Source: https://www.osapublishing.org/ome/abstract.cfm?URI=ome-9-1-95
Timestamp: 2019-04-20 00:56:01+00:00

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We introduce the concept of a photonic Dirac monopole, appropriate for photonic crystals, metamaterials and 2D materials, by utilizing the Dirac-Maxwell correspondence. We start by exploring the vacuum where the reciprocal momentum space of both Maxwell’s equations and the massless Dirac equation (Weyl equation) possess a magnetic monopole. The critical distinction is the nature of magnetic monopole charges, which are integer valued for photons but half-integer for electrons. This inherent difference is directly tied to the spin and ultimately connects to the bosonic or fermionic behavior. We also show the presence of photonic Dirac strings, which are line singularities in the underlying Berry gauge potential. While the results in vacuum are intuitively expected, our central result is the application of this topological Dirac-Maxwell correspondence to 2D photonic (bosonic) materials, as opposed to conventional electronic (fermionic) materials. Intriguingly, within dispersive matter, the presence of photonic Dirac monopoles is captured by nonlocal quantum Hall conductivity–i.e., a spatiotemporally dispersive gyroelectric constant. For both 2D photonic and electronic media, the nontrivial topological phases emerge in the context of massive particles with broken time-reversal symmetry. However, the bulk dynamics of these bosonic and fermionic Chern insulators are characterized by spin-1 and spin-½ skyrmions in momentum space, which have fundamentally different interpretations. This is exemplified by their contrasting spin-1 and spin-½ helically quantized edge states. Our work sheds light on the recently proposed quantum gyroelectric phase of matter and the essential role of photon spin quantization in topological bosonic phases.
22 February 2019: A typographical correction was made to the title.
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Fig. 1: (a) Linear dispersion (light cone) of the 3D massless photon and electron ω± = E± = ±k. At the origin of the momentum space k⃗ = 0 sits a magnetic monopole with quantized charge. This singularity is often called a Weyl point and is quantized to the spin of the particle Qs = s. Integer and half-integer spin quantization is connected to bosonic and fermionic statistics respectively. (b) Dirac monopoles (Berry curvature) F⃗s = ∇⃗ k × A⃗s of the massless electron Q 1 2 = 1 2 and photon Q1 = 1 in momentum space. The monopole charge acts as a source for the magnetic field ∇⃗ k · F⃗s = 4πQsδ3(k⃗) and arises due to the discontinuous behavior in the spin eigenstates. Notice that the flux through any surface enclosing the monopole is necessarily quantized Q s = ( 4 π ) − 1 ∯ F → s ⋅ d 2 k → . This monopole is accompanied by a string of singularities in the underlying gauge potential A⃗s. Any closed path around the equator of the string produces a quantized Berry phase γ s = ∮ A → s ⋅ d k → = 2 π Q s . The accumulated phase in k⃗-space is fundamentally tied to the spin of the particle ℛs(2π) = exp(iγs) = (−1)2s.
Fig. 2: Spin expectation value ℳ̂z(k) as a function of k. (a) N = 0 skyrmion with no band inversion Λ0Λ2 < 0. The spin returns to initial state ℳ̂z(0) = ℳ̂z(∞) and the total winding is trivial. (b) N = 1 skyrmion with band inversion Λ0Λ2 > 0. In this case, the spin flips direction ℳ̂z(0) ≠ ℳ̂z(∞) and the total winding is nontrivial. ki labels the band inversion point where ℳ̂z(ki) = 0 passes through zero. This point must occur for the spin to flip directions and can only be removed at a topological phase transition.
