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t s tt - - PowerPoint PPT Presentation

Jul 06, 2023 424 likes •1.34k views

t s tt r r t rs

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SLIDE 1

◗✉❛♥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✱ ✷✵✶✽

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SLIDE 2

❙❡♠✐♥❛r ♦✉t❧✐♥❡

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

ij✿ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛ ❛♥❞ ❜❡②♦♥❞

❙♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t②✿ ❛♥❛❧②s✐s ♦❢ ❑✉❜♦✲❧✐❦❡ t❡r♠s

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SLIDE 3

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ❇

✶ ✷ ✶ ✷ ✶✷ ✶ ✷ ✶✷ ✶ ✷

❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿

✶✷ ✶✷ ✷

✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t ❇ ❇

✶ ✶ ✷ ✶ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

slide-4
SLIDE 4

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ⊙ ❇

✶ ✷ ✶ ✷

❇✿ ❡①t❡r♥❛❧ ♠❛❣♥❡t✐❝ ✜❡❧❞

✶✷ ✶ ✷ ✶✷ ✶ ✷

❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿

✶✷ ✶✷ ✷

✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t ❇ ❇

✶ ✶ ✷ ✶ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

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SLIDE 5

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

✶✳ ◗✉❛♥t✉♠ ❍❛❧❧ ❝❤❛r❣❡ ❡✛❡❝t ⊙ ❇ j✶ E✷ I✶ ∆V✷ ❈♦♥❞✉❝t❛♥❝❡ G✶✷ := − I✶ ∆V✷ ❈♦♥❞✉❝t✐✈✐t② σ✶✷ := j✶ E✷

✶✷ ✶ ✷ ✶✷ ✶ ✷

❬❑❉P ✬✽✵❪ ❛♥❞ ❜② t❤❡ ❝♦♥t✐♥✉✐t② ❡q✉❛t✐♦♥✿

✶✷ ✶✷ ✷

✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t ❇ ❇

✶ ✶ ✷ ✶ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

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SLIDE 6

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

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

✶✷ ✶✷ ✷

✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t ❇ ❇

✶ ✶ ✷ ✶ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

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SLIDE 7

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

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

✶✷ ✶✷ ✷

✷✳ ◗✉❛♥t✉♠ ❍❛❧❧ s♣✐♥ ❡✛❡❝t ❇ ❇

✶ ✶ ✷ ✶ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

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SLIDE 8

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

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

✶ ✶ ✷ ✶ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

slide-9
SLIDE 9

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

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

✶ ✶ ✷ ✶ ✷

❇↑✱ ❇↓✿ ❢r♦♠ s♣✐♥✲♦r❜✐t ❝♦✉✲ ♣❧✐♥❣

✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

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SLIDE 10

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

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

✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

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SLIDE 11

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

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

✶✷ := − I sz ✶

∆V✷ ❙♣✐♥ ❝♦♥❞✉❝t✐✈✐t② σsz

✶✷ := jsz ✶

E✷

✶✷ ✶ ✷ ✷ ✶✷ ✶ ✷ ✷ ✶✷ ✶✷

✷

slide-12
SLIDE 12

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

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

✶✷ := − I sz ✶

∆V✷ = − ? ∆V✷ , σsz

✶✷ := jsz ✶

E✷ = ? E✷

✶✷ ✶✷

✷

slide-13
SLIDE 13

❊①♣❡r✐♠❡♥t❛❧ s❡t✉♣ ✭s❝❤❡♠❛t✐❝✮✿

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

✶✷ := − I sz ✶

∆V✷ = − ? ∆V✷ , σsz

✶✷ := jsz ✶

E✷ = ? E✷ σsz

✶✷ ?

= G sz

✶✷ ?

∈ e ✷πZ

slide-14
SLIDE 14

▲✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥ts✿ σij

❲❡ ❛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✐❛❧ −εXj ✇✐t❤ ε ≪ ✶✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥❞✉❝t✐✈✐t② σij

◮ ❢♦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✳

slide-15
SLIDE 15

▲✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥ts✿ σij

❲❡ ❛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✐❛❧ −εXj ✇✐t❤ ε ≪ ✶✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥❞✉❝t✐✈✐t② σij

◮ ❢♦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✳

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SLIDE 16

▲✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥ts✿ σij

❲❡ ❛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✐❛❧ −εXj ✇✐t❤ ε ≪ ✶✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥❞✉❝t✐✈✐t② σij

◮ ❢♦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✳

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SLIDE 17

▲✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥ts✿ σij

❲❡ ❛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✐❛❧ −εXj ✇✐t❤ ε ≪ ✶✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥❞✉❝t✐✈✐t② σij

◮ ❢♦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✳

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SLIDE 18

▲✐♥❡❛r r❡s♣♦♥s❡ ❝♦❡✣❝✐❡♥ts✿ σij

❲❡ ❛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✐❛❧ −εXj ✇✐t❤ ε ≪ ✶✱ ✐♥ t❡r♠s ♦❢ t❤❡ ❝♦♥❞✉❝t✐✈✐t② σij

◮ ❢♦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✳

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SLIDE 19

❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt

❆ss✉♠♣t✐♦♥ ✭❍✮ ♦♥ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♠♦❞❡❧ ◮ H := L✷(X) ⊗ CN✱ X = Rd ♦r X = ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧ ⊂ Rd✳ ◮ H✵ ✐s ❛ ♦♣❡r❛t♦r ♦♥ H ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇

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SLIDE 20

❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt

❆ss✉♠♣t✐♦♥ ✭❍✮ ♦♥ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♠♦❞❡❧ ◮ H := L✷(X) ⊗ CN✱ X = Rd ♦r X = ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧ ⊂ Rd✳ ◮ H✵ ✐s ❛ ♣❡r✐♦❞✐❝ ❣❛♣♣❡❞ ♦♣❡r❛t♦r ♦♥ H ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇

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SLIDE 21

❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt

❆ss✉♠♣t✐♦♥ ✭❍✮ ♦♥ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♠♦❞❡❧ ◮ H := L✷(X) ⊗ CN✱ X = Rd ♦r X = ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧ ⊂ Rd✳ ◮ H✵ ✐s ❛ ♣❡r✐♦❞✐❝ ❣❛♣♣❡❞ ♦♣❡r❛t♦r ♦♥ H ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇

◮ ❇r❛✈❛✐s ❧❛tt✐❝❡ ♦❢ tr❛♥s❧❛t✐♦♥s = Γ ≃ Zd [H✵, Tγ] = ✵ ∀γ ∈ Γ. ◮ ✈✐❛ ❇❧♦❝❤✕❋❧♦q✉❡t r❡♣r❡s❡♥t❛t✐♦♥ H✵ ≃ ⊕

Td ❞k H✵(k)✱

H✵(k) ❛❝ts ♦♥ H❢ := L✷(C✶) ⊗ CN✱ C✶ ≃ Rd/Γ.

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SLIDE 22

❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt

❆ss✉♠♣t✐♦♥ ✭❍✮ ♦♥ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♠♦❞❡❧ ◮ H := L✷(X) ⊗ CN✱ X = Rd ♦r X = ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧ ⊂ Rd✳ ◮ H✵ ✐s ❛ ♣❡r✐♦❞✐❝ ❣❛♣♣❡❞ ♦♣❡r❛t♦r ♦♥ H ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇

◮ Π✵ ❂ ❋❡r♠✐ ♣r♦❥❡❝t✐♦♥ ♦♥ ♦❝❝✉♣✐❡❞ ❜❛♥❞s ❜❡❧♦✇ t❤❡ s♣❡❝tr❛❧ ❣❛♣ ✐s ✐♥ Bτ

✶✳

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SLIDE 23

❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt

❆ss✉♠♣t✐♦♥ ✭❍✮ ♦♥ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♠♦❞❡❧ ◮ H := L✷(X) ⊗ CN✱ X = Rd ♦r X = ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧ ⊂ Rd✳ ◮ H✵ ✐s ❛ ♣❡r✐♦❞✐❝ ❣❛♣♣❡❞ ♦♣❡r❛t♦r ♦♥ H ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✱ s✉❝❤ t❤❛t t❡❝❤♥✐❝❛❧ ❜✉t ♠✐❧❞ ❤②♣♦t❤❡s❡s ♦♥ H✵

◮ H✵ : Rd → L(D❢, H❢) , k → H✵(k) ✐s ❛ s♠♦♦t❤ ❡q✉✐✈❛r✐❛♥t ♠❛♣ t❛❦✐♥❣ ✈❛❧✉❡s ✐♥ t❤❡ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦rs ✇✐t❤ ❞❡♥s❡ ❞♦♠❛✐♥ D❢ ⊂ H❢✳ L(D❢, H❢) ✐s t❤❡ s♣❛❝❡ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ❢r♦♠ D❢✱ ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❣r❛♣❤ ♥♦r♠ ♦❢ H✵(✵)✱ t♦ H❢✳

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SLIDE 24

❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt

❆ss✉♠♣t✐♦♥ ✭❍✮ ♦♥ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♠♦❞❡❧ ◮ H := L✷(X) ⊗ CN✱ X = Rd ♦r X = ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧ ⊂ Rd✳ ◮ H✵ ✐s ❛ ♣❡r✐♦❞✐❝ ❣❛♣♣❡❞ ♦♣❡r❛t♦r ♦♥ H ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✱ s✉❝❤ t❤❛t t❡❝❤♥✐❝❛❧ ❜✉t ♠✐❧❞ ❤②♣♦t❤❡s❡s ♦♥ H✵✳ ❘❡♠❛r❦ ❚❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s ❛r❡ s❛t✐s✜❡❞

◮ ✐♥ ♠♦st t✐❣❤t✲❜✐♥❞✐♥❣ ♠♦❞❡❧s ✭❞✐s❝r❡t❡ ❝❛s❡✮ ◮ ❜② ❣❛♣♣❡❞✱ ♣❡r✐♦❞✐❝ ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs H✵ = ✶ ✷(−✐∇ − A(x))✷ + V (x) ✉♥❞❡r st❛♥❞❛r❞ ❤②♣♦t❤❡s❡s ♦❢ r❡❧❛t✐✈❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ t❤❡ ♣♦t❡♥t✐❛❧s ✇✳r✳t✳ −∆ ✭❝♦♥t✐♥✉✉♠ ❝❛s❡✮✳

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SLIDE 25

❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt

❆ss✉♠♣t✐♦♥ ✭❍✮ ♦♥ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♠♦❞❡❧ ◮ H := L✷(X) ⊗ CN✱ X = Rd ♦r X = ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧ ⊂ Rd✳ ◮ H✵ ✐s ❛ ♣❡r✐♦❞✐❝ ❣❛♣♣❡❞ ♦♣❡r❛t♦r ♦♥ H ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✱ s✉❝❤ t❤❛t t❡❝❤♥✐❝❛❧ ❜✉t ♠✐❧❞ ❤②♣♦t❤❡s❡s ♦♥ H✵✳ ❘❡♠❛r❦ ❚❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s ❛r❡ s❛t✐s✜❡❞

