Hubbard spinon and charge dispersions¶

For a worked example with plots and downloadable data, start with the half-filled tutorial.

bethe-hubbard-dispersion evaluates elementary excitation lines of the infinite, repulsive Hubbard chain at zero field, at or below half filling. Unlike the finite-ring and free-end solvers, it takes no length or boundary-condition argument. It requires \(U \gt 0\), hopping \(t =1\), and \(0\lt n \le 1\). The default is half filling (--density 1), described below. For --density n with \(n \lt 1\), see the doped dispersion guide. Attraction, nonzero magnetic fields, spectral weights, and multiparticle continuum thresholds are not implemented in this frontend. A separate two-spinon continuum tool now provides the half-filled spinon and charge-containing family edges.

bethe-hubbard-dispersion --u 4 --points 101 --format csv
bethe-hubbard-dispersion --u 4 --branch spinon --momentum 1
bethe-hubbard-dispersion --u 4 --convention unshifted --format tsv
bethe-hubbard-dispersion --u 4 --precision fp128 --points 33 --format csv

Calculations and output retain the selected fp64 (default), long-double, or optional fp128 precision, or explicit MPFR bits-N/digits-N precision. MPFR curves require UNI20_ENABLE_MPREAL=ON. Doped backgrounds use Uni20's normal LU dispatch, with CPU MPFR available and MPLAPACK preferred when enabled. Arithmetic bits do not replace convergence of the quadrature, tail cutoff, momentum inversion or doped meshes. Grids are uniform in dressed physical momentum, not the auxiliary Bethe parameter. --points includes both endpoints; charge endpoints \(-\pi\) and \(\pi\) are the same point in the Brillouin zone. --momentum instead selects one point in radians. With --branch all it must belong to every selected branch.

Use --threads N to evaluate independent grid points concurrently (default 1), for either half-filled or doped backgrounds. Rows and numerical diagnostics keep their serial order: branch order first, then momentum. Selected branches share one bounded work window, so a later branch can start before all queries from the previous one have finished. This also permits overlap between branches with --momentum and --branch all; a single-branch single-point request has no query parallelism. See parallel observable grids.

The doped frontend prepares its background once and shares the meshes across all queries. Background preparation remains sequential and finishes before the report validates momenta and opens output files. C++ clients can instead submit background preparation and several dependent curves together; see the Async Hubbard example and shared-background contract.

Use --help or \(-h\) for Uni20-rendered options and examples, and --references for the literature and its applicability. --version and --build-info report the compiled application and dependency identity without calculating or opening files. This frontend accepts both --u 4 and --u=4, rejects repeated scalar options, and retains repeatable file-export options. See shared CLI behavior.

Which excitation?¶

Branch	Particle change	Total spin	Momentum interval
`spinon`	0	1/2	`[0,pi]`
`holon`	-1	0	\([-\pi,\pi]\)
`antiholon`	+1	0	\([-\pi,\pi]\)

“Chargon” describes the charge excitations, not a fourth independent branch. Spin projection does not split the spinon line at zero field. At the same bare charge parameter, holon and antiholon have equal symmetric energies but p_antiholon = p_holon - pi (mod 2*pi).

Spinons are gapless at both endpoints. Holon and antiholon minima occur at \(p =-\pi /2\) and \(p =\pi /2\) respectively. The common symmetric minimum is half the charge gap \(E (N +1)+E (N -1)-2E (N)\). At \(U =4\) this single-charge minimum is approximately 0.64336351100645219736629488464487225.

Hamiltonian and energy zero¶

The default --convention symmetric uses the SO(4)-symmetric Hamiltonian

\[ \begin{aligned} H_{\mathrm{symmetric}}&=-\sum_{j,s}\left(c^\dagger_{j,s}c_{j+1,s}+\mathrm{h.c.}\right) +U\sum_j\left(n_{j,\uparrow}-\frac12\right)\left(n_{j,\downarrow}-\frac12\right),\\{} H_{\mathrm{symmetric}}&=H_{\mathrm{unshifted}}-\frac{UN}{2}+\frac{UL}{4}. \end{aligned} \]

--convention unshifted matches \(U \,n_{\mathrm{up}} \,n_{\mathrm{down}}\) in our finite Hubbard tools. For excitation energies above the same background at fixed length,

\[ E_{\mathrm{unshifted}}=E_{\mathrm{symmetric}}+\frac U2\Delta N. \]

The extensive background cancels. Spinons are unchanged, holons shift down by \(U /2\), and antiholons up by \(U /2\). Negative unshifted removal energies are legitimate: half filling is centered at chemical potential \(U /2\) in that Hamiltonian, not at zero chemical potential.

The default --reference hamiltonian reports these Hamiltonian differences. --reference fermi instead reports E_H-mu_H*DeltaN, independent of the chosen interaction convention. At half filling we choose the middle of the Mott plateau: \(\mu_{\mathrm{symmetric}} =0\), \(\mu_{\mathrm{unshifted}} =U /2\). Thus the Fermi-referenced energies equal the symmetric energies here, including the nonzero charge gap. Both chemical potentials appear in the metadata. For doping they are solved from the requested density, not fixed at these half-filled values.

iMPS comparisons¶

These are elementary lines, not necessarily sector minima at every momentum. A holon with two spinons can lie below the elementary holon at the same total momentum. No continuum minimization is performed here.

Fractional excitations need an appropriate domain-wall/topological iMPS ansatz; they are not arbitrary isolated finite-ring states. Match hopping, interaction convention, particle change and spin first. Our momenta are one-site momenta. An enlarged unit cell folds the Brillouin zone, and a domain-wall gauge can shift the momentum origin; no universal shift is imposed. The MPSKit Hubbard example illustrates the comparison.

