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

Towards a Rigorous Proof of Haldane’s Photonic Bulk-edge Correspondence

joint work with Giuseppe De Nittis

Max Lein

Advanced Institute of Materials Research, Tohoku University 2018.09.04@ETH Zürich

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Quantum Hall Efgect for Light

Predicted theoretically by Raghu & Haldane (2005) ...

(𝜁 𝜈) ≠ (𝜁 𝜈) symmetry breaking ⎫ } ⎬ } ⎭ ⟹

a b c

A A B l a Ez Negative Positive

Joannopoulos, Soljačić et al (2009)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Quantum Hall Efgect for Light

... and realized experimentally by Joannopoulos et al (2009)

Joannopoulos, Soljačić et al (2009) Joannopoulos, Soljačić et al (2009)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Insight Topological efgects are wave, not quantum phenomena!

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Topological Efgects: Phenomenological Similarities

Light Coupled Oscillators Quantum

Periodic structure ⇝ bulk band gap Breaking of time-reversal symmetries Unidirectional edge modes Robust under perturbations

Difgerent manifestations of the same underlying physical principles!

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Photonic Bulk-Edge Correspondence

Conjecture

In a two-dimensional photonic crystals with boundary the difgerence of the number of left- and right-moving boundary modes Chbulk = 𝑈edge = net ♯ of edge modes in bulk band gaps is a topologically protected quantity and equals the Chern number associated to the frequency bands below the bulk band gap.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Photonic Bulk-Edge Correspondence

Conjecture

𝑈bulk = 𝑈edge = net ♯ of edge modes My Main Goal Make the statement mathematically precise and provide a proof.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Proof of Haldane’s Photonic Bulk-Edge Correspondence

Seems simple enough …

1

Rewrite Maxwell’s equations in the form of a Schrödinger equation.

De Nittis & L., Comm. Math. Phys. 332, pp. 221–260, 2014

2

Classify electromagnetic media using the Cartan-Altland-Zirnbauer scheme.

De Nittis & L., Annals of Physics 350, pp. 568–587, 2014

3

Adapt existing techniques to prove bulk-boundary correspondences

(relying on e. g. Hatsugai, Graf & Porta, Hayashi; Kellendonk & Schulz-Baldes)

Easy! … No!

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Proof of Haldane’s Photonic Bulk-Edge Correspondence

Seems simple enough …

1

Rewrite Maxwell’s equations in the form of a Schrödinger equation.

De Nittis & L., Comm. Math. Phys. 332, pp. 221–260, 2014

2

Classify electromagnetic media using the Cartan-Altland-Zirnbauer scheme.

De Nittis & L., Annals of Physics 350, pp. 568–587, 2014

3

Adapt existing techniques to prove bulk-boundary correspondences

(relying on e. g. Hatsugai, Graf & Porta, Hayashi; Kellendonk & Schulz-Baldes)

Easy! … No!

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Main Messages of This Talk

Explain why things are not so simple. Explain how to deal with the complications in Steps 1 & 2. Explain the obstacles to be overcome in Step 3.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation 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

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation 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

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

The Real-Valuedness Condition

Classical waves such as (E, H) = (E, H) are real-valued!

Our earlier works start with the standard equations used in the physics community. These equations violate real-valuedness condition. We were aware of this problem and discussed it in one of our earlier works.

(Section 6 of De Nittis & L., Annals of Physics 350, pp. 568–587, 2014)

Clarifjed thanks to discussions with Duncan Haldane and Kostya Bliokh. ⟹ Mathematically correct results about unphysical equations.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

The Real-Valuedness Condition

Classical waves such as (E, H) = (E, H) are real-valued!

Our earlier works start with the standard equations used in the physics community. These equations violate real-valuedness condition. We were aware of this problem and discussed it in one of our earlier works.

(Section 6 of De Nittis & L., Annals of Physics 350, pp. 568–587, 2014)

Clarifjed thanks to discussions with Duncan Haldane and Kostya Bliokh. ⟹ Mathematically correct results about unphysical equations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

The Real-Valuedness Condition

Classical waves such as (E, H) = (E, H) are real-valued!

Our earlier works start with the standard equations used in the physics community. These equations violate real-valuedness condition. We were aware of this problem and discussed it in one of our earlier works.

(Section 6 of De Nittis & L., Annals of Physics 350, pp. 568–587, 2014)

Clarifjed thanks to discussions with Duncan Haldane and Kostya Bliokh. ⟹ Mathematically correct results about unphysical equations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media 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

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Quantum vs. Classical Maxwell’s Equations in Linear Media 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

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Quantum vs. Classical Equations: a Single Spin

Idea Start with the same equations of motion, i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( −i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(𝑢)) ,

• nce in the quantum and then in the classical context.

Math is trivial, everything is explicit Immediately transfers to many other hamiltonian equations as 𝐾 = i𝜏2 is the canonical symplectic form Conceptually applies to all classical wave equations

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Equations

Purpose Anticipate symmetry classifjcation of electromagnetic media i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( −i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(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

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Equations

Purpose Anticipate symmetry classifjcation of electromagnetic media i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( −i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(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

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Systems

Quantum

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: ( 𝜔1(𝑢)

𝜔2(𝑢) ) ∈ ℋℂ = ℂ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Systems

Quantum

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: ( 𝜔1(𝑢)

𝜔2(𝑢) ) ∈ ℋℂ = ℂ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Systems

Quantum

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: ( 𝜔1(𝑢)

𝜔2(𝑢) ) ∈ ℋℂ = ℂ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries? 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Systems

Quantum

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: ( 𝜔1(𝑢)

𝜔2(𝑢) ) ∈ ℋℂ = ℂ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(???) (i𝜏2) 𝐼 (i𝜏2)−1 = +𝐼 (???) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Systems

Quantum

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: ( 𝜔1(𝑢)

𝜔2(𝑢) ) ∈ ℋℂ = ℂ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝐷 not defjned on ℋℝ = ℝ2 𝜏1, i𝜏2 and 𝜏3 are real matrices ⇝ classical spin transformations

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Systems

Quantum

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: ( 𝜔1(𝑢)

𝜔2(𝑢) ) ∈ ℋℂ = ℂ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(chiral) 𝜏2 𝐼 𝜏−1

2

= +𝐼 (ordin.) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝜏1,3 𝐼 𝜏−1

1,3 = −𝐼

(???) (i𝜏2) 𝐼 (i𝜏2)−1 = +𝐼 (???) 𝐷 𝐼 𝐷 = 𝐼 = −𝐼 (+PH) (𝜏1,3 𝐷) 𝐼 (𝜏1,3 𝐷)−1 = +𝐼 (+TR) (𝜏2 𝐷) 𝐼 (𝜏2 𝐷)−1 = −𝐼 (-PH)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Symmetries of Classical and Quantum Spin Systems

Quantum

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔1(𝑢) 𝜔2(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝜔1(𝑢) 𝜔2(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: ( 𝜔1(𝑢)

𝜔2(𝑢) ) ∈ ℋℂ = ℂ2

Symmetries 𝑉1 = 𝜏1 (chiral) 𝑈1 = 𝜏1 𝐷 (+TR) 𝑉2 = 𝜏2 (ordin.) 𝑈2 = 𝜏2 𝐷 (-PH) 𝑉3 = 𝜏3 (chiral) 𝑈3 = 𝜏3 𝐷 (+TR) 𝐷 (+PH)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝑊 ℝ

1 = 𝜏1

(???) 𝑊 ℝ

2 = i𝜏2 (???)

