pdf file

For the special case of the energy, we have
∑
ɛ λ (ϕ)e −βɛ λ(ϕ)
E β,λ [ɛ] =
states
Z(β, λ)
= − ∂ ln Z(β, λ)
∂β
It is thus an important goal to compute the partition function. To this end, we introduce
Boltzmann weights
We then get
R kl
ij (β, λ) = e −βɛkl ij (λ) .
e −βɛ λ(ϕ) = ∏
Consider the contribution of the first line to the partition function where we temporarily allow
different values for the leftmost and rightmost bond:
atoms
R kl
ij .
It equals
i 1
k 1 k 2 k 3 k N−1 k N
. . .
r 1 r 2 r 3 r N−1 i ′ 1
l
. . .
1 l 2 l 3 l N−1 l N
T i′ 1 l 1...l N
i 1 k 1 ...k N
= ∑
R r 1l 1
i 1 k 1
R r 2l 2
r 1 ...r N−1
r 1 k 2
. . . R i′ 1 l N
r N−1 k N
(12)
To eliminate indices, we introduce a complex vector space V freely generated on the set
{1, . . . , n} with basis {v 1 , . . . v n } and a family of endomorphisms
R = R(β, λ) : V ⊗ V → V ⊗ V
v i ⊗ v j ↦→ ∑ k,l Rkl ij v k ⊗ v l .
Definition (12) leads to the definition of the endomorphism T ∈ End (V ⊗ V N ) with
T = R 01 R 02 R 03 . . . R 0n .
Here we understand that the endomorphism R ij acts on the i-th and j-th copy of V in the
tensor product V ⊗ V N .
Periodic boundary conditions imply for the sum over the first line
Tr V (T ) l 1...l N
k 1 ...k N
.
This endomorphism is called the row-to-row transfer matrix. To sum over all M lines, we take
the matrix product and then the trace so that we find:
Z = tr V ⊗N (tr V (T )) M .
This raises the problem of understanding the eigenvalues of the endomorphism Tr V (T ) ∈
End(V ⊗N ): in the thermodynamic limit, we take M → ∞ so that Z ∼ κ M N with κ N the
eigenvalue with the largest modulus.
As usual in eigenvalue problems, we try to find as many endomorphisms of V ⊗N as possible
commuting with Tr V (T ) which allows us to solve the eigenproblem separately on eigenspaces
of these operators.
Definition 4.3.1
98