#  Topological band theory of a generalized eigenvalue problem with Hermitian
matrices: Symmetry-protected exceptional rings with emergent symmetry

Takuma Isobe1 Tsuneya Yoshida1,2 Yasuhiro Hatsugai1,2 1Graduate School of Pure
and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
2Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571,
Japan

###### Abstract

So far, topological band theory is discussed mainly for systems described by
eigenvalue problems. Here, we develop a topological band theory described by a
generalized eigenvalue problem (GEVP). Our analysis elucidates that non-
Hermitian topological band structures may emerge for systems described by a
GEVP with Hermitian matrices. The above result is verified by analyzing a two-
dimensional toy model where symmetry-protected exceptional rings (SPERs)
emerge although the matrices involved are Hermitian. Remarkably, these SPERs
are protected by emergent symmetry, which is unique to the systems described
by the GEVP. Furthermore, these SPERs elucidate the origin of the
characteristic dispersion of hyperbolic metamaterials which is observed in
experiments.

Introduction.— After discovery of topological insulators, topological band
structures have been studied as one of the central issues in condensed matter
systems C.L.Kane_E.J.Mele_PRL.2005 ; C.L.Kane_E.J.Mele_PRL.2005_Z2 ;
L.Fu_C.L.Kane_PRL.2007 ; M.Z.Hasan_C.L.Kane_RevModPhys.2010 ;
X.L.Qi_S.C.Zhang_RevModPhys.2011 ; Y.Ando_JPSJ_2013 ;
B.A.Bernevevig_T.L.Huglhes_S.C.Zhang_Science_2006 ; M.Knig_Science_2007 ;
L.Fu_C.L.Kane_PRB.2007 ; L.Fu_C.L.Kane_PRB.2006 ; D.J.Thouless_PRB.1983 ;
Schnyder_PRB.2008 ; A.Y.Kitaev_AIP_Conf_2009 ;
S.Ryu_A.P.Schnyder_A.Furusaki_New.J.Phys_2010 ;
X.L.Qi_T.L.Hughes_S.C.Zhang_PRB_2008 ;
A.M.Essen_J.E.Moore_D.Vanderbilt_PRL_2009 . In these systems, robust gapless
modes emerge around the boundary due to topological properties in the bulk
Hatsugai_PRL93 . In addition, the topological band theory is also applicable
to semimetals which host robust band touching in the bulk S.Murakami_IOP_2007
; W.Xiang_PRB_2011 ; Yang_PRB_2011 ; A.Birkov_PRL_2011 ; Xu_PRL_2011 ;
Kurebayashi_JPSJ_2014 ; N.Armitage_RevModPhys_2018 ; Koshino_PRB_2016 . For
instance, the Chern number can be assigned to the gapless points in the bulk
of Weyl semimetals, which elucidates novel transport properties
W.Xiang_PRB_2011 ; Yang_PRB_2011 ; A.Birkov_PRL_2011 ; Xu_PRL_2011 .
Mathematically speaking, these topological band structures are described by a
standard eigenvalue problem with a Hermitian matrix.

Recently, the topological band theory has been extended to non-Hermitian
systems K.Esaki_PRB_2011 ; M.Sato_Progress_of_Theoretical_Science_2012 ;
S.-D.Liang_PRA_2013 ; D.Leykam_PRL_2017 ; Xu_PRL(2017)_WeylEP ;
Shunyu_PRL(2018)_SkinEffect ; Gong_class_PRX18 ; Yao_nHChern_PRL2019 ;
Kawabata_nHclass_PRX19 ; Yoshida_nHFQH19 ; xiao_NatPhysics(2020)_Observsation-
nHBBC ; xiao_arXiv(2020)_observation-nBPTsymm ; Dibyendu_PRB(2021)_nH-SSH
which are described by the standard eigenvalue problem with a non-Hermitian
matrix. The platforms of the non-Hermitian topological band theory are
extended to even beyond quantum systems such as photonic crystals with
gain/loss Bo_zhen_nature_2015_Ering ; Takata_pSSH_PRL18 ;
Ozawa_TopoPhoto_RMP19 ; Junpeg_PRL_2020_HMMs , electric circuits
Hofmann_ExpRecipSkin_19 ; Helbig_ExpSkin_19 ; Hofmann_EleCirChern_PRL19 ;
Yoshida_MSkinPRR20 , mechanical metamaterials Yoshida_SPERs_mech19 ;
Scheibner_nHmech_PRL2020  and so on. Remarkably, the non-Hermiticity of such
systems enriches topological properties. For instance, non-Hermitian systems
may host exceptional points where both for the real- and imaginary-parts of
energy bands touch H.Shen_PRL_2018 ; VKozii_nH_arXiv17 ; T.Yoshida_PRB_2018 ;
A.A.Zyuzin_PRB_2018 ; Michishita_EP_PRB2020 ; Yoshida_nHReview_PTEP20 . It has
also been elucidated that symmetry of non-Hermitian systems results in
symmetry-protected exceptional rings (SPERs) Bergholtz_PRB_2019 ;
T.Yoshida_PRB_2019 ; Delplace_EP3_arXiv2021  and symmetry-protected
exceptional surfaces (SPESs) Okugawa_SPERs_PRB19 ; T.Yoshida_PRB_2019 ;
Zhou_SPERs_Optica19 ; K.Kimura_PRB_2019  in two- and three- dimensions,
respectively, although preserving the relevant symmetry requires fine-tuning.

