Ising model with next nearest neighbors

Yamashita and Miyazima [58] and Yamashita [59], [60] adopted the modified ANNNI (axial next-nearest-neighbor Ising) model with the third-neighboring interaction (ANNNI + Jj, model) and wrote the Hamiltonian... 

In this section we extend the treatment of the one-dimensional Ising model in two directions first, for units having more than two states second, for units interacting with both nearest neighbors as well as with next-nearest neighbors. 

Ising Model with Nearest- and Next-Nearest-Neighbor Interactions... 

Fig. 5. Spin configurations near nonmagnetic atoms (circles) of a diluted ferromagnet with competing nearest (/,) and next-nearest (Z ) neighbor interactions, R = J Ui < 0- Only the spins which are not aligned in the direction of the spontaneous magnetization (pointing upwards) are indicated by arrows. The most simple cases of both the Ising and XY or Heisenberg model are shown (from Maletta 1981b).

We shall start in the next section with the simplest Ising model. This is a linear sequence of particles, or units, each of which can attain one of two possible states. Interactions between units extend to the nearest neighbor units only. 

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