Neural-network quantum states have matured into a quantitative variational method for strongly correlated systems. I will present two recent developments: Hamiltonian-conditioned foundation NQS and neural-network-augmented Pfaffian states, which extend accurate variational treatment across families of spin and fermionic models. I will then discuss applications to frustrated antiferromagnets, on lattices large enough for controlled finite-size scaling, and to the doped two-dimensional Hubbard model, where competing stripe and pairing orders can be resolved and cross-validated against established methods. I will close with a brief outlook on continuum fermions and real-time dynamics.

Modern first-principles materials theory has two foundational gaps at the level of its most basic low-energy observables. At the one-electron level, Kohn-Sham bands are routinely used as quasiparticle bands, yet in simple alkali and alkaline-earth metals their bandwidths disagree with ARPES measurements by 20-35%, a discrepancy that persists across standard exchange-correlation functionals. At the two-electron level, calculations of electron-phonon superconductivity rely on the Coulomb pseudopotential, whose microscopic definition and dynamical relation to electron-phonon coupling remain phenomenological. I will present an effective field theory of the inhomogeneous electron gas that addresses both problems within a single downfolding framework. Integrating out high-energy electronic degrees of freedom shows how the Kohn-Sham Hamiltonian emerges as the tree-level quasiparticle Hamiltonian, while frozen-core dynamics generates a frequency renormalization that narrows Kohn-Sham bands and resolves the experimental bandwidth discrepancy. Applying the same effective-field-theory logic to the Cooper channel yields a microscopic construction of the Coulomb pseudopotential and incorporates dynamical Coulomb corrections to electron-phonon pairing through vertex functions. These results suggest that Kohn-Sham band theory and the Coulomb pseudopotential are not separate empirical ingredients, but two low-energy consequences of the same effective theory of the inhomogeneous electron gas: one governing single-particle propagation, the other governing two-particle pairing. The framework provides a controlled route from electronic structure to superconducting transition temperatures beyond the conventional separation of DFT, DFPT, and Eliashberg theory.

Long-range interactions can qualitatively reorganize correlated-electron ground states. In the square-lattice Hubbard model, on-site repulsion produces antiferromagnetic spin and charge stripes upon doping. We show that including a nearest-neighbor repulsion V can dramatically alter this behavior. Using auxiliary-field quantum Monte Carlo and density matrix renormalization group methods, we find that, above a critical ratio V/U (∼ 0.25), the system develops a modulated ferrimagnetic order intertwined with checkerboard charge-density-wave. Inside the ferrimagnetic domains, spin density alternates between positive (or negative) and nearly zero values. When the total spin is fixed to zero, positive and negative domains alternate in space; when spins are unconstrained, a ferrimagnetic state emerges with finite magnetization. Including a next-nearest-neighbor hopping t′ changes the modulation wavelength but leaves the order robust. Our results demonstrate that even short-range nonlocal interactions can stabilize qualitatively new magnetic textures, with implications for cuprate materials and programmable quantum simulators.

Using artificial neural networks to approximate ground-state many-electron wave functions is gaining in popularity. Most electronic structure methods rely on uncontrollable approximations, such as the choice of exchange-correlation functional in density functional theory or the form of the parameterized trial wavefunction in conventional quantum Monte Carlo simulations. Neural wave functions, by contrast, are universal approximators in principle and remarkably flexible in practice. Furthermore, by making use of the variational formulation of the Schrödinger equation, the weights and biases of the network can be optimised without the use of externally generated data. Neural quantum Monte Carlo simulations give results of consistently high quality across a range of diverse systems, including molecules, solids, quantum dots, and even nuclei. After giving a simple introduction to the field, I will discuss applications to molecules, solids, superconductivity, positron binding and annihilation, quantum dots, and quantum phase transitions.

Relaxation and condensation of an isolated low-energy Bose gas provide a controlled setting for the study of the universal features of many-body dynamics and the emergence of large-scale order. Here, by probing the gas across all relevant length scales, we experimentally study regimes of order formation during far-from-equilibrium Bose–Einstein condensation. We find that the first stage, associated with the formation of small-scale coherence can be understood within the framework of weak wave turbulence. Specifically, the initial transport of particles to low momenta corresponds to an inverse turbulent cascade that is, in agreement with the theory, characterized by a power-law momentum distribution and transport times set by the strength of the interactions. The second stage features spreading of coherence over length scales much larger than any microscopic scale, at a universal rate that is independent of interparticle interactions and determined solely by the ratio of Planck’s constant to the atom mass. Theoretically, this process has been associated with the relaxation of vortex and wave excitations. By magnifying the gas and imaging only a slice, we reveal randomly oriented vortex lines, quantify their decay, and find that it is consistent with the predictions of ultraquantum Vinen turbulence for the decay of a vortex tangle.

