# Plasmonic detection of the parity anomaly in a two-dimensional Chern
insulator

M. N. Chen<EMAIL_ADDRESS>Yu Zhou<EMAIL_ADDRESS>1School of Science,
Hangzhou Dianzi University, Hangzhou, 310018, China

###### Abstract

In this paper, we present an analytic study on the surface plasmon polaritons
in two-dimensional parity anomaly Chern insulators. The two-dimensional
conductivity derived from the BHZ model are antisymmetric, based on which two
surface plasmon modes each contains two branches of dispersions have been
found. In the absence of parity anomaly, the Hall conductivities with positive
and negative Dirac mass terms differ by a sign; two branches of each surface
plasmon mode are exactly degenerate. However, the parity anomaly can lift such
degeneracy and lead to significant modifications of these dispersion curves or
even the occurrence of an extra branch of surface plasmons under particular
condition. Our investigations pave a possible way for the detection of the
parity anomaly in a two-dimensional Chern insulator via plasmonic responses.

##  I INTRODUCTION

Topological materials have attracted much attention in both theoretical and
experimental aspects in recent years. As a typical class of topological
materials, topological insulators (TIs) have exotic metallic surface
(boundary) states protected by time-reversal symmetry, whose topological
charge is identified as the $\mathbb{Z}_{2}$ invariant TI1 . Unlike TIs, the
Chern insulators (CIs), or named as quantum anomalous Hall insulators, break
the time-reversal symmetry, which belongs to the $\mathbb{Z}$-topological
classification and the corresponding topological charge is the first Chern
number. A spin-conserved TI may be viewed as two copies of CIs carrying
opposite spin polarizations and counter propagating edge states, respectively
TI2 ; TI3 .

Both TIs and CIs have nontrivial responses to external electromagnetic fields.
For TIs, we have
$j^{\mathrm{s}}_{\mu}=\sigma^{\mathrm{s}}_{xy}\epsilon_{\mu\nu\tau}\partial^{\nu}\Omega^{\tau}$
with $\sigma^{\mathrm{s}}_{xy}$ the spin-Hall conductivity and $\Omega$ being
a pure gauge BHZ ; BHZ1 ; res1 ; res2 ; for CIs, it has form
$j_{\mu}=\sigma_{xy}\epsilon_{\mu\nu\tau}A^{\mu}\partial^{\nu}A^{\tau}$ with
$\sigma_{xy}$ the Hall conductivity and $A^{\mu}$ the gauge fields. Therefore,
nontrivial electromagnetic responses originate from topologically-nontrivial
bulk band structures, which may lead to nontrivial collective excitations.

Surface plasmons are collective oscillations of free electrons coupled with
light existing at the metal-dielectric interface, of which the electric fields
are tightly confined and decay exponentially away from the surface SPPbook ;
JacksonBook . The permittivities at two sides usually possess opposite signs,
i.e. one is positive and the other is negative; otherwise, no dispersion
relations can be found for these surface waves. After the discovery of
graphene, researchers have realized that such one-atom thick material can
support exceedingly strong surface plasmons that is detectable through, for
example, scanning near-field infrared microscopy GrapRev1 ; GrapRev2 ;
GrapRev3 ; NatureGrap1 ; NatureGrap2 ; NSOM ; NatureGrap3 . This can be
understood that the conductivity of doped graphene is large enough to cause
significant in-plane currents and charge oscillations under incident light
pushing the corresponding Drude plasma frequency into the infrared region G1 ;
G2 ; G3 ; G4 ; G5 ; G6 ; G7 ; G8 ; G9 ; G10 ; G11 ; G12 . Although the optical
conductivity of graphene is isotropic, its imaginary part can be positive or
negative depending on the Fermi level as well as the photon energy. With a
positive imaginary part, graphene resembles a thin metallic film supporting
transverse magnetic- (TM) polarized surface plasmons; however, with a negative
imaginary part, it is more like a thin dielectric film and the surface
plasmons are transverse electric- (TE) polarized G13 . Due to the isotropy,
TM- and TE-polarized modes are decoupled and their dispersion relations can be
found separately.

Anisotropy can lead to the coupling of these two polarizations. For example,
in phosphorene the electron masses are largely different along the zigzag and
armchair directions due to its puckered structure. After being doped with
electrons, phosphorene can become metallic supporting surface plasmons BP1 ;
BP2 ; BP3 ; BP4 ; the optical conductivities differ along the zigzag and
armchair directions. The iso-frequency contour of the in-plane surface
plasmons is in most cases elliptic. With proper electron doping, the
conductivities along two directions can have opposite signs and the
corresponding iso-frequency contour becomes a hyperbola. In this case, they
are called hyperbolic surface plasmons BP5 . In the calculation of dispersion
relations, one must solve all the field components since the anisotropy
usually mixes the two polarizations mentioned above.

