Three-defect bound droplets on open chains¶

Biquadratic/TL family · Supported models · Ferromagnetic overview · Bound pairs · Exports

Three TL singlet insertions can bind into a droplet, just as two can form a bound pair. --bound-triples follows the one-three-string family in ell=N-6 for H=+sum(S.S)^2, on odd/even N>=6. It solves three real unknowns, independent of chain length, in fp64, native long double or enabled fp128. This is a finite-size Bethe calculation, not an ideal-string energy substituted for a finite chain.

bethe-biquadratic-obc 128 --ferromagnetic --bound-triples 8
bethe-biquadratic-obc 65 --ferromagnetic --bound-triples all --roots
bethe-biquadratic-obc 100000 --ferromagnetic --bound-triples 4 \
  --precision long-double --json triples.json --csv triples.csv

COUNT selects modes 1,...,min(COUNT,N-5); all selects all N-5 candidate three-string labels, not all states of the three-defect module. The --max-candidates row budget (default 10000) is checked before allocation. This option requires --ferromagnetic, fixes ell=N-6, and cannot be mixed with other state selectors, including --bound-pairs.

Explicit precision in the library¶

The triple, quartet, two-pair and triple-plus-defect C++ solvers also accept uni20::mpreal. Enable MPFR/MPC and construct solver controls at the desired precision. Uni20 supplies CPU LU; normal dispatch prefers MPLAPACK when enabled:

using Real = uni20::mpreal;
auto p = uni20::Precision::bits(256);
bethe::SolverOptions<Real> options(p);
auto triple = bethe::biquadratic::ferromagnetic::bound_triple<Real>(128, 1, options);

For the lower quantum-group APIs, construct Delta at the same precision. Constants, complex scattering, small-deviation corrections, Jacobians, Newton steps and returned diagnostics retain that precision. Unset, exact or mismatched numerical inputs are rejected. More bits do not increase iteration budgets or establish a new branch or a complete module. The CLI examples on this page also accept --precision bits-N and --precision digits-N when MPFR is enabled.

Where this branch lies¶

The gap is measured from the entire exact ground space, \(E_0 =N -1\). Its lowest mode approaches E-E0=2, compared with 3 for three widely separated lowest one-defect excitations, or 8/3 for a separated bound pair and one defect. Two separated single defects also have threshold 2, but belong to a different TL module. These are not physical spin counts.

The thermodynamic droplet limit follows from Nachtergaele, Spitzer and Starr, Theorem 2.1. In our normalization, at eta=acosh(3/2),

\[ g_M=2\sinh\eta\,\tanh(M\eta/2),\qquad g_1=1,\quad g_2=\frac53,\quad g_3=2,\quad g_\infty=\sqrt5. \]

That theorem supplies a thermodynamic module-edge limit, not a proof of our finite-N mode ordering. The lowest mode matches the module minimum in independent ED for N=6,...,10. Other modules and scattering levels can interleave these modes. The global positive gap still approaches 1.

At N=6 the three-defect module has dimension five. Its TL characteristic polynomial is \((g -6)(g -7)(g ^{3}-17g ^{2}+80g -106)\). The targeted droplet is the smallest cubic root, g=2.28668884804195..., with physical energy \(5+g\). This also provides a precision-growth check independent of the Bethe equations. At N=128 the first gap is approximately 2.00008198366582.

Why the deviation needs a phase¶

The reference is quantum-group XXZ with opposite end fields, Delta=3/2, not ordinary zero-field open XXZ. A canonical representative of its three roots is

\[ \begin{aligned} u_0&=\frac{ia}{2},\\{} u_+&=\eta+\mathrm{Re}z+i(a/2+\mathrm{Im}z),\qquad u_-=\overline{u_+},\\{} z&=e^{-L+i\phi},\qquad \eta=\mathrm{arcosh}\Delta,\\{} J&=N-4-\mathrm{mode},\quad \mathrm{mode}=1,\ldots,N-5. \end{aligned} \]

The central root and conjugate pair together make the three-string; there is no additional real-root sea. Unlike a two-string's signed real deviation, \(z\) generally has both real and imaginary parts. Setting its phase to zero does not solve the finite-chain equations. \(L =-\log \lvert z \rvert\) and \(\phi\) remain authoritative even when adding z to eta rounds away the correction, or exp(-L) underflows.

