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We present a matrix-product state (MPS) based band-Lanczos method as solverfor quantum cluster methods such as the variational cluster approximation(VCA). While a na\"ive implementation of MPS as cluster solver would barelyimprove its range of applicability, we show that our approach makes it possibleto treat cluster geometries well beyond the reach of exact diagonalizationmethods. The key modifications we introduce are a continuous energy truncationcombined with a convergence criterion that is more robust against approximationerrors introduced by the MPS representation and provides a bound to deviationsin the resulting Green's function. The potential of the resulting clustersolver is demonstrated by computing the self-energy functional for thesingle-band Hubbard model at half filling in the strongly correlated regime, ondifferent cluster geometries. Here, we find that only when treating largecluster sizes, observables can be extrapolated to the thermodynamic limit,which we demonstrate at the example of the staggered magnetization. Treatingclusters sizes with up to $6\times 6$ sites we obtain excellent agreement withquantum Monte-Carlo results.

Matrix-product-state-based band-Lanczos solver for quantum cluster approaches