Non-local topological electromagnetic phases of matter
NNon-local topological electromagnetic phases of matter
Todd Van Mechelen and Zubin Jacob ∗ Birck Nanotechnology Center and Purdue Quantum Center,Department of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA
In 2+1D, nonlocal topological electromagnetic phases are defined as atomic-scale media which host pho-tonic monopoles in the bulk band structure and respect bosonic symmetries (e.g. time-reversal T = +1 ).Additionally, they support topologically protected spin-1 edge states, which are fundamentally different thanspin- ⁄ and pseudo-spin- ⁄ edge states arising in fermionic and pseudo-fermionic systems. The striking featureof the edge state is that all electric and magnetic field components vanish at the boundary - in stark contrastto analogs of Jackiw-Rebbi domain wall states. This surprising open boundary solution of Maxwell’s equa-tions, dubbed the quantum gyroelectric effect [Phys. Rev. A , 023842 (2018)], is the supersymmetric partnerof the topological Dirac edge state where the spinor wave function completely vanishes at the boundary. Thedefining feature of such phases is the presence of temporal and spatial dispersion in conductivity (the linearresponse function). In this paper, we generalize these topological electromagnetic phases beyond the continuumapproximation to the exact lattice field theory of a periodic atomic crystal. To accomplish this, we put forth theconcept of microscopic photonic band structure of solids - analogous to the traditional theory of electronic bandstructure. Our definition of topological invariants utilizes optical Bloch modes and can be applied to naturallyoccurring crystalline materials. For the photon propagating within a periodic atomic crystal, our theory showsthat besides the Chern invariant C ∈ Z , there are also symmetry-protected topological (SPT) invariants ν ∈ Z N which are related to the cyclic point group C N of the crystal ν = C mod N . Due to the rotational symme-tries of light R (2 π ) = +1 , these SPT phases are manifestly bosonic and behave very differently from theirfermionic counterparts R (2 π ) = − encountered in conventional condensed matter systems. Remarkably, thenontrivial bosonic phases ν (cid:54) = 0 are determined entirely from rotational (spin-1) eigenvalues of the photon athigh-symmetry points in the Brillouin zone. Our work accelerates progress towards the discovery of bosonicphases of matter where the electromagnetic field within an atomic crystal exhibits topological properties. I. INTRODUCTION
From a material science standpoint, all known topologicalphases of matter to date have been characterized by electronicphenomena [1, 2]. This is true for both time-reversal brokenphases - often called Chern insulators [3–6] and time-reversalunbroken phases - known as topological insulators [7–9]. Thesignature of time-reversal broken phases is the quantum Hallconductivity σ xy = ne /h , which is quantized in terms ofthe electronic Chern invariant n ∈ Z [10–12]. e being theelementary charge of the electron and h the Planck constant.Only recently has the idea of bosonic Hall conductivity andtopological bosonic phases been put forth [13–21].However, it should be emphasized that the traditional Hallconductivity [22, 23] only has topological significance, withrespect to the electron, in the static ω = 0 and local k = 0 lim-its of the electromagnetic field σ xy (0 ,
0) = ne /h . At highfrequency ω (cid:54) = 0 and short wavelength k (cid:54) = 0 , the Hall con-ductivity σ xy ( ω, k ) acquires new physical meaning. We haveshown that the electromagnetic field itself becomes topolog-ical [24, 25] and nonlocal Hall conductivity functions iden-tically to a photonic mass [26–28] in the low-energy physics ω ≈ . These topological electromagnetic phases of matterdepend on the global behavior of σ xy ( ω, k ) , over all frequen-cies and wave vectors.As of yet, only the continuum topological theory of theaforementioned quantum gyroelectric effect (QGEE) has beensolved [24, 25]. Our goal is to extend this concept beyond the ∗ [email protected] long wavelength approximation to the exact lattice field the-ory of optical Bloch waves. In this regime, we must considernot only the first spatial component σ xy ( ω, k ) = σ xy ( ω, k , but all spatial harmonics of the crystal g (cid:54) = 0 , to infinite order, J Hall x ( ω, k ) = (cid:88) g σ xy ( ω, k , g ) E y ( ω, k + g ) . (1) g · R ∈ π Z are the reciprocal lattice vectors and R is theprimitive vector of the crystal. In this case, E i is the micro-scopic electric field. The electromagnetic field must be de-scribed to the same scale as the electronic wave functions,i.e. for photon momenta on the order of the lattice constant ka = π , with a ≈ ˚A. Since topological invariants arefundamentally global properties, these astronomically deepsubwavelength fields actually play a role in the topologicalphysics.The idea of lattice topologies in electromagnetism was firstproposed by Haldane [29, 30] in the context of photonic crys-tals [31–38]. These are artificial materials composed of two ormore different constituents which form a macroscopic crys-talline structure. A few important examples are gyrotropicphotonic crystals [31–33], Floquet topological insulators [39–41] and bianisotropic metamaterials [42–56]. Instead, we fo-cus on the microscopic domain and utilize the periodicity ofthe atomic lattice itself. Thus, the topological invariants in ourtheory are connected to the microscopic atomic lattice and notartificially engineered macroscopic structures. We stress thatin the microscopic case, the electromagnetic theory is man-ifestly bosonic [57–60] (e.g. time-reversal T = +1 ) andcharacterizes topological phases of matter fundamentally dis-tinct from known fermionic and pseudo-fermionic phases. a r X i