N-vector model

In statistical mechanics, the n-vector model or O(n) model is a simple system of interacting spins on a crystalline lattice. It was developed by H. Eugene Stanley as a generalization of the Ising model, XY model and Heisenberg model.

In statistical mechanics, the n-vector model or O(n) model is a simple system of interacting spins on a crystalline lattice. It was developed by H. Eugene Stanley as a generalization of the Ising model, XY model and Heisenberg model. ==Definition== In the n-vector model, n-component unit-length classical spins \mathbf{s}_i are placed on the vertices of a d-dimensional lattice. The Hamiltonian of the n-vector model is given by: H = K{\sum}_{\langle i,j \rangle}\mathbf{s}_i \cdot \mathbf{s}_j where the sum runs over all pairs of neighboring spins \langle i, j \rangle and \cdot denotes the standard Euclidean inner product. Special cases of the n-vector model are: n=0 : The self-avoiding walk n=1 : The Ising model n=2 : The XY model n=3 : The Heisenberg model n=4 : Toy model for the Higgs sector of the Standard Model The general mathematical formalism used to describe and solve the n-vector model and certain generalizations are developed in the article on the Potts model. The Hamiltonian of the n-vector model is also the same as the potential term of the quantum rotor model. ==Reformulation as a loop model== In a small coupling expansion, the weight of a configuration may be rewritten as e^H \underset{K\to 0}{\sim} \prod_{\langle i,j \rangle}\left(1+K\mathbf{s}_i \cdot \mathbf{s}_j \right) Integrating over the vector \mathbf{s}_i gives rise to expressions such as \int d\mathbf{s}_i\ \prod_{j=1}^4\left(\mathbf{s}_i \cdot \mathbf{s}_j\right) ==Continuum limit== The continuum limit can be understood to be the sigma model. This can be easily obtained by writing the Hamiltonian in terms of the product -\tfrac{1}{2}(\mathbf{s}_i - \mathbf{s}_j) \cdot (\mathbf{s}_i - \mathbf{s}_j) = \mathbf{s}_i \cdot \mathbf{s}_j - 1 where \mathbf{s}_i \cdot \mathbf{s}_i=1 is the "bulk magnetization" term. Dropping this term as an overall constant factor added to the energy, the limit is obtained by defining the Newton finite difference as \delta_h[\mathbf{s}](i,j)=\frac{\mathbf{s}_i - \mathbf{s}_j}{h} on neighboring lattice locations i,j. Then \delta_h[\mathbf{s}]\to\nabla_\mu\mathbf{s} in the limit h\to 0 , where \nabla_\mu is the gradient in the (i,j)\to\mu direction. Thus, in the limit, -\mathbf{s}_i\cdot \mathbf{s}_j\to \tfrac{1}{2}\nabla_\mu\mathbf{s} \cdot \nabla_\mu\mathbf{s} which can be recognized as the kinetic energy of the field \mathbf{s} in the sigma model. One still has two possibilities for the spin \mathbf{s} : it is either taken from a discrete set of spins (the Potts model) or it is taken as a point on the sphere S^{n-1} ; that is, \mathbf{s} is a continuously-valued vector of unit length. In the later case, this is referred to as the O(n) non-linear sigma model, as the rotation group O(n) is group of isometries of S^{n-1} , and obviously, S^{n-1} isn't "flat", i.e. isn't a linear field. ==Conformal field theory== At the critical temperature and in the continuum limit, the model gives rise to a conformal field theory called the critical O(n) model. This CFT can be analyzed using expansions in the dimension d or in n, or using the conformal bootstrap approach. Its conformal data are functions of d and n, on which many results are known. ==References==

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