Application of Kadanoff's lower-bound renormalization transformation to the Blume-Capel model

The critical behavior of the Blume-Capel model for $d=2,3\mathrm{}$ is analyzed using a renormalization transformation based on Kadanoff's lower-bound transformation. There are fixed points associated with first-order, second-order, and tricritical phase transitions. The discontinuity fixed point onto which the line of first-order phase transition is mapped has two relevant eigenoperators with eigenvalue $2^d$, corresponding to the discontinuities in $〈\sigma 〉$ and $〈{\sigma }^2〉$. Quite different values of the variational parameters in the renormalization transformation optimize the free energy at each fixed point, making it difficult to calculate crossover behavior with the present approach.