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# Topological Photonics: A Mathematical Perspective

Communicated by *Notices* Associate Editor Reza Malek-Madani

## 1. Introduction

Topological photonics is a framework that follows both condensed matter physics and topology. It refers to designing the guiding properties of the propagating medium (e.g., a photonic crystal or a waveguide lattice) in such a way that the transport of electromagnetic energy is realized in unique, robust, and sometimes unexpected ways. Consider a simple thought experiment: imagine first the two-dimensional wave equation on a square domain, and assume homogeneous Dirichlet boundary conditions. We know that the accessible modes extend in periodic form throughout the whole domain and, in time, waves can propagate in all directions. This behavior is in response to the inherent symmetries of the medium. Imagine instead that we engineer the medium in such a way that all the energy concentrates in the boundary of the medium and propagates in only one direction. (In the language of optics, this would be seen as inhibiting back reflection and making the bulk medium act like an insulator).

Typically, in describing a photonic system, we refer to physical quantities such as frequency, wave vector, polarization, and dispersion. Instead, in the relatively new field of topological photonics, the term “topology” refers to a property of a photonic material that characterizes global behavior of the wavefunctions on their entire dispersion map; most importantly, this property takes quantized values. Think of this as characterizing the “genus” of an object, like a doughnut, with the “object” being described in wave vector space rather than in physical space. There are analogues in photonics to the topological fact that continuous deformations will not change the genus of an object. As an example, photonic topological insulators that are designed using artificial materials can support topologically nontrivial unidirectional states of light. These states are characterized by a particular “genus-like” number. Since this number is quantized, this unidirectional property will be robust to perturbations in the underlying photonic structure.

Photonics research often parallels or aims at explaining phenomena in other physical contexts. Bose-Einstein condensation in condensed matter physics is governed by the Gross-Pitaevskii equation, which is identical to the nonlinear Schrödinger equation that governs intense laser beam propagation in a dielectric medium such as air. In the quantum realm, nontrivial states of two-dimensional matter (e.g., a periodic lattice of atoms) with broken time-reversal symmetry can have the property that the bulk is an insulator while states (modes) exist that carry current along the sample edges without dissipation. The characteristic “genus-like” integer is called the Chern number (see section 3 for an example), which arises out of topological properties of the material’s band structure (see the discussion in section 2 and section 3 below).

In photonic crystals, a periodic variation of the dielectric properties of the medium affects photons in the same manner as solids modulate electrons (with the caveat that photons are bosons, while electrons are fermions). The question is whether the topological features are replicated in the analogous photonic system. In two foundational papers by Haldane and Raghu

Since then, the field of topological photonics has matured and continues to be very active, both in theory and experiments, as well as in the linear and nonlinear regimes. While we have briefly discussed its origins, it is not our purpose to give a detailed history of the field (for this purpose, we point the interested reader to the review articles

## 2. SSH Model

The Su-Schrieffer-Heeger (SSH) model

where is the unit cell, and th is the strength of the cubic nonlinearity. (A rigorous mathematical derivation can be found in

Our analysis follows that of

where is the frequency and is the wavenumber. Equation (2) is periodic in with period since , for any integer The points . define another linear lattice, which is called the reciprocal lattice (Figure 1, bottom). The first Brillouin zone (BZ) is the set of points closer to the origin than any other point of the reciprocal lattice, which in this case is the interval Due to the . the BZ is topologically equivalent to the unit circle -periodicity, .

Substituting the ansatz (2) into (1) and simplifying, we obtain the eigenvalue problem -dependent where , and , is the Hermitian matrix

Since is we only need to consider -periodic, i.e., in the first BZ. We note that since we are posing the problem on the full integer lattice, , can take any value in The eigenvalues of . are

3which is the dispersion relation relating the frequency and the wavenumber Each eigenvalue . is a continuous function of the wavenumber on the first BZ, and is called a band. Since is a matrix, the SSH model has two bands. All of the bands of the system form its band structure, which is illustrated in the left column of Figure 2. (These terms are borrowed from solid state physics, where the band structure describes the energy levels that electrons can occupy in a solid).

When there is a space between the upper and lower bands, which is known as a band gap. This band gap has size , where , The band gap closes when . Since the eigenvalues .(3) are unchanged if and are exchanged, it appears at first glance that the cases and are identical, i.e., that the problem is symmetric about Interestingly, this is not the case. For a complete picture, we need to look at the eigenvectors of . as well.

The normalized eigenvectors corresponding to the eigenvalues are given by

4Since is a complex number of unit modulus whose argument is the same as that of we can write , as

5As the wavenumber varies from to over the BZ, the complex number traces a clockwise circle in the complex plane with center and radius This circle encloses the origin when . but does not when , (Figure 2, right column). The topological invariant is the winding number of which is the number of times , travels counterclockwise around the origin. We can see from Figure 2 that

6where a winding number of represents a single clockwise trip around the origin. (The winding number is undefined if ).

