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models:spin_models [2018/05/07 07:24] jakobadmin [Overview] |
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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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- | <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> |
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- | [...] [T]he Q-state Potts model (Potts, 1952; Wu, 1982) [and] the n-vector model (Stanley, 1968). | + | |
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 |