SU(3)/ULS: elementary excitations and continuum bounds¶

Overview · Finite rings · Output

Start with the worked ULS tutorial for plots and example exports.

bethe-su3-dispersion gives zero-field, infinite-chain reference curves for the fundamental SU(3) permutation chain. Its principal application is comparing momentum-resolved MPS excitations at the spin-1 Uimin–Lai–Sutherland point. It reports elementary lines and specified-particle-content continuum bounds, not spectral weights or finite-ring excited energies.

bethe-su3-dispersion --points 129 --csv uls.csv
bethe-su3-dispersion --branch bar3 --precision long-double
bethe-su3-dispersion --branch four-soliton --folded --json edges.json
bethe-su3-dispersion --branch 3 --momentum 0.3 --precision fp128

The default is fp64; long-double and optional MPLAPACK-enabled fp128 keep native arithmetic, input parsing and output precision. --branch all samples each branch over its own momentum interval; a single --momentum must lie in every selected interval. --references prints the literature. CSV/TSV/JSON exports include Hamiltonian, precision, momentum conventions, provenance and CPU time.

The only output table is dispersion, which is also the primary table. --csv uls.csv and --csv-table dispersion=uls.csv are equivalent. The 3, bar3, two-soliton and four-soliton branches are rows within this table, selected with --branch, not separate table names.

Unrepresentable continuum bounds or underflowing positive energies are omitted, with precision_limit and exit 2; invalid parameters exit 1. No optimization tolerance is needed: the bounds below are analytic, including all stationary points and endpoint cases.

Hamiltonian and units¶

where each site is a fundamental SU(3) representation. In a spin-1 basis,

Consequently the ULS Hamiltonian with equal bilinear and biquadratic coefficients J has total energy \(E_{\rm ULS}=E_P+JL\) on a ring; excitation gaps are unchanged. For coefficients \(\cos\theta,\sin\theta\) at \(\theta=\pi/4\), use \(J=1/\sqrt{2}\). This is a nearest-neighbour model, not the SU(3) Haldane–Shastry chain.

Elementary branches and representations¶

Let \(a=\pi/3\) and \(A=4\pi J/(3\sqrt{3})\). The two branches are

Both vanish at their endpoints, with endpoint speed \(v=2\pi J/3\). Their maximum energies are \(3A/2\) and \(A/2\), respectively. The implementation uses a product of sines rather than a cosine difference to preserve tiny positive energies near a soft endpoint.

The first nested sea's hole carries highest weight \((1,0)\), representation 3; the second carries \((0,1)\), representation bar3. The branch ranges follow Eq. (64) of Vörös–Penc, with twice their energies. The prose following their Eq. (65) interchanges the ranges; we use Eq. (64), independently checked against the nested-sea densities and representation assignment of Doikou–Nepomechie. Our positive momentum convention reverses the negative hole momenta of the latter reference; reflection leaves these bands and continuum envelopes invariant.

An isolated elementary line is not a local-response quasiparticle pole. The SU(3) centre charge (triality) of a periodic L-site fundamental chain is L mod 3. A configuration of \(n_3,n_{\bar3}\) holes must therefore obey \(n_3+2n_{\bar3}=L\pmod3\). In particular, neither single branch is by itself a state of the balanced L=0 mod 3 ring.

The smallest neutral pair has \(3\otimes\bar3=1\oplus8\). Longer neutral combinations include three equal representations and two such pairs. The spin-1 spin operators and five quadrupoles together form the SU(3) adjoint 8; from a singlet vacuum they access adjoint states, not the singlet part of the pair. The adjoint decomposes into spin 1 and spin 2 under physical spin rotation. Symmetry permits these channels but does not determine their matrix elements. At exactly Q=0 the summed on-site generators annihilate the singlet, even though the kinematic four-particle continuum has nonzero width.

Two versus four solitons¶

two-soliton means one 3 and one bar3. four-soliton means two of each; its representation product is \((1\oplus8)\otimes(1\oplus8)\) and includes additional adjoint channels. Auxiliary Bethe strings select multiplets; they do not add an independent energy to these thermodynamic hole sums.

All continuum momenta are total one-site momenta modulo \(2\pi\). Set \(q=\min(Q,2\pi-Q)\), so \(0\le q\le\pi\). The two-particle bounds are

To derive them, minimize/maximize the sum at fixed Q over \(p_{\bar3}\in[\max(0,Q-4a),\min(2a,Q)]\). The sum is \(2A\sin(Q/2)\cos[p_{\bar3}-(Q-a)/2]\). Testing its endpoints and stationary point gives the expressions above, without a discretized search.

