Abstract
We studied magnetic orderings, phase transitions, and frustrations in the Ising, 3-state Potts and standard 4-state Potts models on 1D, 2D, and 3D lattices: linear chain, square, triangular, kagome, honeycomb, and body-centered cubic. The main challenge was to find out the causes of frustrations phenomena and those features that distinguish frustrated system from not frustrated ones. The spins may interrelate with one another via the nearest-neighbor, the next-nearest-neighbor or higher-neighbor exchange interactions and via an external magnetic field that may be either competing or not. For problem solving we mainly calculated the entropy and specific heat using the rigorous analytical solutions for Kramers-Wannier transfer-matrix and exploiting computer simulation, par excellence, by Wang-Landau algorithm. Whether a system is ordered or frustrated is depend on the signs and values of exchange interactions. An external magnetic field may both favor the ordering of a system and create frustrations. With the help of calculations of the entropy, the specific heat and magnetic parameters, we obtained the points and ranges of frustrations, the frustration fields and the phase transition points. The results obtained also show that the same exchange interactions my either be competing or noncompeting which depends on the specific model and the lattice topology.
Highlights
Methods and basic formulaeWhere J and J′ mean exchange interactions between nearest-neighbor and next-nearest-neighbor spins; h is an external magnetic field; sums run over lattice sites i, nearest-neighbors ∆, next-nearest-neighbors ∆′, vectors s (with a module equal to unity) take on two, three, or four values, admissible in the Ising, 3-state Potts and 4-state Potts models
We studied magnetic orderings, phase transitions, and frustrations in the Ising, 3-state Potts and standard 4-state Potts models on 1D, 2D, and 3D lattices: linear chain, square, triangular, kagome, honeycomb, and body-centered cubic
Where J and J′ mean exchange interactions between nearest-neighbor and next-nearest-neighbor spins; h is an external magnetic field; sums run over lattice sites i, nearest-neighbors ∆, next-nearest-neighbors ∆′, vectors s take on two, three, or four values, admissible in the Ising, 3-state Potts and 4-state Potts models
Summary
Where J and J′ mean exchange interactions between nearest-neighbor and next-nearest-neighbor spins; h is an external magnetic field; sums run over lattice sites i, nearest-neighbors ∆, next-nearest-neighbors ∆′, vectors s (with a module equal to unity) take on two, three, or four values, admissible in the Ising, 3-state Potts and 4-state Potts models. If the analytical solution for maximum eigenvalue λ of Kramers-Wannier transfer-matrix was found, the entropy S , specific heat C, magnetization M, etc. Are only expressed in terms of λ:. In the absence of analytical solution, we performed the computer simulations by Monte-Carlo method, mainly in Wang-Landau algorithm [1]
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