Blume–Emery–Griffiths model

The classical Blume–Emery–Griffiths model uses spins ${\sigma }_i\in \{-1,0,1\}$ with both bilinear ${\sigma }_i{\sigma }_j$ and biquadratic ${\sigma }_i^2{\sigma }_j^2$ nearest-neighbour interactions, together with a single-site term proportional to ${\sigma }_i^2$. One convention is $H=-J\sum {\sigma }_i{\sigma }_j-L\sum {\sigma }_i^2{\sigma }_j^2-D\sum {\sigma }_i^2-h\sum {\sigma }_i$. Signs of the single-site parameter vary across conventions. Setting $L=0$ gives a Blume–Capel model after matching that sign convention. The one-dimensional version admits an exact spin-chain transfer matrix.