In 1979, [scientist:Wu-Pei Su]Su, [scientist:John Robert Schrieffer]Schrieffer, and [scientist:Alan Jay Heeger]Heeger proposed a model that explained the strange electrical properties of polyacetylene, an organic polymer that conducts electricity almost like a metal. Their chain consisted of carbon atoms connected alternately by single and double bonds. An electron hopping along such a chain feels the difference between the bonds: along one it hops more easily, along the other with more difficulty. This simple model turned out to be topological: it has an indestructible knot that cannot be untied without breaking the chain itself. At the edges of such a knot, states with fractional charge are born, as if the electron splits into parts.

For a long time, topological phases were studied in a world without interactions: electrons were considered unsociable loners that do not notice each other. But in a one-dimensional chain, avoiding interactions is impossible: two electrons cannot pass each other, like two pedestrians on a narrow bridge. Each step of one affects the other. In the new work, physicists included this interaction in the SSH model, and the knot did not come undone. It merely tightened differently. The topological phase survived, but its charge changed.

To understand what happened, the authors translated the language of electrons into the language of collective excitations, quasiparticles. The interaction turned individual electrons into new objects with fractional charge and anomalous dimension. Anomalous dimension is a correction to how an excitation depends on scale: it is as if the quasiparticle grew a "fluffy fur coat" of virtual particles that changes its effective charge. The polarization, the displacement of charge within the unit cell, remained quantized, but its amplitude was multiplied by a factor depending on the interaction strength. At zero interaction, this factor equals 1, and the classical result 1/2 is recovered. As the interaction grows, the factor changes smoothly, and at the threshold g = π/2 the gap closes and the theory becomes massless.

This result is a bridge between worlds. Previously, it was unclear whether topology survives interactions in one dimension and whether the polarization can be computed analytically. Now it has been shown: yes, it survives, and yes, it can be computed. The trick is simple in essence: the model was reduced to the integrable Thirring theory, where all needed quantities are known exactly. This provides a benchmark for testing numerical methods and experiments with cold atoms in optical lattices, where interactions are controlled with high precision. There, fractional charge can be measured directly by the charge transfer through the chain.

The knot, tightened by interactions, did not come undone; it became stronger. This is the main lesson: topology in one dimension is not a fragile toy for noninteracting electrons, but a robust structure that withstands the pressure of correlations. Ahead lie generalizations to long-range interactions, to models with broken symmetry, and to ladder structures. Perhaps this way we will learn to weave topological knots in strongly correlated materials, where conventional notions of particles no longer work.