N- vector model or O (n ) model is a model of statistical physics. This is a highly simplified model (or effective model ) for the description of phase transitions, critical behavior and magnetism. In the model are (classical ) spins with n components placed on the lattice points of a crystal lattice. In the original formulation of the model of HE Stanley from 1968 here interact only at the nearest neighboring spins with each other ( nearest neighbor interaction ) and the spins have unit length. The Hamiltonian is given as:
The sum runs over all pairs of neighboring spins. It should be noted that the pins have a dimension n, but that the crystal lattice of which may have a different dimension d. The n- vector model includes as special cases the following intensively studied models in statistical physics:
The discussion of the model is best done in the various special cases.
Generalizations of the model
A common generalization of the model in all special cases to consider not only nearest neighbor interaction, but also to investigate more distant neighbor interactions. In this case, the coupling constant is dependent on location. The Hamiltonian is then given as:
Further generalizations are given in the respective special cases.
Quantum mechanical formulation
In the quantum mechanical formulation is no longer considered classical spins but quantum mechanical spins expressed by spin operators. One of the main differences is that the spin operators of different dimensions n no longer exchange ( commutate ). The special cases of the n- vector model are then:
- Statistical Physics