Along with the above development of the topological band theory, recent
studies have revealed that several systems are described by generalized
eigenvalue problems (GEVPs) Raghu_PhC_PRL2008 ; Raghu_PhC_PRA2008 ;
Shindou_GEV_PRB2013 . The above progress of topological band theory implies
the presence of novel topological phenomena unique to GEVPs. Unfortunately,
however, most of the previous works focus on the case where the problem is
reduced to the ordinary Hermitian systems. Thus, for systems described by
GEVPs, a further development of a topological band theory remains a crucial
issue to be addressed.

In this letter, we discuss the topological band theory for systems of a GEVP
with Hermitian matrices. Our analysis elucidates that such systems may exhibit
non-Hermitian topological phenomena protected by emergent symmetry. As an
example, we demonstrate the emergence of SPERs for a system described by a
GEVP with Hermitian matrices. Notably, no fine-tuning is necessary to realize
these SPERs because they are protected by emergent symmetry. This property is
unique to systems described by the GEVPs. Furthermore, the SPERs with emergent
symmetry explain the origin of the hyperbolic dispersion of hyperbolic
metamaterials (HMMs) Smith_PRL_2003_HMMs ; Smith_APL_2004_HMMs ;
Liu_OptExp(2008)_HMMs ; Fang_PRB(2009)_HMMs ; Poddubny_NatPhoto_2013_HMMs ;
Drachev_OptExp_2013_HMMs ; Shekhar_2014_HMMs ; Ferrari_PiQE_2015_HMMs ;
Guo_AIP_2020_HMMs ; Noginov_APL(2009)_HMMs ; Kanungo_APL(2010)_HMMs ;
Krishnamoorthy_Science(2012)_HMM ; Starko_JOptSocAmB(2015)_HMMs ;
Kruk_NatComm_2016_HMMs ; W.Ji_PhysRevMaterials(2020)_HMMs .

So far, SPERs and SPESs have been reported for non-Hermitian systems described
by standard eigenvalue problem. We would like to stress, however, that SPERs
and SPESs described by the GEVP with Hermitian matrices do not require fine-
tuning in order to preserve the relevant symmetry, which is a striking
difference from the ordinary SPERs.

GEVP with Hermitian matrices.— Here, we analyze a general theory of a GEVP
with Hermitian matrices describing non-Hermitian topology. Specifically, we
elucidate that a complex band structure may emerge despite the Hermiticity of
matrices. We also show that generalized eigenenergies form pairs due to
emergent symmetry.

Let us consider the band theory described by a GEVP, which is defined as
footnote6

$H\psi=ES\psi.$ (1)

Here, $H$ and $S$ are Hermitian matrices, $E$ are generalized eigenvalues and
$\psi$ are generalized eigenvectors footnote7 . Unless otherwise noted,
generalized eigenvalues and generalized eigenvectors are simply referred to as
eigenvalues and eigenvectors in this letter.

In the following, we show that when $H$ and $S$ are indefinite, $E$ can be
complex in spite of the Hermiticity of $H$ and $S$. Without loss of
generality, we assume that $S$ is diagonalized. Here, matrix $S$ can be
decomposed as

$S=S^{\prime}\Sigma S^{\prime},$ (2)

with $S^{\prime}=\mathrm{diag}(\sqrt{|\beta_{1}|},\dots,\sqrt{|\beta_{n}|})$,
$\Sigma=\mathrm{diag}[\mathrm{sgn}(\beta_{1}),\dots,\mathrm{sgn}(\beta_{n})]$,
and $\beta_{i}$ being eigenvalues of matrix $S$. $\mathrm{sgn}(\beta_{i})$
take the sign of $\beta_{i}$. When $S$ is indefinite, $\Sigma$ is not
propotional to the identity matrix. Here, we define $\phi$ and $\tilde{H}$ as
$\phi=S^{\prime}\psi,\tilde{H}=S^{\prime-1}HS^{\prime-1}$. Because of the
relation $\Sigma=\Sigma^{-1}$, Eq. (1) is rewritten as

$H_{\Sigma}\phi=E\phi,$ (3)

with $H_{\Sigma}=\Sigma\tilde{H}$. The matrix $H_{\Sigma}$ is non-Hermitian,
and eigenvalues $E$ are given by complex. Noticing Hermiticity of $\tilde{H}$
and $\Sigma$, we can find emergent symmetry; $H_{\Sigma}$ satisfies the
relation,

$\Sigma^{-1}H_{\Sigma}\Sigma=H_{\Sigma}^{\dagger},$ (4)

which is known as pseudo-Hermiticity. Therefore, eigenvalues $E$ are real or
form complex conjugate pairs Mostafazadeh_pHvsPT1_2002 ;
Mostafazadeh_pHvsPT2_2002 ; Mostafazadeh_pHvsPT3_2002  (for details, see Sec.
I of Supplemental Material supple ). We note that the indefiniteness of $H$
and $S$ is essential for the complex band structure; either $H$ or $S$ is
definite, eigenvalues are always real footnote1 .

In the above, we have shown that system described by the GEVP with Hermitian
matrices may exhibit a complex band structure. The emergence of the complex
band structure can be understood by mapping the GEVP to the standard
eigenvalue problem footnote9 . We stress, however, that such a mapping yields
pseudo-Hermiticity as emergent symmetry, which is a significant difference
from ordinary systems described by the standard eigenvalue problem with a non-
Hermitian matrix.

