# Towards a theory of types III and IV non-Hermitian Weyl fermions

Zaur Z. Alisultanov  Institute of Physics of DFRS, Russian Academy of
Sciences, Makhachkala, Russia  Edvin G. Idrisov  Department of Physics and
Materials Science, University of Luxembourg, Luxembourg

###### Abstract

We develop the non-Hermitian Hamiltonian formalism to describe Weyl fermions
of type III and IV. The spectrum of Hamiltonian has an unusual type of
anisotropy. Namely, the hermiticity of Hamiltonian strongly depends on the
direction in momentum space: for some directions the spectrum is real, in
contrast for other directions it becomes complex. This fact leads to non-
trivial adiabatic evolution and fractional Chern number. Additionally, we
demonstrate that the non-Hermitian Hamiltonian can be regarded as a one-
particle problem in context of topological band theory.

The current theoretical progress in band theory allows to describe
topologically protected quasiparticles in modern experimentally accessible
materials [1]. Due to their unique properties, for instance chiral transport,
the topological materials are expected to be promising for future electronics
[2]. Among all, the topological materials with Dirac [3, 4, 5, 6] and Weyl [7,
8, 9, 10, 11] points, as well as degenerate points [12] and lines [13, 14, 15]
in Brillouin zone are of particular interest. The massless fermions in such
materials are topologically protected, which results in quantum-
electrodynamics effects known from high energy physics [16].

The idea of realization of Weyl points in condensed matter physics belongs
back to C. Herring [17]. According to Herring’s description, the presence of
Weyl points can be understood within two-band model. The corresponding
Hamiltonian of in momentum representation can be presented in Pauli basis

$H(\mathbf{p})=\sigma_{0}f_{0}(\mathbf{p})+\sum^{3}_{i=1}\sigma_{i}f_{i}(\mathbf{p}),$
(1)

where $\sigma_{0}$ is $2\times 2$ unit matrix, $\sigma_{i}$ are Pauli matrices
and $\mathbf{p}$ is a three-dimensional (3D) momentum vector. The Weyl point
$\mathbf{p}_{\text{W}}$ arises due to crossing of bands, which satisfies to
three algebraic equations $f_{i}(\mathbf{p}_{\text{W}})=0$. Materials with
these conditions are known as Dirac and Weyl semimetals [18]. Typically such
kind of situation is easily realizable in 3D systems, and equations
$f_{i}(\mathbf{p}_{\text{W}})=0$ represent the two-dimensional (2D) surfaces
in momentum space: three closed 2D surfaces can have many crossing points. In
2D system this condition corresponds to the crossing of three lines. This is
not trivial and in order to have the Weyl point in 2D system, the certain
types of symmetries (for example, $C_{6\nu}$ point symmetry) are required
[18]. In the vicinity of Weyl point $\mathbf{p}_{W}$ one can expand the
function $f_{i}(\mathbf{p})$ in Taylor series, which results in low-energy
linear spectrum, namely $f_{i}(\mathbf{p})\propto|\mathbf{p}|$, and
corresponding excitations are known as massless (Weyl or Dirac) fermions [18].
The Eq. (1) describes conventional (non-tilted) type I Weyl fermions for zero
and constant values of $f_{0}(\mathbf{p})$.

Apart from Weyl semimetals of type I, it was proposed Weyl semimetals of type
II [19, 20, 21, 22]. Nowadays such kind of materials are accessible
experimentally and WTe2 is one of the candidates for a realization of type II
Weyl fermions [23]. In short the Hamiltonian for such systems can be modeled
at the intersections of Fermi-pockets and the spectrum turns out to be tilted.
The minimal Hamiltonian for this case has the following form

$H(\mathbf{p})=v_{F}\boldsymbol{\sigma}\cdot\mathbf{p}+\sigma_{0}\boldsymbol{\omega}\cdot\mathbf{p},$
(2)

where $v_{F}$ is Fermi velocity and $\boldsymbol{\omega}$ is a tilt vector.
For $v_{F}>\omega$ this Hamiltonian describes the Weyl semimetals of type I
with tilted spectrum, and consequently for $v_{F}<\omega$ the Hamiltonian
corresponds to type II Weyl semimetals, where $\omega=|\boldsymbol{\omega}|$
is a modulus of vector. It is worth mentioning that Weyl semimetals of type II
can be used for ”modeling” of black and white holes, event horizon [24, 25].
Indeed, assuming that the parameter of tilting,
$\boldsymbol{\omega}(\mathbf{x})$, is a function of spatial coordinates, the
Weyl fermions of type II can be described by action of massless spinor field
[24]

$\mathcal{S}_{II}=\int
d^{4}x\left[i\overline{\psi}\left(\upsilon_{F}\gamma^{\mu}\partial_{\mu}+\gamma^{0}\boldsymbol{\omega}(\mathbf{x})\cdot\boldsymbol{\partial}\right)\psi\right],$
(3)

where $\psi$ is a Dirac spinor and $\gamma^{\mu}$ is the Dirac matrix in the
Weyl representation $\gamma^{\mu}=\left(\begin{array}[]{cc}0&\sigma^{\mu}\\\
\tilde{\sigma}^{\mu}&0\end{array}\right)$ with
$\sigma^{\mu}=\left(\sigma_{0},\bm{\sigma}\right)$ and
$\tilde{\sigma}^{\mu}=\left(\sigma_{0},-\bm{\sigma}\right)$ and
$\partial_{\mu}=(v^{-1}_{F}\partial_{t},\boldsymbol{\partial})$. This is the
possible basic (minimalistic) action for the description of massless spinor
field in curved space with metrics
$ds^{2}=\left(|\boldsymbol{\omega}|^{2}-1\right)dt^{2}-2\boldsymbol{\omega}\cdot
d\mathbf{x}dt+d\mathbf{x}\cdot d\mathbf{x}$, and allows to model phenomena in
the vicinity of event horizon.

In this Letter, we make a further step to develop the theory for Weyl fermions
of types III and IV. We show that these fermions can be described using the
Hamiltonian formalism in the framework of non-Hermitian quantum theory.
Moreover, the non-Hermitian Hamiltonian arises as a one-particle problem in
the context of band theory of Weyl semimetals. In order to study the
properties of this Hamiltonian and associated Hilbert space, the acceptable
theory of non-Hermitian systems is presented below. The spectrum of Weyl
Hamiltonian under consideration turns out to be anisotropic in momentum space,
namely for some directions the spectrum is real and for other directions it is
complex quantity. The necessary and sufficient conditions for the real
spectrum of the general Hamiltonian under consideration are formulated. The
main feature of system under consideration is that ”left” and ”right”
eigenvalues of Hamiltonian coincide with each other, which results in non-
trivial adiabatic evolution and fractional Chern number.

