r strt P t - PowerPoint PPT Presentation

❖r❜✐❢♦❧❞ ❞❡❝♦♥str✉❝t✐♦♥ P✳ ❇❛♥t❛② ❉✉❜r♦✈♥✐❦✱ ❏✉♥❡ ✷✵✶✾

■♥tr♦❞✉❝t✐♦♥ ❈♦♠♠♦♥ ❢♦r♠✉❧❛t✐♦♥s ♦❢ ❞②♥❛♠✐❝s ✭❊❖▼✱ ✈❛r✐❛t✐♦♥❛❧ ♣r✐♥❝✐♣❧❡s✱ ✳✳✳✮ ✉s✉❛❧❧② r❡q✉✐r❡ ✭q✉❛s✐✲✮tr✐✈✐❛❧ t♦♣♦❧♦❣② � r❡❛❧✐③❡ ❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡ ❛s ❡✐t❤❡r ❛ s✉❜♠❛♥✐❢♦❧❞ ♦r ❛ q✉♦t✐❡♥t ♦❢ s♦♠❡ ♥✐❝❡ ❣❡♦♠❡tr✐❝ str✉❝t✉r❡✿ ❝♦♥str❛✐♥❡❞ ❞②♥❛♠✐❝s ✈s ❣❛✉❣❡ s②♠♠❡tr✐❡s✳ ❉②♥❛♠✐❝s ❢♦r♠✉❧❛t❡❞ ♦♥ ❝♦✈❡r✐♥❣ s♣❛❝❡ � st❛t❡ ✐❞❡♥t✐✜❝❛t✐♦♥s✳ ❝♦✈❡r✐♥❣ ✭❞❡❝❦✮ tr❛♥s❢♦r♠❛t✐♦♥s = ❣❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥s ◆❡✇ s✉♣❡rs❡❧❡❝t✐♦♥ s❡❝t♦rs ❢r♦♠ ✬t✇✐st❡❞✬ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❉✐✣❝✉❧t ❝♦♠♣✉t❛t✐♦♥s ✭♠♦st❧② ♥✉♠❡r✐❝❛❧✱ ❡✳❣✳ ❧❛tt✐❝❡ ◗❈❉✮✳

❇❡st ✉♥❞❡rst♦♦❞ ❢♦r ✷❉ ❝♦♥❢♦r♠❛❧ ♠♦❞❡❧s ✭♦r❜✐❢♦❧❞✐♥❣✮✳ ❇❛s✐❝ ♣r♦❜❧❡♠s✿ str✉❝t✉r❡ ♦❢ t✇✐st❡❞ s❡❝t♦rs ❛♥❞ ✜①❡❞ ♣♦✐♥t r❡s♦❧✉t✐♦♥ ✭✉s✉❛❧❧② r❡q✉✐r❡ ❛❞ ❤♦❝ t❡❝❤♥✐q✉❡s✮✳ ❯♥❞❡r ❝♦♥tr♦❧ ♦♥❧② ✐♥ s♣❡❝✐❛❧ ❝❛s❡s✿ • t♦r♦✐❞❛❧ ♦r❜✐❢♦❧❞s ✭❝♦♠♣❛❝t✐✜❡❞ ❢r❡❡ ❜♦s♦♥s✱ ✐✳❡✳ ❧❛tt✐❝❡ ♠♦❞❡❧s✮❀ • ❤♦❧♦♠♦r♣❤✐❝ ♦r❜✐❢♦❧❞s ✭s❡❧❢✲❞✉❛❧ ♠♦❞❡❧s✮❀ • ♣❡r♠✉t❛t✐♦♥ ♦r❜✐❢♦❧❞s ✭♣❡r♠✉t❛t✐♦♥ s②♠♠❡tr✐❡s✮✳ ▼♦♦r❡✬s ✬❝♦♥❥❡❝t✉r❡✬✿ ❛❧❧ r❛t✐♦♥❛❧ ❝♦♥❢♦r♠❛❧ ♠♦❞❡❧s ❛r❡ ●❑❖ ❝♦s❡ts ♦r ♦r❜✐❢♦❧❞s t❤❡r❡♦❢✳ ❈❛♥ ♦♥❡ r❡✈❡rs❡ ♦r❜✐❢♦❧❞✐♥❣❄

