# Tutorial: Topology, waves, and the refractive index

S. A. R. Horsley  School of Physics and Astronomy, University of Exeter,
Stocker Road, Devon, EX4 4QL

###### Abstract

This tutorial is divided into two parts: the first examines the application of
topology to problems in wave physics. The origins of the Chern number are
reviewed, where it is shown that this counts the number of critical points of
a complex tangent vector field on the surface. We then show that this quantity
arises naturally when calculating the dispersion of modes in any linear
system, and give examples of its application to find one–way propagating
interface modes in both continuous and periodic materials.

The second part offers a physical interpretation for the Chern number, based
on the idea that the critical points which it records can be understood as
points where the refractive index vanishes. Using the theory of crystal
optics, we show that when the refractive index vanishes in a _complex valued_
direction, the wave is forced to circulate in only one sense, and this is the
origin of the one–way propagation of topological interface states. We conclude
by demonstrating that this idea of ‘zero refractive index in a complex
direction’ can be used as a shortcut to find acoustic and electromagnetic
materials supporting one–way interface states.

##  I Introduction

Topology is the study of whether objects can be smoothly transformed into each
other. Sometimes these ‘objects’ are extremely abstract mathematical ideas,
sometimes they’re not: Can I uncoil this garden hose without removing the end
from the bucket? Can I untangle these necklaces without undoing the clasps?
Can I wrap this map of the world onto a globe? These are all problems of
topology. Related topological questions appear across physics, where wave
dispersion surfaces [1], knotted fluid flow lines [2], and electromagnetic
fields [3] can all be grouped according to whether, or not they can be
smoothly transformed into one another.

In physics and engineering there has been a recent burst of activity, applying
topology to control waves. Specifically, topology has been applied to design
materials, stipulating what happens at their interface without having to know
anything about what the interface is like. For instance, take two homogeneous
lumps of elastic stuff. We can ensure that vibrational waves can be trapped at
the interface formed when we stick them together, irrespective of how messy
our joinery is!

The basic idea is this: take a material where wave propagation can be
specified in terms of a conserved wave vector $\bm{k}$. For topology to be at
all powerful, we need to be able to treat the components of $\bm{k}$ as the
coordinates on a closed surface. In many cases this is possible. We can, for
example, wrap the first Brillouin zone onto a torus, whenever the material is
periodic.

Imagine that attached to each point on this closed surface is a vector
$|\psi\rangle$; the solution to the wave equation for each particular value of
$\bm{k}$. Now, taking two different materials we can construct two closed
surfaces (e.g. two tori), upon each of which there is a different form of the
wave, $|\psi\rangle$ or $|\psi^{\prime}\rangle$. The question is whether it is
possible to smoothly deform one of these wave fields into the other, a
question topologists have already developed the necessary tools to answer, at
least in the negative!

If the two waves _cannot_ be smoothly deformed into one another then something
non–smooth and perhaps ‘interesting’ must happen when we try, something that
will occur for instance at an interface, in the transition region between the
two materials. This ‘interesting’ thing turns out to be the presence of one or
more interface states, where the wave is trapped in the transition region.
Topology therefore guarantees the presence of interface states between two
materials, without the physicist ever having to consider whether the interface
is flat, rough, curved, sharp, narrow, or wide.

This is an odd business for most physicists and engineers, who are used to
caring about details! A graded index fiber optic cable, for instance, must be
made with precision, confining light rays with a particular spatial
distribution of refractive index to minimize dispersion. Here we have a
completely different kind of theory, where the design process doesn’t even
mention the details of the region where the mode is to be confined. This
peculiar insensitivity to the form of the interface (often called ‘topological
protection’) was perhaps first appreciated by Volkov and co–workers, who were
concerned with the physics of electrons around electronic contacts [4, 5]. The
significance of topology was appreciated later [6, 7], in connection with
earlier work on the quantum hall effect [8, 9].

Since the discovery of negative refraction [10] and transformation optics
[11], and the rapid development of metamaterials [12], it has been widely
appreciated that one wave is like any other. Although quantum mechanics has
its peculiarities, there is nothing fundamentally different between the Dirac
equation, Schrödinger equation, the classical Maxwell equations, or the
equations of elasticity. The same topological arguments have therefore been
applied to design of periodic electromagnetic materials supporting
unidirectional interface states, dubbed ‘photonic topological insulators’
[13], which has led to the fields of topological photonics [14] and acoustics
[15].

The fact that we can use metamaterials to realise a wide range of material
parameters means there is actually more to explore in these classical systems
(the distribution of atoms in an ordinary crystal is not easy to specify on
the scale of an electron wavelength!). Work on topological photonic and
acoustic materials is therefore able to investigate effects that could not be
observed in condensed matter systems (see the discussion in e.g. [16] and
[17]), and even allows the exploration of active topological non–Hermitian
materials, where the material can amplify an absorb the wave in a controlled
way [18] (which is also extremely difficult to mimic in electronic systems).

In the author’s opinion, several things are opaque in this subject. Firstly,
while the Chern classes are by now a familiar tool for classifying the
topology of a wave field $|\psi\rangle$, their origin and connection to other
characteristic classes—such as the more familiar Euler class—is never
explained in terms palatable to the physicist. Secondly, besides giving a
“plausbile” intuitive explanation for its truth, the connection between
integrals of the Chern class and the number of interface states is never
proved in a straightforward way. The first part of the tutorial clarifies both
of these points. Finally there is the problem of a physical interpretation for
these topological calculations. In the final part we connect a non–zero Chern
number to the existence of peculiar points of vanishing refractive index,
where the wave is forced to circulate in only one sense (e.g. only clockwise).

##  II Winding numbers of paths

We’ll begin with the topological classification of curves in terms of a
‘winding number’. Imagine unwinding a ball of string, one end of which (A) is
attached to a wall. With the string we trace out a path, closing it by
returning to A and tying the ends. An example is sketched as the blue curve in
Fig. 1a.

Figure 1: Without any obstacles, any closed path can be deformed into a simple
loop. By pulling along the indicated red arrows, the path in (a) can be
unwound into the simple path shown in (b). In panel (c) an obstacle is placed
in the centre of the path. As the path winds twice around the obstacle, one
reaches an impasse as shown in (d).

If there are no obstacles, we can always move the string until it unwinds into
a single loop. Pulling in the direction indicated in Fig. 1a, the string can
be untwisted into the single loop shown in Fig. 1b. The two configurations are
therefore _topologically equivalent_. This situation changes if the space
contains an obstacle111The obstacle is equivalent to the removal of a point
from the plane.. Suppose there is a tree at the position shown by the black
dot in Fig. 1c. Whether we can deform one arrangement of string into another
is now determined by the _winding number_ , $\nu$: the number of times the
string encircles the tree. As we know from experience, the winding number
cannot be changed by any continuous re–positioning of the string, to change it
we must cut the string (or cut down the tree). A difference in the winding
number between two configurations of string indicates that they are
_topologically in–equivalent_ , and $\nu$ can be used to classify
configurations of string that cannot be continuously changed into one another.

To calculate the winding number, suppose we position the tree at the origin of
the coordinate system, integrating the change in polar angle ${\rm d}\theta$
as we follow the string. After dividing by $2\pi$, the integral counts the
number of times the path encircles the obstacle. Each point on the path thus
picks out an angle $\theta$ on a circle. The winding number counts the number
of times this point covers the circle, adding $+1$ for each anticlockwise
circuit and $-1$ for each clockwise one.

It is simplest to write the change in angle ${\rm d}\theta$ using the complex
number $z=x+{\rm i}y=r{\rm e}^{{\rm i}\theta}$. The polar angle is then simply
$\theta={\rm Im}[\log(z)]$ and the winding number can be written as a line
integral of a vector,

$\nu=\frac{1}{2\pi}\oint_{\rm Path}{\rm d}\theta=\frac{1}{2\pi}\oint_{\rm
Path}\bm{\nabla}\,{\rm Im}\left[\log\left(x+{\rm i}y\right)\right]\cdot\,{\rm
d}\bm{x}$ (1)

where ${\rm d}\bm{x}={\rm d}x\,\bm{e}_{x}+{\rm d}y\,\bm{e}_{y}$. Having
written $\nu$ as a line integral, there is a second equivalent way to write
Eq. (1). Using Stokes’ theorem the one dimensional line integral can be
written as a two dimensional surface integral over the enclosed region
$S_{P}$,

$\nu=\frac{1}{2\pi}\int_{\rm S_{P}}\bm{\nabla}\times\bm{A}\cdot\,{\rm
d}\bm{S}$ (2)

where $S_{P}$ may be quite a strange origami–like surface, like that enclosed
by the blue curve in Fig. 1a. This is an important development that we’ll see
again. The winding number (our topological invariant) now appears as a net
‘magnetic flux’ through the surface $\rm S_{P}$, something which common to all
the topological invariants considered here. The ‘vector potential’ associated
with this flux is defined as

$\bm{A}=\bm{\nabla}\,{\rm Im}\left[\log\left(x+{\rm i}y\right)\right].$ (3)

But having said all this, it now seems as though we made a mistake. The
integral (2) is surely always zero, because the curl of any gradient is zero!

We didn’t make a mistake. We have uncovered an extremely important subtlety
that appears again and again in topology. The concerning result
$\bm{\nabla}\times\bm{\nabla}f=0$ requires $f$ to be a proper function. Every
point $(x,y)$ must be associated with a single number $f(x,y)$. This isn’t
true for the ‘function’, $\theta={\rm Im}[\log(x+{\rm iy})]$, which can take
_any_ value at the origin222Mathematicians refer to ${\rm d}\theta$ as closed
(has zero curl away from the origin) yet inexact (isn’t the gradient of a
proper function) one–form. De Rham cohomology relates the topology of a space
to the existence of such forms.. This defect in the vector potential (3) is
known as a critical point. The ‘flux’ in equation (2) records the presence of
such critical points, and is confined to the obstacle, where both $\theta$ and
$\bm{A}$ are undefined. The winding number thus only depends on the number of
times the surface $\rm S_{P}$ cuts through the critical point $\bm{x}=\bm{0}$.
The fact that the curl of (3) is zero at all points _except_ where there is an
obstacle is actually essential for the winding number to be insensitive to
deformations of the path, and hence for $\nu$ to be a topological invariant.

##  III The Euler characteristic: winding numbers of surfaces

Having given a topological categorization of curves, we move up a dimension to
classify the ‘winding’ of closed two dimensional surfaces, like those shown in
Fig. 2. Just as we classified the path of our string by mapping it to a point
on a circle and counting the total number of revolutions, we associate each
point on a surface $\rm S$ to an equivalent point on a _sphere_ , and count
the number of times the sphere is covered as we move over the surface.

Figure 2: Two dimensional surfaces can be classified by counting the number of
times the surface normal $\bm{n}$ covers a sphere, which equals half of the
_Euler characteristic_ , $\chi$. Panels (b) and (d) show arbitrary surface
normal vectors $\bm{n}$ and $\bm{n}^{\prime}$ on a sphere, picking out points
on the two surfaces in (a) and (c) respectively. Every surface normal on the
sphere corresponds to only one point on the egg–like surface in (a), meaning
$\chi=2$. Meanwhile a point on the sphere corresponds to _two_ points on the
torus: one ‘inside’, and one ‘outside’. Following $\bm{n}$ around the torus we
see that the outside and inside regions cover the sphere in opposite
directions, one undoing the other such that $\chi=0$.

We make the connection between a point on an arbitrary surface $\rm S$ and a
point on a sphere through the surface normal $\bm{n}$. For each coordinate
$(x_{1},x_{2})$ on our surface we find those coordinates $(\theta,\phi)$ on
the sphere where the surface normal vector takes the same value,

$\bm{n}(x_{1},x_{2})=\sin(\theta)\left[\cos(\phi)\bm{e}_{x}+\sin(\phi)\bm{e}_{y}\right]+\cos(\theta)\bm{e}_{z}.$
(4)

as indicated in Fig. 2. In Sec. II we mapped the curve onto a circle and
calculated the winding number through integrating the angle swept out around
the circle. Here we instead integrate up the _solid_ angle ${\rm
d}\Omega=\sin(\theta)\,{\rm d}\theta\,{\rm d}\phi$ swept out on the sphere as
we move over the area ${\rm d}x_{1}\,{\rm d}x_{2}$ on our arbitrary surface
$\rm S$. A useful expression for ${\rm d}\Omega$ can be found through
transforming the expression for the solid angle from spherical to surface
coordinates,

$\displaystyle{\rm d}\Omega$
$\displaystyle=\bm{n}\cdot\left(\frac{\partial\bm{n}}{\partial\theta}\times\frac{\partial\bm{n}}{\partial\phi}\right)\,{\rm
d}\theta\,{\rm
d}\phi=\bm{n}\cdot\left(\frac{\partial\bm{n}}{\partial\theta}\times\frac{\partial\bm{n}}{\partial\phi}\right)\left(\frac{\partial\theta}{\partial
x_{1}}\frac{\partial\phi}{\partial x_{2}}-\frac{\partial\theta}{\partial
x_{2}}\frac{\partial\phi}{\partial x_{1}}\right)\,{\rm d}x_{1}\,{\rm d}x_{2}$
$\displaystyle=\bm{n}\cdot\left(\frac{\partial\bm{n}}{\partial
x_{1}}\times\frac{\partial\bm{n}}{\partial x_{2}}\right)\,{\rm d}x_{1}\,{\rm
d}x_{2}.$ (5)

By analogy with our calculation of the winding number of a curve (1), we count
the number of times $\nu$ the surface normal $\bm{n}$ wraps around the sphere,
simply integrating the solid angle element (5) over the surface $\rm S$, and
dividing the result by $4\pi$

$\nu=\frac{\chi}{2}=\frac{1}{4\pi}\int_{\rm S}\,{\rm
d}\Omega=\frac{1}{4\pi}\int_{\rm
S}\bm{n}\cdot\left(\frac{\partial\bm{n}}{\partial
x_{1}}\times\frac{\partial\bm{n}}{\partial x_{2}}\right)\,{\rm d}x_{1}\,{\rm
d}x_{2}.$ (6)

The winding number is half the _Euler characteristic_ , $\chi=2\nu$ which is
the topological invariant typically used to classify closed surfaces. We
should remember that this is simply a way of expressing the winding number.
Equation (6) is the essence of the topological classification of surfaces:
there is no way to smoothly change one surface into another if their normal
vectors cover a sphere a different number of times. Two such incompatible
surfaces are shown in Fig. 2a and c.

Figure 3: At any point on a surface the infinitesimal element of solid angle
${\rm d}\Omega$ appearing in (6) can be re–written in terms of the Hessian of
the surface height function $h(x,y)$. Rotating the in–plane coordinates to
$x^{\prime},y^{\prime}$ to align with the principal axes of curvature (white
lines), the Hessian is diagonalized with eigenvalues equal to the inverses of
the two radii of curvature (the radii of the two circles that approximate the
white curves in the diagram).

There is a more interesting way to write the Euler characteristic (6), that
replaces the change in the surface normal with the local surface curvature. To
make this transition we take some point on the surface, and use it as origin
of a Cartesian coordinate system, where the $z=0$ plane is tangent to the
surface, as shown in Fig. 3. Close to this point, the surface shape satisfies

$z-h(x,y)=0$ (7)

where $h(x,y)$ is the height of the surface above the tangent plane. By
definition the height and its gradient vanish at the origin. The surface
normal is now proportional to the gradient of the above equation for the
surface height (7),

$\bm{n}=N\left(\bm{e}_{z}-\bm{\nabla}h(x,y)\right).$ (8)

where the scalar $N$ ensures normalization, $\bm{n}\cdot\bm{n}=1$ and equals
unity at $x=y=0$, where $\bm{\nabla}h=0$. Using the $x$ and $y$ coordinates in
the formula for the element of solid angle (5), and substituting the above
expression for the surface normal vector (8), we can re–write the element of
solid angle at $x=y=z=0$ in terms of the Hessian of the surface height,
$\partial^{2}h/\partial x_{i}\partial x_{j}$,

$\displaystyle{\rm d}\Omega=\bm{n}\cdot\frac{\partial\bm{n}}{\partial
x}\times\frac{\partial\bm{n}}{\partial y}\,{\rm d}x\,{\rm d}y$
$\displaystyle=\bm{e}_{z}\cdot\left(\bm{\nabla}\partial_{x}h\times\bm{\nabla}\partial_{y}h\right)\,{\rm
d}x\,{\rm d}y$ $\displaystyle={\rm det}\left(\frac{\partial^{2}h}{\partial
x_{i}\partial x_{j}}\right)\,{\rm d}x\,{\rm d}y.$ (9)

The determinant of the Hessian is positive for surfaces that are locally
elliptic paraboloids, and negative for hyperbolic paraboliods. It equals the
inverse product of the two principal radii of curvature, ${\rm
det}[\partial^{2}h/\partial x_{i}\partial x_{j}]=(R_{1}R_{2})^{-1}$, which is
known as the surface’s _Gaussian curvature_ $K$ (see Fig. 3). This argument
can be carried out at every point on the surface. Summing the results we find
the Euler characteristic (6) can also be re–written as an integral of the
curvature of the surface

$\chi=\frac{1}{2\pi}\int_{\rm S}K\,{\rm d}A$ (10)

where ${\rm d}A$ is an infinitesimal element of surface area. Equation (10) is
the famous _Gauss–Bonnet theorem_ , and is quite a remarkable expression. The
surface curvature integrated over _any_ closed surface always equals a
multiple of $2\pi$. We can continuously deform the surface, changing the
distribution of surface curvature, but—so long as we don’t tear a new hole in
the surface—every region of increased curvature is unavoidably balanced by
regions where it is reduced.

