Incommensurability and edge states in the one-dimensional S=1 bilinear-biquadratic model

TL;DR

This paper investigates the transition between commensurate and incommensurate phases in the one-dimensional S=1 bilinear-biquadratic model, analyzing static structure factors and energy gaps to understand the nature of this change.

Contribution

It provides a detailed analysis of the commensurate-incommensurate transition using analytical and numerical methods, clarifying the behavior of static structure factors near the AKLT point.

Findings

Numerical support for one of the analytical predictions.

Identification of the transition as a pair of poles moving in the complex plane.

Clarification of the static structure factor behavior near the AKLT point.

Abstract

Commensurate-incommensurate change on the one-dimensional S=1 bilinear-biquadratic model (${\cal H}(\alpha)=\sum_i \{{\bf S}_i\cdot {\bf S}_{i+1} +\alpha ({\bf S}_i\cdot{\bf S}_{i+1})^2\}$) is examined. The gapped Haldane phase has two subphases (the commensurate Haldane subphase and the incommensurate Haldane subphase) and the commensurate-incommensurate change point (the Affleck-Kennedy-Lieb-Tasaki point, $\alpha=1/3$). There have been two different analytical predictions about the static structure factor in the neighborhood of this point. By using the S{\o}rensen-Affleck prescription, these static structure factors are related to the Green functions, and also to the energy gap behaviors. Numerical calculations support one of the predictions. Accordingly, the commensurate-incommensurate change is recognized as a motion of a pair of poles in the complex plane.