# Magnetic field effects on the Kitaev model coupled to environment

Kiyu Fukui<EMAIL_ADDRESS>Department of Applied Physics, The
University of Tokyo, Bunkyo, Tokyo 113-8656, Japan  Yasuyuki Kato  Department
of Applied Physics, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan
Yukitoshi Motome  Department of Applied Physics, The University of Tokyo,
Bunkyo, Tokyo 113-8656, Japan

###### Abstract

Open quantum systems display unusual phenomena not seen in closed systems,
such as new topological phases and unconventional phase transitions. An
interesting example was studied for a quantum spin liquid in the Kitaev model
[K. Yang, S. C. Morampudi, and E. J. Bergholtz, Phys. Rev. Lett. 126, 077201
(2021)]; an effective non-Hermitian Kitaev model, which incorporates
dissipation effects, was shown to give rise to a gapless spin liquid state
with exceptional points in the Majorana dispersions. Given that an external
magnetic field induces a gapped Majorana topological state in the Hermitian
case, the exceptional points may bring about intriguing quantum phenomena
under a magnetic field. Here we investigate the non-Hermitian Kitaev model
perturbed by the magnetic field. We show that the exceptional points remain
gapless up to a finite critical magnetic field, in stark contrast to the
Hermitian case where an infinitesimal field opens a gap. The gapless state is
stable over a wide range of the magnetic field for some particular parameter
sets, and in special cases, undergoes topological transitions to another
gapless state with different winding number around the exceptional points
without opening a gap. In addition, in the system with edges, we find that the
non-Hermitian skin effect is induced by the magnetic field, even for the
parameters where the skin effect is absent at zero field. The chirality of
edge states is switched through the exceptional points, similarly to the
surface Fermi arcs connected by the Weyl points in three-dimensional Weyl
semimetals. Our results provide a new possible route to stabilize topological
gapless quantum spin liquids under the magnetic field in the presence of
dissipation.

##  I Introduction

The quantum spin liquid (QSL), which is a quantum disordered state in magnets
with prominent features such as quantum entanglement and fractional
excitations, has been studied as one of the major subjects in modern condensed
matter physics from both theoretical and experimental points of view [1, 2, 3,
4]. Despite the long history of research, well-established examples of the QSL
are still limited, and the realization of the QSL in most of the candidate
models and materials is still under debate. The celebrated Kitaev model,
however, has brought a pivotal breakthrough to this situation, by providing a
rare example of QSLs in more than one dimension [5]. Despite strong
frustration arising from the bond-dependent anisotropic interactions on a
honeycomb lattice, the model is exactly solvable and yields the ground state
of a QSL with fractional excitations of itinerant Majorana fermions and
localized $\mathbb{Z}_{2}$ fluxes. The Majorana fermion excitation constitutes
a gapless Dirac-like node at zero magnetic field, but it is gapped out
immediately by introducing a magnetic field. This gapped state is
topologically nontrivial, exhibiting chiral edge modes and non-Abelian anyons
under a magnetic field that would be utilized for topological quantum
computation [6, 7, 5, 8, 9]. Moreover, since the feasibility of the model was
proposed for spin-orbit coupled Mott insulators [10], a number of intensive
searches for the candidate materials have been carried out [11, 12, 13, 14,
15, 16], for instance, for Na2IrO3 [17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27], $\alpha$-Li2IrO3 [19, 21, 27], and $\alpha$-RuCl3 [28, 29, 27, 30, 31].
In recent years, such exploration has been extended to new family of
candidates, such as cobalt compounds [32, 33, 34, 35], iridium ilmenites [36,
37, 38], and $f$-electron compounds [39, 40, 41, 42, 43]. In addition to the
solid state materials, realization of the model in ultracold atoms [44, 45,
46, 47, 48, 49] and Rydberg atoms [50, 51] has also been proposed.

In most studies to date, QSLs have been discussed for closed systems. However,
effects of dissipation in open systems can be crucially important for both
fundamental physics and applied science perspectives. For instance,
dissipation is not negligible in the implementation of the Kitaev model by the
ultracold atoms. In quantum computation devices, the system is essentially
open due to the input and readout of quantum information as well as the effect
of substrates. Such open quantum systems can display unusual phenomena not
seen in closed systems, such as new topological phases [52, 53, 54, 55] and
unconventional critical phenomena [56, 57, 58, 59]. While theoretical
treatment of open quantum systems coupled to the environment is in general
complicated, one of the efficient methods is to study an effective non-
Hermitian Hamiltonian which incorporates the effects of dissipation. In recent
years, the non-Hermitian systems have been studied intensively, especially
from the topological point of view [60, 61, 62, 63, 64, 65, 66]. For the QSL
in the Kitaev model, non-Hermitian effects have been investigated on the
honeycomb lattice [67, 68], honeycomb bilayer [69], and two-leg ladder [70].
The non-Hermitian Kitaev model defined on the honeycomb lattice was shown to
give rise to an unconventional gapless spin liquid state, in which the Dirac-
like node with linear dispersion in the Majorana excitation spectrum in the
Hermitian case splits into two exceptional points with square-root dispersions
[67]. In the Hermitian case, the effect of the magnetic field is crucial to
make the system gapped and topologically nontrivial as mentioned above, but it
has not been clarified yet in the non-Hermitian case, even though the
exceptional points may bring about unprecedented quantum phenomena.

In this paper, we investigate the effect of the magnetic field on the non-
Hermitian Kitaev model. Based on the perturbation with respect to the field
strength, we show that the exceptional points remain gapless up to a finite
critical effective magnetic field, in stark contrast to the Hermitian case
where the Dirac-like nodes are gapped out by an infinitesimal magnetic field.
We also show that the critical field becomes large beyond the field range
where the perturbation theory is justified, and even diverges for some
particular parameter sets on the complex planes of the coupling constants. By
calculating the winding number of the exceptional points, we find that the
system at certain special parameters exhibits an unconventional gapless-
gapless topological transition by the magnetic field. In addition, in the
system with edges, we find that the non-Hermitian skin effect is induced by
the magnetic field even for the parameters where it does not occur at zero
magnetic field. We also clarify that the chirality of the edge states is
switched at the wave numbers corresponding to the exceptional points, like the
Fermi arc surface states connected by the projections of the Weyl points in
three-dimensional Weyl semimetals. Our results provide a new possible route to
stabilize and control topological gapless QSLs under the magnetic field in the
presence of dissipation.

The structure of this paper is as follows. In Sec. II, we introduce the non-
Hermitian Kitaev model under a magnetic field and briefly review the previous
studies for the zero field case. We present our results for the critical
effective magnetic field for the gap opening at the exceptional points in Sec.
III.1, the topological transitions with changes in the winding numbers of the
exceptional points in Sec. III.2, and the non-Hermitian skin effect in Sec.
III.3. In Sec. IV.1, we discuss the topological transition found in this study
in comparison with the precedented examples. We also discuss the skin effect
in comparison with the surface states in the three-dimensional Hermitian Weyl
semimetals in Sec. IV.2. In Sec. IV.3, we remark on the effects of the
perturbation terms not considered in the present calculations. Finally, we
summarize our findings in Sec. V.

##  II Model

Figure 1: Schematic picture of the Kitaev model coupled to an environment.
$J_{\mu}$ and ${\bf h}$ represent the coupling constants of the Kitaev
interactions on the $\mu$ bond ($\mu=x$, $y$, and $z$) of the honeycomb
lattice and the external magnetic field, respectively; see Eq. (1).
$\gamma_{\mu}$ represent the coupling to the environment. The dashed arrows
represent the sublattice-dependent second-neighbor hoppings of Majorana
fermions stemming from the magnetic field in the perturbation theory; see Eq.
(4). The white (black) spheres represent sites belonging to the A (B)
sublattice. $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ denote the primitive
translation vectors.

We start from the Kitaev model under a uniform magnetic field. The Hamiltonian
is given by [5]

$\mathcal{H}_{0}=-\sum_{\langle
i,j\rangle_{\mu}}J_{\mu}\sigma^{\mu}_{i}\sigma^{\mu}_{j}-\tilde{h}\sum_{\langle\\!\langle
i,j,k\rangle\\!\rangle_{\mu,\nu,\lambda}}\sigma_{i}^{\mu}\sigma_{j}^{\nu}\sigma_{k}^{\lambda},$
(1)

where $\sigma_{i}^{\mu}$ is the $\mu$ component of the Pauli matrix at site
$i$ ($\mu=x$, $y$, and $z$). The first term represents the Kitaev interactions
on the $\mu$ bond with the coupling constant $J_{\mu}$; the summation of
$\langle i,j\rangle_{\mu}$ runs over nearest-neighbor pairs $i$ and $j$
connected by the $\mu$ bond; see Fig. 1. The second term represents effective
three-spin interactions derived by third-order perturbations with respect to
the Zeeman coupling; the coupling strength $\tilde{h}$ is proportional to
$h_{x}h_{y}h_{z}$ for an external magnetic field ${\bf h}=(h_{x},h_{y},h_{z})$
[5]. The summation of $\langle\\!\langle
i,j,k\rangle\\!\rangle_{\mu,\nu,\lambda}$ runs over neighboring three sites
$i$, $j$, and $k$ such that $i$ and $j$ ($j$ and $k$) are connected by $\mu$
($\lambda$) bond and $\nu$ is chosen as the remaining bond
($\mu\neq\nu\neq\lambda\neq\mu$). The effects of other corrections within the
third-order perturbation will be discussed in Sec. IV.3.

