# Length of domain wall in Ising model

A subquestion of a homework for my statistical mechanics class this week asked of the 2d Domain wall Ising model approximation:

"Now argue that for the formation of a domain wall separating the system into two distinct regions we must have $$L ≥ √N$$ " where L is the length of the domain wall and N is the lattice side length.

I do not see where there cannot exist a smaller wall. Surely this smaller the length, the more stable it is?In my understanding from reading, between a domain wall is 2 magnetised regions with opposite values. The exchange interaction between the dipoles which creates the magnetisation is a force which tends to align nearby dipoles so they point in the same direction - and so forcing adjacent dipoles to point in different directions requires energy. Therefore, a domain wall requires extra energy, called the domain wall energy, which is proportional to the area of the wall. (source: wikipedia)

So my feeling is that the limit for the size of L must come from the energy but i cannot argue any further. help?

• You should look up "Peierls argument". The point is that a domain wall of length $\ell$ costs $\exp(-2\beta\ell)$, while the number of such walls (starting from a given point) only grows like $c^\ell$ for some constant $c$ (depending on the lattice). Therefore, once $\beta$ is sufficiently large, it becomes very unlikely to have a long wall. (Making this argument precise, one can actually show that, for large $\beta$, there are no walls of length larger than $K(\beta)\log N$ for some finite constant $K(\beta)$). Commented May 27, 2019 at 14:47
• That being said, I have no clue why they made you assume that $L\geq\sqrt{N}$, as such a condition seems entirely irrelevant. Commented May 27, 2019 at 14:54
• @YvanVelenik thanks for the tip! is c the lattice constant, or number of lattice points on the side? yea, i have no iclue about that either.. Commented May 27, 2019 at 15:55
• How exactly are they defining a "distinct region"? If I have a large sqare lattice of $N$ sites and I want to cut it in half then the boundry between the 2 halves will be at least as long as one side of my lattice, that is $\sqrt{N}$. By angling my cut or adding wiggles to it I can make the boundry longer. This gives me $L \ge \sqrt{N}$. This argument does, however require you to define the phrase "separating the system into two distinct regions" in an appropriate way Commented May 27, 2019 at 16:22
• d'oh..i was thinking it was an NxN lattice, i just completely misread! thank you! Commented May 27, 2019 at 16:37