Direct two-spinon moments¶

Two-spinon spectrum · Selected sectors

bethe/xxx_two_spinon_moments.hpp integrates the two-spinon static structure factor and first frequency moment directly. It supplies a reference alongside the four- and six-spinon moment calculations. No displayed frequency grid or sum-rule rescaling enters the calculation.

Resolve the spectral edges with the rapidity gap¶

Let g be the positive difference between the two rapidities, rho=g/(2*pi), and F(g)=exp(-I(rho))/cosh(g/2)^2, using the same transition-rate function as the spectral evaluator. Fold momentum into 0<q<=pi, and write s=sin(q/2), c=cos(q/2). Energy on the two-spinon surface is

\[ \omega(g)=\pi J s\sqrt{c^2+s^2/\cosh^2(g/2)}. \]

The upper edge is g=0, and the lower edge is approached as g tends to infinity. Changing variables from frequency to this gap absorbs the spectral edge Jacobian. The longitudinal moments are

\[ m_{0,2}(q)=\int_0^G dg\,F(g)\frac{J s^2}{8\omega(g)}, \qquad m_{1,2}(q)=\int_0^G dg\,F(g)\frac{J s^2}{8}. \]

The frequency measure domega/(2*pi) is already included. The chosen finite gap cutoff G omits the lower-edge tail; it is neither a frequency cutoff shared between momenta nor an absolute-rapidity box. The common shift remains unrestricted. At q=pi the zeroth moment diverges as G grows, so a finite answer there is explicitly flagged as a truncated estimate.

Integrating the momentum dependence analytically gives the global moments:

\[ I_{0,2}=\int_0^G dg\,F(g)\frac{g}{8\pi^2\tanh(g/2)}, \qquad I_{1,2}=\int_0^G dg\,F(g)\frac{J}{16}. \]

These coefficients follow from the conservation-law measure, rather than inserting a known spinon fraction. In particular the first moment's momentum dependence separates, while its coefficient is still calculated numerically. The tests independently integrate the spectrum in frequency and the fixed moments in momentum to check these reductions and their normalization.

C++ interface and diagnostics¶

#include <bethe/xxx_two_spinon_moments.hpp>
#include <uni20/async/tbb_scheduler.hpp>

using R = double;
bethe::heisenberg::StructureFactorOptions<R> options;
bethe::heisenberg::ThermodynamicTwoSpinonMoments<R> model(1.0, options);
uni20::async::TbbScheduler scheduler(2);
auto pending = model.fixed_moments(1.5, 16, 48.0).evaluate_async(
    {.scheduler = &scheduler, .max_in_flight = 4});
auto result = pending.get_wait(scheduler);
// global_moments(16, 48.0) uses the same owned query interface.

The arguments are momentum (for fixed moments), Gauss order per gap panel, gap cutoff, and an optional total node budget. Panels have width at most two. Arithmetic precision comes from the supplied scalar/options context; MPFR requires an explicit finite precision throughout. The owned query survives destruction of its model, while the scheduler must survive completion.

On success, result.integral.sums contains m0, m1 and their two integrated transition-kernel diagnostics. Global results contain I0 and I1 instead. The diagnostics exclude gap quadrature and the omitted large-gap region: vary order and cutoff independently before interpreting the result as converged. Failed kernels, exponent-range loss and cancellation produce no partial-prefix estimate.

sum_rule_denominators and sum_rule_fractions contain the global references I0=1/4 and I1=(2J/3)(log(2)-1/4), or the known fixed-q first moment. There is no invented fixed-q static denominator. At zero momentum both inelastic moments vanish without kernel evaluations, and the first-moment fraction is undefined. Channel conversion belongs to the common frontend; these library values are longitudinal throughout.