Single-spinon dispersion¶

XXX family · Supported models

There are two related but different calculations here: an odd periodic XXX chain supplies a finite-size one-spinon branch, while analytic functions give the infinite-chain dispersion. Their energy references are different.

bethe-xxx-pbc 65 --spinons
bethe-xxx-pbc 15 --spinons --roots --precision long-double

The finite branch is not the complete Sz=1/2 spectrum. Begin with periodic XXX conventions if the lattice momentum or quantum-number labels are unfamiliar.

Finite odd-chain branch¶

For odd N, set \(M =(N -1)/2\). The one-spinon family occupies M of the M+1 slots -M/2, -M/2+1, ..., M/2, leaving a hole \(I_{h}\). Use one_spinon_state<Real>(N, hole, options) for one member, or one_spinon_branch<Real>(N, options) for all \((N +1)/2\) members in ascending k. Each SpinonState contains its RealState in .state, the exact .hole, and:

\[ \begin{aligned} k&=\frac{\pi}{2}-\frac{2\pi I_h}{N} &&\text{(spinon momentum)},\\{} E_{\mathrm{bulk\text{-}subtracted}}&=E_N-Ne_\infty, &e_\infty&=\frac14-\log 2. \end{aligned} \]

The allowed finite-size k values run from \(\pi /(2N)\) to \(\pi -\pi /(2N)\) in steps of \(2\,\pi /N\). This is a hole-based spinon convention, with P = pi*M + pi/2 - k (mod 2*pi), not momentum relative to the odd-chain ground state. Both lattice P and spinon k are reported. The bulk-subtracted energy retains finite-size corrections; it is not obtained by subtracting the odd-chain ground-state energy, which itself contains a spinon.

Thermodynamic dispersion¶

For the analytic zero-field thermodynamic dispersion, include <bethe/spinon.hpp> (also included by <bethe/heisenberg.hpp>):

long double k = 1.0L;
auto xxx = bethe::heisenberg::spinon_energy(k);          // J=1
auto xxz = bethe::xxz::spinon_energy(k, 0.5L);           // Delta=0.5, J=1
auto scaled = bethe::xxz::spinon_energy(k, 0.5L, 2.0L); // J=2
auto e_inf = bethe::heisenberg::bulk_energy_density<long double>();

These implement \(\epsilon (k)=J \,\pi \,\sin (k)/2\) for XXX and \(\epsilon (k)=J \,\pi \,\sin (\gamma)\,\sin (k)/(2\,\gamma)\), \(\gamma =\arccos (\Delta)\), for gapless XXZ. They require finite \(k\) in [0,pi], positive finite J, and for XXZ \(-1\lt \Delta \le 1\). The isotropic endpoint uses the explicit XXX limit; k=0 and k=pi return exactly zero. These functions are analytic, distinct from the finite-size XXZ solver. See References and provenance for the derivations.

The separate XXZ dispersion frontend and library extend this thermodynamic coverage to Δ>1 and add bulk energy, the single-spinon gap, and two-spinon continuum edges. That guide explains the topological vacuum sectors and two-site-unit-cell momentum folding needed for MPS comparisons.