# Boundary-dependent dynamical instability of bosonic Green’s function:
Dissipative Bogoliubov-de Gennes Hamiltonian and its application to non-
Hermitian skin effect

Nobuyuki Okuma<EMAIL_ADDRESS>Yukawa Institute for
Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

The energy spectrum of bosonic excitations from a condensate is given by the
spectrum of a non-Hermitian Hamiltonian constructed from a bosonic Bogoliubov-
de Gennes (BdG) Hamiltonian in general even though the system is essentially
Hermitian. In other words, two types of non-Hermiticity can coexist: one from
the bosonic BdG nature and the other from the open quantum nature. In this
paper, we propose boundary-dependent dynamical instability. We first define
the bosonic dissipative BdG Hamiltonian in terms of Green’s function in Nambu
space and discuss the correct particle-hole symmetry of the corresponding non-
Hermitian Hamiltonian. We then construct a model of the boundary-dependent
dynamical instability so that it satisfies the correct particle-hole symmetry.
In this model, an anomalous term that breaks the particle number conservation
represents the non-Hermiticity of the BdG nature, while a normal term is given
by a dissipative Hatano-Nelson model. Thanks to the competition between the
two types of non-Hermiticity, the imaginary part of the spectrum can be
positive without the help of the amplification of the normal part and the
particle-hole band touching that causes the Landau instability. This leads to
the boundary-dependent dynamical instability under the non-Hermitian skin
effect, -strong dependence of spectra on boundary conditions for non-Hermitian
Hamiltonians-, of the Bogoliubov spectrum.

##  I Introduction

At low temperature, the excitations from a Bose-Einstein condensate are well
described by a bosonic Bogoliubov-de Gennes (BdG) Hamiltonian [1, 2, 3, 4]. As
well as the systems of boson particles such as cold atoms [4] and photons [5],
the BdG Hamiltonian has been extensively studied in the systems of emergent
bosonic excitations from ordered states such as the magnons [6] and phonons
[7]. Remarkably, the excitation spectrum of a bosonic BdG system is given by
the eigenspectrum of a non-Hermitian matrix constructed from the bosonic BdG
Hamiltonian, in the presence of $anomalous$ terms violating the particle
number conservation [1, 3, 4, 2]. In this sense, the bosonic BdG Hamiltonian
has been interested in terms of non-Hermitian physics [8, 9, 10, 11, 12, 13],
even when the system is essentially Hermitian. In other words, there can
coexist two types of non-Hermiticity in bosonic systems: one from the bosonic
BdG nature and the other from the open quantum nature.

In this paper, we consider dissipative bosonic systems under the non-Hermitian
skin effect [14], which is the extreme sensitivity of the non-Hermitian
spectrum to the boundary condition. For this sake, we define the dissipative
BdG Hamiltonian in terms of the Green’s function [15] and determine the
symmetry of it. While one can define two equivalent particle-hole symmetries
in general non-dissipative BdG Hamiltonians, we find that the natural
extension to the Green’s function corresponds to only one of them. By noticing
this symmetry, we construct a dissipative bosonic model in which the normal
part is given by the dissipative system of the non-Hermitian skin effect.
Although the non-Hermiticity in the normal part does not correspond to an
amplification, this model shows a dynamical instability [4] that occurs under
the periodic boundary condition (PBC) and does not occur under the open
boundary condition (OBC) for some parameter region, without the help of a band
touching between the particle and hole bands of Bogoliubov excitations. This
result can be understood in terms of a competition between an anomalous term
representing the BdG non-Hermiticity and the skin effect representing the open
quantum nature.

This paper is organized as follows. In Sect. II, we define the frequency-
dependent dissipative BdG Hamiltonian in terms of the Green’s function in
Nambu space. We find that the correct particle-hole symmetry is given by a
conjugate-type symmetry in this formalism. In Sect. III, we propose a
dynamical instability that occurs under the PBC and does not occur under the
OBC, without the help of the band touching between the particle and hole
bands. We construct a dissipative BdG Hamiltonian that shows the boundary-
dependent dynamical instability and analyze the complex Bogoliubov spectrum.
We also discuss the dynamical instabilities induced by infinitesimal
perturbations. In Sect. IV, we point out some subtle points about the Landau
instability in terms of the density of states.

##  II Formalism

In this section, we define the dissipative Bogoliubov-de Gennes Hamiltonian.
We first review the free bosonic case and discuss two types of non-Hermitian
particle-hole symmetry: the conjugate-type and the transpose-type symmetries,
which are identical in the absence of dissipation. We then define the
frequency-dependent BdG Hamiltonian in terms of the Green’s function in Nambu
space and discuss the complex energy spectrum. In this formalism, the correct
particle-hole symmetry is given by the conjugate-type symmetry.

###  II.1 Bosonic Bogoliubov-de Gennes Hamiltonian without dissipation

Before introducing the self-energy, we consider free bosonic Hamiltonians,
i.e., quadratic Hamiltonians of bosonic annihilation/creation operators
($a_{i},a_{i}^{\dagger}$) with $i$ being the quantum degrees of freedom such
as the site index and the orbital. In many applications, $anomalous$ terms
that violate the particle number conservation can exist. In such a case, the
corresponding free Hamiltonian is called the bosonic Bogoliubov-de Gennes
(BdG) Hamiltonian and written in terms of the Nambu spinor
$\bm{\Phi}=(\bm{a},\bm{a}^{\dagger})$:

$\displaystyle\hat{H}=\bm{\Phi}^{\dagger}H^{(0)}_{\rm
BdG}\bm{\Phi}=(\bm{a}^{\dagger},\bm{a})H^{(0)}_{\rm
BdG}\begin{pmatrix}\bm{a}\\\ \bm{a}^{\dagger}\end{pmatrix}.$ (1)

In general, the bosonic BdG Hamiltonian matrix has the following form:

$\displaystyle H^{(0)}_{\rm BdG}=\begin{pmatrix}h&s\\\
s^{\dagger}&h^{T}\end{pmatrix}=\begin{pmatrix}h&s\\\
s^{*}&h^{*}\end{pmatrix},$ (2)

where $h$ and $s$ are Hermitian and symmetric matrices, respectively. We have
assumed the Hermiticity of $H^{(0)}_{\rm BdG}$. Owing to this constraint, one
can define the bosonic analogy of particle-hole symmetry [16]:

$\displaystyle\sigma_{x}[H^{(0)}_{\rm BdG}]^{*}\sigma_{x}=H^{(0)}_{\rm BdG},$
(3)

where $\sigma$ denotes the Pauli matrices in Nambu space henceforth.

Unlike a conventional quadratic Hamiltonian, one-particle energy spectrum of
Eq. (1) is not given by the eigenvalues of the matrix representation
$H^{(0)}_{\rm BdG}$ in the presence of the anomalous terms. This is because
the unitary matrix that diagonalizes $H^{(0)}_{\rm BdG}$ can ruin the bosonic
commutation. In general, one-particle energy spectrum is given by the bosonic
Bogoliubov transformation [3, 17, 18, 4]. We here introduce the mapping to the
fermionic non-Hermitian Hamiltonin:

$\displaystyle H^{(0)}_{\sigma\rm BdG}=\sigma_{z}H^{(0)}_{\rm BdG}.$ (4)

Corresponding to the bosonic particle-hole symmetry (3), this non-Hermitian
Hamiltonian has the fermionic particle-hole symmetry [16]:

$\displaystyle\sigma_{x}[H^{(0)}_{\sigma\rm
BdG}]^{*}\sigma_{x}=-H^{(0)}_{\sigma\rm BdG}.$ (5)

In addition, the Hermiticity of $H^{(0)}_{\rm BdG}$ leads to the pseudo-
Hermiticity of $H^{(0)}_{\sigma\rm BdG}$ [16]:

$\displaystyle\sigma_{z}[H^{(0)}_{\sigma\rm
BdG}]^{\dagger}\sigma_{z}=H^{(0)}_{\sigma\rm BdG}.$ (6)

Actually, the eigenvalue problem of $H^{(0)}_{\sigma\rm BdG}$ is directly
related to the bosonic excitation spectrum:

$\displaystyle P^{-1}H^{(0)}_{\sigma\rm
BdG}P=\frac{1}{2}\begin{pmatrix}\Lambda&0\\\ 0&-\Lambda\end{pmatrix},$ (7)

where $\Lambda$ is a diagonal matrix whose elements give the one-particle
energy spectrum of Eq. (1). In particular, $P$ can be taken as a paraunitary
matrix if the spectrum of $H^{(0)}_{\sigma\rm BdG}$ is real [19]:

$\displaystyle P\sigma_{z}P^{\dagger}=\sigma_{z}.$ (8)

In this case, the particle (hole) eigenstates of $H^{(0)}_{\sigma\rm BdG}$ are
also eigenstates of $\sigma_{z}$ with an eigenvalue $+1~{}(-1)$ [19]. Using
the paraunitary condition, the bosonic Bogoliubov transformation is written in
terms of the bosonic BdG Hamiltonian:

$\displaystyle P^{\dagger}H^{(0)}_{\rm
BdG}P=\frac{1}{2}\begin{pmatrix}\Lambda&0\\\ 0&\Lambda\end{pmatrix}.$ (9)

Note that the presence of the anomalous terms changes the Fock vacuum of the
eigenvectors from that of $a_{i}$’s. This change of the vacuum is called
quantum correction.