As for the systems with nonzero Hall conductivities such as CIs, the off-
diagonal terms of the conductivity tensor induce currents orthogonal to the
applied electric fields, which immediately leads to the situation where all
the field components are interrelated and should be all considered
simultaneously during the calculation of the surface plasmons TIs1 ; TIs2 ;
TIs3 ; TIs4 ; TIs5 ; TIs6 . The dispersion relation strongly depends on the
conductivity as well as the permittivity of the surrounding materials. Most 2D
CIs are encapsulated with optically anisotropic dielectrics showing different
permittivities parallel and perpendicular to the conductive surface. Such
anisotropy can significantly modify the dispersion relations of the surface
plasmons as well.

In this paper, we intend to reveal the connections between the parity anomaly
in a two-dimensional Chern insulator and the dispersion relations of the
surface plasmons. Given by the symmetry of the BHZ model, two surface plasmon
modes have been found, each of which contains two branches of dispersion
relations. Two modes are respectively characterized by $E_{z}=0$ and
$H_{z}=0$; the expressions of their dispersion relations indicate that two
branches of each mode are exactly degenerate without parity anomaly. One
arrives at the same dispersion curve with positive and negative Dirac mass
terms. However, in the presence of the parity anomaly, such degeneracy would
be lifted; the Hall conductivities in the topologically trivial and non-
trivial situations no longer just differ by a sign and an extra branch of
surface plasmons might be found under particular condition, as schematically
shown in Fig.1. Our investigations pave a possible way for the detection of
the parity anomaly in a two-dimensional Chern insulator via plasmonic
responses.

This paper has been organized as follows. In Sec.II, details regarding the BHZ
model describing the two-dimensional Chern insulator and the calculations of
the optical conductivities are given. The real and imaginary parts of both the
longitudinal and Hall conductivities have been derived. In Sec.III, the
expressions of the dispersion relations of the surface plasmons are presented
based on the two dimensional conductivity tensor given by the BHZ model. Two
modes each with two branches have been found. In Sec.IV, the dispersion
relations or equivalently the effective indices of the surface plasmons have
been numerically calculated for all the cases. In Sec.V, a conclusion has been
given.

##  II Model and Optical Conductivities

Figure 1: Schematic illustration of the detection of the parity anomaly in 2D
CIs. An extra branch of the surface plasmons may occur under particular
condition with parity anomaly. Left/right panel corresponds to the the case
without/with parity anomaly.

Let us start from the minimal Hamiltonian for the Bernevig-Hughes-Zhang (BHZ)
model, which can be written as BHZ ; BHZ1

$\hat{H}=v\hbar(k_{x}\hat{\sigma}_{x}+k_{y}\hat{\sigma}_{y})+m_{\mathbf{k}}\hat{\sigma}_{z}\
,$ (1)

where $v$ stands for the Fermi velocity,
$m_{\mathbf{k}}=mv^{2}-b\hbar^{2}k^{2}$ is the regularized Dirac mass term
with $k^{2}=k^{2}_{x}+k^{2}_{y}$, and $\\{\hat{\sigma}_{i}\\}$ are Pauli
matrices with $i=x,y,z$. The Hamiltonian (1) is usually used to describe the
Chern insulator, where we have assumed that the spin is fully polarized and
thus the spin freedoms can be ignored.

After straightforward diagonalization, one can obtain the eigenvalues:

$\epsilon_{\pm}(k)=\pm\sqrt{v^{2}\hbar^{2}k^{2}+m^{2}_{\mathbf{k}}}\ ,$ (2)

and the corresponding eigenvectors are

$|u_{-}\rangle=\begin{pmatrix}\sin\theta_{k}\mathrm{e}^{-\mathrm{i}\varphi_{k}}\\\
-\cos\theta_{k}\end{pmatrix},\quad|u_{+}\rangle=\begin{pmatrix}\cos\theta_{k}\mathrm{e}^{-\mathrm{i}\varphi_{k}}\\\
\sin\theta_{k}\end{pmatrix},$ (3)

where $\varphi_{k}=\arg(k_{x}+\mathrm{i}k_{y})$ and
$2\theta_{k}=\mathrm{arccot}(m_{\mathbf{k}}/v\hbar k)$.

