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Half-filled Hubbard continuum edges

Hubbard family · Supported models

Status: two-spinon and three charge-containing families implemented in the library and the shared frontend.

This extends the elementary dispersion lines to a specified scattering family, not to the entire spectrum at fixed quantum numbers. The current scope is the infinite half-filled repulsive chain, zero magnetic field, hopping t=1, U>0.

bethe-hubbard-continuum --u 4 --points 33 --csv edges.csv
bethe-hubbard-continuum --u 4 --momentum -1 --precision long-double
bethe-hubbard-continuum --u 4 --points 9 --precision fp128 --threads 4 --json edges.json
bethe-hubbard-continuum --references

The default grid has 33 equally spaced total momenta on [-pi,pi], including both endpoints (the same Brillouin-zone point). --momentum selects one point and excludes --points. The two_spinon table reports both energies, constituent momenta, separate error estimates, work counts and status. --tolerance, --max-evaluations, --max-levels and --max-iterations control the solver independently of arithmetic precision. The default is fp64; long-double, enabled fp128 and explicit MPFR are also available. The current fp128 build uses MPLAPACK scalar support; these scalar continuum calculations do not require its linear-algebra kernels.

--threads N evaluates independent total momenta using Uni20's scheduler (default 1), for both the two-spinon and charge-containing families. Each momentum owns its quadrature, inversion and extrema-search work; their internal dependencies remain sequential. The bounded observable grid allows at most 128 active or completed-but-uncollected results, and collects rows on the calling thread in momentum order. Full diagnostics and failed rows are preserved. A single --momentum has no grid parallelism. Use OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 to avoid nested threading.

The common output options support screen output and additional JSON/CSV/TSV files. Exports stream each completed point; --no-retain discards delivered rows. Metadata includes conventions and numerical controls, with CPU time and overall outcome in the final summary. Failed points have missing energies (JSON null, empty CSV/TSV cells) and yield exit status 2; other requested points are still attempted. Invalid arguments exit with status 1 before output files are opened, even with --force. CPU timing covers the whole momentum grid, including ordered row output; the exported CPU timing scope records this explicitly.

Two spinons

Essler–Korepin, Eq. (18) and the following paragraph give additive two-spinon energies and momenta, an endpoint spinon at the lower boundary, and equal rapidities at the upper boundary. For a requested total momentum P in [-pi,pi], put q=|P|. In terms of the existing spinon line:

\[ \begin{aligned} E_{\mathrm{lower}}(P)&=\epsilon_s(q),\\{} E_{\mathrm{upper}}(P)&=2\epsilon_s(q/2). \end{aligned} \]

For P>=0, constituent momenta are (0,q) and (q/2,q/2). For P<0 they are (pi,pi-q) and (pi-q/2,pi-q/2), respectively; their sums equal P modulo 2*pi. Endpoints denote limiting scattering states. No finite-ring multiplicity or spectral weight is implied.

Both edges have particle change zero. Consequently the symmetric and unshifted Hubbard conventions, and Hamiltonian and Fermi references, give the same excitation energies. The upper edge bounds two spinons only; it is not an upper bound on all multiparticle excitations.

API and numerical contract

#include <bethe/hubbard_continuum.hpp>
auto edges = bethe::hubbard::thermo::two_spinon_continuum(4.0, 1.0);
if (edges.converged) {
  auto lower = *edges.lower_energy;
  auto upper = *edges.upper_energy;
}

The implementation reuses hubbard_thermo.hpp without another quadrature implementation or a numerical optimizer. Native fp64, long-double and fp128 follow the same path. Options<Real> has the elementary-line semantics, except its evaluation budget is shared across both edges; inversion updates remain limited per constituent point. Work counts include both solves. At P=0 both edges are analytically zero and need no quadrature.

Neither energy is published if either solve fails. The underlying quadrature_limit, momentum_limit or precision_limit status is retained. Energy quadrature errors and total-momentum errors are separate: quadrature error does not include propagation of momentum inversion error. Unknown errors remain infinite. These are numerical estimates, not certified bounds.

