Monte Carlo method for critical systems in infinite volume: The planar Ising model

Area of Science:

• Statistical Physics
• Computational Physics
• Critical Phenomena

Background:

• Simulating infinite statistical systems on finite lattices is challenging due to boundary effects.
• Existing methods struggle with long-range correlations at critical points.
• Accurate generation of critical distributions is crucial for understanding phase transitions.

Purpose of the Study:

• To develop a novel Monte Carlo method for generating finite-domain marginals of critical distributions in infinite volume.
• To address and correct for boundary effects in simulations of critical phenomena.
• To validate the method using the planar Ising model.

Main Methods:

• Proposed a Monte Carlo algorithm incorporating scale invariance and renormalization group concepts.
• Developed a "holographic" boundary condition to simulate infinite volume effects.
• Applied the method to the planar Ising model, comparing observables to theoretical predictions.

Main Results:

• Accurately reproduced planar two-, three-, and four-point functions for spin and energy operators.
• Defined and numerically obtained a lattice stress-energy tensor.
• Verified conformal Ward identities and calculated the Ising central charge.

Conclusions:

• The proposed Monte Carlo method effectively generates critical distributions for infinite systems.
• Holographic boundary conditions provide an accurate way to handle finite lattice limitations.
• The method's success with the Ising model demonstrates its potential for studying critical phenomena.