n vector model

The n vector model or O ( n ) model is a model of statistical physics . It is a greatly simplified (or effective) model for describing phase transitions , critical behavior, and magnetism .

Classic formulation

In the model, ( classical ) spins with n  components are placed on the grid points of a crystal lattice . In the original formulation of the model of H. E. Stanley in 1968 while interacting only the nearest neighboring spins and with each other ( nearest neighbor interaction ), and the spins have unit length. The Hamilton function is given as: ${\displaystyle {\vec {S}}}$ ${\displaystyle {\vec {S}}_{i}}$ ${\displaystyle {\vec {S}}_{j}}$

${\displaystyle H=-J\sum _{\langle i,j\rangle }{\vec {S}}_{i}\cdot {\vec {S}}_{j}}$

with the coupling constant . ${\displaystyle J}$

The spins have one dimension , but the crystal lattice can have a different dimension . ${\displaystyle n}$ ${\displaystyle d}$

The vector model contains the following intensively investigated models of statistical physics as special cases, in which the discussion of the model is best done: ${\displaystyle n}$

${\displaystyle n=0}$- Self-Avoiding Walks (SAW)

${\displaystyle n=1}$- the Ising model

${\displaystyle n=2}$- the (classic) XY model

${\displaystyle n=3}$- the (classic) Heisenberg model .

Generalizations

A common generalization of the model in all special cases is to consider not only the interaction of the closest neighbors, but also the interactions between more distant neighbors. The coupling constant can also depend on the location. The Hamiltonian is then given as:

${\displaystyle H=\sum _{i,j}J_{ij}{\vec {S}}_{i}\cdot {\vec {S}}_{j}\qquad i,j:\,\mathrm {Gitterpl{\ddot {a}}tze} }$

Further generalizations are given in the respective special cases.

Quantum mechanical formulation

In the quantum mechanical formulation one no longer considers classical, but quantum mechanical spins, expressed using spin operators . One of the main differences between them is that the spin operators no longer interchange ( commute ) in different dimensions . The special cases of the -vector model are then: ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle n=0}$- Self-Avoiding Walks (SAW)

${\displaystyle n=1}$- the Ising model

${\displaystyle n=2}$- the (quantum mechanical) XY model

${\displaystyle n=3}$- the (quantum mechanical) Heisenberg model .

swell

1. ^ HE Stanley, Phys. Rev. Lett. 20,589 (1968); Phys Rev. 176, 718 (1968)