We compute quasi-exact ab initio path-integral Monte Carlo results for the Matsubara Green's function of the uniform electron gas (UEG) at finite temperature over a broad range of coupling strengths (r_s=1,…,10). This allows us to present approximation-free results for the static self-energy Σ_∞(p) and spectral function A(p,ω), and to benchmark previous approximate results for the UEG. In addition, our work opens up intriguing avenues to study the single-particle spectrum and density of states of real warm dense matter systems based on truly first principles.

Correlated real-time dynamics in large, spatially inhomogeneous quantum systems remain difficult to access with nonequilibrium many-body methods. Two-time nonequilibrium Green functions (NEGF) retain dynamical correlations but their computational runtime grows cubically with the number of time steps N_t. This scaling bottleneck could recently be overcome by introducing the G1–G2 scheme that is linear in N_t, but requires propagation of a two-particle correlation function and may suffer from numerical instabilities. This has restricted simulations to small systems with N_b∼ 10^2 basis states. Here we introduce a quantum-fluctuation formulation of nonequilibrium Green functions, denoted δNEGF, that represents dynamical two-particle correlations through fluctuations of field-operator products, δĜ. This guarantees stable dynamics by preserving the positivity of the reduced density matrices, avoids the explicit storage of the two-particle Green function, and reduces the propagation to a finite ensemble of Hartree-Fock-like trajectories. Combined with a stochastic low-rank decomposition of the correlation functions, the method retains time-linear scaling while extending dynamical GW and particle-particle and particle-hole T-matrix simulations to basis sizes of order N_b∼ 10^4. We benchmark δNEGF against exact and HF-GKBA results for lattice systems, finding stable correlated dynamics also at strong coupling. We further demonstrate large-scale simulations of diffusion in two-dimensional Hubbard lattices and ultrafast relaxation in graphene nanoribbon heterostructures with long-range Coulomb interactions. These results establish δNEGF as a scalable route to dynamical self-energy simulations of large, spatially inhomogeneous correlated quantum systems beyond the reach of existing NEGF implementations.

Electronic spectra provide direct insight into the excitations and correlations of condensed matter systems. Their description requires electron correlations beyond mean field. In equilibrium, the GW approximation has become the method of choice for many materials. Extending this approximation to nonequilibrium, however, is challenging. Full two-time GW simulations within the nonequilibrium Green's functions approach scale at least cubically with propagation time, whereas improved scaling schemes such as time-local (adiabatic) approximations or the generalized Kadanoff–Baym ansatz usually retain only mean-field character in the spectra. The recently introduced real-time Dyson expansion recovers dynamical correlations in the spectrum at time-linear cost, but has so far been restricted to the second-order Born approximation with mean-field reference propagators. Here we formulate and implement the RT-DE for the nonequilibrium GW self-energy with reference propagators of general form whose off-diagonal evolution is time local. In particular, we assess Hartree–Fock propagators, propagators with statically screened exchange with nonequilibrium screening, and a correlated propagator based on the Hartree–Fock GKBA. Benchmarks against exact diagonalization for a driven two-band lattice model with long-range interactions show that the mean-field and statically screened references yield the most accurate spectra, including satellite structures absent at mean-field level, whereas the GKBA-based approach best captures scattering-induced occupation dynamics, but artificially broadens and splits spectral peaks. When applied to large systems that are beyond the reach of exact methods, the scheme resolves the excitonic replica of the valence band and satellites identified as exciton shake-up, as well as their reshaping with increasing excitation density.

Ionization potential depression (IPD) is of crucial importance to understand and accurately predict the properties of dense, partially ionized plasmas. Many models of IPD have been developed that, however, exhibit largely varying results. Recently, a novel approach was proposed that is based on first-principles quantum Monte Carlo simulations and that was demonstrated for hydrogen. Here, this concept is discussed in more detail. Particular attention is devoted to the Fermi barrier that electrons have to overcome upon ionization and that significantly contributes to IPD when the nuclear charge increases.

This roadmap presents the state-of-the-art, current challenges and near future developments anticipated in the thriving field of warm dense matter physics. Originating from strongly coupled plasma physics, high pressure physics and high energy density science, the warm dense matter physics community has recently taken a giant leap forward. This is due to spectacular developments in laser technology, diagnostic capabilities, and computer simulation techniques. Only in the last decade has it become possible to perform accurate enough simulations & experiments to truly verify theoretical results as well as to reliably design experiments based on predictions. Consequently, this roadmap discusses recent developments and contemporary challenges that are faced by theoretical methods, and experimental techniques needed to create and diagnose warm dense matter. A large part of this roadmap is dedicated to specific warm dense matter systems and applications in astrophysics, inertial confinement fusion and novel material synthesis.

