In $\tr(\hat{m}_{1}\mathcal{L}[\hat{\rho}])$, we have four terms corresponding
to the four terms in the Lindbladian (130). The first term is

$h_{\downarrow}\tr[\hat{m}_{1}\Big{(}\hat{m}_{1}\hat{\rho}\hat{m}_{1}^{\dagger}-\frac{1}{2}\\{\hat{m}_{1}^{\dagger}\hat{m}_{1},\hat{\rho}\\}\Big{)}\Big{]}=h_{\downarrow}\Big{(}\langle\hat{m}_{1}^{\dagger}\hat{m}_{1}\hat{m}_{1}\rangle-\frac{1}{2}\langle\hat{m}_{1}\hat{m}_{1}^{\dagger}\hat{m}_{1}\rangle-\frac{1}{2}\langle\hat{m}_{1}^{\dagger}\hat{m}_{1}\hat{m}_{1}\rangle\Big{)}=\frac{h_{\downarrow}}{2}\langle[\hat{m}_{1}^{\dagger},\hat{m}_{1}]\hat{m}_{1}\rangle=-\frac{h_{\downarrow}}{2}\langle\hat{m}_{1}\rangle.$
(133)

It describes the damping of $\langle\hat{m}_{1}\rangle$, where
$h_{\downarrow}$ is the local Gilbert damping. It is important to note that,
unlike in Sec. 3.2.4, the effect of the quantum jump is not disregarded in
this treatment. The second term is

$h_{\downarrow}\tr[\hat{m}_{1}\Big{(}\hat{m}_{2}\hat{\rho}\hat{m}_{2}^{\dagger}-\frac{1}{2}\\{\hat{m}_{2}^{\dagger}\hat{m}_{2},\hat{\rho}\\}\Big{)}\Big{]}=h_{\downarrow}\Big{(}\langle\hat{m}_{2}^{\dagger}\hat{m}_{2}\hat{m}_{1}\rangle-\frac{1}{2}\langle\hat{m}_{2}^{\dagger}\hat{m}_{2}\hat{m}_{1}\rangle-\frac{1}{2}\langle\hat{m}_{2}^{\dagger}\hat{m}_{2}\hat{m}_{1}\rangle\Big{)}=0,$
(134)

which implies that the local decay process of the second magnon mode does not
impact the dynamics of $\langle\hat{m}_{1}\rangle$, as one may expect. The
third term is given by

$G_{\downarrow}\tr[\hat{m}_{1}\Big{(}\hat{m}_{1}\hat{\rho}\hat{m}_{2}^{\dagger}-\frac{1}{2}\\{\hat{m}_{2}^{\dagger}\hat{m}_{1},\hat{\rho}\\}\Big{)}\Big{]}=G_{\downarrow}\Big{(}\langle\hat{m}_{2}^{\dagger}\hat{m}_{1}\hat{m}_{1}\rangle-\frac{1}{2}\langle\hat{m}_{2}^{\dagger}\hat{m}_{1}\hat{m}_{1}\rangle-\frac{1}{2}\langle\hat{m}_{2}^{\dagger}\hat{m}_{1}\hat{m}_{1}\rangle\Big{)}=0,$
(135)

and the fourth term is

$G^{*}_{\downarrow}\tr[\hat{m}_{1}\Big{(}\hat{m}_{2}\hat{\rho}\hat{m}_{1}^{\dagger}-\frac{1}{2}\\{\hat{m}_{1}^{\dagger}\hat{m}_{2},\hat{\rho}\\}\Big{)}\Big{]}=G^{*}_{\downarrow}\Big{(}\langle\hat{m}_{1}^{\dagger}\hat{m}_{1}\hat{m}_{2}\rangle-\frac{1}{2}\langle\hat{m}_{1}\hat{m}_{1}^{\dagger}\hat{m}_{2}\rangle-\frac{1}{2}\langle\hat{m}_{1}^{\dagger}\hat{m}_{1}\hat{m}_{2}\Big{)}=\frac{G_{\downarrow}^{*}}{2}\langle[\hat{m}_{1}^{\dagger},\hat{m}_{1}]\hat{m}_{2}\rangle=-\frac{G_{\downarrow}^{*}}{2}\langle\hat{m}_{2}\rangle.$
(136)

Therefore, the dynamics of the mean-field $\langle\hat{m}_{1}\rangle$ is
governed by

$\derivative{\langle\hat{m}_{1}\rangle}{t}=\Big{(}-i\omega_{m}-\frac{h_{\downarrow}}{2}\Big{)}\langle\hat{m}_{1}\rangle+\Big{(}-iJ-\frac{G^{*}_{\downarrow}}{2}\Big{)}\langle\hat{m}_{2}\rangle,$
(137)

and, similarly, the dynamics of $\langle\hat{m}_{2}\rangle$ is

$\derivative{\langle\hat{m}_{2}\rangle}{t}=\Big{(}-i\omega_{m}-\frac{h_{\downarrow}}{2}\Big{)}\langle\hat{m}_{2}\rangle+\Big{(}-iJ^{*}-\frac{G_{\downarrow}}{2}\Big{)}\langle\hat{m}_{1}\rangle.$
(138)

We point out that they are the linearized Landau-Lifshitz-Gilbert equations
[330]. We can recast the two equations above into a more compact Schorödinger-
like equation with a non-Hermitian Hamiltonian:

$i\derivative{\vec{\psi}}{t}=\mathcal{H}\vec{\psi},$ (139)

with $\vec{\psi}=(\langle\hat{m}_{1}\rangle,\langle\hat{m}_{2}\rangle)^{T}$
and

$\mathcal{H}=\matrixquantity[\omega_{m}-\frac{ih_{\downarrow}}{2}&J-\frac{iG^{*}_{\downarrow}}{2}\\\
J^{*}-i\frac{G_{\downarrow}}{2}&\omega_{m}-\frac{ih_{\downarrow}}{2}].$ (140)

Finally, we remark that the spin pumping process can also be modeled by a
Lindbladian. For example, let us pump the first macroscopic magnetic spin by
subjecting it to spin-transfer [331, 332] or spin Seebeck torques [333], which
can be described by

$\mathcal{L}^{\text{pump}}[\hat{\rho}]=\gamma\Big{[}\hat{m}_{1}^{\dagger}\hat{\rho}\hat{m}_{1}-\frac{1}{2}\\{\hat{m}_{1}\hat{m}_{1}^{\dagger},\hat{\rho}\\}\Big{]},$
(141)

with the pumping rate being $\gamma>0$. This would effectively shift the local
damping $h_{\downarrow}$ to a smaller value, which can be seen from its
contribution to the equation of motion of $\langle\hat{m}_{1}\rangle$:

$\tr(\hat{m}_{1}\mathcal{L}^{\text{pump}}[\hat{\rho}])=\gamma\Big{[}\langle\hat{m}_{1}\hat{m}_{1}\hat{m}_{1}^{\dagger}\rangle-\frac{1}{2}\langle\hat{m}_{1}\hat{m}_{1}\hat{m}_{1}^{\dagger}\rangle-\frac{1}{2}\langle\hat{m}_{1}\hat{m}_{1}^{\dagger}\hat{m}_{1}\rangle\Big{]}=\frac{\gamma}{2}\langle\hat{m}_{1}[\hat{m}_{1},\hat{m}_{1}^{\dagger}]\rangle=\frac{\gamma}{2}\langle\hat{m}_{1}\rangle.$
(142)

Then the equation for $\langle\hat{m}_{1}\rangle$ is given by

$\derivative{\langle\hat{m}_{1}\rangle}{t}=\Big{(}-i\omega_{m}-\frac{h_{\downarrow}-\gamma}{2}\Big{)}\langle\hat{m}_{1}\rangle+\Big{(}-iJ-\frac{G^{*}_{\downarrow}}{2}\Big{)}\langle\hat{m}_{2}\rangle,$
(143)

where we see that the local damping $h_{\downarrow}$ is reduced to
$h_{\downarrow}-\gamma$ as expected [334, 335]. We highlight that the signs
before the parameters $h_{\downarrow}$ and $\gamma$ are directly linked to the
jump terms in the respective Lindbladians. The opposite signs of these
parameters are dictated by their physical interpretations, with
$h_{\downarrow}$ representing the decay process and $\gamma$ corresponding to
the pumping process.

###  3.3 Green-function approach for magnon non-Hermitian dynamics

In this review article, we focus on the non-Hermitian topological states of
magnons when they interact with other degrees of freedom. Thus we mainly
discuss Green’s function of bosons, which allows us to describe the response
of the system at any point due to the excitation at any other point. By
employing the Green-function approach, we can effectively capture the effects
of magnon-bath interactions through the self-energy, which can be viewed as an
effective Hamiltonian arising from the interaction of the magnon subsystem
with the bath. In Sec. 3.3.1, we first review the magnon Green’s function,
where an effective non-Hermitian Hamiltonian is introduced by considering the
magnon self-energy. Following this, in Sec. 3.3.2, we study an example with
bilinear magnon-quasiparticle interaction, and compare this Green’s function
approach with the Lindblad master equation approach that we discussed before.

####  3.3.1 Green’s function of magnons

We consider the following Hamiltonian:

$\hat{H}=\hat{H}_{0}+\hat{H}_{E}+\hat{H}_{\text{SE}},$ (144)

where
$\hat{H}_{0}=\sum_{\mathbf{k}}h(\mathbf{k})\hat{m}_{\mathbf{k}}^{\dagger}\hat{m}_{\mathbf{k}}$
is the free magnonic Hamiltonian, $\hat{H}_{E}$ describes the environment, and
$\hat{H}_{\text{SE}}$ stands for their interactions. Let us focus on the
retarded Green’s function of single-magnon defined as [336, 337]

$G^{R}(\mathbf{r},\mathbf{r}^{\prime};t,t^{\prime})=-i\theta(t-t^{\prime})\langle[\hat{m}_{{\bf
r}}(t),\hat{m}_{{\bf r}^{\prime}}^{\dagger}(t^{\prime})]\rangle,$ (145)

where $\theta(t)$ is the step function, the magnon operator is in the
Heisenberg picture
$\hat{m}_{\mathbf{r}}(t)=e^{i\hat{H}t}\hat{m}_{\mathbf{r}}e^{-i\hat{H}t}$, and
$\langle\hat{\mathcal{O}}\rangle\equiv\tr(e^{-\beta\hat{H}}\hat{\mathcal{O}})/Z$
with $Z=\tr e^{-\beta\hat{H}}$. We will refer to such retarded Green’s
function simply as “Green’s function". It is worth noting that we have defined
Green’s function in the real space and time domain. When the system exhibits
time and space translational symmetries, it is convenient to work with
$G^{R}(\mathbf{k},\omega)$ in momentum and frequency space through Fourier
transformations. It is well-known that the complex poles of Green’s function
$G^{R}(\mathbf{k},\omega)$ of magnons determine the magnon spectrum. For
example, in the case of free magnon gas $\hat{H}=\hat{H}_{0}$, we can evaluate
the Green’s function easily with the method of the equation of motion [336,
337]:

$i\partial_{t}G^{R}(\mathbf{k};t,t^{\prime})=\delta(t-t^{\prime})-i\theta(t-t^{\prime})\langle[i\partial_{t}\hat{m}_{\mathbf{k}}(t),\hat{m}^{\dagger}_{\mathbf{k}}(t^{\prime})]\rangle=\delta(t-t^{\prime})+h(\mathbf{k})G^{R}(\mathbf{k},t,t^{\prime}),$
(146)

where
$G^{R}(\mathbf{k};t,t^{\prime})=-i\theta(t-t^{\prime})\langle[\hat{m}_{\mathbf{k}}(t),\hat{m}_{\mathbf{k}}^{\dagger}(t^{\prime})]\rangle,$
and we have used
$i\partial_{t}\hat{m}_{\mathbf{k}}(t)=[\hat{m}_{\mathbf{k}}(t),\hat{H}_{0}]$.
When we perform the Fourier transformation for the equation above, we obtain

$(\omega+i\eta)G^{R}(\mathbf{k},\omega)=1+h(\mathbf{k})G^{R}(\mathbf{k},\omega),$
(147)

which gives us the free Green’s function of magnons:

$G^{R}(\mathbf{k},\omega)=\frac{1}{\omega-h(\mathbf{k})+i\eta}.$ (148)

Here $G^{R}(\mathbf{k},\omega)=\int^{\infty}_{-\infty}dt\,e^{i\omega
t}G^{R}(\mathbf{k},t)$, and we remark that in order to ensure convergence of
the Fourier transform of the Green’s function, which may suffer from some
ringing in the future, we need to make the replacement
$\omega\rightarrow\omega+i\eta$, where $\eta$ is a positive infinitesimal.

In general, in the presence of the environment and interactions, the magnon
Green’s function takes the form of

$G^{R}(\mathbf{k},\omega)=\frac{1}{\omega-h(\mathbf{k})-\Sigma^{R}(\mathbf{k},\omega)},$
(149)

where $\Sigma^{R}(\mathbf{k},\omega)$ is known as self-energy, whose real part
renormalizes the bare magnon spectrum and the imaginary part gives rise to
finite magnon lifetimes. Evaluating the self-energy exactly is typically a
difficult task. Instead, a perturbative expansion can be employed in terms of
the magnon-bath interaction Hamiltonian. The self-energy is defined as the sum
of all diagrams in the perturbative expansion that have two external magnon
lines and any number of internal bath lines. The self-energy diagram can be
represented as a loop with internal magnon and bath propagators, which can be
computed using standard diagrammatic techniques such as Feynman diagrams [336,
337].

We can now introduce an effective Hamiltonian:

$H_{\text{eff}}(\mathbf{k},\omega)=h(\mathbf{k})+\Sigma^{R}(\mathbf{k},\omega),$
(150)

which is non-Hermitian in general. Then the magnon Green’s function can be
written as

$G^{R}(\mathbf{k},\omega)=\frac{1}{\omega-H_{\text{eff}}(\mathbf{k},\omega)}.$
(151)

In contrast to the Lindblad master equation approach, which relies on the
Born-Markov approximation, we point out that the Green-function approach
discussed here can capture non-Markovian effects of the environment by noting
the frequency dependence in the self-energy. It can also go beyond the second
order in the magnon-environment coupling in general. However, it is important
to note that the Green-function approach presented here only captures the
spectrum properties of magnons. To also capture the statistical properties,
one needs to invoke the Keldysh Green’s function, with which one can establish
the equivalence between the Lindblad master equation approach and the Keldysh
formalism in the limit of weak magnon-environment coupling [338, 339].

####  3.3.2 Example with bilinear magnon-quasiparticle interaction

To illustrate the Green-function method and compare it with the Lindblad
master equation approach, we consider the same model that we previously
studied using the Lindblad master equation method in Sec. 3.2.2, consisting of
a single magnon mode with frequency $\omega_{m}$ that interacts with an
environment with Hamiltonian:

$\hat{H}=\omega_{m}\hat{m}^{\dagger}\hat{m}+\sum_{\mathbf{k}}(g_{\mathbf{k}}\hat{m}^{\dagger}\hat{b}_{\mathbf{k}}+g^{*}_{\mathbf{k}}\hat{m}\hat{b}^{\dagger}_{\mathbf{k}})+\sum_{\mathbf{k}}\epsilon_{\mathbf{k}}\hat{b}^{\dagger}_{\mathbf{k}}\hat{b}_{\mathbf{k}}.$
(152)

We introduce two Green’s functions:

$G^{R}(t)=-i\theta(t)\langle[\hat{m}(t),\hat{m}^{\dagger}]\rangle,\;\;\;\text{and}\;\;\;\tilde{G}^{R}(\mathbf{k},t)=-i\theta(t)\langle[\hat{b}_{\mathbf{k}}(t),\hat{m}^{\dagger}]\rangle.$
(153)

Since the Hamiltonian is quadratic we can solve the Green’s function of
magnons exactly. To this end, let us write down the equations of motion for
these two Green’s functions:

$i\partial_{t}G^{R}(t)=\delta(t)+\omega_{m}G^{R}(t)+\sum_{\mathbf{k}}g_{\mathbf{k}}\tilde{G}^{R}(\mathbf{k},t),\;\;\;\;\;\;i\partial_{t}\tilde{G}^{R}(\mathbf{k},t)=\epsilon_{\mathbf{k}}\tilde{G}^{R}(\mathbf{k},t)+g^{*}_{\mathbf{k}}G^{R}(t).$
(154)

We perform Fourier transformations on these two equations, which results in
two algebraic equations that can be exactly solved:

$G^{R}(\omega)=\frac{1}{\omega-\omega_{m}-\Sigma^{R}(\omega)},\;\;\;\text{with}\;\;\;\Sigma^{R}(\omega)=\sum_{\mathbf{k}}\frac{|g_{\mathbf{k}}|^{2}}{\omega-\epsilon_{\mathbf{k}}+i\eta}\equiv\Sigma^{\prime}(\omega)-i\Sigma^{\prime\prime}(\omega),$
(155)

with

$\Sigma^{\prime}(\omega)=P\sum_{\mathbf{k}}\frac{|g_{\mathbf{k}}|^{2}}{\omega-\epsilon_{k}},\;\;\;\Sigma^{\prime\prime}(\omega)=\pi\sum_{\mathbf{k}}|g_{\mathbf{k}}|^{2}\delta(\omega-\epsilon_{k}).$
(156)

So far, we do not make any approximation. Assuming the coupling between the
magnon mode and the environment is weak, we can approximate
$\Sigma^{R}(\omega)\approx\Sigma^{R}(\omega_{m})$. Then we can write the
effective Hamiltonian of the magnon mode as:

$\hat{\mathcal{H}}=\big{[}\omega_{m}+\Sigma^{\prime}(\omega_{m})-i\Sigma^{\prime\prime}(\omega_{m})\big{]}\hat{m}^{\dagger}\hat{m},$
(157)

implying that the lamb shift of the frequency is $\Sigma^{\prime}(\omega_{m})$
and the decay rate of this mode is $\Sigma^{\prime\prime}(\omega_{m})$. We
point out that this result is consistent with what we have obtained from the
Lindblad master equation in the zero temperature case by noting that
$\Sigma^{\prime}(\omega_{m})=\omega_{1}$ and
$\Sigma^{\prime\prime}(\omega_{m})=\Gamma_{\downarrow}/2$. As discussed in the
previous section, the Green-function approach accurately captures the spectral
properties of the magnon mode subjected to an environment.