Figure 2: Plots of conductivities (a) $\mathrm{Re}\sigma_{xy}$, (b)
$\mathrm{Im}\sigma_{xy}$, (c) $\mathrm{Re}\sigma_{xx}$, and (d)
$\mathrm{Im}\sigma_{xx}$ as functions of the photon energy $\hbar\omega$ in
units of $e^{2}/h$. Solid and dashed curves correspond to the cases without
and with parity anomaly, respectively. The conductivities with positive and
negative Dirac mass terms are plotted in blue and red, respectively. The cut-
off energy is $\epsilon_{\mathrm{c}}=4|m|v^{2}$ in (d). Other parameters are
$\hbar v=0.5$eV, $\hbar^{2}b=0.2$eV$\cdot$Å2. Figure 3: Propagation constants
of the surface plasmon mode with $H_{z}=0$ for the situations with (a)
$mv^{2}=0.05$ and (b) $mv^{2}=-0.05$. Insets are the enlarge plots in the low
photon energy ranges. The solid and imaginary parts are plotted in blue and
red, respectively. Propagation constants correspond to the case with/without
parity anomaly are plotted with dashed/solid curves. Figure 4: (a)
$\mathrm{Re}\sigma_{xy}-\mathrm{Im}\sigma_{xx}$ and (b)
$\mathrm{Re}\sigma_{xy}+\mathrm{Im}\sigma_{xx}$ as functions of the photon
energy. Two branches of dispersion relations given by Eq.(27) and Eq.(29)
require $\mathrm{Re}\sigma_{xy}-\mathrm{Im}\sigma_{xx}>0$ and
$\mathrm{Re}\sigma_{xy}+\mathrm{Im}\sigma_{xx}<0$, respectively. (c) and (d)
are $\gamma^{(1)}+\gamma^{(2)}$ corresponding to Eq.(27) and Eq.(29) as
functions of the photon energy, respectively. It must be satisfied that
$\gamma^{(1)}+\gamma^{(2)}>0$. Figure 5: Effective indices as functions of the
photon energy for the two branches of surface plasmons given by (a) Eq.(27)
and (b) Eq.(29). The ratios $\omega_{i}/\omega_{r}$ of the two branches of
surface plasmons given by (c) Eq.(27) and (d) Eq.(29).

The optical conductivity tensor can be obtained by the standard Kubo formula
Rammer  for the $d$-dimensional system,

$\sigma_{ij}(\omega)=-\frac{\mathrm{i}}{L^{d}}\frac{e^{2}}{\hbar}\sum_{n,m}\frac{[n_{\mathrm{F}}(\epsilon_{n})-n_{\mathrm{F}}(\epsilon_{m})]v^{i}_{nm}v^{j}_{mn}}{(\epsilon_{n}-\epsilon_{m})(\epsilon_{n}-\epsilon_{m}+\hbar\omega)}$
(4)

where $v^{i}_{nm}=\langle u_{n}|\frac{\partial\hat{H}}{\partial
k_{i}}|u_{m}\rangle$ is the velocity operator in the $i$-th direction with
$i=1,2,\ldots,d$. The indices $m,n=+,-$ represent the conduction band and
valence band, respectively,
$n_{\mathrm{F}}(\epsilon_{n})=1/\big{(}1+\mathrm{e}^{\beta(\epsilon_{n}-\mu)}\big{)}$
is the Fermi-Dirac distribution function with $\mu$ being the chemical
potential, $\beta=1/(k_{\mathrm{B}}T)$ with $k_{\mathrm{B}}$ being the
Boltzmann constant and $T$ is the tmeperature, and $L$ is the length of the
system. For the model we have considered in this work, $d=2$. In what follows,
we will include the impurity scattering processes, therefore, the frequency
$\omega$ should be replaced by $\omega+\mathrm{i}/2\tau$ with $\tau$ the
elastic scattering time. Substituting Eq.(3) into Eq.(4) and after doing some
algebras, the expression for the optical conductivity becomes

$\begin{split}\sigma_{ij}(\omega)&=-\mathrm{i}e^{2}\hbar\int\frac{\mathrm{d}^{2}\mathbf{k}}{(2\pi)^{2}}\frac{n_{\mathrm{F}}(\epsilon_{+})-n_{\mathrm{F}}(-\epsilon_{+})}{2\epsilon_{+}}\\\
&\quad\times\bigg{[}v^{i}_{+-}v^{j}_{-+}\frac{\hbar\omega-2\epsilon_{+}}{(\hbar\omega-2\epsilon_{+})^{2}+\hbar^{2}/4\tau^{2}}-\mathrm{i}\pi
v^{i}_{+-}v^{j}_{-+}\delta(\hbar\omega-2\epsilon_{+})+v^{i}_{-+}v^{j}_{+-}\frac{1}{2\epsilon_{+}+\hbar\omega}\bigg{]}\
.\end{split}$ (5)

Using Eq.(5), we can obtain the real part for the longitudinal conductivity
$\sigma_{xx}$ at zero temperature as (in units of $e^{2}/h$)

$\begin{split}\mathrm{Re}\,\sigma_{xx}(\omega)=\frac{\pi
v^{2}}{8}\frac{1}{v^{2}-2b(mv^{2}-b\hbar^{2}k^{2})}\bigg{[}1+\frac{4(mv^{2}+b\hbar^{2}k^{2})^{2}}{(\hbar\omega)^{2}}\bigg{]}\Theta(\hbar\omega-2|m|v^{2})\end{split}$
(6)

where $k^{2}$ is solved via the equation

$\frac{(\hbar\omega)^{2}}{4}=v^{2}\hbar^{2}k^{2}+(mv^{2}-b\hbar^{2}k^{2})^{2}\
,$ (7)

the Fermi energy $\epsilon_{\mathrm{F}}$ is set to zero (i.e. stays in the
band gap and hence the intraband contribution to the conductivity is zero),
and the imaginary part of $\sigma_{xx}$ is given by

