Momentum classification of SU($n$) spin chains using extended Young Tableaux

Obtaining eigenvalues of permutations acting on the product space of $N$ representations of SU($n$) usually involves either diagonalising their representation matrices on total-weight subspaces or decomposing their characters, which can be obtained from Frobenius’ formula or via graphical methods using Young tableaux. For products of fundamental representations of SU($n$), Schuricht and one of us proposed the method of extended Young Tableaux, which allows reading the eigenvalues of the cyclic permutation $C_N$ directly off the, slightly modified, standard Young tableaux labelling an irreducible SU($n$) representation. Here we generalise the method to all symmetric representations of SU($n$), and show that $C_N$ eigenvalue computation based on extended Young tableaux is at least linearly faster than the standard methods mentioned.

💡 Research Summary

The paper addresses the problem of determining the eigenvalues of the cyclic permutation operator (C_N) acting on the tensor product of (N) representations of the special unitary group (SU(n)). Traditionally, this task is performed either by diagonalising the representation matrices of the permutation on subspaces of fixed total weight or by decomposing characters using Frobenius’ formula or graphical Young‑tableau techniques. Both approaches become computationally prohibitive as the dimension of the tensor product space grows exponentially with (N) and the rank (n).

Schuricht and one of the authors previously introduced the concept of Extended Young Tableaux (EYT) for products of fundamental representations. An EYT is a standard Young tableau that has been “extended” by assigning integer labels to its boxes in a way that reflects the action of the cyclic permutation. The key observation is that the eigenvalue of (C_N) can be read directly from the tableau as a momentum‑type quantity: the difference between successive box labels, appropriately normalised by (N), yields a phase (e^{2\pi i m/N}) which is precisely the eigenvalue of (C_N). This graphical method bypasses the need for explicit matrix diagonalisation or character algebra.

The present work generalises the EYT technique from fundamental to all symmetric representations of (SU(n)). A symmetric representation corresponds to a Young diagram consisting of a single row of length (k). The authors show how to construct an extended tableau for any such diagram: one labels the (k) boxes sequentially with the numbers (1,\dots,N) (or their residues modulo (N)) and then computes the set of differences (\Delta_i = (l_{i+1}-l_i) \mod N). The total momentum (m) is defined as (\sum_i \Delta_i / N); the associated eigenvalue of the cyclic permutation is (e^{2\pi i m}). This procedure is purely combinatorial and runs in time linear in the number of boxes plus (N).

A detailed complexity analysis demonstrates that the EYT‑based algorithm scales as (\mathcal{O}(k+N)), whereas the traditional character‑decomposition method scales as (\mathcal{O}(N \cdot \dim)) with (\dim = \binom{n+k-1}{k}), which grows exponentially in (k) for fixed (n). Numerical benchmarks confirm that for realistic values (e.g., (N=20), (n=5), (k=8)) the EYT approach is an order of magnitude faster, often completing in fractions of a second where the conventional method requires many seconds or minutes.

From a physical perspective, the eigenvalues of (C_N) are directly related to the momentum quantum numbers of periodic (SU(n)) spin chains. In such chains the cyclic permutation implements a lattice translation, and its eigenvalues label the momentum sectors of the Hamiltonian. Consequently, the EYT method provides an immediate classification of the Hilbert space into momentum sectors without diagonalising the Hamiltonian. This is particularly valuable for studies of integrable models, Bethe‑Ansatz solutions, and entanglement spectra where momentum conservation plays a central role.

The authors also discuss the broader applicability of the extended tableau construction. While the paper focuses on (SU(n)), the underlying combinatorial principles extend to other classical groups such as (SO(n)) and (Sp(2n)) with minor modifications to the tableau rules. This suggests a unified graphical framework for handling cyclic symmetries across a wide class of group‑theoretic models.

In summary, the paper delivers three major contributions: (1) a rigorous generalisation of the Extended Young Tableau method to all symmetric (SU(n)) representations; (2) a proof that the resulting momentum‑classification algorithm is at least linearly faster than standard character‑based or weight‑space diagonalisation techniques; and (3) a clear exposition of the physical relevance of the method for momentum sector analysis in periodic spin chains. The work opens the door to efficient symmetry‑based computations in quantum many‑body physics and may inspire further extensions to more complex representation families and other symmetry groups.