Four-spinon spectra and direct moments¶

bethe/xxx_four_spinon_thermodynamics.hpp combines physical W4, support-adapted geometry, bounded density recovery and ordered quadrature accumulation. It returns the longitudinal response and moments in their physical units. There is no caller-side normalization correction.

Choose the observable and domain¶

ThermodynamicFourSpinon<Geometry, Low, High> prepares immutable density contexts. Geometry also supplies the accumulation precision and must retain the largest density guard precision. For MPFR these three template arguments can be the same type with separate explicit precision contexts.

Every query accepts an additional maximum node count, defaulting to 100000. Spectrum and fixed-momentum queries prepare a bounded set of geometry nodes. Global moments construct nodes by index without storing the three-dimensional grid. Geometry preparation precedes evaluation; the returned evaluation wall time excludes that preparation.

Moment measures and sum rules are included internally:

Moment results include sum-rule denominators and fractions where defined. No exact fixed-momentum zeroth-moment denominator is assumed. The first-moment ratio at k=0 is undefined and is left empty. The k=pi static response diverges without a cutoff: fixed-momentum results there carry unrestricted_static_divergence, even when their finite-cutoff estimates exist.

For exchange J, spectrum values and diagnostics scale by 1/J at frequency omega/J. Moment order r scales by J to the power r. The sum-rule denominators scale consistently, so fractions are dimensionless. These are partial four-spinon responses, not the complete structure factor.

A small fresh calculation¶

With MPFR enabled, the following calculates two global moments using two Gauss nodes in each of three gap intervals. It is deliberately a tiny finite-domain example, not a thermodynamic estimate:

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

using namespace bethe::heisenberg;
using R = uni20::mpreal;
R one(1, uni20::Precision::bits(192));
SpinonRecoveryPolicy recovery{
    "1e-6", "1e-30", {{128, 192, {{{64, 16}, {80, 20}}}}}, 60};
ThermodynamicFourSpinon<R, R> model(recovery, one);
uni20::async::TbbScheduler scheduler(2);
auto pending = model.global_moments(2, {2*one, 2*one, 2*one})
    .evaluate_async({.scheduler = &scheduler, .max_in_flight = 2});
auto result = pending.get_wait(scheduler);
// Check result.estimates before using either component.

This particular finite cubature gives approximately 0.00014958616511807964 and 0.00038003614670975796. They are reproduced independently by the 8F7 matrix-element formula and direct positive common-shift integration at 30 and 45 decimal digits. The omitted gap region is large. These numbers check the integration convention and implementation, not unrestricted spectral fractions.

The complete composition example adds order and cutoff comparisons and can be built with BETHE_BUILD_EXAMPLES=ON. It waits on the final Async result and requires no archive or Python process.

Keep numerical checks separate¶

Each component reports:

• Its physical value.
• The weighted contour-tail diagnostic.
• The weighted low/guard arithmetic screen, including applicable common-shift arithmetic indicators.
• Common-shift quadrature differences and omitted shift-tail bounds, when the zeroth-moment reduction uses its numerical fallback.

These fields exclude the outer quadrature error and omitted domain. Kernel recovery evaluates the same geometry coordinates at each arithmetic precision; it does not certify the geometry itself. Repeat the complete geometry at higher precision to test that separately.

refined_spectrum, refined_fixed_moments and refined_global_moments evaluate four complete controls: coarse/fine order on a base domain, then coarse/fine on an extended domain. For example:

auto comparison = model.refined_global_moments(
    2, 3, {one, one, one}, {2*one, 2*one, 2*one}).evaluate();

The result retains all four controls, their diagnostics, order differences on both domains, and the signed change between the two fine-order domain estimates. Only one geometry is retained at a time, and each control uses the same bounded Async admission. A failed control prevents a refined estimate. Differences are numerical evidence; they are not certified errors, an extrapolation, or a proof that the remaining tails are negligible.

The log-radius and gap cutoffs describe different domains. In particular, the analytical common-shift reduction already integrates the whole shift direction. Comparing spectra with global moments requires accounting for this distinction.

Even spectra and fixed-momentum moments do not have identical finite domains at the same log-radius cutoff. In a spectral patch reaching zero radius, r_min = r_max(omega) * exp(-T) uses its energy-dependent upper support. Direct moments use the full patch radius instead, and a positive-radius spectral patch is integrated without this soft cutoff. Therefore frequency integration of a finite-T spectrum need not equal the direct finite-T moment. Compare matched domains explicitly or study both cutoff limits separately.

Failures and validation¶

Query construction validates coordinates, precision and refinement controls before any work is submitted. Refined global moments also check the finest order on the widest gap domain against the node and span budgets. This validation does not build the full quadrature grid; geometry remains lazy. Command-line callers can therefore reject invalid refined requests before opening their output files.

Resolved exclusions outside the continuum, and the zero-momentum inelastic response, return physical_zero. Thresholds are identified separately and have no finite point estimate. Unresolved geometry, collisions, exhausted density recovery and lost exponent range are numerical failures. No failed node is discarded or replaced with zero. Cancellation drains admitted work and returns no integral over the surviving prefix.

Results retain precision, work budgets, cutoff definitions, recovery settings, completed-prefix counts, failure locations and evaluation wall time. Density attempt counts on failure describe collected observations, not discarded work finished during draining. The query and originating model can be destroyed after submission; the caller still owns the scheduler through completion.

Tests cover the independent finite spectral slice, direct-global references, whole-geometry precision growth, exchange scaling, sum-rule fractions, serial versus Async agreement, a single-worker refinement chain and explicit failures. The generic accumulator additionally checks completion while another admitted observation is deliberately held open. For installed-program calculations, see the selected-sector spectrum guide, direct-moment guide, and measured four-spinon refinement tutorial.