$\mathrm{Im}\,\sigma_{xx}(\omega)=\frac{v^{2}}{4}\int\mathrm{d}\epsilon_{+}\frac{1}{\sqrt{(1-4bm)v^{4}+4b^{2}\epsilon^{2}_{+}}}\bigg{[}1+\frac{(mv^{2}+b\hbar^{2}k^{2})^{2}}{\epsilon^{2}_{+}}\bigg{]}\Bigg{[}\frac{\hbar\omega-2\epsilon_{+}}{(\hbar\omega-2\epsilon_{+})^{2}+\frac{\hbar^{2}}{4\tau^{2}}}+\frac{1}{2\epsilon_{+}+\hbar\omega}\Bigg{]}$
(8)

The real part of the Hall conductivity $\sigma_{xy}$ can be derived as

$\mathrm{Re}\,\sigma_{xy}(\omega)=\frac{v^{2}}{4b\hbar\omega\xi}\sum_{s=+,-}(1-4bm+s\xi)\mathrm{arccoth}\Bigg{[}\frac{2\frac{(b\hbar)^{2}k\omega}{v^{3}}\sqrt{1+\big{(}\frac{mv}{\hbar
k}-\frac{b\hbar
k}{v}\big{)}^{2}}}{s(1-4bm)+(1-2bm+\frac{2b^{2}k^{2}\hbar^{2}}{v^{2}})\xi}\Bigg{]}\Bigg{|}^{\infty}_{0}$
(9)

and the imaginary part of $\sigma_{xy}$

$\mathrm{Im}\,\sigma_{xy}(\omega)=-\frac{\pi
v^{2}}{2\hbar\omega}\frac{mv^{2}+bk^{2}\hbar^{2}}{v^{2}-2b(mv^{2}-b\hbar^{2}k^{2})}\Theta(\hbar\omega-2|m|v^{2})$
(10)

where $\xi=\sqrt{1-4bm+(b\hbar\omega/v^{2})^{2}}$. Using the relation
$\mathrm{arccoth}(x)=[\ln((x+1)/x)-\ln((x-1)/x)]/2$, Eq.(9) becomes

$\begin{split}\mathrm{Re}\,\sigma_{xy}(\omega)=&\frac{v^{2}}{8\xi
b\hbar\omega}\times\\\
&\Bigg{[}2(1-4bm)\ln\bigg{|}\frac{b\hbar\omega/v^{2}+\xi}{b\hbar\omega/v^{2}+\xi}\bigg{|}-\sum_{s=+,-}g_{s}(\omega)\Bigg{]}\
,\end{split}$ (11)

where we have defined

$\begin{split}g_{s}(\omega)=&\big{(}1-4bm+s\xi\big{)}\times\\\
&\ln\bigg{|}\frac{2b^{2}|m|\hbar\omega/v^{2}+(1-4bm)s+\xi(1-2bm)}{2b^{2}|m|\hbar\omega/v^{2}-(1-4bm)s-\xi(1-2bm)}\bigg{|}\
.\end{split}$ (12)

In the DC limit ($\omega\rightarrow 0$), the real part of $\sigma_{xy}$
becomes

$\mathrm{Re}\,\sigma_{xy}=\frac{e^{2}}{h}C\ ,$ (13)

where $C=[\mathrm{sgn}(m)+\mathrm{sgn}(b)]/2$ is the first Chern number, which
characterizes the topological properties of the Hamiltonian (1), as expected.
Equation (9) agrees with the results obtained in Ref.Shen1 . All of the
conductivities as functions of the photon energy $\hbar\omega$ are shown in
Fig.2. One can find that there exist peaks at $\hbar\omega=2mv^{2}$, which are
due to the Rabi resonance. One may also see from Figs.2(c) and 2(d) that the
sign of $m$ does not affect the longitudinal conductivities qualitatively,
while it can significantly modify the Hall conductivities. This can be
understood by fact that the topological nature for the Hall conductivity
depends on the relative sign between the values of $m$ and $b$, which also
determines the values of the Chern number $C$.

In the absence of the parity anomaly term ($b=0$), the conductivities can be
reduced to

$\mathrm{Re}\,\sigma_{xy}(\omega)=\frac{e^{2}}{h}\frac{mv^{2}}{2\hbar\omega}\ln\bigg{|}\frac{2v^{2}|m|+\hbar\omega}{2v^{2}|m|-\hbar\omega}\bigg{|}\
,$ (14)
$\mathrm{Im}\,\sigma_{xy}(\omega)=-\frac{e^{2}}{h}\frac{\pi}{2}\frac{mv^{2}}{\hbar\omega}\Theta(\hbar\omega-2|m|v^{2})\
,$ (15)
$\mathrm{Re}\,\sigma_{xx}=\frac{e^{2}}{h}\frac{\pi}{8}\bigg{(}1+\frac{4m^{2}v^{4}}{(\hbar\omega)^{2}}\bigg{)}\Theta(\hbar\omega-2|m|v^{2})$
(16)

and

$\begin{split}\mathrm{Im}\,\sigma_{xx}(\omega)&=\frac{e^{2}}{h}\frac{1}{4}\int\mathrm{d}\epsilon_{+}\bigg{(}1+\frac{m^{2}v^{4}}{\epsilon^{2}_{+}}\bigg{)}\\\
&\quad\times\Bigg{[}\frac{\hbar\omega-2\epsilon_{+}}{(\hbar\omega-2\epsilon_{+})^{2}+\frac{\hbar^{2}}{4\tau^{2}}}+\frac{1}{2\epsilon_{+}+\hbar\omega}\Bigg{]}\
.\end{split}$ (17)

