Student Seminar: Classical and Quantum Integrable Systems

To write down the fundamental commutation relations we have used the shorthand
notations A λ ≡ A(λ) and α = λ − µ. The relevant commutation relations are
[B(λ), B(µ)] = 0 ,
A(λ)B(µ) = λ − µ − i
λ − µ B(µ)A(λ) + i B(λ)A(µ) , (5.5)
λ − µ
D(λ)B(µ) = λ − µ + i
λ − µ B(µ)D(λ) − i
λ − µ B(λ)D(µ) .
The main idea of the algebraic Bethe ansatz is that there exists a pseudo-vacuum
|0〉 such that C(λ)|0〉 = 0 and the eigenvectors of τ(λ) with M spins down have the
form
|λ 1 , λ 2 , · · · , λ M 〉 = B(λ 1 )B(λ 2 ) · · · B(λ M )|0〉 ,
where {λ i } are “Bethe roots” which we will compare later on with the pseudomomenta
p i of the magnons in the coordinate Bethe ansatz approach. One can see
that the pseudo-vacuum can be identified with the state
Indeed, since we have
we find that
L n (λ)| ↑ n 〉 =
T (λ)|0〉 =
|0〉 = ⊗ L n=1| ↑ n 〉 .
( (λ +
i
)| ↑ )
2 n〉 i| ↓ n 〉
0 (λ − i )| ↑ 2 n〉
( )
(λ +
i
2 )L |0〉 ∗
0 (λ − i ,
2 )L |0〉
where ∗ stands for irrelevant terms. Thus, we indeed have
(
C(λ)|0〉 = 0 , A(λ)|0〉 = λ + i ) L|0〉 (
, D(λ)|0〉 = λ − i L|0〉
.
2
2)
Comparing with the coordinate Bethe ansatz we see that |0〉 ≡ |F 〉. We also see that
|0〉 is an eigenstate of the transfer matrix. The algebraic Bethe ansatz states that
the other eigenstates are of the form
|λ 1 , λ 2 , · · · , λ M 〉 = B(λ 1 )B(λ 2 ) · · · B(λ M )|0〉
provided the Bethe roots {λ i } satisfy certain restrictions.
restrictions.
We compute
A(λ)B(λ 1 )B(λ 2 ) · · · B(λ M )|0〉 =
Let us now find these
(
λ + i ) L ( ∏ M
λ − λ n − i
)
B(λ 1 )B(λ 2 ) · · · B(λ M )|0〉
2 λ − λ
n=1
n
M∑
M∏
+ Wn A (λ, {λ i })B(λ) B(λ j )|0〉 .
n=1
j=1
j≠n
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