XXX chains: states, spinons and spectral weights¶
All models · First calculation
The spin-\(1/2\) Heisenberg chain is a useful starting point for both finite-size energies and MPS excitation comparisons. The guides use \(J=1\) and spin operators \(\mathbf S=\boldsymbol\sigma/2\). Match those conventions before comparing energies or intensities.
| What to calculate | Program | Guide |
|---|---|---|
| Periodic ground state, sector minima and explicit Bethe states | bethe-xxx-pbc |
Periodic XXX |
| Free-end ground state, sector minima and real-root excitations | bethe-xxx-obc |
Open XXX |
| Selected periodic real-root excitations | bethe-xxx-pbc --excitations |
Excitation families and enumeration |
| Odd-ring one-spinon branch or thermodynamic spinon kinematics | bethe-xxx-pbc --spinons; library formulas |
Single spinons |
| Finite-ring two-spinon Lehmann spectrum | bethe-xxx-structure-factor |
Weights, momentum and sum rules |
| Infinite-chain selected spinon spectral densities | bethe-xxx-structure-factor-thermo |
Two-spinons, selected sectors |
| Direct fixed-momentum and global spinon moments | bethe-xxx-moments-thermo |
Moment definitions, controls and exports |
Choose a benchmark¶
Start with finite chains to understand quantum numbers, open versus periodic boundaries and the selected excitation family. For precision-sensitive gaps, use the precision tutorial.
The four-spinon tutorial adds a physically normalized contribution and measures quadrature/cutoff refinement and cost. Continue with six-spinon sampling and the 2/4/6 comparison to examine individual sectors, partial sums and the response above the two-spinon upper edge.
For a dynamical comparison, distinguish the finite-ring spectral lines from the unbroadened thermodynamic density. The finite-ring tool computes the supported two-spinon family. The thermodynamic tool defaults to two spinons and can include four and six; these remain partial responses. Broadening discrete lines is a plotting choice; it does not supply the missing state families.
Interpret the labels¶
Total spin, magnetization and a particular root family are different labels.
The excitation guide explains highest-weight states and what all enumerates.
An open chain has standing waves rather than a lattice momentum. For a periodic
chain, the physical momentum and any unit-cell folding must be matched to the
comparison calculation. Every model result still needs its convergence status
and numerical work limits; choosing MPFR does not enlarge the physical family.
For thermodynamic four- and six-spinon spectra, see selected sectors, explicit proposals and diagnostics.