# Non-Bloch band theory in bosonic Bogoliubov–de Gennes systems

Kazuki Yokomizo  Department of Physics, Tokyo Institute of Technology, 2-12-1
Ookayama, Meguro-ku, Tokyo, 152-8551, Japan  Shuichi Murakami  Department of
Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo,
152-8551, Japan  TIES, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-
ku, Tokyo, 152-8551, Japan

###### Abstract

In recent research, it has been shown that non-Hermitian systems exhibit
sensitivity to boundaries, and it is caused by the non-Hermitian skin effect.
In this work, we construct the non-Bloch band theory in bosonic Bogoliubov–de
Gennes (BdG) systems. From our theory, we can calculate the generalized
Brillouin zone and the energy spectrum in such systems with open boundary
conditions in the thermodynamic limit, and we can thus discuss its non-
Hermitian nature, despite Hermiticity of an original Hamiltonian. In fact, we
find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the non-
Hermitian skin effect, such as instability against infinitesimal perturbations
and reentrant behavior, in terms of the non-Bloch band theory. This result
indicates that our theory is powerful tool for studying non-Hermitian nature
in bosonic BdG systems.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

In condensed matter physics, bosons describing low-energy excitations have
been studied in various fields of physics. Such bosons include, for example,
magnons, phonons, and photons. In many bosonic systems, a Hamiltonian is
described in the form of the Bogoliubov–de Gennes (BdG) Hamiltonian. For
example, in a magnon system, the dipolar interaction or the Dzyaloshinskii-
Moriya interaction between spins leads to the bosonic BdG Hamiltonian with
terms not preserving particle numbers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10].

In recent years, non-Hermitian physics of such bosonic BdG systems has been
focused on and vigorously studied. For example, in a cold-atom system, the
previous works [11] experimentally observed the dynamical instability for the
Bose-Einstein condensate caused by the contact interaction between bosons.
From the theoretical perspective, the complex energy eigenvalues in this
system lead to such instability [12]. In general cases, it was shown that when
the bosonic BdG Hamiltonian is not positive definite, the energy eigenvalues
of bosonic BdG systems become complex [13]. Thus non-Hermiticity emerges in
many bosonic BdG systems.

In many previous works, non-Hermiticity in one-dimensional (1D) bosonic BdG
systems has been investigated mainly under periodic boundary conditions. On
the other hand, as long as we focus on a periodic chain, we cannot clarify
non-Hermitian phenomena unique to such systems. For example, exponential
growth of topological edge states occurs because these states can have complex
energies, whereas bulk eigenstates have real energy eigenvalues [14, 15, 16,
17, 18]. Besides, in Ref. [19], it was theoretically proposed that in a
bosonic BdG system, one can realize the non-Hermitian skin effect [20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] in which
bulk states are localized at either edge of an open chain. These novel
phenomena are caused by the existence of boundaries, and hence we should study
non-Hermitian open chains. However, while some previous works studied non-
Hermiticity in finite systems under open boundary conditions by numerical
calculation, non-Hermitian physics in bosonic BdG systems in the thermodynamic
limit has been unrevealed yet.

In our previous works [40, 41, 42], we established the non-Bloch band theory
and studied non-Hermitian free-fermion systems with open boundary conditions
in the limit of a large system size. In this work, we construct the non-Bloch
band theory in bosonic BdG systems. By using our theory, one can investigate
general non-Hermitian properties in bosonic BdG systems with open boundary
conditions in the thermodynamic limit. In this study, we use the bosonic
Kitaev-Majorana chain proposed in Ref. [19]. We find that this system exhibits
rich aspects of the non-Hermitian skin effect, i.e., instability against
infinitesimal perturbations and reentrant behavior. Interestingly, non-
Hermiticity appears despite the Hermiticity of the original bosonic BdG
Hamiltonian, and this behavior is well understood by the non-Bloch band
theory.

This paper is organized as follows. In Sec. II, we review an eigenvalue
problem of a bosonic BdG system in real space and construct the non-Bloch band
theory in such a system. Based on this discussion, in Sec. III, we investigate
non-Hermiticity and show a way to construct the Bogoliubov transformation in
the bosonic Kitaev-Majorana chain. Finally we summarize the contents of this
paper and discuss the applications of our theory to real physical systems in
Sec. IV.

##  II Bosonic Bogoliubov–de Gennes systems

In this section, we construct the non-Bloch band theory in bosonic systems
described by the BdG Hamiltonian. To this end, we review the formulation of
the eigenvalue problem of the bosonic BdG Hamiltonian in real space in the
previous work.

###  II.1 Real-space Bogoliubov–de Gennes Hamiltonian

First of all, we review the formulation of the eigenvalue problem of the
bosonic BdG Hamiltonian in real space [43, 2, 3, 4, 44]. We start with a
Hermitian bosonic BdG Hamiltonian representing 1D bosonic tight-binding
systems. The Hamiltonian in real space is written as

$H=\frac{1}{2}\left({\bm{a}}^{\dagger}~{}{\bm{a}}\right)H_{\rm
BdG}\left(\begin{array}[]{c}{\bm{a}}\\\ {\bm{a}}^{\dagger}\end{array}\right),$
(1)

where
${\bm{a}}=\left(\dots,a_{1,1},\dots,a_{1,q},\dots,a_{L,1},\dots,a_{L,q},\dots\right)$.
We note that a unit cell is composed of $q$ degrees of freedom, and
$a_{j,\sigma}~{}\left(\sigma=1,\dots,q\right)$ represents a bosonic
annihilation operator at the $j$th unit cell. Then the BdG Hamiltonian $H_{\rm
BdG}$ preserves the particle-hole symmetry (PHS) [13] defined as

