# Bosonic Andreev bound state

Nobuyuki Okuma<EMAIL_ADDRESS>Graduate School of
Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan

###### Abstract

A general free bosonic system with a pairing term is described by a bosonic
Bogoliubov-de Gennes (BdG) Hamiltonian. The representation is given by a
pseudo-Hermitian matrix, which is crucially different from the Hermitian
representation of a fermionic BdG Hamiltonian. In fermionic BdG systems, a
topological invariant of the whole particle (hole) bands can be nontrivial,
which characterizes the Andreev bound states (ABS) including Majorana
fermions. In bosonic cases, on the other hand, the corresponding topological
invariant is thought to be trivial owing to the stability condition of the
bosonic ground state. In this Letter, we consider a two-dimensional model that
realizes a bosonic analogy of the ABS. The boundary states of this model are
located outside the bulk bands and are characterized by a nontrivial Berry
phase (or polarization) of the hole band. Furthermore, we investigate the
zero-energy flat-band limit in which the Bloch Hamiltonian is defective, where
the particle and hole states are identical to each other. In this limit, the
Berry phase is $\mathbb{Z}_{2}$ quantized thanks to an emergent parity-time
symmetry. This is an example of a topological invariant that uses the
defective nature as a projection structure. Thus, boundary states in our model
are essentially different from Hermitian topological modes and their variants.

The bosonic excitations from a Bose-Einstein condensate are well described by
a quadratic Hamiltonian called bosonic Bogoliubov-de Gennes (BdG) Hamiltonian
Bogoliubov (1947); Colpa (1978); Kawaguchi and Ueda (2012); Altland and Simons
(2010). As well as the systems that consist of bosons such as photons Ozawa
_et al._ (2019); Braunstein and Van Loock (2005) and bosonic atoms Kawaguchi
and Ueda (2012), the bosonic BdG Hamiltonian can describe emergent bosonic
quasiparticles in ordered states such as magnons McClarty (2022) and phonons
Zhang  _et al._ (2010). Unlike the fermionic counterpart, its excitation
spectrum is related to the eigenspectrum of a pseudo-Hermitian Hamiltonian
matrix with a particle-hole symmetry Lieu (2018) if there exists a paring
term, which breaks the particle-number conservation. This is an example of the
non-Hermitian system whose non-Hermiticity originates not from an open quantum
nature but from the linear approximation of a non-linear equation.

Recently, a lot of concepts in topological physics Qi and Zhang (2011); Hasan
and Kane (2010) have been generalized to bosonic BdG Hamiltonians even though
the representation matrix is pseudo-Hermitian McClarty (2022). For example,
the Chern number is defined by using a para-unitary matrix, and it
characterizes the bulk-boundary correspondence Shindou  _et al._ (2013) as in
the case of Hermitian topological physics Hatsugai (1993). Similar
generalizations for other topological numbers such as $\mathbb{Z}_{2}$
invariant have been extensively studied Kondo  _et al._ (2019). In the
language of non-Hermitian topological physics Ashida  _et al._ (2020); Okuma
and Sato (2023); Lin  _et al._ (2023), this is a manifestation of the line-gap
topology, which is adiabatically connected to the Hermitian topology without
closing the gap and changing the symmetry K. Kawabata, and K. Shiozaki, and M.
Ueda, and M. Sato (2019).

One interesting direction is to seek topological boundary states that reflect
the BdG nature. In fermionic cases, a topological number of the whole particle
(hole) bands can be non-trivial, and it describes the Andreev bound states
(ABS) including Majorana fermions Tanaka  _et al._ (2011); Sato and Fujimoto
(2016). In bosonic cases, on the other hand, the corresponding topological
invariant can not be nontrivial if we limit our discussion to a topological
phase transition in Kitaev’s periodic table Schnyder  _et al._ (2008); Kitaev
(2009), which requires a non-trivial band inversion process. For the stability
of the ground state, the bosonic excitation energies should be nonnegative.
Owing to this stability condition, the BdG Hamiltonian is adiabatically
connected to a trivial Hamiltonian without closing the gap between the
particle and hole bands Shindou  _et al._ (2013).

