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Consider a quantum system of $2N$ identical spin 1/2 particles, each having a 2D Hilbert space $V$. The total Hilbert space is the tensor product $V^{\otimes 2N} \equiv V \otimes \cdots \otimes V$. Define the total spin operator as

$$ \mathbf{S} = \sum_{i=1}^{2N} \mathbf{S}_i $$

Let $V_0$ be the subspace of $V^{\otimes 2N}$ which has zero total spin:

$$ V_0 = \{|\psi\rangle \in V^{\otimes 2N} | \ \mathbf{S}^2 |\psi\rangle = 0\} $$

What is the dimension of $V_0$? In this article, it is stated that (below Eq. 23)

$$ \dim V_0 = \frac{1}{N+1} \binom{2N}{N} = \frac{(2N)!}{N!(N+1)!} $$

The combination number $\binom{2N}{N}$ can be naively understood as requiring the total $S^z = 0$ (number of different ways to choose $N$ spins with spin-up; the other $N$ spins have spin down). But I am having trouble in understanding the additional $1/(N+1)$ factor. Can someone please explain it? In addition, from the point of view of reducing the tensor product of $N$ copies of the fundamental (spin-1/2) representation of SU(2), what are the basis vectors in $V^{\otimes 2N}$ that furnish these $(\dim V_0)$ copies of the trivial (spin-0) representation?

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You can already see you miscounted for N = 1, just two particles. There is only one singlet in their tensor product, not two! You need the denominator of 2 to cut down on the superfluous/fake multiplicity you inferred. For N =2, 4 particles, you need to divide by 3 to get the answer, one singlet as a composition of two intermediate singlets, and one singlet as a composition of two intermediate triplets.

Your reference's well-known formula is just equation (12) in this reference, for s =0 and n = 2N, following from (7), or, easier, the Catalan triangle structure in the bottom formula of this section in Wikipedia, an ancient result: Zachos, 1992.

The Catalan coefficient for the correct multiplicity, in agreement with your reference, is $$ C(2N,N)= \binom{2N}{N}-\binom{2N}{N-1}, $$ as explained; or you might just evaluate the easy integral (7), a direct approach.

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  • $\begingroup$ Thanks for the references. Wikipedia says $C(x,y)$ is the number of strings consisting of $x$ X's and $y$ Y's such that no initial segment of the string has more Y's than X's. But how is the case $x = 2N$ and $y = N$ related to the singlet state? $\endgroup$ Oct 24, 2023 at 7:48
  • $\begingroup$ Think of the random walk utilized... $\endgroup$ Oct 24, 2023 at 13:50

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