Figure 4: Instead of the surface normal $\bm{n}$, we can use tangent vector
fields $\bm{e}_{1}$, $\bm{e}_{2}$ to calculate the Euler characteristic. The
Euler characteristic $\chi$ can then be written as the surface integral of a
‘magnetic flux’ as in Eq. (15). The topology of the surface is determined by
the number of critical points, where either of the tangent vectors are
undefined. When $\chi\neq 0$, as for the sphere in panels (a) and (b), every
choice of tangent vector field will exhibit critical points, shown here as the
black dots in the zoomed in regions. Only when $\chi=0$ is it possible to have
a tangent vector that is well defined at all points. An example is shown on
the surface of the torus in panel (c).

In parallel with our earlier discussion of one dimensional curves, we can
write the Gauss–Bonnet theorem in a third equivalent form, as integral of an
effective magnetic flux passing through the closed surface. This is achieved
through introducing a pair of orthonormal tangent vectors on the surface,
$\bm{e}_{1}$ and $\bm{e}_{2}$. The surface normal is everywhere given by the
cross product between these tangent vectors

$\bm{n}=\bm{e}_{1}\times\bm{e}_{2}.$ (11)

Substituting expression (11) for the surface normal into our expression for
the solid angle element (5), we see that it equals the curl of a vector

$\displaystyle{\rm d}\Omega=\bm{n}\cdot\frac{\partial\bm{n}}{\partial
x_{1}}\times\frac{\partial\bm{n}}{\partial x_{2}}{\rm d}x_{1}{\rm d}x_{2}$
$\displaystyle=\left[\left(\bm{n}\cdot\frac{\partial\bm{e}_{2}}{\partial
x_{2}}\right)\left(\bm{n}\cdot\frac{\partial\bm{e}_{1}}{\partial
x_{1}}\right)-\left(\bm{n}\cdot\frac{\partial\bm{e}_{1}}{\partial
x_{2}}\right)\left(\bm{n}\cdot\frac{\partial\bm{e}_{2}}{\partial
x_{1}}\right)\right]{\rm d}x_{1}{\rm d}x_{2}$
$\displaystyle=\left[\frac{\partial\bm{e}_{1}}{\partial
x_{1}}\cdot\frac{\partial\bm{e}_{2}}{\partial
x_{2}}-\frac{\partial\bm{e}_{1}}{\partial
x_{2}}\cdot\frac{\partial\bm{e}_{2}}{\partial x_{1}}\right]{\rm d}x_{1}{\rm
d}x_{2}$ $\displaystyle=\left(\frac{\partial A_{2}}{\partial
x_{1}}-\frac{\partial A_{1}}{\partial x_{2}}\right){\rm d}x_{1}{\rm d}x_{2}.$
(12)

where the components of this ‘vector potential’ $A_{i}$ are defined as

$A_{j}=\bm{e}_{1}\cdot\frac{\partial\bm{e}_{2}}{\partial x_{j}}.$ (13)

Note that, despite appearances there is no bias between the two tangent
vectors, and the vector potential can be written equivalently as
$A_{i}=-\bm{e}_{2}\cdot\partial_{i}\,\bm{e}_{1}$, due to the normalization
condition $\bm{e}_{1}\cdot\bm{e}_{2}=0$. Although we have called $A_{j}$ a
‘vector potential’ due to the analogous quantity in physics, more precisely
the quantity $A_{j}$ is a _connection_ on the surface, a quantity from
differential geometry that characterises how the basis vectors change from
point to point (see Appendix A for details).

Our three different expressions for ${\rm d}\Omega$ show that the Gaussian
curvature both expresses a change in the surface normal $\bm{n}$, and an
effective ‘magnetic field’ (the curl of the connection) due to the change in
the surface tangent vectors $\bm{e}_{i}$

$K=\bm{n}\cdot(\bm{\nabla}\times\bm{A})=\bm{n}\cdot\left(\frac{\partial\bm{n}}{\partial
x_{1}}\times\frac{\partial\bm{n}}{\partial x_{2}}\right)$ (14)

Using the above expression for the solid angle in terms of the vector
potential (12) we can also write the Gauss–Bonnet theorem as the integral of a
‘magnetic flux’ passing through the surface

$\chi=\frac{1}{2\pi}\int_{S}\bm{\nabla}\times\bm{A}\cdot{\rm d}\bm{S}.$ (15)

where the vector surface area element is given by ${\rm d}\bm{S}={\rm
d}x_{1}\,{\rm d}x_{2}\,\bm{n}$. Note that the Euler characteristic (15) now
takes an identical form to the winding number of a one dimensional curve (2).
Just as we saw there, the integral (15) doesn’t seem right. Stokes’ theorem
tells us that the integral of a curl over a surface equals a line integral
around the surface boundary. But these closed surfaces have no boundary! So
surely the integral is zero.

But, after Sec. II we are prepared for this puzzle. Stokes’ theorem can only
be applied if the vector potential $\bm{A}$ is well defined over the whole
surface. We have an integral of something that _looks_ like a curl over a
closed surface, but it isn’t the curl of anything at some discrete points
(critical points) on the surface. These critical points are familiar for the
polar and azimuthal unit vectors on a sphere, which are both undefined at the
poles: it is not always possible to have tangent vectors $\bm{e}_{i}$ that are
normalized, orthogonal, and everywhere well defined333The impossibility of
having an everywhere well defined tangent vector field on the surface of an
even–dimensional sphere is known as the _Hairy ball theorem_. We are familiar
with this in everyday life: it is impossible to comb the hair on a sphere to
lie flat without introducing a crown, around which the hair swirls or
diverges.. In general, tangent vectors on a closed surface exhibit _critical
points_ where they do not have a well defined direction, as illustrated in
Fig. 4. If we apply Stokes’ theorem to the region of the surface where
$\bm{A}$ is well defined, we can transform (15) to a sum of line integrals
encircling the critical points, and each of these points will contribute a
multiple of $2\pi$,

$\oint A_{j}\,{\rm
d}x_{j}=\oint\bm{e}_{1}\cdot\frac{\partial\bm{e}_{2}}{\partial x_{j}}\,{\rm
d}x_{j}=\oint\bm{e}_{1}\cdot{\rm d}\bm{e}_{2}=\oint{\rm d}\theta=2\pi{\rm n}.$
(16)

The last three steps follow from the infinitesimal change in the tangent
vector, $\bm{e}_{1}\cdot{\rm d}\bm{e}_{2}={\rm d}\theta$, where ${\rm
d}\theta$ is angle by which the vector $\bm{e}_{2}$ rotates due to the
infinitesimal displacement ${\rm d}x_{j}$. The integer $n$ is known as the
_index_ of the critical point. Examples of critical points of different index
are shown in Fig. 5.

Figure 5: A critical point of a normalized vector field is a point where its
direction is undefined. Critical points are classified in terms of their
index, which equals the number of times the vector rotates as we move around
the critical point, being positive for an anti–clockwise rotation. In (a–c) we
show three critical points (black dots), each of a different index.

From Eqns. (15) and (16) we conclude that the Euler characteristic of a
surface records the critical points of any tangent vector field on the
surface. The sum of the indices ${\rm n}_{\bm{x}_{c}}$ of all the critical
points $\bm{x}_{c}$ of any surface tangent vector field equals the Euler
characteristic

$\chi=\sum_{\bm{x}_{c}}{\rm n}_{\bm{x}_{c}}.$ (17)

This result is known as the _Poincaré–Hopf theorem_. As the Euler
characteristic is a topological invariant we can thus conclude that however we
deform the tangent vector field on a closed surface, the critical points
cannot all be eliminated, unless the surface has the topology of a torus,
$\chi=0$.

### Example: The Euler characteristics of the torus and the sphere

Points on the surfaces of both a torus and a sphere can be parameterized in
terms of two cyclic coordinates, $x_{1}$ and $x_{2}\in[0,2\pi]$,

$\bm{r}(x_{1},x_{2})=a\cos(x_{1})\,\bm{e}_{z}+\left(R+a\sin(x_{1})\right)\left(\cos(x_{2})\bm{e}_{x}+\sin(x_{2})\bm{e}_{y}\right),$
(18)

where $R$ is the distance from the origin to the centre of the torus ‘tube’ of
radius $a$. The surface makes a topological transition when the distance of
the centre of the tube from the origin equals its radius $R=a$, at which point
the innermost circle of points on the torus becomes a single point at the
origin, and the topology changes to that of a sphere. For $R<a$ the $x_{1}$
coordinate has the reduced range $x_{1}\in[-\arcsin(R/a),\pi+\arcsin(R/a)]$,
which becomes $x_{1}\in[0,\pi]$ when $R=0$, where Eq. (18) describes the
surface of a sphere.

Tangent vectors on the surface can be found through differentiating Eq. (18)
with respect to the two coordinates, which after normalization gives the
orthogonal pair of vectors

$\displaystyle\bm{e}_{1}$
$\displaystyle=-\sin(x_{2})\bm{e}_{x}+\cos(x_{2})\bm{e}_{y}$
$\displaystyle\bm{e}_{2}$
$\displaystyle=-\sin(x_{1})\bm{e}_{z}+\cos(x_{1})\left(\cos(x_{2})\bm{e}_{x}+\sin(x_{2})\bm{e}_{y}\right).$
(19)

When the surface has the topology of a torus, the tangent vectors (19) are
uniquely defined at all points, and therefore the Euler characteristic equals
zero

$\chi=\frac{1}{2\pi}\sum_{n}\oint_{C_{n}}A_{j}\,{\rm
d}x_{j}=0\qquad\text{(torus)}$ (20)

where the sum runs over the critical points (of which there are non in the
case of a torus), each encircled by $C_{n}$.

When the surface becomes a sphere ($R=0$), the two points $x_{1}=0$ and
$x_{1}=\pi$ are critical points of the tangent vectors (19), being isolated
points where the tangent vectors take many possible values, depending on how
we approach the point. Using expressions (19) we can calculate the ‘vector
potential’ from our earlier formula (13),

$\displaystyle A_{1}$
$\displaystyle=\bm{e}_{1}\cdot\frac{\partial\bm{e}_{2}}{\partial x_{1}}=0$
$\displaystyle A_{2}$
$\displaystyle=\bm{e}_{1}\cdot\frac{\partial\bm{e}_{2}}{\partial
x_{2}}=\cos(x_{1}).$ (21)

The Euler characteristic then equals the sum of the line integrals of $A_{j}$
around the critical points

$\chi=\frac{1}{2\pi}\left[\int_{0}^{2\pi}A_{2}(x_{1}=0)\,{\rm
d}x_{2}+\int_{2\pi}^{0}A_{2}(x_{1}=\pi)\,{\rm
d}x_{2}\right]=2\qquad\text{(sphere)}$ (22)

where the line integral around the south pole is taken in the opposite
direction, due to the reversal of the surface normal. We have shown the Euler
characteristic of a sphere equals $2$, as expected from the observation that
$\chi$ is twice the winding number of the surface normal around a sphere.

##  IV The Berry connection and the Euler characteristic

So far we’ve illustrated something of the basics of topology, but have not
made much connection with physics, which is supposed to be why we’re here! We
can begin to see the connection by re–writing the formula for the Euler
characteristic (15) in terms of a single _complex_ tangent vector field (a
so–called complex line bundle)

$|\psi\rangle=\frac{1}{\sqrt{2}}\left(\bm{e}_{1}+{\rm i}\bm{e}_{2}\right),$
(23)

where the state is normalized such that $\langle\psi|\psi\rangle=1$ and we
have adopted the bra–ket notation for vectors and inner products. In terms of
this complex vector, the ‘vector potential’ (13) appearing in the Gauss–Bonnet
theorem takes a simpler form

$A_{j}=\bm{e}_{1}\cdot\frac{\partial\bm{e}_{2}}{\partial x_{j}}=-{\rm
i}\frac{1}{2}(\bm{e}_{1}-{\rm i}\bm{e}_{2})\frac{\partial}{\partial
x_{j}}(\bm{e}_{1}+{\rm i}{\bm{e}_{2}})=-{\rm
i}\langle\psi|\frac{\partial}{\partial x_{j}}|\psi\rangle$ (24)

where we used the normalization conditions $\bm{e}_{i}\cdot\bm{e}_{i}=1$,
which implies $\bm{e}_{i}\cdot\partial_{j}\bm{e}_{i}=0$. The final expression
on the right of Eq. (24) can be recognised at once as the _Berry connection_
[22]. An analogous quantity appears in quantum theory, and that tells us how
to transport a quantum mechanical state vector $|\psi\rangle$ around a space
of parameters $x_{i}$. For our tangent vector $|\psi\rangle$, the curl of the
coresponding Berry connection is—via Eq. (14)—simply the Gaussian curvature of
the surface.

Using the complex vector $|\psi\rangle$, we can—again, analogous to quantum
theory—understand it as an eigenvector of a Hermitian operator $\hat{L}={\rm
i}\,\bm{n}\,\times$,

$\hat{L}={\rm i}\,\bm{n}\,\times\rightarrow\hat{L}|\psi\rangle=|\psi\rangle..$
(25)

To understand the origins of this operator note that
$\bm{n}\times(\bm{e}_{1}+{\rm i}\bm{e}_{2})=-{\rm i}(\bm{e}_{1}+{\rm
i}\bm{e}_{2})$. Having introduced the operator $\hat{L}$ we can calculate the
Gauss–Bonnet theorem in yet another way! Not only does the Euler
characteristic record critical points of the complex vector $|\psi\rangle$
through the Berry connection (24), but these critical points arise from the
properties of the operator $\hat{L}$, of which $|\psi\rangle$ is an
eigenfunction.

To see this we calculate the curl of the Berry connection (24)—which can be
used in the formula for the Euler characteristic (15)—and use the eigenvalue
relation (25) to replace derivatives of the vector with those of the operator
$\hat{L}$,

$\displaystyle\frac{\partial A_{2}}{\partial x_{1}}-\frac{\partial
A_{1}}{\partial x_{2}}$ $\displaystyle=-{\rm
i}\left[\bigg{\langle}\frac{\partial\psi}{\partial
x_{1}}\bigg{|}\chi\bigg{\rangle}\bigg{\langle}\chi\bigg{|}\frac{\partial\psi}{\partial
x_{2}}\bigg{\rangle}-\bigg{\langle}\frac{\partial\psi}{\partial
x_{2}}\bigg{|}\chi\bigg{\rangle}\bigg{\langle}\chi\bigg{|}\frac{\partial\psi}{\partial
x_{1}}\bigg{\rangle}\right]$ $\displaystyle=-{\rm
i}\left[\bigg{\langle}\psi\bigg{|}\frac{\partial\hat{L}}{\partial
x_{1}}\bigg{|}\chi\bigg{\rangle}\bigg{\langle}\chi\bigg{|}\frac{\partial\hat{L}}{\partial
x_{2}}\bigg{|}\psi\bigg{\rangle}-\bigg{\langle}\psi\bigg{|}\frac{\partial\hat{L}}{\partial
x_{2}}\bigg{|}\chi\bigg{\rangle}\bigg{\langle}\chi\bigg{|}\frac{\partial\hat{L}}{\partial
x_{1}}\bigg{|}\psi\bigg{\rangle}\right]$
$\displaystyle=\bm{n}\cdot\left(\frac{\partial\bm{n}}{\partial
x_{1}}\times\frac{\partial\bm{n}}{\partial x_{2}}\right)=K$ (26)

where the final line demonstrates the consistency of our different approaches
to the Euler characteristic. We also defined $|\phi\rangle=\bm{e}_{1}-{\rm
i}\bm{e}_{2}$ (eigenvalue $-1$) and $|\chi\rangle=\bm{n}$ (eigenvalue $0$),
used the completeness relation
$|\psi\rangle\langle\psi|+|\phi\rangle\langle\phi|+|\chi\rangle\langle\chi|=1$,
and applied the result
$\partial_{i}\hat{L}|\chi\rangle+\hat{L}\partial_{i}|\chi\rangle=0$.

Eq. (26) shows that the curl of the Berry connection is related to the
‘winding’ of the operator $\hat{L}$. In general this is difficult to picture,
but here it is simply another way of telling us the element of solid angle
swept out by the surface normal $\bm{n}$ on the sphere. We have thus found yet
another method for calculating the Euler characteristic! Not only is it the
integral of the curl of the Berry connection (_the Berry curvature_)
associated with the complex tangent vector field $|\psi\rangle$, divided by
$2\pi$, this can also be written in terms of matrix elements of the operator
$\hat{L}$ (25) defined over the surface.

In physics, the integral of the Berry connection $\oint A_{i}\,{\rm d}x_{i}$
is the phase shift a quantum mechanical wave function obtains after being
adiabatically moved through a parameter space with coordinates $x_{i}$ [22].
Our surface tangent ‘state vector’ $|\psi\rangle$, undergoes the same phase
shift as we encircle a critical point. We can see this by examining how the
vector changes as we move a small distance on the surface

$\langle\psi|{\rm
d}|\psi\rangle=\langle\psi|\frac{1}{\sqrt{2}}\left(-\bm{e}_{2}+{\rm
i}\bm{e}_{1}\right){\rm d}\theta=\left(1+{\rm i}{\rm d}\theta\right)={\rm
e}^{{\rm i}{\rm d}\theta}.$ (27)

Using our earlier equation for the line integral of the vector potential
around a critical point (16), we see from the above that the phase accumulated
around a critical point is $\oint A_{j}\,{\rm d}x^{j}=2\pi n$, where $n$ is
the critical point index.

### Example: Electromagnetic polarization

Free space electromagnetic waves are transverse, with the Fourier amplitudes
of the fields obeying $\bm{k}\cdot\tilde{\bm{E}}=\bm{k}\cdot\tilde{\bm{H}}=0$.
For a monochromatic field of frequency $\omega$, the length of the wavevector
is also fixed by the free space dispersion relation
$\bm{k}^{2}=k_{0}^{2}=\omega^{2}/c^{2}$, which defines the surface of a sphere
of radius $k_{0}$. Monochromatic radiation is therefore defined by a set of
Fourier amplitudes that both live on, and are tangent to a sphere in
$\bm{k}$–space.