In this Letter, we investigate a BdG Hamiltonian on the two-dimensional square
lattice and find a bosonic analogy of ABS that is induced by a non-trivial
Berry phase of the particle (hole) bands defined in an unconventional manner.
In an extreme limit, the Berry phase is $\mathbb{Z}_{2}$-quantized owing to an
emergent parity-time symmetry. This quantization can be understood as the non-
Hermitian topology that uses the defective nature as a projection structure.

### Basics of bosonic BdG Hamiltonian.—

First, we review the basic properties of BdG Hamiltonians and define some
notations. A translation-invariant lattice BdG Hamiltonian with $N$ internal
degrees of freedom is given by Shindou  _et al._ (2013)

$\displaystyle\hat{H}$
$\displaystyle=\frac{1}{2}\sum_{\bm{k}}(\bm{a}^{\dagger}_{\bm{k}},\bm{a}_{-\bm{k}})H_{\bm{k}}\begin{pmatrix}\bm{a_{\bm{k}}}\\\
\bm{a}^{\dagger}_{-\bm{k}}\end{pmatrix},$ (1)

where
$\bm{a}^{\dagger}_{\bm{k}}=(a^{\dagger}_{1,\bm{k}},\cdots,a^{\dagger}_{N,\bm{k}})$
denote creation operators of bosons with crystal momentum $\bm{k}$.
$H_{\bm{k}}$ is a $2N\times 2N$ Hermitian matrix with the following form
Shindou  _et al._ (2013):

$\displaystyle H_{\bm{k}}=\begin{pmatrix}h_{\bm{k}}&s_{\bm{k}}\\\
s^{*}_{-\bm{k}}&h^{*}_{-\bm{k}}\end{pmatrix},$ (2)

where $h_{\bm{k}}$ and $s_{\bm{k}}$ are $N\times N$ matrices that represent
the normal term and pairing term (anomalous term), respectively. It is well
known that the excitation spectrum of the bosonic BdG Hamiltonian is not given
by the eigenspectrum of $H_{\bm{k}}$ if the pairing term is nonzero, unlike in
the case of fermions. Interestingly, the true excitation spectrum is related
to the eigenspectrum of a pseudo-Hermitian matrix with a particle-hole
symmetry Lieu (2018):

$\displaystyle H^{\sigma}_{\bm{k}}:=\sigma_{z}H_{\bm{k}},$ (3)
$\displaystyle\sigma_{z}[H^{\sigma}_{\bm{k}}]^{\dagger}\sigma_{z}=H^{\sigma}_{\bm{k}},$
(4)
$\displaystyle\sigma_{x}[H^{\sigma}_{-\bm{k}}]^{*}\sigma_{x}=-H^{\sigma}_{\bm{k}},$
(5)

where $\sigma$’s denote the Pauli matrices in Nambu space. This pseudo-
Hermitian matrix is diagonalized by a paraunitary matrix $P_{\bm{k}}$ Shindou
_et al._ (2013):

$\displaystyle
P_{\bm{k}}^{-1}H^{\sigma}_{\bm{k}}P_{\bm{k}}=\begin{pmatrix}E_{\bm{k}}&0\\\
0&-E_{-\bm{k}}\end{pmatrix},$ (6) $\displaystyle
P_{\bm{k}}\sigma_{z}P^{\dagger}_{\bm{k}}=P^{\dagger}_{\bm{k}}\sigma_{z}P_{\bm{k}}=\sigma_{z}.$
(7)

Here, $E_{\bm{k}}$ is the diagonal matrix whose elements
$\\{\epsilon_{\bm{k},a}~{}|~{}a=1,\cdots,N\\}$ are the excitation energies.
Note that the excitation energies can be negative or complex without further
assumptions. Since the former/latter leads to the Landau/dynamical instability
of the ground state Kawaguchi and Ueda (2012), the positive semidefiniteness
of the Hermitian matrix $H_{\bm{k}}$ is assumed to realize the nonnegative
excitation energies in conventional condensed matter physics 111In some
isolated cold atomic systems, negative- or complex-energy states are relevant
in physics. See Refs.Kawaguchi and Ueda (2012); Ohashi  _et al._ (2020). . In
the following, we call $\\{\epsilon_{\bm{k},a}\\}$ and
$\\{\epsilon_{\bm{k},-a}:=-\epsilon_{-\bm{k},a}\\}$ the particle and hole
bands, respectively. Owing to the non-Hermiticity (non-normality), the bra
(left) eigenvectors are not always the hermitian conjugate of the ket (right)
eigenvectors:

$\displaystyle\langle\\!\langle\bm{k},i|H^{\sigma}_{\bm{k}}=\epsilon_{\bm{k},i}\langle\\!\langle\bm{k},i|,$
$\displaystyle
H^{\sigma}_{\bm{k}}\ket{\bm{k},i}=\epsilon_{\bm{k},i}\ket{\bm{k},i},$ (8)

where $i$ takes both $a$ and $-a$. If we take the biorthonormal convention,
the bra and ket eigenvectors are in the following relations Ohashi  _et al._
(2020); K. Kawabata, and K. Shiozaki, and M. Ueda, and M. Sato (2019):

$\displaystyle|\bm{k},i\rangle\\!\rangle=\mathrm{sgn}(i)\sigma_{z}|\bm{k},i\rangle,$
(9)
$\displaystyle\langle\\!\langle\bm{k},i|\bm{k},j\rangle=\mathrm{sgn}(i)\bra{\bm{k},i}\sigma_{z}|\bm{k},j\rangle=\delta_{i,j}.$
(10)

In topological physics of bosonic BdG systems, the topological invariants are
usually defined by using both the right and left eigenvectors. For example,
the Berry connection defined in Ref. Shindou  _et al._ (2013); Ohashi  _et
al._ (2020); Engelhardt and Brandes (2015) is rewritten as follows:

$\displaystyle A^{LR}_{i,\nu}(\bm{k}):$
$\displaystyle=i\mathrm{Tr}[\Gamma_{i}\sigma_{z}P^{\dagger}_{\bm{k}}\sigma_{z}(\partial_{k_{\nu}}P_{\bm{k}})]=i\mathrm{Tr}[\Gamma_{i}P^{-1}_{\bm{k}}(\partial_{k_{\nu}}P_{\bm{k}})]$
$\displaystyle=i\langle\\!\langle\bm{k},i|\partial_{k_{\nu}}\ket{\bm{k},i},$
(11)

where $\Gamma_{i}$ is a diagonal matrix taking $+1$ for the $i$-th diagonal
component and zero otherwize. We have used the para-unitary condition (7) and
assumed the biorthonormal convention. The Chern number is defined by using
$A^{LR}_{i,\nu}$, which describes the topological physics of the bulk-boundary
correspondence. As mentioned in the introduction part, these topological
invariants cannot be nontrivial for the whole particle (hole) bands if we
assume the positive definiteness of $H_{\bm{k}}$, which ensures the positivity
of the excitation energies. This is because the Hamiltonian
$H^{\sigma}_{\bm{k}}$ under this condition is adiabatically connected to
$1_{N\times N}\otimes\sigma_{z}$ without closing the gap between the particle
and hole bands Shindou  _et al._ (2013). In the following, we seek another
possibility: the boundary states induced by the polarization of the particle
(hole) bands.

Figure 1: (a) Band structure for $r=0.8$ and $k_{y}$-resolved dispersion on a
cylinder for various $r$. The system size is $32\times 32$. A $\pi$ flux is
inserted in the cylinder to avoid $k_{y}=0,\pi$. (b) Berry phases
$\gamma^{RR}$ and $\gamma^{LR}$ for various $r$. Upper panel: calculation
using a Bloch wave that is a continuous function of $\bm{k}$. Lower panel: the
remainder of the Berry phase divided by $2\pi$. The momenta $k_{y}=0,\pi$ are
avoided. The system size is $200\times 50$. For numerical integration, the
size in the $x$ direction is taken much larger than that in the $y$ direction.

### Model with nontrivial Berry phase.—

According to the “modern theory” of polarization Spaldin (2012), the bulk
polarization is given by the Berry phase (divided by $2\pi$) that is defined
as the integration of the Berry connection on a non-contractible loop in the
Brillouin zone. In one dimension, the Berry phase is given by

$\displaystyle\gamma_{i}=i\int^{\pi}_{-\pi}dk~{}\bra{k,i}\partial_{k}\ket{k,i},$
(12)

where $k$ is the one-dimensional crystal momentum with lattice constant $a=1$.
In recent topological physics, bulk polarization is known as another route to
induce the boundary states. We here generalize this idea to the particle
(hole) bands of a bosonic BdG Hamiltonian.