Weyl fermions of type III and IV. In recent seminal papers, Ref. [26, 27],
authors have suggested Weyl fermions of type III and IV. Briefly, using the
tetrad formalism, there was provided the general action of massless spinor
field of form

$\mathcal{S}=\int
d^{4}x\sqrt{-g}\left[i\overline{\psi}\gamma^{\nu}e_{\nu}^{\mu}\partial_{\mu}\psi\right],$
(4)

where $g$ is the determinant of the metric tensor and $e_{\nu}^{\mu}$ is the
tetrad tensor. One can see from Eq. (4), that Weyl fermions of type I and II
are particular cases of the general tetrad action theory. Indeed, if one keeps
only diagonal elements of tetrad tensor
$e_{\nu}^{\mu}=v_{F}\delta_{\nu}^{\mu}$, then we get the action for Weyl
femions of type I. If one set
$e_{\nu}^{\mu}=v_{F}\delta_{\nu}^{\mu}+\delta_{\nu}^{0}\omega^{i}\delta_{i}^{\mu}$,
then we get the action for type II Weyl fermions presented in Eq. (3).
However, in more general case the Eq. (4) can contain other non-zero
components of tetrad tensor, namely
$e_{\nu}^{\mu}=v_{F}\delta_{\nu}^{\mu}+\delta_{\nu}^{0}\omega^{i}\delta_{i}^{\mu}+\vartheta_{i}\delta_{\nu}^{i}\delta_{0}^{\mu}$,
where $\vartheta_{i}$ are the components of additional tilt parameter vector,
$\boldsymbol{\vartheta}$. The presence of last term restores the symmetry with
respect to rearrangement of indices $\mu$ and $\nu$. Note that this form
corresponds to a spatially isotropic tetrad when there is no
$\delta_{\nu}^{i}\delta_{j}^{\mu}$ term. According to Ref. [26], the case
$v_{F}>\omega$ and $v_{F}<\vartheta$ corresponds to Weyl fermions of type III,
and $v_{F}<\omega$ and $v_{F}<\vartheta$ is associated with type IV, where
$\vartheta=|\boldsymbol{\vartheta}|$. Due to the last term,
$\vartheta_{i}\delta_{\nu}^{i}\delta_{0}^{\mu}$, the Lagrangian contains a new
term with zero component of momentum $\sigma^{i}\vartheta_{i}p_{0}$ and
$p_{\mu}=(p_{0},p_{i})$ with $p_{0}=\varepsilon/v_{F}$. Particularly, the
possible origin of this term and consequently Weyl points of type III and IV
were associated with many-particle effects and thus can be worked out from
self-energy [26]. It is worth mentioning, that in framework of Hermitian
quantum mechanics the Hamilton formalism does not allow to take into account
this term.

Nevertheless, we insist that the Hamiltonian formalism can be constructed, but
in the framework of non-Hermitian quantum mechanics. Moreover, the non-
Hermitian Hamiltonian formalism allows the alternative explanation, which does
not necessarily include many-particle effects as the possible mechanism of
explanation. In order to introduce the Hamiltonian, we first construct the
wave equation

$G^{-1}(p_{0},p_{i})\left|\Psi\right\rangle=0,$ (5)

where for the spatially isotropic case, which contains all important physics,
the Greens function associated with action, Eq. (4) is given by

$G^{-1}\left(p_{0},p_{i}\right)=v_{F}\sigma^{0}p_{0}-v_{F}\sigma^{i}p_{i}-\omega^{i}p_{i}-\sigma^{i}\vartheta_{i}p_{0}.$
(6)

Using the above expression, the Eq. (5) can be easily rewritten in
conventional form, where the Hamiltonian at left hand side is the function of
only three dimensional momentum. After simple algebra one arrives to the
following two wave equations

$\mathcal{H}\left|\Psi^{R}\right\rangle=\varepsilon_{R}\left|\Psi^{R}\right\rangle,\quad\mathcal{H}^{\dagger}\left|\Psi^{L}\right\rangle=\varepsilon_{L}\left|\Psi^{L}\right\rangle,$
(7)

where the non-Hermitian Hamiltonian takes form

$\begin{split}&\mathcal{H}=(i\Gamma+K)/(1-\beta^{2}),\quad\Gamma=\boldsymbol{\sigma}\cdot\mathbf{\left[\mathbf{p}\times\boldsymbol{\vartheta}\right]},\\\
&K=\left[v_{F}\boldsymbol{\sigma}+\sigma_{0}(\boldsymbol{\omega}+\boldsymbol{\vartheta})\right]\cdot\mathbf{p}+\left(\boldsymbol{\sigma}\cdot\boldsymbol{\vartheta}\right)\left(\boldsymbol{\omega}\cdot\mathbf{p}\right)/v_{F},\end{split}$
(8)

and $\beta=\vartheta/v_{F}$. This is the Hamiltonian of type III and IV non-
Hermitian Weyl fermions and is our main result. This Hamiltonian can be also
obtained directly from the action, Eq. (4), if one writes the wave equation as
the Euler-Lagrange equation for the Dirac field. In Eq. (7) we have defined
the $R$ (right) and $L$ (left) wave functions. It is worth mentioning, that
use of so called dual basis $(R,L)$ is a convenient step for non-Hermitian
systems, which allows to restore the common structure of Hilbert space [28,
29, 30].

Now one can easily obtain the spectrum of non-Hermitian and conjugated
Hamiltonians from Eq. (7). The spectra are coincides with each other, namely

$\begin{split}&\varepsilon=\varepsilon_{R}=\varepsilon_{L}=\frac{\mathcal{A}(\mathbf{p})\pm\sqrt{\mathcal{B}(\mathbf{p})}}{1-\beta^{2}},\quad\mathcal{A}(\mathbf{p})=(\boldsymbol{\vartheta}+\boldsymbol{\omega})\cdot\mathbf{p},\\\
&\mathcal{B}(\mathbf{p})=[(\boldsymbol{\vartheta}+\boldsymbol{\omega})\cdot\mathbf{p}]^{2}+(1-\beta^{2})\left[v_{F}^{2}p^{2}-(\boldsymbol{\omega}\cdot\mathbf{p})^{2}\right],\end{split}$
(9)

where $p=|\mathbf{p}|$. Note, that the Hamiltonian of form
$\mathcal{H}=a\sigma_{0}+\left(\mathbf{b}+i\mathbf{c}\right)\boldsymbol{\sigma}$
has the spectrum
$\varepsilon=a\pm\sqrt{\mathbf{b}^{2}-\mathbf{c}^{2}+2i\mathbf{b}\cdot\mathbf{c}}$,
which contains exceptional point due to term $\mathbf{b}\cdot\mathbf{c}$. In
our case $\mathbf{b}$ is proportional (collinear) to $\mathbf{p}$ and
$\mathbf{c}$ is proportional to vector product
$[\mathbf{p}\times\boldsymbol{\vartheta}]$, thus $\mathbf{b}\cdot\mathbf{c}=0$
and the expression under the root always is real.

Figure 1:  Top: It is shown the segments of angles between vectors
$\bm{\vartheta}$ (red arrow) and $\mathbf{p}$ (blue arrow), where the
imaginary part of spectrum, $\textrm{Im}(\varepsilon)$ is zero (green) and
non-zero (yellow). The angle $\chi_{0}=\arcsin(1/\beta)$ corresponds to the
direction of the momentum vector relative to vector $\bm{\vartheta}$, at which
the transition from the real to complex spectrum occurs. The black dots
represent different directions of the momentum relative to vector
$\bm{\vartheta}$. Bottom: It is shown the evolution (black arrows) of the real
and imaginary parts of the spectrum Eq. (9) when the momentum direction
changes from position $1$ to position $9$ (see the upper part of the figure).
The brown and blue dots correspond to the hole and electronic states, while
the red and green dots correspond to the complex spectrum with positive and
negative imaginary parts. The dots $3$ and $7$ correspond to the transition
points from the real to complex spectrum in Eq. (9) at vanishing
$\mathcal{B}\left(\mathbf{p}\right)$.

