Quantum Brownian Motion: proving that the Schmid transition belongs to the Berezinskii-Kosterlitz-Thouless universality class

TL;DR

This paper demonstrates that the dissipation-driven Schmid transition in a quantum Brownian particle with a periodic potential belongs to the BKT universality class, with critical behavior governed by the environmental spectral function.

Contribution

It establishes the BKT universality class of the Schmid transition using Monte Carlo simulations and finite-size scaling, clarifying the role of dissipation and potential in quantum phase transitions.

Findings

The Schmid transition is in the BKT universality class.

The transition disappears in over- and sub-Ohmic regimes.

Periodic potential does not affect localization in non-Ohmic regimes.

Abstract

We investigate the equilibrium properties of a quantum Brownian particle moving in a periodic potential, specifically addressing the nature of the dissipation-driven Schmid transition in the Ohmic regime. By employing World-Line Monte Carlo in the path-integral formalism and introducing a specific binary order parameter, we demonstrate that the transition belongs to the Berezinskii-Kosterlitz-Thouless universality class. This classification is substantiated through finite-size scaling analysis that reveals the characteristic logarithmic decay of the correlation functions associated with the order parameter at the critical point. Quantum phase transition turns out to be extremely fragile: it disappears in both over- and sub-Ohmic dissipation regimes. Crucially, we find that the presence of the periodic potential does not alter the localization properties in the sub-Ohmic and super-Ohmic regimes, where the system exhibits the same qualitative behavior as the free quantum Brownian particle. These findings highlight that the emergence of critical behavior is strictly governed by the low-frequency form of the environmental spectral function, which determines the long-range temporal decay of the dissipative kernel.