The electric field in free space can thus be written as an integral over the
surface of this sphere,

$\bm{E}(\bm{x})=\int_{0}^{\pi}\sin(\theta_{k})\,{\rm
d}\theta_{k}\,\int_{0}^{2\pi}{\rm
d}\phi_{k}\,\bm{e}(\theta_{k},\phi_{k})\,\mathcal{E}(\theta_{k},\phi_{k})\,{\rm
e}^{{\rm i}k_{0}\bm{n}(\theta_{k},\phi_{k})\cdot\bm{x}}$ (28)

where $\bm{e}$ represents the direction of the electric field, and
$\mathcal{E}$ the scalar amplitude of the wave propagating in direction
$\bm{n}=\bm{k}/k_{0}$. The polarization vector is chosen to satisfy
$\bm{e}\cdot\bm{e}^{\star}=1$ and $\bm{n}\cdot\bm{e}=0$.

Figure 6: The nodes in the radiation pattern from an electric dipole are a
consequence of topology. (a) Snapshot of the oscillating electric field
$\bm{E}$ from a dipole moment $\bm{p}$. Enclosing the dipole in a sphere (the
blue curve, for example) there will always be points where the electric field
is normal to the sphere, making its direction tangent to the sphere undefined
(a critical point). (b) Far away from the dipole, the observation direction
picks out a Fourier component of the field (28) and the electric field is
everywhere tangent to the sphere, with critical points along the axis of the
dipole.

If $\bm{e}$ is real valued, the polarization is linear for all directions of
propagation. In this case we can use it as the tangent vector $\bm{e}_{1}$ in
Eq. (13), with $\bm{e}_{2}=\bm{n}\times\bm{e}_{1}$. The integral of the curl
of the ‘vector potential’ defined in (13), $\bm{\nabla}\times\bm{A}$, over the
entire sphere in $\bm{k}$ space will therefore equal $2$, the Euler
characteristic of the sphere. _This means that linear polarized fields always
have at least one index $2$ critical point in the electric field, as a
function of direction_. The same argument can also be applied to circularly
polarized fields, where at every point the direction of the electric field is
of the form $\bm{e}=(\bm{e}_{1}\pm{\rm i}\bm{e}_{2})/\sqrt{2}$. As described
below Eq. (27), there must be a phase vortex around each of these critical
points, which can be understood as the Berry phase.

As a concrete example, take radiation from a dipole with dipole moment
$\bm{p}$, where the far field electric field takes the form

$\bm{E}(\bm{x})\sim\frac{k_{0}^{2}}{4\pi\epsilon_{0}r}\left(\bm{p}-\bm{e}_{r}(\bm{e}_{r}\cdot\bm{p})\right){\rm
e}^{{\rm i}(k_{0}r-\omega t)}.$ (29)

In the far field ($r\to\infty$), the direction of observation selects one
direction of propagation in the expansion (28) and we can therefore see that
the vector $\bm{e}$ on the sphere is

$\bm{e}=\frac{\bm{p}-\bm{e}_{r}(\bm{e}_{r}\cdot\bm{p})}{\sqrt{\bm{p}^{2}-(\bm{e}_{r}\cdot\bm{p})^{2}}}.$
(30)

This vector has two critical points of index $+1$, each when the radial unit
vector points along the axis of the dipole:
$\bm{e}_{r}(\bm{e}_{r}\cdot\bm{p})=\bm{p}$. We can thus see that the nodes in
the radiation pattern from a simple dipole—something we are familiar with from
a first course in electromagnetism and shown in Fig. 6—can be understood to be
a necessary consequence of the topology of a sphere. Were the dispersion
surface to have a different topology (in a hyperbolic material, for example
[23]), these properties would change. This line of argument has far reaching
consequences for the polarization of scattered light, governing the
polarization of sunlight [24] and multipole scattering in metamaterials [25].

##  V Characteristic classes and physics

We have now developed several methods, all for calculating the same thing: the
Euler characteristic (winding number) of a two dimensional surface. This
limitation in part occurs because we always considered _tangent_ vectors
$\bm{e}_{1,2}$; the so–called _tangent bundle_. As the critical points of the
tangent vectors are a direct reflection of the number of times the surface
normal wraps around a sphere, we were stuck with the Euler characteristic. But
there is nothing stopping us from adapting the same formulae to calculate
_different_ topological invariants both for higher dimensional surfaces and
for vectors on the surface that are not related to the tangent vectors in any
way.

To understand the generalization to higher dimensions we need to introduce
some terminology. Although we didn’t name them as such, we have so–far been
looking at integrals of _characteristic classes_. In (15) we integrated the
tangent vectors’ _Euler class_ , $\bm{\nabla}\times\bm{A}/2\pi$ where
$A_{i}=\bm{e}_{1}\cdot\partial_{i}\bm{e}_{2}$, which yields a topological
invariant from the _real_ vectors $\bm{e}_{1,2}$ on the surface. Meanwhile,
when we did the same thing for the single _complex_ vector field
$|\psi\rangle=(\bm{e}_{1}+{\rm i}\bm{e}_{2})/\sqrt{2}$ (23) (a so–called line
bundle) we integrated the tangent vectors’ _first Chern class_ ,

$\text{{\hbox{1^{\rm st}}} Chern class:
}\;c_{1}=\frac{1}{2\pi}\left(\frac{\partial A_{2}}{\partial
x_{1}}-\frac{\partial A_{1}}{\partial x_{2}}\right),\qquad\text{where
}A_{j}=-{\rm i}\langle\psi|\partial_{j}|\psi\rangle$ (31)

which is the Berry curvature divided by $2\pi$, and yields a topological
invariant from _complex_ vectors on the surface. These are two examples of
_characteristic classes_. For the special case of tangent vectors on a two
dimensional surface, integrals of the Euler and first Chern classes give the
same result: the Euler characteristic.

The _Chern classes_ are just one type of characteristic class. Each Chern
class is a ‘closed but not exact’ expression depending on the Berry
connection. This is a generalization of what we’ve already seen in two
dimensions, namely $\bm{\nabla}\times\bm{A}$ ‘looks like a curl’ (it is
closed), but fails to be the curl of any properly defined vector at the
critical points on the surface (it is not exact). These critical points are a
direct reflection of the topology of both the vector field, and the shape of
the surface. The non–zero winding number in both our 1D (1) and 2D (15)
examples is equivalent to summing the indices of these critical points. Each
of the _Chern classes_ beyond the first is a ‘closed but not exact’ expression
that does exactly the same thing in higher dimensions, each being integrated
over ever higher dimensional regions: the first Chern class being associated
with two dimensional integrals, the second with four dimensions, the third
with six dimensions, and so on.

To illustrate the point, let’s look at the second Chern class. We consider a
four dimensional surface, to which we attach a pair of complex vector fields,
$|1\rangle$ and $|2\rangle$ (as opposed to the single vector $|\psi\rangle$
used in two dimensions). As explained in Appendix A, when dealing with a set
of $N$ complex vector fields, each component of the Berry connection $A_{i}$
becomes an $N\times N$ matrix. The Berry curvature is also replaced by the two
index ‘curvature form’ $\Omega_{ij}$, each component of which is—in this
particular case—a $2\times 2$ matrix

$\Omega_{ij}=\frac{1}{2}\left(\frac{\partial A_{j}}{\partial
x_{i}}-\frac{\partial A_{i}}{\partial x_{j}}+{\rm i}[A_{i},A_{j}]\right).$
(32)

The commutator is defined as $[A_{i},A_{j}]=A_{i}\,A_{j}-A_{j}\,A_{i}$, and
represents the difference between $\Omega_{ij}$ and the Berry curvature
encountered in the previous section.

Expressed in terms of the curvature (32), the _second_ Chern class—which is to
be integrated over a four dimensional surface—is simply required to be a
‘closed but not exact’ scalar expression that can be integrated over the
surface444Note that the higher order Chern classes are more concisely
expressed in terms of _differential forms_ [lovelock1990]. Here the equivalent
of the curvature of the connection (32) is a two–form $\Omega={\rm d}A+{\rm
i}\,A\wedge A$, and the second Chern class (33) is written
$c_{2}=\Omega\wedge\Omega/8\pi^{2}$. The Chern–Simons form (35) is almost
always written in this language as $S={\rm d}A\wedge A+\frac{2}{3}A\wedge
A\wedge A$.. As $\Omega_{ij}$ has only two of the requisite four spatial
indices we must therefore consider the _square_ of the Berry curvature,

$\text{{\hbox{2^{\rm nd}}} Chern class:}\;c_{2}=\frac{1}{8\pi^{2}}{\rm
tr}[\Omega^{2}]=\frac{1}{8\pi^{2}}\epsilon_{ijkl}{\rm
tr}[\Omega_{ij}\Omega_{kl}]$ (33)

where $\epsilon_{ijkl}$ is the completely anti–symmetric unit tensor, and
‘${\rm tr}$’ is a trace over the matrix left after the sum over the spatial
indices $i,j,k,l$. The pre–factor of $1/8\pi^{2}$ is analogous to the factor
of $1/2\pi$ in Eq. (31), ensuring the result of integrating (33) over the
surface is an integer.

Substituting the curvature (32) into the definition of the second Chern class
(33) we see that $c_{2}$ can be written as a divergence

$\displaystyle c_{2}$ $\displaystyle=\frac{1}{32\pi^{2}}\epsilon_{ijkl}\,{\rm
tr}\bigg{[}\frac{\partial A_{j}}{\partial x_{i}}-\frac{\partial
A_{i}}{\partial x_{j}}+{\rm i}[A_{i},A_{j}]\bigg{]}\bigg{[}\frac{\partial
A_{l}}{\partial x_{k}}-\frac{\partial A_{k}}{\partial x_{l}}+{\rm
i}[A_{k},A_{l}]\bigg{]}$
$\displaystyle=\frac{1}{8\pi^{2}}\epsilon_{ijkl}\,\frac{\partial}{\partial
x_{i}}{\rm tr}\left[A_{j}\frac{\partial A_{l}}{\partial x_{k}}+\frac{2{\rm
i}}{3}A_{j}A_{k}A_{l}\right].$ (34)

As we hoped, we have something that is ‘closed but not exact’. As with the
expressions we have met for the winding number of a curve (2), and the Euler
characteristic (15), the integral of $c_{2}$ over any closed four dimensional
surface _appears_ to be zero. Eq. (34) takes the form of a divergence, and the
divergence theorem tells us its integral equals an integral over a boundary,
which vanishes for any closed surface! Yet again this is not the case: $c_{2}$
is non–zero due to the critical points of the so–called _Chern–Simons_ form
[27]

$S_{i}=\epsilon_{ijkl}\left[A_{j}\partial_{k}A_{l}+\frac{2{\rm
i}}{3}A_{j}A_{k}A_{l}\right]$ (35)

which exhibits critical points where the basis vectors $|1\rangle$ and
$|2\rangle$ become undefined. We should note that the Chern–Simons form
appears in several places in physics, including in the next section, and is
e.g. an important object in topological field theory [28].

The second Chern class is another example of a quantity that is ‘closed but
not exact’. The integral of the second Chern class depends on the integer
number of critical points of the Chern Simons form and is thus a number that
cannot be continuously changed. The pattern evident in the first two Chern
classes, (31) and (33) continues into higher dimensions, with e.g. the $3^{\rm
rd}$ Chern class given in terms of the _cube_ of $\Omega_{ij}$, which can be
written as the divergence of a Chern–Simons form containing higher powers of
the matrices $A_{i}$ and their derivatives. Note that there are no Chern
classes associated with odd dimensional surfaces, simply because they are all
zero!

In recent years, Chern classes above the first have been used to design
waveguides [29] and acoustic lattices [30], where the dimensions are typically
‘synthetic’, being system parameters such as the resonant frequency.

### Example: the first Chern class applied to distinguish eigenmodes

As an illustration of the application of characteristic classes in physics, we
can calculate the first Chern class for an eigenmode $|\psi\rangle$ of a
system parameterized by coordinates, $x_{1}$ and $x_{2}$ that cover a closed
surface. Suppose we have two such eigenmodes, $|\psi\rangle$ and
$|\phi\rangle$, of a linear operator $\hat{L}$,

$\displaystyle\hat{L}(x_{1},x_{2})\,|\psi\rangle$
$\displaystyle=\lambda\,|\psi\rangle$
$\displaystyle\hat{L}(x_{1},x_{2})\,|\phi\rangle$
$\displaystyle=\lambda^{\prime}\,|\phi\rangle.$ (36)

By analogy with the discussion of the Sec. IV we can define separate Berry
connections for each of these eigenmodes,

$A_{\psi,i}=-{\rm i}\langle\psi|\frac{\partial}{\partial
x_{i}}|\psi\rangle,\qquad A_{\phi,i}=-{\rm
i}\langle\phi|\frac{\partial}{\partial x_{i}}|\phi\rangle.$ (37)

and ask the question of whether the state $|\psi\rangle$ can be continuously
deformed into $|\phi\rangle$. If the integral of the first Chern class, known
as the first Chern number ${\rm Ch}_{1}$,

${\rm Ch}_{1}=\frac{1}{2\pi{\rm
i}}\int\bm{\nabla}\times\langle\psi|\bm{\nabla}|\psi\rangle\,{\rm d}x_{1}{\rm
d}x_{2}=\frac{1}{2\pi}\int\bm{\nabla}\times\bm{A}_{\psi}\,{\rm d}x_{1}{\rm
d}x_{2}$ (38)

is a different integer for modes $|\phi\rangle$ and $|\psi\rangle$ then the
answer to this question is no.

We can gain some useful insights if, as in Sec. IV, we relate the first Chern
class to derivatives of the operator $\hat{L}$, rather than the state
$|\psi\rangle$. To do this we first differentiate the eigenvalue equation (36)
to find to overlap between an arbitrary eigenmode $|n\rangle$ and the
derivative of $|\psi\rangle$, in terms of the derivative of $\hat{L}$,

$\bigg{\langle}n\bigg{|}\frac{\partial\psi}{\partial
x_{i}}\bigg{\rangle}=\frac{\langle n|\frac{\partial\hat{L}}{\partial
x_{i}}|\psi\rangle}{\lambda-\lambda_{n}}$ (39)

where we assume that $|n\rangle$ and $\psi\rangle$ are non–degenerate
eigenstates. Taking the curl of the first of the two Berry connections (37)
and applying the above identity (39) then leads to the following expression
for the Berry curvature,

$\displaystyle\frac{\partial A_{2}}{\partial x_{1}}-\frac{\partial
A_{1}}{\partial x_{2}}$ $\displaystyle=-{\rm
i}\sum_{|n\rangle\neq|\psi\rangle}\left[\bigg{\langle}\frac{\partial\psi}{\partial
x_{1}}\bigg{|}n\bigg{\rangle}\bigg{\langle}n\bigg{|}\frac{\partial\psi}{\partial
x_{2}}\bigg{\rangle}-\bigg{\langle}\frac{\partial\psi}{\partial
x_{2}}\bigg{|}n\bigg{\rangle}\bigg{\langle}n\bigg{|}\frac{\partial\psi}{\partial
x_{1}}\bigg{\rangle}\right]$ $\displaystyle=-{\rm
i}\sum_{|n\rangle\neq|\psi\rangle}\left[\frac{\langle\psi|\frac{\partial\hat{L}}{\partial
x_{1}}|n\rangle\langle n|\frac{\partial\hat{L}}{\partial
x_{2}}|\psi\rangle-\langle\psi|\frac{\partial\hat{L}}{\partial
x_{2}}|n\rangle\langle n|\frac{\partial\hat{L}}{\partial
x_{1}}|\psi\rangle}{(\lambda-\lambda_{n})^{2}}\right]$ (40)

where for brevity we dropped the subscript ‘$\psi$’ from the vector potential.

In Eq. (26) of Sec. IV we related the Gaussian curvature to derivatives of a
Hermitian operator $\hat{L}={\rm i}\,\bm{n}\times$. In the same way, here we
find the more general concept of the Berry curvature is given in terms of
derivatives of an arbitrary linear operator $\hat{L}$, of which $|\psi\rangle$
is an eigenstate. We can see from Eq. (40) that an eigenvalue degeneracy,
$\lambda=\lambda_{n}$ between any of the levels $|n\rangle$ and the state of
interest $|\psi\rangle$, leads to points of singular Berry curvature. Such
points are critical points of the Berry connection, arising due to the
indeterminacy in the eigenvector $|\psi\rangle$. However, degeneracies are not
the only points of non–zero Berry curvature. Equation (40) shows that the
curvature is generally non–zero whenever the derivatives of the linear
operator, $\partial\hat{L}/\partial x_{i}$, have _complex_ off diagonal matrix
elements with the state of interest $|\psi\rangle$.

An important corollary of Eq. (40) is that, if we sum the Berry curvature over
all system states $|\psi\rangle$, we obtain zero

$-{\rm
i}\sum_{|\psi\rangle}\sum_{|n\rangle\neq|\psi\rangle}\left[\frac{\langle\psi|\frac{\partial\hat{L}}{\partial
x_{1}}|n\rangle\langle n|\frac{\partial\hat{L}}{\partial
x_{2}}|\psi\rangle-\langle\psi|\frac{\partial\hat{L}}{\partial
x_{2}}|n\rangle\langle n|\frac{\partial\hat{L}}{\partial
x_{1}}|\psi\rangle}{(\lambda-\lambda_{n})^{2}}\right]=0$ (41)

because the two terms in the numerator are now equal, cancelling due to the
summation. _This simple observation means that the sum of the first Chern
numbers for all the eigenstates of a system is always zero_.

### Example: Polarization eigenstates in anisotropic materials

Take an electromagnetic wave propagating through an anisotropic, non–magnetic
material. We shall use the Chern number to characterize a family of these
materials, to tell us how many times _all_ possible polarizations are explored
as we run through the material parameters.