We present extensive new ab initio path integral Monte Carlo (PIMC) simulation results for the chemical potential of the warm dense uniform electron gas (UEG), spanning a broad range of densities and temperatures. This is achieved by following two independent routes, (i) based on the direct estimation of the free energy [Dornheim et al., Phys. Rev. B 111, L041114 (2025)] and (ii) using a histogram estimator in PIMC simulations with a varying number of particles. We empirically confirm the expected inverse linear dependence of the exchange-correlation (XC) part of the chemical potential on the simulated number of electrons, which allows for a reliable extrapolation to the thermodynamic limit without the necessity for an additional finite-size correction. We find very good agreement (within Delta mu xc less than or similar to 0.5%) with the previous parametrization of the XC-free energy by Groth et al. [Phys. Rev. Lett. 119, 135001 (2017)], which constitutes an important cross validation of current state-of-the-art UEG equations of state. In addition to being interesting in its own right, our study constitutes the basis for the future PIMC based investigation of the chemical potential of real warm dense matter systems starting with hydrogen.

The exploration of ultrafast phenomena is a frontier of condensed matter research, where the interplay of theory, computation, and experiment is unveiling new opportunities for understanding and engineering quantum materials. With the advent of advanced experimental techniques and computational tools, it has become possible to probe and manipulate nonequilibrium processes at unprecedented temporal and spatial resolutions, providing insights into the dynamical behavior of matter under extreme conditions. These capabilities have the potential to revolutionize fields ranging from optoelectronics and quantum information to catalysis and energy storage. This roadmap captures the collective progress and vision of leading researchers, addressing challenges and opportunities across key areas of ultrafast science and condensed matter. Contributions in this roadmap span the development of ab initio methods for time-resolved spectroscopy, the dynamics of driven correlated systems, the engineering of materials in optical cavities, and the adoption of FAIR principles for data sharing and analysis. Together, these efforts highlight the interdisciplinary nature of ultrafast research and its reliance on cutting-edge methodologies, including quantum electrodynamical density-functional theory, correlated electronic structure methods, nonequilibrium Green’s function approaches, quantum and ab initio simulations.

When heteronuclear molecules are illuminated by light of spatially uniform intensity, electronic excitations may, nevertheless, be restricted to parts of the system, depending on the absorption properties of its constituents. Here, we show that this effect is observed also in homogeneous carbon-based systems, such as graphene nanoribbons (GNRs): A spatially uniform laser pulse can create strongly localized carrier excitations, including excitons, on the subnanometer scale within a few femtoseconds. The origin of this effect is the unusual topological-based electronic structure of the GNRs. This opens exciting avenues for nanoelectronics and brings petahertz switching within reach. Using nonequilibrium Green’s function simulations, we demonstrate this effect by exciting small GNR heterostructures of suitable geometry with a laser pulse with carefully chosen photon energy, polarization, and carrier-envelope phase.

The interaction of highly-charged ions with monolayers of graphene and MoS is theoretically investigated based on nonequilibrium Green Functions. In a recent paper [Niggas et al., Phys. Rev. Lett. 129, 086802 (2022)] dramatic differences in the response of the two materials to an impacting slow ion were reported. Here, this analysis is extended, focusing on the effect of electron-electron correlations in the monolayer on the electronic response to the ion. We apply the recently developed time-linear G1-G2 scheme [Schluenzen et al., Phys. Rev. Lett. 124, 076601 (2020)] combined with an embedding approach [Balzer et al., Phys. Rev. B 107, 155 141 (2023)]. We demonstrate that, while electronic correlations have a minor effect in graphene, they significantly influence the electron dynamics in the case of MoS. Our key results are the ultrafast dynamics of the charge density and induced electrostatic potential in the vicinity of the impact point of the ion.

Warm dense matter (WDM) is an active field of research, with applications ranging from astrophysics to inertial confinement fusion. Ionization degree and continuum lowering are important quantities to understand how materials behave under these conditions, but can be difficult to diagnose since experimental campaigns are limited and often require model-dependent analysis. This is especially true for hydrogen, which has a comparably low scattering cross section, making high-quality data particularly difficult to obtain. Consequently, building equation of state tables often relies on ab initio simulations in combination with untested approximations to extract properties from experiments. Here, we investigate an approach for extracting the ionization potential depression and ionization degree—quantities which are otherwise not directly accessible from the physical model—from first-principles path integral Monte Carlo (PIMC) simulations utilizing a chemical model. In contrast to experimental measurements, where noise and nonequilibrium effects add to the uncertainty of the inferred parameters, PIMC simulations provide a clean signal with well-defined thermodynamic conditions. Comparisons against commonly used models show a qualitative agreement, but we find deviations primarily for the high-density and high-temperature cases. We also demonstrate the decreasing sensitivity of the dynamic structure factor with respect to both ionization and continuum lowering for increasing scattering angles in x-ray Thomson scattering experiments. Our work has important implications for the design of future experiments, but also offers qualitative understanding of structure factors and the imaginary-time correlation function obtained from first-principles quantum Monte Carlo simulations.