For an elementary chargon embedded in a multiparticle continuum, see Osborne–McCulloch and the tutorial discussion. The MPS variance-targeting method is not part of this Bethe frontend.

Method and accuracy¶

The reference formulas are the Fourier-Bessel dispersions in Essler–Korepin, Eqs. (4)–(5). Set \(u =U /4\) in that paper's notation. Our spin rapidity is \(\lambda =2\,u \,\beta /\pi\); \(k\) denotes bare charge momentum.

Direct oscillatory integration is costly at weak coupling, and subtracting order-one energies can erase the exponentially small charge gap. Instead we use a nonoscillatory spinon convolution with sech and a positive-integrand resummation of the exact modified-Bessel series in Melzer. His attractive massive spin branch maps to our repulsive charge branch: his interaction parameter becomes our \(u\), and energies multiply by two to change hopping from 1/2 to 1.

For reproducibility, set \(a =\pi /(2\,u)\) and \(A (z)=\mathrm{artanh} (\exp (-z))\). Our resummation is

\[ \begin{aligned} E_{\mathrm{inner}}(k)&=\frac4\pi\int_0^\infty\cosh t\, \left[A\!\left(a(\cosh t-\sin k)\right)+A\!\left(a(\cosh t+\sin k)\right)\right]\,dt,\\{} Q(k)&=\frac2\pi\int_0^\infty \left[A\!\left(a(\cosh t-\sin k)\right)-A\!\left(a(\cosh t+\sin k)\right)\right]\,dt,\\{} E_h(k)&=E_{\mathrm{inner}}(k)+4\max(\cos k,0). \end{aligned} \]

Unwrapped holon momentum is \(\pi /2-2\,k +Q\) for \(\lvert k \rvert \lt \pi /2\), \(-\pi /2+Q\) for \(k \ge \pi /2\), and \(3\,\pi /2+Q\) for \(k \le -\pi /2\). Stable trigonometric differences retain the integrable logarithmic endpoint at \(k =+-\pi /2\).

Tanh-sinh quadrature compares successive meshes; infinite-domain tails are bounded separately. Reported errors are estimates, not interval certificates:

energy_quad_error estimates quadrature, tail and shift-roundoff error at the returned parameter. It excludes propagation of momentum-inversion error.
momentum_error includes the dressed-momentum residual and quadrature error. The spinon target scales with distance to the nearest gapless endpoint; the charge target is absolute in radians.
parameter is rapidity or bare momentum, not plotted momentum. Analytic spinon endpoints have infinite rapidity and zero energy.

Defaults: tolerance \(256\,\epsilon\) of the selected scalar, 1,000,000 quadrature samples per entire point, 12 refinement levels and 160 inversion updates. Use --tolerance, --max-evaluations, --max-levels, and --max-iterations to change them. Weak coupling/fp128 can need substantially more work. Gaps and endpoint distances below the scalar's representable range are not promised.

quadrature_limit, momentum_limit, or precision_limit mark failed points: their energies are omitted, never replaced with a claimed converged zero. Exit status is 0 for complete success, 2 for failed points, and 1 for invalid arguments, runtime errors or output failures. Inspect status before increasing a budget or changing precision.

Output and C++ API¶

auto, pretty, and plain use Uni20 presentation. Add --csv FILE, --tsv FILE or --json FILE to export alongside the screen report, or use --quiet for files only. --format selects stdout, including machine formats. The output guide explains streaming, memory retention, provenance, overwrite protection, and precision-preserving JSON.

CSV/TSV have one header row, unwrapped selected-precision tokens, decimal spins (0.5), empty failed-energy cells, and initial # metadata. Status and solver CPU time are trailing comments. --no-preamble produces strict CSV/TSV. JSON has typed column metadata, decimal strings for real values, numerical half-integers, and null for missing values. energy follows the selected convention/reference; symmetric_energy always means the symmetric Hamiltonian difference and fermi_energy always means the chemical-potential-subtracted energy. Citations appear in --references and CITATIONS.md, not ordinary help or numeric output.

Include <bethe/hubbard_thermo.hpp> and use bethe::hubbard::thermo:

auto point = dispersion(Branch::holon, 4.0, -1.0, Convention::symmetric);
auto spin = spinon_at_rapidity(4.0, 0.5);
auto charge = charge_at_bare_momentum(Branch::antiholon, 4.0, 1.0);

Point<Real> contains optional energies, exact particle change and half-integer spin, parameters, error estimates, work counts and status. Tests use independent Fourier-Bessel references at all three precisions, a weak-coupling relative-gap reference, convention shifts, symmetry, budgets and the large-U XXX limit.

Explicit precision in C++¶

Construct Options<uni20::mpreal>(precision) or DopedOptions<uni20::mpreal>(precision) and supply finite inputs and tolerances at that same precision. Native options retain their default construction. Mixed, exact and unset runtime inputs are rejected before numerical work. Result diagnostics, analytic endpoints and failed iterates retain the selected precision; unconverged physical energies remain absent.

Half-filled integrals use Uni20's finite tanh–sinh quadrature. The weak-coupling spinon convolution is split at its square-root support boundaries, and the integrand uses independently computed endpoint distances there. The omitted physical rapidity tail remains a separate model-specific estimate. Quadrature mesh agreement and endpoint diagnostics are numerical estimates, not rigorous interval bounds or a guarantee against unresolved interior features.