$\tau_{x}H_{\rm BdG}^{\rm T}\tau_{x}^{-1}=H_{\rm BdG}.$ (2)

Here $\tau_{x}$ and $\tau_{z}$ are defined as

$\displaystyle\tau_{x}=\left(\begin{array}[]{cc}O&\mbox{1}\mbox{l}\\\
\mbox{1}\mbox{l}&O\end{array}\right),~{}\tau_{z}=\left(\begin{array}[]{cc}\mbox{1}\mbox{l}&O\\\
O&-\mbox{1}\mbox{l}\end{array}\right),$ (7)

respectively, where $O$ and 1l are a zero matrix and an identity matrix with
the dimension equal to that of the vector ${\bm{a}}$, respectively. Hence a
matrix form of $H_{\rm BdG}$ is given by [13]

$\displaystyle H_{\rm BdG}=\left(\begin{array}[]{cc}M&\Delta\\\
\Delta^{\dagger}&M^{\rm T}\end{array}\right).$ (10)

In analogy with the fermionic BdG Hamiltonian representing Bardeen-Cooper-
Schrieffer superconductors, the Hermitian matrix $M=M^{\dagger}$ represents
the normal parts, and the symmetric matrix $\Delta=\Delta^{\rm T}$ does
pairing terms. In order to get the energy eigenvalues of the systems, we
should diagonalize $H_{\rm BdG}$ by basis transformation
$\left({\bm{a}}^{\dagger}~{}{\bm{a}}\right)=\left({\bm{\alpha}}^{\dagger}~{}{\bm{\alpha}}\right)T^{\dagger}$,
where ${\bm{\alpha}}$ is another set of bosonic annihilation operators. Here,
since ${\bm{a}}$ and ${\bm{\alpha}}$ satisfy bosonic commutation relations,
$T$ must be a paraunitary matrix [43, 2, 3, 4, 44] defined as

$T^{\dagger}\tau_{z}T=\tau_{z}.$ (11)

Here we can explicitly write the eigenvalue equation of the BdG Hamiltonian
$\tau_{z}H_{\rm BdG}$ as

$\displaystyle\left(\tau_{z}H_{\rm
BdG}\right)\left(\begin{array}[]{cc}U&V^{\ast}\\\
V&U^{\ast}\end{array}\right)=\left(\begin{array}[]{cc}U&V^{\ast}\\\
V&U^{\ast}\end{array}\right)\left(\begin{array}[]{cc}\Lambda&O\\\
O&-\Lambda\end{array}\right),$ (18) (19)

where $\Lambda$ is a diagonal matrix, and

$\displaystyle T=\left(\begin{array}[]{cc}U&V^{\ast}\\\
V&U^{\ast}\end{array}\right)$ (22)

is a paraunitary matrix. This indicates that since $\tau_{z}H_{\rm BdG}$ is
non-Hermitian, some features of non-Hermitian physics may arise, although the
original bosonic BdG Hamiltonian is Hermitian. We note that if $H_{\rm BdG}$
is positive definite, all the eigenenergies of the systems become real, and
the systems can be regarded as Hermitian systems [13]. On the other hand, if
not, the bosonic BdG systems are essentially non-Hermitian, and the
eigenenergies are complex in general. Importantly, in this case, we can get
the energy eigenvalue in an open chain by applying the non-Bloch band theory
to such systems, as we will discuss in this paper.

###  II.2 Non-Bloch band theory

Now we construct the non-Bloch band theory in 1D bosonic BdG systems with open
boundary conditions. In the following, we assume that the ranges of the
hopping in the normal terms $M$ and the pairing terms $\Delta$ are up to
$N_{s}$ and $N_{p}$ unit cells, respectively. Then Eq. (19) can be explicitly
written as

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\sum_{j}\sum_{\tau=1}^{q}\left[\sum_{i=-N_{s}}^{N_{s}}M_{i,\sigma\tau}u_{j+i,\tau}^{\kappa}+\sum_{i=-N_{p}}^{N_{p}}\Delta_{i,\sigma\tau}v_{j+i,\tau}^{\kappa}\right]=E^{\kappa}u_{j,\sigma}^{\kappa}\vspace{8pt}\\\
\displaystyle\sum_{j}\sum_{\tau=1}^{q}\left[-\sum_{i=-N_{p}}^{N_{p}}\Delta_{-i,\tau\sigma}^{\ast}u_{j+i,\tau}^{\kappa}-\sum_{i=-N_{s}}^{N_{s}}M_{-i,\tau\sigma}v_{j+i,\tau}^{\kappa}\right]=E^{\kappa}v_{j,\sigma}^{\kappa}\end{array}\right.~{}\left(\sigma=1,\dots,q\right),$
(25)

where $u_{j,\sigma}^{\kappa}$ and $v_{j,\sigma}^{\kappa}$ are the
$(\left(j,\sigma\right),\kappa)$ components of the matrices $U$ and $V$,
respectively, and $M_{i,\sigma\tau}$ and $\Delta_{i,\sigma\tau}$ are the
$(\sigma,\tau)$ components of the matrices $M$ and $\Delta$ representing the
hopping to the $\left(-i\right)$th nearest unit cell. Here, thanks to spatial
periodicity in the bulk, the eigenvectors
$\left(u_{j,\sigma}^{\kappa},v_{j,\sigma}^{\kappa}\right)$ can be given by the
linear combination as

$\left(u_{j,\sigma}^{\kappa},v_{j,\sigma}^{\kappa}\right)=\sum_{m=1}^{4N}\left(u_{\sigma}^{\kappa,m},v_{\sigma}^{\kappa,m}\right)\left(\beta_{m}\right)^{j}~{}\left(\sigma=1,\dots,q\right),$
(26)

where $N=q\max\left(N_{s},N_{p}\right)$, and $\beta=\beta_{m}$ are the
solutions of the characteristic equation