𝑊 ℝ

3 = 𝜏3

(???)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 Insulators (and many other techniques from quantum mechanics)

• nly 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

Work with complex Ψ which represent real states 𝑁 = 2Re Ψ (establish 1-to-1 correspondence ℋℂ ↔ ℋℝ)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 Insulators (and many other techniques from quantum mechanics)

• nly 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

Work with complex Ψ which represent real states 𝑁 = 2Re Ψ (establish 1-to-1 correspondence ℋℂ ↔ ℋℝ)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Complexifying the Classical Equations

Complexifjcation

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: 𝑁 = Ψ+ + Ψ− ∈ ℋℝ ⊂ ℋℂ Symmetries 𝑊 ℂ

1 = ???

𝑊 ℂ

2 = ???

𝑊 ℂ

3 = ???

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝑊 ℝ

1 = 𝜏1

𝑊 ℝ

2 = i𝜏2

𝑊 ℝ

3 = 𝜏3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Complexifying the Classical Equations

Complexifjcation

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: 𝑁 = Ψ+ + Ψ− ∈ ℋℝ ⊂ ℋℂ Symmetries 𝟚 ∣ℋℝ = 𝐷 ∣ℋℝ none vs. +PH 𝜏1,3 ∣ℋℝ = 𝜏1,3 𝐷 ∣ℋℝ chiral vs. +TR i𝜏2 ∣ℋℝ = i𝜏2 𝐷 ∣ℋℝ

• rdin. vs. -PH

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝑊 ℝ

1 = 𝜏1

𝑊 ℝ

2 = i𝜏2

𝑊 ℝ

3 = 𝜏3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Complexifying the Classical Equations

Complexifjcation

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: 𝑁 = Ψ+ + Ψ− ∈ ℋℝ ⊂ ℋℂ Symmetries

• Redundant symmetry operations
• Difgerent choices ⇒ difgerent

topological classifjcations!?

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝑊 ℝ

1 = 𝜏1

𝑊 ℝ

2 = i𝜏2

𝑊 ℝ

3 = 𝜏3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 Insulators (and many other techniques from quantum mechanics)

• nly 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

Work with complex Ψ which represent real states 𝑁 = 2Re Ψ (establish 1-to-1 correspondence ℋℂ ↔ ℋℝ)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 Insulators (and many other techniques from quantum mechanics)

• nly 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

Work with complex Ψ which represent real states 𝑁 = 2Re Ψ (establish 1-to-1 correspondence ℋℂ ↔ ℋℝ)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Schrödinger Formalism of Classical Spin Waves

Complexifjcation

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: 𝑁 = Ψ+ + Ψ− ∈ ℋℝ⊂ ℋℂ Symmetries 𝟚 ∣ℋℝ = 𝐷 ∣ℋℝ 𝜏1,3 ∣ℋℝ = 𝜏1,3 𝐷 ∣ℋℝ i𝜏2 ∣ℋℝ = i𝜏2 𝐷 ∣ℋℝ

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝑊 ℝ

1 = 𝜏1

𝑊 ℝ

2 = i𝜏2

𝑊 ℝ

3 = 𝜏3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Schrödinger Formalism of Classical Spin Waves

𝜕 > 0 Representation

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔+,1(𝑢) 𝜔+,2(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝜔+,1(𝑢) 𝜔+,2(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: 𝑁 = 2Re Ψ+ ∈ ℋℝ Symmetries 𝑊 ℂ

1 = ???

𝑊 ℂ

2 = ???

𝑊 ℂ

3 = ???

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (𝑁𝑦(𝑢) 𝑁𝑧(𝑢)) = ( − i 𝜕0 +i 𝜕0 ) (𝑁𝑦(𝑢) 𝑁𝑧(𝑢))

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: (

𝑁𝑦(𝑢) 𝑁𝑧(𝑢) ) ∈ ℋℝ = ℝ2

Symmetries 𝑊 ℝ

1 = 𝜏1

𝑊 ℝ

2 = i𝜏2

𝑊 ℝ

3 = 𝜏3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 ⇝ Systematically identify ℝ2 ≅ ℂ 𝑁(𝑢) ⏟ real wave = 2Re Ψ(𝑢) ⏟ complex 𝜕 > 0 wave 1-to-1 correspondence

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 ⇝ Systematically identify ℝ2 ≅ ℂ 𝑁(𝑢) = (cos 𝜕0𝑢 − sin 𝜕0𝑢 sin 𝜕0𝑢 cos 𝜕0𝑢 ) (𝑏 𝑐) = 2Re Ψ(𝑢) = 2Re ((𝑏 − i𝑐) e−i𝜕0𝑢Ψ+) where Ψ+ = ( 1

+i ) is the eigenvector of 𝐼 = 𝜕0 𝜏2 to +𝜕0 > 0.

1-to-1 correspondence

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Schrödinger Formalism of Classical Spin Waves

𝜕 > 0 Representation

Fundamental equation i 𝜖

𝜖𝑢Ψ(𝑢) = 𝜕0 𝜏2 Ψ(𝑢)

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: Ψ(𝑢) ∈ ℋ+ = spanℂ {( 1

+i )}

Symmetries 𝑊 ℂ

1 = ???

𝑊 ℂ

2 = ???

𝑊 ℂ

3 = ???

Classical

Fundamental equation i 𝜖

𝜖𝑢𝑁(𝑢) = 𝜕0 𝜏2 𝑁(𝑢)

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: 𝑁(𝑢) = 2Re Ψ(𝑢) ∈ ℝ2 Symmetries 𝑊 ℝ

1 = 𝜏1

𝑊 ℝ

2 = i𝜏2

𝑊 ℝ

3 = 𝜏3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Translating Real Symmetries to 𝜕 > 0 Representation

Requirements

1

𝑁 = 2Re Ψ, then 𝑊 ℝ

𝑘 𝑁 = 2Re (𝑊 ℂ 𝑘 Ψ) 2

𝑊 ℂ

𝑘 is a (anti)unitary on ℋ+, i. e. it

maps 𝜕 > 0 waves onto 𝜕 > 0 waves.

