This repository contains the PIMC data shown in the publication "Static linear density response from X-ray Thomson scattering measurements: a case study of warm dense beryllium".

We present density response estimators for Monte Carlo simulations that are based on a reweighting procedure, where the samples of an unperturbed system are used to estimate the properties of a system perturbed by an external harmonic potential. This allows the linear and nonlinear static density response to be estimated purely from simulations of the unperturbed system. The method is demonstrated for the uniform electron gas under warm dense matter and strongly coupled conditions using ab initio path integral Monte Carlo simulations. The performance of the method with respect to the number of particles and the number of imaginary time slices is investigated. The scheme is generalised to consider multiple external perturbations, acting on different species and with different wavenumbers, giving one access to additional cross-species density response functions and the complete quadratic response function resolved for both wave number arguments through mode coupling. The flexibility of the methodology opens the possibility to investigate numerous new density response properties to further advance our understanding of interacting quantum many-body systems across a broad range of applications.

Conditional probability density functional theory has recently been used to derive the temperature dependence of the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) for the exchange-correlation (XC) free energy. We implement and systematically benchmark thermal PBE within Kohn-Sham density functional theory calculations of warm dense matter. Comparisons with the local density approximation (LDA) and PBE functionals, as well as thermal LDA, show that thermal PBE significantly improves the description of warm dense matter properties, including energies, forces, pressures, and electronic charge densities. In particular, thermal PBE exhibits close agreement with path integral Monte Carlo (PIMC) reference data at negligible additional computational cost. This work demonstrates the practical utility of thermal PBE as an accurate semilocal functional for simulations in the warm dense regime.

This repository contains the PIMC raw data for the publication "Taylor series perspective on ab initio path integral Monte Carlo simulations with Fermi-Dirac statistics" using the same units and formatting as in the plots.

Matter under extreme densities, temperatures, and pressures is abundant throughout our universe and occurs in a variety of celestial objects such as giant planets, brown dwarfs, and the outer layer of neutron stars. On Earth, such warm dense matter (WDM) [1] plays an important role for cutting-edge technological applications such as material science and discovery. However, the holy grail of contemporary high energy density science is given by inertial confinement fusion, where both the fusion fuel and the ablator material have to traverse the WDM regime in a controlled way during the initial phase of the compression. Unfortunately, the rigorous theoretical description of WDM is notoriously difficult. Ab initio density functional theory (DFT) simulations have emerged as the workhorse in the field, but their accuracy decisively depends on the utilized exchange—correlation (XC) free energy functional, which has to be supplied as an external input. The gold standard is given by quasi-exact path integral Monte Carlo (PIMC) simulations, which are, in principle, capable of giving exact results for a great variety of material properties.
Here, I will give an overview of recent developments in the direct PIMC estimation of the free energy [2,3]---the key
quantity for thermal DFT simulations and equation-of-state tables. This includes novel technical developments to deal efficiently with the fermion sign problem [4,5], as well as new opportunities to directly estimate the chemical potential [6]. From a physics perspective, I will show recent results for the uniform electron gas as well as for warm dense hydrogen. Finally, I will give an outlook on upcoming developments and opportunities for improved equation-of-state tables and XC functionals.
[1] J. Vorberger et al., Roadmap for warm dense matter physics, arXiv:2505.02494
[2] T. Dornheim et al. Phys. Rev. B 111, L041114 (2025)
[3] T. Dornheim et al., Phys. Rev. Research 7, 023250 (2025)
[4] T. Dornheim et al., J. Chem. Theory. Comput. 21, 7290–7303 (2025)
[5] P. Svensson et al., J. Phys. Chem. Lett. 16, 10639–10646 (2025)
[6] T. Dornheim et al., Phys. Rev. B 111, 115149 (2025)

The imaging platform developed at the High Energy Density - Helmholtz International Beamline for Extreme Fields (HED-HiBEF) instrument at the European XFEL and its applications to high energy density and fusion related research are presented. The platform combines the XFEL beam with the high-intensity short-pulse laser ReLaX and the high-energy nanosecond-pulse laser DiPOLE-100X. The spatial resolution is better than 500 nm and the temporal resolution of the order of 50 fs. The influence of the XFEL source in the x-ray imaging method is discussed. Free-propagation x-ray phase contrast imaging and Talbot-Lau imaging setups are shown. We show examples of blast waves and converging cylindrical shocks in aluminium, resonant absorption measurements of specific charged states in copper with ReLaX and planar shocks in polystyrene material generated by DiPOLE-100X. For the first time, we show the application of Talbot-Lau interferometry to convergent cylindrical shocks as well as resonant absorption processes. We also discuss the possibilities introduced by combining this imaging platform with a kJ-class laser.

X-ray Thomson scattering (XRTS) constitutes an essential technique for diagnosing material properties under extreme conditions, such as high pressures and intense laser heating. Time-dependent density functional theory (TDDFT) is one of the most accurate available ab initio methods for modeling XRTS spectra, as well as a host of other dynamic material properties. However, strong thermal excitations, along with the need to account for variations in temperature and density as well as the finite size of the detector significantly increase the computational cost of TDDFT simulations compared to ambient conditions. In this work, we present a broadly applicable method for optimizing and enhancing the efficiency of TDDFT calculations. Our approach is based on a one-to-one mapping between the dynamic structure factor and the imaginary time density--density correlation function, which naturally emerges in Feynman’s path integral formulation of quantum many-body theory. Specifically, we combine rigorous convergence tests in the imaginary time domain with a constraints-based attenuation of narrow-band fluctuations to improve the efficiency of TDDFT modeling without the introduction of any significant bias. As a result, we can report a speed-up by up to an order of magnitude, thus substantially reducing the burden of computational cost required for XRTS analysis.

