A bound triple scattering with a single defect¶

Biquadratic/TL family · Supported models · Ferromagnetic overview · Isolated triples · Two pairs

A three-string plus one real root is another four-defect family, in the TL module ell=N-8. The selected-state library API accepts odd/even N>=8:

#include <bethe/biquadratic_ferromagnetic.hpp>
namespace ferro = bethe::biquadratic::ferromagnetic;
auto state = ferro::triple_defect<long double>(128, 125, 121);
if (state.reference.converged) {
    auto gap = state.tl_energy; // E-(N-1), evaluated directly
    auto alpha = state.reference.rapidity;
    auto a = state.reference.center;
    auto L = state.reference.log_deviation;
    auto phi = state.reference.deviation_phase;
}

The labels are \(1\le I \le N -3\) for the real root and \(1\le J \le N -7\) for the triple. They are neither momenta nor energy ranks. The corner \((I,J)=(N -3,N -7)\) approaches gap 3 = 2+1, the separated triple-plus-single threshold, not the minimum of the full four-defect module. These are TL singlet insertions, not physical spin flips; no SU(2) decomposition or spectral weights are implied.

The frontend accepts --precision bits-N and digits-N when Uni20 MPFR scalars are configured. The C++ API follows the shared cluster precision contract. Arithmetic precision and solver work limits remain independent.

Command-line selections and scans¶

bethe-biquadratic-obc 128 --ferromagnetic --triple-defect 125,121 --roots
bethe-biquadratic-obc 128 --ferromagnetic --triple-defects all --mixed-window 3
bethe-biquadratic-obc 128 --ferromagnetic --triple-defects 4 --mixed-window 3 \
  --precision fp128 --json triples.json --csv triples.csv

--triple-defects COUNT|all scans \((N -3)\,(N -7)\) label pairs unless --mixed-window WIDTH selects only the highest WIDTH values of I and J. For this family 1<=WIDTH<=N-7, giving WIDTH^2 candidates. The default --max-candidates 10000 bounds all solves, not just retained output rows; overflow-safe counts are checked before allocation. --real-defects is not applicable: this family has exactly one real root alongside its triple.

The shared cluster reporter sorts by direct gap, with ties ordered by (I,J), and retains the lowest COUNT converged candidates. Tables are states, reference, labels (I, J, alpha), and string (a, L, phi), plus optional roots. The four root indices are 0 for the central triple root, 1 and 2 for its conjugate pair, and 3 for the additional real root. The usual JSON/CSV/TSV exports, streaming, and no-retain output are available.

If any candidate fails, exit status is 2 and the summary remains partial. One failed diagnostic row follows the retained converged levels; its verified gap is null, while energy and tl_energy are only estimates. Metadata records all scanned, converged, failed, and retained counts. Neither all nor a successful scan certifies a complete module spectrum.

Reusing the isolated triple¶

The reference is still quantum-group XXZ at Delta=3/2 with opposite end fields, not the zero-end-field chain. Let eta=acosh(Delta). Its roots are

We retain the isolated triple's three regularized equations and analytic Jacobian. Only the external scattering is new. Define the logarithm of the direct and reflected factors in Bajnok et al., Eq. (5.12):

The shared detail/open_string_scattering.hpp evaluates T and its two analytic complex derivatives. It is only for nonsingular external factors; the triple's singular internal factor is still evaluated using L and phi, never by subtracting rounded roots.

To choose the continuous product-phase branch, put

Theta and wrap have the same definitions as in the isolated-triple guide. Our coupled residuals subtract

The fourth equation is

The ideal fused phase fixes a multiple of 2pi only; it does not replace finite-deviation scattering. Its derivative cancels inside/outside the local lift, so the Jacobian uses derivatives of the original factors. Both sides of \(\alpha =a\) are tested. The actual coincident-root point is excluded, and the inherited triple domain requires \(\lvert z \rvert \lt \eta /4\).

A linear initial triple center and a bare-phase real-root seed start the solve. During initialization, the two center equations use the analytically fused phase F, while L and phi use the isolated triple equations. This avoids the removable external-factor poles at \(\alpha =a\) in the ideal limit. The finite stage then restores all coupled equations above; convergence is never accepted using the initializer alone. The common Newton driver uses one iteration budget and the requested arithmetic precision throughout. It solves four real unknowns, independent of N. Failed iterates retain consistent diagnostics and estimates, not verified energies.

Once exp(-L) underflows, the two internal deviation equations are linear in L and phi. After a finite-stage Newton trial rounds the angles, those two rows are solved at the represented angles. This prevents sub-ulp angular steps near coincident centers from inducing a spurious modulus stall. It is part of the trial normalization and line search, not extra iterations, a tolerance relaxation, or a change of precision. It is never used during ideal initialization.

The lower API is bethe::xxz::quantum_group::triple_defect::solve(N,Delta,I,J,options). Finite Delta>1 is accepted, but neither convergence nor branch existence is guaranteed for every parameter or near unresolved root collisions.

Coverage and validation¶

At Delta=3/2, combining the four-real-root, pair-plus-two-real-root, two-pair, and triple-plus-real-root families matches independent module ED as a multiset at N=8,9,10. The last family contributes \((N -3)\,(N -7)\) levels:

The remaining counts are filled by the separately implemented four-string droplet family. The combined multiset now matches the complete small-chain module spectra, but this counting is not a general completeness proof.

Additional tests check the original outer-, central-, and real-root equations, every Jacobian column, iteration-budget failures, and the threshold through N=100000 in fp64, long double, and enabled fp128. L and phi remain meaningful even when exp(-L) underflows. All selected labels at N=8,...,10 also match distinct ED levels at Delta=1.25, 2 and 3. The initializer's Jacobian is checked even at coincident ideal centers, while the final solver rejects coincident physical roots.