##  4 Exceptional points, lines, and surfaces in magnonic devices

As a general physical phenomenon, exceptional points (EPs) widely exist in
nearly all kinds of coupled systems, including the optic [67, 5, 7, 6],
acoustic [70, 71, 72, 232], spintronic [23, 26, 107, 98, 110], as well as
electronic systems [233, 234, 28, 29, 32]. Many properties, such as the
mutation of eigenfrequencies [2, 5, 6, 7, 12], the topological property of
eigenvalue surfaces [3, 260, 261, 97], and the spontaneous symmetry breaking
[75, 76, 77, 1, 74], have been discovered when the system is driven to be
around or at the EPs [340, 8, 9, 12, 16] (Sec. 2.1). In this context,
searching for exceptional points, lines, and surfaces in magnonic systems is
an attractive and significant topic for the following reasons. First, the
magnonic systems are highly tunable by diverse techniques, such as the
magnetic/electric field, the thermal gradient, and even the stress, which
makes the realization and manipulation of the EPs in magnonic systems
fertilized with other coupled systems. In other words, the magnonic systems
provide us with a powerful platform for engineering the EPs and the unique
functionalities associated with them. Second, magnons are excellent
information carriers that can interact with different particles, such as
electrons, phonons, photons, and other magnons. Given this, researchers are
motivated to study the variation of the properties of magnons near the EPs,
such as their eigenfrequencies and dissipation, which have impacts on the
electrical, mechanical, or optical properties of the magnonic devices.

Thus far, we have elucidated the concept of EPs (Sec. 2.1) and addressed
unified approaches to handle the non-Hermitian dynamics of magnons from the
master equation (Sec. 3.2) and Green’s function (Sec. 3.3) approaches. In this
Section, we review the magnonic realization of EPs and the potential
applications aroused by the singularity in spectrum and eigenvector, as
summarized in Table 3 for typical examples. We start our discussion with
recent theoretical studies on EPs, in which several systems, for instance, the
dissipative coupled spin systems (Sec. 4.1), systems of qubits coupled with
magnon-mediated dissipative couplings (Sec. 4.2), and spin systems with
balanced gain and loss (Sec. 4.3). We discuss several experimental paths to
realize the EPs in magnonics (Sec. 4.3.2 and 4.4). The first technical path is
to construct EPs in magnetic heterostructures (Sec. 4.3.2). These devices are
composed of two or more adjacent ferromagnetic layers separated by such as a
normal metal layer [104, 190, 198]. Two ferromagnetic layers with different
dissipative properties strongly couple with each other via the spin transfer
torque or the magnetic dipolar interaction. Precisely controlling their
separations can control the coupling strength between them, such that EPs can
be achieved. Another path to experimentally realize the EPs is based on cavity
magnonics (Sec. 4.4) [204, 205, 167, 207, 169, 40, 212, 341, 11, 199], which
possess excellent versatility and high tunability, leading the study of EPs in
cavity magnonics to be feasible. Consequently, digging exotic properties
around or at EPs has become one of the central tasks in cavity magnonics. We
finally discuss the realization of exceptional surfaces in the magnonic system
(Sec. 4.5) [98, 99].

Table 3: Potential applications of EPs in magnonics. Key features | Model and Results | Functionalities
---|---|---
Absorption |  |  • Polaritonic coherent perfect absorption [26] • Enhanced magnon current [190]
Amplification |  |  • Stimulated emitted magnon number [185] • Enhancement of magnonic permeability [198] • Magnon amplification [198] • Transmission enhancement [195]
Sensitivity |  |  • Giant enhancement of magnonic frequency comb [203, 199] • Third-order EP [107] • Sensitivity approaches fetotesla [107]

###  4.1 Exceptional points in dissipatively coupled spins

We start with the simplest case proposed by Tserkovnyak [197] as shown in Fig.
11, where only two classical spins are considered. The generalization to a
spin chain and thus EPs in momentum space appears to be straightforward [197].
These classical spins can be regarded as the simplification of magnetic moment
in the ferromagnetic layer, while the coupling between them can be mediated by
the normal-metal spacer. Typically, this coupling includes coherent coupling,
which offers a real coupling strength and leads to a reactive coupling effect,
and dissipative coupling due to damping torque caused by spin pumping.

Ferromagnetic alignment. When the two classical spins are coupled
ferromagnetically as shown in Fig. 11(a), the linearized equations of motion
for the Larmor precession of two spins can be cast in the following
Schrödinger-like form:

$\displaystyle i\frac{d}{dt}\left(\begin{array}[]{c}\hat{\mathscr{L}}_{1}\\\
\hat{\mathscr{L}}_{2}\end{array}\right)=\mathscr{H}\left(\begin{array}[]{c}\hat{\mathscr{L}}_{1}\\\
\hat{\mathscr{L}}_{2}\end{array}\right),$ (162)

where
$\hat{\mathscr{L}}_{l}\equiv(\hat{S}_{l}^{x}+i\hat{S}_{l}^{y})/\sqrt{2S_{l}}$
obey the canonical algebra
$i\\{\hat{\mathscr{L}}_{l},\hat{\mathscr{L}}_{l}^{*}\\}=1$. Here,
$\hat{S}^{i}_{l}$ with $i=\\{x,y,z\\}$ and $l=\\{1,2\\}$ stands for the three
components of the $l$-th spin, and $S_{l}$ is the magnitude of spins. With
local applied magnetic field $b_{l}$ along the $\hat{\bf z}$-direction
$\omega_{1}\equiv b_{1}+JS_{2}$ and $\omega_{2}\equiv b_{2}+JS_{1}$ are
effective frequencies when coupled to each other coherently with strength $J$.
The effective Hamiltonian matrix takes the form of [197]

$\mathscr{H}=(\gamma_{-}-i\alpha_{-})\omega_{+}{\sigma}_{z}-(\gamma^{\prime}-i\alpha^{\prime})\omega_{+}{\sigma}_{x},$
(163)

where terms proportional to $\hat{\sigma}_{0}\equiv I_{2\times 2}$ are
disregarded since they do not affect the following analysis. Here,
$\omega_{+}\equiv(\omega_{1}+\omega_{2})/2$ is the symmetrized spin frequency.
$\gamma_{-}\equiv\omega_{-}/\omega_{+}$ is the normalized detuning with
$\omega_{-}=(\omega_{1}-\omega_{2})/2$ being the asymmetry in the local
frequencies. $\alpha_{-}=(\alpha_{1}-\alpha_{2})/2$ is the damping asymmetry,
and $\alpha^{\prime}$ is the dissipative coupling between the two spins.
$\gamma^{\prime}\equiv\omega^{\prime}/\omega_{+}$ is the normalized coherent
coupling with $\omega^{\prime}\equiv J\sqrt{S_{1}S_{2}}$. In the derivation of
the above Hamiltonian, one has assumed the smallness of the damping
parameters, $\alpha_{l}\ll 1$, and of the interspin detuning and coupling,
$\\{\omega_{-},\omega^{\prime}\\}\ll\omega_{+}$.

Figure 11: Schematic representation of two (a) ferromagnetically and (b)
antiferromagnetically coupled spins, denoted by $\mathbf{S}_{1}$ and
$\mathbf{S}_{2}$, precessing in their respective effective frequency
$\omega_{i}$ with effective local damping $\alpha_{i}$. The spins couple
through both coherent interaction $\omega^{\prime}$ and dissipative coupling
$\alpha^{\prime}$.

Upon diagonalization of the Hamiltonian (163), we obtain the eigenvalues

$\lambda_{\pm}=\pm\sqrt{(\gamma_{-}-i\alpha_{-})^{2}+(\gamma^{\prime}-i\alpha^{\prime})^{2}}.$
(164)

The EPs occur when the diagonalization of the Hamiltonian breaks down, which
takes place when $\lambda_{\pm}=0$, while the individual terms under the
square root are not both zero. In this scenario, $\mathscr{H}\neq 0$, while
$\mathscr{H}^{2}=0$, thereby confirming that the matrix cannot be
diagonalized. This EP condition translates into

$\gamma_{-}-i\alpha_{-}=\pm i(\gamma^{\prime}-i\alpha^{\prime})\neq 0.$ (165)

We address two fundamental scenarios.

First, when the coupling between two spins is purely reactive, the dissipative
coupling $\alpha^{\prime}=0$. Further, when the resonance condition
$\omega_{1}=\omega_{2}$ is assumed, $\gamma_{-}=0$. Hence, the eigenvalues
(164) reduce to $\lambda_{\pm}=\pm\sqrt{\gamma^{\prime 2}-\alpha_{-}^{2}}$.
The EPs occur when the damping asymmetry $\alpha_{-}$ and the normalized
coherent coupling $\gamma^{\prime}$ have the same magnitude, i.e.,
$\alpha_{-}=\pm\gamma^{\prime}$. It is evident that the system is in the
$\mathcal{PT}$-exact regime when the coherent coupling dominates,
$|\gamma^{\prime}|>|\alpha_{-}|$. In this case, the eigenvalues are real,
suggesting two oscillating modes. In contrast, when the damping asymmetry is
relatively large compared to the coherent coupling,
$|\alpha_{-}|>|\gamma^{\prime}|$, we are in the $\mathcal{PT}$-broken regime.
Here, the eigenvalues are purely imaginary, and the spin on one site is more
damped compared to the other (depending on the sign of $\alpha_{-}$).

Second, when the coupling between two spins is purely dissipative, the
coherent coupling $\gamma^{\prime}=0$. Further, we require the local
dissipation to be symmetric with $\alpha_{-}=0$. Thereby, the eigenvalues
(164) reduce to $\lambda_{\pm}=\pm\sqrt{\gamma_{-}^{2}-\alpha^{\prime 2}}$.
When the detuning $\gamma_{-}$ dominates over the dissipative coupling, i.e.,
$|\gamma_{-}|>|\alpha^{\prime}|$, the system is in the $\mathcal{PT}$-exact
regime. Conversely, the $\mathcal{PT}$ symmetry is broken when the dissipative
coupling is large, i.e., $|\alpha^{\prime}|>|\gamma_{-}|$. In the latter case,
the eigenvalues are purely imaginary, with the antisymmetric mode more damped
relative to the symmetric one. Additionally, both modes are perfectly
synchronized in regard to their real frequency components, which is referred
to as level attraction [342, 34], as opposed to the usual level repulsion
observed in a hybridized Hermitian system. The EPs are achieved when
$\alpha^{\prime}=\pm\gamma_{-}$ in this scenario, a natural occurrence in a
magnetic bilayer system with a diffusive normal-meta spacer [343, 174, 344].

Antiferromagnetic alignment. We consider two antiferromagnetically ($J<0$)
coupled spins with equal magnitudes as shown in Fig. 11(b),
$\mathbf{S}_{i}\approx(-1)^{i}S\hat{\bf z}$, for sites $i=\\{1,2\\}$. Let us
suppose the local magnetic fields to be $\mathbf{b}_{i}=(-1)^{i}b_{i}\hat{\bf
z}$, where $b_{i}=(-1)^{i}b+K$. Here, $K\geq 0$ represents the easy-axis
anisotropy, and $b$ is the applied uniform magnetic field that is collinear
with the anisotropy axis. We define the canonical transverse coordinates as
$\hat{\mathscr{L}}_{j}\equiv(-1)^{j}(\hat{S}_{j}^{x}+i\hat{S}^{y}_{j})/\sqrt{2S}$,
which satisfy
$i\\{\hat{\mathscr{L}}_{j},\hat{\mathscr{L}}_{j}^{*}\\}\approx(-1)^{j}$. By
using these coordinates, the linearized equations of motion of the two spins
can be cast into the form of Eq. (4.1) with the following Hamiltonian matrix

$\mathscr{H}=i(\alpha\omega^{\prime}-\alpha^{\prime}\kappa){\sigma}_{x}+(i\omega^{\prime}{\sigma}_{y}-\kappa{\sigma}_{z}),$
(166)

where we have dropped the constant part and assumed the smallness of all
damping parameters as before. Here, $\alpha\equiv g+\alpha^{\prime}$ is the
effective local damping, consisting of the intrinsic local damping $g$
(assumed to be equal for the two sites) and the dissipative coupling
$\alpha^{\prime}$ due to spin pumping. $\kappa\equiv K+\omega^{\prime}$ is the
local Larmor frequency consisting of the anisotropy $K$ and the exchange
coupling $\omega^{\prime}=|J|S$. Thus we have the constraints
$\alpha\geq\alpha^{\prime}$ and $\kappa\geq\omega^{\prime}$. To locate the EP,
we set $\mathscr{H}^{2}=0$, giving rise to the condition

$\alpha\omega^{\prime}-\alpha^{\prime}\kappa=\pm\sqrt{\kappa^{2}-\omega^{\prime
2}},$ (167)

which, in terms of the intrinsic local damping $g$, can be rewritten as

$g\omega^{\prime}=\alpha^{\prime}K\pm\sqrt{K(K+2\omega^{\prime})}.$ (168)

Assuming a weak dissipative coupling $\alpha^{\prime}\ll 1$ and a dominant
exchange interaction $\omega^{\prime}\gg K$, we obtain
$g\approx\sqrt{2K/\omega^{\prime}}=\omega/\omega^{\prime}\ll 1$ with
$\omega=\sqrt{2K\omega^{\prime}}$ being the intrinsic antiferromagnetic
resonance frequency. This is consistent with the small value of the damping
parameter $g$.

We observe that two antiferromagnetically coupled spins naturally lead to an
EP when the Hamiltonian is solely determined by the exchange coupling, with
the Hamiltonian taking the form of

$\mathscr{H}_{\text{AF}}=\omega^{\prime}(i\hat{\sigma}_{y}-\hat{\sigma}_{z}),$
(169)

which is not diagonalizable. However, in the ferromagnetic case, one needs to
introduce damping and additional fields to achieve EPs. This is evident from
the Hamiltonian taking the form of

${\mathscr{H}}_{\text{F}}=\omega^{\prime}(1-\hat{\sigma}_{x}),$ (170)

in the presence of only ferromagnetic exchange. To gain insights into such
behavior, we decompose the small-angle dynamics into symmetric and
antisymmetric components, given by
$\mathscr{L}\pm=(\mathscr{L}_{1}\pm\mathscr{L}_{2})/2$. In both scenarios,
there exists a zero mode $\mathscr{L}_{+}$, corresponding to a reorientation
of the overall order parameter. In the ferromagnetic case, the distortion of
the order parameter triggers its small-angle precession with frequency
$2\omega^{\prime}$, while in the antiferromagnetic case, it results in an
unbounded growth of $\mathscr{L}_{+}$. This suggests that the
antiferromagnetic EP leads to a breakdown of the linearized treatment. In the
context of spin dynamics, it is simply the precession of the Néel order
parameter in the plane perpendicular to the distortion $\mathscr{L}_{-}$ with
frequency $\propto\omega^{\prime}\mathscr{L}_{-}$.

###  4.2 EPs and Bell-state generation between qubits with magnon-mediated
dissipative couplings

In this section, we utilize the previously introduced master equation approach
(Sec. 3.2) to derive the magnon-mediated dissipative couplings between two
qubits [41, 345]. Specifically, we consider two qubits that are coupled to the
same magnetically-active medium [346, 347], as depicted in Fig. 12(a). We
demonstrate that the resulting dissipative couplings can be harnessed for
generating steady entanglement between the two qubits. Moreover, through
appropriate post selections as illustrated in Fig 12(b), the time evolution of
the entanglement can be described by a non-Hermitian Hamiltonian featuring
dynamical phases separated by an EP. This EP enables the generation of a Bell
state, which can play a critical role in quantum information processing.

Figure 12: (a) A system composed of two spin qubits is coupled with a magnetic
environment, which induces local relaxation $\\{a,\tilde{a}\\}$, mediates
dissipative couplings $\\{A,\tilde{A}\\}$, as well as coherent couplings
between two qubits. The two qubits may achieve a stable entangled state with
large enough $A$ and $\tilde{A}$, and even a Bell state with the help of
measurement and postselection. (b) Schematic of dynamics of qubits under
measurements and postselection. This monitored evolution is governed by a non-
Hermitian Hamiltonian (181). The figure is reproduced with permission from
Ref. [41].

Lindblad master equation description. Let us consider an illustrative model
consisting of two spin qubits weakly coupled to a magnet, with the following
Hamiltonian

$\hat{H}=\hat{H}_{\text{S}}+\hat{H}_{\text{E}}+\hat{H}_{\text{SE}}.$ (171)

Here,
$\hat{H}_{\text{S}}=-(\Delta_{1}\hat{\sigma}_{1}^{z}+\Delta_{2}\hat{\sigma}_{2}^{z})/2$
is the Hamiltonian for the system with two qubits subjected to two local
magnetic fields $\Delta_{1}$ and $\Delta_{2}$, respectively, along the
$\hat{\bf z}$-direction, $\hat{H}_{\text{E}}$ is an unspecified Hamiltonian of
the medium as an environment for the system, and
$\hat{H}_{\text{SE}}=\lambda(\hat{\boldsymbol{\sigma}}_{1}\cdot\hat{\bf
S}_{1}+\hat{\boldsymbol{\sigma}}_{2}\cdot\hat{\bf S}_{2})$ describes the
system-environment interaction with coupling strength $\lambda$, where
$\hat{\boldsymbol{\sigma}}_{i}$ stands for the Pauli matrice of the $i$-th
qubit, and $\hat{\bf S}_{i}$ represents local spin density operator within the
medium. Without loss of generality, we assume $\Delta_{1}\geq\Delta_{2}\geq
0$. We focus on an axially-symmetric environment $\hat{H}_{\text{E}}$ in spin
space, while a generalization is straightforward. It would also be interesting
to generalize the physics to the dipolar coupling between the qubit and the
medium [348].