The Hubbard model is a paradigm of the `strong correlation problem’, with relevance to high-Tc superconductors and ultra-cold atoms in optical lattices. Key aspects of its physics in two dimensions can now be established beyond doubt, thanks to the development of controlled and accurate computational methods working in synergy (such as quantum embedding, tensor networks, various flavors of quantum Monte Carlo and, recently, neural quantum states). I will review some of these recent developments in this presentation and highlight the questions that remain to be answered. I will emphasize how the recent development of Neural Quantum States allows to address the question: Does the Hubbard model host a high-Tc superconducting phase?

For many years, the superconductivity of Sr2RuO4 was thought to be chiral p-wave, but experiments in recent years strongly favour a single-component, d_{x^2-y^2} order parameter. But that does not mean that the superconductivity is understood. Some key experimental results, that had a straightforward interpretation under a hypothesis of chiral p-wave order, now require alternative explanation. Experiments under uniaxial stress present some new mysteries. In this talk, I will discuss things about the superconductivity of Sr2RuO4 that we do not know – experimental findings that lack explanation, and places where the experimental situation remains unclear.

Strongly-correlated electrons in transition-metal oxides give rise to intriguing emergent phenomena, including high-temperature superconductivity in cuprates. While simplified one-band Hubbard models capture some aspects, explicitly describing the interplay of copper and oxygen orbitals -- as in the three-band Emery model -- is essential to capture the full phenomenology of cuprates. Quantum simulators based on ultracold atoms offer a promising route to study such systems in a controlled setting, but realizing realistic multi-orbital Hubbard models remains challenging. In my talk, I propose an optical superlattice architecture that implements the three-band Emery model with ultracold fermions. By combining lattice beams with controllable interference, we engineer orbital degrees of freedom that reproduce key features of the cuprate band structure, while enabling independent control of orbital-dependent interactions and charge-transfer energy. Using determinant quantum Monte Carlo, we further investigate thermodynamic properties in the undoped regime and find a finite-temperature metal-insulator crossover accompanied by the onset of antiferromagnetic correlations accessible in current experiments. Finally, we apply a Hamiltonian learning protocol enabling to infer effective single-band Hubbard models from experimental realizations of Emery models. Our results provide a practical pathway to simulate multi-orbital Hubbard physics with quantum gas microscopes.

The far-from-equilibrium dynamics of interacting quantum systems has been the subject of extensive research in recent years. Advancements in quantum simulators have enabled the experimental observation of numerous non-equilibrium phenomena, such as many-body scars, time crystals, Hilbert space fragmentation, and dynamical quantum phase transitions. In this work, we use the time-evolving block decimation (TEBD) method to investigate the dynamics of an anisotropic spin-1 Heisenberg chain for a wide range of experimentally accessible initial states. By adjusting the parameter Jq that controls the quadrupolar interaction strength, we can tune the system from a non-integrable SU(2) Heisenberg model to an integrable SU(3) Heisenberg model. We examine the local magnetization, entanglement entropy, and spin correlations, and characterize their dependence on Jq. We identify a new conserved quantity at the SU(3) symmetric point and provide a theoretical framework to explain our numerical observations in terms of the number of accessible states permitted by this conservation law. Our results provide a route to realize a rich array of non-equilibrium behavior in spin-1 lattice models, which can be realized with ultracold atoms in optical lattices.

Electron-phonon coupling and its effects on phenomena such as transport lifetimes and superconductivity are central to materials physics. In conventional materials, density functional and GW-based methodologies provide predictive theories. In "quantum" materials characterized by strong electron correlations, density functional and GW methods do not by themselves provide an adequate treatment of electronic properties, while accumulating experimental evidence indicates electron-phonon couplings in substantial disagreement with conventional theory. This talk presents a density-functional plus dynamical mean field (DFT+DMFT) methodology for computing electron-phonon coupling in quantum materials and indicates some physical consequences. Model system (Hubbard-Holstein) calculations are used to illustrate some basic issues. A frozen phonon method is used to determine (within DFT+DMFT) the physics associated with representative phonon modes in the three-orbital correlated metal SrVO3, where intra-V-t2g-band correlation significantly increases the coupling of electrons to a Jahn-Teller phonon mode that splits the degenerate orbital energies, while reducing the coupling associated with a breathing phonon that couples to the charge on each V atom. The predicted electron-phonon coupling has a significant dependence on the electronic frequency, showing the inadequacy of the simple picture in which correlations change static local susceptibilities. Finally, a general method for computing the coupling is presented and a preliminary application to barium-potassium-bismuth oxide is presented. The findings shed light on the material- and mode-specific role of dynamical electronic correlation in electron-phonon coupling and highlight the importance of developing efficient computational methods for treating electron-phonon coupling in correlated materials.
This work is performed in collaboration with David Abramovitch, Olivier Parcollet and Jennifer Coulter and is based in part on Physical Review B112, 075113 (2025) and arXiv:2505.08081 (PRB in press).