v : . [ c ond - m a t . s t r- e l ] D ec With that in mind, this paper is dedicated to solving twolongstanding problems, which is of interest to both photonicsand condensed matter physics. The first, is developing the rig-orous theory of optical Bloch modes in natural crystal solids.This problem gained significant interest in the 60’s and 70’sin the context of spatial dispersion (nonlocality) as it lead toqualitatively new phenomena - such as natural optical activity(gyrotropy) [61–64]. The current paper builds on our recentdiscovery of the quantum gyroelectric effect [24, 25] wherewe have shown that nonlocality is also essential for topologi-cal phenomena and is a necessary ingredient in any long wave-length theory. However, since topological field theories areglobal constructs, a complete picture can only be achieved inthe microscopic domain of Bloch waves. Most of the founda-tions have been summarized by Agronovich and Ginzburg intheir seminal monograph on crystal optics [65]. Nevertheless,topological properties have never been tackled to date and afew fundamental quantities, such as the Bloch energy density,have not been defined.This leads to the second problem - deriving the electro-magnetic topological invariants of these systems given onlythe atomic lattice. We solve this problem and also provide asystematic bosonic classification of all 2+1D topological pho-tonic matter. Utilizing the optical Bloch modes, we show thata Chern invariant C ∈ Z can be found for any two-dimensionalcrystal and characterizes distinct topological phases. We thengo one step further and classify these topological phases withrespect to the symmetry group of the crystal - the cyclic pointgroups C N . These are known as symmetry-protected topo-logical (SPT) phases [66–77] and the spin of the photon iscritical to their definition. The rotational symmetries of light R (2 π ) = +1 impart an intrinsically bosonic nature to thesephases, which are fundamentally different than their fermioniccounterparts R (2 π ) = − encountered in conventional con-densed matter systems. We illustrate this fact by directly com-paring SPT bosonic and fermionic phases side-by-side. Ourrigorous formalism of microscopic photonic band structureprovides an immediate parallel with the traditional theory ofelectronic band structure in crystal solids.This article is organized as follows. In Sec. II we developthe general formalism of 2+1D lattice electromagnetism. Firstwe derive the generalized linear response function - account-ing for spatiotemporal dispersion to infinite order in the crys-tal’s spatial harmonics g . Thereafter, we find the equivalentHamiltonian that governs all light-matter Bloch excitationsof the material. In Sec. III we study the discrete rotationalsymmetries (point groups) of the crystal and the implicationson spin-1 quantization [78–83] of the photon. The followingSec. IV discusses the electromagnetic Chern number and itsrelationship to symmetry-protected topological (SPT) bosonicphases. The bosonic classification of each phase is related di-rectly to integer quantization of the photon [Tab. I] and thisis compared alongside their fermionic counterparts [Tab. II].Sec. V presents our conclusions.The focus of this paper is 2+1D topological electromag-netic (bosonic) phases of matter C (cid:54) = 0 which requires break-ing time-reversal symmetry. These bosonic Chern insulatorsare ultimately related to nonlocal gyrotropic response (Hall conductivity) and show unidirectional, completely transverseelectro-magnetic (TEM) edge states [24, 25]. However, time-reversal symmetric topological phenomena can arise in higherdimensional systems in the context of nonlocal magnetoelec-tricity [84]. These time-reversal symmetric phases possesscounter-propagating TEM edge states and are interpreted astwo copies of a bosonic Chern insulator. Features of topo-logical phenomena, such as spin-momentum locking [85–89],have also been reported in conventional surface state prob-lems - surface plasmon-polaritons (SPPs), Dyakonov waves,etc. However, these traditional surface properties are not con-nected to any topologically protected edge states or nontrivialphases. Note:
Due to the frequency of integral formulas, all dif-ferential elements are assumed to be correctly normalized,such as Fourier transforms dω/ √ π → dω and Fourier series e i g · r / √ V → e i g · r , where V is the unit cell area of a crystal. II. LATTICE ELECTROMAGNETISMA. 2+1D electrodynamics
In this paper we focus on two-dimensional materials andthe topological electromagnetic phases associated with them.The preliminaries for 2+1D electromagnetism can be found inAppendix A of Ref. [24]. Conveniently, the restriction to 2Dlimits the degrees of freedom of both the electromagnetic fieldand the induced response of the material, such that strictlytransverse-magnetic (TM) waves propagate. The correspond-ing wave equation reads, H f = i∂ t g, f = E x E y H z , g = D x D y B z . (2) f is the TM polarization state of the electromagnetic field andthe material response is captured by the displacement field g . H ( p ) = p · S are the vacuum Maxwell equations in realspace and describe the dynamics of the free photon, H ( p ) = p x ˆ S x + p y ˆ S y = − p y p x − p y p x . (3) p = − i ∇ is the two-dimensional momentum operator. ˆ S x and ˆ S y are spin-1 operators that satisfy the angular momentumalgebra [ ˆ S i , ˆ S j ] = i(cid:15) ijk ˆ S k , ˆ S z = − i i . (4)Here, ( ˆ S z ) ij = − i(cid:15) ijz is the generator of rotations in the x - y plane and is represented by the antisymmetric matrix. Intwo dimensions, ˆ S z governs all rotational symmetries of theelectromagnetic field. TABLE I. Summary of 2+1D topological electromagnetic (bosonic) phases. Symmetry-protected topological (SPT) bosonic phases exist inall cyclic point groups C N =2 , , , . The continuous group C ∞ describes the long wavelength theory k ≈ . The topological phases arecharacterized by their Chern invariant C ∈ Z and SPT invariant ν ∈ Z N . These numbers are not independent - but intimately related by thesymmetries of the crystal: ν = C mod N . ν is protected by N -fold rotational symmetry and determines the Chern number up to a factor of N . The bosonic classification of ν represents the direct product of rotational eigenvalues ( η N ) N = +1 (roots of unity) of the electromagneticfield at high-symmetry points (HSPs) in the Brillouin zone. For the spin-1 photon, this classification is more intuitively understood in terms ofmodulo integers m N ∈ Z N , which determine the N possible eigenvalues of η N = exp (cid:2) i πN m N (cid:3) .Point group, C N Symmetry, Z N Bosonic classification, ( η N ) N = +1 Boson SPT invariant, ν = C mod NC - - - C Z exp ( i π C /
2) = η (Γ) η ( X ) η ( Y ) η ( M ) ν = m (Γ) + m ( X ) + m ( Y ) + m ( M ) mod 2 C Z exp ( i π C /
3) = η (Γ) η ( K ) η ( K (cid:48) ) ν = m (Γ) + m ( K ) + m ( K (cid:48) ) mod 3 C Z exp ( i π C /
4) = η (Γ) η ( M ) η ( Y ) ν = m (Γ) + m ( M ) + 2 m ( Y ) mod 4 C Z exp ( i π C /
6) = η (Γ) η ( K ) η ( M ) ν = m (Γ) + 2 m ( K ) + 3 m ( M ) mod 6 C ∞ Z exp ( iθ C ) = η θ (0) η ∗ θ ( ∞ ) , η θ = exp( iθm ) ν = C = m (0) − m ( ∞ ) TABLE II. Summary of 2+1D SPT fermionic phases for comparison. The fermionic classification of ν represents the direct product ofrotational eigenvalues ( ζ N ) N = − (roots of negative unity) of the spinor field at HSPs in the Brillouin zone. For the spin- ⁄ electron, thisclassification is more intuitively understood in terms of modulo half-integers m N ∈ Z N + / , which determine the N possible eigenvaluesof ζ N = exp (cid:2) i πN m N (cid:3) .Point group, C N Symmetry, Z N Fermionic classification, ( ζ N ) N = − Fermion SPT invariant, ν = C mod NC - - - C Z exp ( i π C /
2) = ζ (Γ) ζ ( X ) ζ ( Y ) ζ ( M ) ν = m (Γ) + m ( X ) + m ( Y ) + m ( M ) mod 2 C Z exp ( i π C /
3) = − ζ (Γ) ζ ( K ) ζ ( K (cid:48) ) ν = m (Γ) + m ( K ) + m ( K (cid:48) ) + / mod 3 C Z exp ( i π C /
4) = − ζ (Γ) ζ ( M ) ζ ( Y ) ν = m (Γ) + m ( M ) + 2 m ( Y ) + 2 mod 4 C Z exp ( i π C /
6) = − ζ (Γ) ζ ( K ) ζ ( M ) ν = m (Γ) + 2 m ( K ) + 3 m ( M ) + 3 mod 6 C ∞ Z exp ( iθ C ) = ζ θ (0) ζ ∗ θ ( ∞ ) , ζ θ = exp( iθm ) ν = C = m (0) − m ( ∞ ) B. 2+1D linear response theory
The effective electromagnetic properties of a material arevery accurately described by a linear response theory - assum-ing nonlinear interactions are negligible. This is true for lowintensity light | f | (cid:47) V / m that is sufficiently weak com-pared to the atomic fields governing the binding of the crystalitself. Our goal is to characterize the entire topological fieldtheory in this regime. With this in mind, the most general lin-ear response of a 2D material is nonlocal in both space andtime coordinates, g ( t, r ) = ˆ d r (cid:48) ˆ t −∞ dt (cid:48) M ( t, t (cid:48) , r , r (cid:48) ) f ( t (cid:48) , r (cid:48) ) . (5) M is the response function and compactly represents the con-stitutive relations in space-time, M ( t, t (cid:48) , r , r (cid:48) ) = ε xx ε xy χ x ε yx ε yy χ y ζ x ζ y µ . (6)Note that M is a × dimensional matrix and we include allpossible material responses as a generalization, for instancemagnetism µ and magnetoelectricity χ i , ζ i .If the properties of the crystal are not changing temporally(no external modulation), the response function is translation-ally invariant in time, M ( t, t (cid:48) , r , r (cid:48) ) = M ( t − t (cid:48) , r , r (cid:48) )= ˆ dω M ( ω, r , r (cid:48) ) e − iω ( t − t (cid:48) ) . (7) Equation (7) implies energy conservation in Hermitian sys-tems ω (cid:48) = ω . However, a crystal is not translationally invari-ant in space - momentum is not conserved k (cid:48) (cid:54) = k . Instead, thecrystal is periodic and possesses discrete translational symme-try [65, 90], M ( ω, r , r (cid:48) ) = M ( ω, r + R , r (cid:48) + R ) , (8)where R is the primitive lattice vector of the crystal. Thisadmits a Fourier decomposition in the spatial harmonics ofthe crystal g , M ( ω, r , r (cid:48) ) = (cid:88) g M g ( ω, r − r (cid:48) ) e − i r (cid:48) · g , (9)with g · R ∈ π Z arbitrary integer combinations of the recip-rocal lattice vectors.Due to nonlocality, it is necessary to convert to the recipro-cal space, M ( ω, k , k (cid:48) ) = ¨ d r d r (cid:48) M ( ω, r , r (cid:48) ) e − i k · r e i k (cid:48) · r (cid:48) . (10) M ( ω, k , k (cid:48) ) determines the linear transformation propertiesof an input wave with momentum k (cid:48) to an output wave withmomentum k . In a periodic crystal, the momentum is con-served up to a reciprocal vector k (cid:48) = k + g and represents adiscrete spectrum, M ( ω, k , k (cid:48) ) = (cid:88) g M g ( ω, k ) δ ( k + g − k (cid:48) ) . (11) δ ( k + g − k (cid:48) ) is the momentum conserving delta function.Each Fourier element of the response function M g ( ω, k ) de-termines the polarization dependent scattering amplitude from k + g → k . These are essentially the photonic structure fac-tors of the two-dimensional crystal.In this case, k is the crystal momentum and is only uniquelydefined within the Brillouin zone (BZ). Hence, the electro-magnetic eigenstates of the medium are Bloch waves, H ( k ) f k = ω ˆ d k (cid:48) M ( ω, k , k (cid:48) ) f k (cid:48) = ω (cid:88) g M g ( ω, k ) f k + g . (12) H ( k ) = k · S are the vacuum Maxwell equations in mo-mentum space. The Bloch photonic wave function f ( k , r ) = (cid:104) r | f k (cid:105) corresponds to the net propagation of all k + g scatteredwaves in the medium, f ( k , r ) = (cid:88) g f k + g e i g · r , (13)where f ( k , r + R ) = f ( k , r ) is periodic in the crystal lattice.Note that Eq. (12) and (13) reduce to the continuum theory[24, 25] when considering only the 0th order harmonic g = 0 . C. Generalized response function
