MerminWagnerHohenberg

The Mermin-Wagner-Hohenberg theorem is a fundamental result in the theory of phase transitions and critical phenomena in two-dimensional systems. It was independently proven by Norman Mermin and John Michael Kosterlitz in 1966, and later extended by Michael Hohenberg in 1967. The theorem states that continuous symmetry breaking cannot occur in two-dimensional systems at finite temperatures, except in the case of the XY model with a vortex-antivortex pair unbinding transition. This means that systems with continuous symmetries, such as the two-dimensional Ising model, cannot exhibit a true phase transition at any finite temperature. Instead, they exhibit a Kosterlitz-Thouless transition, where topological defects (vortices) play a crucial role.

The theorem is based on the concept of topological defects and the behavior of the correlation functions in two-dimensional systems. In systems with continuous symmetries, the correlation functions decay algebraically at long distances, rather than exponentially as in higher-dimensional systems. This algebraic decay is a signature of the absence of long-range order and the presence of topological defects. The defects, such as vortices in the XY model, are pairs of topological charges that can unbind and move freely in the system, leading to a transition that is distinct from the conventional phase transitions observed in higher-dimensional systems.

The Mermin-Wagner-Hohenberg theorem has significant implications for our understanding of phase transitions and critical phenomena. It highlights the fundamental differences between two-dimensional and higher-dimensional systems and provides insights into the role of topological defects in these systems. The theorem also has practical implications, as it helps to explain the behavior of certain two-dimensional systems, such as thin films and monolayers, which are often used in modern technologies. Understanding the Mermin-Wagner-Hohenberg theorem is essential for researchers in condensed matter physics, statistical mechanics, and materials science.