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XX 0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ❈♦♠❜✐♥❛t♦r✐❛❧ ❛s♣❡❝ts ♦❢ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ♦❢ ✐♥t❡❣r❛❜❧❡ ♠♦❞❡❧s ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈ ❙❛✐♥t P❡t❡rs❜✉r❣ ❉❡♣❛rt♠❡♥t ♦❢ ❱✳❆✳ ❙t❡❦❧♦✈ ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡ ❘❆❙✱ ❛♥❞ ■❚▼❖ ❯♥✐✈❡rs✐t② ❏✉♥❡ ✷✵✶✺✱ ●●■✱ ❋❧♦r❡♥❝❡ ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX 0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ❚❤❡ ❍❡✐s❡♥❜❡r❣ XX 0 ♠♦❞❡❧ ♦♥ t❤❡ ❝❤❛✐♥ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❍❛♠✐❧t♦♥✐❛♥ � H ≡ − 1 ( σ − k +1 σ + k + σ + k +1 σ − k ) . 2 k ± iσ y ❚❤❡ ❧♦❝❛❧ s♣✐♥ ♦♣❡r❛t♦rs σ ± k = 1 2 ( σ x k ) ❛♥❞ σ z k ♦❜❡② t❤❡ ❝♦♠♠✉t❛t✐♦♥ r✉❧❡s✿ [ σ + k , σ − l ] = δ kl σ z l ✱ [ σ z k , σ ± l ] = ± 2 δ kl σ ± l ✭ δ kl ✐s t❤❡ ❑r♦♥❡❝❦❡r s②♠❜♦❧✮✳ ❚❤❡ s♣✐♥ ♦♣❡r❛t♦rs ❛❝t ✐♥ t❤❡ s♣❛❝❡ H M +1 s♣❛♥♥❡❞ ♦✈❡r t❤❡ st❛t❡s � M k =0 | s � k ✱ ✇❤❡r❡ | s � k ✐♠♣❧✐❡s ❡✐t❤❡r s♣✐♥ ✏✉♣✑✱ |↑� ✱ ♦r � 1 � � 0 � s♣✐♥ ✏❞♦✇♥✑✱ |↓� ✱ st❛t❡ ❛t ❦ th s✐t❡✳ ❚❤❡ st❛t❡s |↑� ≡ ❛♥❞ |↓� ≡ 0 1 ♣r♦✈✐❞❡ ❛ ♥❛t✉r❛❧ ❜❛s✐s ♦❢ t❤❡ ❧✐♥❡❛r s♣❛❝❡ C 2 ✳ ❚❤❡ st❛t❡ |⇑� ✇✐t❤ ❛❧❧ s♣✐♥s ✏✉♣✑✿ |⇑� ≡ � M n =0 |↑� n ✐s ❛♥♥✐❤✐❧❛t❡❞ ❜② t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✭✷✮✿ ˆ H |⇑� = 0 ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX 0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ❚❤❡ N ✲♣❛rt✐❝❧❡ st❛t❡✲✈❡❝t♦rs✱ t❤❡ st❛t❡s ✇✐t❤ N s♣✐♥s ✏❞♦✇♥✑✱ ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❡①♣r❡ss ❜② ♠❡❛♥s ♦❢ t❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥s✿ � N � � � S λ ( ✉ 2 σ − | Ψ( ✉ N ) � = N ) |⇑� . µ k k =1 λ ⊆{M N } ❚❤❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛❧❧ ♣❛rt✐t✐♦♥s λ s❛t✐s❢②✐♥❣ M ≡ M + 1 − N ≥ λ 1 ≥ λ 2 ≥ · · · ≥ λ N ≥ 0 ✳ ❚❤❡ s✐t❡s ✇✐t❤ s♣✐♥ ✏❞♦✇♥✑ st❛t❡s ❛r❡ ❧❛❜❡❧❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡s