Generalized Berry phase for a bosonic Bogoliubov system with exceptional points
We discuss the topology of Bogoliubov excitation bands from a Bose-Einstein condensate in an optical lattice. Since the Bogoliubov equation for a bosonic system is non-Hermitian, complex eigenvalues often appear and induce dynamical instability. As a function of momentum, the onset of appearance and disappearance of complex eigenvalues is an exceptional point (EP), which is a point where the Hamiltonian is not diagonalizable and hence the Berry connection and curvature are ill-defined, preventing defining topological invariants. In this paper, we propose a systematic procedure to remove EPs from the Brillouin zone by introducing an imaginary part of the momentum. We then define the Berry phase for a one-dimensional bosonic Bogoliubov system. Extending the argument for Hermitian systems, the Berry phase for an inversion-symmetric system is shown to be . As concrete examples, we numerically investigate two toy models and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. The invariant associated with particle-hole symmetry and the winding number for a time-reversal-symmetric system are also discussed.
Topological phases of matter have attracted much attention in solid-state physics in more than one decade Hasan and Kane (2010); Qi and Zhang (2011); Sato and Ando (2017); Cooper et al. (2018). Although most of such studies treat Hermitian Hamiltonians, there has been growing interest in non-Hermitian systems Hu and Hughes (2011); Esaki et al. (2011); Shen et al. (2018). Non-Hermitian systems exhibit unique phenomena with no counterpart in Hermitian ones, such as non-reciprocal transport Lin et al. (2011a); Regensburger et al. (2012); Feng et al. (2013); Peng et al. (2014a), enhanced sensitivity Wiersig (2014); Liu et al. (2016); Hodaei et al. (2017); Chen et al. (2017); Zhang et al. (2018); Ge (2017), high-performance lasers Longhi (2010); Chong et al. (2011); Jing et al. (2014); Peng et al. (2014b); Feng et al. (2014); Hodaei et al. (2014), and unconventional quantum criticality Lee and Chan (2014); Kawabata et al. (2017); Ashida et al. (2017); Nakagawa et al. (2018); Xiao et al. (2018). Non-Hermiticity also leads to richer topological properties absent in Hermitian systems. For example, appearance of an edge state localized at only one side of the system is predicted Lee (2016); Leykam et al. (2017); Kawabata et al. (2018a); Martinez Alvarez et al. (2018). It is also pointed out that the conventional bulk-edge correspondence breaks down in certain non-Hermitian models, and there have been great efforts to establish modified bulk-edge correspondences Leykam et al. (2017); Martinez Alvarez et al. (2018); Ezawa (2019); Lee (2016); Martinez Alvarez et al. (2018); Kawabata et al. (2018a); Philip et al. (2018); Liu et al. (2019); Xiong (2018); Kunst et al. (2018); Leykam et al. (2017); Yao et al. (2018); Yao and Wang (2018); Ezawa (2019); Edvardsson et al. (2019); Ezawa (2018); Wang et al. (2019); Yokomizo and Murakami (2019); Kawabata et al. (2019a). More recently, topological classification of non-Hermitian systems has been done Gong et al. (2018); Kawabata et al. (2018b); Zhou and Lee (2018); Kawabata et al. (2019b).
Existence of exceptional points (EPs) is also a specific feature absent in Hermitian systems. Due to non-Hermiticity of the system, a non-Hermitian Hamiltonian cannot be diagonalized for a certain set of parameters that describe the Hamiltonian. That is an EP Heiss (2012); Bender (2007); Berry (2004). At an EP, two eigenstates coalesce and become identical to each other. Hence, topological invariants cannot be defined in the presence of an EP below Fermi energy, even though the system is gapfull. So far, topological charge of an EP and deformation of a Weyl point to an exceptional ring (EPs aligned in a ring shape) accompanied with the appearance of bulk Fermi arcs has been discussed Zhen and Hsu (2015); González and Molina (2016); Xu et al. (2017); González and Molina (2017); Zyuzin and Zyuzin (2018); Cerjan et al. (2018); Rosenthal et al. (2018); Shen et al. (2018); Zhou et al. (2018); Carlström and Bergholtz (2018); Luo et al. (2018a); Lee et al. (2018); Zhou et al. (2019); Molina and González (2018); Moors et al. (2019); Budich et al. (2019); Okugawa and Yokoyama (2019); Yang and Hu (2019); Zhou et al. (2019); Wang et al. (2019); Yoshida et al. (2019); Bergholtz and Budich (2019); Zyuzin and Simon (2019).
