For example, let's consider a 1D spin-1/2 ferromagnetic (FM) Heisenberg chain $H=-J\sum\_{i=1}^{N}\mathbf{S}\_i\cdot\mathbf{S}\_{i+1}$ with periodic boundary conditions. Now we want to study its low energy excitations via the following two approaches:

(1)Jordan-Wigner (JW) transformation, we get $H\approx H\_f=\sum \omega\_kf\_k^\dagger f\_k-\frac{NJ}{4}$, where $f\_k$ are JW fermionic operators;

(2)Holstein and Primakoff (HP) transformation, we get $H\approx H\_b=\sum \omega\_kb\_k^\dagger b\_k-\frac{NJ}{4}$, where $b\_k$ are HP bosonic operators.

Where in both of the above two expressions, $\omega\_k=J(1-\cos k)$ (setting lattice constant unity). Then the ground states of $H\_f$ and $H\_b$ are the vacuum states and both **exactly equal to** the *exact* FM ground state of the original Hamiltonian $H$. And the ground state energy of $H\_f$ and $H\_b$ (the constant terms $-\frac{NJ}{4}$) is **exactly equal to** the *exact* ground state energy of the original Hamiltonian $H$. Note that the approximation $\approx$ indicates that we have assumed the spin fluctuation (i.e., $\left \langle \hat{n}\_i\right \rangle\ll 1$) to be small and hence dropped the higher order terms (i.e., interactions between JW fermions or HP bosons), which is justified since the ground states are the vacuum states containing no JW fermions or HP bosons ($\left \langle \hat{n}\_i\right \rangle=0$), implying that this approximation is at least self consistant. Furthermore, $\omega\_k\approx Jk^2/2$ as $k\rightarrow 0$ corresponds to the Goldstone mode.

If we believe that the above two pictures are both correct, then I get some questions: (1) Can the elementary excitations of a system be either fermions or bosons which may depends on the theory we adopt? Is there some *duality relation* or deep connection between the fermionic and bosonic approaches? (2) From $H\_f$, can we infer that the gapless Goldstone mode is a fermion? As we always say a Goldstone boson instead of a Goldstone fermion.

Thank you very much.