On the complexity of the steady-state of weakly symmetric open quantum lattices

Abstract: 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 determinining the \emph{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. As far as we know, this is the first time one is capable of determining the steady-state structure of such model for the 12 sites cluster exactly. Altough in this work we consider explicitly only spin-1/2 systems, our approach can be exploited in the characterisation of arbitrary spin systems, fermion and boson systems (with truncated Fock space), as well as many-particle systems with degrees of freedom having different statistical properties.