SPERs in a two-band model.— We have seen that systems, described by the GEVP
with Hermitian matrices show complex eigenvalues. In the following, we
specifically analyze a two-band model to demonstrate SPERs, a unique
topological band touching of non-Hermitian topological bands.

Here, we consider a two-band model of the honeycomb lattice [see Fig. 1(a)].
We note that such a model described by $2\times 2$ Hermitian matrices is the
minimal model to investigate the SPERs. The GEVP of the model is written as

$\begin{pmatrix}m_{L}&vf_{\bm{k}}\\\
vf_{\bm{k}}^{*}&-m_{L}\end{pmatrix}\psi=E\begin{pmatrix}1+m_{R}&0\\\
0&1-m_{R}\end{pmatrix}\psi,$ (5)

with the hopping $v$. Here, $m_{L}$ and $m_{R}$ are real footnote8 . The
eigenvalues (eigenvectors) are denoted by $E$ ($\psi$). $f_{\bm{k}}$ is
defined as
$f_{\bm{k}}=(1+e^{i\bm{k}\cdot\bm{t_{1}}}+e^{i\bm{k}\cdot\bm{t_{2}}})$ with
$\bm{t}_{1}$, $\bm{t}_{2}$, and $\bm{t}_{3}$ being vectors connecting
neighboring sites [see Fig. 1(a)]. Here, we take the matrix in the left-hand
(right-hand) side of Eq. (5) as $H$ ($S$).

Figure 1:  (a): Sketch of the honeycomb lattice model and its Brillouin zone.
The gray (orange) dashed lines describe the unitcell (first Brillouin zone).
$\bm{t}_{1}$,$\bm{t}_{2}$, and $\bm{t}_{3}$ are vectors connecting neighboring
site. These vectors are defined as $\bm{t}_{1}=(0,-1)$,
$\bm{t}_{2}=(-\sqrt{3},1)/2$, and $\bm{t}_{3}=(\sqrt{3},1)/2$. (b): The
dispersion relation of overlaped honeycomb lattice model with $m_{L}=0$,
$m_{R}=0$. $v$ is fixed at $1$. Dirac points at $K$ and $K^{\prime}$ points
are denoted by arrows. (c)[(d)]: The real [imaginary] part of the dispersion
relation of overlaped honeycomb lattice model with $m_{L}=0.3$, $m_{R}=1.1$.
SPERs (red ring) based on the GEVP emerges around the $K$ and $K^{\prime}$
points.

Now let us discuss the band structure. Eigenvalues of this model are described
by

$\displaystyle E=E_{0}\pm\sqrt{M^{2}-|g_{\bm{k}}|^{2}},$ (6)

with $E_{0}=[m_{L}/(1+m_{R})-m_{L}/(1-m_{R})]/2$,
$M=[m_{L}/(1+m_{R})-m_{L}/(1-m_{R})$]/2, and
$g_{\bm{k}}=(vf_{\bm{k}})/\sqrt{1-m_{R}^{2}}$. Details of analysis are
provided in Sec. II of Supplemental Material supple . In the following, we
discuss the band structure for three cases; (i)$|m_{R}|<1$, (ii) $|m_{R}|>1$,
and (iii) $|m_{R}|=1$. We note that in cases (i), (ii), and (iii), matrix $S$
is definite, indefinite, and not invertible, respectively.

We start with case (i) where the model shows Dirac cones at $K$ and
$K^{\prime}$ points [see Fig. 1(b) which is obtained for $m_{L}=0$ and
$m_{R}=0$]. Here, the band structure has a mass gap with non-zero $m_{L}$ and
$m_{R}$ for $|m_{R}|<1$.

Next, we discuss the case (ii). Figures 1(c) and 1(d) show the band structure
for $m_{L}=0.3$ and $m_{R}=1.1$. In these figures, we can find the SPERs where
both of the real- and the imaginary-parts show band touching around $K$ and
$K^{\prime}$ points. The topological characterization of SPERs can be done by
computing the zero-th Chern number $N_{0\mathrm{Ch}}$, which is the number of
negative eigenvalues of Hermitian matrix $\Sigma(H_{\Sigma}-E_{\mathrm{ref}})$
with $E_{\mathrm{ref}}=1.57$, $\Sigma=\mathrm{diag}(1,-1)$, and
$H_{\Sigma}=\left(\begin{array}[]{cc}{m_{L}}/{|1+m_{R}|}&{vf_{\bm{k}}}/{\sqrt{|1-m_{R}^{2}|}}\\\
{-vf^{*}_{\bm{k}}}/{\sqrt{|1-m_{R}^{2}|}}&{m_{L}}/{|1-m_{R}|}\end{array}\right)$.
Specifically, inside (outside) of the ring, the zero-th Chern number takes
$N_{0\mathrm{Ch}}=$2 (1) which topologically protects the non-Hermitian band
touching. We note that the above SPERs are protected by the emergent symmetry;
they are robust against perturbations preserving Hermiticity of $H$ and $S$,
and translational symmetry. For case (iii) (i.e., for $m_{R}=1$), $S$ is not
invertible, and one of the eigenvalues diverges.

In the above, we have demonstrated the emergence of SPERs for the system
described by a GEVP with $2\times 2$ Hermitian matrices. Intriguingly, the
topology of SPERs is protected by emergent symmetry. Here, we stress that for
the emergence of SPERs, the indefiniteness of both matrices $H$ and $S$ are
essential. Although the overlap matrix $S$ cannot be indefinite for quantum
systems described by Schrödinger equation, it can be indefinite for optical
systems footnote5 . In the following, we apply the above results to an optical
system.

