towards a rigorous proof of haldane s photonic bulk edge
play

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

Feb 12, 2024 •230 likes •1.46k views

Towards a Rigorous Proof of Haldanes Photonic Bulk-edge Correspondence joint work with Giuseppe De Nittis Max Lein Advanced Institute of Materials Research, Tohoku University 2018.09.04@ETH Zrich . . . . . .. . . . . . . .. . . . .

  1. . 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

  2. . 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)

  3. . 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)?

  4. . 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

  5. . ) (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

  6. . ) (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

  7. . 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

  8. . 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

  9. . 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

  10. . 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

  11. . 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 )

  12. . 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 )

  13. . 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

  14. . ) (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

  15. . 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

  16. . 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 )

  17. . 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 )

  18. . 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

  19. . 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

  20. . 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

  21. . . . . . . . . . . . . . . . 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 .

  22. . 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

  23. . 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

  24. . 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

  25. . . . . . . . . . . . . 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

  26. . 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

  27. . 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

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

  29. . 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

  30. . 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

  31. . 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 } .

  32. . 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

  33. . 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

  34. . 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

  35. . 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

  36. . 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

  37. . 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

  38. . 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

  39. . 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

  40. . 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 ∇ × ∇⋅ )

  41. . 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 ∇ × ∇⋅ )

  42. . 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 ∇ × ∇⋅ )

  43. . 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 ∇ × ∇⋅ )

  44. . 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!

  45. . 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!

  46. . 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!

  47. . 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!

  48. . 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

  49. . 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

  50. . 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

  51. . . . . . . . . . . . . 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)

  52. . . . . . . . . . . . . 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)

  53. . . . . . . . . . . . . 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 + )

  54. . . . . . . . . . . . . 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 + )

  55. . . . . . . . . . . . . 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 + )

  56. . . . . . . . . . . . . . . . . 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

  57. . 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 )

  58. . ⎫ . . . 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.)

  59. . 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

  60. . 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

  61. . 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

  62. . . . . . . . . . . . . . . . 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

  63. . . . . . . . . . . . . . . . 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

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

  65. . 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

  66. . 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

  67. . . . . . . . . . . . . . . . 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.

  68. . 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

  69. . 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))

  70. . 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))

  71. . . . . . . . . . . . . . . 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

  72. . 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

  73. . 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

  74. . . . . . . . . . . . . . . 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 )

  75. . . . . . . . . . . . . . . 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!

  76. . 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

  77. . 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

  78. . 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

  79. . 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

  80. . 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

  81. . 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

  82. . . . . . . . . . . . . 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!)

  83. . 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)

Recommend

Si nanopar tic le s: Ne w photonic and e le c tr onic mate r ial Dr. Munir H. Na yfe h De

Si nanopar tic le s: Ne w photonic and e le c tr onic mate r ial Dr. Munir H. Na yfe h De

Si nanopar tic le s: Ne w photonic and e le c tr onic mate r ial Dr. Munir H. Na yfe h De pa rtme nts o f Physic s, Unive rsity o f I llino is NanoSi L e t the r e be L ight Natur e 2 2 Nano S i F r om bulk to nanopar tic

484 views • 12 slides
Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Symposium on

Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Symposium on

Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Symposium on Statistical Mechanics: Many-Body Quantum Systems University of Warwick 17-21 March 2014 Bulk-edge duality for topological insulators Gian Michele Graf

1.91k views • 178 slides
Universality of transport coefficients in the Haldane-Hubbard model Alessandro Giuliani, Univ.

Universality of transport coefficients in the Haldane-Hubbard model Alessandro Giuliani, Univ.

Overview Introduction The model and the main results Sketch of the proof Universality of transport coefficients in the Haldane-Hubbard model Alessandro Giuliani, Univ. Roma Tre Joint work with V. Mastropietro, M. Porta and I. Jauslin

1.13k views • 89 slides
Photonic Crystals Derek Stewart CNF Fall Workshop What are photonic crystals? Photonic crystals

Photonic Crystals Derek Stewart CNF Fall Workshop What are photonic crystals? Photonic crystals

Photonic Crystals Derek Stewart CNF Fall Workshop What are photonic crystals? Photonic crystals are periodic systems that consist of separate high dielectric and low dielectric regions. The periodicity or spacing determines the relevant light

283 views • 11 slides
Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Statistical Mechanics

Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Statistical Mechanics

Bulk-edge duality for topological insulators Gian Michele Graf ETH Zurich Statistical Mechanics Seminar University of Warwick 8 November, 2012 joint work with Marcello Porta Topological insulators: first impressions Insulator in the Bulk:

