Topology, symmetry, phase transitions, and noncollinear spin structures

Area of Science:

• Condensed Matter Physics
• Statistical Mechanics
• Topology in Physics

Background:

• The infinite-range XY model on the AB2 chain exhibits frustration- and field-induced phase transitions.
• Understanding these transitions, especially noncollinear spin structures, requires advanced theoretical frameworks.
• Topological approaches offer a novel perspective on phase transitions in magnetic systems.

Purpose of the Study:

• To describe frustration- and field-induced phase transitions in the infinite-range XY model using a topological approach.
• To investigate the role of topological invariants, such as Morse number and Euler characteristic, in characterizing these transitions.
• To establish a connection between the system's thermodynamics and the topology of its configuration space.

Main Methods:

• Computation of topological invariants: Morse number, Euler characteristic, and other invariants.
• Application of a statistical mechanics analogy to compute the Euler characteristic.
• Introduction of topological energies to analyze transition properties at zero and finite temperatures.
• Exact solution of system thermodynamics via the saddle-point approach.

Main Results:

• Topological invariants exhibit similar behavior as a function of energy level, consistent with Morse theory.
• A feasible method for computing the Euler characteristic using a statistical mechanics analogy was developed.
• A direct connection between thermodynamics and configuration space topology was established.
• Conditions for suppressing the divergence of Jacobian's critical points at critical energy were identified.
• Cusplike singularities in Euler characteristic and topological entropy, alongside the divergence of Jacobian's critical points, are proposed as necessary and sufficient conditions for finite-temperature topology-induced phase transitions.

Conclusions:

• The study successfully employs topological methods to characterize phase transitions in the infinite-range XY model.
• Topological properties, particularly Euler characteristic and entropy contributions, are crucial for understanding finite-temperature transitions.
• The findings suggest a general framework for identifying topology-induced phase transitions, warranting further rigorous investigation.