µ i ✱ 1 ≤ i ≤ N ✳ ❚❤❡s❡ ❝♦♦r❞✐♥❛t❡s ❝♦♥st✐t✉t❡ ❛ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ♣❛rt✐t✐♦♥ M ≥ µ 1 > µ 2 > . . . > µ N ≥ 0 ✳ ❚❤❡ r❡❧❛t✐♦♥ λ j = µ j − N + j ✱ ✇❤❡r❡ 1 ≤ j ≤ N ✱ ❝♦♥♥❡❝ts t❤❡ ♣❛rts ♦❢ λ t♦ t❤♦s❡ ♦❢ µ ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ✇r✐t❡✿ λ = µ − δ N ✱ ✇❤❡r❡ δ N ✐s t❤❡ str✐❝t ♣❛rt✐t✐♦♥ ( N − 1 , . . . , 1 , 0) ✳ ❚❤❡ ♣❛r❛♠❡t❡rs u 2 N ≡ ( u 2 1 , . . . , u 2 N ) ❛r❡ ❛r❜✐tr❛r② ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❚❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥s S λ ❛r❡ ❞❡✜♥❡❞ ❜② t❤❡ ❏❛❝♦❜✐✲❚r✉❞✐ r❡❧❛t✐♦♥✿ det( x λ k + N − k ) 1 ≤ j,k ≤ N j S λ ( ① N ) ≡ S λ ( x 1 , x 2 , . . . , x N ) ≡ , V ( ① N ) ✐♥ ✇❤✐❝❤ V ( ① N ) ✐s t❤❡ ❱❛♥❞❡r♠♦♥❞❡ ❞❡t❡r♠✐♥❛♥t � V ( ① N ) ≡ det( x N − k ) 1 ≤ j,k ≤ N = ( x l − x m ) . j 1 ≤ m<l ≤ N ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX 0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ❚❤❡ ❝♦♥❥✉❣❛t❡❞ st❛t❡✲✈❡❝t♦rs ❛r❡ ❣✐✈❡♥ ❜② � N � � � S λ ( ✈ − 2 σ + � Ψ( v N ) | = �⇑| N ) . µ k k =1 λ ⊆{M N } ❚❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ s♣✐♥ ✏❞♦✇♥✑ st❛t❡s µ ❛♥❞ t❤❡ ♣❛rt✐t✐♦♥ λ ❡①♣r❡ss❡❞ ❜② t❤❡ ❨♦✉♥❣ ❞✐❛❣r❛♠ Ðèñ✳✿ ❘❡❧❛t✐♦♥ ♦❢ t❤❡ s♣✐♥ ✏❞♦✇♥✑ ❝♦♦r❞✐♥❛t❡s µ = (8 , 5 , 3 , 2) ❛♥❞ ♣❛rt✐t✐♦♥ λ = (5 , 3 , 2 , 2) ❢♦r M = 8 ✱ N = 4 ✳ ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX 0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ❋♦r t❤❡ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s σ # k +( M +1) = σ # k ✐❢ ♣❛r❛♠❡t❡rs j ≡ e iθ j ✭ 1 ≤ j ≤ N ✮ s❛t✐s❢② t❤❡ ❇❡t❤❡ ❡q✉❛t✐♦♥s✱ u 2 e i ( M +1) θ j = ( − 1) N − 1 , 1 ≤ j ≤ N , t❤❡♥ t❤❡ st❛t❡✲✈❡❝t♦rs ❜❡❝♦♠❡ t❤❡ ❡✐❣❡♥ ✈❡❝t♦rs ♦❢ t❤❡ ❍❛♠✐❧t♦♥✐❛♥✿ H | Ψ( θ N ) � = E N ( θ N ) | Ψ( θ N ) � . ❚❤❡ s♦❧✉t✐♦♥s θ j t♦ t❤❡ ❇❡t❤❡ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ♣❛r❛♠❡tr✐③❡❞ s✉❝❤ t❤❛t � � 2 π I j − N − 1 θ j = , 1 ≤ j ≤ N , M + 1 2 ✇❤❡r❡ I j ❛r❡ ✐♥t❡❣❡rs ♦r ❤❛❧❢✲✐♥t❡❣❡rs ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r N ✐s ♦❞❞ ♦r ❡✈❡♥✳ ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX 0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ❚❤❡ ❡✐❣❡♥ ❡♥❡r❣✐❡s ♦❢ t❤❡ ♠♦❞❡❧ ❛r❡ ❡q✉❛❧ t♦ � �� N N � � � 2 π I j − N − 1 E N ( θ N ) = − cos θ j = − cos . M + 1 2 j =1 j =1 ❚❤❡ ❣r♦✉♥❞ st❛t❡ ♦❢ t❤❡ ♠♦❞❡❧ ✐s t❤❡ ❡✐❣❡♥✲st❛t❡ t❤❛t ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❧♦✇❡st ❡✐❣❡♥ ❡♥❡r❣② E N ( θ g N ) ✳ ■t ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❇❡t❤❡ ❡q✉❛t✐♦♥s ❛t I j = N − j ✿ � � 2 π N + 1 θ g j ≡ , 1 ≤ j ≤ N , − j M + 1 2 ❛♥❞ ✐s ❡q✉❛❧ t♦ πN sin E N ( θ g M +1 N ) = − . π sin M +1 ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX 0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ❲❡ s❤❛❧❧ ❝♦♥s✐❞❡r t❤❡ t✇♦ t②♣❡s ♦❢ t❤❡ t❤❡ t❤❡r♠❛❧ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ✐♥ ❛ s②st❡♠ ♦❢ ✜♥✐t❡ s✐③❡ ✇✐❧❧ ❜❡ ❝♦♥s✐❞❡r❡❞✳ ❲❡ ❝❛❧❧ t❤❡♠ t❤❡ ♣❡rs✐st❡♥❝❡ ♦❢ ❢❡rr♦♠❛❣♥❡t✐❝ str✐♥❣ ❛♥❞ t❤❡ ♣❡rs✐st❡♥❝❡ ♦❢ ❞♦♠❛✐♥ ✇❛❧❧ ✳ ❚❤❡ ♣❡rs✐st❡♥❝❡ ♦❢ ❢❡rr♦♠❛❣♥❡t✐❝ str✐♥❣ ✐s r❡❧❛t❡❞ t♦ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♣❡r❛t♦r ¯ Π n t❤❛t ❢♦r❜✐❞s s♣✐♥ ✏❞♦✇♥✑ st❛t❡s ♦♥ t❤❡ ✜rst n s✐t❡s ♦❢ t❤❡ ❝❤❛✐♥✿ Π n e − t H ¯ n − 1 N , n, t ) ≡ � Ψ( θ g N ) | ¯ Π n | Ψ( θ g � σ 0 j + σ z N ) � j T ( θ g ¯ , Π n ≡ , N ) | e − t H | Ψ( θ g � Ψ( θ g N ) � 2 j =0 ✇❤❡r❡ t ✐s t❤❡ ✐♥✈❡rs❡ t❡♠♣❡r❛t✉r❡ t = 1 /T ✳ ❚❤❡ ♣❡rs✐st❡♥❝❡ ♦❢ ❞♦♠❛✐♥ ✇❛❧❧ ✐s r❡❧❛t❡❞ t♦ t❤❡ ♦♣❡r❛t♦r ¯ F n t❤❛t ❝r❡❛t❡s ❛ s❡q✉❡♥❝❡ ♦❢ s♣✐♥ ✏❞♦✇♥✑ st❛t❡s ♦♥ t❤❡ ✜rst n s✐t❡s ♦❢ t❤❡ ❝❤❛✐♥✿ n e − t H ¯ N − n , n, t ) ≡ � Ψ( � θ g N − n ) | ¯ F n | Ψ( � θ g n − 1 F + � N − n ) � ¯ F ( � θ g σ − , F n ≡ j . � Ψ( � θ g N − n ) | e − t H | Ψ( � θ g N − n ) � j =0 ❍❡r❡ � θ g N − n ✐s t❤❡ s❡t ♦❢ ❣r♦✉♥❞ st❛t❡ s♦❧✉t✐♦♥s t♦ t❤❡ ❇❡t❤❡ ❡q✉❛t✐♦♥s ❢♦r t❤❡ s②st❡♠ ♦❢ N − n ♣❛rt✐❝❧❡s✳ ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