Fidelity susceptibility of Su-Schrieffer-Heeger model with further neighbour hopping term

In the quiet world of quantum materials, scientists look for the moments when matter fundamentally changes its nature without any heat or external push. These are called quantum phase transitions, and they happen at the coldest possible temperature, where the strange rules of quantum mechanics take over. Unlike the familiar freezing of water or the boiling of a pot, these transitions are driven by the invisible fluctuations of particles themselves. To understand these shifts, researchers often study simple models that capture the essence of complex materials. One such model is the Su-Schrieffer-Heeger model, a theoretical chain of atoms that helps explain how electrons move through certain plastics and why some materials act as insulators while others conduct electricity. A key feature of this model is its ability to exist in two different states: a "trivial" state where electrons are stuck in place, and a "topological" state where the material's internal structure allows for special, protected pathways for electrons to travel along its edges.

To detect exactly when a material flips from one of these states to the other, physicists use a tool called fidelity susceptibility. Think of this as a highly sensitive measure of how much a system's internal arrangement changes when you tweak a single control knob, like the strength of the connection between atoms. If the system is stable, a small tweak causes only a tiny change. But right at the moment of a phase transition, the system becomes incredibly fragile; a tiny nudge causes a massive reorganization, and the fidelity susceptibility spikes dramatically. This makes it a powerful way to find the exact point where the material's identity changes, without needing to know beforehand what that change looks like.

In a recent study, researchers Surajit Mandal and Asim Kumar Ghosh from Jadavpur University in India applied this method to a more complex version of the standard chain. They added a new feature: a "further neighbor" hopping term. In the original model, electrons could only jump to the immediate next atom. In this extended version, they introduced a rule allowing electrons to occasionally jump over one atom to land on the next-nearest neighbor. The scientists wanted to see if this extra freedom for the electrons would change the way the material transitions between its different states, and whether the sharp spike in fidelity susceptibility would still appear to mark the boundary.

The team built a mathematical description of this extended chain and ran detailed computer simulations to see how the system behaved. They calculated the fidelity susceptibility for chains of varying lengths, checking both even and odd numbers of atoms to ensure their results were robust. They found that just as in the simpler, original model, the fidelity susceptibility did indeed shoot up to a sharp peak at specific points. These peaks marked the exact locations where the material switched from a trivial insulator to a topological one, and even where it switched between two different types of topological phases. The researchers discovered that adding this extra hopping term created a new transition point that did not exist in the original model, effectively splitting the topological region into two distinct parts.

To be certain of their findings, the researchers compared their computer simulations with a precise mathematical formula they derived for the system. The two approaches matched almost perfectly, confirming that the peaks were real physical features of the model and not just artifacts of the calculation. They also examined how the height of these peaks grew as the chain got longer, a process known as scaling. They found that the peaks grew in a predictable way, following a specific mathematical pattern that was identical to the pattern seen in the standard model without the extra hopping. This suggests that while the extra hopping term changes where the transitions happen and adds a new type of phase, it does not alter the fundamental nature of the transition itself.

The study also explored what happens when the environment around the chain is disturbed by adding a staggered potential, which is like placing a slightly different energy cost on alternating atoms in the chain. In the standard model, this disturbance destroys the special topological properties. The researchers confirmed that this remains true even with the extra hopping term: the sharp peaks in fidelity susceptibility disappeared, and the material stayed in a simple, non-topological state. This tells us that while the extra hopping term adds complexity, it does not make the system immune to environmental disruptions.

Ultimately, this work provides a clearer map of how quantum materials behave when their internal connections are made more complex. By showing that the fidelity susceptibility remains a reliable and sharp indicator of phase transitions, even in these extended models, the researchers have reinforced a powerful tool for future discoveries. They demonstrated that the fundamental rules governing these quantum shifts are robust, surviving the addition of new pathways for electrons, while also revealing that the delicate nature of topological states means they can still be easily broken by the wrong kind of disturbance. The findings offer a deeper understanding of the geometry of quantum states, helping scientists predict how new materials might behave before they are ever built in a lab.