XY model

The XY model is a generalization of the Ising model of statistical mechanics , with which magnetism and other physical phenomena can be described. The XY model is the special case of the more general n-vector model (the other special cases of this model are the Ising model with and the Heisenberg model with ). ${\displaystyle n=2}$ ${\displaystyle n=1}$ ${\displaystyle n=3}$

It was already considered in 1950 by Yōichirō Nambu in connection with the two-dimensional Ising model. In 1961 Elliott Lieb , Daniel Mattis and T. Schultz gave an exact solution to the XY model of spin 1/2 particles in one dimension. They used the Jordan-Wigner transformation .

The XY model consists of spins that are represented by unit vectors. They are arranged on the points of a grid of any dimension, but can only be aligned in one plane; hence the designation  XY and the special case . ${\displaystyle N}$ ${\displaystyle {\vec {s_{i}}}}$ ${\displaystyle n=2}$

The Hamiltonian for the XY model is given by:

${\displaystyle H=-J\sum _{\langle i,j\rangle }{\vec {s}}_{i}\cdot {\vec {s}}_{j}-{\vec {H}}\sum _{i=1}^{N}{\vec {s}}_{i}}$

in which

• is summed over the next neighboring spins
• " " The standard scalar product for two-dimensional Euclidean space and ${\displaystyle \cdot }$
• ${\displaystyle J}$the coupling constant
• ${\displaystyle {\vec {H}}}$ is an external magnetic field.

The order parameter of the XY model is the magnetization and thus a vector in the XY plane. A phase transition can occur for two- and higher-dimensional lattices. In two dimensions, this is not a normal continuous phase transition or first-order phase transition , but the Kosterlitz-Thouless transition that cannot be described by any conventional local order parameter . This is the main reason why the XY model is interesting for theoretical physics . ${\displaystyle {\vec {M}}=(M_{x},M_{y})}$

See also

• Hexatic phase

Web links

• Java applet for visualizing the Ising and XY models

Individual evidence

1. ^ Daniel Mattis, The many-body problem, World Scientific 1993, p. 683
2. ^ Lieb, Schultz, Mattis, Annals of Physics, Volume 16, 1961, p. 407