In this paper, we consider topology of excitation bands from a Bose-Einstein condensate (BEC) in the presence of EPs. We find that in some special cases EPs can be removed from the Brillouin zone (BZ) and topological invariants can be defined by a simple generalization of the ones defined in Hermitian systems.
In general, a non-Hermitian Hamiltonian describes an open quantum system in which loss and gain of particles coexist. Examples include photons in nanostructures such as photonic crystals St-Jean et al. (2017); Zhao et al. (2018); Parto et al. (2018); Harari et al. (2018); Bandres et al. (2018), micro cavities Aspelmeyer et al. (2014); Jing et al. (2015); Xu et al. (2016), and electric circuits Schindler et al. (2011); Rosenthal et al. (2018), quantum walks with non-unitary time evolution Kim et al. (2016); Xiao et al. (2017), cold atomic gases with controlled loss and gain Barontini et al. (2013); Tomita et al. (2017); Li et al. (2019); Pan et al. (2019); Kreibich et al. (2013, 2014), and solid-state systems with finite quasiparticle lifetimes Shen and Fu (2018); Zyuzin and Zyuzin (2018); Kozii and Fu (2017); Yoshida et al. (2018). Here, we note that bosonic Bogoliubov quasi-particles, which are elementary excitations from a Bose-Einstein condensate (BEC), are also described with a non-Hermitian Hamiltonian, where a BEC works as a particle bath. The characteristic of the bosonic Bogoliubov system is that the non-Hermitian Hamiltonian always satisfies pseudo-Hermiticity Mostafazadeh (2002a, b, c, 2003) and particle-hole symmetry (PHS). In addition to this, we consider inversion symmetry (IS) and/or time-reversal symmetry (TRS) and discuss how topological invariants are calculated in the presence of EPs.
There are several previous works that discuss topology of bosonic Bogoliubov systems and the resulting bulk-edge correspondence, such as magnon excitations Wang et al. (2009); Onose et al. (2010); Katsura et al. (2010); Shindou et al. (2013a, b); Joshi and Schnyder (2017); Kondo et al. (2019), photons Onoda et al. (2004); Raghu and Haldane (2008); Haldane and Raghu (2008), phonons Strohm et al. (2005); Sheng et al. (2006), and Bose atoms Engelhardt and Brandes (2015); Furukawa and Ueda (2015); Galilo et al. (2015); Oshima and Kawaguchi (2016); Barnett (2013); Bardyn et al. (2016); Galilo et al. (2017); Luo et al. (2018b); Lieu (2018a); McDonald et al. (2018); Wang and Clerk (2019). In these works, the definition of topological invariants is extended for bosonic Bogoliubov systems, and the bulk-edge correspondence is numerically confirmed. However, despite non-Hermiticity of bosonic Bogoliubov systems, the works mentioned above discuss only the case when all eigenvalues are real. On the other hand, the appearance of complex eigenvalues is known to cause dynamical instability and observed in various situations Pu et al. (1999); Mueller and Baym (2000); Wu and Niu (2001); Fallani et al. (2004); Kawaguchi and Ohmi (2004); Sadler et al. (2006).
