arX

iv:c

ond-

mat

/060

2501

v3 [

cond

-mat

.mes

-hal

l] 1

8 A

pr 2

006

Analogs of quantum Hall effect edge states in photonic crystals

S. Raghu∗ and F. D. M. HaldaneDepartment of Physics, Princeton University, Princeton NJ 08544-0708

(Dated: February 25, 2006)

“Photonic crystals” built with time-reversal-symmetry-breaking Faraday-effect media can exhibit“chiral” edge modes that propagate unidirectionally along boundaries across which the Faradayaxis reverses. These modes are precise analogs of the electronic edge states of quantum Hall effect(QHE) systems, and are also immune to backscattering and localization by disorder. The “Berrycurvature” of the photonic bands plays a role analogous to that of the magnetic field in the QHE.Explicit calculations demonstrating the existence of such unidirectionally-propagating photonic edgestates are presented.

I. INTRODUCTION

The control of the flow of light using photonic band-gap(PBG) materials has received considerable attention overthe past decade1. Moreover, the potential for using arti-ficially structured “metamaterials”, such as the recentlydiscovered “left-handed media”2, has shown considerabletechnological promise. In the past, significant progresshas been achieved in the field of photonics by making useof analogies with electronic systems. For instance, thetypical PBG material, a system with a spatially vary-ing and periodic dielectric function, was motivated bythe well known physics of electronic Bloch states; thedielectric scattering of light in periodic media presentsthe same formal solutions as those for the scattering ofelectrons in periodic potentials.

Previous photonic bandstructure calculations have fo-cused on the frequency dispersion of the photon bands;it has been usually been assumed that a knowledge ofthe spectrum alone represents a complete understandingof dynamics of the system. A primary goal of such cal-culations has been the quest for a PBG material with acomplete bandgap throughout the Brillouin zone in somefrequency range, which would prevent the transmissionof light with frequency in the range of the Band gap.Both two and three dimensional bandstructures possess-ing these properties have now been discovered3.

Recently, however, in the study of electronic systems, ithas become apparent that, even in the absence of interac-tion effects, the dispersion relations of the energy bandsdo not fully characterize the semiclassical dynamics ofwavepackets, unless both spatial inversion symmetry andtime-reversal symmetry are unbroken4. The additionalinformation, which is not obtainable from knowledge ofthe energy bands ǫn(k) alone, is the variation of the Berrycurvature5 Fab

n (k) = ǫabcΩnc(k), which is an antisym-metric tensor in k-space, where Ωn(k) is analogous to a“magnetic field” (flux density) in k-space. The “Berrycurvature” in k-space is related to the Berry phase6 inthe same way that the Bohm-Aharonov phase of an elec-tronic wavepacket is related to the magnetic flux densityin real space.

While the uniform propagation of wavepackets in per-fectly translationally-invariant systems does not involve

the Berry curvature, the “semiclassical” description ofthe acceleration of wavepackets in media with spatial in-homogeneity of lengthscales large compared to the under-lying lattice spacing is incomplete if it is not taken intoaccount. Recently Onoda et al.7 have pointed out the roleof Berry curvature in photonic crystals without inver-sion symmetry; while these authors characterize this as a“Hall effect of light”, the Hall effect in electronic systemsis associated with broken time-reversal symmetry ratherthan broken spatial inversion symmetry, and we have re-cently discussed8 some of the at-first-sight-surprising ef-fects that broken time-reversal symmetry could producein photonic systems.

In the presence of non-vanishing Berry curvature, theusual “semiclassical” expression for the group velocityof the wavepacket is supplemented by an additional“anomalous” contribution proportional to its accelera-tion and the local Berry curvature of the Bloch band.(The semiclassical treatment of electron dynamics be-comes ray optics in the photonic context). This “anoma-lous velocity” has played an important role in under-standing recent experiments on the anomalous Hall effectof ferromagnets9, for example.

Perhaps the most remarkable among the “exotic” ef-fects associated with Berry curvature, however, is thequantum Hall effect10, which has been the focus of in-tensive experimental and theoretical study in condensedmatter physics for over two decades. The physics of thequantum Hall regime and its connection with Berry Cur-vature phenomena is now well understood. The possibil-ity of transcribing some of the main features of the quan-tum Hall effect to photonic systems, which brings intoplay new possibilities in photonics, is the topic of this pa-per. Specifically, we shall concern ourselves with analogsof “chiral” (unidirectional) quantum Hall edge states inphotonic systems with broken time-reversal symmetry.

The quantum Hall effect is usually associated with twodimensional electron systems in semiconductor hetero-junctions in strong applied magnetic fields. By treat-ing the plane of the heterojunction as a featureless two-dimensional (2D) continuum, and considering the quan-tum mechanical motion of electrons in the presence of amagnetic field, one obtains the electronic Landau levels.The key feature giving rise to the quantization of the Hallconductance is the incompressibility of the electron fluid:

2

either due to the Pauli principle at integer Landau levelfillings, with the spectral gap to the next empty levelgiven by the cyclotron frequency, or when a gap opensdue to strong electron-electron interactions at fractionalfillings11.

While in the experimentally-realized systems, thequantum Hall effect derives from a strong uniform com-ponent of magnetic flux density normal to the 2D planein which the electrons move, the integer quantum Halleffect can also in principle derive from the interplay of aperiodic bandstructure with a magnetic field. Externallyapplied in periodic structures give rise to the celebratedHofstadter model of Bloch bands with an elegant fractalspectral structure depending on the rational value of themagnetic flux through the unit cell1213. The influence ofthe lattice on the quantum Hall effect was further investi-gated in an important paper by Thouless et al (TKNN)14,who discovered a topological invariant of 2D bandstruc-tures known as the “Chern Number” a quantity that waslater interpreted in terms of Berry curvature5.

At first sight, it seems implausible that any of the phe-nomena associated with the quantum Hall effect can betranscribed to photonics. Incompressibility and Landaulevel quantization require fermions and charged particles,respectively, and it is not clear how one could introducean effect similar to the Lorentz force due to a magneticfield on a system of neutral bosons. However, a hint thatpossible analogs could exist in photonics, comes from thefact that a “zero-field” quantum Hall effect without any

net magnetic flux density (and hence without Landau lev-els) could occur in systems consisting of “simple” Blochstates with Broken time-reversal symmetry, as was shownsome time ago by one of us15. The explicit “graphene-like” model investigated in Ref. 15 exploits the topo-logical properties of Bloch states, which motivated us toconstruct its photonic counterpart. This model has alsoturned out to be a very useful for modeling the anoma-lous Hall effect in ferromagnetic metals16, and a recentlyproposed “quantum spin Hall effect”17.

While incompressibility of the fluid in the bulk quan-tizes the Hall conductance, perhaps the most remarkablefeature of quantum Hall systems is the presence of zeroenergy excitations along the edge of a finite system. Inthese edge states, electrons travel along a single direction:this “one-way” propagation is a symptom of broken time-reversal symmetry. In the case that the integer quantumHall effect is exhibited by Bloch electrons, as in the Hof-stadter problem studied by TKNN14, it is related to thetopological Chern invariant of the one-particle bands andtherefore. The edge states necessarily occur at the inter-face between bulk regions in which there is a gap at theFermi energy, which have different values of the sum ofthe Chern invariants of the fully occupied bands belowthe Fermi level. While the integer quantum Hall effectin such a system itself involves the filling of these bandsaccording to the Pauli principle, and hence is essentiallyfermionic in nature, the existence of the edge states isa property of the one-electron band structure, without

reference to the Pauli principle, which suggests that thisfeature is not restricted to fermionic systems. We havefound that they indeed have a direct photonic counter-part, and this leads to the idea that topologically non-trivial unidirectionally propagating photon modes can oc-cur along domain walls separating two PBG regions hav-ing different Chern invariants of bands below the bandgap frequency. In this paper, we present the formal ba-sis of such modes, along with explicit numerical exam-ples, simple model Hamiltonians, and semiclassical cal-culations confirming the concept.

We note finally, that while Berry phases are usually as-sociated with quantum mechanical interference it can inprinciple occur wherever phase interference phenomenaexist and are governed by Hermitian eigenvalue prob-lems, as in the case of classical electromagnetic waves inloss-free media.

This paper is organized as follows. In section II, wepresent the basic formalism of the Maxwell normal modeproblem in periodic, loss-free media, discuss the Berrycurvature of the photonic bandstructure problem, andconsider the effects of broken time-reversal symmetry.In section III, we provide explicit numerical examples ofbandstructures containing non-trivial topological prop-erties, and show the occurrence of edge states along do-main wall configurations. Motivated by the numerical re-sults, in section IV, we derive a simple Dirac Hamiltonianfrom the Maxwell equations using symmetry argumentsas the guiding principle, and we show how under certainconditions the zero modes of this Dirac Hamiltonian ex-hibit anomalous currents along a single direction due tothe breaking of time-reversal symmetry. It is these zeromodes that play the role of the “gapless” edge excita-tions, as we shall consider in detail. Section V containssemiclassical analysis, and we end with a discussion andpoint out possible future directions in section VI.

II. BERRY CURVATURE IN THE MAXWELL

NORMAL-MODE PROBLEM

In this section, we discuss the formal basis of Berry cur-vature in the photon band problem. We begin with thebasic formulation of the photonic bandstructure prob-lem, which is somewhat more complicated than the elec-tronic counterpart, due to the frequency response of di-electric media. We then briefly review the connectionbetween Chern numbers, Berry curvature, and the occur-rence of gapless edge modes along the boundary whereChern numbers of a given band change.

A. Basic Formalism

We will be solving the source-free Maxwell equationsfor propagating electromagnetic wave solutions in lin-ear, loss-free media, and will ignore absorption, nonlinearphoton-photon interactions, and other processes which

3

do not conserve photon number. We also assume thatthe constitutive relations, reflecting the response of themedia to the electromagnetic waves, are given by local,but spatially varying tensors with generalized frequencydependence. The Berry phase, and the associated Berrycurvature, are commonly identified with quantum me-chanics, but in fact are more generally associated withthe adiabatic variation of the complex eigenvectors of aHermitian eigenvalue system as the Hermitian matrix isvaried.

In quantum mechanics, this Hermitian eigenvalueproblem is the time-independent Schrodinger equation;in the photonic context, it is the classical eigenvalueequation for the normal modes of the Maxwell equa-tions. In order to make the correspondence to the stan-dard quantum-mechanical formulation of Berry curva-ture clearer, we will use a somewhat unfamiliar Hamil-tonian formulation of Maxwell’s equations, which is ap-propriate for loss-free linear media. However, we shouldemphasize that that our results are in no way dependenton the use of such a formalism, and are properties of theMaxwell equations, however they are written.

