A bound pair scattering with real defects¶

Biquadratic/TL family · Supported models · Ferromagnetic overview · Real-root scattering · Bound triples

Three TL singlet insertions need not all bind, or all remain separate. A two-string plus one real root describes the intermediate pair-plus-defect branch in ell=N-6. Selected states and CLI family scans support this branch on odd/even N>=6, in fp64, native long double, enabled fp128 and explicit MPFR. See precision controls. --excitations still means the purely real-root family; it does not contain these mixed-string levels. The same CLI also supports one pair plus several real roots, as described below.

Command line and bounded scans¶

# One explicitly selected pair of Bethe labels:
bethe-biquadratic-obc 128 --ferromagnetic --pair-defect 125,123 --roots
# Complete small mixed family, not the entire three-defect module:
bethe-biquadratic-obc 8 --ferromagnetic --pair-defects all
# Lowest eight converged levels among the 36 selected label combinations:
bethe-biquadratic-obc 129 --ferromagnetic --pair-defects 8 --mixed-window 6 \
  --precision long-double --json mixed.json --csv mixed.csv \
  --tsv-table labels=mixed-labels.tsv

--pair-defects COUNT|all scans the full rectangle \(1\le I \le N -3, 1\le J \le N -5\) by default. --mixed-window WIDTH instead selects the highest WIDTH labels on each axis: I=N-2-WIDTH,...,N-3 and J=N-4-WIDTH,...,N-5. This gives WIDTH squared candidates and requires 1<=WIDTH<=N-5. A high-label window makes long-chain low-energy searches practical, but does not certify that excluded labels have higher energy.

COUNT retains at most that many converged levels after solving every candidate in the selected rectangle. all retains every converged candidate in that rectangle, not the full excited spectrum. The --max-candidates budget (default 10000) limits the work before allocation and checks the product without overflowing. Requesting one output level does not bypass the scan budget. A selected --pair-defect I,J requires no enumeration. These options require --ferromagnetic and exclude other state selectors. The default single-real-root family fixes ell=N-6. --mixed-window applies only to --pair-defects.

Sorting uses the direct gap, not the extensive total energy: the latter can round to the same value for distinct states on long chains. Exact numerical gap ties use ascending (I,J). These are rankings within the computed set, not general module minima or a global excitation ranking.

The shared cluster reporter supplies states, reference, string, labels, and optional roots tables. JSON exports all tables; --csv and --tsv export states, while --csv-table NAME=FILE and --tsv-table NAME=FILE select auxiliary tables. Metadata accompanies the exports, and --no-retain uses the same streaming table interface. The solver still stores the budgeted candidate set for sorting.

labels records I, J and alpha in one row per real root, grouped by state_id and ordered by I; string records the pair center, sign, L and optional representable deviation. roots lists the real roots first, followed by the conjugate pair. The states.row column is a display row number, not a physical mode label. L and the sign retain the deviation when exported rounded roots no longer resolve it; an underflowed deviation is null, not exact zero.

If any candidate fails, the command returns exit status 2 and records the total converged/failed counts. Retained converged levels come first, followed by one unconverged diagnostic sample, whose gap is null. Its energy and TL energy are estimates only. That extra diagnostic row does not count toward COUNT. Missing candidates could lie below the retained ones: a partial result is not a verified lowest-level list. The exact ground reference remains valid even if every candidate fails.

Library API¶

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

std::size_t const n = 128;
auto state = ferro::pair_defect<long double>(n, n - 3, n - 5);
if (state.reference.converged) {
    auto gap = state.tl_energy; // E-E0, E0=N-1; evaluated directly
    auto energy = state.energy;
    auto alpha = state.reference.rapidities[0];
    auto a = state.reference.center;
    auto L = state.reference.log_deviation;
}

The two arguments after N are integer Bethe labels I,J, not energy ranks or physical lattice momenta:

The high-label corner \((N -3,N -5)\) approaches the separated-cluster threshold 5/3 + 1 = 8/3. This is above the three-defect droplet edge 2 and below the three-unbound-defect edge 3. It is not the global first excitation, and the labels do not imply a universal ordering of finite-size levels. All of these counts refer to TL insertions, not physical spin flips. Operator spectral weights remain separate work. Physical-spin decomposition resolves the module multiplicity space; it does not compute operator matrix elements.

Same equations, different labels¶

The auxiliary model is quantum-group XXZ with opposite end fields at Delta=3/2, not the usual zero-field open XXZ chain. We reuse the two-string equations with one real root:

Replace the real-root right-hand side by \(2\,\pi \,I\), and the pair-phase right-hand side by \(2\,\pi \,J\). The modulus equation is unchanged. Retaining the unsquared pair equation fixes sign(d)=(-1)^(N-J-1); the product-phase equation alone loses this sign information. The real root may cross the pair center: no additional ordering condition \(\alpha \lt a\) is imposed. In the continuous logarithmic convention, the alpha=pi endpoint has I=N-2 and the a=pi endpoint has J=N-4; those excluded endpoint roots give the label bounds above.