Assuming a permeability $\mu_{0}\bm{1}$, a Hermitian (lossless) permittivity,

$\bm{\epsilon}=\epsilon_{0}\left(\begin{matrix}\epsilon_{xx}&\epsilon_{xy}&0\\\
\epsilon_{yx}&\epsilon_{yy}&0\\\
0&0&\epsilon_{zz}\end{matrix}\right)=\epsilon_{0}\left(\begin{matrix}\bm{\epsilon}_{\parallel}&\bm{0}\\\
\bm{0}&\epsilon_{zz}\end{matrix}\right),$ (42)

and propagation along the $z$–axis, Maxwell’s equations for a wave of fixed
frequency $\omega$ reduce to

$\displaystyle\frac{k}{\omega}\bm{e}_{z}\times\bm{E}$
$\displaystyle=\mu_{0}\bm{H}$
$\displaystyle\frac{k}{\omega}\bm{e}_{z}\times\bm{H}$
$\displaystyle=-\epsilon_{0}\bm{\epsilon}_{\parallel}\cdot\bm{E}$ (43)

where $k$ the propagation constant of the wave. The problem is now to find the
propagation constant and the electric field vector for a given set of material
parameters $\bm{\epsilon}_{\parallel}$. Eliminating the magnetic field from
the two equations (43) we can reduce the problem to an eigenvalue equation

$\bm{\epsilon}_{\parallel}\cdot\bm{E}={\rm n}^{2}\bm{E}$ (44)

where ${\rm n}=ck/\omega$ is the refractive index. Therefore the eigenvalues
and eigenvectors of the in–plane permittivity tensor determine the
polarization and refractive index of the wave, respectively. Interestingly,
Eq. (44) is equivalent to the Schrödinger equation for a spin $1/2$ particle
in a magnetic field (see e.g. [22]). With this in mind we re–write the
in–plane permittivity tensors in terms of the Pauli spin matrices,
$\bm{\sigma}=\sigma_{x}\bm{e}_{x}+\sigma_{y}\bm{e}_{y}+\sigma_{z}\bm{e}_{z}$,
as follows

$\left(\begin{matrix}\epsilon_{xx}&\epsilon_{xy}\\\
\epsilon_{yx}&\epsilon_{yy}\end{matrix}\right)=\left(\begin{matrix}V_{0}+V\,\cos(\theta)&V\,\sin(\theta)\,{\rm
e}^{-{\rm i}\phi}\\\ V\,\sin(\theta)\,{\rm e}^{{\rm
i}\phi}&V_{0}-V\,\cos(\theta)\end{matrix}\right)=V_{0}\bm{1}+V\,\bm{n}\cdot\bm{\sigma}.$
(45)

where $\bm{n}$ is a unit vector, here parameterized by the spherical
coordinates, $\theta$ and $\phi$. While the angle $\theta$ determines the
orientation of the principal axes of the permittivity, $\phi$ governs the
gyrotropy of the medium [31]. We can thus visualise these forms of lossless
in–plane permittivity in terms of two real numbers, $V_{0}$ and $V$, and the
coordinates on a sphere, $\theta$ and $\phi$. Interestingly, the _eigenvalues_
of (45) (the refractive index squared) are independent of the coordinates
$\theta$ and $\phi$,

${\rm n}^{2}=V_{0}\pm V.$ (46)

while the _eigenvectors_ —defining the _polarization_ of the electric
field—depend _only_ on the spherical coordinates

$|\psi_{+}\rangle=\left(\begin{matrix}\cos(\theta/2)\\\ \sin(\theta/2){\rm
e}^{{\rm
i}\phi}\end{matrix}\right),\qquad|\psi_{-}\rangle=\left(\begin{matrix}\sin(\theta/2){\rm
e}^{-{\rm i}\phi}\\\ -\cos(\theta/2)\end{matrix}\right).$ (47)

These two states can be pictured as points on the _Bloch sphere_ shown in Fig.
7. Keeping the refractive index fixed, the two polarizations can be
interchanged by modifying the material parameters (45) according to the
substitution $\theta\to\pi-\theta$ and $\phi\to\phi+\pi$.

Figure 7: (a) An electromagnetic wave propagating through a dielectric (here
along $z$), normal to the plane of anisotropy (here the $x$–$y$ plane) will
generally have two different propagation constants $k={\rm n}\omega/c$, each
corresponding to a different, generally elliptical, polarization, as given by
(46) and (47). (b) We can visualise the polarizations on the Bloch sphere,
with $\theta$ and $\phi$ labelling the different material parameters in Eq.
(45). For $\theta=0,\pi$ the principal axes of the permittivity tensor align
with the $x$ and $y$ axes, and the wave is either $x$ or $y$ polarized.
Meanwhile, for $\theta=\pi/2$ the principal axes are diagonal and the wave
polarization varies between diagonal and circular, depending on the gyrotropy
$\phi$.

For both states (47), the Berry connection contains only a single component,

$A_{+,2}=-{\rm
i}\langle\psi_{+}|\frac{\partial}{\partial\phi}|\psi_{+}\rangle=\sin^{2}(\theta/2)=-A_{-,2}$
(48)

which has an $n=1$ critical point at the south pole, $\theta=\pi$. The result
$A_{+,2}=-A_{-,2}$ is also in agreement with Sec. IV, where we found the sum
of the Berry curvature over all eigenstates should vanish. From the above
equation for the Berry connection, the Berry curvature is found to equal

$\frac{\partial A_{+,2}}{\partial\theta}-\frac{\partial
A_{+,1}}{\partial\phi}=\sin(\theta)/2.$ (49)

The integral of the curvature (49) over all values of $\theta$ and $\phi$
equals the first Chern number, here ${\rm Ch}_{+}$, which in this system is
the number of times the polarization covers the Bloch sphere

${\rm Ch}_{+}=\frac{1}{4\pi}\int_{0}^{2\pi}{\rm d}\phi\int_{0}^{\pi}{\rm
d}\theta\sin(\theta)=1=-{\rm Ch}_{-}.$ (50)

A Chern number of unity tells us that, for a fixed value of the refractive
index squared, ${\rm n}^{2}=V_{0}\pm V$, the family of in–plane permittivity
tensors (45) cover all possible electromagnetic polarizations exactly once.
The difference in sign between ${\rm Ch}_{+}$ and ${\rm Ch}_{-}$ means that
the two polarization eigenstates (47) wind in opposite senses around the Bloch
sphere as we change the material parameters. This argument fails when we look
at isotropic materials where $V=0$, in which case the ‘gap’ between the two
values of refractive index (46) closes, and the permittivity (45) no longer
depends on the spherical coordinates. At this point the two polarizations (47)
have degenerate values of the refractive index, meaning all polarizations are
solutions to Eq. (44), whatever the value of $V_{0}$.

Had we parameterized the permittivity differently, as e.g.

$\left(\begin{matrix}\epsilon_{xx}&\epsilon_{xy}\\\
\epsilon_{yx}&\epsilon_{yy}\end{matrix}\right)=\left(\begin{matrix}V_{0}\cos(\phi)+V\cos(\theta)&V\sin(\theta)\\\
V\sin(\theta)&V_{0}\cos(\phi)-V\cos(\theta)\end{matrix}\right)$ (51)

the refractive index would have been ${\rm n}^{2}=V_{0}\cos(\phi)\pm V$, with
eigenstates independent of $\phi$, e.g.
$|\psi_{+}\rangle=\left(\cos(\theta/2),\sin(\theta/2)\right)^{\rm T}$. For
this family of permittivity tensors the Berry connection (48) therefore
vanishes, and no area is covered on the Bloch sphere. The Chern numbers thus
vanish, ${\rm Ch}_{+}={\rm Ch}_{-}=0$. This tells us that for anisotropic
media, gyrotropy is therefore essential to realise all possible polarization
eigenstates.

##  VI Chern numbers and dispersion relations

We introduced the ‘characteristic classes’ to set up what is probably the most
striking application of topology in wave physics: the prediction of interface
modes, and in particular the possibility of these interface modes being
constrained to propagate in one direction only. In this section we prove the
relation between the integral of the first Chern class, and the prediction of
interface modes between different materials.

Take some planar material that supports waves. Perhaps electromagnetic waves
in a periodic array of pillars, or elastic waves in a plate. Suppose the
system is homogeneous, or at least periodic, so the modes $|n_{\bm{k}}\rangle$
can be labelled with a wave–vector $\bm{k}$. The modes will generally be the
solution to some eigenvalue equation

$\hat{L}(\bm{k})|n_{\bm{k}}\rangle=\lambda_{n,\bm{k}}|n_{\bm{k}}\rangle$ (52)

where the integer $n$ labels the different branches of the dispersion
relation. At this stage we make no assumption about the meaning of
$\lambda_{n,\bm{k}}$; it could be a frequency, a wave vector component, or a
material parameter.

In the case of homogeneous media, the wave–vector ranges over all pairs of
real values, which can be mapped to a sphere via the stereographic projection
(so long as care is taken at infinity [32], see Fig. 8a). For periodic media
the Bloch vector has components ranging over the first Brillouin zone, the
edges of which can be connected to each other through the addition of a
reciprocal lattice vector, thus forming the surface of a torus (see Fig. 8b).
In both cases we are dealing with the situation described in the previous
sections: a single complex vector field $|n_{\bm{k}}\rangle$ defined over a
closed two dimensional surface, the points of which are specified by the
vector $\bm{k}$.

Figure 8: The modes $|n_{\bm{k}}\rangle$ of both homogeneous and periodic two
dimensional media can be understood as a set of vectors on a closed surface,
as discussed in Sec. V. For homogeneous media the wave–vector $\bm{k}$ ranges
over all pairs of real values, which—via the stereographic projection—can be
mapped onto the surface of a sphere, as shown in (a). For periodic media, the
Bloch vector $\bm{K}$ is equivalent after the addition of a reciprocal lattice
vector. The edges of the first Brillouin zone (differing as they do by a
reciprocal lattice vector) can thus be identified, e.g. the two blue lines in
panel (b) represent the same physical states. Connecting the edges of the
Brillouin zone yields a torus, as shown in panel (b).

#### In a nutshell:

Suppose we have a system where the different branches of the dispersion
relation (the $\lambda_{n,\bm{k}}$ for different $n$) are separated by gaps.
For instance $\lambda_{2,\bm{k}}$ takes values in some range, separated above
and below by a gap before the next lower range of values $\lambda_{1,\bm{k}}$,
and larger range of values $\lambda_{3,\bm{k}}$. The idea is that we focus on
one of these gaps in the spectrum. We classify all the branches of the
dispersion relation below this gap in terms of the integral of the first Chern
class (the first Chern number) over all wave–vectors

${\rm Ch}_{1}=\frac{1}{2\pi{\rm i}}\int_{S}\bm{\nabla}_{\bm{k}}\times\langle
n_{\bm{k}}|\bm{\nabla}_{\bm{k}}|n_{\bm{k}}\rangle\,{\rm d}k_{1}\,{\rm
d}k_{2}=\frac{1}{2\pi}\int_{S}\bm{\nabla}\times\bm{A}_{n}\,{\rm d}k_{1}\,{\rm
d}k_{2}.$ (53)

As shown in Fig. 8, the surface $S$ is a sphere for homogeneous media, and a
torus for periodic materials.

Now take two _different_ materials, $A$ and $A^{\prime}$ that have a shared
gap in their eigenvalue spectrum. If we sum the first Chern numbers ${\rm
Ch}_{1}$ for the all the modes $|n_{\bm{k}}\rangle$ and
$|n^{\prime}_{\bm{k}}\rangle$ below the gap of interest and find the answers
for the two materials are not the same, _there is no way to smoothly change
the material from $A$ to $A^{\prime}$, without closing the gap of interest in
the eigenvalue spectrum_.

To see this, note that to change the Chern number for any one of the branches
of the dispersion relation we must introduce or remove critical points from
the Berry connection $\bm{A}_{n}$. From our formula for the Berry connection
(40) we know that new critical points arise whenever another branch of the
dispersion relation becomes degenerate with branch $n$. But we know from Eq.
(41) that the total Berry curvature is zero for all the branches of the
dispersion relation. Therefore the sum of the Chern numbers below the gap
cannot be changed through the closure of any gap, except the one we are
interested in. This argument holds, however we choose to change material $A$
into $A^{\prime}$.

Therefore, if we join the two materials together, in the transition region
between the two there must be a region of space where the gap in the spectrum
closes such that propagation is allowed. This region is where interface modes
can be trapped. To add some meat onto the bones of this idea, we now give an
argument based around a paper of the author [33], which was in turn adapted to
Maxwell’s equations from [34].

###  VI.1 Volovik’s mode counting argument:

Suppose we have a system where the material varies smoothly as a function of
$x$, as shown in Fig. 9. Asymptotically we have material $A$ as $x$ becomes
large and positive and material $A^{\prime}$ when $x$ is large and negative.
Along the $y$–axis, the system remains translationally invariant so that we
can replace $y$ with the Fourier variable $k$. Due to the lack of
translational invariance along $x$, the wave operator cannot be written in
terms of the Fourier variable $k_{x}$, but depends on both $-{\rm
i}\partial_{x}$ and $x$. The eigenvalue equation is thus changed from (52) to

$\hat{L}(-{\rm i}\partial_{x},x,k)|n_{k}\rangle=\lambda_{n,k}|n_{k}\rangle.$
(54)

The Green function for this equation obeys

$\left[\hat{L}(-{\rm
i}\partial_{x},x,k)-\lambda\right]\bm{G}(x,x^{\prime},k,\lambda)=\bm{1}\,\delta(x-x^{\prime})$
(55)

and contains information about all of the eigenmodes. Here we make use of the
Green function to characterize how the eigenvalues change with respect to the
$y$ component of the wavevector, $k$. Using the completeness of the eigenmodes
of Eq. (54), $\sum_{n}\langle x|n\rangle\langle
n|x^{\prime}\rangle=\bm{1}\,\delta(x-x^{\prime})$ we can expand the Green
function in the eigenmode basis, yeilding the ‘bilinear expansion’ [35]

$\bm{G}(x,x^{\prime},k,\lambda)=\sum_{n}\frac{\langle x|n\rangle\langle
n|x^{\prime}\rangle}{\lambda_{n,k}-\lambda}..$ (56)

For us $\lambda$ is a complex number so that the denominator in (56) is
typically non–zero.

Figure 9: The spectral asymmetry $\nu(k)$ defined in (57) is half the
difference between the number of modes with positive and negative eigenvalue,
$\lambda$ for a fixed value of $k_{y}=k$. If a single mode crosses $\lambda=0$
with increasing $k$ (shown here in red) then the difference $\nu(+K)-\nu(-K)$
equals $\pm 1$. If the asymptotic materials $A$ and $A^{\prime}$ both have a
gap in their eigenvalue spectrum around $\lambda=0$, the mode that crosses
$\lambda=0$ must be an interface mode, confined to the inhomogeneous region
between the blue and yellow regions indicated in the figure.

As discussed above, we assume that when taken as homogeneous media, the two
asymptotic materials, $A$ and $A^{\prime}$ have a shared gap in their
eigenvalue spectrum that includes $\lambda=0$. Therefore, if we find a
solution to (54) with $\lambda=0$ we know that it must be confined to the
region in space where the material is in the process of changing from $A$ to
$A^{\prime}$, i.e. it is an interface mode. We count these modes through
introducing the _spectral asymmetry_ , which records the difference in the
number of modes with positive and negative eigenvalues (see Fig. 9),

$\nu(k)=\frac{1}{2}\sum_{n}{\rm sign}[\lambda_{n,k}]$ (57)

this number is a kind of tripwire, changing whenever a mode of the system
crosses $\lambda=0$. The spectral asymmetry can be calculated in terms of the
Green function as follows,

$\displaystyle\nu(k)$ $\displaystyle={\rm Re}\int_{-\infty}^{\infty}\frac{{\rm
d}\lambda}{2\pi}\,{\rm Tr}[G(x,x,k,{\rm i}\lambda)]=\sum_{n}{\rm Tr}[\langle
x|n\rangle\langle n|x\rangle]\,\int_{-\infty}^{\infty}\frac{{\rm
d}\lambda}{2\pi}\frac{\lambda_{n,k}}{\lambda_{n,k}^{2}+\lambda^{2}}$
$\displaystyle=\frac{1}{2}\sum_{n}{\rm sign}[\lambda_{n,k}]$ (58)

where the capitalized ‘${\rm Tr}$’ has been used for the sake of brevity. It
means both a trace over matrix indices, _and_ an integration over position
$x$.

