Experimental finite-q XXZ vertex spectra¶

Experimental APIs · Massive two-spinon density

This opt-in package evaluates the original two- and four-spinon complex vertex amplitudes for the infinite, zero-field massive XXZ chain. It also provides fixed-order spectral-angle quadrature for moments and Gaussian frequency cuts. It complements the supported, unbroadened two-spinon evaluator; the four-spinon quadrature is an experimental calculation with independent product, residue and outer-grid controls.

Use the spin convention and exchange of the other XXZ tools:

\[ H=J\sum_j(S_j^xS_{j+1}^x+S_j^yS_{j+1}^y+\Delta S_j^zS_{j+1}^z), \qquad S^\alpha=\sigma^\alpha/2,\quad J\gt 0,\quad\Delta\gt 1. \]

The observable is the transverse raising response, with physical momentum k=π−Q in the vertex convention. There is no fitted amplitude, normalization switch or rescaling to a sum rule. Two and four spinons remain separate sectors.

Build and run a small example¶

Configure from a separate build directory:

cmake /path/to/bethe -DBETHE_BUILD_EXPERIMENTAL=ON -DBETHE_BUILD_EXAMPLES=ON
cmake --build . --target bethe_example_xxz_vertex
./experimental/bethe_example_xxz_vertex

The example evaluates both sectors at Δ=1.5 with order 4, displays three broadened frequencies and the corresponding fixed-momentum and full-zone moments. This deliberately small grid demonstrates the API, not convergence. Its two-worker executor uses the shared observable-grid final result; it also checks equality with serial evaluation. There are no raw worker threads or scheduler-wide drains in the example.

Link consumers to bethe::experimental. This target and its examples are absent by default and are not part of the normal application installation.

Amplitudes and arithmetic¶

#include <bethe/experimental/xxz_vertex.hpp>

namespace model = bethe::experimental::xxz;
model::VertexEvaluator<double> vertex(1.5);
std::array<double, 4> angles{-1.2, -.2, .6, 1.3};
auto amplitude = vertex(angles);
auto controls = vertex.options();

The input angles φ lie in [−π/2,π/2], with ξ=i exp(−iφ), u=−ξ² and q=−exp(−acosh Δ). value[vacuum][polarization] contains complex amplitudes for both Néel vacua. There is one polarization for two spinons and four for four spinons. Retaining phases matters: a resolved momentum uses coherent combinations of the two vacua, rather than a sum of their individual intensities.

VertexOptions{products, residues} sets independent truncation counts. Zero selects a count from Δ and arithmetic precision; options() reports the actual counts. Products must be 4–20000 and residue levels 1–20000. An automatic count outside those limits is rejected. Refining arithmetic alone does not establish convergence in either count.

Products retain a separate binary exponent until prefactors cancel. The first residue levels use literal q-products; later levels use a cancelled recurrence. Current Uni20 scalar math and scaled theta functions supply kinematics. The complementary elliptic modulus is retained directly so that a small gap is not lost by subtracting two numbers near one.

Binary64, configured 80-bit extended precision and binary128 are tested, alongside explicit MPFR/MPC arithmetic. Construct Δ and all query coordinates at the same finite precision; exact default exchange 1 and default grid offset 0 are accepted. Other mixed, unset or exact continuous inputs are rejected. For example, use uni20::mpreal("1.5", precision) and VertexEvaluator<uni20::mpreal> for a selected working precision. No ambient precision is read or changed.

Coincident or unresolved spectral nodes require confluent formulas and are rejected. Unresolved soft momenta, a physical momentum shift lost when forming π−k, exhausted scalar range and nonfinite sums also fail explicitly. These exceptions are not successful zero contributions.

Moments and broadened cuts¶

#include <bethe/experimental/xxz_vertex_spectrum.hpp>

namespace model = bethe::experimental::xxz;
model::ThermodynamicVertexSpectrum<double> spectrum(1.5);
auto full = spectrum.integrated(4, 12);
std::array<double, 3> omega{1.8, 2.2, 2.6};
auto cut = spectrum.broadened_cut(4, 1.5, omega, .06, 12);

integrated(n, order, offset) uses staggered periodic spectral-angle grids. The measure includes the two-vacuum average and the completeness factor:

\[ W_n=\frac{1}{2n!}\int\prod_{j=1}^{n}\frac{d\phi_j}{\pi} \sum_{i,\ell}|X_{i,\ell}(\phi)|^2. \]

This is the full-zone integral with dk/(2π) and dω/(2π). The returned first_moment additionally weights each node by its total excitation energy. The complete transverse zeroth-moment sum is 1/2; a selected sector does not exhaust it. An order-N calculation uses Nⁿ amplitude nodes.

broadened_cut(n, k, omega, width, order, offset) resolves momentum first, retains both coherent vacuum combinations, then applies a Gaussian with standard deviation width. Its density is S itself: the Gaussian integrates to 2π over the entire frequency axis. The cut's moments and sector_weights are computed directly from the momentum measure, independently of the requested frequency grid. Gaussian tails extend to negative frequencies; clipping them or integrating a finite plotted interval will not reproduce those moments.

Orders are 2–512, even for cuts. Four-spinon cuts use at most 4N³ amplitude nodes. The offset shifts staggered free-angle grids and is a useful quadrature diagnostic; it is not randomized automatically. A cut accepts an optional executor that invokes every row exactly once and joins all accepted work before returning, including on failure. Reduction remains in fixed row order. The example supplies that contract through Bethe's existing Async grid.

What the diagnostics establish¶

Every amplitude and moment carries an arithmetic_indicator. This is a conditioning screen based on roundoff and absolute residue sums. It is not a certified bound, a product/residue tail estimate, an outer-quadrature error, or an estimate of omitted sectors. No single fixed order receives a convergence status. Increase products, residues, precision and outer order independently; also vary offsets and inspect near-collision/soft regions. This becomes especially important approaching Δ=1.

Tests compare eight complex amplitudes with an independent 100-digit residue implementation, check product/residue refinement and precision ownership, and compare fixed-order integrated weights and cuts with independent Python calculations. Two-spinon cuts also agree with a Gaussian convolution of the supported unbroadened formula. Ising and near-isotropic checks are limited controls, not a proof of general quadrature convergence.

The reference derivation is Abada–Bougourzi–Si-Lakhal, Eqs. (2.6)–(2.7). The independent scripts, point fixture and fixed-order quadrature controls are in experimental/reference/xxz. They do not use the printed isotropic four-spinon expression to fix a prefactor.