Kosterlitz-Thouless transition: Difference between revisions

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:<math>\frac{\pi J}{k_BT_c}-1 \approx \pi \tilde{y}_c(0) \exp\left(\frac{-\pi^2J}{k_BT_c} \right) \approx 0.12</math>
:<math>\frac{\pi J}{k_BT_c}-1 \approx \pi \tilde{y}_c(0) \exp\left(\frac{-\pi^2J}{k_BT_c} \right) \approx 0.12</math>
 
==See also==
*[[Universality classes#XY | XY universality class]]
==References==
==References==
<references/>
<references/>

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The Kosterlitz-Thouless transition (also known as the Berezinskii-Kosterlitz-Thouless (BKT) phase transition)[1] [2] [3] [4] is a phase transition found in the two-dimensional XY model. Below the transition temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{KT}} , the system plays host to a 'liquid' of vortex-antivortex pairs that have zero total vorticity. Above Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{KT}} these pairs break up into a gas of independent vortices.

For the XY model the critical temperature is given by (Eq.4 in [4]):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c = \frac{\pi J}{k_B}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} is the spin-spin coupling constant. This can be obtained as (Eq.58 in [4]):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi J}{k_BT_c}-1 \approx \pi \tilde{y}_c(0) \exp\left(\frac{-\pi^2J}{k_BT_c} \right) \approx 0.12}

See also[edit]

References[edit]

  1. ↑ V. L. Berezinskii "Destruction of Long-range Order in One-dimensional and Two-dimensional Systems having a Continuous Symmetry Group I. Classical Systems", Journal of Experimental and Theoretical Physics 32 pp. 493 (1971)
  2. ↑ V. L. Berezinskii "Destruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems", Journal of Experimental and Theoretical Physics 34 pp. 610 (1972)
  3. ↑ J. M. Kosterlitz and D. J. Thouless "Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory)", Journal of Physics C: Solid State Physics 5 pp. L124-L126 (1972)
  4. ↑ 4.0 4.1 4.2 J. M. Kosterlitz and D. J. Thouless "Ordering, metastability and phase transitions in two-dimensional systems", Journal of Physics C: Solid State Physics 6 pp. 1181-1203 (1973)
Related reading
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