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Topological quantum and classical materials can exhibit robust propertiesthat are protected against disorder, for example for noninteracting particlesand linear waves. Here, we demonstrate how to construct topologically protectedstates that arise from the combination of strong interactions and thermalfluctuations inherent to soft materials or miniaturized mechanical structures.Specifically, we consider fluctuating lines under tension (e.g., polymer orvortex lines), subject to a class of spatially modulated substrate potentials.At equilibrium, the lines acquire a collective tilt proportional to an integertopological invariant called the Chern number. This quantized tilt is robustagainst substrate disorder, as verified by classical Langevin dynamicssimulations. This robustness arises because excitations in this system ofthermally fluctuating lines are gapped by virtue of inter-line interactions. Weestablish the topological underpinning of this pattern via a mapping that wedevelop between the interacting-lines system and a hitherto unexploredgeneralization of Thouless pumping to imaginary time. Our work points to a newclass of classical topological phenomena in which the topological signaturemanifests itself in a structural property observed at finite temperature ratherthan a transport measurement.

Topological protection can arise from thermal fluctuations and interactions