Let us consider the following two-dimensional bosonic BdG Hamiltonian:

$\displaystyle H^{\sigma}_{\bm{k}}=$ $\displaystyle 2(1-\cos k_{x}\cos
k_{y})\sigma_{z}$ $\displaystyle+2ir\left[(\cos k_{y}-\cos
k_{x})\sigma_{x}+\sin k_{x}\sin k_{y}\sigma_{y}\right],$ (13)

where $r\geq 0$ describes the strength of the paring term. This Hamiltonian
satisfies Eqs. (4) and (5). The eigenspectrum is given by

$\displaystyle E_{\pm}(\bm{k})=\pm 2\sqrt{1-r^{2}}(1-\cos k_{x}\cos k_{y}),$
(14)

where $\pm$ denotes particle and hole bands [Fig. 1(a)]. From this expression,
we further assume $r\leq 1$ to ensure the nonnegativity of particle energies,
$E_{+}(\bm{k})\geq 0$, which is the condition for the stable ground state. At
the extreme limit $r=1$, the energy spectrum becomes flat.

The Bloch Hamiltonian (13) describes a system with periodic boundary
conditions in both the $x$ and $y$ directions. To discuss the corresponding
boundary states, we impose the open/periodic boundary condition in the $x/y$
direction. In Fig. 1(a), we plot the eigenspectrum with respect to the
momentum in $y$ direction, $k_{y}$. While there are no isolated modes for
$r=0$, we find the isolated modes outside the bulk particle and hole bands for
a large $r$. This behavior is very different from that of Hermitian boundary
states, which are located inside the band gap. At some $k_{y}$, the isolated
modes are absorbed into the bulk bands. The isolated modes do not degenerate
at each momentum $k_{y}$ and are localized at one boundary. The side of this
localization depends on the sign of $k_{y}$ and the particle-hole band index.
At the extreme limit $r=1$, the low-energy dispersion of the boundary states
becomes linear and gapless, and the boundary states are connected to the bulk
states at momenta $k_{y}=0,\pi$. In other words, the boundary states look like
chiral boundary modes around these symmetric points.

These boundary modes are induced by the bulk polarization defined for the
Bloch states. In the non-Hermitian (pseudo-Hermitian) cases, however, one can
define two types of Berry connections at each momentum $k_{y}$:

$\displaystyle\gamma_{i}^{LR}(k_{y})=i\int^{\pi}_{-\pi}dk_{x}\langle\\!\langle\bm{k},i|\partial_{k_{x}}\ket{\bm{k},i},$
(15)
$\displaystyle\gamma_{i}^{RR}(k_{y})=i\int^{\pi}_{-\pi}dk_{x}\langle\bm{k},i|\partial_{k_{x}}\ket{\bm{k},i}.$
(16)

The former definition Ohashi  _et al._ (2020); Engelhardt and Brandes (2015),
which uses both the right and left eigenvectors, reflects the conventional
manner in the line-gap topology. In this definition, we have assumed the
biorthonormal conventions (9) and (10). The latter definition, which uses only
the right eigenvectors, looks like the Hermitian Berry phase. The crucial
difference from the Hermitian one is that the set of the right eigenvectors
can not span the whole Hilbert space if the paring term is nonzero. In this
definition, we have assumed the normalization
$\bra{\bm{k},i}\bm{k},i\rangle=1$. The Berry connections in both definitions
take real values under the present biorthonormal/normal conventions. Owing to
the gauge degree of freedom, the Berry phases (15) and (16) are determined
modulo $2\pi$. In Fig. 1(b), we plot two types of the Berry phases of the hole
band at each $k_{y}$. In the calculation, we have used a Bloch wave that is a
continuous function of momentum. As $r$ is increased, $\gamma^{RR}$ converges
to $\pm\pi$ for positive/negative $k_{y}$. In Hermitian topological physics,
the value $\pi$ indicates the emergence of the boundary states, which
indicates that $\gamma^{RR}$ can characterize the observed boundary states.
Another definition $\gamma^{LR}$, on the other hand, diverges as for the
increase of $r$. Moreover, one cannot define it at the extreme limit $r=1$
because $\bra{\bm{k},i}\sigma_{z}\ket{\bm{k},i}=0$, which leads to the failure
of the biorthonormal convention. Thus, we conclude that $\gamma^{RR}$ is the
true definition to characterize the present boundary states.

