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Physical review. B, Condensed matter

Exact phase diagram of a generalized Kagomé Ising lattice: Reentrance and disorder lines

M. Debauche, H. T. Diep, P. Azaria, Héctor Giacomini

Published Aug 1, 1991• 35 citations
Publisher

Abstract

We solve exactly a generalized Kagom\'e Ising lattice with three kinds of interactions by transforming the system into a 16-vertex model that satisfies the free-fermion condition. The phase diagram shows several interesting phenomena due to the frustration generated by the competing interactions: successive transitions with reentrance, partial disorder, and disorder lines. The reentrance region in the space of interaction parameters is found to be infinite, unlike previous exactly solved models. Two disorder solutions with interesting behavior are found.

DOI
10.1103/physrevb.44.2369

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We solve exactly a generalized Kagom'e Ising lattice with three kinds of interactions by transforming the system into a 16-vertex model that satisfies the free-fermion condition. The phase diagram shows several interesting phenomena due to the frustration generated by the competing interactions: successive transitions with reentrance, partial disorder, and disorder lines. The reentrance region in the space of interaction parameters is found to be infinite, unlike previous exactly solved models. Two disorder solutions with interesting behavior are found.
After a short introduction on frustrated spin systems, we study in this chapter several two-dimensional frustrated Ising spin systems which can be exactly solved by using vertex models. We show that these systems contain most of the spectacular effects due to the frustration: high ground-state degeneracy, existence of several phases in the ground-state phase diagram, multiple phase transitions with increasing temperature, reentrance, disorder lines, partial disorder at equilibrium. Evidences of such effects in non solvable models are also shown and discussed.
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Through the direct decoration transformation approach, we obtain a general solution for the pentagonal Ising model, showing its equivalence to the isotropic free-fermion eight-vertex model. We study the ground-state phase diagram, in which one ferromagnetic (FM) state, one ferrimagnetic (FIM) state, and one frustrated state are found. Using the exact solution of the pentagonal Ising model, we discuss the finite-temperature phase diagrams and find a phase transition between the FIM state and the disordered state as well as a phase transition between the disordered state and the FM state. We also discuss some additional remarkable properties of the model, such as the magnetization, entropy, and specific heat, at finite temperature and at its low-temperature asymptotic limit. Because of the influence of the second-order phase transition between the frustrated and ferromagnetic phases, we obtain surprisingly low values of the entropy and the specific heat until the critical temperature is reached.
The Ising model is famous in condensed matter and statistical physics. In this work we present a free-fermion formulation of the two-dimensional classical Ising models on honeycomb, triangular and Kagomé lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field i(π/2)kBT. We employ the decorated lattice technique, star-triangle transformation, and weak-graph expansion method to exactly map each Ising model in both cases into an eight-vertex model on the square lattice. The resulting vertex weights are shown to satisfy the free-fermion condition. In the zero-field case, each Ising model is an even free-fermion model. In the case of the imaginary field, the Ising model on the honeycomb lattice is an even free-fermion model, while the models on the triangular and Kagomé lattices are odd free-fermion models. We obtain the exact solution of the Kagomé lattice Ising model under the imaginary field i(π/2)kBT, a result not previously reported in the literature. We also show that the frustrated Ising models on the triangular and Kagomé lattices in the imaginary field still exhibit a non-zero residual entropy.