Consequences

1

𝑊 ℂ

𝑘 = {𝑊 ℝ 𝑘

(unitary) 𝑊 ℝ

𝑘 𝐷

(antiunitary)

2

𝑊 ℂ

𝑘 must commute with

𝐼 = 𝜕0 𝜏2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Translating Real Symmetries to 𝜕 > 0 Representation

Requirements

1

𝑁 = 2Re Ψ, then 𝑊 ℝ

𝑘 𝑁 = 2Re (𝑊 ℂ 𝑘 Ψ) 2

𝑊 ℂ

𝑘 is a (anti)unitary on ℋ+, i. e. it

maps 𝜕 > 0 waves onto 𝜕 > 0 waves.

Consequences

1

𝑊 ℂ

𝑘 = {𝑊 ℝ 𝑘

(unitary) 𝑊 ℝ

𝑘 𝐷

(antiunitary)

2

𝑊 ℂ

𝑘 must commute with

𝐼 = 𝜕0 𝜏2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 𝑊 ℝ

1 = 𝜏1

𝑊 ℂ

1 = 𝜏1 𝐷

+TR 𝑊 ℝ

2 = i𝜏2

𝑊 ℂ

2 = i𝜏2

• rdinary

𝑊 ℝ

3 = 𝜏3

𝑊 ℂ

3 = 𝜏3 𝐷

+TR

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Translating Real Symmetries to 𝜕 > 0 Representation

𝜕 > 0 Representation

Fundamental equation i 𝜖

𝜖𝑢Ψ(𝑢) = 𝜕0 𝜏2 Ψ(𝑢)

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 = 𝐼∗ States: Ψ(𝑢) ∈ ℋ+ = spanℂ {( 1

+i )}

Symmetries 𝑊 ℂ

1 = 𝜏1 𝐷

(+TR) 𝑊 ℂ

2 = i𝜏2

(ordinary) 𝑊 ℂ

3 = 𝜏3 𝐷

(+TR)

Classical

Fundamental equation i 𝜖

𝜖𝑢𝑁(𝑢) = 𝜕0 𝜏2 𝑁(𝑢)

Building blocks Hamiltonian: 𝐼 = 𝜕0 𝜏2 States: 𝑁(𝑢) = 2Re Ψ(𝑢) ∈ ℝ2 Symmetries 𝑊 ℝ

1 = 𝜏1

𝑊 ℝ

2 = i𝜏2

𝑊 ℝ

3 = 𝜏3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media 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 ⇝ Incompatible with the real-valuedness of classical waves “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!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Applies directly to vacuum Maxwell equations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Spin ⟶ In Vacuo Maxwell Equations

𝐼 = 𝜕0 𝜏2 ⟶ Rot = −𝜏2 ⊗ ∇× 𝑊 ℝ

1,3 = 𝜏1,3

⟶ 𝑊 ℝ

1,3 = 𝜏1,3 ⊗ 𝟚

𝑊 ℝ

2 = i𝜏2

⟶ 𝑊 ℝ

2 = i𝜏2 ⊗ 𝟚

Same Strategy

1

Complexify classical equations

2

Eliminate superfmuous states in complex Hilbert space

3

Identify complex implementation of the three symmetries

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Schrödinger Formalism of In Vacuo Maxwell Equations

Complexifjcation

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔𝐹(𝑢) 𝜔𝐼(𝑢)) = ( + i ∇× −i ∇× ) (𝜔𝐹(𝑢) 𝜔𝐼(𝑢))

Building blocks Hamiltonian: 𝑁 = −𝜏2 ⊗ ∇× = 𝑁∗ States: Ψ(𝑢) ∈ 𝑀2(ℝ3, ℂ6) Symmetries 𝑊 ℂ

1 = (𝜏1 ⊗ 𝟚) 𝐷

(+TR) 𝑊 ℂ

2 = i𝜏2 ⊗ 𝟚

(ordinary) 𝑊 ℂ

3 = (𝜏3 ⊗ 𝟚) 𝐷

(+TR)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = ( + i ∇× −i ∇× ) (E(𝑢) H(𝑢))

Building blocks Hamiltonian: 𝑁 = −𝜏2 ⊗ ∇× States: ( E(𝑢)

H(𝑢) ) ∈ 𝑀2(ℝ3, ℝ6)

Symmetries 𝑊 ℝ

1 = 𝜏1 ⊗ 𝟚

𝑊 ℝ

2 = i𝜏2 ⊗ 𝟚

𝑊 ℝ

3 = 𝜏3 ⊗ 𝟚

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Schrödinger Formalism of In Vacuo Maxwell Equations

Representing real, transversal EM Fields as complex 𝜕 > 0 waves (E(𝑢), H(𝑢)) ⏟ ⏟ ⏟ ⏟ ⏟ real wave = 2Re Ψ(𝑢) ⏟ complex 𝜕 > 0 wave ⟹ Ψ ∈ ℋ+ = {complex 𝜕 > 0 waves}.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Schrödinger Formalism of In Vacuo Maxwell Equations

𝜕 > 0 Representation

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔𝐹(𝑢) 𝜔𝐼(𝑢)) = ( + i ∇× −i ∇× ) (𝜔𝐹(𝑢) 𝜔𝐼(𝑢))

Building blocks Hamiltonian: 𝑁 = −𝜏2 ⊗ ∇× = 𝑁∗ States: Ψ(𝑢) ∈ {compl. 𝜕 > 0 waves} Symmetries 𝑊 ℂ

1 = (𝜏1 ⊗ 𝟚) 𝐷

(+TR) 𝑊 ℂ

2 = i𝜏2 ⊗ 𝟚

(ordinary) 𝑊 ℂ

3 = (𝜏3 ⊗ 𝟚) 𝐷

(+TR)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = ( + i ∇× −i ∇× ) (E(𝑢) H(𝑢))

Building blocks Hamiltonian: 𝑁 = −𝜏2 ⊗ ∇× States: ( E(𝑢)

H(𝑢) ) ∈ 𝑀2(ℝ3, ℝ6)

Symmetries 𝑊 ℝ

1 = 𝜏1 ⊗ 𝟚

𝑊 ℝ

2 = i𝜏2 ⊗ 𝟚

𝑊 ℝ

3 = 𝜏3 ⊗ 𝟚

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Schrödinger Formalism of In Vacuo Maxwell Equations

