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XX0 ❍❡✐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❡♥❝❡

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ❍❡✐s❡♥❜❡r❣ XX0 ♠♦❞❡❧ ♦♥ t❤❡ ❝❤❛✐♥ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❍❛♠✐❧t♦♥✐❛♥ H ≡ −1 2

• (σ−

k+1σ+ k + σ+ k+1σ− k ) .

❚❤❡ ❧♦❝❛❧ s♣✐♥ ♦♣❡r❛t♦rs σ±

k = 1 2(σx k ± iσy k) ❛♥❞ σz k ♦❜❡② t❤❡

❝♦♠♠✉t❛t✐♦♥ r✉❧❡s✿ [ σ+

k , σ− l ] = δkl σz l ✱ [ σz k, σ± l ] = ±2 δkl σ± l ✭δkl ✐s t❤❡

❑r♦♥❡❝❦❡r s②♠❜♦❧✮✳ ❚❤❡ s♣✐♥ ♦♣❡r❛t♦rs ❛❝t ✐♥ t❤❡ s♣❛❝❡ HM+1 s♣❛♥♥❡❞ ♦✈❡r t❤❡ st❛t❡s M

k=0 |sk✱ ✇❤❡r❡ |sk ✐♠♣❧✐❡s ❡✐t❤❡r s♣✐♥ ✏✉♣✑✱ |↑✱ ♦r

s♣✐♥ ✏❞♦✇♥✑✱ |↓✱ st❛t❡ ❛t ❦th s✐t❡✳ ❚❤❡ st❛t❡s |↑ ≡ 1

• ❛♥❞ |↓ ≡

1

• ♣r♦✈✐❞❡ ❛ ♥❛t✉r❛❧ ❜❛s✐s ♦❢ t❤❡ ❧✐♥❡❛r s♣❛❝❡ C2✳ ❚❤❡ st❛t❡ |⇑ ✇✐t❤ ❛❧❧ s♣✐♥s

✏✉♣✑✿ |⇑ ≡ M

n=0 |↑n ✐s ❛♥♥✐❤✐❧❛t❡❞ ❜② t❤❡ ❍❛♠✐❧t♦♥✐❛♥ ✭✷✮✿

ˆ H |⇑ = 0

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐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✿ |Ψ(✉N) =

• λ⊆{MN}

Sλ(✉2

N)

N

• k=1

σ−

µk

• |⇑ .

❚❤❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛❧❧ ♣❛rt✐t✐♦♥s λ s❛t✐s❢②✐♥❣ M ≡ M + 1 − N ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ 0✳ ❚❤❡ s✐t❡s ✇✐t❤ s♣✐♥ ✏❞♦✇♥✑ st❛t❡s ❛r❡ ❧❛❜❡❧❡❞ ❜② t❤❡ ❝♦♦r❞✐♥❛t❡s µi✱ 1 ≤ i ≤ N✳ ❚❤❡s❡ ❝♦♦r❞✐♥❛t❡s ❝♦♥st✐t✉t❡ ❛ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ♣❛rt✐t✐♦♥ M ≥ µ1 > µ2 > . . . > µN ≥ 0✳ ❚❤❡ r❡❧❛t✐♦♥ λj = µj − N + j✱ ✇❤❡r❡ 1 ≤ j ≤ N✱ ❝♦♥♥❡❝ts t❤❡ ♣❛rts ♦❢ λ t♦ t❤♦s❡ ♦❢ µ✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ✇r✐t❡✿ λ = µ − δN✱ ✇❤❡r❡ δN ✐s t❤❡ str✐❝t ♣❛rt✐t✐♦♥ (N − 1, . . . , 1, 0)✳ ❚❤❡ ♣❛r❛♠❡t❡rs u2

N ≡ (u2 1, . . . , u2 N) ❛r❡ ❛r❜✐tr❛r② ❝♦♠♣❧❡① ♥✉♠❜❡rs✳

❚❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥s Sλ ❛r❡ ❞❡✜♥❡❞ ❜② t❤❡ ❏❛❝♦❜✐✲❚r✉❞✐ r❡❧❛t✐♦♥✿ Sλ(①N) ≡ Sλ(x1, x2, . . . , xN) ≡ det(xλk+N−k

j

)1≤j,k≤N V(①N) , ✐♥ ✇❤✐❝❤ V(①N) ✐s t❤❡ ❱❛♥❞❡r♠♦♥❞❡ ❞❡t❡r♠✐♥❛♥t V(①N) ≡ det(xN−k

j

)1≤j,k≤N =

• 1≤m<l≤N

(xl − xm) .