I present recent result for ab initio path integral Monte Carlo simulations of warm dense matter.

Understanding warm dense matter: from theory to experiment

Warm dense matter (WDM) [1], an extreme state that simultaneously features high densities, temperatures and pressures, is ubiquitous throughout our universe and occurs in a variety of celestial objects such as giant planet interiors, brown dwarfs and white dwarf atmospheres. In addition, WDM is of high current interest due to its relevance for cutting-edge technological applications such as inertial confinement fusion (ICF), where both the fusion fuel and the ablator material have to traverse the WDM regime in a controlled way to reach ignition. As a direct consequence, there has been a remarkable surge of activity in the field and extreme states of matter are nowadays frequently realized in large research facilities such as the National Ignition Facility (NIF) in the US and the European XFEL in Germany.
Despite these advances, a rigorous and comprehensive theoretical description of WDM has remained elusive as it must feature a holistic treatment of a plethora of physical effects including Coulomb coupling, quantum degeneracy and delocalization, strong thermal excitations, and often also partial ionization. This lack of reliable theory limits our ability to model astrophysical objects and practical applications alike. Moreover, it even limits our ability to diagnose experiments with WDM and the interpretation of experimental measurements is often based on a number of model assumptions and de-facto uncontrolled approximations.
To remedy this unfortunate situation, we have recently introduced a new framework for the model-free interpretation of X-ray Thomson scattering (XRTS) experiments, which allows us to infer key parameters such as the temperature directly from the experimental data [2]. In combination with new ab initio path integral Monte Carlo simulations, these capabilities have allowed us to thoroughly re-evaluate an XRTS measurement on strongly compressed beryllium taken at the NIF [3], leading to a substantially lower mass density [4] than it had been assumed based on previously used chemical models.
In this talk, I will give a comprehensive overview of these developments in both theory and experiment, including a discussion of current capabilities and limitations. In addition, I will outline promising opportunities for future experiments harnessing the unique capabilities for high repetition XRTS experiments with ultrahigh resolution [5] using HIBEF-HED at the European XFEL, and also at the NIF using the new colliding planar shock platform [6].

[1] J. Vorberger et al., Roadmap for warm dense matter physics, arXiv:2505.02494
[2] T. Dornheim et al., Accurate temperature diagnostics for matter under extreme conditions, Nature Comm. 13, 7911 (2022)
[3] T. Döppner et al., Observing the onset of pressure-driven K-shell delocalization, Nature 618, 270-275 (2023)
[4] T. Dornheim et al., Unraveling electronic correlations in warm dense quantum plasmas, Nature Communications 16, 5103 (2025)
[5] Th. Gawne et al., Ultrahigh resolution x-ray Thomson scattering measurements at the European X-ray Free Electron Laser, Physical Review B 109, L241112 (2024)
[6] M.J. MacDonald et al., The colliding planar shocks platform to study warm dense matter at the National Ignition Facility, Physics of Plasmas 30, 062701 (2023)

We present the first systematic experimental validation of return-current-driven cylindrical implosion scaling in micrometer-sized Cu and Al wires irradiated by J-class femtosecond laser pulses. Employing XFEL-based imaging with sub-micrometer spatial and femtosecond temporal resolution, supported by hydrodynamic and particle-in-cell simulations, we reveal how return current density depends precisely on wire diameter, material properties, and incident laser energy. We identify deviations from simple theoretical predictions due to geometrically influenced electron escape dynamics. These results refine and confirm the scaling laws essential for predictive modeling in high-energy-density physics and inertial fusion research.

Presentation about the work: Phys. Rev. Research 7, 033016

Presentation on obtaining ionization potential depression and charge state for hydrogen as published in Phys. Rev. Research 7, 033016.

Presentation about the published work Phys. Rev. Research 7, 033016 on determining ionization potential depression and ionization from ab initio simulations of Hydrogen.

Presentation about the published work "Toward model-free temperature diagnostics of warm dense matter from multiple scattering angles", Appl. Phys. Lett. 126, 044104 (2025).

I provide an overview of recent developments in the ab initio quantum Monte Carlo (QMC) simulation of warm dense matter.

A density functional theory (DFT) framework is presented that links functional derivatives of free-energy functionals to non-linear static density response functions in quantum many-body systems. Within this framework, explicit expressions are derived for various higher-order response functions of systems that are homogeneous on average, including the first theoretical result for the cubic response at the first harmonic χ(1,3) 0 (q). Specifically, our framework includes hitherto neglected mode-coupling effects that are important for the non-linear density response even in the presence of a single harmonic perturbation. We compare these predictions for χ(1,3) 0 (q) to new Kohn-Sham DFT simulations, leading to excellent agreement between theory and numerical results. Exact analytical expressions are also obtained for the long-wavelength limits of the ideal quadratic and cubic response functions. Particular emphasis is placed on the connections between the third- and fourth-order functional derivatives of the non-interacting free-energy functional Fs[n] and the ideal quadratic and cubic response functions of the uniform electron gas, respectively. These relations provide exact constraints that may prove useful for the future construction of improved approximations to Fs[n], in particular for warm dense matter applications at finite temperatures. Here, we use
this framework to assess several commonly employed approximations to Fs[n] through orbital-free DFT simulations of the harmonically perturbed ideal electron gas. The results are compared with Kohn-Sham DFT calculations across temperatures ranging from the ground state to the warm dense regime. Additionally, we analyze in detail the temperature- and wavenumber-dependent nonmonotonic behavior of the ideal quadratic and cubic response functions.

I present a topical overview of current warm dense matter research with a focus on x-ray Thomson scattering, experiments and theory.

I present an overview of recent model-free and path integral Monte Carlo based frameworks for the interpretation of x-ray Thomson scattering experiments with warm dense matter.

