= Peierls transition
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A <Peierls transition>, named for <Rudolf Peierls>, is a <metal-insulator transition> caused by a <Peierls distortion> of an initially metallic chain.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book) by International Union of Pure and Applied Chemistry/Peierls transition>[Entry P04468: electronic transition caused by a structural distortion and opening a gap at the Fermi energy]{ref} A modulation at $2k_F$ mixes states near opposite <Fermi points> and can open a <band gap> at the <Fermi energy>. At half filling with one <atomic orbital> per <chemical atom>, this modulation doubles the original lattice period and alternates strong and weak bonds. In a weak-coupling zero-temperature description, the <Peierls energy gain> is proportional to $-(\delta^2/t)\ln(t/|\delta|)$ whereas the elastic cost is proportional to $\delta^2$, where $t>0$ is the undistorted hopping strength and $\delta$ its small alternating change. The logarithm permits a distortion at arbitrarily weak coupling in the ideal model. Finite <temperature>, finite size, interchain coupling and <thermal fluctuations> matter for a real transition; strictly one-dimensional reasoning alone does not establish finite-temperature long-range order.
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