Nevertheless, Eq. (12) poses a few serious problems; it doesnot represent a proper first-order in time Hamiltonian since allharmonics of the response function M g ( ω, k ) depend on theeigenvalue ω . Moreover, it is not evident that the Bloch wavesin Eq. (13) are normalizable, as the system contains complexspatial and temporal dispersion. Due to these issues, it is ad-vantageous to return to the more general form of M ( ω, k , k (cid:48) ) without assuming discrete translational symmetry. This willallow us to derive very robust properties of the response func-tion that can also be applied to amorphous materials or qua-sicrystals.First, we demand Hermiticity, M ( ω, k , k (cid:48) ) = M † ( ω, k (cid:48) , k ) , (14)such that the response is lossless. To account for normalizableelectromagnetic waves, the energy density must be positivedefinite for all ω , U ( ω ) = ¨ d k d k (cid:48) f † k ¯ M ( ω, k , k (cid:48) ) f k (cid:48) > , (15)where ¯ M describes the inner product space in a dispersivemedium, ¯ M ( ω, k , k (cid:48) ) = ∂∂ω [ ω M ( ω, k , k (cid:48) )] . (16)Notice that U ( ω ) = U ∗ ( ω ) is only real-valued when M isHermitian. For realistic materials, the energy density is alsostable at static equilibrium ω = 0 , U (0) = ¨ d k d k (cid:48) f † k M (0 , k , k (cid:48) ) f k (cid:48) > , (17) with M (0 , k , k (cid:48) ) = ¯ M (0 , k , k (cid:48) ) at zero frequency. To en-sure the electromagnetic field is real-valued, i.e. represents aneutral particle, we always require the reality condition, M ( ω, k , k (cid:48) ) = M ∗ ( − ω, − k , − k (cid:48) ) . (18)Furthermore, the response is transparent at high frequency ω → ∞ , as the material cannot respond to sufficiently fasttemporal oscillations, lim ω →∞ M ( ω, k , k (cid:48) ) = δ k − k (cid:48) . (19) is the × identity matrix and δ k − k (cid:48) = δ ( k − k (cid:48) ) is themomentum conserving delta function. Lastly, the responsemust be causal and satisfy the Kramers-Kronig relations.Combining all the above criteria, we find that M can al-ways be decomposed as a discrete summation of oscillators[29, 52, 91], M ( ω, k , k (cid:48) ) = δ k − k (cid:48) − (cid:88) α ˆ d k (cid:48)(cid:48) C † α k (cid:48)(cid:48) k C α k (cid:48)(cid:48) k (cid:48) ω α k (cid:48)(cid:48) ( ω − ω α k (cid:48)(cid:48) ) . (20)Any Hermitian (lossless) response function can be expressedin this form. Equation (20) is easily extended to 3D materi-als but our focus is on 2D topological field theories. In thiscase, α labels an arbitrary bosonic excitation in the material,such as an exciton or phonon, which couples linearly to theelectromagnetic fields via the × tensor, C α ( k , k (cid:48) ) = ¨ d r d r (cid:48) C α ( r , r (cid:48) ) e − i k · r e i k (cid:48) · r (cid:48) . (21) ω α k is the resonant energy of the oscillator and correspondsto a first-order pole of the response function. Notice that M itself contains an integral over k (cid:48)(cid:48) . Microscopically, this con-stitutes the overlap with the electronic momentum to infinites-imally small scale k → ∞ .Substituting Eq. (20) into Eq. (15), we can exchange theorder of integration U ( ω ) = ´ d k U ( ω, k ) and define, U ( ω, k ) = | f k | + (cid:88) α (cid:12)(cid:12)(cid:12)(cid:12) ˆ d k (cid:48) C α kk (cid:48) f k (cid:48) ( ω − ω α k ) (cid:12)(cid:12)(cid:12)(cid:12) > , (22)which is positive definite for all ω and k . Equation (22) isthe generalized inner product for the electromagnetic field andrepresents the energy density at an arbitrary frequency andwave vector. We will now show that Eq. (20) is derived froma first-order in time Hamiltonian. D. Generalized Hamiltonian
To find the corresponding Hamiltonian, we expand the re-sponse function M in terms of three-component matter oscil-lators ψ α . These represent internal polarization and magneti-zation modes of the material, ωψ α k = ω α k ψ α k + ˆ d k (cid:48) C α kk (cid:48) f k (cid:48) . (23)Substituting Eq. (23) and (20) into Eq. (12) we obtain, ωf k = H ( k ) f k + (cid:88) α ¨ d k (cid:48)(cid:48) d k (cid:48) ω α k (cid:48)(cid:48) C † α k (cid:48)(cid:48) k C α k (cid:48)(cid:48) k (cid:48) f k (cid:48) + (cid:88) α ˆ d k (cid:48) C † α k (cid:48) k ψ α k (cid:48) . (24) The first two terms on the right hand side of Eq. (24) representthe vacuum equations and self-energy of the electromagneticfield. The third term is the linear coupling to the oscillators.Combining Eq. (23) and (24) into a single algebraic matrix,we write the generalized Hamiltonian H ( k , k (cid:48) ) as, H ( k , k (cid:48) ) = H ( k ) δ k − k (cid:48) + (cid:80) α ´ d k (cid:48)(cid:48) ω α k (cid:48)(cid:48) C † α k (cid:48)(cid:48) k C α k (cid:48)(cid:48) k (cid:48) C † k (cid:48) k C † k (cid:48) k . . . C kk (cid:48) ω k δ k − k (cid:48) . . . C kk (cid:48) ω k δ k − k (cid:48) . . . ... ... ... . . . , (25)which is manifestly Hermitian H ( k , k (cid:48) ) = H † ( k (cid:48) , k ) .We now define u k as the generalized state vector of theelectromagnetic problem; accounting for the photon f k andall possible internal excitations ψ α k , ˆ d k (cid:48) H kk (cid:48) u k (cid:48) = ωu k , u k = f k ψ k ψ k ... , (26)which is a first-order wave equation. Notice that contractionof u k naturally reproduces the energy density [Eq. (22)] uponsummation over all degrees of freedom, u † k u k = | f k | + (cid:88) α | ψ α k | = U ( ω, k )= | f k | + (cid:88) α (cid:12)(cid:12)(cid:12)(cid:12) ˆ d k (cid:48) C α kk (cid:48) f k (cid:48) ( ω − ω α k ) (cid:12)(cid:12)(cid:12)(cid:12) . (27)The complete set of eigenvectors and eigenvalues is repre-sented by u k . We must define all relevant electromagneticquantities in terms of this generalized state vector. E. Crystal Hamiltonian