❖r❜✐❢♦❧❞s ✷❉ ❝♦♥❢♦r♠❛❧ ♠♦❞❡❧ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❝❤✐r❛❧ s②♠♠❡tr② ❛❧❣❡❜r❛ V ✭♥✐❝❡ ❱❖❆✮ ❛♥❞ ▲✲❘ ❝♦✉♣❧✐♥❣ ✭♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥s✮✳ Pr✐♠❛r② ✜❡❧❞s ✭s✐♠♣❧❡ V ✲♠♦❞✉❧❡s✱ ❛❦❛✳ s✉♣❡rs❡❧❡❝t✐♦♥ s❡❝t♦rs✮ ❝❤❛r❛❝t❡r✐✲ ③❡❞ ❜② t❤❡✐r ❝♦♥❢♦r♠❛❧ ✇❡✐❣❤t h p ✭❧♦✇❡st ❡✐❣❡♥✈❛❧✉❡ ♦❢ L 0 ✮ ❛♥❞ ❝❤✐r❛❧ ❝❤❛r❛❝t❡r ✭tr❛❝❡ ❢✉♥❝t✐♦♥✮ � q L 0 − c / 24 � χ p ( q ) = Tr p ❞❡s❝r✐❜✐♥❣ t❤❡ s♣❡❝tr✉♠ ♦❢ L 0 ✳ ❋✉s✐♦♥ r✉❧❡s✿ ❝♦♠♣♦s✐t✐♦♥ ♦❢ s✉♣❡rs❡❧❡❝t✐♦♥ s❡❝t♦rs ✭t❡♥s♦r ♣r♦❞✉❝ts✮✳ ❋✉s✐♦♥ r✉❧❡s r❡❧❛t❡❞ t♦ ♠♦❞✉❧❛r ♣r♦♣❡rt✐❡s ✈✐❛ ❱❡r❧✐♥❞❡✬s ❢♦r♠✉❧❛✳

❋♦r G < Aut( V ) ✱ t❤❡ ❝❤✐r❛❧ ❛❧❣❡❜r❛ ♦❢ t❤❡ G ✲♦r❜✐❢♦❧❞ ✐s t❤❡ ✜①❡❞ ♣♦✐♥t s✉❜❛❧❣❡❜r❛ V G = { v ∈ V | gv = v ❢♦r ❛❧❧ g ∈ G } ✳ Pr✐♠❛r✐❡s ♦❢ t❤❡ G ✲♦r❜✐❢♦❧❞ ❢r♦♠ G ✲t✇✐st❡❞ ♠♦❞✉❧❡s ♦❢ V ✳ G ♣❡r♠✉t❡s t❤❡ G ✲t✇✐st❡❞ ♠♦❞✉❧❡s ✭♦✉t❡r ❛❝t✐♦♥✮✱ ✇✐t❤ h ∈ G t❛❦✐♥❣ ❛ g ✲t✇✐st❡❞ ♠♦❞✉❧❡ t♦ ❛ hgh ✲ 1 ✲t✇✐st❡❞ ♠♦❞✉❧❡ � G ✲♦r❜✐ts ♦r❣❛♥✐③❡❞ ✐♥t♦ t✇✐st❡❞ s❡❝t♦rs ❧❛❜❡❧❡❞ ❜② ❝♦♥❥✉❣❛❝② ❝❧❛ss❡s✳ ❚✇✐st❡❞ ♠♦❞✉❧❡s ✐♥ t❤❡ s❛♠❡ G ✲♦r❜✐t ✐❞❡♥t✐✜❡❞ ✇✐t❤ ❡❛❝❤ ♦t❤❡r ✭✬st❛t❡ ✐❞❡♥t✐✜❝❛t✐♦♥✬✮✳ ❙t❛❜✐❧✐③❡r G M = { g ∈ G | gM = M } ♦❢ t❤❡ t✇✐st❡❞ ♠♦❞✉❧❡ M r❡♣r❡s❡♥t❡❞ ♣r♦❥❡❝t✐✈❡❧② ♦♥ M ✭✇✐t❤ ❛ss♦❝✐❛t❡❞ ✷✲❝♦❝②❝❧❡ ϑ M ✮ � M s♣❧✐ts ✐♥t♦ ✐s♦t②♣✐❝ ❝♦♠♣♦♥❡♥ts M φ ❧❛❜❡❧❡❞ ❜② ✐rr❡♣s φ ∈ Irr ( G M | ϑ M ) ✳

