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models:spin_models

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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.

- 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 $n$.

For $n=1$ and $Q=2$ the two types of models overlap and this special model is known as the Ising model.

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

$$\mathcal H(d,s)=-J\sum_{\langle ij\rangle} \chi_i \chi_j$$

where $\delta(\chi_i ,\chi_j)=1$ if $\chi_i=\chi_j$, and is zero otherwise. The angular brackets in Eq. (12a) indicate that he summation is over all pairs of nearest-neighbor sites $\langle ij\rangle$. The interaction energy of a pair of neighboring parallel spins is $-J$, so that if $J>0$, the system should order ferromagnetically at $T=0$.

The second such model is the n-vector model (Stanley, 1968), characterized by spins capable of taking on a continuum of states. The Hamiltonian for the n-vector model is

$$\mathcal H(d,n)=-J\sum_{\langle ij\rangle} \vec S_i \cdot \vec S_j$$ 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 parameters in the n-vector model are the system dimensionality d and the spin dimensionality n. The parameter n is sometimes called the order-parameter symmetry number; both d and n determine the universality class of a system for static exponents. Both the Potts and n-vector hierarchies are generalization of the simple Ising model of a uniaxial ferromagnet. Scaling, universality, and renormalization: Three pillars of modern critical phenomena by H.Eugene Stanley

"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).

Scaling, universality, and renormalization: Three pillars of modern critical phenomena by H.Eugene Stanley

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).Lectures on the Ising and Potts models on the hypercubic lattice by Hugo Duminil-Copin

models/spin_models.1525694474.txt.gz · Last modified: 2018/05/07 12:01 (external edit)

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