models:spin_models
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models:spin_models [2018/05/05 10:50] jakobadmin created |
models:spin_models [2018/05/07 14:01] jakobadmin [Overview] |
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| ====== Spin Models ====== | ====== Spin Models ====== | ||
| - | <tabbox Intuitive> | ||
| - | <note tip> | ||
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | ||
| - | </note> | ||
| - | | ||
| - | <tabbox Concrete> | ||
| - | <note tip> | + | <tabbox Overview> |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | |
| - | </note> | + | |
| - | + | ||
| - | <tabbox Abstract> | + | |
| - | <note tip> | + | {{ :models:spinmodelsoverview.png?nolink&600|}} |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | |
| - | <tabbox Why is it interesting?> | + | Spin models can be classified depending on how many different states are possible at each lattice site and if the set of possible states is continuous or discrete. |
| - | <blockquote>"Empirically, one finds that all systems in nature belong to one of a comparatively small number of such universality classes. Two specific microscopic interaction Hamiltonians appear almost sufficient to encompass the universality classes necessary for static critical phenomena. The first of these is the Q-state Potts model (Potts, 1952; Wu, 1982). | + | * Models with a discrete set of states are called **Potts models**. |
| + | * Models with a continuous set of states are called **n-Vector models**. | ||
| - | One assumes that each spin i can be in one of Q possible discrete orientations $\chi_i$ $(\chi_i =,1,2,\ldots,Q)$. If two neighboring spins i and j are in the same orientation, then they contribute an amount $-J$ to the total energy of a configuration. If i and j are in different orientations, they contribute nothing. Thus the interaction Hamiltonian is | + | Within these subcategories we have different models depending on how many states there are in total. For Potts models, the number of orientations that are possible is denoted by $Q$. For n-vector models, the dimension in which the continuous set of states live at each node is denoted by an $n$. |
| + | |||
| + | For $n=1$ and $Q=2$ the two types of models overlap and this special model is known as the [[models:spin_models:ising_model|Ising model]]. | ||
| + | |||
| + | ---- | ||
| + | |||
| + | <blockquote> | ||
| + | |||
| + | The first of these is the Q-state Potts model (Potts, 1952; Wu, 1982). One assumes that each spin i can be in one of Q possible discrete orientations $\chi_i$ $(\chi_i =,1,2,\ldots,Q)$. If two neighboring spins i and j are in the same orientation, then they contribute an amount $-J$ to the total energy of a configuration. If i and j are in different orientations, they contribute nothing. Thus the interaction Hamiltonian is | ||
| $$\mathcal H(d,s)=-J\sum_{\langle ij\rangle} \chi_i \chi_j$$ | $$\mathcal H(d,s)=-J\sum_{\langle ij\rangle} \chi_i \chi_j$$ | ||
| Line 32: | Line 29: | ||
| $$\mathcal H(d,n)=-J\sum_{\langle ij\rangle} \vec S_i \cdot \vec S_j$$ | $$\mathcal H(d,n)=-J\sum_{\langle ij\rangle} \vec S_i \cdot \vec S_j$$ | ||
| - | Here, the spin $\vec S_i=(S_{i1} ,S_{i2} ,\ldots,S_{in}) is an n-dimensional unit vector with ($\sum_{\alpha=1}^n S_{i\alpha}^2=1$, and $\vec S_i$ interacts isotropically with spin $\vec S_j$ localized on site j. Two | + | Here, the spin $\vec S_i=(S_{i1} ,S_{i2} ,\ldots,S_{in})$ is an n-dimensional unit vector with ($\sum_{\alpha=1}^n S_{i\alpha}^2=1$, and $\vec S_i$ interacts isotropically with spin $\vec S_j$ localized on site j. Two |
| parameters in the n-vector model are the system dimensionality | parameters in the n-vector model are the system dimensionality | ||
| d and the spin dimensionality n. The parameter | d and the spin dimensionality n. The parameter | ||
| Line 41: | Line 38: | ||
| of the simple Ising model of a uniaxial ferromagnet. | of the simple Ising model of a uniaxial ferromagnet. | ||
| <cite>[[https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.71.S358|Scaling, universality, and renormalization: Three pillars of modern critical phenomena]] by H.Eugene Stanley</cite></blockquote> | <cite>[[https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.71.S358|Scaling, universality, and renormalization: Three pillars of modern critical phenomena]] by H.Eugene Stanley</cite></blockquote> | ||
| + | <tabbox Why is interesting?> | ||
| + | |||
| + | |||
| + | <blockquote>"Empirically, one finds that all systems in nature belong to one of a comparatively small number of such universality classes. Two specific microscopic interaction Hamiltonians appear almost sufficient to encompass the universality classes necessary for static critical phenomena. | ||
| + | |||
| + | [...] [T]he Q-state Potts model (Potts, 1952; Wu, 1982) [and] the n-vector model (Stanley, 1968). | ||
| + | |||
| + | |||
| + | <cite>[[https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.71.S358|Scaling, universality, and renormalization: Three pillars of modern critical phenomena]] by H.Eugene Stanley</cite></blockquote> | ||
| + | |||
| + | <blockquote>Phase transitions are a central theme of statistical mechanics, and of probability more | ||
| + | generally. Lattice spin models represent a general paradigm for phase transitions in finite | ||
| + | dimensions, describing ferromagnets and even some fluids (lattice gases).<cite>[[https://arxiv.org/pdf/1707.00520.pdf|Lectures on the Ising and Potts models on the hypercubic lattice]] by Hugo Duminil-Copin</cite></blockquote> | ||
| /*<tabbox FAQ>*/ | /*<tabbox FAQ>*/ | ||
models/spin_models.txt · Last modified: 2020/04/12 14:47 by jakobadmin
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