_{y};$ (S14a) $(f_{1}-\cosh q+\sinh q)(f_{2}-\cosh q+\sinh q)=0.$
(S14b)

## S3 Landau Level repulsion in the MWSM parallel system:

We start with the MWSM parallel case,

$\begin{split}H_{MWSM,||}(\bm{k})=&[t_{1}(2+\gamma_{1}-\cos k_{x}-\cos
k_{y}-\cos k_{z})\tau^{z}+t_{1}^{\prime}\sin k_{y}\tau^{y}+t_{1}^{\prime}\sin
k_{x}\tau^{x}]\\\ &\otimes[-t_{2}(2+\gamma_{2}-\cos k_{x}\cos k_{y}-\cos
k_{z})\sigma^{z}+t_{2}^{\prime}\sin k_{y}\sigma^{y}-t_{2}^{\prime}\sin
k_{x}\sigma^{x}].\end{split}$ (S15)

We expand the Bloch Hamiltonian near the z-axis i.e.
$\bm{k}\rightarrow(0,0,k_{z})$,

$\begin{split}H_{MWSM,||}(\bm{k})\approx&[t_{1}(Q_{1}+\frac{1}{2}(k_{x}^{2}+k_{y}^{2}))\tau^{z}+t_{1}^{\prime}k_{y}\tau^{y}+t_{1}^{\prime}k_{x}\tau^{x}]\\\
&\otimes[-t_{2}(Q_{2}+\frac{1}{2}(k_{x}^{2}+k_{y}^{2}))\sigma^{z}+t_{2}^{\prime}k_{y}\sigma^{y}-t_{2}^{\prime}k_{x}\sigma^{x}],\end{split}$
(S16)

where $Q_{i}=\gamma_{i}-\cos k_{z}$ (i=1,2). Expanding only up to second order
in momenta, we have,

$\begin{split}H_{MWSM,||}(\bm{k})\approx&-t_{1}t_{2}(Q_{1}Q_{2}+(Q_{1}+Q_{2})\frac{1}{2}(k_{x}^{2}+k_{y}^{2}))\tau^{z}\sigma^{z}\\\
&-t_{1}t_{2}^{\prime}Q_{1}\tau^{z}(k_{x}\sigma^{x}-k_{y}\sigma^{x})-t_{1}^{\prime}t_{2}Q_{2}(k_{x}\tau^{x}+k_{y}\tau^{y})\sigma^{z}\\\
&-t_{1}^{\prime}t_{2}^{\prime}(k_{x}^{2}\tau^{x}\sigma^{x}-k_{y}^{2}\tau^{y}\sigma^{y}-\frac{1}{2}(k_{x}k_{y}+k_{y}k_{x})\tau^{x}\sigma^{y}+\frac{1}{2}(k_{x}k_{y}+k_{y}k_{x})\tau^{y}\sigma^{x}).\end{split}$
(S17)

Figure S12: Comparison of the numerically calculated Landau Levels of the
MWSM$||$. system with the analytically calculated lower Landau levels for
first order(blue dashed) and second order(red) expansion in momenta along the
direction perpendicular to $k_{z}$. Level repulsion between two parent graded
lowest Landau levels are only observed if one expands to second order in
momenta.

We consider $\mathbf{B}=B\hat{z}$. After Peierls substitution,
$k_{x}\rightarrow k_{x}^{\prime}=k_{x}$, $k_{y}\rightarrow
k_{y}^{\prime}=k_{y}+eBx$, and $k_{z}\rightarrow k_{z}^{\prime}=k_{z}$. The
position-momenta commutator leads to the commutator,
$[k_{y}^{\prime},k_{x}^{\prime}]=ieB$. Here, $e$ is the charge of the particle
in consideration. One can therefore construct bosonic ladder operators of the
form,

$a=\frac{k_{x}^{\prime}-ik_{x}^{\prime}}{\sqrt{2eB}};\quad
a^{\dagger}=\frac{k_{x}^{\prime}+ik_{y}^{\prime}}{\sqrt{2eB}};\quad[a,a^{\dagger}]=1.$
(S18)

We calculate some important identities via Eqn.S18 which we will be using in
the next few lines,

$\begin{split}&\frac{1}{2}({k_{x}^{\prime}}^{2}+{k_{y}^{\prime}}^{2})=eB(a^{\dagger}a+\frac{1}{2});\quad
k_{x}^{\prime}\sigma^{x}+k_{y}^{\prime}\sigma^{y}=\sqrt{2eB}(a\sigma^{+}+a^{\dagger}\sigma^{-});\quad
k_{x}^{\prime}\sigma^{x}-k_{y}^{\prime}\sigma^{y}=\sqrt{2eB}(a\sigma^{-}+a^{\dagger}\sigma^{+}),\\\
&{k_{x}^{\prime}}^{2}-{k_{y}^{\prime}}^{2}=eB(a^{2}+{a^{\dagger}}^{2});\quad
i[k_{x}^{\prime}k_{y}^{\prime}+k_{y}^{\prime}k_{x}^{\prime}]=eB({a^{\dagger}}^{2}-a^{2}),\end{split}$
(S19)

where we have used $\tau^{\pm}=\frac{1}{2}(\tau^{x}\pm i\tau^{y})$ and
$\sigma^{\pm}=\frac{1}{2}(\sigma^{x}\pm i\sigma^{y})$, which are spin ladder
operators in the basis $\\{\left|+\right>,\left|-\right>\\}$ in both the
$\tau$ and $\sigma$ spaces. Now, substituting $\bm{k}$ for $\bm{k^{\prime}}$
in Eqn. S17 and then transforming them via Eqn. S19, we get the following
expression,