For simplicity, we set $\boldsymbol{\omega}=0$ for further discussion. This
does not affect on outcomes, since the non-hermiticity is described by term
$i\Gamma$. Then Hamiltonian from Eq. (8) can be rewritten in the form

$\mathcal{H}=\left[\boldsymbol{\sigma}\cdot\left(v_{F}\mathbf{e}_{p}+i\vartheta\sin\chi\mathbf{e}_{\chi}\right)p+\boldsymbol{\vartheta}\cdot\mathbf{p}\right]/(1-\beta^{2}),$
(10)

where the unit vector $\mathbf{e}_{p}$ is directed along the momentum
$\mathbf{p}$ and the unit vector $\mathbf{e}_{\chi}$ is directed along the
direction of cross product $[\mathbf{p}\times\boldsymbol{\vartheta}]$. For
values $\vartheta<v_{F}$ ($\beta<1$) the parameter $\vartheta$ renormalizes
the velocity $\partial\varepsilon/\partial\boldsymbol{p}$ due to coefficient
$1/(1-\beta^{2})$ and results in slope (tilt). For $\vartheta>v_{F}$
($\beta>1$), the spectrum in Eq. (9) is real in segment
$\left|\sin\chi\right|<\beta^{-1}$. In contrast, for
$\left|\sin\chi\right|>\beta^{-1}$ the spectrum consists of complex-conjugated
branches. Therefore, we get the anisotropy with respect to spectrum’s
(Hamiltonian’s) hermiticity (see Fig. 1). The eigenstates
$\left|\Psi^{R}\right\rangle$ and $\left|\Psi^{L}\right\rangle$ are orthogonal
in the segment $\left|\sin\chi\right|>\beta^{-1}$, where $\chi$ is the angle
between vectors $\boldsymbol{\vartheta}$ and $\mathbf{p}$, i.e.
$\left\langle\Psi^{L}|\Psi^{R}\right\rangle=0$. Indeed, from Eqs. (7), one has
$\left\langle\Psi^{L}\right|\mathcal{H}\left|\Psi^{R}\right\rangle=\varepsilon_{R}\left\langle\Psi^{L}|\Psi^{R}\right\rangle=\varepsilon_{L}^{\ast}\left\langle\Psi^{L}|\Psi^{R}\right\rangle$.
Since $\varepsilon_{R}\neq\varepsilon_{L}^{\ast}$, then
$\left\langle\Psi^{L}|\Psi^{R}\right\rangle=0$. In the opposite segment
$\left|\sin\chi\right|<\beta^{-1}$, where spectrum is real, generally
speaking, these states are not orthogonal. Additionally, in Weyl point these
states are not orthogonal as well, since $\varepsilon_{R}=\varepsilon_{L}=0$.
Further, it is easy to show, that in case of real spectrum
$\left\langle\varPsi_{\pm}^{L}|\varPsi_{\mp}^{R}\right\rangle=0$ and
$\left\langle\varPsi_{\pm}^{L}||\varPsi_{\pm}^{R}\right\rangle\neq 0$, where
”$\pm$” corresponds to electron hole states in Eq. (9). At the same time, in
the segment, where spectrum is complex one has
$\left\langle\varPsi_{\pm}^{L}||\varPsi_{\pm}^{R}\right\rangle=0$ and
$\left\langle\varPsi_{\pm}^{L}|\varPsi_{\mp}^{R}\right\rangle\neq 0$. Such a
situation leads to non-trivial adiabatic evolution (see Sec. A and B in Ref.
[31]). The existence of segment with real spectrum, despite the non-Hermitian
nature of Hamiltonian, is surprising. Nevertheless, it can be proven the
following statement: If the non-Hermitian Hamiltonian, $\mathcal{H}$ can be
brought to hermitian $\mathcal{H}_{h}$ one, using the similarity
transformations with non-degenerate matrix, then the eigenvalues of
$\mathcal{H}$ are real. Indeed, let apply the similarity transformation,
$S\mathcal{H}S^{-1}$ with the linear operator $S$ in the from of non-
degenerate matrix, $(S^{-1})^{\dagger}=(S^{\dagger})^{-1}$, to Hamiltonian
$\mathcal{H}$. Next, we assume that the following equality holds

$S\mathcal{H}S^{-1}=\mathcal{H}_{h}.$ (11)

Then
$S\mathcal{H}S^{-1}\left|\Psi\right\rangle=\mathcal{H}_{h}\left|\Psi\right\rangle=\mathcal{E}\left|\Psi\right\rangle$
, where $\mathcal{E}$ is real since the Hamiltoanin $\mathcal{H}_{h}$ is
hermitian. On the other hand due to linearity of operator $S$, this equation
can be written as
$\mathcal{H}\left|\overline{\Psi}\right\rangle=\mathcal{E}\left|\overline{\Psi}\right\rangle$,
where $\left|\overline{\Psi}\right\rangle=S^{-1}\left|\Psi\right\rangle$.
Thus, the eigenvalues of non-Hermitian Hamiltonian, which satisfies the
condition Eq. (11), are real. The Eq. (11) can be rewritten in new form

$\mathcal{H}^{\dagger}=\left(S^{\dagger}S\right)\mathcal{H}\left(S^{\dagger}S\right)^{-1}=\eta\mathcal{H}\eta^{-1},$
(12)

where we have introduced the Hermitian operator $\eta=S^{\dagger}S$. The Eq.
(12) is the necessary and sufficient condition for the spectrum of Hamiltonain
$\mathcal{H}$ to be real. If $S$ is unitary matrix, then $S^{\dagger}S=1$ and
Eq. (12) recovers the hermicity of Hamiltonian from hermitian quantum
mechanics. One can show that the Hamiltonian Eq. (10) satisfies the condition
(12) for $|\sin\chi|<\beta^{-1}$ segment (see Sec. C of Ref. [31]). For non-
degenerate matrix
$\left(S^{-1}\right)^{\dagger}=\left(S^{\dagger}\right)^{-1}$, the condition
in Eq. (12) coincides with the condition of pseudo-hermiticity [32, 33, 34].
This is the necessary condition for spectrum to be real. The sufficient
condition for spectrum to be real is the existence of operator $\eta$ in the
form of $S^{\dagger}S$. The condition in Eq. (12) at $S^{\dagger}S=\eta$ is
known as $\eta-$ hermiticity. We call the Hamiltonains $\mathcal{H}$ and
$\mathcal{H}_{h}$ from Eq. (11) equivalent, since they have the same real
spectrum even if one of them is non-Hermitian. The equivalence of two
Hamiltonians in that sense is the necessary and sufficient condition to have
real valued spectrum.

