Sabine Andergassen

The functional renormalization group (fRG) is an established tool in the treatment of correlated electron systems, notably for the description of competing instabilities. In recent years, methodological advancements led to the multiloop extension of the fRG, which systematically includes loop corrections beyond the conventional one-loop truncation and yields a quantitatively accurate description of two-dimensional lattice systems. At the same time, the single-boson exchange (SBE) decomposition of the two-particle vertex has been shown to offer both computational and interpretative advantages, paving the way to more affordable approximation schemes. We here apply their combination coined as multiloop SBE fRG to the two-dimensional Hubbard model at weak coupling. After providing a detailed account of the underlying formalism in physical channels, we analyze the results for the frequency- and momentum-dependent vertex functions. We find that the SBE approximation, i.e., neglecting the flow of the multiboson exchange contributions, accurately reproduces the parquet approximation at loop convergence. The presented algorithmic improvement opens the route for the treatment of more challenging parameter regimes and more realistic models.

Out‑of‑equilibrium quantum many‑body systems – such as multi-electron atoms and molecules driven by strong laser fields, quenched ultracold gases, and ultrafast-excited solids – exhibit rapid correlation buildup that underlies many emerging phenomena. Exact wave‑function methods to describe these effects scale exponentially with particle number; simpler mean‑field approaches neglect essential particle correlations. The time-dependent two-particle reduced density matrix (TD2RDM) formalism offers a middle ground by propagating the two‑particle density matrix and closing the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy with a reconstruction of the three‑particle cumulant, which carries information about three-particle correlations. But the validity and existence of time‑local reconstruction functionals ignoring memory effects remain unclear across different dynamical regimes. We show that a neural ordinary differential equation (ODE) model trained on exact two-particle reduced density matrix data (no dimensionality reduction) can reproduce its full dynamics without any explicit three‑particle information – but only in parameter regions where the Pearson correlation between the two‑ and three‑particle cumulants is large. In contrast, in the anti‑correlated or uncorrelated regime, the neural ODE fails, indicating that no simple time‑local functional of the instantaneous two‑particle cumulant can capture the evolution. The magnitude of the time‑averaged three‑particle‑correlation buildup appears to be the primary predictor of successful extrapolation: For a moderate correlation buildup, both neural ODE predictions and existing TD2RDM reconstructions are accurate, whereas stronger values lead to systematic breakdowns. These findings pinpoint the need for memory‑dependent kernels in the three‑particle cumulant reconstruction for the latter regime. Our results place the neural ODE as a model‑agnostic diagnostic tool that maps the regime of applicability of cumulant expansion methods and guides the development of non‑local closure schemes. More broadly, the ability to learn high‑dimensional reduced‑density‑matrix dynamics from limited data opens a pathway to fast, data‑driven simulation of correlated quantum matter, complementing traditional numerical and analytical techniques.

We formulate a quantum embedding algorithm in real-space for the simultaneous theoretical treatment of nonlocal electronic correlations and disorder, the coherent cellular dynamical mean-field theory (C-CDMFT). This algorithm combines the molecular coherent potential approximation (CPA) with the cellular dynamical mean-field theory. After a pedagogical review of quantum embedding theories for disordered and interacting electron systems, and a detailed discussion of its work flow, we present first results from C-CDMFT for the half-filled two-dimensional Anderson–Hubbard model on a square lattice: (i) the analysis of its Mott metal–insulator transition as a function of disorder strength, and (ii) the impact of different types of disorder on its magnetic phase diagram. For the latter, by means of a ‘disorder diagnostics’, we are able to precisely identify the contributions of different disorder configurations to the system’s magnetic response.

Electron-electron and electron-phonon interactions are responsible for the formation of spin, charge, and superconducting correlations in layered quantum materials. A paradigmatic model for such materials that captures both kinds of interactions is the two-dimensional Hubbard-Holstein model with a dispersionless Einstein phonon. In this work, we provide a detailed analysis of the magnetic, density, and superconducting fluctuations at and away from half filling. To that end, we employ the functional renormalization group using the recently introduced extension of the single-boson exchange formulation. More precisely, we go beyond previous approaches to the model by resolving the full frequency dependence of the two-particle vertex and taking into account the feedback from the electronic self-energy. We perform broad parameter scans in the space of Hubbard repulsion, electron-phonon coupling strength, and phonon frequency to explore the leading magnetic, density, and superconducting susceptibilities from the adiabatic to the anti-adiabatic regime. Our numerical data reveal that self-energy effects lead to an enhancement of the d -wave superconducting susceptibility toward larger phonon frequencies, in contrast to earlier isotope-effect studies. At small phonon frequencies, large density contributions to the s -wave superconducting susceptibility change sign and eventually lead to a reduction of s -wave superconductivity with increasing electron-phonon coupling, signaling the breakdown of Migdal-Eliashberg theory. We analyze our findings systematically, employing detailed diagnostics of the intertwined fluctuations and pinning down the various positive and negative isotope effects of the physical susceptibilities. Our results pave the way for a more comprehensive understanding of the interplay of electron-electron and electron-phonon interactions in correlated quantum materials.

The destruction of metallicity due to the mutual Coulomb interaction of quasiparticles gives rise to fascinating phenomena of solid state physics such as the Mott metal-insulator transition and pseudogap. A key observable characterizing their occurrences is the single-particle spectral function, determined by the fermionic self-energy. In this paper, we investigate in detail how real space fluctuations are responsible for a self-energy that drives the Mott-Hubbard metal-insulator transition. To this aim, we first introduce a real space fluctuation diagnostics approach to the Hedin equation, which connects the fermion-boson coupling vertex λ to the self-energy Σ . Second, by using cellular dynamical mean-field theory calculations, we unambiguously identify nearest-neighbor antiferromagnetic excitations as the leading physical processes responsible for the destruction of metallicity across the transition.

We extend the recently introduced single-boson exchange formulation to the computation of the self-energy from the Schwinger–Dyson equation (SDE). In particular, we derive its expression both in diagrammatic and in physical channels. The simple form of the single-boson exchange SDE, involving only the bosonic propagator and the fermion-boson vertex, but not the rest function, allows for an efficient numerical implementation. We furthermore discuss its implications in a truncated unity solver, where a restricted number of form factors introduces an information loss in the projection of the momentum dependence that in general affects the equivalence between the different channel representations. In the application to the functional renormalization group, we find that the convergence in the number of form factors depends on the channel representation of the SDE. For the two-dimensional Hubbard model at weak coupling, the pseudogap opening driven by antiferromagnetic fluctuations is captured already by a single ( s s -wave) form factor in the magnetic channel representation, differently to the density and superconducting channels.

We develop a machine learning algorithm to infer the emergent stochastic equation governing the evolution of an order parameter of a many-body system. We train our neural network to independently learn the directed force acting on the order parameter as well as an effective diffusive noise. We illustrate our approach using the classical Ising model endowed with Glauber dynamics, and the contact process as test cases. For both models, which represent paradigmatic equilibrium and nonequilibrium scenarios, the directed force and noise can be efficiently inferred. The directed force term of the Ising model allows us to reconstruct an effective potential for the order parameter which develops the characteristic double-well shape below the critical temperature. Despite its genuine nonequilibrium nature, such an effective potential can also be obtained for the contact process and its shape signals a phase transition into an absorbing state. Also, in contrast to the equilibrium Ising model, the presence of an absorbing state renders the noise term dependent on the value of the order parameter itself.

DOI: https://doi.org/10.1103/PhysRevResearch.5.029004