A result showing that in a strictly two-dimensional material, heat at any temperature above absolute zero destroys the long-range order of atomic magnets that are free to point in any direction. Real exist because their atomic magnets prefer a particular direction, which breaks the theorem’s assumptions.
Going deeper
The theorem is about how cheap long-wavelength spin waves are. With a continuous symmetry there is no minimum energy to excite them, and in two dimensions there are enough such modes at any finite temperature to destroy long-range order. Anisotropy opens a gap, and the argument no longer applies.
The argument in one line
In a magnet whose can point in any direction, rotating all of them together costs nothing, so a very slow spatial rotation – a long-wavelength – costs almost nothing. The energy of such a mode goes to zero as its wavevector does. At any temperature above absolute zero these modes are thermally excited, and the question is whether the total disruption they cause is finite.
In two dimensions it is not. Counting modes gives a logarithmically divergent contribution, so the spins’ average direction wanders without limit as the system grows and long-range order is destroyed. Mermin and Wagner turned that argument into a rigorous proof: at any nonzero temperature, a one- or two-dimensional isotropic model with finite-range exchange can be neither ferromagnetic nor antiferromagnetic. The same method rules out other kinds of ordering that break a continuous symmetry in low dimensions.
What it does not say
It is a statement about a model, and every assumption is a way out. It requires a continuous symmetry: an Ising magnet, whose spins choose between up and down, has a discrete symmetry and is untouched – two-dimensional Ising models order perfectly well, as Onsager showed. It requires short-range interactions: dipolar coupling falls off slowly enough to escape. It requires the thermodynamic limit: a real is finite, and order that would decay over kilometres is order as far as any experiment is concerned.
It also does not forbid every kind of transition. The in a two-dimensional system with continuous symmetry is a true to a state with quasi-long-range order – correlations that decay as a power law rather than exponentially – which the theorem permits, because it is not long-range order. For years, the theorem was quoted loosely as proving that two-dimensional magnets cannot exist, which is not what it says.
Why 2D magnets exist anyway
Real layered magnets escape through . and the crystal field make some directions cheaper than others, so rotating all the spins is no longer free and the spin-wave spectrum acquires a gap. Modes below that gap cannot be thermally excited, the divergent integral is cut off, and order survives to a finite temperature. This is why CrI3 – strongly anisotropic, effectively Ising-like – remains ferromagnetic as a , while more isotropic materials lose their order as they are thinned.
That is also what makes the theorem practically useful rather than merely restrictive: it identifies anisotropy as the quantity that decides whether a 2D magnet exists and at what temperature. of anisotropic Heisenberg models reproduce this directly, converging to a finite critical temperature when anisotropy is present and losing their order as the simulated sheet grows when it is not.
For specialists
No spontaneous breaking of a continuous symmetry at finite temperature in 2D with short-range interactions; magnetic anisotropy evades it.