In fact, from the point view of the topological field theory, the Hamiltonian
(1) is reduced to a (2+1)-dimensional massive Dirac one when $b=0$, whose mass
term $mv^{2}\hat{\sigma}_{z}$ plays an important role in Chern-Simons field
Lagrangian 1996Aspects

$\mathcal{L}_{\mathrm{CS}}=\frac{\mathrm{sgn}(m)}{2}\int\mathrm{d}^{2}x\mathrm{d}t\,\epsilon^{\mu\nu\tau}A_{\mu}\partial_{\nu}A_{\tau}\
,$ (18)

where $A_{\mu}$ is the gauge field with the space-time indices $\mu=t,x,y$.
The presence of the regulating term $b\hbar^{2}k^{2}$ results in an additional
term $\mathrm{sgn}(b)/2$ to the coefficient of the Chern-Simons term (18),
which leads to the parity anomaly and then Eq.(18) becomes
$\mathcal{L}_{\mathrm{CS}}=C\int\mathrm{d}^{2}x\mathrm{d}t\,\epsilon^{\mu\nu\tau}A_{\mu}\partial_{\nu}A_{\tau}$.
In the following sections, we will consider the surface plasmonic responses in
both $C=1$ and $C=0$ cases, with and without parity anomaly.

##  III Surface plasmons

The BHZ model gives us a two dimensional conductivity tensor as shown above
which should support surface plasmons. Surface plasmons are coupled states of
light and collective electron oscillations; one should work with Maxwell’s
equations at each side of the conductive layer and at same time consider the
current density within the layer given by the conductivity tensor. The
dispersion relations of the surface plasmons can be derived based on the
following two boundary conditions: (i) the tangential electric fields are
continuous across the 2D CIs; (ii) the current densities satisfy Ampere’s law
which causes discontinuity of the tangential magnetic fields at two sides. Due
to the existence of the Hall conductivity, one cannot separate the surface
plasmons into transverse electric (TE) and transverse magnetic (TM) polarized
modes; in fact, they are coupled through $\sigma_{xy}$. The Hall term has
nothing to do with the ohmic losses but can seriously modify the dispersion
relations of the surface plasmons as shown below. Considering the fact that
the surrounding dielectrics at two sides are often composed of layered
materials, anisotropy is allowed where the in-plane and our-of-plane
permittivities are respectively denoted as $\varepsilon_{\mathrm{in}}$ and
$\varepsilon_{\mathrm{out}}$. The wave number $\mathbf{k}=(k_{x},k_{y},k_{z})$
in the Cartesian coordinates can be separated into an in-plane part
$\bm{\beta}=(k_{x},k_{y})$ and a out-of-plane part $k_{z}$. Since we are
interested in the surface waves, it is common to set $k_{z}=\mathrm{i}\gamma$
with $\gamma$ being real. In the surrounding dielectrics, one can derive the
following expression from Maxwell’s equations considering the rotation
symmetry of system implied by the BHZ model:

$[(\gamma^{2}+k^{2}_{0}\varepsilon_{\mathrm{in}})(\beta^{2}-k^{2}_{0}\varepsilon_{\mathrm{out}})-\beta^{2}\gamma^{2}](\beta^{2}-\gamma^{2}-k^{2}_{0}\varepsilon_{\mathrm{out}})=0\
,$ (19)

where $\beta=|\bm{\beta}|$ is the magnitude of the in-plane wave number,
$k_{0}=\omega/c$ is the wave number in vacuum with $\omega$ being the angular
frequency. The above expression leads to two modes with either
$\beta^{2}-\gamma^{2}-k^{2}_{0}\varepsilon_{\mathrm{out}}=0$ or
$(\gamma^{2}+k^{2}_{0}\varepsilon_{\mathrm{in}})(\beta^{2}-\gamma^{2}-k^{2}_{0}\varepsilon_{\mathrm{out}})=\beta^{2}\gamma^{2}$.
Assuming $\gamma>0$, we have
$\gamma=\sqrt{\beta^{2}-k^{2}_{0}\varepsilon_{\mathrm{out}}}$ or
$\gamma=\sqrt{(\beta^{2}/\varepsilon_{\mathrm{out}}-k^{2}_{0})\varepsilon_{\mathrm{in}}}$
depending on the mode we are considering. The first expression clearly shows
that $\varepsilon_{\mathrm{out}}$ is not involved, thus $E_{z}=0$. Further
calculations of the polarization indicate that $\bm{\beta}\cdot\mathbf{E}=0$.
The second expression shows that $E_{x}$, $E_{y}$ and $E_{z}$ are all
involved, while further calculations indicate that $H_{z}=0$ and
$\bm{\beta}\cdot\mathbf{H}=0$. In searching the surface plasmons, we have
chosen to solve $E_{x}$ and $E_{y}$.