When the system is coupled to an environment, the time evolution is described
by the Lindblad equation for the density matrix $\rho$ given by [71, 72]

$\frac{\mathrm{d}}{\mathrm{d}t}\rho=-\mathrm{i}[\mathcal{H}_{0},\rho]+\sum_{\langle
i,j\rangle_{\mu}}\gamma_{\mu}\left(L_{ij}^{\mu}\rho
L_{ij}^{\mu\dagger}-\frac{1}{2}\\{L_{ij}^{\mu\dagger}L_{ij}^{\mu},\rho\\}\right),$
(2)

where $L_{ij}^{\mu}$ is a jump operator describing the interactions with
environment; we set the reduced Planck constant $\hbar=1$. $\gamma_{\mu}$ is a
damping rate and thus it takes a non-negative value. Here, however, we treat
it as a real number which also takes a negative value to simplify the
mathematical treatment. In the following, we choose
$L_{ij}^{\mu}=\sigma^{\mu}_{i}+\sigma^{\mu}_{j}$, which was introduced in the
previous study for zero field [67]. A similar type of dissipation was
considered on the Kitaev model on a two-leg ladder [70]. Since solving this
equation directly is complicated and laborious, we deal with an effective non-
Hermitian Hamiltonian given by

$\displaystyle\mathcal{H}_{\mathrm{eff}}$
$\displaystyle=\mathcal{H}_{0}-\frac{\mathrm{i}}{2}\sum_{\langle
i,j\rangle_{\mu}}\gamma_{\mu}L_{ij}^{\mu\dagger}L_{ij}^{\mu}$
$\displaystyle=-\sum_{\langle
i,j\rangle_{\mu}}G_{\mu}\sigma^{\mu}_{i}\sigma^{\mu}_{j}-\tilde{h}\sum_{\langle\\!\langle
i,j,k\rangle\\!\rangle_{\mu,\nu,\lambda}}\sigma_{i}^{\mu}\sigma_{j}^{\nu}\sigma_{k}^{\lambda},$
(3)

where $G_{\mu}=J_{\mu}+\mathrm{i}\gamma_{\mu}=\lvert
G_{\mu}\rvert\mathrm{e}^{\mathrm{i}\phi_{\mu}}$ is the coupling constant of
the Kitaev interactions extended to the complex numbers under the influence of
dissipation. Note that the first term in Eq. (3) is the same as the non-
Hermitian Hamiltonian in the previous study at zero field [67].

The dissipative dynamics interspersed with quantum jumps to different states
triggered by $L_{ij}^{\mu}\rho L_{ij}^{\mu\dagger}$ in Eq. (2) can be
interpreted as the deterministic time evolution by the effective non-Hermitian
Hamiltonian in Eq. (3) [73, 74, 75]. The feasibility of such a time domain was
also discussed in the previous study [67]. This non-Hermitian model is exactly
solvable through the Majorana fermion representation introduced in the
Kitaev’s seminal paper [5], in which spins are represented by four types of
Majorana fermions ($c_{i}$, $b_{i}^{x}$, $b_{i}^{y}$, $b_{i}^{z}$) as
$\sigma^{\mu}_{i}=\mathrm{i}b_{i}^{\mu}c_{i}$. In this representation, the
spin degree of freedom is fractionalized into itinerant Majorana fermions
$c_{i}$ and the localized $\mathbb{Z}_{2}$ fluxes defined as
$W_{p}=\prod_{\langle i,j\rangle_{\mu}\in p}u_{ij}^{\mu}$ on each hexagonal
plaquette labeled $p$ of the honeycomb lattice;
$u_{ij}^{\mu}=\mathrm{i}b_{i}^{\mu}b_{j}^{\mu}$ is a gauge variable taking
$\pm 1$ on the $\mu$ bond connecting site $i$ and $j$ belonging to the A and B
sublattice, respectively. The latter $\mathbb{Z}_{2}$ fluxes are conserved
quantities even in the non-Hermitian case, which enables us to group the
eigenstates into different sectors of the flux configuration as in the
Hermitian case. Although Lieb’s theorem [76] guarantees that the ground state
is flux free ($W_{p}=1$ for all plaquettes) in the Hermitian case, there is no
longer a mathematical theorem for the non-Hermitian case that tells us the
flux configuration in the ground state, which is defined as a state with the
lowest energy and the longest lifetime. In the previous study at zero magnetic
field, it is numerically confirmed that the flux-free state has the lowest
energy and the longest lifetime for some parameters [67]. We assume that this
holds for the present case perturbed by a weak magnetic field. Then, by
choosing all the gauge variables as $u_{ij}^{\mu}=1$, the Hamiltonian in Eq.
(3) is written by a quadratic form of the Majorana fermion $c_{i}$ as

$\mathcal{H}_{\mathrm{eff}}=\mathrm{i}\sum_{\langle
i,j\rangle_{\mu}}G_{\mu}c_{i}c_{j}-\mathrm{i}\tilde{h}\sum_{\\{i,j\\}}c_{i}c_{j},$
(4)

where the second term denotes sublattice-dependent second-neighbor hoppings of
the Majorana fermions; the summation $\\{i,j\\}$ runs over second-neighbor
pairs shown by the dashed arrows from $j$ to $i$ in Fig. 1.

In Secs. III.1 and III.2, we study this model under the periodic boundary
condition (PBC). In this case, we consider the Fourier transform of Eq. (4)
summarized into

$\mathcal{H}_{\mathrm{eff}}=\sideset{}{{}^{{}^{\prime}}}{\sum}_{\mathbf{k}}\begin{pmatrix}c^{\dagger}_{\mathbf{k},\mathrm{A}}&&c^{\dagger}_{\mathbf{k},\mathrm{B}}\end{pmatrix}H(\mathbf{k})\begin{pmatrix}c_{\mathbf{k},\mathrm{A}}\\\
c_{\mathbf{k},\mathrm{B}}\end{pmatrix},$ (5)

where

$\displaystyle
H(\mathbf{k})=\begin{pmatrix}\Delta(\mathbf{k})&&\mathrm{i}\varepsilon(\mathbf{k})\\\
-\mathrm{i}\varepsilon(-\mathbf{k})&&-\Delta(\mathbf{k})\end{pmatrix},$ (6)

with

$\displaystyle\varepsilon(\mathbf{k})$
$\displaystyle=2(G_{x}\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{a}_{1}}+G_{y}\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{a}_{2}}+G_{z}),$
(7) $\displaystyle\Delta(\mathbf{k})$
$\displaystyle=4\tilde{h}[-\sin(\mathbf{k}\cdot\mathbf{a}_{1})+\sin(\mathbf{k}\cdot\mathbf{a}_{2})+\sin\\{\mathbf{k}\cdot(\mathbf{a}_{1}-\mathbf{a}_{2})\\}].$
(8)

Here, $\mathbf{a}_{1}=(1/2,\ \sqrt{3}/2)$ and $\mathbf{a}_{2}=(-1/2,\
\sqrt{3}/2)$ denote the primitive translation vectors as shown in Fig. 1, and
the Fourier transformations of the Majorana fermion operators are defined as

$\displaystyle c^{\dagger}_{\mathbf{k},\mathrm{A}(\mathrm{B})}$
$\displaystyle=\frac{1}{\sqrt{2N}}\sum_{\mathbf{r}}\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{r}}c^{\dagger}_{\mathbf{r},\mathrm{A}(\mathrm{B})},$
(9) $\displaystyle c_{\mathbf{k},\mathrm{A}(\mathrm{B})}$
$\displaystyle=\frac{1}{\sqrt{2N}}\sum_{\mathbf{r}}\mathrm{e}^{-\mathrm{i}\mathbf{k}\cdot\mathbf{r}}c_{\mathbf{r},\mathrm{A}(\mathrm{B})},$
(10)

where $\mathbf{r}$ represents the position of the unit cell to which the
Majorana fermion belongs, and $N$ is the number of the unit cell. The primed
summation in Eq. (5) runs over a half of the first Brillouin zone. The
Hamiltonian is easily diagonalized and the eigenvalues are obtained as

$E_{\pm}(\mathbf{k})=\pm\sqrt{\varepsilon(\mathbf{k})\varepsilon(-\mathbf{k})+\Delta(\mathbf{k})^{2}}.$
(11)

Note that $E_{\pm}(\mathbf{k})\in\mathbb{C}$ because
$\varepsilon(-\mathbf{k})\neq\varepsilon^{*}(\mathbf{k})$, unlike the
Hermitian case where the eigenvalues are always real. The square root of a
complex function is defined as its principal value.

In the absence of the magnetic field ($\tilde{h}=0$), it was shown that the
model has two kinds of phases in the ground state [67]: a gapped quantum spin
liquid phase (A phase) and a gapless quantum spin liquid phase (B phase). The
latter appears in the region where

$\lvert G_{x}\rvert\leq\lvert G_{y}\rvert+\lvert G_{z}\rvert,\ \lvert
G_{y}\rvert\leq\lvert G_{z}\rvert+\lvert G_{x}\rvert,\ \lvert
G_{z}\rvert\leq\lvert G_{x}\rvert+\lvert G_{y}\rvert.$ (12)

Within the gapless phase, a Dirac-like node with the linear dispersion in the
Hermitian case is split into two exceptional points with the square-root
dispersions. We will investigate the effects of the magnetic field in this
phase in the following sections.

We also study the model under the open boundary condition (OBC) in one
direction while keeping the PBC in the other direction. In the non-Hermitian
system with edges, the so-called skin effect can occur by accumulating the
eigenfunctions near the edges [77, 78, 79, 80]. The condition for the non-
Hermitian skin effect was studied for the model in Eq. (4) at zero field [67].
The details of the setup and calculations will be shown in the beginning of
Sec. III.3. We will show that the skin effect appears in a richer form in the
presence of the magnetic field.