In most applications to solid-state physics, the (semi) positive-definiteness
of $H^{(0)}_{\rm BdG}$ is further assumed, which ensures nonnegative
excitation energies. The instability that comes from the presence of negative
excitation energy is called the Landau instability [4]. In isolated quantum
systems, on the other hand, Landau instability is suppressed, and this
assumption is not always imposed. Such a situation is thought to be realized
in cold atom systems. In the absence of the positive definiteness of
$H^{(0)}_{\rm BdG}$, the excitation energy can be negative or complex. Owing
to the pseudo-Hermiticity, the presence of a complex energy is equivalent to
the presence of an energy with positive imaginary part. Such an excitation
mode grows in time and leads to the instability of the vacuum. This type of
the instability is called dynamical instability [4].

Note that the following particle-hole symmetry is also valid in the presence
of the pseudo-Hermiticity [19]:

$\displaystyle\sigma_{y}[H^{(0)}_{\sigma\rm
BdG}]^{T}\sigma_{y}=-H^{(0)}_{\sigma\rm BdG}.$ (10)

In the absence of the pseudo-Hermiticity that comes from the Hermiticity of
$\hat{H}$, however, the symmetries (5) and (10) are not identical. The correct
particle-hole symmetry may depend on the situation. As we will discuss later,
a natural extension of Eq. (5) is the correct particle-hole symmetry for the
effective Hamiltonian defined in terms of the Green’s function.

###  II.2 Bosonic Green’s function in Nambu representation

There are several ways to treat open quantum systems. A typical mathematical
tool is the Lindblad equation [20, 21]. For example, a bosonic quadratic
Lindblad equation has been studied in terms of topology and skin effect [22].
Another useful mathematical tool is the Green’s function [15, 23]. Several
studies have focused on the relation between the non-Hermitian skin effect and
the Green’s functions [24, 25, 26].

In the following, we consider the retarded/advanced Green’s function for
bosonic excitations to include the dissipation/interaction/disorder effects.
In the presence of the anomalous term known as $anomalous$ $Green^{\prime}s$
$function$, it is convenient to introduce the Nambu representation, as in the
case of the previous subsection. The retarded Green’s function in Nambu
representation is defined in terms of the Heisenberg operators [15]:

$\displaystyle G_{ij}^{R}(t,t^{\prime})$
$\displaystyle=-i\theta(t-t^{\prime})\langle[\Phi_{i}(t),\Phi_{j}^{\dagger}(t^{\prime})]\rangle$
$\displaystyle=-i\theta(t-t^{\prime})\begin{pmatrix}\langle[a_{i}(t),a_{j}^{\dagger}(t^{\prime})]\rangle&\langle[a_{i}(t),a_{j}(t^{\prime})]\rangle\\\
\langle[a^{\dagger}_{i}(t),a^{\dagger}_{j}(t^{\prime})]\rangle&\langle[a^{\dagger}_{i}(t),a_{j}(t^{\prime})]\rangle\end{pmatrix}$
$\displaystyle=:\begin{pmatrix}A^{R}_{ij}(t,t^{\prime})&B^{R}_{ij}(t,t^{\prime})\\\
C^{R}_{ij}(t,t^{\prime})&D^{R}_{ij}(t,t^{\prime})\end{pmatrix},$ (11)

where $\langle\cdot\rangle$ denotes the statistical average. The following
discussion does not depend on the choice of the statistical average, and one
can choose both equilibrium [15] and non-equilibrium statics [23, 27].
Similarly, the advanced Green’s function is defined as

$\displaystyle G_{ij}^{A}(t,t^{\prime})$
$\displaystyle=i\theta(t^{\prime}-t)\langle[\Phi_{i}(t),\Phi_{j}^{\dagger}(t^{\prime})]\rangle$
$\displaystyle=i\theta(t^{\prime}-t)\begin{pmatrix}\langle[a_{i}(t),a_{j}^{\dagger}(t^{\prime})]\rangle&\langle[a_{i}(t),a_{j}(t^{\prime})]\rangle\\\
\langle[a^{\dagger}_{i}(t),a^{\dagger}_{j}(t^{\prime})]\rangle&\langle[a^{\dagger}_{i}(t),a_{j}(t^{\prime})]\rangle\end{pmatrix}$
$\displaystyle=:\begin{pmatrix}A^{A}_{ij}(t,t^{\prime})&B^{A}_{ij}(t,t^{\prime})\\\
C^{A}_{ij}(t,t^{\prime})&D^{A}_{ij}(t,t^{\prime})\end{pmatrix}.$ (12)

In the following, we derive relationships between the matrices
$A^{R/A},B^{R/A},C^{R/A},$ and $D^{R/A}$. By definition, the following
relationships hold:

$\displaystyle D^{R}_{ij}(t,t^{\prime})$
$\displaystyle=i\theta(t-t^{\prime})\langle[a_{j}(t^{\prime}),a_{i}^{\dagger}(t)]\rangle=A^{A}_{ji}(t^{\prime},t),$
$\displaystyle D^{A}_{ij}(t,t^{\prime})$
$\displaystyle=-i\theta(t^{\prime}-t)\langle[a_{j}(t^{\prime}),a_{i}^{\dagger}(t)]\rangle=A^{R}_{ji}(t^{\prime},t),$
$\displaystyle B^{R}_{ij}(t,t^{\prime})$
$\displaystyle=i\theta(t-t^{\prime})\langle[a_{j}(t^{\prime}),a_{i}(t)]\rangle=B^{A}_{ji}(t^{\prime},t),$
$\displaystyle C^{R}_{ij}(t,t^{\prime})$
$\displaystyle=i\theta(t-t^{\prime})\langle[a^{\dagger}_{j}(t^{\prime}),a^{\dagger}_{i}(t)]\rangle=C^{A}_{ji}(t^{\prime},t).$
(13)

In the steady state, the Green’s functions have the time translation
invariance, and these relations are written in the Fourier form:

$\displaystyle D^{R}(\omega)$
$\displaystyle=[A^{A}(-\omega)]^{T},D^{A}(\omega)=[A^{R}(-\omega)]^{T},$
$\displaystyle B^{R}(\omega)$
$\displaystyle=[B^{A}(-\omega)]^{T},C^{R}(\omega)=[C^{A}(-\omega)]^{T}.$ (14)

By combining these equations with $[G^{R}(\omega)]^{\dagger}=G^{A}(\omega)$,
we obtain

$\displaystyle
G^{R/A}(\omega)=\begin{pmatrix}A^{R/A}(\omega)&B^{R/A}(\omega)\\\
[B^{R/A}(-\omega)]^{*}&[A^{R/A}(-\omega)]^{*}\end{pmatrix}.$ (15)

From this expression, one can find the following symmetry for the
retarded/advanced Green’s function:

$\displaystyle\sigma_{x}[G^{R/A}(-\omega)]^{*}\sigma_{x}=G^{R/A}(\omega).$
(16)

###  II.3 Dissipative Bogoliubov-de Gennes Hamiltonian defined for Green’s
function

For a free bosonic system, the Green’s function in Nambu space is defined via

$\displaystyle G^{-1}_{0}=\omega\sigma_{z}-H^{(0)}_{\rm BdG}.$ (17)

