140
21 Symmetry Breaking in the Ising Model
where, for the spin 1/2 case, s i ≡ σ i /2 denotes the z− component of the spin at the
i-th lattice site. The inversion of the spins γ(σ i ) = −σ i is an internal symmetry and
we shall see that it is spontaneously broken at zero temperature.
117
The model can also be used to describe a lattice gas, with the choice n i ≡
(1 + σ i )/2 = 1 if the i-th site is occupied by a “molecule”, and n i = 0 otherwise.
The Hamiltonian models an interaction between the “molecules” by a square well
potential of the form U (r ) = ∞ for r < a ≡ the lattice spacing, U (r ) = −U for
a < r < 2a, and U (r ) = 0 for r > 2a. In this interpretation J is related to U and
h = 2μ + d, where μ is the chemical potential and d is the number of nearest
neighbours per site; the fluid phase would correspond to < n i >= 1 and the gas
to < n i >= 0.
The calculation of the spin correlation functions at non-zero temperature T = 1/β,
in the thermodynamical limit, is a very simple but instructive example of how the
general wisdom of statistical mechanics works in concrete examples and as such can
be regarded as a prototype of the functional integral approach to quantum field theory
models. We shall discuss the model in the case of a one-dimensional lattice, with
the purpose of illustrating the various strategies of constructive symmetry breaking,
discussed in the previous section.
1. Free Boundary Conditions
We consider the case h = 0 with free boundary conditions (i.e. no boundary condition) in finite volume, i.e. for N sites.
The partition function is
Z N =
σ 1 =±1
. . .
σ N =±1
e
β J
N −1
i=1 σ i σ i+1
(21.2)
and can be easily computed by noting that, since σ i takes only the values ±1 and
cosh is an even function,
σ N =±1
e
β J σ N −1 σ N = 2 cosh(β J σ N −1 ) = 2 cosh β J.
Thus, a recursive application of the argument gives
Z N = 2
N
(cosh β J )
N −1
.
117 For the basic elements of statistical mechanics, see, e.g. K. Huang, Statistical Mechanics, Wiley
1987; a brief account is given in the following section.
21 Symmetry Breaking in the Ising Model
where, for the spin 1/2 case, s i ≡ σ i /2 denotes the z− component of the spin at the
i-th lattice site. The inversion of the spins γ(σ i ) = −σ i is an internal symmetry and
we shall see that it is spontaneously broken at zero temperature.
117
The model can also be used to describe a lattice gas, with the choice n i ≡
(1 + σ i )/2 = 1 if the i-th site is occupied by a “molecule”, and n i = 0 otherwise.
The Hamiltonian models an interaction between the “molecules” by a square well
potential of the form U (r ) = ∞ for r < a ≡ the lattice spacing, U (r ) = −U for
a < r < 2a, and U (r ) = 0 for r > 2a. In this interpretation J is related to U and
h = 2μ + d, where μ is the chemical potential and d is the number of nearest
neighbours per site; the fluid phase would correspond to < n i >= 1 and the gas
to < n i >= 0.
The calculation of the spin correlation functions at non-zero temperature T = 1/β,
in the thermodynamical limit, is a very simple but instructive example of how the
general wisdom of statistical mechanics works in concrete examples and as such can
be regarded as a prototype of the functional integral approach to quantum field theory
models. We shall discuss the model in the case of a one-dimensional lattice, with
the purpose of illustrating the various strategies of constructive symmetry breaking,
discussed in the previous section.
1. Free Boundary Conditions
We consider the case h = 0 with free boundary conditions (i.e. no boundary condition) in finite volume, i.e. for N sites.
The partition function is
Z N =
σ 1 =±1
. . .
σ N =±1
e
β J
N −1
i=1 σ i σ i+1
(21.2)
and can be easily computed by noting that, since σ i takes only the values ±1 and
cosh is an even function,
σ N =±1
e
β J σ N −1 σ N = 2 cosh(β J σ N −1 ) = 2 cosh β J.
Thus, a recursive application of the argument gives
Z N = 2
N
(cosh β J )
N −1
.
117 For the basic elements of statistical mechanics, see, e.g. K. Huang, Statistical Mechanics, Wiley
1987; a brief account is given in the following section.