Student Seminar: Classical and Quantum Integrable Systems

Thus, the transfer matrix is also polynomial of degree L:
∑L−2
τ(λ) = tr a T a (λ) = A(λ) + D(λ) = 2λ L + Q j λ j .
Note that the subleading term of order λ L−1 is absent because Pauli matrices are
traceless. The coefficients Q j mutually commute
[Q i , Q j ] = 0 .
j=0
Hamiltonian and Momentum. It remains to find the Hamiltonian among the commuting
family generated by the transfer matrix. The L-operator has two special
points on the spectral parameter plane.
• λ = i 2 , where L i,a(i/2) = iP ia .
• λ = ∞. We see that
1 (λ)
ResT
i λ = ∑ L L
n=1
S α ⊗ σ α = S α
}{{}
su(2)
⊗ σ α .
This point will be related to the realization of the global su(2) symmetry of
the model.
Let us investigate the first point. We have
T a (i/2) = i L P L,a P L−1,a · · · P 1,a = i L P L−1,L P L−2,L · · · P 1,L P L,a =
= i L P L−2,L−1 P L−3,L−1 · · · P 1,L−1 P L−1,L P L,a = · · · = i L P 12 P 23 · · · P L−1,L P L,a .
Thus, we have managed to isolate a single permutation carrying the index of the
auxiliary subspace. Taking the trace and recalling that tr a P L,a = I L we obtain the
transfer matrix
τ(i/2) = i L P 12 P 23 · · · P L−1,L = U ← shift operator
Operator U is unitary U † U = UU † = I and it generates a shift along the chain:
U −1 X n U = X n−1 .
By definition an operator of the infinitezimal shift is the momentum and on the
lattice it is introduced as
U = e ip .
Now we differentiate the logarithm of the transfer matrix
dT a (λ)
dλ
| λ=i/2 = i ∑ L−1 n
P L,a · · · P
}{{} n,a · · · P 1,a = i ∑ L−1 n
absent
P 12 P 23 · · · P n−1,n+1 · · · P L−1,L .
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