Towards a Rigorous Proof of Haldanes Photonic Bulk-edge - PowerPoint PPT Presentation

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Quantum vs. Classical Equations: a Single Spin Idea Start with the same equations of motion, 𝜔 2 (𝑢)) = ( 0 0 ) (𝜔 1 (𝑢) 𝜔 2 (𝑢)) , once in the quantum and then in the classical context. Math is trivial, everything is explicit Immediately transfers to many other hamiltonian equations as . . . . . . . . . . . . . . . . . . . . . . . . Conceptually applies to all classical wave equations . . . . . − i 𝜕 0 i 𝜖 + i 𝜕 0 𝜖𝑢 (𝜔 1 (𝑢) 𝐾 = i 𝜏 2 is the canonical symplectic form

. Bulk-Edge Correspondence . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Da Capo . Symmetries of Classical and Quantum Spin Equations Purpose Anticipate symmetry classifjcation of electromagnetic media 𝜔 2 (𝑢)) = ( 0 0 ) (𝜔 1 (𝑢) 𝜔 2 (𝑢)) , What is the difgerence between the quantum and classical equations when it comes to symmetries? What types of symmetries does the classical equation possess (in the context of the Cartan-Altland-Zirnbauer classifjcation)? . . . . . . . . . . . . . . . . . . . . . . . . . ⟹ Requires us to work with complex Hilbert spaces . . . . . − i 𝜕 0 i 𝜖 + i 𝜕 0 𝜖𝑢 (𝜔 1 (𝑢)

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Symmetries of Classical and Quantum Spin Equations Purpose Anticipate symmetry classifjcation of electromagnetic media 𝜔 2 (𝑢)) = ( 0 0 ) (𝜔 1 (𝑢) 𝜔 2 (𝑢)) , What is the difgerence between the quantum and classical equations when it comes to symmetries? What types of symmetries does the classical equation possess . . . . . . . . . . . . . . . . . . . . . . . . ⟹ Requires us to work with complex Hilbert spaces . . . . . − i 𝜕 0 i 𝜖 + i 𝜕 0 𝜖𝑢 (𝜔 1 (𝑢) (in the context of the Cartan-Altland-Zirnbauer classifjcation)?

. 0 = +𝐼 2 (chiral) Symmetries Building blocks 𝜔 2 (𝑢)) ) (𝜔 1 (𝑢) . 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 𝜔 2 (𝑢)) = ( Fundamental equation Quantum Symmetries of Classical and Quantum Spin Systems Da Capo Bulk-Edge Correspondence Topological Classifjcation (ordin.) (+PH) Quantum vs. Classical 𝑁 𝑦 (𝑢) (+PH) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (ordin.) = +𝐼 2 (chiral) Symmetries Building blocks (-PH) 𝑁 𝑧 (𝑢)) ) (𝑁 𝑦 (𝑢) 0 0 𝑁 𝑧 (𝑢)) = ( Fundamental equation Classical Maxwell’s Equations in Linear Media 0 . . . . . . . . . . . . . . . . . . . . . . . . (-PH) . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝜔 1 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ States: ( 𝜔 1 (𝑢) States: ( 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 𝜔 2 (𝑢) ) ∈ ℋ ℂ = ℂ 2 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝜏 2 𝐼 𝜏 −1 𝜏 2 𝐼 𝜏 −1 (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼 (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼

. ) (𝜔 1 (𝑢) (ordin.) = +𝐼 2 (chiral) Symmetries Building blocks 𝜔 2 (𝑢)) 0 (+PH) . 𝜔 2 (𝑢)) = ( Fundamental equation Quantum Symmetries of Classical and Quantum Spin Systems Da Capo Bulk-Edge Correspondence 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (-PH) Maxwell’s Equations in Linear Media 𝑁 𝑦 (𝑢) (+PH) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (ordin.) = +𝐼 2 (chiral) Symmetries States: ( Classical Building blocks 𝑁 𝑧 (𝑢)) ) (𝑁 𝑦 (𝑢) 0 0 𝑁 𝑧 (𝑢)) = ( Fundamental equation Topological Classifjcation 0 Quantum vs. Classical . . . . . . . . . . . . . . . . . . . . . . . . (-PH) . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝜔 1 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ States: ( 𝜔 1 (𝑢) 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 𝜔 2 (𝑢) ) ∈ ℋ ℂ = ℂ 2 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝜏 2 𝐼 𝜏 −1 𝜏 2 𝐼 𝜏 −1 (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼 (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼

. ) (𝜔 1 (𝑢) (ordin.) = +𝐼 2 (chiral) Symmetries Building blocks 𝜔 2 (𝑢)) 0 (+PH) . 𝜔 2 (𝑢)) = ( Fundamental equation Quantum Symmetries of Classical and Quantum Spin Systems Da Capo Bulk-Edge Correspondence 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (-PH) Maxwell’s Equations in Linear Media 𝑁 𝑦 (𝑢) (+PH) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (ordin.) = +𝐼 2 (chiral) Symmetries? States: ( Classical Building blocks 𝑁 𝑧 (𝑢)) ) (𝑁 𝑦 (𝑢) 0 0 𝑁 𝑧 (𝑢)) = ( Fundamental equation Topological Classifjcation 0 Quantum vs. Classical . . . . . . . . . . . . . . . . . . . . . . . . (-PH) . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝜔 1 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ States: ( 𝜔 1 (𝑢) 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 𝜔 2 (𝑢) ) ∈ ℋ ℂ = ℂ 2 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝜏 2 𝐼 𝜏 −1 𝜏 2 𝐼 𝜏 −1 (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼 (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼

. 0 (chiral) Symmetries Building blocks 𝜔 2 (𝑢)) ) (𝜔 1 (𝑢) 0 . = +𝐼 Fundamental equation Quantum Symmetries of Classical and Quantum Spin Systems Da Capo Bulk-Edge Correspondence Topological Classifjcation Maxwell’s Equations in Linear Media 2 (ordin.) . 𝑁 𝑧 (𝑢)) (+PH) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (???) (???) Symmetries 𝑁 𝑦 (𝑢) Building blocks ) (𝑁 𝑦 (𝑢) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 0 0 𝑁 𝑧 (𝑢)) = ( Fundamental equation Classical (-PH) (+PH) Quantum vs. Classical 𝜔 2 (𝑢)) = ( . . . . . . . . . . . . . . . . . . . . . . . (-PH) . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝜔 1 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ States: ( 𝜔 1 (𝑢) States: ( 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 𝜔 2 (𝑢) ) ∈ ℋ ℂ = ℂ 2 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 ( i 𝜏 2 ) 𝐼 ( i 𝜏 2 ) −1 = +𝐼 𝜏 2 𝐼 𝜏 −1 (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼 (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼

. Quantum 𝜔 2 (𝑢)) ) (𝜔 1 (𝑢) 0 0 𝜔 2 (𝑢)) = ( . Symmetries of Classical and Quantum Spin Systems Symmetries Da Capo Bulk-Edge Correspondence Topological Classifjcation Maxwell’s Equations in Linear Media Quantum vs. Classical . Building blocks (chiral) . 0 Symmetries 𝑁 𝑦 (𝑢) Building blocks 𝑁 𝑧 (𝑢)) ) (𝑁 𝑦 (𝑢) 0 𝑁 𝑧 (𝑢)) = ( 2 Fundamental equation Classical (-PH) (+PH) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (ordin.) = +𝐼 . Fundamental equation . . . . . . . . . . . . . . . . . . . . ⇝ classical spin transformations . . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝜔 1 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ States: ( 𝜔 1 (𝑢) States: ( 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 𝜔 2 (𝑢) ) ∈ ℋ ℂ = ℂ 2 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝐷 not defjned on ℋ ℝ = ℝ 2 𝜏 1 , i 𝜏 2 and 𝜏 3 are real matrices 𝜏 2 𝐼 𝜏 −1 (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼

