= Domain wall
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A <domain wall> is a <topological defect> separating regions whose <order parameter> approaches different disconnected components of a <vacuum manifold>. In $d$ spatial dimensions, a flat <domain wall> extends in $d-1$ directions: a surface in three dimensions, a line in two, and a localized defect in one. A <scalar quartic domain wall> provides an explicit smooth interpolation between two degenerate minima.
For the one-dimensional, zero-field <Ising model> with nearest-neighbor coupling $J>0$, a reversed bond costs $2J$. A single <domain wall> can occupy order $N$ positions on a chain of $N$ sites, giving a free-energy balance $\Delta F\simeq2J-k_BT\log N$. At every positive <temperature>, positional <entropy> defeats a fixed wall cost as $N\to\infty$. Thermally excited <domain walls> then destroy <long-range order>, despite two ordered zero-temperature ground states. This argument assumes short-range interactions and finite wall cost; it does not apply unchanged to long-range interacting chains. In higher dimensions the wall cost grows with its <area>, allowing different ordering behavior.<Statistical Field Theory by David Tong/1 From Spins to Fields>[Sections 1.3.2–1.3.3, wall profile and energy, and the positional-entropy argument]{ref}
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