In the presence of the self-energy part $\Sigma^{R}(\omega)$, the retarded
Green’s function is given by

$\displaystyle G^{R}(\omega)$
$\displaystyle=\frac{1}{G^{-1}_{0}-\Sigma^{R}(\omega)}=\frac{1}{\omega\sigma_{z}-[H^{(0)}_{\rm
BdG}+\Sigma^{R}(\omega)]}$ $\displaystyle=:\frac{1}{\omega\sigma_{z}-H^{(\rm
eff)}_{\rm BdG}(\omega)}.$ (18)

Here we have defined the dissipative BdG Hamiltonian $H^{(\rm eff)}_{\rm
BdG}(\omega)$, which is a frequency-dependent non-Hermitian matrix. From the
symmetry of the retarded Green’s function (16), one can still define the
bosonic particle-Hole symmetry for $H^{(\rm eff)}_{\rm BdG}(\omega)$:

$\displaystyle\sigma_{x}[H^{(\rm eff)}_{\rm
BdG}(-\omega)]^{*}\sigma_{x}=H^{(\rm eff)}_{\rm BdG}(\omega).$ (19)

###  II.4 Eigenvalue problem of dissipative Bogoliubov-de Gennes Hamiltonian

The complex energy spectrum of the quasi-particles is defined by the set of
poles of the retarded Green’s function [28] up to the factor $2$:

$\displaystyle 0=\det[G^{R}(z)^{-1}]$
$\displaystyle=\det[\sigma_{z}]\det[z-\sigma_{z}H^{(\rm eff)}_{\rm BdG}(z)]$
$\displaystyle=-\det[z-H^{(\rm eff)}_{\sigma\rm BdG}(z)],$ (20)

where $G^{R}(z)$ is the analytic continuation of $G^{R}(\omega)$. We have
defined the fermionic counterpart $H^{(\rm eff)}_{\sigma\rm
BdG}=\sigma_{z}H^{(\rm eff)}_{\rm BdG}$. As in the case of the free bosons,
the eigenvalues are calculated in terms of the fermionic BdG Hamiltonian. Note
that there is no longer pseudo-Hermiticity in the fermionic counterpart owing
to the non-Hermiticity of $H^{(\rm eff)}_{\rm BdG}(\omega)$. This implies that
the equivalence between the conjugate-type symmetry and the transpose-type
symmetry does not hold for the fermionic BdG Hamiltonian. In the present
formalism, the proper particle-hole symmetry of the fermionic BdG Hamiltonian
is given by the following conjugate-type symmetry:

$\displaystyle\sigma_{x}[H^{(\rm eff)}_{\sigma\rm
BdG}(-\omega)]^{*}\sigma_{x}=-H^{(\rm eff)}_{\sigma\rm BdG}(\omega).$ (21)

In the absence of the frequency dependence of the self-energy, this symmetry
is reduced to the symmetry (5). Note that we can also define the dissipative
fermionic BdG Hamiltonian $H^{(\rm eff)}_{\rm fBdG}$ for the fermionic Green’s
function via $G^{R}(\omega)=[\omega-H^{(\rm eff)}_{\rm fBdG}]^{-1}$, which
describes superconductors. The same symmetry (21) holds for that case. The
derivation is almost equal to the bosonic case except for the difference
between commutation and anti-commutation relations. This symmetry was
discussed in terms of the dissipative Majorana fermions [29].

Finally, we remark the symmetry constraints for poles of the retarded Green’s
function. For an eigenvalue of $H^{(\rm eff)}_{\sigma\rm BdG}$, $E(\omega)$,
one can find the particle-hole counterpart $-E^{*}_{-\omega}$ owing to the
particle-hole symmetry (21). Correspondingly, for a pole of retarded Green’s
function, $z_{\rm p}$, one can also find the particle-hole counterpart
$-z_{\rm p}^{*}$. This is because the followings hold for a pair of poles:

$\displaystyle z$ $\displaystyle=E(z),$ (22) $\displaystyle z$
$\displaystyle=-[E(-z^{*})]^{*},$ (23)

where $E(z)$ and $-[E(-z^{*})]^{*}$ are the analytic continuations of
$E(\omega)$ and $-E^{*}_{-\omega}$, respectively. If $z=z_{p}$ satisfies Eq.
(22), then $z=-z_{p}^{*}$ does Eq. (23), and vice versa.

###  II.5 Dynamical instability in bosonic steady state

In bosonic systems with nonzero self-energy, the presence or absence of the
dynamical instability can be predicted by using the imaginary part of the
complex energy spectrum defined by Eq. (20). If there is a complex energy with
positive imaginary part, such an excitation grows in time and indicates the
instability of the steady state that is assumed in the quadratic analysis.

###  II.6 Particle and hole bands

The definitions of particle and hole bands are not obvious in dissipative
systems. In the following, we limit our discussion to the poles of
$\sigma_{z}G^{R}(z)$ that is adiabatically connected to $(z-\sigma_{z})^{-1}$
without closing the real energy gap. If a pole is mapped to the pole
$z=+1~{}(-1)$ under this process, we say it is in a particle (hole) hole band.

##  III Boundary-dependent dynamical instability under non-Hermitian skin
effect

In this section, we propose a dynamical instability that occurs under the PBC
and does not occur under the OBC, without the help of the band touching
between the particle and hole bands. We first review the basics of the non-
Hermitian skin effect. We then construct a dissipative BdG Hamiltonian that
shows the boundary-dependent dynamical instability, by noticing the conjugate-
type particle-hole symmetry of the BdG Hamiltonian. We analyze the complex
Bogoliubov spectrum and discuss the competition between two types of non-
Hermiticity: one from the anomalous term and the other from the self-energy.

###  III.1 Hatano-Nelson model and non-Hermitian skin effect

We here briefly review the non-Hermitian skin effect. Non-Hermitian skin
effect is a phenomenon in which the OBC spectrum of a non-Hermitian
Hamiltonian is drastically different from the PBC one [14]. The simplest
example is the Hatano-Nelson model without disorder [30, 31]:

$\displaystyle\hat{H}^{\rm(HN)}=\sum_{i}\left[(t+g)c^{\dagger}_{i+1}c_{i}+(t-g)c^{\dagger}_{i}c_{i+1}\right],$
(24)

where $i$ is the site index, $(c,c^{\dagger})$ are creation/annihilation
operators, and $t\in\mathbb{R}$ and $g\in\mathbb{R}$ are the Hermitian
symmetric and non-Hermitian asymmetric hopping terms, respectively. Regardless
of the statistics of field operators, the one-particle energy is simply given
by the eigenvalues of the matrix form of the Hamiltonian:

$\displaystyle H^{\rm(HN)}:=\begin{pmatrix}0&t-g&0&\cdots\\\
t+g&0&t-g&\cdots\\\ 0&t+g&0&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix}.$ (25)

By using the Fourier transform, one obtain the PBC spectrum parameterized by a
crystal momentum $k$:

$\displaystyle E^{\rm(HN,PBC)}_{k}=2t\cos k+2ig\sin k.$ (26)

Under the infinite-volume limit, the PBC spectrum for $tg\neq 0$ is an ellipse
in the complex plane. Unlike the Hermitian tight-binding models, the OBC
spectrum is drastically different from the PBC one. By using a similarity
transformation called $imaginary$ $gauge$ $transformation$, one can map the
Hatano-Nelson model to a Hermitian matrix:

$\displaystyle H^{(\rm Herm)}$ $\displaystyle:=V_{r}^{-1}H^{\rm(HN)}V_{r}$
$\displaystyle=\begin{pmatrix}0&\sqrt{t^{2}-g^{2}}&\cdots\\\
\sqrt{t^{2}-g^{2}}&0&\cdots\\\ \vdots&\vdots&\ddots\end{pmatrix},$ (27)

where $[V_{r}]_{i,j}=r^{i}\delta_{ij}$ with $r=\sqrt{t+g/t-g}$. We have
assumed $t>g>0$ for simplicity. Since a similarity transformation preserves
the spectrum of any finite-dimensional matrix, the OBC spectrum of $H^{(\rm
HN)}$ is given by that of $H^{(\rm Herm)}$, which is on the real axis owing to
the Hermiticity. Remarkably, the OBC spectrum is very different from the
ellipse-like PBC spectrum. Note that the OBC eigenstates of the Hatano-Nelson
model are localized at one end because they are given by acting $V_{r}$ on the
delocalized solutions of $H^{(\rm Herm)}$.