In such a loss-free, linear medium, the coupling of elec-tromagnetic modes having different frequencies can beignored, and the electromagnetic fields and flux densitiesX(r, t), X ∈ D,B,E,H will be of the form

X(r, t) =(

X∗(r, ω)eiωt + X(r, ω)e−iωt)

, (1)

where the quantities X are in general complex-valuedfunctions of position and frequency with the property:

(

X(r, ω))∗

= X(r,−ω). (2)

The dynamics of these fields are governed by the source-free Maxwell equations:

∇× E = iωB , ∇× H = −iωD, (3)

∇ · D = 0 , ∇ · B = 0. (4)

Consider a single normal mode λ propagating at fre-quency ωλ. For the moment, ignore any internal polar-ization or magnetization modes of the medium, and as-sume instantaneous, frequency-independent response ofthe dielectric material. In this limit, the permeabilityand permittivity tensors, defined by the relations

Ba(r, ωλ) = µab(r)Hb(r, ωλ), (5)

Da(r, ωλ) = ǫab(r)Eb(r, ωλ), (6)

are both positive-definite Hermitian tensors and havewell-defined, positive definite Hermitian inverses ǫ−1

ab (r),

µ−1ab (r). Since we have assumed a linear, loss-free medium

in which photon number is conserved, it is convenient towork with a Hamiltonian formalism: the time-averagedenergy density of the electromagnetic radiation field isgiven by

Hem(r) = ue(r) + um(r), (7)

where

ue(r) =1

2

(

D∗λ, ǫ

−1(r)Dλ

)

(8)

um(r) =1

2

(

B∗λ, µ

−1(r)Bλ

)

. (9)

Then, if Hem is the spatial integral of the energy density,the fields E and H are given by its functional derivativeswith respect to the divergence-free flux densities D andB:

δHem =

∫

d3rEaδBa +HaδB

a. (10)

In the local Hamiltonian formalism, the flux density fieldsD(r) and B(r) are the fundamental degrees of free-dom, and they obey the following non-canonical PoissonBracket relations:

Da(r), Bb(r′)PB = ǫabc∇cδ3(r − r′). (11)

This Poisson bracket generates the Faraday-Maxwellequations d(3):

dD

dt= D(r), HemPB,

dB

dt= B(r), HemPB. (12)

Note that these equations do not generate the Gauss lawequations (4), but merely ensure that any divergences∇aD

a and ∇aBa are constants of the motion; the Gauss

laws are additional constraints that are compatible withthe Faraday-Maxwell equations of motion.

If internal polarization and magnetization modes of themedium are ignored, a discretized form of the electro-magnetic Hamiltonian is formally identical in structureto that of a collection of real oscillator variables xi withnon-canonical Poisson brackets

xi, xjPB = Sij , (13)

where Sij is a real antisymmetric matrix, and the Hamil-tonian energy function is

H =1

2

∑

ij

Bijxixj , (14)

where Bij is a real-symmetric positive-definite matrix. Itis useful to introduce the imaginary antisymmetric Her-mitian matrix Aij = iSij ; The canonical normal modesare given by

qλ ± ipλ = (γλ)−1∑

i

u±iλxi, (15)

where γλ is an arbitrary scale factor, and where (uσiλ)∗

= u−σiλ , σ = ±, which obeys the generalized Hermitian

eigenvalue equation

∑

j

Aiju±jλ = ±ωλ

∑

j

B−1ij u

±jλ, (16)

4

with ωλ > 0, and the orthogonality condition

∑

ij

(uσiλ)∗B−1

ij uσ′

jλ′ =γ2

λ

ωλδσσ′δλλ′ . (17)

Because of the antisymmetric Hermitian property ofthe matrix Aij , and the positive-definite real-symmetricproperty of the matrix Bij , this eigenproblem has realeigenvalues that either come in pairs, ±ωλ, or vanish;these equations provide a straightforward transformationto canonical form only if the generalized eigenvalue prob-lem has no zero-frequency eigenvalues, which is only thecase if Aij is non-singular.

The coefficients uiλ are the analogs of the electromag-netic fields E(r, ω) and H(r, ω). It is also useful to in-troduce the conjugate quantities

vσiλ =

∑

j

B−1ij u

σjλ,

∑

i

(vσiλ)∗uσ′

iλ′ = δσσ′δλλ′ ; (18)

these are the analogs of the flux densities D(r, ω) and

B(r, ω), and Bij encodes the “constitutive relations” be-tween analogs of fluxes and fields.

The Hamiltonian formulation of the Maxwell equationsindeed presents the difficulty of having a null space ofzero-frequency eigenvalues: by themselves, the Faraday-Maxwell equations have static (zero-frequency) solutions

B(r) = ∇f(r), D(r) = ∇g(r); the role of the additionalGauss law constraints is precisely to eliminate these zeromodes. The zero-mode problem in the Hamiltonian for-mulation is the counterpart of the gauge ambiguity ofthe solutions of Maxwell’s equations in the Lagrangianformulation.

In the Maxwell equations, Bij becomes the followingpositive-definite 6 × 6 Hermitian matrix,

Bij →(

ǫ−1ab (r) 0

0 µ−1ab (r)

)

. (19)

More precisely, this a 6 × 6 block of an infinite-dimensional “matrix” that is block-diagonal in terms ofthe spatial coordinate r. (The “A” and “B” matrix no-tation is common in the context of generalized Hermitianeigenvalue problems, where the positive-definite charac-ter of the “B” matrix guarantees reality of the eigenval-ues; hopefully the context should distinguish our use ofthe symbol B for such a matrix from the symbol B(r)used for the magnetic flux density.) In this continuumlimit, the antisymmetric Hermitian matrix Aij becomesa 6 × 6 matrix block of differential operators:

Aac =

(

0 iǫabc∇b

−iǫabc∇b 0

)

. (20)

This A-matrix can be also be elegantly expressed usingthe 3 × 3 spin-1 matrix representations of the angularmomentum algebra,

(

Lb)ac

= iǫabc:

A =

(

0 La∇a

−La∇a 0

)

. (21)

From the antisymmetry of A, it again follows that itseigenvalues come either in ± pairs, or are zero modes,corresponding to static field configurations. Due to thepresence of a huge band of zero modes (one third of thespectrum), the A matrix cannot be written in canonicalform.

Using the Poisson-Brackets, we see that the equation ofmotion of the electric and magnetic fields is a generalizedHermitian eigenvalue problem of the form

(

0 La∇a

−La∇a 0

) [

Eλ

Hλ

]

= ωλ

(

ǫ(r) 00 µ(r)

) [

Eλ

Hλ

]

.

In this formalism, the energy-density of the normal mode(time-averaged over the periodic cycle) is

u(r) =1

2

[

E∗λ H∗

λ

]

(

ǫ(r) 00 µ(r)

) [

Eλ

Hλ

]

, (22)

and the period-averaged averaged energy-current density(Poynting vector) is

ja(r) =1

2

[

E∗λ H∗

λ

]

(

0 −iLa

iLa 0

) [

Eλ

Hλ

]

. (23)

For practical real-space-based calculations of the pho-tonic normal mode spectrum with inhomogeneous localconstitutive relations, it is very convenient to discretizethe continuum Maxwell equations on a lattice (or net-work) in a way that they in fact have the matrix form(16), where the matrix Aij reproduces the zero-mode(null space) structure of the continuum equations, andHij represents the local constitutive relations at networknodes (which come in dual types, electric and magnetic).In such a scheme, divergence-free electric and magneticfluxes flow along the links of the interpenetrating dualelectric and magnetic networks, while electromagneticenergy flows between electric and magnetic nodes, alongthe links between nodes of the network, satisfying a localcontinuity equation (see Appendix B). However, there isone further conceptual ingredient that needs to be addedto the formalism before we can discuss the Maxwell nor-mal modes in “non-reciprocal” media which have brokentime-reversal symmetry.

B. Frequency dependence of the dielectric media

The formalism discussed so far treats the constitutiverelations as static. In general, although we will treatthem as spatially local, we cannot also treat them as in-stantaneous, and must in principle treat the local permit-tivity and permeability tensors as frequency-dependent,ǫ(r) → ǫ(r, ω), µ(r) → µ(r, ω). This is because a non-dissipative time-reversal-symmetry breaking componentof these tensors is both imaginary and antisymmetric (asopposed to real symmetric) and is an odd function offrequency, making frequency-dependence inescapable inprinciple.

5

These effects can on the one hand be treated in aHamiltonian formulation by adding extra local harmonicoscillator modes representing local polarization and mag-netization degrees of freedom of the medium that coupleto the electromagnetic fields. The full description of thisis again a set of harmonic oscillator degrees of freedomdescribed by equations of the form (16). On the otherhand, with the assumption that we are treating the elec-tromagnetic modes in a frequency range that is not res-onant with any internal modes of the medium (i.e., ina frequency range where the loss-free condition is sat-isfied), we can eliminate the internal modes to yield apurely-electromagnetic description, but with frequencydependent constitutive relations.

The details are given in Appendix A, but the resultcan be simply stated. If all oscillator degrees of free-dom are explicitly described, the eigenvalue problem forthe normal modes has the structure (16), where B−1

ij

is positive-definite and real-symmetric. This guaranteesthat the eigenvalues ωλ are real. However, the normalmodes in some positive frequency range ω1 < ω < ω2 canbe treated by eliminating modes with natural frequen-cies outside that range, to give an matrix-eigenvalue-likeproblem of much smaller rank of the form

∑

j

Aiju±jλ = ±ωλ

∑

j

B−1ij (ωλ)u±jλ, (24)

where Bij(ω) is now a frequency-dependent matrix witha Kramers-Kronig structure. The matrix Bij(ω) is nolonger in general real-symmetric, but provided the elim-inated modes are not resonant in the specified frequencyrange it instead is generically complex Hermitian. The“eigenvalue equation” is now a self-consistent equation:

∑

j

Aiju±jλ(ω) = ±ωλ(ω)

∑

j

B−1ij (ω)u±jλ(ω). (25)

This must be solved by varying ω till it coincides withan eigenvalue. Unfortunately, while B−1

ij (ω) is Hermi-tian and non-singular in the dissipationless frequencyrange ω1 < ω < ω2, it is not necessarily positive defi-nite, so a priori, the eigenvalues ωλ(ω) are not guaran-teed to be real, except for the fact that we know thatthese equations were derived from a standard frequency-independent eigenvalue problem which does have realeigenvalues. As shown in Appendix A, the Kramer-Kronig structure of Bij(ω) reflects the stability of the un-derlying full oscillator system, giving the condition thata modified matrix

B−1ij (ω)) =

d

dω

(

ωB−1ij (ω)

)

(26)

is positive-definite Hermitian in the specified frequencyrange, which is sufficient to guarantee reality of the eigen-values in that range. Furthermore, the quadratic expres-sion for the energy of a normal mode solution is givenin terms of B−1

ij (ωλ) rather than B−1ij (ωλ), reflecting the

fact that the total energy of the mode resides in both the

explicitly-retained degrees of freedom (the “electromag-netic fields”) and the those which have been “integratedout” (the non-resonant polarization and magnetizationmodes of the medium):

xi(t) = B−1ij (ωλ)u+

jλeiωλt + c.c., ωλ > 0, (27)

H =1

2

∑

ij

B−1ij (ωλ)(u+

iλ)∗u+jλ,

dH

dt= 0. (28)

If the frequency dependence of the positive-definiteHermitian matrix Bij(ω) is negligible in the range ω1 <

ω < ω2, so Bij(ω) ≈ Bij(ω0), with ω0 = (ω1 + ω2)/2,

one can replace B−1ij (ω) in (24) by the positive-definite

frequency-independent Hermitian matrix B−1ij (ω0). This

in turn allows the eigenvalue problem to be transformedinto the standard Hermitian eigenvalue problem

Hijw(λ)j = ωλw

(λ)i ,

(

w(λ),w(λ′))

= δλλ′ , (29)

with scalar product

(w,w′) ≡∑

j

w∗jw

′j , (30)

valid for positive ωλ in the frequency range where Bij(ω)

≈Bij(ω0), with

Hij =(

B1/2(ω0)AB1/2(ω0)

)

ij, (31)

and

u+iλ ∝

∑

j

B1/2ij (ω0)wjλ. (32)

This allows well-known Berry-curvature formulas fromthe standard Hermitian eigenproblem5 to be quicklytranslated into the generalized problem. It turns outthat when the full problem with frequency-dependentconstitutive relations is treated, the standard formula forthe Berry connection remains correct with the simple re-placement Bij(ω0) → Bij(ωλ) ( the Berry curvature andBerry phase can both be expressed in terms of this Berryconnection).