Regression tests compare with independent Fourier–Bessel spinon values, verify momentum witnesses and interior scattering energies, check the large-U Heisenberg limit, and exhaust the shared budget after the lower-edge solve to ensure a partial pair of edges is not published.

Charge-containing channels

Select --channel spinon-holon, spinon-antiholon or holon-antiholon; the default is two-spinon. Their particle changes are respectively -1, +1 and 0, and their total spins are 1/2, 1/2 and 0. For example:

bethe-hubbard-continuum --u 4 --channel spinon-holon --momentum 0
bethe-hubbard-continuum --u 4 --channel holon-antiholon --points 9 --csv charge.csv
bethe-hubbard-continuum --u 4 --channel spinon-antiholon --momentum 0 \
  --convention unshifted --reference fermi

--convention symmetric (default) uses the SO(4)-symmetric interaction; unshifted adds U*DeltaN/2 to the symmetric excitation energy. --reference hamiltonian (default) reports that Hamiltonian's energy change. --reference fermi subtracts the half-filled chemical potential times DeltaN, recovering the symmetric energy for either convention. Negative unshifted removal energies are therefore not a negative symmetric gap. These switches do not change two-spinon energies.

The charge_continuum table includes selected and symmetric energies, constituent momentum witnesses, separate vertical and horizontal error estimates, quadrature and search counters, and distinct search_status and constituent_status. All energies, witnesses and errors are missing on failure. An objective_failure means a constituent failed; inspect constituent_status for its reason. A search budget failure can occur even when every sampled constituent converged.

Charge channels reuse the selected-precision elementary-line solver and a shared bounded-extrema search. Controls are separated by layer:

  • --tolerance, --max-evaluations (default 1000000), --max-levels and --max-iterations apply per constituent. Unlike two spinons, charge searches need many constituent evaluations.
  • --max-total-evaluations (default 200000000) caps all quadrature work for one requested total momentum, across both edges.
  • --search-tolerance defaults to 65536 epsilon and applies to E/max(1,U). --position-tolerance defaults to sqrt(epsilon), multiplied by the search domain's coordinate scale (pi) for the local bracket width.
  • --max-search-evaluations (20000), --max-search-iterations (256 per local bracket), --initial-intervals (16) and --max-intervals (128) limit the extrema search. Mesh sizes must satisfy 4<=initial<=max<=4096.

Search-specific options are rejected for the two-spinon channel rather than silently ignored. Budgets reset at each requested total momentum. Native fp128 charge searches can be expensive: a default-tolerance U=4, P=0 holon–antiholon point took about 15 minutes in GCC 13 Debug. Start with a single point; use explicitly looser tolerances for exploratory grids, for example --tolerance 1e-10 --search-tolerance 1e-7 --position-tolerance 1e-5. No lower-precision fallback is used.

The charge-continuum library and independent oracle now cover spinon–holon, spinon–antiholon and holon–antiholon families. Their channels, conventions, search limitations and reference points are documented there, including the independent reference calculations.

The charge-containing library searches the corresponding sums at fixed total momentum, refining candidate stationary branches and endpoints and propagating constituent failures. Its mesh-convergence evidence is not a global certificate, nor is a chosen particle family's lower edge a minimum over arbitrary additional particles. Doped continua and spectral weights remain separate work.

Arbitrary precision

--precision bits-N and digits-N use MPFR for constituent dispersions and extrema searches, without requiring MPLAPACK. In C++, construct Options<uni20::mpreal>(precision) for two-spinon edges or ChargeContinuumOptions<uni20::mpreal>(precision) for charge-containing families. Search and constituent tolerances and all inputs must use the same precision. Mesh, search and total quadrature budgets remain independent of arithmetic bits. A high-precision result is still a heuristic numerical family boundary, with vertical errors excluding propagation of momentum-inversion error.