◮ ✐♥ ♠♦st t✐❣❤t✲❜✐♥❞✐♥❣ ♠♦❞❡❧s ✭❞✐s❝r❡t❡ ❝❛s❡✮ ◮ ❜② ❣❛♣♣❡❞✱ ♣❡r✐♦❞✐❝ ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs H✵ = ✶ ✷(−✐∇ − A(x))✷ + V (x) ✉♥❞❡r st❛♥❞❛r❞ ❤②♣♦t❤❡s❡s ♦❢ r❡❧❛t✐✈❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ t❤❡ ♣♦t❡♥t✐❛❧s ✇✳r✳t✳ −∆ ✭❝♦♥t✐♥✉✉♠ ❝❛s❡✮✳

slide-26
SLIDE 26

❆ ♠♦❞❡❧ ❢♦r q✉❛♥t✉♠ tr❛♥s♣♦rt

❆ss✉♠♣t✐♦♥ ✭❍✮ ♦♥ t❤❡ ✉♥♣❡rt✉r❜❡❞ ♠♦❞❡❧ ◮ H := L✷(X) ⊗ CN✱ X = Rd ♦r X = ❞✐s❝r❡t❡ d✲❞✐♠❡♥s✐♦♥❛❧ ❝r②st❛❧ ⊂ Rd✳ ◮ H✵ ✐s ❛ ♣❡r✐♦❞✐❝ ❣❛♣♣❡❞ ♦♣❡r❛t♦r ♦♥ H ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✱ s✉❝❤ t❤❛t t❡❝❤♥✐❝❛❧ ❜✉t ♠✐❧❞ ❤②♣♦t❤❡s❡s ♦♥ H✵✳ ❘❡♠❛r❦ ❚❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s ❛r❡ s❛t✐s✜❡❞

◮ ✐♥ ♠♦st t✐❣❤t✲❜✐♥❞✐♥❣ ♠♦❞❡❧s ✭❞✐s❝r❡t❡ ❝❛s❡✮ ◮ ❜② ❣❛♣♣❡❞✱ ♣❡r✐♦❞✐❝ ❙❝❤rö❞✐♥❣❡r ♦♣❡r❛t♦rs H✵ = ✶ ✷(−✐∇ − A(x))✷ + V (x) ✉♥❞❡r st❛♥❞❛r❞ ❤②♣♦t❤❡s❡s ♦❢ r❡❧❛t✐✈❡ ❜♦✉♥❞❡❞♥❡ss ♦❢ t❤❡ ♣♦t❡♥t✐❛❧s ✇✳r✳t✳ −∆ ✭❝♦♥t✐♥✉✉♠ ❝❛s❡✮✳

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SLIDE 27

❚❤❡ ❢♦❧❧♦✇✐♥❣ s♣❛❝❡s ♦❢ ♦♣❡r❛t♦rs t✉r♥ ♦✉t ✉s❡❢✉❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❉❡✜♥✐t✐♦♥ ▲❡t H✶, H✷ ∈ {D❢ , H❢} P(H✶, H✷) := { Γ✲♣❡r✐♦❞✐❝ A ✇✐t❤ s♠♦♦t❤ ✜❜r❛t✐♦♥ Rd → L(H✶, H✷) } ❛♥❞ P(H✶) := P(H✶, H✶)✳ ❇② ❆ss✉♠♣t✐♦♥ ✭❍✮ H✵ ∈ P(D❢, H❢)

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SLIDE 28

❚❤❡ ❢♦❧❧♦✇✐♥❣ s♣❛❝❡s ♦❢ ♦♣❡r❛t♦rs t✉r♥ ♦✉t ✉s❡❢✉❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❉❡✜♥✐t✐♦♥ ▲❡t H✶, H✷ ∈ {D❢ , H❢} P(H✶, H✷) := { Γ✲♣❡r✐♦❞✐❝ A ✇✐t❤ s♠♦♦t❤ ✜❜r❛t✐♦♥ Rd → L(H✶, H✷) } ❛♥❞ P(H✶) := P(H✶, H✶)✳ ❇② ❆ss✉♠♣t✐♦♥ ✭❍✮ H✵ ∈ P(D❢, H❢)

  • [Π✵, Xj] ∈ P(H❢, D❢) ❛♥❞ [H✵, Xj] ∈ P(D❢, H❢)
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SLIDE 29

❚❤❡ ❢♦❧❧♦✇✐♥❣ s♣❛❝❡s ♦❢ ♦♣❡r❛t♦rs t✉r♥ ♦✉t ✉s❡❢✉❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❉❡✜♥✐t✐♦♥ ▲❡t H✶, H✷ ∈ {D❢ , H❢} P(H✶, H✷) := { Γ✲♣❡r✐♦❞✐❝ A ✇✐t❤ s♠♦♦t❤ ✜❜r❛t✐♦♥ Rd → L(H✶, H✷) } ❛♥❞ P(H✶) := P(H✶, H✶)✳ ❇② ❆ss✉♠♣t✐♦♥ ✭❍✮ H✵ ∈ P(D❢, H❢)

  • [Π✵, Xj](k)
  • ≡−✐∂kj Π✵(k)

∈ P(H❢, D❢) ❛♥❞ [H✵, Xj](k)

  • ≡−✐∂kj H✵(k)

∈ P(D❢, H❢)

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SLIDE 30

P❡rt✉r❜❡❞ ♠♦❞❡❧ ❆❞❞ ❛♥ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥ ❞✐r❡❝t✐♦♥ j ♦❢ s♠❛❧❧ ✐♥t❡♥s✐t② ε ∈ [✵, ✶]✿ Hε := H✵−εXj ❈✉rr❡♥t ♦♣❡r❛t♦r S = ■❞L✷(X) ⊗ s s❡❧❢✲❛❞❥♦✐♥t✱ ❛❝t✐♥❣ ♦♥❧② ♦♥ CN ✭✐♥t❡r♥❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✮ ◮ s = ■❞ − → ❝❤❛r❣❡ ❝✉rr❡♥t ✭◗❍❊✮ ◮ s = sz = σz/✷ − → s♣✐♥ ❝✉rr❡♥t ✭◗❙❍❊✮ ♣r♦♣♦s❡❞ ❜② ❬❙❩❳◆ ✬✵✻❪

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SLIDE 31

P❡rt✉r❜❡❞ ♠♦❞❡❧ ❆❞❞ ❛♥ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥ ❞✐r❡❝t✐♦♥ j ♦❢ s♠❛❧❧ ✐♥t❡♥s✐t② ε ∈ [✵, ✶]✿ Hε := H✵−εXj ❈✉rr❡♥t ♦♣❡r❛t♦r Jε

i := ✐[Hε, SXi]

S = ■❞L✷(X) ⊗ s s❡❧❢✲❛❞❥♦✐♥t✱ ❛❝t✐♥❣ ♦♥❧② ♦♥ CN ✭✐♥t❡r♥❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✮ ◮ s = ■❞ − → ❝❤❛r❣❡ ❝✉rr❡♥t ✭◗❍❊✮ ◮ s = sz = σz/✷ − → s♣✐♥ ❝✉rr❡♥t ✭◗❙❍❊✮ ♣r♦♣♦s❡❞ ❜② ❬❙❩❳◆ ✬✵✻❪

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SLIDE 32

P❡rt✉r❜❡❞ ♠♦❞❡❧ ❆❞❞ ❛♥ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥ ❞✐r❡❝t✐♦♥ j ♦❢ s♠❛❧❧ ✐♥t❡♥s✐t② ε ∈ [✵, ✶]✿ Hε := H✵−εXj ❈✉rr❡♥t ♦♣❡r❛t♦r Ji = ✐[H✵, SXi] S = ■❞L✷(X) ⊗ s s❡❧❢✲❛❞❥♦✐♥t✱ ❛❝t✐♥❣ ♦♥❧② ♦♥ CN ✭✐♥t❡r♥❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✮ ◮ s = ■❞ − → ❝❤❛r❣❡ ❝✉rr❡♥t ✭◗❍❊✮ ◮ s = sz = σz/✷ − → s♣✐♥ ❝✉rr❡♥t ✭◗❙❍❊✮ ♣r♦♣♦s❡❞ ❜② ❬❙❩❳◆ ✬✵✻❪

slide-33
SLIDE 33

P❡rt✉r❜❡❞ ♠♦❞❡❧ ❆❞❞ ❛♥ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥ ❞✐r❡❝t✐♦♥ j ♦❢ s♠❛❧❧ ✐♥t❡♥s✐t② ε ∈ [✵, ✶]✿ Hε := H✵−εXj ❈✉rr❡♥t ♦♣❡r❛t♦r Ji = ✐[H✵, SXi] S = ■❞L✷(X) ⊗ s s❡❧❢✲❛❞❥♦✐♥t✱ ❛❝t✐♥❣ ♦♥❧② ♦♥ CN ✭✐♥t❡r♥❛❧ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✮ ◮ s = ■❞ − → ❝❤❛r❣❡ ❝✉rr❡♥t ✭◗❍❊✮ ◮ s = sz = σz/✷ − → s♣✐♥ ❝✉rr❡♥t ✭◗❙❍❊✮ ♣r♦♣♦s❡❞ ❜② ❬❙❩❳◆ ✬✵✻❪

slide-34
SLIDE 34
  • J ✈❡rs✉s ♣❡r✐♦❞✐❝✐t②

Pr♦❜❧❡♠

  • J ✐s ♥♦t ♣❡r✐♦❞✐❝
  • J = ✐[H✵, S

X] ✐❢ ❛♥❞ ♦♥❧② ✐❢ [H✵, S] = ✵ ✭❢♦r S = ■❞L✷(X) ⊗ sz ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✿ λRashba = ✵✮✳ ❙✐♠♣❧❡ ❜✉t ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ✐♥ ❬▼✳✱ P❛♥❛t✐✱ ❚❛✉❜❡r ✬✶✽❪✿ T

γ

J T −✶

  • γ

= J − γ ✐[H✵, S] ∀ γ ∈ Γ. t❤❡ ♣❡r✐♦❞✐❝✐t② ✐s r❡st♦r❡❞ ♦♥ ♠❡s♦s❝♦♣✐❝ s❝❛❧❡✦

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SLIDE 35
  • J ✈❡rs✉s ♣❡r✐♦❞✐❝✐t②

Pr♦❜❧❡♠

  • J ✐s ♥♦t ♣❡r✐♦❞✐❝
  • J = ✐[H✵, S

X]= ✐ X [H✵, S] + ✐[H✵, X] S ✐❢ ❛♥❞ ♦♥❧② ✐❢ [H✵, S] = ✵ ✭❢♦r S = ■❞L✷(X) ⊗ sz ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✿ λRashba = ✵✮✳ ❙✐♠♣❧❡ ❜✉t ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ✐♥ ❬▼✳✱ P❛♥❛t✐✱ ❚❛✉❜❡r ✬✶✽❪✿ T

γ

J T −✶

  • γ

= J − γ ✐[H✵, S] ∀ γ ∈ Γ. t❤❡ ♣❡r✐♦❞✐❝✐t② ✐s r❡st♦r❡❞ ♦♥ ♠❡s♦s❝♦♣✐❝ s❝❛❧❡✦

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SLIDE 36
  • J ✈❡rs✉s ♣❡r✐♦❞✐❝✐t②

Pr♦❜❧❡♠

  • J ✐s ♥♦t ♣❡r✐♦❞✐❝
  • J = ✐[H✵, S

X]= ✐ X [H✵, S]

♣❡r✐♦❞✐❝

+✐ [H✵, X] S

  • ♣❡r✐♦❞✐❝

✐❢ ❛♥❞ ♦♥❧② ✐❢ [H✵, S] = ✵ ✭❢♦r S = ■❞L✷(X) ⊗ sz ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✿ λRashba = ✵✮✳ ❙✐♠♣❧❡ ❜✉t ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ✐♥ ❬▼✳✱ P❛♥❛t✐✱ ❚❛✉❜❡r ✬✶✽❪✿ T

γ

J T −✶

  • γ

= J − γ ✐[H✵, S] ∀ γ ∈ Γ. t❤❡ ♣❡r✐♦❞✐❝✐t② ✐s r❡st♦r❡❞ ♦♥ ♠❡s♦s❝♦♣✐❝ s❝❛❧❡✦

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SLIDE 37
  • J ✈❡rs✉s ♣❡r✐♦❞✐❝✐t②

Pr♦❜❧❡♠

  • J ✐s ♥♦t ♣❡r✐♦❞✐❝
  • J = ✐[H✵, S

X]= ✐ X [H✵, S]

♣❡r✐♦❞✐❝

+✐ [H✵, X] S

  • ♣❡r✐♦❞✐❝

✐❢ ❛♥❞ ♦♥❧② ✐❢ [H✵, S] = ✵ ✭❢♦r S = ■❞L✷(X) ⊗ sz ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✿ λRashba = ✵✮✳ ❙✐♠♣❧❡ ❜✉t ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ✐♥ ❬▼✳✱ P❛♥❛t✐✱ ❚❛✉❜❡r ✬✶✽❪✿ T

γ

J T −✶

  • γ

= J − γ ✐[H✵, S] ∀ γ ∈ Γ. t❤❡ ♣❡r✐♦❞✐❝✐t② ✐s r❡st♦r❡❞ ♦♥ ♠❡s♦s❝♦♣✐❝ s❝❛❧❡✦

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SLIDE 38

❚r❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡

τ(A) := lim

L→∞ L∈✷N+✶

✶ |CL| Tr(χLAχL), |CL| = Ld |C✶|

▲❡♠♠❛ ✶✳

▲❡t A ❜❡ ♣❡r✐♦❞✐❝ ❛♥❞ χKAχK ∈ B✶(H) ∀ ❝♦♠♣❛❝t s❡t K✳ ❚❤❡♥ τ(A) ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ τ(A) = ✶ |C✶| Tr(χ✶Aχ✶).