Similar to the example that we discussed in Sec. 3.2.2, we can derive the
Lindblad master equation of the density matrix of the two-qubit system based
on Born-Markov approximation:

$\derivative{t}\hat{\rho}=-i\big{[}\hat{H}_{\text{S}}+\hat{H}_{\text{eff}},\hat{\rho}\big{]}-\mathcal{L}[\hat{\rho}].$
(172)

Here, $\hat{H}_{\text{eff}}$ refers to the medium-induced effective coherent
coupling between qubits, which leads to the unitary evolution of the system.
However, we will disregard this factor in the ensuing discussion as we aim to
investigate the impact of induced dissipative couplings. It is worth noting
that the coherent interactions can also be derived using the Schrieffer-Wolff
transformation [349, 350, 351, 352, 353, 348]. $\mathcal{L}[\hat{\rho}]$ is
the dissipative Lindbladian expanded in the usual form:

$\mathcal{L}[\hat{\rho}]=\sum_{nm}h_{nm}\big{(}\hat{\mathcal{O}}^{\dagger}_{m}\hat{\mathcal{O}}_{n}\hat{\rho}+\hat{\rho}\hat{\mathcal{O}}^{\dagger}_{m}\hat{\mathcal{O}}_{n}-2\hat{\mathcal{O}}_{n}\hat{\rho}\hat{\mathcal{O}}^{\dagger}_{m}\big{)},$
(173)

where the coefficient matrix $h$ is Hermitian and positive-semidefinite, and
$\hat{\mathcal{O}}\\!\\!=\\!\\!(\hat{\sigma}_{1}^{-},\hat{\sigma}_{2}^{-},\hat{\sigma}_{1}^{+},\hat{\sigma}_{2}^{+},\hat{\sigma}_{1}^{z},\hat{\sigma}_{2}^{z})$
comprises qubit operators. In the dissipative Lindblad part, $h$ is block
diagonal due to the axial symmetry. In general terms, we have

$h=\matrixquantity(\tilde{a}&\tilde{A}\\\
\tilde{A}^{*}&\tilde{a})\oplus\matrixquantity(a&A^{*}\\\
A&a)\oplus\matrixquantity(d&\mathfrak{D}\\\ \mathfrak{D}^{*}&d),$ (174)

where $\tilde{a},a,d$ and $\tilde{A},A,\mathfrak{D}$ are real and complex
phenomenological parameters, respectively. These parameters represent three
types of dissipative effects: $a$ and $\tilde{a}$ are associated with local
decay and the reverse process. They govern the local relaxation of individual
qubits, giving rise to the relaxation time $T_{1}$ and contributing to the
decoherence time $T_{2}$ of a single qubit. In contrast, $A$ and $\tilde{A}$
are related to cooperative decay and the reverse process involving both qubits
and are referred to as dissipative couplings, which depend on the distance
between the two qubits. $d$ and $\mathfrak{D}$ are pure-dephasing parameters,
originating from those terms in $\hat{H}_{\text{SE}}$ that commute with
$\hat{H}_{\text{S}}$, namely
$\lambda\sum_{i=1,2}\hat{\sigma}_{i}^{z}\hat{S}^{z}_{i}$. They only cause
information but not energy exchange between the system and the medium, and in
practice may be mitigated by dynamic decoupling [354, 355, 356]. We neglect
pure-dephasing effects in the following discussion, though they may also lead
to entanglement between multiple qubits, as shown recently [357]. The
Lindbladian (173) can then be brought into a diagonal form with four quantum-
jump operators

$\hat{J}_{1}=\sqrt{\cfrac{\tilde{a}+|\tilde{A}|}{2}}\left(\hat{\sigma}_{1}^{-}+\hat{\sigma}_{2}^{-}\right),\,\hat{J}_{2}=\sqrt{\cfrac{\tilde{a}-|\tilde{A}|}{2}}\left(\hat{\sigma}_{1}^{-}-\hat{\sigma}_{2}^{-}\right),\,\hat{J}_{3}=\sqrt{\cfrac{a+|A|}{2}}\left(\hat{\sigma}_{1}^{+}+\hat{\sigma}_{2}^{+}\right),\,\hat{J}_{4}=\sqrt{\cfrac{a-|A|}{2}}\left(\hat{\sigma}_{1}^{+}-\hat{\sigma}_{2}^{+}\right),$
(175)

yielding
$\underline{\mathcal{L}}[\hat{\rho}]=\sum_{i=1}^{4}\mathcal{D}_{\hat{J}_{i}}[\hat{\rho}]$,
where the dissipator is defined as
$\mathcal{D}_{\hat{J}}[\hat{\rho}]\equiv\hat{J}^{\dagger}\hat{J}\hat{\rho}+\hat{\rho}\hat{J}^{\dagger}\hat{J}-2\hat{J}\hat{\rho}\hat{J}^{\dagger}$.
Here, we have gauged out the phases of $A,\tilde{A}$ by absorbing them into
the definition of $\hat{\sigma}^{\pm}$.

Microscopically, all parameters are given by Green’s functions of the medium
in equilibrium [41], for example,

$a=\frac{i\lambda^{2}}{2}G^{>}_{\hat{S}_{1}^{+}\hat{S}_{1}^{-}}(\Delta),\;\;\tilde{a}=\frac{i\lambda^{2}}{2}G^{<}_{\hat{S}^{+}_{1}\hat{S}^{-}_{1}}(\Delta),\;\;A=\frac{i\lambda^{2}}{2}G^{>}_{\hat{S}_{1}^{+}\hat{S}_{2}^{-}}(\Delta),\;\;\tilde{A}=\frac{i\lambda^{2}}{2}G^{<}_{\hat{S}^{+}_{1}\hat{S}_{2}^{-}}(\Delta).$
(176)

Here, the greater and lesser Green’s functions follow the conventional
definitions:

$G^{>}_{\hat{X},\hat{Y}}(t)\equiv-i\langle\hat{X}(t)\hat{Y}\rangle,\;\;\;G^{<}_{\hat{X},\hat{Y}}(t)\equiv-i\langle\hat{Y}\hat{X}(t)\rangle,$
(177)

and the Fourier transformation is given by $G(\omega)=\int dt\;G(t)e^{i\omega
t}$. These parameters are not independent, as dictated by the fluctuation-
dissipation theorem $\tilde{a}=e^{-\beta\Delta}a$ and
$\tilde{A}=e^{-\beta\Delta}A$, where $\beta=1/(k_{B}T)$ and
$\Delta\equiv(\Delta_{1}+\Delta_{2})/2$. The zero temperature, therefore,
corresponds to $\tilde{a}=\tilde{A}=0$, where only the decay processes
survive. Also, the thermodynamic stability of the magnetic medium imposes
$a\geq|A|$ and $\tilde{a}\geq|\tilde{A}|$, which ensures the matrix $h$ is
positive-semidefinite.

To explore the dynamics of the entanglement between the two spin qubits in the
presence of the dissipative couplings, we neglect the induced coherent
interaction and focus on the zero-temperature case. Assuming the initial state
is a trivial product state $\ket{\uparrow\downarrow}$, the master equation is
reduced to an equation for
$x\equiv\real\bra{\uparrow\downarrow}\hat{\rho}\ket{\downarrow\uparrow}$:

$\ddot{x}+4a\dot{x}+4\left(\delta^{2}+a^{2}-|A|^{2}\right)x=0,$ (178)

where $\delta\equiv(\Delta_{1}-\Delta_{2})/2$ is the local field asymmetry.
This equation resembles a damped oscillator with complex characteristic
frequencies

$\omega_{\pm}=\pm 2\omega_{0}-i2a,$ (179)

where $\omega_{0}\equiv\sqrt{\delta^{2}-|A|^{2}}$. We identify three distinct
parameter regimes for this dynamics.

In the underdamped regime, $\delta>|A|$, $\omega_{0}$ is real-valued. To
quantify the time evolution of the entanglement between the two qubits, we
calculate the concurrence [358] as a function of time:
$\mathcal{C}(t)=2e^{-2at}|A\sin\omega_{0}t|\sqrt{\delta^{2}-|A\cos\omega_{0}t|^{2}}/\omega_{0}^{2}$.
The entanglement oscillates with frequency $2\omega_{0}$ as the system decays
rapidly to the ground state $\ket{\uparrow\uparrow}$ on the time scale
$\tau=1/2a$, as shown in Fig 13(a).

At the critical point $\delta=|A|$, $\omega_{0}=0$, there is no oscillation,
corresponding to the orange curve in Fig 13(a). The concurrence evolves as
$\mathcal{C}(t)\propto te^{-2at}$, where the final steady state is also
$\ket{\uparrow\uparrow}$. We have a larger transient-peak entanglement, and
the decay process is slowed down moderately compared with the underdamped
regime.

In the overdamped regime, $\delta<|A|$, $\omega_{0}$ becomes purely imaginary
and $\omega_{\pm}=-2i(a\pm\kappa_{0})$, with
$\kappa_{0}=\sqrt{|A|^{2}-\delta^{2}}$. The time-dependent concurrence is
$\mathcal{C}(t)=2e^{-2at}|A|\sinh\kappa_{0}t\sqrt{|A\cosh\kappa_{0}t|^{2}-\delta^{2}}/\kappa_{0}^{2}$.
In addition to a larger transient peak entanglement, the decay process has
been slowed down dramatically, see Fig 13(a). On a long time scale $t\gg
1/\kappa_{0}$, $\mathcal{C}(t)\propto e^{-2(a-\kappa_{0})t}$. The entanglement
can last for $\tau=1/2(a-\kappa_{0})$, which becomes $\tau=1/2(a-|A|)$ when
the two local fields are the same, $\delta=0$. It is clear from this
expression of the lifetime $\tau$ that the dissipative coupling $A$ and the
local relaxation $a$, though both originating from the qubits-magnet coupling,
have opposite effects on the quantum entanglement in the nonunitary evolution.
The local dissipation tends to destroy any quantum coherence whereas the
dissipative coupling can be exploited to extend the lifetime of entanglement
and even realize steady entangled states. With equal local fields $\delta=0$,
a finite entanglement can persist for a long time before eventually decaying
to zero in the large dissipative coupling regime $|A|\lesssim a$. Based on
their (greater) Green’s function expressions, this regime physically
corresponds to the scenario with two qubits placed within a lengthscale
dictated by the relevant excitations responsible for dissipation. For example,
for qubits coupled to a magnetically ordered medium via processes of magnon
absorption and emission, this length scale is set by the wavelength of the
magnon at frequency $\Delta$. Furthermore, the concurrence lifetime extends to
infinity $\tau\rightarrow\infty$ when $|A|$ reaches its maximal allowed value
$|A|=a$, and thus a steady entanglement is achieved, $\mathcal{C}(\infty)=1/2$
for the final state
$\hat{\rho}_{\infty}=\left(\ket{\uparrow\uparrow}\bra{\uparrow\uparrow}+\ket{00}\bra{00}\right)/2$,
where $\ket{00}$ stands for the singlet state.

Figure 13: (a) Concurrence of two qubits as a function of time, with initial
state $\ket{\uparrow\downarrow}$, where we set both local dissipation $a$ and
dissipative coupling $|A|$ to be 1. The black curve corresponds to the
underdamped quantum regime. The orange curve is at the critical point
$\delta=1$, where entanglement decays as $\mathcal{C}(t)\propto te^{-2at}$.
The cyan, $\delta=0.3$, and the red, $\delta=0$, curves are in an overdamped
quantum regime, where the lifetime of entanglement is extended dramatically.
(b) Concurrence of qubits as a function of time with initial state
$\ket{\uparrow\downarrow}$ under continuous measurements and post selections.
We set $\Gamma=2.$ The black curve $\delta=5$ is in the
$\mathcal{P}\mathcal{T}$-exact regime, where entanglement oscillates and its
maximal value is less than 1. At the exceptional point (cyan curve), there is
no oscillation and its maximal value is 1. In the
$\mathcal{P}\mathcal{T}$-broken regime (red curve), entanglement is
$\mathcal{C}(t)=\tanh 2\Gamma t$. The inset shows the maximal concurrence as a
function of $\delta/\Gamma$. The figure is reproduced with permission from
[41].

Non-Hermitian Hamiltonian description. When the Lindblad master equation
governing a quantum system does not involve quantum jumps, the quantum
dynamics can be described using an effective non-Hermitian Hamiltonian. This
can be accomplished through the measurement of the system and the subsequent
post-selection of the measurement results, as discussed in Sec. 3.2.3. To this
end, we rewrite the master equation in the following form:

$\derivative{\hat{\rho}}{t}=-i[\hat{\mathcal{H}}_{\text{eff}},\hat{\rho}]+2\sum_{i=1}^{4}\hat{J}_{i}\hat{\rho}\hat{J}_{i}^{\dagger},$
(180)

where we combine the first two terms in the Lindbladian (173) and the bare
Hamiltonian of the qubits:

$\hat{\mathcal{H}}_{\text{eff}}=\hat{H}_{S}-i\sum_{i}\hat{J}_{i}^{\dagger}\hat{J}_{i}.$
(181)

It is a non-Hermitian Hamiltonian, and the commutator in the master equation
(180) thus should be understood as
$[\hat{\mathcal{H}}_{\text{eff}},\hat{\rho}]\equiv\hat{\mathcal{H}}_{\text{eff}}\hat{\rho}-\hat{\rho}\hat{\mathcal{H}}_{\text{eff}}^{\dagger}$.
By subjecting the two qubits to continuous measurements of the absolute value
of their total spin $z$ component
$\hat{\underline{\sigma}}^{z}=\hat{\sigma}_{1}^{z}+\hat{\sigma}_{2}^{z}$ and
subsequently conditioning the postselection on zero outcomes, we can
effectively forbid all quantum jump processes, as
$[\hat{J}_{i},\hat{\underline{\sigma}}^{z}]\neq 0$. As a possible low-
temperature implementation of the proposed post-selection scheme, we may post-
select on the absence of any emitted magnons. This monitored dynamics of the
two-qubit system formally eliminates the last term in Eq. (180) and reduces
quantum dynamics to a non-Hermitian Hamiltonian form in the
$\hat{\underline{\sigma}}^{z}$ subspace. In the basis
$\\{\ket{\uparrow\downarrow},\ket{\downarrow\uparrow}\\}$, this non-Hermitian
Hamiltonian is [22, 74]

$\hat{\mathcal{H}}_{\text{eff}}=\matrixquantity[-\delta-i(a+\tilde{a})&-i\Gamma\\\
-i\Gamma&\delta-i(a+\tilde{a})],$ (182)

where $\Gamma\equiv A+\tilde{A}$ is the dissipative coupling. It is clear that
there are three different regimes: $\mathcal{PT}$ symmetry broken regime with
$\delta<\Gamma$, the exceptional point with $\delta=\Gamma$, and
$\mathcal{PT}$-exact regime with $\delta>\Gamma$.

In the $\mathcal{PT}$-broken regime, the eigenvalues of this Hamiltonian are
purely imaginary. Assuming the initial state is $\ket{\uparrow\downarrow}$ and
$\delta=0$, the entanglement between the two qubits is
$\mathcal{C}(t)=\tanh(2\Gamma t)$. The final state is maximally entangled, as
shown in Fig. 13(b). At the EP, $\hat{\mathcal{H}}_{\text{eff}}$ is non
diagonalizable, since two eigenstates coalesce into
$\left(\ket{\uparrow\downarrow}+i\ket{\downarrow\uparrow}\right)/\sqrt{2}$.
The two qubits will gradually evolve into this sole state where they are
maximally entangled. For example, starting with a trivial state
$\ket{\uparrow\downarrow}$, the concurrence $\mathcal{C}(t)=2\Gamma
t\sqrt{1+\Gamma^{2}t^{2}}/(1+2\Gamma^{2}t^{2}),$ algebraically approaching 1,
see Fig. 13(b). In the $\mathcal{PT}$-exact regime, two eigenenergies have
nonzero real parts. The entanglement will oscillate with frequency
$2\sqrt{\delta^{2}-\Gamma^{2}}$, as shown in Fig. 13(b). The maximal
entanglement one can achieve is
$\mathcal{C}_{\text{max}}(\eta)=\sqrt{2-1/\eta^{2}}/\eta,$ with
$\eta=\delta/\Gamma$, which is less than 1. Notably, the second derivative of
$\mathcal{C}_{\text{max}}$ is discontinuous across the EP ($\eta=1$),
reflecting a phase transition [see Fig. 13(b)].

###  4.3 EPs by balanced gain and loss of spin dynamics

####  4.3.1 Theoretical proposals

Second-order EPs.—Let us start with a simplified model to address how the EPs
are realized in the collective dynamics of macro spins and how the magnonic
functionality is improved by the EPs. The model of two macroscopic magnetic
structures with balanced gain and loss that are coupled with ferromagnetic
exchange constant $J>0$ [359] is shown in Fig. 14(a). Similar non-Hermitian
magnetic models were extensively studied later in Refs. [23, 188, 104, 197,
191].

Figure 14: Realization of EPs in coupled magnetic structure by balanced gain
and loss. (a) is the configuration. (b) and (c) plot the mutation of complex
energy $\omega$, amplitude ratio $R$, and phase difference $\Delta\psi$ of the
two modes around the EP. $\alpha_{cr}\approx 0.3$, denoted by the green dashed
line in (b) and (c), implies the EP. (d) shows the dependence of critical
damping coefficient $\alpha_{cr}$ on $\gamma M_{s}J/\gamma B_{z}$. The figures
are adapted with permission from Ref. [23].