. 0 (chiral) Symmetries Building blocks 𝜔 2 (𝑢)) ) (𝜔 1 (𝑢) 0 . = +𝐼 Fundamental equation Quantum Symmetries of Classical and Quantum Spin Systems Da Capo Bulk-Edge Correspondence Topological Classifjcation Maxwell’s Equations in Linear Media 2 (ordin.) . 𝑁 𝑧 (𝑢)) (+PH) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (???) (???) Symmetries 𝑁 𝑦 (𝑢) Building blocks ) (𝑁 𝑦 (𝑢) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 0 0 𝑁 𝑧 (𝑢)) = ( Fundamental equation Classical (-PH) (+PH) Quantum vs. Classical 𝜔 2 (𝑢)) = ( . . . . . . . . . . . . . . . . . . . . . . . (-PH) . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝜔 1 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ States: ( 𝜔 1 (𝑢) States: ( 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 𝜔 2 (𝑢) ) ∈ ℋ ℂ = ℂ 2 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 𝜏 1,3 𝐼 𝜏 −1 1,3 = −𝐼 ( i 𝜏 2 ) 𝐼 ( i 𝜏 2 ) −1 = +𝐼 𝜏 2 𝐼 𝜏 −1 (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 1,3 𝐷) 𝐼 (𝜏 1,3 𝐷) −1 = +𝐼 (+TR) (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼 (𝜏 2 𝐷) 𝐼 (𝜏 2 𝐷) −1 = −𝐼

. Quantum ) (𝜔 1 (𝑢) 0 0 𝜔 2 (𝑢)) = ( . Fundamental equation Symmetries of Classical and Quantum Spin Systems Building blocks Da Capo Bulk-Edge Correspondence Topological Classifjcation Maxwell’s Equations in Linear Media Quantum vs. Classical . 𝜔 2 (𝑢)) Symmetries . ) (𝑁 𝑦 (𝑢) (???) Symmetries 𝑁 𝑦 (𝑢) States: ( Building blocks 𝑁 𝑧 (𝑢)) 0 (ordin.) 0 𝑁 𝑧 (𝑢)) = ( Fundamental equation Classical (+PH) 𝐷 (-PH) . (???) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝜔 1 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ States: ( 𝜔 1 (𝑢) 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 𝜔 2 (𝑢) ) ∈ ℋ ℂ = ℂ 2 𝑊 ℝ 𝑉 1 = 𝜏 1 (chiral) 𝑈 1 = 𝜏 1 𝐷 (+TR) 1 = 𝜏 1 2 = i 𝜏 2 (???) 𝑊 ℝ 𝑉 2 = 𝜏 2 𝑈 2 = 𝜏 2 𝐷 𝑊 ℝ 𝑉 3 = 𝜏 3 (chiral) 𝑈 3 = 𝜏 3 𝐷 (+TR) 3 = 𝜏 3

. Quantum vs. Classical . . . . . . . . . . Maxwell’s Equations in Linear Media . Topological Classifjcation Bulk-Edge Correspondence Da Capo Consider Classical Equation on Complex Hilbert Space Cartan-Altland-Zirnbauer classifjcation scheme for Topological only work for operators acting on complex Hilbert spaces Two Ways to Work With Complex Hilbert Spaces 1 Complexify classical equations (introduces unphysical degrees of freedom) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Insulators (and many other techniques from quantum mechanics) Work with complex Ψ which represent real states 𝑁 = 2 Re Ψ (establish 1-to-1 correspondence ℋ ℂ ↔ ℋ ℝ )

. Quantum vs. Classical . . . . . . . . . . Maxwell’s Equations in Linear Media . Topological Classifjcation Bulk-Edge Correspondence Da Capo Consider Classical Equation on Complex Hilbert Space Cartan-Altland-Zirnbauer classifjcation scheme for Topological only work for operators acting on complex Hilbert spaces Two Ways to Work With Complex Hilbert Spaces 1 Complexify classical equations (introduces unphysical degrees of freedom) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Insulators (and many other techniques from quantum mechanics) Work with complex Ψ which represent real states 𝑁 = 2 Re Ψ (establish 1-to-1 correspondence ℋ ℂ ↔ ℋ ℝ )

. 0 . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo . Complexifjcation Fundamental equation 𝑁 𝑧 (𝑢)) = ( 0 ) (𝑁 𝑦 (𝑢) . 𝑁 𝑧 (𝑢)) Building blocks Symmetries Classical Fundamental equation 𝑁 𝑧 (𝑢)) = ( 0 0 ) (𝑁 𝑦 (𝑢) 𝑁 𝑧 (𝑢)) Building blocks States: ( 𝑁 𝑦 (𝑢) Symmetries . Complexifying the Classical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝑁 𝑦 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 States: 𝑁 = Ψ + + Ψ − ∈ ℋ ℝ ⊂ ℋ ℂ 𝑊 ℂ 𝑊 ℝ 1 = ??? 1 = 𝜏 1 2 = i 𝜏 2 𝑊 ℂ 𝑊 ℝ 2 = ??? 𝑊 ℂ 𝑊 ℝ 3 = ??? 3 = 𝜏 3

. ) (𝑁 𝑦 (𝑢) . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence . Complexifying the Classical Equations Complexifjcation Fundamental equation 𝑁 𝑧 (𝑢)) = ( 0 0 𝑁 𝑧 (𝑢)) . 0 Symmetries 𝑁 𝑦 (𝑢) States: ( Building blocks 𝑁 𝑧 (𝑢)) ) (𝑁 𝑦 (𝑢) 0 Building blocks 𝑁 𝑧 (𝑢)) = ( Fundamental equation Classical ordin. vs. -PH none vs. +PH Symmetries . Da Capo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝑁 𝑦 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 States: 𝑁 = Ψ + + Ψ − ∈ ℋ ℝ ⊂ ℋ ℂ 𝑊 ℝ 𝟚 ∣ ℋ ℝ = 𝐷 ∣ ℋ ℝ 1 = 𝜏 1 2 = i 𝜏 2 𝑊 ℝ 𝜏 1,3 ∣ ℋ ℝ = 𝜏 1,3 𝐷 ∣ ℋ ℝ chiral vs. +TR 𝑊 ℝ i 𝜏 2 ∣ ℋ ℝ = i 𝜏 2 𝐷 ∣ ℋ ℝ 3 = 𝜏 3

. 𝑁 𝑧 (𝑢)) . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation . Da Capo Complexifying the Classical Equations Complexifjcation Fundamental equation 𝑁 𝑧 (𝑢)) = ( 0 0 ) (𝑁 𝑦 (𝑢) Building blocks . 0 Symmetries 𝑁 𝑦 (𝑢) States: ( Building blocks 𝑁 𝑧 (𝑢)) ) (𝑁 𝑦 (𝑢) 0 Symmetries 𝑁 𝑧 (𝑢)) = ( Fundamental equation Classical topological classifjcations!? • Difgerent choices ⇒ difgerent • Redundant symmetry operations . Bulk-Edge Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝑁 𝑦 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 States: 𝑁 = Ψ + + Ψ − ∈ ℋ ℝ ⊂ ℋ ℂ 𝑊 ℝ 1 = 𝜏 1 2 = i 𝜏 2 𝑊 ℝ 𝑊 ℝ 3 = 𝜏 3