$\det\left[s_{z}{\cal H}_{\rm BdG}\left(\beta\right)-E\right]=0$ (27)

of the non-Bloch BdG matrix

$\displaystyle\left[{\cal H}_{\rm
BdG}\left(\beta\right)\right]_{\sigma\tau}=\left(\begin{array}[]{cc}\displaystyle\sum_{i=-N_{s}}^{N_{s}}M_{i,\sigma\tau}\beta^{i}&\displaystyle\sum_{i=-N_{p}}^{N_{p}}\Delta_{i,\sigma\tau}\beta^{i}\vspace{8pt}\\\
\displaystyle\sum_{i=-N_{p}}^{N_{p}}\Delta_{-i,\tau\sigma}^{\ast}\beta^{i}&\displaystyle\sum_{i=-N_{s}}^{N_{s}}M_{-i,\tau\sigma}\beta^{i}\end{array}\right),$
(30) (31)

where $s_{z}$ is the $2q\times 2q$ matrix expressed as $s_{z}={\rm
diag}\left(\mbox{1}\mbox{l},-\mbox{1}\mbox{l}\right)$. We note that the
characteristic equation (27) is an algebraic equation for $\beta$ of degree
$4N$. From the non-Bloch band theory [40, 41, 42], we show that the continuum
energy band in a long open chain can be obtained from the condition

$\left|\beta_{2N}\right|=\left|\beta_{2N+1}\right|,$ (32)

where the solutions of Eq. (27) are ordered in the following way:

$\left|\beta_{1}\right|\leq\left|\beta_{2}\right|\leq\cdots\leq\left|\beta_{4N-1}\right|\leq\left|\beta_{4N}\right|.$
(33)

Then, according to the non-Bloch band theory, the trajectories of $\beta_{2N}$
and $\beta_{2N+1}$ on the complex plane give the generalized Brillouin zone
(GBZ) for the complex wave number $\beta={\rm e}^{ik},~{}k\in{\mathbb{C}}$. In
particular, one can show $\left|\beta_{2N}\right|=\left|\beta_{2N+1}\right|=1$
in Hermitian systems, leading to the conventional Brillouin zone,
$k\in{\mathbb{R}}$. We note that in free-fermion systems, the previous works
[40, 28, 30, 41, 42, 31, 45] found various features of the GBZ unique to non-
Hermitian systems. We expect that the bosonic BdG systems inherit such
features of the GBZ.

Thus Eq. (26) indicates that the paraunitary matrix $T$ can be given by the
linear combination of the terms
$\left(\beta_{m}\right)^{j}~{}\left(m=1,\dots,4N\right)$. In fact, since from
the basis transformation
$\left({\bm{a}}^{\dagger}~{}{\bm{a}}\right)=\left({\bm{\alpha}}^{\dagger}~{}{\bm{\alpha}}\right)T^{\dagger}$,
we can get

${\bm{\alpha}}^{\dagger}={\bm{a}}^{\dagger}U-{\bm{a}}V,$ (34)

the quasiparticle-creation operator $\alpha^{\dagger}_{\kappa,\sigma}$ can be
explicitly written as

$\displaystyle\alpha_{\kappa,\sigma}^{\dagger}$ $\displaystyle=$
$\displaystyle\sum_{j}\left(u_{j,\sigma}^{\kappa}a_{j,\sigma}^{\dagger}-v_{j,\sigma}^{\kappa}a_{j,\sigma}\right)$
$\displaystyle=$
$\displaystyle\sum_{j}\sum_{m=1}^{4N}\left[u_{\sigma}^{\kappa,m}\left(\beta_{m}\right)^{j}a_{j,\sigma}^{\dagger}-v_{\sigma}^{\kappa,m}\left(\beta_{m}\right)^{j}a_{j,\sigma}\right]$

for $\sigma=1,\dots,q$. It is important that the coefficients
$u_{\sigma}^{\kappa,m}$ and $v_{\sigma}^{\kappa,m}$ are determined by the open
boundary conditions given as

$\alpha_{\kappa,\sigma}\left(j=0\right)=\alpha_{\kappa,\sigma}\left(j=L+1\right)=0$
(36)

and by the condition that the operators $\alpha_{\kappa,\sigma}$ and
$\alpha_{\kappa,\sigma}^{\dagger}$ satisfy the boson statistics given as

$\left[\alpha_{\kappa,\sigma},\alpha_{\kappa^{\prime},\tau}^{\dagger}\right]=\delta_{\kappa,\kappa^{\prime}}\delta_{\sigma,\tau}.$
(37)

Therefore, from Eqs. (LABEL:eq15)–(37), we can get the Bogoliubov
transformation diagonalizing the BdG Hamiltonian in terms of the non-Bloch
band theory.

##  III Bosonic Kitaev-Majorana chain

In this section, we investigate the bosonic Kitaev-Majorana chain proposed in
Ref. [19]. While the occurrence of the non-Hermitian skin effect in this
system was proposed in the previous work, we show that the non-Hermitian skin
effect is fragile against infinitesimal perturbations that couple between two
skin modes. Furthermore, in a special case, we can derive the analytical
representation of the Bogoliubov transformation from Eq. (LABEL:eq15).