Fig. 3: Left: spin texture ℳ̂(k) as a function of k for trivial and nontrivial skyrmions. (a) N = 0 skyrmion with no band inversion Λ0Λ2 < 0. As an example, we have let v = 0.5, Λ0 = 4 and Λ2 = −2. Since the spin returns to initial state within the dispersion ℳ̂z(0) = ℳ̂z(∞), the total winding is trivial. (b) N = 1 skyrmion with band inversion Λ0Λ2 > 0. To demonstrate, we have let v = 0.5, Λ0 = 4 and Λ2 = 2. In this case, the spin flips direction within the dispersion ℳ̂z(0) ≠ ℳ̂z(∞) and the total winding is nontrivial. Right: spin texture ℳ̂ of the skyrmion projected on the unit sphere. As the momentum varies over all possible values, ℳ̂(k) can perform either a (c) retracted or (d) full evolution over the unit sphere. This corresponds to a total solid angle of Ω = 0 or 4π respectively.
Fig. 4: Dispersion relation of the bulk and gapless edge bands (black lines) of the topologically massive 2D particles. (a) The conventional fermionic Chern insulator is characterized by a spin-½ skyrmion (Dirac equation). (b) The bosonic Chern insulator is described by a spin-1 skyrmion (Maxwell’s equations). The bulk Chern number Cs = 2QsN depends on both the magnetic charge (spin) Qs = s and the skyrmion number N ∈ 𝕑. This corresponds to integer phases for electrons C 1 2 ∈ 𝕑 but even integer phases for photons C1 ∈ 2𝕑. At low energy, a band gap is formed at E = ω = 0 by a spatially dispersive effective mass Λ(k) = Λ0 − Λ2k2. (a) For the 2D electron, this is simply the Dirac mass. (b) For the 2D photon, this mass is equivalent to a nonlocal Hall conductivity εΛ(k) = σH(k). In the presence of band inversion Λ0Λ2 > 0, there is a point where the effective mass changes sign Λ(ki) = 0, precisely at k i = Λ 0 / Λ 2 . The massless helically quantized edge states touch the bulk bands at this point. This is known as the quantum anomalous Hall effect (QAHE) for electrons and the quantum gyroelectric effect (QGEE) for photons . The flat longitudinal band ω0 = 0 is shown for completeness and represents the electrostatic limit (irrotational fields). However, this band can be removed from the spectrum by requiring that all static charges vanish.
(1) 2 e Q h ∈ 핑 .
(4) H 1 ( k → ) = k → ⋅ S → = k x S x + k y S y + k z S z .
(5) S x = [ 0 0 0 0 0 − i 0 i 0 ] , S y = [ 0 0 i 0 0 0 − i 0 0 ] , S z = [ 0 − i 0 i 0 0 0 0 0 ] .
(7) H 1 2 ( k → ) = k → ⋅ σ → = k x σ x + k y σ y + k z σ z .
(8) σ x = [ 0 1 1 0 ] , σ y = [ 0 − i i 0 ] , σ z = [ 1 0 0 − 1 ] .
(10) H 1 2 ψ ± = ± k ψ ± , ψ + ( k → ) = [ cos ( θ / 2 ) sin ( θ / 2 ) e i ϕ ] , ψ − ( k → ) = [ sin ( θ / 2 ) − cos ( θ / 2 ) e i ϕ ] .
(11) F → 1 ± = − i ∇ → k × [ e → ± * ⋅ ( ∇ → k e → ± ) ] .
(12) F → 1 2 ± = − i ∇ → k × [ ψ ± † ∇ → k ψ ± ] .
(13) F → s = Q s F → .
(14) F → ( k → ) = k → k 3 .
(15) Q s = s .
(16) Q s = 1 4 π ∯ F → s ⋅ d 2 k → .
(18) A → 1 ± = − i e → ± * ⋅ ( ∇ → k e → ± ) , A → 1 2 ± = − i ψ ± † ∇ → k ψ ± .
(20) γ s = ∮ A → s ⋅ d k → = ∬ F → s ⋅ d 2 k → .
(22) γ s = 2 π Q s .
(23) exp ( i γ s ) = ( − 1 ) 2 Q s .
(24) ℛ s ( 2 π ) = ( − 1 ) 2 s .
