Bath parameterization in multi-band cluster Dynamical Mean-Field Theory

Accurate and reliable algorithms to solve complex impurity problems are instrumental to a routine use of quantum embedding methods for material discovery. In this context, we employ an efficient selected configuration interaction impurity solver to investigate the role of bath discretization, specifically, bath size and parameterization, in Hamiltonian-based cluster dynamical mean field theory (CDMFT) for the one- and two-orbital Hubbard models. We consider two- and four-site clusters for the single-orbital model and a two-site cluster for the two-orbital model. Our results demonstrate that, for small bath sizes, the choice of parameterization can significantly influence the solution, highlighting the importance of systematic convergence checks. Comparing different bath parameterizations not only reveals the robustness of a given solution but can also provide insights into the nature of different solutions and potential instabilities of the paramagnetic state. We present an extensive analysis of the zero-temperature Mott transition of the paramagnetic half-filled single-band Hubbard model, benchmarking our findings against previous literature. We find that for the single-band model the dependence on parameterization is weak for the largest bath sizes accessible with ASCI, while a tendency towards a nematic solution can be seen when the bath size is small. Building on this, we extend our study to the multi-band regime, where we present the first systematic analysis at zero temperature for two orbitals and a two-site cluster. This setup allows us to assess the effect of nearest-neighbor dynamical correlations on the multi-orbital Mott transition. In this case, some quantitative dependence on the parameterization is retained for the two-orbital model, for instance in the value of the critical interaction for a Mott transition.

💡 Research Summary

This paper presents a comprehensive investigation into the critical role of bath discretization in Hamiltonian-based cluster dynamical mean-field theory (CDMFT), focusing on how the choice of bath size and parameterization scheme influences the physical outcomes for strongly correlated electron systems. The study employs an efficient selected configuration interaction impurity solver, the Adaptive Sampling Configuration Interaction (ASCI) method, which allows access to larger bath sizes than conventional exact diagonalization, enabling a more controlled analysis of discretization errors.

The core problem addressed is the fitting of the continuous bath hybridization function, obtained from the DMFT self-consistency condition, onto a finite set of non-interacting bath orbitals. This high-dimensional, non-convex optimization is prone to getting trapped in local minima. To mitigate this, various constrained bath parameterization schemes are used: the Semi-Definite Relaxation (SDR) method, which convexifies the problem; symmetry-adapted parameterization based on cluster irreps (Irreps); and a simple Replica scheme where the bath is a direct copy of the cluster geometry. The research systematically compares these schemes.

The analysis begins with the paramagnetic, half-filled single-band Hubbard model on a square lattice, using 2-site and 4-site clusters. A key finding is that for small bath sizes (e.g., 8 bath orbitals), the different parameterizations can lead to quantitatively and sometimes qualitatively different solutions, such as favoring a nematic (symmetry-broken) tendency in some cases. This highlights that with limited bath degrees of freedom, the discretization process itself can artificially bias the solution. However, as the bath size is increased to the maximum accessible by ASCI (e.g., 12-16 orbitals), the results from different parameterizations converge, demonstrating that a sufficiently large bath can reliably represent the continuous environment and that the discretization artifact diminishes.

Building on this, the study extends the framework to the multi-band regime, presenting the first systematic zero-temperature CDMFT analysis of the two-orbital Hubbard-Kanamori model on a two-site cluster. This minimal setup captures both multi-site and multi-orbital correlations. Here, some quantitative dependence on the bath parameterization persists even for larger baths, for instance in the estimated critical interaction strength for the Mott transition. This indicates that the additional complexity from orbital degrees of freedom (like Hund’s coupling) makes the fitting and representation of the bath more subtle.

The paper provides extensive benchmarking of the single-band Mott transition against established literature, validating the methodology. It also computes key observables like quasiparticle weights, double occupancies, and spin-spin correlations to compare solutions.

In conclusion, the work emphasizes that rigorous convergence checks with respect to both bath size and parameterization are essential for reliable CDMFT calculations, especially when aiming for quantitative predictions in material discovery. It also suggests that comparing results from different bath parameterizations is not just a robustness test but can offer valuable insights into potential alternative ground states or instabilities of the assumed paramagnetic solution. The use of advanced impurity solvers like ASCI is crucial in enabling these necessary systematic studies.