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 nonunitary counterpart of the universality class of long-range interacting Ising models. For $\alpha <1$ we find a nonequilibrium critical point ruled by a dynamical field theory described by a Langevin model with coexisting inertial ($\sim {\partial }_t^2$) 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 Letter lays out perspectives for a revision of universality in driven open systems by employing dark states tailored by programmable dissipation.

Figure 1

Schematic portrait of a spin lattice subject to nonlocal losses, ${\gamma }_{i,j}$, acting on pairs of spins at positions $i$ and $j$.