Classical lattice spin models involving singular interactions isotropic in spin space.

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Quantum Field Theory

Background:

• The Mermin-Wagner theorem prohibits orientational order in continuous spin models at finite temperatures.
• Generalizations of the theorem also exclude phase transitions in one dimension (D=1).

Purpose of the Study:

• Investigate classical lattice spin models with n-component unit vectors (n=2,3) on D-dimensional lattices (D=1,2).
• Analyze models with potentials that are bounded below and have integrable singularities.
• Determine the presence or absence of phase transitions and orientational order.

Main Methods:

• Exact solutions for D=1 lattice spin models.
• Extensive numerical simulations for D=2 lattice spin models.
• Analysis of potentials defined by scalar products of interacting spins.

Main Results:

• For D=1, exact solutions confirm the absence of phase transitions and orientational order at all finite temperatures in the thermodynamic limit.
• For D=2, simulations indicate the absence of orientational order at finite temperatures.
• Simulations for D=2 suggest the potential existence of a Berezinskiĭ-Kosterlitz-Thouless transition.

Conclusions:

• The studied lattice spin models exhibit distinct behavior compared to models with continuous potentials.
• The findings contribute to understanding phase transitions and critical phenomena in lower-dimensional systems.
• The results highlight the importance of potential characteristics in determining thermodynamic properties.