Note that these boundary states are essentially different from the bosonic
topological boundary modes in previous studies that are defined for the gap
between the particle bands. In our case, the boundary states are characterized
by a quantity that is defined for the gap between the particle and hole bands.
We call the present boundary states the bosonic ABS.

### Non-Hermitian topology at flat-band limit.—

For general $r$, the bosonic ABS is geometrical rather than topological
because the Berry phase $\gamma^{RR}$ varies continuously and is adiabatically
connected to zero. At the extreme limit $r=0$, however, one can find a non-
Hermitian topology in the following sense. In this limit, the band becomes
completely flat, and its energy is exactly zero. Owing to the particle-hole
symmetry, $\sigma_{x}\ket{-\bm{k},i}^{*}$ is an eigenstate of
$H^{\sigma}_{\bm{k}}$ with an eigenenergy $-\epsilon_{-\bm{k},i}$. In the
present case, both $\ket{\bm{k},i}$ and $\sigma_{x}\ket{-\bm{k},i}^{*}$ are
the zero-energy states. Moreover, the Hamiltonian $H^{\sigma}_{\bm{k}}$ is
defective (i.e. not diagonalizable), and the hole eigenstate is identical to
the particle one, except for at $\bm{k}=(0,0)$ and $(\pi,\pi)$. Thus,
$\ket{\bm{k},i}$ is identical to $\sigma_{x}\ket{-\bm{k},i}^{*}$ up to the
phase, which means that the particle-hole symmetry effectively acts as if the
time-reversal symmetry at $r=1$. In addition, $H^{\sigma}_{\bm{k}}$ is
invariant under the inversion $\bm{k}\rightarrow-\bm{k}$. In total, we can
define an effective parity-time symmetry, and $\ket{\bm{k},i}$ is identical to
$\sigma_{x}\ket{\bm{k},i}^{*}$ up to the phase. It is known that the Berry
phase under this type of antiunitary symmetry is $\mathbb{Z}_{2}$ quantized
Hatsugai (2006). In the present case, it is checked by the following
calculation:

$\displaystyle\gamma_{i}^{RR}(k_{y})$ $\displaystyle\equiv
i\int^{\pi}_{-\pi}dk_{x}\langle\bm{k},i|^{*}\sigma_{x}\partial_{k_{x}}\sigma_{x}\ket{\bm{k},i}^{*}~{}(\mathrm{mod}~{}2\pi)$
$\displaystyle=-\gamma_{i}^{RR}(k_{y}).$ (17)

From this relation, the Berry phase is quantized to $0$ or $\pi$. The bosonic
ABS corresponds to $\gamma^{RR}_{i}=\pi$.

Thanks to the two-band nature, the above physics is also explained by a
$\mathbb{Z}$ topology of the Hamiltonian itself. The Bloch Hamiltonian in the
defective region takes the following form:

$\displaystyle H^{\sigma}_{\bm{k}}\propto\begin{pmatrix}1&ih^{*}_{\bm{k}}\\\
ih_{\bm{k}}&-1\end{pmatrix},$ (18)

where $h_{\bm{k}}\in\mathbb{C}$ is on the unit circle in the complex plane. In
other words, the degree of freedom of the Bloch Hamiltonian is limited to U(1)
if we impose that the matrix is defective. Thus, one can define the winding
number on a closed loop $C$ in the Brillouin zone:

$\displaystyle W=\frac{1}{2\pi i}\oint_{C}d\ln h.$ (19)

This winding number (19) characterizes two related properties. First, it
describes the polarization at each $k_{y}$:

$\displaystyle W(k_{y})=\frac{1}{2\pi i}\int_{-\pi}^{\pi}dk_{x}\frac{d\ln
h}{dk_{x}}=\int_{-\pi}^{\pi}\frac{dk_{x}}{2\pi}\theta_{\bm{k}},$ (20)

where $\theta_{\bm{k}}=\arg(h_{\bm{k}})$. At special points $k_{y}=\pm\pi/2$,
the winding number is $\pm 1$, which is easily calculated by
$\theta_{\bm{k}}=\pm k_{x}$. At general $k_{y}$ except for $k_{y}=0,\pi$,
$W(k_{y})=\mathrm{sgn}(k_{y})$. Remarkably, $\mathbb{Z}$ topological number
can distinguish $\pm 1$, while the Berry phase cannot. Second, the winding
number (19) detects the non-defective points in the Brillouin zone. Near the
non-defective point $\bm{k}=(0,0)$, the Hamiltonian takes the following form:

$\displaystyle H^{\sigma}_{\bm{k}}$
$\displaystyle\simeq(k_{x}^{2}+k_{y}^{2})\sigma_{z}+i(k_{x}^{2}-k_{y}^{2})\sigma_{x}+2ik_{x}k_{y}\sigma_{y}$
$\displaystyle\propto\begin{pmatrix}1&ie^{-i2\phi}\\\
ie^{i2\phi}&-1\end{pmatrix},$ (21)

where $\bm{k}=k(\cos\phi,\sin\phi)$ with $k=|\bm{k}|$. Thus, $\bm{k}=(0,0)$ is
characterized by $W=+2$. Similarly, $\bm{k}=(\pi,\pi)$ is characterized by
$W=-2$. The total winding number around non-defective points is 0, which is a
non-Hermitian analogy of the Nielsen-Ninomiya theorem Nielsen and Ninomiya
(1981). Note that the paring term is proportional to an effective model of the
quadratic band touching in Hermitian topological physics. The connection
between our non-Hermitian topology and the physics of quadratic band touching,
including the geometry-induced surface states Yang and Nagaosa (2014) and the
Euler number Ahn  _et al._ (2019), may be an interesting future work. Note
also that the winding number itself can be defined for general $r$ except for
$0$. In that case, however, the degree of freedom of the Hamiltonian is larger
than $U(1)$.

### Discussion.—

We here investigate several remaining topics and discuss related future works.
The bosonic ABS is expected to be found in various dimensions. In one
dimension, however, the realization of a model with one positive- and one
negative-energy boundary states localized at opposite boundaries seems to be
difficult. If possible, these boundary states are related to each other by the
particle-hole symmetry, which does not act on real-space coordinates. Thus,
the boundary states are localized at the same boundary, which conflicts with
the localization at the opposite boundaries. In our two-dimensional model, the
particle-hole symmetry is absent at each $k_{y}$ except for $0,\pi$, and the
localization at the opposite boundary is allowed in each momentum sector.
Instead, one can consider the bosonic ABS in a model with a unitary symmetry.
For example, let us consider the following one-dimensional model:

$\displaystyle H^{\sigma}_{k}=1_{2\times 2}\otimes\sigma_{z}+ir(\cos
k\tau_{x}+\sin k\tau_{y})\otimes\sigma_{y},$ (22)

where $\tau$’s are the Pauli matrices in orbital space, and $0\leq r\leq 1$.
This model commutes with $\tau_{z}\otimes\sigma_{z}$ and is block-diagonalized
into $\tau_{z}\otimes\sigma_{z}=\pm 1$ sectors. Each block is given by

$\displaystyle\begin{pmatrix}1&ire^{\mp ik}\\\ ire^{\pm ik}&-1\end{pmatrix},$
(23)

which means the non-trivial polarization. A generalization of the bosonic ABS
may be an interesting future work. Since particle and hole states coincide at
an extreme parameter, $\mathbb{Z}_{2}$ nature is expected to be essential if
we impose that the total topological number of the particle and hole bands is
zero.

Another interesting issue is the entanglement property of the ground state of
a model with the bosonic ABS. In free bosonic systems, the entanglement
entropy of the ground state is induced only by the paring term. Such a term is
naturally introduced by squeezed states of light and plays an important role
in continuous-variable (CV) quantum computing Braunstein and Van Loock (2005).
In our calculation 222See Supplemental Material., boundary states similar to
those in our model (but with degeneracy) are found in the CV surface codes
(CVSC) Zhang  _et al._ (2008); Milne  _et al._ (2012); Morimae (2013); Demarie
_et al._ (2014). Reference Demarie  _et al._ (2014) claims that topological
entanglement entropy Kitaev and Preskill (2006); Levin and Wen (2006) is not
quantized in a physical CVSC. In Supplemental Material, we investigate the
entanglement structure of the ground state of the model (13) by using the
formula in Ref. Demarie (2012) and find behaviors similar to those in CVSC. A
relationship between CV topological order and the bosonic ABS is also an
interesting remaining topic.