###  III.1 Non-Hermitian property

First of all, we start with the real-space Hamiltonian of the bosonic Kitaev-
Majorana chain. It is given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{j}\left[\mu
a_{j}^{\dagger}a_{j}+\frac{t}{2}\left(e^{i\phi}a_{j+1}^{\dagger}a_{j}+e^{-i\phi}a_{j}^{\dagger}a_{j+1}\right)\right.$
(38)
$\displaystyle\left.+\frac{i\Delta}{2}\left(a_{j+1}^{\dagger}a_{j}^{\dagger}-a_{j}a_{j+1}\right)\right],$

where $a_{j}$ is a bosonic annihilation operator at the $j$th site, and all
the parameters are set to be positive real numbers, for simplicity. This model
corresponds to the case of $q=N=1$ in Sec. II. Remarkably, although
Hamiltonian (38) is Hermitian, it exhibits non-Hermitian physics as we will
discuss later. We note that the previous work only studied non-Hermitian
properties in this model with the case of $\mu=0$ and $\phi=\pi/2$. In the
following, when $\phi=0$ and $\phi=\pi/2$, we only focus on the case of
$t>\Delta$ because the system is dynamically unstable if $t<\Delta$ [19].

The non-Bloch BdG matrix for Eq. (38) is written as

$\displaystyle{\cal H}_{\rm BdG}\left(\beta\right)$ $\displaystyle=$
$\displaystyle\left[\mu+\frac{t}{2}\cos\phi\left(\beta+\beta^{-1}\right)\right]\sigma_{0}-\frac{\Delta}{2}\left(\beta+\beta^{-1}\right)\sigma_{y}$
(39)
$\displaystyle-\frac{it}{2}\sin\phi\left(\beta-\beta^{-1}\right)\sigma_{z},$

where $\beta=e^{ik},~{}k\in{\mathbb{C}}$, $\sigma_{0}$ is a $2\times 2$
identity matrix, and $\sigma_{i}~{}\left(i=x,y,z\right)$ are the Pauli
matrices. As mentioned in Sec. II, we can get the GBZ and the continuum energy
bands by the characteristic equation

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\det\left[\sigma_{z}{\cal
H}_{\rm BdG}\left(\beta\right)-E\right]$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\Delta^{2}-t^{2}\right)\left(\beta^{2}+\beta^{-2}\right)-\mu^{2}+E^{2}+\frac{1}{2}\left(\Delta^{2}-t^{2}\cos
2\phi\right)$ $\displaystyle-\left(\mu t\cos\phi-
itE\sin\phi\right)\beta-\left(\mu t\cos\phi+itE\sin\phi\right)\beta^{-1}$

by applying condition (32), i.e.,
$\left|\beta_{2}\right|=\left|\beta_{3}\right|$. The resulting GBZ is not a
unit circle in the complex $\beta$ plane in general, leading to non-Hermitian
physics in this system. Importantly, since this system is intrinsically non-
Hermitian, it exhibits the non-Hermitian skin effect.

When $\mu=0$, it is shown that $\sigma_{z}{\cal H}_{\rm
BdG}\left(\beta\right)$ can be transformed into a block-diagonal matrix form
as

$\displaystyle P\left(\sigma_{z}{\cal H}_{\rm
BdG}\left(\beta\right)\right)P^{-1}=\left(\begin{array}[]{cc}{\cal
H}_{+}\left(\beta\right)&O\\\ O&{\cal
H}_{-}\left(\beta\right)\end{array}\right)$ (43)

by the similarity transformation given by

$\displaystyle
P=\frac{1}{\sqrt{2C\left(C+t\cos\phi\right)}}\left(\begin{array}[]{cc}C+t\cos\phi&-i\Delta\\\
i\Delta&C+t\cos\phi\end{array}\right),$ (46) (47)

where $C=\sqrt{t^{2}\cos^{2}\phi-\Delta^{2}}$. Then the diagonal elements of
Eq. (43) are given by

${\cal
H}_{\pm}\left(\beta\right)=\frac{t}{2i}\sin\phi\left(\beta-\beta^{-1}\right)\pm\frac{1}{2}\sqrt{t^{2}\cos^{2}\phi-\Delta^{2}}\left(\beta+\beta^{-1}\right).$
(48)

In other words, Eq. (LABEL:eq20) can be factorized into two irreducible
algebraic equations for $\beta$ and $E$. Therefore, in this case, this two-
band model is decoupled into two systems described by ${\cal
H}_{\pm}\left(\beta\right)$, which are Hatano-Nelson models without disorder
[46]. We note that the condition for continuum energy bands is also split for
each band. Namely, they are given by
$\left|\beta_{1}^{\pm}\right|=\left|\beta_{2}^{\pm}\right|$ for each band,
where $\beta_{1,2}^{\pm}$ are the solutions of the equation $\det\left[{\cal
H}_{\pm}\left(\beta\right)-E\right]=0$, respectively.

Before going to the discussion of non-Hermitian properties in this system, let
us introduce the energy winding number [23, 30], which tells us the
topological origin of the non-Hermitian skin effect. This topological
invariant in a non-Hermitian system with periodic boundary conditions can be
defined as

$W\left(E\right)=\frac{1}{2\pi i}\int_{0}^{2\pi}{\rm d}k\frac{{\rm d}}{{\rm
d}k}\log\det\left[H\left(k\right)-E\right],$ (49)

where $k$ is the conventional Bloch wave number, and $H\left(k\right)$ is the
conventional Bloch Hamiltonian written as $H\left(k\right)\equiv{\cal
H}\left(\beta=e^{ik}\right),~{}k\in{\mathbb{R}}$ from our non-Bloch matrix
${\cal H}\left(\beta\right)$. For a given value of the reference energy
$E\in{\mathbb{C}}$, when the energy winding number $W\left(E\right)$ takes
nonzero values, the non-Hermitian skin effect occurs as a manifestation of
non-Hermitian topology, and furthermore, there exists a difference between
energy eigenvalues in a periodic chain and those in an open chain. In other
words, when these energy eigenvalues are identical, $W\left(E\right)=0$ for
any reference energies, and the non-Hermitian skin effect does not occur.
Thus, by using Eq. (49), we can investigate the behavior of non-Hermitian
systems. In the following, we discuss the physics of the model for three
different cases of the value of the parameter $\phi$.