##  III Result

###  III.1 Critical effective magnetic field

Figure 2: Majorana excitation spectra of the model in Eq. (5) with
$G_{x}=2.0$, $G_{y}=1.0$, and $G_{z}=2.5\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$
under the effective magnetic field of (a) $\tilde{h}=0.0$, (b)
$\tilde{h}=0.5$, and (c) $\tilde{h}=1.0$. The left panels show the absolute
value of the complex energy eigenvalue, $\lvert E_{+}(\mathbf{k})\rvert$,
together with the real part $\mathrm{Re}E_{+}(\mathbf{k})$ on the bottom plane
[see Eq. (11)]. The right panels show $\mathrm{Re}E_{+}(\mathbf{k})$. The
white and blue lines show the real and imaginary Fermi arcs, which are defined
by $\mathrm{Re}E_{+}(\mathbf{k})=0$ and $\mathrm{Im}E_{+}(\mathbf{k})=0$,
respectively. The black hexagon indicates the first Brillouin zone, and
$\Gamma$, $\mathrm{K}$, $\mathrm{K}^{\prime}$, $\mathrm{M}$, and
$\mathrm{M}^{\prime}$ denote the high symmetry points.  Figure 3: Contour
plots of the critical effective magnetic field $\tilde{h}_{\mathrm{c}}$ on the
complex plane of $G_{z}$ at (a) $G_{x}=2.0$ and $G_{y}=1.0$ and (b)
$G_{x}=2.0\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$ and
$G_{y}=1.0\mathrm{e}^{\mathrm{i}\frac{\pi}{6}}$. The gray and white regions
indicate the gapped A phase at zero magnetic field and the regions where
$\tilde{h}_{\mathrm{c}}>5.0$, respectively. The blue, green, and red rings
show the analytical results in Eq. (13). The black dots indicate the
parameters at which $\varepsilon(\mathbf{k})=\varepsilon(-\mathbf{k})=0$; see
Eqs. (14) and (15). The black dot with an asterisk in (b) denotes the
parameters used in Fig. 4. The black horizontal line at ${\rm Im}G_{z}=0$ in
(a) indicates the Hermitian case.

First, we study how the applied magnetic field modifies the Majorana
excitation spectrum. Figure 2 displays the typical spectra by plotting $\lvert
E_{+}(\mathbf{k})\rvert$ and $\mathrm{Re}E_{+}(\mathbf{k})$ for three values
of the effective magnetic field. Here we choose $G_{x}=2.0$, $G_{y}=1.0$, and
$G_{z}=2.5\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$. These parameters satisfy the
conditions in Eq. (12), i.e., the system is in the gapless B phase at zero
field, as shown in Fig. 2(a); the Majorana fermion dispersion shows
exceptional points with square-root ${\bf k}$ dependence that appear at the
boundaries between the real and imaginary Fermi arcs defined as
$\mathrm{Re}E_{+}(\mathbf{k})=0$ and $\mathrm{Im}E_{+}(\mathbf{k})=0$, shown
by the white and blue lines, respectively. When $\tilde{h}$ is introduced, the
system remains gapless at the exceptional points, as shown in Fig. 2(b). While
increasing $\tilde{h}$, the exceptional points move to shrink the real Fermi
arcs, and finally, annihilate in pairs at a critical effective magnetic field
$\tilde{h}_{\mathrm{c}}$. For $\tilde{h}>\tilde{h}_{\mathrm{c}}$, the Majorana
spectrum is gapped, as shown in Fig. 2(c).

Thus, in the non-Hermitian case, the exceptional points are gapped out at a
nonzero effective magnetic field. This is in stark contrast to the Hermitian
case where the Dirac-like linear dispersion is gapped out by an infinitesimal
effective magnetic field [5]. The qualitative difference stems from the
different number of conditions for the gapless states between the Hermitian
and non-Hermitian cases [81, 61]: In the Hermitian case, three conditions are
required in the $2\times 2$ Bloch Hamiltonian matrix for gap closing, but they
are reduced to two in the non-Hermitian case, thus stabilizing the gapless
state more easily.

The value of the critical field $\tilde{h}_{\mathrm{c}}$ depends on the
parameters $G_{\mu}$. We plot $\tilde{h}_{\mathrm{c}}$ for two sets of $G_{x}$
and $G_{y}$ on the complex $G_{z}$ plane in Fig. 3. Here, we numerically
estimate $\tilde{h}_{\mathrm{c}}$ by the value for which the minimum of
$\lvert E_{+}(\mathbf{k})\rvert^{4}$ exceeds a threshold of $5.0\times
10^{-4}$ while increasing $\tilde{h}$ from $0.0$ to $5.0$ in $0.001$
increments; we calculate $E_{+}(\mathbf{k})$ on $500\times 500$
$\mathbf{k}$-point mesh. The gray regions indicate the A phase, i.e., in these
regions the Majorana spectrum is already gapped at $\tilde{h}=0$. The white
regions indicate the parameter regions where $\tilde{h}_{\mathrm{c}}>5.0$. We
find that $\tilde{h}_{\mathrm{c}}$ is nonzero in all regions where the system
is in the gapless B phase at zero field, and exhibits complicated dependences
on the coupling constants $G_{\mu}$, except for the black line in Fig. 3(a)
corresponding to the Hermitian case.

In Fig. 3, the white regions where $\tilde{h}_{\mathrm{c}}$ becomes large
appear in specific regions on the $G_{z}$ plane. To understand the origin of
these behaviors, we analyze the $G_{\mu}$ dependence of the critical field
$\tilde{h}_{c}$, and find that $\tilde{h}_{c}$ diverges, within the
perturbation theory, for the parameters

$\displaystyle\lvert G_{x}\rvert=\lvert G_{y}+G_{z}\rvert,\ \lvert
G_{y}\rvert=\lvert G_{z}+G_{x}\rvert,\ \lvert G_{z}\rvert=\lvert
G_{x}+G_{y}\rvert.$ (13)

See Appendix A for the derivation. These divergence conditions are shown by
the blue, green, and red rings in Fig. 3; the white regions appear along these
three rings. Furthermore, assuming that $\varepsilon(\mathbf{k})=0$ and
$\varepsilon(-\mathbf{k})=0$ are satisfied simultaneously at the same
$\mathbf{k}$, we obtain four more special solutions from Eq. (13):

$\displaystyle G_{x}$ $\displaystyle=G_{y}+G_{z},\ G_{y}=G_{z}+G_{x},\
G_{z}=G_{x}+G_{y},$ (14) $\displaystyle G_{z}$ $\displaystyle=-G_{x}-G_{y}.$
(15)

We also show these solutions by the black dots in Fig. 3. Equation (14)
correspond to the three black dots on the blue, green, and red ring, while Eq.
(15) corresponds to the dot at the intersection of the three rings. We note
that the white regions along the rings become wide near these four dots,
especially the last one for Eq. (15), indicating strong divergence around
these parameters.

###  III.2 Topological transitions by the effective magnetic field

Figure 4: Topological transitions caused by increasing $\tilde{h}$. In (a) and
(c), the left panels show $\mathrm{Re}E_{+}(\mathbf{k})$, and the right panels
show $\lvert E_{+}(\mathbf{k})\rvert$ with $\mathrm{Re}E_{+}(\mathbf{k})$ on
the bottom plane within the regions indicated by the white dashed rectangles
in the left panels. In (b) and (d), the left panels show
$\mathrm{Arg}[\mathrm{det}H(\mathbf{k})]$, and the right panels show enlarged
views of the regions indicated by the black dashed squares in the left panels.
The data are plotted for the parameters $G_{\mu}$ indicated by the black dot
with an asterisk in Fig. 3(b) at $\tilde{h}=0.2$ in (a) and (b), and
$\tilde{h}=0.3$ in (c) and (d). The notations in (a) and (c) are common to
those in Fig. 2. The black dots in (b) and (d) indicate the exceptional
points, and the white ones in (d) indicate those generated in pairs by the
topological transition. The white triangles indicate the exceptional points
that change the associated winding numbers $W$ in Eq. (16) through the
topological transition.  Figure 5: $\tilde{h}$ dependences of the absolute
values of poles $\lvert w_{+}\rvert$ (green) and $\lvert w_{-}\rvert$ (blue)
in Eq. (17) for the same $G_{\mu}$ as Fig. 4. The red vertical line indicates
the effective magnetic field $\tilde{h}_{*}$ at which the topological
transition occurs with the change in the winding number $W$ from $2$ to $0$ by
$|w_{-}|$ exceeding $1$.

The exceptional points correspond to the zeros of the complex functions
$E_{\pm}(\mathbf{k})$. One can define the winding number around each of them
as [82]

$W=\oint_{C}\frac{\mathrm{d}\mathbf{k}}{2\pi\mathrm{i}}\nabla_{\mathbf{k}}\ln\det[H(\mathbf{k})-E(\mathbf{k}_{\mathrm{EP}})],$
(16)

where $H(\mathbf{k})$ and $E(\mathbf{k}_{\mathrm{EP}})$ represent the Bloch
Hamiltonian with the wave vector $\mathbf{k}$ in Eq. (6) and the energy
eigenvalue at an exceptional point $\mathbf{k}=\mathbf{k}_{\mathrm{EP}}$,
respectively; the contour $C$ is taken to enclose an exceptional point
counterclockwise. Performing the integration, we obtain $W=\pm 1$ for the
exceptional points with the square-root dispersion in Figs. 2(a) and 2(b).
Similarly, the winding number takes $\pm 1$ up to $\tilde{h}_{\mathrm{c}}$ for
generic parameters included in the B phase, whereas it is no longer well
defined for $\tilde{h}>\tilde{h}_{\mathrm{c}}$ since the exceptional points
are gapped out. Details of the calculations of the winding number are
described in Appendix B.