Figure 2:  (a): A sketch of the HMM consist of a metallo-wire structure. (b)
and (c): The real- and imaginary-parts of the dispersion relation of Eq. (7),
respectively. In these panels, red lines indicate the SPERs. (d): Zero-th
Chern number on the $\omega=0$ plane. (e): Hyperbolic dispersion with
$\tilde{\varepsilon}_{xx}=-0.36$ and $\tilde{\omega}=\omega a/2\pi c=0.071$
(blue solid line) and eliptic dispersion with $\tilde{\varepsilon}_{xx}=0.1$
(green dashed line). (f):Dispersion relation with square of the dimensionless
frequency for $k_{x}a=\pi$. Data denoted by blue solid line are obtained for
$\tilde{\varepsilon}_{xx}=-0.36$, $\tilde{\varepsilon}_{yy}=0.54$, and
$\tilde{\mu}_{zz}=1$. Data denoted by orange solid line are obtained for
$\tilde{\varepsilon}_{xx}=-0.36+i0.016$,
$\tilde{\varepsilon}_{yy}=0.54+i0.025$, and $\tilde{\mu}_{zz}=1$. We note that
the data of the blue line are almost the same as those of the orange line. The
red dashed line indicates the dimensionless frequency in which the parameter
takes the above value. The black dashed line is a guide to eye. (g): The
Exceptional surface in three-dimensional HMMs with the zero-th Chern number.

Application to HMMs.— The SPERs described by the GEVP elucidate the origin of
the experimentally observed hyperbolic dispersion of the HMMs.

The HMMs are optical metamaterials with extreme anisotropy. One of the
examples is a system of a metallo-wire structure [Fig. 2(a)]. In this case,
permittivity for the $x$-direction (permittivity for the $y$\- and
$z$-direction) satisfies $\epsilon_{xx}<0$ ( $\varepsilon_{yy}>0$ and
$\varepsilon_{zz}>0$) footnote3 .

In these optical systems, the band structure is described by Maxwell
equations; eigenvalues (eigenvectors) correspond to angular frequencies
(electromagnetic field).

Here, we consider the two-dimensional system, whose permittivity and
permeability are anisotropic. In this case, EM modes are separable to the TM
modes and TE modes. The former is defined as $\bm{E}=(0,0,E_{z})^{T}$ and
$\bm{H}=(H_{x},H_{y},0)^{T}$ while the latter is defined as
$\bm{E}=(E_{x},E_{y},0)^{T}$ and $\bm{H}=(0,0,H_{z})^{T}$. Here, electric
(magnetic) filed is denoted by $\bm{E}$ ($\bm{H}$).

Maxwell equations of the TE modes are given by

$\displaystyle\begin{pmatrix}0&-k_{y}&k_{x}\\\ -k_{y}&0&0\\\
k_{x}&0&0\end{pmatrix}\begin{pmatrix}H_{z}\\\ E_{x}\\\
E_{y}\end{pmatrix}=\omega\begin{pmatrix}\mu_{zz}&&\\\ &\varepsilon_{xx}&\\\
&&\varepsilon_{yy}\end{pmatrix}\begin{pmatrix}H_{z}\\\ E_{x}\\\
E_{y}\end{pmatrix},$ (7)

with permittivity $\varepsilon$ and permeability $\mu$. We have assumed that
the EM wave is a normal mode which is proportional to
$\mathrm{exp}(i\bm{k}\cdot\bm{r}-i\omega t)$.

The eigenvalues of Eq. (7) are given by

$\displaystyle\omega_{0}=0,$ $\displaystyle\omega_{\pm}=\pm\sqrt{k_{x}^{\prime
2}-k_{y}^{\prime 2}},$ (8)

with $k_{x}^{\prime}=k_{x}/\sqrt{\varepsilon_{yy}\mu_{zz}}$ and
$k_{y}^{\prime}=k_{y}/\sqrt{|\varepsilon_{xx}|\mu_{zz}}$. Eigenvectors of each
eigenvalue are given by

$\displaystyle\bm{v}_{\omega_{0}}=\begin{pmatrix}0\\\ 1\\\
k_{y}/k_{x}\end{pmatrix},\ \ \
\bm{v}_{\omega_{\pm}}=\frac{\varepsilon_{yy}}{k_{x}}\begin{pmatrix}\omega_{\pm}\\\
k_{y}/\varepsilon_{xx}\\\ 1\end{pmatrix}.$ (9)

We discurd the eigenvector $\bm{v}_{\omega_{0}}$ because it does not satisfy
the Gauss’s law. We also discurd the eigenvector $\bm{v}_{\omega_{-}}$ in the
region of $|k_{x}^{\prime}|<|k_{y}^{\prime}|$, in which $\bm{v}_{\omega_{-}}$
is amplified because of $\mathrm{Im}(\omega_{-})<0$. In the case of
$|k_{x}^{\prime}|=|k_{y}^{\prime}|$, eigenvectors satisfy
$\bm{v}_{\omega_{0}}=\bm{v}_{{\omega_{+}}}=\bm{v}_{{\omega_{-}}}$ and all
eigenspaces coalesce footnote2 .

