Generalized Solution of Inverse Problem for Ising Connection Matrix on d-Dimensional Hypercubic Lattice

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Computational Physics

Background:

• The Ising model is a fundamental model in statistical mechanics for understanding magnetism.
• Analyzing spin interactions is crucial for predicting material properties.
• Eigenvalue spectra contain rich information about system Hamiltonians.

Purpose of the Study:

• To develop a method for determining spin interaction constants from the eigenvalue spectrum of a d-dimensional Ising system.
• To investigate the influence of boundary conditions on interaction range.

Main Methods:

• Analysis of the connection matrix of a d-dimensional Ising system.
• Solving the inverse problem using the known eigenvalue spectrum.
• Comparison of results under periodic and free boundary conditions.

Main Results:

• Successfully restored spin interaction constants based on the eigenvalue spectrum.
• Demonstrated that periodic boundary conditions permit interactions between arbitrarily distant spins.
• Showed that free boundary conditions restrict interactions to the first d coordination spheres.

Conclusions:

• The eigenvalue spectrum provides a viable route to reconstruct spin interactions in Ising systems.
• Boundary conditions significantly dictate the nature and range of spin-spin interactions.
• This inverse problem approach offers insights into complex magnetic systems.