Topological protection. We continue to investigate the spectrum of Weyl
fermions of types III and IV. It is worth mentioning, that Weyl points are
topologically protected with respect to external perturbations. The spectrum
remains gapless, for the case of unit matrix perturbation, $H\to H+IU_{0}$,
which causes the shift of Weyl points with respect to energy and momentum,
$\varepsilon\to\varepsilon(p-p_{0})+\varepsilon(p_{0})$, where $p_{0}$
satisfies to the condition
$\left[\left(\boldsymbol{\vartheta}+\boldsymbol{\omega}\right)\cdot\mathbf{p_{0}}+U_{0}\right]^{2}+\left[1-\beta^{2}\right]\left[v_{F}^{2}p_{0}^{2}-\left(\boldsymbol{\omega}\cdot\mathbf{p_{0}}+U_{0}\right)^{2}\right]=0$.
Next, the perturbation in the form of Pauli matrix, $H\to
H+\boldsymbol{\sigma}\mathbf{U}$ results in the following spectrum

$\varepsilon=\left[\mathcal{E}_{1}(\mathbf{p})\pm\mathcal{E}_{2}(\mathbf{p})\right]/\left(1-\beta^{2}\right),\\\
$ (13)

where
$\mathcal{E}_{1}(\mathbf{p})=\mathcal{A}(\mathbf{p})+\boldsymbol{\vartheta}\cdot\mathbf{U}/v_{F}$
and
$\mathcal{E}_{2}(\mathbf{p})=\sqrt{\mathcal{E}^{2}_{1}(\mathbf{p})+(1-\beta^{2})\xi(\mathbf{p})}$,
with
$\xi(\mathbf{p})=(v_{F}\mathbf{p}+\mathbf{U})^{2}-\left(\boldsymbol{\omega}\cdot\mathbf{p}\right)^{2}$
and $\mathcal{A}(\mathbf{p})$ is given in Eq. (9). This spectrum has a shift
with respect to initial one, Eq. (9), but does not contain a gap.

Figure 2:  It is shown a closed surface in momentum space enclosing the Weyl
point. The surface is divided into two regions: The spectrum of Hamiltonian,
Eq. (10) is real in the region $S_{\textrm{R}}$ and complex in the region
$S_{\textrm{C}}$. The lines $\bm{l}_{1}$ and $\bm{l}_{2}$ represent the
boundaries between $S_{\textrm{R}}$ and $S_{\textrm{C}}$. The vector
$\bm{\vartheta}$ is directed along $Z$-axis.

It is as well useful to investigate the stability of spectrum with the help of
topological invariant-Chern number. For non-Hermitian Hamiltonian the Chern
number can be written as [35, 30]

$\mathcal{N}=\frac{1}{4\pi}\oint
i\partial_{\mathbf{p}}\times\left\langle\varPsi^{L}\right|\partial_{\mathbf{p}}\left|\varPsi^{R}\right\rangle\cdot
d\mathbf{S},$ (14)

where the integration is carried out over a closed surface enclosing the Weyl
point. The Chern number defined in Eq. (14) must vanish, if the Hamiltonian
has certain symmetries, namely
$\mathcal{H}\left(\mathbf{p}\right)=\mathcal{H}^{T}\left(\mathbf{p}\right)$,
$\sigma_{x}\mathcal{H}\left(\mathbf{p}\right)\sigma_{x}=\mathcal{H}^{T}\left(\mathbf{p}\right)$
or
$\mathcal{H}\left(\mathbf{p}\right)=\mathcal{H}^{T}\left(-\mathbf{p}\right)$.
The Hamiltonian does not satisfies any of these conditions, which indicates of
topological protection of Chern invariant for the Hamiltonian under
consideration.

The integration domain in Eq. (14) can be divided into two regions,
$S_{\textrm{C}}$ and $S_{\textrm{R}}$, which correspond to complex and real
spectrum of Hamiltonian in Eq. (10) (see Fig. 2). The region $S_{\textrm{C}}$
does not contribute to integral in Eq. (14) (see Sec.D of Ref. [31]),
therefore
$\mathcal{N}=\frac{1}{4\pi}\int_{S_{\textrm{R}}}i\partial_{\mathbf{p}}\times\left\langle\varPsi^{L}\right|\partial_{\mathbf{p}}\left|\varPsi^{R}\right\rangle\cdot
d^{2}\mathbf{p}$, which results in fractional Chern number. This means that
Weyl fermions of types III and IV are characterized with fractional Chern
number. Fractional topological indexes have been considered earlier in
literature, and typically are associated with exceptional points in spectrum
[36], in contrast to the case under consideration with no exceptional points.

Discussion. Let investigate the spectrum in phase transition point
$v_{F}=\vartheta$. For simplicity, we assume that vectors
$\boldsymbol{\vartheta},\boldsymbol{\omega}$ are parallel to $\mathbf{p}$.
Then one has
$\left(\boldsymbol{\vartheta}+\boldsymbol{\omega}\right)\cdot\mathbf{p}=\pm\left(\vartheta+\omega\right)p$,
where ”$\pm$” correspond to positive and negative directions of momentum
$\mathbf{p}$. Let be $v_{F}>\omega$. This corresponds to phase transition
between type I and III. The electron states with positive momentum and hole
states with negative momentum in phase transition point can be associated with
infinite group velocity $\partial\varepsilon/\partial p$. Indeed, in this case
from Eq. (9) one has

$\lim_{\vartheta\to v_{F}}\varepsilon\rightarrow\infty.$ (15)

Such situation is described by vertical line. The second branch of spectrum
originates from electron states with negative momentum and hole states with
positive momentum. For these states from Eq. (9) we have

$\lim_{\vartheta\to
v_{F}}\varepsilon=-\frac{v^{2}_{F}p^{2}-(\boldsymbol{\omega}\cdot\mathbf{p})^{2}}{2(\boldsymbol{\vartheta}+\boldsymbol{\omega})\cdot\mathbf{p}}.$
(16)

These two branches for the cases $v_{F}>\vartheta$, $v_{F}=\vartheta$ and
$v_{F}<\vartheta$ are provided at Fig. 3 in case of type III (the qualitative
picture for the case of type IV is similar). Let discuss these transitions in
details. To do this, let consider the linear function $y=(a-b)x$. For $a>b$
this function describes the line with positive slope. At point $a=b$ the
continuous transition from positive to negative slope occurs. At the same time
the point $a=b$ corresponds to horizontal line, i.e. the transition occurs
through $X$ axis. Is it possible to make a phase transition from positive
slope to negative through $Y$ axis? This transition correspond to type I-III
(II-IV) and it is completely different compared to previous one. The main
difference is that during the phase transition there is the swapping of the
electron and hole states. Such kind of transition can be described with
function of form $y=x/(a-b)$. For $a>b$ this function describes the straight
line with positive slope. For $a<b$ the slope is negative. The point $a=b$
correspond to transition from positive to negative slope. However, compared to
previous example, $(a-b)x$, this is the singularity point. Thus, the
transition from positive to negative slope is not continuous. In other words,
in second example conditions $a>b$ and $a<b$ correspond to fundamentally
different phases, and the continuous transition between them is forbidden.
Now, it become clear that types I and II are not topologically different since
the continuous transition is allowed. Consequently, in this context, types I
and III, I and IV are topologically different phases. Note that the discussion
above is true only for one branch (in our case it is red line). For blue
branch there is no inversion of electron and hole parts of spectrum. Indeed,
the slope of blue branch of spectrum does not depend on the ratio between
$v_{F}$ and $\vartheta$.

Figure 3:  The electron spectrums of type I (left) and III (right). The
vertical axis is energy and the horizontal axis is momentum. The spectrum
during the phase transition I-III (middle).