####  III.1.1 Case I: $\phi=0$

In the case of $\mu=0$, the energy winding number (49) for each band is always
zero because from Eq. (48), the energy eigenvalues take either of the real
values $\pm\sqrt{t^{2}-\Delta^{2}}\cos k$ for $k\in{\mathbb{R}}$. Furthermore,
Eq. (49) of the system is also zero in the case of $\mu\neq 0$. Therefore, the
non-Hermitian skin effect does not occur, and the resulting GBZ is always a
unit circle, identical with Hermitian cases.

####  III.1.2 Case II: $\phi=\pi/2$

First of all, we begin with the case of $\mu=0$. Then, from Eq. (48), the
energy winding number (49) for each band with ${\cal
H}_{\pm}\left(\beta\right)$ is $\pm 1$, respectively, for any values of the
reference energy within the region surrounded by an ellipse $E=t\sin
k+i\Delta\cos k,~{}k\in{\mathbb{R}}$, which is the energy eigenvalue for a
periodic chain, shown in black in Fig. 1(a). Therefore, the non-Hermitian skin
effect occurs in an open chain. In fact, we can confirm that the energy
eigenvalue in an open chain completely differs from that in a periodic chain
as shown in Fig. 1(a), and the GBZ for each band is not a unit circle but a
circle with the radius

$\left|\beta^{+}\right|=\sqrt{\frac{t+\Delta}{t-\Delta}},~{}\left|\beta^{-}\right|=\sqrt{\frac{t-\Delta}{t+\Delta}},$
(50)

for the bands with ${\cal H}_{\pm}\left(\beta\right)$, respectively, as shown
in Fig. 1(b). It means that the eigenstates from ${\cal
H}_{+}\left(\beta\right)$ and those from ${\cal H}_{-}\left(\beta\right)$ are
localized at the opposite ends of an open chain.

While the non-Hermitian skin effect appears when $\mu=0$, it becomes unstable
against perturbations by the infinitesimal values of the parameter $\mu$ [47,
32]. When $\mu$ becomes nonzero, the matrix $\sigma_{z}{\cal H}_{\rm
BdG}\left(\beta\right)$ cannot be written as the block-diagonal matrix form
(43), which means that the two localized eigenstates couple with each other.
Namely, this perturbation couples the two bands with the energy winding number
$W\left(E\right)$ equal to $\pm 1$, leading to a trivial value of
$W\left(E\right)$, and thus the non-Hermitian skin effect is immediately
suppressed. As a result, the GBZ becomes a unit circle when $\mu\neq 0$ as
shown in Fig. 1(b), which means that the Bloch wave number takes real values,
and the energy spectrum in an open chain coincides with that in a periodic
chain. The transition from the nontrivial phase to the trivial phase
discontinuously occurs as $\mu$ becomes nonzero. On the other hand, in a
finite open chain, the energy levels continuously approach those in a finite
periodic chain as the value of $\mu$ continuously increases, with a rapid
change around the critical value $\mu_{0}$, as shown in Fig. 1(c). Here, from
Eq. (50), the order of $\mu_{0}$ is expected to be [47]

$\mu_{0}/t\simeq{\cal O}\left(\left|\beta^{-}\right|^{L}\right),$ (51)

where $L$ is the system size. For example, when $t=1,\Delta=0.7$, and $L=50$
adopted in Fig. 1(c), we get $\mu_{0}\simeq 10^{-12}$, and indeed, one can
confirm this value of $\mu_{0}$ in a finite open chain; the energy levels with
$\mu=10^{-15}<\mu_{0}$ largely deviate from those with $\mu=10^{-8}$ and
$10^{-3}$ exceeding $\mu_{0}$. Thus the non-Hermitian skin effect exhibits
instability against infinitesimal values of perturbation by $\mu$.

Figure 1: Eigenenergies and generalized Brillouin zone (GBZ) in the bosonic
Kitaev-Majorana chain with various values of the parameter $\mu$ in the case
of $\phi=\pi/2$. We set the parameters to be $t=1$ and $\Delta=0.7$. (a)
Energy eigenvalues in an open chain (red) and in a periodic chain (black) with
$\mu=0$. (b) Generalized Brillouin zone in the case of $\mu=0$ (green and
blue) and of $\mu\neq 0$ (black). The green (blue) circle describes the GBZ
for the band with ${\cal H}_{+}\left(\beta\right)$ $\left[{\cal
H}_{-}\left(\beta\right)\right]$ and has the radius $\left|\beta^{+}\right|$
($\left|\beta^{-}\right|$). (c) Energy levels in a finite periodic chain and
in a finite open chain with $\mu=10^{-15},10^{-8}$, and $10^{-3}$. We set the
system size to be $L=50$. In the case of $L=50$, the critical value can be
obtained as $\mu_{0}\simeq 10^{-12}$. We note that the energy levels for a
periodic chain with $\mu=10^{-15},10^{-8}$, and $10^{-3}$ almost overlap in
the figure.

