models:spin_models
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models:spin_models [2018/05/07 07:22] jakobadmin [Overview] |
models:spin_models [2018/05/07 07:24] jakobadmin [Why is interesting?] |
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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?> | + | <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. | <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). 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 | + | [...] [T]he Q-state Potts model (Potts, 1952; Wu, 1982) [and] the n-vector model (Stanley, 1968). |
| - | $$\mathcal H(d,s)=-J\sum_{\langle ij\rangle} \chi_i \chi_j$$ | ||
| - | 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 $T=0$. | ||
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| - | 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 | ||
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| - | $$\mathcal H(d,n)=-J\sum_{\langle ij\rangle} \vec S_i \cdot \vec S_j$$ | ||
| - | 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. | ||
| <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.txt · Last modified: 2020/04/12 14:47 by jakobadmin
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