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ❝♦♥❥✉❣❛t❡❞ st❛t❡✲✈❡❝t♦rs ❛r❡ ❣✐✈❡♥ ❜② Ψ(vN) | =

• λ⊆{MN}

⇑| N

• k=1

σ+

µk

• Sλ(✈−2

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✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐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

u2

j ≡ eiθj ✭1 ≤ j ≤ N✮ s❛t✐s❢② t❤❡ ❇❡t❤❡ ❡q✉❛t✐♦♥s✱

ei(M+1)θj = (−1)N−1 , 1 ≤ j ≤ N , t❤❡♥ t❤❡ st❛t❡✲✈❡❝t♦rs ❜❡❝♦♠❡ t❤❡ ❡✐❣❡♥ ✈❡❝t♦rs ♦❢ t❤❡ ❍❛♠✐❧t♦♥✐❛♥✿ H |Ψ(θN) = EN(θN) |Ψ(θN) . ❚❤❡ s♦❧✉t✐♦♥s θj t♦ t❤❡ ❇❡t❤❡ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ♣❛r❛♠❡tr✐③❡❞ s✉❝❤ t❤❛t θj = 2π M + 1

• Ij − N − 1

2

• ,

1 ≤ j ≤ N , ✇❤❡r❡ Ij ❛r❡ ✐♥t❡❣❡rs ♦r ❤❛❧❢✲✐♥t❡❣❡rs ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r N ✐s ♦❞❞ ♦r ❡✈❡♥✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ❡✐❣❡♥ ❡♥❡r❣✐❡s ♦❢ t❤❡ ♠♦❞❡❧ ❛r❡ ❡q✉❛❧ t♦ EN(θN) = −

N

• j=1

cos θj = −

N

• j=1

cos

• 2π

M + 1

• Ij − N − 1

2

• .

❚❤❡ ❣r♦✉♥❞ st❛t❡ ♦❢ t❤❡ ♠♦❞❡❧ ✐s t❤❡ ❡✐❣❡♥✲st❛t❡ t❤❛t ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❧♦✇❡st ❡✐❣❡♥ ❡♥❡r❣② EN(θ g

N)✳ ■t ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡

❇❡t❤❡ ❡q✉❛t✐♦♥s ❛t Ij = N − j✿ θ g

j ≡

2π M + 1

• N + 1

2 − j

• ,

1 ≤ j ≤ N , ❛♥❞ ✐s ❡q✉❛❧ t♦ EN(θ g

N) = −

sin

πN M+1

sin

π M+1

.

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐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 ¯ Πn t❤❛t ❢♦r❜✐❞s s♣✐♥ ✏❞♦✇♥✑ st❛t❡s ♦♥ t❤❡ ✜rst n s✐t❡s ♦❢ t❤❡ ❝❤❛✐♥✿ T (θ g

N, n, t) ≡ Ψ(θ g N) | ¯

Πn e−tH ¯ Πn |Ψ(θ g

N)

Ψ(θ g

N) | e−tH |Ψ(θ g N)

, ¯ Πn ≡

n−1

• j=0

σ0

j + σz j

2 , ✇❤❡r❡ t ✐s t❤❡ ✐♥✈❡rs❡ t❡♠♣❡r❛t✉r❡ t = 1/T✳ ❚❤❡ ♣❡rs✐st❡♥❝❡ ♦❢ ❞♦♠❛✐♥ ✇❛❧❧ ✐s r❡❧❛t❡❞ t♦ t❤❡ ♦♣❡r❛t♦r ¯ Fn t❤❛t ❝r❡❛t❡s ❛ s❡q✉❡♥❝❡ ♦❢ s♣✐♥ ✏❞♦✇♥✑ st❛t❡s ♦♥ t❤❡ ✜rst n s✐t❡s ♦❢ t❤❡ ❝❤❛✐♥✿ F( θ g

N−n, n, t) ≡ Ψ(

θ g

N−n) | ¯

F+

n e−tH ¯

Fn |Ψ( θ g

N−n)

Ψ( θ g

N−n) | e−tH |Ψ(

θ g

N−n)

, ¯ Fn ≡

n−1

• j=0

σ−

j .

❍❡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✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ✐♥tr♦❞✉❝❡❞ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ✇✐❧❧ ❜❡ ❜❛s❡❞ ♦♥ t❤❡ ❇✐♥❡t✕❈❛✉❝❤② ❢♦r♠✉❧❛ ❛❞❛♣t❡❞ ❢♦r t❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥s✿ PL/n(yN, xN) ≡

• λ⊆{(L/n)N}

Sλ(②N)Sλ(①N) = N

• l=1

yn

l xn l

• det(Tkj)1≤k,j≤N

V(②N)V(①N) , ✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛❧❧ ♣❛rt✐t✐♦♥s λ s❛t✐s❢②✐♥❣✿ L ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ n✱ ❛♥❞ t❤❡ ❡♥tr✐❡s Tkj ❛r❡ ❣✐✈❡♥ ❜②✿ Tkj = 1 − (xkyj)N+L−n 1 − xkyj .

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❈♦♥s✐❞❡r t❤❡ s✐♠♣❧❡st ♦♥❡✲♣❛rt✐❝❧❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ G (j, m|t) ≡ ⇑| σ+

j etHσ− m |⇑.

❚❤❡ ❍❛♠✐❧t♦♥✐❛♥ ♠❛② ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❤❡ ❢♦r♠ H ≡ −1 2

• (σ−

k+1σ+ k + σ+ k+1σ− k ) = −

• n,m

∆nmσ−

n σ+ m ,

✇❤❡r❡ ∆nm ✐s t❤❡ ❤♦♣♣✐♥❣ ♠❛tr✐① ✇✐t❤ t❤❡ ❡♥tr✐❡s ❡q✉❛❧ t♦ ∆nm = δn+1,m + δn−1,m. ❉✐✛❡r❡♥t✐❛t✐♥❣ G (j, m|t) ✇✐t❤ r❡s♣❡❝t t♦ t ❛♥❞ ❛♣♣❧②✐♥❣ t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥ [H, σ−

m] =

• n

∆nmσ−

n σz m ,

✇❡ ♦❜t❛✐♥ t❤❡ ❡q✉❛❧✐t② d dtG (j, m|t) = ⇑| σ+

j etHHσ− m |⇑

=

• n

∆nm⇑| σ+

j etHσ− n |⇑ =

• n

∆nmG (j, n|t) .

