Z_{4}-symmetric perturbations to the XY model from functional renormalization

Area of Science:

• Condensed Matter Physics
• Statistical Mechanics
• Quantum Field Theory

Background:

• The XY model is a fundamental model in statistical mechanics, crucial for understanding phase transitions.
• Perturbations to the XY model, particularly Z_{4}-symmetric ones, are essential for exploring critical phenomena and universality classes.
• Renormalization group (RG) techniques are powerful tools for studying systems at critical points and understanding their behavior across different dimensions.

Purpose of the Study:

• To investigate the effects of cubic Z_{4}-symmetric perturbations on the classical XY model in dimensions ranging from 2 to 4.
• To accurately estimate critical exponents, specifically the leading irrelevant eigenvalue (y_{4}), in three dimensions.
• To trace the evolution of physical behavior as dimensionality is reduced towards the 2D Kosterlitz-Thouless transition and analyze the interplay of different symmetry-breaking perturbations.

Main Methods:

• Employing the second-order derivative expansion of the nonperturbative renormalization group.
• Calculating accurate estimates for the leading irrelevant eigenvalue (y_{4}) in d=3.
• Analyzing the behavior of irrelevant eigenvalues under changes in dimensionality, particularly approaching d=2.
• Comparing different implementations of the derivative expansion for systems with one and two symmetry invariants.

Main Results:

• Accurate estimation of the leading irrelevant eigenvalue y_{4} in d=3.
• Observation of the approximate collapse of leading irrelevant eigenvalues for both O(2)- and Z_{4}-symmetric perturbations as dimensionality approaches d=2.
• Recovery of the onset of Kosterlitz-Thouless physics in d=2, indicating the importance of these perturbations in understanding the transition.
• Discussion of the impact of different derivative expansion implementations on the results.

Conclusions:

• The study provides crucial insights into the behavior of the XY model under Z_{4}-symmetric perturbations across various dimensions.
• The observed collapse of irrelevant eigenvalues near d=2 highlights a significant feature of the critical behavior leading to the Kosterlitz-Thouless transition.
• The findings contribute to a deeper understanding of universality classes and critical phenomena in (2+1)-dimensional systems.