Local and global magnetization on the Sierpiński carpet.

Area of Science:

• Statistical physics
• Condensed matter physics
• Complex systems

Background:

• The classical Ising model is a fundamental tool for studying magnetism and phase transitions.
• Fractal lattices, like the Sierpiński carpet, exhibit complex geometries that can alter critical phenomena.
• Understanding phase transitions on fractal structures is crucial for materials science and complex systems theory.

Purpose of the Study:

• To investigate the phase transition of the classical Ising model on a Sierpiński carpet fractal lattice.
• To analyze the position dependence of local functions and critical exponents.
• To determine the global critical exponent for spontaneous magnetization.

Main Methods:

• Utilized an adapted higher-order tensor renormalization group method.
• Employed impurity tensors to study position-dependent local functions.
• Applied automatic differentiation to calculate free energy derivatives for magnetization.

Main Results:

• Observed a second-order phase transition at a critical temperature (Tc) of approximately 1.478.
• Found that the critical exponent β for local magnetization varied significantly with lattice location.
• Determined the global critical exponent β for average spontaneous magnetization to be approximately 0.135.

Conclusions:

• The critical temperature of the Ising model on a Sierpiński carpet is robust against local variations.
• Lattice location critically influences local magnetic properties and associated critical exponents.
• This study provides insights into the behavior of magnetic systems on complex fractal geometries.