In this paper, we consider the topology of bosonic Bogoliubov bands from an atomic BEC confined in an optical lattice, in particular, in the presence of complex eigenvalues. We first show that the onset of appearance and disappearance of complex conjugate eigenvalues is actually an EP. We then introduce an imaginary part of the momentum around an EP and remove the EP from the BZ (which is defined for the real part of the momentum). Note that this idea is distinguished from non-Bloch wave number in that we are considering not the bulk-edge correspondence Yao and Wang (2018); Yao et al. (2018) but bulk topology. Due to the square root singularity of an EP, we have to be careful to choose the sign of the imaginary part of the momentum. With this procedure, we define topological invariants in one-dimensional (1D) systems with various symmetries. For the case of a system with IS, the band topology is characterized by a topological invariant, which is the Berry phase modulo . We can also define a topological invariant associated with PHS for some special cases. When the system is invariant under time-reversal operation, we can define a winding number, which is however found to be always trivial. As concrete examples, we consider two toy models, Kitaev-chain-like model and Su-Schrieffer-Heeger-like (SSH-like) model. Although these models are known to have PHS, we mainly focus on the topological property related to IS. We numerically obtain the eigen-spectrum with open boundary condition and confirm the bulk-edge correspondence even in the presence of complex eigenvalues. As an interesting result, we find an edge state whose energy is located in a gap of pure-imaginary bulk eigenvalues [see Fig. 9 (d)].
The organization of this paper is as follows. In Sec. II, we introduce the bosonic Bogoliubov formalism which describes the quasiparticle excitations from a condensed state. The symmetry property of the system is also given in this section. In Sec. III, we introduce the bi-orthogonal basis and discuss the properties of EPs in the bosonic Bogoliubov system. In particular, the extension to the complex momentum and how to remove EPs from the BZ are explained. In Sec. IV, we define the generalized topological invariant for a 1D inversion-symmetric system that can be calculated in the presence of EPs. In Sec. V, we discuss two other topological invariants, the topological invariant associated with PHS and the winding number associated with chiral symmetry (CS) which is a combined symmetry of PHS and TRS. In Sec. VI, we numerically solve two toy models and confirm the bulk-edge correspondence. In Sec. VII, we conclude the paper and give some discussions.
Ii Bogoliubov formalism
Bogoliubov theory well describes quasi-particle excitations from a condensate at low temperature Bogolyubov (1947); Kawaguchi and Ueda (2012). In this section, we review the general properties of the Bogoliubov theory. We assume a spatially uniform or periodic system and discuss in the momentum space. However, the properties derived below, i.e., the PHS and the pseudo-Hermiticity, hold in the absence of the continuous nor discrete translational symmetries. A general Hamiltonian is given by , where
are the single-particle Hamiltonian and the inter-particle interaction, respectively, with being a bosonic operator that annihilates (creates) an atom with momentum in internal state , and is the chemical potential. For the case of a periodic system, is a quasi-momentum and may include orbital degrees of freedom, as well as spin degrees of freedom. From the commutation relation between bosonic operators, the coefficients satisfy the following relations:
Due to the momentum conservation, is nonzero only when is equal to zero (a reciprocal lattice vector) in a uniform (periodic) system.
For simplicity, we consider a case when a macroscopic number of atoms are condensed in a state. Generalization for a condensate with a nonzero momentum is straightforward. The creation and annihilation operators corresponding to the condensed state can be replaced with c-numbers, say and , where ’s are determined so that the condensate is stationary. Expanding the Hamiltonian up to the second order of and , we rewrite the Hamiltonian in the Nambu description as follows:
where , , , and with being the number of internal states. Here, is a Hermitian matrix whose elements are given by
being an Hermitian matrix and an symmetric matrix, respectively.
To diagonalize , we perform the Bogoliubov transformation:
where and are the annihilation and creation operators of quasi-particles, and is a matrix with and being matrices. Since both and satisfy the bosonic commutation relation, is not a unitary matrix but a para-unitary matrix that satisfies
where are the Pauli matrices in the Nambu space. Equation (10) is a distinctive feature of a bosonic system: the Bogoliubov transformation for a fermionic superconducting system is a unitary transformation. The Hamiltonian is then diagonalized when satisfies the following Bogoliubov equation:
is a non-Hermitian matrix and is an diagonal matrix. In this paper, we refer to defined in Eq. (12) as the bosonic Bogoliubov Hamiltonian, or simply Bogoliubov Hamiltonian, in the sense that its eigenvalues describe the excitation spectrum. Since is non-Hermitian, its eigenvalues are in general complex. In Eq. (11), we have used PHS, i.e.,
where is the antiunitary particle-hole operator defined by
with being the complex conjugate operator: From Eq. (13), one can see that if is an eigenstate of with an eigenvalue , is an eigenstate of with an eigenvalue , obtaining Eq. (11). The particle-hole symmetry is a generic property of the Bogoliubov equation originating from condensation, and always holds independently from other symmetries.