Real Symmetry Complex Representative TI Classifjcation Meaning 𝑊 ℝ

1 = 𝜏1 ⊗ 𝟚

𝑊 ℂ

1 = (𝜏1 ⊗𝟚) 𝐷

+TR Flips helicity and arrow of time 𝑊 ℝ

2 = i𝜏2 ⊗ 𝟚

𝑊 ℂ

2 = i𝜏2 ⊗ 𝟚

• rdinary

Dual symmetry 𝑊 ℝ

3 = 𝜏3 ⊗ 𝟚

𝑊 ℂ

3 = (𝜏3 ⊗𝟚) 𝐷

+TR Ordinary EM time-reversal

Media selectively break or preserve these symmetries!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Schrödinger Formalism of In Vacuo Maxwell Equations

𝜕 > 0 Representation

Fundamental equation

i 𝜖 𝜖𝑢 (𝜔𝐹(𝑢) 𝜔𝐼(𝑢)) = ( + i ∇× −i ∇× ) (𝜔𝐹(𝑢) 𝜔𝐼(𝑢))

Building blocks Hamiltonian: 𝑁 = −𝜏2 ⊗ ∇× = 𝑁∗ States: Ψ(𝑢) ∈ {compl. 𝜕 > 0 waves} Symmetries 𝑊 ℂ

1 = (𝜏1 ⊗ 𝟚) 𝐷

(+TR) 𝑊 ℂ

2 = i𝜏2 ⊗ 𝟚

(ordinary) 𝑊 ℂ

3 = (𝜏3 ⊗ 𝟚) 𝐷

(+TR)

Classical

Fundamental equation

i 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = ( + i ∇× −i ∇× ) (E(𝑢) H(𝑢))

Building blocks Hamiltonian: 𝑁 = −𝜏2 ⊗ ∇× States: ( E(𝑢)

H(𝑢) ) ∈ 𝑀2(ℝ3, ℝ6)

Symmetries 𝑊 ℝ

1 = 𝜏1 ⊗ 𝟚

𝑊 ℝ

2 = i𝜏2 ⊗ 𝟚

𝑊 ℝ

3 = 𝜏3 ⊗ 𝟚

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media 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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation 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

Maxwell’s Equations in Linear, Dispersionless Media

( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = (+∇ × H(𝑢) −∇ × E(𝑢)) − (0 0) (dynamical) (∇⋅ ∇⋅) ( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) (E(𝑢) H(𝑢)) = (0 0) (constraint) Constituent Parts Material weights phenomenologically describe properties of the medium Absence of sources

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Maxwell’s Equations in Linear, Dispersionless Media

( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = (+∇ × H(𝑢) −∇ × E(𝑢)) − (0 0) (dynamical) (∇⋅ ∇⋅) ( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) (E(𝑢) H(𝑢)) = (0 0) (constraint) Constituent Parts Material weights phenomenologically describe properties of the medium Absence of sources

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Maxwell’s Equations in Linear, Dispersionless Media

( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = (+∇ × H(𝑢) −∇ × E(𝑢)) − (0 0) (dynamical) (∇⋅ ∇⋅) ( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) (E(𝑢) H(𝑢)) = (0 0) (constraint) Constituent Parts Material weights phenomenologically describe properties of the medium Absence of sources

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Maxwell’s Equations in Linear, Dispersionless Media

( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = (+∇ × H(𝑢) −∇ × E(𝑢)) − (0 0) (dynamical) (∇⋅ ∇⋅) ( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) (E(𝑢) H(𝑢)) = (0 0) (constraint) Abbreviations and Notation Multiply both sides of dynamical Maxwell equations by i 𝑋(𝑦) = (

𝜁(𝑦) 𝜓𝐹𝐼(𝑦) 𝜓𝐼𝐹(𝑦) 𝜈(𝑦) )

Introduce Rot ∶= (

+i∇× −i∇×

) and Div ∶= ( ∇⋅

∇⋅ )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Maxwell’s Equations in Linear, Dispersionless Media

( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 )i 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = (+i∇ × H(𝑢) −i∇ × E(𝑢)) − (0 0) (dynamical) (∇⋅ ∇⋅) ( 𝜁 𝜓𝐹𝐼 𝜓𝐼𝐹 𝜈 ) (E(𝑢) H(𝑢)) = (0 0) (constraint) Abbreviations and Notation Multiply both sides of dynamical Maxwell equations by i 𝑋(𝑦) = (

𝜁(𝑦) 𝜓𝐹𝐼(𝑦) 𝜓𝐼𝐹(𝑦) 𝜈(𝑦) )

Introduce Rot ∶= (

+i∇× −i∇×

) and Div ∶= ( ∇⋅

∇⋅ )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Maxwell’s Equations in Linear, Dispersionless Media

𝑋 i 𝜖 𝜖𝑢 (E(𝑢) H(𝑢)) = (+i∇ × H(𝑢) −i∇ × E(𝑢)) − (0 0) (dynamical) (∇⋅ ∇⋅) 𝑋 (E(𝑢) H(𝑢)) = (0 0) (constraint) Abbreviations and Notation Multiply both sides of dynamical Maxwell equations by i 𝑋(𝑦) = (

𝜁(𝑦) 𝜓𝐹𝐼(𝑦) 𝜓𝐼𝐹(𝑦) 𝜈(𝑦) )

Introduce Rot ∶= (

+i∇× −i∇×

) and Div ∶= ( ∇⋅

∇⋅ )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Maxwell’s Equations in Linear, Dispersionless Media

𝑋 i 𝜖

𝜖𝑢 ( E(𝑢) H(𝑢) ) = Rot ( E(𝑢) H(𝑢) )

(dynamical) Div 𝑋( E(𝑢)

H(𝑢) ) = 0

(constraint) Abbreviations and Notation Multiply both sides of dynamical Maxwell equations by i 𝑋(𝑦) = (

𝜁(𝑦) 𝜓𝐹𝐼(𝑦) 𝜓𝐼𝐹(𝑦) 𝜈(𝑦) )

Introduce Rot ∶= (

+i∇× −i∇×

) and Div ∶= ( ∇⋅

∇⋅ )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Commonly Used, But Unphysical Maxwell’s Equations

𝑋 i 𝜖

𝜖𝑢 ( E(𝑢) H(𝑢) ) = Rot ( E(𝑢) H(𝑢) )

(dynamical) Div 𝑋( E(𝑢)

H(𝑢) ) = 0

(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 𝑀2(ℝ3, ℂ6) Even if initial conditions are real, solutions (E(𝑢) , H(𝑢)) ≠ (E(𝑢) , H(𝑢)) acquire imaginary part over time!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Commonly Used, But Unphysical Maxwell’s Equations

𝑋 i 𝜖

𝜖𝑢 ( E(𝑢) H(𝑢) ) = Rot ( E(𝑢) H(𝑢) )

(dynamical) Div 𝑋( E(𝑢)

H(𝑢) ) = 0

(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 𝑀2(ℝ3, ℂ6) Even if initial conditions are real, solutions (E(𝑢) , H(𝑢)) ≠ (E(𝑢) , H(𝑢)) acquire imaginary part over time!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Commonly Used, But Unphysical Maxwell’s Equations

