18
F. Finkel et al.
in terms of the largest eigenvalue 3 λ(x) of the transition matrix A(ε(x)) as
f (μ, T ) = −
1
β
1
0
log λ(x) dx .
(34)
We shall outline the derivation of this formula in the simplest nontrivial case m =
n = 1 in the next section.
7.1 Example: The su(1|1) Case
Although the free energy per site of the chain (24) can in principle be computed
from Eq. (34) for arbitrary values of m and n, this is almost impossible in practice
unless m and n are small enough. We shall only present here a brief description of
the simplest case m = n = 1, in which Eq. (34) yields a simple explicit expression
for the free energy per site. In this case we have only one chemical potential μ 1 ≡ μ
(for the bosonic degree of freedom), and the transfer matrix is simply
A(ε) =
q −μ q −μ/2
q ε−μ/2 q ε
.
The matrices A (i) = A(ε(x i )) are easily diagonalized:
A
(i)
= P i D
(i) P
−1
i , D
(i)
=
λ i 0
0 0
, P i =
q
−ε i −
μ
2
1
1
−q
−
μ
2
,
with ε i ≡ ε(x i ) and λ i = q ε i + q −μ . We can thus write
A
(0) A
(1) A
(2)
· · · A
(N −1)
= A
(0) P 1 D
(1) P
−1
1 P 2 D
(2) P
−1
2 · · · P N −1 D
(N −1) P
−1
N −1 .
Let us take, as an example, the PF spin chain. In this case
lim
N →∞
(ε i+1 − ε i ) = K lim
N →∞
1
N
(i + 1 − i) = 0, P
−1
i P i+1 →
N →∞
1 0
0 1
.
Thus when N → ∞ we have
A
(0) A
(1) A
(2)
· · · A
(N −1)
A
(0) P 1
N −1
i=1 λ i 0
0
0
P
−1
N −1 ,
3 Since all the matrix elements of the matrix A(ε) are positive, it follows from the Perron–Frobenius
theorem that its largest eigenvalue in module is positive and simple.