The macroscopic magnetic structures extended in the $y$-$z$ plane are applied
by the external field ${\bf B}=B_{z}\hat{\bf z}$ along the $\hat{\bf
z}$-direction, with the free energy

$F=-{\bf M}_{1}\cdot{\bf B}-{\bf M}_{2}\cdot{\bf B}-J{\bf M}_{1}\cdot{\bf
M}_{2},$ (183)

where ${\bf M}_{1}$ and ${\bf M}_{2}$ are the magnetization of two magnetic
layers that are coupled with ferromagnetic exchange coupling $J>0$. The
magnetization dynamics are governed by the LLG equation [310, 309]

$\displaystyle\frac{d{\bf M}_{1}}{dt}$ $\displaystyle=-\gamma{\bf
M}_{1}\times(B_{z}\hat{\bf z}+J{\bf M}_{2})+\frac{\alpha}{\left|{\bf
M}_{1}\right|}{\bf M}_{1}\times\frac{d{\bf M}_{1}}{dt},$ (184)
$\displaystyle\frac{d{\bf M}_{2}}{dt}$ $\displaystyle=-\gamma{\bf
M}_{2}\times(B_{z}\hat{\bf z}+J{\bf M}_{1})-\frac{\alpha}{\left|{\bf
M}_{2}\right|}{\bf M}_{2}\times\frac{d{\bf M}_{2}}{dt},$

where $\gamma$ is the gyromagnetic ratio. The positive Gilbert damping
coefficient $\alpha$ describes the intrinsic damping of magnetization, while a
negative one implies the amplification or “gain", which may be realized
effectively by such as parametric driving [360, 23, 361], spin transfer torque
[362, 363, 192, 194], and optomagnonic interaction [364, 365, 11].

In the linear regime, the magnetization

${\bf M}_{1,2}(t)\approx M_{s}\hat{\bf z}+\left({m}_{1,2x}\hat{\bf
x}+{m}_{1,2y}\hat{\bf y}\right)e^{-i\omega t},$ (185)

where $M_{s}=\left|{\bf M}_{1}\right|=\left|{\bf M}_{2}\right|$ is the
saturated magnetization and $\left|{m}_{1,2x,y}\right|\ll M_{s}$. With
${m}_{1,2}^{-}={m}_{1,2x}-i{m}_{1,2y}$, the LLG equation (184) reduces to

$\omega\left(\begin{array}[]{c}m_{1}^{-}\\\ m_{2}^{-}\\\
\end{array}\right)=\frac{\gamma}{1+\alpha^{2}}\left(\begin{array}[]{cc}B_{z}+M_{s}J-i\alpha\left(B_{z}+M_{s}J\right)&-M_{s}J+i\alpha
M_{s}J\\\ -M_{s}J-i\alpha
M_{s}J&B_{z}+M_{s}J+i\alpha\left(B_{z}+M_{s}J\right)\\\
\end{array}\right)\left(\begin{array}[]{c}m_{1}^{-}\\\ m_{2}^{-}\\\
\end{array}\right),$ (186)

with the eigenvalues and eigenvectors for the two branches

$\displaystyle\omega_{\pm}$
$\displaystyle=\frac{\gamma}{1+\alpha^{2}}\left({B_{z}+M_{s}J\pm\sqrt{(M_{s}J)^{2}-\alpha^{2}(B_{z}^{2}+2B_{z}M_{s}J)}}\right),$
(187a) $\displaystyle\left(\begin{array}[]{cc}m_{1}^{-}\\\
m_{2}^{-}\end{array}\right)_{\pm}$
$\displaystyle=\left(\begin{array}[]{c}\frac{i}{(1+i\alpha)M_{s}J}\left(\alpha(B_{z}+M_{s}J)\pm
i\sqrt{(M_{s}J)^{2}-\alpha^{2}(B_{z}^{2}+2B_{z}M_{s}J)}\right)\\\ 1\\\
\end{array}\right).$ (187f)

According to Eq. (187a), the EP appears with the specific Gilbert damping
coefficient $\alpha_{\rm cr}=M_{s}J/\sqrt{B_{z}^{2}+2B_{z}M_{s}J}$. The
eigenvalue is purely real when $\alpha<\alpha_{\rm cr}$, which, on the other
hand, contains the imaginary component when $\alpha>\alpha_{\rm cr}$. As shown
in Fig. 14(b) with a specific external field $B_{z}=2.5M_{s}J$, the two
eigenvalues merge at $\alpha=\alpha_{\rm cr}\approx 0.3$. The amplitude ratio
$R=\left|{m_{1}^{-}}/{m_{2}^{-}}\right|_{\pm}$ and the phase difference
$\Delta\psi={{\rm Arg}(m_{1}^{-})}/{{\rm Arg}(m_{2}^{-})}$ of the two modes
experience a sudden change around the EP, as shown in Fig. 14(c). The two
eigenmodes share the same amplitude when $\alpha<\alpha_{cr}$ as in Fig.
14(c), implying an equal distribution of energy. On the other hand, when
$\alpha>\alpha_{cr}$, the relative amplitude becomes asymmetric with a fixed
phase difference, suggesting that the energy tends to accumulate in one of the
two eigenmodes. With an increase of ratio $M_{s}J/B_{z}$, $\alpha_{\rm cr}$
for the EP increases as in Fig. 14(d). It also implies that the condition for
the emergence of EP becomes a specific line in the two-dimensional parameter
space constituted by $M_{s}J$ and $B_{z}$. By analogy, it can be envisioned
that the exceptional lines and surfaces can emerge in higher dimensional
parameter space when more freedoms are introduced such as the frequency
asymmetry.

A similar design is also proposed in the system with two magnetic layers
embedded with a metallic Pt layer, where the injected current in the Pt layer
exerts opposite torques for two adjacent magnetic layers [198]. For such a
$\mathcal{PT}$-symmetric system, Wang et.al. found many unique functionalities
near the EPs such as enhanced magnetic susceptibility, magnon trapping, and
magnon enhancement [198].

Different from the indirect coupling between two ferromagnetic layers mediated
by a metallic spacer, Temnaya et al. studied a direct coupling by the magneto-
dipolar interaction in two identical planar ferromagnetic layers and found
that the EPs can be realized by injected spin current from thin layers of
heavy metal [191], as shown in Fig. 15(a). The magnon in the ferromagnetic
layer is dissipated governed by the Gilbert damping, while the injected spin
current can compensate or enhance this dissipation via tuning the sign of the
voltage in the heavy metal layer, thus with the opposite voltage for the two
heavy-metal layers achieving controllable gain or loss for the ferromagnetic
layer. Let us denote one magnetization of the ferromagnetic layer as “${\bf
M}_{1}$" and the other one “${\bf M}_{2}$". The magnetic dynamics of the
coupled ferromagnetic layers are governed by

$\displaystyle\frac{d{\bf M}_{1}}{dt}$ $\displaystyle=-\mu_{0}\gamma{\bf
M}_{1}\times\mathbf{H}_{\mathrm{eff},1}+\frac{\alpha}{\left|{\bf
M}_{1}\right|}{\bf M}_{1}\times\frac{d{\bf M}_{1}}{dt}+\frac{\sigma
J_{s}}{\left|{\bf M}_{1}\right|}\left({\bf M}_{1}\times\left({\bf
M}_{1}\times{{\bf{p}}_{1}}\right)\right),$ (188) $\displaystyle\frac{d{\bf
M}_{2}}{dt}$ $\displaystyle=-\mu_{0}\gamma{\bf
M}_{2}\times\mathbf{H}_{\mathrm{eff},2}+\frac{\alpha}{\left|{\bf
M}_{2}\right|}{\bf M}_{2}\times\frac{d{\bf M}_{2}}{dt}+\frac{\sigma
J_{s}}{\left|{\bf M}_{2}\right|}\left({\bf M}_{2}\times\left({\bf
M}_{2}\times{{\bf{p}}_{2}}\right)\right),$

where $\mu_{0}$ is the vacuum magnetic permeability and
$\mathbf{H}_{\mathrm{eff,1}}$ and $\mathbf{H}_{\mathrm{eff,2}}$ are,
respectively, the effective field for ${\bf M}_{1}$ and ${\bf M}_{2}$, which
is dominated by the external field and mutual magnetic dipolar field. The
third term phenomenologically describes the extra gain or loss introduced by
the injected spin current $J_{s}$ with ${\bf p}$ denoting the polarization
direction of electron spin and $\sigma$ characterizing the efficiency of spin
transfer torque. As shown in Fig. 15(a), the balanced extra gain or loss by
the injected spin current is controlled by applying the opposite voltage on
the heavy-metal layer.

Figure 15: Realization of EPs in planar $\mathcal{PT}$-symmetric
heterostructures of coupled ferromagnetic–heavy-metal layer. (a) is the
configuration. (b) shows the dependence of critical voltage for the emergence
of EP on the distance between two ferromagnetic layers. The inset in (b) shows
the frequency variation near the EP as s function of voltage. The figure is
reproduced with permission from Ref. [191].

Similar to the previous analysis, the EPs emerge in the competition between
the relative gain and loss of two ferromagnetic layers, and their coupling
strength. At a fixed distance, the EPs emerge at a certain voltage accompanied
by a sudden change of eigenvalue in the parameter space. An increase in
distance causes weaker magneto-dipolar coupling, leading to a small required
voltage for the emergence of EP as shown in Fig. 15(b). In addition, the non-
contact magneto-dipolar coupling and the injected spin current can be
modulated by the distance between ferromagnetic layers and applied voltage,
respectively, providing a feasible way of producing and modulating EPs
experimentally.

High-order EPs.—As addressed above, the second-order EPs exist with two
coupled ferromagnetic layers or macro spins with balanced gain and loss. Yu et
al. predicted the third-order EPs that can be realized in a three-layer
structure composed of a ferromagnetic layer without net damping (“neutral")
sandwiched by two ferromagnetic layers with gain and loss of the same
magnitude governed by parameter $\alpha$ [104], as depicted in Fig. 16. The
dependence of the three eigenvalues $\omega_{n}$ with the mode indexes
$\\{n=1,2,3\\}$ on the gain and loss parameter $\alpha$ is shown in Fig.
16(b), where the second-order EPs (“EP2") and third-order EPs (“EP3") exist
with different $\alpha$ when choosing suitable exchange-coupling strength and
external magnetic field. Similar to the merging of two modes at EP2, EP3
implies that three modes share the same eigenvalue and eigenvectors.

Figure 16: Realization of high-order EPs in a ferromagnetic trilayer. (a) is
the configuration, where the red, gray, and blue layers imply the
ferromagnetic film with a gain of rate $\alpha$, without net dissipation, and
with a loss of the same rate $\alpha$. (b) The dependence of frequency spectra
on the gain-loss parameter $\alpha$, where “EP2" and “EP3" represent the
second-order and third-order EPs, respectively. (c) shows the dependence of
the ferromagnetic resonance frequency on the perturbation $\epsilon$. (d)
depicts the frequency splitting on the logarithmic scale around the third-
order EPs. The figure is reproduced with permission from Ref. [104].

As addressed in Sec. 2.1.2, one application of the EPs is the enhanced
sensitivity when a perturbation is implemented in the vicinity of EPs. Here
for the ferromagnetic trilayer, a small perturbation $\epsilon$ can be induced
by an external magnetic field. As shown in Fig. 16(c), the dependence of
ferromagnetic resonance frequency $\Omega_{n}$ on $\epsilon$ in the vicinity
of “EP3" exhibits non-linear behavior, different from the conventional linear
response. The frequency splitting can be fitted by a logarithmic function
$\text{Re}\left(\omega_{2}-\omega_{3}\right)\propto\epsilon^{1/3}$,
corresponding to the third-order EPs [Fig. 16(d)], i.e., a superior response
by small perturbations.

####  4.3.2 Observation of EPs in passive $\mathcal{PT}$-symmetric magnonic
devices

As mentioned above, the heterostructures composed of two ferromagnetic metal
(FM) layers and one normal metal (NM) spacer are the choices to investigate
non-Hermitian topological magnonics [104]. Despite the intensive studies among
theories [23, 188, 103, 104, 191, 200, 182, 265], an inconvenient fact is a
lag in experimental research. Recently, Liu et al. [190] made efforts to
achieve $\mathcal{PT}$-symmetry in a magnonic device of sandwiched structure,
as shown in Fig. 17(a). In their device, two FM layers are separated by a Pt
layer of thickness $d$. The coupling between two FM layers is governed by the
Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction, parametrized by the
constant $J$. As $d$ increases, $J$ decreases and alternates between the
ferromagnetic and antiferromagnetic couplings [190], which affects the
effective magnetic field and the damping torque of each FM layer (due to the
spin pumping effect), such that the experiment observes the variation of both
the resonant frequencies and damping rates of two FM layers with changing $d$.
Based on such tunability, the EPs via the passive $\mathcal{PT}$-symmetry is
experimentally possible to be achieved.

Figure 17: (a) A sandwiched magnetic trilayer consisting of two coupled FM
layers and a nonmagnetic layer. (b) Frequency differences between two
eigenmodes and the corresponding frequencies of each FM layer without
couplings, where $\alpha_{1}=\alpha_{0}$ and $\alpha_{2}=5/2\alpha_{0}$ are
the Gilbert damping rates of the Co and NiFe layers, respectively. (c) and (d)
Experimental results of the frequency changes and damping constants of two
eigenmodes. The figures are reproduced with permission from Ref. [190].

Besides precisely controlling the coupling strength, the Gilbert damping rates
of two FM layers need to differ to achieve the $\mathcal{PT}$-symmetry.
Accordingly, the two FM layers are chosen to be different materials with
cobalt (30 nm) and permalloy (5 nm). Their Gilbert damping rates
$\alpha_{0}({\rm Co})=0.0103$ and $\alpha_{0}({\rm NiFe})=0.0250$. With two
macro spins, there are two eigenmodes, i.e., the acoustic and optical modes
with low and high frequencies, which repel each other and form an anti-
crossing when $J$ changes. The frequency differences between the eigenmodes
and the individual frequencies of each FM layer are defined as
$\Delta\nu_{1,2}=\nu_{1,2}-\omega_{1,2}$, where $\nu_{1,2}$ are the
frequencies of the acoustic and optical modes, $\omega_{1,2}$ are the
individual frequencies of two FM layer. The frequency differences and the
damping constants $\beta$ of these two eigenmodes cross each other at $J>0$,
as shown in Fig.17(b). Thereby, the authors addressed that EP and the
$\mathcal{PT}$ symmetric transition is theoretically allowed in their device.

To experimentally test the expectation, the authors fabricate a series of
devices with the thickness of the Pt layer varying from 0 to 12 nm. An
external in-plane magnetic field $B=700$ G is applied. Figure 17(c)-(d) shows
a comparison between the experimental data and theoretical calculations. As
$J$ decreases, the eigenfrequencies of acoustic and optical modes get close.
Fitted by the Kittel formula
$\nu=\mu_{0}\gamma\sqrt{H_{\rm{eff}}(H_{\rm{eff}}+M_{\rm{eff}})}$ at a fixed
magnetic field of $700$ G, where $\gamma$, $H_{\rm{eff}}=H_{0}+H_{\rm{ex}}$,
and $M_{\rm{eff}}=J/(H_{\rm{ex}}d)$ are the gyromagnetic ratio, effective
field, and effective magnetization, respectively, the two eigenmodes cross
twice at $J=0$ and $0.13$ $\mathrm{erg/cm^{2}}$. The authors define the area
between the two intersections as the $\mathcal{PT}$ symmetry regime, while the
other areas as the $\mathcal{PT}$-symmetry broken phase. At the second
intersection with $J=0.13$ $\mathrm{erg/cm^{2}}$, the damping rates of two
eigenmodes are equal, which appears to be evidence of EPs. This experimental
finding may inspire further exploration of the exotic properties around EPs in
magnonic devices.

###  4.4 Observation of exceptional points in cavity magnonic systems

Coupled magnon-photon system in “cavity magnonics" [204, 205, 167, 206, 207,
366, 169, 213, 208, 209, 211, 210, 212, 40, 214, 11] is an alternative choice
to achieve the EPs. A cavity magnonic system involves discrete photon modes in
a microwave cavity and magnons in a bulk magnet. They can strongly couple to
the photons in a microwave cavity via the magnetic dipole-dipole interaction.
The coupling strength is proportional to the square root of the magnet’s spin
number such that it is enhanced by increasing the material volume. In
addition, changing the overlap coefficient of the two modes can also tune the
photon-magnon coupling strength.

Both the cavity photon and magnon can be treated as harmonic oscillators, with
annihilation operators denoted by $\hat{p}$ and $\hat{m}$, respectively. The
Hamiltonian for bare cavity photon
$\hat{H}_{c}=\hbar\widetilde{\omega}_{c}(\hat{p}\hat{p}^{\dagger}+1/2)$ and
magnon
$\hat{H}_{m}=\hbar\widetilde{\omega}_{m}(\hat{m}\hat{m}^{\dagger}+1/2)$, where
$\tilde{\omega}_{c}=\omega_{c}-i(\kappa+\beta)$ and
$\tilde{\omega}_{m}=\omega_{m}-i(\gamma+\alpha)$ are the complex frequencies
of the photon and magnon modes with their intrinsic (external) damping rates
$\beta$ ($\kappa$) and $\alpha$ ($\gamma$). The magnetic dipolar interaction
produces a coherent photon-magnon coupling $\hat{H}_{int}=\hbar
J(\hat{p}+\hat{p}^{\dagger})(\hat{m}+\hat{m}^{\dagger})$ with a coupling
strength $J$. Additional effects appear when the cavity mode and magnon
interact with the same bath that supports traveling waves, which offer a
common reservoir for both modes with additional dissipative rates $\kappa$ and
$\gamma$. An indirect “dissipative" coupling $i\Gamma=i\sqrt{\kappa\gamma}$ is
induced due to the cooperative radiation of two modes into the common
reservoir [184, 367, 172]. Including both the coherent and dissipative
couplings, the dynamics of a cavity magnonic system is governed by [11]

$\displaystyle i\frac{d}{dt}\left(\begin{array}[]{c}\hat{p}\\\
\hat{m}\end{array}\right)=\hbar\left(\begin{array}[]{cc}\omega_{c}-i(\kappa+\beta)&J+i\Gamma\\\
J+i\Gamma&\omega_{m}-i(\gamma+\alpha)\end{array}\right)\left(\begin{array}[]{c}\hat{p}\\\
\hat{m}\end{array}\right).$ (195)

The two eigenvalues

$\tilde{\omega}_{\pm}=\omega_{0}-i\kappa_{0}\pm\frac{1}{2}\sqrt{(\Delta\omega-i\Delta\kappa)^{2}+4(J+i\Gamma)^{2}},$
(196)

where $\omega_{0}=(\omega_{c}+\omega_{m})/2$,
$\kappa_{0}=(\kappa+\beta+\gamma+\alpha)/2$,
$\Delta\omega=\omega_{c}-\omega_{m}$, and
$\Delta\kappa=(\kappa+\beta)-(\gamma+\alpha)$. When the square root in Eq.
(196) becomes zero, the two eigenvalues and eigenfunctions of the coupled
photon-magnon system coalesce into one, indicating the emergence of the EP.

