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Bethe's Ansatz is a method to find the eigenenergies and eigenstates of the Heisenberg ferromagnet (see also spin waves). For a general n-excitation state it involves solving rather complicated equations, nevertheless it does not involve any expansions/approximations in solving the problem, but yields the exact eigenstates.

The Holstein-Primakoff transformation is a mapping from the spin-operators in the above problem to bosonic operators. It is usually employed to successively perform a large spin expansion of the Hamiltonian. To lowest non-trivial order this Hamiltonian can be diagonalised easily by means of unitary transformations.

The two approaches therefore work in two different ways to solve the same problem. Intriguingly the Holstein-Primakoff transformation method applied to lowest order yields the same eigenenergies as the exact result from Bethe's Ansatz in the one-excitation case, namely

$$E = E_0 + J(1-\cos(k))$$

Motivated by that I was wondering if the Holstein-Primakoff transformation can maybe reproduce the higher excitation terms too, if manipulated correctly at higher order. Or are they simply two completely different things and coincidentally reproduce the same low order/excitation behaviour?

TL;DR: What is the relation between the two approaches?

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