We note that the Bogoliubov equation for a bosonic system is categorized to a pseudo-Hermitian eigenvalue equation. A non-Hermitian matrix is called pseudo-Hermitian when it satisfies the following relation Mostafazadeh (2002a, b, c, 2003):
where is an Hermitian matrix and referred to as a metric operator. For the case of defined in Eq. (12), it satisfies the pseudo-Hermiticity with , i.e.,
From this relation, the orthonormal condition is obtained, which differs for real-eigenvalue modes and complex-eigenvalue states: By evaluating , we obtain , which means, the normalization condition for a real-eigenvalue state can be defined as
whereas a complex-eigenvalue state is orthonormal to its conjugate
The above normalization condition suggests that Eq. (10) no longer holds in the presence of complex eigenvalues and that the corresponding quasi-particles do not satisfy the bosonic commutation relation. In such a case, however, bosonic quasi-particles can be defined by constructing linear combinations of complex eigenstates Kawaguchi and Ohmi (2004). We also note that when satisfies Eq. (17), the corresponding hole excitation has a negative norm:
In other words, for the case of bosonic Bogoliubov equation, we can classify real-eigenvalue eigenstates into particle and hole excitations according to the sign of the “norm”, , although they describe physically the same excitation. In the following, we refer to an eigenstate satisfying Eq. (17) as a positive-norm state and to that satisfying Eq. (19) as a negative-norm state.
The rest of this paper discusses topological properties of the eigen-spectrum of the Bogoliubov Hamiltonian (12). In order to classify the system in terms of symmetries, we here summarize the related symmetry operations:
pseudo-Hermiticity: as seen in the above, always satisfies the pseudo-Hermiticity relation:
particle-hole symmetry (PHS): as a consequence of condensation, always preserves particle-hole symmetry:
time-reversal symmetry (TRS): a system preserves time-reversal symmetry when satisfies
where is the antiunitary time-reversal operator. Due to Bose statistics, it satisfies , and therefore, there is no Kramers degeneracy. Since the time-reversal operation does not mix the particle and hole sectors, can be written as
with being an unitary matrix.
chiral symmetry (CS): a system preserves chiral symmetry when satisfies
where is the chiral operator. Since the bosonic Bogoliubov Hamiltonian always preserves PHS, the presence of CS and TRS are equivalent and they are related to each other via .
inversion symmetry (IS): a system preserves the inversion symmetry when satisfies
where is the inversion operator.
Iii Mathematical framework relating to an exceptional point
A salient feature of a non-Hermitian system that is distinct from an Hermitian system is the existence of an EP Heiss (2012); Bender (2007); Berry (2004); Zhen and Hsu (2015); González and Molina (2016); Xu et al. (2017); González and Molina (2017); Zyuzin and Zyuzin (2018); Cerjan et al. (2018); Rosenthal et al. (2018); Shen et al. (2018); Zhou et al. (2018); Carlström and Bergholtz (2018); Luo et al. (2018a); Lee et al. (2018); Zhou et al. (2019); Molina and González (2018); Moors et al. (2019); Budich et al. (2019); Okugawa and Yokoyama (2019); Yang and Hu (2019); Zhou et al. (2019); Wang et al. (2019); Yoshida et al. (2019); Bergholtz and Budich (2019); Zyuzin and Simon (2019). An EP is a point in a parameter space where the Hamiltonian cannot be diagonalized. In this section, we introduce some mathematical framework relating to EPs and discuss how to avoid EPs in calculation of topological invariants in bosonic Bogoliubov systems.
