Holstein-Primakoff boson

Holstein-Primakoff bosons or the Holstein-Primakoff transformation are a method to describe spins .

The method comes from Henry Primakoff and Theodore Holstein .

The spin operators are replaced by bosonic creation and annihilation operators as follows :

${\displaystyle S_{+}=\left({\sqrt {2S-b^{\dagger }b}}\right)b}$

${\displaystyle S_{-}=b^{\dagger }{\sqrt {2S-b^{\dagger }b}}}$

${\displaystyle S_{z}=S-b^{\dagger }b}$

As one can easily show, the operators defined in this way are sufficient for angular momentum algebra . ${\displaystyle [S_{i},S_{j}]=\epsilon _{ijk}S_{k}}$

The Holstein- Primakoff representation is used, among other things, in the theory of spin waves , here one often considers the limiting case of a small number of bosons , in which the roots can be developed. ${\displaystyle b^{\dagger }b\ll S}$

There are also other methods of mapping angular momentum operators to boson creation and annihilation operators ( Freeman Dyson , Julian Schwinger ).

literature

• T. Holstein, H. Primakoff, Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet, Phys. Rev. 58, 1098-1113 (1940)