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ s❛t✐s✜❡s t❤❡ ❞✐✛❡r❡♥❝❡ ❡q✉❛t✐♦♥✿ d dtG (j, m|t) = G (j, m − 1|t) + G (j, m + 1|t) , ❢♦r t❤❡ ✜①❡❞ s✉❜✐♥❞❡① j✱ ❛♥❞ t❤❡ s❛♠❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ s✉❜✐♥❞❡① j ✇✐t❤ t❤❡ ✜①❡❞ m✳ ❚❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❡q✉❛❧✐t② G (j, m|0) = δjm✳ ❚❤❡ ❝♦rr❡❧❛t♦r G (j, l|t) ♠❛② ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✇❛❧❦s✳ ❊①♣❛♥❞✐♥❣ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ✐♥ ♣♦✇❡rs ♦❢ t ♦♥❡ ❤❛s G (j, m|t) =

• K

tK K!⇑| σ+

j (H)Kσ− m |⇑.

❆♣♣❧②✐♥❣ t❤❡♥ t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s✱ ♦♥❡ ♦❜t❛✐♥s HKσ−

m |⇑ = HK−1[ ˆ

H, σ−

m] |⇑ = ˆ

HK−1

n1

∆n1mσ−

n1 |⇑

=

• n1,...,nK

∆nKnK−1...∆n2n1∆n1mσ−

nK |⇑.

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ ⇑| σ+

j ✇✐❧❧ ✜① t❤❡ ❡♥❞✐♥❣ ♣♦✐♥t ♦❢ t❤❡ tr❛❥❡❝t♦r②✿

⇑| σ+

j (H)Kσ− m |⇑ = G(j, m|K) =

• n1,...,nK−1

∆jnK−1...∆n2n1∆n1m . ❚❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ✇❛❧❦❡r ♦♥ ❛ ❧❛tt✐❝❡ ✐s ❧❛❜❡❧❧❡❞ ❜② t❤❡ s♣✐♥ ❞♦✇♥ st❛t❡✱ ✇❤✐❧❡ t❤❡ s♣✐♥ ✉♣ st❛t❡s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❡♠♣t② s✐t❡s✱ ∆ps ✐s ❛♥ ❡❧❡♠❡♥t❛r② st❡♣✳ ❚❤❡ ♦❜t❛✐♥❡❞ ❡q✉❛❧✐t② ❡♥✉♠❡r❛t❡s ❛❧❧ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s ♦❢ t❤❡ ✇❛❧❦❡r st❛rt✐♥❣ ❢r♦♠ t❤❡ s✐t❡ j ❛♥❞ t❡r♠✐♥❛t✐♥❣ ❛t m✳ ❚❤❡ ❢✉♥❝t✐♦♥ G(j, m|K) s❛t✐s✜❡s ❡q✉❛t✐♦♥✿ G(j, m|K + 1) = G(j, m − 1|K) + G(j, m + 1|K) .

Ðèñ✳✿ ❆ r❛♥❞♦♠ ✇❛❧❦ ✐♥ K = 17 st❡♣s✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ♠✉❧t✐✲♣❛rt✐❝❧❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ G(j1, j2, ..., jN; l1, l2, ..., lN|t) = ⇑| σ+

j1σ+ j2...σ+ jN e−tHσ− l1σ− l2...σ− lN |⇑ .

❆♣♣❧②✐♥❣ t❤❡ ♦❢ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥ [H, σ−

l1σ− l2...σ− lN ] = N

• k=1

σ−

l1...σ− lk−1[H, σ− lk]σ− lk+1...σ− lN

✇❡ ♦❜t❛✐♥ t❤❡ ❡q✉❛t✐♦♥ d dtG(j1, ..., jN; l1, ..., lN|t) =

N

• k=1

(G(j1, ..., jN; l1, l2, ..., lk + 1, ..., lN|t) + G(j1, ..., jN; l1, l2, ..., lk − 1, ..., lN|t)) ❚❤❡ ❝♦♥❞✐t✐♦♥ G(j1, ..., jN; l1, l2, ..., lN|t) = 0 ✐❢ lk = lp, jk = jp ❢♦r ❛♥② 1 ≤ j, l ≤ N ✐s ❣✉❛r❛♥t❡❡❞ ❜② t❤❡ ♣r♦♣❡rt② ♦❢ t❤❡ P❛✉❧✐ ♠❛tr✐❝❡s (σ±

k )2 = 0✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❞❡t❡r♠✐♥❛♥t ❢♦r♠ G(j1, ..., jN; l1, ..., lN|t) = det {G(jr, ls|t)}r,s=1,...,N . ✇❤❡r❡ G(j, l|t) ❛r❡ t❤❡ ♦♥❡✲♣❛rt✐❝❧❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s✳ ❚❤❡ ❛✈❡r❛❣❡ ⇑| σ+