▲❡♠♠❛ ✷✳

▲❡t A ❜❡ ♣❡r✐♦❞✐❝ ❛♥❞ χKAχK ∈ B✶(H) ∀ ❝♦♠♣❛❝t s❡t K✳ ❚❤❡♥ t❤❡ ♦♣❡r❛t♦r XiA ❤❛s ✜♥✐t❡ tr❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡ ❛♥❞ τ(XiA) = ✶ |C✶| Tr (χ✶XiAχ✶).

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SLIDE 39

❚r❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡

τ(A) := lim

L→∞ L∈✷N+✶

✶ |CL| Tr(χLAχL), |CL| = Ld |C✶|

▲❡♠♠❛ ✶✳

▲❡t A ❜❡ ♣❡r✐♦❞✐❝ ❛♥❞ χKAχK ∈ B✶(H) ∀ ❝♦♠♣❛❝t s❡t K✳ ❚❤❡♥ τ(A) ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ τ(A) = ✶ |C✶| Tr(χ✶Aχ✶).

▲❡♠♠❛ ✷✳

▲❡t A ❜❡ ♣❡r✐♦❞✐❝ ❛♥❞ χKAχK ∈ B✶(H) ∀ ❝♦♠♣❛❝t s❡t K✳ ❚❤❡♥ t❤❡ ♦♣❡r❛t♦r XiA ❤❛s ✜♥✐t❡ tr❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡ ❛♥❞ τ(XiA) = ✶ |C✶| Tr (χ✶XiAχ✶).

slide-40
SLIDE 40

❚r❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡

τ(A) := lim

L→∞ L∈✷N+✶

✶ |CL| Tr(χLAχL), |CL| = Ld |C✶|

▲❡♠♠❛ ✶✳

▲❡t A ❜❡ ♣❡r✐♦❞✐❝ ❛♥❞ χKAχK ∈ B✶(H) ∀ ❝♦♠♣❛❝t s❡t K✳ ❚❤❡♥ τ(A) ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ τ(A) = ✶ |C✶| Tr(χ✶Aχ✶).

▲❡♠♠❛ ✷✳

▲❡t A ❜❡ ♣❡r✐♦❞✐❝ ❛♥❞ χKAχK ∈ B✶(H) ∀ ❝♦♠♣❛❝t s❡t K✳ ❚❤❡♥ t❤❡ ♦♣❡r❛t♦r XiA ❤❛s ✜♥✐t❡ tr❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡ ❛♥❞ τ(XiA) = ✶ |C✶| Tr (χ✶XiAχ✶).

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SLIDE 41

❚r❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡

τ(A) := lim

L→∞ L∈✷N+✶

✶ |CL| Tr(χLAχL), |CL| = Ld |C✶|

▲❡♠♠❛ ✶✳

▲❡t A ❜❡ ♣❡r✐♦❞✐❝ ❛♥❞ χKAχK ∈ B✶(H) ∀ ❝♦♠♣❛❝t s❡t K✳ ❚❤❡♥ τ(A) ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ τ(A) = ✶ |C✶| Tr(χ✶Aχ✶).

▲❡♠♠❛ ✷✳

▲❡t A ❜❡ ♣❡r✐♦❞✐❝ ❛♥❞ χKAχK ∈ B✶(H) ∀ ❝♦♠♣❛❝t s❡t K✱ s✉❝❤ t❤❛t τ(A) = ✵✳ ❚❤❡♥ t❤❡ ♦♣❡r❛t♦r XiA ❤❛s ✜♥✐t❡ tr❛❝❡ ♣❡r ✉♥✐t ✈♦❧✉♠❡ ❛♥❞ τ(XiA) = ✶ |C✶| Tr (χ✶XiAχ✶).

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SLIDE 42

P❡rt✉r❜❡❞ ♠♦❞❡❧ ❆❞❞ ❛♥ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥ ❞✐r❡❝t✐♦♥ j ♦❢ s♠❛❧❧ ✐♥t❡♥s✐t② ε ∈ [✵, ✶]✿ Hε := H✵−εXj

❚❤❡♦r❡♠ ✸ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

❖♥❡ ❝❛♥ ❝♦♥str✉❝t ❛ ♥♦♥✲❡q✉✐❧✐❜r✐✉♠ ❛❧♠♦st✲st❛t✐♦♥❛r② st❛t❡ ✭◆❊❆❙❙✮ Πε ❢♦r Hε s✉❝❤ t❤❛t H✵ ❡♥❥♦②s ❆ss✉♠♣t✐♦♥ ✭❍✮✿ ✶✳ Πε = ❡−✐εSΠ✵❡✐εS ❢♦r s♦♠❡ ❜♦✉♥❞❡❞✱ ♣❡r✐♦❞✐❝ ❛♥❞ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r S❀ ✷✳ Πε ❛❧♠♦st✲❝♦♠♠✉t❡s ✇✐t❤ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ Hε✱ ♥❛♠❡❧② [Hε, Πε] = O(ε✷)✳

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SLIDE 43

P❡rt✉r❜❡❞ ♠♦❞❡❧ ❆❞❞ ❛♥ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥ ❞✐r❡❝t✐♦♥ j ♦❢ s♠❛❧❧ ✐♥t❡♥s✐t② ε ∈ [✵, ✶]✿ Hε := H✵−εXj

❚❤❡♦r❡♠ ✸ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

❖♥❡ ❝❛♥ ❝♦♥str✉❝t ❛ ♥♦♥✲❡q✉✐❧✐❜r✐✉♠ ❛❧♠♦st✲st❛t✐♦♥❛r② st❛t❡ ✭◆❊❆❙❙✮ Πε ❢♦r Hε s✉❝❤ t❤❛t H✵ ❡♥❥♦②s ❆ss✉♠♣t✐♦♥ ✭❍✮✿ ✶✳ Πε = ❡−✐εSΠ✵❡✐εS ❢♦r s♦♠❡ ❜♦✉♥❞❡❞✱ ♣❡r✐♦❞✐❝ ❛♥❞ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r S❀ ✷✳ Πε ❛❧♠♦st✲❝♦♠♠✉t❡s ✇✐t❤ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ Hε✱ ♥❛♠❡❧② [Hε, Πε] = O(ε✷)✳

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SLIDE 44

P❡rt✉r❜❡❞ ♠♦❞❡❧ ❆❞❞ ❛♥ ❡❧❡❝tr✐❝ ✜❡❧❞ ✐♥ ❞✐r❡❝t✐♦♥ j ♦❢ s♠❛❧❧ ✐♥t❡♥s✐t② ε ∈ [✵, ✶]✿ Hε := H✵−εXj

❚❤❡♦r❡♠ ✸ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

❖♥❡ ❝❛♥ ❝♦♥str✉❝t ❛ ♥♦♥✲❡q✉✐❧✐❜r✐✉♠ ❛❧♠♦st✲st❛t✐♦♥❛r② st❛t❡ ✭◆❊❆❙❙✮ Πε ❢♦r Hε s✉❝❤ t❤❛t H✵ ❡♥❥♦②s ❆ss✉♠♣t✐♦♥ ✭❍✮✿ ✶✳ Πε = ❡−✐εSΠ✵❡✐εS ❢♦r s♦♠❡ ❜♦✉♥❞❡❞✱ ♣❡r✐♦❞✐❝ ❛♥❞ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r S❀ ✷✳ Πε ❛❧♠♦st✲❝♦♠♠✉t❡s ✇✐t❤ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ Hε✱ ♥❛♠❡❧② [Hε, Πε] = O(ε✷)✳

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SLIDE 45

Pr♦♦❢✳

◮ I( · ) ❂ ✐♥✈❡rs❡ ▲✐♦✉✈✐❧❧✐❛♥✿ ❢♦r A = A❖❉ := Π✵AΠ⊥

✵ + Π⊥ ✵ AΠ✵ ∈ P(H❢)

I(A) := ✐ ✷π

  • C

❞z (H✵−z■❞)−✶ [A, Π✵] (H✵−z■❞)−✶ ∈ P(H❢, D❢) s✉❝❤ t❤❛t ✐t s♦❧✈❡s [H✵, I(A)] = A ❢♦r A = A❖❉✳ ◮ ❉❡✜♥✐♥❣ S := ✐ I(X ❖❉

j

) t❤❡♥ Πε = Π✵ + εΠ✶ + O(ε✷) ∈ P(H❢, D❢)✱ ✇✐t❤ Π✶ = I([Xj, Π✵])✱ s❛t✐s✜❡s [Hε, Πε] = O(ε✷)✳

slide-46
SLIDE 46

Pr♦♦❢✳

◮ I( · ) ❂ ✐♥✈❡rs❡ ▲✐♦✉✈✐❧❧✐❛♥✿ ❢♦r A = A❖❉ := Π✵AΠ⊥

✵ + Π⊥ ✵ AΠ✵ ∈ P(H❢)

I(A) := ✐ ✷π

  • C

❞z (H✵−z■❞)−✶ [A, Π✵] (H✵−z■❞)−✶ ∈ P(H❢, D❢) s✉❝❤ t❤❛t ✐t s♦❧✈❡s [H✵, I(A)] = A ❢♦r A = A❖❉✳ ◮ ❉❡✜♥✐♥❣ S := ✐ I(X ❖❉

j

) t❤❡♥ Πε = Π✵ + εΠ✶ + O(ε✷) ∈ P(H❢, D❢)✱ ✇✐t❤ Π✶ = I([Xj, Π✵])✱ s❛t✐s✜❡s [Hε, Πε] = O(ε✷)✳

slide-47
SLIDE 47

Pr♦♦❢✳

◮ I( · ) ❂ ✐♥✈❡rs❡ ▲✐♦✉✈✐❧❧✐❛♥✿ ❢♦r A = A❖❉ := Π✵AΠ⊥

✵ + Π⊥ ✵ AΠ✵ ∈ P(H❢)