. Quantum vs. Classical . . . . . . . . . . Maxwell’s Equations in Linear Media . Topological Classifjcation Bulk-Edge Correspondence Da Capo Consider Classical Equation on Complex Hilbert Space Cartan-Altland-Zirnbauer classifjcation scheme for Topological only work for operators acting on complex Hilbert spaces Two Ways to Work With Complex Hilbert Spaces 1 Complexify classical equations (introduces unphysical degrees of freedom) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Insulators (and many other techniques from quantum mechanics) Work with complex Ψ which represent real states 𝑁 = 2 Re Ψ (establish 1-to-1 correspondence ℋ ℂ ↔ ℋ ℝ )

. Quantum vs. Classical . . . . . . . . . . Maxwell’s Equations in Linear Media . Topological Classifjcation Bulk-Edge Correspondence Da Capo Consider Classical Equation on Complex Hilbert Space Cartan-Altland-Zirnbauer classifjcation scheme for Topological only work for operators acting on complex Hilbert spaces Two Ways to Work With Complex Hilbert Spaces 1 Complexify classical equations (introduces unphysical degrees of freedom) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Insulators (and many other techniques from quantum mechanics) Work with complex Ψ which represent real states 𝑁 = 2 Re Ψ (establish 1-to-1 correspondence ℋ ℂ ↔ ℋ ℝ )

. 0 . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation . Da Capo Schrödinger Formalism of Classical Spin Waves Complexifjcation Fundamental equation 𝑁 𝑧 (𝑢)) = ( 0 ) (𝑁 𝑦 (𝑢) . 𝑁 𝑧 (𝑢)) Building blocks Symmetries Classical Fundamental equation 𝑁 𝑧 (𝑢)) = ( 0 0 ) (𝑁 𝑦 (𝑢) 𝑁 𝑧 (𝑢)) Building blocks States: ( 𝑁 𝑦 (𝑢) Symmetries . Bulk-Edge Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝑁 𝑦 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 States: 𝑁 = Ψ + + Ψ − ∈ ℋ ℝ ⊂ ℋ ℂ 𝑊 ℝ 𝟚 ∣ ℋ ℝ = 𝐷 ∣ ℋ ℝ 1 = 𝜏 1 2 = i 𝜏 2 𝑊 ℝ 𝜏 1,3 ∣ ℋ ℝ = 𝜏 1,3 𝐷 ∣ ℋ ℝ 𝑊 ℝ i 𝜏 2 ∣ ℋ ℝ = i 𝜏 2 𝐷 ∣ ℋ ℝ 3 = 𝜏 3

. 0 . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo . 𝜕 > 0 Representation Fundamental equation 𝜔 +,2 (𝑢)) = ( 0 ) (𝜔 +,1 (𝑢) . 𝜔 +,2 (𝑢)) Building blocks Symmetries Classical Fundamental equation 𝑁 𝑧 (𝑢)) = ( 0 0 ) (𝑁 𝑦 (𝑢) 𝑁 𝑧 (𝑢)) Building blocks States: ( 𝑁 𝑦 (𝑢) Symmetries . Schrödinger Formalism of Classical Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − i 𝜕 0 − i 𝜕 0 i 𝜖 i 𝜖 + i 𝜕 0 + i 𝜕 0 𝜖𝑢 (𝑁 𝑦 (𝑢) 𝜖𝑢 (𝜔 +,1 (𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ States: 𝑁 = 2 Re Ψ + ∈ ℋ ℝ 𝑁 𝑧 (𝑢) ) ∈ ℋ ℝ = ℝ 2 𝑊 ℂ 𝑊 ℝ 1 = ??? 1 = 𝜏 1 2 = i 𝜏 2 𝑊 ℂ 𝑊 ℝ 2 = ??? 𝑊 ℂ 𝑊 ℝ 3 = ??? 3 = 𝜏 3

. Quantum vs. Classical . . . . . . . . . . Maxwell’s Equations in Linear Media . Topological Classifjcation Bulk-Edge Correspondence Da Capo Schrödinger Formalism of Classical Spin Waves Eliminate superfmuous degree of freedom in complexifjed equations 𝑁(𝑢) ⏟ real wave Ψ(𝑢) ⏟ complex 𝜕 > 0 wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-to-1 correspondence ⇝ Systematically identify ℝ 2 ≅ ℂ = 2 Re

. . . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Schrödinger Formalism of Classical Spin Waves Eliminate superfmuous degree of freedom in complexifjed equations 𝑐) . . . . . . . . . . . . . . . . . . . . 1-to-1 correspondence . . . . . ⇝ Systematically identify ℝ 2 ≅ ℂ 𝑁(𝑢) = ( cos 𝜕 0 𝑢 − sin 𝜕 0 𝑢 sin 𝜕 0 𝑢 cos 𝜕 0 𝑢 ) (𝑏 = 2 Re Ψ(𝑢) = 2 Re ((𝑏 − i 𝑐) e − i 𝜕 0 𝑢 Ψ + ) where Ψ + = ( 1 + i ) is the eigenvector of 𝐼 = 𝜕 0 𝜏 2 to +𝜕 0 > 0 .

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo Schrödinger Formalism of Classical Spin Waves 𝜕 > 0 Representation Fundamental equation Building blocks Symmetries Classical Fundamental equation Building blocks Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 𝜖 i 𝜖 𝜖𝑢 Ψ(𝑢) = 𝜕 0 𝜏 2 Ψ(𝑢) 𝜖𝑢 𝑁(𝑢) = 𝜕 0 𝜏 2 𝑁(𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 States: Ψ(𝑢) ∈ ℋ + = span ℂ {( 1 States: 𝑁(𝑢) = 2 Re Ψ(𝑢) ∈ ℝ 2 + i )} 𝑊 ℂ 𝑊 ℝ 1 = ??? 1 = 𝜏 1 2 = i 𝜏 2 𝑊 ℂ 𝑊 ℝ 2 = ??? 𝑊 ℂ 𝑊 ℝ 3 = ??? 3 = 𝜏 3

. Bulk-Edge Correspondence . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Da Capo . Translating Real Symmetries to 𝜕 > 0 Representation Requirements 1 2 maps 𝜕 > 0 waves onto 𝜕 > 0 waves. Consequences 1 𝑘 (unitary) ( anti unitary) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑁 = 2 Re Ψ , then 𝑘 = {𝑊 ℝ 𝑘 𝑁 = 2 Re (𝑊 ℂ 𝑊 ℂ 𝑊 ℝ 𝑘 Ψ) 𝑊 ℝ 𝑘 𝐷 𝑊 ℂ 𝑘 is a (anti)unitary on ℋ + , i. e. it 𝑊 ℂ 𝑘 must commute with 𝐼 = 𝜕 0 𝜏 2

. Bulk-Edge Correspondence . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Da Capo . Translating Real Symmetries to 𝜕 > 0 Representation Requirements 1 2 maps 𝜕 > 0 waves onto 𝜕 > 0 waves. Consequences 1 𝑘 (unitary) ( anti unitary) 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑁 = 2 Re Ψ , then 𝑘 = {𝑊 ℝ 𝑘 𝑁 = 2 Re (𝑊 ℂ 𝑊 ℂ 𝑊 ℝ 𝑘 Ψ) 𝑊 ℝ 𝑘 𝐷 𝑊 ℂ 𝑘 is a (anti)unitary on ℋ + , i. e. it 𝑊 ℂ 𝑘 must commute with 𝐼 = 𝜕 0 𝜏 2

