Is there a nematic phase in the spin-1 bilinear-biquadratic chain?
In plain words
The spin-1 chain with both ordinary and squared spin couplings has a known phase diagram except near its boundary with ferromagnetism. There a spin-nematic phase was proposed, and numerics have not cleanly confirmed or excluded it.
Precise statement
$H = \sum_i [\cos(\theta) S_i . S_{i+1} + \sin(\theta) (S_i . S_{i+1})^2]$, $S = 1$, for $\theta$ just above $-3\pi/4$, next to the ferromagnetic phase. Determine whether the dimerized phase extends all the way to $\theta = -3\pi/4$ with an exponentially small gap, or a gapless or gapped nematic phase intervenes. An answer gives the phase boundary and the dimerization order parameter as $\theta \to -3\pi/4$.
What would settle it
High-precision DMRG or field-theory analysis that resolves the exponentially small gap and dimerization near $\theta = -3\pi/4$.
Status in the literature
Proposed by Chubukov (1991); numerics favor dimerization up to the ferromagnetic boundary but cannot resolve the exponentially small gap.