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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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Let me first begin by introducing myself. My title is Boyd Butts although it is not the name on my beginning certificate. For many years he's been residing in North Dakota and his family loves it. To gather cash is what his family and him appreciate. Managing people has been his working day job for a while.
Feel free to visit my site - http://Chatbook.biz