A quantum particle in a periodic potential, coupled to an Ohmic bath, undergoes a phase transition. Below a critical dissipation strength, the particle tunnels freely between wells. Above it, dissipation pins the particle in place. This is the Schmid transition, predicted decades ago but only now proven to belong to the BKT universality class.

The proof comes from Monte Carlo simulations revealing logarithmic decay of correlation functions at the critical point — the signature of BKT. The transition is driven by vortex unbinding in the imaginary-time representation, exactly as in the classical 2D XY model. Dissipation maps onto temperature: stronger dissipation corresponds to higher effective temperature, and the vortex-unbinding threshold determines whether the particle is localized or delocalized.

The result is exquisitely sensitive. It occurs only for Ohmic dissipation — where the spectral function of the bath is linear at low frequencies. Any deviation from linearity (sub-Ohmic or super-Ohmic) destroys the transition or changes its nature. The universality class depends not on the strength of the coupling but on its frequency dependence.

This is a case where the environment’s spectral shape, not its amplitude, determines the physics. A strongly coupled bath with the wrong spectrum does less than a weakly coupled bath with the right one. The organizing principle isn’t how much the environment disturbs the system, but how it distributes that disturbance across frequencies.