144
21 Symmetry Breaking in the Ising Model
largest eigenvalue λ + is non-degenerate.
119 Hence, if |λ + > denotes the eigenstate
with the highest eigenvalue and P the corresponding projection, one has for large
N , if < σ B |λ + > = 0, (
√
2|λ + >= |+ > +|− >)
Z N ∼ λ
N −1
+
< σ B |P|σ B > .
To compute the (average) magnetization < σ >, we consider a spin chain of 2N + 1
sites, centred at the origin; then, for large N ,
< σ 0 > 2N +1 = Z
−1
2N +1
σ 0 =±
< σ B |T
N
|σ 0 > σ 0 < σ 0 |T
N
|σ B >∼
∼< σ B |P|σ B >
−1
< σ B |P τ 3 P|σ B >= 0,
where we have used that
lim
N →∞
T
N
/λ
N
+ = P,
and that, in terms of the spin Pauli matrices τ i , one has
P = (1 + τ 1 )/2,
σ 0
|σ 0 > σ 0 < σ 0 | = τ 3 , Pτ 3 P = 0.
On the other hand, for β → ∞, T is no longer strictly positive, actually
T = e
β J 1, λ + = λ − ≡ λ, Z N = λ
N −1
and
< σ 0 > 2N +1 =< σ B |τ 3 |σ B >= ±1, if σ B = ±1.
Thus, at zero temperature the magnetization equals the spin value at the boundary.
By the same technique, one may compute, e.g. the two- point function and check
that the cluster property is satisfied. In conclusion, with Ruelle’s strategy one gets
pure phases and symmetry breaking at zero temperature.
4. Bogoliubov Strategy
It is not difficult to check the Bogoliubov strategy in this model, by working with a
non-zero magnetic field. In this case, the thermodynamical limit is independent of
the boundary conditions and the computation is particularly simple if one uses cyclic
boundary conditions.
119 For the proof of this result, and its relevance in the functional integral approach to quantum
theories, see J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd
ed., Springer 1987, p. 51.
21 Symmetry Breaking in the Ising Model
largest eigenvalue λ + is non-degenerate.
119 Hence, if |λ + > denotes the eigenstate
with the highest eigenvalue and P the corresponding projection, one has for large
N , if < σ B |λ + > = 0, (
√
2|λ + >= |+ > +|− >)
Z N ∼ λ
N −1
+
< σ B |P|σ B > .
To compute the (average) magnetization < σ >, we consider a spin chain of 2N + 1
sites, centred at the origin; then, for large N ,
< σ 0 > 2N +1 = Z
−1
2N +1
σ 0 =±
< σ B |T
N
|σ 0 > σ 0 < σ 0 |T
N
|σ B >∼
∼< σ B |P|σ B >
−1
< σ B |P τ 3 P|σ B >= 0,
where we have used that
lim
N →∞
T
N
/λ
N
+ = P,
and that, in terms of the spin Pauli matrices τ i , one has
P = (1 + τ 1 )/2,
σ 0
|σ 0 > σ 0 < σ 0 | = τ 3 , Pτ 3 P = 0.
On the other hand, for β → ∞, T is no longer strictly positive, actually
T = e
β J 1, λ + = λ − ≡ λ, Z N = λ
N −1
and
< σ 0 > 2N +1 =< σ B |τ 3 |σ B >= ±1, if σ B = ±1.
Thus, at zero temperature the magnetization equals the spin value at the boundary.
By the same technique, one may compute, e.g. the two- point function and check
that the cluster property is satisfied. In conclusion, with Ruelle’s strategy one gets
pure phases and symmetry breaking at zero temperature.
4. Bogoliubov Strategy
It is not difficult to check the Bogoliubov strategy in this model, by working with a
non-zero magnetic field. In this case, the thermodynamical limit is independent of
the boundary conditions and the computation is particularly simple if one uses cyclic
boundary conditions.
119 For the proof of this result, and its relevance in the functional integral approach to quantum
theories, see J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd
ed., Springer 1987, p. 51.