  Recent years have seen the development of two types of non-local extensions
to the single-site dynamical mean field theory. On one hand, cluster
approximations, such as the dynamical cluster approximation, recover
short-range momentum-dependent correlations non-perturbatively. On the other
hand, diagrammatic extensions, such as the dual fermion theory, recover long
ranged corrections perturbatively. The correct treatment of both strong
short-ranged and weak long-ranged correlations within the same framework is
therefore expected to lead to a quick convergence of results, and offers the
potential of obtaining smooth self-energies in non-perturbative regimes of
phase space. In this paper, we present an exact cluster dual fermion method
based on an expansion around the dynamical cluster approximation. Unlike
previous formulations, our method does not employ a coarse graining
approximation to the interaction, which we show to be the leading source of
error at high temperature, and converges to the exact result independent of the
size of the underlying cluster. We illustrate the power of the method with
results for the second-order cluster dual fermion approximation to the
single-particle self-energies and double occupancies.
