Persistent Haldane phase in carbon tetris chains

We introduce the concept of “tetris chains,” which are linear arrays of four-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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>U</a:mi><a:mo>≫</a:mo><a:mi>t</a:mi></a:mrow></a:math>, weak coupling <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>U</b:mi><b:mo>≪</b:mo><b:mi>t</b:mi></b:mrow></b:math>, and the weak intermolecular hopping limit <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msup><c:mi>t</c:mi><c:mo>′</c:mo></c:msup><c:mo>≪</c:mo><c:mi>t</c:mi></c:mrow></c:math>. 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msup><d:mi>t</d:mi><d:mo>′</d:mo></d:msup><d:mtext>−</d:mtext><d:mi>U</d:mi></d:mrow></d:math> plane due to the cooperative nature of the three limits. Secondly, we study a near-homogeneous “<e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msup><e:mrow><e:mi mathvariant="normal">Y</e:mi></e:mrow><e:mo>′</e:mo></e:msup></e:math>-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 band-structure winding number.