Four-defect bound droplets¶

Biquadratic/TL family · Supported models · Ferromagnetic overview · Triples · Triple plus defect

Four TL singlet insertions can bind into one droplet, or quartet. This is the last of the five four-defect string topologies, alongside four real roots, one pair plus two real roots, two pairs, and one triple plus a real root. The selected-state library API accepts odd/even N>=8 in fp64, native long double, enabled fp128, or explicit MPFR precision. See the shared precision contract. For example, in native precision:

#include <bethe/biquadratic_ferromagnetic.hpp>
namespace ferro = bethe::biquadratic::ferromagnetic;
auto state = ferro::bound_quartet<long double>(128, 1);
if (state.reference.converged) {
    auto gap = state.tl_energy; // E-(N-1), computed without extensive subtraction
    auto inner_L = state.reference.inner_log_deviation;
    auto outer_L = state.reference.outer_log_deviation;
    auto outer_phi = state.reference.outer_deviation_phase;
}

mode=1,...,N-7 selects the four-string branch, not a rank in the full module or excitation spectrum. The module is \(\ell =N -8\); these are TL singlet insertions, not four physical spin flips. The lower API is bethe::xxz::quantum_group::four_string::bound_quartet(N,Delta,mode,options) for finite Delta>1, with no blanket convergence or branch-existence guarantee.

Command line and output¶

bethe-biquadratic-obc 128 --ferromagnetic --bound-quartets 4
bethe-biquadratic-obc 9 --ferromagnetic --bound-quartets all --roots
bethe-biquadratic-obc 128 --ferromagnetic --bound-quartets 4 --precision fp128 --json quartets.json

COUNT selects the first modes, capped at the available \(N -7\); all means all quartet labels, not all four-defect excitations. The usual --max-candidates limit applies. This selection cannot be combined with other state families or an explicit through-line sector.

The shared bound-cluster frontend writes states, the exact ground-space reference, and string tables; --roots adds the four reconstructed roots per state. The string table preserves both logarithmic deviations, the inner sign and the outer phase. Underflowed deviation components are null, not a claim of an exact ideal string. JSON/CSV/TSV exports and --no-retain streaming use the same tables as the pair and triple frontends. For example, --csv-table string=quartets.string.csv exports the string parameters. Failed solves return exit status 2, keep diagnostic estimates, and leave the verified gap null.

Two deviations, four real unknowns¶

The reference is quantum-group XXZ with opposite end fields, at Delta=3/2 for the biquadratic model. Define eta=acosh(Delta), and use canonical roots

\[ \begin{aligned} u&=\frac{\eta+d}{2}+\frac{ia}{2},\qquad \overline u,\\{} w&=u+\eta+z,\qquad \overline w,\\{} d&=\sigma e^{-L_{\mathrm{inner}}},\qquad z=e^{-L_{\mathrm{outer}}+i\phi},\\{} J&=N-6-\mathrm{mode},\qquad \sigma=(-1)^{N-J-1}=(-1)^{\mathrm{mode}+1}. \end{aligned} \]

The inner pair has a signed real deviation, while the outer pair needs a complex relative deviation. The solver retains both logarithms, the sign, and phi even when the deviations underflow or round away in the roots. It requires \(0\lt a \lt \pi\), 0<a+2 Im(z)<pi, and \(\lvert d \rvert,\lvert z \rvert \lt \eta /4\).

Our regularization of Bajnok et al., Eq. (5.12) cancels the singular internal scattering in the sum of the u and w equations. Write

\[ \begin{aligned} D_0&=\log\sinh(u+\eta/2)-\log\sinh(u-\eta/2),\\{} D_1&=\log\sinh(w+\eta/2)-\log\sinh(w-\eta/2),\qquad D=D_0+D_1,\\{} A&=\log\sinh(2\eta+z),\\{} B&=\log\sinh(3\eta+d+ia+z)-\log\sinh(\eta+d+ia+z),\\{} K&=\log\sinh(2\eta+ia+z)-\log\sinh(ia+z),\\{} R&=\log\sinh(3\eta+d+z)-\log\sinh(\eta+d+z),\\{} q&=\log\sinh(2\eta+d),\\{} Q&=\log\sinh(4\eta+d+2\mathrm{Re}z)-\log\sinh(2\eta+d+2\mathrm{Re}z),\\{} c_d&=\log\frac{\sinh d}{d},\qquad c_z=\log\frac{\sinh z}{z},\\{} t_0&=\Theta(2a;\eta),\qquad t_1=\Theta(2a+4\mathrm{Im}z;\eta). \end{aligned} \]

Theta and wrap are as in the pair/triple guides. The four residuals are

\[ \begin{aligned} f_0&=2\mathrm{Im}D+\pi-\frac{t_0+t_1+2\mathrm{Im}B+2\mathrm{Im}K+\pi(J+1)}N,\\{} f_1&=\mathrm{Re}D-\frac{q+L_{\mathrm{inner}}-\mathrm{Re}c_d+Q+2\mathrm{Re}B+2\mathrm{Re}R}{2N},\\{} f_2&=\mathrm{Re}D_1-\frac{\mathrm{Re}A+L_{\mathrm{outer}}-\mathrm{Re}c_z+\mathrm{Re}B+\mathrm{Re}K+\mathrm{Re}R+Q}{2N},\\{} f_3&=\frac{\mathrm{wrap}(2N\mathrm{Im}D_1-\mathrm{Im}A+\phi+\mathrm{Im}c_z-\mathrm{Im}B-\mathrm{Im}K-\mathrm{Im}R-t_1+\pi)}{2N}. \end{aligned} \]

The unsquared equations fix sigma; the product phase alone cannot recover it. The analytic Jacobian differentiates these regularized expressions. The ideal-string initializer reduces the center equation to

\[ N\Theta(a;2\eta)-\sum_{k=1}^{3}\Theta(2a;k\eta)=\pi J. \]

Initialization and the full finite-deviation solve share one Newton budget; only the latter can certify convergence. Work and storage per step do not grow with N. This reuses the common string Newton driver and correction functions, plus the stable inner-pair energy. The outer conjugate pair adds its real energy contribution directly. Failed results retain consistent estimates and residuals, not verified energies.

Threshold and coverage¶

The thermodynamic droplet formula \(g_{M} =2\,\sinh (\eta)\,\tanh (M \,\eta /2)\) gives g4=15/7 at Delta=3/2; see Nachtergaele, Spitzer and Starr, Theorem 2.1. The lowest quartet branch approaches this value from above. It lies below the separated triple-plus-single threshold 3 and the two-pair threshold 10/3, but the global positive gap remains the one-defect edge, approaching 1.

All five families together reproduce independent four-defect TL-module ED as a multiset, leaving no unmatched levels at these tested sizes:

N	Four real	Pair + two real	Two pairs	Triple + real	Quartet	Module dimension
8	1	6	1	5	1	14
9	5	20	3	12	2	42
10	15	45	6	21	3	90

This is finite-size numerical evidence, not a general completeness proof or a full spin-chain spectrum. Individual quartet branches match distinct ED levels at Delta=1.25,1.5,2,3; mode 1 is the module minimum in the ferro mapping at these sizes. Tests also verify the original inner/outer complex equations with both inner signs, every Jacobian column, the threshold and both logarithmic deviations through N=100000, and failure diagnostics.