We assume the material parameters change on a length scale that is large
compared to all other system scales. Equation (58) also shows us we only need
the behaviour of the Green function in the neighbourhood of $x=x^{\prime}$.
The differential equation (55) can therefore be expanded to leading order in
the distance $x-x^{\prime}$. Making the change of variables from $x$ and
$x^{\prime}$ to the average position $X=(x+x^{\prime})/2$, and separation
$\xi=x-x^{\prime}$, we expand the linear operator $\hat{L}$ to leading order
in $\xi$ and $\partial_{X}$,

$\displaystyle\hat{L}(-{\rm i}\partial_{x},x,k)$
$\displaystyle=\hat{L}\left(-{\rm i}\partial_{\xi}-{\rm
i}\frac{1}{2}\partial_{X},X+\frac{1}{2}\xi,k\right)$
$\displaystyle\sim\hat{L}(-{\rm i}\partial_{\xi},X,k_{y})-\frac{\rm
i}{4}\left[\frac{\partial}{\partial X}\frac{\partial\hat{L}}{\partial
k_{x}}+\frac{\partial\hat{L}}{\partial k_{x}}\frac{\partial}{\partial
X}\right]+\frac{1}{4}\left[\xi\frac{\partial\hat{L}}{\partial
X}+\frac{\partial\hat{L}}{\partial X}\xi\right].$ (59)

where derivatives with respect to $k_{x}$ indicate a derivative of the
operator with respect to the first argument of $\hat{L}$. Everywhere in (59)
the operator $\hat{L}$ depends only on derivatives $\partial_{\xi}$ and
average coordinate $X$. Performing a Fourier transform of (59) over $\xi$, the
wave operator $\hat{L}$ becomes a simpler object involving only two
differential operators

$\int{\rm d}\xi\,{\rm e}^{-{\rm i}k_{x}\xi}\,\hat{L}(-{\rm
i}\partial_{x},x,k)\sim L(k_{x},X,k)+\frac{\rm i}{2}\left[\frac{\partial
L}{\partial X}\frac{\partial}{\partial k_{x}}-\frac{\partial L}{\partial
k_{x}}\frac{\partial}{\partial X}\right].$ (60)

Staying in $k_{x}$ space, we can now solve for the Green function of (60) to
the same order, writing it as the sum of a zeroth and first order term $G\sim
G_{0}+G_{1}$. The zeroth order Green function is a solution to (55) with the
operator $\hat{L}$ replaced by the first term in Eq. (60). This is simply the
Green function for an infinite homogeneous medium with the local material
properties at the average position $X$,

$G_{0}(k_{x},X,k_{y},\lambda)=\left[L(k_{x},X,k_{y})-\lambda\right]^{-1}$ (61)

Meanwhile the first order correction to $G$ is the solution to

$\left[L(k_{x},X,k)-\lambda\right]G_{1}=-\frac{\rm i}{2}\left[\frac{\partial
L}{\partial X}\frac{\partial}{\partial k_{x}}-\frac{\partial L}{\partial
k_{x}}\frac{\partial}{\partial X}\right]G_{0}$ (62)

which can be solved using the zeroth order Green function (61), giving the
correction

$G_{1}(k_{x},X,k,\lambda)=-\frac{\rm i}{2}G_{0}\left[\frac{\partial
L}{\partial X}\frac{\partial G_{0}}{\partial k_{x}}-\frac{\partial L}{\partial
k_{x}}\frac{\partial G_{0}}{\partial X}\right].$ (63)

Summing Eqns. (61) and (63) we have an approximate expression for the Green
function, valid when there is a slow change of material parameters with
respect to position.

With our approximate expression for the Green function we can now compute the
spectral asymmetry via Eq. (58). To keep the expressions relatively simple, we
assume the spectral asymmetry computed from $G_{0}$ is zero. This assumption
is easily relaxed, and means that we only consider materials on the way from
$A$ to $A^{\prime}$ that–were they a homogeneous medium—would have a symmetric
eigenvalue spectrum around $\lambda_{n,k}=0$555In the more general case, where
the eigenvalues are not symmetrically distributed around the gap at
$\lambda\sim 0$ we have to consider _two_ contributions to the spectral
asymmetry.. The spectral asymmetry is then determined by the first order
correction to the Green function. Substituting the first order correction to
the Green function (63) into (58) yields

$\displaystyle\nu(k)$ $\displaystyle={\rm Im}\int_{-\infty}^{\infty}\frac{{\rm
d}\lambda}{2\pi}\int_{-\infty}^{\infty}\frac{{\rm
d}k_{x}}{2\pi}\int_{-\infty}^{\infty}\frac{{\rm d}X}{2}\,{\rm
tr}\,\left[G_{0}\frac{\partial G_{0}^{-1}}{\partial X}\frac{\partial
G_{0}}{\partial k_{x}}-G_{0}\frac{\partial G_{0}^{-1}}{\partial
k_{x}}\frac{\partial G_{0}}{\partial X}\right]$ $\displaystyle={\rm
Im}\int\frac{{\rm d}{}^{3}x}{8\pi^{2}}\,{\rm tr}\left[G_{0}\frac{\partial
G_{0}^{-1}}{\partial x_{1}}G_{0}\,\frac{\partial G_{0}^{-1}}{\partial
x_{2}}G_{0}\frac{\partial G_{0}^{-1}}{\partial x_{0}}-G_{0}\frac{\partial
G_{0}^{-1}}{\partial x_{2}}G_{0}\,\frac{\partial G_{0}^{-1}}{\partial
x_{1}}G_{0}\frac{\partial G_{0}^{-1}}{\partial x_{0}}\right]$

where we introduced the four coordinates
$(x_{0},x_{1},x_{2},x_{3})=(\lambda,X,k_{x},k)$., with the integration carried
out over the first three coordinates, ${\rm d}{}^{3}x={\rm d}x_{0}{\rm
d}x_{1}{\rm d}x_{2}$.

Figure 10: Illustration of the integrals (LABEL:eq:G1_spectral_asymmetry) and
(66). The spectral asymmetry is calculated as a three dimensional surface
integral (here shown as a shaded 2D plane), at fixed $k$. The number of modes
crossing $\lambda_{n,k}=0$ equals the difference in the spectral asymmetry
$\nu(+K)-\nu(-K)$ (the two shaded planes). Provided the Green function
vanishes for large $X$, $k_{x}$, and $\lambda$ the integral can be replaced
with the closed cubic surface integral shown above. As the only contributions
to the integral are the critical points of $A_{i}$ on the $\lambda=0$ surface
(occurring whenever the band gap closes, and shown as red dots here), we can
deform the closed surface into two parallel surfaces at fixed $X$, where the
material parameters no longer change.

Now the topology appears! Equation (LABEL:eq:G1_spectral_asymmetry) is
actually an integral of the Chern–Simons form (35) in disguise. To see this,
we define a connection $A_{i}$ with matrix components, as described in Sec. IV
and Appendix A,

$A_{i}={\rm i}G_{0}\frac{\partial G_{0}^{-1}}{\partial x_{i}}={\rm
i}H_{0}^{-1}\frac{\partial H_{0}}{\partial x_{i}}+{\rm
i}U_{0}^{\dagger}\frac{\partial U_{0}}{\partial x_{i}}.$ (65)

where we have written the matrix $G_{0}^{-1}$ in polar form,
$G_{0}^{-1}=H_{0}U_{0}$, where $H_{0}$ is Hermitian and $U_{0}$ unitary. Note
that the connection (65) has zero curvature (167), and that the final term in
Eq. (65) involving the unitary matrix is that found in Eq. (175) of Appendix
A, corresponding to a rotation of the basis vectors. It is this final term
that winds around the critical points of the connection $A_{i}$, analogous to
the winding of the Berry phase around critical points of a tangent vector on a
surface.

The difference between the spectral asymmetry between the fixed values $k=+K$
and $-K$ tells us the number of modes $N$ that cross $\lambda_{n,k}=0$. After
taking this difference, the open surface integral in Eq.
(LABEL:eq:G1_spectral_asymmetry) can be replaced with an integral over a
closed three dimensional surface see Fig. 10), and the number of modes $N$ can
be written as666To form the closed surface integral the integral of the
‘vector potential’ (65) must vanish over the bounding surfaces of constant
$X$, $k_{x}$, and $\lambda$. For large fixed $\lambda$ this is automatic. For
the large fixed $k_{x}$ boundary we require periodic boundary conditions or
diverging operator $L$ with increasing $k_{x}$. But for the large fixed $X$
boundary we require the operator $L$ to diverge as $|X|\to\infty$. Thus, e.g.
increasing $x$ we assume material $A$ is continuously changed into material
$A^{\prime}$, then remaining unchanged over a long distance, before $L$
ultimately diverges at infinity. (

$\displaystyle N=\nu(+K)-\nu(-K)$ $\displaystyle=\frac{\rm i}{24\pi^{2}}{\rm
Im}\oint{\rm d}{}^{3}x\,\epsilon_{ijk3}\,{\rm
tr}\left[A_{i}\,A_{j}\,A_{k}\right].$ $\displaystyle=\frac{1}{8\pi^{2}}{\rm
Im}\oint{\rm d}{}^{3}x\,\epsilon_{ijk3}\,{\rm tr}\left[\frac{\partial
A_{j}}{\partial x_{i}}A_{k}+\frac{2{\rm i}}{3}A_{i}\,A_{j}\,A_{k}\right]$ (66)

Note the sign of $N$ indicates the direction in which the modes cross
$\lambda_{n,k}=0$.

After comparison with Sec. IV we can see that Eq. (66) is the integral of the
Chern–Simons form over a closed three dimensional surface, divided by
$8\pi^{2}$. As described there, this boundary integral is equal to the four
dimensional bulk integral of the second Chern class (33)! However, due to the
form of the vector potential (65), the ‘curvature’ $\Omega_{ij}$ defined in
Eq. (32) appears to vanish identically everywhere, and once again we have an
integral that looks like it should be exactly zero!

It’s the same old story: the integral is recording the discrete critical
points of $A_{i}$, which occur when $G_{0}^{-1}=[L-{\rm i}\lambda]$ cannot be
inverted (i.e. both the coordinate $\lambda$ is zero, and one or more pairs of
eigenvalues $\lambda_{n,k}$ are zero). _This means that the number of
interface modes $N$ in our inhomogeneous system that pass through zero
eigenvalue is determined by the number of times the ‘gap’ in the eigenvalue
spectrum of $L(k_{x},X,k)$ closes._ This is a satisfying result: as we deform
material $A$ into material $A^{\prime}$ we must close the gap $N$ times in
order to have $N$ interface modes crossing $\lambda_{n,k}=0$.

So long as our integration volume encloses all of the critical points of
$A_{i}$ we won’t change the predicted number of interface modes, $N$.
Therefore—as illustrated in Fig. 10—we can replace the integral over the
constant $k$ surfaces in (66) with an equivalent one over constant $X$
surfaces,

$N=\frac{1}{24\pi^{2}}{\rm Im}\oint{\rm d}{}^{3}x\epsilon_{i1jk}\,{\rm
tr}\left[A_{i}A_{j}A_{k}\right]$ (67)

We now only need understand the dependence of the operator $L$ on the
wave–vector $\bm{k}$ in the two asymptotic homogeneous materials $A$ and
$A^{\prime}$, and no longer need to consider the interface. It is also clear
from the above argument that our integral over the coordinate $\lambda$ on
which the Green function depends is somewhat redundant. The critical points of
$A_{i}$ always occur in the plane of $\lambda=0$. Substituting the expression
(65) for $A_{i}$ into (67), we can perform the integral over $\lambda$ exactly
using the following result

$\int_{-\infty}^{\infty}{\rm d}\lambda\,{\rm tr}\left[\left(L-{\rm
i}\lambda\right)^{-2}\frac{\partial L}{\partial k_{x}}\left(L-{\rm
i}\lambda\right)^{-1}\frac{\partial L}{\partial k}\right]\\\
=2\pi\sum_{n<0,m\neq n}\frac{\langle n|\frac{\partial L}{\partial
k_{x}}|m\rangle\langle m|\frac{\partial L}{\partial k}|n\rangle-\langle
n|\frac{\partial L}{\partial k}|m\rangle\langle m|\frac{\partial L}{\partial
k_{x}}|n\rangle}{(\lambda_{n,k}-\lambda_{m,k})^{2}}$ (68)

which comes from an expansion of the operator $L$ in terms of its eigemodes
$L=\sum_{n}\lambda_{n,k}|n\rangle\langle n|$ and an application of Cauchy’s
integral formula. After applying (68) to our integral (67), the number of
interface modes equals the difference

$N=\bar{\nu}(X_{A^{\prime}})-\bar{\nu}(X_{A})$ (69)

where we have defined

$\bar{\nu}(x)=\frac{1}{2\pi{\rm i}}\sum_{n<0,m\neq n}\int_{S}{\rm
d}{}^{2}k\,\frac{\langle n|\frac{\partial L}{\partial k_{x}}|m\rangle\langle
m|\frac{\partial L}{\partial k}|n\rangle-\langle n|\frac{\partial L}{\partial
k}|m\rangle\langle m|\frac{\partial L}{\partial
k_{x}}|n\rangle}{(\lambda_{n,k}-\lambda_{m,k})^{2}}$ (70)

and e.g. $X_{A}$ indicates a position away from the inhomogeneity where the
material parameters are those of material $A$.

Comparison with our expression for the Berry curvature (40), and the first
Chern number (38) shows that $\bar{\nu}(x)$ is equal to the sum of all the
Chern numbers for the bands below $\lambda_{n,k}=0$,

$\bar{\nu}(x)=\sum_{n<0}{\rm Ch}_{1,n}(x)$ (71)

which will be an integer so long as the surface $S$ appearing in (70) can be
made closed (as shown in Fig. 8). The number of interface modes $N$ arising
from changing the material from $A$ to $A^{\prime}$ is thus equal to the
difference in Chern numbers between the two media, summed over all the bands
below the gap in the eigenvalue spectrum,

$N=\sum_{n<0}\left[{\rm Ch}_{1,n}(X_{A^{\prime}})-{\rm
Ch}_{1,n}(X_{A})\right].$ (72)

We now have the striking result mentioned at the beginning of this section.
The number of interface modes trapped between materials $A$ and $A^{\prime}$
is determined by the difference in their Chern numbers. These are computed
over a closed surface parameterized by the wave–vector, $\bm{k}$, and summed
over all branches of the dispersion relation below the gap in the spectrum
where there are the interface modes of interest.

So far in the tutorial we have gradually introduced the machinery of topology
in order to get to this amazing result, the origin of which is not often
discussed in metamaterials textbooks or papers. We now consider a few simple
examples.

### Example: Counting one mode

As our first example we give a simple one dimensional calculation of the
spectral asymmetry (58) in terms of the Green function, and relate it to a
winding number.

Suppose we have a mode that satisfies the linear dispersion relation $k=\alpha
k_{0}$. We can see immediately that exactly one mode crosses from negative
$k_{0}$ to positive $k_{0}$ as $k$ is increased from $-\infty$ to $+\infty$.
One choice of linear equation that gives such a dispersion relations is,

$-{\rm i}\alpha^{-1}\frac{\partial\phi}{\partial y}=k_{0}\phi.$ (73)

Comparing to the previous section we see that the linear operator is
$\hat{L}=-{\rm i}\alpha^{-1}\partial_{y}$, and the eigenvalue is the
wavenumber, $\lambda_{k}=k_{0}$.

We can now calculate the spectral asymmetry as a function of the wave–vector,
$k$. For a fixed value of $k$ the Green function is simple

$\left(\alpha^{-1}k-\lambda\right)G(k,\lambda)=1\to
G(k,\lambda)=\frac{1}{\alpha^{-1}k-\lambda}.$ (74)

According to Eq. (58), the spectral asymmetry equals the real part of the
integral of the Green function over purely imaginary values of $\lambda$. For
our simple Green function (74), this is simply the derivative of the phase of
$G$

${\rm Re}[G(k,{\rm i}\lambda)]={\rm Im}\frac{{\rm d}}{{\rm
d}\lambda}\log\left(\frac{1}{\alpha^{-1}k-{\rm i}\lambda}\right)=\frac{{\rm
d}}{{\rm d}\lambda}{\rm arg}[G(k,{\rm i}\lambda)]$ (75)

The integral of (75) over $\lambda$ is thus simply the change in the phase of
the Green function between the end points of the integral. The phase angle is
defined relative to the critical point in Eq. (75) that is at $k,\lambda=0$
(see Fig. 11), where the Green function diverges, and is exactly where our
mode crosses $\lambda_{k}=0$!

Integrating from $\lambda=-\infty$ to $+\infty$, this change of angle is
$\pm\pi$ depending on the sign of $\alpha^{-1}k$. Applying Eq. (58) we thus
find the spectral asymmetry equals

$\nu(k)={\rm Re}\int_{-\infty}^{\infty}\frac{{\rm d}\lambda}{2\pi}\,G({\rm
i}\lambda)=\int_{-\infty}^{\infty}\frac{{\rm d}\lambda}{2\pi}\,\frac{{\rm
d}}{{\rm d}\lambda}{\rm arg}[G(k,{\rm i}\lambda)]=\frac{1}{2}{\rm
sign}[\alpha^{-1}k]$ (76)

Assuming $\alpha>0$ and taking the difference between the spectral asymmetry
at fixed values $k=+K$ and $k=-K$, the number of modes equals the winding
number of the argument of the Green function

$\displaystyle N$ $\displaystyle=\nu(+K)-\nu(-K)$
$\displaystyle=\frac{1}{2\pi}\bigg{[}{\rm arg}[G(K,{\rm i}\infty)]-{\rm
arg}[G(K,-{\rm i}\infty)]+{\rm arg}[G(-K,{\rm i}\infty)]-{\rm arg}[G(-K,-{\rm
i}\infty)]\bigg{]}$ $\displaystyle=1$ (77)

This is a simple case of the winding number given in Sec. VI: it is the
topological invariant counting the critical points of the Green function,
which here occurs where the mode crosses $\lambda_{k}=0$.

Figure 11: Counting one mode: the first order equation (73) has wave–like
solutions with a dispersion relation $k=\alpha k_{0}=\alpha\omega/c$. Panel
(a) shows the dispersion relation for $\alpha>0$, indicating the spectral
asymmetry, which is $+1/2$ for positive $k$ and $-1/2$ for negative $k$. (b)
The spectral asymmetry $\nu(K)$ can be written as the change in the argument
($\Delta\theta$) of the Green function (74) as $\lambda$ varies from $-{\rm
i}\infty$ to $+{\rm i}\infty$, divided by $2\pi$. The difference
$\nu(K)-\nu(-K)$ thus equals the number of times the argument of the Green
function winds around the critical point at $\alpha^{-1}k+{\rm i}\lambda=0$.

### Example: A lattice of resonators

Figure 12: In some cases wave equations can be approximated as a discrete
system of coupled resonators. A periodic honeycomb lattice of resonators
(lattice constant $d$, lattice vectors $\bm{a}_{1}$ and $\bm{a}_{2}$) is drawn
in panel (a) where there are two resonators, ‘a’ and ‘b’ per unit cell,
separated by a distance $d/\sqrt{3}$. In the Haldane model each resonator is
coupled to both its three nearest neighbours (red lines) and its six next
nearest neighbours (green dashed lines). The corresponding reciprocal space is
sketched in panel (b), where the grey circles indicate multiples of the
reciprocal lattice vectors $\bm{b}_{1}$ and $\bm{b}_{2}$. The first Brillouin
zone is shaded in blue, with the dashed and solid boundaries of the same
colour connected by the addition of a reciprocal lattice vector. The six
corners of the Brillouin zone where the coupling $g(\bm{K})$ vanishes are
indicated as blue dots.