For the special parameters satisfying one of Eqs. (14) and (15), however, we
find a topological transition associated with the change of $W$ within the
gapless phase. In these cases, the system exhibits two types of exceptional
points at zero field: one with square-root dispersion as for the other
parameters, and the other with linear dispersion satisfying
$\varepsilon(\mathbf{k}_{\mathrm{EP}})=\varepsilon(-\mathbf{k}_{\mathrm{EP}})=0$.
In the latter, the linear dispersion appears as $\varepsilon(\mathbf{k})$ and
$\varepsilon(-\mathbf{k})$ become zero simultaneously at the same
$\mathbf{k}=\mathbf{k}_{\mathrm{EP}}$, and hence, these exceptional points
have the winding number $W=\pm 2$ each at zero field. Interestingly, we find
that the winding number changes from $W=\pm 2$ to $W=0$ while increasing
$\tilde{h}$, indicating the topological transition.

We demonstrate this in Fig. 4 for the parameters indicated by the black dot
with an asterisk in Fig. 3(b). In this case, below the critical field
$\tilde{h}_{*}$ (in this specific case, $\tilde{h}_{*}=0.25$; see Appendix C),
we find the exceptional point with linear dispersion at the crossing point of
the real and imaginary Fermi arcs as shown in Fig. 4(a), in addition to a pair
of the exceptional points with square-root dispersion. The winding number for
the former takes $W=+2$, while that for the latter is $W=\pm 1$, as shown in
the plot of the complex phase of the Bloch Hamiltonian in Fig. 4(b). With the
increase of $\tilde{h}$ above $\tilde{h}_{*}$, the former turns into $W=0$
through the topological transition. This is caused by pair creations of
additional exceptional points indicated by the white dots in the left panel of
Fig. 4(d), leading to the splitting of the real Fermi arcs indicated by the
white lines in Figs. 4(a) and 4(c). We note that the exceptional point with
$W=0$ retains linear dispersion, as shown in Fig. 4(c).

The changes of the winding number at the topological transition can be
understood from the analysis of the poles in the integrand of Eq. (16). By
taking the contour $C$ as a sufficiently small circle of radius $r$ around the
exceptional point with parametrizing $\mathbf{k}$ as
$\mathbf{k}=\mathbf{k}_{\mathrm{EP}}+r(\cos\theta,\ \sin\theta)$
($0\leq\theta<2\pi$) and transforming the integral variable as
$z=\mathrm{e}^{2\mathrm{i}\theta}$, we obtain a formula for the winding number
as

$\displaystyle W=\oint_{\lvert
z\rvert=1}\frac{\mathrm{d}z}{2\pi\mathrm{i}}\frac{2}{z}\frac{z^{2}-w_{+}w_{-}}{(z-w_{+})(z-w_{-})},$
(17)

where $r$ appears in both the denominator and numerator and thus they cancel
each other out. See Appendix B for the detailed calculations. The integrand
has three poles: $z=0$, $w_{+}$, and $w_{-}$ with residues $-2$, $+2$, and
$+2$, respectively. The pole at $z=0$ is always in the unit circle, while the
positions of the other two $z=w_{\pm}$ depend on $\tilde{h}$ as well as
$G_{\mu}$. When both are inside the unit circle at $\tilde{h}=0$, one of them
moves outside the unit circle at a critical value $\tilde{h}=\tilde{h}_{*}$ as
$\tilde{h}$ is increased. Conversely, when they are both outside the unit
circle at $\tilde{h}=0$, one of the poles moves inside the unit circle at
$\tilde{h}_{*}$. In the former (latter) case, the winding number changes from
$+2$ ($-2$) to $0$ at $\tilde{h}_{*}$, leading to the topological transition.
Figure 5 shows the $\tilde{h}$ dependences of $\lvert w_{\pm}\rvert$ of the
exceptional point indicated by the white triangle in Figs. 4(b) and 4(d),
which corresponds to the former case. We derive the concrete formula of
$\tilde{h}_{*}$ analytically and discuss the conditions for the topological
transition in Appendix C.

Thus, our non-Hermitian Hamiltonian in Eq. (5) exhibits topological
transitions with the increase of the magnetic field. This provides an
interesting example of topological transitions induced by the magnetic field,
where the winding number of the exceptional points changes within the gapless
phase. We will discuss this issue in Sec. IV.1.

###  III.3 Non-Hermitian skin effect

Figure 6: Schematic picture of the system with the open boundary condition
(OBC) in the $y$ direction, which leaves zigzag edges. We take the periodic
boundary condition (PBC) in the $x$ direction. $M$ denotes the number of the
layers.  Figure 7: $k_{x}$ dependences of the absolute value of the complex
energy eigenvalue, $\lvert E\rvert$, for the system under the PBC in the $x$
direction and the OBC in the $y$ direction at $G_{x}=2.0$, $G_{y}=1.0$, and
$G_{z}=2.5\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$ with (a) $\tilde{h}=0.0$, (b)
$\tilde{h}=0.1$, (c) $\tilde{h}=0.5$, and (d) $\tilde{h}=1.0$. We take $M=240$
and $1000$ $k_{x}$-mesh in the first Brillouin zone. The color bar shows the
center of mass of the wave function in the $y$ direction, $\bar{m}$ in Eq.
(24). The data for the system with PBCs in both directions are shown in gray,
where we take 1000 $k_{x}$\- and 480 $k_{y}$-mesh in the first Brillouin zone.
Figure 8: (Upper panels) $k_{x}$ dependences of $\lvert E\rvert$ at
$G_{x}=2.0\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$,
$G_{y}=1.0\mathrm{e}^{\mathrm{i}\frac{\pi}{6}}$, and $G_{z}=2.5$, and (a)
$\tilde{h}=0.0$, (b) $\tilde{h}=0.05$, (c) $\tilde{h}=0.1$, and (d)
$\tilde{h}=0.2$. The notations are common to those in Fig. 7. We take the same
$M$ and $\mathbf{k}$-mesh as Fig. 7. Due to numerical precision, we obtain the
zero-energy mode in (a) by taking $M=220$ and plot it with $\bar{m}$ rescaled
to $M=480$. (Lower panels) $\mathrm{Re}E_{+}(\mathbf{k})$ for the system at
the same parameters with the PBCs in both directions. The notations are common
to those in Fig. 2. The gray dashed lines connect the position of the
exceptional points in the lower panels and the zeros in the upper panels. At
these values of $k_{x}$, the edge modes switch their chirality, as shown by
the color changes between blue and red in the upper panels.

Next, we study the effect of a magnetic field on the non-Hermitian skin effect
by introducing the OBC in one direction and retaining the PBC in the other
direction. The setup is shown in Fig. 6. By cutting the $z$ bonds colored red,
we make zigzag open edges at the top and bottom of the system in the $y$
direction. Performing the Fourier transformation in the $x$ direction, we can
write the Hamiltonian in Eq. (4) as

$\mathcal{H}_{\mathrm{eff}}=\sum_{k_{x}>0}\sum_{m,m^{\prime}=1}^{M}H_{m^{\prime},m}(k_{x})c_{-k_{x},m^{\prime}}c_{k_{x},m},$
(18)

where

$H(k_{x})=\begin{pmatrix}m(k_{x})&\mathrm{i}r(k_{x})&d(k_{x})&0&\cdots\\\
-\mathrm{i}r(-k_{x})&-m(k_{x})&-\mathrm{i}t&-d(k_{x})&\cdots\\\
-d(-k_{x})&\mathrm{i}t&m(k_{x})&\mathrm{i}r(k_{x})&\ddots\\\
0&d(-k_{x})&-\mathrm{i}r(-k_{x})&-m(k_{x})&\ddots\\\
\vdots&\vdots&\ddots&\ddots&\ddots\end{pmatrix}.$ (19)

Here, the matrix elements are given by

$\displaystyle r(k_{x})$
$\displaystyle=2(G_{x}\mathrm{e}^{\mathrm{i}k_{x}/2}+G_{y}\mathrm{e}^{-\mathrm{i}k_{x}/2}),$
(20) $\displaystyle t$ $\displaystyle=2G_{z},$ (21) $\displaystyle d(k_{x})$
$\displaystyle=-4\tilde{h}\sin(k_{x}/2),$ (22) $\displaystyle m(k_{x})$
$\displaystyle=4\tilde{h}\sin(k_{x}).$ (23)

In the previous study [67], the condition for the non-Hermitian skin effect
for the zero magnetic field case was clarified: The skin effect occurs when
$\phi_{x}-\phi_{y}\neq 0$ (mod $\pi$), where $\phi_{\mu}$ is the complex phase
of $G_{\mu}$, in the setting shown in Fig. 6.

Figure 7 shows the absolute value of the complex energy eigenvalues of the
system, $\lvert E\rvert$, at $G_{x}=2.0$, $G_{y}=1.0$, and
$G_{z}=2.5\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$, obtained by the numerical
diagonalization of Eq. (18). The accuracy of the diagonalization of the non-
Hermitian matrices is significantly reduced when the skin effect occurs, and
hence, we perform our calculations with quadruple accuracy by using QEISPACK
package [83]. Due to the particle-hole symmetry of the system, both states
with the energy $E$ and $-E$ are eigenstates, but here we only show the states
with $\mathrm{Re}E\geq 0$. The color indicates the average localization of the
wave function defined as

$\bar{m}(n,k_{x})=\sum_{m=1}^{M}m\lvert\psi_{n}(k_{x},m)\rvert^{2},$ (24)

where $\psi_{n}(k_{x},m)$ is the normalized wave function of the $n$th
eigenvalue at ($k_{x}$, $m$). For comparison, we also plot $\lvert
E_{+}\rvert$ for the system with PBCs in both directions in gray. For this
parameter setting with $\phi_{x}=\phi_{y}=0$, the skin effect does not occur
for $\tilde{h}=0$ as shown in the previous study [67]. Indeed, in our result
for $\tilde{h}=0$ shown in Fig. 7(a), no skin effect occurs as all the
eigenstates are colored green. In this case, the spectra for the OBC-PBC and
PBC-PBC cases coincide with each other, except for the modes lying at $\lvert
E\rvert=0$ in the former case. We confirm that the zero-energy modes are edge
modes by calculating the average square position; see Appendix D. For
$\tilde{h}>0$, however, the skin effect occurs as shown in Figs. 7(b)–7(d),
and the difference between the OBC-PBC and PBC-PBC cases becomes pronounced
with the increase of $\tilde{h}$. The eigenstates colored blue are localized
at the lower edge of the system, while those colored red are localized at the
upper edge. This indicates that the skin effect is induced by the application
of the magnetic field, even for the parameters where it is absent at zero
magnetic field.

