# What if we used "Schwinger Fermions" to study spin waves?

When studying spin waves excitations in the Heisenberg Hamiltonian people often use Schwinger Bosons representation or Holstein-Primakoff which is a specific case of Schwinger Bosons. This leads you naturally to the description of the magnon. What if instead we used instead of a bosonic representation we used a fermionic representation for spins, like this: $$S^z=1/2[c^†_↑c_↑−c^†_↓c_↓]$$ $$S^+=c^†_↑c_↓$$ $$S^−=c^†_↓c_↑$$

This is just the Schwinger Boson representation with fermionic operators in the place of bosonic operators, that's why I nicknamed it "Schwinger Fermions", don't know if this is a thing. It would still obeys SU(2) algebra though, so no problems with this representation.

My question is, where this representation would lead? People told me you would still arrive at the same description and physics for magnons that you find using Schwinger Bosons, is it true?

This is known is Abrikosov's pseudo-fermion representation, which is compactly written $$\mathbf{S}_j = \frac{1}{2} \sum_{\sigma,\sigma'} f_{j\sigma}^\dagger \vec{\tau}_{\sigma,\sigma'} f_{j\sigma'},$$ where $$\vec{\tau}$$ is the vector of Pauli matrices, and I use the symbol $$f$$ for the annihilation operator to distinguish the particle from physical fermions in the system — $$f$$ fermions do not carry charge and are often called spinons. Similar to the Schwinger boson case, we need a local constraint $$\sum_\sigma f_{j\sigma}^\dagger f_{j\sigma}$$=1.
As you say, it is a good spin representation. If handled exactly, it will provide the same results as other representations. However, the requirement of a constraint complicates its treatment somewhat, which is why representations like the Holstein-Primakoff one are generally used in symmetry-broken (i.e. magnetically ordered) states. However, both Schwinger bosons and Abrikosov fermions are commonly used to study spin liquid states. If we rewrite the Heisenberg Hamiltonian in terms of spinons, $$H = -J \sum_{ij} \mathbf{S}_i\cdot \mathbf{S}_j = J \sum_{ij} \sum_{\alpha\beta} \left( \frac{1}{4} f_{i\alpha}^\dagger f_{i\alpha} f_{j\beta}^\dagger f_{j\beta} + \frac{1}{2} f_{i\alpha}^\dagger f_{j\alpha} f_{j\beta}^\dagger f_{i\beta} \right),$$ we get a Hamiltonian of interacting fermions. Note that there is no clear small parameter that can be used for a perturbative study. This is in contrast with the Holstein-Primakoff representation, which allows a (at least) formal expansion in orders of $$1/S$$. Hence, we typically introduce a mean-field approximation.
• @PedroDM I'm not sure you'll find this useful, but... If you use an exact treatment, then they're just two different ways of expressing the spin algebra. You can think of it as a set of identities to rewrite say the spin operator $S^+$ as expressions $A$ and $B$. Either one still behaves the same way as the original $S^+$ despite being formed from constituents of some statistics. You're not really putting in any new physics unless some approximation is made. Jul 12, 2020 at 6:36