C. Berry curvature in Hermitian eigenproblems

Let Hij(g) be a family of complex Hermitian matri-ces defined on a manifold parameterized by a set g ofindependent coordinates gµ, µ = 1, . . .D It is assumedthat the matrix is generic, so its eigenvalues are all dis-tinct; as it is well known, three independent parametersmust be “fine-tuned” to produce a “accidental degener-acy” between a pair of eigenvalues. Thus if the paramet-ric variation of the Hermitian matrix is confined to a two-parameter submanifold, each eigenvalue ωλ(g) will gener-ically remain distinct. Under these circumstances, the

6

corresponding eigenvector is fully defined by the eigen-value equation and normalization condition, up to mul-tiplication by a unimodular phase factor, that can varyon the manifold:

wλi(g) → eiφ(g)wλi(g). (33)

This is the well-known “U(1) gauge ambiguity” of thecomplex Hermitian eigenproblem. Associated with eacheigenvector is a gauge-field (vector potential in the pa-rameter space), called the “Berry connection”:

A(λ)µ (g) = −i (wλ(g)), ∂µwλ(g)) , ∂µ ≡ ∂

∂gµ. (34)

This field on the manifold is gauge-dependent, like theelectromagnetic vector potential A(r), but its curvature(the “Berry curvature”), the analog of the magnetic fluxdensity B(r) = ∇ × A(r), is gauge invariant and givenby

F (λ)µν (g) = ∂µA(λ)

ν (g) − ∂νA(λ)µ (g). (35)

The Berry phase associated with a closed path Γ in pa-rameter space is given (modulo 2π) by

exp(

−iφ(λ)(Γ))

= exp

(

−i∮

Γ

Aµ(g)dgµ

)

. (36)

.Ignoring frequency-dependence, the oscillator system

has

H(g) = B1/2(g)AB1/2(g), (37)

where the positive-definite Hermitian matrix B(g) cancontinuously vary as a function of some parameters g,but A is invariant. Then converting to the oscillatorvariables gives

A(λ)µ (g) = Im.

(

u(λ), B−1(g, ωλ)∂µu(λ))

(

u(λ), B−1(g, ωλ)u(λ))

. (38)

Here B−1(g) has been replaced by B−1(g, ωλ) whichis the correct result when frequency dependence ofB−1(g, ω) is taken into account (see Appendix A).

D. Photonic bands and Berry curvature

In the case of periodic systems, the normal modes havediscrete translational symmetry classified by a Bloch vec-tor k defined in the Brillouin zone, i.e., defined moduloa reciprocal vector G. For fixed k, the spectrum of nor-mal mode frequencies ωn(k) is discrete, labeled by bandindices n, and, as emphasized by Sundaram and Niu4 inthe electronic context, the Bloch vector of a wavepacketplays the role of the control-parameter vector g.

In order to compute the Berry curvature of the photonband Bloch states, we shall find it convenient to work

in a fixed Hilbert space for all Bloch vectors k, and wedo this by performing a unitary transformation on theA “matrix” (which becomes the 6× 6 matrix of differen-tial operators (20) in the continuum formulation of theMaxwell equations) as

A(k,∇) ≡ e−ik·rA(∇)eik·r = A(∇ + ik). (39)

Note that parametric dependence on the Bloch vector k isa little different from parametric dependence on param-eters g that control the Hamiltonian. After projectioninto a subspace of fixed k, the “A” matrix also becomesparameter-dependent, while (if the constitutive relationsare taken to be completely local) the “B” matrix in thephotonics case is only implicitly k-dependent through itsself-consistent dependence on the frequency eigenvalue.(Parameter-dependence of the “A” matrix does not af-fect the expression (38) for the Berry connection.)

The discrete eigenvalue spectrum ωn(k) is then ob-tained by solving the self-consistent matrix-differential-equation eigenvalue problem:

A(k,∇)un(k, r) = ωn(k)B−1(r, ωn(k))un(k, r), (40)

where B−1(r, ω) is the 6×6 block-diagonal permittivity-permeablity tensor diag(ǫ(r, ω),µ(r, ω)), andun(k, r) exp ik · r represents the 6-component com-

plex vector (En(k, r), Hn(k, r)) of electromagneticfields of the normal mode with Bloch vector k and fre-quency ωn(k). The eigenfunction satisfies the periodicboundary condition un(k, r + R) = un(k, r), whereR is any lattice vector of the photonic crystal, whereB−1(r + R, ω) = B−1(r, ω).

The transcription of Eq.(38) to the case of periodicmedia then gives the three-component Berry connection(vector potential) in k-space as

Aa(n)(k) = Im.

(

un(k), B−1(ωn(k))∇akun(k)

)

(

un(k), B−1(ωn(k))un(k))

.

(41)The scalar products in Eq.(41) are defined by the traceover the six components of un(k, r), plus integration ofthe spatial coordinate r over a unit cell of the photoniccrystal. By construction, if a “Berry gauge transforma-tion” un(k, r) → un(k, r) exp iχn(k) is made, Aa

(n)(k)

→ Aa(n)(k) + ∇a

kχn(k).

In three-dimensional k-space, the antisymmetric Berrycurvature tensor Fab

(n)(k) = ∇akAb

(n)(k) −∇bkAa

(n)(k) can

also be represented as the three-component “Berry flux

density” Ω(n)a (k) = ǫabc∇b

kAc(n)(k) (the k-space curl of

the Berry connection), to emphasize the duality betweenr-space and k-space, and the analogy between Berry fluxin k-space and magnetic flux in r-space,

If a wavepacket travels adiabatically (without inter-band transitions) through a region with slow spatial vari-ation of the properties of the medium, so the photonicnormal-mode eigenvalue spectrum can be represented as

7

a position-dependent dispersion relation ωn(k, r), thewavepacket must be accelerated as its mean Bloch vectork slowly changes to keep its frequency constant. Whentranslated into the language of photonics, the semiclassi-cal electronic equations of motion then become the equa-tions of ray optics:

na dka

dt= −na∇aωn(k, r), (42)

dra

dt= ∇a

kωn(k, r) + Fabn (k, r)

dkb

dt, (43)

where n ∝ dr/dt is parallel to the ray path, ∇a ≡ ∂/∂ra

and ∇ak ≡ ∂/∂ka (it is useful to use covariant and con-

travariant indices to distinguish components of spatialcoordinates ra from the dual Bloch vector componentska). The Bloch-space Berry curvature Fab

n (k, r) controlsthe additional “anomalous velocity18” correction in (43)to the familiar group velocity of a wave packet va

n(k) =∇a

kωn(k), which is active only when the wavepacket isbeing accelerated by the inhomogeneity of the medium.

Before we conclude our general discussion on Berrycurvature in photon band systems, we must state theconstraints inversion and time-reversal symmetries placeon the tensor Fab

n (k). In what follows, we will use theBloch state wn defined in Eq.(32). If inversion symmetry(I) is present, the periodic part of the Bloch state wn(khas the following property: wn(k) = wn(−k) whereasif time-reversal symmetry (T) is present, wn(k) =w∗

n(−k). If only (I) is present, it then follows thatFab

n (k) = Fabn (−k), whereas if only (T) is present,

Fabn = −Fab

n (−k). If both symmetries are present, thenthe Berry Curvature is identically zero everywhere exceptat isolated points of “accidental degeneracy”, where it isnot well-defined. When Fab

n is non-zero, the phases ofthe Bloch vectors cannot all be chosen to be real. Theseproperties will be crucial when we consider the effects ofvarious symmetry breaking perturbations on the photonband structure.

E. Topological structure of the photon bands

The main consequence of having bands with non-zeroBerry curvature field is that if the path C is closed andencloses an entire Brillouin zone, the single-valuedness ofthe state wn requires that

exp

(∮

Aandka

)

= exp

(∫ ∫

dka ∧ dkbFabn

)

= 1

or,

∫ ∫

SC

dka ∧ dkbFabn = 2πC(1)

n , (44)

where C(1)n is an integer, known as the Chern invariant

associated with the nth band, and have well-known con-sequences in the quantum Hall effect: in the integer quan-tum Hall effect, where the interactions among electrons

are weak, the Hall conductance is expressed in terms ofthe sum of all Chern invariants of bands below the Fermilevel14:

σH =e2

2π~

∑

i,ǫi<ǫf

C(1)i . (45)

The gauge structure of the photon band problem out-lined above is formally analogous to the local U(1) gaugeinvariance of ordinary electromagnetism. Note that thegauge invariance refers to the phase of the of the six-component electromagnetic fields as a whole; adding ar-bitrary phase k-dependent phase factors to each field sep-arately will in general not preserve the Maxwell equationconstraint.

A phase convention can be specified, for instance,by arbitrarily choosing real and imaginary axes of thephases; the local gauge-dependent phase fields of the elec-tromagnetic Bloch states are then represented as two-component rotor variables at each point of the Brillouinzone. In addition, a gauge choice may be made sepa-rately for each band so long as the spectrum remainsnon-degenerate.

By representing the phase covering on the Bloch mani-fold this way, the possibility of the occurrence of topolog-ical defects of the gauge field become transparent. Localgauge transformations correspond to local smooth defor-mations of the rotor variables, and the Chern invariantcorresponds to the total winding number of theses rotorvariables along a closed path enclosing the entire Bril-louin zone.

In the case of two dimensional Bloch bands, the de-fects of the phase field are point singularities, having azero-dimensional “core” region where a phase conventionis not well defined, due to quasi-degeneracies with neigh-boring bands. It is clear that Bands can have non-zeroChern numbers only if time-reversal symmetry is broken.Otherwise, the Berry curvature will be an odd functionof k, and it’s integral over the entire 2D Brillouin zonevanishes.

In three dimensions, the defects of the phase fieldare line defects or “vortices” and their stability requiresquasi- degeneracies to occur along isolated lines in recip-rocal space.

In the photonic system of interest, even if photon bandscan have non-zero Chern numbers, there can be no Hallconductance as given above due to their Bose statistics(and hence, to their finite compressibility). However, theconnection between edge modes and Chern invariants isindependent of statistics: if the Chern number of a bandchanges at an interface, the net number of unidirection-ally moving modes localized at the interface is given bythe difference of the Chern numbers of the band at theinterface. We shall consider the problem of how Chernnumbers can change across an interface in the next sec-tion.

Since the Chern invariant of a band is a topologicalnumber, it therefore cannot vary smoothly as we vary

8

(a)

(b)

+

FIG. 1: A representation of the phase fields of the photonicBloch states in a two dimensional Brillouin zone using twocomponent rotors. The entire set of six electric and mag-netic fields is associated with a single phase at each point inthe Brillouin zone. The Chern invariant simply representsthe winding number of this phase along the Brillouin zoneboundary and is also given by the integral of the Berry cur-vature Fxy over the two dimensional Brillouin zone. Thephases in (a) correspond to bands with both inversion andtime-reversal symmetries, and the phases of the band can bechosen to be real every where in the Brillouin zone. For bandshaving non-zero Chern invariants (b), the phase around thezone boundary winds by an integer multiple of 2π and thereis a phase vortex-like singularity somewhere in the Brillouinzone where the Berry connection cannot be defined, due tothe occurrence of quasi-degeneracies.

CRCLB

F

FIG. 2: The number of forward minus the number of back-ward moving edgemodes equals the difference of the Chernnumber of the band across the interface.

some parameter of the periodic eigenproblem. So longas a band remains non-degenerate, it’s Chern numbercannot vary. However, if we tune some parameter λ ofthe Hamiltonian to a critical value λc such that two bandshaving non-zero Chern invariants touch at some isolatedpoint in the Brillouin zone when λ = λc, the two bandscan exchange their Chern numbers at these degeneratepoints; if we tuned λ beyond its critical value, the bandswould emerge with different Chern invariants. Since thetotal Berry “magnetic flux” of all bands remains fixed

always, if only two isolated bands exchange their Chernnumbers at points of degeneracy, the sum of their Chernnumbers must remain invariant5.

