Renormalization group for first-order phase transitions: Equation of state of the compressible Ising magnet

Thermodynamic functions of an $n$-component magnetic system coupled to an isotropic elastic medium in $d=4-\epsilon$ dimensions are calculated to leading order in $\epsilon$. Solutions to renormalization-group recursion relations correct to $O\left(\epsilon \right)$ along with trajectory-integral methods are employed in these calculations. For $n>~2$ the finite compressibility modifies the critical-point behavior producing new corrections to scaling in the thermodynamic functions which are given. In the $n=1\mathrm{}$ case renormalization-group predictions of a first-order transition are verified. Thermodynamic functions are given and analyzed in this case. In particular the transition temperature and discontinuity in order parameter are determined and the phase diagram is constructed.