# Topological study of a Bogoliubov-de Gennes system of pseudo-spin-$1/2$
bosons with conserved magnetization in a honeycomb lattice

Hong Y. Ling  Department of Physics and Astronomy, Rowan University,
Glassboro, New Jersey 08028, USA  Ben Kain  Department of Physics, College of
the Holy Cross, Worcester, Massachussets 01610, USA

###### Abstract

We consider a Bogoliubov-de Gennes (BdG) Hamiltonian, which is a non-Hermitian
Hamiltonian with pseudo-Hermiticity, for a system of (pseudo) spin-$1/2$
bosons in a honeycomb lattice under the condition that the population
difference between the two spin components, i.e., magnetization, is a
constant. Such a system is capable of acting as a topological amplifier, under
time-reversal symmetry, with stable bulk bands but unstable edge modes which
can be populated at an exponentially fast rate. We quantitatively study the
topological properties of this model within the framework of the 38-fold way
for non-Hermitian systems. We find, through the symmetry analysis of the Bloch
Hamiltonian, that this model is classified either as two copies of symmetry
class $\mbox{AIII}+\eta_{-}$ or two copies of symmetry class $\mbox{A}+\eta$
depending on whether the (total) system is time-reversal symmetric, where
$\eta$ is the matrix representing pseudo-Hermiticity and $\eta_{-}$ indicates
that pseudo-Hermiticity and chiral symmetry operators anticommute. We prove,
within the context of non-Hermitian physics where eigenstates obey the bi-
orthonormality relation, that a stable bulk is characterized by a single
topological invariant, the Chern number for the Haldane model, independent of
pairing interactions. We construct a convenient analytical description for the
edge modes of the Haldane model in semi-infinite planes, which is expected to
be useful for models built upon copies of the Haldane model across a broad
array of disciplines. We adapt the theorem in our recent work [Ling and Kain,
Phys. Rev. A 104, 013305 (2021)] to pseudo-Hermitian Hamiltonians that are
less restrictive than BdG Hamiltonians and apply it to highlight that the
vanishing of an unconventional commutator between number-conserving and
number-nonconserving parts of the Hamiltonian indicates whether a system can
be made to act as a topological amplifier.

###### pacs:

Valid PACS appear here

##  I introduction

Symmetry principles play a pivotal role in our understanding of topological
matter [1, 2], i.e., systems whose edge modes are guaranteed to be robust
against disorder by bulk topological invariants. Haldane in 1988 [3] first
recognized that it is not so much a net magnetic field but rather the breaking
of time-reversal symmetry that is responsible for the integer quantum Hall
(IQH) effect [4, 5]. Kane and Melee in 2005 [6, 7] showed that the anomalous
quantum spin Hall (QSH) effect [8] can arise from a time-reversal invariant
system of spin-$1/2$ fermions. The formulation of the QSH effect in terms of
topology [6] along with the experimental realization of the ideas behind the
Kane-Melee model [6] in HgTe quantum wells [9] following the theoretical
proposal by Bernevig et al. [10] has motivated the creation of a periodic
table, called the 10-fold way [11, 12, 13], of topological insulators (and
superconductors) for closed systems of fermions described by quadratic
Hermitian Hamiltonians [14, 15].

A topological insulator has a bulk characterized by a topologically non-
trivial invariant, which is determined globally by the bulk Bloch band
structure. Since Bloch bands are universal in periodic systems, topological
phenomena are ubiquitous. This realization has generated an enormous interest
in topological phases of matter across a wide variety of systems, including
photons [16, 17, 18, 19, 20], phonons (and mechanical metamaterials) [21, 22,
23, 24, 25, 26], magnons [27, 28], and ultracold atoms [29, 30, 31, 32, 33]
(see recent review articles [34, 35, 36, 37] for additional references). In
fact, a large body of studies in recent years has been devoted to topological
phenomena in non-Hermitian Hamiltonians for systems that may not be closed,
fermionic, or even in thermal equilibrium [38, 39, 40, 41, 42, 43, 44, 45]
(see Refs. [46, 47, 48, 49, 50] and references therein).

The prospect of topological classes unique to non-Hermitian Hamiltonians [39,
51, 46] calls for systematic classification of topological symmetry classes
beyond the 10-fold way paradigm. The 10-fold way is founded on the celebrated
Altland-Zirnbauer (AZ) theory [52] which groups quadratic Hermitian
Hamiltonians according to three fundamental symmetries, i.e., time-reversal,
particle-hole, and chiral symmetry, into 10 classes [11, 12, 13], a direct
generalization of the Wigner-Dyson symmetry classes [53] in random matrix
theory [54]. There exists, also in random matrix theory, a non-Hermitian
analog of the AZ classification, due to Bernard and LeClair (BL) [55], based
on four fundamental symmetries [56, 57] — three AZ symmetries plus a so-called
Q symmetry which distinguishes between Hermitian and non-Hermitian matrices.
As such, the BL classification contains two analogs of time-reversal and
particle-hole symmetries (C and K symmetries) which are identical for
Hermitian matrices but are distinct for non-Hermitian matrices, leading, in
large part, to the proliferation of the symmetry classes in non-Hermitian
systems. In fact, following recent studies [39, 57, 46] which traced some
symmetry classes of non-Hermitian Hamiltonians to the BL scheme, Kawabara et
al. [47] and Zhou and Lee [48] categorized non-Hermitian Hamiltonians into a
total of 38 classes, thereby giving rise to the so-called (BL-based) 38-fold
way, a non-Hermitian generalization of the (AZ-based) 10-fold way [11, 12,
13].

In this paper, we focus on a quadratic bosonic Hamiltonian, which, as in other
quadratic bosonic models [58, 59, 60, 61], contains pairing terms (or
interactions) that break particle number conservation, making it a bosonic
analog of the Bogoliubov-de Gennes (BdG) Hamiltonian for fermions. However, in
contrast to the fermionic BdG model, where the matrix to be diagonalized is
Hermitian, the matrix to be diagonalized in the bosonic BdG model is non-
Hermitian. As we shall see in later sections, our bosonic BdG Hamiltonian and
the sub-Hamiltonians that are descended from it belong to a special class of
non-Hermitian Hamiltonians which obey pseudo-Hermiticity, whereby a
Hamiltonian differs from its Hermitian conjugate by a similarity
transformation characterized by a unitary and Hermitian matrix $\eta$. This is
a special version of the Q symmetry in the BL classification and thus our
Hamiltonian and its associated sub-Hamiltonians are grouped according to the
38-fold way classification of Hamiltonians with pseudo-Hermiticity.

The purpose of the present work is to conduct, within the framework of the
38-fold way, a systematic study of the topological properties of such a
quadratic bosonic Hamiltonian. Our paper is organized as follows. In Sec. II,
we present our model Hamiltonian — a generic BdG Hamiltonian for spin-$1/2$
bosons in a honeycomb lattice with both periodic boundaries (Sec. II.1) and
open boundaries (Sec. II.2). We give a brief description of a cold atom
realization in an optical lattice with a spin-1 condensate, while relegating
the details to Appendix A.

In Sec. III, we visit the Haldane model to which our system traces its
topological origin. We review the Chern number in Sec. III.1. We derive in
III.2 a set of (relatively) simple analytical results describing edge modes in
the Haldane model in semi-infinite planes, which are expected to be useful for
understanding edge modes not only in our model but also in many other models
that are built on copies of the Haldane model.

In Sec. IV, we generalize the study in Sec. III to our model in the absence of
time-reversal symmetry. A unitary symmetry due to magnetization conservation
in our model partitions the Hilbert space into two independent sectors so that
the total Hamiltonian inherits the topologies of the sub-Hamiltonians in the
smaller sectors. In Sec. IV.1, we show that these sub-Hamiltonians are class
A$+\eta$ systems with pseudo-Hermiticity. We also show, with the help of the
bi-orthonormality relation involving left- and right-eigenstates, that the
bulk topology is indeed characterized by a single bulk invariant, the Chern
($\mathbb{Z}$) number, as predicted by the 38-fold way. Furthermore, we prove
that for a stable bulk, this single Chern number is independent of pairing
interactions and corresponds to the Chern number for the Haldane model. In
Sec. IV.3, we investigate the energy spectrum of a sub-Hamiltonian for systems
with open boundaries and apply the simple results in Sec. III.2 to gain
analytical insights into the edge modes that lie inside the gaps of particle
and hole bulk bands.

In Sec. V, we include time-reversal symmetry in our system, which has been
investigated previously in connection with exponential amplification of the
edge modes with [59] and without [62] inversion symmetry. In Ref. [62], we
used this model to illustrate a theorem for creating topological amplifiers
with stable bulks in BdG Hamiltonians. In Appendix D, we generalize this
theorem to sub-Hamiltonians which are not BdG Hamiltonians, and apply it in
Sec. V to prove that, as long as time-reversal symmetry is preserved, our
system can act as a topological amplifier with its bulk spectrum free of
instabilities. Furthermore, we offer a detailed explanation of why the total
system should now be classified as two copies of class AIII$+\eta_{-}$
systems, where $\eta_{-}$ indicates that the matrix representing pseudo-
Hermiticity anticommutes with the matrix representing chiral symmetry. Here,
the bulk topology is described again by the Chern ($\mathbb{Z}$) number for
the Haldane model, independent of pairing interactions.

In Sec. VI, we conclude our work.

##  II Our Model: Tight binding Hamiltonians

Figure 1: (a) A honeycomb lattice model consisting of an A sublattice and a B
sublattice where $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are the base vectors
for a unit cell, $\delta_{1},\delta_{2}$, $\delta_{3}$ are the vectors between
a site and its three nearest neighbors. The separation between two adjacent
sites on the same sublattice is chosen as the distance unit. (b) The Brillouin
zone of a honeycomb lattice where $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ are
the base vectors of a unit cell in the reciprocal (momentum) space. Indicated
are high symmetric points in the Brillouin zone: $\Gamma$, $M$, and two
inequivalent Dirac points $K$ and $K^{\prime}$.

We consider a system of pseudo-spin-$1/2$ bosons in a honeycomb lattice
described, in position space, by the tight-binding Hamiltonian,

$\begin{split}\hat{H}_{1/2}=&-t_{1}\sum_{\left<\mathbf{ij}\right>}\hat{b}_{\mathbf{i}}^{\dagger}s_{0}\hat{b}_{\mathbf{j}}+t_{2}\sum_{\left<\left<\mathbf{ij}\right>\right>}e^{-i\nu_{\mathbf{ij}}\phi}\hat{b}_{\mathbf{i}}^{\dagger}s_{z}\hat{b}_{\mathbf{j}}\\\
&+q\sum_{\mathbf{i}}\hat{b}_{\mathbf{i}}^{\dagger}s_{0}\hat{b}_{\mathbf{i}}+m\sum_{\mathbf{i}}\xi_{\mathbf{i}}\hat{b}_{\mathbf{i}}^{\dagger}s_{0}\hat{b}_{\mathbf{i}}\\\
&+\frac{c}{2}\sum_{\mathbf{i}}(\hat{b}_{\mathbf{i}}s_{x}\hat{b}_{\mathbf{i}}+\hat{b}^{\dagger}_{\mathbf{i}}s_{x}\hat{b}^{\dagger}_{\mathbf{i}}),\end{split}$
(1)

where
$\hat{b}_{\mathbf{j}}=(\hat{b}_{\mathbf{j},\uparrow},\hat{b}_{\mathbf{j},\downarrow})$
is the two-component field operator at site $\mathbf{j}$ and $s_{x,y,z}$ and
$s_{0}$ are, respectively, the Pauli matrices and the identity matrix in
(pseudo) spin space,

$\mathscr{H}_{s}=(\ket{\uparrow},\ket{\downarrow}).$ (2)

In Eq. (1), $t_{1}$ is the nearest-neighbor hopping amplitude,
$t_{2}e^{-i\nu_{\mathbf{ij}}\phi}$ is the next-nearest-neighbor hopping
amplitude introduced by Haldane [3], where $\nu_{\mathbf{ij}}=\pm 1$
alternates periodically in the manner of Kane and Mele [6], $c$ denotes the
strength of pairing interactions, and $q$ and $m$ measure, respectively, the
nonstaggered and staggered onsite potential with $\xi_{\mathbf{i}}=+1$ for
sites on sublattice A and $-1$ for sites on sublattice B.

Equation (1) is a generic Hamiltonian and may be engineered in different
physical systems. Galilo et al. [59] proposed in 2015 a realization in an
ultracold atomic gas of interacting spin-1 bosons in a honeycomb lattice,
which was inspired, in large part, by the cold atom implementation of the
Haldane model by Esslinger group [32]. The present study is based on a
generalization of their model described in Appendix A. Here, the spin-$1/2$
system is a subsystem consisting of spin-$\pm 1$ excitations which are
generated by quenching the onsite potential in such a manner that it does not
affect the spin-$0$ component, which is stable and decoupled from the
spin-$1/2$ system. In this context, Eq. (1) is the postquench Hamiltonian for
the spin-$1/2$ system, which we recently employed to test a criterion for
creating a topological amplifier with a bulk free of dynamical instability
[62].

###  II.1 The model with periodic boundaries

To explore the bulk band topology, we consider a honeycomb lattice with
periodic boundaries and move from position space in Fig. 1(a) to momentum
$\mathbf{k}$ space in Fig. 1(b) where the Hamiltonian is diagonal.

