Persistent Haldane Phase in Carbon Tetris Chains

TL;DR

This paper explores the fermionic Haldane phase in novel tetris chain systems, demonstrating its robustness across various models and parameters through advanced numerical and analytical methods.

Contribution

It introduces tetris chains as a new platform for studying the Haldane phase and shows its stability in experimentally relevant nanographene and related molecular systems.

Findings

Haldane phase is robust in Y-chain of triangulenes across the entire parameter space.

Haldane phase persists in Y'-chains related to poly(p-phenylene vinylene).

Numerical results confirm the topological phase stability using tensor network methods.

Abstract

We introduce the concept of tetris chains, which are linear arrays of 4-site molecules that differ by their intermolecular hopping geometry. We investigate the fermionic symmetry-protected topological Haldane phase in these systems using Hubbard-type models. The topological phase diagrams can be understood via different competing limits and mechanisms: strong-coupling $U\gg t$, weak-coupling $U\ll t$, and the weak intermolecular hopping limit $t'\ll t$. Our particular focus is on two tetris chains that are of experimental relevance. First, we show that a Y-chain of coarse-grained nanographene molecules (triangulenes) is robustly in the Haldane phase in the whole $t'-U$ plane due to the cooperative nature of the three limits. Secondly, we study a near-homogeneous Y$^{\prime}$-chain that is closely related to the electronic model for poly(p-phenylene vinylene). In the latter case, the above mechanisms compete, but the Haldane phase manifests robustly and is stable when long-ranged Pariser-Parr-Popple interactions are added. The site-edged Hubbard ladder can also be viewed as a tetris chain, which gives a very general perspective on the emergence of its fermionic Haldane phase. Our numerical results are obtained by large-scale, SU(2)-symmetric tensor network calculations. We employ the density-matrix-renormalization group as well as the variational uniform matrix-product state (VUMPS) algorithms for finite and infinite systems, respectively. The numerics are supplemented by analytical calculations of the bandstructure winding number.