High-order diagrammatic strong coupling expansion of the Hubbard model

• Applicant: Dr. Kirill Alpin
• Subject Area: Theoretical Condensed Matter Physics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 575566778

The main project of this proposal is a high-order diagrammatic expansion of the Hubbard model around the strongly interacting limit via Quantum Monte Carlo (QMC). To do so, a new unitary transformation is used, that is able to map the interaction part of the Hubbard Hamiltonian to a non-interacting chemical potential term. Doing so, the hopping term transforms into a new interaction term. This transformation is exact and results in a Hamiltonian consisting of unconstrained fermions (unlike the t-J model). Therefore, ordinary diagrammatic methods can be applied to perform a 1/U expansion. Such an expansion has already been performed exactly up to second order, which resulted in the observation of d-wave superconductivity at non-zero doping. Then, a similar transformation has been applied to the Heisenberg model. A QMC study on the resulting Hamiltonian was successful. Therefore, this project, being a combination of both, is highly likely to be successful itself and is expected to result in a, for the first time, unbiased, numerically exact description of the Hubbard model in the strongly correlated regime and therefore also for cuprates. Such a description is expected to also for the first time stabilize the d-wave superconductivity phase in a QMC calculation. Such a result is of very high importance for the condensed matter community. Another project aims to generalize the alpha-trick and counter-terms in DiagMC/DQMC calculations. Here, a general free Hamiltonian is used as a counter-term ansatz, which is optimized end-to-end during a QMC calculation. There, the pole-structure is optimized, such that poles are pushed as far away from the origin as possible. This is realizable by the use of an auto-differentiation library like pyTorch. At every step of a QMC calculation, not only the observable is accumulated but also its gradient with respect to counter-term parameters. This should also lead to QMC algorithms that remedy the sign problem at strong correlations, such that parameter-ranges where QMC is applicable are extended. Lastly, not only exact solutions of the d-wave superconductivity phase are studied, but also of the pseudogap phase. We consider local 4-point correlation-functions, able to encode information about the existence of so called preformed pairs. These composite particles are thought to be the reason for the pseudogap phase and are bound states of electrons before they condense via a Bose Einstein condensation into the superconducting phase at lower temperatures. Results can already be achieved with existing methods like Cluster-DMFT. This way results can be compared to the previous QMC calculations. We expect that such an analysis will give important insights about the nature of the pseudogap phase.

• DFG Programme: Fellowship
• International Connection: France
• Hosts: Dr. Jean-René Chazottes; Professor Dr. Antoine Georges