𝑋 i 𝜖

𝜖𝑢 ( E(𝑢) H(𝑢) ) = Rot ( E(𝑢) H(𝑢) )

(dynamical) Div 𝑋( E(𝑢)

H(𝑢) ) = 0

(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 𝑀2(ℝ3, ℂ6) Even if initial conditions are real, solutions (E(𝑢) , H(𝑢)) ≠ (E(𝑢) , H(𝑢)) acquire imaginary part over time!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Commonly Used, But Unphysical Maxwell’s Equations

𝑋 i 𝜖

𝜖𝑢 ( E(𝑢) H(𝑢) ) = Rot ( E(𝑢) H(𝑢) )

(dynamical) Div 𝑋( E(𝑢)

H(𝑢) ) = 0

(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!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Commonly Used, But Unphysical Maxwell’s Equations

𝑋 i 𝜖

𝜖𝑢 ( E(𝑢) H(𝑢) ) = Rot ( E(𝑢) H(𝑢) )

(dynamical) Div 𝑋( E(𝑢)

H(𝑢) ) = 0

(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!

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Commonly Used, But Unphysical Maxwell’s Equations

𝑋 i 𝜖

𝜖𝑢 ( E(𝑢) H(𝑢) ) = Rot ( E(𝑢) H(𝑢) )

(dynamical) Div 𝑋( E(𝑢)

H(𝑢) ) = 0

(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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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: (E, H) = Ψ+ + Ψ− = 2Re Ψ± Pair of equations (derived from Maxwell’s equations for linear, dispersive media!) 𝜕 > 0 ∶ {𝑋+ i𝜖𝑢Ψ+ = Rot Ψ+ Div 𝑋+ Ψ+ = 0 𝜕 < 0 ∶ {𝑋− i𝜖𝑢Ψ− = Rot Ψ− Div 𝑋− Ψ− = 0 𝑋(𝑢, 𝑦) = 𝑋(𝑢, 𝑦) ⟺ 𝑋−(𝑦) = 𝑋+(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: (E, H) = Ψ+ + Ψ− = 2Re Ψ± Pair of equations (derived from Maxwell’s equations for linear, dispersive media!) 𝜕 > 0 ∶ {𝑋+ i𝜖𝑢Ψ+ = Rot Ψ+ Div 𝑋+ Ψ+ = 0 𝜕 < 0 ∶ {𝑋+ i𝜖𝑢Ψ− = Rot Ψ− Div 𝑋+ Ψ− = 0 𝑋(𝑢, 𝑦) = 𝑋(𝑢, 𝑦) ⟺ 𝑋−(𝑦) = 𝑋+(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: (E, H) = Ψ+ + Ψ− = 2Re Ψ+ Pair of equations (derived from Maxwell’s equations for linear, dispersive media!) 𝜕 > 0 ∶ {𝑋+ i𝜖𝑢Ψ+ = Rot Ψ+ Div 𝑋+ Ψ+ = 0 𝜕 < 0 ∶ {𝑋− i𝜖𝑢Ψ− = Rot Ψ− Div 𝑋− Ψ− = 0 𝑋(𝑢, 𝑦) = 𝑋(𝑢, 𝑦) ⟺ 𝑋−(𝑦) = 𝑋+(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 Real solutions (E(𝑢) , H(𝑢)) = 2Re Ψ+(𝑢) where Ψ+(𝑢) solves 𝜕 > 0 ∶ {𝑋+ i𝜖𝑢Ψ+ = Rot Ψ+ Div 𝑋+ Ψ+ = 0 Compatibility with reality-condition baked in! Difgerence between physical and unphysical equations: defjned on difgerent subspaces of Banach space 𝑀2(ℝ3, ℂ6) ℋ+ = {complex 𝜕 > 0 states} ⊊ ℋℂ,⟂ = 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 Real solutions (E(𝑢) , H(𝑢)) = 2Re Ψ+(𝑢) where Ψ+(𝑢) solves 𝜕 > 0 ∶ {𝑋+ i𝜖𝑢Ψ+ = Rot Ψ+ Div 𝑋+ Ψ+ = 0 Compatibility with reality-condition baked in! Difgerence between physical and unphysical equations: defjned on difgerent subspaces of Banach space 𝑀2(ℝ3, ℂ6) ℋ+ = {complex 𝜕 > 0 states} ⊊ ℋℂ,⟂ = 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 Real solutions (E(𝑢) , H(𝑢)) = 2Re Ψ+(𝑢) where Ψ+(𝑢) solves 𝜕 > 0 ∶ {𝑋+ i𝜖𝑢Ψ+ = Rot Ψ+ Div 𝑋+ Ψ+ = 0 Compatibility with reality-condition baked in! Difgerence between physical and unphysical equations: defjned on difgerent subspaces of Banach space 𝑀2(ℝ3, ℂ6) ℋ+ = {complex 𝜕 > 0 states} ⊊ ℋℂ,⟂ = 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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Relevant Electromagnetic Media

Assumption (Material weights) 𝑋+(𝑦) = ( 𝜁(𝑦) 𝜓(𝑦) 𝜓(𝑦)∗ 𝜈(𝑦))

1

The medium is lossless. (𝑋 ∗

+ = 𝑋+)

2

𝑋+ describes a positive index medium. (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

Theorem (De Nittis & L. (2018))

Real transversal states (E, H) = 2Re Ψ

(𝜁 𝜓 𝜓∗ 𝜈) 𝜖 𝜖𝑢 (𝜔𝐹 𝜔𝐼) = (+∇ × 𝜔𝐹 −∇ × 𝜔𝐼)

⎫ } ⎬ } ⎭ ⟷ ⎧ { { ⎨ { { ⎩ Complex states with 𝜕 > 0 Ψ = 𝑄+(E, H) 𝑁 = 𝑋 −1 Rot |𝜕>0 = 𝑁∗𝑋 i 𝜖𝑢Ψ = 𝑁Ψ ℋ = {Ψ ∈ 𝑀2(ℝ3, ℂ6) ∣ Ψ is 𝜕 > 0 state} ⟨Φ, Ψ⟩𝑋 = ∫

ℝ3 d𝑦 Φ(𝑦) ⋅ 𝑋(𝑦)Ψ(𝑦)

Energy scalar product (All subscripts + dropped to simplify notation.)

