Two-Dimensional Phase Transitions in Classical Systems: 60 Years after the Hohenberg-Mermin-Wagner Theorem

Abstract

In 1966, Hohenberg, Mermin and Wagner proved that long-wavelength fluctuations destabilize the long-range order of continuous symmetry in two-dimensional (2D) systems. Later in the 1970s, Berezinskii, Kosterlitz and Thouless developed the BKT theory describing an unconventional phase transition between quasi-long-range and short-range order in 2D systems driven by the binding-unbinding of topological defects, which has become a fundamental topic in statistical mechanics, condensed matter physics, and soft matter physics. One of the most important applications of the BKT theory is the melting of 2D crystals, whose mechanisms are not yet fully understood. Recently, this topic has been extended to the area of active matter, where the non-equilibrium nature leads to novel phenomena that deviate from the Hohenberg-Mermin-Wagner theorem. In this review, we first focus on the recent theoretical and computational progress in the 2D melting problem in passive systems, and then summarize the inspiring results obtained from non-equilibrium systems. The review closes with comments on several promising directions for predicting 2D melting scenarios and for understanding the non-equilibrium nature in 2D active matter systems.