# Finite Size Scaling at the Topological Transition: Bilinear-Biquadratic Spin-1 Chain

## TL;DR

This study demonstrates the universality of a finite size scaling function at a topological phase transition in a spin-1 chain, aligning it with non-interacting fermion models and revealing edge state effects.

## Contribution

It verifies the universality of the finite size scaling function for a topological transition in a spin-1 chain using high-accuracy simulations, extending understanding beyond non-interacting models.

## Key findings

- Scaling function matches that of non-interacting Majorana fermions.
- Universality holds even away from the conformal critical point.
- Differences between even and odd chains are explained by edge state interactions.

## Abstract

We consider a finite size scaling function across a topological phase transition in 1D models. For models of non-interacting fermions it was shown to be universal for all topological symmetry classes and markedly asymmetric between trivial and topological sides of the transition (Gulden et al 2016). 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 $L/\xi$, where $L$ is the chain length and $\xi$ is the correlation length, coincides with that of three species of non-interacting 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 and odd size chains, which may be fully accounted for by residual interactions of the edge states.

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Source: https://tomesphere.com/paper/1901.02107