Finally, let us discuss the possible origin of non-Hermitian Weyl Hamiltonian
of types III and IV. The authors of Ref. [26] suggest to associate the Weyl
fermions of types III and IV with many-particle effects. However, it can be
provided the alternative interpretation as well. The alternative
interpretation does not depend directly on the many-particle effects. We
present the explanation in a general form, keeping in mind two-band model of
band structure. The wave function in two-band model has the form
$\left|\Psi_{\mathbf{p}}\right\rangle=C_{u}\left|u_{\mathbf{p}}\right\rangle+C_{v}\left|v_{\mathbf{p}}\right\rangle$,
where amplitude $C_{u/v}$ corresponds to
$\left|u_{\mathbf{p}}\right\rangle/\left|u_{\mathbf{p}}\right\rangle$ Bloch’s
function. Multiplying the wave equation
$H\left|\Psi_{\mathbf{p}}\right\rangle=E_{\mathbf{p}}\left|\Psi_{\mathbf{p}}\right\rangle$
with $\left\langle u_{\mathbf{p}}\right|$ and
$\left\langle\upsilon_{\mathbf{p}}\right|$, we obtain the system of equations
for the amplitudes

$\begin{split}&\left(\begin{array}[]{cc}H_{uu}^{\mathbf{p}}&H_{uv}^{\mathbf{p}}\\\
H_{vu}^{\mathbf{p}}&H_{vv}^{\mathbf{p}}\end{array}\right)\left(\begin{array}[]{c}C_{u}\\\
C_{v}\end{array}\right)=E_{\mathbf{p}}\left(\begin{array}[]{cc}S_{uu}^{\mathbf{p}}&S_{uv}^{\mathbf{p}}\\\
S_{vu}^{\mathbf{p}}&S_{vv}^{\mathbf{p}}\end{array}\right)\left(\begin{array}[]{c}C_{u}\\\
C_{v}\end{array}\right)\\\
&=\left[f_{0}(\mathbf{p})+\sigma_{i}f_{i}(\mathbf{p})\right]\left(\begin{array}[]{cc}C_{u}&C_{v}\\\
\end{array}\right)^{T},\end{split}$ (17)

where $H_{ab}^{\mathbf{p}}=\left\langle
a_{\mathbf{p}}\right|H\left|b_{\mathbf{p}}\right\rangle$ is the matrix element
of Hamiltonian, with $a,b=u,v$ and $S_{ab}^{\mathbf{p}}=\left\langle
a_{\mathbf{p}}|b_{\mathbf{p}}\right\rangle=\int
a_{\mathbf{p}}^{\ast}\left(\mathbf{r}\right)b_{\mathbf{p}}\left(\mathbf{r}\right)d^{3}\mathbf{r}$
is the overlap integral. Typically, the Bloch functions are taken to be
orthogonal, thus in Eq. (17) the overlap matrix becomes unit matrix,
$S^{\mathbf{p}}_{ab}=\delta_{ab}$. However, in real materials, this is not
always true and off-diagonal elements of overlap matrix do not vanish (for
instance, one of the reason is indeed many-particle effects) [37, 38, 39, 40].
Let introduce the Bloch functions
$u_{\mathbf{p}}(\mathbf{r})=\sum_{\mathbf{R}}e^{i\mathbf{p}\mathbf{R}}\phi_{u}(\mathbf{r}-\mathbf{R})$
and
$v_{\mathbf{p}}(\mathbf{r})=\sum_{\mathbf{R}}e^{i\mathbf{p}\mathbf{R}}\phi_{v}(\mathbf{r}-\mathbf{R})$,
where $\phi_{u}$, $\phi_{v}$ atomic orbitals, corresponding to bands $u$, $v$
and $\mathbf{R}$ is a radius vector of a given atom at lattice site. Then the
off-diagonal overlap integral can be written as $\int
u^{\ast}_{\mathbf{p}}(\mathbf{r})v_{\mathbf{p}}(\mathbf{r})d^{3}\mathbf{r}=\sum_{\mathbf{R}\mathbf{R}^{\prime}}e^{i\mathbf{p}(\mathbf{R}-\mathbf{R}^{\prime})}\int\phi^{\ast}_{u}(\mathbf{r}-\mathbf{R}^{\prime})\phi_{v}(\mathbf{r}-\mathbf{R})d^{3}\mathbf{r}$.
Atomic orbitals corresponding to the same atom
($\mathbf{R}=\mathbf{R}^{\prime}$) are orthogonal,
$\int\phi^{\ast}_{a}(\mathbf{r}-\mathbf{R}^{\prime})\phi_{b}(\mathbf{r}-\mathbf{R})d^{3}\mathbf{r}=\delta_{ab}$.
However, at different $\mathbf{R}\neq\mathbf{R}^{\prime}$, generally speaking
these orbitals are not orthogonal, which results in non-zero off-diagonal
elements of overlap matrix, $S^{\mathbf{p}}_{ab}$. At the same time, we note,
that the Bloch functions at the Weyl point will be orthogonal, since at this
point the time reversal symmetry is not broken. This means that $\left\langle
u_{\mathbf{p}_{W}}|\mathcal{T}u_{\mathbf{p}_{W}}\right\rangle=0$, where
$\mathcal{T}$ is the time reversal operator [17]. Since
$\mathcal{T}\left|u_{\mathbf{p}_{W}}\right\rangle=\left|\upsilon_{\mathbf{p}_{W}}\right\rangle$
then $\left\langle
u_{\mathbf{p}_{W}}|\upsilon_{\mathbf{p}_{W}}\right\rangle=0$. Further, the Eq.
(17) can be rewritten as
$(S^{\mathbf{p}})^{-1}\left[f_{0}(\mathbf{p})+\sigma_{i}f_{i}(\mathbf{p})\right]\left(\begin{array}[]{cc}C_{u}&C_{v}\\\
\end{array}\right)^{T}=\tilde{\mathcal{H}}\left(\begin{array}[]{cc}C_{u}&C_{v}\\\
\end{array}\right)^{T}=\tilde{\varepsilon}\left(\begin{array}[]{cc}C_{u}&C_{v}\\\
\end{array}\right)^{T}$, where $S^{\mathbf{p}}_{ab}$ is a matrix on the right
hand side of Eq. (17). In the vicinity of Weyl point one has
$f_{i}(\mathbf{p})\approx v_{F}p_{i}$ and
$f_{0}(\mathbf{p})\approx\boldsymbol{\omega}\cdot\mathbf{p}$. If one requires
the overlap matrix in Pauli basis has components
$S_{uu(vv)}^{\mathbf{p}}=1\mp\vartheta_{z}/v_{F}$,
$S_{uv}^{\mathbf{p}}=(i\vartheta_{y}-\vartheta_{x})/v_{F}=(S_{vu}^{\mathbf{p}})^{\ast}$,
then the effective Hamiltonian,
$\tilde{\mathcal{H}}=(S^{\mathbf{p}}_{ab})^{-1}\left[f_{0}(\mathbf{p})+\sigma_{i}f_{i}(\mathbf{p})\right]$
completely coincides with the Hamiltonian from Eq. (8) with the same prefactor
$1/(1-\beta^{2})$. Thus the origin of this Hamiltonian can be related with the
overlap between Bloch functions. For 1D chain such a Hamiltonian has been
recently introduced by generalizing the Su-Schrieffer-Heeger model on non-
hermitian case [41].

To summarize, we have provided the theory of type III and IV Weyl semimetals
within non-Hermitian Hamiltonian formalism. The spectrum of this Weyl
Hamiltonian exhibits an unusual type of anisotropy in momentum space, namely
for some directions the spectrum is real, in contrast for other directions it
is complex. The necessary and sufficient conditions for spectrum to be real is
provided. Moreover, the anisotropy in spectrum results in anisotropy of the
inner product of states, and thus leads to non-trivial adiabatic evolution and
fractional Chern number. Additionally, we speculate the possible origin of
non-Hermitian Hamiltonian.

