  Using the recently introduced time-local formulation of the nonequilibrium
cluster perturbation theory (CPT), we construct a generalization of the
approach such that macroscopic conservation laws are respected. This is
achieved by exploiting the freedom for the choice of the starting point of the
all-order perturbation theory in the inter-cluster hopping. The proposed
conserving CPT is a self-consistent propagation scheme which respects the
conservation of energy, particle number and spin, which treats short-range
correlations exactly up to the linear scale of the cluster, and which
represents a mean-field-like approach on length scales beyond the cluster size.
Using Green's functions, conservation laws are formulated as local continuity
equations for the local spin-dependent particle and the doublon density. We
consider them as conditional equations to self-consistently fix the
time-dependent intra-cluster one-particle parameters. Thanks to the intrinsic
causality of the CPT, this can be set up as a step-by-step time propagation
scheme with a computational effort scaling linearly with the maximum
propagation time and exponentially in the cluster size. As a proof of concept,
we consider the dynamics of the two-dimensional, particle-hole-symmetric
Hubbard model following a weak interaction quench by simply employing two-site
clusters only. Conservation laws are satisfied by construction. We demonstrate
that enforcing them has strong impact on the dynamics. While the doublon
density is strongly oscillating within plain CPT, a monotonic relaxation is
observed within the conserving CPT.