###### Acknowledgements.

I thank Tomonari Mizoguchi for the fruitful discussions. This work was
supported by JSPS KAKENHI Grant No. JP20K14373 and No. JP23K03243.

## References

  * Bogoliubov (1947) N. Bogoliubov, J. Phys 11, 23 (1947).
  * Colpa (1978) J. Colpa, Physica A: Statistical Mechanics and its Applications 93, 327 (1978).
  * Kawaguchi and Ueda (2012) Y. Kawaguchi and M. Ueda, Physics Reports 520, 253 (2012).
  * Altland and Simons (2010) A. Altland and B. D. Simons,  _Condensed matter field theory_ (Cambridge university press, 2010).
  * Ozawa  _et al._ (2019) T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, _et al._ , Reviews of Modern Physics 91, 015006 (2019).
  * Braunstein and Van Loock (2005) S. L. Braunstein and P. Van Loock, Reviews of modern physics 77, 513 (2005).
  * McClarty (2022) P. A. McClarty, Annual Review of Condensed Matter Physics 13, null (2022), https://doi.org/10.1146/annurev-conmatphys-031620-104715 .
  * Zhang  _et al._ (2010) L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev. Lett. 105, 225901 (2010).
  * Lieu (2018) S. Lieu, Phys. Rev. B 98, 115135 (2018).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics 83, 1057 (2011).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Reviews of modern physics 82, 3045 (2010).
  * Shindou  _et al._ (2013) R. Shindou, R. Matsumoto, S. Murakami, and J.-i. Ohe, Phys. Rev. B 87, 174427 (2013).
  * Hatsugai (1993) Y. Hatsugai, Physical review letters 71, 3697 (1993).
  * Kondo  _et al._ (2019) H. Kondo, Y. Akagi, and H. Katsura, Phys. Rev. B 99, 041110 (2019).
  * Ashida  _et al._ (2020) Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249 (2020).
  * Okuma and Sato (2023) N. Okuma and M. Sato, Annual Review of Condensed Matter Physics 14, 83 (2023).
  * Lin  _et al._ (2023) R. Lin, T. Tai, L. Li, and C. H. Lee, Frontiers of Physics 18, 53605 (2023).
  * K. Kawabata, and K. Shiozaki, and M. Ueda, and M. Sato (2019) K. Kawabata, and K. Shiozaki, and M. Ueda, and M. Sato, Phys. Rev. X 9, 041015 (2019).
  * Tanaka  _et al._ (2011) Y. Tanaka, M. Sato, and N. Nagaosa, Journal of the Physical Society of Japan 81, 011013 (2011).
  * Sato and Fujimoto (2016) M. Sato and S. Fujimoto, Journal of the Physical Society of Japan 85, 072001 (2016).
  * Schnyder  _et al._ (2008) A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).
  * Kitaev (2009) A. Kitaev, in  _AIP conference proceedings_ , Vol. 1134 (American Institute of Physics, 2009) pp. 22–30.
  * Note (1) In some isolated cold atomic systems, negative- or complex-energy states are relevant in physics. See Refs.Kawaguchi and Ueda (2012); Ohashi  _et al._ (2020).
  * Ohashi  _et al._ (2020) T. Ohashi, S. Kobayashi, and Y. Kawaguchi, Phys. Rev. A 101, 013625 (2020).
  * Engelhardt and Brandes (2015) G. Engelhardt and T. Brandes, Physical Review A 91, 053621 (2015).
  * Spaldin (2012) N. A. Spaldin, Journal of Solid State Chemistry 195, 2 (2012).
  * Hatsugai (2006) Y. Hatsugai, Journal of the Physical Society of Japan 75, 123601 (2006).
  * Nielsen and Ninomiya (1981) H. B. Nielsen and M. Ninomiya,  _No-go theorum for regularizing chiral fermions_ , Tech. Rep. (Science Research Council, 1981).
  * Yang and Nagaosa (2014) B.-J. Yang and N. Nagaosa, Physical review letters 112, 246402 (2014).
  * Ahn  _et al._ (2019) J. Ahn, S. Park, and B.-J. Yang, Physical Review X 9, 021013 (2019).
  * Note (2) See Supplemental Material.
  * Zhang  _et al._ (2008) J. Zhang, C. Xie, K. Peng, and P. van Loock, Physical Review A 78, 052121 (2008).
  * Milne  _et al._ (2012) D. F. Milne, N. V. Korolkova, and P. van Loock, Physical Review A 85, 052325 (2012).
  * Morimae (2013) T. Morimae, Physical Review A 88, 042311 (2013).
  * Demarie  _et al._ (2014) T. F. Demarie, T. Linjordet, N. C. Menicucci, and G. K. Brennen, New Journal of Physics 16, 085011 (2014).
  * Kitaev and Preskill (2006) A. Kitaev and J. Preskill, Physical review letters 96, 110404 (2006).
  * Levin and Wen (2006) M. Levin and X.-G. Wen, Physical review letters 96, 110405 (2006).
  * Demarie (2012) T. F. Demarie, arXiv preprint arXiv:1209.2748 (2012).