As a first step to discuss the topological origin of the hyperbolic
dispersion, let us show the emergence of the SPERs in this system. The
emergence of SPERs can be confirmed in the band structure of Eq. (7) [see Fig.
2(b) and 2(c)]. These data are computed for $\tilde{\varepsilon}_{xx}=-0.36$,
$\tilde{\varepsilon}_{yy}=0.54$, and $\tilde{\mu}_{zz}=1$ which are obtained
in Ref. W.Ji_PhysRevMaterials(2020)_HMMs . $\tilde{\varepsilon}$ and
$\tilde{\mu}$ indicate relative permittivity and relative permeability
respectively. At $\omega=0$, in the regions where the condition
$|k_{x}^{\prime}|>|k_{y}^{\prime}|$ is satisfied, the eigenvalues become real.
In contrast, for regions where the condition
$|k_{y}^{\prime}|>|k_{x}^{\prime}|$ is satisfied, the eigenvalues become pure
imaginary. The real- and imaginary-parts of the eigenvalues become zero on the
$|k_{x}^{\prime}|=|k_{y}^{\prime}|$ lines. This line corresponds to the SPER
in the lattice system. In HMMs, SPERs separate the metallic region and non-
metallic region. These SPERs are robust for perturbation without breaking the
Hermiticity of the Maxwell equation because of the topological nature (see
below).

The emergence of the above SPERs explains the origin of the hyperbolic
dispersion of HMMs. At $\omega=0$, the isofrequency surface forms the straight
lines corresponding to SPERs. Increasing $\omega$ from zero, the structure of
the isofrequency surface continuously changes into a hyperbolic structure
which is observed in experiments [see blue solid line of Fig. 2(e)]. These
results explains that SPERs are the origin of the hyperbolic dispersion of
HMMs. We note that for $\varepsilon_{xx}>0$, $\varepsilon_{yy}>0$, and
$\mu_{zz}>0$, the hyperbolic dispersion is not observed [see green dashed line
of Fig. 2(e)] corresponding to the absence of the SPERs.

At $\omega=0$, topological characterization of the metallic region and the
non-metallic region of the HMMs is done by computing the zero-th Chern number
$N_{0\mathrm{Ch}}$ T.Yoshida_PRB_2019 . Figure 2(c) indicates the zero-th
Chern number on the zero-frequency plane. $\Sigma$ denotes the pseudo-
Hermitian operator, and $H_{\Sigma}$ denotes the pseudo-Hermitian matrix (for
more details, see Sec. III of Supplemental Material supple ). The zero-th
Chern number takes $N_{0\mathrm{Ch}}=2$ in the non-metallic region and takes
$N_{0\mathrm{Ch}}=1$ in the metallic region.

We also expect that the square-root dispersion is one of the experimental
signatures of SPERs. Here, prior to the experimental observations, we provide
theoretical data by approximating that $\varepsilon$ and $\mu$ are independent
of $\omega$. The blue line in Fig. 2(f) indicates square of dimensionless
frequency footnote4  for the $\tilde{\varepsilon}_{xx}=-0.36$,
$\tilde{\varepsilon}_{yy}=0.54$, and $\tilde{\mu}_{zz}=1$ which are obtained
by Ref. W.Ji_PhysRevMaterials(2020)_HMMs . In Ref.
W.Ji_PhysRevMaterials(2020)_HMMs , microwave around 8[GHz] is used. Near the
EP, the dispersion of the square of dimensionless frequency is linear. Dashed
line indicates the dimensionless frequency at which the permittivity and the
permeability take the above values. Therefore, we expect to be able to observe
the linear dispersion with the square of the frequency experimentally around
the dashed line. The above band structures exist out of a light cone. As the
experimental method observe out of the light cone, attenuated total reflection
(ATR) method is employed otto(1976)_ATR ; kretschmann(1968)_ATR ;
Futamata_ApplOpt(1997)_ATR . We note that, for more quantitative prediction,
the frequency dependence of $\varepsilon$ and $\mu$ should be taken into
account. Such theoretical works are desired prior to the experimental
observation.

Finally, we note that when these HMMs are treated as three-dimensional
systems, SPESs form a cone structure in the three-dimensional momentum space.
Figure 2(g) shows the SPES when the $z$-direction is the anisotropic axis.
These SPESs also explain the characteristic dispersion of three-dimensional
HMMs (for details, see Sec. IV of Supplemental Material supple ).

In the above, we have applied the topological band theory to HMMs, which
explains the origin of the experimentally observed hyperbolic dispersion of
these systems. Decreasing $\omega$ from a finite value to zero, the hyperbolic
dispersion asymptotically changes to SPERs. These SPERs emerging at
$\omega=0$, separate metallic region and non-metallic region due to the
indefiniteness of the Maxwell equation describing HMMs.

Previous works have analyzed non-Hermitian band structures of HMMs
Junpeg_PRL_2020_HMMs . We stress, however, that SPERs and SPESs described by
the GEVP reveal the origin of the experimentally observed hyperbolic
dispersion of HMMs.

Conclusion.— In this letter, we have investigated the systems described by the
GEVP with Hermitian matrices. Our analysis has elucidated that non-Hermitian
topological band structures emerge in spite of the Hermiticity of the
matrices. For the non-Hermitian topological band structure, the indefiniteness
of matrices appearing the left- and right- hand sides is essential.
Remarkably, these SPERs are protected by emergent symmetry; any fine-tuning is
not necessary to preserve the symmetry as long as $H$ and $S$ are Hermitian,
which is a striking difference of ordinary SPERs.

