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This study introduces a free-fermion formulation for 2D Ising models on honeycomb, triangular, and Kagomé lattices. Exact solutions were found, including a novel result for the Kagomé lattice under an imaginary field.

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Area of Science:

  • Condensed matter physics
  • Statistical physics
  • Quantum field theory

Background:

  • The Ising model is a fundamental model in statistical physics.
  • Understanding its behavior on different lattices and under various field conditions is crucial.

Purpose of the Study:

  • To develop a free-fermion formulation for 2D classical Ising models on honeycomb, triangular, and Kagomé lattices.
  • To analyze these models under zero and imaginary field conditions.
  • To obtain exact solutions and investigate residual entropy.

Main Methods:

  • Decorated lattice technique
  • Star-triangle transformation
  • Weak-graph expansion method
  • Mapping to the eight-vertex model on a square lattice

Main Results:

  • All studied Ising models were mapped to free-fermion eight-vertex models.
  • Ising models in zero field are even free-fermion models.
  • Under an imaginary field, honeycomb lattice models are even, while triangular and Kagomé models are odd free-fermion models.
  • The exact solution for the Kagomé lattice Ising model with an imaginary field was obtained.

Conclusions:

  • The free-fermion condition is satisfied by the mapped vertex weights.
  • The Kagomé lattice Ising model under an imaginary field exhibits a non-zero residual entropy.
  • Frustrated Ising models on triangular and Kagomé lattices retain residual entropy even in an imaginary field.