User Tools

Site Tools


models:spin_models

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Next revision
Previous revision
models:spin_models [2018/05/05 10:50]
jakobadmin created
models:spin_models [2020/04/12 14:47] (current)
jakobadmin
Line 1: Line 1:
 ====== Spin Models ====== ====== Spin Models ======
  
-<​tabbox ​Intuitive+<​tabbox ​Overview
  
-<note tip> +{{ :​models:​spinmodelsoverview.png?​nolink&​600|}}
-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> +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.
-In this section things should ​be explained by analogy ​and with pictures and, if necessary, some formulas. +
-</​note>​ +
-  +
-<tabbox Abstract> ​+
  
-<note tip> +  * Models with a discrete set of states are called **Potts models**.  
-The motto in this section is: //the higher the level of abstraction,​ the better//+  * Models with a continuous set of states are called **n-Vector models**.
-</​note>​+
  
-<tabbox Why is it interesting?> ​  +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 $n$. 
  
-<​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).+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]].
  
-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 +---- 
 + 
 +<​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 27:
  
 $$\mathcal H(d,​n)=-J\sum_{\langle ij\rangle} \vec S_i \cdot \vec S_j$$ $$\mathcal H(d,​n)=-J\sum_{\langle ij\rangle} \vec S_i \cdot \vec S_j$$
-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 36:
 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.1525510240.txt.gz · Last modified: 2018/05/05 08:50 (external edit)