✐s♦t②♣✐❝ ❝♦♠♣♦♥❡♥ts � ♣r✐♠❛r✐❡s ♦❢ t❤❡ ♦r❜✐❢♦❧❞ G ✲♦r❜✐ts ♦❢ ✭t✇✐st❡❞✮ ♠♦❞✉❧❡s � ❜❧♦❝❦s ♦❢ ♣r✐♠❛r✐❡s � � 1 ϑ M ( x, y ) ♥✉♠❜❡r ♦❢ ♣r✐♠❛r✐❡s = | G | ϑ M ( y, x ) xy = yx M ∈ ❋✐① ( x,y ) � 1 ♥✉♠❜❡r ♦❢ ❜❧♦❝❦s = [ G : G M ] M ❊❛❝❤ ❜❧♦❝❦ b ❝❤❛r❛❝t❡r✐③❡❞ ❜② ✐♥❡rt✐❛ s✉❜❣r♦✉♣ I b ✭st❛❜✐❧✐③❡r G M ♦❢ ❛♥② ♠♦❞✉❧❡ M ✐♥ t❤❡ ♦r❜✐t ❝♦rr❡s♣♦♥❞✐♥❣ t♦ b ✮ ❛♥❞ ✷✲❝♦❝②❝❧❡ ϑ b ∈ Z 2 ( I b , C ) ✳ ■♥t❡❣r❛❧❧② s♣❛❝❡❞ L 0 s♣❡❝tr✉♠ ❢♦r ✉♥t✇✐st❡❞ ♠♦❞✉❧❡s � t❤❡ ❝♦♥❢♦r♠❛❧ ✇❡✐❣❤ts ♦❢ ♣r✐♠❛r✐❡s ❢r♦♠ ❛ ❜❧♦❝❦ ✐♥ t❤❡ ✉♥t✇✐st❡❞ s❡❝t♦r ❞✐✛❡r ❜② ✐♥t❡❣❡rs✳