I present DMA highlights related to the ab initio description of warm dense matter and corresponding x-ray scattering experiments.

Warm dense matter (WDM) is an extreme state that is characterized by the complex
interplay of Coulomb correlations, strong thermal excitations, quantum delocalization and
degeneracy effects, and often also partial ionization and overlapping bound-state
wavefunctions [1]. These conditions are fairly ubiquitous throughout nature and occur in a
variety of compact astrophysical objects such as giant planet interiors, brown dwarfs and
white dwarf atmospheres. In addition, matter under extreme conditions is starting to play an
increasingly important role for cutting-edge technological applications, such as the
discovery and synthesis of novel and exotic materials. The holy grail of contemporary high
energy density science is given by inertial confinement fusion (ICF), which holds the
tantalizing promise of a potentially unlimited source of clean energy in the future. In ICF
experiments, both the fusion fuel and the ablator have to traverse the WDM regime in a
controlled way during the early segments of the compression path while avoiding the
formation of any notorious instabilities. The further optimization that is required to realize
an actual ICF power plant thus makes a more rigorous understanding of WDM
indispensable.
Ab initio path integral Monte Carlo (PIMC) methods are, in principle, uniquely suited to
capture the full complexity of WDM, but their application is severely limited by the
notorious fermion sign problem [2]; it constitutes an exponential computational bottleneck
with respect to increasing numbers of electrons and decreasing temperature. In a seminal
recent work, Xiong and Xiong [3] have suggested to partially avoid the sign problem by
carrying out path integral simulations of fictitious identical particles that are guided by a
continuous spin-statistics parameter ξ.
In this presentation, I give a broad overview about a variety of advances in PIMC
simulations of WDM using this ξ-extrapolation technique, including the estimation of
different observables [4], large-scale simulations of up to 1000 electrons [5], and free
energy calculations [6]. Moreover, I discuss recent methodological developments such as re-
weighting [7] and a generalized Taylor series perspective onto the simulation of fictitious
identical particles [8]. As the capstone of our work, I demonstrate how the ξ-extrapolation
method can be used to compute the properties of real WDM systems [9,10], facilitating for
the first time the direct comparison with x-ray scattering experiments with strongly
compressed beryllium taken at the National Ignition Facility in the USA [11,12,13].
[1] J. Vorberger et al., Roadmap for warm dense matter physics, arXiv:2505.02494
[2] T. Dornheim, Physical Review E 100, 023307 (2019)
[3] Y. Xiong and H. Xiong, The Journal of Chemical Physics 157, 094112 (2022)
[4] T. Dornheim et al., The Journal of Chemical Physics 159, 164113 (2023)
[5] T. Dornheim et al., The Journal of Physical Chemistry Letters 15, 1305 (2024)
[6] T. Dornheim et al., Journal of Chemical Theory and Computation 21, 7290 (2025)
[7] T. Dornheim et al., arXiv:2508.12323 (under review)
[8] T. Dornheim et al., arXiv:2509.11317 (under review)
[9] T. Dornheim et al., The Journal of Chemical Physics 160, 164111 (2024)
[10] T. Dornheim et al., Matter and Radiation at Extremes 9, 057401 (2024)
[11] T. Dornheim et al., Nature Communications 16, 5103 (2025)
[12] T. Dornheim et al., Physics of Plasmas 32, 052712 (2025)
[13] S. Schwalbe et al., arXiv:2504.13611 (under review)

X-ray Thomson scattering (XRTS) has emerged as a valuable diagnostic for matter under extreme conditions, as it captures the intricate many-body physics of the probed sample. Recent advances, such as the model-free temperature diagnostic of Dornheim et al. [this http URL. 13, 7911 (2022)], have demonstrated how much information can be extracted directly within the imaginary-time formalism. However, since the imaginary-time formalism is a concept often difficult to grasp, we provide here a systematic overview of its theoretical foundations and explicitly demonstrate its practical applications to temperature inference, including relevant subtleties. Furthermore, we present recent developments that enable the determination of the absolute normalization, Rayleigh weight, and density from XRTS measurements without reliance on uncontrolled model assumptions. Finally, we outline a unified workflow that guides the extraction of these key observables, offering a practical framework for applying the method to interpret experimental measurements.

