Math Conjectures

Haldane gap for the spin-11 Heisenberg chain

OPENLandmarkConjectureProposed 1983 · Canonical special case

Canonical statement

For every integer L≥2L\ge2, on (C3)⊗L(\mathbb C^3)^{\otimes L} with periodic boundary conditions let
HL=∑j=1LSj⋅Sj+1,SL+1=S1, H_L=\sum_{j=1}^{L}\mathbf S_j\cdot\mathbf S_{j+1}, \qquad \mathbf S_{L+1}=\mathbf S_1,
where Sj\mathbf S_j are the spin-11 SU(2)SU(2) generators at site jj. If E0(L)<E1(L)E_0(L)<E_1(L) are its two lowest distinct eigenvalues, then
lim inf⁡L→∞(E1(L)−E0(L))>0. \liminf_{L\to\infty}\bigl(E_1(L)-E_0(L)\bigr)>0.
View source LaTeX
For every integer \(L\ge2\), on \((\mathbb C^3)^{\otimes L}\) with periodic boundary conditions let \[ H_L=\sum_{j=1}^{L}\mathbf S_j\cdot\mathbf S_{j+1}, \qquad \mathbf S_{L+1}=\mathbf S_1, \] where \(\mathbf S_j\) are the spin-\(1\) \(SU(2)\) generators at site \(j\). If \(E_0(L)<E_1(L)\) are its two lowest distinct eigenvalues, then \[ \liminf_{L\to\infty}\bigl(E_1(L)-E_0(L)\bigr)>0. \]
MATH-PHYS-002:[Haldane1983NonlinearField][AffleckKennedyLiebTasaki1987RigorousResults][Nachtergaele1996SpectralGap][Tasaki2025HeisenbergTopology]
Mathematical physicsAnalysis and functional analysis

Notes

In 1983 Haldane argued, via a mapping of large-spin Heisenberg antiferromagnets to a nonlinear field theory, that one-dimensional integer-spin chains behave fundamentally differently from half-integer ones: integer-spin chains should have a spectral gap above the ground state [Haldane1983NonlinearField]. The canonical case is the spin-11 chain with nearest-neighbor coupling HL=∑jSj⋅Sj+1H_L=\sum_j\mathbf S_j\cdot\mathbf S_{j+1} on a periodic ring; the conjecture asserts that the gap E1(L)−E0(L)E_1(L)-E_0(L) stays bounded away from zero as L→∞L\to\infty.

The prediction ran against naive extrapolation from the gapless spin-1/21/2 chain but is now firmly supported. Affleck, Kennedy, Lieb and Tasaki constructed a deformed spin-11 chain with an exact valence-bond ground state and a rigorously proven gap [AffleckKennedyLiebTasaki1987RigorousResults], and rigorous gaps are known for broad frustration-free and perturbative classes [Nachtergaele1996SpectralGap]. Numerics and experiment place the Heisenberg gap near 0.41050.4105 in units of the coupling, and recent work shows the Heisenberg ground state is topologically nontrivial [Tasaki2025HeisenbergTopology].

What is missing is a proof for the exact Heisenberg Hamiltonian itself, which lies outside the rigorously gapped classes; a uniform positive lower bound on its gap remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.