$\begin{split}H_{MWSM,||}(\bm{k}^{\prime})\approx&-t_{1}t_{2}(Q_{1}Q_{2}+(Q_{1}+Q_{2})eB(a^{\dagger}a+\frac{1}{2}))\tau^{z}\sigma^{z}-t_{1}t_{2}^{\prime}Q_{1}\sqrt{2eB}\tau^{z}(a\sigma^{-}+a^{\dagger}\sigma^{+})-t_{1}^{\prime}t_{2}Q_{2}\sqrt{2eB}(a\tau^{+}+a^{\dagger}\tau^{-})\sigma^{z}\\\
&-t_{1}^{\prime}t_{2}^{\prime}(2eB)(a^{\dagger}a+\frac{1}{2})(\tau^{+}\sigma^{+}+\tau^{-}\sigma^{-})-t_{1}^{\prime}t_{2}^{\prime}(2eB)(a^{2}\tau^{+}\sigma^{-}+{a^{\dagger}}^{2}\tau^{-}\sigma^{+}).\end{split}$
(S20)

Let us ignore the second order perturbations not in the mass term (i.e.
$\tau^{z}\sigma^{z}$) and simplify the Hamiltonian,

$H_{MWSM,||}(\bm{k}^{\prime})\approx-(Q_{1}Q_{2}+(Q_{1}+Q_{2})eB(a^{\dagger}a+\frac{1}{2}))\tau^{z}\sigma^{z}-Q_{1}\sqrt{2eB}\tau^{z}(a\sigma^{-}+a^{\dagger}\sigma^{+})-Q_{2}\sqrt{2eB}(a\tau^{+}+a^{\dagger}\tau^{-})\sigma^{z}.$
(S21)

We obtain one of the lowest Landau levels,
$\left|\psi\right>_{1,LLL}=\left|0;-,+\right>$ with energy
$E_{1,LLL}=(Q_{1}Q_{2}+\frac{eB}{2}(Q_{1}+Q_{2}))$ which match exactly both
numerically and analytically in first and second order expansions. For the
other lowest Landau level, we observe an amalgamation of chiral Landau levels
obtained from each parent which cause level repulsion at the intersection
point.

## S4 Euler space topology calculation

In the main text, we have already reported that the MWSM system possesses both
time reversal, $\mathcal{T}$ and inversion symmetry, $\mathcal{I}$ and hence
the combined symmetry, $\mathcal{T}^{\prime}$ denoted by
$\tau^{y}\sigma^{y}\kappa$, where $\kappa$ refers to complex conjugation.
However, here ${\mathcal{T}^{\prime}}^{2}=1$, so that a $\mathbb{Z}_{2}$
invariant is not possible. Instead, it is possible to find a basis, where
$\mathcal{T}^{\prime}=\kappa$. Here we provide the unitary transformation
which makes this possible,

$V=\frac{1}{2}[(1+i)\tau^{0}\sigma^{0}+(1-i)\tau^{y}\sigma^{y}].$ (S22)

Based on the method provided in the appendix in a previous work Guan  _et al._
(2022), the above unitary transformation satisfies,
$V\tau^{y}\sigma^{y}V^{T}=1$, so that we get a Hamiltonian,
$\tilde{H}(\bm{k})=VH(\bm{k})V^{\dagger}$ which satisfies,
$\tilde{H}^{(}\bm{k})=\tilde{H}^{*}(\bm{k})$, and is real and symmetric.
Denoting the MWSM in a condensed notation,

$H=(M_{1}\tau^{z}+Q_{1}\tau^{x}+R_{1}\tau^{y})\otimes(-M_{2}\sigma^{z}-Q_{2}\sigma^{x}+R_{2}\sigma^{y}),$
(S23)

we obtain after the transformation,

$\begin{split}\tilde{H}=\quad&M_{1}(-M_{2}\tau^{z}\sigma^{z}-Q_{2}\tau^{z}\sigma^{x}+R_{2}\tau^{x}\sigma^{0})\\\
&-Q_{1}(M_{2}\tau^{x}\sigma^{z}+Q_{2}\tau^{x}\sigma^{x}+R_{2}\tau^{z}\sigma^{0})\\\
&-R_{1}(M_{2}\tau^{0}\sigma^{x}-Q_{2}\tau^{0}\sigma^{z}-R_{2}\tau^{y}\sigma^{y}).\end{split}$
(S24)

Comparing with the method introduced in Guan  _et al._ (2022), it is possible
to view the real Hamiltonian as an element of a Real oriented Grassmannian,
$\tilde{\mathbb{G}}_{2,4}^{\mathbb{R}}$ which is diffeomorphic to $S^{2}\times
S^{2}$. For a given $k_{z}$, then it is possible to define a mapping from the
2d BZ spanned by $k_{x}$ and $k_{y}$ (for MWSM $||$) into
$(\mathbf{n_{1}},\mathbf{n}_{2})\in S^{2}\times S^{2}$ and the topology of
$\tilde{H}$ is then determined by the two skyrmion numbers,
$Q[\mathbf{n_{1}}]=q_{1}$ and $Q[\mathbf{n_{2}}]=q_{2}$ of parent 1 and parent
2, respectively. The Euler class topology is then found from these skyrmion
numbers as follows,

$E_{I}=q_{2}-q_{1};\quad E_{II}=q_{2}+q_{1}.$ (S25)

The Euler numbers are unique up to the mapping
$(E_{I},E_{II})\rightarrow(-E_{I},-E_{II})$.