(De Nittis & L., Annals of Physics 396, pp. 221–260, 2018)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media 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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation 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

Symmetries of the In Vacuo Maxwell Equations

𝜕 > 0 ∶ {i𝜖𝑢Ψ = Rot Ψ Div Ψ = 0

Real Symmetry Complex Representative TI Classifjcation Meaning 𝑊 ℝ

1 = 𝜏1 ⊗ 𝟚

𝑊 ℂ

1 = (𝜏1 ⊗𝟚) 𝐷

+TR Flips helicity and arrow of time 𝑊 ℝ

2 = i𝜏2 ⊗ 𝟚

𝑊 ℂ

2 = i𝜏2 ⊗ 𝟚

• rdinary

Dual symmetry 𝑊 ℝ

3 = 𝜏3 ⊗ 𝟚

𝑊 ℂ

3 = (𝜏3 ⊗𝟚) 𝐷

+TR Ordinary EM time-reversal

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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 ⟺ {[Rot , 𝑊 ℂ] = 0 (vac. symm.) 𝑊 ℂ (anti)unitary on ℋ

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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

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Photonic Crystals: Periodic Electromagnetic Media

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Material vs. Crystallographic Symmetries

Material

𝑋 = ( 𝜁 𝜓 𝜓∗ 𝜈) Properties of and relations between 𝜁, 𝜈 and 𝜓 𝑊 ℂ

1 , 𝑊 ℂ 2 and 𝑊 ℂ 3

Crystallographic

Wu & Hu (2015)

az ˆ ax ˆ ay ˆ a2

a

a3 a1

• r

Lu et al (2013)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Material vs. Crystallographic Symmetries

Material

𝑋 = ( 𝜁 𝜓 𝜓∗ 𝜈) Properties of and relations between 𝜁, 𝜈 and 𝜓 𝑊 ℂ

1 , 𝑊 ℂ 2 and 𝑊 ℂ 3

Crystallographic

Wu & Hu (2015)

az ˆ ax ˆ ay ˆ a2

a

a3 a1

• r

Lu et al (2013)

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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.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Topological Classifjcation of EM Media

Theorem (De Nittis & L. (2017)) Non-gyrotropic

𝑋 = ( 𝜁 0

0 𝜈 ) = ( 𝜁 0 0 𝜈 )

𝑊 ℂ

3 = (𝜏3 ⊗ 𝟚) 𝐷

Dual-symmetric, non-gyrotr.

𝑋 = (

𝜁 −i𝜓 +i𝜓 𝜁 ) = ( 𝜁 −i 𝜓 +i 𝜓 𝜁 )

𝑊 ℂ

1 = (𝜏1 ⊗ 𝟚) 𝐷, 𝑊 ℂ 3 = (𝜏3 ⊗ 𝟚) 𝐷

Gyrotropic

𝑋 = ( 𝜁 0

0 𝜈 ) ≠ ( 𝜁 0 0 𝜈 )

No symmetries

Magneto-electric

𝑋 = ( 𝜁 𝜓

𝜓 𝜁 ) = ( 𝜁 𝜓 𝜓 𝜁 )

𝑊 ℂ

1 = (𝜏1 ⊗ 𝟚) 𝐷

(De Nittis & L., arxiv:1710.08104 (2017))

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Topological Classifjcation of EM Media

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))

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Topological Classifjcation of EM Media

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))

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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 (e. g. fermionic time-reversal symmetries 𝑊f = (𝜏2 ⊗ 𝟚) 𝐷) Class AII cannot occur via material symmetries alone ⇝ No ℤ2-valued Kane-Mele-type topological invariants supported! Tight-binding operators cannot have incompatible symmetries!

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Quantum vs. Classical Maxwell’s Equations in Linear Media 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

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation 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

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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

• f class A

(i. e. 𝑋 breaks 𝑊 ℂ

1 and 𝑊 ℂ 3 )

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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!

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Photonic Bulk-Edge Correspondence

Conjecture

𝑈bulk = 𝑈edge= net ♯ of edge modes Tasks

1

Defjne topological bulk invariant 𝑈bulk

2

Defjne edge system (⇝ boundary conditions can break +TR symmetries!)

3

Proof of “mathematical” bulk-edge correspondence

4

Identify the topological observable ⇝ Poynting vector?

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Photonic Bulk-Edge Correspondence

Conjecture

𝑈bulk = 𝑈edge= net ♯ of edge modes Tasks

1

Defjne topological bulk invariant 𝑈bulk

2

Defjne edge system (⇝ boundary conditions can break +TR symmetries!)

3

Proof of “mathematical” bulk-edge correspondence

4

Identify the topological observable ⇝ Poynting vector?

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Photonic Bulk-Edge Correspondence

Conjecture

𝑈bulk = 𝑈edge= net ♯ of edge modes Tasks

1

Defjne topological bulk invariant 𝑈bulk

2

Defjne edge system (⇝ boundary conditions can break +TR symmetries!)

3

Proof of “mathematical” bulk-edge correspondence

4

Identify the topological observable ⇝ Poynting vector?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Photonic Bulk-Edge Correspondence

Conjecture

𝑈bulk = 𝑈edge= net ♯ of edge modes Tasks

1

Defjne topological bulk invariant 𝑈bulk

2

Defjne edge system (⇝ boundary conditions can break +TR symmetries!)

3

Proof of “mathematical” bulk-edge correspondence

4

Identify the topological observable ⇝ Poynting vector?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Photonic Bulk-Edge Correspondence

Conjecture

𝑈bulk = 𝑈edge= net ♯ of edge modes Tasks

1

Defjne topological bulk invariant 𝑈bulk

2

Defjne edge system (⇝ boundary conditions can break +TR symmetries!)

3

Proof of “mathematical” bulk-edge correspondence

4

Identify the topological observable ⇝ Poynting vector?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Haldane’s Photonic Bulk-Edge Correspondence

Conjecture

𝑈bulk = 𝑈edge= net ♯ of edge modes Tasks

1

Defjne topological bulk invariant 𝑈bulk

2

Defjne edge system (⇝ boundary conditions can break +TR symmetries!)

3

Proof of “mathematical” bulk-edge correspondence

4

Identify the topological observable ⇝ Poynting vector?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

The Frequency Band Picture

A+

n2 n 1 n1

n3

n4

A B B+

• p

p k w

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

𝑄gs(𝑙) discontinuous at 𝑙 = 0 (jump in dimensionality!)

(Theorem 1.4 and Lemma 3.7 in De Nittis & L., Documenta Math. 19, pp. 63–101, 2014)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

The Bloch Vector Bundle

A+

n2 n 1 n1

n3

n4

A B B+

• p

p k w

Proceed as Usual

1

Select bulk frequency band gap.

2

Defjne the “Fermi projection” 𝑄(𝑙) ∶= ∑

𝑜 𝑘=1 |𝜒𝑘(𝑙)⟩⟨𝜒𝑘(𝑙)|.