Supplemental Material for “Bosonic Andreev bound state”

Figure S1: (a) The Bogoliubov spectrum of the physical CVSC Demarie  _et al._
(2014) on the cylinder. $s$ is the squeezing parameter. The system size is
$32\times 32$. (b) Size dependence of the entanglement entropy of the ground
state of the model (S1) on the cylinder.

##  I Entanglement entropy of a model with bosonic Andreev bound states

The physical continuous-variable surface code (CVSC) Demarie  _et al._ (2014),
which is a generalization of the CVSC Zhang  _et al._ (2008); Milne  _et al._
(2012); Morimae (2013), is characterized by the “topological” entanglement
entropy whose value is a continuous function of a system parameter Demarie
_et al._ (2014). In our calculation [Fig.S1 (a)], this model (with
$45^{\circ}$ rotation from the original one, $4$ sites in 1 unit cell) has
boundary modes similar (but with degeneracy) to our model analyzed in the main
text. Motivated by this fact, we investigate the entanglement entropy of our
model.

Our model on torus configuration is restated as follows:

$\displaystyle\hat{H}=\frac{1}{2}\sum_{\bm{k}}(\bm{a}^{\dagger}_{\bm{k}},\bm{a}_{-\bm{k}})H_{\bm{k}}\begin{pmatrix}\bm{a_{\bm{k}}}\\\
\bm{a}^{\dagger}_{-\bm{k}}\end{pmatrix}=\frac{1}{2}\sum_{\bm{k}}(\bm{a}^{\dagger}_{\bm{k}},-\bm{a}_{-\bm{k}})H^{\sigma}_{\bm{k}}\begin{pmatrix}\bm{a_{\bm{k}}}\\\
\bm{a}^{\dagger}_{-\bm{k}}\end{pmatrix},$ $\displaystyle
H^{\sigma}_{\bm{k}}=2(1-\cos k_{x}\cos k_{y})\sigma_{z}+2ir\left[(\cos
k_{y}-\cos k_{x})\sigma_{x}+\sin k_{x}\sin k_{y}\sigma_{y}\right],$ (S1)

where $0\leq r\leq 1$. We here investigate the cylindrical configuration ($x$:
open, $y$: periodic) and measure the entanglement entropy $S$ of the ground
state for half of the system with $L_{x}=2L_{y}$. A formula for the
entanglement entropy of the quantum harmonic oscillator, which is identical to
the bosonic BdG system, is given in Ref. Demarie (2012). Using this formula
with the Fourier transform, we calculate the size dependence of the
entanglement entropy [Fig. S1(b)]. While the dominant term obeys the area law
[i.e., $S_{\rm dom}\propto L_{y}$], there is a negative constant subleading
term. The value depends on the model parameter $r$, which is similar to the
behavior of the physical CVSC. In gapped systems, this type of subleading term
is called topological entanglement entropy because it characterizes the
topological order Kitaev and Preskill (2006); Levin and Wen (2006). Unlike in
the gapped systems, the constant subleading term is not quantized in the
physical CVSC and our model.