Furthermore, the above SPERs described by the GEVP reveal the origin of the
hyperbolic dispersion of HMMs which is observed in experiments. In HMMs, SPERs
separate the momentum space to the metallic region and the non-metallic
region. Because a hyperbolic isofrequency surface has been observed by angle-
resolved reflection spectrum measurements, we also expect that the square-root
dispersion of SPERs can be observed which is unique to dispersion of SPERs.

Acknowledgement.— We thank Satoshi Iwamoto, Shun Takahashi, and Atsushi Kubo
for fruitful discussions. This work is supported by JSPS Grant-in-Aid for
Scientific Research on innovative Areas ”Discrete Geometric Analysis for
Materials Design”: Grants No. JP20H04627. This work is also supported by JSPS
KAKENHI Grants No. JP17H06138, and No. JP19K21032.

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  * (83) An example of a system described by the GEVP is a photonic system as we see below. Quantum systems are described by the standard eigenvalue problem. For such systems, applying the ordinary topological band theory is more appropriate.
  * (84) This matrix $S$ naturally appears in photonic systems.
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  * (86) Consider open quantum systems described by the non-Hermitian eigenvalue problems. If the non-Hermitian Hamiltonian preserves the pseudo-Hermiticity [see Eq. (4)] for such systems, we can mathematically map the non-Hermitian eigenvalue problem to the GEVP. We note, however, that we mainly focus on photonic systems which are essentially described by the GEVP.
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  * (91) We note that the square-root dispersion survives even when the imaginary part of $\varepsilon$ and $\mu$ are taken into account [see the orange line of Fig. 2(f)].
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Supplemental Materials:
Topological band theory of a generalized eigenvalue problem with Hermitian
matrices: Symmetry-protected exceptional rings with emergent symmetry

##  S1 Complex conjugate eigenvalues of pseudo-Hermitian matrix

In this section, we show that the complex eigenvalues of the pseudo-Hermitian
matrix appear as complex conjugate pairs. The condition of pseudo-Hermite is
defined as

$\Sigma^{-1}H^{\dagger}\Sigma=H,$ (S1)

with operator $\Sigma$. Here, let us consider the eigenvalue problem

$\displaystyle H|R_{n}\rangle=E_{n}|R_{n}\rangle,$ (S2) $\displaystyle\langle
L_{n}|H=E_{n}\langle L_{n}|,$

where $|R_{n}\rangle$ ($\langle L_{n}|$) are right (left) eigenstates, and
$E_{n}$ are eigenvalues. $n$ indicate the label of the eigenvalues. From Eq.
(S1) and Eq. (S2), we can be also obtained eigenvalue problem with eigenstates
$\Sigma|L_{n}\rangle$,

$H(\Sigma|L_{n}\rangle)=\Sigma H^{\dagger}|L_{n}\rangle=\Sigma(\langle
L_{n}|H)^{\dagger}=E^{*}_{n}(\Sigma|L_{n}\rangle),$ (S3)

From Eq. (S3), we can see the fact that $\Sigma$ maps the eigenstate of
eigenvalue $E_{n}$ to the eigenstate of eigenvalue $E_{n}^{*}$. Therefore,
complex eigenvalues of the pseudo-Hermitian matrix are given by complex
conjugate pairs.

##  S2 Eigenvalue of the two-band model

In the main text, we have introduced the $2\times 2$ model of a GEVP with
diagonal overlap matrix $S$. In this section, we consider the $2\times 2$
model with the off-diagonal term of $S$,

$\begin{pmatrix}\epsilon+m_{L}&vf_{\bm{k}}\\\ vf_{\bm{k}}^{*}&\epsilon-
m_{L}\end{pmatrix}\psi=E\begin{pmatrix}1+m_{R}&wf_{\bm{k}}\\\
wf_{\bm{k}}^{*}&1-m_{R}\end{pmatrix}\psi,$ (S4)

with the onsite potential $\epsilon$ and hopping $v$ and $w$. The difference
of the onsite potentials are parametrized by $m_{L}$ and $m_{R}$. The
eigenvalue (eigenvector) are denoted by $E(\psi)$. where $E$ is eigenvalue,
$\psi$ is eigenvector, $\epsilon$ is onsite term, $v$ and $w$ are hopping term
of left and right matrix respectively, $m_{L}$ and $m_{R}$ describe the
potential defference of red site and blue site (Fig. 1(a)), and $f_{\bm{k}}$
is defined as
$f_{\bm{k}}=(1+e^{i\bm{k}\cdot\bm{t_{1}}+e^{\bm{k}\cdot\bm{t_{2}}}})$.

Let us discuss the band structure. First, we take left side matrix $H$, right
side matrix $S$ and diagonalize matrix $S$ by the unitary matrix $U_{S}$,

$U_{S}^{-1}HU_{S}\psi=EU_{S}^{-1}SU_{S}\psi.$ (S5)

$U_{S}^{-1}HU_{S}$ and $U_{S}^{-1}SU_{S}$ are taken $\tilde{H},\tilde{S}$ to
simplify the symbols. Here, eigenvalues of matrix $S$ are given as
$s_{\pm}=1\pm\sqrt{|wf(\bm{k})|^{2}+m_{R}^{2}}$.