3

Defjne the Bloch bundle ℰ𝕌∗(𝑄) ∶ ⨆

𝑙∈𝕌∗

ran 𝑄(𝑙)

𝜌

⟶ 𝕌∗

In Bloch-Floquet representation.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

The Bloch Vector Bundle

A+

n2 n 1 n1

n3

n4

A B B+

• p

p k w

Proposition

1

ℰ𝕌∗(𝑄) is only a bundle over the Brillouin torus 𝕌∗, but not a vector bundle.

2

The restriction ℰ𝕌∗ {0}(𝑄) ∶= ℰ𝕌∗(𝑄)∣𝕌∗ {0} is a vector bundle.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

The Bloch Vector Bundle

Proposition

1

ℰ𝕌∗(𝑄) is only a bundle over the Brillouin torus 𝕌∗, but not a vector bundle.

2

The restriction ℰ𝕌∗ {0}(𝑄) ∶= ℰ𝕌∗(𝑄)∣𝕌∗ {0} is a vector bundle. Origin of the Problem 𝑙 ↦ 𝑄gs(𝑙) is not continuous at 𝑙 = 0. 𝑙 ↦ 𝑄(𝑙) − 𝑄gs(𝑙) is analytic at 𝑙 = 0. (The ground state bands are responsible for the bad behavior.) ⟹ 𝑙 ↦ 𝑄(𝑙) is continuous (in fact, analytic) only on 𝕌∗ {0}.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Classifjcation of the Bloch Bundle

Idea 1 Classify the bundle over the entire Brillouin torus 𝕌∗ ℰ𝕌∗(𝑄) ∶ ⨆

𝑙∈𝕌∗

ran 𝑄(𝑙)

𝜌

⟶ 𝕌∗ Only a bundle, not a vector bundle! Classifjcation theory not well-developed What is the topological invariant here?!?

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Classifjcation of the Bloch Bundle

Idea 2 Classify the bundle over 𝕌∗ {0} ℰ𝕌∗ {0}(𝑄) ∶ ⨆

𝑙∈𝕌∗ {0}

ran 𝑄(𝑙)

𝜌

⟶ 𝕌∗ {0} Bona fjde vector bundle 𝕌∗ {0} deformation retracts to 𝕋1 ⟹ Vec𝑙(𝕌∗ {0}) ≅ Vec𝑙(𝕋1) = 0 Chern number Ch1(ℰ𝕌∗ {0}(𝑄)) = 0 well-defjned, but always 0!

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Classifjcation of the Bloch Bundle

Idea 3 Extend the vector bundle over 𝕌∗ 𝐶𝜁 ℰ𝕌∗ 𝐶𝜁(𝑄) ∶ ⨆

𝑙∈𝕌∗ 𝐶𝜁

ran 𝑄(𝑙)

𝜌

⟶ 𝕌∗ 𝐶𝜁 to a vector bundle over 𝕌∗ Relative cohomology theory? Might work, but we need to construct a “natural” extension It seems that the Chern charge ∫

|𝑙|=𝜁

d𝑙 Tr(𝒝gs(𝑙)) = 0

• vanishes. ⇝ Extension by vector bundle surgery possible?

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Classifjcation of the Bloch Bundle

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

Chern charges of ground state bands 0

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Classifjcation of the Bloch Bundle

Idea 3 Extend the vector bundle over 𝕌∗ 𝐶𝜁 ℰ𝕌∗ 𝐶𝜁(𝑄) ∶ ⨆

𝑙∈𝕌∗ 𝐶𝜁

ran 𝑄(𝑙)

𝜌

⟶ 𝕌∗ 𝐶𝜁 to a vector bundle over 𝕌∗ Relative cohomology theory? Might work, but we need to construct a “natural” extension It seems that the Chern charge ∫

|𝑙|=𝜁

d𝑙 Tr(𝒝gs(𝑙)) = 0

• vanishes. ⇝ Extension by vector bundle surgery possible?

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Thank you for your attention!

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Interaction of Time-Reversal & Crystallographic Symmetries

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Talk Based On

De Nittis & L., On the Role of Symmetries in Photonic Crystals, Annals of Physics 350, pp. 568–587, 2014 De Nittis & L., The Schrödinger Formalism of Electromagnetism and Other Classical Waves — How to Make Quantum-Wave Analogies Rigorous, Annals of Physics 396, pp. 221–260, 2018 De Nittis & L., Symmetry Classifjcation of Topological Photonic Crystals, arXiv 1710.08104, 1–49, 2017 De Nittis & L., Equivalence of Electric, Magnetic and Electromagnetic Chern Numbers for Topological Photonic Crystals, arxiv 1806.07783, pp. 1–33, 2018 L., Taking Inspiration from Quantum-Wave Analogies — Recent Results for Photonic Crystals, Macroscopic Limits of Quantum Systems — Munich, Germany, March 20–April 1, 2017, to appear in Springer Proceedings in Mathematics and Statistics, 2018

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Interaction of Material and Crystallographic Symmetries

Idea of real vs. complex implementation of symmetries works also for other symmetries (e. g. rotations, parity) ⟹ Crystallographic symmetries can be handled within the Schrödinger formalism of classical electromagnetism Recent works from condensed matter physics on crystallographic TIs (e. g. by Shiozaki, Sato & Gomi, arxiv:1802.06694 (2018))

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Example: “Spin-Valley Hall Efgect”

Wu & Hu (2015)

Edge modes topological Pseudospin degree of freedom in a time-reversal-symmetric medium Time-reversal symmetry 𝑈3 ≠ 𝑈↑ ⊕ 𝑈↓ not blockdiagonal ⟹ 𝑁↑/↓ class A (no symmetry) Chern numbers 𝐷↑ = −𝐷↓ ≠ 0 possible Not in contradiction, edge modes come in ↑ / ↓ pairs Topologically protected against perturbations which preserve 𝑈3 symmetry and honeycomb structure

Wu & Hu (2015)

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Derivation of Maxwell’s Equations in Linear Media

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Fundamental Equations

Maxwell’s Equations in Media

1

Maxwell’s equations i 𝜖 𝜖𝑢 (D B) = ( +i∇× −i∇× ) (H E) − i (𝐾 𝐸 𝐾 𝐶) (dynamical) (∇ ⋅ D ∇ ⋅ B) = (𝜍𝐸 𝜍𝐶) (constraint)

2

Constitutive relations (D B) = 𝒳 (E H)

3

Conservation of charge ∇ ⋅ 𝐾 ♯ + 𝜍♯ = 0, ♯ = 𝐸, 𝐶

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Fundamental Equations

Maxwell’s Equations in Media

1

Maxwell’s equations i 𝜖 𝜖𝑢 (D B) = ( +i∇× −i∇× ) (H E) (dynamical) (∇ ⋅ D ∇ ⋅ B) = (0 0) (constraint)

2

Constitutive relations (D B) = 𝒳 (E H)