Firstly, it is assumed that $E_{z}\neq 0$, The electric fields are tightly
confined near the conductive surface, thus they are proportional to
$\mathrm{e}^{\mathrm{i}k_{x}x}\mathrm{e}^{\mathrm{i}k_{y}y}\mathrm{e}^{-\gamma
z}$, and
$E_{z}=\mathrm{i}\bm{\beta}\cdot\mathbf{E}\varepsilon_{\mathrm{in}}/\gamma\varepsilon_{\mathrm{out}}$.
Consequantly, $H_{x}$ and $H_{y}$ are given by

$\begin{split}-\frac{\varepsilon_{\mathrm{in}}}{\varepsilon_{\mathrm{out}}}\bigg{(}\frac{k_{x}k_{y}}{\gamma}E_{x}+\frac{k^{2}_{y}}{\gamma}E_{y}\bigg{)}+\gamma
E_{y}&=\mathrm{i}\omega\mu_{0}H_{x}\\\ -\gamma
E_{x}+\frac{\varepsilon_{\mathrm{in}}}{\varepsilon_{\mathrm{out}}}\bigg{(}\frac{k^{2}_{x}}{\gamma}E_{x}+\frac{k_{x}k_{y}}{\gamma}E_{y}\bigg{)}&=\mathrm{i}\omega\mu_{0}H_{y}\end{split}$
(20)

where $\mu_{0}$ is the permeability of vacuum. Based on the boundary
conditions mentioned above, the equations regarding $E_{x}$ and $E_{y}$ can be
derived which finally leads to the following expression for the dispersion
relation of this surface plasmon mode

$\begin{split}&\big{(}\gamma^{(1)}+\gamma^{(2)}-k^{2}_{x}\Gamma-\mathrm{i}\omega\mu_{0}\sigma_{xx}\big{)}\big{(}\gamma^{(1)}+\gamma^{(2)}-k^{2}_{y}\Gamma-\mathrm{i}\omega\mu_{0}\sigma_{yy}\big{)}\\\
&=\big{(}k_{x}k_{y}\Gamma+\mathrm{i}\omega\mu_{0}\sigma_{xy}\big{)}\big{(}k_{x}k_{y}\Gamma+\mathrm{i}\omega\mu_{0}\sigma_{yx}\big{)}\
,\end{split}$ (21)

where
$\Gamma=\varepsilon^{(1)}_{\mathrm{in}}/(\gamma^{(1)}\varepsilon^{(1)}_{\mathrm{out}})+\varepsilon^{(2)}_{\mathrm{in}}/(\gamma^{(2)}\varepsilon^{(2)}_{\mathrm{out}})$.
$\gamma^{(i)}=\sqrt{\beta^{2}-k^{2}_{0}\varepsilon^{(i)}_{\mathrm{in}}}$,
where the superscripts $i=1,2$ denote the space above and below the conductive
layer, respectively. The expression of the dispersion relation given above can
be simply written as

$\begin{split}&\big{(}\gamma^{(1)}+\gamma^{(2)}-\mathrm{i}\omega\mu_{0}\sigma_{xx}\big{)}^{2}-(\omega\mu_{0}\sigma_{xy})^{2}\\\
&=\big{(}\gamma^{(1)}+\gamma^{(2)}-\mathrm{i}\omega\mu_{0}\sigma_{xx}\big{)}\beta^{2}\Gamma\end{split}$
(22)

Solving $\beta$ in the complex plane with $\mathrm{Re}\,\beta\geq 0$ and
$\mathrm{Im}\,\beta\geq 0$ at each frequency one can find the dispersion
curves of the surface plasmons. Below the gap, $\sigma_{xx}$ is purely
imaginary and $\sigma_{xy}$ is purely real, hence one only needs to solve
$\beta$ on the real axis.

The above equation contains two branches of dispersion relations, which can be
written as

$\frac{\omega}{c}=\frac{Z_{0}(\mathrm{i}\sigma_{xx})[2(\gamma^{(1)}+\gamma^{(2)})-\beta^{2}\Gamma]\pm\sqrt{Z^{2}_{0}\sigma^{2}_{xy}[2(\gamma^{(1)}+\gamma^{(2)})-\beta^{2}\Gamma]^{2}+Z^{2}_{0}\beta^{4}\Gamma^{2}[(\mathrm{i}\sigma_{xx})^{2}-\sigma^{2}_{xy}]}}{2Z^{2}_{0}[(\mathrm{i}\sigma_{xx})^{2}-\sigma^{2}_{xy}]}$
(23)

where $Z_{0}$ is the vacuum impedance. Equation (23) is one of our main
results concerning the dispersion relations of the surface plasmons.

