Boundary Fidelity and Entanglement in the symmetry protected topological phase of the SSH model

TL;DR

This paper investigates boundary effects and entanglement properties in the topological phase of the SSH model, revealing boundary-localized entanglement and phase transitions using analytical and numerical methods.

Contribution

It provides a detailed analysis of boundary contributions to fidelity susceptibility and entanglement in the SSH model, including effects of interactions and phase transitions.

Findings

Boundary fidelity susceptibility differs between topological and trivial phases.

Entanglement spectrum and entropy match field theory predictions for bulk and boundary cases.

Interaction induces a phase transition to a trivial charge-density wave phase.

Abstract

We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermions---a simplified Su-Schrieffer-Heeger (SSH) model---with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a scaling analysis of these data for essentially semi-infinite chains we obtain the fidelity susceptibility and show that it contains a boundary contribution which is different in the topologically ordered than in the topologically trivial phase. For the entanglement spectrum and entropy we confirm predictions from massive field theory for a block in the middle of an infinite chain but also consider blocks containing the edge of the chain. For the latter case we show that in the SPT phase additional entanglement---as compared to the trivial phase---is present which is localized at the boundary. Finally, we extend our study to the dimerized chain with a nearest-neighbour interaction using exact diagonalization, Arnoldi, and density-matrix renormalization group methods and show that a phase transition into a topologically trivial charge-density wave phase occurs.