####  4.4.1 Topological properties around EPs in cavity magnonic systems

As addressed in Sec. 2.1.2, the topological mode switching near the EP means
that encircling an EP in a three-dimensional parameter space enables the
dynamic switching of the eigenmodes of the system. Such a topological property
was first demonstrated in cavity magnonics by Harder et al. [193]. In the
experiment, a yttrium iron garnet (YIG) sphere of diameter 0.3-mm was placed
at the outer and bottom edge of a cylindrical microwave cavity made of oxygen-
free copper (diameter=25 cm, height=33 cm), as sketched in Fig. 18(a). The
TM011 cavity mode is exploited in the experiment, with resonant frequency
$\omega_{c}/2\pi=10.2$ GHz and quality factor $Q=660$ [Fig. 18(b)]. The
frequency of the magnon mode, on the other hand, can be tuned by the static
magnetic field $H$ following a linear dependence, i.e.,
$\omega_{m}=\gamma_{e}(H+H_{A})$, where $\gamma_{e}$ and $H_{A}$ represent the
gyromagnetic ratio and anisotropy field, respectively.

Figure 18: (a) Experiment setup with an in situ control of the photon-magnon
coupling strength by changing the direction of the static magnetic field with
an angle $\theta$ with respect to the local microwave field. (b) Transmission
spectrum of microwaves for the TM011 mode. (c) and (d) Calculated frequencies
and line widths of two eigenmodes at different magnetic-field strengths and
directions. The blue and yellow curves show the measured mode frequencies and
linewidths when $\theta>\theta_{\rm EP}$ and $\theta<\theta_{\rm EP}$,
respectively, where $\theta_{\rm EP}$ is the critical angle for the
realization of EPs. (e) and (f) Path taken to encircle or avoid encircling the
EPs. The figure is reproduced with permission from Ref. [193].

When the damping rates of the cavity photon and magnon modes are fixed, a
method for achieving the EPs is to continuously tune the coupling strength
$J$. According to Eq. (196), the EP can be realized as long as this cavity
magnonic system possesses $\mathcal{PT}$ symmetry, i.e., $J=\Delta\kappa/2$
and $\Delta\omega=0$. Via changing the relative direction $\theta$ between the
local microwave and static magnetic fields, the coupling strength
$J=J_{M}|\sin\theta|$ [193]. Figure 18(c) and (d) address the calculated
eigenfrequencies [real part of Eq. (196)] and linewidths [imaginary part of
Eq. (196)] at different magnetic-field directions and strengths. The solid
curves are experimental data measured at $\theta=90^{\circ}$ and $23^{\circ}$,
corresponding to the strong and weak coupling cases, respectively. It is noted
that according to the calculation, the EP occurs at $\theta=33^{\circ}$, at
which the eigenfrequencies of two modes merge with each other.

To demonstrate the geometric mode switching in this cavity magnonic system,
the coupling strength is tuned to encircle the EP in the $\theta$-$H$
parameter space [193]. Two different evolution paths are addressed in Fig.
18(e). Along the evolution path, the mode initially at low (high) frequencies
can continuously evolve to high (low) frequencies after a closed path around
the EP, i.e., a mode switching. Such switching cannot be achieved when the EP
is not encircled as plotted in Fig. 18(f). Because of this mutation, one
eigenstate of the photon-magnon system can be adiabatically transitioned to
the other without crossing the coupling gap.

####  4.4.2 Coherent perfect absorption at EPs in cavity magnonics

As a singularity in a non-Hermitian system, the EP relates to a “phase"
transition in a coupled system and leads to novel phenomena, for instance, the
spontaneous symmetry breaking across the EPs (refer to Sec. 2.1). In a cavity
magnonic system, besides the abrupt change of the system’s eigenfrequencies
near EPs, the dissipation rates of eigenmodes are dramatically altered as
well. Since the dissipation rates have a significant impact on the
electromagnetic scattering of the coupled photon-magnon modes, the energy
absorption of the cavity magnonic system exhibits a significant change near
EPs. Zhang et al. observed the polaritonic coherent perfect absorption at the
EP [26].

In the experiment [26], a YIG sphere was placed near the bottom of a
rectangular microwave cavity as shown in Fig. 19(a). A wooden rod attached to
the YIG sphere is used to precisely control its position in the cavity and the
photon-magnon coupling strength $g_{m}$ as well, noting $g_{m}=J$ in Eq.
(196). To achieve the EP in this system, the authors tuned the damping rates
of the cavity modes, while the damping rate of the magnon mode $\gamma_{m}$ is
not affected. For the cavity modes, besides the intrinsic damping rate
$\kappa_{int}$ that arises from the material and surface roughness, the cavity
has two external damping rates $\kappa_{1,2}$ because of the photon leakage at
two ports. These two external damping rates can be precisely controlled by
adjusting the lengths of the pins.

Figure 19: (a) shows the cavity magnonic system, where a YIG sphere glued on a
wooden rod is placed at the bottom of a rectangular cavity. (b) illustrates
the two feedings, indicated by the blue and red arrows, and the position of
the YIG sphere, which is horizontally moved to tune the photon-magnon coupling
strength. (c) and (d) Total output microwave spectra when the
$\mathcal{PT}$-symmetry is unbroken and broken. (e) Measured total microwave
output spectra vs. the position of the YIG sphere in the cavity. The white
circles indicate the two photon-magnon hybridized modes that merge with each
other at $\pm 1$ mm, corresponding to the EPs in the system. (f) Different
magnon-photon coupling scenarios, including the weak coupling, the MIT, and
the strong coupling regimes, indicated by different colors. The EPs determine
the phase transition boundary between the weak coupling and the MIT regime.
The black circles represent the minimal microwave output spectra measured at
different positions of the YIG sphere. The red solid curve indicates the
magnon-photon coupling strengths. The figure is reproduced with permission
from [26].

Further, to construct an effective Hamiltonian with $\mathcal{PT}$-symmetry,
the authors input two coherent microwave tones into the cavity from the two
ports to drive the coupled photon-magnon system into a steady state. The
authors treat the two input tones as effective gains and phenomenologically
model their system as

$\hat{H}_{\rm
CPA}/\hbar=[\omega_{c}+i(\kappa_{1}+\kappa_{2}-\kappa_{int})]\hat{p}^{\dagger}\hat{p}+(\omega_{m}-i\gamma_{m})\hat{m}^{\dagger}\hat{m}+g_{m}(\hat{p}^{\dagger}\hat{m}+\hat{p}\hat{m}^{\dagger}).$
(197)

When $\omega_{c}=\omega_{m}$ and
$\kappa_{1}+\kappa_{2}-\kappa_{int}=\gamma_{m}$ are satisfied by tuning the
magnetic field and the lengths of pins, respectively, a non-Hermitian
Hamiltonian with the $\mathcal{PT}$-symmetry can be achieved. The two
eigenfrequencies $\omega_{\pm}=\omega_{0}\pm\sqrt{g^{2}_{m}-\gamma^{2}_{m}}$
of the system become real as long as $g_{m}>\gamma_{m}$. Furthermore, by
setting the power ratio as $\kappa_{1}/\kappa_{2}=1.23$ and the phase
difference as $\Delta\phi=0$ in their experiment, coherent perfect absorption
of electromagnetic waves is achieved: in the measured total output spectrum
$|S_{tot}^{-}|^{2}=|S_{1}^{-}|^{2}+|S_{2}^{-}|^{2}$, two zero dips, at which
there is nearly no microwave leakage, indicate that all feeding microwave
photons from the two ports are absorbed by the system.

On the other hand, in this coupled system the EPs can be achieved by tuning
the photon-magnon coupling strength $g_{m}$. Because of the field profile of
TE102 mode, $g_{m}$ almost linearly decreases from 5.5 MHz to zero when the
YIG sphere is moved from $x=\pm 4$ mm to the center. The total microwave
output spectra measured at each YIG position are plotted in Fig. 19(e). The
two eigenmodes merge at $x=\pm 1$ mm, indicating the emergence of EPs. Between
the two EPs, only one absorption dip occurs, which is much smaller in
magnitude than that of two dips outside the EPs. It implies that microwave
absorption is suppressed when the modes coalesce, and EPs imply a transition
boundary. This transition was explained as the $\mathcal{PT}$-symmetry
breaking. The authors argued that, between the two EPs, the eigenfrequencies
evolve from purely real numbers to complex pairs, such that additional energy
dissipation is produced and breaks the balance between the effective gain and
loss.

In addition, these two EPs indicate the boundary between the magnetic induce
transparency (MIT) ($\kappa_{int}+\kappa_{1}+\kappa_{2}>g_{m}>\gamma_{m}$) and
weak-coupling ($\kappa_{int}+\kappa_{1}+\kappa_{2},\gamma_{m}>g_{m}$) regimes,
as indicated in Fig. 19(f). The black circles show the minimum of
$|S_{tot}^{-}|^{2}$ as a function of the position of the YIG sphere. Two
abrupt changes of $|S_{tot}^{-}|^{2}$ at $x=\pm 1$ mm correspond to the two
EPs. This work combines three intriguing phenomena, i.e., coherent perfect
absorption, EPs, and $\mathcal{PT}$-symmetry breaking, in a single system,
which demonstrates the potential of hybridized magnonic devices in the
exploration of non-Hermitian physics.

####  4.4.3 EPs and singularities in a dissipatively coupled photon-magnon
system

The “level attraction" that exhibits the coalescence of real components and
repulsion of imaginary components of eigenfrequencies of a coupled magnon-
photon system was theoretically proposed by Grigoryan et al. [210]. The
experimental observation was first reported in a microwave optomechanical
circuit [368]. Shortly after it, a similar phenomenon was found in a cavity
magnonic system and attributed to the cavity Lenz effect [34] [see Fig. 20(a)
and (b) for similar spectra]. These motivate efforts to explore the physical
origin and the potential functionalities of level attraction in cavity
magnonic devices [369, 211, 370, 241, 248, 184, 371, 367]. One mechanism that
can produce level attraction is the multi-channel interference in a cavity
magnonic system, which was experimentally demonstrated in Ref. [40]. Another
mechanism is the dissipative photon-magnon coupling mediated by a high-damping
mode or photon reservoirs, such as traveling waves in cavity magnonics systems
[211, 184, 367, 172]. The coupling constant is real in the coherent photon-
magnon coupling case but is purely imaginary number
$i\Gamma=i\sqrt{\kappa\gamma}$ for the dissipative photon-magnon coupling,
noting $\kappa$ and $\gamma$ are defined in Eq. (195). This imaginary coupling
strength leads, respectively, to the coalescence of the real parts and the
repulsion of the imaginary parts of eigenfrequencies of the system [see Fig.
20(c) and (d)].

Figure 20: (a) A YIG sphere biased by a perpendicular magnetic field $H$ is
placed at the center of the cross cavity. A loop antenna above the YIG sphere
can control the damping rate of the magnon mode. (b) Measured transmission
spectra as a function of the frequency detuning
$\Delta_{\omega}=\omega-\omega_{\rm ref}$ and the field detuning $\Delta_{H}$
at the condition of $\alpha=\beta$. The two red spots indicate two
unconventional BICs, whose transmission amplitude approaches zero. The black
curves represent the calculated eigenfrequencies. (c) and (d) The real and
imaginary parts of the eigenvalues at the condition $\alpha=\beta$. The blue
and red parts are dominated, respectively, by the cavity and magnon modes. The
green arrows in (d) indicate the zero imaginary parts of an eigenmode,
implying the unconventional BICs. When $|\Delta_{H}/(2\Gamma)|<1$, the system
lies in the anti-$\mathcal{PT}$ symmetry phase, as shaded by the yellow color.
(e) The Bloch sphere is constructed from the two eigenvectors of the system
when $\alpha=\beta$. The orange circles and the green stars represent the EPs
and two unconventional BICs. The equator of the Bloch sphere with
$\theta=\pi/2$ indicates the anti-$\mathcal{PT}$ symmetry phase. The figure is
reproduced with permission from Ref. [93].

Level attraction is a typical non-Hermitian phenomenon, which makes searching
for the EPs in a purely dissipative cavity magnonic system possible. However,
to reach such a goal, a paradox must be properly solved. In order to achieve
an observable dissipative photon-magnon coupling, $\Gamma=\sqrt{\kappa\gamma}$
should be as large as possible, i.e., it requires either $\kappa$ or $\gamma$
to be sufficiently large, noting $\kappa$ and $\gamma$ are the photon and
magnon radiation rates via the traveling waves. Enhancing $\gamma$ in
experiments appears to be a challenge, but adjusting the design of microwave
cavities can produce a large radiation damping rate $\kappa$. The band-stop
cavity such as the planar cross-cavity shown in Fig. 20(a) can support a large
$\kappa$ that is several orders in magnitude larger than $\gamma$. Placing a
YIG sphere in the center of the cross cavity enables an observable dissipative
photon-magnon coupling. However, this huge $\kappa$ also renders a nonzero
square root in Eq. (196), indicating the difficulty to achieve the EPs in a
dissipative cavity magnonic system. To obtain a similar phenomenology to the
EPs, a trade-off is needed. Yang et al. reduces the full Hamiltonian (195) to
an effective one that specifically describes the anti-resonances in the
coupled system [93], i.e., disregarding the photon background radiation
$\kappa$ in Eq. (195). The reduced Hamiltonian in a rotating frame with
respect to the reference frequency $\omega_{\rm
ref}=(\omega_{m}+\omega_{c})/2$ reads

$\hat{H}_{\rm
rot}/\hbar=\left(\begin{array}[]{cc}-\Delta_{H}/2-i\beta&i\Gamma\\\
i\Gamma&\Delta_{H}/2-i(\alpha+\gamma)\end{array}\right),$ (198)

where $\Delta_{H}=\omega_{m}-\omega_{c}$ is the field detuning tuned by an
external magnetic field. From this effective Hamiltonian (198), two pseudo-EPs
are expected to occur at $\Delta\omega=\pm\Gamma$, when $\beta=\alpha+\gamma$.

Figure 20(a) shows the experimental setup for studying the EPs in a
dissipative cavity magnonic device. A loop antenna is used to enhance the
damping rate $\gamma$ of the YIG sphere to meet the criteria
$\beta=\alpha+\gamma$ for EPs. The transmission spectra of the system measured
at different field detuning are plotted in Fig. 20(b), where the frequencies
and line widths of two anti-resonances are extracted by curve fitting. The
real and imaginary parts of the eigenvalues calculated from the reduced
Hamiltonian (198) can well reproduce the measured results, as shown by the
black solid lines in Fig. 20(c) and (d). At $\Delta\omega=\pm\Gamma$, the
frequencies of two anti-resonances coalesce, indicating the occurrence of EPs.
By contrast, the linewidths of two anti-resonances bifurcate to the upper and
lower branches. When $\Delta\omega=\pm\Gamma\sqrt{1-(\beta/\Gamma)^{2}}$, the
lower branch can reach zero, as indicated by green arrows in Fig. 20 (d), such
that two singularities occur, which is referred to as unconventional bound
state in the continuum (BIC). Back to the transmission spectra Fig. 20(b),
there indeed exist two ultra-sharp dips symmetrically occurring with respect
to $\Delta_{H}=0$, as shown by the red spots. These two unconventional BICs
are very sensitive to the damping rates of the dissipative cavity magnonic
system.

The two EPs define the boundaries between the phases preserving and breaking
the anti-$\mathcal{PT}$ symmetry. Mathematically, a Hamiltonian with an
anti-$\mathcal{PT}$ symmetry is obtained by multiplying the $\mathcal{PT}$
symmetric Hamiltonian by $i$, such that $\\{\hat{H}^{({\rm
APT})},\mathcal{PT}\\}=0$. The authors calculated the eigenvector evolution in
this system with the anti-$\mathcal{PT}$ symmetry [93]. From Eq. (198), the
two eigenvectors

$|\lambda_{\pm}\rangle=\left(\begin{array}[]{cc}v_{1\pm}\\\
v_{2\pm}\end{array}\right)=\left(\begin{array}[]{cc}i\Gamma\\\
\frac{\Delta_{H}}{2}-i\frac{\alpha-\beta}{2}\pm\sqrt{\left(\frac{\Delta_{H}}{2}-i\frac{\alpha-\beta}{2}\right)^{2}-\Gamma^{2}}\end{array}\right).$
(199)

By using the notation $v_{1,2}=|v_{1,2}|e^{i\phi_{1,2}}$, a Bloch sphere when
$\alpha=\beta$ can be constructed from the two eigenvectors as follows. The
relative intensity and phase difference of the two eigenvectors is defined,
respectively, as $\theta=2\arctan(|v_{2}|/|v_{1}|)$ and
$\phi=\phi_{1}-\phi_{2}$, which represent the polar and azimuthal angles in
the Bloch sphere. Mapping $(\theta,\phi)$ of the calculated eigenvectors at
different detuning, a Bloch sphere is constructed [see Fig. 20(e)], where the
north and south poles represent the uncoupled cavity mode $|\hat{p}\rangle$
and magnon mode $|\hat{m}\rangle$, respectively. The arrows indicate the
direction of increasing $\Delta_{H}$. Two eigenvectors coalesce at
$\Delta_{H}=\pm 2\Gamma$, i.e., two EPs on the equator, and one of them turns
into BICs at $\Delta_{H}=\pm\Gamma\sqrt{1-(\beta/\Gamma)^{2}}$. From the Bloch
sphere, it indicates that in the anti-$\mathcal{PT}$ symmetry-preserved phase,
both eigenvectors locate at the “equator modes" with equal contributions from
cavity and magnon modes. As addressed by the authors, these equator modes are
nontrivial and may find applications in quantum information processing.