Generically, 2D bands with both time-reversal and in-version symmetry touch at isolated points of accidentaldegeneracy in a linear conical fashion, forming “Diraccones” in the vicinity of which the spectrum is deter-mined by a massless Dirac Hamiltonian:

H ≡ ω − ωD = vD

(

δk1σ1 + δk2σ2

)

, (46)

where vD, is a parameter that gives the slope of the coneclose to the accidental degeneracy.

C(1) =− −1

C(1)+ = 1 C(1)

+

kx

ky

ωD

ω

λ λc

= −1

C(1)=− 1

FIG. 3: As we tune some parameter λ of the Hamiltonianacross a critical point where “accidental degeneracies” occur,and two bands touch in a linear fashion forming a “Diracpoint”, Chern numbers of bands may be exchanged.

III. BROKEN T AND I IN PHOTONICS

In this section, we shall discuss our strategy for con-structing photon bands with non-zero Chern invariants,and “chiral” edge states, whose existence is confirmed inthe following sections.

To break time-reversal symmetry in photonics, we shallneed magneto-optic materials (i.e. a Faraday rotation ef-fect). Such materials are characterized by their ability torotate the plane of polarization of light, when placed in amagnetic field, and are used in conventional optical iso-lators. The amount of rotation per length is known as aVerdet coefficient, which depends on temperature as wellas on the wavelength of light. Materials known to havelarge Verdet coefficients (∼ 100mm−1 at wavelengths ofthe order of microns) are the iron garnet crystals such asHo3Fe5012

20. Due to the breaking of time-reversal sym-metry in this materials, the eigenfrequency degeneracy islifted for light characterized by different states of circularpolarization.

While such magneto-optic devices employ magneticfields in the direction of travel of the light beam, we shallbe interested in two dimensional photonic crystals withthe magnetic fields placed perpendicular to the plane of

9

r)(

H

H

ε

(b)(a)

FIG. 4: In the conventional Faraday effect used in optical iso-lators, light travels in the same direction as the applied field,resulting in the rotation of its polarization plane. However, inthe photonic analog of a 2DEG heterojunction, light travelsin F

propagation of light, as shown in Fig. 4. We shall call theaxis perpendicular to the 2D photon bands the Faradayaxis, and the setup here is reminiscent of a 2D electrongas placed in a perpendicular magnetic field.

Although we now have a means of introducing bro-ken time reversal symmetry, we still need a strategy forthe nucleation of equal and opposite pairs of Chern in-variants on bands near points of accidental degeneracy.To do this, we choose hexagonal lattice geometry. Thethreefold rotation symmetry of such a system guaranteesthe existence of Dirac points in the Brillouin zone cor-ners when both inversion and time-reversal symmetry arepresent ; in this case the only irreducible representationsof the space group of three-fold rotations correspond tonon-degenerate singlets and degenerate doublets. As asimple example consider the case of free photon “bands”with dispersion ω = c|k| in the first Brillouin zone of atriangular lattice. The eigenfrequencies of the photonsare six-fold degenerate at the zone corners. Adding aweak periodic perturbation in the constitutive relationswill lift the degeneracy and the bands will now be eithernon-degenerate or will form degenerate doublets, as de-manded by the symmetry of the perturbation. Due tothe 6-fold rotation symmetry, the doublets are allowedto have a linear dispersion close to the zone corners andshall be our Dirac points of interest, whereas the non-degenerate singlet bands disperse quadratically. We shallprovide explicit examples of hexagonal photonic band-structures having Dirac points in section V.

While the existence of such Dirac points are virtuallyguaranteed in triangular lattice systems, their stability

has little to do with lattice geometry. Such points arestable in two dimensions only because of the presence oftime-reversal symmetry and inversion symmetry, whenthe eigenvalue problem is a real-symmetric one: in thiscase it is possible to find “accidental” degeneracies byvarying just two parameters, according to the Wigner-von Neumann Theorem. Thus, if the perfect hexago-nal geometry of the constitutive relations is slightly dis-torted, the Dirac points will simply move elsewhere in thetwo dimensional Brillouin zone. Provided that such dis-tortions are not too strong such that an axis of two-fold

rotations is introduced, in which case the linear disper-sion characteristic of a Dirac point is no longer allowed,or if the distortion is so great that the Dirac points meetand annihilate at a point of inversion symmetry, Diracpoints will still exist in the system.

If, however, inversion or time-reversal symmetry is bro-ken in the system, the eigenvalue problem becomes com-plex Hermitian, and according to the Wigner-von Neu-mann theorem, three parameters are required to ensurestability of the Dirac points - in this case, the Dirac pointdegeneracy of the 2D bandstructure is immediately lifted.In both cases, the two bands which split apart each ac-quire a non-zero Berry curvature field Fxy(k).

If inversion symmetry alone is broken, Fxy(k) is anodd function of k, as discussed above. While the bandsdo have interesting semiclassical dynamics due to theiranomalous velocity, they do not have any interestingtopological properties since their Chern invariants areidentically zero.

On the other hand, if time-reversal symmetry alone isbroken, via the Faraday coupling, the Berry curvaturefield will be an even function of k, and each band whichsplit apart due to the Faraday coupling will have equaland opposite non-zero Chern invariants.

Finally, if we can slowly tune the Faraday coupling inspace, from a positive value, across the critical value ofzero, where the local bandstructure problem would per-mit Dirac spectra, to a negative value, we would gen-erate a system of photonic bands with non-zero Chernnumbers, that get exchanged at the region of space cor-responding to the critical zero Faraday coupling. It thenfollows, that modes with exact correspondence to the in-teger quantum Hall edge states would arise in such asystem. In the following section, we shall display thisexplicitly using an example Bandstructure.

IV. EXPLICIT REALIZATION OF EDGE

MODES

An example of a photonic bandstructure with the de-sired properties is shown in Fig. 5. It consists of a trian-gular lattice of dielectric rods (ǫa = 14) placed in a back-ground of air (ǫ = 1) with a area filling ratio of f = 0.431.The authors of ref. 22, in a quest for optimal photonicbandgap materials, first studied this system. They com-puted the TE mode spectrum and found a full band gapin the TE spectrum. We have reproduced numericallytheir calculation and have also computed the spectrumfor the TM modes.

The key feature of this particular system which is ofimportance to us are the presence of a pair of Dirac pointsin the spectrum of the TE modes which are well isolatedfrom both the remaining TE and TM modes. Each ofthe six zone corners contains the Dirac cone spectrum,but there are only two distinct Dirac points, the othersbeing related by reciprocal lattice translations of thesepoints. In this particular system, the two Dirac points

10

FIG. 5: Photon bands in the kz = 0 plane of a 2D hexagonallattice of cylindrical dielectric rods. Electromagnetic wavesare propagating only in the x−y plane (Brillouin zone shownin the lower right). As in Ref. 22, the rods have a fillingfraction f = 0.431, ǫ = 14, and the background has ǫ = 1.The band structure contains a pair of Dirac points at the zonecorners (J).

are related by inversion in k-space.As we have discussed, a gap immediately opens when

either inversion or time-reversal symmetries are brokenin this system. We break inversion symmetry in the sim-plest possible way by introducing a slight imbalance inthe value of the dielectric tensor in the rods at oppositeends of the unit cell, and we parameterize the inversionbreaking by defining the quantity

MI = log

(

ǫ+ǫ−

)

, (47)

where ǫ+(ǫ−) is the value of the permittivity inside therods in the upper (lower) half of the unit cell depicted inFig. 5.

To break time-reversal symmetry, we add a faradayeffect term in the region outside the rods. This is done bygiving the dielectric tensor a slight imaginary componentwithout varying the constitutive relations inside the rods:

outside rods: ǫ−1ij (x) =

(

ǫ−1b iΛ

−iΛ ǫ−1b

)

, (48)

inside rods: ǫ−1ij (x) =

(

ǫ−1a 00 ǫ−1

a

)

. (49)

We also define a parameter

MT = Λ, (50)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

MT

MI

I

II

III

IV

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

MT

MI

I

II

III

IV

FIG. 6: Phase diagram of the photonic system as a functionof inversion and time-reversal symmetry breaking. In regionsI and III, the gap openings of both Dirac points are primarilydue to inversion symmetry breaking, whereas in regions II andIV, the breaking of time-reversal symmetry lifts the degener-acy of the bands which formed the Dirac point. In all fourregions, the two bands of interest have non-zero Berry curva-ture, but only in regions II and IV do they contain non-zeroChern numbers.

to define the strength of the time-reversal symmetrybreaking perturbation.

We first determined the phase diagram of the systemin the (mI ,mT ) plane by breaking both inversion and

time-reversal symmetry, and locating special values of thesymmetry breaking parameters that result in the closingof the bandgap at one or more Dirac points (Fig. 6). Thephase diagram separates regions characterized by bandsjust below the band gap having a non-zero Chern numberfrom regions with all bands having zero Chern numbers.The boundary between these phases are where the gap atone or more of the Dirac points vanishes, as shown in Fig.6. Since there are two Dirac points, each phase bound-ary corresponds to the locus of parameters for which thegap at one of the Dirac points closes. Thus, both Diracpoints close only when both lines intersect, namely at thepoint (mI = 0,mT = 0), where both inversion and time-reversal symmetries are simultaneously present. Wheninversion symmetry alone is broken, the Berry curvaturefield of Dirac point 1 is equal in magnitude and oppo-site in sign of the Berry curvature at the second Diracpoint. When time-reversal symmetry is broken, on theother hand, each Dirac point has an identical (both inmagnitude and sign) Berry curvature field. In this case,the photon bands which split apart at the Dirac pointeach have Non-zero Chern number, which depends onlyon the direction of the Faraday axis (±z).

We have also studied numerically the frequency gapas a function of the time-reversal breaking perturbationabove and found that so long as the dielectric tensor re-mains positive-definite, the gap increases linearly with

11

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

∆

Λ

FIG. 7: The bandgap opened by time-reversal breaking as afunction of the strength of the Faraday coupling shows thatthe gap is linearly proportional to Λ.

ǫxy (7). This will be important when we consider ef-fective Dirac Hamiltonians for this problem: as we shallsee, the fact that exactly at the zone corner, the gaprises linearly with MT is consistent with the spectrum ofa massive Dirac Hamiltonian with mass MT . Thus, wehave shown an example of a bandstructure which con-tains Dirac points whose gaps can be tuned using time-reversal and inversion symmetry breaking perturbations.We can now show the existence of “chiral” edge states inthis system.

To study edge states in this system, we introduce a“domain wall” configuration across which the Faradayaxis reverses. As we shall now show numerically, (andjustify analytically in the following section), the edgemodes that occur along the domain wall are bound statesthat decay exponentially away from the wall while prop-agating freely in the direction parallel to the interface. Inorder to study the exponential decay of these modes, weglue together N copies of a single hexagonal unit cellalong a single lattice translation direction R⊥, whichshall be the direction perpendicular to the domain wall.We treat this composite cell as a unit cell with periodicboundary conditions. Since a domain corresponds to acertain direction of the Faraday axis, we study a configu-ration here in which the axis changes direction abruptlyacross the domain wall from the +z to the −z direction.

