Direct thermodynamic spinon moments¶

XXX family · Selected spectra · C++ two-spinon moments

bethe-xxx-moments-thermo integrates moments directly, without first sampling a frequency plot. Choose --momentum Q for a fixed momentum in radians/site, or --global to integrate over momentum too. The default is the two-spinon sector; --sectors 2,4,6 includes the requested contributions and their partial sum. Every contribution already has its physical normalization.

Definitions and reference denominators¶

For the longitudinal channel, the output moments are

\[ m_n^{(r)}(q)=\int_0^\infty\frac{d\omega}{2\pi}\, \omega^n S_r^{zz}(q,\omega),\qquad I_n^{(r)}=\int_0^{2\pi}\frac{dq}{2\pi}\,m_n^{(r)}(q). \]

Here \(r\) is the spinon sector and \(n=0,1\). The full-response reference values, summed over all sectors, are

\[ I_0=\frac14,\qquad I_1=\frac{2J}{3}\left(\ln2-\frac14\right),\qquad m_1(q)=I_1(1-\cos q). \]

The fraction column divides the computed contribution by the corresponding full-response denominator. No multiplier is fitted to a sum rule. A fixed momentum has no supplied exact zeroth-moment denominator: sum_rule and fraction are absent for that row. In particular, \(m_0(q)\) is not generally \(1/4\).

--channel raising multiplies both moments and their reference denominators by two; covariance scales by four and fractions stay unchanged. Changing --exchange J leaves the zeroth moment unchanged and scales the first moment by \(J\).

Start with a finite-domain calculation¶

bethe-xxx-moments-thermo --global --diagnostics

The two-spinon calculation uses relative rapidity \(g=\beta_2-\beta_1\gt0\), with a default cutoff of 48 and order 16 in panels of width at most two. The common rapidity shift is integrated over its full range. Compare higher orders at the same cutoff, then extend the cutoff; the transition-kernel diagnostic alone does not estimate either outer-quadrature error or the omitted tail.

A small two-plus-four execution check is:

bethe-xxx-moments-thermo --momentum 1.5 --sectors 2,4 \
  --precision long-double --two-order 4 --two-gap-cutoff 4 \
  --four-order 2 --four-cutoff 2 \
  --density-relative-tolerance 1e-4 --density-absolute-tolerance 1e-24 \
  --threads 2 --diagnostics

This deliberately short calculation checks the interface, not convergence. The nonzero absolute density tolerance permits a meaningful check near a vanishing density, where a relative-only test can fail. It is an explicit accuracy control, recorded in metadata; it does not skip failed nodes.

The cutoff definitions differ:

Sector and calculation	Integration domain
Two, fixed or global	Relative gap below `--two-gap-cutoff`; common shift unrestricted
Four, fixed momentum	Soft log-radius controlled by `--four-cutoff`
Four, global	Three positive ordered gaps bounded by `--four-gap-cutoffs A,B,C`; common shift unrestricted
Six, fixed or global	Every absolute rapidity lies in `(-R,R)`, with `--six-cutoff R`

Equal numbers in those options do not describe equal domains. The reported sum combines the requested finite-domain estimates; it contains no automatic rapidity-tail correction.

The same --four-cutoff also need not select identical domains in a spectrum and a fixed-momentum moment: the spectral radius bound depends on frequency. See the domain comparison before using frequency-integrated spectral data to check a direct moment.

Refine quadrature and sampling separately¶

For four-spinons, --four-refine-order requests four controls: coarse and fine orders on the original and extended domains. Pair it with --four-extended-cutoff at fixed momentum, or with --four-extended-gap-cutoffs A,B,C for global moments. The result uses the fine order on the extended domain. quadrature_difference is the fine/coarse change there; cutoff_change is the signed change between domains at fine order. They are separate comparisons, not a certified total error bar.

Six-spinons use the same explicit mixture and recovery controls as the selected-spectrum frontend. One configuration supplies both moment observations, and their covariance is retained. For example:

bethe-xxx-moments-thermo \
  --momentum 1.57079632679489661923132169163975 --sectors 6 \
  --precision long-double --six-cutoff 8 --replicates 2 --samples 4 \
  --seed 42 --threads 2 --diagnostics --replicate-output --covariance

As in the spectrum frontend, --samples-per-task (default 16) adjusts Async block size independently of the sampling budget. Small blocks expose parallelism within replicates; see parallel calculations.

These eight draws are only an execution check. Increase independent work, inspect influential observations and change proposal weights at fixed domain. Separately extend the rapidity cutoff: a small replicate error does not bound the missing tail. Changes in arithmetic precision and contour budgets provide further checks, distinct from both sampling and domain truncation.

For these exact eight draws, native 80-bit arithmetic agrees with the 384-bit MPFR reference to better than \(4\times10^{-16}\) relative in the moments, replicate values and maxima, sampling errors and covariance. Binary64 fails the arithmetic screen. The precision audit retains both failures and successes, effective widths and measured costs. Check the actual long-double format on your platform and repeat the checks when changing the seed or numerical controls.

With MPFR, the shared --density-guard-bits option can reduce W4/W6 density cost while preserving a higher geometry/output precision. Its bounds, lower comparison precision and validation procedure are described in the spectrum precision controls. It applies to both fixed-momentum and global moments; the recovery checks and finite-domain interpretation are unchanged.

Tables and failures¶

The frontend declares four exportable tables:

Table	Contents	Availability / screen option
`moments`	Sector, moment, value, denominator, fraction and status	Always; primary output
`diagnostics`	Kernel, contour, arithmetic, quadrature, cutoff and sampling diagnostics	Always / `--diagnostics`
`replicates`	Six-spinon replicate means, sample counts, maxima and failures	Sector 6 / `--replicate-output`
`covariance`	The 2-by-2 covariance of six-spinon moment means	Sector 6 / `--covariance`

Use, for example, --csv moments.csv --csv-table diagnostics=checks.csv. Named table exports do not enable their screen display. Unknown or unavailable tables are errors, detected before opening export files. JSON/CSV/TSV exports record the numerical controls and run metadata.

A failed requested sector has no value, and its partial sum is marked incomplete; successful sectors remain visible. No failed observation or replicate is removed to manufacture a survivor-only estimate. Numerical incompleteness exits with status 2; invalid arguments exit with status 1.

At zero momentum the response vanishes and the first-moment fraction is undefined (\(0/0\)). At \(q=\pi\) the unrestricted static response diverges; a finite-cutoff zeroth moment must not be interpreted as its converged value. The result carries unrestricted_static_divergence when the momentum equals \(\pi\) at working precision.

For library composition, use the owned moment queries' evaluate_async methods with a shared execution policy and wait for the final returned result. Independent sectors share the scheduler without draining unrelated work. See Async execution, four-spinon integrals and six-spinon moments.