Strongly correlated electron systems are characterized by a manifold of complex, competing phenomena, which emerge in the thermodynamic limit. The underlying mechanisms involve correlations on all length scales. Currently no single feasible numerical method exists to accurately address these correlations at all lengths. Both finite size and mean-field calculations alike are faced with these limitations. However, interest in areas such as quantum phase transitions and magnetically-driven superconductivity point to the need for numerical schemes that accurately bridge the short and long length scales.

Quantum cluster techniques(1), constitute a good staring point in addressing the entire range of correlations by dividing the problem into two length regimes; explicitly solving for short ranged correlations and approximately for the remaining longer length-scales. One such method, the dynamical cluster approximation (DCA)(2), maps the lattice problem onto an embedded cluster problem. In doing so, short-ranged correlations within a given cluster are treated accurately, while the remaining longer-ranged correlations are approximated on a dynamical mean-field level. However, the extent of length-scales which can thus be accurately addressed beyond the mean-field level is severely limited by the numerical expense involved.

This limitation results in an inadequate treatment of medium ranged correlations which are outside the scope of explicit calculations. We introduce a Multi-scale Many-Body (MSMB) approach which addresses each length-scale using approximations adequate for the strength of the correlations on the respective scale. The strongest, local and short ranged correlations are well accounted for in traditional, numerically exact implementations of the DCA. Correlations, except for in the vicinity of phase transitions, fall off rapidly with distance and are hence considered weaker in the intermediate length-regime. However, these correlations remain significant and will hence be approximated diagrammatically. Only the remaining third regime of the longest length scale will be treated at the dynamical mean-field level.

The perturbative inclusion of correlations on an intermediate
length-scale within
a multi-scale approach has previously been explored by Hague
*et. al.* (3). Contributions to the single-particle
self-energy on various scales were linked in a hybrid approach.
However, the inherent perturbative nature of the approach limited it
to high temperatures and/or weak coupling strengths.
In this work, we present a non-perturbative two-particle diagrammatic
approach to the intermediate length-regime.

To illustrate and test this scheme, this MSMB is applied to the one dimensional Hubbard model(4). While a formally simple model, the Hubbard model contains a multitude of the underlying physics of correlated electron systems. It therefore lends itself ideally as a benchmark for the method. We show that the MSMB approach yields results in very good quantitative agreement with explicit large cluster calculations.

Before we proceed to provide detailed results in section 4.3 we first establish the theoretical basis for the method. A subsequent outlook on further developments in MSMB techniques is provided in section 3.6, concluding with a brief summary.