As a second example, we take an array of coupled resonators, as sketched in
Fig. 12a. These ‘resonators’ are a discrete approximation to a continuous
system. For instance, the amplitude of a single resonator can represent the
pressure in an acoustically resonant hole; the electric polarization of a
dielectric particle; the displacement of an elastic rod; or simply the
extension of a spring.

As we are dealing with a _discrete_ lattice of resonators, the continuum
theory given in Sec. VI does not obviously apply. Yet the reasoning can be
simply adapted. For instance we can expand the continuous wave in an
orthonormal basis of $N$ functions,
$|\psi\rangle=a_{0}|0\rangle+a_{1}|1\rangle+\dots+a_{N-1}|N-1\rangle$, with
the basis functions representing e.g. the modes of the lattice before a
perturbation is applied, or the modes of individual resonators in the tight
binding approximation [38]. Assuming the basis itself obeys
$\bm{\nabla}_{\bm{K}}\times\langle n|\bm{\nabla}_{\bm{K}}|m\rangle=0$, the
Berry curvature of a given mode $|\psi\rangle$ is given entirely in terms of
the expansion coefficients, $-{\rm
i}\langle\psi|\bm{\nabla}_{\bm{K}}|\psi\rangle=-{\rm
i}\bm{\nabla}_{\bm{K}}\times\sum_{n}a_{n}^{\star}\bm{\nabla}_{\bm{K}}a_{n}$.
This is the expression we would obtain for the Berry curvature of an $N$
component complex vector $|\psi\rangle=(a_{0},a_{1},\dots a_{N-1})^{\rm T}$ in
a discrete system.

In our simplified model we assume a tight binding approximation, where each
resonator only has one possible mode (i.e. frequencies are such that higher
order modes do not contribute). Due to the periodicity of the lattice, the
unit cell contains all the degrees of freedom. Therefore a unit cell
containing a single resonator has only one effective degree of freedom and
will therefore exhibit a dispersion relation $\omega(\bm{K})$ with only one
branch (band). As the theory given in Sec. VI depends on the closure of a
_gap_ between two or more bands, we consider a lattice with _two_ resonators
(labelled $a$ and $b$) per unit cell.

Labelling the amplitudes of the two resonators in each point in the lattice as
$a_{n,m}$ and $b_{n,m}$, the equations of motion can be written in the general
form

$\displaystyle\ddot{a}_{n,m}+\omega_{a}^{2}\,a_{n,m}$
$\displaystyle=\sum_{n^{\prime},m^{\prime}}\left[\alpha_{n^{\prime}-n,m^{\prime}-m}b_{n^{\prime},m^{\prime}}+\beta_{n^{\prime}-n,m^{\prime}-m}a_{n^{\prime},m^{\prime}}\right]$
$\displaystyle\ddot{b}_{n,m}+\omega_{b}^{2}\,b_{n,m}$
$\displaystyle=\sum_{n^{\prime},m^{\prime}}\left[\alpha_{n-n^{\prime},m-m^{\prime}}a_{n^{\prime},m^{\prime}}+\gamma_{n^{\prime}-n,m^{\prime}-m}b_{n^{\prime},m^{\prime}}\right]$
(78)

where the ‘$a$’ and ‘$b$’ resonant frequencies are $\omega_{a}$ and
$\omega_{b}$ respectively. The amplitude of the cross–coupling between
resonators is given by $\alpha_{n,m}$, and the coupling between like
resonators is given by $\beta_{n,m}$ and $\gamma_{n,m}$. It is assumed that
the self coupling between like resonators vanishes
$\gamma_{0,0}=\beta_{0,0}=0$, as these terms are equivalent to a modification
of the resonant frequencies $\omega_{a,b}$. From hereon we will work at a
fixed frequency $\omega$, where the coupling constants can take complex
values.

We now compute the first Chern number for an infinite periodic system of
resonators, writing the resonator amplitudes in accordance with Bloch’s
theorem, and taking a fixed frequency of oscillation $\omega$

$\displaystyle a_{n,m}=e^{{\rm i}[\bm{K}\cdot(n\bm{a}_{1}+m\bm{a}_{2})-\omega
t]}a$ $\displaystyle b_{n,m}=e^{{\rm
i}[\bm{K}\cdot(n\bm{a}_{1}+m\bm{a}_{2})-\omega t]}b$ (79)

where $\bm{K}$ is the Bloch vector, and $\bm{a}_{1}$ and $\bm{a}_{2}$ are the
real space lattice vectors (see Fig. 12). Substituting Eq. (79) into Eq. (78),
the infinite set equations of motion reduce to a set of two coupled linear
equations

$\displaystyle[\omega_{a}^{2}-f(\bm{K})]\,a-g(\bm{K})b$
$\displaystyle=\omega^{2}a$
$\displaystyle[\omega_{b}^{2}-h(\bm{K})]\,b-g^{\star}(\bm{K})a$
$\displaystyle=\omega^{2}b$ (80)

where we have defined the three Bloch–vector dependent coupling functions

$\displaystyle f(\bm{K})$ $\displaystyle=\sum_{n^{\prime},m^{\prime}\neq
0,0}\beta_{n^{\prime},m^{\prime}}{\rm e}^{{\rm
i}\bm{K}\cdot(n^{\prime}\bm{a}_{1}+m^{\prime}\bm{a}_{2})}$ $\displaystyle
g(\bm{K})$ $\displaystyle=\alpha_{0,0}+\sum_{n^{\prime},m^{\prime}\neq
0,0}\alpha_{n^{\prime},m^{\prime}}{\rm e}^{{\rm
i}\bm{K}\cdot(n^{\prime}\bm{a}_{1}+m^{\prime}\bm{a}_{2})}$ $\displaystyle
h(\bm{K})$ $\displaystyle=\sum_{n^{\prime},m^{\prime}\neq
0,0}\gamma_{n^{\prime},m^{\prime}}{\rm e}^{{\rm
i}\bm{K}\cdot(n^{\prime}\bm{a}_{1}+m^{\prime}\bm{a}_{2})}.$ (81)

Equation (80) is almost in the form required by the theory described in Sec.
VI. One wrinkle is that we do not yet satisfy the assumption that the
eigenvalues of the operator $\hat{L}$ are symmetrically distributed around
zero (the trace of $\hat{L}$ does not vanish). To avoid complicating the
discussion, we assume the two resonators in the unit cell are identical:
$\omega_{a}=\omega_{b}$ and $h(\bm{K})=f(\bm{K})$. Subtracting
$\omega_{a}^{2}$ from both sides of (80) and writing
$\lambda=\omega^{2}-\omega_{a}^{2}$ then gives us

$\left(\begin{matrix}-f(\bm{K})&-g(\bm{K})\\\
-g^{\star}(\bm{K})&f(\bm{K})\end{matrix}\right)\left(\begin{matrix}a\\\
b\end{matrix}\right)=\lambda\left(\begin{matrix}a\\\ b\end{matrix}\right)$
(82)

The operator $\hat{L}$ is now traceless and can be written in terms of Pauli
matrices, as in the example application of the Chern number given in Sec. V,

$\hat{L}=-f(\bm{K})\,\sigma_{z}-g_{1}(\bm{K})\,\sigma_{x}+g_{2}(\bm{K})\,\sigma_{y}$
(83)

where $f$ is a real function, and $g=g_{1}+{\rm i}g_{2}$. The Chern number for
the operator (83) records the same thing as in the example of Sec. V, namely
the number of times the vector

$\bm{n}=\frac{-g_{1}\bm{e}_{x}+g_{2}\bm{e}_{y}-f\bm{e}_{z}}{\sqrt{f^{2}+g_{1}^{2}+g_{2}^{2}}}$
(84)

covers the unit sphere as $\bm{K}$ is varied over the first Brillouin zone,
i.e. the winding of the Hamiltonian around the point $f=g=0$. Comparison with
the example of Sec. V shows that the two eigenstates of $\hat{L}$ are given by
Eq. (47) with eigenvalues
$\lambda=\omega^{2}-\omega_{a}^{2}=\pm[f^{2}+g_{1}^{2}+g_{2}^{2}]$ . As in
that example, the eigenstates depend only on the spherical angles $\theta$ and
$\phi$ of the vector $\bm{n}$, which are here identified as
$\cos(\theta)=-f/[f^{2}+g_{1}^{2}+g_{2}^{2}]^{1/2}$ and ${\rm exp}({\rm
i}\phi)=(-g_{1}+{\rm i}g_{2})/[g_{1}^{2}+g_{2}^{2}]^{1/2}$. The Berry
connection is also given in Sec. V, by Eq. (48).

From the example of Sec. V, we know that the points of vanishing cross
coupling between the two resonators in the unit cell, $g(\bm{K})=0$ are the
critical points of the Berry connection. These correspond to the North or
South pole of the unit sphere defined by (84), depending on the sign of $f$.
We therefore see that the Chern number depends in an important way on the
coupling functions between the same resonators in each unit cell, $f$ and $g$.
If, for example $f$ is always positive then $\bm{n}$ will only every explore
the lower half of the unit sphere and the Chern number will therefore always
be zero. Between such lattices of resonators there can never be the
non–trivial interface states predicted above.

Figure 13: Coupling functions and Berry curvature for the honeycomb lattice
shown in Fig. 12, for parameters $\beta=0.01$ and $\alpha=1$. The hexagon
indicated in all four plots shows the boundary of the first Brillouin zone.
Panel (a) shows a phase plot of the complex coupling $g(\bm{K})$ between ‘a’
and ‘b’ resonators (color indicates phase and saturation, magnitude). Panel
(b) shows the real valued coupling $f(\bm{K})$ between next–nearest–neighbour
resonators. Panel (c) shows the Berry curvature, evaluated within the first
Brillouin zone. Panel (d) indicates the ‘folding’ rules for mapping the first
Brillouin zone onto a torus, with the total Berry curvature around each corner
of the zone indicated.

To ensure that the Chern number does not vanish, we take an approach that is
close to the famous example known as the ‘Haldane model’ developed by F.
Duncan Haldane in the late 1980s [8]. We consider the special case of a
lattice with hexagonal symmetry and lattice constant $d$777Lattice vectors
$\bm{a}_{1}=\frac{d}{2}(\bm{e}_{x}+\sqrt{3}\bm{e}_{y})$, and
$\bm{a}_{2}=\frac{d}{2}(-\bm{e}_{x}+\sqrt{3}\bm{e}_{y})$. Taking two identical
resonators per unit cell, we construct a honeycomb lattice where each ‘ _a_ ’
resonator is separated by $d/\sqrt{3}$ from three nearest neighbour ‘ _b_ ’
resonators, as shown in Fig. 12a. Assuming only nearest neighbour coupling
between the $a$ and $b$ resonators,
$\alpha_{0,0}=\alpha_{1,0}=\alpha_{0,1}=\alpha\,{\rm exp}(-{\rm
i}K_{y}d/\sqrt{3})$ of equal strength $\alpha_{1,0}=\alpha_{0,1}=\alpha$, the
coupling function $g(\bm{K})$ reduces to

$\displaystyle g(\bm{K})$ $\displaystyle=\alpha\,{\rm e}^{-{\rm
i}K_{y}d/\sqrt{3}}\left[1+{\rm e}^{{\rm i}\bm{K}\cdot\bm{a}_{1}}+{\rm e}^{{\rm
i}\bm{K}\cdot\bm{a}_{2}}\right]$ $\displaystyle=\alpha\,{\rm e}^{-{\rm
i}K_{y}d/\sqrt{3}}\left[1+2\cos(K_{x}d/2){\rm e}^{\frac{{\rm
i}\sqrt{3}d}{2}K_{y}}\right]$ (85)

This quantity vanishes at the $6$ corner points on the boundary of the first
Brillouin zone (see Fig. 13a): when the $K_{x}$ component of the Bloch vector
equals $\pm 2\pi/3d$, or $\pm 4\pi/3d$, with the $K_{y}$ component equal to
$\pm 2\pi/\sqrt{3}d$, or $0$ respectively. These $6$ points can be grouped
into two lots of three, where one group can be obtained from the other via the
substitution $\bm{K}\to-\bm{K}$, equivalent to a complex conjugation of Eq.
(85). Being equivalent to a complex conjugation, the phase of $g(\bm{K})$
winds in an opposite sense around these two sets of points. Therefore, if
$f(\bm{K})=f(-\bm{K})$ (as it does if e.g. there is no coupling beyond nearest
neighbour, $f=0$), the winding number around these critical points will
cancel, yielding a Chern number of zero!

To ensure that the Chern number does not vanish, we introduce a complex
next–nearest neighbour coupling that breaks time reversal symmetry. We suppose
that each resonator couples to its six equivalents at a distance $a$ away with
strength $\beta_{1,0}=-\beta_{-1,0}={\rm i}\beta$,
$\beta_{0,1}=-\beta_{0,-1}=-{\rm i}\beta$, and
$\beta_{1,-1}=\beta_{-1,1}=-{\rm i}\beta$

$\displaystyle f(\bm{K})$ $\displaystyle=-{\rm i}\beta\left[-{\rm e}^{{\rm
i}\bm{K}\cdot\bm{a}_{1}}+{\rm e}^{-{\rm i}\bm{K}\cdot\bm{a}_{1}}+{\rm e}^{{\rm
i}\bm{K}\cdot\bm{a}_{2}}-{\rm e}^{-{\rm i}\bm{K}\cdot\bm{a}_{2}}+{\rm e}^{{\rm
i}\bm{K}\cdot(\bm{a}_{1}-\bm{a}_{2})}-{\rm e}^{-{\rm
i}\bm{K}\cdot(\bm{a}_{1}-\bm{a}_{2})}\right]$
$\displaystyle=2\beta\left[\sin(K_{x}d)-\sin((K_{x}+\sqrt{3}K_{y})d/2)-\sin((K_{x}-\sqrt{3}K_{y})d/2)\right]$
(86)

which is an odd function of $\bm{K}$, meaning—via the argument above—that the
total Berry curvature does not vanish.

Figure 13 shows the Berry curvature in the first Brillouin zone for the
coupling functions (85) and (86) (positive $\alpha$ and $\beta$), which now
corresponds to a Chern number of $+1$ . From the relationship between
interface states and Chern numbers derived above we can thus see that there
will always be an interface state trapped between a lattice with non–zero
next–nearest neighbour coupling (86), and one with $f(\bm{K})=0$, where the
Chern number vanishes.

### Example: Electromagnetic waves in a gyrotropic medium

As a final example we consider a continuous system where the eigenvalue
$\lambda$, appearing in Sec. VI, is a material parameter rather than
frequency. This example is based on the results in [33].

Take an electromagnetic material where the relative magnetic permeability
$\mu$ is a real scalar, and the permittivity $\bm{\epsilon}$ is a tensor of
the form

$\bm{\epsilon}=\left(\begin{matrix}\bm{\epsilon}_{\parallel}&\bm{0}\\\
\bm{0}&\epsilon_{\perp}\end{matrix}\right)$ (87)

where $\bm{\epsilon}_{\parallel}$ is a $2\times 2$ matrix representing the
anisotropic permittivity in the $x$–$y$ plane of propagation. Taking
propagation in the $x$–$y$ plane, and assuming TM polarization, where
$\bm{H}=(h/\eta_{0})\bm{e}_{z}$, where $\eta_{0}=\sqrt{\mu_{0}/\epsilon_{0}}$,
Maxwell’s equations take the form

$\displaystyle-{\rm i}\bm{\nabla}\times\bm{E}$ $\displaystyle=k_{0}\mu
h\bm{e}_{z}$ $\displaystyle{\rm i}\bm{\nabla}h\times\bm{e}_{z}$
$\displaystyle=k_{0}\bm{\epsilon}_{\parallel}\cdot\bm{E}.$ (88)

To simplify the discussion, consider media where the two diagonal elements of
$\bm{\epsilon}_{\parallel}$ are equal to each other and also equal to the
scalar permeability $\mu$. This allows us to write
$\bm{\epsilon}_{\parallel}=\bm{\epsilon}^{\prime}_{\parallel}+\bm{1}\lambda$
and $\mu=\lambda$, where $\bm{\epsilon}_{\parallel}^{\prime}$ has zeros on the
diagonal. When $\bm{\epsilon}_{\parallel}^{\prime}=0$, the material is
_impedance matched_ ($\mu=\epsilon=\lambda$), and the field behaves as if the
distance has been rescaled by a factor of $\lambda$. This is a simple example
of ‘transformation optics’ [40, 11]. When the impedance matching condition is
satisfied, there is a propagating wave with some wave number $|\bm{k}|$ for
every value of $\lambda$, with positive $\lambda$ corresponding to positive
index media, and negative $\lambda$ negative index media. There is thus no
‘gap’ in the $\lambda$ spectrum. Through introducing a _gyrotropy_ [31]
parameterized by $\alpha$

$\bm{\epsilon}_{\parallel}^{\prime}=\left(\begin{matrix}0&-{\rm i}\alpha(x)\\\
{\rm i}\alpha(x)&0\end{matrix}\right)$ (89)

we break time reversal symmetry and open up a ‘gap’, where a range of
$\lambda$ values correspond to materials where no wave can propagate. We now
use the mode counting argument given above to count the number of interface
modes that cross this spectral gap in an inhomoeneous material.