3

Conservation of charge ⇝ neglect sources for simplicity ∇ ⋅ 𝐾 ♯ + 𝜍♯ = 0, ♯ = 𝐸, 𝐶

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Constitutive Relations for a Linear, Dispersive Medium

For a linear medium the constitutive relations maps a trajectory (−∞, 𝑢] ∋ 𝑡 ↦ (E(𝑡), H(𝑡))

• nto

(D(𝑢, 𝑦) B(𝑢, 𝑦)) ∶= ∫

𝑢 −∞

d𝑡 𝑋(𝑢 − 𝑡, 𝑦) (E(𝑡, 𝑦) H(𝑡, 𝑦)) ⇝ reaction of medium to impinging em wave depends on the past

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Constitutive Relations for a Linear, Dispersive Medium

(D(𝑢), B(𝑢)) ∶= ∫

𝑢 −∞

d𝑡 𝑋(𝑢 − 𝑡) (E(𝑡), H(𝑡))

Assumption (Constitutive relations)

We assume that 𝑋(𝑢, 𝑦) = ( 𝜁(𝑢, 𝑦) 𝜓𝐹𝐼(𝑢, 𝑦) 𝜓𝐼𝐹(𝑢, 𝑦) 𝜈(𝑢, 𝑦) ) ∈ Matℂ(6)

1

is real, 𝑋 = 𝑋, and

2

satisfjes the causality condition 𝑋(𝑢) = 0 for all 𝑢 > 0.

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Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Constitutive Relations for a Linear, Dispersive Medium

(D(𝑢), B(𝑢)) = (𝑋 ∗ (E, H))(𝑢)∫

𝑢 −∞

Assumption (Constitutive relations)

We assume that 𝑋(𝑢, 𝑦) = ( 𝜁(𝑢, 𝑦) 𝜓𝐹𝐼(𝑢, 𝑦) 𝜓𝐼𝐹(𝑢, 𝑦) 𝜈(𝑢, 𝑦) ) ∈ Matℂ(6)

1

is real, 𝑋 = 𝑋, and

2

satisfjes the causality condition 𝑋(𝑢) = 0 for all 𝑢 > 0.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Fundamental Equations for Linear, Dispersive Media

1

Maxwell’s equations i 𝜖

𝜖𝑢(𝑋 ∗ (E, H))(𝑢) = Rot(E(𝑢), H(𝑢))

(dynamical) Div(𝑋 ∗ (E, H))(𝑢) = 0 (constraint)

2

Constitutive relations (D(𝑢), B(𝑢)) = (𝑋 ∗ (E, H))(𝑢)

3

Conservation of charge ⇝ neglect sources for simplicity ∇ ⋅ 𝐾 ♯ + 𝜍♯ = 0, ♯ = 𝐸, 𝐶

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Heuristically Neglecting Dispersion in Maxwell’s Equations

i 𝜖

𝜖𝑢𝑋 ∗ Ψ(𝑢) = Rot Ψ(𝑢)

𝜕 ̂ 𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

ℱ−1

• i 𝜖

𝜖𝑢𝑋 ∗ Ψ(𝑢) = Rot Ψ(𝑢)

𝜕 ̂ 𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

ℱ−1

• 𝜕 ̂

𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕) ± 𝜕 > 0 ∶ 𝜕 ̂ 𝑋(±𝜕0) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

≈

• ± 𝜕 > 0 ∶ 𝜕 ̂

𝑋(±𝜕0) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕) ̂ 𝑋(±𝜕0) i 𝜖

𝜖𝑢Ψ±(𝑢) = Rot Ψ±(𝑢) ℱ

• 1

Apply inverse Fourier transform in time to go from time-dependent to frequency-dependent equations.

2

Approximate material weights ̂ 𝑋(±𝜕) ≈ ̂ 𝑋(±𝜕0) = 𝑋± for frequencies ±𝜕 ≈ ±𝜕0. +𝜕0 and −𝜕0 contributions necessary to reconstruct real solutions.

3

Undo Fourier transform to obtain dynamical equations in the absence of dispersion.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Heuristically Neglecting Dispersion in Maxwell’s Equations

i 𝜖

𝜖𝑢𝑋 ∗ Ψ(𝑢) = Rot Ψ(𝑢)

𝜕 ̂ 𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

ℱ−1

• i 𝜖

𝜖𝑢𝑋 ∗ Ψ(𝑢) = Rot Ψ(𝑢)

𝜕 ̂ 𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

ℱ−1

• 𝜕 ̂

𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕) ± 𝜕 > 0 ∶ 𝜕 ̂ 𝑋(±𝜕0) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

≈

• ± 𝜕 > 0 ∶ 𝜕 ̂

𝑋(±𝜕0) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕) ̂ 𝑋(±𝜕0) i 𝜖

𝜖𝑢Ψ±(𝑢) = Rot Ψ±(𝑢) ℱ

• 1

Apply inverse Fourier transform in time to go from time-dependent to frequency-dependent equations.

2

Approximate material weights ̂ 𝑋(±𝜕) ≈ ̂ 𝑋(±𝜕0) = 𝑋± for frequencies ±𝜕 ≈ ±𝜕0. +𝜕0 and −𝜕0 contributions necessary to reconstruct real solutions.

3

Undo Fourier transform to obtain dynamical equations in the absence of dispersion.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Quantum vs. Classical Maxwell’s Equations in Linear Media Topological Classifjcation Bulk-Edge Correspondence Da Capo

Heuristically Neglecting Dispersion in Maxwell’s Equations

i 𝜖

𝜖𝑢𝑋 ∗ Ψ(𝑢) = Rot Ψ(𝑢)

𝜕 ̂ 𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

ℱ−1

• i 𝜖

𝜖𝑢𝑋 ∗ Ψ(𝑢) = Rot Ψ(𝑢)

𝜕 ̂ 𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

ℱ−1

• 𝜕 ̂

𝑋(𝜕) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕) ± 𝜕 > 0 ∶ 𝜕 ̂ 𝑋(±𝜕0) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕)

≈

• ± 𝜕 > 0 ∶ 𝜕 ̂

𝑋(±𝜕0) ̂ Ψ(𝜕) = Rot ̂ Ψ(𝜕) ̂ 𝑋(±𝜕0) i 𝜖

𝜖𝑢Ψ±(𝑢) = Rot Ψ±(𝑢) ℱ

• 1

Apply inverse Fourier transform in time to go from time-dependent to frequency-dependent equations.

2

Approximate material weights ̂ 𝑋(±𝜕) ≈ ̂ 𝑋(±𝜕0) = 𝑋± for frequencies ±𝜕 ≈ ±𝜕0. +𝜕0 and −𝜕0 contributions necessary to reconstruct real solutions.

3

Undo Fourier transform to obtain dynamical equations in the absence of dispersion.