Two-defect bound pairs on long open chains¶

Biquadratic/TL family · Supported models · Ferromagnetic overview · Signed-string equations · Exports

For the spin-1 chain H=+sum(S.S)^2, two singlet insertions can bind. The targeted solver follows the one-two-string family in ell=N-4, without enumerating scattering states or searching for every Q polynomial. It has only two unknowns, independent of chain length, and supports odd/even N>=4 in fp64, native long double, enabled fp128 and explicit MPFR. See precision and validation for runtime controls.

# First eight modes of the bound-pair family, not the first eight levels
# of the entire physical spectrum:
bethe-biquadratic-obc 128 --ferromagnetic --bound-pairs 8
# Every supported two-string label at this size:
bethe-biquadratic-obc 65 --ferromagnetic --bound-pairs all --roots
# Long chain, native higher precision, with reusable exports:
bethe-biquadratic-obc 100000 --ferromagnetic --bound-pairs 4 \
  --precision long-double --json pairs.json --csv pairs.csv

COUNT selects modes 1,...,min(COUNT,N-3); all means all N-3 candidate two-string labels, not all states of the two-defect module. A row budget --max-candidates (default 10000) is checked before allocating results. Selecting four modes on a large chain needs only four solves, not a scan of all modes. --bound-pairs requires --ferromagnetic and cannot be combined with other state selectors; it fixes the through-line sector automatically.

Excitation structure¶

The gap is measured from the entire exact ground manifold, \(E_0 =N -1\). For N=4 there is one bound-pair mode:

\[ E=\frac{15-\sqrt{17}}2,\qquad E-E_0=\frac{9-\sqrt{17}}2=2.438447187\ldots \]

This is the complex-root singlet missed by the all-real scan. At larger N, ell=N-4 need not be a physical singlet: each TL eigenvector carries the representation multiplicity m_(N-4), with several possible physical spins.

The lowest two-defect branch approaches 5/3, below the energy 2 needed for two widely separated lowest one-defect excitations (each has limiting gap 1). This is a bound-pair threshold, not the global positive gap, which remains 1. The droplet theorem of Nachtergaele, Spitzer and Starr, Theorem 2.1, gives the thermodynamic module-edge limit. Rescaling their Hamiltonian by \(2\,\Delta =3\), with eta=acosh(3/2), gives \(2\,\sinh (\eta)\,\tanh (\eta)=5/3\) for two defects.

The lowest mode matches the module minimum in independent ED checks for N=4,...,10. We do not use that finite validation as a proof of finite-N ordering at arbitrary N. Other modules and scattering levels can interleave the modes shown. The mode number and string center are not physical lattice momentum; free ends have no translation quantum number. No form factors or coupling strengths to a chosen ground state are supplied.

Equations and branch labels¶

Use the auxiliary XXZ chain with Delta=3/2 and opposite end fields, not the ordinary zero-field open XXZ chain. Our regularization of Bajnok et al., Eq. (5.12) writes the two roots as

\[ \begin{aligned} u_\pm&=\frac{\eta+d}{2}\pm\frac{ia}{2},\quad 0\lt a\lt \pi,\quad |d|\lt \eta,\\{} d&=\sigma e^{-L},\qquad L=-\log|d|,\\{} J&=N-2-\mathrm{mode},\quad \mathrm{mode}=1,\ldots,N-3,\\{} \sigma&=(-1)^{\mathrm{mode}+1}. \end{aligned} \]

There is no real-root sea. Using Theta, C and G from the two-string derivation, the two equations are

\[ \begin{aligned} 2N[\Theta(a;\eta+d/2)+C(a;d)]-2\Theta(2a;\eta)&=2\pi J,\\{} 2N[G(a;\eta+d/2)-G(a;d/2)]-\log\sinh(2\eta+d)-L+\log\frac{\sinh d}{d}&=0. \end{aligned} \]

Both signs of d are essential. The reflected scattering factor has a negative denominator when d<0, changing its phase by pi. Consequently positive d requires N-J odd, negative d requires N-J even. The lowest-edge mode has positive deviation and the next has negative deviation. Keeping only positive deviations would omit every second mode.

The limit log(sinh(d)/d)=d^2/6+... is regular for either sign. Retaining L avoids losing the finite-size deviation when adding it to eta rounds it away, or when exp(-L) underflows. An ideal-string stage supplies only the initializer; convergence always checks the finite-deviation equations. Both stages share the same iteration budget and selected precision. The two-variable Newton correction is equilibrated for the very different angle and logarithm scales; this does not rescale the convergence tolerance. Work and storage per Newton step are O(1); output storage is O(COUNT).

Diagnostics and library API¶

Output tables are states, reference, string, and optionally roots. string records the sign and L separately; deviation is null on underflow, not an exact zero. Rounded root coordinates must not be used to reconstruct the deviation. In states, tl_energy=E-E0 is evaluated directly from the pair contribution, without subtracting extensive total energies.

Unconverged modes remain as estimates with converged=false, no published gap, and exit status 2. The exact ground reference remains valid. Residuals are phase/log-modulus errors divided by 2N, not energy-error bounds. Tight tolerances do not certify more digits than the chosen arithmetic resolves. Very large N can make distinct root coordinates unresolvable; such cases must not be interpreted as a certified set of distinct modes. An unresolvable initial center is rejected explicitly.

#include <bethe/biquadratic_ferromagnetic.hpp>
auto s = bethe::biquadratic::ferromagnetic::bound_pair<long double>(128, 1);
if (s.reference.converged) {
  auto gap = s.tl_energy;
  auto energy = s.energy;
}

The lower API is xxz::quantum_group::two_string::bound_pair(N,Delta,mode). It returns auxiliary-sign energies, not physical ferro energies. Delta>1 is accepted, but away from 3/2 an individual finite-size branch can collapse or fail to converge; no general-anisotropy completeness claim is made.

Precision and validation¶

The bound-pair library and its biquadratic mapping also accept explicit MPFR controls, as described in the two-string precision guide. The CLI supports the same explicit precision, for example bethe-biquadratic-obc 32 --ferromagnetic --bound-pairs 1 --precision bits-256. All biquadratic frontend modes accept bits-N/digits-N in an MPFR-enabled build; their state-family and convergence restrictions still apply.

Validation covers independent ED on N=4,...,10, both deviation signs in the original complex equations, analytic Jacobians and native-precision exact values. Long-chain tests reach N=100000, including deviation underflow. The three-defect droplet family reuses the same logarithmic-string Newton driver and CLI reporting machinery. Real-root scattering windows cover the unbound family. Bound quartets, mixed cluster families and physical-spin multiplicities are also implemented. Droplets of five or more defects, general string mixtures and operator-dependent spectral weights remain outside the supported families.