We are now ready to enforce crystal periodicity. Instead ofexpanding M directly, we utilize the periodicity of the cou-pling tensors C α ( r , r (cid:48) ) = C α ( r + R , r (cid:48) + R ) , which is a dis-crete spectrum in g , C α ( k , k (cid:48) ) = (cid:88) g C α g ( k ) δ ( k + g − k (cid:48) ) . (28) C α g ( k ) tells us the scattering amplitude of a photon f k + g withmomentum k + g into an internal mode of the material ψ α k at momentum k , and vice versa. The crystal Hamiltonian ac-counts for all such scattering events, H ( k , k (cid:48) ) = (cid:88) g H g ( k ) δ ( k + g − k (cid:48) ) , (29)with Hermiticity H g ( k ) = H †− g ( k + g ) satisfied by defini-tion. Note, the resonant energies ω α ( k + g ) = ω α ( k ) are generally periodic in k , since they correspond to energy gapsin the electronic band structure. However, we do not need toassume this to define the optical Bloch excitations. A periodiccoupling is sufficient.The quasiparticle eigenstates of this Hamiltonian describethe complete spectrum of Bloch waves, (cid:88) g H g ( k ) u n k + g = ω n k u n k , ω n ( k + g ) = ω n ( k ) , (30)and the eigenenergies ω n k are periodic Bloch bands. n la-bels a particular energy band of the material with its associ-ated Bloch eigenstate | u n k (cid:105) . The total wave function | u n k (cid:105) contains the photon | f n k (cid:105) and all internal degrees of freedomdescribing the linear response | ψ nα k (cid:105) . This is expressed com-pactly in the Fourier basis u n ( k , r ) = (cid:104) r | u n k (cid:105) , u n ( k , r ) = (cid:88) g u n k + g e i g · r , u n k + g = f n k + g ψ n k + g ψ n k + g ... , (31)where u n ( k , r + R ) = u n ( k , r ) is periodic in the crystal lat-tice. In this basis, | u n k (cid:105) is normalized to the energy densityas, (cid:104) u n k | u n k (cid:105) = (cid:88) g u † n k + g u n k + g = (cid:88) g (cid:32) f † n k + g f n k + g + (cid:88) α ψ † nα k + g ψ nα k + g (cid:33) = (cid:88) gg (cid:48) f † n k + g ¯ M g (cid:48) − g ( ω n k , k + g ) f n k + g (cid:48) . (32)The bra-ket notation (cid:104)|(cid:105) implies integration over the 2D unitcell and we have utilized the linear response theory to express ψ α in terms of the driving field f , ψ nα k + g = (cid:80) g (cid:48) C α g (cid:48) ( k + g ) f n k + g (cid:48) + g ω n k − ω α k + g . (33) ¯ M g ( ω, k ) = ∂ ω [ ω M g ( ω, k )] is the contribution to the en-ergy density arising from each spatial harmonic of the crystal.Finally, the eigenenergies ω n k are the n nontrivial roots ofthe characteristic wave equation, H ( k ) f n k = ω n k (cid:88) g M g ( ω n k , k ) f n k + g , (34)which generates all possible photonic bands of the crystal.Note, the response function M g ( ω, k ) is now expressed interms of C α g ( k ) and describes the net summation of all scat-tering and back-scattering events in the material, M g ( ω, k ) = δ g − (cid:88) α g (cid:48) C † α − g (cid:48) ( k + g (cid:48) ) C α g − g (cid:48) ( k + g (cid:48) ) ω α k + g (cid:48) ( ω − ω α k + g (cid:48) ) . (35)This proves that the wave equation is derived from a first-orderHamiltonian, has real eigenvalues ω = ω n k for all momenta,and is normalizable in terms of | u n k (cid:105) . III. DISCRETE ROTATIONAL SYMMETRYA. Point groups in 2D
Point groups are the discrete analogs of continuous rota-tions and reflections. They represent the number of ways theatomic lattice can be transformed into itself [92, 93]. Due tothe crystallographic restriction theorem (CRT), there are tensuch point groups in 2D. The first five are the cyclic groups C N , C , C , C , C , C . (36)For instance, C implies threefold cyclic symmetry while C is no symmetry. The last five are the dihedral groups D N , D , D , D , D , D . (37)The dihedral group D N contains C N plus reflections. How-ever, it can be proven that the Chern number for all D N pointgroups vanish [69]. Therefore, we concern ourselves withonly the cyclic groups C N . The Brillouin zone of each pointgroup is displayed in Fig. 1.The defining characteristic of each cyclic group is thefermionic or bosonic representation. When we rotate the fieldsby π , we take the particle into itself and acquire a phase, R (2 π ) = ( − F . (38) F is twice the total spin of particle, or equivalently, thefermion number. Fermions with half-integer spin are anti-symmetric under rotations R (2 π ) = − , while bosons withinteger spin are symmetric R (2 π ) = +1 . Depending on thesymmetries of the lattice, the topology fundamentally changesfor fermions and bosons. We will understand the implicationsthis has for spin-1 photons. FIG. 1. Brillouin zone of each cyclic point group C N . (a), (b),(c), (d), and (e) correspond to N = 2 , , , , and ∞ respectively.Due to rotational symmetry, the total Brillouin zone is equivalentto N copies of the irreducible Brillouin zone (IBZ), which is rep-resented by the blue quadrant. For continuous symmetry N = ∞ ,this is simply a line. The yellow circles label high-symmetry points R k i = k i where the crystal Hamiltonian is invariant under a certainrotation ˆ R . At these specific momenta, a Bloch photonic wave func-tion ˆ R N | f ( k i ) (cid:105) = η N ( k i ) | f ( k i ) (cid:105) is an eigenstate of a an N -foldrotation η N ( k i ) = (cid:2) i πN m N ( k i ) (cid:3) such that the photon possessesquantized integer eigenvalues m N ( k i ) ∈ Z N . Since m N are dis-crete quantum numbers, their values cannot vary continuously if thecrystal symmetry is preserved - they can only be changed at a topo-logical phase transition. B. Spin-1 discrete symmetries