iii.1 Bi-orthogonal basis
For a non-Hermitian matrix , the right and left eigenstates are defined as
where and are conjugate transpose of each other. In this notation, the eigenstate appearing in Sec. II should be replaced with . By evaluating and , one can see that two eigenstates with different eigenvalues are orthogonal to each other: for . In particular, when all eigenvalues are different, the following orthonormal and completeness conditions hold:
When two eigenvalues are the same for a certain , say , there are two possibilities: a degenerate point or an EP. The difference is whether the non-Hermite matrix can be diagonalized or not. In general, a simplest form of a non-Hermitian matrix a Jordan normal form. For the case of a degenerate point, the block corresponding to and 2 eigenstates is given by
whereas that for the case of an EP is given by
Here, has two linearly independent right eigenstates and , which means, by choosing a proper basis, the whole set of the eigenstates satisfies Eq. (28) at a degenerate point. On the other hand, has only one right eigenstate and one left eigenstate . Accordingly, at an EP, two of left and right eigenstates of are linearly dependent, and , and they are orthogonal to their adjoint:
which is often referred to as self-orthogonality Bender (2007); Heiss (2012). The vanishment of the independent eigenstates means that the Bogoliubov transformation matrix , which is identical to defined in Eq. (9), has zero determinant at an EP.
iii.2 Properties derived from pseudo-Hermiticity
For the case when the non-Hermitian Hamiltonian satisfies the pseudo-Hermiticity (16), we have some more restrictions on the eigenstates. Though we use in the following discussion, the derived properties except for Eq. (32) hold for a generic metric operator.
First, we show that complex eigenvalues appear as a complex conjugate pair: When is a right eigenstate of with eigenvalue , is also a right eigenstate of with eigenvalue . It follows that when is real and not degenerate, and are identical up to a phase factor. Actually for the case of a bosonic Bogoliubov system, Eqs. (17), (19) and (28a) lead to
where the plus (minus) sign in the right-hand side is for positive-norm (negative-norm) states. On the other hand, when , and are linearly independent. In this case, there exists such that and .
An EP appears where two real eigenvalues coalesce and change into a pair of complex conjugate eigenvalues. To see this, we consider a one-dimensional system and assume that two eigenvalues and coalesce at and change from two real values at to a complex conjugate pair at [Fig. 1 (a)]. From the above argument, linearly dependent pairs among the four eigenvectors, , and , are
for , and
for . Hence, exchange of the linearly dependent partners occurs at and and are identical at this point, i.e., is an EP. A more rigorous proof will be given in Appendix A. In particular, for the case of a bosonic Bogoliubov system, the two coalescing real-eigenvalue states should be a pair of positive-norm and negative-norm states, which is also explained in Appendix A.
We also note that since is real (pure imaginary) for (), the energy spectra exhibit the square root behavior at around :
where is a real number. Moreover, since EPs arise at the points where sign change of occurs, they appear as an exceptional line (surface) in a two-dimensional (three-dimensional) system.
iii.3 Extension to a complex momentum plane to avoid exceptional points
Topological invariants are often defined by an integral of the Berry curvature and/or connection in the BZ, where the system is assumed to be gapped in the domain of the integral. In the absence of an EP, we can define the Berry curvature and connection for a non-Hermitian system, which will be given in Sec. IV, and follow the argument for Hermitian systems. The only assumption to define a topological invariant is that the energy bands are separated. Emergence of an EP, however, causes two difficulties: (i) the fact that makes it impossible to derive quantization of the topological charge, and (ii) coalescence of two energy bands makes it difficult to classify “occupied” and “unoccupied” bands. (Since there is no Fermi energy for a bosonic system, we define “occupied” and “unoccupied” bands by introducing an index below which the bands are occupied. is defined such that the “occupied” and “unoccupied” bands do not touch, and we take the summation over “occupied” bands to calculate the topological invariants. See, e.g., Eqs. (46) and (55).) Here, we show that these difficulties can be overcome when we consider a system defined on a complex space. For simplicity, we assume a 1D system and use with to explicitly distinguish complex () and real () momentum. Generalization to higher dimensions is straightforward.