Actually, the spectral winding number of the PBC curve indicates the emergence
of the the non-Hermitian skin effect. Several observations about this
correspondence were indicated in Refs. [32, 33, 24], and related theorems were
proven in Refs. [34, 35]. While the Hatano-Nelson model requires asymmetric
hopping (or momentum-dependent dissipation), on-site dissipation can also
induce the spectral winding number [36]. For example, the combination of the
spin-momentum locked band and spin-dependent momentum-independent dissipation
gives an effective momentum-dependence of the dissipation [37]. For
applications in condensed matter physics, such an on-site dissipation may be
more useful than the momentum-dependent dissipation. This construction is
analogous to an implementation of a topological superconductor in which the
$p$-wave paring is effectively realized by the combination of the spin-
momentum-locked band structure and the $s$-wave paring [38, 39, 40, 41].

###  III.2 Boundary-dependent dynamical instability under non-Hermitian skin
effect

We here propose a dynamical instability that occurs under the PBC and does not
occur under the OBC. The non-Hermitian skin effect discussed in the previous
subsection plays a central role in this proposal. For simplicity, we consider
the frequency-independent self-energy henceforth.

If anomalous terms are absent, there is no need to introduce the Nambu
representation, and the bosonic Green’s function is expressed in terms of a
conventional form:

$\displaystyle
G^{R}(\omega)=\frac{1}{\omega-H-\Sigma^{R}}=:\frac{1}{\omega-H^{(\rm eff)}}.$
(28)

In that case, however, $H^{(\rm eff)}$ cannot have eigenvalues with positive
imaginary part, which means the absence of the dynamical instability. Thus,
the anomalous term is essential to observe the dynamical instability. If the
self-energy is negligibly small, there emerges the pseudo-Hermiticity. In that
case, the complex energy can be found only when the particle and hole bands
experience the band crossing at the zero energy. This means that one should
give up the positive definiteness of $H^{(0)}_{\rm BdG}$ in order to observe
the non-Hermitian skin effect because one cannot find a PBC spectral winding
for the real spectrum. Such an assumption is relevant in specific cases such
as cold atom systems. See Ref. [42, 43, 44] for the dynamical instability in
such cases.

For the relevance in various applications including solid-state physics, we
consider the case where the particle and hole bands do not experience the band
crossing. For this sake, we use the combination of the Hatano-Nelson model
(25) and the anomalous terms. Let us consider the following BdG Hamiltonian:

$\displaystyle H^{(\rm eff)}_{\sigma\rm BdG}=\begin{pmatrix}H^{(\rm
HN)}-\alpha&i\Delta\\\ i\Delta&-[H^{(\rm HN)}-\alpha]^{*}\end{pmatrix},$ (29)

where $\Delta\in\mathbb{R}$ and $\alpha\in\mathbb{C}$ represent the anomalous
term and a constant shift, respectively. This Hamiltonian is constructed to
satisfy the particle-hole symmetry (21). Note that the non-Hermiticity of the
anomalous term originates from the Hermitian term of the $H^{(\rm eff)}_{\rm
BdG}$, and the essential non-Hermiticity is only in the normal term. Our
motivation is to discuss the competition between these two types of non-
Hermiticity.

Figure 1: Bogoliubov spectra of the dissipative Bogoliubov-de Gennes
Hamiltonian (29) for $\Delta=0$ and $0.5$. The other model parameters are the
followings: $t=1$, $g=0.2$, and $\alpha=-2(t+0.3)+2ig$. The PBC spectra are
plotted by using the analytical expression (32), while the OBC spectra are
plotted by numerical diagonalization for size $L=50$.

Under the PBC, the Bloch Hamiltonian for Eq.(29) is given by

$\displaystyle H^{(\rm eff)}_{\sigma\rm
BdG}(k)=-i~{}\mathrm{Im}~{}\alpha+\left[E^{\rm(HN,PBC)}_{k}-\mathrm{Re}~{}\alpha\right]\sigma_{z}+i\Delta\sigma_{x},$
(30)

where $E^{\rm(HN,PBC)}_{k}$ is given in Eq.(26). Since the non-Hermiticity in
the normal part should correspond to dissipation, we assume

$\displaystyle\max_{k\in[0,2\pi)}\mathrm{Im}~{}E_{k}^{\rm(HN,PBC)}=2g\leq\mathrm{Im}~{}\alpha.$
(31)

In the absence of the anomalous term, the BdG Hamiltonian is reduced to the
Hatano-Nelson model with a constant shift, and there is no dynamical
instability owing to Eq. (31). In the presence of the anomalous term, on the
other hand, there is a possibility for the dynamical instability. To see this,
we consider the explicit form of the Bogoliubov spectrum:

$\displaystyle E^{(\rm BdG)}_{k,\pm}$
$\displaystyle=-i~{}\mathrm{Im}~{}\alpha\pm\sqrt{\left[E^{\rm(HN,PBC)}_{k}-\mathrm{Re}~{}\alpha\right]^{2}-\Delta^{2}}$
$\displaystyle=:-i~{}\mathrm{Im}~{}\alpha\pm\sqrt{A(|\Delta|)+iB},$ (32)

where $A,B\in\mathbb{R}$. The imaginary part of the square root is expressed
as

$\displaystyle\frac{\mathrm{Im}\sqrt{A+iB}}{B}=\frac{(A^{2}+B^{2})^{\frac{1}{4}}}{\sqrt{(A+\sqrt{A^{2}+B^{2}})^{2}+B^{2}}}.$
(33)

Since this form is a monotonically decreasing function of $A$ and
$A(|\Delta|)$ is a monotonically decreasing function of $|\Delta|$, the
absolute value of the square root is a monotonically increasing function of
$|\Delta|$. Thus, the imaginary part of the Bogoliubov spectrum can be
positive for sufficiently large $|\Delta|$, which leads to the dynamical
instability. Remarkably, one can choose a parameter so that the dynamical
instability does not occur under the OBC because the OBC spectrum can be
drastically different from the PBC one under the non-Hermitian skin effect. In
Fig.1, we plot the PBC curve (32) and the OBC curve that is obtained by a
numerical diagonalization for $L=50$, where $L$ is the number of sites. We
have set the largest imaginary part of PBC eigenvalues to be infinitesimally
small value $0^{-}$ in the absence of the anomalous term. Figure 1 indicates
that the introduction of the anomalous term leads to the dynamical instability
under the PBC without the help of the band crossing, while the largest
imaginary part of the OBC eigenvalues remains negative for finite $\Delta$
owing to the non-Hermitian skin effect, which means the absence of the
dynamical instability. Under the PBC, the complex energies with positive
imaginary part are around $k\sim\pi/2$, which are the chiral modes (left
movers) with finite real energies. Thus, the collapse of the steady state is
accompanied by a current circulating the PBC system. If we set the largest
imaginary part of PBC eigenvalues is a finite negative value, there is a
threshold for $\Delta$ to induce the dynamical instability. Note that too
large $\Delta$ leads to the band crossing at the imaginary axis. Note also
that our proposal does not depend on the specific properties of the model and
the situation. One can discuss the boundary-dependent dynamical instability of
the system with the non-Hermitian skin effect, which does not require the
momentum-dependent dissipation as discussed in the previous subsection. In
addition, one can define the retarded Green’s function both for equilibrium
and non-equilibrium steady states, and the above discussion may be applicable
to various systems.