We are grateful to G. E. Volovik, J. Nissinen and T. L. Schmidt for fruitful
discussions. Z. A. acknowledges the financial support from President Grant (MD
647.2020.2). E. I. acknowledges financial support from the National Research
Fund Luxembourg under Grant CORE C19/MS/13579612/HYBMES.

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Supplementary Material for “Towards a theory of types III and IV non-Hermitian
Weyl fermions”

##  I A: Anisotropy of orthogonality conditions

We write the spectrum from Eq. (9) in main text with additional subscripts as

$\varepsilon_{R,\pm}=\varepsilon_{L,\pm}=\frac{\mathcal{A}\left(\mathbf{p}\right)\pm\sqrt{\mathcal{B}\left(\mathbf{p}\right)}}{1-\beta^{2}},$
(S-1)

where ”$\pm$” are associated with electron and hole states. The corresponding
eigenvectors are denoted by $\left|\varPsi_{\pm}^{R,L}\right\rangle$, which
satisfy wave equations
$\mathcal{H}\left|\varPsi_{\pm}^{R}\right\rangle=\varepsilon_{R,\pm}\left|\varPsi_{\pm}^{R}\right\rangle$,
and
$\mathcal{H}^{\dagger}\left|\varPsi_{\pm}^{L}\right\rangle=\varepsilon_{L,\pm}\left|\varPsi_{\pm}^{L}\right\rangle$.
It is worth mentioning that $\varepsilon_{R}=\varepsilon_{L}=\varepsilon$,
where ”$\pm$” is omitted. In phases III and IV the spectrum is anysotropic
(see Eq. (9) from main text) with respect to space orientation of momentum
vector: for some directions the spectrum is real, in contrast, for other
directions it is complex. The existence of domain, where
$\text{Im}(\varepsilon)\neq 0$ is the main distinguishing feature of the new
phases, III and IV. Apart from this, in domain of complex spectrum
$\varepsilon_{\pm}^{*}=\varepsilon_{\mp}$, which leads to important
consequences. The main one is related with scalar product
$\left\langle\varPsi_{\alpha}^{L}|\varPsi_{\beta}^{R}\right\rangle$. Let
consider the following $2\times 2$ matrix

$\left(\begin{array}[]{cc}\left\langle\varPsi_{+}^{L}|\mathcal{H}|\varPsi_{+}^{R}\right\rangle&\left\langle\varPsi_{+}^{L}|\mathcal{H}|\varPsi_{-}^{R}\right\rangle\\\
\left\langle\varPsi_{-}^{L}|\mathcal{H}|\varPsi_{+}^{R}\right\rangle&\left\langle\varPsi_{-}^{L}|\mathcal{H}|\varPsi_{-}^{R}\right\rangle\end{array}\right).$
(S-2)

Applying the Hamiltonian inside matrix elements on right and left hand sides
one obtains

$\left(\begin{array}[]{cc}\varepsilon_{+}\left\langle\varPsi_{+}^{L}||\varPsi_{+}^{R}\right\rangle&\varepsilon_{-}\left\langle\varPsi_{+}^{L}||\varPsi_{-}^{R}\right\rangle\\\
\varepsilon_{+}\left\langle\varPsi_{-}^{L}||\varPsi_{+}^{R}\right\rangle&\varepsilon_{-}\left\langle\varPsi_{-}^{L}||\varPsi_{-}^{R}\right\rangle\end{array}\right)=\left(\begin{array}[]{cc}\varepsilon_{+}^{*}\left\langle\varPsi_{+}^{L}||\varPsi_{+}^{R}\right\rangle&\varepsilon_{+}^{*}\left\langle\varPsi_{+}^{L}||\varPsi_{-}^{R}\right\rangle\\\
\varepsilon_{-}^{*}\left\langle\varPsi_{-}^{L}||\varPsi_{+}^{R}\right\rangle&\varepsilon_{-}^{*}\left\langle\varPsi_{-}^{L}||\varPsi_{-}^{R}\right\rangle\end{array}\right).$
(S-3)

For real spectrum from the above equation it is obvious that
$\left\langle\varPsi_{\pm}^{L}|\varPsi_{\mp}^{R}\right\rangle=0$ and
$\left\langle\varPsi_{\pm}^{L}||\varPsi_{\pm}^{R}\right\rangle\neq 0$. At the
same time, for the domain, where the spectrum is complex one gets
$\left\langle\varPsi_{\pm}^{L}||\varPsi_{\pm}^{R}\right\rangle=0$ and
$\left\langle\varPsi_{\pm}^{L}|\varPsi_{\mp}^{R}\right\rangle\neq 0$.

##  II B: The form of eigenstates

The Hamiltonian under consideration has the form

$\mathcal{H}=A\left(\mathbf{p}\right)\sigma_{0}+\left[\bm{D}\left(\mathbf{p}\right)+i\bm{C}\left(\mathbf{p}\right)\right]\cdot\bm{\sigma},$
(S-4)

where $\bm{D}\cdot\bm{C}=0$. The spectrum of this Hamiltonian is given by

$\varepsilon_{R,\pm}=\varepsilon_{L,\pm}=\varepsilon_{\pm}=A\pm\sqrt{D^{2}-C^{2}},$
(S-5)

where $C=\left|\bm{C}\right|$ and $D=\left|\bm{D}\right|$. Particularly, for
the case under consideration in main text, the parameters are
$A=(\bm{\omega}\cdot\mathbf{p}+\bm{\vartheta}\cdot\mathbf{p})/(1-\beta^{2})$,
$\bm{D}=(\upsilon_{F}\mathbf{p}+\frac{1}{\upsilon_{F}}\bm{\vartheta}\left(\bm{\omega}\cdot\mathbf{p}\right))/(1-\beta^{2})$
and $\bm{C}=\left[\mathbf{p}\times\bm{\vartheta}\right]/(1-\beta^{2})$. The
eigenstates of Hamiltonian are given by

$\left|\varPsi_{\pm}^{R}\right\rangle=\left(\begin{array}[]{c}\frac{C_{z}-iD_{z}\mp
i\sqrt{D^{2}-C^{2}}}{C_{x}+i\left(C_{y}-D_{x}\right)+D_{y}}\\\
1\end{array}\right),\,\,\,\,\left|\varPsi_{\pm}^{L}\right\rangle=\left(\begin{array}[]{c}\frac{C_{z}+iD_{z}\pm
i\sqrt{D^{2}-C^{2}}}{C_{x}+i\left(C_{y}+D_{x}\right)-D_{y}}\\\
1\end{array}\right),$ (S-6)

and

$\left\langle\varPsi_{\pm}^{R}\right|=\left(\begin{array}[]{cc}\frac{C_{z}+iD_{z}\pm
i(\sqrt{D^{2}-C^{2}})^{*}}{C_{x}-i\left(C_{y}-D_{x}\right)+D_{y}}&1\end{array}\right),\,\,\,\,\left\langle\varPsi_{\pm}^{L}\right|=\left(\begin{array}[]{cc}\frac{C_{z}-iD_{z}\mp
i(\sqrt{D^{2}-C^{2}})^{*}}{C_{x}-i\left(C_{y}+D_{x}\right)-D_{y}}&1\end{array}\right),$
(S-7)