. . . . . . . . . . . . Quantum vs. Classical . Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Translating Real Symmetries to 𝜕 > 0 Representation Real Symmetry Complex Representative TI Classifjcation +TR ordinary . . . . . . . . . . . . . . . . . . . . . +TR . . . . . . 𝑊 ℝ 𝑊 ℂ 1 = 𝜏 1 1 = 𝜏 1 𝐷 2 = i 𝜏 2 2 = i 𝜏 2 𝑊 ℝ 𝑊 ℂ 𝑊 ℝ 𝑊 ℂ 3 = 𝜏 3 3 = 𝜏 3 𝐷

. Bulk-Edge Correspondence . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Da Capo . Translating Real Symmetries to 𝜕 > 0 Representation 𝜕 > 0 Representation Fundamental equation Building blocks Symmetries (+TR) (ordinary) (+TR) Classical Fundamental equation Building blocks Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 𝜖 i 𝜖 𝜖𝑢 Ψ(𝑢) = 𝜕 0 𝜏 2 Ψ(𝑢) 𝜖𝑢 𝑁(𝑢) = 𝜕 0 𝜏 2 𝑁(𝑢) Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 = 𝐼 ∗ Hamiltonian: 𝐼 = 𝜕 0 𝜏 2 States: Ψ(𝑢) ∈ ℋ + = span ℂ {( 1 States: 𝑁(𝑢) = 2 Re Ψ(𝑢) ∈ ℝ 2 + i )} 𝑊 ℂ 𝑊 ℝ 1 = 𝜏 1 𝐷 1 = 𝜏 1 2 = i 𝜏 2 2 = i 𝜏 2 𝑊 ℂ 𝑊 ℝ 𝑊 ℂ 𝑊 ℝ 3 = 𝜏 3 𝐷 3 = 𝜏 3

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo Translating Real Symmetries to 𝜕 > 0 Representation Moral of the Story Not all “quantum” symmetries are symmetries of the classical equations “Schrödinger” form of classical equations necessary to identify the nature of these symmetries in the context of TIs 𝐷 is not a meaningful symmetry of the “Schrödinger” form of the classical equations! No fermionic time-reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideas apply to all classical wave equations! ⇝ Incompatible with the real-valuedness of classical waves

. . . . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo . . . . . . . . . . . . . . . . . . . . . . . . Applies directly to vacuum Maxwell equations

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Spin ⟶ In Vacuo Maxwell Equations ⟶ ⟶ ⟶ Same Strategy 1 Complexify classical equations 2 Eliminate superfmuous states in complex Hilbert space 3 . . . . . . . . . . . . . . . . . . . . . . . . . Identify complex implementation of the three symmetries . . . . Rot = −𝜏 2 ⊗ ∇ × 𝐼 = 𝜕 0 𝜏 2 𝑊 ℝ 𝑊 ℝ 1,3 = 𝜏 1,3 1,3 = 𝜏 1,3 ⊗ 𝟚 2 = i 𝜏 2 2 = i 𝜏 2 ⊗ 𝟚 𝑊 ℝ 𝑊 ℝ

. 0 . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Schrödinger Formalism of In Vacuo Maxwell Equations . Fundamental equation 𝜔 𝐼 (𝑢)) = ( 0 . ) (𝜔 𝐹 (𝑢) 𝜔 𝐼 (𝑢)) Building blocks States: Ψ(𝑢) ∈ 𝑀 2 (ℝ 3 , ℂ 6 ) Symmetries (+TR) (ordinary) (+TR) Classical Fundamental equation 0 0 Building blocks Symmetries . Complexifjcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) + i ∇ × ) ( E (𝑢) + i ∇ × i 𝜖 i 𝜖 H (𝑢)) = ( − i ∇ × H (𝑢)) − i ∇ × 𝜖𝑢 (𝜔 𝐹 (𝑢) Hamiltonian: 𝑁 = −𝜏 2 ⊗ ∇ × = 𝑁 ∗ Hamiltonian: 𝑁 = −𝜏 2 ⊗ ∇ × States: ( E (𝑢) H (𝑢) ) ∈ 𝑀 2 (ℝ 3 , ℝ 6 ) 𝑊 ℝ 𝑊 ℂ 1 = 𝜏 1 ⊗ 𝟚 1 = (𝜏 1 ⊗ 𝟚) 𝐷 2 = i 𝜏 2 ⊗ 𝟚 2 = i 𝜏 2 ⊗ 𝟚 𝑊 ℝ 𝑊 ℂ 𝑊 ℝ 𝑊 ℂ 3 = 𝜏 3 ⊗ 𝟚 3 = (𝜏 3 ⊗ 𝟚) 𝐷

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Schrödinger Formalism of In Vacuo Maxwell Equations Representing real, transversal EM Fields as complex 𝜕 > 0 waves ⏟ ⏟ ⏟ ⏟ ⏟ real wave Ψ(𝑢) ⏟ complex 𝜕 > 0 wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( E (𝑢), H (𝑢)) = 2 Re ⟹ Ψ ∈ ℋ + = { complex 𝜕 > 0 waves } .

. 𝜔 𝐼 (𝑢)) = ( . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Schrödinger Formalism of In Vacuo Maxwell Equations . Fundamental equation 0 . 0 ) (𝜔 𝐹 (𝑢) 𝜔 𝐼 (𝑢)) Building blocks Symmetries (+TR) (ordinary) (+TR) Classical Fundamental equation 0 0 Building blocks Symmetries . 𝜕 > 0 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) + i ∇ × ) ( E (𝑢) + i ∇ × i 𝜖 i 𝜖 H (𝑢)) = ( − i ∇ × H (𝑢)) − i ∇ × 𝜖𝑢 (𝜔 𝐹 (𝑢) Hamiltonian: 𝑁 = −𝜏 2 ⊗ ∇ × = 𝑁 ∗ Hamiltonian: 𝑁 = −𝜏 2 ⊗ ∇ × States: ( E (𝑢) H (𝑢) ) ∈ 𝑀 2 (ℝ 3 , ℝ 6 ) States: Ψ(𝑢) ∈ { compl. 𝜕 > 0 waves } 𝑊 ℝ 𝑊 ℂ 1 = 𝜏 1 ⊗ 𝟚 1 = (𝜏 1 ⊗ 𝟚) 𝐷 2 = i 𝜏 2 ⊗ 𝟚 2 = i 𝜏 2 ⊗ 𝟚 𝑊 ℝ 𝑊 ℂ 𝑊 ℝ 𝑊 ℂ 3 = 𝜏 3 ⊗ 𝟚 3 = (𝜏 3 ⊗ 𝟚) 𝐷

. Schrödinger Formalism of In Vacuo Maxwell Equations . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Real Symmetry . Complex Representative TI Classifjcation Meaning +TR Flips helicity and arrow of time ordinary Dual symmetry +TR Ordinary EM time-reversal . . . . . . . . . . . . . . . . . Media selectively break or preserve these symmetries! . . . . . . . . . . . . . . 𝑊 ℝ 𝑊 ℂ 1 = 𝜏 1 ⊗ 𝟚 1 = (𝜏 1 ⊗𝟚) 𝐷 2 = i 𝜏 2 ⊗ 𝟚 2 = i 𝜏 2 ⊗ 𝟚 𝑊 ℝ 𝑊 ℂ 𝑊 ℝ 𝑊 ℂ 3 = 𝜏 3 ⊗ 𝟚 3 = (𝜏 3 ⊗𝟚) 𝐷