###  III.3 Dynamical instability related to infinitesimal instability

In terms of the mathematics of the Toeplitz matrices, a small $nonlocal$
perturbation that couples the different edges of a tight-binding model
drastically changes the OBC spectrum [45, 46]. In particular, it becomes
identical to the PBC curve with a spectral winding for an infinitesimally
small perturbation after taking the infinite-volume limit. In finite systems,
the threshold for this spectral instability is taken to be an exponentially
small value in terms of the system size. In Ref. [47], the author and a
coworker discussed an infinitesimal instability against a $local$ perturbation
of a spin-dependent non-Hermitian skin effect in which the direction of the
asymmetric hopping depends on the spin. The simplest model for this phenomenon
is given by the stack of the Hatano-Nelson model:

$\displaystyle H_{\rm
spin}\left(k\right)=\left(\begin{array}[]{@{\,}cc@{\,}}H^{(\rm
HN)}\left(k\right)&0\\\ 0&H^{(\rm HN)}\left(-k\right)\\\ \end{array}\right).$
(36)

While the OBC spectrum of this model is on the real axis, the introduction of
the transverse magnetic field that couples the different blocks of $H_{\rm
spin}$ drastically changes the OBC spectrum. In Ref. [34], the author and
coworkers used the notion of infinitesimal instability to characterize the
$\mathbb{Z}_{2}$ skin effect protected by the time-reversal symmetry, which is
characterized not by the spectral winding nubmer but by the $\mathbb{Z}_{2}$
topological invariant. In general, symmetry-protected skin effects should be
unstable to symmetry-breaking perturbations.

The notion of infinitesimal instability would be relevant for the physics of
dynamical instability. If we introduce a small perturbation that couples the
edges of Eq. (29), we can expect a dynamical instability against it. If we
replace the normal part of Eq. (29) with the system with a symmetry-protected
skin effect, we can expect a dynamical instability due to a small symmetry-
breaking local perturbation under the OBC. Note that a non-dissipative
Bogoliubov-de Gennes Hamiltonian treated in Ref. [44] shows an infinitesimal
instability, which was also discussed in terms of a dynamical instability.

##  IV Discussion

We remark several points about the band crossing between particle and hole
bands and the Landau instability. Strictly speaking, the absence of the band
crossing does not directly mean the absence of the Landau instability. While
we have assumed the frequency-independent dissipation, it means that the
density of states for the particle bands calculated by $D_{\rm
part}(\omega)=1/\pi\mathrm{Im}\sum_{i:\mathrm{particle}}(\omega-E_{i})^{-1}$ ,
where $E_{i}$’s are eigenvalues of $H^{(\rm eff)}_{\sigma\rm BdG}$, takes a
finite value at $\omega<0$, which indicates the presence of the Landau
instability. In this sense, $\omega$-dependence of the dissipative BdG
Hamiltonian should be considered for more realistic calculations. For this
sake, one should treat $\omega$-dependent equation (20) to get the Bogoliubov
spectrum. Let $z_{0}$ be a Bogoliubov eigenvalue with positive imaginary part
of some frequency-independent $H^{(\rm eff)}_{\sigma\rm BdG}$. From this
Hamiltonian, we can construct a frequency-dependent Hamiltonian with the
dynamical instability and without the Landau instability by choosing
$\Sigma^{R}(\omega)$ so that $\Sigma^{R}(z_{0})=\sigma_{z}(H^{(\rm
eff)}_{\sigma\rm BdG}-H^{(0)}_{\sigma\rm BdG})$ and $D_{\rm
part}(\omega<0)=0$, at least mathematically. Thus, the essence of our result
does not suffer from the Landau instability.

We also mention the relationship between the non-Hermitian skin effect and the
Landau instability. Although one might expect that the skin effect can also
induce the sensitivity of the Landau instability to the boundary condition,
such an expectation may be incorrect. If the smallest real part of the PBC
curve is negative and that of the OBC curve is positive, the Landau
instability is induced for both cases. This is because the relevant quantity
for the Landau instability may not be the Bogoliubov complex energy but the
density of states around zero energy. As shown in Ref. [25], the density of
states is insensitive to the non-Hermitian skin effect in the absence of
complex eigenvalues with positive imaginary part, which implies the
insensitivity of the Landau instability to the boundary condition.

###### Acknowledgements.

I thank Masatoshi Sato for a fruitful discussion about free bosonic BdG
Hamiltonians. I also thank Terufumi Yamaguchi for a fruitful discussion about
Green’s functions. This work was supported by JST CREST Grant No. JPMJCR19T2,
Japan. N.O. was supported by JSPS KAKENHI Grant No. JP20K14373.

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Andrei E. Egorov

# Updated constraints on WIMP dark matter annihilation by radio observations
of M31 – all annihilation channels

###### Abstract

The present work derived the robust constraints on annihilating WIMP
parameters utilizing new radio observations of M31, as well as new studies of
its DM distribution and other properties. The characteristics of emission due
to DM annihilation were computed in the frame of 2D galactic model employing
GALPROP code adapted specifically for M31. This enabled us to refine various
inaccuracies of previous studies on the subject. DM constraints were obtained
for all possible annihilation channels except
$\chi\chi\rightarrow\gamma\gamma,\nu\bar{\nu}$. A wide variety of radio data
was utilized in the frequency range $\approx$(0.1–10) GHz. As the result the
thermal WIMP lighter than fiducially $\approx$(40–70) GeV was excluded in the
case of light primary annihilation products
$\chi\chi\rightarrow\tau^{+}\tau^{-},\mu^{+}\mu^{-},gg,c\overline{c},u\overline{u},d\overline{d},s\overline{s},e^{+}e^{-},b\overline{b}$
(and an arbitrary combination of them). Heavier WIMP, which can annihilate to
$W^{+}W^{-},Z^{0}Z^{0},t\overline{t},hh$; can not be probed at the level of
thermal cross section, unless one assumes the optimistic cases of DM density
and magnetic field distributions in M31. In summary, $m_{x}\gtrsim 40$ GeV
represents the fiducial channel-independent mass limit for the thermal WIMP
with the full uncertainty range estimated to be $\approx$(20–90) GeV. The
obtained exclusions are competitive to those from Fermi-LAT observations of
dwarfs and AMS-02 measurements of antiprotons. Our constraints significantly
restrict the opportunity to explain the gamma-ray outer halo of M31 by
annihilating DM. And, finally, we questioned the possibility claimed in other
studies to robustly constrain heavy thermal WIMP with $m_{x}>100$ GeV by radio
data on M31.

##  1 Introduction and motivation

The physical nature of dark matter (DM) remains to be one of the biggest
puzzles in modern physics and astronomy. Currently, we may outline three most
popular candidates for the role of DM: weakly interacting massive particles
(WIMPs), axionlike particles (ALPs) and sterile neutrinos (see e.g. [1] for a
review). This work is dedicated to indirect searches of WIMPs, which
traditionally have been the most anticipated. In our case we consider DM halo
of the big neighbor galaxy M31 (Andromeda galaxy). Annihilating WIMPs in its
halo would produce relativistic electrons and positrons ($e^{\pm}$), which in
turn would generate synchrotron emission in the galactic magnetic field (MF)
at radio frequencies. Hence, radio observations of the galaxy may infer some
constraints on annihilating DM or even hints of its signal. Meanwhile,
historically, the first attempt (known to the author) to constrain DM particle
properties by radio observations was made in [2].

Traditionally the gamma-ray band has been the most promising for WIMP indirect
searches: space-based gamma-ray telescopes, particularly Fermi-LAT, are able
to probe the thermal WIMPs up to $m_{x}\approx 100$ GeV by observations of
Milky Way (MW) dwarf satellites [3]; and ground-based facilities, particularly
CTA, will be able to probe the thermal WIMPs above this mass scale by
observations of the Galactic halo [4]. However, the mass range around
$m_{x}\approx 100$ GeV is poorly reachable by both techniques. And here it is
very relevant to employ alternative methodologies – particularly, radio
observations – in order to complement the gamma-ray window and explore the
whole WIMP mass range evenly.

M31 is one of the best targets in the sky for DM searches in radio due to its
proximity, which allows detailed imaging and studies of the galactic medium.
Also the central region of M31 is relatively faint in radio, which implies
higher sensitivity to the potential DM signal due to lower hindering by usual
astrophysical emissions. Being motivated by these advantages we started our
work on the subject by [5]. The current work refined and updated the former
(and also the works by other authors) in a variety of aspects. All the details
are thoroughly described in [6] (sec. II there detailizes inaccuracies of the
previous works on the subject). This letter provides a brief summary and
generalization for all possible annihilation channels (in [6] only popular
$\chi\chi\rightarrow b\bar{b},\tau^{+}\tau^{-}$ channels were computed).

