Finite-size scaling in anisotropic systems

Area of Science:

• Condensed matter physics
• Statistical mechanics

Background:

• Anisotropy in physical systems leads to direction-dependent critical exponents.
• Systems with long-range interactions or near quantum critical points often exhibit anisotropy.
• Geometric confinement and boundary conditions influence scaling behavior.

Purpose of the Study:

• To present analytical results for finite-size scaling in anisotropic O(N) systems.
• To investigate how direction-dependent critical exponents affect various physical systems.
• To analyze the impact of specific geometric confinements on scaling properties.

Main Methods:

• Analytical calculations in the N --> infinity limit for O(N) systems.
• Utilizing the properties of generalized Mittag-Leffler functions to overcome computational challenges.
• Considering systems confined to a d-dimensional layer with specific boundary conditions.

Main Results:

• Direction-dependent critical exponents (nu{ ||} and nu{ perpendicular}) were derived for anisotropic O(N) systems.
• The study provides a framework for understanding finite-size scaling in systems with anisotropic long-range interactions.
• Analytical results were obtained for systems with Lifshitz points and space-time anisotropy.

Conclusions:

• Finite-size scaling in anisotropic systems is complex and direction-dependent.
• The employed mathematical techniques are effective for analyzing such systems.
• The findings are relevant for diverse physical systems exhibiting anisotropy and specific geometries.