Takhtajan–Babujian: thermodynamic spinons and continua¶

Overview · Finite-ring ground states · Output

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

bethe-tb-dispersion supplies zero-field excitation references for the critical spin-1 Takhtajan–Babujian (TB) chain: an elementary spinon line and two-/four-spinon energy bounds. These are infinite-chain kinematic curves, not finite-ring levels, form factors or a complete spectrum.

bethe-tb-dispersion --points 129 --csv tb.csv
bethe-tb-dispersion --branch spinon --momentum 0.3 --precision long-double
bethe-tb-dispersion --branch four-spinon --folded --json quadrupole-bounds.json
bethe-tb-dispersion --points 65 --precision fp128 --tsv tb.tsv

The default is fp64, with native long-double, optional fp128, and explicit MPFR precision (bits-N or digits-N). MPFR needs Uni20 scalar support only, without MPLAPACK. Each branch's grid spans its own momentum interval. --momentum replaces the grid and must be valid for every selected branch. --references prints the literature; normal reports retain the Hamiltonian, conventions, provenance and CPU time. Spin labels use Uni20's half_int, exported as decimal 0.5 rather than a plotting-unfriendly fraction. CSV/TSV/JSON, streaming and retention use the same output facilities as the ULS tool.

Explicit precision and independent queries¶

bethe-tb-dispersion --branch spinon --momentum 0.3 --precision digits-100
bethe-tb-dispersion --branch four-spinon --folded --precision bits-256 --csv tb.csv

Exchange and momentum are parsed directly at the requested precision. The band owns that precision for pi, its amplitude and its endpoint zeros; all subsequent momenta must have the same finite precision. For example:

auto precision = uni20::Precision::bits(256);
bethe::takhtajan_babujian::SpinonDispersion<uni20::mpreal> band(
    uni20::mpreal("1", precision));
auto energy = band.energy(uni20::mpreal("0.3", precision));
auto four = band.four_spinon_continuum(uni20::mpreal("1.5", precision));

Mixed, exact or unset runtime-precision inputs are rejected. Native convenience construction is unchanged. Independent owning band/momentum requests compose through the existing Async observable grid; capture the band and momenta by value and await the final result. These elementary formulas are cheap, so the frontend keeps per-point evaluation serial.

The shared sine/elliptic-band helper carries explicit precision too. The XXZ and XYZ models have their own explicit-MPFR elliptic implementations and validation; support for a shared helper alone would not establish those model-level capabilities. Tests retain their native massive-band checks and add scalar-only MPFR tests for typed endpoints, tiny positive momenta and massive-band extrema. TB accuracy grows at 128/256/333 bits against an independent 512-bit dressed-sea reference; a separate Decimal series checks CLI normalization and folded bounds. Mixed-precision Async comparisons include complete results and failure isolation. No quadrature or root-solving convergence error is introduced.

A factor of four worth keeping explicit¶

Our Hamiltonian is

The spins are 1. There is no additive constant, and the bond energies in total-spin sectors 0, 1 and 2 are -6J, -2J and 0. The existing finite-ring bethe-tb-pbc tool uses J=1.

Vlijm–Caux, Eq. (1.2), has a prefactor \(J_{\mathrm{paper}}/4\): our J=1 means their J=4, not J=1. For the common Hamiltonian with coefficients cos(theta), sin(theta), use \(J=1/\sqrt{2}\) at \(\theta=-\pi/4\). This is not the generic bilinear spin-1 Heisenberg chain or the positive-biquadratic ULS point.

Spinons and physical sectors¶

With lattice spacing and hbar equal to one,

The spinon is gapless at both endpoints and carries spin 1/2, although each physical site has spin 1. A periodic chain of integer-spin sites has only integer total spin: an isolated spinon is not one of its eigenstates.