If the two-dimensional crystal belongs to a cyclic pointgroup C N , the Hamiltonian possesses discrete rotational sym-metry about the z -axis, R − H R g ( R k ) R = H g ( k ) , ω n ( R k ) = ω n ( k ) , (39)where R is any rotation in C N . It is important to note that R is diagonal in u , meaning the photon and each oscillator isrotated individually, f → R f and ψ α → R ψ α . This impliesthere is no mixing of fields. The symmetries of the Hamilto-nian are endowed by the coupling tensors, which dictates thedegrees of freedom of the material response, R − C α R g ( R k ) R = C α g ( k ) , ω α ( R k ) = ω α ( k ) . (40)After summation over all C α g ( k ) , we can prove that the re-sponse function transforms identically under such a rotation, R − M R g ( ω, R k ) R = M g ( ω, k ) . (41)Therefore, the photon inherits all symmetries of the crystal.In this case, the R matrix represents a discrete rotation andcan be expressed as the exponential of the spin-1 generator ( ˆ S z ) ij = − i(cid:15) ijz , R N = exp (cid:18) i πN ˆ S z (cid:19) = cos πN sin πN − sin πN cos πN
00 0 1 , (42)where π / N is an N -fold rotation. We stress that every cyclicgroup for the photon is a vector representation, which is bosonic , R (2 π ) = R ( N θ N ) = ( R N ) N = + . (43)The electromagnetic field returns in phase under cyclic revo-lution. C. High-symmetry points
The Bloch eigenstates | u n k (cid:105) are essentially a collection ofperiodic vector fields. To rotate the fields, we must performan operation on both the coordinates r and the polarizationstates f and ψ α . In real space, the operation of a rotation ˆ R ispreformed as, (cid:104) r | ˆ R| u n k (cid:105) = R u n ( k , R − r ) = η n ( k ) u n ( R k , r ) , (44)where R is a discrete rotation defined in Eq. (42). This impliesthe Fourier coefficients obey, R u n k + R − g = η n ( k ) u n R k + g . (45)It follows from symmetry that the operation of ˆ R takes a wavefunction at k to R k with the same energy ω n ( k ) = ω n ( R k ) -but with a possibly different phase | η n ( k ) | = 1 . Utilizing thelinear response theory, we notice that the phase factor η n ( k ) is governed entirely by the photon, R ψ nα k + R − g = (cid:80) g (cid:48) RC α g (cid:48) ( k + R − g ) f n k + g (cid:48) + R − g ω n k − ω α k + R − g = (cid:80) g (cid:48) C α R g (cid:48) ( R k + g ) R f n k + g (cid:48) + R − g ω n k − ω α k + R − g = (cid:80) g (cid:48) C α g (cid:48) ( R k + g ) η n ( k ) f n R k + g (cid:48) + g ω n k − ω α R k + g = η n ( k ) ψ nα R k + g . (46)This is an incredibly convenient simplification and implies theprecise coordinates of the matter oscillations ψ α are superflu-ous when discussing symmetries. The electromagnetic field f tells us everything. Importantly, there are specific points in the Brillouin zonewhere k is invariant under a discrete rotation, R k i = k i . (47)This is because the crystal momentum only differs by a latticetranslation at these points R k i = k i + g , which leaves a Blochwave function unchanged, e i R k i · r u n ( R k i , r ) = e i ( k i + g ) · r u n ( k i + g , r )= e i k i · r u n ( k i , r ) . (48)These are called high-symmetry points (HSPs); they occur atthe center and certain vertices of the Brillouin zone. The crys-tal Hamiltonian is rotationally invariant at these momenta -i.e. it commutes with ˆ R . Therefore, the wave functions aresimultaneous eigenstates of ˆ R at HSPs, ˆ R| u n ( k i ) (cid:105) = η n ( k i ) | u n ( k i ) (cid:105) , (49)which immediately implies, ˆ R| f n ( k i ) (cid:105) = η n ( k i ) | f n ( k i ) (cid:105) . (50)Here, η n ( k i ) is the eigenvalue of ˆ R at k i for the n th band. D. Spin-1 eigenvalues
Depending on the point group and the precise HSP, η n ( k i ) = η N,n ( k i ) can represent any N th root of unity cor-responding to the rotation operator ˆ R N , η N,n ( k i ) = exp (cid:20) i πN m N,n ( k i ) (cid:21) , ( η N,n ) N = +1 . (51) m N,n ( k p ) ∈ Z N is a modulo integer - it labels the N pos-sible spin-1 eigenvalues at k i . In C for example, the Γ andM points are invariant under ˆ R rotations, while the X andY points are invariant under ˆ R rotations (inversion). Thismeans there are 4 possible spin-1 charges located at m ,n (Γ) & m ,n ( M ) ∈ Z respectively and 2 possible charges locatedat m ,n ( X ) = m ,n ( Y ) ∈ Z . A visualization of these topo-logical charges is presented in Fig. 2 and this is contrastedwith their fermionic counterparts in Fig. 3. In Sec. IV we willconnect these rotational eigenvalues directly to the topologicalinvariants. IV. TOPOLOGICAL ELECTROMAGNETIC (BOSONIC)PHASES OF MATTERA. Electromagnetic Chern number
The Berry connection for a band n is found by varying thetotal Bloch wave function | u n k (cid:105) with respect to the momen-tum, A n ( k ) = − i (cid:104) u n k | ∂ k u n k (cid:105) = − i (cid:88) g u † n k + g ∂ k u n k + g . (52) FIG. 2. The collection of spin-1 (bosonic) charges for the C point group. (a) Fourfold rotations ( R ) = +1 ; there are four uniqueeigenvalues η = exp (cid:2) i π m (cid:3) corresponding to the roots of unity ( η ) = +1 . These represent the modulo 4 integers m ∈ Z . Note that m = 3 = − can also be interpreted as a left-handed eigenstate. (b) Bosonic inversion ( R ) = +1 ; there are two unique eigenvalues η = exp (cid:2) i π m (cid:3) corresponding to the roots of unity ( η ) = +1 . These represent the modulo 2 integers m ∈ Z .FIG. 3. The collection of spin- ⁄ (fermionic) charges for the C point group. (a) Fourfold rotations ( R ) = − ; there are four uniqueeigenvalues ζ = exp (cid:2) i π m (cid:3) corresponding to the roots of negative unity ( ζ ) = − . These represent the modulo 4 half-integers m ∈ Z + / . Note that m = / = − / can be interpreted as a spin-down fermion while m = / = / + 1 and m = / = − / + 3 constitute a fermion plus a boson. (b) Fermionic inversion ( R ) = − ; there are two unique eigenvalues ζ = exp (cid:2) i π m (cid:3) correspondingto the roots of negative unity ( ζ ) = − . These represent the modulo 2 half-integers m ∈ Z + / . Note that m = / = − / can alsobe interpreted as a spin-down fermion under modulo 2. This can be simplified slightly to obtain, A n ( k ) = − i (cid:88) gg (cid:48) f † n k + g ¯ M g (cid:48) − g ( ω n k , k + g ) ∂ k f n k + g (cid:48) + (cid:88) gg (cid:48) f † n k + g AAA g (cid:48) − g ( ω n k , k + g ) f n k + g (cid:48) . (53)The first term gives the Berry connection of the photon, whilethe second term AAA g ( ω, k ) arises solely from the matter oscil-lations, AAA g ( ω, k ) = − i (cid:88) α g (cid:48) C † α − g (cid:48) ( k + g (cid:48) ) ∂ k C α g − g (cid:48) ( k + g (cid:48) )( ω − ω α k + g (cid:48) ) . (54)Due to nonlocality, Eq. (54) does not generally vanish. Thisadditional contribution to the Berry phase corresponds to vor-tices in the response function itself - independent of the Berrygauge of the photon. This means the Chern number can benonzero C n (cid:54) = 0 even if the winding of electromagnetic fieldis trivial. However, we will show in the proceeding sectionsthat all symmetry constraints on the Chern number can be es-tablished entirely in terms of the photon. As can be seen from Eq. (53), the Berry connection is onlydefined within the Brillouin zone A n k + g = A n k + ∂ k χ n k ,up to a possible U(1) gauge. Hence, the gauge invariant Berrycurvature is periodic F n k + g = F n k , F n ( k ) = ˆz · [ ∂ k × A n ( k )] . (55)The Chern number is found by integrating the Berry curvatureover the two-dimensional Brillouin zone, C n = 12 π ˆ BZ F n ( k ) d k , C n ∈ Z , (56)which determines the winding number of the collective light-matter excitations over the torus T = S × S . Equation (56)is one of the central results of this paper. An electromagneticChern invariant can be found for any 2D crystal and charac-terizes distinct topological phases of matter C n (cid:54) = 0 . B. Symmetry-protected topological bosonic phases
Nevertheless, even if we knew the specifics of the mate-rial, evaluating the Chern number by brute force would be a
FIG. 4. Examples of SPT bosonic phases in a crystal with C symmetry. These phases are characterized by their SPT invariant ν = m (Γ) + m ( M ) + 2 m ( Y ) mod 4 which determines the electromagnetic Chern number up to a multiple of 4. Here, m ∈ Z and m ∈ Z aremodulo integers. (a), (b), (c) and (d) correspond to SPT bosonic phases of ν = 3 , , and 0 respectively. For bosons, we simply add upall the integer charges within the irreducible Brillouin zone. For instance, the ν = 2 phase has eigenvalues of m (Γ) = 1 at the center and m ( M ) = 3 = − at the vertices, with inversion eigenvalues of m ( Y ) = m ( X ) = 1 at the edge centers: ν = 1 + 3 + 2 × .FIG. 5. Examples of SPT fermionic phases in a crystal with C symmetry. These phases are characterized by their SPT invariant ν = m (Γ) + m ( M ) + 2 m ( Y ) + 2 mod 4 which determines the electronic Chern number up to a multiple of 4. In this case, m ∈ Z + / and m ∈ Z + / are modulo half-integers. (a), (b), (c) and (d) correspond to SPT fermionic phases of ν = 3 , , and 0 respectively.The problem is more complicated for fermions because the charges are fractional and we must also account for the antisymmetric phases of aspinor wave function. As an example, the ν = 2 phase has eigenvalues of m (Γ) = m ( M ) = / at the center and vertices, with inversioneigenvalues of m ( Y ) = m ( X ) = / = − / at the edge centers: ν = / + / + 2 × / + 2 = 2 . herculean task. Instead, we invoke constraints of the pointgroups, which constitute a type of symmetry-protected topo-logical (SPT) phase [66–77]. SPT phases are protected by the N -fold rotational symmetry of C N and this gives rise to anadditional topological invariant ν n ∈ Z N . Remarkably, ν n is classified entirely from η n ( k i ) eigenvalues at HSPs and re-quires no complicated integration to compute. This invariantis related to the Chern number up to a multiple of N , ν n = C n mod N, C n ∈ N Z + ν n . (57)The interpretation of ν n is quite simple - it tells us the geo-metric phase around the irreducible Brillouin zone (IBZ) ofthe crystal, exp (cid:18) i πN C n (cid:19) = exp (cid:18) i ˆ IBZ F n ( k ) d k (cid:19) = exp (cid:18) i ˛ ∂ IBZ A n ( k ) · d k (cid:19) , (58)where ∂ IBZ is the path around IBZ. This follows from ro-tational symmetry of the Berry curvature F n ( k ) = F n ( R k ) .For instance, the path in C is ∂ ( IBZ ) = Γ XMY Γ . Applyingthe logarithm, ν n is equivalent to, ν n = N π ˛ ∂ RBZ A n ( k ) · d k mod N. (59)As we will see more explicitly, ν n is tied entirely to η n . Thereason is subtle - any vortex within the interior of the IBZ contributes a Berry phase of π , and by symmetry, there are N such vortices within the total Brillouin zone C n → C n + N .However, this has no effect on ν n → ν n . Only the vorticeslying at HSPs contribute to ν n because these come in fractionsof π .In the following sections we will discuss the bosonic classi-fication of ν n for each cyclic point group and the SPT phasesassociated with them. We do not present the full derivationshere since the rigorous proofs have been carried out by others(see Ref. [69]) - we simply state the salient results. For com-pleteness, in Appendix A we also discuss the SPT fermionic phases associated with each point group. We do this to em-phasize that fermionic and bosonic systems represent distincttopological field theories, with fundamentally different inter-pretations. These differences are highlighted with a few ex-amples [Fig. 4 and 5]. C. Twofold (inversion) symmetry: C For the C point group, or simply inversion symmetry, theSPT phase is related to the Chern number by ν n = C n mod 2 which is a Z invariant. There is only one nontrivial SPTphase and it can be found modulo 2 from, exp (cid:18) i π C n (cid:19) = η ,n (Γ) η ,n ( X ) η ,n ( Y ) η ,n ( M ) . (60)0Applying the logarithm, this classification can be expressedequivalently in terms of m ,n ∈ Z inversion eigenvalues, ν n = m ,n (Γ) + m ,n ( X ) + m ,n ( Y ) + m ,n ( M ) mod 2 . (61)If the summation of m ,n eigenvalues is odd, the