At first we modeled theoretically the characteristics of anticipated emission
due to DM annihilation, i.e. the emission intensity dependence on all the
parameters. This modeling procedure is described in the next sec. 2. Then the
relevant observational data were collected and compared with the theoretically
calculated DM emission intensities in order to derive DM constraints of
interest – this is described in sec. 3. And sec. 4 provides a summary and
relation with the gamma-ray band.

##  2 Modeling the emission due to DM annihilation by GALPROP

For this purpose one has to solve the transport equation for $e^{\pm}$
produced by DM annihilation (DM $e^{\pm}$ hereafter) and then to calculate the
synchrotron emission from them assuming certain MF distribution. This task was
fully solved by employment of GALPROP package [7] (v56, specifically adapted
for M31) together with my addition [8], which precisely calculates DM source
term for the transport equation. The former has the following form (in general
case of an arbitrary combination of the annihilation channels):

$q(R,E)=\frac{1}{2}\langle\sigma
v\rangle\left(\frac{\rho(R)}{m_{x}}\right)^{2}\xi(R)\sum_{i}BR_{i}\frac{dN_{e,i}}{dE}(E,m_{x}),$
(1)

where $\langle\sigma v\rangle$ denotes WIMP annihilation cross section,
$\rho(R)$ is DM density distribution, $m_{x}$ is WIMP mass, $\xi(R)$ is the DM
annihilation rate boost factor due to substructures, $BR_{i}$ is the branching
ratio to the $i$-th channel (summation goes over all channels and
$\sum_{i}BR_{i}=1$) and $\frac{dN_{e,i}}{dE}(E,m_{x})$ represents the energy
spectra of $e^{\pm}$ per annihilation. These spectra were taken from PPPC 4 DM
ID resource [9]. The transport equation for DM $e^{\pm}$ was solved in 2D,
which is a proper approximation for a spiral galaxy. All the previous works
employed a deficient 1D (i.e. spherical symmetry) approximation.

Indeed there are significant uncertainties in the emission modeling procedure.
The main sources of uncertainties are DM density and MF distributions.
$e^{\pm}$ propagation parameters are uncertain too, but they play a secondary
role. In order to quantify the uncertainties thoroughly I employed the quite
traditional MIN-MED-MAX paradigm, when MIN and MAX model setups provide
respectively the lowest and highest DM emission intensities and, hence, the
weakest and strongest possible constraints. And MED setup represents some
middle or "average expectation" scenario (but not necessarily the average
intensity). MIN-MED-MAX models were built separately for DM density profile
and MF distribution together with the propagation parameters (MF/prop. with
"/" meaning "and") – i.e. three independent density profiles and three
independent MF/prop. configurations were tested. MF and propagation parameters
were treated jointly in order to reduce a computational heaviness of the task
taking into account a subdominant role of the propagation uncertainties. Thus,
for each discrete WIMP mass and annihilation channel 3 DM density profiles
$\times$ 3 MF/prop. configurations = 9 independent models were computed,
providing a good coverage of the parameter space. 1161 separate GALPROP runs
were made in order to explore all the relevant WIMP masses and annihilation
channels.

In order to set MIN-MED-MAX DM density profiles I conducted a comprehensive
analysis of various profile determinations available in the literature with
especial emphasis to newer works, which emerged in recent years. Several
papers on the subject agree that very cuspy or cored profiles are not favored
for M31. Finally I chose Einasto density profiles from [10] and [11] for MIN
and MAX cases respectively, and NFW profile from [10] for MED. I refer readers
to [6] for the fully-detailed description. Here the description is very brief
due to a paper length limit. Regarding the MF distribution model, it is 2D and
comprises piecewise-linear dependence on the radius $r$ multiplied by the
exponential dependence on the vertical coordinate $z$. MED and MAX MF models
have additional central cusps, which were motivated by MF determination in MW
central region [12] and possible similarity between two galaxies. MF strength
in the central region of M31 has the biggest importance for our work, since
the potential DM signal is very concentrated around the galactic center. I set
the central field values based mainly on the results of [13, 12]. These values
are the following: for MIN – 14 $\mu$G, for MED – 50 $\mu$G and for MAX – 100
$\mu$G. $e^{\pm}$ propagation parameters were modeled based on CR diffusion
studies for both M31 and MW. One important peculiarity in M31 nucleus is that
$e^{\pm}$ there have much faster cooling rate through the inverse Compton
scattering losses due to denser (by $\approx$2 times) radiation field in
comparison with MW nucleus. And the free-free absorption of the synchrotron
emission of interest (at $\nu\gtrsim$ 0.1 GHz) was proved to be negligible.

Putting together all the ingredients described above, DM emission maps and
spectra were computed. Their examples and parameter dependencies can be seen
in [6, sec. IV].

##  3 Derivation of DM constraints from various radio data

M31 images at eight frequencies in the range $\approx$(0.1–10) GHz were
employed for the derivation of constraints. These maps are mainly clean of the
projected discrete sources. On some images even the diffuse thermal emission
was subtracted. The main role (by constraining power) belongs to very new and
sensitive data from LOFAR Two-meter Sky Survey (LoTSS) [14]. Regarding a
choice of the region of interest (ROI) on the sky for derivation of
constraints, the central (i.e. bulge) region with $R\lesssim 3$ kpc was
employed, since it comprises a vast majority of the emission flux due to DM.
Outside of this radius DM emission intensity becomes too low w.r.t. the usual
astrophysical backgrounds. Intrinsically this bulge region was divided to
concentric annuli individually at each frequency according to the
corresponding image resolution. And the observed emission intensity in each
annulus at each frequency entered the likelihood function individually and
independently. Thus the latter has 33 components/multipliers in total. An
important aspect in our model is an absence of any specific assumptions about
the intensity of standard astrophysical emission, i.e. the flat independent
priors were set for it at all frequencies. This essentially means that the
upper limit of the total observed intensity defines the limit for DM emission.
This makes our DM constraints quite robust and model-independent. And our
likelihood function includes also the systematic uncertainties for the
observed intensity besides the statistical uncertainties in the form of map
noise.

After construction of the likelihood, which is a function of DM parameters and
the observed intensities with their uncertainties, it was marginalized over
nuisance parameters in order to get the probability density for DM
annihilation cross section. DM emission intensity is linearly proportional to
the cross section, if other DM parameters are fixed. Thus the limiting
$\langle\sigma v\rangle$ values were numerically calculated for each of 1161
DM models. The results are presented in fig. 1. Overall, we do not see any
drastic differences between the exclusion bands for various channels. A
general trend is that the exclusion lines for the hadronic channels have
smaller slope (i.e. $\langle\sigma v\rangle_{\text{lim}}$ increases with the
mass slower) and wider uncertainty range w.r.t. the leptonic channels.
Considering heavier thermal WIMP annihilating to
$W^{+}W^{-},Z^{0}Z^{0},t\overline{t},hh$, it might manifest itself only in the
case of optimistic configurations of DM density and MF/prop. A typical width
of the whole uncertainty bands in fig. 1 in the vertical direction (i.e., over
$\langle\sigma v\rangle$) slightly exceeds one order of magnitude.

We also naturally would like to have some representative constraints, which
are averaged over the uncertainties. For this purpose the geometric mean of
all nine DM density and MF/prop. models was calculated for each channel as the
fiducial or effective average constraint. The result is shown in fig. 2 for
all the channels. We see that the exclusions for all channels form a
relatively narrow band – narrower than one order of magnitude over the cross
section at any WIMP mass. For $m_{x}\gtrsim 60$ GeV the exclusions for all
channels are mainly confined between those for $b\overline{b}$ and
$\tau^{+}\tau^{-}$. And $\tau^{+}\tau^{-}$ appears to be the least constrained
channel over almost the whole mass range considered. Table 1 lists WIMP mass
limits and their uncertainty ranges for the thermal cross section. The "…"
symbol in the table means that the corresponding exclusion curve does not
reach the thermal level at any WIMP mass; i.e. our setup is not sensitive
enough to probe the corresponding channel, and any WIMP mass is allowed.