j1σ+ j2...σ+ jN (−H)Kσ− l1σ− l2...σ− lN |⇑ ✐s ❡q✉❛❧ t♦ t❤❡

♥✉♠❜❡r ♦❢ ❝♦♥✜❣✉r❛t✐♦♥s t❤❛t ❤❛✈❡ t❤❡ N r❛♥❞♦♠ t✉r♥s ✇❛❧❦❡rs ❜❡✐♥❣ ✐♥✐t✐❛❧❧② ❧♦❝❛t❡❞ ♦♥ t❤❡ ❧❛tt✐❝❡ s✐t❡s l1 > l2 > ... > lN ❛♥❞ ❛❢t❡r K st❡♣s ❛rr✐✈❡❞ ❛t t❤❡ ♣♦s✐t✐♦♥s j1 > j2 > ... > jN✳

Ðèñ✳✿ ❘❛♥❞♦♠ t✉r♥s ✇❛❧❦❡rs✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❋♦r t❤❡ ✐♥✜♥✐t❡ ✐♥ t❤❡ ❜♦t❤ s✐❞❡s ❧❛tt✐❝❡ t❤❡ ♦♥❡✲♣❛rt✐❝❧❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐s G(j, m|t) = Im−j(2t) = 1 2π π

−π

e2t cos θei(m−j)θdθ. ❊①♣❛♥❞✐♥❣ ♠♦❞✐✜❡❞ ❇❡ss❡❧ ❢✉♥❝t✐♦♥ ✐♥ ♣♦✇❡rs ♦❢ t Im−j(2t) =

∞

• k≥|l−j|

tk

• k−j+m

2

• !
• k+j−m

2

• !

✇❤❡r❡ t❤❡ s✉♠ ✐s t❛❦❡♥ ♦✈❡r k s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ k + |j − m| = 0(mod2)✱ ✇❡ ♦❜t❛✐♥ G(j, m|K) = K!

• K−j+m

2

• !
• K+j−m

2

• !

. ■t ✐s ❛ ✇❡❧❧ ❦♥♦✇♥ ❜✐♥♦♠✐❛❧ ❢♦r♠✉❧❛ ❢♦r ❛ ♥✉♠❜❡r ♦❢ ❛❧❧ ❧❛tt✐❝❡ ♣❛t❤s ❢r♦♠ m t♦ j ♦❢ ❧❡♥❣t❤ K ♦♥ t❤❡ ✐♥✜♥✐t❡ ❧❛tt✐❝❡✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ N r❛♥❞♦♠ t✉r♥s ✇❛❧❦❡rs ❜❡✐♥❣ ✐♥✐t✐❛❧❧② ❧♦❝❛t❡❞ ♦♥ t❤❡ ❧❛tt✐❝❡ s✐t❡s l1 > l2 > ... > lN ≥ 0 ❛♥❞ ❛rr✐✈❡❞ ❛t t❤❡ ♣♦s✐t✐♦♥s j1 > j2 > ... > jN ≥ 0 ✐s G(j1, ..., jN; l1, ..., lN|t) = 1 2π N π

−π

dθ1... π

−π

dθNe2t N

m=1 cos θm det

• ei(ls−jr)θr

r,s=1,...,N .

▼❛❦✐♥❣ ✉s❡ ♦❢ t❤❡ s②♠♠❡tr② ♦❢ t❤❡ ✐♥t❡❣r❛♥❞ ✇✐t❤ r❡s♣❡❝t t♦ ♣❡r♠✉t❛t✐♦♥s ♦❢ t❤❡ ✈❛r✐❛❜❧❡s θ1, . . . , θN G(j1, ..., jN; l1, ..., lN|t) = 1 (2π)NN! π

−π

dθ1... π

−π

dθNe2t N

m=1 cos θm

×sλ(eiθ1, eiθ2, ..., eiθN )sµ(e−iθ1, e−iθ2, ..., e−iθN )

• 1≤j<k≤N

|eiθj−eiθk|2 , ✇❤❡r❡ λk = jk − N + k ❛♥❞ µk = lk − N + k✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ♦♥❡✲♣❛rt✐❝❧❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ s❡♠✐❛①✐s (0 ≤ j, m < ∞) s❛t✐s✜❡s t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ G(j, m|t) = 0 ❢♦r j, m = −1✿ G(j, m|t) = 1 π π

−π

e2t cos θ sin [(j + 1)θ] sin [(m + 1)θ] dθ = Ij−m(2t) − Ij+m+2(2t) . ❚❤✐s ❢✉♥❝t✐♦♥ ✐s t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❈❛t❛❧❛♥ ♣❛t❤s✳ ❚❤❡ ❈❛t❛❧❛♥ ♣❛t❤ ✐s ❛ ❧❛tt✐❝❡ ♣❛t❤ t❤❛t st❛rts ❛t (0, j)✱ ❡♥❞s ✐♥ (K, m)✱ ❛♥❞ ♦♥❧② ❝♦♥t❛✐♥s ✉♣st❡♣s (1, 1) ❛♥❞ ❞♦✇♥st❡♣s (1 − 1)✱ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐t ♥❡✈❡r ❣♦❡s ❜❡❧♦✇ t❤❡ t✲❛①✐s✳