1.23k views • 33 slides
Recent Advances in Photonic Recent Advances in Photonic effect employing IP- based distributed

Recent Advances in Photonic Recent Advances in Photonic effect employing IP- based distributed

Development of Photonic Network Optical burst switching Next Generation Enhancement of Networks Dynamic (Adaptive) Optical Path Control statistical &gt;10Tb/s multiplexing Photonic MPLS Network Recent Advances in Photonic Recent Advances

762 views • 8 slides
Violation of bulk-edge correspondence in a hydrodynamic model Gian Michele Graf ETH Zurich PhD

Violation of bulk-edge correspondence in a hydrodynamic model Gian Michele Graf ETH Zurich PhD

Violation of bulk-edge correspondence in a hydrodynamic model Gian Michele Graf ETH Zurich PhD School: September 16-20, 2019 @Universit` a degli Studi Roma Tre Violation of bulk-edge correspondence in a hydrodynamic model Gian Michele Graf

1.39k views • 93 slides
Bulk-edge corresponding revisited Yasuhiro Hatsugai Department of physics Univ. Tsukuba Sep.3,

Bulk-edge corresponding revisited Yasuhiro Hatsugai Department of physics Univ. Tsukuba Sep.3,

Workshop, ETH Zrich, 3rd-6th September 2018 Bulk-edge corresponding revisited Yasuhiro Hatsugai Department of physics Univ. Tsukuba Sep.3, 2018 Workshop, ETH Zrich, 3rd-6th September 2018 Plan Topological phases and bulk-edge

738 views • 63 slides
Bulk-Edge correspondence and Fractionalization As a topological (spin) insulator with strong

Bulk-Edge correspondence and Fractionalization As a topological (spin) insulator with strong

Dec. 10, 2008 at KITP (3,1) (1,1) Bulk-Edge correspondence and Fractionalization As a topological (spin) insulator with strong interaction Y. Hatsugai i C ( A ) = A C Institute of Physics Univ. of Tsukuba JAPAN Plan With

1.32k views • 49 slides
How the colour is created? Semiconductors vs Photonic Crystals (PCs) Semiconductors vs Photonic

How the colour is created? Semiconductors vs Photonic Crystals (PCs) Semiconductors vs Photonic

Physical phenomena in nature Denise Montesdeoca Cardenes (Lancaster University) How the colour is created? Semiconductors vs Photonic Crystals (PCs) Semiconductors vs Photonic Crystals (PCs) Band Gap Photonic crystals Controlling: -

218 views • 6 slides
Topological Aspects of Graphene Dirac Fermions and Bulk-Edge Correspondence in a Magnetic Field

Topological Aspects of Graphene Dirac Fermions and Bulk-Edge Correspondence in a Magnetic Field

20 COQUSY06, Sep28, 2006 10 -3 -2 -1 1 2 3 -10 -20 -30 Topological Aspects of Graphene Dirac Fermions and Bulk-Edge Correspondence in a Magnetic Field Department of Applied Physics University of Tokyo Y. Hatsugai with T. Fukui (

792 views • 76 slides
1960-15 ICTP Conference Graphene Week 2008 25 - 29 August 2008 Bulk--edge correspondence in

1960-15 ICTP Conference Graphene Week 2008 25 - 29 August 2008 Bulk--edge correspondence in

1960-15 ICTP Conference Graphene Week 2008 25 - 29 August 2008 Bulk--edge correspondence in graphene with/without magnetic field Topological aspects of Dirac fermions in real materials Y. Hatsugai Institute of Physics, University of Tsukuba,

1.7k views • 49 slides
Unitary Matrix Models: universality conjecture in the bulk and orthogonal and on the edge of the

Unitary Matrix Models: universality conjecture in the bulk and orthogonal and on the edge of the

Unitary Matrix Models Poplavskyi M. Random matrices Unitary Matrix Models: universality conjecture in the bulk and orthogonal and on the edge of the spectrum. polynomials Overview of previous results Global M. Poplavskyi regime

354 views • 22 slides
Haldane Discovery Process and Strategic Planning Meeting with the Garrison Board of Education

Haldane Discovery Process and Strategic Planning Meeting with the Garrison Board of Education

Haldane Discovery Process and Strategic Planning Meeting with the Garrison Board of Education February 4, 2015 The Discovery Process The Haldane School District considered three different questions related to College and Career Readiness during

638 views • 14 slides
Self-Assembly of Metal-Organic Framework Photonic Sensors Nanyang Research Programme Loi Si

