Complexity of the steady state of weakly symmetric open quantum lattices

We investigate the properties of Lindblad equations on d-dimensional lattices supporting a unique steady-state configuration. We consider the case of a time evolution weakly symmetric under the action of a finite group G, which is also a symmetry group for the lattice structure. We show that in such case the steady state belongs to a relevant subspace, and provide an explicit algorithm for constructing an orthonormal basis of such set. As explicitly shown for a spin-1/2 system, the dimension of such subspace is extremely smaller than the dimension of the set of square operators. As a consequence, by projecting the dynamics within such set, the steady-state configuration can be determined with a considerably reduced amount of resources. We demonstrate the validity of our theoretical results by determining the exact structure of the steady-state configuration of the two-dimensional XYZ model in the presence of uniform dissipation, with and without magnetic fields, up to a number of sites equal to 12. Although in this work we consider explicitly only spin-1/2 systems, our approach can be exploited in the characterization of arbitrary spin systems and fermion and boson systems (with truncated Fock space), as well as many-particle systems with degrees of freedom having different statistical properties.