We note that the zero-energy edge modes at $\tilde{h}=0$ are colored green,
meaning that they are equally localized at both the upper and lower edges of
the system (see Appendix D). When we turn on $\tilde{h}$, these modes are
lifted from zero energy except for $k_{x}=\pi$, and colored blue and red for
$k_{x}<\pi$ and $k_{x}>\pi$, respectively. This indicates that they are chiral
edge modes whose localization direction is different for $k_{x}<\pi$ and
$k_{x}>\pi$. These behaviors are similar to those of the edge modes in the
Hermitian case. When the dissipation vanishes, these chiral edge modes are
continuously connected to those in the Hermitian systems.

Next, we show the results for a parameter where the skin effect occurs for
$\tilde{h}=0$ in Fig. 8. Here we choose
$G_{x}=2.0\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$,
$G_{y}=1.0\mathrm{e}^{\mathrm{i}\frac{\pi}{6}}$, and $G_{z}=2.5$, and thus
$\phi_{x}-\phi_{y}\neq 0$. We can see in Fig. 8(a) that the skin effect
already occurs for $\tilde{h}=0$ as expected, and that the OBC-PBC and PBC-PBC
spectra are clearly different. When $\tilde{h}$ is introduced, the bulk
spectrum does not change significantly, but the edge modes are modified; not
only they are lifted from zero energy except at $k_{x}=\pi$, but also the
localization direction switches at certain $k_{x}\neq\pi$, as shown in Figs.
8(b)–8(d). See also Appendix D. This is regarded as the switching of chirality
of the edge modes. Notably, the wavenumbers for this chirality switching
correspond to the exceptional points of the PBC-PBC spectra, as shown by the
gray dashed lines in Fig. 8. We will discuss this interesting correspondence
in analogy with three-dimensional Hermitian Weyl semimetals in Sec. IV.2.

##  IV Discussion

###  IV.1 Gapless-gapless topological transitions by a magnetic field

In Sec. III.2, we found interesting topological transitions with changes in
the winding number of the exceptional points caused by the magnetic field. It
is worth noting that the exceptional points whose winding numbers change
[indicated by the white triangles in Figs. 4(b) and 4(d)] are gapless
throughout the topological transition, while new exceptional points are
created in pairs [indicated by the white dots in Fig. 4(d)]. This is
qualitatively different from the topological transitions studied for other
non-Hermitian systems [84, 85, 86, 87]. In these cases, while the winding
number or a topological number called vorticity [84, 88] changes between two
gapless phases, the topological transitions take place only when a gap opens
once between them.

It is also worth highlighting that the topological transition in this study is
driven by the external magnetic field. This is in contrast to other non-
Hermitian systems where the topological transitions are caused by changes in
the internal model parameters, such as the tight-binding parameters [84, 85,
86, 87]. This difference stems from the fact that for the Kitaev model the
magnetic field effectively generates the second-neighbor hoppings of the
Majorana fermions, causing a topological transition through the modulation of
the tight-binding parameters. Thus, our finding offers an interesting example
of non-Hermitian topological transitions driven by an external field.

###  IV.2 Analogy with the three-dimensional Weyl semimetals

Figure 9: Schematics of (a) edge modes in the two-dimensional non-Hermitian
Kitaev model and (b) surface modes in a three-dimensional Hermitian Weyl
semimetal, both under a magnetic field. The upper panels of (a) and (b) show
the position of EPs and Weyl points in momentum space, respectively. The lower
panels of (a) and (b) show switching of the edge and surface modes,
respectively. In (a), only one of two chiral edge modes is shown. The gray
dashed lines in (a) and (b) show the correspondence between the positions of
EPs and Weyl points in the upper panels and those of switching points of the
edge and surface modes in the lower panels, respectively.

In Sec. III.3, we found that the chirality of the edge modes is switched at
the wavenumbers corresponding to the exceptional points in the bulk spectra.
This is similar to the switching of surface Fermi arcs in three-dimensional
Hermitian Weyl semimetals under a magnetic field. In those systems, the
surface Fermi arc states switch from one surface to the opposite at
wavenumbers corresponding to the projection of the Weyl points in the bulk
spectra [89, 90, 91]. A schematic picture of this similarity is shown in Fig.
9. Thus, this similarity provides an example of the analogy between two-
dimensional non-Hermitian systems and three-dimensional Hermitian topological
systems [61]. While this analogy is well known in the PBC cases for general
spatial dimensions [92, 93] due to the number of gap-closing conditions and
degree of freedom [94, 61], our result gives a rare example where the analogy
holds for the boundary modes in the OBC cases as well.

###  IV.3 Other contributions within the third-order perturbation

In the derivation of the Hamiltonian in Eq. (1) by the third-order
perturbation in terms of the Zeeman coupling, we neglect the second-order
contribution and the third-order contribution that gives rise to the four-
Majorana interaction terms [5]. Here we discuss the effects of these
additional contributions, following the previous study [95].

The second-order contribution is given by

$\displaystyle\mathcal{H}_{2}=-\delta\sum_{\langle
i,j\rangle_{\mu}}h_{\mu}^{2}\sigma^{\mu}_{i}\sigma^{\mu}_{j}=\mathrm{i}\delta\sum_{\langle
i,j\rangle_{\mu}}h_{\mu}^{2}c_{i}c_{j},$ (25)

where $\delta$ is assumed to be a constant. This correction gives a bond-
dependent change in the Kitaev interaction $J_{\mu}$. Meanwhile, the third-
order contribution that we neglect in Eq. (1) is given by

$\displaystyle\mathcal{H}_{4}=-\tilde{h}\sum_{[i,j,k]_{\mu,\nu,\lambda}}\sigma^{\mu}_{i}\sigma^{\nu}_{j}\sigma^{\lambda}_{k}=-\tilde{h}\sum_{[i,j,k;l]}(-1)^{l}c_{i}c_{j}c_{k}c_{l},$
(26)

where the factor $(-1)^{l}$ takes $+1$ ($-1$) when site $l$ belongs to A (B)
sublattice. The summation of $[i,j,k]_{\mu,\nu,\lambda}$ runs over triplets of
second-neighbor sites $i$, $j$, and $k$ connected to the common neighboring
site by $\mu$, $\nu$, and $\lambda$ bond, respectively
($\mu\neq\nu\neq\lambda\neq\mu$), while the summation of $[i,j,k;l]$ runs over
triplets of second-neighbor sites $i$, $j$, and $k$ connected to the common
neighboring site $l$. Equation (26) includes the four-Majorana interaction
terms, which hamper the exact solvability of the problem. Within a mean-field
approximation, it leads to modifications of both $J_{\mu}$ and $\tilde{h}$ in
Eq. (1). Another possible effect of the four-Majorana interactions is an
instability of the Majorana Fermi surfaces [96], which appear in the presence
of the non-Kitaev interactions [97], electric fields [96], or staggard
magnetic fields [95].

Summarizing Eqs. (25) and (26) within the mean-field approximation, we obtain
the Hamiltonian in the form of Eq. (4) with normalized $G_{\mu}$ and
$\tilde{h}$ as

$\displaystyle G_{\mu}$ $\displaystyle\rightarrow G_{\mu}+\delta
h_{\mu}^{2}-\tilde{h}(\Delta_{\mathrm{AA}}+\Delta_{\mathrm{BB}}),$ (27)
$\displaystyle\tilde{h}$
$\displaystyle\rightarrow\tilde{h}(1+\Delta_{\mathrm{AB}}),$ (28)

where we assume spatially uniform mean fields
$\Delta_{\mathrm{AA}}=\mathrm{i}\langle
c_{i\in\mathrm{A}}c_{j\in\mathrm{A}}\rangle$,
$\Delta_{\mathrm{BB}}=\mathrm{i}\langle
c_{i\in\mathrm{B}}c_{j\in\mathrm{B}}\rangle$, and
$\Delta_{\mathrm{AB}}=\mathrm{i}\langle
c_{i\in\mathrm{A}}c_{j\in\mathrm{B}}\rangle$ to decouple the four-Majorana
interactions in Eq. (26). We note that $\Delta_{\mathrm{AB}}$ is estimated as
$\sim-0.52$ in the zero magnetic field [98, 99, 100, 101], and
$\Delta_{\mathrm{AA}}=\Delta_{\mathrm{BB}}$ are estimated as between $-0.30$
and $-0.20$ at $\tilde{h}=1.0$ [101, 102], for the Hermitian case with
isotropic $J_{x}=J_{y}=J_{z}=1$. We emphasize that incorporating these
corrections does not qualitatively change our results, only shifts the values
of $J_{\mu}$ and $\tilde{h}$.