Self-consistent approaches in many-electron problems typically converge to an unphysical solution in strongly correlated regimes. By deriving the mathematical condition for the stability of the physical solution, we unveil [1] the precise link between two distinct issues previously considered equivalent: (i) the misleading convergence in self-consistent schemes and (ii) the multivaluedness of the Luttinger-Ward functional. Although these problems are fundamentally linked through the divergences of the irreducible vertex function, we show that misleading convergence can occur even in the absence of such divergences. Eventually, a systematic procedure for stabilizing the physical solution is proposed and applied to different iterative algorithms, in particular to parquet-based [2] and single-boson-exchange-based [3] calculations.
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[1] H. Eßl, M. Reitner, E. Kozik, and A. Toschi, arXiv:2502.01420, Phys. Rev. Lett. in press (2026).
[2] H. Eßl, S. Rohshap, M. Gievers, M. Wallerberger, A. Toschi, and A. Kauch, "Stabilizing the parquet problem", in preparation.
[3] M.Gievers, H. Eßl, S. Rohshap, A. Kauch, and A. Toschi, "Instabilities in self-consistent diagrammatic approaches and how to cure them", in preparation.

Two-dimension van-der-Waals materials provide a versatile platform with a high degree of tunability, that has led to a multitude of realizations of correlated and topological phases in the recent years. However, theoretical modeling remains a challenge: Strong correlations call for tools beyond mean-field theory. In this talk I will explain how variational Monte Carlo, in particular neural quantum states, addresses this challenge. I will use these tools to piece together features such as anisotropy or quantum geometry in 2D quantum materials and explain their effect on the emerging phases.

I will give an overview of recent methodological progress in the computation of real-frequency dynamical correlation functions in various contexts: symmetric improved estimators for the self-energy and 4-point vertex; computing the discrete self-energy of discrete many-body systems without invoking broadening; causality-preserving Liouvillian interpolation of cluster-DMFT self-energy; and a tangent-space Krylov method for computing of dynamical 2-point correlators.

The development of nonequilibrium impurity solvers is relevant for the study of nonequilibrium lattice systems within the framework of dynamical mean field theory (DMFT). A promising approach for strongly correlated systems is the self-consistently renormalized hybridization expansion. The cost of this method increases rapidly with diagram order because of the growing number of diagrams, and because the calculations involve multiple integrations over the Keldysh contour. One possible strategy is to replace these integrations by Monte Carlo sampling, as is done in the inchworm algorithm. Here, we explore a different route and use (quantics) tensor cross interpolation to factorize the integrands. This approach enables efficient simulations up to third order, and potentially beyond, which is sufficient for accurate simulations in the strong-coupling regime. The method is demonstrated with applications to photo-doped Mott systems in long lived (quasi-)steady states, where it enables DMFT investigations of the eta-pairing superconducting phase with unprecedented accuracy.

Understanding how fermions form pairs and develop long-range phase coherence is a central question in strongly interacting Fermi systems. In this talk, I will present two experiments that probe this physics in complementary settings: a homogeneous unitary Fermi gas in the continuum and a homogeneous attractive fermionic Hubbard system in an optical lattice. First, I will discuss the observation of a pairing pseudogap in a homogeneous unitary Fermi gas. By developing high-resolution momentum-resolved microwave spectroscopy and suppressing final-state effects, we directly measure the single-particle spectral function in a unitary Fermi gas. These measurements reveal a clear suppression of spectral weight near the Fermi surface above the superfluid transition, providing direct evidence for a pairing pseudogap. From the spectra, we further extract the pairing gap, inverse pair lifetime, and single-particle scattering rate, allowing us to characterize the microscopic properties of the system. I will then turn to the realization of single-band superfluidity in the attractive fermionic Hubbard model. Using high-precision measurements of the doublon fraction in a low-temperature homogeneous optical lattice, we resolve clear signatures of lattice superfluidity. In the BCS regime, the doublon fraction exhibits a pronounced dip as the system is cooled through the superfluid transition, while its temperature dependence evolves systematically with increasing attraction, reflecting the physics of the BCS-BEC crossover. Together, these results highlight the power of ultracold Fermi gases as quantum simulators for addressing central questions in strongly correlated matter.

Quantum gas microscopy enables the probing of dilute quantum matter with single-atom resolution. Initially developed to study lattice and spin chain systems—most notably the Hubbard model and its generalizations—this technique has recently been extended to the continuum. In this talk, I will provide an overview of the tools we have developed and present our recent work on strongly interacting Fermi gases, demonstrating how they can be characterized at previously inaccessible levels of resolution. We obtain direct access to spatial density and spin correlations up to high order, opening a new vista onto the microscopic inner workings of these strongly correlated systems.

Atomic quantum gases in the continuum realize paradigmatic states of matter. However, we were so far lacking single-atom resolution imaging of "bulk" quantum gases, while such resolution has been commonplace in optical lattice experiments for more than a decade. We recently realized such a continuum gas microscope for bosons and fermions, including spin resolution of fermions. A 2D Fermi gas with attractive interactions realizes the BEC-BCS crossover in 2D. Single-atom and spin-resolved images directly reveal fermion pairing, all thermodynamic quantities including the contact measuring short-range correlations, and the "charge" and spin structure factor. The latter directly reveal sound and pseudo-gap behavior. Our technique is general and should be applicable in the future also to e.g. dipolar atomic or molecular gases.