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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Corner transfer matrix ： ウィキペディア英語版 Corner transfer matrix In statistical mechanics, the corner transfer matrix describes the effect of adding a quadrant to a lattice. Introduced by Rodney Baxter in 1968 as an extension of the Kramers-Wannier row-to-row transfer matrix, it provides a powerful method of studying lattice models. Calculations with corner transfer matrices led Baxter to the exact solution of the hard hexagon model in 1980. ==Definition== Consider an IRF (interaction-round-a-face) model, i.e. a square lattice model with a spin σ''i'' assigned to each site ''i'' and interactions limited to spins around a common face. Let the total energy be given by :$E=\backslash sum\_\backslash epsilon\backslash left(\backslash sigma\_,\backslash sigma\_,\backslash sigma\_,\backslash sigma\_\backslash right),$ where for each face the surrounding sites ''i'', ''j'', ''k'' and ''l'' are arranged as follows: For a lattice with ''N'' sites, the partition function is :$Z\_=\backslash sum\_\backslash prod\_w\backslash left(\backslash sigma\_,\backslash sigma\_,\backslash sigma\_,\backslash sigma\_\backslash right),$ where the sum is over all possible spin configurations and ''w'' is the Boltzmann weight :$w\backslash left(\backslash sigma\_,\backslash sigma\_,\backslash sigma\_,\backslash sigma\_\backslash right)=\backslash exp\backslash left(-\backslash epsilon\backslash left(\backslash sigma\_,\backslash sigma\_,\backslash sigma\_,\backslash sigma\_\backslash right)/k\_T\backslash right).$ To simplify the notation, we use a ferromagnetic Ising-type lattice where each spin has the value +1 or −1, and the ground state is given by all spins up (i.e. the total energy is minimised when all spins on the lattice have the value +1). We also assume the lattice has 4-fold rotational symmetry (up to boundary conditions) and is reflection-invariant. These simplifying assumptions are not crucial, and extending the definition to the general case is relatively straightforward. Now consider the lattice quadrant shown below: The outer boundary sites, marked by triangles, are assigned their ground state spins (+1 in this case). The sites marked by open circles form the inner boundaries of the quadrant; their associated spin sets are labelled and , where σ1 = σ'1. There are 2''m'' possible configurations for each inner boundary, so we define a 2''m''×2''m'' matrix entry-wise by :$A\_=\backslash delta\backslash left(\backslash sigma\_,\backslash sigma\_\text{'}\backslash right)\backslash sum\_\backslash prod\_w\backslash left(\backslash sigma\_,\backslash sigma\_,\backslash sigma\_,\backslash sigma\_\backslash right).$ The matrix ''A'', then, is the corner transfer matrix for the given lattice quadrant. Since the outer boundary spins are fixed and the sum is over all interior spins, each entry of ''A'' is a function of the inner boundary spins. The Kronecker delta in the expression ensures that σ1 = σ'1, so by ordering the configurations appropriately we may cast ''A'' as a block diagonal matrix: :$\backslash begin$ & & \begin \sigma_'=+1 & & & & \sigma_'=-1\end\\ A & = & \left(& & & |\\ & A_ & & | & & 0\\ & & & |\\ - & - & - & | & - & - & -\\ & & & |\\ & 0 & & | & & A_\\ & & & |\end\right ) & \begin \sigma_=+1\\ \\\\\\\sigma_=-1\end\end Corner transfer matrices are related to the partition function in a simple way. In our simplified example, we construct the full lattice from four rotated copies of the lattice quadrant, where the inner boundary spin sets σ, σ', σ" and σ'" are allowed to differ: The partition function is then written in terms of the corner transfer matrix ''A'' as :$Z\_=\backslash sum\_A\_A\_A\_A\_=\backslash textrmA^.$ 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「Corner transfer matrix」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.