##  V Summary

To summarize, we have studied the effect of a magnetic field on the non-
Hermitian Kitaev model, which effectively describes the Kitaev model coupled
to the environment, based on the perturbation theory with respect to the field
strength. We showed that the exceptional points remain gapless up to a nonzero
critical field, in stark contrast to the Hermitian case where the Dirac-like
nodes are gapped out immediately by applying the magnetic field. We also
clarified that the value of the critical effective field depends on the
coupling constants, grows beyond the field range where the perturbation theory
is valid, and even diverges for some particular parameter conditions on the
complex plane of the coupling constants. By calculating the winding number of
the exceptional points, we found the topological transition caused by the
effective magnetic field within the gapless phase. This offers an interesting
example of non-Hermitian topological transitions between two gapless states
without opening a gap driven by an external field. In addition, in the system
with edges, we showed that the non-Hermitian skin effect can be induced by
applying the magnetic field even for the parameters where it does not occur at
zero field. We also clarified that the chirality of the edge modes is switched
at the wavenumbers corresponding to the exceptional points in the bulk
spectrum. This is analogous to the switching of the Fermi arc surface states
through the projections of the Weyl points in three-dimensional Weyl
semimetals, suggesting a correspondence between the $d$-dimensional non-
Hermitian systems and the $(d+1)$-dimensional Hermitian topological systems.

Our results provide us with a new possible route to stabilize gapless
topological quantum spin liquids under the magnetic field in the presence of
dissipation. In the particular case, it contains an intriguing example of the
topological transitions without gap opening. Our results also provide new
insight into the connection between the topology of the quantum spin liquids
and their edge modes in the presence of dissipation. In particular, the
finding of the chirality switching of the edge modes suggests the possibility
of controlling the direction of the edge modes by dissipation. Thus, our
results pave the way for extending the physics of quantum spin liquids to open
quantum systems. Further research, especially on other types of quantum spin
liquids, is desired.

###### Acknowledgements.

K.F. thanks K. Kawabata, S. Kitahama, H. Obuse, K. Shimizu, K. Sone, and M.
Ueda for constructive suggestions. The authors thank T. Misawa, J. Nasu, and
T. Okubo for fruitful discussions. Parts of the numerical calculations have
been done using the facilities of the Supercomputer Center, the Institute for
Solid State Physics, the University of Tokyo, the Information Technology
Center, the University of Tokyo, and the Center for Computational Science,
University of Tsukuba. This work was supported by Japan Society for the
Promotion of Science (JSPS) KAKENHI Grant Nos. 19H05825, 20H00122, and
22K03509.

##  Appendix A Conditions for divergence of $\tilde{h}_{\mathrm{c}}$

In this Appendix, we derive the divergence conditions of
$\tilde{h}_{\mathrm{c}}$ in Eqs. (13), (14), and (15). The conditions are
obtained by noting that the energy dispersion in Eq. (11) remains gapless at
the exceptional points satisfying $\varepsilon(\mathbf{k})=0$ or
$\varepsilon(-\mathbf{k})=0$, if $\Delta(\mathbf{k})=0$ for $\tilde{h}>0$.
There are three possibilities to fulfill this situation: (i)
$\tilde{k}_{1}=\tilde{k}_{2}$ (mod $2\pi$), (ii) $\tilde{k}_{1}\neq 0$ and
$\tilde{k}_{2}=0$ (mod $2\pi$), and (iii) $\tilde{k}_{1}=0$ and
$\tilde{k}_{2}\neq 0$ (mod $2\pi$), where
$\tilde{k}_{1}=\mathbf{k}\cdot\mathbf{a}_{1}$ and
$\tilde{k}_{2}=\mathbf{k}\cdot\mathbf{a}_{2}$.

For the case (i), from $\varepsilon(\mathbf{k})=0$ in Eq. (7), i.e.,
$\mathrm{Re}\varepsilon(\mathbf{k})=0$ and
$\mathrm{Im}\varepsilon(\mathbf{k})=0$, we obtain

$\begin{pmatrix}c&-s\\\ s&c\end{pmatrix}\begin{pmatrix}\cos\tilde{k}_{1}\\\
\sin\tilde{k}_{1}\end{pmatrix}+\begin{pmatrix}u\\\ v\end{pmatrix}=0,$ (29)

where $c=\lvert G_{x}\rvert\cos\phi_{x}+\lvert G_{y}\rvert\cos\phi_{y}$,
$s=\lvert G_{x}\rvert\sin\phi_{x}+\lvert G_{y}\rvert\sin\phi_{y}$, and
$G_{z}=u+\mathrm{i}v$. This leads to

$\cos\tilde{k}_{1}=-\frac{cu+sv}{c^{2}+s^{2}},\ \sin\tilde{k}_{1}=\frac{su-
cv}{c^{2}+s^{2}}.$ (30)

Combing these relations with $\cos^{2}\tilde{k}_{1}+\sin^{2}\tilde{k}_{1}=1$,
we obtain the condition

$u^{2}+v^{2}=c^{2}+s^{2},$ (31)

which corresponds to the ring given by $\lvert G_{z}\rvert=\lvert
G_{x}+G_{y}\rvert$ in Eq. (13). Similarly, for $\varepsilon(-\mathbf{k})=0$,
we obtain

$\cos\tilde{k}_{1}=-\frac{cu+sv}{c^{2}+s^{2}},\ \sin\tilde{k}_{1}=-\frac{su-
cv}{c^{2}+s^{2}},$ (32)

which leads to the same condition. When $\varepsilon(\mathbf{k})=0$ and
$\varepsilon(-\mathbf{k})=0$ are satisfied simultaneously, Eqs. (30) and (32)
give $su-cv=0$. This condition and Eq. (31) leads to

$(u,v)=(\pm c,\pm s).$ (33)

The upper and lower signs correspond to the last equation in Eq. (14) and Eq.
(15), respectively.

For the case (ii), we can rewrite Eq. (7) as
$\varepsilon(\mathbf{k})=2\mathrm{e}^{\mathrm{i}\phi_{y}}\varepsilon^{\prime}$
with

$\displaystyle\varepsilon^{\prime}=\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\tilde{k}_{1}+\mathrm{i}\Delta\phi}+\lvert
G_{y}\rvert+u^{\prime}+\mathrm{i}v^{\prime},$ (34)

where $\Delta\phi=\phi_{x}-\phi_{y}$ and $\lvert
G_{z}\rvert\mathrm{e}^{\mathrm{i}\phi_{z}-\mathrm{i}\phi_{y}}=u^{\prime}+\mathrm{i}v^{\prime}$.
To satisfy $\varepsilon(\mathbf{k})=0$, we need
$\mathrm{Re}\varepsilon^{\prime}=0$ and $\mathrm{Im}\varepsilon^{\prime}=0$,
which result in

$\displaystyle\cos(\tilde{k}_{1}+\Delta\phi)=-\frac{\lvert
G_{y}\rvert+u^{\prime}}{\lvert G_{x}\rvert},$ (35)

and

$\displaystyle\sin(\tilde{k}_{1}+\Delta\phi)=-\frac{v^{\prime}}{\lvert
G_{x}\rvert},$ (36)

respectively. From these two and
$\cos^{2}(\tilde{k}_{1}+\Delta\phi)+\sin^{2}(\tilde{k}_{1}+\Delta\phi)=1$, we
obtain the condition

$(u^{\prime}+\lvert G_{y}\rvert)^{2}+v^{\prime 2}=\lvert G_{x}\rvert^{2},$
(37)

which corresponds to the ring given by $\lvert G_{x}\rvert=\lvert
G_{y}+G_{z}\rvert$ in Eq. (13). For the case with
$\varepsilon(-\mathbf{k})=0$, we obtain similar relations

$\displaystyle\cos(-\tilde{k}_{1}+\Delta\phi)$ $\displaystyle=-\frac{\lvert
G_{y}\rvert+u^{\prime}}{\lvert G_{x}\rvert},$ (38)
$\displaystyle\sin(-\tilde{k}_{1}+\Delta\phi)$
$\displaystyle=-\frac{v^{\prime}}{\lvert G_{x}\rvert},$ (39)

which leads to the same condition as Eq. (37). When
$\varepsilon(\mathbf{k})=0$ and $\varepsilon(-\mathbf{k})=0$ are both
satisfied, Eqs. (35), (36), (38), and (39) give

$(u^{\prime},v^{\prime})=(-\lvert G_{y}\rvert\pm\lvert
G_{x}\rvert\cos\Delta\phi,\pm\lvert G_{x}\rvert\sin\Delta\phi).$ (40)

The upper and lower signs give the first equation in Eq. (14) and Eq. (15),
respectively.

Finally, for the case (iii), by replacing $\tilde{k}_{1}$ by $\tilde{k}_{2}$
and interchanging $G_{x}$ and $G_{y}$ in the derivation for the case (ii), we
obtain

$(u^{\prime}+\lvert G_{x}\rvert)^{2}+v^{\prime 2}=\lvert G_{y}\rvert^{2},$
(41)

which corresponds to the ring given by $\lvert G_{y}\rvert=\lvert
G_{z}+G_{x}\rvert$ in Eq. (13). In addition, we can also obtain the condition
for the point satisfying $\varepsilon(\mathbf{k})=\varepsilon(-\mathbf{k})=0$:

$(u^{\prime},v^{\prime})=(-\lvert G_{x}\rvert\pm\lvert
G_{y}\rvert\cos\Delta\phi,\mp\lvert G_{y}\rvert\sin\Delta\phi).$ (42)

The upper and lower signs correspond to the second equation in Eq. (14) and
Eq. (15), respectively.