####  III.1.3 Case III: $\phi\neq 0,\pi/2$

Figure 2: Eigenenergies and generalized Brillouin zone in the bosonic Kitaev-
Majorana chain with various values of the parameter $\mu$ in the case of
$\phi=\pi/3$. We set the parameters to be $t=1$. (a1–a6) Continuum energy
bands in an open chain (red) and the energy spectrum in a periodic chain
(black). In (a3) and (a6), the region in green (blue) represents that the
energy winding number $W\left(E\right)$ takes $+1$ ($-1$). (b1–b6) Generalized
Brillouin zone with the colored lines. The black broken line expresses a unit
circle, meaning the conventional Brillouin zone. (c1–c3) Schematic figures of
the reentrant behavior of the energy winding number $W\left(E\right)$ through
an increase of the value of the parameter $\mu$. At (c1) $\mu=0$, the two
bands with $W\left(E\right)=\pm 1$ are decoupled. When we increase $\mu$, they
couple with each other, leading to $W\left(E\right)=0$ as shown in (c2). A
further increase of $\mu$ leads to nonzero values of $W\left(E\right)$ because
of the deviation between two energy spectra for the two bands in a periodic
chain as shown in (c3). The region in green (blue) on the complex energy plane
describes the region with $W\left(E\right)$ being $1$ ($-1$).

First of all, when $\mu=0$, whether the non-Hermitian skin effect occurs or
not depends on the values of the system parameters. When
$t\left|\cos\phi\right|>\Delta$, the non-Hermitian skin effect disappears
because the energy winding number (49) becomes zero for any reference
energies. This is because the energy eigenvalues for $k\in{\mathbb{R}}$ take
the real values $t\sin\phi\sin k\pm\sqrt{t^{2}\cos^{2}\phi-\Delta^{2}}\cos k$.
On the other hand, when $t\left|\cos\phi\right|<\Delta$, the system exhibits
the non-Hermitian skin effect. In fact, the above discussion is consistent
with the numerical results as shown in Figs. 2(a1) and 2(b1) and in Figs.
2(a4) and 2(b4).

Next we focus on the case of $\mu\neq 0$. When $\mu$ is a positive
infinitesimal, $\mu=+0$, the GBZ is a unit circle, regardless of the value of
$\Delta$. It is seen in Figs. 2(b2) and 2(b5) for $\mu=0.01$, where we note
that because $\mu$ is small but finite, the GBZ slightly deviates from a unit
circle, but this deviation is tiny when $\mu\ll{\cal O}\left(t,\Delta\right)$.
Accordingly, the eigenenergies in an open chain are almost identical with
those in a periodic chain, shown in Figs. 2(a2) and 2(a5), and the non-
Hermitian skin effect is almost absent. As the value of $\mu$ increases, the
deviation of the GBZ from a unit circle becomes prominent [Figs. 2(b3) and
2(b6)]. Therefore, in particular, when $t\left|\cos\phi\right|<\Delta$, as the
value of $\mu$ increases, the non-Hermitian skin effect disappears once and it
reoccurs as shown in Figs. 2(a4)–2(a6) and (b4)–2(b6). In this sense, we call
this phenomenon reentrant behavior. It is noted that the reappearance of the
deviation of the GBZ and that of the non-Hermitian skin effect occurs in
$\mu>0$ as a crossover, and not a phase transition.

We note that the GBZ in Figs. 2(b3) and 2(b6) partially overlaps a unit circle
shown in a black broken line. This means that the Bloch wave number takes real
values because the BdG Hamiltonian (38) is positive definite in some regions
on the complex $\beta$ plane. On the other hand, the positive definiteness of
Eq. (38) is partially broken in the other regions, and the shape of the GBZ
deviates from a unit circle in such regions. Thus the Hermiticity and the non-
Hermiticity can coexist in a set of the system parameters.

Now we explain the reason for the reentrant behavior of the non-Hermitian skin
effect upon changing the value of $\mu$ as shown in Figs. 2(c1)–2(c3). As
mentioned above, when $\mu=0$, this system can be regarded as two decoupled
Hatano-Nelson models [Fig. 2(c1)]. In this case, for any values of the
reference energy $E$ in the region surrounded by the energy band, the values
of the energy winding number $W\left(E\right)$ are equal to $\pm 1$ for the
two decoupled bands, leading to one skin mode each. On the other hand, for the
infinitesimal values of $\mu$, the two bands couple with each other, and the
total energy winding number in this system is merely summed over the two bands
to approximately become zero [Fig. 2(c2))]. Therefore the non-Hermitian skin
effect almost disappears, as we have already seen in Figs. 2(a5) and 2(b5).
This disappearance gradually occurs when the system size is not large, and it
becomes sharp at $\mu$ being infinitesimal in the limit of a large system
size. Furthermore, as the value of $\mu$ increases, the splitting of the two
bands becomes significant, and the region with $W\left(E\right)\neq 0$ on the
complex energy plane appears due to the deviation of the two bands [Fig.
2(c3)]. Therefore, these regions give rise to the two skin modes localized at
both ends of an open chain. We note that in the case of $\phi=\pi/2$, this
deviation does not occur because the two bands are accidentally degenerate for
any values of $\mu$.

Finally, in order to see how the non-Hermitian skin effect appears in real
space, we show the numerical result of the real-space distribution of the
coefficients $u_{j}^{\kappa}$ and $v_{j}^{\kappa}$ included in the Bogoliubov
transformation with $t=1,\Delta=0.7,\phi=\pi/3$, and $\mu=0.01$ adopted in
Figs. 2(a5) and 2(b5), and $\mu=0.1$ adopted in Figs. 2(a6) and 2(b6). When
$\mu=0.01$, the non-Hermitian skin effect almost does not occur, and their
distribution is almost uniform [Fig. 3(a)]. In contrast, when $\mu=0.1$, the
components of $u_{j}^{\kappa}$ and $v_{j}^{\kappa}$ at either end of an open
chain have larger values than any other sites [Fig. 3(b)]. This indicates that
the non-Hermitian skin effect reoccurs as the values of $\mu$ increase from
$\mu=0.01$, and therefore, this numerical result is consistent with the above
discussion.