. 0 . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Schrödinger Formalism of In Vacuo Maxwell Equations . Fundamental equation 𝜔 𝐼 (𝑢)) = ( 0 . ) (𝜔 𝐹 (𝑢) 𝜔 𝐼 (𝑢)) Building blocks States: Ψ(𝑢) ∈ { compl. 𝜕 > 0 waves } Symmetries (+TR) (ordinary) (+TR) Classical Fundamental equation 0 0 Building blocks Symmetries . 𝜕 > 0 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) + i ∇ × ) ( E (𝑢) + i ∇ × i 𝜖 i 𝜖 H (𝑢)) = ( − i ∇ × H (𝑢)) − i ∇ × 𝜖𝑢 (𝜔 𝐹 (𝑢) Hamiltonian: 𝑁 = −𝜏 2 ⊗ ∇ × = 𝑁 ∗ Hamiltonian: 𝑁 = −𝜏 2 ⊗ ∇ × States: ( E (𝑢) H (𝑢) ) ∈ 𝑀 2 (ℝ 3 , ℝ 6 ) 𝑊 ℝ 𝑊 ℂ 1 = 𝜏 1 ⊗ 𝟚 1 = (𝜏 1 ⊗ 𝟚) 𝐷 2 = i 𝜏 2 ⊗ 𝟚 2 = i 𝜏 2 ⊗ 𝟚 𝑊 ℝ 𝑊 ℂ 𝑊 ℝ 𝑊 ℂ 3 = 𝜏 3 ⊗ 𝟚 3 = (𝜏 3 ⊗ 𝟚) 𝐷

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo 1 Quantum vs. Classical 2 Maxwell’s Equations in Linear Media 3 Topological Classifjcation of Electromagnetic Media 4 Obstacles For Proving the Photonic Bulk-Edge Correspondence 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Da Capo

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Main Messages of This Talk 1 Rewrite Maxwell’s equations in the form of a Schrödinger equation. De Nittis & L., Annals of Physics 396 , pp. 221–260, 2018 2 Classify electromagnetic media using the Cartan-Altland-Zirnbauer scheme. De Nittis & L., arXiv:1710.08104, 2017; De Nittis & L., arXiv:1806.07783, 2018 3 Adapt existing techniques to prove bulk-boundary correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . … in progress

. Da Capo . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Maxwell’s Equations in Linear, Dispersionless Media . 𝜓 𝐹𝐼 𝜓 𝐼𝐹 0) (dynamical) (∇⋅ 𝜓 𝐹𝐼 𝜓 𝐼𝐹 0) (constraint) Constituent Parts Material weights phenomenologically describe properties of the medium . . . . . . . . . . . . . . . . . . . . . . . . . . Absence of sources . . . . 𝜖𝑢 ( E (𝑢) H (𝑢)) = (+∇ × H (𝑢) 𝜈 ) 𝜖 ( 𝜁 −∇ × E (𝑢)) − (0 𝜈 ) ( E (𝑢) ∇⋅) ( 𝜁 H (𝑢)) = (0

. Da Capo . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Maxwell’s Equations in Linear, Dispersionless Media . 𝜓 𝐹𝐼 𝜓 𝐼𝐹 0) (dynamical) (∇⋅ 𝜓 𝐹𝐼 𝜓 𝐼𝐹 0) (constraint) Constituent Parts Material weights phenomenologically describe properties of the medium . . . . . . . . . . . . . . . . . . . . . . . . . . Absence of sources . . . . 𝜖𝑢 ( E (𝑢) H (𝑢)) = (+∇ × H (𝑢) 𝜈 ) 𝜖 ( 𝜁 −∇ × E (𝑢)) − (0 𝜈 ) ( E (𝑢) ∇⋅) ( 𝜁 H (𝑢)) = (0

. Da Capo . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Maxwell’s Equations in Linear, Dispersionless Media . 𝜓 𝐹𝐼 𝜓 𝐼𝐹 0) (dynamical) (∇⋅ 𝜓 𝐹𝐼 𝜓 𝐼𝐹 0) (constraint) Constituent Parts Material weights phenomenologically describe properties of the medium . . . . . . . . . . . . . . . . . . . . . . . . . . Absence of sources . . . . 𝜖𝑢 ( E (𝑢) H (𝑢)) = (+∇ × H (𝑢) 𝜈 ) 𝜖 ( 𝜁 −∇ × E (𝑢)) − (0 𝜈 ) ( E (𝑢) ∇⋅) ( 𝜁 H (𝑢)) = (0

. 𝜓 𝐼𝐹 . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Maxwell’s Equations in Linear, Dispersionless Media 𝜓 𝐹𝐼 0) . (dynamical) (∇⋅ 𝜓 𝐹𝐼 𝜓 𝐼𝐹 0) (constraint) Abbreviations and Notation 𝑋(𝑦) = ( 𝜁(𝑦) 𝜓 𝐹𝐼 (𝑦) 𝜓 𝐼𝐹 (𝑦) 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢)) = (+∇ × H (𝑢) 𝜈 ) 𝜖 ( 𝜁 −∇ × E (𝑢)) − (0 𝜈 ) ( E (𝑢) ∇⋅) ( 𝜁 H (𝑢)) = (0 Multiply both sides of dynamical Maxwell equations by i 𝜈(𝑦) ) + i ∇ × Introduce Rot ∶= ( ) and Div ∶= ( ∇⋅ − i ∇ × ∇⋅ )

. 𝜓 𝐼𝐹 . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Maxwell’s Equations in Linear, Dispersionless Media 𝜓 𝐹𝐼 0) . (dynamical) (∇⋅ 𝜓 𝐹𝐼 𝜓 𝐼𝐹 0) (constraint) Abbreviations and Notation 𝑋(𝑦) = ( 𝜁(𝑦) 𝜓 𝐹𝐼 (𝑦) 𝜓 𝐼𝐹 (𝑦) 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢)) = (+ i ∇ × H (𝑢) 𝜈 ) i 𝜖 ( 𝜁 − i ∇ × E (𝑢)) − (0 𝜈 ) ( E (𝑢) ∇⋅) ( 𝜁 H (𝑢)) = (0 Multiply both sides of dynamical Maxwell equations by i 𝜈(𝑦) ) + i ∇ × Introduce Rot ∶= ( ) and Div ∶= ( ∇⋅ − i ∇ × ∇⋅ )

. Da Capo . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Maxwell’s Equations in Linear, Dispersionless Media . 0) (dynamical) (∇⋅ 0) (constraint) Abbreviations and Notation 𝑋(𝑦) = ( 𝜁(𝑦) 𝜓 𝐹𝐼 (𝑦) 𝜓 𝐼𝐹 (𝑦) 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢)) = (+ i ∇ × H (𝑢) 𝑋 i 𝜖 − i ∇ × E (𝑢)) − (0 ∇⋅) 𝑋 ( E (𝑢) H (𝑢)) = (0 Multiply both sides of dynamical Maxwell equations by i 𝜈(𝑦) ) + i ∇ × Introduce Rot ∶= ( ) and Div ∶= ( ∇⋅ − i ∇ × ∇⋅ )

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Maxwell’s Equations in Linear, Dispersionless Media (dynamical) (constraint) Abbreviations and Notation 𝑋(𝑦) = ( 𝜁(𝑦) 𝜓 𝐹𝐼 (𝑦) 𝜓 𝐼𝐹 (𝑦) 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢) ) = Rot ( E (𝑢) 𝑋 i 𝜖 H (𝑢) ) Div 𝑋( E (𝑢) H (𝑢) ) = 0 Multiply both sides of dynamical Maxwell equations by i 𝜈(𝑦) ) Introduce Rot ∶= ( + i ∇ × ) and Div ∶= ( ∇⋅ − i ∇ × ∇⋅ )

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo Commonly Used, But Unphysical Maxwell’s Equations (dynamical) (constraint) Usually material weights are 𝑋 ≠ 𝑋 complex! ⇝ e. g. gyrotropic media (QHE of Light!) Immediate Consequences Equations must be considered on subspaces complex Banach space 𝑀 2 (ℝ 3 , ℂ 6 ) Even if initial conditions are real, solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢) ) = Rot ( E (𝑢) 𝑋 i 𝜖 H (𝑢) ) Div 𝑋( E (𝑢) H (𝑢) ) = 0 ( E (𝑢) , H (𝑢)) ≠ ( E (𝑢) , H (𝑢)) acquire imaginary part over time!