I(A) := ✐ ✷π

  • C

❞z (H✵−z■❞)−✶ [A, Π✵] (H✵−z■❞)−✶ ∈ P(H❢, D❢) s✉❝❤ t❤❛t ✐t s♦❧✈❡s [H✵, I(A)] = A ❢♦r A = A❖❉✳ ◮ ❉❡✜♥✐♥❣ S := ✐ I(X ❖❉

j

) t❤❡♥ Πε = Π✵ + εΠ✶ + O(ε✷) ∈ P(H❢, D❢)✱ ✇✐t❤ Π✶ = I([Xj, Π✵])✱ s❛t✐s✜❡s [Hε, Πε] = O(ε✷)✳

slide-48
SLIDE 48

❘❡♠❛r❦✿ ❏✉st✐✜❝❛t✐♦♥ ❢♦r ✉s✐♥❣ t❤❡ ◆❊❆❙❙ ✭✐♥ ♣r♦❣r❡ss✮ ❈♦♥s✐❞❡r t❤❡ t✐♠❡✲❞❡♣❡♥❞❡♥t ❍❛♠✐❧t♦♥✐❛♥ Hε

s✇✐t❝❤(t) := H✵−f (t) ε Xj,

✇❤❡r❡ f : R → [✵, ✶] ✐s ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ✿ f (t) = ✵ ❢♦r ❛❧❧ t ≤ ✵ ❛♥❞ f (t) = ✶ ❢♦r ❛❧❧ t ≥ T > ✵✳ ρε(t) ✿ ♣❡rt✉r❜❡❞ st❛t❡ ✐ ε ❞ ❞t ρε(t) = [Hε

s✇✐t❝❤(t), ρε(t)],

ρε(✵) = Π✵. ❚❤❡♥ ρε(t) − Πε = O(ε✷) ✉♥✐❢♦r♠❧② ♦♥ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧s ✐♥ t✐♠❡. ❚❤✐s st❛t❡♠❡♥t ✐s ❛❧r❡❛❞② ♣r♦✈❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✐♥t❡r❛❝t✐♥❣ ♠♦❞❡❧s ♦♥ ❧❛tt✐❝❡s ❬❚❡✉❢❡❧✱ ✬✶✼❪✳ ◆❊❆❙❙ ❜②♣❛ss❡s t❤❡ ▲❘❆ ❛♥❞ t❤❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ ✐ts ✈❛❧✐❞✐t②✱ ❛♥❞ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ s❤❛♣❡ ♦❢ t❤❡ s✇✐t❝❤✐♥❣ ❢✉♥❝t✐♦♥✦

slide-49
SLIDE 49

❘❡♠❛r❦✿ ❏✉st✐✜❝❛t✐♦♥ ❢♦r ✉s✐♥❣ t❤❡ ◆❊❆❙❙ ✭✐♥ ♣r♦❣r❡ss✮ ❈♦♥s✐❞❡r t❤❡ t✐♠❡✲❞❡♣❡♥❞❡♥t ❍❛♠✐❧t♦♥✐❛♥ Hε

s✇✐t❝❤(t) := H✵−f (t) ε Xj,

✇❤❡r❡ f : R → [✵, ✶] ✐s ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ✿ f (t) = ✵ ❢♦r ❛❧❧ t ≤ ✵ ❛♥❞ f (t) = ✶ ❢♦r ❛❧❧ t ≥ T > ✵✳ ρε(t) ✿ ♣❡rt✉r❜❡❞ st❛t❡ ✐ ε ❞ ❞t ρε(t) = [Hε

s✇✐t❝❤(t), ρε(t)],

ρε(✵) = Π✵. ❚❤❡♥ ρε(t) − Πε = O(ε✷) ✉♥✐❢♦r♠❧② ♦♥ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧s ✐♥ t✐♠❡. ❚❤✐s st❛t❡♠❡♥t ✐s ❛❧r❡❛❞② ♣r♦✈❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✐♥t❡r❛❝t✐♥❣ ♠♦❞❡❧s ♦♥ ❧❛tt✐❝❡s ❬❚❡✉❢❡❧✱ ✬✶✼❪✳ ◆❊❆❙❙ ❜②♣❛ss❡s t❤❡ ▲❘❆ ❛♥❞ t❤❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ ✐ts ✈❛❧✐❞✐t②✱ ❛♥❞ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ s❤❛♣❡ ♦❢ t❤❡ s✇✐t❝❤✐♥❣ ❢✉♥❝t✐♦♥✦

slide-50
SLIDE 50

❘❡♠❛r❦✿ ❏✉st✐✜❝❛t✐♦♥ ❢♦r ✉s✐♥❣ t❤❡ ◆❊❆❙❙ ✭✐♥ ♣r♦❣r❡ss✮ ❈♦♥s✐❞❡r t❤❡ t✐♠❡✲❞❡♣❡♥❞❡♥t ❍❛♠✐❧t♦♥✐❛♥ Hε

s✇✐t❝❤(t) := H✵−f (t) ε Xj,

✇❤❡r❡ f : R → [✵, ✶] ✐s ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ✿ f (t) = ✵ ❢♦r ❛❧❧ t ≤ ✵ ❛♥❞ f (t) = ✶ ❢♦r ❛❧❧ t ≥ T > ✵✳ ρε(t) ✿ ♣❡rt✉r❜❡❞ st❛t❡ ✐ ε ❞ ❞t ρε(t) = [Hε

s✇✐t❝❤(t), ρε(t)],

ρε(✵) = Π✵. ❚❤❡♥ ρε(t) − Πε = O(ε✷) ✉♥✐❢♦r♠❧② ♦♥ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧s ✐♥ t✐♠❡. ❚❤✐s st❛t❡♠❡♥t ✐s ❛❧r❡❛❞② ♣r♦✈❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✐♥t❡r❛❝t✐♥❣ ♠♦❞❡❧s ♦♥ ❧❛tt✐❝❡s ❬❚❡✉❢❡❧✱ ✬✶✼❪✳ ◆❊❆❙❙ ❜②♣❛ss❡s t❤❡ ▲❘❆ ❛♥❞ t❤❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ ✐ts ✈❛❧✐❞✐t②✱ ❛♥❞ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ s❤❛♣❡ ♦❢ t❤❡ s✇✐t❝❤✐♥❣ ❢✉♥❝t✐♦♥✦

slide-51
SLIDE 51

❘❡♠❛r❦✿ ❏✉st✐✜❝❛t✐♦♥ ❢♦r ✉s✐♥❣ t❤❡ ◆❊❆❙❙ ✭✐♥ ♣r♦❣r❡ss✮ ❈♦♥s✐❞❡r t❤❡ t✐♠❡✲❞❡♣❡♥❞❡♥t ❍❛♠✐❧t♦♥✐❛♥ Hε

s✇✐t❝❤(t) := H✵−f (t) ε Xj,

✇❤❡r❡ f : R → [✵, ✶] ✐s ❛ s♠♦♦t❤ ❢✉♥❝t✐♦♥ ✿ f (t) = ✵ ❢♦r ❛❧❧ t ≤ ✵ ❛♥❞ f (t) = ✶ ❢♦r ❛❧❧ t ≥ T > ✵✳ ρε(t) ✿ ♣❡rt✉r❜❡❞ st❛t❡ ✐ ε ❞ ❞t ρε(t) = [Hε

s✇✐t❝❤(t), ρε(t)],

ρε(✵) = Π✵. ❚❤❡♥ ρε(t) − Πε = O(ε✷) ✉♥✐❢♦r♠❧② ♦♥ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧s ✐♥ t✐♠❡. ❚❤✐s st❛t❡♠❡♥t ✐s ❛❧r❡❛❞② ♣r♦✈❡❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✐♥t❡r❛❝t✐♥❣ ♠♦❞❡❧s ♦♥ ❧❛tt✐❝❡s ❬❚❡✉❢❡❧✱ ✬✶✼❪✳ ◆❊❆❙❙ ❜②♣❛ss❡s t❤❡ ▲❘❆ ❛♥❞ t❤❡ ❥✉st✐✜❝❛t✐♦♥ ♦❢ ✐ts ✈❛❧✐❞✐t②✱ ❛♥❞ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ s❤❛♣❡ ♦❢ t❤❡ s✇✐t❝❤✐♥❣ ❢✉♥❝t✐♦♥✦

slide-52
SLIDE 52

❘❡s♣♦♥s❡ ❝✉rr❡♥ts ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ r❡s♣♦♥s❡ ♦❢ ❛ ❝✉rr❡♥t t♦ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ✇❡❛❦ ❡❧❡❝tr✐❝ ✜❡❧❞✱ ✐♥ t❤❡ r❡❣✐♠❡ ♦❢ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② t❡♥s♦r σε

ij

σε

ij : = ✶

ε Re τ (Ji Πε) = ✶ ε Re τ (✐[H✵, SXi] Πε) = ✶ ε Re τ (✐Xi[H✵, S] Πε) + ✶ ε Re τ (✐[H✵, Xi]S Πε). ❇② ▲❡♠♠❛ ✷ ❛♥❞ ❚❤❡♦r❡♠ ✸✱ σε

ij ✐s ✇❡❧❧✲❞❡✜♥❡❞ ✭❡✈❡♥ ✐❢ t❤❡ ❝✉rr❡♥t

♦♣❡r❛t♦r ✐s ♥♦t ♣❡r✐♦❞✐❝✦✮ ❊①♣❛♥s✐♦♥ ✐♥ ε σε

ij = ✶

ε Re τ (✐[H✵, SXi]Π✵)

  • =:♣❡rs✐st❡♥t ❝✉rr❡♥t

+Re τ (✐[H✵, SXi]Π✶) + O(ε)

slide-53
SLIDE 53

❘❡s♣♦♥s❡ ❝✉rr❡♥ts ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ r❡s♣♦♥s❡ ♦❢ ❛ ❝✉rr❡♥t t♦ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ✇❡❛❦ ❡❧❡❝tr✐❝ ✜❡❧❞✱ ✐♥ t❤❡ r❡❣✐♠❡ ♦❢ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② t❡♥s♦r σε

ij

σε

ij : = ✶

ε Re τ (Ji Πε) = ✶ ε Re τ (✐[H✵, SXi] Πε) = ✶ ε Re τ (✐Xi[H✵, S] Πε) + ✶ ε Re τ (✐[H✵, Xi]S Πε). ❇② ▲❡♠♠❛ ✷ ❛♥❞ ❚❤❡♦r❡♠ ✸✱ σε

ij ✐s ✇❡❧❧✲❞❡✜♥❡❞ ✭❡✈❡♥ ✐❢ t❤❡ ❝✉rr❡♥t

♦♣❡r❛t♦r ✐s ♥♦t ♣❡r✐♦❞✐❝✦✮ ❊①♣❛♥s✐♦♥ ✐♥ ε σε

ij = ✶

ε Re τ (✐[H✵, SXi]Π✵)

  • =:♣❡rs✐st❡♥t ❝✉rr❡♥t

+Re τ (✐[H✵, SXi]Π✶) + O(ε)

slide-54
SLIDE 54

❘❡s♣♦♥s❡ ❝✉rr❡♥ts ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ r❡s♣♦♥s❡ ♦❢ ❛ ❝✉rr❡♥t t♦ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ✇❡❛❦ ❡❧❡❝tr✐❝ ✜❡❧❞✱ ✐♥ t❤❡ r❡❣✐♠❡ ♦❢ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② t❡♥s♦r σε

ij

σε

ij : = ✶

ε Re τ (Ji Πε) = ✶ ε Re τ (✐[H✵, SXi] Πε) = ✶ ε Re τ (✐Xi[H✵, S] Πε) + ✶ ε Re τ (✐[H✵, Xi]S Πε). ❇② ▲❡♠♠❛ ✷ ❛♥❞ ❚❤❡♦r❡♠ ✸✱ σε

ij ✐s ✇❡❧❧✲❞❡✜♥❡❞ ✭❡✈❡♥ ✐❢ t❤❡ ❝✉rr❡♥t

♦♣❡r❛t♦r ✐s ♥♦t ♣❡r✐♦❞✐❝✦✮ ❊①♣❛♥s✐♦♥ ✐♥ ε σε

ij = ✶

ε Re τ (✐[H✵, SXi]Π✵)