Over the last decades, there has been a remarkable surge
of interest in the properties of matter at extreme
densities, temperatures and pressures. Such warm dense
matter (WDM) naturally occurs in a host of compact
astrophysical objects such as giant planet interiors, brown
and white dwarfs, and the outer layer of neutron stars. In
addition, WDM plays an important role in cutting-edge
technological applications such as material science and
inertial confinement fusion.
From a theoretical perspective, WDM is characterized by
the complex interplay of effects such as Coulomb
coupling between the electrons and nuclei, quantum
degeneracy and delocalization, strong thermal excitation,
and partial ionization. This makes its description
challenging even for ab-initio methods [1]. An additional
challenge is given by the diagnostics of experiments with
WDM as corresponding measurements are often being
interpreted based on a number of model assumptions and
approximations.
Here, I give an overview of a number of recent
developments that promise to overcome these limitations.
Very recently, a new framework for the interpretation of
x-ray Thomson scattering (XRTS) experiments has been
suggested that gives one direct access to important
parameters such as the temperature [2] and absolute
intensity [3] without the need for any model calculations.
The combination of this idea with novel approximation-
free ab-initio path integral Monte Carlo (PIMC)
simulation capabilities [4] allows for the rigorous
interpretation of XRTS measurements of WDM states;
this is demonstrated on the example of a dataset for
spherically compressed beryllium [5] that has been taken
at the National Ignition Facility (NIF) in Livermore,
USA. Interestingly, the utilization of state-of-the-art ab-
initio methods such as PIMC or density functional theory
(DFT) leads to a substantially lower mass density
compared to the original analysis that was based on the
widely assumed Chihara decomposition [see Fig. 1]. This
has important implications for the diagnostics of future
experiments with WDM, and for the modeling of inertial
confinement fusion applications.
Beyond their value for XRTS diagnostics, these new
PIMC capabilities open up avenues for a number of
research directions. Recently, first results have been
presented for the density response of warm dense
hydrogen [6,7], which is of key importance for the
modeling of laserfusion applications and astrophysical
objects alike. In addition to being interesting in their own
right, such datasets are also useful for the benchmarking
of other simulation methods such as static [8] and time-
dependent density functional theory (TD-DFT) [9] and
the ubiquitous Chihara models [10], and for the
construction of improved equation-of-state tables.
The talk is concluded by a summary of current
capabilities and of future challenges for the improved
description of extreme states of matter on a true ab-initio
level [1].
Figure 1: Ionization degree of warm dense beryllium; the
solid red cross shows novel PIMC results. Taken from
[4].
References
[1] M. Bonitz et al., Phys. Plasmas 31, 110501 (2024)
[2] T. Dornheim et al., Nature Commun. 13, 7911 (2022)
[3] T. Dornheim et al., Sci. Rep. 14, 14377 (2024)
[4] T. Dornheim et al., arXiv:2402.19113 (2024)
[5] T. Döppner et al., Nature 618, 270-275 (2023)
[6] M. Böhme et al., Phys. Rev. Lett. 129, 066402 (2022)
[7] T. Dornheim et al., Matt. Rad. Extr. 9, 057401 (2024)
[8] Zh. Moldabekov et al., J. Chem. Theor. Comp. 20,
68-78 (2024)
[9] Zh. Moldabekov et al., arXiv:2502:04921 (2025)
[10] H. Bellenbaum et al., arXiv:2503:14014 (2025)

This repository contains the PIMC raw data presented in the publication "Re-weighting estimator for ab initio path integral Monte Carlo simulations of fictitious identical particles" in the same format and units as in the main text.

The rigorous description of warm dense matter (WDM)---an extreme state that is characterized
by high densities, temperatures, and pressures---is of high importance for integrated radiation
hydrodynamics simulations of inertial confinement fusion applications, in particular during the
initial stage of the compression path. In addition, WDM conditions abound in a host of astro-
physical objects such as giant planet interiors and white dwarf atmospheres.
In the laboratory, WDM can be created using different techniques, but accurately diagnosing
even basic parameters such as the density or temperature is difficult and usually relies on
various model assumptions and approximations [1]. Very recently, we have shown that it is
possible to extract a wealth of information such as the temperature [2] or degree of non-
equilibrium [3] directly from x-ray Thomson scattering (XRTS) measurements
without the need for any models or simulations. The combination of this new paradigm
with highly accurate path integral Monte Carlo (PIMC) simulations [4] allows us to rigorously
diagnose an experiment with spherically compressed beryllium carried out at the National
Ignition Facility (NIF) [5], leading to a substantially lower estimate for the mass density
(ρ = 22 ± 2 g/cc) compared to the Chihara model used in the original analysis
(ρ = 34 ± 4 g/cc). Our work has important implications for radiation hydrodynamics
simulations of implosion dynamics and equation-of-state measurements.
References
[1] T. Dornheim, Z. Moldabekov, K. Ramakrishna et al., Phys. Plasmas 30, 032705 (2023)
[2] T. Dornheim, M. Böhme, D. Kraus, T. Döppner et al., Nature Commun. 13, 7911 (2022)
[3] J. Vorberger, T. Preston, N. Medvedev et al., Phys. Lettl. A 499, 129362 (2024)
[4] T. Dornheim, T. Döppner, P. Tolias, M. Böhme, L. Fletcher et al., arXiv:2402.19113
[5] T. Döppner, M. Bethkenhagen, D. Kraus et al., Nature 618, 270-275 (2023)

We report X-ray Thomson scattering measurements of shock-compressed aluminium at densities between 3.75 and
4.5 g/cm3 and a temperature of approximately 0.6 eV, performed at the HED-HiBEF instrument of the European XFEL using the DiPOLE-100X laser drive. By probing plasmon dispersion across momentum transfers k = 0.99 – 2.57 ˚A−1 with high statistical fidelity, we directly test competing theoretical descriptions of electron dynamics under extreme conditions. Time-dependent density functional theory reproduces both the observed plasmon energies and spectral shapes across the full k-range, whereas standard mean-field and static local field correction models systematically overestimate the plasmon frequency, even for aluminium – a canonical uniform electron gas metal. By considering the localisation of electrons around ions and the loss of crystalline symmetry due to disorder in liquid aluminium, our results provide direct experimental evidence for the failure of simple uniform electron gas models in warm dense matter and establish time-dependent density functional theory as a reliable approach for describing electronic correlations under these conditions.

The fermion sign problem constitutes a fundamental computational bottleneck across a plethora of research fields in physics, quantum chemistry and related disciplines. Recently, it has been suggested to alleviate the sign problem in \emph{ab initio} path integral Molecular Dynamics and path integral Monte Carlo (PIMC) calculations based on the simulation of fictitious identical particles that are represented by a continuous quantum statistics variable [\textit{J.~Chem.~Phys.}~\textbf{157}, 094112 (2022)]. This idea facilitated a host of applications including the interpretation of an x-ray scattering experiment with strongly compressed beryllium at the National Ignition Facility [\textit{Nature Commun.}~\textbf{16}, 5103 (2025)]. In the present work, we express the original isothermal -extrapolation method as a special case of a truncated Taylor series expansion around the limit of distinguishable particles. We derive new PIMC estimators that allow us to evaluate the Taylor coefficients up to arbitrary order and we carry out extensive new PIMC simulations of the warm dense electron gas to systematically analyze the sign problem from this new perspective. This gives us important insights into the applicability of the -extrapolation method for different levels of quantum degeneracy in terms of the Taylor series radius of convergence. Moreover, the direct PIMC evaluation of the -derivatives, in principle, removes the necessity for simulations at different values of and can facilitate more efficient simulations that are designed to maximize compute time in those regions of the full permutation space that contribute most to the final Taylor estimate of the fermionic expectation value of interest.

