Functional Renormalization with Interaction Flows Enables Dressed Regulators for Electron-Phonon Systems

The functional renormalization group offers physicists a powerful method for understanding complex systems, but conventional approaches typically focus on regulating only the basic building blocks of a problem. Aiman Al-Eryani, Marcel Gievers, and Kilian Fraboulet, working at Ruhr-Universität Bochum, TU Wien, and the Max-Planck-Institut für Festkörperforschung respectively, now present a significant advance by developing a formulation that regulates both the fundamental components and the interactions between them. This new approach, built upon established equations, simplifies the complex flow equations by interpreting them as adding regulation to the bosonic propagator, and crucially, opens the door to previously inaccessible calculations, such as modelling temperature-dependent effects in systems with delayed interactions. The team demonstrates the reliability of their scheme by applying it to established models, and successfully implements a novel temperature-flow method for an impurity model coupled to vibrations, paving the way for more accurate and comprehensive investigations of interacting quantum systems.

Scientists have now expanded this technique with a new formulation that dresses both the bare propagator and the bare interaction with regulators, offering greater flexibility than previous methods which typically regulated only the propagator. This advancement allows researchers to tackle problems previously inaccessible to standard fRG approaches, such as systems with complex interactions and temperature-dependent behaviour.

Functional Renormalization Group for Correlated Electrons

Research focuses on understanding strongly correlated electron systems, materials where interactions between electrons play a crucial role in determining their properties. Scientists employ the functional renormalization group (fRG) to investigate these systems, a method that gradually simplifies complex interactions to reveal underlying physics. A key innovation is the single-boson exchange (SBE) formalism, which emphasizes the role of bosonic excitations, such as vibrations in the material, in mediating interactions between electrons. This approach simplifies calculations and improves the accuracy of results.

The fRG method works by progressively integrating out degrees of freedom, effectively flowing towards a simplified model. Different schemes exist within the fRG framework, each with its strengths and weaknesses. The truncated unity scheme improves convergence and efficiency, while the SBE scheme focuses on bosonic excitations. Choosing the right regulator function is crucial for the fRG, influencing the accuracy and stability of the calculations. Researchers are developing more efficient truncation schemes and combining the fRG with other computational methods to tackle increasingly complex problems.

Applications of this research span a wide range of materials and phenomena. The fRG can be used to study metal-insulator transitions, the properties of Fermi polarons (quasiparticles formed by impurities interacting with a sea of electrons), and the mechanisms of superconductivity. It is also being applied to investigate charge density waves and the electronic properties of moiré materials, formed by stacking two-dimensional materials with a slight twist.

Dressed Propagators Expand Renormalization Group Capabilities

Scientists have developed a new functional renormalization group (fRG) scheme that significantly expands the capabilities of existing methods for studying complex many-electron systems. This work introduces a novel approach where both the bare propagator and the bare interaction within the theory are dressed with regulators, offering greater flexibility than conventional fRG implementations which typically regulate only the propagator. The team derived flow equations based on the Schwinger-Dyson and Bethe-Salpeter equations, demonstrating that adding a regulator to the bosonic propagator scarcely alters the fundamental structure of the flow equations. This advancement enables the treatment of problems previously inaccessible to standard fRG methods, such as systems with retarded interactions and temperature flows.

The researchers successfully formulated a new interaction flow, deriving flow equations for correlation functions within a bosonization approach utilizing the single-boson exchange (SBE) decomposition of the two-particle vertex. This SBE decomposition simplifies the analysis of complex frequency and momentum dependencies by representing them in terms of fermion-boson couplings and bosonic propagators, providing direct access to susceptibilities relevant for studying competing orders. To validate the new scheme, the team applied it to three concrete models: the Hubbard atom, the Anderson impurity model, and the Anderson-Holstein impurity model. Numerical tests demonstrated the loop convergence of a simple flow scheme featuring a cutoff only in the bare interaction, confirming its validity when compared to conventional schemes utilizing a cutoff solely in the propagator. Furthermore, they devised a temperature flow scheme applicable to electron-phonon systems, supplementing the conventional approach with a flow in the bare interaction and successfully applying it to the Anderson-Holstein impurity model. These results demonstrate the power of the extended multiloop SBE fRG scheme for accurately describing many-electron systems and opening new avenues for research in condensed matter physics and beyond.

Multiloop Functional Renormalization Group with Regulators

This work presents a significant advancement in the functional renormalization group (fRG) technique, extending its capabilities by incorporating regulators for both bare propagators and interactions. Researchers developed a generalized multiloop fRG formulation, demonstrating that this extension minimally alters the structure of the fundamental flow equations. The team successfully implemented this approach, devising a new temperature-flow scheme that includes cutoffs in both the propagator and the bare interaction, and validating it using models of an Anderson impurity coupled to a phonon. Notably, the researchers demonstrated that the derived flow equations can be obtained through two distinct pathways: starting from Schwinger-Dyson and Bethe-Salpeter equations, or alternatively, utilizing the parquet decomposition with the Bethe-Salpeter equations.

This provides flexibility in applying the method to various physical systems. The analysis confirms preservation of spin rotation invariance throughout the calculations for the Hubbard atom, Anderson impurity model, and Anderson-Holstein impurity model. The authors acknowledge that the current formulation relies on an approximation concerning the two-particle irreducible vertex, which may introduce limitations in certain scenarios. Future work will likely focus on refining this approximation or exploring alternative approaches to fully capture the complex interplay between interactions and propagators in strongly correlated systems. This research expands the toolkit available to physicists studying many-body problems and paves the way for more accurate investigations of complex materials.