Figure 3:  Real-space distribution of the coefficients $u_{j}^{\kappa}$ and
$v_{j}^{\kappa}$ for $\kappa=1,\dots,50$ in the Bogoliubov transformation in a
finite open chain with $t=1,\Delta=0.7,\phi=\pi/3$, and (a) $\mu=0.01$ and (b)
$\mu=0.1$, respectively. We set the system size to be $L=50$.

###  III.2 Analytical representation of Bogoliubov transformation

In this section, we study the Bogoliubov transformation in a special case,
i.e., $\mu=0$ and $\phi=\pi/2$. For simplicity, we assume $t>\Delta$ in the
following. In this case, from Eq. (50), we can get an analytical form of the
Bogoliubov transformation in a finite open chain with the system size $L$.
From Eq. (48), the values of $\beta_{m}^{\pm}~{}(m=1,2)$ are explicitly
written as

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\beta_{1}^{+}=i\sqrt{\frac{t+\Delta}{t-\Delta}}e^{ik},~{}\beta_{2}^{+}=i\sqrt{\displaystyle\frac{t+\Delta}{t-\Delta}}e^{-ik},\vspace{8pt}\\\
\displaystyle\beta_{1}^{-}=i\sqrt{\frac{t-\Delta}{t+\Delta}}e^{ik},~{}\beta_{2}^{-}=i\sqrt{\displaystyle\frac{t-\Delta}{t+\Delta}}e^{-ik},\end{array}\right.$
(54)

where $k\in{\mathbb{R}}$. We note that theses complex Bloch wave numbers form
the continuum energy band as $E=\sqrt{t^{2}-\Delta^{2}}\cos k$. Then we can
determine the coefficients included in Eq. (LABEL:eq15) so as to satisfy
conditions (36) and (37) and can get the analytical representation of the
quasiparticle-creation operator $\alpha_{n}^{\dagger}$ as

$\displaystyle\alpha_{n}^{\dagger}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{L}\sum_{m=1}^{4}\left[u^{\kappa,m}\left(\beta_{m}\right)^{j}a_{j}^{\dagger}-v^{\kappa,m}\left(\beta_{m}\right)^{j}a_{j}\right]$
(55) $\displaystyle=$ $\displaystyle\frac{1}{2i\sqrt{2\left(L+1\right)}}$
$\displaystyle\times\sum_{j=1}^{L}\left\\{\left[\left(\beta_{1}^{+}\right)^{j}-\left(\beta_{2}^{+}\right)^{j}+\left(\beta_{1}^{-}\right)^{j}-\left(\beta_{2}^{-}\right)^{j}\right]a_{j}\right.$
$\displaystyle\left.-\left[\left(\beta_{1}^{+}\right)^{j}-\left(\beta_{2}^{+}\right)^{j}-\left(\beta_{1}^{-}\right)^{j}+\left(\beta_{2}^{-}\right)^{j}\right]a_{j}^{\dagger}\right\\}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2}{L+1}}\sum_{j=1}^{L}i^{j}\left[\sin\left(k_{n}j\right)\cosh\left(rj\right)a_{j}^{\dagger}\right.$
$\displaystyle\left.-\sin\left(k_{n}j\right)\sinh\left(rj\right)a_{j}\right],$

where

$e^{2r}=\frac{t+\Delta}{t-\Delta},$ (56)

and $k_{n}=n\pi/\left(L+1\right)~{}(n=1,\dots,L)$. We note that these results
obtained here agree with those in the previous work [19]. We can
systematically get the Bogoliubov transformation in terms of the non-Bloch
band theory in general cases. Thus the non-Bloch band theory proves to be a
powerful tool in the study of bosonic BdG systems.

##  IV Summary

In our work, we construct the non-Bloch band theory and study non-Hermitian
nature in bosonic BdG systems. In such bosonic systems, although the original
BdG Hamiltonian is Hermitian, the systems are intrinsically non-Hermitian.
This is because the BdG Hamiltonian is locally a Hermitian operator but not a
self-adjoint operator due to boundary conditions. In other words, since
boundary conditions break the self-adjoint nature of the operator, eigenvalues
of the BdG Hamiltonian can take complex values.

In fact, we analyze the non-Hermitian properties of the bosonic Kitaev-
Majorana chain (38) and find that this system exhibits rich aspects of the
non-Hermitian skin effect with various values of the system parameters. Here
we emphasize that one can analyze this non-Hermiticity only via the non-Bloch
band theory. In this theory, in terms of the complex wave number
$\beta=e^{ik},~{}k\in{\mathbb{C}}$, we can transform the matrix
$\sigma_{z}{\cal H}_{\rm BdG}\left(\beta\right)$ into the block-diagonal form
(43) when $\mu=0$. Furthermore, based on this theory, interestingly, this
model with $\phi\neq 0,\pi/2$ exhibits the reentrant behavior of the non-
Hermitian skin effect when $\mu\neq 0$, which has not been found. It is a
promising direction to observe this reentrant phenomenon in experiments, such
as the Bose-Einstein condensate in cold-atom systems [11, 48] and in magnon
systems [49].

We comment on a bulk-edge correspondence between a topological invariant and
existence of topological edge states in bosonic BdG systems. Since the bosonic
Kitaev-Majorana chain is classified as class A in Altland-Zirnbauer classes,
it is topologically trivial, and it does not show the conventional bulk-edge
correspondence.