When we consider the spectrum on a torus, there arenecessarily two domain walls. Furthermore, since manyunit cells are copied in this system, there are as manyduplicates of the bands in the enlarged system under con-sideration. We study the bandgap precisely at the Diracpoint as a function of the fractional distance between thetwo domain walls on the torus x (Fig. 8) for a compositeunit cell consisting of N = 30 unit cells copied along theR⊥ direction. When x = 0 or x = 1, the two domainwalls are at the same point, and this corresponds to asingle domain with a single Faraday coupling Λ. For all

x

+

++++

+

+

+ ___

___

_

FIG. 8: In a system with periodic boundary conditions, thereare necessarily two domain walls separating regions with dif-ferent faraday axes. We study the gap of the spectrum atthe (now non-degenerate) Dirac points as a function of thedistance x between the two walls.

other values of x, the “unit cell” comprises a two domainsystem with non-equivalent lengths. In Fig. 9, the gapbetween the two bands closest to the Dirac frequency de-cays exponentially as a function of the distance betweenthe two domain walls. We shall show that the exponen-tial decay in the gap corresponds to the localization ofthe edge modes along each domain wall. The small gapat intermediate values of x, when the two walls are farapart corresponds to the fact that each edge mode hasa small amplitude, and therefore hardly mix with eachother at those length scales.

0 0.005 0.01

0.015 0.02

0.025 0.03

0.035

0 0.2 0.4 0.6 0.8 1

∆

x

FIG. 9: The spectral gap between the two bands which splitapart due to the breaking of time-reversal symmetry. Thespectrum is computed on a torus for the extended systemconsisting of 30 copies of the hexagonal unit cell. Further-more, domain walls, across which the sign of the Faraday axisflips are introduced, and the spectrum is plotted as a functionof the separation x between the walls (see also Fig. 8).

When the domain wall is introduced, translationalsymmetry is still preserved along the direction parallel

12

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

10 20 30 40 50 60 70

ωa/

2πc

Kparallel

FIG. 10: The spectrum of the composite system consisting30 copies of a single hexagonal unit cell duplicated along adirection R⊥. Both inversion and time-reversal symmetriesare present, and the Dirac points are clearly visible. Whilethe composite system has a spectrum containing many bands,only two bands touch at the Dirac point. The dispersion iscomputed in k space along the direction parallel to the wall.

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

10 20 30 40 50 60 70

ωa/2

πc

Kparallel

FIG. 11: The same system as above, but with broken time-reversal symmetry without a domain wall. There is a singleFaraday axis in the rods of the entire system.

to the wall, and the states of the composite system of30 unit cells can be labeled by k‖, Bloch vectors in thedirection parallel to the wall. Figures 10, 11,and 12 con-sist of a spectral series of a system without any brokentime-reversal symmetry (Fig. 10), with uniformly bro-ken time-reversal symmetry (Fig 11), and a domain wallconfiguration (Fig. 12) for the 30 unit cell composite sys-tem. The bands are plotted along a trajectory in k-spacein the k‖ direction which contains the two distinct Bril-louin zone corners. It is clear that in the Domain wall,there are two additional modes formed in the bandgapthat arose from the Faraday coupling. Since the domain

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

10 20 30 40 50 60 70

ωa/2

πc

Kparallel

FIG. 12: Same system as above, but with a domain wall in-troduced corresponding to maximum separation of the wallson the torus. The two additional modes present in the gapcorrespond to edge modes with a “free photon” linear disper-sion along the wall. There are two modes, since across thedomain wall, the Chern number of the band just below theband gap changes by 2.

walls are duplicated on the torus, the spectrum of edgemodes will also be doubled; in Fig. 12, only the two non-equivalent modes are shown. Each mode in the band gaphas a free photon linear dispersion along the direction ofthe wall; moreover, both have positive group velocities,and therefore propagate unidirectionally.

To be certain, however, that these “chiral” modes areindeed localized near the interface, we have numericallycomputed 〈u(r)|B−1|u(r)〉, the electromagnetic energydensity (the B matrix, defined in section II, is not tobe confused with the magnetic flux density), the pho-ton probability density in real space. We have computedthis quantity along with all the spectra of the compositesystem using the real space bandstructure algorithms de-scribed in Appendix B. As shown in Fig. 13, the energydensity is a gaussian function, peaked at the position ofthe domain wall, decaying exponentially away from thewall. From this calculation, we extract a localizationalso approximately 5 unit translations in the directionperpendicular to the interface.

We have therefore shown here using explicit numeri-cal examples that photonic analogs of the “chiral” edgestates of the integer quantum Hall effect can exist alongdomain walls of Hexagonal photonic systems with brokentime-reversal symmetry. We have studied the unphysicalcase in which such domain walls are abrupt changes in theaxis of the Faraday coupling. However, due to the topo-logical nature of these modes, a smoother domain wallin which the Faraday axis slowly reverses over a lengthscale much larger than a unit cell dimension would alsoproduce such modes. The most important requirementfor the existence of these modes, is that at some spa-tial location, the Faraday coupling is tuned across its

13

critical value. How this particular tuning is effected isirrelevant. In the following section, after deriving the ef-fective Hamiltonians for these modes, we shall considera smoothly varying Faraday coupling, which correspondsto an exactly soluble system, and shall show the evolutionof these modes as the smoothness of the Faraday couplingis varied towards the step function limit considered here.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

-10 -5 0 5 10

Energy density

Unit cells

FIG. 13: The real-space electromagnetic energy density pro-file associated with the edge modes in Fig. 12 plotted as afunction of the direction perpendicular to the domain wall(and “integrated” over the direction parallel to the interface)and fit to a gaussian profile. The integrated energy densitydepicted here plays the role of the photon probability densitywhich confirms that light is confined to the interface.

V. MODEL HAMILTONIAN APPROACH

The crucial feature exploited in the previous sectionswas the possibility of tuning bandgaps at Dirac pointsby adding time-reversal breaking perturbations. Beforeadding these perturbations, the linear conical spectrumat these Dirac points are governed by two dimensionalmassless Dirac Hamiltonians, and time-reversal or inver-sion symmetry breaking perturbations contribute mass

terms to the Hamiltonian. In this section, we shallconstruct these Dirac Hamiltonians starting from theMaxwell equations for two dimensional photonic systemswith hexagonal symmetry.

To motivate Dirac Hamiltonians in photonic systems,we begin this section by considering a “nearly-free pho-ton” approach in which a two dimensional “free photon”spectrum consisting of plane waves is perturbed by aweak periodic and hexagonal modulation of ǫ(r). Due tothe underlying symmetry of the perturbation, the planewaves mix in a manner to generate Dirac points in thezone corners of this system. We then consider the effectadding time-reversal and inversion symmetry breakingperturbations in this system and derive an expression forthe Dirac mass. Having motivated the Dirac points, we

revert to our photon band problem and derive expressionsfor the Dirac mass in these systems.

In analogy with the “nearly-free electron” approxima-tion, we consider the photon propagation problem in theweak-coupling regime, in which the dielectric propertiesof the medium act as a weak perturbation. We solvethe Maxwell normal mode problem for Bloch state solu-tions, and work out corrections to the free photon dis-persion relations in the Brillouin zone boundaries. Weshall assume continuous translational invariance in the z-direction, and study the propagation of electromagneticwaves in the x-y plane.

The free photon constitutive relations are isotropic anduniform in the plane:

B0 =

(

ǫ0 00 µ0

)

. (51)

We consider the “free photon bands” in the first hexag-onal Brillouin zone depicted in Fig. 14. Let Gi, i = 1, 2, 3be the three equal-length reciprocal lattice vectors eachrotated 1200 with respect to one another. The hexag-onal zone corners correspond to the points ±Ki, whereK1 = (G2 − G3) /3, etc., and |K| = |G|/

√3. At each of

the zone corners, the free-photon spectrum is six-fold de-generate with ω0 = c0K. In two dimensions, the modesdecouple into TE (Ex, Ey, Hz), and TM (Hx, Hy, Ez)sets, and we shall focus only on the TE modes and con-sider the 3-fold TE mode symmetry at the zone corners(the TE and TM modes do not mix in 2 dimensions).The eigenvalue equation for the free photon plane wavemodes at the zone corners is A|u0〉 = ω0B

−10 |u0〉, or

equivalently, B1/20 AB

1/20 |z0〉 = ω0|z0〉 and the states

|z0〉 = B−1/20 |u0〉 satisfy 〈z(λ)

0 |z(λ′)0 〉 = δλλ′ .

1G

2G

K2

K1

K3

G3

FIG. 14: In the weak coupling approach, the free photon TEmode plane waves are perturbed by a periodic modulationin the permittivity. The plane wave frequency at the threeequivalent zone corners (Ki, i=1,2,3) is lifted by the permit-tivity in “k · p” perturbation theory into a non-degeneratesinglet and a degenerate doublet.

Next, keeping the uniform isotropic permeability fixed,we add a weak periodic perturbation to the permittivity

14

of the form

λB−11 =

(

ǫ0λVG(r) 00 0

)

, (52)

with

VG(r) = 2

3∑

n=1

cos (Gn · r) . (53)

After this perturbation is added, the TE and TMmodes no longer remain degenerate; while the TM modesremain 3-fold degenerate at the zone corners at the fre-quency ω = c0|K|, the TE modes split apart into asinglet and a degenerate doublet. We now determinethe splitting to leading order in λ with within a weak-coupling “nearly-free photon” approach.

With the periodic perturbation, the eigenvalue prob-lem is

A|u〉 = ω(

B−10 + λB−1

1

)

|u〉, (54)

which is equivalent to

B1/20

(

A − λωB−11

)

B1/20 |z〉 = (ω0 + δω) |z〉. (55)

The energy splittings are worked out in degenerate per-turbation theory (see Appendix C) as

δωn

ω0= −λ〈zn|B1/2

0 B−11 B

1/20 |zn〉

= −λ〈un|B−11 |un〉,

where |zn〉 are appropriate combinations of the threefree photon plane-plane waves that diagonalizes the pe-riodic potential. These states are obtained by requir-ing them to be invariant under 3-fold rotations in theplane. Instead of writing the fields in the coordinatebasis, It is convenient to use a redundant basis of thethree vectors (eiK1·r, eiK2·r, eiK3·r), with

∑

n Kn = 0,and Ki ·Kj = −K2/2, i 6= j. In this basis, the magnetic

field of the TE modes is written as (η = e2πi/3):

Hz1 = (1, 1, 1) , (56)

Hz2 = (1, η∗, η) , (57)

and

Hz3 = (1, η, η∗) . (58)

The corresponding electric flux densities are easily ob-tained:

D‖1 =

1

ω(z × K1, z × K2, z × K3) , (59)

D‖2 =

1

ω(z × K1, η

∗z × K2, ηz × K3) , (60)

D‖3 =

1

ω(z × K1, ηz × K2, η

∗z × K3) , (61)

and

|zi〉 =

(

E‖i

Hzi

)

. (62)

Clearly, these are the plane wave solutions that satisfyMaxwell equations and transform appropriately under 3-fold rotations in the plane. We are therefore led to thesimple result that the splitting at the zone corners dueto the mixing of the three plane waves is related to theintegral over the unit-cell of the electric fields and theperiodic potential, which is a traceless, real-symmetric3 × 3 problem. It is easy to see that the problem istraceless because diagonal terms of the form 〈ui|B1|ui〉vanish identically since ui are plane waves.

To leading order in λ, the three photon bandssplit to form a singlet band at frequency ω0 =c0|K|

(

1 + λ/2 +O(λ2))

and a degenerate doublet at fre-quency

ωD = c0|K|(

1 − λ/4 +O(λ2))

. (63)

Exactly at the zone corners, the singlet and doubletstates above diagonalize the perturbation in Eq.(52). Toleading order in λ and δk ≡ k−Ki, the deviation in theBloch vector from the zone corners, the states |z2(δk)〉and |z3(δk)〉, (where |zi(δk)〉 = exp(iδk · r)|zi〉), whichare degenerate at δk = 0 mix and split apart linearly asa function of |δk|, forming a “Dirac point”. To leadingorder, the Dirac point doublet does not mix with the sin-glet state |z1(δk)〉. The effective Hamiltonian governingthe spectrum of the doublet, to leading order in δk is a2D massless Dirac equation:

ω±(δk) = ωD ± vD (δkxσx + δkyσ

y) , (64)

where vD = c0/2 + O(λ), and σi are the Pauli matriceswritten in the subspace of the doublet states. The lineardispersion of the doublet in the neighborhood of the zonecorners is immediately obtained by solving Eq.(64):

ω = ωD ± vD|δk|. (65)

The singlet band’s frequency remains unchanged toleading order in δk: ω0(δk) = ω0 + O(|δk|2). Thus, wehave shown that the periodic modulation of the permit-tivity having 3-fold rotational symmetry gives rise to aquadratically dispersing singlet band and a “Dirac point”with linear dispersion.