Figure 1: The limits on annihilation cross section vs. WIMP mass for various
annihilation channels marked at each panel ($q\equiv u,d,s$). Each panel shows
all nine computed configurations of DM density profiles and MF/prop. The
(almost) horizontal dashed line shows the thermal relic cross section value
(taken from [15]). Figure 2: The fiducial or effective average exclusions for
all the channels together. These exclusions represent the geometric averages
of all nine DM density and MF/prop. models.

Another natural opportunity, which should be taken into account, is the
annihilation through several channels simultaneously. It is easy to prove
algebraically, that in the case of an arbitrary mixture of channels the limit
for cross section is confined between the limits for the least and most
constrained channels in the mixture, when the latter channels are considered
alone. In general, WIMPs may also annihilate directly to photons or neutrinos,
although these channels are highly suppressed in many models [9].
$\chi\chi\rightarrow\gamma\gamma$ channel is probed effectively by gamma-ray
observations [16], which severely exclude the thermal WIMP, up to $m_{x}\sim
10$ TeV!

Table 1: The obtained lower limits on the thermal WIMP mass (in GeV) and their uncertainty ranges. |  | Range due to | Range due to
---|---|---|---
| Fiducial | DM density | DM density
Annihilation | (geometric) | uncertainties | and MF/prop.
channel | average | (MED MF/prop.) | uncertainties
$\chi\chi\rightarrow\tau^{+}\tau^{-}$ | 39 | 27–65 | 18–89
$\chi\chi\rightarrow\mu^{+}\mu^{-}$ | 54 | 41–81 | 27–100
$\chi\chi\rightarrow gg$ | 62 | 36–110 | 19–190
$\chi\chi\rightarrow c\overline{c}$ | 63 | 36–110 | 19–190
$\chi\chi\rightarrow u\overline{u},d\overline{d},s\overline{s}$ | 64 | 37–110 | 20–190
$\chi\chi\rightarrow e^{+}e^{-}$ | 67 | 54–89 | 37–100
$\chi\chi\rightarrow b\overline{b}$ | 72 | 40–120 | 20–210
$\chi\chi\rightarrow W^{+}W^{-}$ | … | …–98 | …–170
$\chi\chi\rightarrow Z^{0}Z^{0}$ | … | …–99 | …–170
$\chi\chi\rightarrow t\overline{t}$ | … | … | …–210
$\chi\chi\rightarrow hh$ | … | … | …–220

##  4 Conclusions and discussion

The robust WIMP annihilation constraints were derived here based on radio
observations of the central (bulge) region of M31 in the wide frequency range
$\approx$(0.1–10) GHz. Our model does not assume anything specific about the
intensities of usual astrophysical emissions in M31. Table 1 summarizes the
thermal WIMP mass lower limits for all annihilation channels. Thus, we may
finally conclude that $m_{x}\gtrsim 20$ GeV represents the hard and channel-
independent limit for the thermal WIMP; i.e. the former holds under any
reasonable assumptions about DM density and MF distributions, and propagation
parameters. $m_{x}\gtrsim 40$ GeV is the fiducial (effective average over the
uncertainties) limit, and $m_{x}\gtrsim 90$ GeV is the most optimistic limit.

It was reported recently in [17], that M31 possess the extended gamma-ray
outer halo, which may originate from annihilating WIMPs with
$m_{x}\approx(45-72)$ GeV ($\chi\chi\rightarrow b\bar{b}$ channel). Our limits
severely constrain this interpretation. The latter requires $\langle\sigma
v\rangle\sim(10^{-26}-10^{-23})$ cm3/s depending on the assumed DM density
distribution. One can see from the corresponding panel of fig. 1 above, that
such high cross section values are majorly excluded even for MIN configuration
of MF/prop.

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$F_{\pm}(q_{1},q_{2})=\left(1\pm\frac{q_{1}}{q_{2}}\cos\theta+\frac{q_{1}^{2}}{4q_{2}^{2}}\right)^{-1/2}.$
(196)

Since the large separation region dominates the integrals, both $q_{1}$ and
$q_{2}$ are large and have approximately same size. Then, expanding $F_{\pm}$
around $q_{1}=q_{2}$ we get

$F_{\pm}(q_{1},q_{2})=f_{\pm}(\theta)+O\left(\frac{q_{1}}{q_{2}}\right)$ (197)

and at the leading order, up to a $\theta$ dependent pre-factor, the spatial
integral reduces to

$J_{3}(g_{3}^{2})\simeq{\cal
V}\int_{0}^{\infty}{\mathrm{d}}q_{1}\,q_{1}^{2}e^{-\frac{2\pi|\alpha_{i}\alpha_{j}|}{g_{3}^{2}\,q_{1}}}\int_{0}^{\infty}{\mathrm{d}}q_{2}\,q_{2}^{2}\,e^{-\frac{2\pi\left(|\alpha_{j}\alpha_{k}|+|\alpha_{i}\alpha_{k}|\right)}{g_{3}^{2}\,q_{2}}}$
(198)

These are in the same form with the integrals we get at 2 instanton level in
Section 3. Thus, we can immediately deduce the partition function from (29) as

$Z_{3}=\frac{\xi^{3}}{3!}\,{\cal V}\left({\cal
V}+I_{1}(g^{2})\right)\left({\cal V}+I_{2}(g^{2})\right)$ (199)

where $I_{1}$ and $I_{2}$ are

$\displaystyle I_{1}(g^{2})$ $\displaystyle=\frac{4\pi
v^{-3}}{6}\left(\frac{2\pi|\alpha_{i}\cdot\alpha_{j}|}{g^{2}}\right)^{3}\left(\ln\Big{(}\frac{2\pi|\alpha_{i}\cdot\alpha_{j}|}{g^{2}}\Big{)}+\gamma-\frac{11}{6}\right),$
(200) $\displaystyle I_{2}(g^{2})$ $\displaystyle=\frac{4\pi
v^{-3}}{6}\left(\frac{2\pi}{g^{2}}\Big{(}|\alpha_{j}\alpha_{k}|+|\alpha_{i}\alpha_{k}|\Big{)}\right)^{3}\Bigg{(}\ln\bigg{[}\frac{2\pi}{g^{2}}\Big{(}|\alpha_{j}\alpha_{k}|+|\alpha_{i}\alpha_{k}|\Big{)}\bigg{]}+\gamma-\frac{11}{6}\Bigg{)}.$
(201)

#### Attractive Interactions:

Now, we turn our attention to the cases with attractive Coulomb potentials. An
example suitable to the our choice of Jacobi coordinates in (191) is the case
which there are 2 monopoles located at $r_{i}$ and $r_{j}$ and 1 anti-monopole
at $r_{k}$. (See Fig. 9.) The associated interaction potential is given as

$V_{\mathrm{int}}=\frac{\pi}{g_{3}^{2}}\left[\frac{|\alpha_{i}\alpha_{j}|}{|{\mathbf{r}}_{i}-{\mathbf{r}}_{j}|}-\frac{|\alpha_{j}\alpha_{k}|}{|{\mathbf{r}}_{k}-{\mathbf{r}}_{j}|}-\frac{|\alpha_{i}\alpha_{k}|}{|{\mathbf{r}}_{k}-{\mathbf{r}}_{i}|}\right]$
(202)

The partition function in this case can be probed by an analytical
continuation of the $q_{2}$ integral in (198) as we did for attractive two
instanton cases. This leads to a two fold ambiguous result in the form of (34)
and we get

$Z_{3}=\frac{\xi^{3}}{3!}\,{\cal V}\left({\cal
V}+I_{1}(g^{2})\right)\left({\cal V}+I_{2}(g^{2})\pm
i\pi\left(\frac{\pi}{g^{2}}\big{(}|\alpha_{j}\alpha_{k}|+|\alpha_{i}\alpha_{k}|\big{)}\right)^{3}\right)$
(203)

Remark: There are other possible sign combinations for the interaction
potential in (202). In general, for any 3 instanton anti-instanton event,
either all the signs are $(+)$ as in (190) or there are 2 $(-)$ sign along
with 1 $(+)$ sign as in (202). In the latter case, the Jacobi coordinates
should be chosen appropriately so that the signs in the exponent of the
$q_{2}$ integral should be the same. Otherwise, it is possible that they could
cancel out and higher-order terms arising from (197) could lead to
$\frac{1}{q^{2}_{2}}$ terms, i.e. dipole interaction. In these cases, the
logarithmic behavior arising from the incomplete gamma function would be
obscured. With appropriate coordinate choices, all of them lead to the same
form with (203) up to modifications in charge-dependent terms.

