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models:spin_models [2018/05/07 07:14]
jakobadmin [Overview]
models:spin_models [2020/04/12 14:47] (current)
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 ====== Spin Models ====== ====== Spin Models ======
  
 +<tabbox Overview> ​
  
 +{{ :​models:​spinmodelsoverview.png?​nolink&​600|}}
  
-<tabbox Overview> ​+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:​spinmodelsoverview.png?​nolink&​600 |}}+  * Models with a discrete set of states are called **Potts ​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 $n$. 
  
-<tabbox Why is interesting?>  ​+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>​"​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?> ​
 + 
 +
 +<​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>​ <​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.1525670069.txt.gz · Last modified: 2018/05/07 05:14 (external edit)