The two-string sea's density and dressed hole energy give an independent way to identify the scale:

Two spinons combine into S=0 or S=1, while four allow S=0,1,2. From the singlet ground state, a local spin operator selects S=1, a scalar bond/dimer operator selects S=0, and a local rank-two quadrupole selects S=2. The last therefore requires at least four spinons, even though its kinematic lower threshold can coincide with the two-spinon one. These selection rules permit matrix elements; they do not prove nonzero weight at every momentum or on an edge. For example, the Q=0 total-spin operator annihilates the singlet.

Identical sine dispersions do not make this the XXX spectrum with a rescaled J. TB's low-energy theory is SU(2) at level 2 (central charge 3/2), and its spinons have additional non-Abelian state-counting structure. We do not infer finite-size degeneracies, fusion multiplicities, or spectral weights from the curves. See Frahm–Stahlsmeier.

Two- and four-spinon bounds¶

For total one-site momentum \(0\le Q\le2\pi\), the two-spinon envelope is

It follows by extremizing \(\epsilon(p)+\epsilon(Q-p)\) over allowed individual momenta. The minimum puts one spinon at a soft endpoint; the maximum shares the total momentum equally. The shared sine/elliptic-band helper also used by XXX/XXZ supplies this calculation; the TB layer owns normalization and sectors.

For four spinons, let \(q=\min(Q,2\pi-Q)\). Including total-momentum aliases gives

Soft spectator spinons preserve the minimum. Concavity of sin(p) on [0,pi] puts the maximum at four equal momenta, choosing the alias nearest a total of 2pi. At Q=0, the two-spinon interval collapses to zero but the four-spinon upper edge is 4v: ignoring the modulo-2pi alias would miss it entirely. At Q=pi their upper edges are 2v and \(2\sqrt{2}v\) respectively. Neither upper edge bounds the full spectrum with arbitrarily many spinons.

--folded takes the envelope of Q and Q+pi, corresponding to a two-site iMPS unit cell. p remains in radians/site, while cell_momentum is \(K=2p\pmod{2\pi}\) in radians/cell. The lower edge is unchanged. With \(r=\min(q,\pi-q)\), the folded upper edges are \(2v\cos(r/2)\) and \(4v\cos(r/4)\). Folding is bookkeeping for the chosen cell, not evidence of a gapped or spontaneously dimerized ground state at the critical TB point.

Finite chains are checks, not the thermodynamic answer¶

Independent ED constructs the spin-1 bond matrix, projects lattice translations, and resolves total S using the Casimir. It does not use the Bethe equations or dispersion formulas. Fixed Sz alone is insufficient to label total S. For J=1, selected gaps are:

The small-momentum triplet velocity approaches 2pi. The staggered triplet has scaling dimension 3/8, with significant logarithmic finite-size corrections; its positive finite-L gap should not be mistaken for a thermodynamic gap. Tests compare the existing finite-root ground solver against the independent ED normalization while retaining its finite string deviations. Replacing those roots by ideal two-strings is not an exact finite-size excitation method.

Native-precision regressions also check an 80-digit dispersion reference, independent rapidity-space expressions, exchange scaling, direct momentum scans, four-particle convolutions and folding. Unrepresentable positive energies or bounds produce precision_limit, missing values (JSON null), and exit 2. Invalid parameters exit 1; an exchange that overflows the whole elementary band is rejected before opening exports. The implementation evaluates the soft endpoints exactly and does not silently replace underflowing positive energies by gapless zeros. There are no numerical quadrature or root-solving tolerances for these analytic curves.

Library interface¶

#include <bethe/tb_dispersion.hpp>
bethe::takhtajan_babujian::SpinonDispersion<long double> band(1.0L);
auto spin = band.spin(); // Uni20 half_int, exactly 1/2
auto energy = band.energy(0.3L);
auto pair = band.continuum(3.0L);
auto four = band.four_spinon_continuum(3.0L, bethe::SpinonMomentum::folded);

Finite-ring excited-string enumeration, fields, open ends, spectral intensities and general spin-s chains are not implemented by this frontend.