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| The '''Kramers–Wannier duality''' is a [[symmetry]] in [[statistical physics]]. It relates the [[Thermodynamic free energy|free energy]] of a two-dimensional [[square-lattice Ising model]] at a low temperature to that of another Ising model at a high temperature. It was discovered by [[Hendrik Anthony Kramers|Hendrik Kramers]] and [[Gregory Wannier]] in 1941. With the aid of this duality Kramers and Wannier found the exact location of the [[critical point (thermodynamics)|critical point]] for the Ising model on the square lattice.
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| Similar dualities establish relations between free energies of other statistical models. For instance, in 3 dimensions the Ising model is dual to an Ising gauge model.
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| ==Intuitive idea==
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| The 2-dimensional Ising model exists on a lattice, which is a collection of squares in a chessboard pattern. With the finite lattice, the edges can be connected to form a torus. In theories of this kind, one constructs an [[Involution (mathematics)|involutive transform]]. For instance, [[Lars Onsager]] suggested that the [[Star-Triangle transformation]] could be used for the triangular lattice.<ref>Somendra M. Bhattacharjee, and Avinash Khare, ''Fifty Years of the Exact Solution of the Two-Dimensional Ising Model by Onsager (1995)'', arxiv:cond-mat/9511003</ref> Now the dual of the ''discrete'' torus is [[dual lattice|itself]]. Moreover, the dual of a highly disordered system (high temperature) is a well ordered system (low temperature). This is because the fourier transform takes a high [[Bandwidth (signal processing)|bandwidth]] signal (more [[standard deviation]]) to a low one (less standard deviation). So one has essentially the same theory with an inverse temperature.
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| When one raises the temperature in one theory, one lowers the temperature in the other. If there is only one [[phase transition]], it will be at the point at which they cross, at which the temperature is equal. Because the 2D Ising model goes from a disordered state to an ordered state, there is a near [[one-to-one mapping]] between the disordered and ordered phases.
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| The theory has been generalized, and is now blended with many other ideas. For instance, the square lattice is replaced by a circle,<ref>arXiv:cond-mat/9805301, '' Self-dual property of the Potts model in one dimension'', F. Y. Wu</ref> random lattice,<ref>arXiv:hep-lat/0110063, ''Dirac operator and Ising model on a compact 2D random lattice'', L.Bogacz, Z.Burda, J.Jurkiewicz, A.Krzywicki, C.Petersen, B.Petersson</ref> nonhomogenous torus,<ref>arXiv:hep-th/9703037, ''Duality of the 2D Nonhomogeneous Ising Model on the Torus'', A.I. Bugrij, V.N. Shadura</ref> triangular lattice,<ref>arXiv:cond-mat/0402420, ''Selfduality for coupled Potts models on the triangular lattice'', Jean-Francois Richard, Jesper Lykke Jacobsen, Marco Picco</ref> labyrinth,<ref>arXiv:solv-int/9902009, '' A critical Ising model on the Labyrinth'', M. Baake, U. Grimm, R. J. Baxter</ref> lattices with twisted boundaries,<ref>arXiv:hep-th/0209048, '' Duality and conformal twisted boundaries in the Ising model'', Uwe Grimm</ref> chiral potts model,<ref>arXiv:0905.1924, ''Duality and Symmetry in Chiral Potts Model'', Shi-shyr Roan</ref> and many others.
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| ==Derivation==
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| Define these variables.
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| The low temperature expansion for (K<sup>*</sup>,L<sup>*</sup>) is
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| :::<math> Z_N(K^*,L^*) = 2 e^{N(K^*+L^*)} \sum_{ P \subset \Lambda_D} (e^{-2L^*})^r(e^{-2K^*})^s </math>
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| which by using the transformation
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| :::<math> \tanh K = e^{-2L*}, \ \tanh L = e^{-2K*} </math>
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| gives
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| :::<math> Z_N(K^*,L^*) = 2(\tanh K \; \tanh L)^{-N/2} \sum_{P} v^r w^s </math>
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| :::<math> = 2(\sinh 2K \; \sinh 2L)^{-N/2} Z_N(K,L) </math>
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| where ''v = tanh K'' and '' w = tanh L''. This yields a relation with the high-temperature expansion. The relations can be written more symmetrically as
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| :::<math>\, \sinh 2K^* \sinh 2L = 1</math>
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| :::<math>\, \sinh 2L^* \sinh 2K = 1</math>
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| With the free energy per site in the [[thermodynamic limit]]
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| :::<math> f(K,L) = \lim_{N \rightarrow \infty} f_N(K,L) = -kT \lim_{N\rightarrow \infty} \frac{1}{N} \log Z_N(K,L) </math>
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| the Kramers–Wannier duality gives
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| :::<math> f(K^*,L^*) = f(K,L) + \frac{1}{2} kT \log(\sinh 2K \sinh 2L) </math>
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| In the isotropic case where ''K = L'', if there is a critical point at ''K = K<sub>c</sub>'' then there is another at ''K = K<sup>*</sup><sub>c</sub>''. Hence, in the case of there being a unique critical point, it would be located at ''K = K<sup>*</sup> = K<sup>*</sup><sub>c</sub>'', implying ''sinh 2K<sub>c</sub> = 1'', yielding ''kT<sub>c</sub> = 2.2692J''.
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| ==See also==
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| *[[Ising model]]
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| *[[S-duality]]
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| ==References==
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| <references/>
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| ==External links==
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| * {{cite journal | author=H. A. Kramers and G. H. Wannier | title=Statistics of the two-dimensional ferromagnet| journal = Physical Review | volume=60 | pages=252–262 | year=1941 | doi=10.1103/PhysRev.60.252|bibcode = 1941PhRv...60..252K }}
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| * {{cite journal | author=J. B. Kogut | title=An introduction to lattice gauge theory and spin systems| journal = Reviews of Modern Physics | volume=51 | pages=659–713 | year=1979 | doi=10.1103/RevModPhys.51.659 | bibcode=1979RvMP...51..659K}}
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| {{DEFAULTSORT:Kramers-Wannier duality}}
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| [[Category:Statistical mechanics]]
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| [[Category:Exactly solvable models]]
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| [[Category:Lattice models]]
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Hi there. Allow me start by introducing the author, her title is Myrtle Cleary. My working day job is a meter reader. One of the things he loves most is ice skating but he is struggling to find time for it. California is our beginning place.
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