  • =:♣❡rs✐st❡♥t ❝✉rr❡♥t

+Re τ (✐[H✵, SXi]Π✶) + O(ε)

slide-55
SLIDE 55

❘❡s♣♦♥s❡ ❝✉rr❡♥ts ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ r❡s♣♦♥s❡ ♦❢ ❛ ❝✉rr❡♥t t♦ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ✇❡❛❦ ❡❧❡❝tr✐❝ ✜❡❧❞✱ ✐♥ t❤❡ r❡❣✐♠❡ ♦❢ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② t❡♥s♦r σε

ij

σε

ij : = ✶

ε Re τ (Ji Πε) = ✶ ε Re τ (✐[H✵, SXi] Πε) = ✶ ε Re τ (✐Xi[H✵, S] Πε) + ✶ ε Re τ (✐[H✵, Xi]S Πε). ❇② ▲❡♠♠❛ ✷ ❛♥❞ ❚❤❡♦r❡♠ ✸✱ σε

ij ✐s ✇❡❧❧✲❞❡✜♥❡❞ ✭❡✈❡♥ ✐❢ t❤❡ ❝✉rr❡♥t

♦♣❡r❛t♦r ✐s ♥♦t ♣❡r✐♦❞✐❝✦✮ ❊①♣❛♥s✐♦♥ ✐♥ ε σε

ij = ✶

ε Re τ (✐[H✵, SXi]Π✵)

  • =:♣❡rs✐st❡♥t ❝✉rr❡♥t

+Re τ (✐[H✵, SXi]Π✶) + O(ε)

slide-56
SLIDE 56

❘❡s♣♦♥s❡ ❝✉rr❡♥ts ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ r❡s♣♦♥s❡ ♦❢ ❛ ❝✉rr❡♥t t♦ t❤❡ ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ✇❡❛❦ ❡❧❡❝tr✐❝ ✜❡❧❞✱ ✐♥ t❤❡ r❡❣✐♠❡ ♦❢ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② t❡♥s♦r σε

ij

σε

ij : = ✶

ε Re τ (Ji Πε) = ✶ ε Re τ (✐[H✵, SXi] Πε) = ✶ ε Re τ (✐Xi[H✵, S] Πε) + ✶ ε Re τ (✐[H✵, Xi]S Πε). ❇② ▲❡♠♠❛ ✷ ❛♥❞ ❚❤❡♦r❡♠ ✸✱ σε

ij ✐s ✇❡❧❧✲❞❡✜♥❡❞ ✭❡✈❡♥ ✐❢ t❤❡ ❝✉rr❡♥t

♦♣❡r❛t♦r ✐s ♥♦t ♣❡r✐♦❞✐❝✦✮ ❊①♣❛♥s✐♦♥ ✐♥ ε σε

ij = ✶

ε Re τ (✐[H✵, SXi]Π✵)

  • =:♣❡rs✐st❡♥t ❝✉rr❡♥t

+Re τ (✐[H✵, SXi]Π✶) + O(ε)

slide-57
SLIDE 57

[H✵, S] = ✵ ✿ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✹ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ ♥♦ ♣❡rs✐st❡♥t ❝✉rr❡♥t ✢♦✇s ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ st❛t❡ Π✵✳ ❚❤❡♥ σε

ij = ✐τ

  • [SXi, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+ Re τ

  • ✐[H✵, (SXi)❉]Π✶ + ✐[H✵, (SXi)❖❉Π✶] + ✐
  • [SXi, Π✵], Π✵[Π✵, Xj]
  • =:❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s

+ O(ε).

slide-58
SLIDE 58

[H✵, S] = ✵ ✿ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✹ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ ♥♦ ♣❡rs✐st❡♥t ❝✉rr❡♥t ✢♦✇s ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ st❛t❡ Π✵✶✳ ❚❤❡♥ σε

ij = ✐τ

  • [SXi, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+ Re τ

  • ✐[H✵, (SXi)❉]Π✶ + ✐[H✵, (SXi)❖❉Π✶] + ✐
  • [SXi, Π✵], Π✵[Π✵, Xj]
  • =:❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s

+ O(ε).

✶■❢ H✵ ❡♥❥♦②s s♣❛t✐❛❧ s②♠♠❡tr✐❡s ❤②♣♦t❤❡s✐s ✷ ✐s s❛t✐s✜❡❞ ✭❡✳ ❣✳ t❤❡

❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r ✷π/✸ r♦t❛t✐♦♥✮✳

slide-59
SLIDE 59

[H✵, S] = ✵ ✿ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✹ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ ♥♦ ♣❡rs✐st❡♥t ❝✉rr❡♥t ✢♦✇s ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ st❛t❡ Π✵✳ ❚❤❡♥ σε

ij = ✐τ

  • [SXi, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+ Re τ

  • ✐[H✵, (SXi)❉]Π✶ + ✐[H✵, (SXi)❖❉Π✶] + ✐
  • [SXi, Π✵], Π✵[Π✵, Xj]
  • =:❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s

+ O(ε).

slide-60
SLIDE 60

[H✵, S] = ✵ ✿ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✹ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ ♥♦ ♣❡rs✐st❡♥t ❝✉rr❡♥t ✢♦✇s ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ st❛t❡ Π✵✳ ❚❤❡♥ σε

ij = ✐τ

  • [SXi, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+ Re τ

  • ✐[H✵, (SXi)❉]Π✶ + ✐[H✵, (SXi)❖❉Π✶] + ✐
  • [SXi, Π✵], Π✵[Π✵, Xj]
  • =:❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ❞♦ ♥♦t ✈❛♥✐s❤ ❜❡❝❛✉s❡ τ( · ) ✐s ♥♦t ❝②❝❧✐❝ ✐♥ ❣❡♥❡r❛❧✦

+ O(ε).

slide-61
SLIDE 61

[H✵, S] = ✵ ✿ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✺ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ [H✵, S] = ✵✳ ❚❤❡♥ σε

ij = ✐τ

  • S
  • [Xj, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+O(ε) = − ✐ (✷π)d

  • Bd ❞k TrH❢
  • SΠ✵(k)
  • ∂kiΠ✵(k), ∂kjΠ✵(k)
  • + O(ε).

❘❡♠❛r❦

slide-62
SLIDE 62

[H✵, S] = ✵ ✿ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✺ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ [H✵, S] = ✵✳ ❚❤❡♥ σε

ij = ✐τ

  • S
  • [Xj, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+O(ε) = − ✐ (✷π)d

  • Bd ❞k TrH❢
  • SΠ✵(k)
  • ∂kiΠ✵(k), ∂kjΠ✵(k)
  • + O(ε).

❘❡♠❛r❦

slide-63
SLIDE 63

[H✵, S] = ✵ ✿ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✺ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ [H✵, S] = ✵✳ ❚❤❡♥ σε

ij = ✐τ

  • S
  • [Xj, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+O(ε) = − ✐ (✷π)d

  • Bd ❞k TrH❢
  • SΠ✵(k)
  • ∂kiΠ✵(k), ∂kjΠ✵(k)
  • + O(ε).

❘❡♠❛r❦

slide-64
SLIDE 64

[H✵, S] = ✵ ✿ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✺ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ [H✵, S] = ✵✳ ❚❤❡♥ σε

ij = ✐τ

  • S
  • [Xj, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+O(ε) = − ✐ (✷π)d

  • Bd ❞k TrH❢
  • SΠ✵(k)
  • ∂kiΠ✵(k), ∂kjΠ✵(k)
  • + O(ε).

❘❡♠❛r❦ ❝♦♥❞✐t✐♦♥❛❧ ❝②❝❧✐❝✐t② ♦❢ τ( · ) = ⇒ ♣❡rs✐st❡♥t ❝✉rr❡♥t ✈❛♥✐s❤❡s ❛✉t♦♠❛t✐❝❛❧❧② ❛♥❞ t❤❡ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ✈❛♥✐s❤✳ ■♥ d = ✷ t❤❡ ❑✉❜♦✲t❡r♠ ✐s ❡q✉❛❧ t♦ t❤❡ ✭❙♣✐♥✮ ❈❤❡r♥ ♥✉♠❜❡r ❢♦r ✭S = ■❞ ⊗ sz✮ S = ■❞ ✭✇❤❡♥❡✈❡r H✵ ✐s t✐♠❡✲r❡✈❡rs❛❧ s②♠♠❡tr✐❝✮✳

slide-65
SLIDE 65

[H✵, S] = ✵ ✿ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛

❚❤❡♦r❡♠ ✺ ✭▼✳✱ ▼♦♥❛❝♦✱ P❛♥❛t✐✱ ❚❡✉❢❡❧ ✬✶✽✮✳

✶✳ ▲❡t H✵ s❛t✐s❢② ❆ss✉♠♣t✐♦♥ ✭❍✮ ❛♥❞ ❧❡t Hε = H✵−εXj✳ ✷✳ ❆ss✉♠❡ [H✵, S] = ✵✳ ❚❤❡♥ σε

ij = ✐τ

  • S
  • [Xj, Π✵], [Xj, Π✵]
  • Π✵
  • =:❑✉❜♦✲❧✐❦❡ t❡r♠

+O(ε) = − ✐ (✷π)d

  • Bd ❞k TrH❢
  • SΠ✵(k)
  • ∂kiΠ✵(k), ∂kjΠ✵(k)
  • + O(ε).

❘❡♠❛r❦ ❋♦r S = ■❞ t❤✐s r❡s✉❧t ❛❣r❡❡s ✇✐t❤ ❬❆● ✬✾✽✱ ❇❊❙ ✬✾✹✱ ❇●❑❙ ✬✵✺✱ ❆❲ ✬✶✺ . . . ❪ ❛♥❞ ❢♦r S = ■❞ ⊗ sz ✐t ❛❣r❡❡s ✇✐t❤ ❬Pr ✬✵✾✱ ❙❝❤ ✬✶✸❪✳

slide-66
SLIDE 66

■♥s♣✐r❡❞ ❜② t❤❡ ❑✉❜♦ t❤❡♦r② ♦❢ ❝❤❛r❣❡ tr❛♥s♣♦rt ❬❆❙❙✱ ✬✾✹❪ ◮ ✇❡ ❞❡✜♥❡ t❤❡ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ G sz

K ❛♥❞ ♣r♦✈❡ t❤❛t

◮ ❢♦r ❛♥② ❣❛♣♣❡❞✱ ♣❡r✐♦❞✐❝✱ ♦♥❡✲♣❛rt✐❝❧❡ ❛♥❞ ♥❡❛r✲s✐❣❤t❡❞ ❞✐s❝r❡t❡ ❍❛♠✐❧t♦♥✐❛♥ G sz

K ✐s ✇❡❧❧✲❞❡✜♥❡❞

◮ t❤❡ ❡q✉❛❧✐t② G sz

K

  • r❛t✐♦ ♦❢ ❡①t❡♥s✐✈❡ q✉❛♥t✐t✐❡s

= σsz

K

  • r❛t✐♦ ♦❢ ✐♥t❡♥s✐✈❡ q✉❛♥t✐t✐❡s

❤♦❧❞s tr✉❡ ✭✐♥ t❤❡ ♥♦♥ tr✐✈✐❛❧ ❝❛s❡ [H✵, ■❞ ⊗ sz] = ✵✮✳

slide-67
SLIDE 67

■♥s♣✐r❡❞ ❜② t❤❡ ❑✉❜♦ t❤❡♦r② ♦❢ ❝❤❛r❣❡ tr❛♥s♣♦rt ❬❆❙❙✱ ✬✾✹❪ ◮ ✇❡ ❞❡✜♥❡ t❤❡ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ G sz

K ❛♥❞ ♣r♦✈❡ t❤❛t

◮ ❢♦r ❛♥② ❣❛♣♣❡❞✱ ♣❡r✐♦❞✐❝✱ ♦♥❡✲♣❛rt✐❝❧❡ ❛♥❞ ♥❡❛r✲s✐❣❤t❡❞ ❞✐s❝r❡t❡ ❍❛♠✐❧t♦♥✐❛♥ G sz