In the absence of pairing interactions, the (Bloch) Hamiltonian in momentum
space simply consists of two independent copies of the Haldane model, with

$h_{\uparrow}(\mathbf{k})=h_{0}(\mathbf{k})\sigma_{0}+\mathbf{h}(\mathbf{k})\cdot\mathbf{\sigma}$
(3)

for the spin-$\uparrow$ copy and

$h_{\downarrow}(\mathbf{k})=h^{\prime}_{0}(\mathbf{k})\sigma_{0}+\mathbf{h^{\prime}}(\mathbf{k})\cdot\mathbf{\sigma}$
(4)

for the spin-$\downarrow$ copy, where $\sigma_{x},\sigma_{y},\sigma_{z}$, and
$\sigma_{0}$ are the Pauli and identity matrices, respectively, in sublattice
space,

$\mathscr{H}_{\sigma}=(\ket{A},\ket{B}).$ (5)

In Eqs. (3) and (4),

$h_{0}(\mathbf{k})=q+\epsilon(\mathbf{k}),\quad
h^{\prime}_{0}(\mathbf{k})=q-\epsilon(\mathbf{k}),$ (6)

are purely real, where

$\epsilon(\mathbf{k})=2t_{2}\cos\phi\real\Lambda_{\mathbf{k}},$ (7)

and

$\begin{split}h_{x}(\mathbf{k})=&h^{\prime}_{x}(\mathbf{k})=-t_{1}\real\Gamma_{\mathbf{k}},\\\
h_{y}(\mathbf{k})=&h^{\prime}_{y}(\mathbf{k})=+t_{1}\imaginary\Gamma_{\mathbf{k}},\\\
h_{z}(\mathbf{k})=&h^{\prime}_{z}(-\mathbf{k})=m-2t_{2}\sin\phi\imaginary\Lambda_{\mathbf{k}},\end{split}$
(8)

are the components of $\mathbf{h}(\mathbf{k})$ and
$\mathbf{h}^{\prime}(\mathbf{k})$, which are also purely real, where

$\displaystyle\Gamma_{\mathbf{k}}$
$\displaystyle=e^{i\frac{k_{y}}{\sqrt{3}}}+2\cos\frac{k_{x}}{2}e^{-i\frac{k_{y}}{2\sqrt{3}}},$
(9) $\displaystyle\Lambda_{\mathbf{k}}$
$\displaystyle=e^{ik_{x}}+2e^{-i\frac{k_{x}}{2}}\cos\frac{\sqrt{3}k_{y}}{2}.$
(10)

In the presence of pairing interactions, the Hamiltonian can no longer be
separated into two independent spin copies. Let
$\hat{\psi}_{\mathbf{k}}=(\hat{b}_{\mathbf{k}},\hat{b}^{\dagger}_{-\mathbf{k}})$
be the Nambu spinor, where $\hat{b}_{\mathbf{k}}$ and its Hermitian conjugate
are the vector fields in the tensor product of spin and sublattice spaces:
$\mathscr{H}_{s}\otimes\mathscr{H}_{\sigma}$. The total Hamiltonian up to a
constant reads

$\hat{H}_{1/2}=\frac{1}{2}\sum_{\mathbf{k}}\hat{\psi}^{\dagger}_{\mathbf{k}}\Sigma_{z}H_{BdG}(\mathbf{k})\hat{\psi}_{\mathbf{k}},$
(11)

where $\Sigma_{z}$ is a matrix defined directly below Eq. (13) and
$H_{BdG}(\mathbf{k})$ is given explicitly in Eq. (14). In this work, we call
matrix $H_{BdG}(\mathbf{k})$ the bosonic BdG Hamiltonian, which is typically
non-Hermitian, and matrix $\Sigma_{z}H_{BdG}(\mathbf{k})$ the “first
quantized” Hamiltonian, which is guaranteed to be Hermitian by Bose
statistics.

$\hat{H}_{1/2}$ in Eq. (11), which is a bosonic analog of the fermionic BdG
Hamiltonian that lies at the heart of the Bardeen-Cooper-Schrieffer
description of superconductivity [63], can be diagonalized in terms of the
quasiparticle field operator $\hat{\phi}_{\mathbf{k}}$ defined as

$\hat{\phi}_{\mathbf{k}}=T\hat{\psi}_{\mathbf{k}}$ (12)

with the help of the generalized Bogoliubov transformation $T$. The
requirement that $T$ be canonical, i.e., $T$ leaves the Bose commutator
invariant, dictates that $T$ be paraunitary, i.e.,

$T\Sigma_{z}T^{\dagger}=\Sigma_{z},$ (13)

where $\Sigma_{z}=\tau_{z}\otimes s_{0}\otimes\sigma_{0}$. Here, we have
defined $\tau_{x},\tau_{y},\tau_{z}$, and $\tau_{0}$ as the Pauli and identity
matrices, respectively, in Nambu space.

The quadratic bosonic Hamiltonian (11) is thus diagonalized by a paraunitary
transformation, in sharp contrast with its fermionic counterpart which is
diagonalized by a unitary transformation. As a consequence, it is the matrix

${H}_{BdG}(\mathbf{k})=\begin{pmatrix}A(\mathbf{k})&B(\mathbf{k})\\\
-B^{*}(-\mathbf{k})&-A^{*}(-\mathbf{k})\end{pmatrix},$ (14)

that is to be diagonalized, where

$A(\mathbf{k})=\mathbb{P}_{\uparrow}\otimes
h_{\uparrow}(\mathbf{k})+\mathbb{P}_{\downarrow}\otimes
h_{\downarrow}(\mathbf{k}),\quad B(\mathbf{k})=cs_{x}\otimes\sigma_{0},$ (15)

and

$\mathbb{P}_{\uparrow/\downarrow}=\frac{1\pm
s_{z}}{2},\quad\mathbb{P}_{+/-}=\frac{1\pm\tau_{z}}{2},$ (16)

are, respectively, the projection operators in spin and Nambu space (we shall
use $\mathbb{P}_{+/-}$ in later sections). One may easily verify that

$A(\mathbf{k})=A^{\dagger}(\mathbf{k}),\quad
B(\mathbf{k})=B^{T}(-\mathbf{k}),$ (17)

which are guaranteed by the Hermiticity of $\Sigma_{z}H_{BdG}(\mathbf{k})$.

###  II.2 The model with open boundaries

Topologically protected edge modes for a system with open boundaries follow
from the existence of a topologically nontrivial invariant, which itself
follows from the bulk band structure. To examine edge properties, we consider
the same honeycomb lattice but in a stripe geometry with open boundaries along
$y$ and periodic (zigzag) boundaries along $x$ so that the $x$-momentum
$k_{x}$ remains a good quantum number. Let $N_{x}$ ($N_{y}$) be the number of
unit cells along $x$ ($y$) and $X_{\mathbf{j}}$ be the x-component of the
position vector at site $\mathbf{j}$. Applying a partial Fourier
transformation along $x$,

$\hat{b}_{\mathbf{j}\equiv(j_{x},j_{y})}=\sum_{k_{x}}\hat{b}_{k_{x},j_{y}}\frac{e^{ik_{x}X_{\mathbf{j}}}}{\sqrt{N_{x}}},\quad
j_{y}=1,2,\cdots,N_{y},$ (18)

we change Eq. (1) to the corresponding Hamiltonian in the stripe geometry,

$\hat{H}_{1/2}=\frac{1}{2}\sum_{k_{x}}\hat{\psi}^{\dagger}_{k_{x}}\Sigma_{z}{H}_{BdG}(k_{x})\hat{\psi}_{k_{x}},$
(19)

where $\hat{\psi}_{k_{x}}=(\hat{b}_{k_{x}},\hat{b}^{\dagger}_{-k_{x}})$ is the
Nambu spinor with $\hat{b}_{k_{x}}$ and its Hermitian conjugate the vector
fields in $\mathscr{H}_{I}\otimes\mathscr{H}_{s}\otimes\mathscr{H}_{\sigma}$,
where

$\mathscr{H}_{I}=(\ket{1},\ket{2},\cdots,\ket{N_{y}})$ (20)

is the ($y$) unit-cell space and $\mathscr{H}_{s}$ and $\mathscr{H}_{\sigma}$
are spin and sublattice spaces previously defined in Eq. (2) and Eq. (5),
respectively. In Nambu space, $H_{BdG}(k_{x})$ in Eq. (19) takes the form,

${H}_{BdG}(k_{x})=\begin{pmatrix}{A}(k_{x})&{B}(k_{x})\\\
-{B}(-k_{x})&-{A}(-k_{x})\end{pmatrix},$ (21)

where ${B}(k_{x})$ and ${A}(k_{x})$ are two real symmetric $4N_{y}\times
4N_{y}$ matrices in
$\mathscr{H}_{I}\otimes\mathscr{H}_{s}\otimes\mathscr{H}_{\sigma}$ defined as,

${B}(k_{x})=cI_{0}\otimes\sigma_{0}\otimes s_{x},$ (22)

and

${A}(k_{x})=\mathbb{P}_{\uparrow}\otimes{h}_{\uparrow}(k_{x})+\mathbb{P}_{\downarrow}\otimes{h}_{\downarrow}(k_{x}),$
(23)

where ${h}_{s=\uparrow,\downarrow}(k_{x})$ are matrices in
$\mathscr{H}_{I}\otimes\mathscr{H}_{\sigma}$ given by

${h}_{s=\uparrow,\downarrow}(k_{x})=I_{-}\otimes\alpha_{s}(k_{x})+I_{0}\otimes\beta_{s}(k_{x})+I_{+}\otimes\gamma_{s}(k_{x}).$
(24)

In Eq. 24 , $I_{0}$, $I_{-}$ and $I_{+}$ are the main-, sub- and super-
diagonal identities in $\mathscr{H}_{I}$, e.g.,

$I_{0}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},I_{-}=\begin{pmatrix}0&0&0\\\ 1&0&0\\\
0&1&0\end{pmatrix},I_{+}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 0&0&0\end{pmatrix},$
(25)

for the case where $N_{y}=3$, and $\alpha_{s},\beta_{s}$, and $\gamma_{s}$ are
$2\times 2$ (real) matrices in $\mathscr{H}_{\sigma}$ defined as

$\begin{split}\alpha_{\uparrow}(k_{x})=&2t_{2}\cos\phi_{+}(k_{x})\sigma_{z}-t_{1}\sigma_{+},\\\
\beta_{\uparrow}(k_{x})=&[2t_{2}\cos\phi_{-}(k_{x})+m]\sigma_{z}\\\
&+q\sigma_{0}-2t_{1}\cos\frac{k_{x}}{2}\sigma_{x},\\\
\gamma_{\uparrow}(k_{x})=&2t_{2}\cos\phi_{+}(k_{x})\sigma_{z}-t_{1}\sigma_{-},\end{split}$
(26)

and

$\begin{split}\alpha_{\downarrow}(k_{x})=&-2t_{2}\cos\phi_{+}(k_{x})\sigma_{z}-t_{1}\sigma_{+},\\\
\beta_{\downarrow}(k_{x})=&[-2t_{2}\cos\phi_{-}(k_{x})+m]\sigma_{z}\\\
&+q\sigma_{0}-2t_{1}\cos\frac{k_{x}}{2}\sigma_{x},\\\
\gamma_{\downarrow}(k_{x})=&-2t_{2}\cos\phi_{+}(k_{x})\sigma_{z}-t_{1}\sigma_{-},\end{split}$
(27)

where $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$ and

$\phi_{+}(k_{x})=\phi+\frac{k_{x}}{2},\quad\phi_{-}(k_{x})=\phi-k_{x}.$ (28)

##  III Haldane model

As explained in Sec. II, our system described in the absence of pairing
interactions is reduced to two independent copies of the Haldane model, a
spin-$\uparrow$ copy and a spin-$\downarrow$ copy. As a preparation for the
topological study of our system which includes pairing interactions, we review
the Chern number in Sec. III.1, and conduct an analytical study of the edge
modes in semi-infinite plane geometries in Sec. III.2, using, without loss of
generality, the spin-$\uparrow$ copy of the Haldane model.

###  III.1 Bulk Topology: Chern Number

In 1988, Haldane [3] demonstrated, by subjecting a 2D graphene model to a
periodic local magnetic field with no net magnetic flux, that the IQH effect
can appear as a result of time reversal symmetry breaking, rather than from an
external magnet field (for that matter, Landau levels). The Haldane model
according to the 10-fold way is a class A insulator whose bulk is
characterized by the Chern number. The Chern number is an intrinsic property
of the bulk band structure, which is determined from the eigenvalue equation
of $h_{\uparrow}(\mathbf{k})$,

$h_{\uparrow}(\mathbf{k})\ket{n}=\left[\omega_{n}(\mathbf{k})+\epsilon(\mathbf{k})\right]\ket{n},$
(29)

where

$\omega_{1}(\mathbf{k})=q-h(\mathbf{k}),\quad\omega_{2}(\mathbf{k})=q+h(\mathbf{k}),$
(30)

are the bulk band energies [relative to $\epsilon(\mathbf{k})]$ with

$h(\mathbf{k})=\sqrt{h^{2}_{x}(\mathbf{k})+h^{2}_{y}(\mathbf{k})+h^{2}_{z}(\mathbf{k})},$
(31)

and

$\begin{split}\ket{1}=&\cos\theta_{\mathbf{k}}\ket{A}-\sin\theta_{\mathbf{k}}e^{i\phi_{\mathbf{k}}}\ket{B},\\\
\ket{2}=&\sin\theta_{\mathbf{k}}e^{-i\phi_{\mathbf{k}}}\ket{A}+\cos\theta_{\mathbf{k}}\ket{B},\end{split}$
(32)

are the corresponding eigenvectors in sublattice space with

$\begin{split}\theta_{\mathbf{k}}=&\tan^{-1}\sqrt{\frac{h(\mathbf{k})+h_{z}(\mathbf{k})}{h(\mathbf{k})-h_{z}(\mathbf{k})}},\\\
e^{i\phi_{\mathbf{k}}}=&\frac{h_{x}(\mathbf{k})+ih_{y}(\mathbf{k})}{\sqrt{h^{2}_{x}(\mathbf{k})+h^{2}_{y}(\mathbf{k})}}.\end{split}$
(33)

Phase transitions cannot take place unless the upper band,
$\omega_{2}(\mathbf{k})$, and the lower band, $\omega_{1}(\mathbf{k})$, cross,
which may only occur at Dirac points $K$ and $K^{\prime}$ defined in the first
Brillouin zone as shown in Fig. 1(b). The low-energy physics near $K$ and
$K^{\prime}$ are modeled by the Dirac Hamiltonians, which are defined,
respectively, as

$\displaystyle h_{\uparrow}(\mathbf{K}+\delta\mathbf{k})$
$\displaystyle=\tilde{m}_{+}\sigma_{z}+\frac{\sqrt{3}t_{1}}{2}(\delta
k_{x}\sigma_{x}+\delta k_{y}\sigma_{y}),$ (34) $\displaystyle
h_{\uparrow}(\mathbf{K}^{\prime}+\delta\mathbf{k})$
$\displaystyle=\tilde{m}_{-}\sigma_{z}-\frac{\sqrt{3}t_{1}}{2}(\delta
k_{x}\sigma_{x}-\delta k_{y}\sigma_{y}),$ (35)

(apart from an overall constant) where

$\tilde{m}_{\pm}=m\pm 3\sqrt{3}t_{2}\sin\phi.$ (36)

Note that, in arriving at Eq. (35), we used, instead of $K^{\prime}$ marked in
Fig. 1 (b), a Dirac point equivalent to $K^{\prime}$ but that lies opposite to
$K$.

