This calculation works through an example of a system in which long-range order (in a sense made precise below) appears in spatial dimensions but not in dimensions :
Suppose that a real scalar field is defined on a -dimensional cube of side length , with boundary condition and an upper bound on the frequency components of the field, as would be the case if we were trying to approximate a scalar field defined on a lattice. Let the system be governed by a Hamiltonian given by
At nonzero temperatures, for points and close to the center of the cube, the thermal average of the squared field variation has behavior varying with dimension:
so that for dimensions greater than two, the field is roughly constant no matter how far away you look, whereas for dimension one, the thermal fluctuations destroy this long-range order. The case apparently gives logarithmic dependence on separation (see e.g. these lecture notes), but I don’t show it here.
We’ll demonstrate that this behavior exists in a few steps:
1. Mode decomposition of Hamiltonian
We can start by decomposing the scalar field into Fourier modes:
where is a -dimensional vector of integers ranging from 1 to some cutoff . In this basis, evaluating the Hamiltonian is simple:
The first equality resulted from the given Hamiltonian by integration by parts. The modes are non-interacting, which simplifies the following calculation.
2. Calculation of frequency-space correlation function
Consider the following thermal correlation function:
By the symmetry of the integrand, it’s clear that the correlation function vanishes unless . In this case:
so that overall we get
Since the factorization of the partition function is a direct consequence of the absence of mode-coupling terms in the Hamiltonian, we just had to do a single-mode calculation and then write this answer down.
3. Calculation of average squared disorder
Expanding the squared difference between field values at points and in terms of the Fourier modes, we get
Taking the expectation using the frequency-space correlation function:
Note that up to this point, everything has been exact (for the continuum theory defined in the problem statement, which itself could be used as an approximation of a lattice system).
4. Approximations for different values of
For very large and , close to on the scale of , the summand varies slowly in , so we can approximate the sum by an integral. Since for sufficiently large , which assuming a fixed frequency cutoff is achieved by taking large enough, the integrand is heavily suppressed, we can extend the upper limit to infinity without too much error and evaluate the resulting integral (I did it in Mathematica):
For the cutoff is significant, as the integrand grows with increasing (due to the factor of in the measure). The largest the numerator of the integrand can be is four, since it’s the square of a difference between products of sines. Fixing the integrand to its maximum value and integrating up to the cutoff frequency gives an upper bound independent of and .