As for the mode with $E_{z}=0$, Eq.(20) is reduced to

$\gamma E_{y}=\mathrm{i}\omega\mu_{0}H_{x}\ ,\quad-\gamma
E_{x}=\mathrm{i}\omega\mu_{0}H_{y}\ .$ (24)

Following the calculations as shown above, one can derive the expression of
the dispersion relation of the surface plasmons as

$\begin{split}&\big{(}\gamma^{(1)}+\gamma^{(2)}-\mathrm{i}\omega\mu_{0}\sigma_{xx}\big{)}\big{(}\gamma^{(1)}+\gamma^{(2)}-\mathrm{i}\omega\mu_{0}\sigma_{yy}\big{)}\\\
&+(\omega\mu_{0})^{2}\sigma_{xy}\sigma_{yx}=0\ .\end{split}$ (25)

Considering the symmetry of the BHZ model, this expression can be further
written as

$\big{(}\gamma^{(1)}+\gamma^{(2)}-\mathrm{i}\omega\mu_{0}\sigma_{xx}\big{)}^{2}=(\omega\mu_{0}\sigma_{xy})^{2}\
$ (26)

which can be easily solved. We have found that it is more convenient working
with complex angular frequency. Replacing $\omega$ with
$\omega_{\mathrm{R}}-\mathrm{i}\omega_{\mathrm{I}}$, where
$\omega_{\mathrm{R}}$ and $\omega_{\mathrm{I}}$ are respectively the real and
imaginary parts, Eq.(26) can be broken down into two branches of dispersion
relations written as follows:

$\begin{split}&\gamma^{(1)}+\gamma^{(2)}\\\
&=\frac{[(\mathrm{Re}(\sigma_{xy})-\mathrm{Im}(\sigma_{xx}))^{2}+(\mathrm{Im}(\sigma_{xy})+\mathrm{Re}(\sigma_{xx}))^{2}]\omega_{\mathrm{R}}\mu_{0}}{\mathrm{Re}(\sigma_{xy})-\mathrm{Im}(\sigma_{xx})}\end{split}$
(27)

with

$\omega_{\mathrm{I}}=\frac{\mathrm{Im}(\sigma_{xy})+\mathrm{Re}(\sigma_{xx})}{\mathrm{Re}(\sigma_{xy})-\mathrm{Im}(\sigma_{xx})}\omega_{\mathrm{R}}\
,$ (28)

and

$\begin{split}&\gamma^{(1)}+\gamma^{(2)}\\\
&=-\frac{[(\mathrm{Re}(\sigma_{xy})+\mathrm{Im}(\sigma_{xx}))^{2}+(\mathrm{Im}(\sigma_{xy})-\mathrm{Re}(\sigma_{xx}))^{2}]\omega_{\mathrm{R}}\mu_{0}}{\mathrm{Re}(\sigma_{xy})+\mathrm{Im}(\sigma_{xx})}\end{split}$
(29)

with

$\omega_{\mathrm{I}}=\frac{\mathrm{Im}(\sigma_{xy})-\mathrm{Re}(\sigma_{xx})}{\mathrm{Re}(\sigma_{xy})+\mathrm{Im}(\sigma_{xx})}\omega_{\mathrm{R}}\
.$ (30)

These two solutions are related by time-reversal symmetry. Since
$\gamma^{(1)}+\gamma^{(2)}>0$,
$\mathrm{Re}(\sigma_{xy})-\mathrm{Im}(\sigma_{xx})>0$ and
$\mathrm{Re}(\sigma_{xy})+\mathrm{Im}(\sigma_{xx})<0$ must be satisfied in
Eqs.(27) and (29), respectively. Also, $\omega_{\mathrm{I}}$ must be positive.

##  IV Results

The Fermi energy locates within the gap, thus only interband transition of
electrons needs to be considered during the calculations of the
conductivities. Due to the absence of the intraband transitions, such
optically conductive surface resembles a dielectric thin film rather than a
metallic one. This fact can also be known from the imaginary parts of the
longitudinal conductivities as shown in Fig.2 which is negative leading to
positive effective permittivities. Without parity anomaly, the Hall
conductivities with positive and negative Dirac mass terms differ just by a
sign, as shown by the solid red and blue solid curves in Fig.2(a). In the
presence of parity anomaly, the Hall conductivities are respectively zero and
integer-valued in the topologically trivial and non-trivial situations.

