Heisenberg Spinons: Dispersion, des Cloizeaux–Pearson Continuum, Müller Ansatz

This page collects the closed-form excitation kinematics of the spin-$\tfrac{1}{2}$ isotropic antiferromagnetic Heisenberg chain in the thermodynamic limit that are exposed as Phase-1 helpers and fetch methods on Heisenberg1D.

Main results

For

\[H = J\sum_{i}\mathbf{S}_i\cdot\mathbf{S}_{i+1},\qquad J>0,\]

the elementary excitations are massless spinons (half-odd-integer spin) which always come in pairs in any physical observable. The single-spinon dispersion (Faddeev–Takhtajan 1981) is

\[\boxed{\;\varepsilon(k) \;=\; \frac{\pi J}{2}\,|\sin k|,\qquad k\in[0,\pi].\;}\]

The two-spinon continuum, parameterised by the total momentum $q$, is bounded by the des Cloizeaux–Pearson (1962) edges

\[\boxed{\; \varepsilon_L(q) \;=\; \frac{\pi J}{2}\,|\sin q|,\qquad \varepsilon_U(q) \;=\; \pi J\,\bigl|\sin(q/2)\bigr|. \;}\]

The lower edge coincides with the single-spinon dispersion and the continuum is gapless at $q=0$ and $q=\pi$ (Umklapp).

The longitudinal dynamic structure factor inside the continuum is approximated by the Müller ansatz (Müller–Thomas–Beck–Bonner 1981):

\[\boxed{\; S^{zz}_{\rm Müller}(q,\omega) \;=\; \frac{\Theta\!\bigl[\omega-\varepsilon_L(q)\bigr]\, \Theta\!\bigl[\varepsilon_U(q)-\omega\bigr]} {2\,\sqrt{\omega^2 - \varepsilon_L(q)^2}}, \;}\]

with $S^{zz}=0$ outside $[\varepsilon_L,\varepsilon_U]$.

Derivation sketch

Spinon dispersion (Faddeev–Takhtajan 1981)

The Bethe-ansatz solution of the spin-$\tfrac{1}{2}$ XXX antiferromagnet admits a thermodynamic state — the antiferromagnetic Dirac sea — built from a continuous distribution of real rapidities $\lambda\in\mathbb{R}$ with density $\rho_0(\lambda)=1/(2\cosh(\pi\lambda))$. Holes in this distribution carry spin $\tfrac{1}{2}$ — these are the spinons.

The dispersion of a single spinon is obtained by adding one hole of rapidity $\lambda$ on top of the sea. The resulting energy and momentum, relative to the ground state, are

\[\varepsilon(\lambda) \;=\; \frac{\pi J}{2}\, \frac{1}{\cosh(\pi\lambda)},\qquad p(\lambda) \;=\; \frac{\pi}{2} \;-\; \arctan\bigl(\sinh(\pi\lambda)\bigr).\]

Eliminating $\lambda$ via $\cosh(\pi\lambda) = 1/\sin p$ — itself a direct consequence of the second relation — yields the closed-form dispersion stated above:

\[\varepsilon(p) \;=\; \frac{\pi J}{2}\,\sin p,\qquad p\in[0,\pi].\]

The absolute value $|\sin p|$ in the boxed formula extends the result by the periodicity of the Brillouin zone.

Two-spinon continuum and des Cloizeaux–Pearson edges (1962)

A pair of spinons with momenta $k_1, k_2 \in [0,\pi]$ carries total momentum $q = k_1 + k_2$ (mod $2\pi$) and total energy $E = \varepsilon(k_1) + \varepsilon(k_2)$. At fixed $q$ the spinon-pair energy ranges over an interval whose endpoints are extracted by Lagrange-multiplying $\varepsilon(k_1)+\varepsilon(k_2)$ with the constraint $k_1 + k_2 = q$:

\[\partial_{k_1}\varepsilon(k_1) \;=\; \partial_{k_2}\varepsilon(k_2) \quad\Longleftrightarrow\quad \cos k_1 \;=\; \cos k_2.\]

Two solutions emerge.

$k_1 = k_2 = q/2$ — both spinons share the momentum, giving the upper edge
```math

\varepsilon_U(q) \;=\; 2\,\varepsilon(q/2) \;=\; \pi J\,\bigl|\sin(q/2)\bigr|.


* ``k_1 = 0,\ k_2 = q`` (one spinon at the gapless point) —
  giving the **lower edge**

  ```math
\varepsilon_L(q) \;=\; \varepsilon(0) + \varepsilon(q)
                       \;=\; \frac{\pi J}{2}\,|\sin q|.

Hence $\varepsilon_L(q) \equiv \varepsilon(q)$, and the continuum collapses ($\varepsilon_U = \varepsilon_L = 0$) at the gapless points $q = 0, \pi$.