An exact expression is derived for the kinetic contribution to the odd (arbitrary order) frequency moments of the dynamic structure factor via a finite summation that features averages of even (all lower orders) powers of the momentum over the exact momentum distribution. The derivation is carried out for the non-interacting Fermi gas and generalized to the interacting case based on the conjecture that averages over the Fermi distribution can be substituted with averages over the exact distribution. The expression is validated against known results (first, third frequency moments) and new explicit calculations (fifth, seventh frequency moments).

Ab initio path integral Monte Carlo (PIMC) simulations constitute the gold standard for the estimation of a broad range of equilibrium properties of a host of interacting quantum many-body systems spanning conditions from ultracold atoms to warm dense quantum plasmas. A key practical limitation is given by the notorious fermion sign problem, which manifests as an exponential computational bottleneck with respect to system size and inverse temperature. In practice, the sign problem is particularly severe in the grandcanonical ensemble, where the bosonic and fermionic configuration spaces differ not only with respect to the symmetry of the thermal density matrix but, crucially, also with respect to the particle number distribution for a given chemical potential [T. Dornheim, J. Phys. A 54, 335001 (2021)]. Here, we present a simple reweighting scheme that basically allows one to retain access to grandcanonical expectation values at the cost of fermionic PIMC simulations in the canonical ensemble for the largest significant particle number in the fermionic sector. As a practical example, we consider the warm dense electron gas, which has attracted considerable recent attention due to its relevance, e.g., for the modeling of compact astrophysical objects and inertial fusion energy applications

The fermion sign problem constitutes one of the most fundamental obstacles in quantum many-body theory. Recently, it has been suggested to circumvent the sign problem by carrying out path integral simulations with a fictitious quantum statistics variable , which allows for a smooth interpolation between the bosonic and fermionic limits [\textit{J.~Chem.~Phys.}~\textbf{157}, 094112 (2022)]. This -extrapolation method has subsequently been applied to a variety of systems and has facilitated the analysis of an x-ray scattering measurement taken at the National Ignition Facility with unprecedented accuracy [\textit{Nature Commun.}~\textbf{16}, 5103 (2025)]. Yet, it comes at the cost of performing an additional simulations, which, in combination with the required small error bars, can pose a serious practical limitation. Here, we remove this bottleneck by presenting a new re-weighting estimator, which allows the study of the full -dependence from a single path integral Monte Carlo (PIMC) simulation. This is demonstrated for various observables of the uniform electron gas and also warm dense beryllium. We expect our work to be useful for future PIMC simulations of Fermi systems, including ultracold atoms, electrons in quantum dots, and warm dense quantum plasmas.

Over the last years, there has emerged a remarkable interest in warm dense matter (WDM) [1], an extreme state that occurs in astrophysical objects and which is relevant for technological applications such as material science and inertial confinement fusion. WDM theory must take into account the complex interplay of effects such as Coulomb coupling, thermal excitations, quantum degeneracy and partial ionization. Here, I present a number of recent advances in ab initio path integral Monte Carlo (PIMC) simulations, that allow for quasi-exact simulations of light elements over significant parts of the WDM parameter space. In addition to being interesting in their ownright, these capabilities inform computationally less expensive thermal density functional theory calculations and are directly
useful for the interpretation of x-ray Thomson scattering experiments, e.g., at the National Ignition Facility (NIF) [2].

[1] J. Vorberger et al., arXiv:2505.02494 [2] T. Dornheim et al.,
arXiv:2402.19113

I discuss recent developments in the ab initio path integral Monte Carlo simulation of warm dense matter, and how they apply to practical applications such as the interpretation of x-ray scattering experiments at the National Ignition Facility in Livermore, California.

We compare the predictions of the dynamic structure factor (DSF) of ambient polycrystalline aluminium from time-dependent density functional theory (TDDFT) in the pair continuum regime to recent ultrahigh resolution x-ray Thomson scattering measurements, collected at the European XFEL. TDDFT predicts strong anisotropy in the DSF at the wavenumber examined here, even with $q$-blurring accounted for. The experimental spectrum has more than sufficient resolution and signal-to-noise levels to resolve these orientation dependencies, and therefore the orientational averaging of the polycrystalline sample is observed rigorously. Once the orientation averaging is accounted for, TDDFT is able to reproduce the experimental spectrum adequately. Finally, comparisons of predicted DSFs from jellium to experiment demonstrates the importance of accounting for lattice effects in modelling the spectrum from a polycrystal.

We present a methodology for accelerating the estimation of the free energy from path integral Monte Carlo simulations by considering an intermediate artificial reference system where interactions are inexpensive to evaluate numerically. Using the spherically averaged Ewald interaction as this intermediate reference system for the uniform electron gas, the interaction contribution for the free energy was evaluated up to 18 times faster than the Ewald-only method. Furthermore, a ξ-extrapolation technique was tested and applied to alleviate the fermion sign problem and to resolve the sign for large particle numbers. Combining these two techniques enabled the evaluation of the free energy for a system of 1000 electrons, where both finite-size and statistical errors are below chemical accuracy. The general procedure can be applied to systems relevant for planetary and inertial confinement fusion modeling with low to moderate levels of quantum degeneracy.