Accurate knowledge of the properties of hydrogen at high compression is crucial for astrophysics (e.g., planetary and stellar interiors, brown dwarfs, atmosphere of compact stars) and laboratory experiments, including inertial confinement fusion. There exists experimental data for the equation of state, conductivity, and Thomson scattering spectra. However, the analysis of the measurements at extreme pressures and temperatures typically involves additional model assumptions, which makes it difficult to assess the accuracy of the experimental data rigorously. On the other hand, theory and modeling have produced extensive collections of data. They originate from a very large variety of models and simulations including path integral Monte Carlo (PIMC) simulations, density functional theory (DFT), chemical models, machine-learned models, and combinations thereof. At the same time, each of these methods has fundamental limitations (fermion sign problem in PIMC, approximate exchange–correlation functionals of DFT, inconsistent interaction energy contributions in chemical models, etc.), so for some parameter ranges accurate predictions are difficult. Recently, a number of breakthroughs in first principles PIMC as well as in DFT simulations were achieved which are discussed in this review. Here we use these results to benchmark different simulation methods. We present an update of the hydrogen phase diagram at high pressures, the expected phase transitions, and thermodynamic properties including the equation of state and momentum distribution. Furthermore, we discuss available dynamic results for warm dense hydrogen, including the conductivity, dynamic structure factor, plasmon dispersion, imaginary-time structure, and density response functions. We conclude by outlining strategies to combine different simulations to achieve accurate theoretical predictions that are based on first principles.

Yuri L'vovich Klimontovich (September 28, 1924–October 26, 2002) was an outstanding theoretical physicist who made major contributions to kinetic theory. On the occasion of his 100th birthday, we recall his main scientific achievements.

The theory of nonequilibrium Green functions (NEGF) has seen a rapid development over the recent three decades. Applications include diverse correlated many‐body systems in and out of equilibrium. Very good agreement with experiments and available exact theoretical results could be demonstrated if the proper selfenergy approximations were used. However, full two‐time NEGF simulations are computationally costly, as they suffer from a cubic scaling of the computation time with the simulation duration. Recently the G1–G2 scheme that exactly reformulates the generalized Kadanoff–Baym ansatz with Hartree–Fock propagators (HF‐GKBA) into time‐local equations is introduced, which achieves time‐linear scaling and allows for a dramatic speedup and extension of the simulations (Schluenzen et al. Phys. Rev. Lett. 2020, 124, 076601). Remarkably, this scaling is achieved quickly, and also for high‐level selfenergies, including the nonequilibrium GW and T‐matrix approximations (Joost et al. Phys. Rev. B 2020, 101, 245101). Even the dynamically screened ladder approximation is now feasible (Joost et al. Phys. Rev. B 2022, 105, 165155), and also applications to electron‐boson systems are demonstrated. Herein, an overview on recent results that are achieved with the G1–G2 scheme is presented. Problems and open questions are discussed and further ideas of how to overcome the current limitations of the scheme and present are presented. The G1–G2 scheme is illustrated by presenting applying it to the excitation dynamics of Hubbard clusters, to optical excitation of graphene, and to charge transfer during stopping of ions by correlated materials.

Accurate knowledge of the properties of hydrogen at high compression is crucial for astrophysics (e.g. planetary and stellar interiors, brown dwarfs, atmosphere of compact stars) and laboratory experiments, including inertial confinement fusion. There exists experimental data for the equation of state, conductivity, and Thomson scattering spectra. However, the analysis of the measurements at extreme pressures and temperatures typically involves additional model assumptions, which makes it difficult to assess the accuracy of the experimental data. rigorously. On the other hand, theory and modeling have produced extensive collections of data. They originate from a very large variety of models and simulations including path integral Monte Carlo (PIMC) simulations, density functional theory (DFT), chemical models, machine-learned models, and combinations thereof. At the same time, each of these methods has fundamental limitations (fermion sign problem in PIMC, approximate exchange-correlation functionals of DFT, inconsistent interaction energy contributions in chemical models, etc.), so for some parameter ranges accurate predictions are difficult. Recently, a number of breakthroughs in first principle PIMC and DFT simulations were achieved which are discussed in this review. Here we use these results to benchmark different simulation methods. We present an update of the hydrogen phase diagram at high pressures, the expected phase transitions, and thermodynamic properties including the equation of state and momentum distribution. Furthermore, we discuss available dynamic results for warm dense hydrogen, including the conductivity, dynamic structure factor, plasmon dispersion, imaginary-time structure, and density response functions. We conclude by outlining strategies to combine different simulations to achieve accurate theoretical predictions.