Complex momentum has also been introduced to discuss the bulk-edge correspondence in a non-Hermitian system Lee (2016); Martinez Alvarez et al. (2018); Kawabata et al. (2018a); Philip et al. (2018); Liu et al. (2019); Xiong (2018); Kunst et al. (2018); Leykam et al. (2017); Yao et al. (2018); Yao and Wang (2018); Liu et al. (2019); Ezawa (2019); Edvardsson et al. (2019); Ezawa (2018); Wang et al. (2019); Yokomizo and Murakami (2019); Kawabata et al. (2019a), where wave functions in open-boundary systems are expanded with respect to plane waves with complex wave numbers. Our system is different from the previous works in that we consider only a bulk and that a complex momentum is introduced just for avoiding EPs in the BZ.
We first define on a complex plane as
Accordingly, the definitions for PHS, TS, CS, and IS are extended as
respectively. The proof of Eq. (37) will be given in Appendix B. The Hamiltonian differs from the one for a real two-dimensional system in that and do not change the sign of . In this sense, is unphysical and introduced just for a theoretical procedure.
On the other hand, the pseudo-Hermiticity holds only on the real axis in a complex plane:
This means that a complex conjugate of an eigenvalue is not necessarily an eigenvalue of when . Now, we reconsider the situation discussed in Fig. 1 (a). Since we are interested in the square root singularity of the spectrum, we can set without loss of generality. Then we obtain
When is analytically continued to a complex plane as , the analytic continuations of and are two branches of
From this form, one can see that the EP at is isolated in the complex plane and the two eigenvalues are exchanged when one goes around the EP Heiss (2012).
The above argument indicates that an EP can be circumvented by taking a path in the complex plane. In the following discussions, we introduce an infinitesimal imaginary part , which is a function of , and perform the integration along a path in the complex plane:
where in the right-hand side is the analytic continuation of the integrand defined on the real axis except for at EPs according to Eq. (36). The path , i.e., which path we choose in the upper- or lower-half plane at each EP, is defined such that the eigenvalues labeled in a certain manner are continuously change along the path. The labeling rule for eigenvalues depends on what topological invariant we discuss, and the detailed manner is given in the following sections. Once how the real- and complex-eigenvalue states are continued at EPs is determined, we can draw a path in the complex plane. Examples are shown in Fig. 2.
iii.4 Nambu-Goldstone modes
Before closing this section, we have to comment on EPs coming from Nambu-Goldstone modes. Nambu-Goldstone mode is a gapless mode associated with the spontaneous breaking of a continuous symmetry due to condensation. A reduction of occurs at the gap closing point, and hence, this is an EP. However, this type of EPs can be removed by adding an infinitesimal external field that breaks the continuous symmetry. For example, for the case of the Nambu-Goldstone phonon, which comes from a spontaneous U(1) symmetry breaking, an infinitesimal gap opens when one shifts the chemical potential in the negative direction: . With this procedure, we treat Nambu-Goldstone modes as if it is gapped and do nothing special for this type of EPs.
Iv invariant associated with inversion symmetry
From now on, we restrict ourselves to a 1D periodic system. In particular in this section, we consider an inversion-symmetric BEC in a 1D optical lattice and derive a topological invariant associated with IS.
iv.1 Berry connection
One of the fundamental field that characterizes topology of quantum states is the Berry connection. For the case of non-Hermitian systems, quantization of Berry phase is predicted by various authors Berry (1985); Garrison and Wright (1988); Mostafazadeh (1999); Schomerus (2013a); Menke and Hirschmann (2017); Leykam et al. (2017); Weimann et al. (2017); Yuce (2018); Lieu (2018b); Shen et al. (2018), where the Berry connection is defined by using the bi-orthogonal basis. Here, we define the Berry connection matrix as
Because of the relation , is an Hermitian matrix and is always real. Although the four kinds of matrices, , and , have been introduced in the context of topological phases Shen et al. (2018), we consider only two of them since the natural inner product is defined between the right- and left-eigenstates.