where ”$\ast$” means the complex conjugation. Using the form of eigenstates
one can show that
$\left\langle\varPsi_{\pm}^{L}|\varPsi_{\mp}^{R}\right\rangle=0$ for $D>C$,
and $\left\langle\varPsi_{\pm}^{L}|\varPsi_{\pm}^{R}\right\rangle=0$ for
$D<C$. Further, the direct calculations give
$\left\langle\varPsi_{\pm}^{L}|\varPsi_{\pm}^{R}\right\rangle=C^{2}-D^{2}-\left|D^{2}-C^{2}\right|\mp
2i\left(C_{z}-iD_{z}\right)\textrm{Re}\sqrt{D^{2}-C^{2}}$. Therefore, for
$D<C$ we get $\left\langle\varPsi_{\pm}^{L}|\varPsi_{\pm}^{R}\right\rangle=0$.
On the other hand,
$\left\langle\varPsi_{\mp}^{L}|\varPsi_{\pm}^{R}\right\rangle=C^{2}-D^{2}+\left|D^{2}-C^{2}\right|\pm
2\left(C_{z}-iD_{z}\right)\textrm{Im}\sqrt{D^{2}-C^{2}}$. Thus, for $D>C$ we
have $\left\langle\varPsi_{\mp}^{L}|\varPsi_{\pm}^{R}\right\rangle=0$.

##  III C: Real spectrum of pseudo-Hermitian Hamiltonian

From the main text, Eq. (12), it is clear, that
$\left[\eta,\mathcal{H}\right]=\eta\mathcal{H}-\mathcal{H}\eta\neq 0$, for
non-Hermitian $\mathcal{H}$. Indeed, otherwise $[\eta,\mathcal{H}]=0$ and one
arrives to contradiction, namely the conjugated Hamiltonian
$\mathcal{H}^{\dagger}=\eta\mathcal{H}\eta^{-1}=\mathcal{H}\eta\eta^{-1}=\mathcal{H}$
is Hermitian. As an example, let consider the simple non-Hermitian matrix
$M=\left(\begin{array}[]{cc}1&1+\lambda\\\ 1-\lambda&-1\end{array}\right)$,
where $\lambda$ is the arbitrary number. Next, let find the matrix $\eta$,
which satisfies the condition $M^{\dagger}=\eta M\eta^{-1}$. One can show that
the $\eta$ has the following form
$\eta=\left(\begin{array}[]{cc}\frac{r+q}{1+\lambda}&q\\\
q&\frac{r-q}{1-\lambda}\end{array}\right)$, where $p$, $q$ are arbitrary
numbers as well. Further, in order to have the real eigenvalues for matrix
$M$, there must be possibility to present $\eta=S^{\dagger}S$. From these
condition, one consequently obtains that $q,r\in\mathbb{R}$ and
$\eta^{\dagger}=\eta$. Further, it is obvious, that the condition
$\det\left(SS^{\dagger}\right)>0$ must be satisfied. This condition set limits
on the choice of elements of matrix $\eta$. In our case, this condition brings
us to inequality for $p$, $q$ and $\lambda$, namely
$\frac{q^{2}-r^{2}}{\lambda^{2}-1}-q^{2}>0$. This inequality does not work for
$\lambda^{2}>2$. Thus, the eigenvalues of matrix $M$ are real for
$\lambda^{2}<2$.

Let now investigate our Hamiltonian in the same manner as the example above.
For simplicity, we consider the case of
$\boldsymbol{\vartheta}=(0,0,\vartheta)$. In this case, the Hamiltonian in Eq.
(10) from main text is written as

$\mathcal{H}=\frac{\upsilon_{F}}{1-\beta^{2}}\left[\left(\begin{array}[]{cc}p_{z}&\left(1-\beta\right)\left(p_{x}-ip_{y}\right)\\\
\left(1+\beta\right)\left(p_{x}+ip_{y}\right)&-p_{z}\end{array}\right)+\beta
p_{z}\sigma_{0}\right],$ (S-8)

where $\beta=\vartheta/v_{F}$ and $\sigma_{0}$ is the identity matrix. The
matrix $\eta$ has the form $\eta=\left(\begin{array}[]{cc}a&b\\\
b^{\ast}&d\end{array}\right)$, where
$a=\frac{r+bp_{z}}{\left(1-\beta\right)\left(p_{x}-ip_{y}\right)}$ and
$d=\frac{r-bp_{z}}{\left(1+\beta\right)\left(p_{x}-ip_{y}\right)}$,
$r,b\in\mathbb{C}$. From condition $\eta=S^{\dagger}S$ one gets that
$q,r>0\in\mathbb{R}$ and the condition $\det(S^{\dagger}S)>0$ results in
inequality for other parameters, $ad-\left|b\right|^{2}>0\in\mathbb{R}$. Using
these conditions at $\beta<1$, one can show that matrix $\eta=S^{\dagger}S$
exists always. For example,
$\eta=\left(\begin{array}[]{cc}\frac{q}{1-\beta}&0\\\
0&\frac{q}{1+\beta}\end{array}\right)$, where $q>0\in\mathbb{R}$. For
$\beta>1$ the matrix $\eta=S^{\dagger}S$ exists only for
$\frac{p_{z}^{2}}{p_{x}^{2}+p_{y}^{2}}<\beta^{2}-1$, which turns out to be
$|\sin\chi|<\beta^{-1}$ in spherical coordinates, and this condition coincides
with the one we have provided in Eq. (9) from main text. In this case the
matrix has the form
$\eta=\left(\begin{array}[]{cc}\frac{p_{z}}{\beta-1}&-p_{x}+ip_{y}\\\
-p_{x}-ip_{y}&\frac{p_{z}}{\beta+1}\end{array}\right)$. In other words, for
$\left|\sin\chi\right|<\beta^{-1}$ the Hamiltonian above, Eq. (S-8) can be
brought to hermitian Hamiltonian using similarity transformations. Namely
$S^{-1}\mathcal{H}S$ is the Hermitian operator, despite the fact that
$\mathcal{H}$ is non-Hermitian. Additionally, it is worth pointing out, that
redefinition of scalar product in Hilbert space (unitary theorem) in the
following form
$\left\langle\left\langle\Psi|\Psi\right\rangle\right\rangle_{S}\equiv\left\langle\Psi|S^{\dagger}S|\Psi\right\rangle$
does depend on time, if Hamiltonian satisfies the condition Eq. (12) from main
text. Thus, this means that the probability density does not depend on time,
and the unitary condition is suited.

As one can observe from Eq. (12) from main text, in general the realness of
spectrum is defined not only by the properties of operator by itself, but as
well it influences on Hilbert space. It worth mentioning, that the
$\mathcal{PT}$ symmetry for non-Hermitian quantum systems introduced by Bender
[1], says that $\mathcal{PT}$ operator and Hamiltonian have the same
eigenvectors. In this case Hamiltonian has the real spectrum, despite the fact
that it is non-Hermitian. This requirements is the particular case of Eq. (12)
from main text. Namely, to show this, let change a bit the matrix we have
considered above, $M\Rightarrow\left(\begin{array}[]{cc}1&\lambda-1\\\
\lambda+1&-1\end{array}\right)$. This matrix is not hermitian and it does not
have $\mathcal{PT}$ symmetry. However, the eigenvalues of this matrix are real
for any real $\lambda$. This is related with the fact that matrix
$\left(\begin{array}[]{cc}1&\lambda-1\\\ \lambda+1&-1\end{array}\right)$ is
equivalent to hermitian matrix $\left(\begin{array}[]{cc}0&\lambda\\\
\lambda&0\end{array}\right)$. Namely, there always exists the similarity
transformation $R$, therefore $R\left(\begin{array}[]{cc}0&\lambda\\\
\lambda&0\end{array}\right)R^{-1}=\left(\begin{array}[]{cc}1&\lambda-1\\\
\lambda+1&-1\end{array}\right)$.