For two pairs, convolution of those intervals, including total-momentum aliases, gives

For the lower bound, periodically extend the narrow branch with period 2a. It is subadditive under momentum addition; each elementary energy bounds it from above. A soft pair at momentum 0, 2a or 4a saturates the required shifted two-particle minimum. For the upper bound, each pair obeys \(E_{2,+}(Q)\le2A\sin(Q/2)\); sharing the appropriate total-momentum alias equally between the pairs saturates the maximum.

In particular, at Q=pi the two-particle onset is A but the four-particle onset is A/2. Plotting only the former misses lower-energy adjoint excitations between the soft momenta. Conversely, neither upper edge bounds the full spectrum: additional particles extend it. These outputs do not calculate spectral intensity, threshold exponents, or claim a finite-size state count.

Three-site iMPS momentum¶

The table always retains one-site p (radians/site). cell_momentum is \(K=3p\pmod{2\pi}\) (radians/three-site cell). Three distinct one-site images map to the same cell momentum. --folded takes their envelope for each continuum; it does not relabel a single elementary band as a complete cell spectrum.

If r is the distance to the nearest multiple of 2a, \(0\le r\le a\), the folded lower bound is \(\epsilon_{\bar3}(r)\) for both particle contents. Their upper bounds are \(2A\cos[(a-r)/2]\) and \(4A\cos(r/4)\) respectively. A folded envelope does not by itself specify internal gaps, translation eigenvectors or weights.

Validation and finite-size caveats¶

Typed tests compare 80-digit momentum-space references, independently expressed nested-sea densities, endpoint velocities, exchange scaling, direct constrained momentum scans, pair-continuum convolutions, and three-image folding. Tests cover fp64, long-double and fp128, including energies at momenta below machine epsilon. MPFR tests at 128, 256 and 333 bits additionally compare independent nested-sea densities, verify accuracy growth against a 512-bit reference, and retain positive energies at momenta as small as 1e-7000. The CLI tests use an independent decimal cosine series, non-dyadic grids, folded envelopes and full-precision exports. These tests require only the scalar provider, including in the sanitizer configuration.

The independent ED script swaps colour words, projects one-site momentum, and uses the total SU(3) Casimir to isolate the adjoint. Fixed colour populations alone do not isolate a representation. For J=1 the lowest adjoint gaps include:

The small-momentum velocity approaches 2pi/3. The soft tower has conformal dimension 2/3, with logarithmic finite-size corrections: its finite-ring gap is positive even though the thermodynamic threshold vanishes. At L=12, Q=pi, the adjoint level is already below the thermodynamic two-soliton lower edge 2.418399152312, consistent with the lower multiparticle branch. It is not equal to the thermodynamic four-soliton onset 1.209199576156; small rings are checks of normalization, sectors and scaling, not exact continuum-bound fixtures.

C++ API¶

#include <bethe/su3_dispersion.hpp>
bethe::su3::ExcitationDispersion<long double> band(1.0L);
auto energy = band.energy(bethe::su3::Particle::antifundamental, 0.3L);
auto edges = band.continuum(3.0L, bethe::su3::Continuum::four_soliton,
                           bethe::su3::Momentum::three_site_folded);

With arbitrary precision enabled, all branches and both momentum conventions also support bits-N and digits-N:

bethe-su3-dispersion --branch four-soliton --folded --precision digits-100 --csv uls.csv

This model needs GMP/MPFR/MPC scalar support, without MPLAPACK. The band retains the finite precision of its exchange parameter; C++ callers must supply momenta at the same precision, including zero. Exact-only or mixed-precision inputs are rejected instead of selecting a default or silently rounding:

using Real = uni20::mpreal;
auto precision = uni20::Precision::bits(256);
bethe::su3::ExcitationDispersion<Real> band(Real("1", precision));
auto energy = band.energy(bethe::su3::Particle::antifundamental, Real("0.3", precision));

Endpoint zeros, continuum bounds and exported coordinates keep that precision. The product-of-sines formula preserves small positive energies that a difference of cosines would lose to cancellation. Additional arithmetic precision does not add finite-size levels or spectral weights.

These analytic formulas are cheap, so the frontend evaluates its grid serially. For a larger application, an immutable band and its inputs can be captured by value in the existing Async observable grid. No separate SU(3) scheduler or preparation API is needed. Tests exercise owning queries at different precisions on the same scheduler, await each final operation result, and check that a failed grid leaves an independent one unaffected. This is composition support, without a claimed speedup for individual band evaluations.

The existing finite-ring ground-state API is unchanged. Other populations, finite-ring excited roots, fields, open ends and general SU(N) excitation tools remain separate extensions.