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The Kramers–Wannier duality is a symmetry in statistical physics. It relates the 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 Kramers and Gregory Wannier in 1941. With the aid of this duality Kramers and Wannier found the exact location of the critical point for the Ising model on the square lattice. 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. ==Intuitive idea== 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 involutive transform. For instance, Lars Onsager suggested that the Star-Triangle transformation could be used for the triangular lattice.〔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〕 Now the dual of the ''discrete'' torus is 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 (more standard deviation) to a low one (less standard deviation). So one has essentially the same theory with an inverse temperature. 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. The theory has been generalized, and is now blended with many other ideas. For instance, the square lattice is replaced by a circle,〔arXiv:cond-mat/9805301, '' Self-dual property of the Potts model in one dimension'', F. Y. Wu〕 random lattice,〔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〕 nonhomogeneous torus,〔arXiv:hep-th/9703037, ''Duality of the 2D Nonhomogeneous Ising Model on the Torus'', A.I. Bugrij, V.N. Shadura〕 triangular lattice,〔arXiv:cond-mat/0402420, ''Selfduality for coupled Potts models on the triangular lattice'', Jean-Francois Richard, Jesper Lykke Jacobsen, Marco Picco〕 labyrinth,〔arXiv:solv-int/9902009, '' A critical Ising model on the Labyrinth'', M. Baake, U. Grimm, R. J. Baxter〕 lattices with twisted boundaries,〔arXiv:hep-th/0209048, '' Duality and conformal twisted boundaries in the Ising model'', Uwe Grimm〕 chiral potts model,〔arXiv:0905.1924, ''Duality and Symmetry in Chiral Potts Model'', Shi-shyr Roan〕 and many others. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kramers–Wannier duality」の詳細全文を読む スポンサード リンク
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