##  Appendix B Calculations of the winding number

In this Appendix, we calculate the winding number of the exceptional points in
Eq. (16) analytically. From the explicit form of the Majorana Bloch
Hamiltonian in Eq. (5), the integrand can be expressed as

$\displaystyle\nabla_{\mathbf{k}}\mathrm{ln}\det[H(\mathbf{k})]$
$\displaystyle=\frac{[\nabla_{\mathbf{k}}\varepsilon(\mathbf{k})]\varepsilon(-\mathbf{k})+\varepsilon(\mathbf{k})\nabla_{\mathbf{k}}\varepsilon(-\mathbf{k})+2\Delta(\mathbf{k})\nabla_{\mathbf{k}}\Delta(\mathbf{k})}{\varepsilon(\mathbf{k})\varepsilon(-\mathbf{k})+\Delta(\mathbf{k})^{2}}.$
(43)

Note that $E(\mathbf{k}_{\mathrm{EP}})=0$. We expand Eq. (43) near the
exceptional point at $\mathbf{k}=\mathbf{k}_{\mathrm{EP}}$ as

$\displaystyle\varepsilon(\mathbf{k})$
$\displaystyle\simeq\varepsilon(\mathbf{k}_{\mathrm{EP}})+\mathrm{i}A_{+}(k_{x}-k_{\mathrm{EP},x})+\mathrm{i}B_{+}(k_{y}-k_{\mathrm{EP},y}),$
(44) $\displaystyle\varepsilon(-\mathbf{k})$
$\displaystyle\simeq\varepsilon(-\mathbf{k}_{\mathrm{EP}})+\mathrm{i}A_{-}(k_{x}-k_{\mathrm{EP},x})+\mathrm{i}B_{-}(k_{y}-k_{\mathrm{EP},y}),$
(45) $\displaystyle\Delta(\mathbf{k})$
$\displaystyle\simeq\Delta(\mathbf{k}_{\mathrm{EP}})+D_{1}(k_{x}-k_{\mathrm{EP},x})+D_{2}(k_{y}-k_{\mathrm{EP},y}),$
(46)

where

$\displaystyle A_{\pm}$ $\displaystyle=\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}(\pm\mathbf{k}_{\mathrm{EP}}\cdot\mathbf{a}_{1}+\phi_{x})}-\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}(\pm\mathbf{k}_{\mathrm{EP}}\cdot\mathbf{a}_{2}+\phi_{y})},$
(47) $\displaystyle B_{\pm}$ $\displaystyle=\sqrt{3}[\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}(\pm\mathbf{k}_{\mathrm{EP}}\cdot\mathbf{a}_{1}+\phi_{x}}+\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}(\pm\mathbf{k}_{\mathrm{EP}}\cdot\mathbf{a}_{2}+\phi_{y})}],$
(48) $\displaystyle D_{1}$
$\displaystyle=2\tilde{h}[-\cos(\mathbf{k}_{\mathrm{EP}}\cdot\mathbf{a}_{1})-\cos(\mathbf{k}_{\mathrm{EP}}\cdot\mathbf{a}_{2})+2\cos(k_{\mathrm{EP},x})],$
(49) $\displaystyle D_{2}$
$\displaystyle=2\sqrt{3}\tilde{h}[-\cos(\mathbf{k}_{\mathrm{EP}}\cdot\mathbf{a}_{1})+\cos(\mathbf{k}_{\mathrm{EP}}\cdot\mathbf{a}_{2})].$
(50)

First, we calculate the winding number of the exceptional points at a generic
parameter in the B phase except for the rings described by Eq. (13).
Substituting Eqs. (44)–(46) into Eq. (43) and neglecting the regular part, Eq.
(16) can be written as

$W=\oint_{C}\frac{1}{2\pi}\frac{A^{\prime}\mathrm{d}k_{x}+B^{\prime}\mathrm{d}k_{y}}{A^{\prime}(k_{x}-k_{\mathrm{EP},x})+B^{\prime}(k_{y}-k_{\mathrm{EP},y})},$
(51)

where

$\displaystyle A^{\prime}$
$\displaystyle=\mathrm{i}[\varepsilon(-\mathbf{k}_{\mathrm{EP}})A_{+}-\varepsilon(\mathbf{k}_{\mathrm{EP}})A_{-}]+2\Delta(\mathbf{k}_{\mathrm{EP}})D_{1},$
(52) $\displaystyle B^{\prime}$
$\displaystyle=\mathrm{i}[\varepsilon(-\mathbf{k}_{\mathrm{EP}})B_{+}-\varepsilon(\mathbf{k}_{\mathrm{EP}})B_{-}]+2\Delta(\mathbf{k}_{\mathrm{EP}})D_{2}.$
(53)

By parametrizing the integration path $C$ with a circle of radius $r$ around
$\mathbf{k}_{\mathrm{EP}}$ as
$\mathbf{k}=\mathbf{k}_{\mathrm{EP}}+r(\cos\theta,\sin\theta)$
$(0\leq\theta<2\pi)$, we can rewrite the winding number as

$\displaystyle W$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{d}\theta}{2\pi}\frac{-A^{\prime}\sin\theta+B^{\prime}\cos\theta}{A^{\prime}\cos\theta+B^{\prime}\sin\theta}$
$\displaystyle=\oint_{\lvert
z\rvert=1}\frac{\mathrm{d}z}{2\pi\mathrm{i}}\frac{1}{\mathrm{i}z}\frac{-A^{\prime}(z^{2}-1)+\mathrm{i}B^{\prime}(z^{2}+1)}{\mathrm{i}A^{\prime}(z^{2}+1)+B^{\prime}(z^{2}-1)}$
$\displaystyle=\oint_{\lvert
z\rvert=1}\frac{\mathrm{d}z}{2\pi\mathrm{i}}\frac{z^{2}-\frac{A^{\prime}+\mathrm{i}B^{\prime}}{A^{\prime}-\mathrm{i}B^{\prime}}}{z\left(z-\sqrt{\frac{B^{\prime}-\mathrm{i}A^{\prime}}{B^{\prime}+\mathrm{i}A^{\prime}}}\right)\left(z+\sqrt{\frac{B^{\prime}-\mathrm{i}A^{\prime}}{B^{\prime}+\mathrm{i}A^{\prime}}}\right)}.$
(54)

At the second equality, we change the integral variable from $\theta$ to
$z=\mathrm{e}^{\mathrm{i}\theta}$. The integrand in Eq. (54) has three poles
at $z=0$,
$+\sqrt{\frac{B^{\prime}-\mathrm{i}A^{\prime}}{B+\mathrm{i}A^{\prime}}}$, and
$-\sqrt{\frac{B^{\prime}-\mathrm{i}A^{\prime}}{B+\mathrm{i}A^{\prime}}}$ with
residues $-1$, $+1$, and $+1$, respectively. Therefore, we obtain

$W=\begin{cases}+1&\left(\left\lvert\sqrt{\frac{B^{\prime}-\mathrm{i}A^{\prime}}{B^{\prime}+\mathrm{i}A^{\prime}}}\right\rvert<1\right)\\\
-1&\left(\left\lvert\sqrt{\frac{B^{\prime}-\mathrm{i}A^{\prime}}{B^{\prime}+\mathrm{i}A^{\prime}}}\right\rvert>1\right).\end{cases}$
(55)

Next, we consider a parameter on the rings in Eq. (13) except for the points
where $\varepsilon(\mathbf{k}_{\mathrm{EP}})=0$ and
$\varepsilon(-\mathbf{k}_{\mathrm{EP}})=0$ are satisfied simultaneously. When
$\varepsilon(\mathbf{k}_{\mathrm{EP}})=0$ and
$\varepsilon(-\mathbf{k}_{\mathrm{EP}})\neq 0$, Eq. (16) can be written as

$W=\oint_{C}\frac{1}{2\pi}\frac{A_{+}\mathrm{d}k_{x}+B_{+}\mathrm{d}k_{y}}{A_{+}(k_{x}-k_{\mathrm{EP},x})+B_{+}(k_{y}-k_{\mathrm{EP},y})}.$
(56)

Note that $\Delta(\mathbf{k}_{\mathrm{EP}})=0$ on the rings described by Eq.
(13). Equation (56) is equivalent to Eq. (51) by replacing $A^{\prime}$ by
$A_{+}$ and $B^{\prime}$ by $B_{+}$. Hence, similar calculations to Eq. (54)
yield

$W=\begin{cases}+1&\left(\left\lvert\sqrt{\frac{B_{+}-\mathrm{i}A_{+}}{B_{+}+\mathrm{i}A_{+}}}\right\rvert<1\right)\\\
-1&\left(\left\lvert\sqrt{\frac{B_{+}-\mathrm{i}A_{+}}{B_{+}+\mathrm{i}A_{+}}}\right\rvert>1\right).\end{cases}$
(57)

Similarly, when $\varepsilon(\mathbf{k}_{\mathrm{EP}})\neq 0$ and
$\varepsilon(-\mathbf{k}_{\mathrm{EP}})=0$, by replacing $A^{\prime}$ by
$A_{-}$ and $B^{\prime}$ by $B_{-}$ in Eq. (51), we obtain

$W=\begin{cases}+1&\left(\left\lvert\sqrt{\frac{B_{-}-\mathrm{i}A_{-}}{B_{-}+\mathrm{i}A_{-}}}\right\rvert<1\right)\\\
-1&\left(\left\lvert\sqrt{\frac{B_{-}-\mathrm{i}A_{-}}{B_{+}+\mathrm{i}A_{-}}}\right\rvert>1\right).\end{cases}$
(58)

Finally, we calculate the winding number of the exceptional points described
by Eqs. (14) and (15), where
$\varepsilon(\mathbf{k}_{\mathrm{EP}})=\varepsilon(-\mathbf{k}_{\mathrm{EP}})=0$.
In this case, using the integral variable $z=\mathrm{e}^{2\mathrm{i}\theta}$,
we obtain

