Finite-size scaling at a topological transition: Bilinear-biquadratic spin-1 chain

We consider a finite-size scaling function across a topological phase transition in one-dimensional models. For models of noninteracting fermions it was shown to be universal for all topological symmetry classes and markedly asymmetric between trivial and topological sides of the transition [T. Gulden, M. Janas, Y. Wang, and A. Kamenev, Phys. Rev. Lett. 116, 026402 (2016)10.1103/PhysRevLett.116.026402]. Here we verify its universality for the topological transition between dimerized and Haldane phases in bilinear-biquadratic spin-1 chain. To this end we perform high-accuracy variational matrix product state simulations. We show that the scaling function, expressed in terms of L/ζ, where L is the chain length and ζ is the correlation length, coincides with that of three species of noninteracting massive Majorana fermions. The latter is known to be a proper description of the conformal critical theory with central charge c=3/2. We have shown that it still holds away from the conformal point, including the finite-size corrections. We have also observed peculiar differences between even- A nd odd-size chains, which may be fully accounted for by residual interactions of the edge states.