One reason for the lack of global symmetry breaking is, that one can easily excite long wavelength fluctuations which destroy perfect order. "Easily excited" means, that the energy for those fluctuations tend to zero for large enough systems. Let's consider a magnetic model (e.g. the XY-model in one dimension). It is a chain of magnetic moments of length L . We consider harmonic approximation, where the forces (torque) between neighbouring moments increase linearly with the angle of twisting γ γ i . This implies, that the energy due to twisting increases quadratically E i ∝ ∝ γ γ i 2 . The total energy is the sum of all twisted pairs of magnetic moments E g e s ∝ ∝ ∑ ∑ i γ γ i 2 . If one considers the excited mode with the lowest energy in one dimension (see figure), then the moments on the chain of length L are tilted by π π along the chain. The relative angle between neighbouring moments is the same for all pairs of moments in this mode and equals γ γ i = π π / N , if the chain consists of N magnetic moments. It follows that the total energy of this lowest mode is E g e s ∝ ∝ N ⋅ ⋅ γ γ i 2 = N π π 2 N 2 ∝ ∝ L π π 2 L 2 . It decreases with increasing system size ∝ ∝ 1 / L and tends to zero in the thermodynamic limit L → → ∞ ∞ , N → → ∞ ∞ , L / N = const. For arbitrary large systems follows, that the lowest modes do not cost any energy and will be thermally excited. Simultaneously, the long range order is destroyed on the chain. In two dimensions (or in a plane) the number of magnetic moments is proportional to the area of the plain N ∝ ∝ L 2 . The energy for the lowest excited mode is then E g e s ∝ ∝ N 2 ⋅ ⋅ γ γ i 2 ∝ ∝ L 2 π π 2 L 2 , which tends to a constant in the thermodynamic limit. Thus the modes will be excited at sufficiently large temperatures. In three dimensions, the number of magnetic moments is proportional to the volume V = L 3 and the energy of the lowest mode is E g e s ∝ ∝ N 3 ⋅ ⋅ γ γ i 2 ∝ ∝ L 3 π π 2 L 2 . It diverges with system size and will thus not be excited for large enough systems. Long range order is not affected by this mode and global symmetry breaking is allowed.