Using the Streda formula [64], Haldane found, from the Hall conductance of
this effective model, the Chern number,

$C=\frac{1}{2}\left[\mathrm{sgn}(\tilde{m}_{+})-\mathrm{sgn}(\tilde{m}_{-})\right],$
(37)

which gives rise to the iconic phase diagram with two loops in the $\phi-m$
space as shown in the inset of Fig. 2(a). The same result can also be obtained
from Eq. (90) in Sec. IV.2, where we describe a generic approach for
determining the Chern number from a surface integral of Berry curvatures,
which, for the Haldane model, are determined by the eigenstates of
$h_{\uparrow}(\mathbf{k})$ summarized in Eq. (32).

Figure 2:  (a) The energy spectrum, $E_{\uparrow}(k_{x})$, of the Hamiltonian,
$h_{\uparrow}(k_{x})$ [Eq. (24)], for the Haldane model in the stripe geometry
for a state located at $m=3^{-1}\times 3\sqrt{3}t_{2}$ and $\phi=\pi/4$ in the
$\phi-m$ phase diagram (inset), where regions with the Chern number $-1$,
$+1$, and $0$ are indicated with blue, orange, and white colors, respectively.
Panel (a) compares numerical edge mode dispersions (dotted black lines) with
their analytical counterparts in semi-infinite geometries from Eqs. (100) and
(101) with $i=b$ [which are reformulations of Eqs. (52) and (57)] for the
bottom edge mode (red dashed line) and from Eqs. (100) and Eq. (102) with
$i=t$ for the top edge mode (blue dashed line). Panels (b) and (c) compare
numerically normalized population distributions,
$\absolutevalue{\psi_{n}}^{2}$ (solid lines), and their analytical
counterparts (dashed lines) for the edge modes at point $b$ and point $c$ with
$k_{x}=\pi$ marked in panel (a). Other parameters are $t_{1}=t_{2}=1$, $q=0$,
and $N_{y}=20$.

###  III.2 Edge modes

For an open system operating such that its bulk is characterized with a
nontrivial Chern number, the bulk-boundary correspondence [65, 66] dictates
that topologically protected modes, robust to defects and disorder, exist
along the system boundaries. Good insight into edge modes can be gained by
analyzing the Haldane model in semi-infinite planes with zigzag boundaries.
For illustrative purposes, we consider the edge mode at the bottom of an
upper-half plane that is localized at $n=1$ where $n$ is the index to the $y$
unit cell. This edge mode, within the context of the present model, is an
eigenstate of $h_{\uparrow}(k_{x})$, a tridiagonal system described by Eq.
(24). Let $\psi_{n}$ describe the amplitude of this state at the $n$th unit
cell, where $\psi_{n}$ is itself a two-component vector in sublattice space.
This state with eigenvalue $\epsilon(k_{x})$ is governed in the bulk where
$n>1$ by

$\alpha_{\uparrow}\psi_{n-1}+\beta_{\uparrow}\psi_{n}+\gamma_{\uparrow}\psi_{n+1}=\epsilon\psi_{n},$
(38)

and in the boundary where $n=1$ by

$\beta_{\uparrow}\psi_{1}+\gamma_{\uparrow}\psi_{2}=\epsilon\psi_{1},$ (39)

where we made use of the open boundary condition $\psi_{0}=0$. Note that when
no confusion is likely to arise, we do not indicate dependence on $k_{x}$. We
choose

$\psi_{n}\propto\lambda^{n}\psi_{1}$ (40)

as the ansatz for the bulk [67, 68], where $\absolutevalue{\lambda}<1$, as
required by the boundary condition $\psi_{n}\rightarrow 0$ in the semi-
infinite limit $n\rightarrow\infty$. This ansatz changes Eq. (38) into a set
of homogeneous equations for $\psi_{1}$,

$\left[\lambda^{-1}\alpha_{\uparrow}+\beta_{\uparrow}+\lambda\gamma_{\uparrow}\right]\psi_{1}=\epsilon\psi_{1}.$
(41)

Just as in the case when $m=0$ [69], the condition for non-trivial solutions
to Eq. (41) gives rise immediately to a quadratic function of
$\lambda+\lambda^{-1}$ (not shown) which guarantees the existence of two
roots, which we call $\lambda_{\pm}$, smaller in magnitude than 1. This
implies that a general edge mode is a linear superposition of the solution
with $\lambda_{+}$ and $\lambda_{-}$. Together with the boundary condition at
$n=0$, a general edge mode solution takes the form,

$\psi_{n}\propto(\lambda_{+}^{n}-\lambda_{-}^{n})f,$ (42)

where $f$ is the solution to Eq. (41). In terms of $f$, the equation at the
boundary $n=1$, Eq. (39), becomes

$\left[\beta_{\uparrow}+\gamma_{\uparrow}(\lambda_{+}+\lambda_{-})\right]f=\epsilon
f,$ (43)

which is a matrix equation in $\mathscr{H}_{\sigma}$,

$\begin{pmatrix}2t_{2}\cos\phi_{-}+m+2t_{2}(\lambda_{+}+\lambda_{-})\cos\phi_{+}&-2t_{1}\cos\frac{k_{x}}{2}\\\
-2t_{1}\cos\frac{k_{x}}{2}-t_{1}(\lambda_{+}+\lambda_{-})&-2t_{2}\cos\phi_{-}-m-2t_{2}(\lambda_{+}+\lambda_{-})\cos\phi_{+}\end{pmatrix}\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}=(\epsilon-q)\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}.$
(44)

Combining Eq. (43) with Eq. (41) where $\psi_{1}$ is replaced with $f$ gives
rise to

$(\alpha_{\uparrow}-\lambda_{+}\lambda_{-}\gamma_{\uparrow})f=0,$ (45)

which is another matrix equation in $\mathscr{H}_{\sigma}$,

$\begin{pmatrix}1-\lambda_{+}\lambda_{-}&-\frac{t_{1}}{2t_{2}\cos\phi_{+}}\\\
+\frac{t_{1}\lambda_{+}\lambda_{-}}{2t_{2}\cos\phi_{+}}&\lambda_{+}\lambda_{-}-1\end{pmatrix}\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix}=0.$ (46)

The edge state and its energy are obtained by solving Eqs. (44) and (46)
simultaneously. We first consider Eq. (46) which is independent of
$\epsilon(k_{x})$. We find that under the condition,

$(2t_{2}\cos\phi_{+})^{2}(\lambda_{+}\lambda_{-}-1)^{2}-t_{1}^{2}(\lambda_{+}\lambda_{-}-1)-t_{1}^{2}=0,$
(47)

Eq. (46) supports a nontrivial solution,

$f\propto\begin{pmatrix}-t_{1}\\\
(\lambda_{+}\lambda_{-}-1)2t_{2}\cos\phi_{+}\end{pmatrix},$ (48)

where

$\begin{split}\lambda_{+}\lambda_{-}&=1+a_{1},\end{split}$ (49)

is the solution of the quadratic equation (47) with
$\absolutevalue{\lambda_{+}\lambda_{-}}<1$, where

$a_{1}(k_{x})=\frac{t_{1}^{2}}{8t_{2}^{2}\cos^{2}\phi_{+}}-\sqrt{\left(1+\frac{t_{1}^{2}}{8t_{2}^{2}\cos^{2}\phi_{+}}\right)^{2}-1}.$
(50)

We next consider Eq. (44), which, because $f_{1}$ and $f_{2}$ are already
known through Eq. (48), is actually an inhomogeneous equation of $\epsilon-q$
and $\lambda_{+}+\lambda_{-}$,

$\begin{pmatrix}f_{1}&,&-2t_{2}\cos\phi_{+}f_{1}\\\
f_{2}&,&2t_{2}\cos\phi_{+}f_{2}-t_{1}f_{1}\end{pmatrix}\begin{pmatrix}\epsilon-q\\\
\lambda_{+}+\lambda_{-}\end{pmatrix}=\begin{pmatrix}-2t_{1}\cos\frac{k_{x}}{2}f_{2}+(2t_{2}\cos\phi_{-}+m)f_{1}\\\
-2t_{1}\cos\frac{k_{x}}{2}f_{1}-(2t_{2}\cos\phi_{-}+m)f_{2}\end{pmatrix}.$
(51)

Solving Eq. (51), we arrive at the desired edge state dispersion,

$\epsilon(k_{x})=q+\epsilon^{\prime}(k_{x}),$ (52)

where

$\epsilon^{\prime}(k_{x})=\absolutevalue{t_{1}}\frac{2t_{2}\cos\phi_{-}+m-8t_{2}\cos\frac{k_{x}}{2}\cos\phi_{+}}{\sqrt{t_{1}^{2}+16t_{2}^{2}\cos^{2}\phi_{+}}},$
(53)

and

$\lambda_{+}+\lambda_{-}=a_{1}a_{2},$ (54)

where

$a_{2}(k_{x})=\frac{2t_{1}^{2}\cos\frac{k_{x}}{2}+4t_{2}\cos\phi_{+}\left(2t_{2}\cos\phi_{-}+m\right)}{\absolutevalue{t_{1}}\sqrt{t_{1}^{2}+16t_{2}^{2}\cos^{2}\phi_{+}}}.$
(55)

Finally, we have by solving simultaneously Eq. (49) and Eq. (54),

$\lambda_{\pm}(k_{x})=\frac{1}{2}\left(a_{1}a_{2}\pm\sqrt{a_{1}^{2}a_{2}^{2}-4(1+a_{1})}\right).$
(56)

If the edge mode exists, it must be limited to the momentum space in which

$\absolutevalue{\lambda_{+}(k_{x})}<1\mbox{ and
}\absolutevalue{\lambda_{-}(k_{x})}<1$ (57)

hold true simultaneously.

Equation (53), which we have just obtained in the same spirit as the transfer
matrix method [65], reduces to the form by Galilo et al. [59] when
$\phi=\pi/2$ and $m=0$ and the form in our earlier publication [62] when
$\phi=\pi/2$ but $m$ remains arbitrary. There also exists in the literature
other analytical studies under limited conditions [70, 71, 72]. While
preparing this paper, we became aware of a recent paper [73] which includes a
similar problem and their result for $\phi=\pi/2$ has the same simple
mathematical form as ours.

Figure 2(a) compares the energy dispersions and Figs. 2(b,c) compare the
population distributions of the edge modes, as obtained analytically for semi-
infinite plane geometries, with those obtained numerically from eigenvalues of
the Hamiltonian $h_{\uparrow}(k_{x})$ in Eq. (24) in the stripe geometry. We
can see that the dispersions agree quite well; discrepancies due to finite-
size effects are not discernible even when $N_{y}$ is as low as $N_{y}=20$.

The edge mode that we have studied so far lies at the bottom edge. There shall
be an edge mode at the top, traveling in the opposite direction, according to
the bulk-boundary correspondence for a Chern insulator [65, 66]. An analogous
set of results can be obtained for this top edge mode by analyzing a lower-
half plane. We will postpone this and other related results until Sec. IV.3
where we apply them to gain insight into the edge modes of an open system with
pairing interactions.

##  IV Our model without time-reversal symmetry:topological properties

Having reviewed the Haldane model in Sec. III, we now turn our attention to
our model in Sec. II which includes pairing interactions and is described by a
bosonic BdG Hamiltonian. We study the topological properties of such a system
in connection with questions such as what topological symmetry class our
system belongs to under the 38-fold way, what topological invariant we use to
characterize the bulk, and how pairing interactions affect the edge modes. We
pursue these questions for systems without time-reversal symmetry in this
section and make a related study for systems with time-reversal symmetry in
the next section, where we study the physics behind topological amplifiers.

To avoid confusion with terminology, we stress at the onset that we follow the
convention set by Kawabata et al. [47], where, for example,
$\mathcal{C}H^{T}_{BdG}(\mathbf{k})\mathcal{C}^{-1}=-H_{BdG}(-\mathbf{k})$,
rather than the usual
$\mathcal{C}H^{*}_{BdG}(\mathbf{k})\mathcal{C}^{-1}=-H_{BdG}(-\mathbf{k})$, is
defined as the particle-hole symmetry. A topological phase describes a group
of Hamiltonians that can be continuously deformed into each other without
causing their energy bands to close, i.e., to cross a reference, which, for
non-Hermitian Hamiltonians, where eigenenergies live in a complex plane, can
be either a point or a line [47]. In the present study, we always deal with
gaps that exist in the real parts of complex energies and topological phases
are thus always classified according to a real line gap in which the imaginary
axis in the complex-energy plane serves as the reference for an energy-band
crossing.