The same topological information can be obtained in a different way by computing a quantity known as the Berry phase

Let be a normalized eigenvector of For any wavenumbers . and we define the relative phase between , and by

7where is the phase, or argument, of the complex number (We are using the Hermitian inner product . where the dagger symbol denotes the conjugate transpose; the complex conjugation is placed on the first component to be consistent with the Dirac notation of quantum mechanics). The relative phase , satisfies the equation

8It is important to note that the eigenvector is not unique. In particular, since it is a unit vector, it is specified only up to multiplication by a constant unit complex number The transformation . is called a gauge transformation. The relative phase is not invariant under a gauge transformation, since if we take , transforms to

thus .

We wish to define the change of phase of in such a way as to be gauge invariant. To do this, we take a sequence of points in ordered in a loop. We then define the discrete Berry phase by -space

which is the phase accumulated by around the loop. Unlike the relative phases the Berry phase is gauge invariant; if we take the gauge transformations , the Berry phase transforms to , which is equal to , since all of the , cancel. We note that the Berry phase is only unique up to an integer multiple of unless we take the principal value of the argument, i.e., restrict to .

We now move from discrete to continuous. In particular, we wish to compute the phase accumulated by along a continuous, closed path. For small let , be the relative phase accumulated between and Following .(8), satisfies the equation

9Since is small and is a unit vector, the denominator in (9) is approximately 1, thus

Expanding both sides in a Taylor series to first order in and and simplifying, we find that is approximately given by

10We define the Berry connection by

11which is the coefficient of on the RHS of (10). We then define the Berry phase to be the integral of Berry connection around a closed contour :

12As in the discrete case, the Berry connection is not gauge invariant, while the Berry phase is invariant under gauge transformations (modulo integer multiples of ).

Returning to the SSH model, we first compute the Berry connection using the eigenvector :

We then compute the Berry phase by integrating the Berry connection from to This is a closed contour in the BZ since the endpoints of . correspond to the same point on the unit circle The Berry phase is .

which is the change in phase of the eigenvector over the BZ. Since it is a constant multiple of (6), it conveys the same information as the winding number.

The fundamental topological difference between the two asymmetric lattice configurations ( and becomes evident when we consider a finite lattice. Specifically, we take a lattice comprising ) waveguides ( unit cells) with Dirichlet boundary conditions at the two ends ( and For the linear system, solutions are standing waves of the form ). where , and represent the and sublattices of the system. Substituting this ansatz into (1) yields the eigenvalue problem where , and is the off-diagonal block matrix

An intuitive understanding of the difference between the two cases can be gained by considering the two extreme configurations, where one of the coupling constants is set to 0. If and the lattice comprises , independent dimers with internal coupling constant The eigenvalues of . are each with multiplicity , On the other hand, if . and the lattice instead comprises , independent dimers (staggered from the ones in the previous case) with internal coupling constant as well as two unconnected nodes at the ends of the lattice. In addition to eigenvalues at , each with multiplicity , the matrix , has two eigenvalues at The eigenvectors corresponding to these zero eigenvalues are . and These are called edge modes, since they are localized at the ends of the lattice. .

Since is a matrix, its spectrum is a discrete set of eigenvalues, as opposed to the two continuous bands of eigenvalues found from the dispersion relation in the infinite lattice case. The eigenvalues of are shown in Figure 3 for , and , The two asymmetric configurations contain a “gap,” which closes when . This eigenvalue gap is analogous to the band gap in the infinite lattice case. When . there are no eigenvalues in this gap, and all of the eigenvectors are nonlocalized. When , however, there are two eigenvalues close to (but not exactly at) 0 which lie within this gap (these eigenvalues approach 0 in the limit , As in the case where ). these eigenvalues correspond to edge modes, since , and are localized to the left and right edges of the lattice, respectively (Figure 3, bottom right). All of the remaining modes are nonlocalized.

Finally, we briefly comment on what occurs when a cubic nonlinearity ( is present (see

## 3. Haldane Model

We now turn to a two-dimensional model. We start with a honeycomb lattice (Figure 4), which is constructed from a two-site unit cell, with sites labeled and These unit cells tile the plane periodically along the two primitive lattice vectors . and to obtain a hexagonal lattice. We note that the and -sites form two offset, triangular sublattices. This structure is similar to that of the material graphene, which is a hexagonal lattice constructed entirely from carbon atoms. The spatial location of a unit cell is specified by the vector -sites where , It is therefore natural to index the unit cells by the vector . the locations of the lattice sites ; and in unit cell are and respectively. Each node in the honeycomb lattice is connected to its three nearest neighbors with coupling strength , which is a coupling between sublattices. The directions of the nearest-neighbor (NN) couplings are given by the vectors , , and , which are depicted in ,Figure 4. The resulting linear model can be written as

13where the angle brackets indicate that the sum is taken over nearest neighbors. The system (13) obeys time-reversal symmetry, i.e., is invariant under the transformation , .

The Haldane model

where the double angle brackets indicate that the sum is taken over next-nearest neighbors. The signs of the NNN couplings are indicated by the arrows in Figure 4, where outward and inward pointing arrows denote couplings of and respectively. The arrangement of the arrows in two staggered, counterclockwise triangles ensures no net flux results from the NNN term. Time-reversal symmetry is broken when , but is unaffected by the on-site term , .

As in the SSH model, the first step is to compute the band structure, which is found by looking for plane wave solutions to (14) of the form

15where