Student Seminar: Classical and Quantum Integrable Systems

ecomes a differential equation for g and d:
g −1 ∂ x g + i∂ x dσ 3 + i λ g−1 Sg = 0 .
We project this equation on the Pauli matrices and get
∂ x v = − i λ (S − + 2vS 3 − S + v 2 )
∂ x d = 1
2λ (−2S 3 + vS + + ¯vS − ) .
The first of these equations is a Riccati equation for v(x).
functions v(x) and d(x) as
Expanding in λ the
∂ x d = − s ∞
λ + ∑
ρ n (x)λ n
v(x) =
n=0
∞∑
v n (x)λ n ,
n=0
v 0 = S 3 − s
S +
,
we rewrite the Riccati equation in the form
n∑
2isv n+1 = −v n ′ + iS + v n+1−m v m
m=1
and
ρ n = 1 2 (v n+1S + + ¯v n+1 S − ) .
Note that v(x) is regular at λ = 0. Equations above recursively determine the
functions v n (x) and ρ n (x) as local functions of the dynamical variables S i (x). This
describes the asymptotic behavior of T(λ) around λ = 0. The asymptotic series
become convergent if we regularize the model by discretizing the space interval!
Concerning the monodromy matrix T(λ), since g(x) is local and if we assume periodic
boundary conditions, we can write
where M(λ) = g(0)σ 3 g(0) −1 and
T(λ) = cos p(λ)I + i sin p(λ)M(λ) ,
p(λ) =
∫ 2π
0
dx ∂ x d .
The trace of the monodromy matrix, called the transfer matrix, is
trT(λ) = 2 cos p(λ) .
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