Ðèñ✳✿ ❆ ❈❛t❛❧❛♥ ♣❛t❤ ✐♥ K = 16 st❡♣s st❛rt✐♥❣ ❢r♦♠ ❛ ♣♦✐♥t (0, 3) ❛♥❞ t❡r♠✐♥❛t✐♥❣ ❛t (16, 5)✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ♥✉♠❜❡r ♦❢ ❈❛t❛❧❛♥ ♣❛t❤s ✐♥ K st❡♣s ❢r♦♠ j t♦ m ✐s ❡q✉❛❧ t♦ G(j, m|K) =

• K

K−j+m 2

• −
• K

K+j+m+2 2

• .

❚❤❡ ❉②❝❦ ♣❛t❤ ✐s t❤❡ ❈❛t❛❧❛♥ ♣❛t❤ t❤❛t st❛rts ❛t t❤❡ ♦r✐❣✐♥ ❛♥❞ ❡♥❞s ❛t (2K, 0)✳ ❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ s✉❝❤ ♣❛t❤s ✐s G(0, 0|t) = 1 t I1(2t) =

∞

• k=0

t2k 2k!Ck , ✇❤❡r❡ Ck ❛r❡ ❈❛t❛❧❛♥ ♥✉♠❜❡rs Ck = 1 k + 1 2k k

• ,

❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❉②❝❦ ♣❛t❤s ✐s G(0, 0|2K) = CK .

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ♠✉❧t✐✲♣❛rt✐❝❧❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ♦♥ ❛ s❡♠✐✲✐♥✜♥✐t❡ ❧❛tt✐❝❡ ✐s ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ r❛♥❞♦♠ t✉r♥ ♣❛t❤s st❛rt✐♥❣ ❛t l1 > l2 > ... > lN ≥ 0 ❛♥❞ t❡r♠✐♥❛t✐♥❣ ❛t j1 > j2 > ... > jN ≥ 0 t❤❛t ❞♦ ♥♦t ♣❛ss ❜❡❧♦✇ t ❛①✐s ✐s ❡q✉❛❧ t♦ G(j1, ..., jN; l1, ..., lN|t) = 1 (π)NN! π

−π

dθ1... π

−π

dθNe2t N

m=1 cos θm

× spλ(eiθ1, eiθ2, ..., eiθN )spµ(e−iθ1, e−iθ2, ..., e−iθN ) ×

• 1≤j<k≤N
• det

1≤r,s≤N{sin sθr}

2 , ✇❤❡r❡ λk = jk − N + k✱ µk = lk − N + k ❛♥❞ spλ(x1, x2, ..., xK) = det1≤j,k≤K(xλk+K−k+1

j

− x−(λk+K−k+1)

j

) det1≤j,k≤K(xK−k+1

j

− x−(K−k+1)

j

) ✐s t❤❡ ❝❤❛r❛❝t❡r ♦❢ t❤❡ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ s②♠♣❧❡❝t✐❝ ▲✐❡ ❛❧❣❡❜r❛ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛ ♣❛rt✐t✐♦♥ λ✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ♦❢ t❤❡ ♠♦❞❡❧ ✐s ❜❛s❡❞ ♦♥ ❇✐♥❡t✲❈❛✉❝❤② ❢♦r♠✉❧❛✿

• λ⊆M N

Sλ(①)Sλ(②) = det(Tkj)1≤k,j≤N VN(①)VN(②) , ✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛❧❧ ♣❛rt✐t✐♦♥s λ s❛t✐s❢②✐♥❣✿ M ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ 0✳ ❚❤❡ ❡♥tr✐❡s Tkj = 1 − (xkyj)M+N 1 − xkyj . ❚♦ st✉❞② t❤❡ ❛s②♠♣t♦t✐❝❛❧ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ✐♥tr♦❞✉❝❡❞ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ✇❡ ♥❡❡❞ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t❤❡ q✲♣❛r❛♠❡t❡r✐③❡❞ ❇✐♥❡t✲❈❛✉❝❤② r❡❧❛t✐♦♥✳ ❲❡ ♣✉t ② = q ≡ (q, q2, . . . , qN)✱ ① = q/q ≡ (1, q, . . . , qN−1) ❛♥❞ ♦❜t❛✐♥

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

• λ⊆M N

Sλ(q)Sλ(q/q) = V−1

N (q)V−1 N (q/q) det

• 1 − q(M+N)(j+k−1)

1 − qj+k−1

• 1≤j,k≤N

= q

NM 2

(1−M) det

2N + i − 1 N + j − 1

• 1≤i,j≤M

= Z(N, N, M) . ❚❤❡ ❡♥tr✐❡s ♦❢ t❤❡ ❧❛st ❞❡t❡r♠✐♥❛♥t ❛r❡ t❤❡ q✲❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✿ R r