####  4.4.4 Giant enhancement of magnonic frequency combs by EPs

So far, researchers exploit two different paths to realize the EPs and
investigate the functionality associated with the EPs in magnonics. The first
one is to fabricate the magnetic heterostructures [191]. A significant
challenge in this path is to achieve high-quality ferromagnet/normal metal
interfaces and proper film thickness. The other path is based on the cavity
magnonics by engineering either the photon-magnon coupling strength or the
individual dissipation. Compared to the former path, the technical obstacle in
the second path is smaller, since the design and fabrication of a centimeter-
scale microwave cavity is easy to implement. So we have witnessed a rapid
development of the experimental study on non-Hermitian physics in cavity
magnonics in recent years [26, 98, 198, 93]. Even so, there still exist
difficulties in the precise control of the photon-magnon coupling inside the
cavity. An alternative technical path that can get rid of the restrictions is
in high demand.

The recent discovery of the pump-induced magnon mode (PIM) provides an
alternative approach to realize the EPs in magnonics [203]. Although the
anticrossing between the Kittel mode and cavity mode is widely demonstrated in
the experiments [241, 92, 249, 11, 372, 203], its emergence when loading a YIG
sphere to the coplanar waveguide driven away from the equilibrium is a
surprise since the continuous waves of the waveguide cause radiative damping
rather than the coherent coupling. Figure 21(a) shows the experimental setup
to excite and detect the PIM of a YIG sphere in a coplanar waveguide. A strong
microwave with a single frequency works as the “pump" to excite the PIM, while
a weak probe across a wide frequency window transmits through the waveguide
(“probe"), which is received by a vector network analyzer and a signal
analyzer, respectively. Figure 21(b) compares the two microwave transmission
spectra without and with the pump: without the pump, the signal is simply the
resonant absorption with a dip of the (2,2,0) Walker mode [373, 374, 375] of
the YIG sphere; with the pump, the absorption dip splits into two, behaving
like an anticrossing between two modes. Phenomenological analysis suggests
that such a splitting is produced by the strong coupling effect between the
PIM and the Walker mode.

Figure 21: (a) Experimental setup with loading a YIG sphere to the coplanar
waveguide. Continuous pumps and probes of microwaves excite and detect the
magnon modes in the YIG sphere. (b) Measured transmission spectra $S_{21}$ of
the (2,2,0) Walker mode with or without the pump. (c) Measured transmission
spectra of the coupled PIM and Walker modes at different pump frequencies. The
frequency of the Walker mode is fixed at $3.4$ GHz. The pink strips in (b) and
(c) indicate the pump. (d) Experimental setup for the realization of EPs in
the driven magnetic sphere when loaded in the cross-shaped coplanar waveguide.
(e) compares the measured and calculated coupling strength $g/2\pi$ between
the PIM and Kittel mode in the YIG sphere. The circles are experimental data,
while the colored surface is a theoretical calculation. Two stars mark the
EPs. (f) Giant enhancement of the magnonic frequency combs near the two EPs.
Top: Eigenfrquencies $\Delta_{\pm}$ of the coupled system under different
polarizations of the pump field. Bottom: Radiation spectra measured at
different polarizations of the pump field. The tone number of the combs
dramatically increases when the system is tuned to approach the EPs. The
figure is reproduced with permission from Refs. [203, 199].

Unlike the normal magnon modes, e.g., the Walker modes, the PIM exhibits
excellent tunability by the pump. Its frequency strictly follows the pump
frequency and is independent of the external magnetic field, such that we can
easily control its coupling with the desired Walker mode by tuning the pump
frequency $\omega_{d}$, as addressed in Fig. 21(c). In addition, the
experiment finds the coupling strength $g$ between the PIM and Walker mode is
tunable by the pump power $P$, i.e., $g\propto P^{1/4}$ [203, 199]. These
properties facilitate the realization of the EPs between the PIM and desired
Walker modes via matching the coupling constants and the dissipation
difference. Compared with conventional methods, the new method based on the
coupling of the PIM and Walker modes merely needs to control the pump signal,
such as its frequency and power, and hence greatly simplifies the experimental
operations.

In addition, when the system is simultaneously driven by the pump and probe,
the coupling between the Walker mode and PIM can exhibit nonlinear behavior.
This nonlinear coupling enables the frequency conversion and hence produces
the magnonic frequency combs [203, 376, 377, 199, 378]. As an intriguing
singularity in the coupled systems, the EPs have been found to exhibit a
significant enhancement of sensitivity to the disturbance. Using the EPs to
amplify the sensitivity in a coupled system with the Kittel modes and PIM may
greatly enhance the magnonic frequency comb generation. Recently, this idea
has been experimentally tested in Ref. [199]. The authors constructed a
coupled system comprising a PIM and the (1,1,0) Walker mode or Kittel mode,
and then precisely tune the coupling strength between the PIM and Kittel mode
by tuning the driving power when varying its polarization. The authors
fabricate a cross-shaped coplanar waveguide with two orthogonal arms to
regulate the polarization of the pump field [Fig. 21(d)]. By adjusting the
relative phase $\phi$ of the polarizations between two coherent pump
microwaves that are, respectively, input from two orthogonal arms of the
cross-shaped coplanar waveguide, the polarization of the pump field can be
tuned from the right-handedness to the left-handedness. As a consequence, by
the angular momentum conservation between magnons, the coupling strength
between the PIM and Kittel mode can be precisely controlled.

Figure 21(e) shows the evolution of the coupling constant $g/(2\pi)$ in the
parameter space constructed by $\phi$ and the frequency difference
$\Delta_{H}$ between the Kittel mode and PIM. The colored surface is given by
the numerical calculation, which nicely supports the experimental data with
the purple circles. When $g$ is adjusted to meet the criterion
$g=\delta_{-}/2$, where $\delta_{-}$ is the damping difference between the two
modes, the EPs occur at $\Delta_{H}=0$ [refer to Eq. (12)], which is marked by
the two stars in Fig. 21(e). Further, when the coupled system is
simultaneously driven by the pump and probe, the coupling strength $g$ between
the PIM and Kittel mode oscillates periodically in time, instead of a constant
under a single-tone drive. Such a nonlinear process reaches its maximum near
the EPs, leading to the giant enhancement of the magnonic frequency combs, as
shown in Fig. 21(f). In the measurement, the Kittel mode is fixed at the pump
frequency, and the detuning between the pump and probe microwaves is set to be
0.1 MHz. When the polarization of the pump field is tuned by $\phi$, the
authors observe the dramatic enhancement of the tone number of the combs from
several to more than 32 near the two EPs [Fig. 21(f)]. This work establishes a
strong link between the non-Hermitian topology and the frequency combs and
provides a new avenue for frequency multiplication by taking advantage of the
EPs. Although it is demonstrated in the microwave regime, it has the potential
to be applied in other frequency ranges.

###  4.5 Exceptional surfaces in cavity magnonics

The EPs are isolated points in the parameter space, as discussed above.
Bringing these ideas to higher dimensions may help to provide chances to
expand EPs to exceptional surfaces, i.e., a collection of EPs on surfaces [98,
99]. It requires additional degrees of freedom and flexible tunability, thus
posing a significant challenge for the experimental demonstration. A cavity
magnonic system can be easily manipulated by either adjusting the geometric
boundaries of microwave cavities or tuning the magnetic field on the magnet.
This prominent tunability provides an ideal solution for the experimental
realization of exceptional surfaces. The experimental setup [98] consists of a
microwave cavity and a YIG sphere of 400 $\mu$m that is placed inside the
cavity, as shown in Fig. 22(a). The cavity is fabricated from a printed
circuit board with a high dielectric constant. The cavity mode volume is much
smaller than that of a conventional three-dimensional resonant cavity, making
it very suitable for the realization of strong photon-magnon coupling. A small
rod attached to the YIG sphere is used to control the position of the YIG
sphere in the cavity such that the coherent photon-magnon coupling strength
$g$ is precisely controlled. When $g$ matches a critical coupling strength
$g_{c}$, i.e., $g=g_{c}=\Delta\kappa/2$, the EP in this coupled system is
realized according to Eq. (196).

Figure 22: (a) Schematic of the microwave cavity with simulated magnetic
fields. (b) and (c) show, respectively, the measured Riemann surfaces for the
real (resonant frequency) and imaginary (resonant linewidth) parts of the
eigenfrequency. (d) show the dependence of the measured Riemann surfaces for
the real (top) and imaginary (bottom) parts of the eigenfrequencies on the
position $x=0.66$, $0.13$, and $0.00$ mm of the YIG sphere. The EP coalescence
is observed at $x=0.13$ mm when $g=g_{c}$. (e) The coupling strength $g$ as a
function of $y$ at different $x$ locations. (f) and (g) Cross-sectional views
of the case with $x=0.13$ mm. (h) Slices of exceptional surface in the 3D
($x$,$y$,$\theta$) parameter space. The figure is reproduced with permission
from Ref. [98].

To realize an exceptional surface, Zhang et al. introduces multiple degrees of
freedom to manipulate the EPs in a four-dimensional synthetic space, including
the magnitude of the bias field ${\bf H}$, the direction $\theta$ of ${\bf H}$
with respect to the $\hat{\bf y}$-direction, the position of YIG sphere in
both the $x$ and $y$ directions [see Fig. 22(d)]. When $x=0.66$ mm, two EPs
are widely separated. As $x$ decreases, they approach and coalesce at $x=0.13$
mm. When $x$ keeps decreasing, no EP can be observed (e.g. at $x=0$) and the
real parts of two eigenfrequencies always cross each other at diabolic points,
while their imaginary parts are separated. Comparing $g$ at different $x$
positions can help to find the relation between coupling strength $g$ and
$g_{c}$, as displayed in Fig. 22(e). It is worth noting that the curve of
coupling strength crosses with the line of $g_{c}$ twice at $x=0.66$ mm,
crosses only once at $x=0.13$ mm, and has no intersections at $x=0.00$ mm.
These results show a good correspondence with Fig. 22(d). When an EP pair
coalesces, no signature of mode coupling can be observed in the Riemann
surface for the real part of the eigenfrequency. However, in the imaginary
part, the singularity condition associated with this EP can be easily
observed. At this point, this coalesced EP behaves as a linear and square-root
dependence respectively [Fig. 22(f) and (g)].

Although it appears that the EP coalescence only applies via tuning the $y$
position of the YIG sphere, by changing the magnetic-field direction $\theta$,
the coalescence of EP pairs can be realized by tuning the $x$ positions as
well. Figure 22(h) plots the distribution of EPs in a 4D synthetic space
defined by $x$, $y$, $\theta$ and $H$. Here the hidden dimension $H$ is fixed
at 3237 Oe for ensuring the zero-detuning condition. White dots in Figure
22(h) are the EPs calculated from experimental data, while the solid lines are
theoretical calculations from extrapolated data. They are consistent with each
other. These white dots depict the outline of the exceptional surface in the
4D synthetic space. By varying $\theta$, the overall amplitude of the saddle
surface of $g$ is tuned. At a critical angle $\theta_{c}$, the saddle surface
intersects the plane at the saddle point, suggesting the coalescence of EPs in
both $x$ and $y$ dimensions. This work points out a novel direction for
creating high-dimensional EPs. Four-dimensional exceptional volumes or even
higher-order EP assembles can be achieved by introducing higher-order
synthetic dimensions, laying the groundwork for magnonic non-Hermitian
physics, and paving a new avenue for magnon-based signal processing.

The experiments above exploit the frequency and dissipation asymmetries
$\Delta\omega$ and $\Delta\kappa$ between the magnon and microwave modes in
Eq. (195) to realize the exceptional surface. An alternative choice is to use
the competition between the dissipative and coherent couplings, as addressed
in Sec. 2.1.1. Grigoryan et. al. [210, 99] propose the very first mechanism to
realize the dissipative coupling in cavity magnonic system: Beyond the
conventional magnon-cavity coupling, they introduce a phase shift “$\phi$” in
the feedback microwaves driven by the magnetization dynamics in the cavity, as
shown in Fig. 23(a). In Fig. 23(b), the magnetization precession induces a
driving voltage in the LCR circuit that models the electromagnetic field in
the cavity, while the induced current is modulated by a phase shift “$\phi$”
and amplitude $\delta_{0}$ that affects the magnetization precession [379,
380, 210, 99]. Recent experiments exploit such a design and realize a regime
transition for the cavity photon-magnon states from conventional “level
repulsion” to “level attraction” by tuning the phase shift $\phi=\pi$ [248,
249]. Such a system is governed by the Hamiltonian

$\hat{H}=\left(\begin{array}[]{cc}\omega_{p}-i\kappa_{p}&g_{a}\\\
g_{a}\left(1+\delta_{0}e^{i\phi}\right)&\omega_{m}-i\kappa_{m}\end{array}\right),$
(200)

where $\omega_{p}$ and $\omega_{m}$ are the frequencies of photon and magnon
modes with damping rates $\kappa_{p}$ and $\kappa_{m}$, respectively, $g_{a}$
is the coupling constant, and the dissipative component is contained in
$\delta_{0}e^{i\phi}$ [381, 382]. Equation. (200) is equivalent to Eq. (10)
when replacing $g_{a}\left(1+\delta_{0}e^{i\phi}\right)=g_{b}$ (refer to Sec.
2.1). The exceptional saddle surface is achievable by the frequency
asymmetries $\Delta\omega$, dissipation asymmetries $\Delta\kappa$, and
coupling between two modes [99]. Figure 23(c) plots the condition
$\left(\Delta\omega\right)^{2}+\left(\Delta\kappa\right)^{2}=r_{0}$ as a new
exceptional surface and reveals features associated with the $\mathcal{PT}$\-
and anti-$\mathcal{PT}$-symmetric breakings.

Figure 23: Proposal of exceptional surface in cavity magnonics via microwave
modulation. In (a) the magnon-photon coupling in the cavity is modulated by
additional phase $\Phi$. (b) is the proposal of the microwave modulation that
realizes (a). (c) plots the exceptional surface, on which the blue and red
dots correspond to the EPs with $\mathcal{PT}$ and anti-$\mathcal{PT}$symmetry
breakings, respectively. Figures (a) and (c) are adapted with permission from
Ref. [99]. Figure (b) is adapted with permission from Ref. [210].

##  5 Exceptional nodal phases of magnons

As addressed in Sec. 4, the EPs, exceptional lines and exceptional surfaces
have been realized in various magnonic systems [188, 29, 98, 190, 93, 31,
166], exhibiting their potential advantages in exploring the non-Hermitian
physics due to the flexible way to engineer the dissipation in these devices.
There are also theoretical efforts for the realization of EPs in the
reciprocal space, and the non-Hermitian bulk Fermi arcs as well in magnonic
system [181, 189, 201]. In this section, we first review the proposal of the
non-Hermitian nodal phase originating from the self-energy corrections from
the ferromagnetic lead to the electronic surface states [181], and from the
Dzyaloshinskii-Moriya interaction (DMI) to the magnon scattering in the
framework of the Green’s function approach [189] (for the approach, refer to
Sec. 3.3). Then we address a new type of spin liquid with EPs in the
reciprocal space in a strongly correlated spin model coupled to the
environment in the framework of the master-equation approach [201] (for the
approach, refer to Sec. 3.2). Finally, we introduce a van der Waals
ferromagnetic bilayer where a multitude of EPs emerges in reciprocal space,
which is beneficial for studying the unique transport properties brought by
EPs [200].

###  5.1 Non-Hermitian Weyl physics

To construct a two-dimensional non-Hermitian Hamiltonian, Bergholtz and Budich
considered the non-Hermitian physics in a three-dimensional (3D) topological
insulator [280, 276] that couples to a proximity ferromagnetic metal [181], as
shown in Fig. 24(a), where the surface states of a topological insulator are
described with Hermitian Hamiltonian, but become non-Hermitian due to the
coupling with the magnons of the ferromagnetic lead.

Figure 24: Non-Hermitian Weyl physics in topological insulator$|$ferromagnet
junctions. (a) is the configuration in which the surface states of a 3D
topological insulator couple to the ferromagnetic metal with magnetization
$\boldsymbol{\mathrm{m}}$ in the direction $\phi$ via the hopping of
electrons. (b) describe the spectrum of a non-Hermitian Weyl phase with
exceptional points connected by the bulk Fermi arc. The figure is reproduced
with permission from Ref. [181].