###  IV.1 Symmetry and topological classification

We begin the topological study of our model by analyzing how the Bloch BdG
Hamiltonian (14) or equivalently,

$\begin{split}&{H}_{BdG}(\mathbf{k})=\mathbb{P}_{+}\mathbb{P}_{\uparrow}h_{\uparrow}(\mathbf{k})+\mathbb{P}_{+}\mathbb{P}_{\downarrow}h_{\downarrow}(\mathbf{k})\\\
&-\mathbb{P}_{-}\mathbb{P}_{\uparrow}h^{*}_{\uparrow}(-{\mathbf{k}})-\mathbb{P}_{-}\mathbb{P}_{\downarrow}h^{*}_{\downarrow}(-{\mathbf{k}})+ic\tau_{y}s_{x}\sigma_{0},\end{split}$
(58)

transforms under various symmetry operations, where
$\mathbb{P}_{\uparrow,\downarrow}$ and $\mathbb{P}_{+,-}$ are the projection
operators in spin and Nambu space defined in Eq. (16). If confusion is
unlikely to arise, we do not explicitly write the tensor product symbol
$\otimes$ (as well as the subspace identity matrices). The bosonic BdG
Hamiltonian (58) obeys the pseudo-Hermiticity condition,

$\eta H^{\dagger}_{BdG}(\mathbf{k})\eta^{-1}=H_{BdG}(\mathbf{k}),$ (59)

where $\eta=\tau_{z}(=\Sigma_{z}$) is the matrix representing pseudo-
Hermiticity. That $\eta$ and $\Sigma_{z}$ being equivalent follows from the
“first quantized” Hamiltonian $\Sigma_{z}H_{BdG}(\mathbf{k})$ being Hermitian.
In addition, the Bose BdG Hamiltonian obeys particle-hole symmetry,

$\mathcal{C}H^{T}_{BdG}(\mathbf{k})\mathcal{C}^{-1}=-H_{BdG}(-\mathbf{k}),$
(60)

where $\mathcal{C}=\tau_{y}$ is the matrix representing particle-hole
symmetry. The pseudo-Hermiticity and particle-hole symmetry are intrinsic to
the bosonic BdG Hamiltonian; they act more as constraints arising from Bose
statistics than as symmetries imposed by additional conditions. As to how it
transforms under time reversal, we note that, for our pseudo-spin-$1/2$
system, the time-reversal operator is given by $\mathcal{T}=s_{x}$, which is a
time-reversal even operator, i.e., $\mathcal{T}^{2}=1$. It can be shown that
Eq. (58) does not possess time-reversal symmetry, i.e.,

$\mathcal{T}H^{*}_{BdG}(\mathbf{k})\mathcal{T}^{-1}=H_{BdG}(-\mathbf{k}),$
(61)

unless $\cos\phi=0$, i.e., $\phi=\pi/2$ (or any half-integral multiple of
$\pi$), a situation which we will deal with separately in Sec. V.

For our model, topological properties are not determined by
$H_{BdG}(\mathbf{k})$ but rather by Hamiltonians in smaller Hilbert spaces as
we now explain. It comes down to magnetization being conserved under
Hamiltonian (1), i.e., $[\hat{Q},\hat{H}_{1/2}]=0$, where

$\begin{split}\hat{Q}&=\sum_{\mathbf{i}}\left(\hat{b}^{\dagger}_{\mathbf{i},\uparrow}\sigma_{0}\hat{b}_{\mathbf{i},\uparrow}-\hat{b}^{\dagger}_{\mathbf{i},\downarrow}\sigma_{0}\hat{b}_{\mathbf{i},\downarrow}\right)\\\
\end{split}$ (62)

is the magnetization operator (often called the charge operator [74]) in
position space. In momentum space, conservation of magnetization amounts to

$[J_{z},H_{BdG}(\mathbf{k})]=0,$ (63)

so that $J_{z}$ and $H_{BdG}(\mathbf{k})$ can be simultaneously diagonalized,
where $J_{z}=\tau_{z}s_{z}$ is the matrix representation of the magnetization
operator in momentum space (also called the $z$-component of the spin rotation
generator represented on Nambu space [52]). In fact, $J_{z}$ partitions
$H_{BdG}(\mathbf{k})$ to

$H_{BdG}(\mathbf{k})=H_{\Uparrow}(\mathbf{k})\oplus
H_{\Downarrow}(\mathbf{k}),$ (64)

a direct sum of $H_{\Uparrow}(\mathbf{k})$ on the degenerate subspace of
$J_{z}$ with eigenvalue $+1$ and $H_{\Downarrow}(\mathbf{k})$ on the
degenerate subspace of $J_{z}$ with eigenvalue $-1$, where

$H_{\Uparrow,\Downarrow}(\mathbf{k})=\begin{pmatrix}h_{\uparrow,\downarrow}(\mathbf{k})&c\sigma_{0}\\\
-c\sigma_{0}&-h^{*}_{\downarrow,\uparrow}(-\mathbf{k})\end{pmatrix},$ (65)

in Nambu space. As a result, our system inherits the topological classes of
and can therefore be classified according to the Hamiltonians in smaller
sectors, $H_{\Uparrow}(\mathbf{k})$ and $H_{\Downarrow}(\mathbf{k})$.

Without loss of generality, we focus on the $J_{z}$-$\Uparrow$ sector and
investigate symmetry as well as energy-band properties of
$H_{\Uparrow}(\mathbf{k})$. We begin by rewriting $H_{\Uparrow}(\mathbf{k})$
in terms of the matrices in Nambu space as

$H_{\Uparrow}(\mathbf{k})=\mathbb{P}_{+}h_{\uparrow}(\mathbf{k})-\mathbb{P}_{-}h^{*}_{\downarrow}(-\mathbf{k})+ic\tau_{y}\sigma_{0},$
(66)

which is perhaps the simplest spinful system built on copies of the Haldane
model that involve pairing interactions.

The Hamiltonian in the $J_{z}$-$\Uparrow$ sector obeys the pseudo-Hermiticity,

$\eta H^{\dagger}_{\Uparrow}(\mathbf{k})\eta^{-1}=H_{\Uparrow}(\mathbf{k}).$
(67)

However, unlike its parent Hamiltonian $H_{BdG}(\mathbf{k})$ in Eq. (58),
$H_{\Uparrow}(\mathbf{k})$ does not preserve particle-hole symmetry,

$\mathcal{C}H^{T}_{\Uparrow}(\mathbf{k})\mathcal{C}^{-1}\neq-
H_{\Uparrow}(-\mathbf{k}),$ (68)

since $\mathcal{C}H^{T}_{\Uparrow}(\mathbf{k})\mathcal{C}^{-1}$ is found to
equal $-H_{\Downarrow}(-\mathbf{k})$ instead of $-H_{\Uparrow}(-\mathbf{k})$.
For this reason, $H_{\Uparrow}(\mathbf{k})$ is not a BdG Hamiltonian.
Furthermore, it possesses neither time-reversal symmetry nor chiral symmetry.
As such, $H_{\Uparrow}(\mathbf{k})$ belongs to symmetry class $A+\eta$ whose
topological invariant in two dimensions is $\mathbb{Z}\oplus\mathbb{Z}$ when
classified by a real line gap. In the next section, we further discuss this
topological invariant and the related Chern numbers for systems with stable
bulk energy bands.

Figure 3: The Bloch energy spectra [relative to $\epsilon(\mathbf{k})$],
sampled at points a, b, c, d, and e as indicated in panel (f), are shown,
respectively, in panels (a)-(e) where $\phi=0$ and $m=0.5\times
3\sqrt{3}t_{2}$ at point a, $\phi=\pi/8$ and $m=\sin(\pi/8)\times
3\sqrt{3}t_{2}$ at point b, $\phi=3\pi/8$ and $m=0.3\times 3\sqrt{3}t_{2}$ at
point c, $\phi=\pi/2$ and $m=0.7\times 3\sqrt{3}t_{2}$ at point d, and
$\phi=25\pi/32$ and $m=0.7\times 3\sqrt{3}t_{2}$ at point e. These Bloch
energy spectra are calculated along a closed path, $\Gamma\rightarrow
K\rightarrow M\rightarrow K^{\prime}\rightarrow\Gamma$, in the first Brillouin
zone as indicated in Fig. 1(b). The red and green curves represent,
respectively, the particle and hole energy bands (of real energies) while the
dark blue curves [only in panel (d)] depict the imaginary parts of complex
energies. (The distinction between the particle and hole bands is clarified in
Sec. IV.3.) Other parameters are the same as in Fig. 4(a).

###  IV.2 Topological invariant: The Chern number

The Chern number traces its root to the geometric phase, which arises
naturally in Hermitian Hamiltonians due to the interplay between unitary
nature of time evolution and orthornormality property of eigenenergy states.
In 1988, Garrison and Wright [75] extended the Berry phase to non-Hermitian
systems with the help of the bi-orthonormality relation constructed from right
and left eigenvectors, which are defined, within the context of our work, as

$\begin{split}H_{\Uparrow}(\mathbf{k})\ket{E}=&\left[E(\mathbf{k})+\epsilon(\mathbf{k})\right]\ket{E},\\\
\left<\left<E\right|\right.H_{\Uparrow}(\mathbf{k})=&\left[E(\mathbf{k})+\epsilon(\mathbf{k})\right]\left<\left<E\right|\right..\end{split}$
(69)

The energy spectrum [relative to $\epsilon(\mathbf{k})$] consists of four
branches given by

$E_{n,\alpha}(\mathbf{k})=\alpha E_{n}(\mathbf{k}),\quad
n=1,2,\quad\alpha=\pm,$ (70)

where

$E_{n}(\mathbf{k})=\sqrt{\omega^{2}_{n}(\mathbf{k})-c^{2}}.$ (71)

Figure 3 showcases several bulk (excitation) energy spectra, which we will
explain with more detail towards the end of this section, along a closed path,
$\Gamma\rightarrow K\rightarrow M\rightarrow K^{\prime}\rightarrow\Gamma$, in
the first Brillouin zone as shown in Fig. 1(b). The eigenvectors of energy
$E_{n,\alpha}(\mathbf{k})$ are given by

$\begin{split}\ket{E_{n,\alpha}}=&\ket{n,\alpha}\otimes\ket{n},\\\
\left<\left<E_{n,\alpha}\right|\right.=&\left<\left<n,\alpha\right|\right.\otimes\bra{n},\end{split}$
(72)

where $\ket{n}$ are the eigenvectors [with eigenvalues
$\omega_{n}(\mathbf{k})$] of $h_{\uparrow}(\mathbf{k})$ given by Eq. (32) and

$\begin{split}\ket{n,\alpha}=&-u_{n\alpha}(\mathbf{k})\ket{+}+v_{n\alpha}(\mathbf{k})\ket{-},\\\
\left<\left<n,\alpha\right|\right.=&\left[-u_{n\alpha}(\mathbf{k})\bra{+}+v_{n\alpha}(\mathbf{k})\bra{-}\right]\tau_{z},\end{split}$
(73)

are the right and left eigenvectors in Nambu space $(\ket{+},\ket{-})$, where

$\begin{split}u_{n\alpha}(\mathbf{k})=&\frac{c}{\sqrt{2\alpha
E_{n}(\mathbf{k})\left[\omega_{n}(\mathbf{k})-\alpha
E_{n}(\mathbf{k})\right]}},\\\
v_{n\alpha}(\mathbf{k})=&\frac{\omega_{n}(\mathbf{k})-\alpha
E_{n}(\mathbf{k})}{\sqrt{2\alpha
E_{n}(\mathbf{k})\left[\omega_{n}(\mathbf{k})-\alpha
E_{n}(\mathbf{k})\right]}}.\end{split}$ (74)

This set of four eigenstates are normalized in accordance with the bi-
orthonormality condition,

$\left<\left<E_{n,\alpha}\right|\ket{E_{m,\beta}}\right.=\delta_{n,m}\delta_{\alpha,\beta},$
(75)

and obey the completeness relation,

$\sum_{n,\alpha}\ket{E_{n,\alpha}}\left<\left<E_{n,\alpha}\right|\right.=\tau_{0}\sigma_{0},$
(76)

which, together with the linear superposition principle, forms the two pillars
upon which phenomena associated with the Berry phase can be generalized from
Hermitian to non-Hermitian systems in a straightforward manner.

The Berry phase for the $E_{n,\alpha}$ energy band is a (closed) line integral
of the Berry potential

$A_{n\alpha}=i\left<\left<E_{n,\alpha}\right|\right.d\ket{E_{n,\alpha}},$ (77)

which is a 1-form involving both the right and left eigenenergy states, where
$d$ is the exterior derivative. The Berry curvature is then a 2-form obtained
by taking the exterior derivative of the Berry potential,

$F_{n\alpha}=dA_{n\alpha}.$ (78)

Let

$A_{n\alpha,j}(\mathbf{k})\equiv
i\left<\left<E_{n,\alpha}(\mathbf{k})\right|\right.\partial_{j}\ket{E_{n,\alpha}(\mathbf{k})},$
(79)

where $\partial_{j}\equiv\partial/\partial k_{j}$ with $j=x,y$. In momentum
space, the Berry potential and the corresponding curvature are given,
respectively, by

$A_{n\alpha}(\mathbf{k})=A_{n\alpha,x}(\mathbf{k})dk_{x}+A_{n\alpha,y}(\mathbf{k})dk_{y},$
(80)

and

$F_{n\alpha}(\mathbf{k})=\left[\partial_{x}A_{n\alpha,y}(\mathbf{k})-\partial_{y}A_{n\alpha,x}(\mathbf{k})\right]dk_{x}\wedge
dk_{y}.$ (81)

The first Chern number is defined as a surface integral over an entire first
Brillouin zone, which is a closed manifold (a torus),

$C_{n\alpha}=\frac{1}{2\pi}\int_{BZ}F_{n\alpha}.$ (82)

Figure 4 is a phase diagram in the $\phi-m$ space which divides into stable
and unstable regions depending on whether all bulk energy bands in Eq. (70)
are free of any complex values. In the stable region, we adapt the gauge-
invariant algorithm developed by Fukui et al. [76] to Eq. (82) where the Berry
connection is formulated in terms of both right and left eigenvectors; we
apply it to numerically determine, without loss of generality, the Chern
number for the $E_{n=1,\alpha=+}(\mathbf{k})$ band. We find that the boundary
separating the topologically trivial phase from the topologically nontrivial
phase is the same as in the Haldane model [see the inset of Fig. 2(a)].

