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Critical behavior in rectangles with mixed boundaries.

E Eisenriegler¹

¹Theoretische Physik der lebenden Materie, Institut für biologische Informationsprozesse (IBI-5), Forschungszentrum Jülich, D-52425 Jülich, Germany.

Physical Review. E

|November 18, 2023

Summary

This study explores density profiles in a 2D critical Ising model within rectangular domains. Exact results for energy and order parameter profiles reveal complex behaviors, especially near mixed boundaries.

Area of Science:

Statistical Physics
Condensed Matter Physics
Critical Phenomena

Background:

The critical Ising model in two dimensions is a fundamental model in statistical mechanics.
Understanding behavior in confined geometries is crucial for realistic physical systems.

Purpose of the Study:

Investigate density profiles of energy and order parameter in a 2D critical Ising model confined to a rectangle.
Analyze the impact of uniform and mixed boundary conditions on these profiles.
Explore the behavior near corners with mixed boundary conditions.

Main Methods:

Utilizing conformal transformations to map results from a half-plane to a rectangular domain.
Applying a "corner-operator-expansion" to describe behavior near mixed boundaries.

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Comparing analytical predictions with simulation data.

Main Results:

Exact results for density profiles of energy and order parameter were derived for specific boundary conditions (+-+- and +f+f+).
Rich and complex behavior was observed, particularly influenced by mixed boundary conditions at the rectangle's corners.
The corner-operator-expansion effectively describes the behavior near these critical points.

Conclusions:

The study provides exact analytical results for density profiles in a confined 2D critical Ising model.
Mixed boundary conditions introduce unique and complex behaviors, especially at corners.
Analytical predictions show excellent agreement with simulations, validating the theoretical framework.