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo Commonly Used, But Unphysical Maxwell’s Equations (dynamical) (constraint) Usually material weights are 𝑋 ≠ 𝑋 complex! ⇝ e. g. gyrotropic media (QHE of Light!) Immediate Consequences Equations must be considered on subspaces complex Banach space 𝑀 2 (ℝ 3 , ℂ 6 ) Even if initial conditions are real, solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢) ) = Rot ( E (𝑢) 𝑋 i 𝜖 H (𝑢) ) Div 𝑋( E (𝑢) H (𝑢) ) = 0 ( E (𝑢) , H (𝑢)) ≠ ( E (𝑢) , H (𝑢)) acquire imaginary part over time!

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo Commonly Used, But Unphysical Maxwell’s Equations (dynamical) (constraint) Usually material weights are 𝑋 ≠ 𝑋 complex! ⇝ e. g. gyrotropic media (QHE of Light!) Immediate Consequences Equations must be considered on subspaces complex Banach space 𝑀 2 (ℝ 3 , ℂ 6 ) Even if initial conditions are real, solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢) ) = Rot ( E (𝑢) 𝑋 i 𝜖 H (𝑢) ) Div 𝑋( E (𝑢) H (𝑢) ) = 0 ( E (𝑢) , H (𝑢)) ≠ ( E (𝑢) , H (𝑢)) acquire imaginary part over time!

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Commonly Used, But Unphysical Maxwell’s Equations (dynamical) (constraint) Usually material weights are 𝑋 ≠ 𝑋 complex! Three Options 1 Take the real part of the complex wave 2 Give up on real-valuedness of electromagnetic fjelds. ⇝ Inconsistent interpretation (complex Lorentz force!?!) 3 . . . . . . . . . . . . . . . . Modify equations of motion. ⇝ Correct choice! . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢) ) = Rot ( E (𝑢) 𝑋 i 𝜖 H (𝑢) ) Div 𝑋( E (𝑢) H (𝑢) ) = 0 ⟹ Breaks conservation of energy!

. Bulk-Edge Correspondence . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Da Capo . Commonly Used, But Unphysical Maxwell’s Equations (dynamical) (constraint) Usually material weights are 𝑋 ≠ 𝑋 complex! Three Options 1 Take the real part of the complex wave ⟹ Breaks conservation of energy! 2 Give up on real-valuedness of electromagnetic fjelds. ⇝ Inconsistent interpretation (complex Lorentz force!?!) 3 . . . . . . . . . . . . . . . . Modify equations of motion. ⇝ Correct choice! . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢) ) = Rot ( E (𝑢) 𝑋 i 𝜖 H (𝑢) ) Div 𝑋( E (𝑢) H (𝑢) ) = 0

. Bulk-Edge Correspondence . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Da Capo . Commonly Used, But Unphysical Maxwell’s Equations (dynamical) (constraint) Usually material weights are 𝑋 ≠ 𝑋 complex! Three Options 1 Take the real part of the complex wave ⟹ Breaks conservation of energy! 2 Give up on real-valuedness of electromagnetic fjelds. ⇝ Inconsistent interpretation (complex Lorentz force!?!) 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝜖𝑢 ( E (𝑢) H (𝑢) ) = Rot ( E (𝑢) 𝑋 i 𝜖 H (𝑢) ) Div 𝑋( E (𝑢) H (𝑢) ) = 0 Modify equations of motion. ⇝ Correct choice

. Quantum vs. Classical . . . . . . . . . . Maxwell’s Equations in Linear Media . Topological Classifjcation Bulk-Edge Correspondence Da Capo Maxwell’s Equations for Gyrotropic Media Real solutions linear combination of complex ±𝜕 waves: Pair of equations ( derived from Maxwell’s equations for linear, dispersive media!) 𝜕 > 0 ∶ 𝜕 < 0 ∶ 𝑋(𝑢, 𝑦) = 𝑋(𝑢, 𝑦) ⟺ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑋 − (𝑦) = 𝑋 + (𝑦) ( E , H ) = Ψ + + Ψ − = 2 Re Ψ ± {𝑋 + i 𝜖 𝑢 Ψ + = Rot Ψ + Div 𝑋 + Ψ + = 0 {𝑋 − i 𝜖 𝑢 Ψ − = Rot Ψ − Div 𝑋 − Ψ − = 0

. . . . . . . . . . . . Quantum vs. Classical . Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Maxwell’s Equations for Gyrotropic Media Real solutions linear combination of complex ±𝜕 waves: Pair of equations ( derived from Maxwell’s equations for linear, dispersive media!) 𝜕 > 0 ∶ 𝜕 < 0 ∶ ⟺ . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑋 − (𝑦) = 𝑋 + (𝑦) ( E , H ) = Ψ + + Ψ − = 2 Re Ψ ± {𝑋 + i 𝜖 𝑢 Ψ + = Rot Ψ + Div 𝑋 + Ψ + = 0 {𝑋 + i 𝜖 𝑢 Ψ − = Rot Ψ − Div 𝑋 + Ψ − = 0 𝑋(𝑢, 𝑦) = 𝑋(𝑢, 𝑦)

. . . . . . . . . . . . Quantum vs. Classical . Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Maxwell’s Equations for Gyrotropic Media Real solutions linear combination of complex ±𝜕 waves: Pair of equations ( derived from Maxwell’s equations for linear, dispersive media!) 𝜕 > 0 ∶ 𝜕 < 0 ∶ ⟺ . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑋 − (𝑦) = 𝑋 + (𝑦) ( E , H ) = Ψ + + Ψ − = 2 Re Ψ + {𝑋 + i 𝜖 𝑢 Ψ + = Rot Ψ + Div 𝑋 + Ψ + = 0 {𝑋 − i 𝜖 𝑢 Ψ − = Rot Ψ − Div 𝑋 − Ψ − = 0 𝑋(𝑢, 𝑦) = 𝑋(𝑢, 𝑦)

. . . . . . . . . . . . Quantum vs. Classical . Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Maxwell’s Equations for Gyrotropic Media Physically Meaningful Equations 𝜕 > 0 ∶ Compatibility with reality-condition baked in! Difgerence between physical and unphysical equations: defjned on difgerent subspaces of Banach space 𝑀 2 (ℝ 3 , ℂ 6 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . Real solutions ( E (𝑢) , H (𝑢)) = 2 Re Ψ + (𝑢) where Ψ + (𝑢) solves {𝑋 + i 𝜖 𝑢 Ψ + = Rot Ψ + Div 𝑋 + Ψ + = 0 ℋ + = { complex 𝜕 > 0 states } ⊊ ℋ ℂ,⟂ = ker ( Div 𝑋 + )