We now work to prove that the Chern number in the stable region should indeed
be that of the Haldane model. We first recall that the integral for the Chern
number [Eq. (82)] has been formulated in momentum $\mathbf{k}=(k_{x},k_{y})$
space. However, states in Eq. (72) depend on $\mathbf{k}$ only via the map
$\mathbf{k}\mapsto\mathbf{h(\mathbf{k})}$, and have nothing to do with
$\epsilon(\mathbf{k})$. As a result, our system can be understood as living in
the $\mathbf{h}=(h_{x},h_{y},h_{z})$ space as far as band topology is
concerned. Thus, we consider the Berry curvature 2-form in the
$\mathbf{h}=(h_{x},h_{y},h_{z})$ space,

$F_{n\alpha}(\mathbf{h})=i\sum_{m\beta\neq
n\alpha}\frac{\left<\left<E_{n,\alpha}\right|\right.\partial_{i}H_{\Uparrow}\ket{E_{m,\beta}}\left<\left<E_{m,\beta}\right|\right.\partial_{j}H_{\Uparrow}\ket{E_{n,\alpha}}}{(E_{n,\alpha}-E_{m,\beta})^{2}}dh_{i}\wedge
dh_{j},$ (83)

where repeated indices are summed over and $\partial_{j}=\partial/\partial
h_{j}$. Equation (83) generalizes the computation of the Berry curvature (in a
manifestly gauge-independent manner) from Hermitian systems where the
completeness relation is founded on the usual orthonormality condition [77] to
non-Hermitian system where the completeness relation (76) is founded on the
bi-orthonormality condition (75). Replacing $\partial_{i}H_{\Uparrow}$ in Eq.
(83) with a separable form

$\partial_{i}H_{\Uparrow}=\tau_{z}\sigma_{i},$ (84)

which we obtain from Eqs (66), (3), and (4), we simplify Eq. (83) to

$F_{n\alpha}=iD_{n\alpha}\bra{n}\sigma_{i}\ket{m}\bra{m}\sigma_{j}\ket{n}dh_{i}\wedge
dh_{j},$ (85)

where

$D_{n\alpha}=\sum_{\beta}\frac{\left<\left<n,\alpha\right|\right.\tau_{z}\ket{m,\beta}\left<\left<m,\beta\right|\right.\tau_{z}\ket{n,\alpha}}{(\alpha
E_{n}-\beta E_{m})^{2}},$ (86)

with $n\neq m$. In Appendix B, we show that the summation in Eq. (86) can be
reduced to

$D_{n\alpha}=\frac{1}{(\omega_{n}-\omega_{m})^{2}},$ (87)

so that

$F_{n\alpha}(\mathbf{h})=i\frac{\bra{n}\partial_{i}h_{\uparrow}\ket{m}\bra{m}\partial_{j}h_{\uparrow}\ket{n}}{(\omega_{n}-\omega_{m})^{2}}dh_{i}\wedge
dh_{j},$ (88)

where $\partial_{i}h_{\uparrow}=\sigma_{i}$ has been used. Equation (88) is
the Berry curvature of the Haldane model described by the two-band Hamiltonian
(3) and therefore has the well-known result [77],

$F_{1\alpha}(\mathbf{h})=-F_{2\alpha}(\mathbf{h})=\epsilon_{ijk}\frac{h_{k}(\mathbf{k})}{4h^{3}(\mathbf{k})}dh_{i}(\mathbf{k})\wedge
dh_{j}(\mathbf{k})$ (89)

where $\epsilon_{ijk}$ is the Levi-Civita symbol. Following standard practice,
we pull the Berry curvature 2-form in Eq. (89) back, via the map
$\mathbf{k}\mapsto\mathbf{h}(\mathbf{k})$, to momentum space where the first
Brillouin zone is well defined so that the Chern number integral (82) becomes

$C_{1\alpha}=-C_{2\alpha}=\int_{BZ}\frac{\mathbf{h}(\mathbf{k})\cdot\left[\partial_{x}\mathbf{h}(\mathbf{k})\times\partial_{y}\mathbf{h}(\mathbf{k})\right]}{4\pi
h^{3}(\mathbf{k})}d^{2}\mathbf{k}.$ (90)

Expressed in this way, the Chern number is recognized as the winding number —
the number of times that the image of the first Brillouin zone in the
$\mathbf{h}$ space wraps around the origin. This can be evaluated analytically
(see, for example, Ref. [78]) to give Haldane’s result [3] in Eq. (37). In
conclusion, we have

$\begin{split}C_{1+}&=-C_{2+}=C,\\\ C_{1-}&=-C_{2-}=C.\end{split}$ (91)

A comment is in order. In the previous section, we have classified
$H_{\Uparrow}(\mathbf{k})$, within the 38-fold way, as symmetry class A+$\eta$
with the $\mathbb{Z}\oplus\mathbb{Z}$ topological invariant. However, as one
may verify, for stable bands, states $\ket{E_{n,-}(\mathbf{k})}$ are connected
to states $\ket{E_{n,+}(\mathbf{k})}$ via a unitary transformation, i.e.,
$\ket{E_{n,-}(\mathbf{k})}=i\tau_{x}\ket{E_{n,+}(\mathbf{k})}$. As a result,
$\mathbb{Z}\oplus\mathbb{Z}$ is reduced to $\mathbb{Z}$. In arriving at Eq.
(91), we have shown, in essence, that this single $\mathbb{Z}$ corresponds to
$C$, the Chern number for the Haldane model in Eq. (37).

Figure 4: Phase diagrams in the $\phi-m$ space for $q=4t_{1}$ and (a)
$c=0.2t_{1}$ and (b) $c=0.8t_{1}$, which are identical to the phase diagram
for the Haldane model in the inset of Fig. 2 apart from the unstable region
with the light-purple color. Other parameters are $t_{1}=1$ and
$t_{2}=0.54t_{1}$.

We now return to Fig. 4 where pairing interactions increase from Figs. 4(a) to
4(b). We can see that an increased pairing interaction leads to an increased
region of instability, but also that, in the stable region, the Chern number
remain unchanged. There are two simple reasons that account for this. First,
pairing terms, i.e., the last line of Eq. (1), do not affect the $C_{3}$
point-group symmetry underlying a honeycomb lattice, which our model inherits
from the Haldane model, since it is built upon two copies of it. Thus, even in
the presence of pairing interations, a gap closing and reopening transition
still occurs at Dirac points as guaranteed by $C_{3}$ point group symmetry
[15]. Second, pairing in our model takes the form $ic\tau_{y}\sigma_{0}$ [the
last term in Eq. (66)] and therefore breaks neither time-reversal symmetry nor
inversion symmetry. Thus, as far as closing and reopening a gap at a Dirac
point is concerned, such a pairing remains a spectator, having no effect on
the topologically trivial-nontrivial phase transition. This is to be
contrasted to the Bose-Hubbard extension of the Haldane model, where pairing
may break inversion symmetry and therefore may affect the topologically
trivial-nontrivial phase transition [79].

We conclude this section by taking a look at Fig. 3(a)-3(e), which show Bloch
energy spectra sampled from states marked, respectively, as points a-e in Fig.
3(f) [where 3(f) a sketch representing the right-half of Fig. 4(a)]. Figures
3(a)-3(c) depict a gap-closing and -reopening transition from the trivial
gapped state at point a to the nontrivial gapped state at point c via the
gapless state at point b where two particle energy bands touch at Dirac point
$K^{\prime}$ [and two hole energy bands are guaranteed to touch also at the
same $K^{\prime}$ by pseudo-Hermiticity of $H_{\Uparrow}(\mathbf{k})$]. Figure
3(d) shows a typical unstable bulk state at point d where energies become
complex near the regions where there are level crossings between particle and
hole energy bands. Finally, Fig. 3(e) displays the energy spectrum at point e
where three phases, i.e., topologically trivial, topologically nontrivial, and
unstable states, meet; this spectrum is characterized by two distinct types of
gapless points, one between two particle energy bands (and two hole energy
bands) as in Fig. 3(b) and the other between a particle and hole energy band,
which signals the onset of dynamical instability.

###  IV.3 Edge modes

We now investigate our system with zigzag boundaries, as first presented in
Sec. II.2. We begin by noting that $J_{z}$ is still a conserved quantity under
$H_{BdG}(k_{x})$ in Eq. (21) and partitions ${H}_{BdG}(k_{x})$ into the direct
sum,

${H}_{BdG}(k_{x})={H}_{\Uparrow}(k_{x})\oplus{H}_{\Downarrow}(k_{x}),$ (92)

where

${H}_{\Uparrow,\Downarrow}(k_{x})=\begin{pmatrix}{h}_{\uparrow,\downarrow}(k_{x})&cI_{0}\sigma_{0}\\\
-cI_{0}\sigma_{0}&-{h}_{\downarrow,\uparrow}(-k_{x})\end{pmatrix}.$ (93)

with ${h}_{s=\uparrow,\downarrow}(k_{x})$ being the matrices defined earlier
in Eq. (24). As in the previous section, we focus on the sub-Hamiltonian in
the $J_{z}-\Uparrow$ sector and investigate the band structure of
${H}_{\Uparrow}(k_{x})$ in Eq. (93) or equivalently

${H}_{\Uparrow}(k_{x})=P_{+}{h}_{\uparrow}(k_{x})-P_{-}{h}_{\downarrow}(-k_{x})+ic\tau_{y}I_{0}\sigma_{0}$
(94)

by analyzing the Schrödinger equation,

${H}_{\Uparrow}(k_{x})\ket{E_{i}}=E_{i}(k_{x})\ket{E_{i}}.$ (95)

A comment is in order concerning the bi-orthonormalization scheme. In the
previous subsection, state $\ket{E_{i}}$ is normalized according to
$\left<\left<E_{i}\right|\ket{E_{j}}\right.=\delta_{i,j}$, which is
traditionally used in the study of non-Hermitian systems with gain and loss
[75]. It is a convenient normalization convention for problems where there is
no need to distinguish between the particle and hole band, as we demonstrated
in the previous section when we used this normalization to prove the result in
Eq. (91). In the present section, we use instead the bi-orthonormalization
scheme, which is traditionally used in the study of the bosonic BdG systems
[74]. A stable state in this scheme is classified either as a particle state
or as a hole state, depending on whether its norm with metric $\eta$ can be
scaled to $+1$ or $-1$, so that particle and hole states obey different
orthonormality relations given, respectively, by
$\bra{E_{i}}\eta\ket{E_{j}}=+\delta_{i,j}$ and
$\bra{E_{i}}\eta\ket{E_{j}}=-\delta_{i,j}$.

Figure 5: A typical (open-boundary) energy spectrum in a stable region with
$\phi=\pi/4$ and $m=0.1\times 3\sqrt{3}t_{2}$. The analytical mode
dispersions, $\epsilon_{p}^{b}(k_{x})$ at the bottom edge (dashed red line)
and $\epsilon_{p}^{t}(k_{x})$ at the top edge (dashed blue line), that lie
inside the gap of the particle bands (solid red lines), are determined from
Eqs. (100) - (102). The analytical mode dispersions, $\epsilon_{h}^{b}(k_{x})$
at the bottom edge (dashed orange line) and $\epsilon_{h}^{t}(k_{x})$ at the
top edge (dashed green line), that lie inside the gap of the hole bands (solid
green lines), are determined from Eqs. (106) -(108). Other parameters are the
same as in Fig. 4(a).

We stress that ${H}_{\Uparrow}(k_{x})$ in Eq. (94) for an open-boundary system
is not a BdG Hamiltonian since it does not obey particle-hole symmetry.
However, as we show in Appendix D, the aforementioned normalization involving
metric $\eta$ holds for any Hamiltonian that is pseudo-Hermitian, irrespective
of whether it obeys particle-hole symmetry; classification of eigenenergy
states into hole and particle types is solely determined by pseudo-
Hermiticity.