• ≡

[R]! [r]! [R − r]! , [n] ≡ 1 − qn 1 − q , ❛♥❞ Z(N, N, M) =

N

• k=1

N

• j=1

1 − qM+j+k−1 1 − qj+k−1 ✐s t❤❡ ▼❛❝▼❛❤♦♥ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ♣❧❛♥❡ ♣❛rt✐t✐♦♥s ✐♥ t❤❡ N × N × M ❜♦①✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❆ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥s ♠❛② ❜❡ ❣✐✈❡♥ ✐♥ t❡r♠s ♦❢ s❡♠✐st❛♥❞❛r❞ ❨♦✉♥❣ t❛❜❧❡❛✉①✳ ❆ ✜❧❧✐♥❣ ♦❢ t❤❡ ❝❡❧❧s ♦❢ t❤❡ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♦❢ λ ✇✐t❤ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs n ∈ N+ ✐s ❝❛❧❧❡❞ ❛ s❡♠✐st❛♥❞❛r❞ t❛❜❧❡❛✉ ♦❢ s❤❛♣❡ λ ♣r♦✈✐❞❡❞ ✐t ✐s ✇❡❛❦❧② ✐♥❝r❡❛s✐♥❣ ❛❧♦♥❣ r♦✇s ❛♥❞ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❛❧♦♥❣ ❝♦❧✉♠♥s✳ ❚❤❡ ✇❡✐❣❤t ①T ♦❢ ❛ t❛❜❧❡❛✉ T ✐s ❞❡✜♥❡❞ ❛s ①T ≡

• i,j

xTij , ✇❤❡r❡ t❤❡ ♣r♦❞✉❝t ✐s ♦✈❡r ❛❧❧ ❡♥tr✐❡s Tij ♦❢ t❤❡ t❛❜❧❡❛✉ T✳ ❆♥ ❡q✉✐✈❛❧❡♥t ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② Sλ(x1, x2, . . . , xm) =

• T

①T , ✇❤❡r❡ m ≥ N✱ ❛♥❞ t❤❡ s✉♠ ✐s ♦✈❡r ❛❧❧ t❛❜❧❡❛✉① T ♦❢ s❤❛♣❡ λ ✇✐t❤ t❤❡ ❡♥tr✐❡s ❜❡✐♥❣ ♥✉♠❜❡rs ❢r♦♠ t❤❡ s❡t {1, 2, . . . , m}✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ ✇❛② ♦❢ r❡♣r❡s❡♥t✐♥❣ ❡❛❝❤ s❡♠✐st❛♥❞❛r❞ t❛❜❧❡❛✉ ♦❢ s❤❛♣❡ λ ✇✐t❤ ❡♥tr✐❡s ♥♦t ❡①❝❡❡❞✐♥❣ N ❛s ❛ ♥❡st ♦❢ s❡❧❢✲❛✈♦✐❞✐♥❣ ❧❛tt✐❝❡ ♣❛t❤s ✇✐t❤ ♣r❡s❝r✐❜❡❞ st❛rt ❛♥❞ ❡♥❞ ♣♦✐♥ts✳

Ðèñ✳✿ ❆ s❡♠✐st❛♥❞❛r❞ t❛❜❧❡❛✉ ♦❢ s❤❛♣❡ λ = (6, 3, 3, 1)✳

❆♥ ❡q✉✐✈❛❧❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥ Sλ(x1, x2, . . . , xN) =

• C

N

• j=1

xlj

j ,

✇❤❡r❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛❧❧ ❛❞♠✐ss✐❜❧❡ ♥❡sts C✱ t❤❡ ♣♦✇❡r lj ♦❢ xj ✐s t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s t♦ ♥♦rt❤ t❛❦❡♥ ❛❧♦♥❣ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ xj✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ ✇❛② ♦❢ r❡♣r❡s❡♥t✐♥❣ ❡❛❝❤ s❡♠✐st❛♥❞❛r❞ t❛❜❧❡❛✉ ♦❢ s❤❛♣❡ λ ✇✐t❤ ❡♥tr✐❡s ♥♦t ❡①❝❡❡❞✐♥❣ N ❛s ❛ ♥❡st ♦❢ s❡❧❢✲❛✈♦✐❞✐♥❣ ❧❛tt✐❝❡ ♣❛t❤s ✇✐t❤ ♣r❡s❝r✐❜❡❞ st❛rt ❛♥❞ ❡♥❞ ♣♦✐♥ts✳

Ðèñ✳✿ ❆ s❡♠✐st❛♥❞❛r❞ t❛❜❧❡❛✉ ♦❢ s❤❛♣❡ λ = (6, 3, 3, 1)✳

❆♥ ❡q✉✐✈❛❧❡♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥ Sλ(x1, x2, . . . , xN) =

• C

N

• j=1

xlj

j ,

✇❤❡r❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛❧❧ ❛❞♠✐ss✐❜❧❡ ♥❡sts C✱ t❤❡ ♣♦✇❡r lj ♦❢ xj ✐s t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s t♦ ♥♦rt❤ t❛❦❡♥ ❛❧♦♥❣ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ xj✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ kth ❧❛tt✐❝❡ ♣❛t❤ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ r❡❝t❛♥❣❧❡ ♦❢ t❤❡ s✐③❡ λk × (N − k)✳ ❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ ❡❛❝❤ ♣❛t❤ ✐s t❤❡ ❧♦✇❡r ❧❡❢t ✈❡rt❡①✳ ❚❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ♣❛t❤ ✐s t❤❡ ♥✉♠❜❡r ♦❢ sq✉❛r❡s ❜❡❧♦✇ ✐t ✐♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡❝t❛♥❣❧❡✳ ❚❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ♥❡st ♦❢ ❧❛tt✐❝❡ ♣❛t❤s C ✐s✿ | ζ |C =

N

• j=1

(N − j)lj =

N

• j=1

(j − 1)lN−j+1. ❚❤❡ q✲♣❛r❛♠❡tr✐③❡❞ ❙❝❤✉r ❢✉♥❝t✐♦♥ ✐s ❛ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥❡st✿ Sλ(q) =

• C

q|ξ|C = q|λ|

C

q|ζ|C , |λ| =

N

• k=1

λk .