. . . . . . . . . . . . Quantum vs. Classical . Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Maxwell’s Equations for Gyrotropic Media Physically Meaningful Equations 𝜕 > 0 ∶ Compatibility with reality-condition baked in! Difgerence between physical and unphysical equations: defjned on difgerent subspaces of Banach space 𝑀 2 (ℝ 3 , ℂ 6 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . Real solutions ( E (𝑢) , H (𝑢)) = 2 Re Ψ + (𝑢) where Ψ + (𝑢) solves {𝑋 + i 𝜖 𝑢 Ψ + = Rot Ψ + Div 𝑋 + Ψ + = 0 ℋ + = { complex 𝜕 > 0 states } ⊊ ℋ ℂ,⟂ = ker ( Div 𝑋 + )

. . . . . . . . . . . . Quantum vs. Classical . Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Maxwell’s Equations for Gyrotropic Media Physically Meaningful Equations 𝜕 > 0 ∶ Compatibility with reality-condition baked in! Difgerence between physical and unphysical equations: defjned on difgerent subspaces of Banach space 𝑀 2 (ℝ 3 , ℂ 6 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . Real solutions ( E (𝑢) , H (𝑢)) = 2 Re Ψ + (𝑢) where Ψ + (𝑢) solves {𝑋 + i 𝜖 𝑢 Ψ + = Rot Ψ + Div 𝑋 + Ψ + = 0 ℋ + = { complex 𝜕 > 0 states } ⊊ ℋ ℂ,⟂ = ker ( Div 𝑋 + )

. . . . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Schrödinger formalism for Maxwell’s . . . . . . . . . . . . . . . . . . . . . . . . equations in non-dispersive media

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo Relevant Electromagnetic Media Assumption (Material weights) 𝜓(𝑦) 𝜓(𝑦) ∗ 𝜈(𝑦)) 1 The medium is lossless . 2 index medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑋 + (𝑦) = ( 𝜁(𝑦) ( 𝑋 ∗ + = 𝑋 + ) 𝑋 + describes a positive ( 0 < 𝑑 𝟚 ≤ 𝑋 + ≤ 𝐷 𝟚 )

. ⎫ . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Schrödinger Formalism of Maxwell’s Equations Real transversal states (𝜁 𝜓 𝜔 𝐼 ) = (+∇ × 𝜔 𝐹 −∇ × 𝜔 𝐼 ) } . ⎬ } ⎭ ⟷ ⎧ { { ⎨ { { ⎩ Complex states with 𝜕 > 0 ℋ = {Ψ ∈ 𝑀 2 (ℝ 3 , ℂ 6 ) ∣ Ψ is 𝜕 > 0 state } Energy scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (De Nittis & L., Annals of Physics 396 , pp. 221–260, 2018) Theorem (De Nittis & L. (2018)) Ψ = 𝑄 + ( E , H ) ( E , H ) = 2 Re Ψ 𝑁 = 𝑋 −1 Rot | 𝜕>0 = 𝑁 ∗ 𝑋 𝜓 ∗ 𝜈) 𝜖 i 𝜖 𝑢 Ψ = 𝑁Ψ 𝜖𝑢 (𝜔 𝐹 ℝ 3 d 𝑦 Φ(𝑦) ⋅ 𝑋(𝑦)Ψ(𝑦) ⟨Φ, Ψ⟩ 𝑋 = ∫ (All subscripts + dropped to simplify notation.)

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo 1 Quantum vs. Classical 2 Maxwell’s Equations in Linear Media 3 Topological Classifjcation of Electromagnetic Media 4 Obstacles For Proving the Photonic Bulk-Edge Correspondence 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Da Capo

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Main Messages of This Talk 1 Rewrite Maxwell’s equations in the form of a Schrödinger equation. De Nittis & L., Annals of Physics 396 , pp. 221–260, 2018 2 Classify electromagnetic media using the Cartan-Altland-Zirnbauer scheme. De Nittis & L., arXiv:1710.08104, 2017; De Nittis & L., arXiv:1806.07783, 2018 3 Adapt existing techniques to prove bulk-boundary correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . … in progress

. Symmetries of the In Vacuo Maxwell Equations . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo 𝜕 > 0 ∶ . Real Symmetry Complex Representative TI Classifjcation Meaning +TR Flips helicity and arrow of time ordinary Dual symmetry +TR Ordinary EM . . . . . . . . . . . . . . . . . . . . . . . . . . time-reversal . . . . . { i 𝜖 𝑢 Ψ = Rot Ψ Div Ψ = 0 𝑊 ℝ 𝑊 ℂ 1 = 𝜏 1 ⊗ 𝟚 1 = (𝜏 1 ⊗𝟚) 𝐷 2 = i 𝜏 2 ⊗ 𝟚 2 = i 𝜏 2 ⊗ 𝟚 𝑊 ℝ 𝑊 ℂ 𝑊 ℝ 𝑊 ℂ 3 = 𝜏 3 ⊗ 𝟚 3 = (𝜏 3 ⊗𝟚) 𝐷

. . . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Media Breaking/Preserving Symmetries ⟺ . . . . . . . . . . . . . . . . . . . . . . . . . {[ Rot , 𝑊 ℂ ] = 0 (vac. symm.) Medium has symmetry 𝑊 ℂ 𝑊 ℂ (anti)unitary on ℋ

. . . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Media Breaking/Preserving Symmetries ⟺ . . . . . . . . . . . . . . . . . . . . . . . . . Medium has symmetry 𝑊 ℂ [𝑋, 𝑊 ℂ ] = 0

. . . . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo . . . . . . . . . . . . . . . . . . . . . . . . Photonic Crystals: Periodic Electromagnetic Media

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Material vs. Crystallographic Symmetries Material 𝜓 𝜓 ∗ 𝜈) Properties of and relations between 𝜁 , 𝜈 and 𝜓 3 Crystallographic Wu & Hu (2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lu et al (2013) 𝑋 = ( 𝜁 a az ˆ r a 2 a 1 𝑊 ℂ 1 , 𝑊 ℂ 2 and 𝑊 ℂ o ax ˆ ay ˆ a 3

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Material vs. Crystallographic Symmetries Material 𝜓 𝜓 ∗ 𝜈) Properties of and relations between 𝜁 , 𝜈 and 𝜓 3 Crystallographic Wu & Hu (2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lu et al (2013) 𝑋 = ( 𝜁 a az ˆ r 𝑊 ℂ 1 , 𝑊 ℂ 2 and 𝑊 ℂ a 2 a 1 o ax ˆ ay ˆ a 3

. . . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Topological Classifjcation of EM Media Assumption . . . . . . . . . . . . . . . . . . . . . . . . . 𝑋 has no crystallographic symmetries.

