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

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