Holstein-Primakoff transformation

There are a variety of formalisms that allow for a mapping of a discrete quantum system onto a continuous analog (for reviews see Refs. 218 and 219). The most prominent examples are Schwinger s theory of angular momentum [98] and the Holstein-Primakoff transformation [97], both of which allow a continuous representation of spin degrees of freedom. To discuss these two theories, we consider a spin DoF that is described by the spin operators Si,S2,Si with commutation relations... 

In the Schwinger representation the identity operator in the spin Hilbert space is mapped onto the constant of motion a a + a a2. The existence of this constant of motion is utilized by the Holstein-Primakoff transformation to eliminate one boson DoF, thus representing the spin DoF by a single oscillator [97] ... 

The Holstein-Primakoff transformation also preserves the commutation relations (70). Due to the square-root operators in Eqs. (78a)-(78d), however, the mutual adjointness of S+ and 5 as well as the self-adjointness of S3 is only guaranteed in the physical subspace 0),..., i- -m) of the transformation [219]. This flaw of the Holstein-Primakoff transformation outside the physical subspace does not present a problem on the quantum-mechanical level of description. This is because the physical subspace again is invariant under the action of any operator which results from the mapping (78) of an arbitrary spin operator A(5i, 2, 3). As has been discussed in Ref. 100, however, the square-root operators may cause serious problems in the semiclassical evaluation of the Holstein-Primakoff transformation. 

Finally, it is noted that there exist alternative mappings of spin to continuous DoF. For example. Ref. 219 discusses various mappings that (like the Holstein-Primakoff transformation) represent a spin system by a single-boson DoF. The possibility of utilizing spin coherent states for this purpose is discussed in Section IX. 

The conventional treatment of spin-waves in magnetically ordered systems uses the Holstein-Primakoff transformation (1940). One transforms to operators at, u7 defined by... 

The basic idea of the mapping approach is to change from the discrete representation employed in Eq. (53) to a continuous representation. There are several ways to do so, most of them are based on the representation of spin operators by boson operators. Well-known examples of such mappings are the Holstein-Primakoff transformation, which represents a spin system by a single nonlinear boson DoF, and Schwinger s theory of angular momentum,which represents a spin system by two independent boson DoF. 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

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