A targeted complex-root singlet on long open chains¶

XXZ family · Supported models · Biquadratic application · Q-system search

This page describes the AF singlet branch. The same regularized solver also supports ferromagnetic empty-sea bound pairs, including signed deviations and other string labels on odd/even chains, and a bound pair plus selected real defects.

The Q-system is useful for discovering small spectra, but finding every polynomial is an expensive way to obtain one low-lying level on a long chain. The separate two-string solver targets an ell=0 branch with one conjugate pair and N/2-2 positive real roots, for even N>=4. In the spin-1 biquadratic chain this is a physical singlet with multiplicity one.

bethe-biquadratic-obc 128 --singlet-excitation
bethe-biquadratic-obc 512 --singlet-excitation --precision long-double --roots
bethe-biquadratic-obc 128 --singlet-excitation --json singlet.json

This branch is the lowest excited singlet in the small-chain ED checks. Following it to longer chains is a targeted branch calculation, not a proof that no other singlet lies below it, and not a global first-excitation claim across TL modules. It is separate from --excitations, whose real-root family does not contain this state, and from the multistart --q-spectrum.

Coordinates that retain the string deviation¶

We use the same opposite-end-field XXZ reference, Delta=cosh(eta)>1, as the TL layer. These are not the ordinary zero-boundary-field XXZ equations. Starting from Bajnok et al., equation (5.12), write

\[ \begin{aligned} u_j&=\frac{i\alpha_j}{2},\qquad 0\lt \alpha_1\lt \cdots\lt \alpha_r\lt \pi,\quad r=N/2-2,\\{} u_\pm&=\frac{\eta+d}{2}\pm\frac{ia}{2},\qquad 0\lt a\lt \pi,\quad 0\lt d\lt \eta,\\{} L&=-\log d. \end{aligned} \]

The real labels are I_j=j and the two-string label is 1. The positive-deviation branch and string topology are fixed for this singlet API. The separate bound-pair API allows negative deviations and other string labels. The equations below are our regularization of the published equations, not a claim that the paper proves its energy ordering.

For a nearly ideal string, forming \(u _+ + u _- - \eta\) can give zero by rounding, although the true deviation is positive. We therefore solve for L, and evaluate the singular factor through \(\log (\sinh (d)) = -L + \log (\sinh (d)/d)\). The latter correction has a regular small-d limit. Even if \(\exp (-L)\) underflows, L remains finite and meaningful. We never set d=0 as a physical approximation to obtain convergence.

Define

\[ \begin{aligned} \Theta(\beta;w)&=2\mathrm{atan2}\!\left(\sin(\beta/2),\tanh w\cos(\beta/2)\right),\\{} G(\beta;w)&=\log|\sinh(w+i\beta/2)|,\\{} w_+&=\frac{3\eta+d}{2},\qquad w_-=\frac{\eta-d}{2},\\{} S(\beta)&=\Theta(\beta;w_+)+\Theta(\beta;w_-),\\{} C(a;d)&=2\mathrm{atan2}\!\left(\tanh(d/2)\cos(a/2),\sin(a/2)\right). \end{aligned} \]

The real-root equations are

\[ 2N\Theta(\alpha_j;\eta/2) -\sum_{k\ne j}\left[\Theta(\alpha_j-\alpha_k;\eta)+\Theta(\alpha_j+\alpha_k;\eta)\right] -S(\alpha_j-a)-S(\alpha_j+a)=2\pi j. \]

The pair's phase and log-modulus equations are

\[ \begin{aligned} 2N[\Theta(a;\eta+d/2)+C(a;d)]-2\Theta(2a;\eta) -\sum_j[S(a-\alpha_j)+S(a+\alpha_j)]&=2\pi,\\{} 2N[G(a;\eta+d/2)-G(a;d/2)]-\log\sinh(2\eta+d)-L+\log\frac{\sinh d}{d} &\\[-1ex] {}-\sum_{j,\,s=\pm1}[G(a+s\alpha_j;w_+)-G(a+s\alpha_j;w_-)]&=0. \end{aligned} \]

These retain the finite deviation. The reflected self-scattering is essential: dropping it changes both equations. The real roots need not all lie below a; the continuous phase convention also covers their crossing of the center.

Algorithm, precision and output¶

A damped Newton solver uses an analytic Jacobian in the selected native fp64, long-double or optional fp128 type; the C++ library also supports explicit MPFR precision. An ideal-string solve initializes the coordinates; acceptance always uses the finite-deviation equations. Both stages share the user's --max-iterations budget, with no hidden extra updates. A zero budget returns the unconverged seed. Residuals are divided by 2N; the default tolerance is 32 machine epsilon, not an energy-error bound.

There is no site cutoff. Storage is O(N^2) and a dense Newton update costs O(N^3); long chains are not cost-free. Conditioning, initialization or limited precision can still prevent convergence. Failures retain consistent estimates and diagnostics, return exit 2, and never emit a verified gap. No fallback to another precision or relaxation of the requested tolerance occurs.

Named tables are states, reference, string, and optionally roots. The string table always carries center=a and log_deviation=L. deviation=d is null on underflow, not a reported exact zero. Displayed u coordinates are rounded; do not reconstruct d from their difference. The state table reports physical energy, TL and auxiliary XXZ energies, E-E0, multiplicity, and separate phase/modulus diagnostics. Gaps are emitted only if both the excitation and global ground reference converge.

The energy and multiplicity maps are unchanged: \(E_{\mathrm{biquadratic}} = 2 E_{\mathrm{ref}} - 7(N -1)/4\), ell=0, multiplicity=1.

#include <bethe/biquadratic_two_string.hpp>
auto state = bethe::biquadratic::two_string::singlet<long double>(128);
if (state.reference.converged) {
  auto energy = state.energy;
  auto L = state.reference.log_deviation;
}

The lower APIs are bethe::temperley_lieb::two_string::singlet(N,loop_weight) and bethe::xxz::quantum_group::two_string::singlet(N,Delta), with loop weight

2 and Delta>1 respectively. The public biquadratic adapter fixes Delta=3/2.

MPFR library calls require finite precision in both the physical input and solver controls. Newton corrections use normal Uni20 dispatch, with CPU MPFR support and an enabled MPLAPACK provider preferred when available:

using R = uni20::mpreal;
auto p = uni20::Precision::bits(160);
bethe::SolverOptions<R> controls(p);
auto singlet = bethe::biquadratic::two_string::singlet<R>(32, controls);
auto pair = bethe::xxz::quantum_group::two_string::bound_pair<R>(256, R(3, p) / 2, 1, controls);

This also covers the existing pair-with-real-roots and biquadratic bound-pair mappings. Seeds, derivatives, regularization thresholds, energy sums and failed states retain the requested precision. Exact, unset and mixed-precision inputs are rejected. Independent requests can use owning Async query grids; each ideal initializer and subsequent finite-deviation solve stays sequential. The work budget and state family are unchanged. The biquadratic frontend also accepts --precision bits-N and digits-N for --singlet-excitation and all supported complex-state modes; see the precision guide.

Tests cover the exact four-site singlet at native precision, analytic Jacobians, independent small-chain ED through N=10, original complex Bethe equations, longer chains through N=512, logarithmic coordinates below the underflow threshold, iteration budgets, failure output and typed exports. MPFR checks demonstrate four-site energy accuracy beyond binary128 at 80/160/256 bits, verify the original multiplicative equations independently, and compare complete mixed-precision serial/Async results, including failures. This API does not enumerate arbitrary complex-root states, other string topologies, singlet excitation ranks, or odd-chain spinons.