models:spin_models
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| ====== Spin Models ====== | ====== Spin Models ====== | ||
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| <tabbox Overview> | <tabbox Overview> | ||
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| 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. | 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. | ||
| - | * Models with a discrete set of states are called Potts models. | + | * Models with a discrete set of states are called **Potts models**. |
| - | * Models with a continuous set of states are called n-Vector models | + | * Models with a continuous set of states are called **n-Vector models**. |
| 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 $n$. | 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 $n$. | ||
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| 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]]. | 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]]. | ||
| - | <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. | + | <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 | 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 | ||
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| Both the Potts and n-vector hierarchies are generalization | Both the Potts and n-vector hierarchies are generalization | ||
| 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> | ||
| + | <tabbox Why is interesting?> | ||
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| + | <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. | ||
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| + | [...] [T]he Q-state Potts model (Potts, 1952; Wu, 1982) [and] the n-vector model (Stanley, 1968). | ||
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| <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> | ||
models/spin_models.1525670543.txt.gz · Last modified: 2018/05/07 05:22 (external edit)
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