Griffiths-like phase close to the Mott transition.

We explore the coexistence region in the vicinity of the Mott critical end point employing a coupled energy-volume spin-$1/2$ Ising-like model. We analyze the case for the spin-liquid candidate $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$, where close to the Mott critical end point metallic puddles coexist with an insulating ferroelectric phase. Our results are fourfold: $i$) a universal divergent-like behavior of the Gr\"uneisen parameter upon crossing the spinodal line; $ii$) based on scaling arguments, we show that in the coexistence region, for $any$ system close to the critical point, the relaxation time is entropy-dependent; $iii$) we propose the electric Gr\"uneisen parameter $\Gamma_E$, which quantifies the electrocaloric effect; $iv$) we identify the metallic/insulating coexistence region as a Griffiths phase. Our findings suggest that $\Gamma_E$ governs the dielectric response close to the critical point and that a Griffiths-like phase emerges in the coexistence region of any system.