As the spatial dimensionality $d$ decreases, fluctuations become larger and the stability of the low-temperature ordered state deteriorates. The dimensionality where long-range order disappears is known as lower critical dimension. For instance, the Ising model in one dimension does not display long-range order at finite temperatures, however in two dimensions Peierls argument explains why the same model has an ordered phase below a certain critical temperature. If the basic variables and symmetries are continuous as in the $XY$ and Heisenberg models, the (long-range) ordered state at any finite temperature disappears already in two dimensions. This is the result of Mermin-Wagner's theorem. The $XY$ model nevertheless undergoes an unusual phase transition without an onset of long-range order in two dimensions, which is known as the Kosterlitz-Thouless transition. Gauge or local symmetries cannot spontaneously be broken as elucidated by Elitzur's theorem when applied to lattice gauge theories.
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