Universality class of Ising critical states with long-range losses

We show that spatial resolved dissipation can act on $d$-dimensional spin systems in the Ising universality class by qualitatively modifying the nature of their critical points. We consider power-law decaying spin losses with a Lindbladian spectrum closing at small momenta as $\propto q^\alpha$, with $\alpha$ a positive tunable exponent directly related to the power-law decay of the spatial profile of losses at long distances, $1/r^{(\alpha+d)}$. This yields a class of soft modes asymptotically decoupled from dissipation at small momenta, which are responsible for the emergence of a critical scaling regime ascribable to the non-unitary counterpart of the universality class of long-range interacting Ising models. For $\alpha<1$ we find a non-equilibrium critical point ruled by a dynamical field theory described by a Langevin model with coexisting inertial ($\sim {\partial^2_t}$) and frictional ($\sim {\partial_t}$) kinetic coefficients, and driven by a gapless Markovian noise with variance $\propto q^\alpha$ at small momenta. This effective field theory is beyond the Halperin-Hohenberg description of dynamical criticality, and its critical exponents differ from their unitary long-range counterparts. Our work lays out perspectives for a revision of universality in driven-open systems by employing dark states taylored by programmable dissipation.