Here is our regularization of the original complex equations Bajnok et al., Eq. (5.12). Put \(u =u +\), b=a+2 Im(z) and define

\[ \begin{aligned} \Theta(\beta;w)&=2\mathrm{atan2}\!\left(\sin(\beta/2),\tanh w\cos(\beta/2)\right),\\{} D&=\log\sinh(u+\eta/2)-\log\sinh(u-\eta/2),\\{} A&=\log\sinh(2\eta+z),\\{} B&=\log\sinh(2\eta+z+ia)-\log\sinh(z+ia),\\{} C&=\log\sinh(\eta+ib)-\log\sinh(-\eta+ib),\\{} R&=\log\sinh(3\eta+2\mathrm{Re}z)-\log\sinh(\eta+2\mathrm{Re}z),\\{} c&=\log\frac{\sinh z}{z},\qquad c(0)=0. \end{aligned} \]

With wrap(t)=atan2(sin(t),cos(t)), the three residuals are

\[ \begin{aligned} f_0&=\Theta(a;\eta/2)+2\mathrm{Im}D -\frac{2\mathrm{Im}B+\Theta(2b;\eta)+\pi(J+1)}{N},\\{} f_1&=\mathrm{Re}D-\frac{\mathrm{Re}A+L-\mathrm{Re}c+\mathrm{Re}B+R}{2N},\\{} f_2&=\frac{\mathrm{wrap}(2N\mathrm{Im}D-\mathrm{Im}A+\phi+\mathrm{Im}c-\mathrm{Im}B-\mathrm{Im}C)}{2N}. \end{aligned} \]

The first equation is the product phase with the singular internal scattering cancelled; the other two retain the original outer-root equation. For tiny z, the regular correction c is evaluated by its Taylor series. The search stays in \(0\lt a \lt \pi\), \(0\lt b \lt \pi\), \(\lvert z \rvert \lt \eta /4\).

An ideal-string stage initializes the solve. Its center satisfies N*Theta(a;3eta/2)-Theta(2a;eta)-Theta(2a;2eta)=pi*J. Final convergence always uses the finite-deviation residuals above; both stages share one iteration budget. Column/row equilibration improves Newton conditioning without relaxing the tolerance. Each Newton step costs O(1).

Output, validation and reuse¶

Tables are states, reference, string, and optional roots. The string table records a, L and phi, with deviation_real/deviation_imag null on magnitude underflow. Root coordinates are rounded, not a substitute for these logarithmic parameters. The center and mode are not lattice momentum.

tl_energy evaluates the gap contribution directly, avoiding subtraction of extensive energies. Multiplicity is m_(N-6), the number of physical states per TL eigenvector, not a physical SU(2) spin label; it is null on uint64 overflow. Unconverged modes retain estimates but have no published gap and give exit status 2. The exact ground reference remains valid. Residuals are normalized equation errors, not certified energy-error bounds. Unresolvable seeds are rejected; an arbitrarily small tolerance cannot add digits beyond the selected arithmetic.

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

The equations live in xxz::quantum_group::three_string::bound_triple(N,Delta,mode). This lower API returns auxiliary-sign XXZ energies; the biquadratic adapter only applies the physical energy convention and representation multiplicity. General finite Delta>1 is accepted, but individual branches can collapse or fail away from 3/2; no general-anisotropy completeness claim is made.

Both string families and the AF two-string singlet use the common logarithmic-string Newton driver, including phase functions, ideal-stage handling, line search, scaling and budget/status rules. Pair/triple CLI scans, state/reference tables, exports and their output regression checks are also shared.

Tests cover original complex equations (including the central root), analytic Jacobians, native-precision N=6 energies, all branch labels at small odd/even sizes, and ED through N=10. XXZ ED checks also exercise Delta=1.25, 2 and 3. Long-chain checks reach N=100000, including both family edges and deviation underflow. Real-root scattering windows cover three unbound defects. Separate guides cover a pair with real roots, a triple with one real root, two pairs, bound quartets and physical-spin multiplicities. General string mixtures, droplets of five or more defects and operator-dependent spectral weights remain separate extensions.