#### Relation to the Cluster Expansion:

Now let us understand our result in terms of the cluster expansion: First note
that if we consider them separately, both $q_{1}$ and $q_{2}$ integrals in
(198) can be interpreted as 2 body interactions. In this sense, the $q_{1}$
integral, along with the volume prefactor ${\cal V}$, corresponds to the
interaction between the monopoles at ${\mathbf{r}}_{i}$ and
${\mathbf{r}}_{j}$. Its contribution is given as $\xi^{2}{\cal V}\big{(}{\cal
V}+I_{1}(g^{2})\big{)}$, which is in parallel with the 2 instanton events in
Section 3. On the other hand, if we took the $q_{2}$ integral independently,
it would correspond to the radial integration of an interaction between the
monopole at ${\mathbf{r}}_{k}$ and a monopole with a total charge
$Q_{ij}=\frac{2\pi}{g_{3}}\left(\alpha_{i}+\alpha_{j}\right)$. In (198), this
means that the monopole at ${\mathbf{r}}_{k}$ coalesce into the 2 body
interaction of the monopoles at ${\mathbf{r}}_{i}$ and ${\mathbf{r}}_{j}$.

This is indeed how we should relate (198) to the cluster expansion.
Specifically, the contribution of the $q_{1}$ integral consists of 2-cluster
connected and disconnected parts, i.e. $\xi^{2}{\cal V}\big{(}{\cal
V}+I_{1}(g^{2})\big{)}\sim{\cal V}^{2}[{\cal C}_{1}]^{2}+{\cal V}[{\cal
C}_{2}]$, as in 2 instanton case. The contribution of the $q_{2}$, i.e.
$\xi({\cal V}+I_{2}(g^{2}))$, on the other hand, should be interpreted in
terms of the connectedness of the monopole at ${\mathbf{r}}_{k}$ to the two
body interaction represented by the $q_{1}$ integral. Then,

  * •

Volume contribution of $q_{2}$ integral refers that the monopole at
${\mathbf{r}}_{k}$ is not connected to the neither of the other monopoles.
Thus, it contributes as an independent 1-cluster, i.e. ${\cal V}[{\cal
C}_{1}]$, onto the expansion of the $q_{2}$ integral, i.e. ${\cal V}^{2}[{\cal
C}_{1}]^{2}+{\cal V}[{\cal C}_{2}]$.

  * •

$I_{2}(g^{2})$ part refers to the cases that the monopole at $r_{k}$ is
attached to one or both of the other monopoles. Therefore, when $\xi
I_{2}(g^{2})$ combines with the two 1-cluster part $\xi^{2}{\cal V}^{2}$, it
results in a product of 1-cluster and 2-cluster, i.e. ${\cal V}^{2}[{\cal
C}_{1}]^{2}\rightarrow{\cal V}^{2}[{\cal C}_{1}][{\cal C}_{2}]$. Similarly,
the combination of $\xi I_{2}(g^{2})$ with $\xi^{2}{\cal V}I_{1}(g^{2})$ turns
the 2-cluster part of the $q_{1}$ integral into a 3-cluster, i.e. ${\cal
V}[{\cal C}_{2}]\rightarrow{\cal V}[{\cal C}_{3}]$.

This indicates that the 3 instanton partition function in (199) indeed takes
the form of $\xi^{3}$ order terms in the cluster expansion (163):

$Z_{3}=\frac{{\cal V}^{3}}{3!}[{\cal C}_{1}]^{3}+\frac{{\cal V}^{2}}{2!}[{\cal
C}_{1}][{\cal C}_{2}]+\frac{{\cal V}}{1!}[{\cal C}_{3}]$ (204)

with the following identifications:

$\displaystyle\xi^{3}\big{(}I_{1}(g^{2})+I_{2}(g^{2})\big{)}$
$\displaystyle\in[{\cal C}_{1}][{\cal C}_{2}],$ (205)
$\displaystyle\xi^{3}I_{1}(g^{2})I_{2}(g^{2})$ $\displaystyle\in[{\cal
C}_{3}].$ (206)

It is also possible to express (204) as a collection of the monopole events.
First note that as stated in the footnote in page 12, the effective
interaction between ${\mathcal{M}}_{\alpha_{i}}$ and
${\mathcal{M}}_{\alpha_{j}}$ vanishes. Therefore, they do not form 2 clusters
which are restricted as in (164). However, contrary to the 2-cluster case,
${\mathcal{M}}_{\alpha_{i}}$ and ${\mathcal{M}}_{\alpha_{j}}$ can participate
in the same 3-cluster term. Then, we get the following characteristics:

  * •

Terms arising from the combination of 1-cluster events, e.g. $\frac{{\cal
V}^{3}}{3!}[{\mathcal{M}}_{\alpha_{i}}]^{3}$

  * •

Unambiguous 2-cluster terms with an additional 1-cluster term, e.g

${\cal
V}^{3}[{\mathcal{M}}_{\alpha_{j}}][{\mathcal{M}}_{\alpha_{i+1}}][\overline{\mathcal{M}}_{\alpha_{i}}]+{\cal
V}^{2}[{\mathcal{M}}_{\alpha_{j}}][{\mathcal{M}}_{\alpha_{i+1}}\overline{\mathcal{M}}_{\alpha_{i}}]\quad,\quad
j\neq i\pm 1$ (207)

which is a direct extension of the second order term.

  * •

Ambiguous 2-cluster terms with an additional 1-cluster term, e.g

${\cal
V}^{3}[{\mathcal{M}}_{\alpha_{j}}][{\mathcal{M}}_{\alpha_{i}}][\overline{\mathcal{M}}_{\alpha_{i}}]+{\cal
V}^{2}[{\mathcal{M}}_{\alpha_{j}}][{\mathcal{M}}_{\alpha_{i}}\overline{\mathcal{M}}_{\alpha_{i}}]_{\pm}\quad,\quad
j\neq i\pm 1$ (208)

which is a direct extension of the second order term.

  * •

Unambiguous 3-cluster terms, e.g.

${\cal
V}^{3}[{\mathcal{M}}_{\alpha_{i+1}}][{\mathcal{M}}_{\alpha_{i+1}}][\overline{\mathcal{M}}_{\alpha_{i}}]+{\cal
V}^{2}[{\mathcal{M}}_{\alpha_{i+1}}][{\mathcal{M}}_{\alpha_{i+1}}\overline{\mathcal{M}}_{\alpha_{i}}]+{\cal
V}[{\mathcal{M}}_{\alpha_{i+1}}{\mathcal{M}}_{\alpha_{i+1}}\overline{\mathcal{M}}_{\alpha_{i}}]$
(209)

  * •

Ambiguous 3-cluster terms, e.g.

${\cal
V}^{3}[{\mathcal{M}}_{\alpha_{i}}][\overline{\mathcal{M}}_{\alpha_{i}}[{\mathcal{M}}_{\alpha_{i}}]]+{\cal
V}^{2}[{\mathcal{M}}_{\alpha_{i}}][\overline{\mathcal{M}}_{\alpha_{i}}{\mathcal{M}}_{\alpha_{i}}]_{\pm}+{\cal
V}[{\mathcal{M}}_{\alpha_{i}}\overline{\mathcal{M}}_{\alpha_{i}}{\mathcal{M}}_{\alpha_{i}}]_{\pm}$
(210)

Since only $3$-cluster terms $[{\cal C}_{3}]$ contribute to the spectrum at 3
instanton order, the ambiguous imaginary part of the spectrum arises from
$\xi^{3}I_{1}(g^{2})I_{2}(g^{2})$ part of the partition function. Then,
considering events with the ambiguity, we get the imaginary contribution to
spectrum as

$\displaystyle\mathrm{Im}\mathcal{E}^{(3)}$
$\displaystyle=\mathrm{Im}[{\mathcal{M}}\overline{\mathcal{M}}{\mathcal{M}}]_{\pm}\simeq\pm\left(\frac{{\mathrm{s}}_{0}}{g^{2}}\right)^{12}\ln\left(\frac{1}{g^{2}}\right)\,e^{-\frac{12\pi}{g^{2}}}.$
(211)

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