翻訳と辞書 Words near each other ・ Heisei College of Health Sciences ・ Heisei College of Music ・ Heisei Fūzoku ・ Heisei International University ・ Heisei Ishingun ・ Heisei no Sato ・ Heisei Ozeki Dam ・ Heisei period ・ Heisei Station ・ Heisei Ultra Seven ・ Heisenberg ・ Heisenberg cut ・ Heisenberg group ・ Heisenberg limit ・ Heisenberg model ・ Heisenberg model (quantum) ・ Heisenberg picture ・ Heisenberg's entryway to matrix mechanics ・ Heisenberg's microscope ・ Heisenberg-Langevin equation ・ Heisenbug ・ Heiser ・ Heiser Ridge ・ Heiser, Rosenfeld, and Strauss Buildings ・ Heisey ・ Heisey Glass Company ・ Heisey House ・ Heishan ・ Heishan bandits ・ Heishan County Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Heisenberg model (quantum) ： ウィキペディア英語版 Heisenberg model (quantum) The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins of the magnetic systems are treated quantum mechanically. In the prototypical Ising model, defined on a ''d''-dimensional lattice, at each lattice site, a spin $\backslash sigma\_i\; \backslash in\; \backslash$ represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction. ==Overview== For quantum mechanical reasons (see exchange interaction or the subchapter "quantum-mechanical origin of magnetism" in the article on magnetism), the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are ''aligned''. Under this assumption (so that magnetic interactions only occur between adjacent dipoles) the Hamiltonian can be written in the form :$\backslash hat\; H\; =\; -J\; \backslash sum\_^\; \backslash sigma\_j\; \backslash sigma\_\; -\; h\; \backslash sum\_^\; \backslash sigma\_j$ where $J$ is the coupling constant for a 1-dimensional model consisting of ''N'' dipoles, represented by classical vectors (or "spins") σj, subject to the periodic boundary condition $\backslash sigma\_\; =\; \backslash sigma\_1$. The Heisenberg model is a more realistic model in that it treats the spins quantum-mechanically, by replacing the spin by a quantum operator (Pauli spin-1/2 matrices at spin 1/2), and the coupling constants $J\_x,\; J\_y,$ and $J\_z$. As such in 3-dimensions, the Hamiltonian is given by :$\backslash hat\; H\; =\; -\backslash frac\; \backslash sum\_^\; (J\_x\; \backslash sigma\_j^x\; \backslash sigma\_^x\; +\; J\_y\; \backslash sigma\_j^y\; \backslash sigma\_^y\; +\; J\_z\; \backslash sigma\_j^z\; \backslash sigma\_^z\; +\; h\backslash sigma\_j^)$ where the $h$ on the right-hand side indicates the external magnetic field, with periodic boundary conditions, and at spin $s=1/2$, the spin matrices are given by :$$ \sigma^x = \begin 0&1\\ 1&0 \end :$$ \sigma^y = \begin 0&-i\\ i&0 \end :$$ \sigma^z = \begin 1&0\\ 0&-1 \end The Hamiltonian then acts upon the tensor product $(\backslash mathbb^2)^$, of dimension $2^N$. The objective is to determine the spectrum of the Hamiltonian, from which the partition function can be calculated, from which the thermodynamics of the system can be studied. A simplified version of Heisenberg model is the one-dimensional Ising model, where the transverse magnetic field is in the x-direction, and the interaction is only in the z-direction: :$\backslash hat\; H\; =\; -J\_z\; \backslash sum\_^\; \backslash sigma\_j^z\; \backslash sigma\_^z\; -\; gJ\_z\; \backslash sum\_^\; \backslash sigma\_j^x$ At small g and large g, the ground state degeneracy is different, which implies that there must be a quantum phase transition in between. It can be solved exactly for the critical point using the duality analysis.〔https://www.kitp.ucsb.edu/sites/default/files/users/mpaf/publications/p109.pdf〕 The duality transition of the Pauli matrices is $\backslash sigma\_i^z\; =\; \backslash prod\_S^x\_j$ and $\backslash sigma\_i^x\; =\; S^z\_i\; S^z\_$, where $S^x$ and $S^z$ are also Pauli matrices which obey the Pauli matrix algebra. Under periodic boundary conditions, the transformed Hamiltonian can be shown is of a very similar form: :$\backslash hat\; H\; =\; -gJ\_z\; \backslash sum\_^\; S\_j^z\; S\_^z\; -\; gJ\_z\; \backslash sum\_^\; S\_j^x$ but for the $g$ attached to the spin interaction term. Assuming that there's only one critical point, we can conclude that the phase transition happens at $g=1$. The most widely known type of Heisenberg model is the Heisenberg XXZ model, which occurs in the case $J\; =\; J\_x\; =\; J\_y\; \backslash neq\; J\_z\; =\; \backslash Delta$. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz,〔http://www.iop.org/EJ/abstract/0305-4470/25/15/007〕 while other approaches do so without Bethe ansatz.〔http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332001000400008〕 The physics of the Heisenberg model strongly depends on the sign of the coupling constant $J$ and the dimension of the space. For positive $J$ the ground state is always ferromagnetic. At negative $J$ the ground state is antiferromagnetic in two and three dimensions, it is from this ground state that the Hubbard model is given.〔http://www.math.ucdavis.edu/~bxn/qs.html〕 In one dimension the nature of correlations in the antiferromagnetic Heisenberg model depends on the spin of the magnetic dipoles. If the spin is integer then only short-range order is present. A system of half-integer spins exhibits quasi-long range order. 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「Heisenberg model (quantum)」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.