K ✐s ✇❡❧❧✲❞❡✜♥❡❞

◮ t❤❡ ❡q✉❛❧✐t② G sz

K

  • r❛t✐♦ ♦❢ ❡①t❡♥s✐✈❡ q✉❛♥t✐t✐❡s

= σsz

K

  • r❛t✐♦ ♦❢ ✐♥t❡♥s✐✈❡ q✉❛♥t✐t✐❡s

❤♦❧❞s tr✉❡ ✭✐♥ t❤❡ ♥♦♥ tr✐✈✐❛❧ ❝❛s❡ [H✵, ■❞ ⊗ sz] = ✵✮✳

slide-68
SLIDE 68

■♥s♣✐r❡❞ ❜② t❤❡ ❑✉❜♦ t❤❡♦r② ♦❢ ❝❤❛r❣❡ tr❛♥s♣♦rt ❬❆❙❙✱ ✬✾✹❪ ◮ ✇❡ ❞❡✜♥❡ t❤❡ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ G sz

K ❛♥❞ ♣r♦✈❡ t❤❛t

◮ ❢♦r ❛♥② ❣❛♣♣❡❞✱ ♣❡r✐♦❞✐❝✱ ♦♥❡✲♣❛rt✐❝❧❡ ❛♥❞ ♥❡❛r✲s✐❣❤t❡❞ ❞✐s❝r❡t❡ ❍❛♠✐❧t♦♥✐❛♥ G sz

K ✐s ✇❡❧❧✲❞❡✜♥❡❞

◮ t❤❡ ❡q✉❛❧✐t② G sz

K

  • r❛t✐♦ ♦❢ ❡①t❡♥s✐✈❡ q✉❛♥t✐t✐❡s

= σsz

K

  • r❛t✐♦ ♦❢ ✐♥t❡♥s✐✈❡ q✉❛♥t✐t✐❡s

❤♦❧❞s tr✉❡ ✭✐♥ t❤❡ ♥♦♥ tr✐✈✐❛❧ ❝❛s❡ [H✵, ■❞ ⊗ sz] = ✵✮✳

slide-69
SLIDE 69

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②✿ ♠❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

◮ ❍✐❧❜❡rt s♣❛❝❡ H❞✐s❝ := ℓ✷(Z✷) ⊗ CN ⊗ C✷ ❋♦r A ∈ B(H❞✐s❝) A♠,♥ :=

  • δ(k)

♠ , A δ(k) ♥

  • {k∈{✶,...,✷N}} ∈ ❊♥❞✷N(C)

◮ ❍❛♠✐❧t♦♥✐❛♥ H✵ ✐s ❜♦✉♥❞❡❞✱ s❡❧❢✲❛❞❥♦✐♥t

✶✳ ♣❡r✐♦❞✐❝✿ H✵♠,♥ = H✵♠−♣,♥−♣ ∀ ♠, ♥, ♣ ∈ Z✷ ✷✳ ♥❡❛r✲s✐❣❤t❡❞✿

  • H✵♠,♥
  • ≤ C❡− ✶

ζ (|m✶−n✶|+|m✷−n✷|)

∀♠, ♥ ∈ Z✷ ✸✳ ❛❞♠✐ts ❛ s♣❡❝tr❛❧ ❣❛♣✿ σ(H) µ

◮ ❋❡r♠✐ ♣r♦❥❡❝t✐♦♥ Π✵ := χ(−∞,µ)(H) ✭❛❧s♦ ♥❡❛r✲s✐❣❤t❡❞✮ ❊①✿ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧

slide-70
SLIDE 70

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②✿ ♠❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

◮ ❍✐❧❜❡rt s♣❛❝❡ H❞✐s❝ := ℓ✷(Z✷) ⊗ CN ⊗ C✷ ❋♦r A ∈ B(H❞✐s❝) A♠,♥ :=

  • δ(k)

♠ , A δ(k) ♥

  • {k∈{✶,...,✷N}} ∈ ❊♥❞✷N(C)

◮ ❍❛♠✐❧t♦♥✐❛♥ H✵ ✐s ❜♦✉♥❞❡❞✱ s❡❧❢✲❛❞❥♦✐♥t

✶✳ ♣❡r✐♦❞✐❝✿ H✵♠,♥ = H✵♠−♣,♥−♣ ∀ ♠, ♥, ♣ ∈ Z✷ ✷✳ ♥❡❛r✲s✐❣❤t❡❞✿

  • H✵♠,♥
  • ≤ C❡− ✶

ζ (|m✶−n✶|+|m✷−n✷|)

∀♠, ♥ ∈ Z✷ ✸✳ ❛❞♠✐ts ❛ s♣❡❝tr❛❧ ❣❛♣✿ σ(H) µ

◮ ❋❡r♠✐ ♣r♦❥❡❝t✐♦♥ Π✵ := χ(−∞,µ)(H) ✭❛❧s♦ ♥❡❛r✲s✐❣❤t❡❞✮ ❊①✿ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧

slide-71
SLIDE 71

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②✿ ♠❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

◮ ❍✐❧❜❡rt s♣❛❝❡ H❞✐s❝ := ℓ✷(Z✷) ⊗ CN ⊗ C✷ ❋♦r A ∈ B(H❞✐s❝) A♠,♥ :=

  • δ(k)

♠ , A δ(k) ♥

  • {k∈{✶,...,✷N}} ∈ ❊♥❞✷N(C)

◮ ❍❛♠✐❧t♦♥✐❛♥ H✵ ✐s ❜♦✉♥❞❡❞✱ s❡❧❢✲❛❞❥♦✐♥t

✶✳ ♣❡r✐♦❞✐❝✿ H✵♠,♥ = H✵♠−♣,♥−♣ ∀ ♠, ♥, ♣ ∈ Z✷ ✷✳ ♥❡❛r✲s✐❣❤t❡❞✿

  • H✵♠,♥
  • ≤ C❡− ✶

ζ (|m✶−n✶|+|m✷−n✷|)

∀♠, ♥ ∈ Z✷ ✸✳ ❛❞♠✐ts ❛ s♣❡❝tr❛❧ ❣❛♣✿ σ(H) µ

◮ ❋❡r♠✐ ♣r♦❥❡❝t✐♦♥ Π✵ := χ(−∞,µ)(H) ✭❛❧s♦ ♥❡❛r✲s✐❣❤t❡❞✮ ❊①✿ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧

slide-72
SLIDE 72

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②✿ ♠❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

◮ ❍✐❧❜❡rt s♣❛❝❡ H❞✐s❝ := ℓ✷(Z✷) ⊗ CN ⊗ C✷ ❋♦r A ∈ B(H❞✐s❝) A♠,♥ :=

  • δ(k)

♠ , A δ(k) ♥

  • {k∈{✶,...,✷N}} ∈ ❊♥❞✷N(C)

◮ ❍❛♠✐❧t♦♥✐❛♥ H✵ ✐s ❜♦✉♥❞❡❞✱ s❡❧❢✲❛❞❥♦✐♥t

✶✳ ♣❡r✐♦❞✐❝✿ H✵♠,♥ = H✵♠−♣,♥−♣ ∀ ♠, ♥, ♣ ∈ Z✷ ✷✳ ♥❡❛r✲s✐❣❤t❡❞✿

  • H✵♠,♥
  • ≤ C❡− ✶

ζ (|m✶−n✶|+|m✷−n✷|)

∀♠, ♥ ∈ Z✷ ✸✳ ❛❞♠✐ts ❛ s♣❡❝tr❛❧ ❣❛♣✿ σ(H) µ

◮ ❋❡r♠✐ ♣r♦❥❡❝t✐♦♥ Π✵ := χ(−∞,µ)(H) ✭❛❧s♦ ♥❡❛r✲s✐❣❤t❡❞✮ ❊①✿ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧

slide-73
SLIDE 73

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②✿ ♠❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

◮ ❍✐❧❜❡rt s♣❛❝❡ H❞✐s❝ := ℓ✷(Z✷) ⊗ CN ⊗ C✷ ❋♦r A ∈ B(H❞✐s❝) A♠,♥ :=

  • δ(k)

♠ , A δ(k) ♥

  • {k∈{✶,...,✷N}} ∈ ❊♥❞✷N(C)

◮ ❍❛♠✐❧t♦♥✐❛♥ H✵ ✐s ❜♦✉♥❞❡❞✱ s❡❧❢✲❛❞❥♦✐♥t

✶✳ ♣❡r✐♦❞✐❝✿ H✵♠,♥ = H✵♠−♣,♥−♣ ∀ ♠, ♥, ♣ ∈ Z✷ ✷✳ ♥❡❛r✲s✐❣❤t❡❞✿

  • H✵♠,♥
  • ≤ C❡− ✶

ζ (|m✶−n✶|+|m✷−n✷|)

∀♠, ♥ ∈ Z✷ ✸✳ ❛❞♠✐ts ❛ s♣❡❝tr❛❧ ❣❛♣✿ σ(H) µ

◮ ❋❡r♠✐ ♣r♦❥❡❝t✐♦♥ Π✵ := χ(−∞,µ)(H) ✭❛❧s♦ ♥❡❛r✲s✐❣❤t❡❞✮ ❊①✿ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧

slide-74
SLIDE 74

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②✿ ♠❛t❤❡♠❛t✐❝❛❧ s❡tt✐♥❣

◮ ❍✐❧❜❡rt s♣❛❝❡ H❞✐s❝ := ℓ✷(Z✷) ⊗ CN ⊗ C✷ ❋♦r A ∈ B(H❞✐s❝) A♠,♥ :=

  • δ(k)

♠ , A δ(k) ♥

  • {k∈{✶,...,✷N}} ∈ ❊♥❞✷N(C)

◮ ❍❛♠✐❧t♦♥✐❛♥ H✵ ✐s ❜♦✉♥❞❡❞✱ s❡❧❢✲❛❞❥♦✐♥t

✶✳ ♣❡r✐♦❞✐❝✿ H✵♠,♥ = H✵♠−♣,♥−♣ ∀ ♠, ♥, ♣ ∈ Z✷ ✷✳ ♥❡❛r✲s✐❣❤t❡❞✿

  • H✵♠,♥
  • ≤ C❡− ✶

ζ (|m✶−n✶|+|m✷−n✷|)

∀♠, ♥ ∈ Z✷ ✸✳ ❛❞♠✐ts ❛ s♣❡❝tr❛❧ ❣❛♣✿ σ(H) µ

◮ ❋❡r♠✐ ♣r♦❥❡❝t✐♦♥ Π✵ := χ(−∞,µ)(H) ✭❛❧s♦ ♥❡❛r✲s✐❣❤t❡❞✮ ❊①✿ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧

slide-75
SLIDE 75

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②

◮ ❙♣✐♥ t♦rq✉❡ r❡s♣♦♥s❡ Tsz := ✐Π✵

  • [Π✵, sz]

∼✐[H✵,sz]

, [Π✵, X✷]

∼E✷

  • ◮ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t②

σsz

K := τ

  • Σsz

K

  • ✇❤❡r❡

Σsz

K := ✐Π✵

  • [Π✵, X✶ ⊗ sz]
  • ∼Jsz

✶ := ✐ [H✵,X✶⊗sz]

, [Π✵, X✷]

∼E✷

  • ◮ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡

G sz

K := ✶✲♣✈❚r

  • Gsz

K

  • ✇❤❡r❡

Gsz

K := ✐Π✵

  • [Π✵, Λ✶ ⊗ sz]
  • ∼I sz

✶ := ✐ [H✵,Λ✶⊗sz]

, [Π✵, Λ✷]

∼∆V✷

slide-76
SLIDE 76

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②

◮ ❙♣✐♥ t♦rq✉❡ r❡s♣♦♥s❡ Tsz := ✐Π✵

  • [Π✵, sz]

∼✐[H✵,sz]

, [Π✵, X✷]