These are our regularized equations derived from Bajnok et al., Eq. (5.12), not an ideal-string energy formula. L and the separate deviation sign remain authoritative when exp(-L) underflows. The existing analytic Jacobian, damped Newton driver, iteration budget, selected arithmetic and direct energy-shift mapping are shared with the AF singlet and isolated bound-pair solvers. There are three unknowns, so storage and work per Newton step do not grow with N.

reference.real_labels stores I and reference.string_label stores J. The common cluster result's mode is zero for this mixed branch: there is no one-dimensional mode ordering. An unconverged result retains estimates and residuals, not verified energies. No tolerance relaxation, hidden extra iterations or change of precision is performed.

The lower API bethe::xxz::quantum_group::two_string::pair_defect(N,Delta,I,J,options) accepts finite Delta>1. Branch collapse or convergence failure can occur; the biquadratic Delta=3/2 coverage below is not a guarantee for other Delta.

Checks and limits of coverage¶

For N=6,...,10, independent exact diagonalization verifies every level of the ell=N-6 module when the three families are combined:

Their sum is the module dimension binomial(N,3)-binomial(N,2). Matching the dimension alone would not establish completeness: the regression compares the entire sorted spectrum including multiplicities. This is small-chain numerical evidence, not a completeness proof for arbitrary N.

Further tests check the original complex equations for both deviation signs and odd/even chains, the analytic Jacobian, native-precision six-site characteristic polynomial, zero/one-iteration failure diagnostics, and the 8/3 threshold up to N=100000. The generic finite-size string description still permits numerical failure and does not provide physical SU(2) labels, scattering amplitudes or form factors.

A pair with several real defects¶

The library and CLI also accept one pair and a selected real-root set, with M=r+2 insertions and ell=N-2M.

# Two real roots I1=123, I2=124, and pair label J=121:
bethe-biquadratic-obc 128 --ferromagnetic --pair-defect 123,124,121 --roots
# All six one-pair-plus-two-real-root candidates on eight sites:
bethe-biquadratic-obc 8 --ferromagnetic --pair-defects all --real-defects 2
# Two real roots in a six-label window, times six pair labels: 90 candidates:
bethe-biquadratic-obc 129 --ferromagnetic --pair-defects 8 \
  --real-defects 2 --mixed-window 6 --precision fp128 --json mixed-four.json

For selected states, --pair-defect I1,...,Ir,J infers r from the labels; the last label belongs to the pair. For scans, --real-defects R sets r (default 1). Both require r>=1 and M=r+2<=N/2. Use --bound-pairs for an isolated pair; the library additionally accepts an empty real-root set.

Without a window, the candidate count is binomial(N-M,r)*(N-2M+1). With --mixed-window WIDTH, it is binomial(WIDTH,r)*WIDTH: choose r distinct real labels from N-M-WIDTH+1,...,N-M, independently of the pair label in N-2M-WIDTH+2,...,N-2M+1. The window requires r<=WIDTH<=N-2M+1. Near the largest allowed M, no such shared-width window may exist; omit the window to scan the full family within the candidate budget, or select explicit labels. No unbudgeted fallback scan occurs.

All candidates are solved before selecting COUNT converged levels. Sorting uses direct gaps, with lexicographic (I1,...,Ir,J) ties, and the same failure-sample convention as the r=1 scan. The shared combination iterator and overflow-checked binomial count are also used by the real-root-only excitation scans. Table schemas are unchanged: several labels rows now share each state ID, and the root table has r+2 rows per state.

std::vector<std::size_t> labels{123, 124};
auto four_defects = ferro::pair_with_real_roots<long double>(128, labels, 121);
// M=4, ell=120: one pair plus two separate defects, not a four-string.
if (four_defects.reference.converged) {
    auto gap = four_defects.tl_energy;
}

The domain is \(2\le M \le N /2\), ordered distinct real labels in 1,...,N-M, and pair label J=1,...,N-2M+1. The endpoint limits of the same logarithmic equations give these ranges: a real root at alpha=pi would have label N-M+1, while the pair center at a=pi would have J=N-2M+2. Both endpoints are excluded. The deviation sign remains \((-1)^{N -J -1}\).

No new Bethe equations or Newton implementation are introduced. Explicit sea labels, a label-dependent pair seed and edge-scaled coordinates extend the existing two-string system. For fixed r the work is independent of N; in r it uses a dense (r+2)-by-(r+2) Jacobian, quadratic storage and cubic linear-solve work per update. Matrix-size checks precede label allocation. Conditioning, branch topology and precision can still prevent convergence.

An empty label set reproduces an isolated pair. One label reproduces pair_defect, which delegates to this API. Consecutive labels 1,...,N/2-2 and J=1 reproduce the even-chain AF singlet's auxiliary solution, though the general API uses a different Newton scaling. Its biquadratic adapter always maps to the ferromagnetic convention. The lower XXZ API is two_string::pair_with_real_roots(N,Delta,labels,J,options).

For Delta=3/2, all pair-plus-two-real-root candidates match distinct module ED levels for N=8,...,10 (6, 20 and 45 candidates respectively). The ten pair-plus-three-real-root candidates at N=10 also match ED. These are subsets, not complete four- or five-defect spectra. A separate two-pair solver and CLI scans cover another four-defect family. A selected triple-plus-one-real-root solver covers another, and bound quartets supply the four-string droplet. Triples with more real roots and droplets of five or more defects remain outside these solvers. Independent original-equation checks cover r=2,3,4, both deviation signs, odd/even chains and all supported precisions. High-label branches approach the separated-cluster threshold \(5/3+r\), checked up to N=100000. This threshold is not a claim about the minimum of the whole TL module.