15

0.5

0.55

0.6

0.65

0.7

0.75

ω a

/2 π

c

J

FIG. 15: Spectrum of photon dispersion in the vicinity ofthe zone corners. We have arbitrarily set λ < 0 so that thesinglet band has a lower frequency than the Doublet. Freephoton spectra are given by dashed lines. Away from the zonecorners, the free spectrum is not affected to leading order inλ.

Next, we add a Faraday term, with an axis normal tothe xy-plane, to the permittivity tensor ǫxy = −ǫyx =iǫ0η(r, ω), where

η(r, ω) = η0(ω) + η1(ω)VG(r). (66)

Both η0(ω) and η1(ω) are odd functions of ω. Inthe limit that the Faraday coupling is much weaker instrength than the periodic modulation, |η0|, |η1| ≪ |λ| ≪1, the mixing between the non-degenerate singlet stateand the doublet remains negligible, and the energy ofthe singlet state is unaffected by the Faraday perturba-tion. However, the doublet states split apart at the Diracpoint. Using the expression for the Dirac point splitting,derived in the Appendix C, we find that the splitting ofthe doublet at the zone corner is given by

ω± − ωD = ±vDκ, κ = |K|(

3

2η1(ωD) − 3λη0(ωD)

)

.

(67)Away from the Dirac point (but still close enough to thezone corners so that the “nearly-free photon” approxima-tion for the three plane wave states remains valid), thedoublet bands acquire a dispersion

ω = ωD ± vD

(

|δk|2 + κ2)1/2

, (68)

which is the spectrum of a 2D massive Dirac Hamilto-nian:

ω±(δk) = ωD ± vD (δkxσx + δkyσ

y + κσz) . (69)

The Dirac points that occur in the “nearly-free pho-ton” approximation are not isolated points of degener-acy, since, away from the zone corners, the two bandswhich formed the Dirac point merge together to resume

their original free-photon form. Consequently, the typeof modes studied in the previous section cannot be re-produced using this type of weak-coupling expansion.

However, we can gain understanding by suppose thatwe have the exact solutions of the electromagnetic Blochstates and eigenfrequencies of a system containing iso-lated Dirac points, such as the one studied numericallyin section IV. We can use precisely the same weak Fara-day coupling approximation to work out the splitting ofthe Dirac point with a Faraday term. Assuming we aregiven example photonic bandstructures of long hexago-nal systems with kz = 0, which contain only isolatedDirac points, a weak Faraday coupling would split apartthe bands that formed the Dirac point, and the split-ting is identical to that in (68). Suppose that the twobands having a Dirac point, otherwise form a PBG witha gap ∆ ≫ vDκ (as in the case of the numerical exam-ple given in the previous section. In this case, since theFaraday term removes all points of degeneracy, the nownon-degenerate bands have a well-defined Berry curva-ture field

F±(δk) = ±1

2κ

(

|δk|2 + κ2)−3/2

, (70)

which decays rapidly away from the Dirac point, andcontributes a total integrated Berry curvature of ±π.Since there are two non-equivalent Dirac points in thehexagonal geometry under consideration, the net Berrycurvature of the system is the sum of the contributionsfrom each Dirac point. If, as in the case under consider-ation, spatial inversion symmetry is preserved, but time-reversal symmetry is broken, the Berry curvature fieldsat each Dirac point of a given band add, giving totalChern numbers ±1 for each of the split bands. However,if time-reversal symmetry were preserved, and inversionsymmetry breaking caused the gap to open, the Berrycurvature field of each Dirac point for a given band areequal in magnitude but opposite in sign, and the Chernnumber would vanish.

As before, to get unidirectional edge modes of light inthis system, the Faraday coupling must be tuned acrossits critical value η(r, ω) = 0. To do this, we consider aFaraday coupling that varies slowly and adiabatically inspace, we shall assume negligible frequency dependenceof the Faraday coupling, and we shall parameterize thelocal value of the Faraday coupling by a smoothly varyingfunction κ(r), which is positive in some regions and neg-ative in other regions of the 2D plane perpendicular tothe cylindrical axis of the hexagonal array of rods. Dueto the adiabatic variation of κ(r), each point in space ischaracterized by a local bandstructure problem, and thesplitting at the Dirac point is given again by the expres-sion in (68), but with the local value of κ. In this limit,the smooth variation of κ(r) leads to a 2D Dirac Hamil-tonian with a adiabatically spatially varying mass gap.At all points where κ(r) = 0, the local bandstructure inthe vicinity of the Dirac point is the massless 2D DiracHamiltonian; provided that |κ(r)| ≪ ∆, the PBG, the

16

spectrum far away from the Dirac points is unaffected byκ(r). In what follows, we assume that when κ = 0, ourbandstructure contains Dirac points which are formedby two isolated bands in a PBG region having no otherpoints of degeneracy.

We neglect the mixing between modes at differentDirac points, and consider the situation in which κ(r)vanishes along a single line ( x = 0 for instance), and weassume translational invariance along the direction par-allel to the interface (y− direction). As before, we con-sider the degenerate perturbation problem of the normalmodes close to the Dirac point. Now, however, the coef-ficients of the degenerate solutions of the Maxwell equa-tions are spatially varying quantities. Let |uσ(±kD)〉,σ = ±, be the degenerate solutions (i.e. the periodicparts of the photon Bloch state wave functions) at a pairof Dirac points when κ = 0. With the local variation, wetake spatially varying linear combination of these Blochstates

u(kD, r) =∑

σ,±

ψσ(r) exp (±ikD · r)uσ(±kD, r), (71)

and arrive at the fact that the local value of the splittingof the two bands at kD is

ω+(kD) − ω−(kD) = 2κ(r). (72)

In the neighborhood of the Dirac point, the degener-ate perturbation problem gives us a 2D massive DiracHamiltonian, with δkx replaced by the operator −i∇x inthe position representation, since translation symmetryin the x-direction is broken by κ(x). We thus obtain an

expression of the form vDK|ψ〉 = δω|ψ〉, and

K = −iσx∇x + δk‖σy + κ(x)σz . (73)

The Bloch vector in the y-direction, which remains con-served due to the preservation of translation invariancealong this direction, is kDy + δk‖.

For the particular choice of κ(x) = κ∞ tanh(x/ξ),ξ > 0, (where κ∞ is the asymptotic value of the Diracpoint splitting at distances ≫ ξ from the interface), theproblem is exactly solvable, since the Dirac HamiltonianK, when squared, becomes a 1D Schrodinger Hamilto-nian K2 corresponding to the integrable Poschl-TellerHamiltonian21.

To see how this comes about, we explicitly work out theoperator K2, making use of the anti-commuting propertyof the Pauli matrices σa, σb = 2δab :

K2 − δk2‖ = −∇2

x + κ(x)2 − σyκ′. (74)

The spatially varying Dirac mass term that changedsign across the interface becomes a “potential well” withbound states given by21

ω0(δk‖) = ωD + sκvDδk‖, sκ = sgn(κ∞) (75)

ωn± = ωD ± vD

(

δk2‖ + κ2

n

)1/2

, n > 0. (76)

where |κn = 2n|κ∞|/ξ, n < |κ∞ξ/2. In the n = 0 mode,light propagates unidirectionally, with velocity vD, in thedirection parallel to the wall. All other bound modesare bidirectional modes. The numerical example of aDirac mass studied in the previous section that changedsign abruptly, as a step function, has the 1D Schrodingerproblem in an attractive delta function potential as itssquare. Consequently, as we have seen, the model per-mitted for only a single bound state, corresponding tothe unidirectional mode.

-20 -15 -10 -5 0 5 10 15 20

δk||

ωD

ω

FIG. 16: Spectrum of the integrable Poschl-Teller model.With the exception of the zero mode, all bound states corre-spond to bi-directionally propagating modes localized at theinterface where the function κ(r) = 0. The zero mode, on theother hand, is unbalanced, and furthermore, it corresponds tounidirectional propagation.

For the generic case, the second order differential equa-tion for the n > 0 bound states can not be solved ana-lytically. However, a formal solution for the zero modeeigenfrequency can be obtained, as it is obtained by solv-ing a first order equation, as we now discuss. Startingfrom the Dirac equation for the more general case

vD (−iσx∇x + −iσy∇y + κ(x)σz) |ψ±〉 = δω|ψ±〉,(77)

by definition, the “zero mode” has the free photon dis-persion along the direction parallel to the wall, whichimplies that the function |ψ〉 ∝ exp(iδk‖y). We are thusleft with the equation

(−iσx∇x + κ(x)σz) |ψ±〉 = 0. (78)

Multiplying both sides with σx, we arrive at the followingfirst order differential equation :

(∇x + κ(x)σy) = 0, (79)

which has as it’s formal solution

|ψ±〉 = exp

(

iδk‖y + α

∫ x

dx′κ(x′)

)

|φ±(α)〉, (80)

where σy|φ±(α)〉 = α|φ±(α)〉. Although there are for-mally two solutions for the zero mode, corresponding to

17

α = ±1, only one can occur; the other is not normal-izable and thus cannot represent a physically observablestate.

VI. SEMICLASSICAL ANALYSIS

Now let the “Dirac mass” term that opens the photonicband gap be a slowly varying function κ(x) that changesmonotonically (and analytically) from −k0 at x = −∞to k0 at x = +∞. The photonic spectrum of modeswith wavenumbers k = kD + δk near the “Dirac point”kD, and which become doubly-degenerate at kD, is anadiabatic function of x:

ω(x, δkx, δky) = ωD ± vD

(

δk2y + k(x, δkx)2

)1/2,

k(x, δkx)2 = δk2x + κ(x)2, (81)

where vD > 0 is the “Dirac speed”. For k(x, δkx)2 < k20 ,

the modes are evanescent as x → ±∞, so are localizedon the wall. In the x − δkx plane, the contours of con-stant k(x, δkx)2 < k2

0 are simple closed curves, enclosinga finite dimensionless area φ(k2), given by

φ(k2) = 2

∫ x+

x−

dx(

k2 − κ(x)2)1/2

, (82)

where x−(k2) < x+(k2) are the two “turning point” so-lutions of κ(x±)2 = k2 . Since κ(x) is assumed to bemonotonic, this can be written as

φ(k2) = 2

∫ |k|

0

dy(

k2 − y2)1/2

(

1

κ′+(y2)+

1

κ′−(y2)

)

,

κ′±(k2) ≡ dκ

dx

∣

∣

∣

∣

x±(k2)

, (83)

Note that this transformation has turned φ(k2) into asigned area, where sgn(φ) = sgn(k0), which is indeedthe correct form (the function φ(k2) vanishes as k0 →0, when its domain k2 ≤ k2

0 shrinks to zero). In thelimit k2 → 0, x±(k2) → x0, the formal location of theinterface. Then κ′±(k2) → κ′(x0), and φ(k2) vanishes as

φ(k2) → πk2

κ′(x0), (k/k0)

2 → 0. (84)

It is very instructive to examine the special case

κ(x) = k0 tanh(α(x − x0)), (85)

which is integrable. In this case,

κ′(x) = αk0sech2(α(x − x0)), (86)

k0sech2(α(x±(k2) − x0)) =

k20 − k2

k0. (87)

Thus the explicit dependence on x±(k2) can be elimi-nated, and

κ′±(k2) = α

(

k20 − k2

k0

)

. (88)

This make the integral for φ(k2) trivial (it becomes ex-pressible in terms of a simple Hilbert transform), andthe asymptotic small-k2 form (84) remains valid for all

values of k2 in the domain of the function:

φ(k2) =πk2

αk0, k2 ≤ k2

0 . (89)

Then the frequency of the interface mode with wavenum-ber δky = δk‖ along the interface can be expressed as

ω(δk‖, φ) = ωD ± vD

(

δk2‖ + κ2

⊥(φ))1/2

,

κ2⊥(φ) ≡ |αk0φ|/π. (90)

A standard “semiclassical” analysis of interference effectson a light ray trapped in a “waveguide” at an interfacewould conclude that the “quantized” values of φ corre-sponding to interface modes were

φn = 2πn+ γ, (91)

where γ is a “Maslov phase”, usually π. In this case, com-parison with the exact solution of the integrable prob-lem confirms that this problem instead has a vanishing“Maslov phase” γ = 0. This can be attributed to an un-derlying “Z2” Berry phase factor of −1 (Berry phase ofπ) for orbiting around the degeneracy point at (x−x0, kx)=(0, 0).