Figure 5 displays a typical open-boundary energy spectrum in the topologically
nontrivial region. There are two particle edge modes inside the gap between
two particle bulk bands (solid red lines) and two hole edge modes inside the
gap between two hole bulk bands (solid green lines). Also plotted in Fig. 5
are the edge mode dispersions of systems with semi-infinite plane geometries
in the limit of weak pairing where $c\rightarrow 0$ (dashed lines). In this
limit, the particle spectrum $\epsilon_{p}(k_{x})$ is the eigenenergy of
$h_{\uparrow}(k_{x})$

$h_{\uparrow}(k_{x})\ket{\psi_{p}}=\epsilon_{p}(k_{x})\ket{\psi_{p}}.$ (96)

while the hole spectrum $\epsilon_{h}(k_{x})$ is the eigenenergy of
$-h_{\uparrow}(-k_{x})$,

$-h_{\uparrow}(-k_{x})\ket{\psi_{h}}=\epsilon_{h}(k_{x})\ket{\psi_{h}}.$ (97)

What we obtained in Sec. III.2 is the particle edge mode of an upper-half
lane, a solution to Eq. (96). The method used there can be generalized to
solve, from Eqs. (96) and (97), all types of edge modes in semi-infinite
planes. We summarize all analytical results (including, for convenience, those
from in Sec. III.2) in terms of three functions of $k_{x}$,

$\begin{split}d(k_{x})&=\cos^{2}\phi_{+}(k_{x})\\\
p(k_{x})&=\cos\phi_{+}(k_{x})\cos\phi_{-}(k_{x}),\\\
r(k_{x})&=\cos\phi_{-}(k_{x})-4\cos(k_{x}/2)\cos\phi_{+}(k_{x}).\end{split}$
(98)

Let $i=b$ or $t$ and define $\text{sgn}(i)$ as

$\text{sgn}(i)=\begin{cases}+1&\mbox{ if }i=b,\\\ -1&\mbox{ if
}i=t,\end{cases}$ (99)

where “$b$” and “$t$” are abbreviations representing, respectively, the bottom
and top edge of a half plane. Then, we find from Eq. (96),

$\epsilon_{p}^{i}(k_{x})=q+\text{sgn}(i)\frac{\absolutevalue{t_{1}}\left[m+2t_{2}r(k_{x})\right]}{\sqrt{t_{1}^{2}+16t_{2}^{2}\cos^{2}\phi_{+}(k_{x})}}$
(100)

gives the particle edge mode dispersion at the bottom (dashed red line),
$\epsilon_{p}^{b}(k_{x})$, under the condition,

$\absolutevalue{\lambda_{p,+}^{b}(k_{x})}<1\mbox{ and
}\absolutevalue{\lambda_{p,-}^{b}(k_{x})}<1,$ (101)

and gives the particle edge mode dispersion at the top (dashed blue line),
$\epsilon_{p}^{t}(k_{x})$, under the condition,

$\absolutevalue{\lambda_{p,+}^{t}(k_{x})}>1\mbox{ and
}\absolutevalue{\lambda_{p,-}^{t}(k_{x})}>1,$ (102)

where

$\lambda^{i}_{p,\pm}(k_{x})=\frac{a_{p}^{i}a^{0}_{p}\pm\sqrt{(a_{p}^{i}a^{0}_{p})^{2}-4(1+a_{p}^{i})}}{2}$
(103)

are functions of

$a_{p}^{i}(k_{x})=\frac{(t_{1}/t_{2})^{2}}{8d(k_{x})}-\text{sgn}(i)\sqrt{\left[1+\frac{(t_{1}/t_{2})^{2}}{8d(k_{x})}\right]^{2}-1},$
(104)

and

$a_{p}^{0}(k_{x})=\frac{t_{1}^{2}\cos\frac{k_{x}}{2}+4t_{2}^{2}p(k_{x})+2t_{2}m\cos\phi_{+}(k_{x})}{0.5t_{1}\sqrt{t_{1}^{2}+16t_{2}^{2}\cos^{2}\phi_{+}(k_{x})}}.$
(105)

Similarly, we find from Eq. (97),

$\epsilon_{h}^{i}(k_{x})=-q-\text{sgn}(i)\frac{t_{1}\left[m-2t_{2}r(-k_{x})\right]}{\sqrt{t_{1}^{2}+16t_{2}^{2}\cos^{2}\phi_{+}(-k_{x})}},$
(106)

gives the hole edge mode dispersion at the bottom (dashed orange line),
$\epsilon_{h}^{b}(k_{x})$, under the condition,

$\absolutevalue{\lambda_{h,+}^{b}(k_{x})}<1\mbox{ and
}\absolutevalue{\lambda_{h,-}^{b}(k_{x})}<1,$ (107)

and gives the hole edge mode dispersion at the top (dashed green line),
$\epsilon_{h}^{t}(k_{x})$, under the condition,

$\absolutevalue{\lambda_{h,+}^{t}(k_{x})}>1\mbox{ and
}\absolutevalue{\lambda_{h,-}^{t}(k_{x})}>1,$ (108)

where

$\lambda_{h,\pm}^{i}(k_{x})=\frac{a_{h}^{i}a^{0}_{h}\pm\sqrt{(a_{h}^{i}a^{0}_{h})^{2}-4(1+a_{h}^{i})}}{2},$
(109)

are functions of

$a_{h}^{i}(k_{x})=a_{p}^{i}(-k_{x}),$ (110)

and

$a_{h}^{0}(k_{x})=\frac{t_{1}^{2}\cos\frac{k_{x}}{2}+4t_{2}^{2}p(-k_{x})-2t_{2}m\cos\phi_{+}(-k_{x})}{0.5t_{1}\sqrt{t_{1}^{2}+16t_{2}^{2}\cos^{2}\phi_{+}(-k_{x})}}.$
(111)

##  V Our model with time reversal symmetry:topological properties

A bosonic system, to which the Pauli exclusion principle is not applicable,
has a fascinating prospect of being made to operate as active topological
matter (“lasers”) where population amplification occurs in modes on boundaries
[80, 81, 82, 83, 84, 85, 86]. A topological amplifier in a bosonic BdG system
has unstable edge modes and a stable bulk, and is founded on the ability to
make dynamical instability occur preferentially in edge modes, which allows
edge modes to be populated at an exponentially fast rate [59, 62]. In Ref.
[62], we showed that a time-reversal-symmetric system ($\phi=\pi/2$) can be
made to behave like a topological amplifier, irrespective of whether the
system is inversion symmetric ($m=0$) [59] or inversion asymmetric ($m\neq
0$). In this section, we study our model with time-reversal symmetry and use
it as an opportunity to shed more light on the physics underlying topological
amplifiers.

An instability in a bosonic BdG model occurs because of a level crossing
between particle and hole states [87, 88, 89]. This insight led us to the
theorem in Ref. [62] which expresses the (purely imaginary) energy splitting
of a pair of degenerate states with energy sufficiently far away from zero in
terms of an unconventional commutator between the normal and abnormal parts of
a BdG Hamiltonian, e.g., $H_{BdG}(k_{x})$ in Eq. (21). The vanishing of this
“commutator” acts as a general and straightforward to use criterion for
creating a topological amplifier from systems described by BdG Hamiltonians.
In Appendix D, we adapt this criterion so that it can be directly applied to
sub-Hamiltonians like $H_{\Uparrow}(k_{x})$ in Eq. (93) which is pseudo-
Hermitian but not necessarily of a BdG type.

We begin by applying Eq. (158) to determine the unconventional commutator for
$\mathcal{H}_{\Uparrow}(k_{x})$ in Eq. (93), which is found to follow

$\begin{split}\left\lceil
A,B\right\rfloor&=cI_{-}[\alpha_{\uparrow}(k_{x})-\alpha_{\downarrow}(-k_{x})]\\\
&+cI_{0}[\beta_{\uparrow}(k_{x})-\beta_{\downarrow}(-k_{x})]\\\
&+cI_{+}[\gamma_{\uparrow}(k_{x})-\gamma_{\downarrow}(-k_{x})].\end{split}$
(112)

We next use Eqs. (26) and (27) to simplify Eq. (112) to

$\begin{split}\left\lceil A,B\right\rfloor=&4ct_{2}\cos\phi\\\
&\times\left[\cos
k_{x}I_{0}+\cos\frac{k_{x}}{2}(I_{-}+I_{+})\right]\sigma_{z},\end{split}$
(113)

which vanishes when $\phi=\frac{\pi}{2}$. Thus, the analysis of the sub-
Hamiltonian (65) arrives at the same conclusion as the analysis of the total
Hamiltonian (14) in [62]: As long as time reversal symmetry is preserved, our
system may act as a topological amplifier for which the bulk is free of
dynamical instabilities.

How does the existence of time-reversal symmetry for the total system affect
the topological classification of the sub-Hamiltonians? We have already shown
in Sec. IV.1 that $H_{\Uparrow}(\mathbf{k})$ is pseudo-Hermitian [Eq. 67)] but
not invariant under particle-hole symmetry [Eq. 68]. Furthermore,
$H_{\Uparrow}(\mathbf{k})$ is not time-reversal-symmetric even though the
total system is. However, it can be shown that when $\phi=\pi/2$,
$H_{\Uparrow}(\mathbf{k})$ is invariant under chiral symmetry,

$\Gamma
H^{\dagger}_{\Uparrow}(\mathbf{k})\Gamma^{-1}=-H_{\Uparrow}(\mathbf{k})$ (114)

where $\Gamma=\tau_{y}$ is a matrix representing chiral symmetry. As a
consequence, the symmetry class of this sub-Hamiltonian must be built upon AZ
symmetry class AIII. (As a side note, we mention that the AZ symmertry class
AIII was known to occur in non-Hermitian systems without pseudo-Hermiticity,
examples of which include a passive dimerized photonic crystal in one
dimension [90] and an active array of microring resonators [91]. Both Refs.
[90] and [91] use variants of the Su-Schrieffer-Heeger model [92] with loss
and gain, whose topological phases, when classified by a real line gap, are
characterized, according to the 38-fold way [47], by a $\mathbb{Z}$
topological invariant identified as the winding number.)

For pseudo-Hermitian Hamiltonians, whether $\eta$ commutes and anticommutes
with other symmetry operators plays an important role in the topological
classification. In our model, $\eta=\tau_{z}$ anticommutes with the chiral
symmetry operator $\Gamma=\tau_{y}$, i.e., $\\{\eta,\Gamma\\}=0$. Thus,
$H_{\Uparrow}(\mathbf{k})$ belongs to class AIII + $\eta_{-}$, whose
topological invariant, when classified by a real line gap, is $\mathbb{Z}$ in
two dimensions, which is again identified as the Chern number of the Haldane
model in Eq. (37), regardless of pairing interactions. In a nutshell, the
criterion that the unconventional commutator be made to vanish may impose
additional symmetries on a topological system, thereby altering the
classification of the topological system, e.g., it changes our model from the
A$+\eta$ class without time-reversal symmetry to the AIII$+\eta_{-}$ class
with time-reversal symmetry.

Figure 6(a) displays the phase diagram in the $q-m$ space for the
$J_{z}-\Uparrow$ subsystem with $\phi=\pi/2$. The topologically nontrivial
(trivial) phase would lie inside (outside) the region between the two
horizontal lines at $m=-3\sqrt{3}t_{2}$ and $m=+3\sqrt{3}t_{2}$, irrespective
of $q$, if pairing interactions were absent. As expected, in the presence of
pairing interactions, an unstable region (light purple) emerges, thereby
reducing the region with a topologically nontrivial phase (orange) as well as
the region with a topologically trivial phase (white).

Figure 6: Panel (a) is a phase diagram in the $q-m$ space for $\phi=\pi/2$ and
$c=0.2t_{1}$. Panels (b)-(d) display the Bloch energy spectra, sampled,
respectively, at points b, c and d as indicated in panel (a), where
$m=0.36\times 3\sqrt{3}t_{2}$ and $q=-4.2t_{1}$ at point b, $q=-2t_{1}$ at
point c, and $q=-0.4t_{1}$ at point d. Panels (e)-(g)[(h)-(j)] are the real
(imaginary) parts of the corresponding open-boundary energy spectra at points
b, c and d, respectively. For panels (e)-(g), red and green indicate,
respectively, hole and particle states just as in Fig. 5 while dark blue
indicates unstable states. The other parameters, $t_{1}$ and $t_{2}$, are the
same as in Fig. 4(a).

The bulk energy spectra for a periodic system can be computed analytically
from Eq. (70) with $\phi=\pi/2$. For models with open boundaries, we note that
for a time-reversal invariant total system,
$h_{\downarrow}(-k_{x})=h_{\uparrow}(k_{x})$ according to Eq. (24), so that
${H}_{\Uparrow}(k_{x})$ in Eq. (94) is invariant under $h_{\uparrow}(k_{x})$,

$[h_{\uparrow}(k_{x}),{H}_{\Uparrow}(k_{x})]=0.$ (115)

This indicates that $\omega_{n}$, which is the eigenenergy of
$h_{\uparrow}(k_{x})$,

$h_{\uparrow}(k_{x})\ket{\omega_{n}(k_{x})}=\omega_{n}(k_{x})\ket{\omega_{n}(k_{x})},$
(116)

is a good quantum number. As such, in the space
$\\{\ket{\omega_{n}(k_{x})}\\}$ where $h_{\uparrow}(k_{x})$ is diagonal,
${H}_{\Uparrow}(k_{x})$ in Eq. (94) is block diagonal,

${H}_{\Uparrow}(k_{x})=\sum_{n}\left[\omega_{n}(k_{x})\tau_{z}+ic\tau_{y}\right]\ket{\omega_{n}(k_{x})}\bra{\omega_{n}(k_{x})},$
(117)

where

$\omega_{n}(k_{x})\tau_{z}+ic\tau_{y}=\begin{pmatrix}\omega_{n}(k_{x})&c\\\
-c&-\omega_{n}(k_{x})\end{pmatrix},$ (118)

is a two state Hamiltonian in Nambu space. The energy spectra are then given
by

$E_{n,\pm}(k_{x})=\pm\sqrt{\omega^{2}_{n}(k_{x})-c^{2}},$ (119)

which appear in pairs of ($E_{n,+}(k_{x})$, $E_{n,-}(k_{x})=-E_{n,+}(k_{x})$)
as anticipated from an underlying sublattice symmetry:
$\sigma_{x}{H}_{\Uparrow}(k_{x})\sigma_{x}^{-1}=-{H}_{\Uparrow}(k_{x})$.

Figures 6(b)-6(d) display Bloch spectra (second column), real open-boundary
spectra in Figs. 6(e)-6(g) (third row), and the corresponding imaginary open-
boundary spectra in Figs. 6(h)-6(j) (fourth row). Figures 6(b), 6(e), and6(h),
made for a stable state at point b in Fig. 6(a) with a relative large
$\absolutevalue{q}$, are free of any complex energies. Figures 6(c),6(f), and
6(i), made for an unstable state at point c in Fig. 6(a) with a reduced
$\absolutevalue{q}$, contain complex energies in its bulk bands, and Figs.
6(d),6(g), and 6(j), made for a topological amplifier state at point d in Fig.
6(a) with a very small $\absolutevalue{q}$, are all real with the exception of
some energies of the edge modes, which are complex.