∼E✷

  • ◮ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t②

σsz

K := τ

  • Σsz

K

  • ✇❤❡r❡

Σsz

K := ✐Π✵

  • [Π✵, X✶ ⊗ sz]
  • ∼Jsz

✶ := ✐ [H✵,X✶⊗sz]

, [Π✵, X✷]

∼E✷

  • ◮ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡

G sz

K := ✶✲♣✈❚r

  • Gsz

K

  • ✇❤❡r❡

Gsz

K := ✐Π✵

  • [Π✵, Λ✶ ⊗ sz]
  • ∼I sz

✶ := ✐ [H✵,Λ✶⊗sz]

, [Π✵, Λ✷]

∼∆V✷

slide-77
SLIDE 77

❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛♥❞ ❝♦♥❞✉❝t✐✈✐t②

◮ ❙♣✐♥ t♦rq✉❡ r❡s♣♦♥s❡ Tsz := ✐Π✵

  • [Π✵, sz]

∼✐[H✵,sz]

, [Π✵, X✷]

∼E✷

  • ◮ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t②

σsz

K := τ

  • Σsz

K

  • ✇❤❡r❡

Σsz

K := ✐Π✵

  • [Π✵, X✶ ⊗ sz]
  • ∼Jsz

✶ := ✐ [H✵,X✶⊗sz]

, [Π✵, X✷]

∼E✷

  • ◮ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡

G sz

K := ✶✲♣✈❚r

  • Gsz

K

  • ✇❤❡r❡

Gsz

K := ✐Π✵

  • [Π✵, Λ✶ ⊗ sz]
  • ∼I sz

✶ := ✐ [H✵,Λ✶⊗sz]

, [Π✵, Λ✷]

∼∆V✷

slide-78
SLIDE 78

❚❤❡♦r❡♠ ✻ ✭▼✳✱ P❛♥❛t✐✱ ❚❛✉❜❡r ✬✶✽✮✳

✶✳ τ(Tsz) = ✵✳ ✷✳ Σsz

K ✐s ♥♦t ♣❡r✐♦❞✐❝✱ σsz K ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ s❛t✐s✜❡s

σsz

K = Tr(χ✶Σsz K χ✶)✳

✸✳ ❋✐① Λ✷✳ ❆ss✉♠❡ t❤❛t G sz

K (Λ✶, Λ✷) ✐s ✜♥✐t❡ ❢♦r ❛t ❧❡❛st ❛ s✇✐t❝❤

❢✉♥❝t✐♦♥ Λ✶✳ ❚❤❡♥ G sz

K (Λ′ ✶, Λ✷) ✐s ✜♥✐t❡ ❢♦r ❛♥② ♦❢ s✇✐t❝❤

❢✉♥❝t✐♦♥ Λ′

✶ ❛♥❞ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ Λ′ ✶✳

✹✳ ❚❤❡ ❡q✉❛❧✐t② G sz

K = σsz K ❤♦❧❞s tr✉❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ G sz K ✐s ✜♥✐t❡

❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ s✇✐t❝❤ ❢✉♥❝t✐♦♥s Λ✶, Λ✷✳ Pr♦♦❢ ✉s❡s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❝②❝❧✐❝✐t② ♦❢ Tr( · ) ❛♥❞ τ( · )✱ ▲❡♠♠❛ ✷✱ ❬❊❧❣❛rt✱ ●r❛❢✱ ❙❝❤❡♥❦❡r ✬✵✹❪✱ . . .

slide-79
SLIDE 79

❚❤❡♦r❡♠ ✻ ✭▼✳✱ P❛♥❛t✐✱ ❚❛✉❜❡r ✬✶✽✮✳

✶✳ τ(Tsz) = ✵✳ ✷✳ Σsz

K ✐s ♥♦t ♣❡r✐♦❞✐❝✱ σsz K ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ s❛t✐s✜❡s

σsz

K = Tr(χ✶Σsz K χ✶)✳

✸✳ ❋✐① Λ✷✳ ❆ss✉♠❡ t❤❛t G sz

K (Λ✶, Λ✷) ✐s ✜♥✐t❡ ❢♦r ❛t ❧❡❛st ❛ s✇✐t❝❤

❢✉♥❝t✐♦♥ Λ✶✳ ❚❤❡♥ G sz

K (Λ′ ✶, Λ✷) ✐s ✜♥✐t❡ ❢♦r ❛♥② ♦❢ s✇✐t❝❤

❢✉♥❝t✐♦♥ Λ′

✶ ❛♥❞ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ Λ′ ✶✳

✹✳ ❚❤❡ ❡q✉❛❧✐t② G sz

K = σsz K ❤♦❧❞s tr✉❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ G sz K ✐s ✜♥✐t❡

❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ s✇✐t❝❤ ❢✉♥❝t✐♦♥s Λ✶, Λ✷✳ Pr♦♦❢ ✉s❡s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❝②❝❧✐❝✐t② ♦❢ Tr( · ) ❛♥❞ τ( · )✱ ▲❡♠♠❛ ✷✱ ❬❊❧❣❛rt✱ ●r❛❢✱ ❙❝❤❡♥❦❡r ✬✵✹❪✱ . . .

slide-80
SLIDE 80

❚❤❡♦r❡♠ ✻ ✭▼✳✱ P❛♥❛t✐✱ ❚❛✉❜❡r ✬✶✽✮✳

✶✳ τ(Tsz) = ✵✳ ✷✳ Σsz

K ✐s ♥♦t ♣❡r✐♦❞✐❝✱ σsz K ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ s❛t✐s✜❡s

σsz

K = Tr(χ✶Σsz K χ✶)✳

✸✳ ❋✐① Λ✷✳ ❆ss✉♠❡ t❤❛t G sz

K (Λ✶, Λ✷) ✐s ✜♥✐t❡ ❢♦r ❛t ❧❡❛st ❛ s✇✐t❝❤

❢✉♥❝t✐♦♥ Λ✶✳ ❚❤❡♥ G sz

K (Λ′ ✶, Λ✷) ✐s ✜♥✐t❡ ❢♦r ❛♥② ♦❢ s✇✐t❝❤

❢✉♥❝t✐♦♥ Λ′

✶ ❛♥❞ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ Λ′ ✶✳

✹✳ ❚❤❡ ❡q✉❛❧✐t② G sz

K = σsz K ❤♦❧❞s tr✉❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ G sz K ✐s ✜♥✐t❡

❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ s✇✐t❝❤ ❢✉♥❝t✐♦♥s Λ✶, Λ✷✳ Pr♦♦❢ ✉s❡s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❝②❝❧✐❝✐t② ♦❢ Tr( · ) ❛♥❞ τ( · )✱ ▲❡♠♠❛ ✷✱ ❬❊❧❣❛rt✱ ●r❛❢✱ ❙❝❤❡♥❦❡r ✬✵✹❪✱ . . .

slide-81
SLIDE 81

❚❤❡♦r❡♠ ✻ ✭▼✳✱ P❛♥❛t✐✱ ❚❛✉❜❡r ✬✶✽✮✳

✶✳ τ(Tsz) = ✵✳ ✷✳ Σsz

K ✐s ♥♦t ♣❡r✐♦❞✐❝✱ σsz K ✐s ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ s❛t✐s✜❡s

σsz

K = Tr(χ✶Σsz K χ✶)✳

✸✳ ❋✐① Λ✷✳ ❆ss✉♠❡ t❤❛t G sz

K (Λ✶, Λ✷) ✐s ✜♥✐t❡ ❢♦r ❛t ❧❡❛st ❛ s✇✐t❝❤

❢✉♥❝t✐♦♥ Λ✶✳ ❚❤❡♥ G sz

K (Λ′ ✶, Λ✷) ✐s ✜♥✐t❡ ❢♦r ❛♥② ♦❢ s✇✐t❝❤

❢✉♥❝t✐♦♥ Λ′

✶ ❛♥❞ ✐t ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ Λ′ ✶✳

✹✳ ❚❤❡ ❡q✉❛❧✐t② G sz

K = σsz K ❤♦❧❞s tr✉❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ G sz K ✐s ✜♥✐t❡

❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ s✇✐t❝❤ ❢✉♥❝t✐♦♥s Λ✶, Λ✷✳ Pr♦♦❢ ✉s❡s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❝②❝❧✐❝✐t② ♦❢ Tr( · ) ❛♥❞ τ( · )✱ ▲❡♠♠❛ ✷✱ ❬❊❧❣❛rt✱ ●r❛❢✱ ❙❝❤❡♥❦❡r ✬✵✹❪✱ . . .

slide-82
SLIDE 82

❈♦♥❝❧✉s✐♦♥

✶✳ ❲❡ ❤❛✈❡ ❛♥❛❧②③❡❞ q✉❛♥t✉♠ tr❛♥s♣♦rt ♦❢ ❝❤❛r❣❡ ❛♥❞ s♣✐♥ ✈✐❛ s♣❛❝❡✲❛❞✐❛❜❛t✐❝ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ✭◆❊❆❙❙✮ ❛✈♦✐❞✐♥❣ t❤❡ ▲❘❆✳ ✷✳ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ ❛ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② r❡❧❛t❡❞ t♦ t❤❡ ❝✉rr❡♥t ♦♣❡r❛t♦r J = ✐[H✵, S X] ✇✐t❤ ❝♦rr❡❝t✐♦♥s ✇❤❡♥ S ✐s ♥♦t ❝♦♥s❡r✈❡❞✳ ✸✳ ■♥ ❝❤❛r❣❡✲ ♦r s♣✐♥✲♣r❡s❡r✈✐♥❣ ♠♦❞❡❧s✱ ✇❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ q✉❛♥t✐③❛t✐♦♥ ♦❢ ❝♦♥❞✉❝t✐✈✐t✐❡s ✈✐❛ ✭s♣✐♥✮ ❈❤❡r♥ ♥✉♠❜❡rs✳ ✹✳ ❊✈❡♥ ✐❢ [H✵, sz] = ✵✱ t❤❡♥ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t② ❛♥❞ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛r❡ ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ ❝♦✐♥❝✐❞❡ ✉s✐♥❣ t❤❡ ✏♣r♦♣❡r✑ s♣✐♥ ❝✉rr❡♥t J✱ ❞✉❡ t♦ τ(Tz) = ✵✱ ❜❡❝❛✉s❡ ♣❡r✐♦❞✐❝✐t② ❛♥❞ s♣✐♥ ❝♦♥s❡r✈❛t✐♦♥ ❛r❡ r❡st♦r❡❞ ♦♥ ❛✈❡r❛❣❡✳

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SLIDE 83

❈♦♥❝❧✉s✐♦♥

✶✳ ❲❡ ❤❛✈❡ ❛♥❛❧②③❡❞ q✉❛♥t✉♠ tr❛♥s♣♦rt ♦❢ ❝❤❛r❣❡ ❛♥❞ s♣✐♥ ✈✐❛ s♣❛❝❡✲❛❞✐❛❜❛t✐❝ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ✭◆❊❆❙❙✮ ❛✈♦✐❞✐♥❣ t❤❡ ▲❘❆✳ ✷✳ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ ❛ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② r❡❧❛t❡❞ t♦ t❤❡ ❝✉rr❡♥t ♦♣❡r❛t♦r J = ✐[H✵, S X] ✇✐t❤ ❝♦rr❡❝t✐♦♥s ✇❤❡♥ S ✐s ♥♦t ❝♦♥s❡r✈❡❞✳ ✸✳ ■♥ ❝❤❛r❣❡✲ ♦r s♣✐♥✲♣r❡s❡r✈✐♥❣ ♠♦❞❡❧s✱ ✇❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ q✉❛♥t✐③❛t✐♦♥ ♦❢ ❝♦♥❞✉❝t✐✈✐t✐❡s ✈✐❛ ✭s♣✐♥✮ ❈❤❡r♥ ♥✉♠❜❡rs✳ ✹✳ ❊✈❡♥ ✐❢ [H✵, sz] = ✵✱ t❤❡♥ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t② ❛♥❞ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛r❡ ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ ❝♦✐♥❝✐❞❡ ✉s✐♥❣ t❤❡ ✏♣r♦♣❡r✑ s♣✐♥ ❝✉rr❡♥t J✱ ❞✉❡ t♦ τ(Tz) = ✵✱ ❜❡❝❛✉s❡ ♣❡r✐♦❞✐❝✐t② ❛♥❞ s♣✐♥ ❝♦♥s❡r✈❛t✐♦♥ ❛r❡ r❡st♦r❡❞ ♦♥ ❛✈❡r❛❣❡✳