We then conclude that the interface modes at a slowly-varying interface are in general given (for small δk‖) by

ω0(δk‖) = ωD + vDsgn(k0)δk‖,

ωn±(δk‖) = ωD ± vD

(

δk2‖ + k2

n

)1/2

, n ≥ 1,

φ(k2n) = 2πn, k2

n ≤ k20 . (92)

The unidirectional “zero mode” persists however sharpthe interface is; the bidirectional modes with n ≥ 1 mustobey 2πn < φ(k2

0), which has fewer and fewer (and even-tually no) solutions as the width of the interface regionshrinks. In the special case of the integrable model (85),this spectrum is exact for small δk‖ without any condi-tion that the wall is slowly varying.

VII. DISCUSSION

The occurrence of zero energy modes in the 2D DiracHamiltonian is well known and represents the sim-plest example of a phenomenon known as the “Chiralanomaly”. The crucial feature, namely, the occurrenceof interfaces where the Dirac mass gap changes sign, cor-responds to tuning our photon band problem across acritical point using a Faraday effect.

We have shown that analogs of quantum Hall effectedge modes can exist in photonic crystals whose bandgaps can be tuned by a Faraday coupling. The crucial

18

new feature we present here here is that photonic sys-tems can have bands with non-trivial topological proper-ties including non-zero Chern invariants. These in turncan be varied in a controlled manner to yield unidirec-tional (“chiral”) edge modes. The edge modes are ro-bust against elastic back-scattering since they are stateswhich are protected by the underlying 2D band structuretopology. However, they are not robust against photonnumber non-conserving processes, such as absorption andother non-linear effects. We believe that this could be anentirely new direction in “photonic band structure engi-neering” due to the absence of scattering at bends andimperfections in the channel.

Ef

C = +1

C = −1E ω

C = −1

PBG

C = +1

D.O.S(b)

D.O.S(a)

FIG. 17: We have shown in this paper that although thephoton bands (b) cannot be filled as in the electronic case(a), they can have no analog of the bulk quantum Hall ef-fect. However, the Chern number is a topological invariantof Bloch states independent of the constituents. With theFaraday term, we are able to tune the system such that thetotal Chern number below a photonic band gap changes acrossin interface, which gives rise to unidirectionally propagatingedge modes of photons localized at the interface. These modesare direct analogues of the “Chiral” edge modes of electronicsystems which occur at interfaces between two regions hav-ing different total Chern invariants below the Fermi level (i.e.with different Hall conductances).

A practical realization of such one-way transmissionchannels in photonics will necessarily have to deal withthe problem of finding a magneto-optic material with astrong enough Faraday effect to confine the light closeto the interface. Furthermore, in a practical design, theproblem of TE/TM mode mixing when light is confinedin the direction perpendicular to the 2d system will haveto be addressed. A practical design could, for instance,make use of PBG materials to confine light in the z-direction. Although there are many obstacles to the re-alization of such interesting effects in photonics, none ofthem are fundamental, and we believe that these unidi-rectional channels could have potentially useful techno-logical applications which could in principle be realizedsomeday through “bandstructure engineering”.

Acknowledgments

This work was supported in part by the U. S. NationalScience Foundation (under MRSEC Grant No. DMR-0213706 at the Princeton Center for Complex Materials).Part of this work was carried out at the Kavli Institute forTheoretical Physics, Santa Barbara, with support fromKITP’s NSF Grant No. PHY99-07949.

∗ Electronic address: sraghu@princeton.edu1 J. D. Joannopoulos et.al. Photonic Crystals: Molding the

Flow of Light. Princeton University Press, 1995.2 R. Marques, J. Martel, F. Mesa, and F. Medina, Phys.

Rev. Lett. 89 183901 (2002).3 Shawn-Yu Lin, Edmund Chow, Vince Hietala, Pierre R.

Villeneuve, Science 282 274 (1998).4 Sundaram and Q. Niu, Phys. Rev. B.5 B. Simon, Phys. Rev. Lett. 51, 2167 (1983).6 M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984).7 M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett.

93, 083901 (2004)

8 F. D. M. Haldane and S. Raghu, cond-mat/0503588.9 Wei-Li Lee, Satoshi Watauchi, V.L. Miller, R. J. Cava, and

N. P. Ong, Science 303, 1647 (2004).10 K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev.

Lett. 45, 494 (1980)11 R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).12 D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).13 M. C. Chang and Q. Niu, Phys. Rev. B 53 7010 (1996).14 D. Thouless, M. Kohmoto, M. Nightingale, M. den Nijs,

Phys. Rev. Lett. 49, 405, (1982).15 F. D. M. Haldane, Phys. Rev. Lett. 61 2015, (1988).16 M. Onoda, N. Nagaosa, Phys. Rev. Lett. 90 206601 (2003).

19

17 C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005)C. L. Kane, E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005)

18 R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154(1954)

19 M. Kohmoto, Ann. Phys. (NY) 160 343 (1985).20 R. C. Booth and E. A. D. White. J. Phys. D: Appl. Phys.,

17 579-587 (1984).21 L. D. Landau and L. P. Lifshitz. Quantum Mechanics:

Non-Relativistic Theory.22 M. Plihal and A. A. Maradudin, Phys. Rev. B 44 16 8565

(1991).23 D. J. Thouless, Topological Quantum Number in Nonrela-

tivistic Physics. World Scientific, 1998.24 We shall temporarily use the inverse permittivity in this

subsection since it shall enable us to derive a simple differ-ential for the scalar magnetic field of the TE modes.

APPENDIX A: FREQUENCY DEPENDENCE OF

THE DIELECTRIC MEDIA

In this section, we shall provide the details of the gen-eralization of the normal mode problem to include thefrequency-dependent response of the media outlined insection II. We shall couple the electromagnetic fields toharmonic oscillator degrees of freedom of the medium.Defining φiσ and πiσ (i = 1, · · ·N , σ = ǫ, µ) to be a setof N independent canonically conjugate oscillator coordi-nates and momenta respectively, which represent internalpolarization and magnetization modes, we consider thetotal Hamiltonian

H = Hem +∑

σ

Hσ , (σ = ǫ, µ), (A1)

where, for instance,

Hǫ =∑

i

Da (αiǫ(r)aπiǫ(r) + βaiǫ(r)φiǫ(r))

+1

2

∑

i

ωiǫ

(

πiǫ(r)2 + φiǫ(r)2)

.

The first term above represents the local coupling be-tween the electric fluxes and the polarization modes,whereas the second term represents the energy of theoscillators themselves. A similar equation exists for themagnetization degrees of freedom coupled with the mag-netic fluxes. The Hamiltonian, as stated in Eq.(A1), isreal-symmetric and positive-definite, and therefore, itseigenvalues are real. The electric and magnetic fields areobtained by varying the Hamiltonian with respect to theassociated flux densities: Ea(r) = δH/δDa(r), Ha(r) =δH/δBa(r)

Ea(r) = ǫ−1ab (r)Db(r)+

∑

n

(αanǫ(r)φnǫ(r) + βa

nǫ(r)πnǫ(r)) ,

(A2)

and similarly for the field Ha. The time-evolution of theoscillator modes are obtained from the Hamilton equa-tions of motion (letting ∂tφnσ = −iωφnσ, etc)

− iωφnǫ(r) =δH

δπnǫ(r)= ωnǫπn(r) + βa

n(r)Da(r) (A3)

iωπnǫ(r) =δH

δφnǫ(r)= ωnǫφnǫ(r) + αa

nǫ(r)Da(r). (A4)

We invert this equation to solve for the oscillator co-ordinates and momenta in terms of the fluxes:

(

φnǫ(r)πnǫ(r)

)

=1

ω2 − ω2n

(

ωn iω−iω ωn

) (

αanǫ(r)βa

nǫ(r)

)

Da(r).

(A5)By substituting Eq.(A5) into the expression for the

electric field (A2), we obtain a correction δǫ−1ab (r ω) to

the permittivity tensor coming from the oscillator modes:

δǫ−1ab (r ω) =

∑

n

(

Γǫab(r)(ω + ωn) − Γ∗ǫ

ab(r)(ω − ωn)

ω2 − ω2n

)

,

(A6)where

Γabǫ (r) = (αa

nǫ(r) − iβanǫ(r))

(

αbnǫ(r) + iβb

nǫ(r))

. (A7)

Finally, the correction term above to the permittivityis expressed in Kramers-Kronig form as

δǫ−1ab (r, ω) =

∑

n

(

Γǫab(r)

ω − ωn− Γǫ∗

ab(r)

ω + ωn

)

. (A8)

The same formal manipulations occur in the frequencydependence of the magnetization modes; in the end, theconstitutive relations are given by a tensor B(r, ω) de-fined by

B(r, ω) =

(

ǫ−1(r, ω) 00 µ−1(r, ω)

)

, (A9)

which is written in Kramers-Kronig form as:

Bab(r, ω) = Sab(r) +∑

n

(

Γab(r)

ω − ωn− Γ∗

ab(r)

ω + ωn

)

. (A10)

The first term, Sab(r) = limω→∞B(r, ω) is the sametensor defining the Hamiltonian in Eq.(14). In the zerofrequency limit,

Bab(r, 0) = Sab(r) −∑

n

(

Γab(r) + Γ∗ab(r)

ωn

)

. (A11)

Stability of the medium imposes the following impor-tant constraint:

B(r, 0) > 0. (A12)

20

Eliminating Sab in Eq.(A10) using Eq.(A11), we get

δB(ω) =∑

n

[

Γ

(

ω

ωn(ω − ωn)

)

+ Γ∗

(

ω

ωn(ω + ωn)

)]

where δB(ω) = B(ω) − B(0). Whereas B(ω) is not apositive-definite matrix, the quantity which is guaranteed

to be positive-definite in lossless frequency ranges is

B(ω) = B(ω) − ω∂

∂ωB(ω) > 0, (A13)

because

B(ω) = B(0) +∑

n

1

ωn

[

Γn

(

ω

ω − ωn

)2

+ Γ∗n

(

ω

ω + ωn

)2]

, (A14)

and Γn, Γ∗n, and B(0) are all positive-definite tensors.

