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◗✉❛♥t✉♠ ✭s♣✐♥✮ ❍❛❧❧ ❝♦♥❞✉❝t✐✈✐t②✿ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛ ✭❛♥❞ ❜❡②♦♥❞✮ ●✐♦✈❛♥♥❛ ▼❛r❝❡❧❧✐ ❥♦✐♥t ✇♦r❦s ✇✐t❤ ❉✳ ▼♦♥❛❝♦ ✭ ❘♦♠❛ ❚r❡ ✱ ❘♦♠❛✮✱ ●✳ P❛♥❛t✐ ✭ ▲❛ ❙❛♣✐❡♥③❛ ✱ ❘♦♠❛✮✱ ❈✳ ❚❛✉❜❡r ✭ ❊❚❍ ✱ ❩ür✐❝❤✮ ❛♥❞ ❙✳ ❚❡✉❢❡❧ ✭❯♥✐✈❡rs✐tät ❚ü❜✐♥❣❡♥✮ ❬▼▼P❚❡❪✿ ✐♥ ♣r♦❣r❡ss ❛♥❞ ❬▼P❚❛❪✿ ❛r❳✐✈✿✶✽✵✶✳✵✷✻✶✶ ❘❡❝❡♥t Pr♦❣r❡ss ✐♥ ▼❛t❤❡♠❛t✐❝s ♦❢ ❚♦♣♦❧♦❣✐❝❛❧ ■♥s✉❧❛t♦rs ✹t❤ ❙❡♣t❡♠❜❡r✱ ✷✵✶✽

❙❡♠✐♥❛r ♦✉t❧✐♥❡ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣s ❢♦r ◗❍❊ ❛♥❞ ◗❙❍❊ ▲✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥ts✿ σ ij ❢♦r ❜♦t❤ ◗❍❊ ❛♥❞ ◗❙❍❊ ❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt ❈❤❛r❣❡ ❛♥❞ s♣✐♥ ❝✉rr❡♥t ♦♣❡r❛t♦r ❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ◆❊❆❙❙ ❆❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② σ ε ij ✿ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛ ❛♥❞ ❜❡②♦♥❞ ❙♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t②✿ ❛♥❛❧②s✐s ♦❢ ❑✉❜♦✲❧✐❦❡ t❡r♠s

✶ ✶ ✷ ✷ ✷ ✷ ❇ ✶ ✶ ✶ ❇ ❇ ✶ ✶ ✶ ✶✷ ✶✷ ✶✷ ✷ ✷ ✷ ✷ ✶ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ✶✷ ✷ ✷ ❡q✉❛t✐♦♥✿ ✷ ✶✷ ✶✷ ✶✷ ✶✷ ✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t

✶ ✶ ✷ ✷ ✷ ✷ ❇ ✶ ✶ ✶ ❇ ✶ ✶✷ ✶ ✶ ✷ ✷ ✶✷ ✶✷ ✷ ✷ ✶ ✶✷ ✷ ✷ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿ ✶✷ ✶✷ ✷ ✷ ✶✷ ✶✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t ⊙ ❇ ❇ ✿ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞

✶ ✷ ✷ ❇ ✶ ✶ ❇ ✶ ✶✷ ✷ ✷ ✶ ✶✷ ✷ ✷ ✶ ✶ ✶✷ ✶✷ ✶✷ ✶✷ ✷ ✷ ✷ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿ ✷ ✶✷ ✶✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ E ✷ ∆ V ✷ j ✶ ⊙ ❇ ❈♦♥❞✉❝t❛♥❝❡ G ✶✷ := − I ✶ ∆ V ✷ ❈♦♥❞✉❝t✐✈✐t② σ ✶✷ := j ✶ E ✷

✶ ✷ ✷ ❇ ✶ ✶ ❇ ✶ ✶✷ ✷ ✷ ✶ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ✶✷ ✷ ✷ ❡q✉❛t✐♦♥✿ ✷ ✶✷ ✶✷ ✶✷ ✶✷ ✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ E ✷ ∆ V ✷ j ✶ ⊙ ❇ G ✶✷ := − I ✶ , σ ✶✷ := j ✶ ∆ V ✷ E ✷

✶ ✷ ✷ ❇ ✶ ✶ ❇ ✶ ✶✷ ✷ ✷ ✶ ✶✷ ✷ ✷ ✶✷ ✶✷ ✷ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿ ✷ ✶✷ ✶✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ E ✷ ∆ V ✷ j ✶ ⊙ ❇ G ✶✷ := − I ✶ , σ ✶✷ := j ✶ ∆ V ✷ E ✷ ❬❑❉P ✬✽✵❪✿ G ✶✷ ≃ ne ✷ h , n ∈ Z

✶ ✷ ✷ ❇ ✶ ✶ ❇ ✶ ✶✷ ✷ ✷ ✶ ✶✷ ✷ ✷ ✶✷ ✶✷ ✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ E ✷ ∆ V ✷ j ✶ ⊙ ❇ G ✶✷ := − I ✶ , σ ✶✷ := j ✶ ∆ V ✷ E ✷ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿ σ ✶✷ = G ✶✷ ≃ ne ✷ h , n ∈ Z