We have searched for the two surface plasmon modes mentioned in Sec.III, and
for simplicity the surrounding dielectrics have been assumed to be isotropic
with refractive indices $n=3$, which is reasonable and will not affect our
main conclusions in this paper. The dispersion relations of the surface
plasmon mode with $H_{z}=0$ are shown in Fig.3, where we have solved the
dispersion relations in the complex plane and plotted the propagation constant
$\beta$ as functions of the photon energy. Figs. 3(a) and 3(b) correspond to
$mv^{2}=0.05$ and $mv^{2}=-0.05$, respectively. The real and imaginary parts
are plotted in blue and red, respectively. Propagation constants correspond to
the case with/without parity anomaly are plotted with dashed/solid curves.
Insets are the enlarged plots of the region below the transition threshold.
Since the conductivities given by the BHZ model are not Drude-type, the
dispersion curves as shown in Fig.3 are either straight lines corresponding to
the light line in surrounding dielectrics or curves with relatively large
imaginary parts. Straight lines indicate the absence of any surface-confined
electromagnetic modes, and $\mathrm{Im}(\beta)\gg\mathrm{Re}(\beta)$ simply
means large energy dissipations. For $mv^{2}=0.05$, the parity anomaly seems
to lower the curves of the propagation constants, as shown in Fig. 3(a); while
for $mv^{2}=-0.05$ the propagation constants are increased, as shown in Fig.
3(b). It is clear that the parity anomaly in a two-dimensional Chern insulator
can seriously modify the dispersion curves of this surface plasmon mode.

As for the surface plasmon mode with $E_{z}=0$, two branches of dispersion
relations have been found by numerically solving Eqs.(27) and (29). The
dominators of these two equations are respectively plotted as functions of the
photon energy in Figs.4(a) and 4(b). From a mathematical point of view,
Eqs.(27) and (29) respectively require
$\mathrm{Re}(\sigma_{xy})-\mathrm{Im}(\sigma_{xx})>0$ and
$\mathrm{Re}(\sigma_{xy})+\mathrm{Im}(\sigma_{xx})<0$. Without parity anomaly,
as shown by the solid curves in Figs. 4(a)and 4(b), $mv^{2}=0.05$ and
$mv^{2}=-0.05$ actually give the same dispersion curves, i.e. they are
degenerate. However, with parity anomaly, such degeneracy is lifted, as shown
by the dashed blue and red curves. For the branch corresponding to Eq.(27),
both $mv^{2}=0.05$ and $mv^{2}=-0.05$ can lead to physical solutions since
$\mathrm{Re}(\sigma_{xy})-\mathrm{Im}(\sigma_{xx})>0$; while for the branch
corresponding to Eq.(29), only $mv^{2}=-0.05$ can lead to meaningful results
where $\mathrm{Re}(\sigma_{xy})+\mathrm{Im}(\sigma_{xx})<0$.

The right hand sides of Eqs.(27) and (29) are solely determined by the photon
energy which are plotted in Figs.4(c) and 4(d). Without parity anomaly,
$\gamma^{(1)}+\gamma^{(2)}$ of the two branches corresponding to Eq.(27)
($mv^{2}=0.05$) and Eq.(29) ($mv^{2}=-0.05$) are the same, as indicated by the
fact that the solid blue curve in Fig.4(c) and solid red curve in Fig.4(d)
coincide. With parity anomaly, as shown by the dashed curves in Figs.4(c) and
4(d), the branch corresponding to Eq.(27) has two solutions, where dashed blue
and red curves in Fig.4(c) respectively denote the situations with
$mv^{2}=0.05$ and $mv^{2}=-0.05$; while the branch corresponding to Eq.(29)
has only one solution as indicated by the red dashed curve in Fig.4(d)
denoting the situation with $mv^{2}=-0.05$.

These surface plasmons are weakly guided based on the observation that the
dispersion relations are quite close to the light line in the surrounding
dielectric material. We have solved the mode effective indices defined as
$\beta/k_{0}$ as functions of the photon energy, which are plotted in
Figs.5(a) and 5(b). Again, without parity anomaly, two branches of this
surface plasmon mode are degenerate and the solid blue and red curves in Figs.
5(a) and 5(b) are identical; such degeneracy can be lifted by introducing the
parity anomaly term in the Hamiltonian. The effective index of the surface
mode should slightly larger than the refractive index of the surrounding
dielectric material. We further plot $\omega_{i}/\omega_{r}$ as functions of
the photon energy in Figs.5(c) and 5(d). $\omega_{i}/\omega_{r}>0$ must be
satisfied since the energy must be damped during propagation. Large ratios
mean that these surface plasmons possess significant losses which can be
ascribed to relatively small conductivities given by the BHZ model.

##  V Conclusion

In this paper, we have investigated the relations between the parity anomaly
in a two-dimensional Chern insulator and the dispersion relations of the
surface plasmons. Given by the symmetry of the model we have considered, two
surface plasmon modes have been found. Each mode contains two branches of
dispersion relations which are degenerate with regard to the sign of the Dirac
mass term in the absence of the parity anomaly. Introducing the parity anomaly
term into the Hamiltonian will lift this degeneracy and significantly modify
the dispersions of the surface plasmons. In the presence of the parity
anomaly, the band topology of the bulk states results in integer-valued Hall
conductivity. Despite the fact that the Hall conductivity is shifted about
$e^{2}/2h$, it can cause significant changes and even leads to the occurrence
of an extra branch of surface plasmons. Our findings have revealed the
connections between the parity of two dimensional materials and the dispersion
relations of their surface plasmons, which might become valuable in, for
example, detection of the parity anomaly via plasmonic responses.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China (Grant
Nos.11804070, 61805062, 11975088).

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