Can Three or More Spin-1/2 Particles Form a Singlet?

A total-spin singlet is a state with total angular momentum quantum number $S=0$. For spin-1/2 particles, the answer depends on whether the number of particles is even or odd: an odd number cannot form a total-spin singlet, while every even number can. The construction is simple—make two-particle singlets and combine them—but the word “singlet” can also refer to a pair inside a larger system, which need not make the whole system a singlet.

What “singlet” means

For particles labeled $1,\ldots,N$, the total spin operator is

$$\mathbf{S}_{\mathrm{tot}}=\sum_{i=1}^N \mathbf{S}_i.$$

A total-spin singlet satisfies $S_{\mathrm{tot}}^2|\psi\rangle=0$. Equivalently, a common rotation applied to every spin leaves the state unchanged. For two spin-1/2 particles, the familiar singlet is

$$|s_{12}\rangle=\frac{1}{\sqrt{2}}\bigl(|\uparrow_1\downarrow_2\rangle-|\downarrow_1\uparrow_2\rangle\bigr).$$

Its two spins cancel: their combined total spin is zero. This is a statement about the total spin of the system, not just about one measurement result or one pairwise correlation.

Why an odd number cannot be a singlet

Adding angular momenta follows the rule that a spin $j$ combined with spin $1/2$ can give $j+1/2$ or $|j-1/2|$. Starting with one spin-1/2 and adding particles one at a time, the total spin is always a half-integer for an odd number of particles and an integer for an even number. Since zero is an integer, an odd number cannot have total spin zero.

For three particles, the steps make this especially clear. The first two spins combine to total spin $0$ or $1$. Adding the third spin-1/2 gives total spin $1/2$ from the first possibility, or $1/2$ and $3/2$ from the second. There is no total-spin-zero state. A pair of the three particles can be in a singlet, but the remaining spin-1/2 still carries nonzero total spin.

How to build a singlet for any even number

For $N=2n$, divide the particles into $n$ pairs and put every pair into its two-particle singlet. Each pair has zero total spin, so the sum over all pairs also has zero total spin. For four particles, one example is

$$|\psi\rangle=|s_{12}\rangle|s_{34}\rangle.$$

In the order $1,2,3,4$, this is

$$|\psi\rangle=\frac{1}{2}\bigl( |\uparrow\downarrow\uparrow\downarrow\rangle- |\uparrow\downarrow\downarrow\uparrow\rangle- |\downarrow\uparrow\uparrow\downarrow\rangle+ |\downarrow\uparrow\downarrow\uparrow\rangle \bigr).$$

This is one four-particle singlet, not the only one. In fact, for $N=2n$ labeled, distinguishable spins, the number of independent total-spin-zero states is

$$\frac{1}{n+1}\binom{2n}{n}.$$

For four spins ($n=2$), this gives two independent singlets. Different ways of pairing the spins can give different states, though the states from all possible pairings are not necessarily independent.

Identical particles and electrons

The parity rule applies to the spin degrees of freedom whether the particles are distinguishable or identical. Identical particles introduce an additional requirement: the full state, including position or orbital degrees of freedom, must have the correct exchange symmetry. For electrons, which are fermions, the full state must be antisymmetric under exchange of any two electrons.

That requirement does not rule out a spin singlet for an even number of electrons. Instead, the symmetry of the spin part must be combined appropriately with the symmetry of the spatial part. For example, two electrons in the same spatial orbital can have a symmetric spatial state and an antisymmetric spin-singlet state, making the combined state antisymmetric. In more complicated systems, check the complete state rather than treating the spin part alone as the whole wavefunction.

Common mistakes

• Calling a singlet pair a singlet system: In a three-particle state, two particles may form a singlet while the third remains. The total system then has spin 1/2, not zero.
• Confusing zero average spin with a singlet: A state can have zero expectation value of a spin component without being an eigenstate of $S_{\mathrm{tot}}^2$ with eigenvalue zero. A total-spin singlet requires the latter.
• Assuming a unique singlet for many particles: Four or more spins can have multiple independent singlet states. Pairing is a way to construct them, not always a unique classification.
• Ignoring exchange symmetry: For identical particles, a valid spin state must fit into a full wavefunction with the required exchange symmetry.