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SLIDE 84

❈♦♥❝❧✉s✐♦♥

✶✳ ❲❡ ❤❛✈❡ ❛♥❛❧②③❡❞ q✉❛♥t✉♠ tr❛♥s♣♦rt ♦❢ ❝❤❛r❣❡ ❛♥❞ s♣✐♥ ✈✐❛ s♣❛❝❡✲❛❞✐❛❜❛t✐❝ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ✭◆❊❆❙❙✮ ❛✈♦✐❞✐♥❣ t❤❡ ▲❘❆✳ ✷✳ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ ❛ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② r❡❧❛t❡❞ t♦ t❤❡ ❝✉rr❡♥t ♦♣❡r❛t♦r J = ✐[H✵, S X] ✇✐t❤ ❝♦rr❡❝t✐♦♥s ✇❤❡♥ S ✐s ♥♦t ❝♦♥s❡r✈❡❞✳ ✸✳ ■♥ ❝❤❛r❣❡✲ ♦r s♣✐♥✲♣r❡s❡r✈✐♥❣ ♠♦❞❡❧s✱ ✇❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ q✉❛♥t✐③❛t✐♦♥ ♦❢ ❝♦♥❞✉❝t✐✈✐t✐❡s ✈✐❛ ✭s♣✐♥✮ ❈❤❡r♥ ♥✉♠❜❡rs✳ ✹✳ ❊✈❡♥ ✐❢ [H✵, sz] = ✵✱ t❤❡♥ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t② ❛♥❞ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛r❡ ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ ❝♦✐♥❝✐❞❡ ✉s✐♥❣ t❤❡ ✏♣r♦♣❡r✑ s♣✐♥ ❝✉rr❡♥t J✱ ❞✉❡ t♦ τ(Tz) = ✵✱ ❜❡❝❛✉s❡ ♣❡r✐♦❞✐❝✐t② ❛♥❞ s♣✐♥ ❝♦♥s❡r✈❛t✐♦♥ ❛r❡ r❡st♦r❡❞ ♦♥ ❛✈❡r❛❣❡✳

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SLIDE 85

❈♦♥❝❧✉s✐♦♥

✶✳ ❲❡ ❤❛✈❡ ❛♥❛❧②③❡❞ q✉❛♥t✉♠ tr❛♥s♣♦rt ♦❢ ❝❤❛r❣❡ ❛♥❞ s♣✐♥ ✈✐❛ s♣❛❝❡✲❛❞✐❛❜❛t✐❝ ♣❡rt✉r❜❛t✐♦♥ t❤❡♦r② ✭◆❊❆❙❙✮ ❛✈♦✐❞✐♥❣ t❤❡ ▲❘❆✳ ✷✳ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ ❛ ❑✉❜♦✲❧✐❦❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② r❡❧❛t❡❞ t♦ t❤❡ ❝✉rr❡♥t ♦♣❡r❛t♦r J = ✐[H✵, S X] ✇✐t❤ ❝♦rr❡❝t✐♦♥s ✇❤❡♥ S ✐s ♥♦t ❝♦♥s❡r✈❡❞✳ ✸✳ ■♥ ❝❤❛r❣❡✲ ♦r s♣✐♥✲♣r❡s❡r✈✐♥❣ ♠♦❞❡❧s✱ ✇❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ q✉❛♥t✐③❛t✐♦♥ ♦❢ ❝♦♥❞✉❝t✐✈✐t✐❡s ✈✐❛ ✭s♣✐♥✮ ❈❤❡r♥ ♥✉♠❜❡rs✳ ✹✳ ❊✈❡♥ ✐❢ [H✵, sz] = ✵✱ t❤❡♥ ❑✉❜♦✲❧✐❦❡ s♣✐♥ ❝♦♥❞✉❝t✐✈✐t② ❛♥❞ s♣✐♥ ❝♦♥❞✉❝t❛♥❝❡ ❛r❡ ✇❡❧❧✲❞❡✜♥❡❞ ❛♥❞ ❝♦✐♥❝✐❞❡ ✉s✐♥❣ t❤❡ ✏♣r♦♣❡r✑ s♣✐♥ ❝✉rr❡♥t J✱ ❞✉❡ t♦ τ(Tz) = ✵✱ ❜❡❝❛✉s❡ ♣❡r✐♦❞✐❝✐t② ❛♥❞ s♣✐♥ ❝♦♥s❡r✈❛t✐♦♥ ❛r❡ r❡st♦r❡❞ ♦♥ ❛✈❡r❛❣❡✳

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SLIDE 86

P❡rs♣❡❝t✐✈❡s

✶✳ ❙t✉❞② ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✐♥ s♣✐♥ tr❛♥s♣♦rt ✭❡✳ ❣✳ ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✮✳ ✷✳ ❉❡✜♥❡ ❛♥❞ ❛♥❛❧②③❡ ♦❢ t❤❡ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ✐♥ t❡r♠s ♦❢ ❝♦♥❞✉❝t❛♥❝❡ ❞♦❡s t❤❡ ❡q✉❛❧✐t② G sz

ij = σsz ij st✐❧❧ ❤♦❧❞❄

✸✳ ❙t✉❞② ❤✐❣❤❡r✲♦r❞❡r ❝♦rr❡❝t✐♦♥s ✐♥ ε t♦ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② σij✳ ✹✳ ❋♦r S = ■❞ ⊗ sz r❡❧❛t❡ tr❛♥s♣♦rt ❝♦❡✣❝✐❡♥ts t♦ Z✷ t♦♣♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts✳ ✺✳ ■♥❝❧✉❞❡ ♦t❤❡r ❡✛❡❝ts✿ ❞✐s♦r❞❡r✱ ✐♥t❡r❛❝t✐♦♥s − → ✉♥✐✈❡rs❛❧✐t②

slide-87
SLIDE 87

P❡rs♣❡❝t✐✈❡s

✶✳ ❙t✉❞② ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✐♥ s♣✐♥ tr❛♥s♣♦rt ✭❡✳ ❣✳ ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✮✳ ✷✳ ❉❡✜♥❡ ❛♥❞ ❛♥❛❧②③❡ ♦❢ t❤❡ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ✐♥ t❡r♠s ♦❢ ❝♦♥❞✉❝t❛♥❝❡ ❞♦❡s t❤❡ ❡q✉❛❧✐t② G sz

ij = σsz ij st✐❧❧ ❤♦❧❞❄

✸✳ ❙t✉❞② ❤✐❣❤❡r✲♦r❞❡r ❝♦rr❡❝t✐♦♥s ✐♥ ε t♦ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② σij✳ ✹✳ ❋♦r S = ■❞ ⊗ sz r❡❧❛t❡ tr❛♥s♣♦rt ❝♦❡✣❝✐❡♥ts t♦ Z✷ t♦♣♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts✳ ✺✳ ■♥❝❧✉❞❡ ♦t❤❡r ❡✛❡❝ts✿ ❞✐s♦r❞❡r✱ ✐♥t❡r❛❝t✐♦♥s − → ✉♥✐✈❡rs❛❧✐t②

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SLIDE 88

P❡rs♣❡❝t✐✈❡s

✶✳ ❙t✉❞② ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✐♥ s♣✐♥ tr❛♥s♣♦rt ✭❡✳ ❣✳ ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✮✳ ✷✳ ❉❡✜♥❡ ❛♥❞ ❛♥❛❧②③❡ ♦❢ t❤❡ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ✐♥ t❡r♠s ♦❢ ❝♦♥❞✉❝t❛♥❝❡ ❞♦❡s t❤❡ ❡q✉❛❧✐t② G sz

ij = σsz ij st✐❧❧ ❤♦❧❞❄

✸✳ ❙t✉❞② ❤✐❣❤❡r✲♦r❞❡r ❝♦rr❡❝t✐♦♥s ✐♥ ε t♦ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② σij✳ ✹✳ ❋♦r S = ■❞ ⊗ sz r❡❧❛t❡ tr❛♥s♣♦rt ❝♦❡✣❝✐❡♥ts t♦ Z✷ t♦♣♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts✳ ✺✳ ■♥❝❧✉❞❡ ♦t❤❡r ❡✛❡❝ts✿ ❞✐s♦r❞❡r✱ ✐♥t❡r❛❝t✐♦♥s − → ✉♥✐✈❡rs❛❧✐t②

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SLIDE 89

P❡rs♣❡❝t✐✈❡s

✶✳ ❙t✉❞② ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✐♥ s♣✐♥ tr❛♥s♣♦rt ✭❡✳ ❣✳ ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✮✳ ✷✳ ❉❡✜♥❡ ❛♥❞ ❛♥❛❧②③❡ ♦❢ t❤❡ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ✐♥ t❡r♠s ♦❢ ❝♦♥❞✉❝t❛♥❝❡ ❞♦❡s t❤❡ ❡q✉❛❧✐t② G sz

ij = σsz ij st✐❧❧ ❤♦❧❞❄

✸✳ ❙t✉❞② ❤✐❣❤❡r✲♦r❞❡r ❝♦rr❡❝t✐♦♥s ✐♥ ε t♦ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② σij✳ ✹✳ ❋♦r S = ■❞ ⊗ sz r❡❧❛t❡ tr❛♥s♣♦rt ❝♦❡✣❝✐❡♥ts t♦ Z✷ t♦♣♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts✳ ✺✳ ■♥❝❧✉❞❡ ♦t❤❡r ❡✛❡❝ts✿ ❞✐s♦r❞❡r✱ ✐♥t❡r❛❝t✐♦♥s − → ✉♥✐✈❡rs❛❧✐t②

slide-90
SLIDE 90

P❡rs♣❡❝t✐✈❡s

✶✳ ❙t✉❞② ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✐♥ s♣✐♥ tr❛♥s♣♦rt ✭❡✳ ❣✳ ✐♥ t❤❡ ❑❛♥❡✕▼❡❧❡ ♠♦❞❡❧✮✳ ✷✳ ❉❡✜♥❡ ❛♥❞ ❛♥❛❧②③❡ ♦❢ t❤❡ ❜❡②♦♥❞✲❑✉❜♦✲❧✐❦❡ t❡r♠s ✐♥ t❡r♠s ♦❢ ❝♦♥❞✉❝t❛♥❝❡ ❞♦❡s t❤❡ ❡q✉❛❧✐t② G sz

ij = σsz ij st✐❧❧ ❤♦❧❞❄

✸✳ ❙t✉❞② ❤✐❣❤❡r✲♦r❞❡r ❝♦rr❡❝t✐♦♥s ✐♥ ε t♦ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❛❞✐❛❜❛t✐❝ ❝♦♥❞✉❝t✐✈✐t② σij✳ ✹✳ ❋♦r S = ■❞ ⊗ sz r❡❧❛t❡ tr❛♥s♣♦rt ❝♦❡✣❝✐❡♥ts t♦ Z✷ t♦♣♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts✳ ✺✳ ■♥❝❧✉❞❡ ♦t❤❡r ❡✛❡❝ts✿ ❞✐s♦r❞❡r✱ ✐♥t❡r❛❝t✐♦♥s − → ✉♥✐✈❡rs❛❧✐t②