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❙❝❤✉r ❢✉♥❝t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝♦♥❥✉❣❛t❡ ♥❡st ♦❢ s❡❧❢✲❛✈♦✐❞✐♥❣ ❧❛tt✐❝❡ ♣❛t❤s Sλ(y1, y2, . . . , yN) =

• B

N

• j=1

y(M−lj)

j

, ✇❤❡r❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛❧❧ ❛❞♠✐ss✐❜❧❡ ♥❡sts B ♦❢ N s❡❧❢✲❛✈♦✐❞✐♥❣ ❧❛tt✐❝❡ ♣❛t❤s✳ ❚❤❡ ✈♦❧✉♠❡ ♦❢ t❤❡ ♥❡st B ♦❢ ❧❛tt✐❝❡ ♣❛t❤s ✐s ❣✐✈❡♥ ❜② | ζ |B =

N

• j=1

(j − 1)(M − lj).

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♠❛② ❜❡ ❣r❛♣❤✐❝❛❧❧② ❡①♣r❡ss❡❞ ❛s ❛ ♥❡st ♦❢ N s❡❧❢✲❛✈♦✐❞✐♥❣ ❧❛tt✐❝❡ ♣❛t❤s st❛rt✐♥❣ ❛t t❤❡ ❡q✉✐❞✐st❛♥t ♣♦✐♥ts Ci ❛♥❞ t❡r♠✐♥❛t✐♥❣ ❛t t❤❡ ❡q✉✐❞✐st❛♥t ♣♦✐♥ts Bi ✭i = 1, . . . , N✮✳ ❚❤✐s ❝♦♥✜❣✉r❛t✐♦♥ ✐s ❦♥♦✇♥ ❛s ✇❛t❡r♠❡❧♦♥

Ðèñ✳✿ ❲❛t❡r♠❡❧♦♥ ❝♦♥✜❣✉r❛t✐♦♥ ❛♥❞ ❝♦rr❡s♣♦♥❞❡♥t ♣❧❛♥❡ ♣❛rt✐t✐♦♥✳

❚❤❡ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ✇❛t❡r♠❡❧♦♥s ✭t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ✇❛t❡r♠❡❧♦♥s✮ ✐s ❡q✉❛❧ t♦ t❤❡ q✲♣❛r❛♠❡t❡r✐③❡❞ s❝❛❧❛r ♣r♦❞✉❝t✿ Z(N, N, M) =

• W

q|ξ|C+|ζ|B =

• λ⊆M N

Sλ(q)Sλ(q/q) = ΨN(q− 1

2 )|ΨN(q/q) 1 2 ) , ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ♠❛tr✐① ❡❧❡♠❡♥t ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦♣❡r❛t♦r Πn ≡

n−1

• j=0

σ0

j +σz j

2

✐s ❡q✉❛❧ t♦ Ψ(✈N) | ¯ Πn e−tH ¯ Πn |Ψ(✉N) =

• λL, λR⊆{(M/n)N}

SλL(✈−2

N )SλR(✉2 N)

× ⇑| N

• l=1

σ+

µL

l

• e−tH

N

• p=1

σ−

µR

p

• |⇑

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

Ψ(✈N) | ¯ Π0 e−tH ¯ Π0 |Ψ(✉N) =

• k=0

tk k!Ψ(✈N) | ¯ Π0 Hk ¯ Π0 |Ψ(✉N)

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ ❛♥s✇❡r ❢♦r t❤❡ ♣❡rs✐st❡♥❝❡ ♦❢ t❤❡ ❢❡rr♦♠❛❣♥❡t✐❝ str✐♥❣ ✐s T (θ g

N, n, t) ≡ Ψ(

θ g

N−n) | ¯

F+

n e−tH ¯

Fn |Ψ( θ g

N−n)

Ψ( θ g

N−n) | e−tH |Ψ(

θ g

N−n)

= etEN(θ g

N)

(M + 1)N det

• M
• k,l=n

G(k, l|t) ei(lθ g

i −kθ g j )

• 1≤i,j≤N

. ❆♥ ❛❧t❡r♥❛t✐✈❡ ❡①♣r❡ss✐♦♥✿ T (θ g

N, n, t) = |V(eiθg

N )|2

(M + 1)2N

• {θN}

e−t(EN(θN)−EN(θ g

N))

×

• V(eiθN ) PM/n(e−iθN , eiθ g

N )

• 2 ,

✇❤❡r❡ M = M + 1 − N ❛♥❞ PM/n(e−iθN , eiθ g

N ) =

• λ⊆{(M/n)N}

Sλ(e−iθN )Sλ(eiθ g

N ) . ◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❋♦r ❛ ❧♦♥❣ ❡♥♦✉❣❤ ❝❤❛✐♥✱ M ≫ 1✱ ❛♥❞ N ♠♦❞❡r❛t❡✿ 1 ≪ N ≪ M✱ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ ♠❛② ❜❡ ❡①♣r❡ss❡❞ ❛s T (θ g

N, n, t) ≃

|V(eiθg

N )|2

(M + 1)NN!

