# Local entanglement and String Order Parameter in dimerized models

## TL;DR

This paper applies String Order Parameter to fermionic systems, specifically a dimerized fermionic chain, to identify topological phases and analyze phase transitions using analytical and entanglement measures.

## Contribution

It introduces a novel application of SOP to fermionic models and analytically characterizes topological phase transitions via Toeplitz determinants and entanglement measures.

## Key findings

- SOP successfully identifies topological phases in fermionic chains.
- Phase transition characterized by non-analytic behavior of concurrence derivative.
- Analytical expressions for SOP and entanglement measures in the model.

## Abstract

In this letter, we propose an application of String Order Parameter (SOP), commonly used in quantum spin systems, to identify symmetry-protected topological phase (SPT) in fermionic systems in the example of the dimerized fermionic chain. As a generalized form of dimerized model, we consider a one-dimensional spin-1/2 XX model with alternating spin couplings. We employ Jordan-Wigner fermionization to map this model to the spinless Su-Schrieffer-Heeger fermionic model (SSH) with generalized hopping signs. We demonstrate a phase transition between a trivial insulating phase and the Haldane phase by the exact analytical evaluation of reconstructed SOPs which are represented as determinants of Toeplitz matrices with the given generating functions. To get more insight into the topological quantum phase transition (tQPT) and microscopic correlations, we study the pairwise concurrence as a local entanglement measure of the model. We show that the first derivative of the concurrence has a non-analytic behaviour in the vicinity of the tQPT, like in the second order trivial QPTs.