The surface states of the 3D topological insulator in the $x$-$y$ plane are
well described by the two-band model [276]

$\displaystyle
H_{\rm{TI}}(\boldsymbol{k})=\lambda(k_{y}\sigma_{x}-k_{x}\sigma_{y}),$ (201)

where ${\boldsymbol{\sigma}}$ is the Pauli matrix in the spin spaces, and
$\lambda$ characterizes the group velocity of Dirac fermion. This Hamiltonian
acts as a Rashba-like spin-orbit interaction for the spintronics study [383,
384, 385]. In the ferromagnetic metal, the itinerant electron interacts with
the local magnetization ${\bf m}$ via the Zeeman interaction, where the
Hamiltonian is described by

$\displaystyle
H_{F}(\boldsymbol{k})=-2t\left[\cos(k_{x})+\cos(k_{y})\right]\sigma_{0}-2t_{z}\cos(k_{z})\sigma_{0}-\mu_{F}\sigma_{0}+\boldsymbol{\mathrm{m\cdot\sigma}},$
(202)

where $t$ is the in-plane hopping parameter in the $x$-$y$ directions, $t_{z}$
is the out-of-plane hopping parameter in the $z$-direction, and $\mu_{F}$ is
the chemical potential.

Due to the interface electron-magnon interaction, the electrons at the
interface acquire the surface self-energy mediated by the magnons in the
ferromagnet, which breaks the time-reversal symmetry and alters the
topological surface state to be a non-Hermitian Weyl phase without specific
symmetries. The surface retarded self-energy

$\displaystyle\Sigma_{F}^{r}(\omega)=V_{\rm SF}G_{F}^{r}(\omega)V_{\rm
SF}^{\dagger}$ (203)

is spin-dependent since the retarded Green’s function $G_{F}^{r}(\omega)$ of
the electrons in the ferromagnet is spin-dependent [386], where $V_{\rm SF}$
is the spin-independent hopping amplitude between the topological insulator
and the ferromagnet. Inclusion of the surface retarded self-energy at the
chemical potential of electrons $\Sigma_{F}^{r}(\omega=0)$ into the
topological insulator Hamiltonian (201), the effective Hamiltonian for the
surface states becomes

$\displaystyle{H}_{\rm eff}$
$\displaystyle=\lambda(k_{y}\sigma_{x}-k_{x}\sigma_{y})+\Sigma_{F}^{r}(\omega=0)-B\sigma_{z}\equiv\epsilon_{0}+\boldsymbol{\mathrm{d}}(\boldsymbol{k})\cdot\boldsymbol{\sigma},$
(204)

where the term $B$, if exists, breaks the time-reversal symmetry,
$\epsilon_{0}$ are a complex number and
$\boldsymbol{\mathrm{d}}(\boldsymbol{k})=\boldsymbol{\mathrm{d}}_{R}(\boldsymbol{k})+i\boldsymbol{\mathrm{d}}_{I}(\boldsymbol{k})$
are complex vectors with real components
$\boldsymbol{\mathrm{d}}_{R}(\boldsymbol{k})$ and
$\boldsymbol{\mathrm{d}}_{I}(\boldsymbol{k})$. Here, the in-plane wave vector
${\bf k}={k}_{x}\hat{\bf x}+k_{y}\hat{\bf y}$,
$\boldsymbol{\mathrm{d}}_{R}(\boldsymbol{k})$ is composed of the spin-orbit
coupling of surface state and the difference of the real part of the self-
energy correction, and $\boldsymbol{\mathrm{d}}_{I}(\boldsymbol{k})$ is the
difference of the imaginary part of the self-energy correction.

Many non-Hermitian properties of the effective Hamiltonian (204) have been
reviewed in Sec. 2.2. In short, its complex spectrum
$E_{\pm}(\boldsymbol{k})=\epsilon_{0}\pm\sqrt{\boldsymbol{\mathrm{d}}_{R}^{2}(\boldsymbol{k})-\boldsymbol{\mathrm{d}}_{I}^{2}(\boldsymbol{k})+2i\boldsymbol{\mathrm{d}}_{R}(\boldsymbol{k})\cdot\boldsymbol{\mathrm{d}}_{I}(\boldsymbol{k})}$
shows exceptional degeneracies at $E(\boldsymbol{k})=\epsilon_{0}$ when
condition Eq. (49) is satisfied. Since ${\bf d}_{R}$ and ${\bf d}_{I}$ are
functions of wave vector ${\bf k}$, the EPs with degenerate
$E(\boldsymbol{k})=\epsilon_{0}$ (and coalesced eigenvectors) solved from Eq.
(49) appear in pairs with a distance in the momentum space, and cannot be
annihilated by weak perturbations, i.e., being stable. When the momentum
evolves away from one EP, the solutions for ${\bf k}$ with ${\rm
Re}\left(E(\boldsymbol{k})\right)={\rm Re}(\epsilon_{0})$ with ${\bf
d}_{R}(\boldsymbol{k})\cdot{\bf d}_{I}(\boldsymbol{k})=0$ connects the pair of
EPs, which is called bulk Fermi arc [97, 181, 9]. Figure 24(b) shows the
exceptional points and the bulk Fermi arc that connects them with specified
magnetization $\boldsymbol{\mathrm{m}}$. This is similar to the Hermitian Weyl
physics in the three dimensions, with the difference that in the Hermitian
case, the Fermi arc is the surface state [122, 123].

The combination of linear Dirac dispersion of the surface state and the non-
Hermitian perturbation from self-energy correction provides an experimentally
feasible platform to study the exceptional nodal phase [181, 9, 387].
Meanwhile, the transition between conventional Dirac dispersion and non-
Hermitian exceptional phase can be readily implemented by tuning the magnetic
direction in the ferromagnetic lead since it affects the magnitude of the
self-energy correction [181].

###  5.2 Non-Hermitian nodal phase by magnon interactions

Different from the hetero-structure as mentioned above [181], McClarty et al.
consider a spin-1/2 ferromagnet on the honeycomb lattice [189] and propose
that the nonlinear perturbation from the DMI can induce magnon-magnon
scattering and thereby bring magnon decay [388]. For such a system, the
Hamiltonian

$\hat{H}=-\sum\limits_{n=1}^{3}\sum\limits_{\langle
i,j\rangle_{n}}J_{n}(\hat{\boldsymbol{S}}_{i}\cdot\hat{\boldsymbol{S}}_{j})+D\sum\limits_{\langle
i,j\rangle_{2}}\nu_{ij}\hat{\boldsymbol{z}}\cdot(\hat{\boldsymbol{S}}_{i}\times\hat{\boldsymbol{S}}_{j}),$
(205)

where $J_{n}$ is the $n$-th neighboring ferromagnetic couplings between spin
$\hat{\boldsymbol{S}}$. The last term in Eq. (205) is the DMI with the index
$\nu_{ij}=-1$ ($+1$) for the clockwise (counterclockwise) hopping with respect
to the normal $\hat{\boldsymbol{z}}$ axis, from the site $i$ to $j$, which
induces spontaneous magnon decay at higher order regime.

In the linear order the Holstein-Primakoff transformation [312, 389] in terms
of the boson operator $\hat{a}_{\boldsymbol{k}\alpha/\beta}$ with wave vector
${\boldsymbol{k}}$ on the sublattices $\alpha$ and $\beta$ leads to the
Hamiltonian of the form

$\hat{H}(\boldsymbol{k})=\sum\limits_{\boldsymbol{k}}\sum\limits_{\alpha\beta}\left(A_{\boldsymbol{k}}^{\alpha\beta}\hat{a}_{\boldsymbol{k}\alpha}^{\dagger}\hat{a}_{\boldsymbol{k}\beta}+\frac{1}{2}\left(B_{\boldsymbol{k}}^{\alpha\beta}\hat{a}_{\boldsymbol{k}\alpha}^{\dagger}\hat{a}_{-\boldsymbol{k}\beta}^{\dagger}+\rm{H.c.}\right)\right).$
(206)

Here, $A_{\boldsymbol{k}}$ and $B_{\boldsymbol{k}}$ represent the spin-
conserving and nonconserving interactions, both of which depend on the detail
of the model. The linear spin-wave spectrum is given by diagonalizing the
matrix [390]

$\sigma_{z}M_{\boldsymbol{k}}\equiv\left(\begin{array}[]{cc}A_{\boldsymbol{k}}&B_{\boldsymbol{k}}\\\
-B_{-\boldsymbol{k}}^{\ast}&-A_{-\boldsymbol{k}}^{\ast}\end{array}\right)$
(207)

with the matrix $M_{\boldsymbol{k}}$ and $\sigma_{z}={\rm diag}(1,-1)$. Under
specified parameters, a linear Dirac point appears in the Brillouin zone
[189].

In the linear regime, the DMI does not strongly affect the magnon dispersion.
But entering the nonlinear regime, the magnon-magnon interactions from the DMI
contribute to the retarded self-energy $\Sigma(\boldsymbol{k},\omega)$ [391,
390], with which the retarded Green function of magnons reads

$G(\boldsymbol{k},\omega)\equiv\Big{\\{}(\omega+i0^{+})\sigma_{z}-[M_{\boldsymbol{k}}+\Sigma(\boldsymbol{k},\omega)]\Big{\\}}^{-1}.$
(208)

Supposing that the self-energy is frequency-independent near the band
degeneracy point $\omega_{0}$, i.e., the linear Dirac touching point,

$\Sigma(\boldsymbol{k},\omega)\approx\Sigma^{\prime}(\boldsymbol{k},\omega_{0})+\Sigma^{\prime\prime}(\boldsymbol{k},\omega_{0}),$
(209)

where the real part $\Sigma^{\prime}(\boldsymbol{k},\omega_{0})$ renormalizes
the spin-wave spectrum and imaginary part
$\Sigma^{\prime\prime}(\boldsymbol{k},\omega_{0})$ leads to spontaneous magnon
decay. Near the band degeneracy point, an explicit non-Hermitian band
structure is constructed that allows for the exploration of non-Hermitian
physics. The two-band model near the band degeneracy point reads

$\displaystyle{H}_{\rm eff}$
$\displaystyle=\lambda(k_{x}\sigma_{x}+k_{y}\sigma_{y})-i\left(a_{0}+\boldsymbol{a}({\bf
k})\cdot\boldsymbol{\sigma}\right),$ (210)

which is similar to Eq. (204). Here $\lambda$ is the group velocity, and the
constant $a_{0}$ and $\boldsymbol{a}({\bf k})$ contribute to the non-Hermitian
perturbation. The real part $E_{\boldsymbol{k}}$ of the spectra from Eq. (210)
and the corresponding inverse of the imaginary part
${\Gamma}_{\boldsymbol{k}}$ denoted by the color bar are shown in Fig. 25(a),
where a pair of EPs emerge in the reciprocal space. Such a nodal phase is
similar to the case of applying a non-Hermitian perturbation on the band
degeneracy point as addressed in Sec. 2.2. These two EPs possess the same real
part and there is a bulk Fermi arc that connects them. Compared to the Fermi
arcs with zero real energy, the only difference is that this bulk arc is
shifted by finite real energy.

The magnon modes below the EPs with the shortest lifetime are skewed to one
side in the reciprocal space but located on the other side when they are above
the EPs, implying that the magnon-magnon interactions are strongly
anisotropic. The calculated spectrum in Fig. 25(a) is further confirmed by the
spectral function as shown in Fig. 25(b), which reveals the density of state
at certain momentum $\boldsymbol{k}$ and energy $\omega$. The spectral
function can be detected by angle-resolved photoemission spectroscopy [392],
providing a way of experimental verification of bulk Fermi arc.

Figure 25: Prediction of bulk Fermi arc between a pair of EPs in the magnon
spectra of the honeycomb lattice model. (a) is the constant slices of the real
part $E_{\boldsymbol{k}}$ of the mode spectra, where the inverse of the
imaginary part $\Gamma_{\boldsymbol{k}}$ is denoted by the color bar. (b) show
the spectral function $A(\boldsymbol{k},\omega)$ of different energies around
the EPs. These figures are reproduced with permission from Ref. [189].

In Sec. 5.1 and Sec. 5.2, The quasiparticle renormalization is captured by the
energy shift and finite lifetime in the framework of the Green-function
approach [119, 181, 189, 393]. Such non-Hermitian perturbation from the self-
energy correction to the quasiparticle spectra renders the band degenerate
points such as the Dirac or Weyl points split into a pair of EPs that is
connected by the bulk Fermi arc, similar to the mechanism addressed in Refs.
[97, 201] (refer to Sec. 2.2). A similar approach is used to model the bulk
Fermi arc in several other systems such as heavy-fermion system [394], Dirac
fermion system [395], superconductors coupled with ferromagnetic lead [396]
and photonic system [97, 397]. A different approach to describe the non-
Hermitian perturbation lies in the framework of the master-equation approach,
which is used to deal with the excitation of the spin liquid in Ref. [201],
which is one non-trivial ground state of the strongly correlated quantum spins
[398, 399, 400]. Recently, Yang et. al. propose a new excitation of the spin
liquid state, which is termed exceptional spin liquid, based on the Kitaev
honeycomb lattice when coupling to the environment, which possesses the EPs in
reciprocal space and has no counterpart in Hermitian systems [201]. The
spectra of the excitation contain the Dirac point as well [401]. When coupling
to the environment, the spin dynamics is approached by the Lindblad master
equation with complex effective coupling between spins, due to which each
Dirac point is split into two EPs. Different from the above-mentioned cases,
these EPs are not only connected by bulk Fermi arcs with $\text{Re}E=0$ but
also the i-Fermi arcs with $\text{Im}E=0$. Moreover, such gapless exceptional
spin liquids are robust to perturbations, making a contrast to the Hermitian
gapless phases in the honeycomb model that requires symmetry protection [201].

Experimentally, the bulk Fermi arc has been observed in periodic photonic
crystals by angle-resolved scattering measurements [97], and can also be
detected in principle by angle-resolved photoemission spectroscopy [394] or
interferometric measurements [397]. Such measurements may be extended to
detect the bulk Fermi arcs for magnonic systems as well. A recent study
demonstrated that the optical conductivity in the topological phase with non-
Hermitian Fermi arc exhibits a different response to the optical field
distinguished from other topological phases [393].

###  5.3 A multitude of EPs in reciprocal space

Although there are several methods available to realize the EPs in the
reciprocal space, the EPs are often isolated. Such isolated EPs may not affect
remarkably the bulk properties such as the thermal conductivity of magnons
since it requires the integral over the whole Brillouin zone where the phase
space around the EPs appears to be quite small. The exceptional line occupies
larger phase space, which may exist when they lie in a two-dimensional
reciprocal space [188]. Recently, Li et. al. [200] investigate the non-
Hermitian topological phase of magnons in a bilayer of AA-stacked
ferromagnetic honeycomb lattice, as shown in Fig. 26(a). They found that a
multitude of exceptional points can be realized in the momentum space when one
layer possesses “gain" from the injected spin current by the adjacent metal
layer while the other layer is dissipated [200].

Figure 26: Prediction of a multitude of EPs in AA-stacked van der Waals
ferromagnetic bilayer. (a) is configuration, where the spin current is
injected to one layer from the adjacent Pt layer. (b) shows the overlap of the
eigenvectors
$\mathcal{P}_{EP}\equiv\left|\left\langle\phi_{1}^{R}|\phi_{2}^{R}\right\rangle\right|^{2}$
for the two lowest bands, where the regions occupied by the EPs are denoted by
blue color. The figure is reproduced with permission from Ref. [200].

The magnetization dynamics are governed by the effective magnetic field
$\mathbf{h}_{\text{eff }}$ involving the intralayer and interlayer couplings
and the intrinsic dissipation (gain) denoted by minus $\alpha$ (positive
$\alpha$):

$\displaystyle\dot{\mathbf{m}}_{i,j}^{(l=1)}=-\gamma\mathbf{m}_{i,j}^{(l=1)}\times\mathbf{h}_{{i,j},\text{
eff
}}^{(l=1)}-\alpha\mathbf{m}_{i,j}^{(l=1)}\times\dot{\mathbf{m}}_{i,j}^{(l=1)},$
(211)
$\displaystyle\dot{\mathbf{m}}_{i,j}^{(l=2)}=-\gamma\mathbf{m}_{i,j}^{(l=2)}\times\mathbf{h}_{{i,j},\text{
eff
}}^{(l=2)}+\alpha\mathbf{m}_{i,j}^{(l=2)}\times\dot{\mathbf{m}}_{i,j}^{(l=2)},$

where $l=1,2$ denote the bottom layer and top layer and $i,j$ represents two
sublattices. Here, the positive $\alpha$ originates from the injected spin
current from the adjacent Pt layer. With balanced gain and loss, such a system
possesses $\mathcal{PT}$-symmetry, where the interlayer coupling is Hermitian
as addressed in Sec. 2.1. Considering that the eigenvectors are coalescent at
the EPs, the authors introduce
$\mathcal{P}_{EP}\equiv\left|\langle\phi_{1}^{R}|\phi_{2}^{R}\rangle\right|^{2}$
to denote the region of EPs for the two lowest bands $1,2$, which becomes
parallel for the right eigenvector at the EPs. As shown in Fig. 26(b), the EPs
cover a large area in the Brillouin zone with suitable parameters $\alpha$,
implying the EPs may affect significantly the macroscopic transport property
for the magnons. Such a multitude of EPs in the van der Waals ferromagnetic
bilayer may provide an ideal platform for exploring the effect of EPs on
magnon transport [200].

##  6 Topological edge state vs. non-Hermitian skin effect in magnonic devices

Above we have addressed many aspects of the non-Hermitian topology associated
with the EPs in the parameter or wave-vector space without resorting to the
lattice collective modes in the absence of the EPs, which, however, can be
governed by the non-Hermitian topology as well [280, 273, 276, 270, 274, 288].
In this section, we start with a brief introduction of the Su-Schrieffer-
Heeger (SSH) model [216, 220, 218, 402, 403, 124, 404] to elucidate the
concept of band topology and the topologically protected edge state. We
compare with its non-Hermitian generalization by emphasizing their difference
in the topological characterization (Sec. 6.1). We then focus on the
realization of the non-Hermitian skin effect in magnonic devices [145, 56, 38,
42, 202] when the bulk-boundary correspondence breaks down. We emphasize the
role of the long-range chiral interaction that can be realized in magnonic
devices (Sec. 6.2), which displays several non-trivial features compared to
the short-range chiral interaction in the Hatano-Nelson model as stated in
Sec. 2.3. We finally emphasize the recent progress of the non-Hermitian skin
effect in a higher dimension with a magnonic realization in van der Waals
ferromagnetic honeycomb lattice and magnetic array with new topological
characterization (Sec. 6.3).