An exact series representation of the even frequency moments of the dynamic structure factor is
derived. Truncations are proposed that allow to evaluate the explicitly unknown second, fourth and
fifth frequency moments for the finite temperature uniform electron gas. Their applicability range
in terms of degeneracy parameter and wavenumber is determined by exploiting the non-interacting
limit and by comparing with the quasi-exact results of path integral Monte Carlo simulations.

I give an overview of the theoretical description of warm dense matter focusing on ab initio path integral Monte Carlo (PIMC) simulations and their utility for the interpretation of x-ray scattering diagnostics.

Understanding the properties of warm dense hydrogen is of key importance for the modeling of compact astrophysical objects and to understand and further optimize inertial confinement fusion (ICF) applications. The work horse of warm dense matter theory is given by thermal density functional theory (DFT), which, however, suffers from two limitations: (i) its accuracy can depend on the utilized exchange–correlation (XC) functional, which has to be approximated and (ii) it is generally limited to single-electron properties such as the density distribution. Here, we present a new ansatz combining time-dependent DFT results for the dynamic structure factor See(q, ω) with static DFT results for the density response. This allows us to estimate the electron–electron static structure factor See(q) of warm dense hydrogen with high accuracy over a broad range of densities and temperatures. In addition to its value for the study of warm dense matter, our work opens up new avenues for the future study of electronic correlations exclusively within the framework of DFT for a host of applications.

Stanislaw Lem Lecture for European Research Award 2024.

Path integral Monte Carlo (PIMC) simulations are a cornerstone for studying quantum many-body systems. The analytic continuation (AC) needed to estimate dynamic quantities from these simulations is an inverse Laplace transform, which is ill-conditioned. If this inversion were surmounted, then dynamical observables (e.g. dynamic structure factor (DSF) ) could be extracted from the imaginary-time correlation functions estimates.
Although of important, the AC problem remains challenging due to its ill-posedness. To address this challenge, we express the DSF as a linear combination of kernel functions with known Laplace transforms that have been tailored to satisfy its physical constraints. We use least-squares optimization regularized with a Bayesian prior to determine the coefficients of this linear combination. We explore various regularization term, such as the commonly used entropic regularizer, as well as the Wasserstein distance and -distance as well as techniques for setting the regularization weight. A key outcome is the open-source package PyLIT (\textbf{Py}thon \textbf{L}aplace \textbf{I}nverse \textbf{T}ransform), which leverages Numba and unifies the presented formulations. PyLIT's core functionality is kernel construction and optimization.
In our applications, we find PyLIT's DSF estimates share qualitative features with other more established methods. We identify three key findings. Firstly, independent of the regularization choice, utilizing non-uniform grid point distributions reduced the number of unknowns and thus reduced our space of possible solutions. Secondly, the Wasserstein distance, a previously unexplored regularizer, performs as good as the entropic regularizer while benefiting from its linear gradient. Thirdly, future work can meaningfully combine regularized and stochastic optimization.
(text cut for char. limit)

We present extensive \emph{ab initio} path integral Monte Carlo (PIMC) results for the dynamic properties of the finite temperature uniform electron gas (UEG) over a broad range of densities, . We demonstrate that the direct analysis of the imaginary-time density--density correlation function (ITCF) allows for a rigorous assessment of the density and temperature dependence of the previously reported roton-type feature [T.~Dornheim, \emph{Phys.~Rev.~Lett.}~\textbf{121}, 255001 (2018)] at intermediate wavenumbers. We clearly resolve the emergence of a second roton at the second harmonic of the original feature for , which we identify as an incipient phonon dispersion. Finally, we use our highly accurate PIMC results for the ITCF as the basis for an analytic continuation to compute the dynamic structure factor, which additionally substantiates the existence of the second roton in the strongly coupled electron liquid. Our investigation further elucidates the complex interplay between quantum delocalization and Coulomb coupling in the UEG. All PIMC results are freely available online and provide valuable benchmarks for other theoretical methodologies and approximations.

Ab initio modeling of dynamic structure factors (DSF) and related density response properties in the warm dense matter (WDM) regime is a challenging computational task. The DSF, convolved with a probing X-ray beam and instrument function, is measured in X-ray Thomson scattering (XRTS) experiments, which allow the study of electronic structure properties at the microscopic level. Among the various ab initio methods, linear-response time-dependent density-functional theory (LR-TDDFT) is a key framework for simulating the DSF. The standard approach in LR-TDDFT for computing the DSF relies on the orbital representation. A significant drawback of this method is the unfavorable scaling of the number of required empty bands as the wavenumber increases, making LR-TDDFT impractical for modeling XRTS measurements over large energy scales, such as in backward scattering geometry. In this work, we consider and test an alternative approach to LR-TDDFT that employs the Liouville–Lanczos (LL) method for simulating the DSF of WDM. This approach does not require empty states and allows the DSF at large momentum transfer values and over a broad frequency range to be accessed. We compare the results obtained from the LL method with those from the solution of Dyson’s equation using the standard LR-TDDFT within the projector augmented-wave formalism for isochorically heated aluminum and warm dense hydrogen. Additionally, we utilize exact path integral Monte Carlo results for the imaginary-time density-density correlation function (ITCF) of warm dense hydrogen to rigorously benchmark the LL approach. We discuss the application of the LL method for calculating DSFs and ITCFs at different wavenumbers, the effects of pseudopotentials, and the role of Lorentzian smearing. The successful validation of the LL method under WDM conditions makes it a valuable addition to the ab initio simulation landscape, supporting experimental efforts and advancing WDM theory.

