Critical exponents can be different on the two sides of a transition: A generic mechanism

We present models where $\gamma_+$ and $\gamma_-$, the exponents of the susceptibility in the high and low temperature phases, are generically different. In these models, continuous symmetries are explicitly broken down by discrete anisotropies that are irrelevant in the renormalization-group sense. The $\mathbb{Z}_q$-invariant models are the simplest examples for two-component order parameters ($N=2$) and the icosahedral symmetry for $N=3$. We compute accurately $\gamma_+ -\gamma_-$ as well as the ratio $\nu/\nu'$ of the exponents of the two correlation lengths present for $T<T_c$.