Peierls contour

A Peierls contour is a dual-lattice boundary separating regions of opposite Ising spins. For a nearest-neighbor ferromagnet with exchange $J>0$ and zero field, each disagreeing bond costs $2J$ relative to alignment. A contour crossing $m$ bonds therefore has energy cost $2Jm$.

Flipping the spins it encloses removes that contour and changes the relative Boltzmann factor by $e^{-2\beta Jm}$. In units $J=1$, this is the common $e^{-2\beta m}$ expression. Counting possible contours compares their configurational entropy with this energy penalty, giving a route to low-temperature ordering. The argument depends on geometry and boundary conditions; an open contour or a nonzero external field requires additional terms rather than the same formula without qualification.