It can be proved that the above defined satisfies the sum rule:
without assuming any symmetry. We note that in the presence of EPs, we define the extended Berry connection matrix just by replacing with and use Eq. (41). Equation (44) always holds independently from how we choose the integration path. The proof of Eq. (44) will be given in Appendix C.
iv.2 topological invariant associated with inversion symmetry
For the case when all eigenvalues are real, the diagonal terms of the matrix defined in Eq. (42) agree with those defined in the previous works Engelhardt and Brandes (2015); Furukawa and Ueda (2015):
where we have assigned positive (negative) index for positive-norm (negative-norm) states. Following the discussion in Ref. Engelhardt and Brandes (2015), it is natural to define the generalized topological invariant by using defined in Eq. (42) as
Here, is a reference index defined such that the bands with and are separated, i.e., for all in the BZ. As shown below, the above defined takes or , given by
since among the negative-index states the numbers of even-parity and odd-parity states are the same at each and , i.e., .
iv.3 Labeling rule
To define the integration path in Eq. (46) in the presence of EPs, we have to properly label the eigenstates. Here, only the requirement is that the indices assigned to a pair of inversion symmetric states have the same sign, i.e., an inversion symmetric counterpart of a particle-band (hole-band) state is in a particle (hole) band. To satisfy the requirement, we assign positive (negative) indices to real-eigenvalue positive-norm (negative-norm) states according to Eq. (32). For complex-eigenvalue states, we assign a positive (negative) index to a state with (). Since and , which are eigenstates of and , respectively, share the same eigenvalue , this labeling rule satisfies our requirement. The positive- and negative-index states are respectively sorted in a certain manner, say in the order of . Then, according to the discussion in Fig. 2, the integration path becomes symmetric with respect to , i.e., the path satisfies . For example, when the structure shown in Fig. 2 (c) [Fig. 2 (d)] appears in the region, the structure of Fig. 2 (e) [Fig. 2 (d)] should appear in the region due to IS. (Note that the dependence of is irrelevant in determining the integration path.) In the case when complex eigenvalues appear at , the band structure around should be those shown in Figs. 2 (d) or 2 (f), for which the integration path is symmetric with respect to the origin. An example of a more complicated band structure is given in Fig 3.
iv.4 Proof of
On the other hand, the integration of the right-hand side of Eq. (49) is given by
Note that since with and commutes with the inversion operator , and are diagonal matrices and identical to each other. The diagonal elements are the eigenvalues of : with or . As a result, we obtain
V Other topological invariants
In this section, we consider topological invariants associated with PHS and CS. Unfortunately, the definition for the former is subtle, whereas the latter is shown to be always trivial.
v.1 topological invariant associated with particle-hole symmetry
Suppose that all eigenstates, including complex-eigenvalue states, are classified into particle and hole excitations and assigned positive and negative indices, respectively. Then, we can divide into the contributions from the positive-index and negative-index bands as
By regarding the negative-index (positive-index) states as occupied (unoccupied) states, the topological invariant is defined as the Berry phase of the negative index bands:
Below, we give the labeling rules for eigenstates and prove or for some special cases.
v.1.1 Labeling rule
Since we have to classify particle and hole bands, the requirement is that a pair of particle-hole conjugate states has indices with opposite signs. The classification for real-eigenvalue states based on Eq. (32) satisfies this requirement. When complex-eigenvalue states exist, four complex-eigenvalue states with eigenvalues and appear all together, which are related to each other via and as summarized in Table 1. As shown in Table 1, when is an eigenvalue with a positive index, and () should be assigned negative indices (a positive index). To be consistent with the above observation, we classify complex-eigenvalue states such that a state with () has a positive (negative) index. It follows that in the case when complex eigenvalues appear at or , we fail to assign indices, which means that we cannot define .