❱❛❝✉✉♠ ❜❧♦❝❦ b 0 ✭✉♥t✇✐st❡❞ s❡❝t♦r✮ ❤❛s tr✐✈✐❛❧ ❝♦❝②❧❡ � ❛❧❧ ❡❧❡♠❡♥ts ♦❢ b 0 ❤❛✈❡ ✐♥t❡❣❡r ❝♦♥❢♦r♠❛❧ ✇❡✐❣❤ts✱ ❛♥❞ ❝♦rr❡s♣♦♥❞ t♦ ✭♦r❞✐♥❛r②✮ ✐rr❡♣s ♦❢ G ✱ ✇✐t❤ ♠❛t❝❤✐♥❣ ❢✉s✐♦♥ r✉❧❡s ❛♥❞ ✭q✉❛♥t✉♠✮ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡ ✈❛❝✉✉♠ ❜❧♦❝❦ ✐s ❛ t✇✐st❡r✿ ❛ s❡t ♦❢ ♣r✐♠❛r✐❡s ✇✐t❤ ✐♥t❡❣❡r ❝♦♥❢♦r♠❛❧ ✇❡✐❣❤ts ✭❛♥❞ q✉❛♥t✉♠ ❞✐♠❡♥s✐♦♥s✮ ❝❧♦s❡❞ ✉♥❞❡r ❢✉s✐♦♥✳ ◗✉❡st✐♦♥ ✿ ❝❛♥ ✇❡ ✐❞❡♥t✐❢② t❤❡ ♦r❜✐❢♦❧❞ ❢r♦♠ ✐ts ✈❛❝✉✉♠ ❜❧♦❝❦❄ ❇r❛✐❞✐♥❣ r❡str✐❝t❡❞ t♦ ❛ t✇✐st❡r ✐s ✐♥✈♦❧✉t✐✈❡ � ❡❧❡♠❡♥ts ♦❢ t❤❡ t✇✐st❡r g ❛r❡ t❤❡ s✐♠♣❧❡ ♦❜❥❡❝ts ♦❢ ❛ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r②✳ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠✿ t❤❡ s✉❜r✐♥❣ ♦❢ t❤❡ ❢✉s✐♦♥ r✐♥❣ ❣❡♥❡r❛t❡❞ ❜② ❛ t✇✐st❡r ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❝❤❛r❛❝t❡r r✐♥❣ ♦❢ s♦♠❡ ✜♥✐t❡ ❣r♦✉♣✳

❋✉s✐♦♥ ♠❛tr✐❝❡s ❋♦r ❛ ♣r✐♠❛r② p ❞❡✜♥❡ t❤❡ ❢✉s✐♦♥ ♠❛tr✐① [ N ( p )] qr = N r pq ❙♣❛♥ ❛ ❝♦♠♠✉t❛t✐✈❡ ♠❛tr✐① ❛❧❣❡❜r❛ ✭t❤❡ ❱❡r❧✐♥❞❡ ❛❧❣❡❜r❛ V ✮ � N r N ( p ) N ( q ) = pq N ( r ) r ❈♦♠♠✉t✐♥❣ ♠❛tr✐❝❡s ✇✐t❤ ♥♦♥✲♥❡❣❛t✐✈❡ ❡❧❡♠❡♥ts � ❝♦♠♠♦♥ P❡rr♦♥✲ ❋r♦❜❡♥✐✉s ❡✐❣❡♥✈❡❝t♦r ✭q✉❛♥t✉♠ ❞✐♠❡♥s✐♦♥✮ d p ≥ 1 ✳ � N r pq d r = d p d q r

▼♦❞✉❧❛r S ✲♠❛tr✐① � 1 N r pq d r exp { 2 π i ( h p + h q − h r ) } S pq = �� r d 2 r r ❱❡r❧✐♥❞❡✬s t❤❡♦r❡♠✿ ρ p ( N ( q )) = S qp S 0 p ✐s ❛♥ ✐rr❡♣ ♦❢ V ❢♦r ❡❛❝❤ ♣r✐♠❛r② p ✱ ✇❤❡r❡ 0 ❞❡♥♦t❡s t❤❡ ✈❛❝✉✉♠ ♣r✐♠❛r②✳ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ V � ♣r✐♠❛r② ✜❡❧❞s P❡rr♦♥✬s t❤❡♦r❡♠ � | ρ p ( N ( q )) | ≤ ρ 0 ( N ( q )) = d q

❚✇✐st❡rs ❆ s❡t g ♦❢ ♣r✐♠❛r✐❡s ✐s ❢✉s✐♦♥ ❝❧♦s❡❞ ✐❢ N r pq > 0 ❢♦r p, q ∈ g ✐♠♣❧✐❡s r ∈ g ✳ ❋♦r♠ ❛ ♠♦❞✉❧❛r ✭❡✈❡♥ ❆r❣✉❡s✐❛♥✮ ❧❛tt✐❝❡ L ✳