It is assumed that the gyrotropy, $\alpha$ varies with position, as discussed
in the theory given above. Maxwell’s equations can then be written as an
eigenvalue problem equivalent to that given in Eq. (54) at the beginning of
the theory of interface modes given in Sec. VI

$\left(\begin{matrix}0&{\rm i}\alpha(x_{1})&{\rm i}\partial_{2}\\\ -{\rm
i}\alpha(x_{1})&0&-{\rm i}\partial_{1}\\\ {\rm i}\partial_{2}&-{\rm
i}\partial_{1}&0\end{matrix}\right)\left(\begin{matrix}E_{x}\\\ E_{y}\\\
h\end{matrix}\right)=\lambda\left(\begin{matrix}E_{x}\\\ E_{y}\\\
h\end{matrix}\right)$ (90)

where we used the dimensionless coordinates $(x_{1},x_{2})=k_{0}(x,y)$. At
large $|x|$, where the material becomes homogeneous, the operator $\hat{L}$
can be written in Fourier space as

$L=\left(\begin{matrix}0&{\rm i}\alpha&-k_{2}\\\ -{\rm i}\alpha&0&k_{1}\\\
-k_{2}&k_{1}&0\end{matrix}\right)$ (91)

which has vanishing trace, as assumed in Sec. VI.

Figure 14: The material parameters $\lambda=\epsilon=\mu$, calculated as a
function of the wavenumber $k=(k_{1}^{2}+k_{2}^{2})^{1/2}$ and fixed gyrotropy
$\alpha=1$. There are three modes with $\lambda=\pm(k^{2}+\alpha^{2})^{1/2}$
and $\lambda=0$. A non–zero value of the gyrotropy opens up a ‘gap’ in the
spectrum: a range of $\lambda$ between positive and negative index materials,
where there are no propagating solutions.

Finding the eigenvalue of Eq. (91) is to ask the question “what is the
material parameter $\lambda$ for a medium with gyrotropy $\alpha$ and a wave
with wave–vector $\bm{k}=(k_{1},k_{2})$?”. From the eigenvectors and
eigenvalues of this operator we can now compute the Chern number of the lower
propagation branch shown in Fig. 14. The eigenvectors and eigenvalues are

$\lambda=\pm\sqrt{k^{2}+\alpha^{2}},\qquad|\psi_{\pm}\rangle=\frac{1}{\sqrt{2}k\sqrt{k^{2}+\alpha^{2}}}\left(\begin{matrix}-\lambda
k_{2}-{\rm i}\alpha k_{1}\\\ \lambda k_{1}-{\rm i}\alpha k_{2}\\\
k^{2}\end{matrix}\right)$ (92)

and

$\lambda=0,\qquad|\psi_{0}\rangle=\frac{1}{\sqrt{k^{2}+\alpha^{2}}}\left(\begin{matrix}k_{1}\\\
k_{2}\\\ {\rm i}\alpha\end{matrix}\right)$ (93)

where $k^{2}=k_{1}^{2}+k_{2}^{2}$888Unlike the theory presented in Sec. VI,
one of the eigenmodes (92–93) has zero eigenvalue. But if the eigenvalues are
all shifted _up_ by a small positive number $\eta$ then the zero eigenvalue is
removed from the spectrum. This makes the zeroth order spectral asymmetry
discussed previously (below Eq. (63)) non–zero, but leaves the _difference_ in
the spectral asymmetry (69) unaffected, and thus all the results derived above
hold..

We now count the interface modes that arise from the change in the gyrotropy
from $-$ve to $+$ve. The final result of Sec. VI says that the number of these
modes is given by the difference in the Chern numbers for the homogeneous
media either side of the interface. To calculate this we use the Berry
connection for the state $|\psi_{-}\rangle$, wrapping the infinite k–space
onto the sphere using the stereographic projection (see the schematic in Fig.
8a)

$\mathcal{K}=k_{x}+{\rm i}k_{y}=k\,{\rm e}^{{\rm
i}\phi}=\frac{\sin(\theta)\,{\rm e}^{{\rm
i}\phi}}{1-\cos(\theta)}=\cot(\theta/2)\,{\rm e}^{{\rm i}\phi}.$ (94)

so that on the sphere the state $|\psi_{-}\rangle$ takes the form

$|\psi_{-}\rangle=\frac{1}{\sqrt{2}\sqrt{\cot^{2}(\theta/2)+\alpha^{2}}}\left(\begin{matrix}\sqrt{\cot^{2}(\theta/2)+\alpha^{2}}\sin(\phi)-{\rm
i}\alpha\cos(\phi)\\\ -\sqrt{\cot^{2}(\theta/2)+\alpha^{2}}\cos(\phi)-{\rm
i}\alpha\sin(\phi)\\\ \cot(\theta/2)\end{matrix}\right)$ (95)

This state is undefined at both North ($\theta=0$) and South ($\theta=\pi$)
poles. However, at the North pole the vector is purely real, a defect that is
associated with zero Berry curvature (31). The Berry connection $A_{j}$
computed from (95) has a single relevant component

$A_{\phi}=-{\rm
i}\langle\psi_{-}|\frac{\partial}{\partial\phi}|\psi_{-}\rangle=\frac{\alpha}{\sqrt{\cot^{2}(\theta/2)+\alpha^{2}}}$
(96)

which vanishes at the North pole, and has a critical point at the South pole.
Integrating (96) around the critical point at the South pole $\theta=\pi$
yields a Chern number of $\pm 1$,

${\rm Ch}_{1}=-\frac{1}{2\pi}\int_{0}^{2\pi}A_{\phi}\,{\rm d}\phi=-{\rm
sign}[\alpha].$ (97)

Thus from Eq. (72) the number of interface modes $N$ supported at an interface
between media where the gyrotropy changes from positive to negative sign is

$N=(+1)-(-1)=2.$ (98)

This pair of interface modes can be understood as those with vanishing
tangential electric and magnetic field at the point where the gyrotropy
changes sign on the interface, i.e. the interface modes associated with a
perfect electric or magnetic conductor placed at this point in the graded
material. An examination of this pair of modes has revealed some of the
subtleties in the application of the theory of Sec. VI to continuous media.
The reader is encouraged to consult [42] and [43] for more details.

##  VII One–way propagation and the refractive index

The application of topology to predict interface modes reveals two remarkable
things. Firstly that the abstract mathematics of topology has a rather direct
and powerful application to the design of materials. Secondly that there exist
interface modes that can propagate in only _one_ direction (see, for example
Fig. 15). This is unusual behaviour for a wave, to say the least. It means
that whatever you put in the way of such an interface mode (a mirror,
chocolate, or an elephant), there is simply no possibility for it to reflect.
This is not entirely true for the example of Fig. 15, which has a fixed
polarization and can thus be reflected by any polarization converting object,
but let’s not let that discourage us.

Figure 15: One–way propagating electromagnetic interface waves, from a source
in a medium with graded gyrotropy, as described in the final example of Sec.
VI (simulated using program given in Appendix B). Panel (a) shows a phase plot
of the out of plane magnetic field for the profile of gyrotropy given in (b),
with constant diagonal part of the permittivity
$\epsilon=\lambda=2.5+0.001{\rm i}$. Away from the interface the value of the
gyrotropy exceeds the permittivity and we are in the gap indicated in Fig. 14.
A close examination of the interface mode shows that it exhibits a beating as
it propagates. This is the interference between the two interface modes
predicted in Eq. (98).

One problem with these topological arguments is that they do not give us an
explanation for _why_ there are such interface modes. All we have to go on is
an integer that took a long time to calculate. What is it about these
particular materials that force the wave to propagate in only one direction?
In this section we more fully explore the final example of Sec. VI, using the
concept of the refractive index rather than topology. This is based on the
findings of [44], and we’ll find that the refractive index concept gives us a
different, but complementary way to understand one–way propagation.

The starting point is the Berry connection (96) for a gyrotropic medium. As we
established in the previous section, the Chern number (97), ${\rm
Ch}_{1}=-{\rm sign}[\alpha]$ records the single critical point in the Berry
connection, which is at the South pole ($\theta=\pi$) of the sphere onto which
$k$–space has been stereographically projected. As shown in Fig. 8), the South
pole of the sphere is the origin of $k$–space. _So what is happening at this
critical point in the Berry connection?_

To answer this, let’s return to Maxwell’s Eqns. (88), setting $\mu=\lambda$.
We use the same set of material parameters as we did our earlier discussion,
$\bm{\epsilon}_{\parallel}=\lambda\bm{1}+{\rm i}\alpha\bm{e}_{z}\times$.
Substituting this in our earlier form of Maxwell’s equations (88), the
gradient of the out of plane magnetic field is governed by

$\bm{\nabla}h=-{\rm i}k_{0}(\lambda\bm{e}_{z}\times\bm{E}-{\rm
i}\alpha\bm{E})$ (99)

As we saw earlier in Sec. IV, the complex vectors
$\bm{e}_{\pm}=(\bm{e}_{x}\pm{\rm i}\bm{e}_{y})/\sqrt{2}$ are eigenvectors of
the cross product $\bm{e}_{z}\times\bm{e}_{\pm}=\mp{\rm i}\bm{e}_{\pm}$.
Therefore, taking the inner product of (99) with $\bm{e}_{+}$ simplifies the
equation to

$\bm{e}_{+}\cdot\bm{\nabla}h=\frac{1}{\sqrt{2}}\left(\frac{\partial
h}{\partial x}+{\rm i}\frac{\partial h}{\partial
y}\right)=k_{0}\left(\lambda-\alpha\right)\bm{e}_{+}\cdot\bm{E}$ (100)

and with $\bm{e}_{-}$ it simplifies to

$\bm{e}_{-}\cdot\bm{\nabla}h=\frac{1}{\sqrt{2}}\left(\frac{\partial
h}{\partial x}-{\rm i}\frac{\partial h}{\partial
y}\right)=-k_{0}\left(\lambda+\alpha\right)\bm{e}_{-}\cdot\bm{E}$ (101)

Equations (100) and (101) are important. The critical point of the Berry
connection is at the point $k=0$, which from the dispersion plot given in Fig.
14, corresponds to the material parameters $\lambda=\pm\alpha$ (depending on
whether we are computing the Chern number of the upper, or lower band of
propagation, respectively). From Eqns. (100) and (101)) we can see that at
these critical points the magnetic field obeys the equation

$\frac{\partial h}{\partial x}\pm{\rm i}\frac{\partial h}{\partial
y}=0\qquad(\lambda=\pm\alpha)$ (102)

These are the _Cauchy–Riemann_ equations from complex analysis (see e.g.
[45])! These are fulfilled by analytic functions of either
$\mathcal{Z}^{\star}=x-{\rm i}y$, in the case $\lambda=\alpha$, or
$\mathcal{Z}=x+{\rm i}y$, when $\lambda=-\alpha$. Therefore the critical
points of the Berry connection in a gyrotropic medium—which are, of course the
reason the Chern number is non–zero—correspond to _those points where the wave
is an analytic function of position_. At these points the wave depends solely
on either the complex number $\mathcal{Z}$, or on $\mathcal{Z}^{\star}$,
depending on the band of interest and the sign of the gyrotropy.

Figure 16: Difference between (a) a random superposition of plane waves of
wavenumber $k_{0}$, and (b) an analytic function of $\mathcal{Z}=x+{\rm i}y$.
Arrows indicate the direction in which the phase increases around the zeros.
In panel (a) we plot the complex function obtained through adding together $8$
plane waves of random complex amplitude, propagating at angles
$\\{0,\pi/4,\pi/2,3\pi/4,\dots\\}$. In panel (b) we plot the function
$\Pi_{n}(z-z_{n})$ for $8$ randomly generated complex numbers $z_{n}$. While
the sum of waves generates zeros around which the phase circulates in either
sense, the analytic function always exhibits circulation in an anti–clockwise
sense.

The physical significance of the wave becoming an analytic function is clear
if we consider a Taylor expansion of the out of plane magnetic field, $h$
around some point $\mathcal{Z}_{0}=x_{0}+{\rm i}y_{0}$ in the plane. Assuming
$h$ is a function of $\mathcal{Z}$, and expressing the complex number in terms
of polar coordinates $(r,\theta)$ centred at the point of expansion

$h(x+{\rm
i}y)=\sum_{n=0}^{\infty}h_{n}\,(\mathcal{Z}-\mathcal{Z}_{0})^{n}=\sum_{n=0}^{\infty}h_{n}\,r^{n}\,{\rm
e}^{{\rm i}n\theta}.$ (103)

This expansion in powers of ${\rm exp}({\rm i}\theta)$ is equivalent to
expanding the wave in terms of its component angular momenta. Noting that the
terms in the series each evolve in time as ${\rm exp}({\rm i}(n\theta-\omega
t))$, we see that each term rotates with a fixed angular velocity
$\dot{\theta}=\omega/n$. As the field must be everywhere finite (assuming the
material is simply connected), $n$ is always positive in the series (103).
Therefore _a wave that is given as an analytic function of position rotates in
only one sense_ ; anti–clockwise in the case of Eq. (103), and as shown in
Fig. 16b.

To emphasize the point, compare this to the expansion of a generic function of
$x$ and $y$,

$h(x,y)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}h_{n,m}x^{n}y^{m}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}h_{n,m}r^{n+m}\left(\frac{{\rm
e}^{{\rm i}\theta}+{\rm e}^{-{\rm i}\theta}}{2}\right)^{n}\left(\frac{{\rm
e}^{{\rm i}\theta}-{\rm e}^{-{\rm i}\theta}}{2{\rm i}}\right)^{m}$ (104)

which—as well as containing two summation indices rather than one—contains
both positive and negative powers of ${\rm exp}({\rm i}\theta)$, meaning that
there are component waves that can rotate in both senses around the origin.
This is illustrated in Fig. 16a. _The critical point of the Berry connection
calculated in Eq. ( 96) therefore corresponds to a set of material parameters
where the wave can only circulate one way._

###  VII.1 Critical points and the refractive index

We have just established that in a gyrotropic medium, the points where
$\lambda=\pm\alpha$ are where the wave behaves as an analytic function of
position. As the length of the wave–vector also vanishes at this point, $k=0$,
it is reminiscent of a point of vanishing refractive index. Indeed, the
behaviour of the field can be connected to the study of wave propagation in
anisotropic materials, and the critical points of the Berry connection can be
understood as an unusual kind of point of vanishing refractive index. We shall
show that this finding can be used as a shortcut to materials where there are
one–way interface states.

Figure 17: Refractive index surfaces in planar media: (a) Isotropic medium
with equal eigenvalues. The dispersion surface closes to a point as the
eigenvalues are reduced to zero. In panel (b) we have an anisotropic medium
with eigenvectors $\bm{e}_{1}=\bm{e}_{x}$ and $\bm{e}_{2}=\bm{e}_{y}$, and
positive eigenvalues. The dispersion surface now forms an ellipse, with
eccentricity approaching unity as one of the eigenvalues approaches zero. In
panel (c) the medium is hyperbolic with the same eigenvectors as (b). The
dispersion surface is now open, with a range of angles where propagation is
not allowed. Finally, panel (d) shows the dispersion surface for complex
eigenvectors $\bm{e}_{1}=(2\bm{e}_{x}+{\rm i}\bm{e}_{y})/\sqrt{5}$ and
$\bm{e}_{2}=(\bm{e}_{x}+2{\rm i}\bm{e}_{y})/\sqrt{5}$. Unlike the case of real
eigenvectors, the dispersion surface closes to a point rather than a line as
only _one_ of the eigenvalues approaches zero.

To simplify the discussion we consider the propagation of a transverse
magnetic field $\bm{H}=H\bm{e}_{z}$ in the $x$–$y$ plane of a homogeneous
non–magnetic ($\mu=1$) material. Combining the two Maxwell equations given by
Eq. (88), the electric field can be eliminated, leaving a second order
equation for the out of plane magnetic field amplitude $h$

$\bm{\nabla}\times\left(\bm{\epsilon}^{-1}\cdot\bm{\nabla}h\times\bm{e}_{z}\right)=k_{0}^{2}h\bm{e}_{z}.$
(105)

As the material is homogeneous, we can write the magnetic field in the form
$h=\bm{e}_{z}\,h_{0}\,{\rm exp}({\rm i}\,k\,\bm{n}\cdot\bm{x})$, where
$\bm{n}(\theta)=\cos(\theta)\bm{e}_{x}+\sin(\theta)\bm{e}_{y}$ and
$k=(k_{x}^{2}+k_{y}^{2})^{1/2}$, thus eliminating the derivatives from (105).
Assuming a material of the same form as in Eq. (87), where the only
off–diagonal elements are $\epsilon_{xy}$ and $\epsilon_{yx}$, Eq. (105)
becomes an equation for the $\theta$ dependent refractive index ${\rm
n}=k/k_{0}$

$(\bm{n}\times\bm{e}_{z})\cdot\bm{\epsilon}^{-1}_{\parallel}\cdot(\bm{n}\times\bm{e}_{z})=\left(\frac{k_{0}}{k}\right)^{2}=\frac{1}{{\rm
n}(\theta)^{2}}$ (106)

This equation defines the refractive index as a function of the propagation
angle $\theta$ in the $x$–$y$ plane. Eq. (106) is a special case of the
defining equation for the _refractive index ellipsoid_ , used in the optics of
three dimensional crystals [46].

It is illustrative to re–write the dispersion relation (106) in terms of the
eigenvalues, $\epsilon_{i}$ and eigenvectors, $\bm{e}_{i}$ of the in–plane
permittivity, obeying
$\bm{\epsilon}_{\parallel}\cdot\bm{e}_{i}=\lambda_{i}\bm{e}_{i}$. In terms of
these quantities, the inverse of the in–plane permittivity is given by,

$\bm{\epsilon}_{\parallel}^{-1}=\frac{1}{\epsilon_{1}}\bm{e}_{1}\otimes\bm{e}_{1}^{\star}+\frac{1}{\epsilon_{2}}\bm{e}_{2}\otimes\bm{e}_{2}^{\star}.$
(107)

where we’ve assumed a Hermitian (and hence lossless) permittivity tensor.
Substituting (107) into the dispersion relation (106), the angle dependent
refractive index can then be written as

${\rm
n}(\theta)=\sqrt{\frac{\epsilon_{1}\epsilon_{2}}{\epsilon_{2}\,|(\bm{e}_{1}\times\bm{e}_{z})\cdot\bm{n}|^{2}+\epsilon_{1}\,|(\bm{e}_{2}\times\bm{e}_{z})\cdot\bm{n}|^{2}}},$
(108)

where we have taken the positive root (although in some important cases the
negative root should be taken [10]).