Although B(ω) is not positive-definite, we will be in-terested in cases where

Det(B(ω)) = 0. (A15)

When this condition is satisfied and B(ω) has no zeromodes corresponding to metallic conditions, there is awell defined inverse tensor B−1(ω)

B−1(r, ω) =

(

ǫ(r, ω) 00 µ(r, ω)

)

. (A16)

From the stability condition stated for B(ω), there existsa similar condition for B−1(ω):

B − ω∂

∂ωB = B

(

B−1 + ω∂

∂ωB−1

)

B > 0,

where we have made use of B−1B = 1 and∂/∂ω

(

B−1B)

= 0. Supplementing the inequality abovewith the condition in Eq.(A15), we obtain

B−1(ω) ≡ ∂

∂ω

(

ωB−1(ω))

> 0. (A17)

The eigenvalue problem is solved for each value of thebloch vector k in the first Brillouin zone, and The formalstrategy for obtaining the energy eigenvalues is to solveA|un(k)〉 = ωn(k)B−1(ω(k))|un(k)〉, and then to vary

ω until it coincides with a frequency of an eigenmode.The stability condition (see Eq.(A17)) guarantees thatsuch a prescription enables us to find the entire spec-trum in a lossless range of real frequencies, where B−1 isHermitian.

Indeed, if we consider for the moment the Hermitianproblem

(

A − ωB−1(ω))

|un〉 = λn(ω)|un〉, (A18)

and vary ω to find the zero modes

λn(ω) = 0, (A19)

the stability of such a prescription is guaranteed only if

∂λn

∂ω< 0, (A20)

so that the eigenvalues are monotonically decreasingfunctions of ω. But from first order perturbation the-ory, we know that the requirement above is satisfied onlyif

〈un|d

dω

(

ωB−1(ω))

|un〉 > 0, (A21)

which is precisely equivalent to the condition in Eq.(A17).

When we eliminate the internal oscillator (polari-ton) modes and explicitly substitute the expressions inEq.(A5) into the total Hamiltonian, Eq. (A1), we ob-tain the following quadratic form that involves only theelectromagnetic flux densities:

H =1

2

∑

ij

B−1ij (ω)(ui)

∗uj. (A22)

Our result can be summarized as follows. We beginwith our total Hamiltonian, Eq.(A1), which can be writ-ten as a positive-definite real-symmetric matrix whosestates live in an enlarged Hilbert space containing elec-tromagnetic flux densities and internal oscillator modes.When we “integrate out” the non-resonant internal os-cillator modes of the media, we are left with a set ofeffective constitutive relations of the form

viλ =∑

j

Bij(ωλ)u+jλe

iωλt + c.c., (A23)

and an effective Hamiltonian (which represents the con-served time-averaged energy density of the electromag-netic fields as well as the oscillator modes) that involves

a different tensor Bij(ω) given in A22. Using the relationin Eq.(A13), we can equivalently write the Hamiltonian

21

as

H =1

2

∑

ij

Bij(vi)∗vj . (A24)

For the case of generalized frequency dependence con-sidered here, the normalization of the electromagneticfields are given (up to a scale factor) in terms of thetime-averaged energy density, Eq.(A22):

∑

µν

(

uµ)∗, B−1(ωµ)uν

)

=1

ωµδµν . (A25)

Finally, the matrix B−1 and not B−1 enters the expres-sion for the Berry connection, since it also defines thenormalization of our states.

APPENDIX B: NUMERICAL ALGORITHMS

FOR BANDSTRUCTURE CALCULATIONS

In this section, we shall describe our formulation of thephotonic bandstructure problem which has been used inthe computations of the edge mode spectra.

Since we always neglect absorption/emmision andother non-linear processes of light (i.e. we work within anapproximation of photon number conservation), we seeka real-space Hamiltonian formulation of the bandstruc-ture problem. A real-space method is desirable over ex-isting Fourier-space methods for our purposes; the modeswe are particularly interested in are obtained in domainwall configurations of the Faraday “mass term” as a func-tion of position, and it is most simple and suitable towork within a real space formulation.

In the numerical implementation of a Hamiltonian for-mulation, we shall treat the continuum flux densities〈y| = (D,B) rather than 〈r| = (E,H) as our funda-mental dynamical variables. The former set obey thesource-free Gauss’ relations:

∇ · |y〉 = 0. (B1)

The Hamiltonian of our system is given by the follow-ing quadratic form:

H =1

2

(

D, ǫ−1D)

+1

2

(

B, µ−1B)

. (B2)

Furthermore, the propagating solutions of Maxwell’sequations require the fields to be coupled in non-canonical Poisson bracket relations:

Da(x), Bb(x′) = ǫabc∇cδ3(x − x′) (B3)

The two sets of fields are related by |y〉 = B|r〉, whereB is the matrix of constitutive relations introduced insection II. The source free Maxwell equations are slightvariants of the ones described in section II. Written as ageneralized eigenmode problem of form

AB|y〉 = ω|y〉. (B4)

The matrix A is the imaginary anti-symmetric matrixintroduced in section II, and B = B−1 is a positive-definite Hermitian matrix. The eigenmode problem hereis formally analogous to the problem of a non-canonicalharmonic oscillator with Hamiltonian

H =1

2

∑

ij

Bijyiyj, (B5)

and Poisson brackets

yi, yj = Aij . (B6)

Since A is imaginary and anti-symmetric, its eigenval-ues are either zero, or come in pairs with opposite sign.It is the presence of zero modes which prevents a canon-ical treatment of the problem. In the Maxwell problem,one third of the A matrix eigenvalues are zero modes.

Φ

ΦD

B

FIG. 18: The generic discretization scheme for the photonBand structure problem. Space is broken up into polyhedra.Local electric energy density is defined at the vertices of eachpolyhedron, and the electric fluxes, defined on the edges of thepolyhedron, connect two electric energy sites. The volume ofthe polyhedron is associated with local magnetic energy den-sity, and magnetic fluxes “live” on the faces of the polyhedron.The scheme here has electric-magnetic duality in that a dualpolyhedron can be defined on the vertices of which magneticenergy density defined, etc. The scheme here is inspired bylattice QED, which ensures the correct long wavelength pho-ton dispersion; the only difference here is absence of sources.

In the spatial discretization of this problem, we di-vide space into polyhedral cells, whose vertices containthe local electrical energy density as well as the inversepermittivity tensor ǫ−1

ij (r). The electrical fluxes, ΦD aredefined on the edges of the polyhedron, while the mag-netic fluxes, ΦB are associated with the faces of eachcell. Finally, magnetic energy and the local inverse per-meability tensor µ−1

ij (r) are defined on the centers of eachpolyhedron.

This discretization scheme preserves the self-duality ofthe source-free Maxwell equations in three dimensions;for each such electric polyhedron described above, thereis a dual magnetic polyhedron whose faces correspond tothe edges of the electric polyhedron, and whose centercorresponds to the vertices of the electric polyhedron.

22

The discretized form the A matrix couples electricfluxes to magnetic ones, and vice-versa. The couplingis (see Fig. 19)

ADBij = ΦD

i ,ΦBj = 0,±i. (B7)

ΦB

ΦDΦB

ΦD

ΦB

ΦD

FIG. 19: The discretized form of the A, which contains thePoisson bracket relations of the fluxes. Shown here are exam-ple configurations of ΦD

i , ΦBj = +i (top), −i (middle), and

0 (bottom).

The B matrix couples fluxes of the same type, and de-pends on the geometry of the polyhedra used to discretizespace. For the case of a simple cubic discretization, andfor the electric fluxes (see Fig. 20),

Bii =1

2

(

ǫ−1ii (r1) + ǫ−1

ii (r2))

(B8)

Bij =1

4ǫ−1ji (r2). (B9)

Identical relations involving the inverse permeabilitytensor are constructed for the magnetic fluxes. Withthe present formulation, the complete Hamiltonian ofthe system is expressed as a sum of local terms, H =∑

n h(rn), with

h(rn) =∑

ij

Bij(rn)yiyj . (B10)

Using this method, we can handle the case where the con-stitutive relations have generalized anisotropy, and varyin space.

ij

x

x

1

2

FIG. 20: The discretized form of the B, which contains thecontains the geometric as well as the dynamics information. Itcouples fluxes of the same type only, and allows for anisotropyin the constitutive relations.

We have made use of a simple cubic discretization ofthis general algorithm (although similar implementationsusing BCC and FCC lattices have given the correct countof the long-wavelength photon modes) to compute pho-tonic band structures. The fluxes ΦD and ΦB are madeto obey the generalized Bloch boundary conditions

Φσ(x + R) = eik·RΦσ(x), σ = D,B, (B11)

where R is a lattice translation of the particular photoniccrystal under consideration, and k is a Bloch vector inthe first Brillouin zone. We compute the bandstructure ofthe system by varying the Bloch vector in the boundaryconditions, which introduces Bloch phase factors into afew off-diagonal elements of the A and B matrices andgive rise to the band dispersions.

Both the A and B matrix are sufficiently sparse andare stored as matrix-vector multipliers and are treatedusing a Lanczos algorithm. However, due to the sig-nificantly large number of zero modes, a conventionalLanczos treatment of Eq.(B4) would not converge. TheLanczos adaptation for the Photonic problem is done bymodifying the A matrix to

A → A′ = ABA − 2ω0A, (B12)

A′B|y〉 = ω(ω − 2ω0)|y〉. (B13)

Here, ω0 is the lowest eigenvalue, and the low lying(negative) eigenvalues of the modified problem are nowthe physically relevant ones which are easily found withthe Lanczos implementation. The dimensions of the ma-trices are d = 6N , where N is the number of points usedto discretize the constitutive relations. We have foundsystem sizes up to 106 to be accessible within this ap-proach. Furthermore, local polarization and magnetiza-tion modes can be added to the algorithm.

23

APPENDIX C: DERIVATION OF THE DIRAC

POINT SPLITTING

In this section, we derive a general expression for thefrequency splitting at the Dirac point caused by inver-sion or time-reversal symmetry breaking perturbations.We will use “Bra-ket” notation to represent our eigen-vectors instead of writing equations for each component.We suppose that we know the exact eigenstates of theproblem

A|u0〉 = ωDB−10 |u0〉, (C1)

and that the solutions are two fold degenerate at theDirac point, as for example, in the numerical exampleswe have considered. Now add a perturbation in the con-stitutive relations:

B−1 = B−10 + λB−1

1 . (C2)

This term represents our inversion or time-reversalbreaking perturbation. To find the splitting of the Diracpoint (our “Dirac mass”), we solve the modified problem

A|u〉 = ω(

B−10 + λB−1

1

)

|u〉. (C3)

Since the B−10 matrix is positive-definite, it has a well

defined positive-definite inverse square root matrix B1/20 ,

and we can rewrite the unperturbed problem in the formof a conventional Hermitian eigenvalue problem

B1/20 AB

1/20 |z0〉 = ωD|z0〉, (C4)

where

|z0〉 = B−1/20 |u0〉. (C5)

The new eigenvalue problem with the symmetry break-ing terms is

A|u〉 = ω(

B−10 + λB−1

1

)

|u〉

= ωB−1/20

(

1 + λB1/20 B−1

1 B1/20

)

B−1/20 |u〉,

which subsequently is rewritten in the canonical form as

B1/20

(

A − λωB−11

)

B1/20 |z〉 = ω|z〉, (C6)

where |z〉 = B−1/20 |u〉. The correction to the spectrum

to first order in perturbation theory in the eigenvalueproblem above is then

δω = −ωDλ〈z0|B1/20 B−1

1 B1/20 |z0〉

= −ωDλ〈u0|B−1

1 |u0〉〈u0|B−1

0 |u0〉.

We have restored the normalization factor for the state|u0〉 in the last line above. Thus, our main result here isa general expression for the splitting of the Dirac pointfrequency spectrum, given by the dimensionless quantity

δω

ωD= −λ〈u0|B−1

1 |u0〉〈u0|B−1

0 |u0〉. (C7)