Next, matrix $S$ is decomposed as,

$\displaystyle\tilde{S}$
$\displaystyle=\tilde{S}^{\frac{1}{2}}\Sigma\tilde{S}^{\frac{1}{2}}=\begin{pmatrix}\sqrt{|s_{+}|}&0\\\
0&\sqrt{|s_{-}|}\\\ \end{pmatrix}\Sigma\begin{pmatrix}\sqrt{|s_{+}|}&0\\\
0&\sqrt{|s_{-}|}\\\ \end{pmatrix}.$ (S6)

where $S^{\frac{1}{2}}$ and $\Sigma$ consist of the square root of the
absolute value of eigenvalue and sign of eigenvalue of matrix $S$. For
$\mathrm{det}S>0$, matrix $\Sigma$ becomes identity matrix. In this case, a
GEVP transformed Hermitian standard eigenvalue problem as follows,

$\left(\tilde{S}^{-\frac{1}{2}}\tilde{H}\tilde{S}^{-\frac{1}{2}}\right)\left(\tilde{S}^{\frac{1}{2}}\psi\right)=E\left(\tilde{S}^{\frac{1}{2}}\psi\right).$
(S7)

where matrix
$\left(\tilde{S}^{-\frac{1}{2}}\tilde{H}\tilde{S}^{-\frac{1}{2}}\right)\left(\tilde{S}^{\frac{1}{2}}\right)$
is

$\displaystyle\tilde{S}^{-\frac{1}{2}}\tilde{H}\tilde{S}^{-\frac{1}{2}}=\begin{pmatrix}\frac{\epsilon+m_{L}}{|s_{+}|}&\frac{vf_{\bm{k}}}{\sqrt{|s_{+}|}\sqrt{|s_{-}|}}\\\
\frac{vf_{\bm{k}}}{\sqrt{|s_{+}|}\sqrt{|s_{-}|}}&\frac{\epsilon-
m_{L}}{|s_{-}|}\end{pmatrix}.$ (S8)

Eigenvalues are given by

$\displaystyle E$
$\displaystyle=\frac{1}{2}\left[\left(\frac{\epsilon+m_{L}}{|s_{+}|}+\frac{\epsilon-
m_{L}}{|s_{-}|}\right)\pm\sqrt{\left(\frac{\epsilon+m_{L}}{|s_{+}|}-\frac{\epsilon-
m_{L}}{|s_{-}|}\right)^{2}+\frac{4v^{2}|f_{\bm{k}}|^{2}}{|s_{+}||s_{-}|}}\right].$
(S9)

However, For $\mathrm{det}S<0$, matrix $\Sigma$ becomes

$\Sigma=\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (S10)

In this case, the GEVP cannot be transformed in to a Hermitian standard
eigenvalue problem but into a non-Hermitian standard eigenvalue problem as
follows,
$\sigma_{3}\left(\tilde{S}^{-\frac{1}{2}}\tilde{H}\tilde{S}^{-\frac{1}{2}}\right)\left(\tilde{S}^{\frac{1}{2}}\psi\right)=E\left(\tilde{S}^{\frac{1}{2}}\psi\right).$
We can easily prove the matrix
$\tilde{S}^{-\frac{1}{2}}\tilde{H}\tilde{S}^{-\frac{1}{2}}$ satisfy the
pseudo-Hermite condition,

$\sigma_{3}\left(\sigma_{3}\tilde{S}^{-\frac{1}{2}}\tilde{H}\tilde{S}^{-\frac{1}{2}}\right)\sigma_{3}=\left(\tilde{S}^{-\frac{1}{2}}\tilde{H}\tilde{S}^{-\frac{1}{2}}\sigma_{3}\right)=\left(\sigma_{3}\tilde{S}^{-\frac{1}{2}}\tilde{H}\tilde{S}^{-\frac{1}{2}}\right)^{\dagger},$
(S11)

where we used Hermiticity of matrix $\tilde{H}$, $\tilde{S}^{\frac{1}{2}}$,
and $\sigma_{3}$.

The eigenvalues of this model are described as

$\displaystyle E$
$\displaystyle=\frac{1}{2}\left[\left(\frac{\epsilon+m_{L}}{|s_{+}|}-\frac{\epsilon-
m_{L}}{|s_{-}|}\right)\pm\sqrt{\left(\frac{\epsilon+m_{L}}{|s_{+}|}+\frac{\epsilon-
m_{L}}{|s_{-}|}\right)^{2}-\frac{4v^{2}|f_{\bm{k}}|^{2}}{|s_{+}||s_{-}|}}\right].$
(S12)

Here, eigenvalue (S12) can be complex though eigenvalue (S9) cannot be complex
[see Fig. 1].

Figure S1:  Band structure of the $2\times 2$ model with the off-diagonal term
of $S$. Parameters are chosed in $\epsilon=2.0$, $v=1.0$, $m_{L}=2.2$,
$w=0.3$, and $m_{R}=1.3$. Red lines indicate the SPERs. These figures
demonstrate the robustness of SPERs described by the GEVP.

