We study the interplay between critical isotropic elasticity and classical Ising criticality using a functional renormalization group (FRG) approach which is implemented such that the volume is fixed during the entire renormalization group flow. For dimensions slightly smaller than four, we use a simple truncation of the FRG flow equations to recover the fixed points of the constrained Ising model: the Gaussian fixed point G, the Ising fixed point I, the renormalized Ising fixed point R, and the spherical fixed point S. We show that the fixed points R and S are both characterized by a finite anomalous dimension $y_{*}<0$ of strain fluctuations, implying that the energy dispersion of longitudinal acoustic phonons exhibits a nonanalytic momentum dependence proportional to $k^{1-y_{*}/2}$ for small momentum $k$. We also derive and solve flow equations for the free energy at constant strain and compute stress-strain relations in the vicinity of the fixed points. As a result, we reaffirm that Ising criticality, controlled by the fixed point I, is preempted by a bulk instability. Beyond that, we find that the stress-strain relation at R and S remains linear to leading order (Hooke's law), as long as the interaction between strain and Ising fluctuations is sufficiently weak. However, the finite anomalous dimension of strain fluctuations $y_{*}$ gives rise to nonanalytic corrections to Hooke's law.

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• Critical exponents
• Critical phenomena
• Elasticity
• Equations of state
• Functional renormalization group
• Phase transitions
• Second order phase transitions
• Ising model
• Path-integral methods