which preserves $C_4$ rotation about this axis (in the absence of Zeeman field, the direction of strain is again unimportant). This leads to the perturbations $$\begin{aligned}
{\mathcal H}' = -\delta (J_z^2-\frac{5}{4}) -H (\cos(\theta) J_z + \sin(\theta) J_z^3) ,\label{pert}\end{aligned}$$ where $\delta$ parametrizes the strain, $h$ is the Zeeman field, and $\theta$ controls the strength of the cubic Zeeman term allowed by the cubic symmetry [@kane_fu; @ewald]. Using the RG results, the dimensions of these perturbations are $[\delta]=z-\eta_1 \approx 2.1$ and $[H]=z-\eta_{12}\approx 1.9$; [*i.e.*]{} strain is slightly enhanced while Zeeman field is slightly suppressed by interactions. However, both dimensions are positive and close to $2$, so that they are strongly [*relevant*]{}. They flow to strong coupling under the RG, and the fate of the system must be re-analyzed in the limit.

To do so, we assume, and check self-consistently, that interactions have weak effects at strong coupling, and simply solve the quadratic Hamiltonian (with $m/M_0=m/M_c=0$) in the presence of the renormalized $\mathcal{H}'$. The result depends upon the dimensionless quantities $\theta$ and the renormalized coupling ratio $\Delta = (\delta/H)_R \sim
\delta/H^{(z-\eta_1)/(z-\eta_{12})}$. For $H=0$ ($\Delta=\infty$), we have time-reversal invariance, and we recover the known result that strain $\delta>0$ induces a gapped, 3d TI phase, 