Numerically, at $q = \pi$ one has $\varepsilon_L = 0$ and $\varepsilon_U = \pi J$, which is the value carried by heisenberg_two_spinon_upper_edge at the AFM ordering wave vector.

Müller ansatz for $S^{zz}(q,\omega)$ (1981)

Müller, Thomas, Beck, and Bonner proposed an explicit closed form for the longitudinal dynamic structure factor that

has the correct support on the two-spinon continuum,
reproduces the integrable square-root singularity at the lower edge $\omega \to \varepsilon_L^+$ (which dominates the spectral weight),
is normalised to give the correct equal-time longitudinal structure factor in leading order.

The ansatz is

\[S^{zz}_{\rm Müller}(q,\omega) \;=\; \frac{\Theta(\omega-\varepsilon_L)\,\Theta(\varepsilon_U-\omega)} {2\,\sqrt{\omega^2 - \varepsilon_L^2}},\]

returning $0$ outside the closed continuum $[\varepsilon_L(q),\varepsilon_U(q)]$. The ansatz is approximate: it captures the lower-edge behaviour and the support exactly but misestimates the spectral weight near the upper edge, where four-spinon contributions become important.

The ansatz value diverges as $\omega \to \varepsilon_L^+$ but remains integrable: $\int_{\varepsilon_L}^{\varepsilon_U} S^{zz}\,d\omega < \infty$. QAtlas returns the raw analytical value without a regulator; downstream callers integrating in $\omega$ should either use a quadrature aware of the square-root singularity (e.g. Gauss–Chebyshev, or the change of variables $\omega^2 = \varepsilon_L^2 + s$) or regulate via $\sqrt{\omega^2 - \varepsilon_L^2 + \eta^2}$ at their own choice of $\eta$.

API

heisenberg_spinon_dispersion(model::Heisenberg1D, k::Real; J::Real = 1.0)::Float64
heisenberg_two_spinon_lower_edge(model::Heisenberg1D, q::Real; J::Real = 1.0)::Float64
heisenberg_two_spinon_upper_edge(model::Heisenberg1D, q::Real; J::Real = 1.0)::Float64

fetch(::Heisenberg1D, ::ZZStructureFactor, ::Infinite;
      q::Real, ω::Real, method::Symbol = :muller, J::Real = 1.0)::Float64

Special values, all at $J = 1$:

quantity	$q = 0$	$q = \pi/2$	$q = \pi$
`heisenberg_spinon_dispersion`	$0$	$\pi/2$	$0$
`heisenberg_two_spinon_lower_edge`	$0$	$\pi/2$	$0$
`heisenberg_two_spinon_upper_edge`	$0$	$\pi/\sqrt{2}$	$\pi$

For the Quantity-based dispatch the routing is:

fetch(Heisenberg1D(), ZZStructureFactor(), Infinite();
      q = π/2, ω = 1.0)                        # → Müller ansatz
fetch(Heisenberg1D(), ZZStructureFactor(), Infinite();
      q = π/2, ω = 1.0, method = :caux_hagemans)
# → Phase-2 placeholder; raises an informative error

Quasiparticle dispersion stays a top-level helper (no QuasiparticleDispersion quantity type is introduced in Phase 1) — this matches the existing TFIM style with tfim_quasiparticle_dispersion and tfim_two_spinon_dos. A unified QuasiparticleDispersion quantity is a candidate refactor target if/when more models grow analogous helpers.

Phase 2 (TODO): exact dynamic structure factor

The Müller ansatz is the standard 1981-vintage approximation. The exact result for the dynamic longitudinal structure factor of the infinite Heisenberg chain is given by the algebraic-Bethe-ansatz form factor sum due to

J.-S. Caux, R. Hagemans, The four-spinon dynamical structure factor of the Heisenberg chain, J. Stat. Mech. P12013 (2006).

Implementing the Caux–Hagemans formula is Phase 2 of issue #154 and is not yet shipped. The current fetch(::Heisenberg1D, ::ZZStructureFactor, ::Infinite; method = :caux_hagemans, …) branch raises an informative ErrorException so downstream code can probe for availability. The Müller branch (method = :muller, default) remains the only Phase-1 implementation.

References

J. des Cloizeaux, J. J. Pearson, "Spin-wave spectrum of the antiferromagnetic linear chain", Phys. Rev. 128, 2131 (1962).
L. D. Faddeev, L. A. Takhtajan, "What is the spin of a spin wave?", Phys. Lett. A 85, 375 (1981).
G. Müller, H. Thomas, H. Beck, J. C. Bonner, "Quantum spin dynamics of the antiferromagnetic linear chain in zero and nonzero magnetic field", Phys. Rev. B 24, 1429 (1981).
J.-S. Caux, R. Hagemans, "The four-spinon dynamical structure factor of the Heisenberg chain", J. Stat. Mech. P12013 (2006). (Phase 2 reference.)