O(N)-universality classes and the Mermin-Wagner theorem.

Area of Science:

• Condensed matter physics
• Quantum field theory
• Statistical mechanics

Background:

• Universality classes describe systems with similar critical behavior.
• The Mermin-Wagner-Hohenberg theorem states no spontaneous symmetry breaking in 2D for O(N)-symmetric systems with N>=1.
• Renormalization group (RG) methods are crucial for understanding critical phenomena.

Purpose of the Study:

• To investigate the continuous dependence of O(N)-symmetric model universality classes on dimension (d) and field components (N).
• To understand the emergence of Mermin-Wagner-Hohenberg theorem implications as d approaches 2.
• To explore multicritical behavior and O(N=0) universality classes.

Main Methods:

• Renormalization group analysis.
• Continuous deformation of theory space.
• Investigation of effective potentials.

Main Results:

• For fractal dimensions 2 < d < 3 and N ≥ 1, a finite family of multicritical effective potentials was found.
• These potentials disappear at d = 2 for N ≥ 1, consistent with the Mermin-Wagner-Hohenberg theorem.
• An infinite family of O(N=0) universality classes was identified in two dimensions.

Conclusions:

• The study provides a continuous picture of universality classes transitioning towards the 2D limit.
• It highlights the critical role of dimension and field components in determining critical behavior.
• The findings offer new insights into the nature of phase transitions and universality in various dimensions.