21 Symmetry Breaking in the Ising Model
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The correlation functions can be computed with the trick of introducing sitedependent couplings, as before, and by taking derivatives of
Z N (J i ) =
N −1
i=1
λ
i
+ +
N −1
i=1
λ
i
− ,
since the T (J i ) are all simultaneously diagonalizable.
For β < ∞, the thermodynamical limit is dominated by the highest eigenvalue
λ + > λ − , for N large
Z N = λ
N −1
+
(1 + (λ − /λ + )
N −1
) ∼ λ
N −1
+ .
In this limit one gets the same results as for the case of free boundary conditions, as
expected.
The partition function can be easily computed also if one imposes cyclic boundary
conditions, by which the open line of the lattice is turned into a circle with the
identification σ N +1 ≡ σ 1 . Then, the Hamiltonian reads
H N = −J
N
i=1
σ i σ i+1 − h
N
i=1
σ i
(21.9)
and one has
Z N = Tr (T (1, 2) . . . T (N , N + 1)) =
σ 1
< σ 1 |T
N
|σ 1 >= λ + (h)
N
+ λ − (h)
N
.
Also in this case, for h = 0, one gets symmetric correlation functions and a violation
of the cluster property at T = 0.
3. Ruelle Strategy. Symmetry Breaking Boundary Conditions
According to the general discussion of the previous section, the pure phases can
be obtained by an appropriate choice of the boundary conditions, in this case by
symmetry breaking boundary conditions.
In fact, for boundary conditions σ 1 = σ N = σ B and for h = 0, one has (by (21.7))
Z N =< σ B |T
N −1
|σ B > .
We start with the case β < ∞ (non-zero temperature). In this case, the transfer
matrix T has strictly positive entries and by the Perron–Frobenius theorem, the