Determine the spin-1 bilinear–biquadratic phase diagram

Determine the complete phase diagram of the spin-1 bilinear–biquadratic family of translation-invariant, rotation-invariant nearest-neighbor quantum spin chains, including the phases and phase boundaries represented in the conjectured diagram.

Background

The paper places the spin-1 singlet model, which is equivalent up to a shift and normalization to the purely biquadratic point, within the broader bilinear–biquadratic family. Several special points and phases are rigorously or exactly understood, including the Uimin–Lai–Sutherland, Takhtajan–Babujian, AKLT, and dimerized regimes.

However, the global phase diagram summarized in the paper is explicitly described as conjectural. Establishing the complete phase structure and rigorously identifying all phase boundaries would resolve the uncertainty represented by that conjectured diagram.

References

That family, $\cos\theta\,\left(\vec S_x\cdot\vec S_{x+1}\right) + \sin\theta\,\left(\vec S_x\cdot\vec S_{x+1}\right)2$, whose conjectured phase diagram is shown in Figure~\ref{fig:Spin1 Diagram}, has been studied intensively.

— On the Spectrum of Some Temperley-Lieb Spin Chains  (2609.38074 - Ferydouni et al., 29 Sep 2026) in Section 1, Introduction, paragraph discussing the bilinear–biquadratic family