SPT phaseis nontrivial ν n = 1 and corresponds to an odd-valued Chernnumber. Likewise, ν n = 0 is an even-valued Chern number. D. Threefold symmetry: C C is unique because it is the only point group with an oddrotational symmetry - i.e. it lacks inversion symmetry. Thismeans the parity of Chern number (odd or even) is not re-stricted by the symmetries of the crystal. For C , the SPTphase is ν n = C n mod 3 which is a Z invariant. There aretwo nontrivial SPT phases and they can be found modulo 3from, exp (cid:18) i π C n (cid:19) = η ,n (Γ) η ,n ( K ) η ,n (cid:0) K (cid:48) (cid:1) . (62)This classification is expressed equivalently in terms of quan-tized modulo 3 integers m ,n ∈ Z at HSPs, ν n = m ,n (Γ) + m ,n ( K ) + m ,n ( K (cid:48) ) mod 3 . (63)Note though, odd and even phases are not distinct ν = − under modulo 3. E. Fourfold symmetry: C For the C point group, the SPT phase is related to theChern number by ν n = C n mod 4 which is a Z invariant.There are three nontrivial SPT phases and they can be foundmodulo 4 from, exp (cid:18) i π C n (cid:19) = η ,n (Γ) η ,n ( M ) η ,n ( Y ) . (64)The classification is expressed equivalently in terms of spin-1eigenvalues, ν n = m ,n (Γ) + m ,n ( M ) + 2 m ,n ( Y ) mod 4 , (65)where m ,n (Γ) & m ,n ( M ) ∈ Z are modulo 4 integers and m ,n ( Y ) ∈ Z is a modulo 2 integer. Examples of all SPTphases of the C point group are displayed in Fig. 4 and theseare compared with their fermionic counterparts in Fig. 5. F. Sixfold symmetry: C For the C point group, the SPT phase is ν n = C n mod 6 which is a Z invariant. There are five nontrivial SPT phasesand they can be found modulo 6 from, exp (cid:18) i π C n (cid:19) = η ,n (Γ) η ,n ( K ) η ,n ( M ) . (66) This is equivalent to the summation of spin-1 eigenvalues atthe HSPs, ν n = m ,n (Γ) + 2 m ,n ( K ) + 3 m ,n ( M ) mod 6 , (67)where m ,n (Γ) ∈ Z is a modulo 4 integer, m ,n ( K ) ∈ Z is a modulo 3 integer and m ,n ( M ) ∈ Z is a modulo 2 inte-ger. This completes the classification of all 2+1D topologicalelectromagnetic (bosonic) phases of matter which is summa-rized in Tbl. I. These are compared alongside their fermioniccounterparts in Tbl. II. G. Continuous symmetry: C ∞ To finish, we briefly discuss the continuum limit g = 0 and the topological phases that can be described by a longwavelength theory k ≈ . The physics is significantlymore tractable here and exactly solvable models are possible[24, 25]. In this limit, the rotational symmetry of the crystalis approximately continuous C ∞ . The SPT invariant ν n andChern number C n are thus equivalent, ν n = C n = m n (0) − m n ( ∞ ) . (68)Note that ν n ∈ Z and m n ∈ Z are not modulo integers inthis limit and do not have the same interpretation as the lat-tice theory. This is because we have gained the full rotationalsymmetry in the continuum approximation. Clearly though,the eigenvalues must change at HSPs m n (0) (cid:54) = m n ( ∞ ) for anontrivial phase to exist C n (cid:54) = 0 . In the continuum regulariza-tion, k i = 0 represents the Γ point and k i = ∞ is interpretedas mapping the vertices of the Brillouin zone into one another. V. CONCLUSIONS
In summary, we have developed the complete 2+1D lat-tice field theory describing all symmetry-protected topolog-ical bosonic phases of the photon. To accomplish this, we an-alyzed the electromagnetic Bloch waves in microscopic crys-tals and derived the Chern invariant of these light-matter ex-citations. Thereafter, the rotational symmetries of the crystalwere examined extensively and the implications these haveon photonic spin. We have studied all two dimensional pointgroups C N with nonvanishing Chern number C (cid:54) = 0 andlinked the topological invariants directly to spin-1 quantizedeigenvalues of the electromagnetic field - establishing thebosonic classification for each topological phase. ACKNOWLEDGEMENTS
This research was supported by the Defense Advanced Re-search Projects Agency (DARPA) Nascent Light-Matter Inter-actions (NLM) Program and the National Science Foundation(NSF) [Grant No. EFMA-1641101].1
APPENDIXAppendix A: Symmetry-protected topological fermionic phases
For completeness, we examine the SPT fermionic phasesassociated with each point group C N and highlight their es-sential differences from bosons. The most important distinc-tion is how they transform under rotations; half-integer parti-cles are antisymmetric R (2 π ) = − . In terms of discrete ro-tations ˆ R N about the z -axis, the eigenstates of a Bloch spinorparticle satisfy, ˆ R N | Ψ( k i ) (cid:105) = ζ N ( k i ) | Ψ( k i ) (cid:105) , (A1)where the eigenvalues at HSPs are related by, ζ N ( k i ) = exp (cid:20) i πN m N ( k i ) (cid:21) , ( ζ N ) N = − . (A2) m N ( k i ) ∈ Z N + / is a modulo half-integer and labels the N possible spin- ⁄ eigenvalues. Notice that ζ N representsthe N th roots of negative unity which is characteristic of afermionic field.The single-particle fermionic classification for C , C , C and C respectively is [69, 76], exp (cid:18) i π C (cid:19) = ζ (Γ) ζ ( X ) ζ ( Y ) ζ ( M ) , (A3a) exp (cid:18) i π C (cid:19) = − ζ (Γ) ζ ( K ) ζ (cid:0) K (cid:48) (cid:1) , (A3b) exp (cid:18) i π C (cid:19) = − ζ (Γ) ζ ( M ) ζ ( Y ) , (A3c) exp (cid:18) i π C (cid:19) = − ζ (Γ) ζ ( K ) ζ ( M ) . (A3d)Although the classification appears similar, the SPT fermionicphases constitute very different physics than their bosoniccounterparts, which is alluded to by the antisymmetric phasefactors R (2 π ) = − . We illustrate this with an example in C . Applying the logarithm - the classification for the SPTfermionic phase ν = C mod 4 can be expressed as, ν = m (Γ) + m ( M ) + 2 m ( Y ) + 2 mod 4 , (A4)where m (Γ) & m ( M ) ∈ Z + / are modulo half-integersand m ( Y ) ∈ Z + / is a modulo half-integer. [1] M. Z. Hasan and C. L. 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