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Matrix product state based band Lanczos solver for quantum cluster approaches

Sebastian Paeckel1,*,†, Thomas Köhler2,3,*,‡, Salvatore R. Manmana4,§, and Benjamin Lenz5,∥

  • *These authors contributed equally to this work.
  • †Contact author: sebastian.paeckel@physik.uni-muenchen.de
  • ‡Contact author: t.kohler@hw.ac.uk
  • §Contact author: salvatore.manmana@theorie.physik.uni-goettingen.de
  • ∥Contact author: benjamin.lenz@sorbonne-universite.fr
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Phys. Rev. B 112, 235108 – Published 2 December, 2025

DOI: https://doi.org/10.1103/tjnr-kqmk

Abstract

We present a matrix-product state (MPS) based band Lanczos method as solver for quantum cluster methods such as the variational cluster approximation. While a naïve implementation of MPS as cluster solver would barely improve its range of applicability, we show that our approach makes it possible to treat cluster geometries well beyond the reach of exact diagonalization methods. The key modifications we introduce are a continuous energy truncation combined with a convergence criterion that is more robust against approximation errors introduced by the MPS representation and provides a bound to deviations in the resulting Green's function. The potential of the resulting cluster solver is demonstrated by computing the self-energy functional for the single-band Hubbard model at half filling in the strongly correlated regime, on different cluster geometries. Here, we find that only when treating large cluster sizes, observables can be extrapolated to the thermodynamic limit, which we demonstrate at the example of the staggered magnetization. Treating clusters sizes with up to 6×6 sites we obtain significant improvement over the extrapolation accessible with exact diagonalization solvers when comparing with quantum Monte Carlo results. Finally, we illustrate the applicability of the MPS cluster solver to more complex models by calculating spectral properties as relevant for the electron-doped cuprate CaCuO2.

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Phys. Rev. B 112, 235108– Published 2 December, 2025
  • Received 15 November 2023
  • Revised 21 September 2025
  • Accepted 23 September 2025
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