Correlated quantum many‐particle systems out of equilibrium are of high interest in many fields, including correlated solids, ultracold atoms, or dense plasmas. Accurate theoretical description of these systems is challenging both, conceptionally and with respect to computational resources. A quantum fluctuations approach is recently presented, which is equivalent to the nonequilibrium GW approximation that promises high accuracy at low computational cost. The method exhibits process time scaling that is linear in the number of time steps, like the G1–G2 scheme, however, at a much reduced computer memory cost. In a second publication, this approach is extended to the two‐time exchange–correlation functions and the dynamic density response properties. Herein, the properties of this approach are analyzed in more detail. The physical meaning of the central approximation, the quantum polarization approximation, is established. It is demonstrated that the method is equivalent to the Bethe–Salpeter equation for the two‐time exchange–correlation function when the generalized Kadanoff–Baym ansatz with Hartree–Fock propagators is applied.

Correlated classical and quantum many-particle systems out of equilibrium are of high interest in many fields, including dense plasmas, correlated solids, and ultracold atoms. Accurate theoretical description of these systems is challenging both, conceptionally and with respect to computational resources. While for classical systems, in principle, exact simulations are possible via molecular dynamics, this is not the case for quantum systems. Alternatively, one can use many-particle approaches such as hydrodynamics, kinetic theory, or nonequilibrium Green functions (NEGF). However, NEGF exhibit a very unfavorable cubic scaling of the CPU time with the number of time steps. An alternative is the G1-G2 scheme [N. Schl & uuml;nzen et al., Phys. Rev. Lett. 124, 076601 (2020)] which allows for NEGF simulations with time linear scaling, however, at the cost of large memory consumption. The reason is the need to store the two-particle correlation function. This problem can be overcome for a number of approximations by reformulating the kinetic equations in terms of fluctuations - an approach that was developed, for classical systems, by Yu.L. Klimontovich [JETP 33, 982 (1957)]. Here, we present an overview of his ideas and extend them to quantum systems.

We explore the non-equilibrium dynamics of a one-dimensional Fermi-Hubbard system as a sensitive testbed for the capabilities of the time-dependent two-particle reduced density matrix (TD2RDM) theory to accurately describe time-dependent correlated systems. We follow the time evolution of the out-of-equilibrium finite-size Fermi-Hubbard model initialized by a quench over extended periods of time. By comparison with exact calculations for small systems and with matrix product state (MPS) calculations for larger systems but limited to short times, we demonstrate that the TD2RDM theory can accurately account for the non-equilibrium dynamics in the regime from weak to moderately strong inter-particle correlations. We find that the quality of the approximate reconstruction of the three-particle cumulant (or correlation) required for the closure of the equations of motion for the reduced density matrix is key to the accuracy of the numerical TD2RDM results. We identify the size of the dynamically induced three-particle correlations and the amplitude of cross correlations between the two- and three-particle cumulants as critical parameters that control the accuracy of the TD2RDM theory when current state-of-the art reconstruction functionals are employed.

In a recent Letter [T. Dornheim \textit{et al.}, Phys. Rev. Lett. \textbf{121}, 255001 (2018)], it was predicted on the basis of \textit{ab initio} quantum Monte Carlo simulations that, in a uniform electron gas, the peak $\omega_0$ of the dynamic structure factor $S(q,\omega)$ exhibits an unusual non-monotonic wave number dependence, where $d\omega_0/dq < 0$, at intermediate $q$, under strong coupling conditions. This effect was subsequently explained by the pair alignment of electrons %at an intermediate range of wave numbers [T. Dornheim \textit{et al.}, Comm. Phys. \textbf{5}, 304 (2022)]. Here we predict that this non-monotonic dispersion resembling the roton-type behavior known from superfluids should be observable in a dense, partially ionized hydrogen plasma. Based on a combination of path integral Monte Carlo simulations and linear response results for the density response function, we present the approximate range of densities, temperatures and wave numbers and make predictions for possible experimental observations.