Thouless pumping and Haldane-model topology in a driven extended SSH chain

In the world of modern physics, scientists have discovered that matter can exist in states defined not by how its atoms are arranged, but by the hidden geometry of the way electrons move through it. These are called topological phases. Imagine a coffee mug and a donut; to a mathematician, they are the same shape because you can stretch one into the other without tearing a hole or gluing a piece on. This idea of shape that survives stretching applies to the invisible paths electrons take. In two-dimensional materials, this geometry creates a special kind of electrical current that flows along the edges without any resistance, a phenomenon known as the quantum Hall effect. To create this effect in a lab, researchers usually need to build complex two-dimensional grids and apply strong magnetic fields, which is difficult to control. However, a clever idea emerged: if you take a one-dimensional chain of atoms and wiggle its properties in a precise, repeating cycle, you can trick the system into behaving as if it were a two-dimensional object. This process, known as Thouless pumping, allows scientists to measure the same hidden geometric properties using a simple, linear setup.

A team of researchers at Sichuan Normal University has taken this concept further by designing a specific type of atomic chain that acts as a bridge between a simple one-dimensional line and the complex two-dimensional world of the Haldane model. The Haldane model is a famous theoretical blueprint for a material that conducts electricity on its edges without a magnetic field, but it is notoriously hard to build in reality because it requires very specific, complex connections between atoms. The researchers proposed a solution using an extended version of a well-known model called the Su-Schrieffer-Heeger chain. In this setup, atoms are arranged in pairs, and the connections between them can be tuned. The team added an extra layer of complexity by allowing atoms to also jump over their immediate neighbors to connect with the next ones. By carefully adjusting the strength of these jumps over time in a repeating loop, they created a "pump" that moves charge in a quantized, or fixed, amount with every cycle.

To understand how this works, the researchers visualized the system not as a physical chain of atoms, but as a set of moving vectors in an abstract space. They found that as they cycled the parameters of the chain, these vectors traced out a closed loop. The key discovery was that the shape of this loop and how it moved through space determined the outcome. If the loop wrapped around a specific point in this abstract space exactly once, the system would pump a precise amount of electric charge. This wrapping behavior is the physical signature of the topological property they were looking for. The researchers showed that by choosing the right path for their adjustments, they could force the system to wrap around this point, effectively turning the one-dimensional chain into a machine that generates a quantized current. This current is robust, meaning it stays the same even if the system is slightly disturbed, because it is protected by the underlying geometry of the loop.

The team then tested this idea in two different ways to prove it would work in real experiments. First, they simulated a scenario where the chain was completely filled with particles, like a full pipe of water. In this case, they calculated the flow of current and found that after one full cycle of their adjustments, exactly two units of charge were transported across the system. This confirmed that the topological property was indeed present and measurable, with the total charge being twice the fundamental Chern number of the underlying two-dimensional model. Second, they addressed a practical problem: in many real-world experiments, especially with cold atoms, it is difficult to fill the entire system uniformly. To solve this, they introduced a gentle tilt to the chain, which causes a single packet of particles to oscillate back and forth while the system is being pumped. This oscillation acts like a scanner, sweeping the packet across the entire range of the system's properties. Their simulations showed that even with just this single packet of particles, the center of the packet moved a precise, quantized distance after a set number of cycles. This distance corresponded directly to the topological number of the system, proving that the effect could be measured without needing a perfectly filled band.

Perhaps the most significant finding of the work is the direct link they established between their one-dimensional chain and the complex two-dimensional Haldane model. By mathematically mapping the time of their pumping cycle to a second dimension of space, they demonstrated that their simple chain is a perfect stand-in for the difficult-to-build two-dimensional model. They showed that the same rules governing the exotic two-dimensional material apply to their one-dimensional pump, though with a crucial scaling factor: because the geometry of the system causes the parameter space to be covered twice during a full cycle, the measured charge is twice the Chern number. This means that scientists can now study the mysterious properties of the Haldane model using a much simpler, one-dimensional setup that is easier to build and control in the lab. The researchers confirmed that the "phase diagram" of their chain—the map of which settings produce which results—matches the Haldane model exactly. They found that by changing the strength of the connections and the timing of the cycle, they could switch the system between conducting and non-conducting states, just as one would in the two-dimensional version.

This work offers a practical and accessible route for detecting high-dimensional topology. Instead of struggling to create complex two-dimensional lattices with magnetic fields, researchers can use a driven one-dimensional chain to read out the same topological numbers. The study suggests that this approach is viable for platforms like photonic lattices, where light moves through glass structures, and cold-atom systems, where atoms are trapped by lasers. By using the simple act of wiggling a chain of atoms or light paths, scientists can now probe the deep, geometric secrets of matter that were previously hidden in complex two-dimensional geometries. The results, derived from detailed simulations and theoretical mapping, provide a clear blueprint for future experiments to observe these topological phenomena in a controlled, one-dimensional environment.