We report on results from an experiment at the European x-ray free electron laser where we measured the x-ray Thomson scattering (XRTS) spectrum of single crystal silicon with ultrahigh resolution. Compared to similar previous experiments, we consider a more complex scattering setup, in which the scattering vector changes orientation through the crystal lattice. In doing so, we are able to observe strong geometric dependencies in the inelastic scattering spectrum of silicon at low scattering angles. Furthermore, the high quality of the experimental data allows us to benchmark state-of-the-art TDDFT calculations, and demonstrate TDDFT’s ability to accurately predict these geometric dependencies. Finally, we note that this experimental data was collected at a much faster rate than another recently reported dataset using the same setup, demonstrating that ultrahigh resolution XRTS data can be collected in more general experimental scenarios.

The accurate estimation of the properties of matter under extreme densities, temperatures, and pressures constitutes a highly active frontier at the interface of plasma physics, quantum chemistry, material science and related disciplines. From a theoretical perspective, such warm dense matter (WDM) is characterized by the complex interplay of Coulomb coupling, quantum degeneracy, thermal excitations, and partial ionization; the holistic treatment of this state thus constitutes a most formidable challenge even for state-of-the-art ab initio methods.

The work horse of modern WDM theory is given by thermal density functional theory (DFT), but its accuracy is decisively determined by the utilized electronic exchange--correlation (XC) functional, which has to be supplied as an external input. In contrast, ab initio path integral Monte Carlo (PIMC) simulations are, in principle, capable of providing exact results without any empirical input, but their range of applicability is substantially hampered by an exponential computational bottleneck: the notorious fermion sign problem. In this work, I will give an overview of recent advances in PIMC simulations of WDM, which have facilitated the first exact results for the density response and XC-kernel of warm dense hydrogen over a broad range of densities. In addition, I will present new PIMC results for warm dense beryllium, which have proven highly useful for the interpretation of an x-ray scattering experiment performed at the National Ignition Facility (NIF) in Livermore, and for the benchmarking of other methods such as DFT.

Understanding matter at extreme densities, temperatures and pressures is important for the modeling of astrophysical objects (e.g. giant planet interiors) and technological applications (most notably inertial confinement fusion) alike. Yet, the intricate interplay of effects such as Coulomb coupling, quantum degeneracy, and strong thermal excitations renders the rigorous theoretical description of such warm dense matter (WDM) challenging.

Here, I present an overview of a number of recent developments in the ab initio path integral Monte Carlo (PIMC) simulation of WDM. While being computationally demanding, PIMC is exact within the given error bars and, thus, constitutes a valuable benchmark for computationally more efficient but potentially less accurate methods such as density functional theory (DFT). Moreover, these simulations open up new avenues for the interpretation of X-ray Thomson scattering (XRTS) measurements, which is a key method of diagnostics for experiments with extreme states of matter. As a practical example, we consider a recent XRTS experiment on strongly compressed beryllium carried out at the National Ignition Facility (NIF) in Livermore, for which we find a significantly lower density based on both ab initio PIMC and DFT simulations compared to previously used chemical models and radiation hydrodynamics calculations.

Spherically averaged periodic pair potentials offer the enticing promise to provide accurate results
at a drastically reduced computational cost compared to the traditional Ewald sum. In this work,
we employ the pair potential by Yakub and Ronchi [J. Chem. Phys. 119, 11556 (2003)] in ab initio
path integral Monte Carlo (PIMC) simulations of the warm dense uniform electron gas. Overall,
we find very accurate results with respect to Ewald reference data for integrated properties such as
the kinetic and potential energy, whereas wavenumber resolved properties such as the static struc-
ture factor S(q), the static linear density response χ(q) and the static quadratic density response
χ(2)(q, 0) fluctuate for small q. In addition, we perform an analytic continuation to compute the
dynamic structure factor S(q, ω) from PIMC results of the imaginary-time density–density corre-
lation function F (q, τ ) for both pair potentials. Our results have important implications for future
PIMC calculations, which can be sped up significantly using the YR potential for the estimation
of equation-of-state properties or q-resolved observables in the non-collective regime, whereas a full
Ewald treatment is mandatory to accurately resolve physical effects manifesting for smaller q, includ-
ing the evaluation of compressibility sum rules, the interpretation of x-ray scattering experiments
at small scattering angles, and the estimation of optical and transport properties.

Warme dichte Materie rückt immer mehr in den Fokus, nicht nur in der Physik und Astronomie, sondern auch in den Ingenieurswissenschaften und sogar der Politik. Es handelt sich um interessante, aber auch komplizierte Zustände von Materie, die unser Verständnis ihrer Funktionsweise herausfordern. Die Erforschung warmer dichter Materie hilft dabei, mit Kernfusion Energie zu erzeugen, neue Materialien mit besonderen Eigenschaften herzustellen oder die Struktur von Planeten und ihre Entwicklung zu erklären.

Understanding the dynamic properties of the uniform electron gas (UEG) is important for numerous applications ranging from semiconductor physics to exotic warm dense matter. In this work, we apply the maximum entropy method (MEM), as implemented in Chuna \emph{et al.}~[arXiv:2501.01869], to \emph{ab initio} path integral Monte Carlo (PIMC) results for the imaginary-time correlation function F(q,τ) to estimate the dynamic structure factor S(q,ω) over an unprecedented range of densities at the electronic Fermi temperature. To conduct the MEM, we propose to construct the Bayesian prior μ from the PIMC data. Constructing the static approximation leads to a drastic improvement in S(q,ω) estimate over using the more simple random phase approximation (RPA) as the Bayesian prior. We find good agreement with existing results by Dornheim \emph{et al.}~[\textit{Phys.~Rev.~Lett.}~\textbf{121}, 255001 (2018)], where they are available. In addition, we present new results for the strongly coupled electron liquid regime with rs=50,...,200, which reveal a pronounced roton-type feature and an incipient double peak structure in S(q,ω) at intermediate wavenumbers. We also find that our dynamic structure factors satisfy known sum rules, even though these sum rules are not enforced explicitly. An advantage of our set-up is that it is not specific to the UEG, thereby opening up new avenues to study the dynamics of real warm dense matter systems based on cutting-edge PIMC simulations in future works.