✶ ✷ ✷ ✶ ✶ ✶ ✶✷ ✷ ✷ ✶ ✶✷ ✷ ✷ ✶✷ ✶✷ ✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ E ✷ ∆ V ✷ j ✶ ⊗ ❇ ↓ ⊙ ❇ ⊙ ❇ ↑ G ✶✷ := − I ✶ , σ ✶✷ := j ✶ ❇ ↑ ✱ ❇ ↓ ✿ ❢r♦♠ s♣✐♥✲♦r❜✐t ❝♦✉✲ ∆ V ✷ E ✷ ♣❧✐♥❣ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿ σ ✶✷ = G ✶✷ ≃ ne ✷ h , n ∈ Z

✶ ✶✷ ✷ ✷ ✶ ✶✷ ✷ ✷ ✶✷ ✶✷ ✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ I ✶ , ↑↓ E ✷ E ✷ ∆ V ✷ ∆ V ✷ j ✶ ⊗ ❇ ↓ j ✶ , ↓ j ✶ , ↑ ⊙ ❇ ⊙ ❇ ↑ G ✶✷ := − I ✶ , σ ✶✷ := j ✶ ∆ V ✷ E ✷ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿ σ ✶✷ = G ✶✷ ≃ ne ✷ h , n ∈ Z

✶ ✶✷ ✷ ✷ ✶ ✶✷ ✷ ✷ ✶✷ ✶✷ ✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ I ✶ , ↑↓ E ✷ E ✷ ∆ V ✷ ∆ V ✷ j ✶ ⊗ ❇ ↓ j ✶ , ↓ j ✶ , ↑ ⊙ ❇ ⊙ ❇ ↑ G ✶✷ := − I ✶ , σ ✶✷ := j ✶ ✶✷ := − I s z ∆ V ✷ E ✷ ❙♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ G s z ✶ ∆ V ✷ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ✶✷ := j s z ❙♣✐♥ ❝♦♥❞✉❝t✐✈✐t② σ s z ✶ ❡q✉❛t✐♦♥✿ E ✷ σ ✶✷ = G ✶✷ ≃ ne ✷ h , n ∈ Z

✶✷ ✶✷ ✷ ❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ I ✶ , ↑↓ E ✷ E ✷ ∆ V ✷ ∆ V ✷ j ✶ ⊗ ❇ ↓ j ✶ , ↓ j ✶ , ↑ ⊙ ❇ ⊙ ❇ ↑ G ✶✷ := − I ✶ , σ ✶✷ := j ✶ ✶✷ := − I s z ? G s z ✶ = − , ∆ V ✷ E ✷ ∆ V ✷ ∆ V ✷ ✶✷ := j s z = ? σ s z ✶ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② E ✷ E ✷ ❡q✉❛t✐♦♥✿ σ ✶✷ = G ✶✷ ≃ ne ✷ h , n ∈ Z

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿ ✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t I ✶ I ✶ , ↑↓ E ✷ E ✷ ∆ V ✷ ∆ V ✷ j ✶ ⊗ ❇ ↓ j ✶ , ↓ j ✶ , ↑ ⊙ ❇ ⊙ ❇ ↑ G ✶✷ := − I ✶ , σ ✶✷ := j ✶ ✶✷ := − I s z ? G s z ✶ = − , ∆ V ✷ E ✷ ∆ V ✷ ∆ V ✷ ✶✷ := j s z = ? σ s z ✶ ❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② E ✷ E ✷ ❡q✉❛t✐♦♥✿ σ ✶✷ = G ✶✷ ≃ ne ✷ ∈ e ? ? σ s z = G s z ✷ π Z h , n ∈ Z ✶✷ ✶✷

▲✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥ts✿ σ ij ❲❡ ❛r❡ ❣♦✐♥❣ t♦ ◮ st✉❞② t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❛ ❣❛♣♣❡❞✱ ♣❡r✐♦❞✐❝ ❛♥❞ ♦♥❡✲♣❛rt✐❝❧❡ q✉❛♥t✉♠ s②st❡♠ t♦ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s♠❛❧❧ ❡❧❡❝tr✐❝ ✜❡❧❞✱ ♠♦❞❡❧❡❞ ❜② ❛ ♣♦t❡♥t✐❛❧ − ε X j ✇✐t❤ ε ≪ ✶✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥❞✉❝t✐✈✐t② σ ij ◮ ❢♦r ❜♦t❤ ❝❤❛r❣❡ ✭◗✉❛♥t✉♠ ❍❛❧❧ ❡✛❡❝t✮ ❛♥❞ s♣✐♥ ✭◗✉❛♥t✉♠ s♣✐♥ ❍❛❧❧ ❡✛❡❝t✮ tr❛♥s♣♦rt✳ ◮ ❞❡r✐✈❡ ❢♦r♠✉❧❛s ✈✐❛ ❛♥ ❛r❣✉♠❡♥t ✇❤✐❝❤ ✐s ❛s ♠♦❞❡❧✲✐♥❞❡♣❡♥❞❡♥t ❛s ♣♦ss✐❜❧❡ ✈✐❛ t❤❡ ♠❡t❤♦❞ ♦❢ ♥♦♥✲❡q✉✐❧✐❜r✐✉♠ ❛❧♠♦st✲st❛t✐♦♥❛r② st❛t❡ ✭◆❊❆❙❙✮ ◮ ❛✈♦✐❞✐♥❣ t❤❡ ❧✐♥❡❛r r❡s♣♦♥s❡ ❛♥s❛t③ ✭▲❘❆✮ ❛♥❞ ❛♥② ❥✉st✐✜❝❛t✐♦♥ ♦❢ ✐ts ✈❛❧✐❞✐t②✳