Take a moment to dwell on the dependence of the refractive index (108) on the
permittivity tensor. As the permittivity is Hermitian, the eigenvectors are
orthonormal $\bm{e}_{i}^{\star}\cdot\bm{e}_{j}=\delta_{ij}$, and form a
complete set
$\bm{1}=\bm{e}_{1}^{\star}\otimes\bm{e}_{1}+\bm{e}_{2}^{\star}\otimes\bm{e}_{2}$.
Therefore, when the two eigenvalues are equal
$\epsilon_{1}=\epsilon_{2}=\epsilon$, the denominator on the right of (108)
simply equals $\epsilon$. Such a medium is an isotropic dielectric in the
plane of propagation, and the refractive index reduces to the textbook
expression, ${\rm n}(\theta)=\sqrt{\epsilon}$, shown as the dashed circle in
Fig. 17a. As shown in the figure, when $\epsilon\to 0$, this dispersion circle
closes to a point and the refractive index vanishes.

Meanwhile, when the two eigenvalues differ $\epsilon_{2}>\epsilon_{1}$ and the
eigenvectors are real, the refractive index varies between its largest value
$\sqrt{\epsilon_{2}}$ (propagation along $\bm{e}_{1}$) and its smallest value
$\sqrt{\epsilon_{1}}$ (propagation along $\bm{e}_{2}$). The angle dependence
of the refractive index ${\rm n}(\theta)$ now either traces out an ellipse
(Fig. 17b), when $\epsilon_{1}$ and $\epsilon_{2}$ are both positive, or a
hyperbola (Fig. 17c), when $\epsilon_{1}$ and $\epsilon_{2}$ have different
signs, constituting a hyperbolic material999See [23] for details about these
fascinating materials.. At the transition between elliptical and hyperbolic
dispersion, the smallest eigenvalue passes through zero $\epsilon_{1}\to 0$.
In this case the eccentricity of the dispersion ellipse tends to unity, and
the ellipse is compressed into a line, as shown in Fig. 17b. Again _the
refractive index vanishes, but now this only occurs for one direction of
propagation_. For instance taking $\bm{e}_{1}=\bm{e}_{x}$ and
$\bm{e}_{2}=\bm{e}_{y}$ we have,

${\rm
n}(\theta)=\sqrt{\frac{\epsilon_{1}\epsilon_{2}}{\epsilon_{2}\sin^{2}(\theta)+\epsilon_{1}\cos^{2}(\theta)}}=\begin{cases}\sqrt{\epsilon_{2}}\to
0&\;(\text{Propagation along }$x$)\\\ \sqrt{\epsilon_{1}}\neq
0&\;(\text{Propagation along }$y$).\end{cases}$ (109)

We can therefore see that setting one of the eigenvalues of the in–plane
permittivity to zero makes the refractive index in the direction perpendicular
to the corresponding eigenvector vanish. In this way it is possible to have
zero refractive index for only one direction of propagation.

The situation becomes more interesting when the eigenvectors $\bm{e}_{i}$ are
complex and the eigenvalues are positive. Now the propagation vector $\bm{n}$,
appearing in the denominator of Eq. (108), can never be completely parallel
(or orthogonal) to either of the complex vectors, $\bm{e}_{1}\times\bm{e}_{z}$
or $\bm{e}_{2}\times\bm{e}_{z}$. As a consequence the denominator can _never_
vanish, even if one of the eigenvalues are zero! Therefore, if we let the
smallest eigenvalue $\epsilon_{1}$ alone tend to zero, the numerator of Eq.
(108) is zero, making the refractive index ${\rm n}(\theta)$ vanish for all
directions of propagation $\theta$! As shown in Fig. 17d, in this limit the
dispersion ellipse closes to a point, like that of an isotropic zero index
medium, where the permittivity tensor as a whole vanishes. However, the
behaviour is more subtle now.

If we return to the defining equation for the refractive index (106) and
multiply through by $k$, we have

$(\bm{k}\times\bm{e}_{z})\cdot\left(\frac{1}{\epsilon_{1}}\bm{e}_{1}\otimes\bm{e}_{1}^{\star}+\frac{1}{\epsilon_{2}}\bm{e}_{2}\otimes\bm{e}_{2}^{\star}\right)\cdot(\bm{k}\times\bm{e}_{z})=k_{0}^{2}.$
(110)

Given that the magnitude of the wave number $k_{0}$ is fixed, as
$\epsilon_{1}\to 0$ the first term in the brackets of Eq. (110) dominates and
we are left with the condition

$\bm{e}_{1}\cdot(\bm{k}\times\bm{e}_{z})\to 0$ (111)

an equation that could be equivalently written as
$(\bm{e}_{z}\times\bm{e}_{1})\cdot\bm{\nabla}h=0$, i.e. the refractive index
is zero in the $\bm{e}_{z}\times\bm{e}_{1}^{\star}$ direction. For real
$\bm{e}_{1}$, this indicates the squashing of the dispersion ellipse into a
line, as shown in Fig. 17. When the first eigenvector is a complex vector e.g.
$\bm{e}_{1}=\bm{e}_{+}=(\bm{e}_{x}+{\rm i}\bm{e}_{y})/\sqrt{2}$, the
refractive index is zero in a complex direction, and our condition reduces to
$\partial_{x}h+{\rm i}\partial_{y}h=0$, which are the Cauchy–Riemann
conditions (102) found earlier101010Other choices of complex vectors also
yield the Cauchy–Riemann conditions, but with the coordinates $x$ and $y$
rescaled..

Therefore, even though the dispersion surface in the limit $\epsilon_{1}\to 0$
shown in Fig. 17d _looks_ like that of an isotropic medium where the
refractive index vanishes, the behaviour of the wave is quite different.
Rather than uniformly stretch the wavelength to infinity as would happen in an
isotropic medium, instead the wave is forced to propagate with only one sense
of circulation. Although analytic functions diverge at infinity and are
therefore inadmissible in a bulk material, this behaviour is revealed at a
boundary with another material (see Sec. VIII), where e.g. an interface state
${\rm exp}(-{\rm i}k(x-{\rm i}y))$ ($y>0$) would be an allowed solution,
whereas the counter propagating wave ${\rm exp}({\rm i}k(x+{\rm i}y))$ would
not. This unusual kind of zero index material exhibits one–way propagation
where the wave obeys the Cauchy–Riemann conditions. This is what the defect in
the Berry connection (96), and the non–zero Chern number (97) is recording.

##  VIII Applications

We now give applications in three different wave physics regimes where we can
enforce one–way propagation through simply demanding that the refractive index
is zero in a complex direction.

###  VIII.1 General electromagnetic media

We can use this idea of a ‘vanishing index in a complex direction’ to extend
the discussion of Sec. VII from gyrotropic media, to general electromagnetic
materials. In an arbitrary material, in the absence of any sources, and at a
fixed frequency $\omega$, Maxwell’s equations are given by

$\displaystyle\bm{\nabla}\times\bm{E}$ $\displaystyle={\rm i}\omega\bm{B}$
$\displaystyle\bm{\nabla}\times\bm{H}$ $\displaystyle=-{\rm i}\omega\bm{D}.$
(112)

We take a general lossless linear material, where the constitutive relations
are given by

$\displaystyle\bm{D}$
$\displaystyle=\epsilon_{0}[\bm{\epsilon}\cdot\bm{E}+\bm{\xi}\cdot\eta_{0}\bm{H}]$
$\displaystyle\bm{B}$
$\displaystyle=\mu_{0}[\bm{\mu}\cdot\bm{H}+\bm{\xi}^{\dagger}\cdot\eta_{0}^{-1}\bm{E}]$
(113)

where the three $3\times 3$ tensors $\bm{\epsilon}$ and $\bm{\mu}$ are
Hermitian, and the bi–anisotropy tensor $\bm{\xi}$ is arbitrary. The Hermitian
property of the permittivity and permeability, and the appearance of
$\bm{\xi}$ and $\bm{\xi}^{\dagger}$ ensure that the material does not absorb
wave energy [49].

A compact and useful way to write Maxwell’s equations (112) is in the form of
a six–vector $(\bm{E},\bm{h})^{\rm T}$, as follows

$\left(\begin{matrix}\bm{0}&{\rm i}\bm{\nabla}\times\\\ -{\rm
i}\bm{\nabla}\times&\bm{0}\end{matrix}\right)\left(\begin{matrix}\bm{E}\\\
\bm{h}\end{matrix}\right)=k_{0}\left(\begin{matrix}\bm{\epsilon}&\bm{\xi}\\\
\bm{\xi}^{\dagger}&\bm{\mu}\end{matrix}\right)\left(\begin{matrix}\bm{E}\\\
\bm{h}\end{matrix}\right)$ (114)

where, as in the previous sections $\bm{h}=\eta_{0}\bm{H}$. As discussed in
[50, 33, 51, 52], equation (114) has a great deal in common with the Dirac
equation [53], where the operator on the left hand side is analogous to the
operator $\bm{\alpha}\cdot\hat{\bm{p}}$, and the right hand side matrix
contains terms analogous to the mass, energy, and an external gauge field.

We now restrict propagation to the $x$–$y$ plane. With this assumption, the
curl of the fields can be written in terms of derivatives of the in–plane
field components e.g. $\bm{E}_{\parallel}$, and the out of plane ones e.g.
$E_{z}=E$. For example,
$\bm{\nabla}\times\bm{E}=\bm{\nabla}\times\bm{E}_{\parallel}+\bm{\nabla}E\times\bm{e}_{z}$.
With this assumption, the in–plane part of the left hand side of (114) depends
only on the gradient of the out of plane field components $E$ and $h$. To
isolate these parts of the field, we take an inner product of Eq. (114) with
the six–vector

$V=\left(\begin{matrix}\bm{v}_{E}\\\ 0\\\ \bm{v}_{H}\\\ 0\end{matrix}\right)$
(115)

where $\bm{v}_{E,H}$ are two arbitrary vectors lying in the $x$–$y$
plane111111Recall that the topological arguments described in Secs. V and VI
are only applicable to planar systems. This is also true for the arguments
based on the refractive index given here.. This inner product yields the
single scalar equation for the derivatives of the out–of–plane field

${\rm
i}\left[\bm{v}_{E}\cdot\bm{\nabla}h\times\bm{e}_{z}-\bm{v}_{H}\cdot\bm{\nabla}E\times\bm{e}_{z}\right]\\\
=k_{0}\left[(\bm{v}_{E}\cdot\bm{\epsilon}+\bm{v}_{H}\cdot\bm{\xi}^{\dagger})\cdot\bm{E}+\left(\bm{v}_{H}\cdot\bm{\mu}+\bm{v}_{E}\cdot\bm{\xi}\right)\cdot\bm{h}\right].$
(116)

This equation can be reduced to a simple gradient of a combination of out of
plane field components if the two vectors $\bm{v}_{E,H}$ are chosen as
parallel. We write $\bm{v}_{E,H}=\alpha_{E,H}\bm{e}\times\bm{e}_{z}$ where
$\alpha_{E,H}$ are scalar quantities, $\bm{e}$ is a unit vector
($\bm{e}\cdot\bm{e}^{\star}=1$). We then have,

${\rm
i}\bm{e}\cdot\bm{\nabla}\left[\alpha_{E}h-\alpha_{H}E\right]=k_{0}(\bm{e}\times\bm{e}_{z})\cdot\left[\left(\alpha_{E}\bm{\epsilon}+\alpha_{H}\bm{\xi}^{\dagger}\right)\cdot\bm{E}+\left(\alpha_{H}\bm{\mu}+\alpha_{E}\bm{\xi}\right)\cdot\bm{h}\right].$
(117)

The left hand side of this equation is a generalization of Eqns. (100) and
(101) discussed in Sec. VII. Setting the right hand side of (117) to zero
picks out a set of material parameters such that the refractive index is zero
in the direction $\bm{e}$. In order for this to hold, we must impose _two_
conditions on the material tensors

$\displaystyle(\bm{e}\times\bm{e}_{z})\cdot(\alpha_{E}\bm{\epsilon}+\alpha_{H}\bm{\xi}^{\dagger})$
$\displaystyle=0$
$\displaystyle(\bm{e}\times\bm{e}_{z})\cdot(\alpha_{H}\bm{\mu}+\alpha_{E}\bm{\xi})$
$\displaystyle=0.$ (118)

For the particular case of $\bm{e}=\bm{e}_{+}=(\bm{e}_{x}+{\rm
i}\bm{e}_{y})/\sqrt{2}$, Eq. (118) provides a large family of material
parameters where the out of plane field component $\alpha_{E}h-\alpha_{H}E$
behaves as an analytic function of position; thus circulating in only one
sense and exhibiting unidirectional interface states. Note that in the
particular case where we take $\alpha_{H}=0$ and $\bm{\xi}=0$, Eq. (118)
reduces to the zero–index condition for gyrotropic media
$\bm{e}_{+}\times\bm{e}_{z}\cdot\bm{\epsilon}=0$ identified above in Eqns.
(99–101).

Figure 18: Free space exhibits a similar dispersion relation to that of a
gyrotropic medium, see e.g. Fig. 14, and in some cases shows ‘zero index in a
complex direction’. (a) For a fixed out of plane propagation constant $k_{z}$
the frequency is $k_{0}=\pm(k^{2}+k_{z}^{2})^{1/2}$, with a ‘gap’ in the
dispersion relation where $k_{0}<|k_{z}|$. The wave becomes an analytic
function of position when $k_{0}=|k_{z}|$ (dashed lines). (b) An imagined
experiment where an oscillating line current $j_{z}=j_{0}{\rm exp}({\rm
ik_{z}z})$ excites electromagnetic waves close to a planar dielectric, a
distance $d$ away, where one–way propagation is evident when $k_{0}<|k_{z}|$.

#### Example: The Cauchy–Riemann conditions in free space

There is an interesting special case of conditions (118), where it can be
applied to electromagnetic near fields propagating in _free space_. At first
this seems counter intuitive: surely we need a material if the wave is to
exhibit something as strange as complex analyticity! But suppose we consider a
free space electromagnetic wave propagating out of the plane, at fixed
wave–vector $k_{z}$. The dispersion relation of such as mode is given by

$k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k_{0}^{2}\to k_{0}=\pm\sqrt{k^{2}+k_{z}^{2}}$
(119)

which is of the same form as for a gyrotropic medium (92), and for a
relativistic particle: we have two bands of propagation, separated by a ‘gap’
$\Delta k_{0}=2k_{z}$ equivalent to a mass, and due to the propagation
constant $k_{z}$, as shown in Fig. 18. The system also has exactly the same
behaviour under time reversal. Just as for the gyrotropy constant $\alpha$, if
we reverse the direction of time the out of plane wave–vector $k_{z}$ changes
sign, although such a sign change does not affect the dispersion relation
(119). As we shall see, in a particular polarization basis, out of plane
propagation is completely equivalent to gyrotropy.

As above, consider a system that is translationally invariant along the $z$
axis. Instead of taking the field as uniform in $z$, we assume propagation
with wave vector component $k_{z}$. Separating out the in–plane derivatives as
$\bm{\nabla}=\bm{\nabla}_{\parallel}+\bm{e}_{z}\partial_{z}$, this modifies
Maxwell’s equations (112) to

$\displaystyle\bm{\nabla}_{\parallel}\times\bm{E}+{\rm
i}k_{z}\bm{e}_{z}\times\bm{E}$ $\displaystyle={\rm i}\omega\bm{B}$
$\displaystyle\bm{\nabla}_{\parallel}\times\bm{H}+{\rm
i}k_{z}\bm{e}_{z}\times\bm{H}$ $\displaystyle=-{\rm i}\omega\bm{D}$ (120)

Comparison with the constitutive relations (113) we can see that out of plane
propagation is equivalent to adding an anti–symmetric contribution to the
bi–anisotropy tensor $\Delta\bm{\xi}=(k_{z}/k_{0})\,\bm{e}_{z}\times$. This
effective contribution to the bi–anisotropy is what allows us to fulfil the
zero index condition (118), even in free space.

Figure 19: Phase plot of the electric $E$, magnetic $h$, and the two ‘circular
polarizations’ $E\pm{\rm i}h$ for an oscillating line source of wave number
$k_{z}$ in front of a metal with $\epsilon=-2$ (see Fig. 18b). As the
wavenumber approaches the free space wave number $k_{0}/k_{z}\to 1$, the
system approaches the ‘zero index’ point shown in Fig. 18a, and the field
components $E\pm{\rm i}h$ behave as analytic functions of $\mathcal{Z}$ and
$\mathcal{Z}^{\star}$.

Taking free space $\bm{\epsilon}=\bm{\mu}=\bm{1}_{3}$, and the effective
bi–anisotropy $\bm{\xi}=(k_{z}/k_{0})\,\bm{e}_{z}\times$, the zero index
conditions become

$(\bm{e}\times\bm{e}_{z})\cdot\left(\alpha_{E}\bm{\epsilon}+\alpha_{H}\bm{\xi}^{\dagger}\right)=\left(\begin{matrix}e_{y}&-e_{x}&0\end{matrix}\right)\left(\begin{matrix}\alpha_{E}&\alpha_{H}\left(\frac{k_{z}}{k_{0}}\right)&0\\\
-\alpha_{H}\left(\frac{k_{z}}{k_{0}}\right)&\alpha_{E}&0\\\
0&0&\alpha_{E}\end{matrix}\right)=0$ (121)

and