The Mermin-Wagner theorem states that a one- or two-dimensional isotropic Heisenberg spin model with finite-range exchange interaction can have neither ferromagnetic nor antiferromagnetic long-range order at any positive temperature.
An isotropic Heisenberg spin model places a fixed-size spin 
at every site 
of a 
-dimensional
point lattice. In a classical model, the spins are
vectors with the same vector
norm. In a quantum model, their components are linear
operators on a Hilbert space. The interaction
energy has the form
 |
where 
specifies the interaction strength between two sites. Finite
range means that 
whenever 
for some fixed 
.
The dot product makes 
invariant under simultaneous continuous rotations
of all spins. With the system size taken to infinity
before an applied symmetry-breaking field is sent to
zero, both the uniform and staggered spin expectation
values vanish (Mermin and Wagner 1966, Friedli and Velenik 2017).
After a Fourier transform, the proof gives a bound whose small-wave-vector part contains the integral
 |
This integral diverges at 
for 
, excluding the assumed order. The continuous rotational
symmetry and the short-range character of the interaction
are essential hypotheses. In particular, the theorem does not apply to the two-dimensional
Ising model, whose spin symmetry
is discrete.