The edge mode dispersions in Fig. 6 are found in excellent agreement (not
shown) with those on half-plane geometries, which, when pairing terms are
taken into consideration, are given by

$E_{edge}^{i}(k_{x})=\pm\sqrt{\epsilon_{p}^{i}-c^{2}},\quad i=b,t,$ (120)

where

$\epsilon^{i}_{p}(k_{x})=q+\text{sgn}(i)\absolutevalue{t_{1}}\frac{6t_{2}\sin
k_{x}+m}{\sqrt{t_{1}^{2}+16t_{2}^{2}\sin^{2}\frac{k_{x}}{2}}}$ (121)

are the particle edge mode dispersions in the absence of pairing interactions,
i.e., Eq. (100) when $\phi=\pi/2$. Here, $E_{edge}^{i}(k_{x})$ become the
particle edge mode dispersions, $E_{p}^{i}(k_{x})$, when
$\omega_{n}(k_{x})+E_{edge}^{i}(k_{x})>0$ and the hole edge mode dispersions,
$E_{h}^{i}(k_{x})$, when $\omega_{n}(k_{x})+E_{edge}^{i}(k_{x})<0$.

Note that complex energies appear only when particle and hole spectra cross
near zero energy, irrespective of whether such states belong to bulk bands
[Fig. 6(f) for point $c$ in Fig. 6(c)] or to edge modes [Fig. 6(g) for point d
in Fig. 6(a)], confirming that the condition $\left\lceil A,B\right\rfloor=0$
in Appendix D [62] cannot prevent any degenerate states, other than those with
energies away from zero, from undergoing dynamical instability. We are thus
led to conclude that a necessary condition for our model to act as a
topological amplifier would be inversion (or crossing) between particle and
hole edge modes near zero energy.

It is worth noting that it is not uncommon for a system to enter an
unconventional phase with novel properties under some forms of inversion. An
optical cavity emits highly coherent laser light when there is a population
inversion between upper and lower energy levels of the atoms inside the cavity
[93]. An insulator becomes a topological insulator with topologically
protected gapless edge modes when there is an inversion between conduction and
valance bands of the insulator [10]. In the present system, we can reduce
$\absolutevalue{q}$ and cause particle and hole edge modes to cross, thereby
exchanging/reversing the particle and hole roles, near the zero energy level,
leading to amplification of the edge modes.

Before concluding this section, we consider how the spectra of
$H_{\Uparrow}(k_{x})$ might modify if the pairing term were to change to
$ic\tau_{y}\sigma_{0}+ic^{\prime}\tau_{y}\sigma_{z}$, which contains a
component proportional to $\sigma_{z}$. We compute the “commutator” using Eq.
(158) and find

$\begin{split}&\left\lceil A,B\right\rfloor=(\cdots)\\\
&+4c^{\prime}t_{2}\cos\phi[I_{0}\cos
k_{x}+I_{-}\cos\frac{k_{x}}{2}+I_{+}\cos\frac{k_{x}}{2}]\sigma_{0}\\\
&+i4c^{\prime}t_{1}\cos\frac{k_{x}}{2}I_{0}\sigma_{y}+2t_{1}c^{\prime}\left(I_{-}\sigma_{+}-I_{+}\sigma_{-}\right).\end{split}$
(122)

where $(\cdots)$ is the right-hand side of Eq. (113). Thus, even when the
total system is time-reversal symmetric, where both $(\cdots)$ and the second
line vanish, the unconventional commutator is nonzero because of the last
line. For a system with such pairing interactions, not only do the boundaries
separating trivial from nontrivial topological phases depend on pairing
interactions, as in Ref. [79], since $ic^{\prime}\tau_{y}\sigma_{z}$ breaks
inversion symmetry, but also the bulk bands are unstable even when the total
system is time-reversal-symmetric, as illustrated in Fig. 7. We use this
example to stress a point in Ref. [62]: It is not so much time-reversal
symmetry as the vanishing of the unconventional commutator that indicates if a
system has the potential to be made into a topological amplifier.

Figure 7: The open-boundary energy spectrum if the pairing term in
$H_{\Uparrow}(k_{x})$ [the last term in Eq. (66)] were to change from
$ic\tau_{y}\sigma_{0}$ to $ic\tau_{y}\sigma_{0}+ic^{\prime}\tau_{y}\sigma_{z}$
(instead of just $ic\tau_{y}\sigma_{0}$) with $c^{\prime}=0.1$. Other
parameters are the same as in Fig. 6(g).

##  VI Conclusion

In this work, we explored the topological properties of quadratic Hamiltonians
for spin-$1/2$ bosons with conserved magnetization in a honeycomb lattice with
both periodic and open boundaries.

We used this study as an opportunity to construct a relatively simple
analytical description for the edge modes of the Haldane model in semi-
infinite planes, which not only lays the foundation for an analytical
understanding of the edge modes in our model, but also will be useful for many
other models across a broad array of disciplines, that are built upon copies
of the Haldane model.

In the spirit of the 38-fold way for topological classification of non-
Hermitian Hamiltonians [47, 48], we carried out an in-depth analysis of the
symmetry properties of the Bloch Hamiltonian and identified our pseudo-
Hermitian system to be made up of either two copies of the AIII $+\eta_{-}$ or
two copies of the A $+\eta$ symmetry classes depending on whether the total
system is time-reversal symmetric. An important goal of the 38-fold way [47,
48] is to help find and design novel symmetry-protected topological lasers.
Indeed, the 38-fold way helped us find that the edge mode amplifier, which
Galilo et al. [59] proposed before the development of the 38-fold way, is
classified as a topological “laser” consisting of two copies of the AIII
$+\eta_{-}$ symmetry class. Furthermore, we verified rigorously, starting from
a surface integral of Berry curvature over the first Brillouin zone, where the
Berry curvature is constructed from right and left eigenvectors within the
context of non-Hermitian physics, that a system with a dynamical stable bulk
is characterized by a single Chern number, independent of pairing
interactions, as predicted from symmetry analysis.

We generalized a theorem presented in Ref. [62] for a BdG Hamiltonian so that
the theorem is now expressed in terms of an unconventional commutator between
the number-conserving and number-nonconserving part of pseudo-Hermitian
Hamiltonians which are similar to but less restrictive than the BdG
Hamiltonian. We applied this generalized theorem to learn directly from the
sub-Hamiltonians that it is not so much time-reversal symmetry but rather the
vanishing of the unconventional commutator which foretells whether a system
can be made to act as a topological amplifier.

##  Appendix A An implementation of Hamiltonian (1) in a spin-1 cold atom
system

A possible realization of the model we explore in this paper is in a honeycomb
optical lattice with interacting spin-1 bosons in the context of ultracold
atoms [94, 95, 96]. (Part of what we present in this Appendix can be found in
Refs. [59, 62].) We describe the bosons by a field vector
$\hat{b}^{\prime}_{\mathbf{i}}=(\hat{b}^{\prime}_{\mathbf{i},+1},\hat{b}^{\prime}_{\mathbf{i},0},\hat{b}^{\prime}_{\mathbf{i},-1})$
with $\hat{b}^{\prime}_{\mathbf{i},\alpha}$ the annihilation operator for a
boson with the spin-$\alpha$ component at site $\mathbf{i}$ and by a spin
vector $\mathbf{S}=S_{x}\hat{i}+S_{y}\hat{j}+S_{z}\hat{k}$ with each of its
components being a $3\times 3$ matrix in the spin-1 internal space.
Furthermore, $S_{0}$ is used to denote the identity matrix in this same spin
space. We model such a system with a grand canonical Hamiltonian

$\hat{H}=\hat{H}_{hop}+\hat{H}_{pot}+\hat{H}_{col}-\mu\sum_{\mathbf{i}}\hat{b}^{\prime{\dagger}}_{\mathbf{i}}S_{0}\hat{b}^{\prime}_{\mathbf{i}}$
(123)

which, in addition to
$\mu\sum_{\mathbf{i}}\hat{b}^{\prime{\dagger}}_{\mathbf{i}}S_{0}\hat{b}^{\prime}_{\mathbf{i}}$
with $\mu$ being the chemical potential introduced to conserve the total atom
number, is made up of three pieces, which we now explain in some details.

The first piece in Eq. (123) is given by

$\hat{H}_{hop}=-t_{1}\sum_{\left<\mathbf{ij}\right>}\hat{b}^{\prime{\dagger}}_{\mathbf{i}}S_{0}\hat{b}^{\prime}_{\mathbf{j}}+t_{2}\sum_{\left<\left<\mathbf{ij}\right>\right>}e^{-i\nu_{\mathbf{ij}}\phi}\hat{b}^{\prime{\dagger}}_{\mathbf{i}}S_{z}\hat{b}^{\prime}_{\mathbf{j}}$
(124)

where the first term describes hopping of an atom between nearest neighbors
(NN) with real amplitude $t_{1}$ and the second one describes hopping of an
atom between the next-nearest neighbors (NNN) with complex amplitude
$t_{2}e^{-i\nu_{\mathbf{ij}}\phi}$ where the phase changes sign,
$\nu_{\mathbf{ij}}=\pm$, periodically in the manner of Kane and Mele [6]. The
latter hopping is the generalization of the hopping first introduced by
Haldane for spinless fermions in graphene [3].

The second piece in Eq. (123) takes the form

$\hat{H}_{pot}=q^{\prime}\sum_{\mathbf{i}}\hat{b}^{\prime{\dagger}}_{\mathbf{i}}S_{z}^{2}\hat{b}^{\prime}_{\mathbf{i}}+m\sum_{\mathbf{i}}\xi_{\mathbf{i}}\hat{b}^{\prime{\dagger}}_{\mathbf{i}}S_{z}^{2}\hat{b}^{\prime}_{\mathbf{i}},$
(125)

which summarizes the onsite potential (with staggering), which can be
generated by a magnetic field through the quadratic Zeeman shift for a spinor
condensate [97] or by eletromagnetic waves (such as the microwaves in Ref.
[98]) through the ac-Stark shift. Here, $\xi_{\mathbf{i}}=+1$ for sites on
sublatice $A$ and $-1$ for sites on sublatice $B$.

The third piece in Eq. (123) accounts for the on-site two-body interaction
that preserves spin rotation invariance [94]

$\hat{H}_{col}=\frac{c_{0}}{2}\sum_{\mathbf{i}}:\left(\hat{b}^{\prime{\dagger}}_{\mathbf{i}}S_{0}\hat{b}^{\prime}_{\mathbf{i}}\right)^{2}:+\frac{c_{2}}{2}\sum_{\mathbf{i}}:\left(\hat{b}^{\prime{\dagger}}_{\mathbf{i}}\mathbf{S}\hat{b}^{\prime}_{\mathbf{i}}\right)^{2}:$
(126)

where $c_{0}$ and $c_{2}$ describe, respectively, the magnitude of the density
and spin interactions, and the terms between $:$ and $:$ are to follow normal
orderings.

This model, for a purely imaginary NNN hopping amplitude ($\phi=\pi/2$) and in
the absence of the staggered potential ($m=0$), reduces to the one proposed by
Galilo et al. [59] for realizing topological atom amplifiers characterized
with nonreciprocal amplification of matter fields in the chiral modes that are
hosted in the system boundaries.

Cold atom systems provide an excellent platform for the exploration of the
dynamics of a system driven out of equilibrium, thanks to the ability to tune
many of its key parameters with high precision. In the present work, we follow
[59] and consider a quench process in which we abruptly change the onsite
potential away from those that support an initial spinor condensate in a polar
state $b^{\prime}_{0}(0,1,0)$, where we assume that the spin-0 component be in
a spatially uniform coherent state $\ket{b^{\prime}_{0}}$, i.e.,
$\hat{b}^{\prime}_{i,0}\ket{b^{\prime}_{0}}=b^{\prime}_{0}\ket{b^{\prime}_{0}}$
(where $b^{\prime}_{0}$ will be taken as a real number without loss of
generality). The polar condensate is then characterized by a chemical
potential

$\mu=-3t_{1}+c_{0}n_{B},$ (127)

which we obtain by minimizing the energy per lattice site:
$(-3t_{1}-\mu)b^{\prime 2}_{0}+c_{0}b^{\prime 4}_{0}/2$, where $n_{B}\equiv
b^{\prime 2}_{0}$ is the filling factor for the condensed bosons. Such a polar
state is energetically favored as long as $q^{\prime}$ is set to a
sufficiently large positive value.