##  S3 Computation of the zeroth-Chern number

In this section, we compute the zeroth-Chern number of HMMs. Here, we consider
the case of TE modes with $\mu_{zz}=1$, $\varepsilon_{xx}=-1$, and
$\varepsilon_{yy}=1$. The matrix of left-hand (right-hand) $H$ ($\Sigma$) is
given by

$\displaystyle H=\begin{pmatrix}0&-k_{y}&k_{x}\\\ -k_{y}&0&0\\\
k_{x}&0&0\end{pmatrix},\ \ \ \Sigma=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&1\end{pmatrix}.$ (S13)

First, we decompose $H_{\Sigma}=\Sigma H$ as follows,

$\displaystyle
H_{\Sigma}=\hat{R}\hat{E}\hat{L}^{\dagger}=\begin{pmatrix}\bm{v}_{R,0},&\bm{v}_{R,\omega_{+}},&\bm{v}_{R,\omega_{-}}\end{pmatrix}\begin{pmatrix}\omega_{0}&0&0\\\
0&\omega_{+}&0\\\
0&0&\omega_{-}\end{pmatrix}\begin{pmatrix}\bm{v}_{L,\omega_{0}}^{\dagger}\\\
\bm{v}_{L,\omega_{+}}^{\dagger}\\\
\bm{v}_{L,\omega_{-}}^{\dagger}\end{pmatrix},$ (S14)

with $\omega_{0}=0$, and $\omega_{\pm}=\pm\sqrt{k_{y}^{2}-k_{x}^{2}}$. On the
SPERs, the condition of $\mathrm{det}(\Sigma H_{\Sigma})=0$ is satisfied.
However, our Hamiltonian is always $\mathrm{det}(\Sigma H_{\Sigma})=0$ due to
the existence of $\omega_{0}$. In order to characterize the SPERs, we define
the modified Hamiltonian $\tilde{H}_{\Sigma}$, which $\omega_{0}$ is shifted
by $\delta$,

$\displaystyle\tilde{H}_{\Sigma}=\hat{R}\hat{\tilde{E}}\hat{L}^{\dagger}=\begin{pmatrix}\bm{v}_{R,0},&\bm{v}_{R,\omega_{+}},&\bm{v}_{R,\omega_{-}}\end{pmatrix}\begin{pmatrix}\omega_{0}+\delta&0&0\\\
0&\omega_{+}&0\\\
0&0&\omega_{-}\end{pmatrix}\begin{pmatrix}\bm{v}_{L,\omega_{0}}^{\dagger}\\\
\bm{v}_{L,\omega_{+}}^{\dagger}\\\
\bm{v}_{L,\omega_{-}}^{\dagger}\end{pmatrix}.$ (S15)

Therefore, Hermitian matrix $\Sigma\tilde{H}_{\Sigma}$ is given by

$\displaystyle\Sigma\tilde{H}_{\Sigma}=\begin{pmatrix}0&-k_{y}&k_{x}\\\
-k_{y}&-\frac{k_{x}^{2}}{k_{x}^{2}-k_{y}^{2}}\delta&\frac{k_{x}k_{y}}{k_{x}^{2}-k_{y}^{2}}\delta\\\
k_{x}&\frac{k_{x}k_{y}}{k_{x}^{2}-k_{y}^{2}}\delta&-\frac{k_{y}^{2}}{k_{x}^{2}-k_{y}^{2}}\delta\end{pmatrix}.$
(S16)

In the main text, we characterized the SPERs by the zeroth-Chern number of
$\Sigma\tilde{H}_{\Sigma}$ which is the number of negative eigenvalues. We
note that the zeroth-Chern number in Figs. 2(b) and 2(e) are computed for
$\delta>0$. In the case for $\delta<0$, the region of $N_{0\mathrm{Ch}}=1$ and
that of $N_{0\mathrm{Ch}}=2$ are fliped.

##  S4 SPES and three-dimensional hyperbolic dispersion

In the main text, we have shown that SPERs at $\omega=0$ are the origin of the
hyperbolic dispersion. In this section, we show that SPESs at $\omega=0$ are
the origin of hyperbolic dispersion in the three-dimensional momentum space.
Here, we analyze three-dimensional Maxwell equations,

$\begin{pmatrix}0&\bm{k}\times\\\
-\bm{k}\times&0\end{pmatrix}\begin{pmatrix}\bm{E}\\\
\bm{H}\end{pmatrix}=\omega\begin{pmatrix}\varepsilon&0\\\
0&\mu\end{pmatrix}\begin{pmatrix}\bm{E}\\\ \bm{H}\end{pmatrix}.$ (S17)

We assume that $z$-direction is anisotropic axis. The permittivity and the
permeability are given by

$\displaystyle\varepsilon=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-1\end{pmatrix},\ \ \ \mu=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}.$ (S18)

In this case, non-zero eigenvalues are given by

$\omega=\pm\sqrt{k_{x}^{2}+k_{y}^{2}-k_{z}^{2}}.$ (S19)

A surface satisfying $k_{x}^{2}+k_{y}^{2}=k_{z}^{2}$ corresponds to a SPES
[see Fig. S2(a)]. Solving the above equation with respect to $k_{z}$ we obtain

$k_{z}=\pm\sqrt{k_{x}^{2}+k_{y}^{2}-\omega^{2}}.$ (S20)

When SPESs emerge at $\omega=0$, hyperbolic dispersion in three-dimensional
momentum space emerges for $\omega\neq 0$ [see Fig. S2(b)]. For
$\varepsilon=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$, the matrix
in the right-hand side of Eq. (S17) is positive definite. In this case, the
SPES vanishes. Correspondingly, hyperbolic dispersion in three-dimensional
momentum space also vanishs [see Fig. S2(c)].

Figure S2:  (a): The SPES on the $\omega=0$ surface in the three-dimensional
momentum space. (b): Hyperbolic dispersion in three-dimensional momentum space
with $\omega=0.5$. (c): Eliptic dispersion with $\varepsilon_{xx}>0$.