(25) ℋ 1 2 ψ = E ψ , ℋ 1 2 ( k ) = v ( k x σ x + k y σ y ) + Λ ( k ) σ z .
(26) Λ ( k ) = Λ 0 − Λ 2 k 2 .
(27) ℋ 1 Ψ → = ω Ψ → , ℋ 1 ( k ) = v ( k x S x + k y S y ) + Λ ( k ) S z .
(28) Ψ → = 1 2 ( ε E x , ε E y , i H z ) = 1 2 ( ε E , i H z ) .
(29) ε Λ ( k ) = σ H ( k ) = σ 0 − σ 2 k 2 .
(30) ℋ 1 = ℳ → ⋅ S → , ℋ 1 2 = ℳ → ⋅ σ → .
(31) S → ˙ = − i [ S → , ℋ 1 ] = g 1 ℳ → × S → .
(32) σ → ˙ = − i [ σ → , ℋ 1 2 ] = g 1 2 ℳ → × σ → .
(34) ℋ 1 e → ± = ± ℳ e → ± , ℋ 1 2 ψ ± = ± ℳ ψ ± .
(37) 〈 σ z / 2 〉 ± = ψ ± † ( σ z / 2 ) ψ ± = ± Q 1 2 ℳ ^ z .
(38) ℋ 1 e → 0 = 0 , e → 0 = ℳ ^ = ℳ → / ℳ .
(39) ℱ 1 ± = − i ( ∂ k x e → ± * ⋅ ∂ k y e → ± − ∂ k y e → ± * ⋅ ∂ k x e → ± ) .
(40) ℱ 1 2 ± = − i ( ∂ k x ψ ± † ∂ k y ψ ± − ∂ k y ψ ± † ∂ k x ψ ± ) .
(41) ℱ s = Q s ℱ .
(42) ℱ = ℳ ^ ⋅ ( ∂ k x ℳ ^ × ∂ k y ℳ ^ ) = F → ⋅ d 2 ℳ → .
(44) N = 1 4 π ∬ ℝ 2 ℱ d k x d k y = 1 4 π ∬ ℝ 2 ℳ ^ ⋅ ( ∂ k x ℳ ^ × ∂ k y ℳ ^ ) d k x d k y , N ∈ 핑 .
(45) C s = 1 2 π ∬ ℝ 2 ℱ s d k x d k y = Q s 2 π ∬ ℝ 2 ℱ d k x d k y = 2 Q s N .
(46) ℱ ( k ) = v k [ Λ ( k ) − v k Λ ′ ( k ) ] [ v 2 k 2 + Λ 2 ( k ) ] 3 / 2 = sin θ ( k ) ∂ k θ ( k ) = − ∂ k ℳ ^ z ( k ) .
(47) N = 1 2 [ cos θ ( 0 ) − cos θ ( ∞ ) ] = 1 2 [ ℳ ^ z ( 0 ) − ℳ ^ z ( ∞ ) ] = 1 2 ( sgn [ Λ 0 ] + sgn [ Λ 2 ] ) .
(48) Ψ → ± e ( x = 0 + ) = ψ ± e ( x = 0 + ) = 0 .
(49) Ψ → ± e ( x ) = [ ε E x ε E y i H z ] ± e = Ψ 0 [ 1 0 ∓ i ] ( e − η 1 x − e − η 2 x ) .
(50) ψ ± e ( x ) = ψ 0 [ 1 ± i ] ( e − η 1 x − e − η 2 x ) .
(51) Λ 0 + Λ 2 ( η 2 − k y 2 ) ∓ v η = 0 .
(52) η 1 , 2 ( k y ) = 1 2 | Λ 2 | [ v ± v 2 + 4 Λ 2 ( Λ 2 k y 2 − Λ 0 ) ] .
(53) S y Ψ → ± e = ± Q 1 Ψ → ± e .
(55) ω ± ( k y ) = E ± ( k y ) = ± v k y , − k i < k y < k i .

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