Following the usual practice (see, for example, Ref. [99]), we apply the
Bogoliubov perturbation ansatz,
$\hat{b}^{\prime}_{\mathbf{i}}\approx(0,\sqrt{n_{B}},0)+\hat{b}_{\mathbf{i}}$,
to the postquench Hamiltonian where the system parameters are fixed at their
quenched values, with $\hat{b}_{\mathbf{i}}$ the bosonic operator describing
the quantum fluctuation around the initial state. The treatment of the two-
body collision, $\hat{H}_{col}$, which is by far the most complicated term, is
significantly simplified thanks to the average spin vector of a spinor
condensate being zero in a polar state [94]. This, along with the fact that
the remaining terms in Eq. (123) preserve the $S_{z}$-rotation invariance,
makes the (postquench) Hamiltonian under the Bogoliubov approximation, where
the Hamiltonian is expanded in the second order in $\hat{b}_{\mathbf{i}}$,
block diagonal,

$\hat{H}=\hat{H}_{0}\oplus\hat{H}_{1/2},$ (128)

where

$\begin{split}\hat{H}_{0}=&-t_{1}\sum_{\left<\mathbf{ij}\right>}\hat{b}_{\mathbf{i},0}^{\dagger}\hat{b}_{\mathbf{j},0}+(2c_{0}n_{B}-\mu)\sum_{\mathbf{i}}\hat{b}^{\dagger}_{i,0}\hat{b}_{i,0}\\\
&+\frac{c_{0}n_{B}}{2}\sum_{\mathbf{i}}(\hat{b}_{\mathbf{i},0}\hat{b}_{\mathbf{i},0}+H.c.)\end{split}$
(129)

is the Hamiltonian in the $\ket{S=1,M=0}$ spin space and $\hat{H}_{1/2}$ is
the Hamiltonian in the (pseudo) spin-$1/2$ space $\mathcal{H}_{s}$ defined in
Eq. (2) with $\ket{\uparrow}\equiv\ket{S=1,M=+1}$ and
$\ket{\downarrow}\equiv\ket{S=1,M=-1})$. Explicitly, $\hat{H}_{1/2}$ is given
by Eq. (1), in which

$c=n_{B}c_{2},$ (130)

and

$q=q^{\prime}+3t_{1}+c.$ (131)

As can be seen, $\hat{H}_{0}$ is the Hamiltonian for a single-component Bose-
Hubbard model in the Bogoliubov approximation, which is known to support a
superfluid state as long as the two-body onsite interaction is repulsive
($c_{0}>0$) and relatively weak. Indeed, a simple analysis shows that its
excitation spectrum consists of two stable positive branches (and their
negative counterparts) with the lower branch

$\sqrt{(3t_{1}-t_{1}\absolutevalue{\Gamma_{\mathbf{k}}})(3t_{1}-t_{1}\absolutevalue{\Gamma_{\mathbf{k}}}+2n_{B}c_{0})}$
(132)

featuring, as expected, a Goldstone mode at $\mathbf{k}=0$, and a phonon
dispersion in the small momentum region near the Goldstone mode, where
$\Gamma_{\mathbf{k}}(\leq 3)$ is defined in Eq. (9).

Thus, within the Bogoliubov approximation, the dynamics of the spin-0
component is decoupled from that of the spin-1/2 Bose system. Moreover,
$\hat{H}_{0}$ does not possess any quench parameters and the spin-0 component
remains a spectator for a quench process involving a sudden change of the
onsite potential. Nevertheless, it is the superfluid in the spin-0 component
that induces the superfluid pairing [i.e., the last line in Eq. (1)] which is
an essential ingredient in this spin-$1/2$ bosonic BdG Hamiltonian. In this
work, we confine our interest to topological properties (i.e., bulk
invariants, edge modes, and symmetry classifications). We refer the readers,
who are interested in how topology affects quench dynamics, to Ref. [59] for
details.

##  Appendix B The steps from Eq. (86) to Eq. (87)

In this appendix, we show that the summation in Eq. (86) in the main text,

$D_{n\alpha}=\sum_{\beta}\frac{\left<\left<n,\alpha\right|\right.\tau_{z}\ket{m,\beta}\left<\left<m,\beta\right|\right.\tau_{z}\ket{n,\alpha}}{(\alpha
E_{n}-\beta E_{m})^{2}}$ (133)

can be simplified to $1/(\omega_{n}-\omega_{m})^{2}$, where $n\neq m$,
$n,m=1,2$, and $\alpha,\beta=+,-$. We begin by noting that, for a given
$\alpha$, $\beta$ can equal either $\alpha$ or $-\alpha$ so that upon
expansion, $D_{n\alpha}$ becomes

$\begin{split}D_{n\alpha}=&\frac{\left<\left<n,\alpha\right|\right.\tau_{z}\ket{m,\alpha}\left<\left<m,\alpha\right|\right.\tau_{z}\ket{n,\alpha}}{(E_{n}-E_{m})^{2}}\\\
+&\frac{\left<\left<n,\alpha\right|\right.\tau_{z}\ket{m,-\alpha}\left<\left<m,-\alpha\right|\right.\tau_{z}\ket{n,\alpha}}{(E_{n}+E_{m})^{2}}.\end{split}$
(134)

We replace the left and right eigenvectors with Eq. (73) and express
$D_{n\alpha}$ in terms of the Bogoliubov parameters

$\begin{split}D_{n\alpha}=&\frac{(u_{n\alpha}u_{m\alpha}+v_{n\alpha}v_{m\alpha})^{2}}{(E_{n}-E_{m})^{2}}\\\
+&\frac{(u_{n\alpha}u_{m\bar{\alpha}}+v_{n\alpha}v_{m\bar{\alpha}})^{2}}{(E_{n}+E_{m})^{2}}.\end{split}$
(135)

where $u_{m\bar{\alpha}}\equiv u_{m(-\alpha)}$ and $v_{m\bar{\alpha}}\equiv
v_{m(-\alpha)}$. In Eq. (135), we find, with the help of Eq. (74), that

$\begin{split}&(u_{n\alpha}u_{m\alpha}+v_{n\alpha}v_{m\alpha})^{2}\\\
=&\frac{\left[c^{2}-(\omega_{n}-\alpha E_{n})(\omega_{m}-\alpha
E_{m})\right]^{2}}{4E_{n}E_{m}(\omega_{n}-\alpha E_{n})(\omega_{m}-\alpha
E_{m})}\\\ &(u_{n\alpha}u_{m\bar{\alpha}}+v_{n\alpha}v_{m\bar{\alpha}})^{2}\\\
=&\frac{\left[c^{2}-(\omega_{n}-\alpha E_{n})(\omega_{m}+\alpha
E_{m})\right]^{2}}{-4E_{n}E_{m}(\omega_{n}-\alpha E_{n})(\omega_{m}+\alpha
E_{m})}\end{split}$ (136)

which, with the help of the identity

$c^{2}=\sqrt{(\omega_{n}^{2}-\alpha^{2}E_{n}^{2})(\omega_{m}^{2}-\alpha^{2}E_{m}^{2})}$
(137)

simplifies to

$\begin{split}(u_{n\alpha}u_{m\alpha}+v_{n\alpha}v_{m\alpha})^{2}=&\frac{1}{2}+\frac{\omega_{n}\omega_{m}+c^{2}}{2E_{n}E_{m}}\\\
(u_{n\alpha}u_{m\bar{\alpha}}+v_{n\alpha}v_{m\bar{\alpha}})^{2}=&\frac{1}{2}-\frac{\omega_{n}\omega_{m}+c^{2}}{2E_{n}E_{m}}\end{split}$
(138)

which are independent of $\alpha$. Inserting Eq. (138) into Eq. (135), we have

$\begin{split}D_{n\alpha}=&\frac{\omega_{n}\omega_{m}+c^{2}}{2E_{n}E_{m}}\times\\\
&\left[\frac{1}{(E_{n}-E_{m})^{2}}-\frac{1}{(E_{n}+E_{m})^{2}}\right]\\\
&+\frac{1}{2}\left[\frac{1}{(E_{n}-E_{m})^{2}}+\frac{1}{(E_{n}+E_{m})^{2}}\right]\end{split}$
(139)

which simplifies to

$D_{n\alpha}=\frac{E^{2}_{n}+E^{2}_{m}+2(\omega_{n}\omega_{m}+c^{2})}{(E_{n}^{2}-E_{m}^{2})^{2}}$
(140)

Finally, when use of Eq. (71) is made, we arrive at Eq. (87) in the main text.

##  Appendix C Pseudo-Hermitian Hamiltonians and particle and hole states

Pseudo-Hermiticity of a Hamiltonian $H$ in non-Hermitian physics is defined as

$\eta H^{\dagger}\eta^{-1}=H,\quad\eta^{2}=1,$ (141)

which is a similarity transformation characterized by a unitary and Hermitian
matrix $\eta$, i.e.,

$\eta\eta^{\dagger}=\eta^{\dagger}\eta=1,\quad\eta=\eta^{\dagger}.$ (142)

Note that because $\eta^{2}=1$, the same pseudo-Hermiticity condition can also
be written as

$H^{\dagger}=\eta H\eta^{-1},\quad\eta^{2}=1.$ (143)

In what follows, we show that states in any pseudo-Hermitian Hamiltonian can
be divided into either particle states or hole states according to whether
their norm with metric $\eta$ is $+1$ or $-1$. Let $\ket{\phi_{n}}$ be the
eigenstate of $H$ with eigenvalue $\mathcal{E}_{n}$,

$H\ket{\phi_{n}}=\mathcal{E}_{n}\ket{\phi_{n}},$ (144)

or

$\bra{\phi_{m}}H^{\dagger}=\mathcal{E}^{*}_{m}\bra{\phi_{m}}.$ (145)

On one hand, multiplying Eq. (144) from the left with $\bra{\phi_{m}}\eta$, we
have

$\bra{\phi_{m}}\eta
H(k)\eta^{-1}\eta\ket{\phi_{n}}=\mathcal{E}_{n}\bra{\phi_{m}}\eta\ket{\phi_{n}},$
(146)

which, when use of the pseudo-Hermiticity condition (143) is made, is changed
to

$\bra{\phi_{m}}H^{\dagger}(k)\eta\ket{\phi_{n}}=\mathcal{E}_{n}\bra{\phi_{m}}\eta\ket{\phi_{n}}.$
(147)

On the other hand, multiplying Eq. (145) from the right with
$\eta\ket{\phi_{n}}$, we have

$\bra{\phi_{m}}H^{\dagger}(k)\eta\ket{\phi_{n}}=\mathcal{E}^{*}_{m}\bra{\phi_{m}}\eta\ket{\phi_{n}}$
(148)

which, when combined with Eq. (147), leads to

$\left(\mathcal{E}^{*}_{m}-\mathcal{E}_{n}\right)\bra{\phi_{m}}\eta\ket{\phi_{n}}=0.$
(149)

From Eq. (149), we conclude that $\bra{\phi_{n}}\eta\ket{\phi_{n}}$, the norm
with metric $\eta$, is zero for an unstable eigenstate (i.e., a state with a
complex eigenvalue) and is either a positive or a negative real number for a
stable state (i.e., a state with a real eigenvalue). In arriving at the latter
conclusion, we have considered the fact that (a) the norm with metric $\eta$
is an average of a Hermitian matrix, $\eta$, and is therefore always real, and
(b) in contrast to the usual norm, which is always positive, the norm with
metric $\eta$ is not guaranteed to be a positive number.

In summary, Eq. (149) means that the eigenstates of a pseudo-Hermitian
Hamiltonian, $H(k)$ in Eq. (150), still obey the same bi-orthonormality
relation as that for a BdG Hamiltonian, in which a stable state is classified
either as a particle state $\ket{p}$ where $\bra{p}\eta\ket{p}=1$ or as a hole
state $\ket{h}$, where $\bra{h}\eta\ket{h}=-1$.

##  Appendix D Generalized theorem

In this appendix, we consider a Hamiltonian

$H(k)=\begin{pmatrix}A_{1}(k)&B_{1}(k)\\\
-B^{*}_{2}(-k)&-A^{*}_{2}(-k)\end{pmatrix}$ (150)

where $k$ stands for the crystal momentum (which may contain several
components) within the first Brillouin zone. The submatrices in Eq. (150)
satisfy the conditions,

$A_{1,2}(k)=A_{1,2}^{\dagger}(k),\quad B_{2}(k)=B_{1}^{T}(-k),$ (151)

which guarantees that $H(k)$ is pseudo-Hermitian,

$\eta H^{\dagger}(k)\eta^{-1}=H(k),$ (152)

which is equivalent to $\eta H(k)$ being Hermitian where $\eta=\tau_{z}$.
$H(k)$ in Eq. (150) generalizes a bosonic BdG Hamiltonian for which

$A_{1}(k)=A_{2}(k)\mbox{ and }B_{1}(k)=B_{2}(k).$ (153)

Here, we consider situations where condition (153) does not hold. Then, $H(k)$
in Eq. (150) is not invariant under particle-hole symmetry,

$\mathcal{C}H^{T}(k)\mathcal{C}^{-1}\neq-H(-k),$ (154)

where $\mathcal{C}=\tau_{y}$ and is therefore not a BdG Hamiltonian.

We showed in Appendix C that a stable state, of a pseudo-Hermitian Hamiltonian
in Eq. (150), is classified either as a particle state or as a hole state,
independent of any other symmetries. This means that the bi-orthonormality
relation built upon particle and hole states, which forms the basis of our
study in Ref. [62], still holds true for a system described by Hamiltonian
(150). We can thus expect to generalize our theorem in [62] straightforwardly
from a BdG Hamiltonian to its non-BdG generalization (150), beginning with the
definition of a pair of a particle state, $\ket{\psi_{p_{0}}}$, and a hole
state, $\ket{\psi_{h_{0}}}$, that are degenerate with energy $E_{0}$ in the
absence of $B_{1}(k)$ and $B_{2}(k)$,

$\begin{split}A_{1}(k)\ket{\psi_{p_{0}}}&=E_{0}\ket{\psi_{p_{0}}},\\\
-A^{*}_{2}(-k)\ket{\psi_{h_{0}}}&=E_{0}\ket{\psi_{h_{0}}}.\\\ \end{split}$
(155)

We do not present the derivation here, but it can be shown that weak pairing
lifts the degeneracy, splitting $E_{0}$ into a pair of complex-conjugate
energies, $E_{0}+i\absolutevalue{E^{(1)}}$ and
$E_{0}-i\absolutevalue{E^{(1)}}$, where

$|E^{(1)}|=\absolutevalue{\bra{\psi_{p_{0}}}B_{1}(k)\ket{\psi_{h_{0}}}}$ (156)

or

$|E^{(1)}|=\absolutevalue{\bra{\psi_{p_{0}}}\left\lceil
A,B\right\rfloor\ket{\psi_{h_{0}}}}/{2\absolutevalue{E_{0}}}$ (157)

with

$\left\lceil A,B\right\rfloor\equiv A_{1}(k)B_{1}(k)-B_{1}(-k)A^{*}_{2}(-k)$
(158)

being defined symbolically as an unconventional commutator between $A$ and $B$
matrices in (150). A sufficient condition for creating stable
high-$\absolutevalue{E_{0}}$ states is that the unconventional commutator
vanishes, i.e., $\left\lceil A,B\right\rfloor=0$.

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