Physical spin content of a TL level¶

Biquadratic/TL family · Supported models · Model overview · Ferromagnetic excitations

A through-line label \(\ell\) is not a physical spin. Each eigenvector of the open TL module carries a whole multiplicity space, which can contain several different SU(2) spins and repeated copies of the same spin. This space is the same for real-root, bound-string and Q-system levels in that module, and for either sign of the biquadratic Hamiltonian.

The shared library now resolves that space:

#include <bethe/temperley_lieb.hpp>
auto counts = bethe::temperley_lieb::spin_one_multiplets(4);
// counts[S] is the number of spin-S multiplets per TL eigenvector.
// Here: one singlet, three triplets, three quintets, three septets,
// and one spin-4 multiplet: 1 + 9 + 15 + 21 + 9 = 55 states.
if (counts) {
    auto triplets = (*counts)[1]; // 3, not 9 magnetic states
}

The argument is \(\ell\), not the chain length (except in the ferro ground space, where \(\ell =N\)). The vector has \(\ell +1\) entries including zeros. It is available for ell=0,...,45; larger labels return nullopt, consistently with overflow of the total representation dimension in the existing spin_chain_multiplicity(3,ell) API. This deliberately does not return a partially representable spin distribution. Long-chain energy solvers are unaffected: a small-ell module can still be resolved on any chain admitting it.

Optional CLI table¶

bethe-biquadratic-obc 8 --ferromagnetic --bound-pairs 2 --spin-content
bethe-biquadratic-obc 6 --sectors --spin-content --json sectors.json
bethe-biquadratic-obc 8 --ferromagnetic --one-defect --spin-content \
  --csv-table spin_content=spins.csv

--spin-content works with every biquadratic state-selection mode and either Hamiltonian sign. It adds a spin_content table, leaving default output unchanged. Each row gives state_id, through_lines, physical spin, multiplets, magnetic_states=(2S+1)*multiplets, and status. Spin uses Uni20's native half-integer column type, exported as a decimal number. Zero-count spins are omitted. Join by state_id to states, reference, or failed; reference rows are included even when they repeat a state already in the main table.

For an overflowing multiplicity space there is one row with null spin and counts, and status total dimension exceeds uint64. This does not change the energy solver's outcome or exit status. For an unconverged state, exact spin counts describe its requested module, not a verified eigenvalue: check the state table's convergence status before interpreting energies. JSON/CSV/TSV and --no-retain streaming use the same writer and schema.

Why the recurrence resolves physical spins¶

The kernel and surjective map in Aufgebauer and Kluemper, Sec. 3.3, Eq. (49) give the usual TL multiplicity recurrence. In the isotropic spin-1 representation the map commutes with physical SU(2), and the removed two-site state is a singlet. We therefore lift that recurrence from dimensions to characters:

\[ W_0=[0],\qquad W_1=[1],\qquad W_{\ell+1}=[1]\otimes W_\ell-W_{\ell-1}. \]

Our implementation expands the tensor product using [1] tensor [S] = [S-1] + [S] + [S+1] for \(S \ge 1\), with the distinct boundary rule [1] tensor [0] = [1]. All final coefficients are nonnegative integers. The first spaces are

ell	Multiplet counts by S=0,1,...,ell	Physical states
0	1	1
1	0, 1	3
2	0, 1, 1	8
3	1, 1, 2, 1	21
4	1, 3, 3, 3, 1	55

Tests reconstruct the entire energy multiset at every physical spin for N=2,...,6, including odd chains. The independent oracle builds \(-(S_{i} .S _{i +1})^{2}\) in fixed physical magnetization spaces, then subtracts the Sz=S+1 spectrum from Sz=S to isolate spin-S multiplets. This checks which spins accompany each energy, not just the sum of degeneracies. Dimension sums and overflow are also tested through the exact-integer limit. CLI tests cover all output paths, references and failures, using an independent magnetization-character expansion to check each spin count.

What this does not say about excitation weights¶

If several TL eigenvectors share an energy, their spin counts must be added; the API is per eigenvector, not per distinct energy. It does not select a particular state in the degenerate ferro ground manifold. Nor does it give matrix elements, form factors, or weights for a chosen iMPS ground state. The presence of a spin-S multiplet is not a guarantee that a particular operator excites it. Periodic TL representations and anisotropic physical spin chains require separate treatment; auxiliary XXZ spin is still distinct.