###  6.1 Hermitian and non-Hermitian edge states in Su-Schrieffer-Heeger model

Hermitian SSH model.—To compare with the non-Hermitian edge states, we first
address the Hermitian SSH model that is first considered in the pioneering
works [216, 220] for the motion of solitons in polyacetylene, where the
hopping is staggered since it is different from sublattice “B" to two
neighboring “A" sites [Fig. 27(a)]. It turns out to be one of the simplest
topological lattice models that are widely used to demonstrate the basic
topological concepts and properties in Hermitian and non-Hermitian scenarios
[218, 402, 403, 124, 404]. From a modern viewpoint, this staggered hopping
potential makes an analogy to a dimer lattice with distinct intercell and
intracell hoppings; there are two topologically distinct phases depending on
whether the intercell hopping is larger than the intracell one. The edge
states exist between topologically distinct phases that are characterized by
the bulk-boundary correspondence [275, 280, 276].

Figure 27: Comparison of the Hermitian and non-Hermitian SSH models. (a) shows
the Hermitian SSH model with staggered coupling $J$ and $\tilde{J}$. When
$\left|J\right|>\left|\tilde{J}\right|$ [(b)]
$\left|J\right|<\left|\tilde{J}\right|$ [(c)], the evolution of $h(\bf k)$
when ${\bf k}$ evolves in the Brillouin zone can or cannot encircle the origin
point, corresponding to the winding number $W=1$ and $W=0$ as shown in (d).
(e) The non-Hermitian SSH model, where the intercell coupling is chiral with
the hopping strength $J+\gamma$ ($J-\gamma$) from right to left (from left to
right). (f) shows the GBZ on the complex plane with $J=2/3$, $\gamma=0.2$, and
$\tilde{J}=1$. As $\beta$ evolves along the GBZ, indicated by the red arrow in
(f), ${h}_{1}(\beta)$ and ${h}_{2}(\beta)$ (defined in the text) encircle the
origin point with the winding number $W=1$ as shown in (g). (h) plots the
dependence of the winding number on $J$ and $\gamma$ when $\tilde{J}=1$.

With the intracell and intercell couplings $J$ and $\tilde{J}$, the Bloch
Hamiltonian for the Hermitian SSH model

$\displaystyle H({\bf k})=h_{x}({\bf k})\sigma_{x}+h_{y}({\bf k})\sigma_{y}$
(212)

is a simplified version of Eq. (43), where $h_{x}({\bf k})=J+\tilde{J}\cos{k}$
and $h_{y}({\bf k})=\tilde{J}\sin{k}$ [402, 275, 124]. The eigenequation
$H({\bf k})\left|\Phi({\bf k})\right>=E({\bf k})\left|\Phi({\bf k})\right>$
solves the eigenvalue $E({\bf k})$ and eigenstates $|\Phi({\bf k})\rangle$:

$\displaystyle E_{\pm}({\bf k})=\pm\left|{\bf h}({\bf
k})\right|=\pm\sqrt{h_{x}^{2}({\bf k})+h_{y}^{2}({\bf k})},$ (213a)
$\displaystyle|\Phi({\bf
k})\rangle_{\pm}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\pm
e^{-i\theta_{\bf k}}\\\ 1\end{array}\right),$ (213d)

where $\tan\theta_{\bf k}=h_{y}({\bf k})/h_{x}({\bf k})$. The topological
property is well characterized by the Berry phase [219], i.e., the geometry
phase factor accumulated in the adiabatic evolution of the system, which acts
as a topological invariant when the wave vector evolves along a closed loop.
For a one-dimensional periodic lattice, the Brillouin zone is periodic by
$2\pi$ [221] with $H({\bf k})=H({\bf k}+2\pi)$, thus forming a closed loop
when ${\bf k}$ evolves in one period. For example, the Berry phase over the
Brillouin zone for state $|\Phi({\bf k})\rangle_{-}$ reads

$\gamma=\frac{i}{2}\oint_{\rm BZ}d{\bf k}\left(-e^{i\theta({\bf
k})},1\right)\frac{\partial}{\partial{\bf
k}}\left(\begin{array}[]{c}-e^{-i\theta_{\bf k}}\\\
1\end{array}\right)=\frac{\theta({\bf k}=2\pi)-\theta({\bf k}=0)}{2},$ (214)

which depending on whether the parameter evolution surrounds the energy
degeneracy point with $h_{x}({\bf k})=h_{y}({\bf k})=0$ can be $0$ or $\pi$,
as plotted in Fig. 27(b) and (c) for the evolution of $\theta_{\bf k}$ denoted
by the red vector. When $\left|J\right|<\left|\tilde{J}\right|$ [Fig. 27(b)]
and $\left|J\right|>\left|\tilde{J}\right|$ [Fig. 27(c)], $\theta_{\bf k}$ is
changed by $2\pi$ and unchanged, respectively. Only the former case
contributes a nonzero Berry phase, which also corresponds to a nonzero winding
number $W=1$. The topological nontrivial and trivial phases are thereby
bounded by $\left|J\right|=\left|\tilde{J}\right|$, as shown in Fig. 27(d).
When $\left|J\right|<\left|\tilde{J}\right|$ and $W=1$, an edge state emerges,
which can be understood as the intracell coupling favors the “dimer" between
adjacent cells, leaving two unpaired boundary states. This implies a bulk-
boundary correspondence in the Hermitian topological description [275, 280,
276].

The SSH model is widely used to study the topological edge states in such as
phononic and photonic crystals [405, 406, 407] and a chain of plasmonic
nanoparticles or cold atoms [222, 408, 223]. Its magnonic version is proposed
in a chain of magnetic spheres loaded in a waveguide [409, 410], where
distinct intracell and intercell couplings can be tuned by the separation
between magnetic spheres and external magnetic inductions, providing an
experimentally feasible and tunable platform for engineering the magnonic
soliton states.

Non-Hermitian SSH model.—The established bulk-boundary correspondence in the
Hermitian scenario is based on a prerequisite that the bulk spectrum and
associated eigenstates under open boundary condition (OBC) can be approximated
by those under periodic boundary condition (PBC). However, in the non-
Hermitian cases, it was reported that the spectrum and eigenstates can be
strongly altered under the OBC and PBC, which comes as a surprise since the
bulk-boundary correspondence breaks down [149, 136, 134, 124, 411]. By
allowing the complex wave vectors under the OBC, the edge state can be again
characterized by the bulk topological quantity [124, 137, 140, 139].

Here we introduce the non-Hermitian band topology by addressing a simple non-
Hermitian SSH model as shown in Fig. 27(e), where the intercell $J$ in Fig.
27(a) is replaced by the non-reciprocal hopping with strength $J+\gamma$ from
the right to left, but $J-\gamma$ from the left to right. We allow the complex
wave vector under the OBC and perform the mapping from ${\bf k}$ to
${\bf\beta}=e^{i{k}}$ on the complex plane. The distribution of $\beta$ on the
complex plane is known as the generalized Brillouin zone (GBZ) [124, 140,
139], while the distribution of real wave vector is composed of a unit circle,
as illustrated in Fig. 27(f). Under OBC
$\left|{\bf\beta}\right|=\sqrt{\left|(J-\gamma)\right|/\left|(J+\gamma)\right|}<1$
when $J>\gamma>0$ [124, 137, 139], implying the amplification of the
eigenstate when approaching the left boundary of the chain, i.e., a
manifestation of skin effect.

In terms of $\beta$ and the non-reciprocal hopping between two neighboring
sites, the Hamiltonian (212) is modified in the non-Hermitian SSH model [Fig.
27(f)] to be [149, 124, 143]

$H({\bf\beta})=\left(\begin{array}[]{cc}0&{h}_{1}({\bf\beta})\\\
{h}_{2}({\bf\beta})&0\end{array}\right)=\left(\begin{array}[]{cc}0&\left|{h}_{1}({\bf\beta})\right|e^{i\theta_{1}({\bf\beta})}\\\
\left|{h}_{2}({\bf\beta})\right|e^{i\theta_{2}({\bf\beta})}&0\end{array}\right),$
(215)

where $h_{1}({\bf\beta})=J+\gamma+\tilde{J}/{\bf\beta}$ and
$h_{2}({\bf\beta})=J-\gamma+\tilde{J}{\bf\beta}$ with $\theta_{1}({\bf\beta})$
and $\theta_{2}({\bf\beta})$ denoting their respective phase angles. The off-
diagonal terms in (215) are not conjugated with each other. The eigenvalue
equation

$\displaystyle E^{2}({\bf\beta})=h_{1}({\bf\beta})h_{2}({\bf\beta}),$ (216)

and with two roots $\beta_{1}$ and $\beta_{2}$,
$\beta_{1}\beta_{2}=\left(J-\gamma\right)/\left(J+\gamma\right)$. The
continuum bands form in the sufficiently long chain when
$\left|\beta_{1}\right|=\left|\beta_{2}\right|$ [124, 137], leading to
$\left|\beta_{1,2}\right|=\sqrt{\left|(J-\gamma)\right|/\left|(J+\gamma)\right|}<1$.

The biorthogonal basis diagonalize the matrix (215) via
$\langle\Psi({\bf\beta})|H({\bf\beta})|\Phi({\bf\beta})\rangle=E({\bf\beta})$
[412], with the eigenvalues and eigenvectors

$\displaystyle E_{\pm}({\bf\beta})=\pm
h({\bf\beta})=\pm\sqrt{\left|{h}_{1}({\bf\beta}){h}_{2}({\bf\beta})\right|e^{i\left[\theta_{1}({\bf\beta})+\theta_{2}({\bf\beta})\right]}},$
(217a)
$\displaystyle\langle\Psi({\bf\beta})|_{\pm}=\frac{1}{\sqrt{2}h({\bf\beta})}\left(\left|{h}_{2}({\bf\beta})\right|e^{i\theta_{2}({\bf\beta})},\pm
h({\bf\beta})\right),$ (217b)
$\displaystyle|\Phi({\bf\beta})\rangle_{\pm}=\frac{1}{\sqrt{2}h({\bf\beta})}\left(\begin{array}[]{c}\left|{h}_{1}({\bf\beta})\right|e^{i\theta_{1}({\bf\beta})}\\\
\pm h({\bf\beta})\end{array}\right).$ (217e)

To characterize the topological invariant, a convenient tool is the so-called
“Q matrix" [279], which for the present two-band model is given by [124, 139]

$Q(\beta)=|\Phi({\bf\beta})\rangle_{+}\langle\Psi({\bf\beta})|_{+}-|\Phi({\bf\beta})\rangle_{-}\langle\Psi({\bf\beta})|_{-}=\left(\begin{array}[]{cc}0&q(\bf\beta)\\\
q^{-1}(\bf\beta)&0\end{array}\right),$ (218)

so $q(\beta)=\sqrt{h_{1}(\beta)/h_{2}(\beta)}$ is governed by two non-
Hermitian couplings $h_{1}({\bf\beta})$ and $h_{2}({\bf\beta})$. The
corresponding winding number defined on the GBZ reads

$W=\frac{i}{2\pi}\oint_{\rm
GBZ}q^{-1}(\beta)dq=-\frac{1}{2\pi}\frac{(\delta\theta_{1}-\delta\theta_{2})|_{\rm
GBZ}}{2},$ (219)

where $\delta\theta_{1}|_{\rm GBZ}$ and $\delta\theta_{2}|_{\rm GBZ}$ are the
phase changes of $\theta_{1}({\beta})$ and $\theta_{2}({\beta})$ when
${\bf\beta}$ evolves on a closed curve in a counterclockwise fashion on the
GBZ. As shown in Fig. 27(g), when $J=2/3$, $\gamma=0.2$, and $\tilde{J}=1$,
the evolution of $h_{1}({\bf\beta})$ and $h_{2}({\bf\beta})$ on closed curves
denoted by the blue arrows acquire the phase accumulations
$\delta\theta_{1}|_{\rm GBZ}=-2\pi$ and $\delta\theta_{2}|_{\rm GBZ}=2\pi$,
respectively, thus contributing a nonzero winding number $W=1$ by Eq. (219).

To determine the critical parameter of the chiral coupling for a topological
phase transition, we solve the zero-energy edge states [216, 220, 124] via
$h_{1}({\bf\beta})h_{2}({\bf\beta})=0$ [Eq. (216)], leading to two solutions
${\beta_{1,2}=-\left(J-r\right)/\tilde{J}}$ and $-\tilde{J}/\left(J+r\right)$.
With $\left|\beta_{1}\right|=\left|\beta_{2}\right|$, we find the critical
chiral coupling $\left|J^{2}-r^{2}\right|=\left|\tilde{J}^{2}\right|$, which
divides the region of $W=0$ and $W=1$ in the parameter space, as shown in Fig.
27(h) for the topological phase diagram for this non-Hermitian SSH model when
$\tilde{J}=1$.

However, the Berry phase in Eq. (214) or the winding number in Eq. (219) is
not well-defined when the evolution path of the couplings contain the energy
degeneracy points in the parameter space, such as the Dirac point with
$h_{1}(\beta)=0$ and $h_{2}(\beta)=0$ [123, 413, 414]. The energy degeneracy
point in the non-Hermitian case corresponds to the EPs [9] such that the
matrix becomes defective and the eigenstates are collapsed. Thereby, the
winding number is not well-defined when the EPs are involved in the parameter
space, an issue well addressed in Ref. [137].

Magnonic non-Hermitian SSH model.—Recently, Flebus et al. proposed a magnonic
realization of the non-Hermitian SSH model in terms of an array of spin-torque
oscillators (STOs) [129, 130]. The authors predicted the topological magnonic
lasing edge modes, which can be excited by the spin current injection [129,
130]. Each STO consists of a trilayer composed of a free ferromagnet, a
metallic spacer, and a ferromagnet with pinned magnetization, as shown in Fig.
28(a), where the injected spin current is polarized by the pinned
magnetization and thus exerts a torque on the misaligned magnetization in the
free layer and induce oscillations [415, 416, 417, 418]. Different STOs are
mediated by the metallic layer (“MS") and coupled by the RKKY exchange
interaction and spin pumping [419, 420, 421, 422], which can even achieve
mutual synchronization when their coupling exceeds their frequency mismatch
[423, 424, 425]. An STO itself is a non-Hermitian system since there exists
competition between an effective “gain" due to the spin-current injection
[331, 332, 426] and the “loss" by the dissipation or Gilbert damping. The
metallic layer between two neighboring STOs mediates the non-Hermitian
coupling including the dissipative component due to the spin pumping. Via
tuning the length of metallic spacers, the distinct intracell and intercell
couplings can be achieved, making a close analogy to the non-Hermitian SSH
model [218, 402, 403, 404].

Figure 28: Magnonic realization of the non-Hermitian SSH model in terms of an
array of STOs that are spaced by the metallic layers. In (a) the metallic
spacers mediate the RKKY exchange coupling, $J$ and $\tilde{J}$, and spin-
pumping induced dissipative coupling, $G$ and $\tilde{G}$, between neighboring
STOs. (b) and (c) plots the numerically calculated energy spectrum for $N=40$
unit cells as a function of $J/\left|\tilde{J}\right|$ under the
$\mathcal{PT}$ symmetry, where the topological modes with Re$E=0$ lie within
the gap. In (d), only the magnetization at the end of the array is driven to
its threshold by the injected spin current, acting as the lasing topological
edge mode. The figures are adapted with permission from Ref. [129].

The STOs in the array are divided into $A$ and $B$ sublattices when they are
spaced by two different metallic spacers, viz. “MS1" and “MS2" in Fig. 27(a).
The magnetization in the $A$ sublattice is exerted by the torque from the spin
current $J_{s}$, while no spin current is injected into the sublattice $B$; on
the other hand, the magnetization for both the $A$ and $B$ sublattices
dissipates with the same rates $\alpha\omega$, parametrized by the Gilbert
coefficient $\alpha$. The RKKY exchange interaction contains the intracell and
intercell Hermitian couplings $J$ and $\tilde{J}$, while the spin pumping
across the metallic spaces mediates the intracell and intercell dissipative
couplings $G$ and $\tilde{G}$ (refer to Sec. 4.1). Thus, the Hamiltonian for
such an array of STOs, in the reciprocal space, reads

$\hat{H}(k)=id_{0}(k)\sigma_{0}+{\bf d}(k)\cdot{\boldsymbol{\sigma}},$ (220)

where $\sigma_{0}$ is the identity matrix and ${\boldsymbol{\sigma}}$ are the
Pauli matrices. $d_{0}(k)=(J_{s}-2\alpha\omega)/2$ is contributed by the on-
site spin torque and damping, and the coupling between STOs governs the
complex vector

${\boldsymbol{d}}({k})=\left(\begin{array}[]{c}-J+i\omega
G-(\tilde{J}-i\omega\tilde{G})\cos k\\\ -(\tilde{J}-i\omega\tilde{G})\sin k\\\
iJ_{s}/2\end{array}\right).$ (221)

The Bloch Hamiltonian preserves the $\mathcal{PT}$ symmetry, as demonstrated
in Sec. 2.1, when the gain by the injected spin current balances exactly the
intrinsic dissipation of one unit cell, viz. $J_{s}=2\alpha\omega$, and the
spin pumping between two neighboring STOs vanishes $G=\tilde{G}=0$. The
numerical calculation for the frequency spectrum with $N=40$ unit cells in
terms of real part Re$E-\omega$ and imaginary part Im$E$ as a function of
$J/\left|\tilde{J}\right|$, as plotted in Figs. 27(b) and (c), contains a real
line gap when
$\left||J/\tilde{J}|-1\right|>\left|\alpha\omega/\tilde{J}\right|$,