# Does 1D Ising model with next-nearest neighbor interaction has solution?

We know that a one-dimensional Ising model that only considers nearest neighbor (NN) interaction can be solved exactly. The $$\mathcal{H}$$ of such model is given by: $$\mathcal{H}=-J \sum_{i=1}^{N} \sigma_i\sigma_{i+1} - B\mu\sum_{i=1}^{N} \sigma_i$$, or the scaled version as:

$$\bar{\mathcal{H}} = -\beta\mathcal{H}= K \sum_{i=1}^{N} \sigma_i\sigma_{i+1} + h \sum_{i=1}^{N} \sigma_i \qquad \qquad ;\;K=-\beta J,\; h=-\beta B \mu$$ Now, I can solve this model analytically using the Transfer matrix method as detailed in here, for example. We also know that 2D Ising model has an exact solution when there is no external magnetic field $$(h=0)$$.

But what I am wondering is if this is the case for NN interaction only. What if we consider next-nearest-neighbor (NNN) interactions? Can we solve it exactly? If not, can we solve it at least in case of no external magnetic field?

For example, the Hamiltonian in that case would be given by

$$\bar{\mathcal{H}} = K_1 \sum_{i=1}^{N} \sigma_i\sigma_{i+1} + K_2 \sum_{i=1}^{N} \sigma_i\sigma_{i+2}$$

where $$K_1$$ and $$K_2$$ are the coupling constants for the NN and the NNN interactions, respectively. Can we apply the Transfer matrix method here? or do we need to apply something else? It seems to me that here I need Transfer tensor, if such thing exists, instead of Transfer matrix since I have to take care of three indices $$(i,i+1,i+2)$$ instead of two $$(i,i+1)$$. Or can we do it using Transfer matrix but maybe that matrix is not square or something?

P.S.: In all cases, I assume a periodic boundary condition of $$\sigma_{N+1}=\sigma_1$$.

• Sure. Any one-dimensional Ising model with finite-range, translation-invariant interactions can be solved using the transfer matrix (but the size of the latter increases with the range, of course). Here's an example with first and second-nearest neighbors. In this case, the transfer matrix is a $4\times4$ matrix. Jun 13, 2023 at 7:27
• When $K_1<0$ and $K_2>0$ and no field you have the rather famous axial next-nearest neighbor Ising model, or ANNNI model, the solution of which should be easy to find in the literature. Jun 15, 2023 at 22:29