Self-Assembly of Metal-Organic Framework Photonic Sensors Nanyang Research Programme Loi Si

Self-Assembly of Metal-Organic Framework Photonic Sensors Nanyang Research Programme Loi Si Xian Raffles Institution INTRODUCTION Photonic Crystals Photonic Crystals Photonic crystals consist of periodic arrangements of regularly

388 views • 22 slides
Edge Caches and Localization Nicholas Weaver International Computer Science Institute

Edge Caches and Localization Nicholas Weaver International Computer Science Institute

Edge Caches and Localization Nicholas Weaver International Computer Science Institute (Apologies for my absence) Bulk data P2P shifts costs By default, bulk-data P2P shifts the deliver costs from the content provider to the retail ISPs

302 views • 6 slides
Coq Proof Assistant Jos Carlos Bacelar Almeida Departamento de Informtica Universidade do

Coq Proof Assistant Jos Carlos Bacelar Almeida Departamento de Informtica Universidade do

Rigorous Software Development Coq Proof Assistant Jos Carlos Bacelar Almeida Departamento de Informtica Universidade do Minho MAP-i, Braga 2010 1 Part II - Deductive Reasoning A Practical Approach to the Coq Proof Assistant

1.04k views • 76 slides
Bulk Density and Void Content Bulk Density Bulk density ( n .) the mass of a unit volume of bulk

Bulk Density and Void Content Bulk Density Bulk density ( n .) the mass of a unit volume of bulk

Bulk Density and Void Content Bulk Density Bulk density ( n .) the mass of a unit volume of bulk aggregate including the volume of the individual particles and the volume of the voids between them. m kg agg bulk density 3 V m

1.45k views • 30 slides
Workflow Plus Bulk Request Actions Tool for Synergy Enterprise What is This Tool ? Allows

Workflow Plus Bulk Request Actions Tool for Synergy Enterprise What is This Tool ? Allows

Workflow Plus Bulk Request Actions Tool for Synergy Enterprise What is This Tool ? Allows Synergy Administrators to manage bulk updates of requests to change statuses. These include: Bulk Approve Bulk Realize Bulk Process

267 views • 8 slides
A Rigorous Curriculum A rigorous curriculum is an inclusive set of intentionally aligned

A Rigorous Curriculum A rigorous curriculum is an inclusive set of intentionally aligned

A Rigorous Curriculum A rigorous curriculum is an inclusive set of intentionally aligned components- clear learning outcomes with matching assessments, engaging learning experiences and instructional strategies - organized into sequenced units

431 views • 22 slides
Why Algorithmic and Rigorous Polynomial Approximations? Rigorous Polynomial Approximation =

Why Algorithmic and Rigorous Polynomial Approximations? Rigorous Polynomial Approximation =

Why Algorithmic and Rigorous Polynomial Approximations? Rigorous Polynomial Approximation = Polynomial + error bound Rigorous methods Algorithmic methods Efficient and accurate 1/16 Multinormval Why Algorithmic and Rigorous Polynomial

1.12k views • 98 slides
Silicon nitride based TriPleX Photonic Integrated Circuits for sensing applications Arne Leinse

Silicon nitride based TriPleX Photonic Integrated Circuits for sensing applications Arne Leinse

Silicon nitride based TriPleX Photonic Integrated Circuits for sensing applications Arne Leinse a.leinse@lionix-int.com 2 11-Dec-18 Micronano Conference 2018 Our chips drive your business 2 What are Photonic ICs (PICs)? Photonic

360 views • 18 slides
CS 225 Data Structures Fe February 15 Tr Tree Proof Wa Wade Fa Fagen-Ul Ulmsch schnei

CS 225 Data Structures Fe February 15 Tr Tree Proof Wa Wade Fa Fagen-Ul Ulmsch schnei

CS 225 Data Structures Fe February 15 Tr Tree Proof Wa Wade Fa Fagen-Ul Ulmsch schnei eider er, , Cra Craig Zi Zilles Tree ee Ter erminology Find an edge that is not on the longest path in the tree. Give that edge a

449 views • 24 slides
Rigorous Evaluation Usability Testing What is Usability Testing? Formal and rigorous testing

Rigorous Evaluation Usability Testing What is Usability Testing? Formal and rigorous testing

Rigorous Evaluation Usability Testing What is Usability Testing? Formal and rigorous testing using a structured process Validate adherence to interaction requirements Actual users who perform realistic and representative tasks

490 views • 38 slides

More recommend