Counterterm Solver

The counterterm solver is a hybrid Hamiltonian-diagrammatic quantum impurity solver. Hamiltonian-based approaches, which rely on an explicit bath discretization, are typically limited to a small number of bath sites or small entanglement, while diagrammatic methods suffer from sign problems, slow convergence, or diagram truncation approximations. The counterterm solver combines the two: augmenting a diagrammatic expansion with a small auxiliary bath (the “counterterms”) reduces the residual problem to a regime where low-order perturbation theory is highly accurate and rapidly converging.

The method is described in:

Yang Yu, Gaurav Harsha, Lei Zhang, Agnieszka Jażdżewska, Dominika Zgid, Xinyang Dong, Emanuel Gull, “Hybrid Hamiltonian-diagrammatic quantum impurity solver”, arXiv:2606.11095 (2026).

In a simple benchmark, the precision of the hybrid approach surpasses bold-line calculations by several orders of magnitude; for a strongly interacting two-orbital model with a severe sign problem, convergence is achieved at three orders of magnitude lower computational cost than competing methods; and convergence to the unknown exact result is rapidly accelerated in a difficult realistic problem.

To get started, head to the repository at green-jl-counterterms and follow the instructions there.

A video tutorial is in preparation.