Boson representation of spin operator

🧠 How to Turn Spins into Bosons?

A Gentle Tour Through Schwinger, Holstein–Primakoff, and Dyson–Maleev Transformations

When it comes to quantum mechanical description of spins, we're often stuck with operators that refuse to behave. They don’t commute, they don’t simplify, and they certainly don’t like being diagonalized. So how do we tame them?

One trick physicists have up their sleeves is to turn spin operators into bosonic operators.

That’s right—replace your spin operators with bosonic creation and annihilation operators, and suddenly, the tangled mess of SU(2) algebra becomes a playground of harmonic oscillators and perturbation theory.

But not all boson representations are created equal.

Depending on whether you’re studying an ordered magnet or a quantum spin liquid, your choice of transformation matters. In this article, we’ll explore three of the most widely used spin-to-boson transformations:

→ Schwinger boson transformation

→ Holstein–Primakoff transformation

→ Dyson–Maleev transformation

Each comes with its own philosophy, structure, and sweet spot of applicability.

🔄 Why Replace Spins with Bosons?

Spin operators obey SU(2) algebra:

This structure is elegant, but not easy to compute with—especially in large, interacting many-body systems.

Bosonic operators, on the other hand, come with:

A well-understood Hilbert space
Clear interpretation (e.g., number of excitations)

By expressing spins in terms of bosons, we can apply a whole arsenal of quantum field theory techniques to magnetic systems.

🧬 Schwinger Bosons: SU(2) Symmetry at its Core

This transformation uses two types of bosons, a and b, representing spin-up and spin-down configurations. The full SU(2) symmetry is preserved, making this method especially powerful in disordered or quantum fluctuating systems—like quantum spin liquids and RVB states.

(But actually, as an experimentalist, I’ve practically never used this formalism for any explicit calculation.)

However, the Schwinger boson formalism shines in another way: it provides a natural route to the Holstein–Primakoff transformation. By condensing one of the boson species—that is, treating one as macroscopically occupied and replacing its operator by a classical number—we recover the HP form.

In particular, under an appropriate approximation, we can reduce the formalism into more practical form for experimentalists, Holstein-Primakoff transformation.

Physical meaning of this process is as follows:

We will assume the situation that spin of each sites are pointing some specific "well-defined" direction. Or more simply, we will assume magnetically ordered phase.

SU(2) structure reduces to a single-boson language around a polarized spin background. This step connects the general symmetry-respecting Schwinger formalism to the more pragmatic H-P approach that’s widely used in linear spin wave theory.

📉 Holstein–Primakoff: Spin Waves from Order

This is a single-boson approach that expands around a fully ordered spin state—ideal for studying low-temperature spin waves in magnetically ordered systems at leading order.

When the excitation density is low, the square-root expressions can be expanded perturbatively, yielding the well-known linear spin wave theory.

This formalism strikes a nice balance between physical insight and computational effort—just enough math to make you feel like you're doing something profound.

If you are interested in a linear spinwave theory from ordered triangular lattice system, you may want to check this paper.

Nicely organized paper for linear spin wave theory.

However, bad news for those (like me) who dream of reinventing the wheel: most of this machinery is already nicely packaged in user-friendly codes like SpinW, or Su(n)ny.

⚙️ Dyson–Maleev: Keep It Simple (Even If It’s Not Hermitian)

Unlike Holstein–Primakoff, the Dyson–Maleev transformation drops the square roots entirely, trading exact Hermiticity for algebraic simplicity.

The spin operators are expressed as:

Note the asymmetry between raising / lowering operator in spin operator. They are not Hermitian conjugate of each other.

This might seem problematic at first—but the key point is that the SU(2) algebra is still exactly preserved, and for many practical purposes, that’s what matters.

So why introduce this non-Hermitian oddity?

Well, the real advantage lies in how easy it becomes to perform higher-order expansions. When you're calculating quantum corrections beyond linear spin wave theory—say, magnon-magnon interactions or 1/S expansions—HP’s nested square roots can get messy fast. D-M formalism avoids all that by giving you pure polynomials in boson operators.

However, if you just want to hang out with the key takeaway, here it is:

Whether you're using the Holstein–Primakoff or Dyson–Maleev transformation, as long as you're working in the leading order of linear spin wave theory, you're essentially using the same thing. Up to harmonic term, they are just the same.

The spin raising operator becomes a boson annihilation operator,
and the spin lowering operator becomes a boson creation operator.

That's it. Everything else is just a matter of how fancy you want to be with the corrections.

Page updated

Report abuse