Student Seminar: Classical and Quantum Integrable Systems

i.e. the
L!
(L−1)!1!
= L allowed values of the pseudo-momenta are
p = 2πk with k = 0, · · · , L − 1 .
L
Further, we have the eigenvalue equation
H|ψ〉 = − JA L∑
]
e
[S ipm n + Sn+1 − + Sn − S n+1 + + 2S 3
2
nSn+1
3 |m〉 = E(p)|ψ〉 .
m,n=1
To work out the l.h.s. we have to use the formulae
as well as
S + n S − n+1|m〉 = δ nm |m + 1〉 , S − n S + n+1|m〉 = δ n+1,m |m − 1〉
2SnS 3 n+1|m〉 3 = 1 |m〉 , for m ≠ n, n + 1 ,
2
2SnS 3 n+1|m〉 3 = − 1 |m〉 , for m = n, or m = n + 1 .
2
Taking this into account we obtain
H|ψ〉 = − JA [ ∑ L (
)
e ipn |n + 1〉 + e ip(n+1) |n〉 + 1 2
2
− 1 2
n=1
L∑
e ipn |n〉 − 1 2
n=1
L∑
n=1
]
e ip(n+1) |n + 1〉 .
Using periodicity conditions we finally get
H|ψ〉 = − JA L∑ (
e ip(n−1) + e ip(n+1) + L − 4
2
2
n=1
From here we read off the eigenvalue
L∑ ( ∑ L
m=1
)
e ipn |n〉 = − J (
2
E − E 0 = J(1 − cos p) = 2J sin 2 p 2 ,
n=1
n≠m,m−1
)
e ipm |m〉
e −ip + e ip + L − 4
2
)
|ψ〉 .
where E 0 = − JL . Excitation of the spin chain around the pseudo-vacuum |F 〉
4
carrying the pseudo-momentum p is called a magnon 12 . Thus, magnon can be viewed
12 The concept of a magnon was introduced in 1930 by Felix Bloch in order to explain the reduction
of the spontaneous magnetization in a ferromagnet. At absolute zero temperature, a ferromagnet
reaches the state of lowest energy, in which all of the atomic spins (and hence magnetic moments)
point in the same direction. As the temperature increases, more and more spins deviate randomly
from the common direction, thus increasing the internal energy and reducing the net magnetization.
If one views the perfectly magnetized state at zero temperature as the vacuum state of the
ferromagnet, the low-temperature state with a few spins out of alignment can be viewed as a gas
of quasiparticles, in this case magnons. Each magnon reduces the total spin along the direction of
magnetization by one unit of and the magnetization itself by , where g is the gyromagnetic ratio.
The quantitative theory of quantized spin waves, or magnons, was developed further by Ted Holstein
and Henry Primakoff (1940) and Freeman Dyson (1956). By using the formalism of second
quantization they showed that the magnons behave as weakly interacting quasiparticles obeying
the Bose-Einstein statistics (the bosons).
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