$\displaystyle W=\oint_{\lvert
z\rvert=1}\frac{\mathrm{d}z}{2\pi\mathrm{i}}\frac{2}{z}\frac{z^{2}+\frac{F_{1}+\mathrm{i}F_{2}}{F_{1}-\mathrm{i}F_{2}}}{z^{2}+2\mathrm{i}\frac{F_{3}+F_{4}}{F_{1}-\mathrm{i}F_{2}}z-\frac{F_{1}+\mathrm{i}F_{2}}{F_{1}-\mathrm{i}F_{2}}},$
(59)

where

$\displaystyle F_{1}$ $\displaystyle=A_{+}B_{-}+A_{-}B_{+}+2D_{1}D_{2},$ (60)
$\displaystyle F_{2}$
$\displaystyle=-A_{+}A_{-}+B_{+}B_{-}-D_{1}^{2}+D_{2}^{2},$ (61)
$\displaystyle F_{3}$ $\displaystyle=A_{+}A_{-}+D_{1}^{2},$ (62)
$\displaystyle F_{4}$ $\displaystyle=B_{+}B_{-}+D_{2}^{2}.$ (63)

This corresponds to Eq. (17) with

$w_{\pm}=-\mathrm{i}\frac{F_{3}+F_{4}}{F_{1}-\mathrm{i}F_{2}}\pm\sqrt{-\left(\frac{F_{3}+F_{4}}{F_{1}-\mathrm{i}F_{2}}\right)^{2}+\frac{F_{1}+\mathrm{i}F_{2}}{F_{1}-\mathrm{i}F_{2}}}.$
(64)

The poles $z=0$, $w_{+}$, and $w_{-}$ have residues $-2$, $+2$, and $+2$,
respectively. Hence we conclude

$W=\begin{cases}+2&\left(\lvert w_{+}\rvert,\ \lvert w_{-}\rvert<1\right)\\\
-2&\left(\lvert w_{+}\rvert,\ \lvert w_{-}\rvert>1\right)\\\
0&\text{otherwise},\end{cases}$ (65)

as discussed in Sec. III.2.

##  Appendix C Derivation of $\tilde{h}_{*}$

In this Appendix, we derive the analytic formulas for $\tilde{h}_{*}$ where
the topological transition occurs with changes in the winding number. The
topological transition occurs at the exceptional points satisfying
$\varepsilon(\mathbf{k}_{\mathrm{EP}})=0$ and
$\varepsilon(-\mathbf{k}_{\mathrm{EP}})=0$ simultaneously, which appear at the
parameters described by Eqs. (14) and (15). We begin with the first equation
in Eq. (14). As discussed in Appendix A, the exceptional points satisfying
$\varepsilon(\mathbf{k}_{\mathrm{EP}})=\varepsilon(-\mathbf{k}_{\mathrm{EP}})=0$
fulfill the conditions $\tilde{k}_{1}\neq 0$ and $\tilde{k}_{2}=0$ (mod
$2\pi$). From this and the conditions in Eqs. (36) and (39), we obtain
$\sin(\tilde{k}_{1}+\Delta\phi)=\sin(-\tilde{k}_{1}+\Delta\phi)$, which is
satisfied only by $\tilde{k}_{1}=\pi$. By substituting $\tilde{k}_{1}=\pi$ and
$\tilde{k}_{2}=0$ to Eqs. (47)–(69), we obtain

$\displaystyle A_{+}$ $\displaystyle=A_{-}=-(\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\phi_{x}}+\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}\phi_{y}}),$ (66) $\displaystyle B_{+}$
$\displaystyle=B_{-}=-\sqrt{3}(\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\phi_{x}}-\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}\phi_{y}}),$ (67) $\displaystyle D_{1}$
$\displaystyle=-4\tilde{h},$ (68) $\displaystyle D_{2}$
$\displaystyle=4\sqrt{3}\tilde{h}.$ (69)

The topological transition occurs when $\lvert w_{+}\rvert$ or $\lvert
w_{-}\rvert$ becomes unity, as discussed in Appendix B. From Eq. (64) with
Eqs. (66)–(69), we can derive the explicit formula of $\tilde{h}_{*}$ as

$\tilde{h}_{*}=\mp\frac{\lvert
G_{y}\rvert}{4}\frac{\sin(\phi_{x}-\phi_{y})}{\cos\phi_{x}}.$ (70)

Similarly, we can obtain the fomulas for $\tilde{h}_{*}$ for the second and
third equations in Eq. (14). For the second equation, Eqs. (36) and (39) with
replacing $\tilde{k}_{1}$ by $\tilde{k}_{2}$ and interchanging $G_{x}$ and
$G_{y}$ lead to $\tilde{k}_{1}=0$ and $\tilde{k}_{2}=\pi$. By substituting
them, we obtain

$\displaystyle A_{+}$ $\displaystyle=A_{-}=\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\phi_{x}}+\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}\phi_{y}},$ (71) $\displaystyle B_{+}$
$\displaystyle=B_{-}=\sqrt{3}(\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\phi_{x}}-\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}\phi_{y}}),$ (72) $\displaystyle D_{1}$
$\displaystyle=-4\tilde{h},$ (73) $\displaystyle D_{2}$
$\displaystyle=-4\sqrt{3}\tilde{h},$ (74)

and also derive

$\tilde{h}_{*}=\mp\frac{\lvert
G_{x}\rvert}{4}\frac{\sin(\phi_{x}-\phi_{y})}{\cos\phi_{y}}.$ (75)

Meanwhile, for the third equation in Eq. (14), we obtain
$\tilde{k}_{1}=\tilde{k}_{2}=\pi$ by substituting the upper sign of Eq. (33)
into Eqs. (30) and (32), and hence,

$\displaystyle A_{+}$ $\displaystyle=A_{-}=-(\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\phi_{x}}-\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}\phi_{y}}),$ (76) $\displaystyle B_{+}$
$\displaystyle=B_{-}=-\sqrt{3}(\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\phi_{x}}+\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}\phi_{y}}),$ (77) $\displaystyle D_{1}$
$\displaystyle=8\tilde{h},$ (78) $\displaystyle D_{2}$ $\displaystyle=0,$ (79)

and

$\tilde{h}_{*}=\mp\frac{\lvert G_{x}\rvert\lvert
G_{y}\rvert\sin(\phi_{x}-\phi_{y})}{4(\lvert G_{x}\rvert\cos\phi_{x}+\lvert
G_{y}\rvert\cos\phi_{y})}.$ (80)

Finally, we consider the case of the intersection of three rings described by
Eq. (15). By substituting the lower sign of Eq. (33) into Eqs. (30) and (32),
we can obtain the location of the exceptional points as
$\tilde{k}_{1}=\tilde{k}_{2}=0$. Hence, we obtain

$\displaystyle A_{+}$ $\displaystyle=A_{-}=\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\phi_{x}}-\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}\phi_{y}},$ (81) $\displaystyle B_{+}$
$\displaystyle=B_{-}=\sqrt{3}(\lvert
G_{x}\rvert\mathrm{e}^{\mathrm{i}\phi_{x}}+\lvert
G_{y}\rvert\mathrm{e}^{\mathrm{i}\phi_{y}}),$ (82) $\displaystyle D_{1}$
$\displaystyle=0,$ (83) $\displaystyle D_{2}$ $\displaystyle=0.$ (84)

Thus, in this case, $w_{\pm}$ are independent of $\tilde{h}$ and the
topological transition does not occur.

##  Appendix D Average square position $\Delta m^{2}$ in the OBC case

Figure 10: $k_{x}$ dependence of the absolute value of the complex energy
eigenvalue with the open boundary condition at $G_{x}=2.0$, $G_{y}=1.0$,
$G_{z}=2.5\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$, and (a) $\tilde{h}=0.0$, (b)
$\tilde{h}=0.1$, (c) $\tilde{h}=0.5$, (d) $\tilde{h}=1.0$, and at
$G_{x}=2.0\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$,
$G_{y}=1.0\mathrm{e}^{\mathrm{i}\frac{\pi}{6}}$, $G_{z}=2.5$, and (e)
$\tilde{h}=0.0$, (f) $\tilde{h}=0.05$, (g) $\tilde{h}=0.1$, and (h)
$\tilde{h}=0.2$. The color bar shows the average square position $\Delta
m^{2}$. The data with the periodic boundary condition are shown in gray.

In this Appendix, we show supplemental data for the localization of the wave
function in the skin effect in Sec. III.3. Figure 10 shows the average square
position corresponding to Figs. 7 and 8, which is defined as [67]

$\Delta
m^{2}(n,k_{x})=\frac{1}{M_{0}}\sum_{m=1}^{M}\left(m-M_{0}\right)^{2}\lvert\psi_{n}(k_{x},m)\rvert^{2},$
(85)

where $M_{0}=\frac{M-1}{2}$. This quantity provide supplementary information
to the average localization $\tilde{m}$ in Eq. (24); it takes $1$ if the wave
function is localized at the edges of the system, and $1/2$ if the wave
function is spread over the entire system. Figures 10(a)–10(d), which
correspond to Fig. 7, show that the isolated low-energy modes are localized at
the edges of the system. Note that this is not evident solely from $\tilde{m}$
for the case of $\tilde{h}=0$ in Fig. 7(a), since $\tilde{m}$ takes the same
value for the state equally localized at both edges and a state spread over
the system. Meanwhile, in Figs. 10(f)–10(h), which corresponds to Fig. 8,
$\Delta m^{2}$ also indicates that the low-energy modes are localized at the
edges. Furthermore, it takes the value of $1/2$ at $k_{x}$ corresponding to
the exceptional points of the PBC spectra. This supports the switching of the
edge modes discussed in Sec. III.3.

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