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Topological Classifjcation of EM Media Theorem (De Nittis & L. (2017)) Non-gyrotropic Dual-symmetric, non-gyrotr. 𝑋 = ( 𝜁 𝜁 Gyrotropic No symmetries Magneto-electric . . . . . . . . . . . . . . . . . . . . . . . . . ( De Nittis & L., arxiv:1710.08104 (2017)) . . . . 𝑋 = ( 𝜁 0 0 𝜈 ) = ( 𝜁 0 𝑋 = ( 𝜁 0 0 𝜈 ) ≠ ( 𝜁 0 0 𝜈 ) 0 𝜈 ) 𝑊 ℂ 3 = (𝜏 3 ⊗ 𝟚) 𝐷 − i 𝜓 − i 𝜓 𝑋 = ( 𝜁 𝜓 𝜓 𝜁 ) = ( 𝜁 𝜓 + i 𝜓 + i 𝜓 𝜁 ) = ( 𝜁 ) 𝜓 𝜁 ) 𝑊 ℂ 𝑊 ℂ 1 = (𝜏 1 ⊗ 𝟚) 𝐷 , 𝑊 ℂ 1 = (𝜏 1 ⊗ 𝟚) 𝐷 3 = (𝜏 3 ⊗ 𝟚) 𝐷

. Topological Classifjcation of EM Media . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Theorem (De Nittis & L. (2017)) . Non-gyrotropic Class AI Realized, e. g. dielectrics Dual-symmetric, non-gyrotr. Two +TR ⟹ 2 × Class AI Realized, e. g. vacuum and YIG Gyrotropic Class A (Quantum Hall Class) Realized, e. g. YIG for microwaves Magneto-electric Class AI Realized, e. g. Tellegen media 4 difgerent topological classes of EM media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( De Nittis & L., arxiv:1710.08104 (2017))

. Topological Classifjcation of EM Media . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Theorem (De Nittis & L. (2017)) . Non-gyrotropic Class AI Realized, e. g. dielectrics Dual-symmetric, non-gyrotr. Two +TR ⟹ 2 × Class AI Realized, e. g. vacuum and YIG Gyrotropic Class A (Quantum Hall Class) Realized, e. g. YIG for microwaves Magneto-electric Class AI Realized, e. g. Tellegen media Only one is topologically non-trivial in 𝑒 ≤ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( De Nittis & L., arxiv:1710.08104 (2017))

. . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Conclusions from Topological Classifjcation Some works proposed to use unphysical symmetries Class AII cannot occur via material symmetries alone supported! Tight-binding operators cannot have incompatible . . . . . . . . . . . . . . . . . . . . . . . . . . symmetries! (e. g. fermionic time-reversal symmetries 𝑊 f = (𝜏 2 ⊗ 𝟚) 𝐷 ) ⇝ No ℤ 2 -valued Kane-Mele-type topological invariants

. Maxwell’s Equations in Linear Media . . . . . . . . . . Quantum vs. Classical Topological Classifjcation . Bulk-Edge Correspondence Da Capo 1 Quantum vs. Classical 2 Maxwell’s Equations in Linear Media 3 Topological Classifjcation of Electromagnetic Media 4 Obstacles For Proving the Photonic Bulk-Edge Correspondence 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Da Capo

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Main Messages of This Talk 1 Rewrite Maxwell’s equations in the form of a Schrödinger equation. De Nittis & L., Annals of Physics 396 , pp. 221–260, 2018 2 Classify electromagnetic media using the Cartan-Altland-Zirnbauer scheme. De Nittis & L., arXiv:1710.08104, 2017; De Nittis & L., arXiv:1806.07783, 2018 3 Adapt existing techniques to prove bulk-boundary correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . … in progress

. . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo Physical Setting Joannopoulos, Soljačić et al (2009) Quasi-2d photonic crystal Topological photonic crystal of class A . . . . . . . . . . . . . . . . . . . . . . . . . . (i. e. 𝑋 breaks 𝑊 ℂ 1 and 𝑊 ℂ 3 )

. . . . . . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo A Physicist’s POV of the Bulk-Edge Correspondence Joannopoulos, Soljačić et al (2009) 0 + 1 = 1 ⇒ 1 edge mode Skirlo et al, PRL 113, 113904, 2014 0 + 0 − 2 + 4 + 2 = 4 ⇒ 4 edge modes . . . . . . . . . . . . . . . . . . . . . . . . . . Works as advertised!

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Haldane’s Photonic Bulk-Edge Correspondence Conjecture Tasks 1 2 Defjne edge system ( ⇝ boundary conditions can break +TR symmetries!) 3 Proof of “mathematical” bulk-edge correspondence 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identify the topological observable ⇝ Poynting vector? 𝑈 bulk = 𝑈 edge = net ♯ of edge modes Defjne topological bulk invariant 𝑈 bulk

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Haldane’s Photonic Bulk-Edge Correspondence Conjecture Tasks 1 2 Defjne edge system ( ⇝ boundary conditions can break +TR symmetries!) 3 Proof of “mathematical” bulk-edge correspondence 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identify the topological observable ⇝ Poynting vector? 𝑈 bulk = 𝑈 edge = net ♯ of edge modes Defjne topological bulk invariant 𝑈 bulk

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Haldane’s Photonic Bulk-Edge Correspondence Conjecture Tasks 1 2 Defjne edge system ( ⇝ boundary conditions can break +TR symmetries!) 3 Proof of “mathematical” bulk-edge correspondence 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identify the topological observable ⇝ Poynting vector? 𝑈 bulk = 𝑈 edge = net ♯ of edge modes Defjne topological bulk invariant 𝑈 bulk

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Haldane’s Photonic Bulk-Edge Correspondence Conjecture Tasks 1 2 Defjne edge system ( ⇝ boundary conditions can break +TR symmetries!) 3 Proof of “mathematical” bulk-edge correspondence 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identify the topological observable ⇝ Poynting vector? 𝑈 bulk = 𝑈 edge = net ♯ of edge modes Defjne topological bulk invariant 𝑈 bulk

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Haldane’s Photonic Bulk-Edge Correspondence Conjecture Tasks 1 2 Defjne edge system ( ⇝ boundary conditions can break +TR symmetries!) 3 Proof of “mathematical” bulk-edge correspondence 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identify the topological observable ⇝ Poynting vector? 𝑈 bulk = 𝑈 edge = net ♯ of edge modes Defjne topological bulk invariant 𝑈 bulk

. Topological Classifjcation . . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Bulk-Edge Correspondence . Da Capo Haldane’s Photonic Bulk-Edge Correspondence Conjecture Tasks 1 2 Defjne edge system ( ⇝ boundary conditions can break +TR symmetries!) 3 Proof of “mathematical” bulk-edge correspondence 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identify the topological observable ⇝ Poynting vector? 𝑈 bulk = 𝑈 edge = net ♯ of edge modes Defjne topological bulk invariant 𝑈 bulk

. . . . . . . . . . . . Quantum vs. Classical . Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo The Frequency Band Picture Theorem (De Nittis & L., 2014) 1 Bloch bands and functions locally analytic away from crossings 2 2 ground state bands with ≈ linear dispersion at 𝑙 = 0 and 𝜕 = 0 3 . . . . . . . . . . . . . . . (Theorem 1.4 and Lemma 3.7 in De Nittis & L., Documenta Math. 19 , pp. 63–101, 2014) . . . . . . . . . . . . w A + n 4 n 3 B + n 2 n 1 k -p p n 1 B A 𝑄 gs (𝑙) discontinuous at 𝑙 = 0 (jump in dimensionality!)

. Da Capo . . . . . . . . Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence The Bloch Vector Bundle . Proceed as Usual 1 Select bulk frequency band gap. 2 Defjne the “Fermi projection” 𝑄(𝑙) ∶= ∑ 𝑜 3 Defjne the Bloch bundle ℰ 𝕌 ∗ (𝑄) ∶ ⨆ 𝑙∈𝕌 ∗ 𝜌 ⟶ 𝕌 ∗ . . . . . . . . . . . . . . . . In Bloch-Floquet representation. . . . . . . . . . . . . . . w A + n 4 n 3 B + n 2 n 1 k -p p n 1 B 𝑘=1 |𝜒 𝑘 (𝑙)⟩⟨𝜒 𝑘 (𝑙)| . A ran 𝑄(𝑙)