When the BdG Hamiltonian is a positive-definite matrix, the system can be
regarded as a Hermitian system, and the Hamiltonian can be diagonalized by
using the Cholesky decomposition [43]. It is exemplified by the bosonic
Kitaev-Majorana chain with a sufficiently large value of the parameter $\mu$.
In this case, energy spectra of an open chain and a periodic chain are
asymptotically the same in the limit of the large system. On the other hand,
if not, the system is essentially a non-Hermitian system, and the calculation
method by using the Cholesky decomposition cannot be applied to this non-
Hermitian system. In this sense, our theory is a powerful tool for studying
non-Hermitian nature in bosonic BdG systems with open boundary conditions.

Finally, we discuss experimental applications of our theory established here.
So far, many previous works proposed that various bosonic systems can be
described by the BdG Hamiltonian, and in such systems, the non-Hermitian
physics has intensively been studied, such as the dynamical instability in
cold-atom systems [14, 15, 16, 17, 18]. Besides, the non-Hermitian nature of
magnon systems has been also focused on in recent years [50]. Hence, in these
systems, we expect experimental observations of localization states induced by
the non-Hermitian skin effect. Furthermore, through the observation of the
skin modes, we can also confirm that the energy winding number (49) takes
nonzero values. We note that Eq. (49) cannot be associated with measurable
quantities, such as transport coefficients. Therefore, the nonzero energy
winding number can be indirectly observed through presence or absence of bulk
skin modes and through observations of a complex energy spectrum in a periodic
chain [51].

Thus bosonic BdG systems are useful for investigating some non-Hermitian
properties. In our work, we show that non-Hermitian phenomena in bosonic BdG
systems are accessible via the non-Bloch band theory, and it paves a way to an
implementation of the non-Hermitian phenomenon, such as the non-Hermitian skin
effect, in many bosonic BdG systems.

###### Acknowledgements.

We are grateful to Ryo Okugawa for valuable discussion. This work was
supported by JSPS KAKENHI (Grant No. JP18H03678) and by the MEXT Elements
Strategy Initiative to Form Core Research Center (TIES) (Grant No.
JPMXP0112101001). K.Y. was also supported by JSPS KAKENHI Grant No.
JP18J22113.

##  Appendix A Particle-hole symmetry of bosonic Bogoliubov–de Gennes
Hamiltonian

In this Appendix, we prove that the BdG Hamiltonian $H_{\rm BdG}$ has the PHS
defined in Eq. (2). We can rewrite the form of the real-space Hamiltonian (1)
as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\bm{a}}~{}{\bm{a}}^{\dagger}\right)\left(\tau_{x}H_{\rm
BdG}\tau_{x}^{-1}\right)\left(\begin{array}[]{c}{\bm{a}}^{\dagger}\\\
{\bm{a}}\end{array}\right)$ (59) $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\bm{a}}^{\dagger}~{}{\bm{a}}\right)\left(\tau_{x}H_{\rm
BdG}\tau_{x}^{-1}\right)^{\rm T}\left(\begin{array}[]{c}{\bm{a}}\\\
{\bm{a}}^{\dagger}\end{array}\right)-\frac{1}{2}{\rm Tr}\left(\tau_{z}H_{\rm
BdG}\right)$ (62) $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\bm{a}}^{\dagger}~{}{\bm{a}}\right)\left(\tau_{x}H_{\rm
BdG}^{\rm T}\tau_{x}^{-1}\right)\left(\begin{array}[]{c}{\bm{a}}\\\
{\bm{a}}^{\dagger}\end{array}\right)-\frac{1}{2}{\rm Tr}\left(\tau_{z}H_{\rm
BdG}\right),$ (65)

where $\tau_{x}$ and $\tau_{z}$ are defined in Eq. (7). Since Eq.
(LABEL:eqappA1) should be equal to Eq. (1), we can obtain Eq. (2).

##  Appendix B Bogoliubov transformation

In this Appendix, we show a way to construct the Bogoliubov transformation to
get the energy eigenvalues in bosonic BdG systems. In this case, we must
diagonalize the real-space Hamiltonian (1) by the basis transformation
$\left({\bm{a}}^{\dagger}~{}{\bm{a}}\right)=\left({\bm{\alpha}}^{\dagger}~{}{\bm{\alpha}}\right)T^{\dagger}$,
where $T$ is the paraunitary matrix defined in Eq. (11). Then Eq. (1) is
diagonalized as

$\displaystyle
H=\frac{1}{2}\left({\bm{\alpha}}^{\dagger}~{}{\bm{\alpha}}\right)\left(\begin{array}[]{cc}\Lambda&O\\\
O&\Lambda\end{array}\right)\left(\begin{array}[]{c}{\bm{\alpha}}\\\
{\bm{\alpha}}^{\dagger}\end{array}\right),$ (71)

where

$\displaystyle T^{-1}\left(\tau_{z}H_{\rm
BdG}\right)T=\left(\begin{array}[]{cc}\Lambda&O\\\
O&-\Lambda\end{array}\right),$ (74)

and $\Lambda$ is a diagonal matrix, and $\tau_{z}$ is given in Eq. (7). We
note that the columns of $T$ are the right eigenvectors of $\tau_{z}H_{\rm
BdG}$. On the other hand, since $\tau_{z}H_{\rm BdG}$ has the pseudo-
Hermiticity [13] defined as

$\tau_{z}\left(\tau_{z}H_{\rm BdG}\right)^{\dagger}\tau_{z}=\tau_{z}H_{\rm
BdG},$ (75)

the rows of $T^{\dagger}\tau_{z}$ are the left eigenvectors of $\tau_{z}H_{\rm
BdG}$. Hence, if a set of eigenvectors of $\tau_{z}H_{\rm BdG}$ forms the
biorthogonal basis [52], we can get the paraunitary matrix $T$. In other
words, the constitution of the biorthogonal basis is equivalent to the
condition (11).

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