= Berezinskii–Kosterlitz–Thouless transition
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The Berezinskii–Kosterlitz–Thouless transition in the two-dimensional <XY model> separates a low-temperature phase with algebraically decaying correlations from a high-temperature phase with unbound vortices and exponentially decaying correlations. It does not produce conventional long-range order.
How can something resist a twist when its orientation never settles down? The distinction is between <long-range order>, a nonzero limiting <correlation function>, and <phase stiffness>, a <free energy> cost for a slow spatial twist. In a uniform two-dimensional <XY model> at positive <temperature>, smooth <spin waves> can erase the first while leaving the second finite. <Vortex-antivortex binding> then decides whether that resistance survives on arbitrarily large scales.
<Vadim Lvovich Berezinskii> developed an earlier description of two-dimensional order. <John Michael Kosterlitz> and <D. J. Thouless> independently developed the defect mechanism; Kosterlitz's 2016 lecture explicitly acknowledges their initial ignorance of Berezinskii's work. The central step was to include rare <vortices> alongside smooth <spin waves>.<Topological Defects and Phase Transitions by John Michael Kosterlitz>[2016 Nobel lecture, pp. 89–94, sections 2–3; independent Berezinskii work, winding sectors and vortex renormalization]{ref}
The <Mermin-Wagner theorem> concerns <continuous symmetry> and sufficiently short-range interactions in the <thermodynamic limit>. It does not say that every two-dimensional system is disordered in every sense, nor that every <continuous symmetry> gives a <Berezinskii–Kosterlitz–Thouless transition>. Here the <order parameter> has a circular phase, permitting integer <winding numbers>. <Finite-scale vortex renormalization> explains why a finite sample can look coherent even when its infinite counterpart lacks <long-range order>.
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