Two bound pairs scattering on a free-end chain¶

Biquadratic/TL family · Supported models · Ferromagnetic overview · One pair plus real roots · Isolated bound pairs

Four TL singlet insertions can form two bound pairs. This selected-state library solver follows that family in ell=N-8, for odd/even N>=8. It retains both finite string deviations in fp64, native long double, enabled fp128 or explicit MPFR precision. --pair-defects means one pair plus real roots; the two-pair family has separate selections and scans:

bethe-biquadratic-obc 128 --ferromagnetic --two-pairs 121,122 --roots
bethe-biquadratic-obc 128 --ferromagnetic --two-pair-states all --pair-window 4
bethe-biquadratic-obc 128 --ferromagnetic --two-pair-states 3 --pair-window 4 \
  --precision fp128 --json pairs.json --csv pairs.csv

--two-pair-states COUNT|all scans all binomial(N-6,2) candidates unless --pair-window WIDTH restricts both labels to the highest WIDTH values (2<=WIDTH<=N-6, giving binomial(WIDTH,2) candidates). The default --max-candidates 10000 bounds all solves, not just retained rows; counts are checked before allocation. The scan ranks direct gaps and retains the lowest COUNT converged candidates, with ties ordered by (J1,J2). This is not a global excitation rank or a claim of module completeness.

Tables are states, reference, and string (two rows per state, zero-based pair_index), plus optional roots (four rows per state). Both signed logarithmic deviations are preserved, with null deviation on underflow. Standard JSON/CSV/TSV exports and streaming/no-retain output work as for the other bound families. If any candidate fails, exit status is 2 and metadata records all candidate counts; one failed diagnostic row follows the retained converged rows, with a null verified gap. Its energy and tl_energy are only estimates.

#include <bethe/biquadratic_ferromagnetic.hpp>
namespace ferro = bethe::biquadratic::ferromagnetic;

auto state = ferro::two_bound_pairs<long double>(128, {121, 122});
if (state.reference.converged) {
    auto gap = state.tl_energy; // E-E0, evaluated directly; E0=N-1
    auto energy = state.energy;
    auto centers = state.reference.centers;
    auto logarithmic_deviations = state.reference.log_deviations;
}

The arguments are ordered Bethe labels 1<=J1<J2<=N-6, not energy ranks or physical lattice momenta. The high-label corner \((N -7,N -6)\) approaches \(E -E_0 =10/3\), the separated-pair threshold 5/3+5/3. It does not give the minimum of the entire four-defect module; other string topologies are missing. These are TL insertions, not spin flips, and no operator spectral weights or physical SU(2) labels are supplied.

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.

Coupling two existing pair systems¶

The reference remains quantum-group XXZ with opposite end fields, Delta=3/2. Its roots are

\[ \begin{aligned} u_{i,\pm}&=\frac{\eta+d_i}{2}\pm\frac{ia_i}{2},\quad i=1,2,\\{} 0&\lt a_1\lt a_2\lt \pi,\qquad \eta=\mathrm{arcosh}\Delta,\\{} d_i&=\sigma_i e^{-L_i},\quad |d_i|\lt \eta,\\{} \sigma_i&=(-1)^{N-J_i-i}\qquad(i\text{ is one-based}). \end{aligned} \]

The ordering-dependent sign is essential. Applying the isolated-pair rule to both strings loses a pi phase in the second pair's equation. \(L_{i}\) and \(\sigma_{i}\) are authoritative even if exp(-L_i) underflows or the deviation disappears when added to eta. Distinct ordered centers prevent coincident pair roots; a seed that cannot resolve them at the selected precision is rejected.

The solver reuses each pair's phase, log-modulus, analytic Jacobian and direct energy contribution from the two-string system. Only the pair-pair coupling is new. The shared damped Newton driver solves four real unknowns, with work and storage per iteration independent of N. The ideal-string stage initializes the solve; convergence always uses the finite-deviation equations and the original iteration/tolerance budget.

Here is our regularization of Bajnok et al., Eq. (5.12). For pair i scattering with j, set \(t =(d_{i} -d_{j})/2\), \(s =(d_{i} +d_{j})/2\) and beta=a_i±a_j. Define

\[ \begin{aligned} \Theta(\beta;w)&=2\mathrm{atan2}\!\left(\sin(\beta/2),\tanh w\cos(\beta/2)\right),\\{} C(\beta;w)&=2\arctan\!\left(\frac{\tanh w}{\tan(\beta/2)}\right),\\{} G(\beta;w)&=\log|\sinh(w+i\beta/2)|,\\{} F(\beta)&=\Theta(\beta;\eta+t)+\Theta(\beta;\eta-t)+\Theta(\beta;2\eta+s)+C(\beta;s),\\{} K(\beta)&=G(\beta;\eta+t)-G(\beta;\eta-t)+G(\beta;2\eta+s)-G(\beta;s). \end{aligned} \]

From each isolated pair's scaled phase residual subtract \([F (a_{i} -a_{j})+F (a_{i} +a_{j})]/(2N)\). From its log-modulus residual subtract the analogous sum of K. The complementary phase C is chosen continuously for negative as well as positive beta; using the unadjusted atan2 complement would introduce a spurious 2pi jump at negative beta.

The original unsquared pair equation includes a 3pi branch contribution when \(a_{i} \gt a_{j}\). This gives the deviation signs above. The upper center's excluded a=pi endpoint has label N-5, leaving the largest admissible candidate label N-6. The energies add through their direct auxiliary shifts, then use the common ferro mapping gap=-2*energy_shift.

The lower API is bethe::xxz::quantum_group::two_pairs::solve(N,Delta,{J1,J2},options). It accepts finite Delta>1, but the string ansatz is not a guarantee of convergence or branch existence for every parameter. Failed iterates retain consistent estimates, signs and residuals, not verified energies.

Validation and remaining four-defect states¶

All binomial(N-6,2) two-pair candidates match distinct module ED levels for N=8,...,10 at Delta=1.25, 1.5, 2 and 3. At the biquadratic value, combining them with the existing four-real-root and one-pair-plus-two-real-root families also matches ED as a multiset, so these are distinct subsets:

N	Four real roots	One pair + two real roots	Two pairs	Module dimension	Still missing
8	1	6	1	14	6
9	5	20	3	42	14
10	15	45	6	90	24

This is finite-size numerical evidence, not a general completeness proof. A three-string plus a real root is now available separately, as is the four-string droplet. Tests additionally check the original complex equations for both deviation parities, every analytic Jacobian column in ideal and finite-deviation modes, failure diagnostics, and the 10/3 threshold through N=100000, where both deviations underflow but their logarithms remain finite.