Universality class for driven interfaces and... integrable spin hydrodynamics?

The Kardar-Parisi-Zhang (KPZ) universality class, originally formulated to describe driven systems such as growing interfaces, has undergone several paradigm shifts [1]. One major breakthrough was the discovery of exact solutions for one-dimensional models within the KPZ class — remarkable given their non-equilibrium and non-linear nature — enabled by underlying integrability. These exact results revealed nontrivial fluctuation properties, some closely linked to random matrix theory, which were subsequently observed in real experiments on driven interfaces. But more recently, the KPZ framework appears to be entering a new phase, extending unexpectedly to integrable spin chains at thermal equilibrium [2,3]. Although this connection was nearly dismissed when clear discrepancy in full counting statistics was reported, the speaker and collaborators numerically found that various two-point quantities agree precisely with KPZ exact solutions, so the KPZ class indeed governs integrable spin chains, yet only their two-point quantities [4]. I will also discuss a recent hydrodynamic theory aiming to bridge spin chains and KPZ, which, currently, falls short of fully explaining the numerical observations and calls for further refinement [2,3].