Duality at Lattice Scale around the Takhtajan-Babujain Point in the Bilinear Biquadratic Spin-1 Chain

In the literature, it is known that near the Takhtajan-Babujain point of the bilinear-biquadratic spin-$1$ chain, the low energy theory is captured by three Majorana fermions with a mass sign change, or three Ising chains near the transition. In this paper, we study how the spin-$1$, the Ising spins, and the fermions are related at lattice scale. We construct the spin-$1$ operator out from the Ising spins and fermions, showing that when the Ising chains are in the ordered phase, the spin-$1$ chain is in the dimer phase, and when the Ising chains are in the paramagnetic phase, the spin-$1$ chain is in the Haldane phase. Through this mapping we can see how unit cell doubling in the dimer phase and the boundary spin-$\frac12$ zero mode in the Haldane phase take place, even though in the Ising transition neither phenomena exist.