Student Seminar: Classical and Quantum Integrable Systems

This allows one to determine the ratio
B
A = −ei(p 1+p 2 ) + 1 − 2e ip 2
e i(p 1+p 2) + 1 − 2e ip 1 .
Problem. Show that for real values of momenta the ratio B A
is the pure phase:
B
A = eiθ(p 2,p 1 ) ≡ S(p 2 , p 1 ) .
This phase is called the S-matrix. We further note that it obeys the following relation
S(p 1 , p 2 )S(p 2 , p 1 ) = 1 .
Thus, the two-magnon Bethe ansatz takes the form
a(n 1 , n 2 ) = e i(p 1n 1 +p 2 n 2 ) + S(p 2 , p 1 )e i(p 2n 1 +p 1 n 2 ) ,
where we factored out the unessential normalization coefficient A.
Let us now substitute the Bethe ansatz in eq.(5.2). We get
(
) [ (
)
2(E − E 0 ) Ae i(p 1n 1 +p 2 n 2 ) + Be i(p 2n 1 +p 1 n 2 )
= J 4 Ae i(p 1n 1 +p 2 n 2 ) + Be i(p 2n 1 +p 1 n 2 )
−
)
)
−
(Ae i(p1n1+p2n2) e ip1 + Be i(p2n1+p1n2) e ip2 −
(Ae i(p1n1+p2n2) e −ip1 + Be i(p2n1+p1n2) e −ip2
)
)]
−
(Ae i(p1n1+p2n2) e ip2 + Be i(p2n1+p1n2) e ip1 −
(Ae i(p1n1+p2n2) e −ip2 + Be i(p2n1+p1n2) e −ip1 .
We see that the dependence on A and B cancel out completely and we get the
following equation for the energy
)
E − E 0 = J
(2 − cos p 1 − cos p 2 = 2J
2∑
k=1
sin 2 p k
2 .
Quite remarkably, the energy appears to be additive, i.e. the energy of a two-magnon
state appears to be equal to the sum of energies of one-magnon states! This shows
that magnons essentially behave themselves as free particles in the box.
Finally, we have to impose the periodicity condition a(n 2 , n 1 + L) = a(n 1 , n 2 ). This
results into
which implies
e i(p 1n 2 +p 2 n 1 ) e ip 2L + B A eip 1L e i(p 2n 2 +p 1 n 1 ) = e i(p 1n 1 +p 2 n 2 ) + B A ei(p 2n 1 +p 1 n 2 )
e ip 1L = A B = S(p 1, p 2 ) , e ip 2L = B A = S(p 2, p 1 ) .
– 64 –