π

• −π

π

• −π

. . .

π

• −π

e

t

N

• l=1

(cos θl−cos θ g

l )

×

• PM/n(e−iθN , eiθ g

N )

• 2
• 1≤k<l≤N
• eiθk − eiθl

2 dθ1dθ2 . . . dθN (2π)N . ■♥ t❤❡ ❧❛r❣❡ t ❧✐♠✐t ✭s♠❛❧❧ t❡♠♣❡r❛t✉r❡ ❧✐♠✐t t = 1/T)✮ ✐♥ t❤❡ ❧❡❛❞✐♥❣ ♦r❞❡r ✐♥ t−1✿ T (θ g ≈ ✵, n, t) ≃ A2(N, N, M − N + 1 − n) eΦ(N,M,t) , Φ(N, M, t) = N 2 log 2π M + 1 − N 2 2 log t + 3φN , ✇❤❡r❡ A(N, N, M − N + 1 − n) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣❧❛♥❡ ♣❛rt✐t✐♦♥s ✐♥ ❛ ❜♦① N × N × (M − N + 1 − n) ❛♥❞ φN = log G(N + 1) − N 2 log 2π ,

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❇❛r♥❡s ❢✉♥❝t✐♦♥ φN = log G(N + 1) − N 2 log 2π . ■♥ t❡r♠s ♦❢ ❇❛r♥❡s ❢✉♥❝t✐♦♥✱ t❤❡ ♥✉♠❜❡r ♦❢ ❜♦①❡❞ ♣❧❛♥❡ ♣❛rt✐t✐♦♥ A(N, N, M−N+1−n) = G2(N + 1) G(M + 2 − n + N) G(M + 2 − n − N) G(2N + 1) G2(M + 2 − n) . ❚❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ♣❡rs✐st❡♥❝❡ ♦❢ ❢❡rr♦♠❛❣♥❡t✐❝ str✐♥❣ ✐s✿ log T (θ g ≈ ✵, n, t) ≃ N 2 log

• C (M − n)2

M(Nt)1/2

• .

❋♦r t❤❡ ♣❡rs✐st❡♥❝❡ ♦❢ ❞♦♠❛✐♥ ✇❛❧❧ ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥ t❤❡ ❛s②♠♣t♦t✐❝ ✇❡ ❤❛✈❡✿ log F(θ g ≈ 0, n, t) ≃ N 2 log

• B N 3/2

Mt1/2

• +2N(N−n) log
• D M − n

2N − n

• .

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

ˆ HXXZ = −1 2

M

• k=0
• σ−

k+1σ+ k + σ+ k+1σ− k + ∆

2 (σz

k+1σz k − 1) + hσz k

• ,

∆ → 0 ∆ → −∞ ■③✐♥❣ ❧✐♠✐t✿ lim

∆→ −∞

1 ∆ ˆ HXXZ = ˆ HIZ ≡ −1 4

M

• k=0

(σz

k+1σz k − 1) .

❙tr♦♥❣ ❛♥✐s♦tr♦♣② ❧✐♠✐t✿ ˆ HSA = −1 2

M

• k=0

P (σ−

k+1σ+ k +σ+ k+1σ− k +h σz k) P ,

P ≡

M

• k=0

(I−ˆ qk+1ˆ qk) , ✇❤❡r❡ ˆ qk ≡

1 2 (I − σz k)✱ ❛♥❞ [ ˆ

HSA, ˆ HIZ] = 0✳ ❚❤❡ ♥❡❛r❡st ♥❡✐❣❤❜♦✉rs ✇✐t❤ s♣✐♥s ✧❞♦✇♥✧❛r❡ ♥♦t ❛❧❧♦✇❡❞✳

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈

XX0 ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❘❛♥❞♦♠ ✇❛❧❦s ♦♥ ❛ ♣❧❛♥❡ ✇✐t❤ ❛ ✇❛❧❧ ❙❝❛❧❛r ♣r♦❞✉❝ts ❈♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s

❚❤❡ st❛t❡ ✈❡❝t♦r✿ |ΨN(✉) =

• λ⊆{(M−2(N−1))N}

S

λ(✉2)

N

• k=1

σ−

• µk
• |⇑ .

❚❤❡ s✉♠♠❛t✐♦♥ ✐s ♦✈❡r ❛❧❧ ♣❛rt✐t✐♦♥s

• λ =

µ − 2δN (M + 2(1 − N) ≥ λ1 ≥ λ2 ≥ · · · ≥ λN ≥ 0)✱ ❛♥❞

• µi >

µi+1 + 1✳ ❋♦✉r ✈❡rt❡① ♠♦❞❡❧

◆✳▼✳ ❇♦❣♦❧✐✉❜♦✈