##  IV D: Fractional Chern number

The Chern number is given by

$\mathcal{N}_{\alpha}=\frac{1}{4\pi}\oint\bm{\Omega}_{\alpha}\cdot
d\mathbf{S},$ (S-9)

where $\bm{\Omega}_{\alpha}$ is Berry curvature of band $\alpha$, and the
integration is carried out with respect to closed surface, enclosing the Weyl
point. We rewrite the Eq. (S-9) as

$\mathcal{N}_{\alpha}=\frac{1}{4\pi}\left(\int_{S_{\textrm{C}}}\bm{\Omega}_{\alpha}\cdot
d\mathbf{S}+\int_{S_{\textrm{R}}}\bm{\Omega}_{\alpha}\cdot
d\mathbf{S}\right),$ (S-10)

where $S_{\textrm{C}}$ and $S_{\textrm{R}}$ are domains of surface
corresponding to complex and real spectra (see Fig. 4 from main text). The
Berry curvature for non-Hermitian Hamiltonian is given by

$\bm{\Omega}_{\alpha}=i\partial_{\mathbf{p}}\times\left\langle\varPsi_{\alpha}^{L}\right|\partial_{\mathbf{p}}\left|\varPsi_{\alpha}^{R}\right\rangle.$
(S-11)

One can show, that $\int_{S_{\textrm{C}}}\bm{\Omega}_{\alpha}\cdot
d\mathbf{S}=0$. To do this, we consider the time-evolution of eigenstates at
$D=C$. In this case the spectrum has only one band,
$\varepsilon_{R}=\varepsilon_{L}=A\left(\mathbf{p}\right)$ and the eigenstates
are given by

$\left|\varPsi_{\pm}^{R}\right\rangle=\left|\varPsi^{R}\right\rangle=\left(\begin{array}[]{c}\frac{C_{z}-iD_{z}}{C_{x}+i\left(C_{y}-D_{x}\right)+D_{y}}\\\
1\end{array}\right),\,\,\,\,\left|\varPsi_{\pm}^{L}\right\rangle=\left|\varPsi^{L}\right\rangle=\left(\begin{array}[]{c}\frac{C_{z}+iD_{z}}{C_{x}+i\left(C_{y}+D_{x}\right)-D_{y}}\\\
1\end{array}\right).$ (S-12)

From the above equations one gets
$\left\langle\varPsi^{L}\right|\left|\varPsi^{R}\right\rangle=\left\langle\varPsi^{R}\right|\left|\varPsi^{L}\right\rangle=0$,
and for adiabatic time evolution we have
$i\partial_{t}\left|\varPsi^{R}\left(t\right)\right\rangle=\mathcal{H}\left(t\right)\left|\varPsi^{R}\left(t\right)\right\rangle=\varepsilon\left(t\right)\left|\varPsi^{R}\left(t\right)\right\rangle$.
Therefore

$i\left\langle\varPsi^{L}\right|\partial_{t}\left|\varPsi^{R}\right\rangle=\varepsilon\left(t\right)\left\langle\varPsi^{L}\right|\left|\varPsi^{R}\right\rangle=0,\quad\text{or}\quad\left\langle\varPsi^{L}\right|\partial_{\mathbf{p}}\left|\varPsi^{R}\right\rangle\cdot\partial_{t}\mathbf{p}=0.$
(S-13)

Consequently, the line integral
$\oint\left\langle\varPsi^{L}\right|\partial_{\mathbf{p}}\left|\varPsi^{R}\right\rangle\cdot
d\mathbf{p}$, corresponding to equation $D=C$ vanishes. Indeed

$\oint\left\langle\varPsi^{L}\right|\partial_{\mathbf{p}}\left|\varPsi^{R}\right\rangle\cdot
d\mathbf{p}=\int_{0}^{T}\left(\left\langle\varPsi^{L}\right|\partial_{\mathbf{p}}\left|\varPsi^{R}\right\rangle\cdot\partial_{t}\mathbf{p}\right)dt=0,$
(S-14)

where the cyclic evolution around closed path is denoted by times $0$ and $T$.
The line (contour) for $D=C$ is presented at Fig. 4 in main text. According to
Stoke’s theorem

$\oint_{\bm{l}_{1}+\bm{l}_{2}}\left\langle\varPsi_{\alpha}^{L}\right|\partial_{\mathbf{p}}\left|\varPsi_{\alpha}^{R}\right\rangle\cdot
d\mathbf{p}=\int_{S_{\textrm{C}}}\left(\partial_{\mathbf{p}}\times\left\langle\varPsi_{\alpha}^{L}\right|\partial_{\mathbf{p}}\left|\varPsi_{\alpha}^{R}\right\rangle\right)\cdot
d^{2}\mathbf{p}=0,$ (S-15)

since the contour $\bm{l}_{1}+\bm{l}_{2}$ is the boundary of domain
$S_{\textrm{C}}$ (see Fig. 4 in main text). It is worth mentioning, that the
Stoke’s theorem can be applied only to domian $S_{\textrm{C}}$, because the
orthogonality condition
$\left\langle\varPsi_{\alpha}^{L}|\varPsi_{\alpha}^{R}\right\rangle=0$ is the
same as for contour $\bm{l}_{1}+\bm{l}_{2}$. Thus, the domain of momentum
space, $S_{\textrm{C}}$, corresponding to complex spectrum does not contribute
to integral, Eq. (S-15). Consequently, it follows that the Chern number is
given by

$\mathcal{N}_{\alpha}=\frac{1}{4\pi}\int_{S_{\textrm{R}}}\bm{\Omega}_{\alpha}\cdot
d^{2}\mathbf{p}.$ (S-16)

Thereby, only part of the sphere in momentum space contributes to integral,
which results in fractional Chern number. Thus, the Weyl fermions of types III
and IV are characterized by fractional Chern number.

## References

  * [1] C. M. Bender, PT Symmetry In Quantum and Classical Physics, (World Scientific, Singapore, 2019)


correlation between $\hat{\theta}_{t_{2}}$ and
$\hat{\theta}_{t_{2}}-\hat{\theta}_{t_{1}}$ is approximately equal to 0.11 and
the correlation between $\hat{\theta}_{t_{3}}$ and
$\hat{\theta}_{t_{3}}-\hat{\theta}_{t_{1}}$ is approximately equal to 0.09.
The orthogonalization ensures that the correlation between
$\widetilde{\theta}_{2}$ and $\widetilde{\theta}_{2}-\widetilde{\theta}_{1}$
and the correlation between $\widetilde{\theta}_{3}$ and
$\widetilde{\theta}_{3}-\widetilde{\theta}_{1}$ are not significantly
different from zero (at the 1% significance level). Specifically, this leads
to a decreased Type I error of 5.31% (Monte Carlo error of approximately
0.0014). This small inflation in Type I error (after orthogonalization) is a
small-sample problem and disappears for larger samples (i.e., higher
information).

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