21 Symmetry Breaking in the Ising Model
145
The magnetization is obtained by taking the derivative of Z N with respect to βh
and one gets in the thermodynamical limit
< σ k > h =
e
β J sinh(βh)
[e 2β J sinh
2
(βh) + e −2β J ] 1/2 .
(21.10)
Now, for any non-zero temperature (i.e. β < ∞), the limit h → 0 vanishes independently of the direction of h.
By the same trick, one may prove that all correlation functions have a limit independent of the direction along which h → 0 and therefore by Bogoliubov criterion,
there is only one phase and no symmetry breaking.
On the other hand, for T = 0 (i.e. β → ∞), one has
< σ k >
T =0
h
= h/|h|.
(21.11)
Thus, the limit h → 0
± depends on the direction of h and there are two possible
values of the magnetization, corresponding to two different phases. In each phase
there is symmetry breaking.
5. Mean Field Approximation
Finally, it is worthwhile to check how the mean field approximation, which is related
to the Goldstone criterion, compares with the exact solution.
The approximation is defined by expanding the spin configurations on the lattice
around a mean magnetization < σ >, to be determined at the end self-consistently,
120
and by keeping only the lowest order terms. This leads to the following finite volume
Hamiltonian
H
mean
V
= −2J
i∈V
σ i < σ > −h
i∈V
σ i = −(2J < σ > +h)
i∈V
σ i ,
(21.12)
where the factor 2 accounts for the number of nearest neighbours for each site.
The corresponding partition function is the same as that of a non-interacting chain
of spins in the presence of an (effective) external field h e f f = h + 2J < σ > and it
is easily computed as
Z N = 2
N
(cosh βh e f f )
N
.
120 This approximation is at the basis of the Curie–Weiss theory of magnetic phase transitions, also
called molecular field approximation; see, e.g. H. E. Stanley, Introduction to Phase Transitions and
Critical Phenomena, Oxford Univ. Press 1974, Chap. 16; C. J. Thompson, Mathematical Statistical
Mechanics, Princeton Univ. Press 1972, Sect. 4.5.
145
The magnetization is obtained by taking the derivative of Z N with respect to βh
and one gets in the thermodynamical limit
< σ k > h =
e
β J sinh(βh)
[e 2β J sinh
2
(βh) + e −2β J ] 1/2 .
(21.10)
Now, for any non-zero temperature (i.e. β < ∞), the limit h → 0 vanishes independently of the direction of h.
By the same trick, one may prove that all correlation functions have a limit independent of the direction along which h → 0 and therefore by Bogoliubov criterion,
there is only one phase and no symmetry breaking.
On the other hand, for T = 0 (i.e. β → ∞), one has
< σ k >
T =0
h
= h/|h|.
(21.11)
Thus, the limit h → 0
± depends on the direction of h and there are two possible
values of the magnetization, corresponding to two different phases. In each phase
there is symmetry breaking.
5. Mean Field Approximation
Finally, it is worthwhile to check how the mean field approximation, which is related
to the Goldstone criterion, compares with the exact solution.
The approximation is defined by expanding the spin configurations on the lattice
around a mean magnetization < σ >, to be determined at the end self-consistently,
120
and by keeping only the lowest order terms. This leads to the following finite volume
Hamiltonian
H
mean
V
= −2J
i∈V
σ i < σ > −h
i∈V
σ i = −(2J < σ > +h)
i∈V
σ i ,
(21.12)
where the factor 2 accounts for the number of nearest neighbours for each site.
The corresponding partition function is the same as that of a non-interacting chain
of spins in the presence of an (effective) external field h e f f = h + 2J < σ > and it
is easily computed as
Z N = 2
N
(cosh βh e f f )
N
.
120 This approximation is at the basis of the Curie–Weiss theory of magnetic phase transitions, also
called molecular field approximation; see, e.g. H. E. Stanley, Introduction to Phase Transitions and
Critical Phenomena, Oxford Univ. Press 1974, Chap. 16; C. J. Thompson, Mathematical Statistical
Mechanics, Princeton Univ. Press 1972, Sect. 4.5.