We introduce a new open-source Python x-ray tracing code for modelling Bragg diffracting mosaic crystal spectrometers: High Energy Applications Ray Tracer (HEART). HEART's high modularity enables customizable workflows as well as efficient development of novel features. Utilizing Numba's just-in-time (JIT) compiler and the message-passing interface (MPI) allows running HEART in parallel leading to excellent performance. HEART is intended to be used for modelling x-ray spectra as they would be seen in experiments that measure x-ray spectroscopy with a mosaic crystal spectrometer. This enables the user to, for example, make predictions about what will be seen on a detector in experiment, perform optimizations on the design of the spectrometer setup, or to study the effect of the spectrometer on measured spectra. However, the code certainly has further uses beyond these example use cases. Here, we discuss the physical model used in the code, and explore a number of different mosaic distribution functions, intrinsic rocking curves, and sampling approaches which are available to the user. Finally, we demonstrate its strong predictive capability in comparison to spectroscopic data collected at the European XFEL in Germany.

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 exact 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 exact ab initio path integral Monte Carlo (PIMC) simulations utilizing a chemical model. In contrast to experimental measurements, where noise and non-equilibrium 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 exact quantum Monte Carlo simulations.

We combine the recent η−ensemble path integral Monte Carlo (PIMC) approach to the free energy [T.~Dornheim \textit{et al.}, \textit{Phys.~Rev.~B} \textbf{111}, L041114 (2025)] with a recent fictitious partition function technique based on inserting a continuous variable that interpolates between the bosonic and fermionic limits [Xiong and Xiong, \textit{J.~Chem.~Phys.}~\textbf{157}, 094112 (2022)] to deal with the fermion sign problem. As a practical example, we apply our set-up to the warm dense uniform electron gas over a broad range of densities and temperatures. We obtain accurate results for the exchange--correlation free energy down to half the Fermi temperature, and find excellent agreement with the state-of-the-art parametrization by Groth \textit{et al.}~[\textit{Phys.~Rev.~Lett.}~\textbf{119}, 135001 (2017)]. Our work opens up new avenues for the future study of a host of interacting Fermi-systems, including warm dense matter, ultracold atoms, and electrons in quantum dots.

Warm dense matter (WDM) is an extreme state that abounds in a host of astrophysical objects such
as giant planet interiors, brown dwarfs, and the outer layer of neutron stars. On Earth, WDM plays
an important role for technological applications such as material science, synthesis and discovery. A
particularly important application is given by inertial confinement fusion, where both the fuel
capsule and the ablator have to traverse the WDM regime in a controlled way to reach ignition.
From a physical perspective, WDM is characterized by the intriguing though highly nontrivial
interplay of effects such as Coulomb coupling, strong thermal excitations, quantum degeneracy and
partial ionization, making its rigorous theoretical description notoriously difficult [1]. A second
challenging aspect of studying WDM is the reliable diagnostics of corresponding experiments.
Here, X-ray Thomson scattering (XRTS) has emerged as a widely used tool that is, in principle,
capable of giving one microscopic insights into the probed sample [2]. In practice, however, the
interpretation of XRTS measurements requires an accurate theoretical description of its electronic
properties, which were usually treated based on a number of de-facto uncontrolled model
assumptions such as the decomposition into effectively bound and free electrons within chemical
models.
Here, I give an overview of a number of recent developments that open up new avenues for the
future study of warm dense matter without previous models and approximations. First, I will show
how we can extract a wealth of information such as the temperature [3] or the absolute intensity [4]
directly from the XRTS measurement by switching to the so-called imaginary-time domain---a well
known concept that naturally emerges in Feynman’s path integral formulation of statistical
mechanics and that involves a simple Laplace transform of the experimental signal. Second, I will
show new ab initio path integral Monte Carlo (PIMC) simulation capabilities [5] that allow us to
simulate light elements without any empirical input such as the usual exchange—correlation
functional in DFT or the nodal structure in restricted PIMC. Taken together, these advances allow
for a true first-principles interpretation of the measured XRTS intensity. As a practical example, we
re-examine an XRTS measurement on warm dense beryllium that was collected at the National
Ignition Facility (NIF) in California. Interestingly, we find that using either PIMC or DFT [6]
simulations leads to a substantially reduced mass density compared to the much simpler chemical
models used in the original work [7], which has important implications for the interpretation of
future XRTS experiments, and which calls into question the accuracy of radiation hydrodynamics
models in the WDM regime.
The talk is concluded by outlining remaining limitations and ongoing efforts to extend current
capabilities [8].
[1] T. Dornheim et al., Physics of Plasmas 30, 032705 (2023)
[2] S. Glenzer and R. Redmer, Reviews of Modern Physics 81, 1625 (2009)
[3] T. Dornheim et al., Nature Communications 13, 7911 (2022)
[4] T. Dornheim et al., Scientific Reports 14, 14377 (2024)
[5] T. Dornheim et al., arXiv:2402.19113 (submitted)
[6] T. Dornheim et al., arXiv.2409.08591 (submitted)
[7] T. Döppner et al., Nature 618, 270-275 (2023)
[8] Th. Gawne et al., Physical Review B 109, L241112 (2024)

We summarize a number of recent developments that allow for the model-free interpretation of x-ray Thomson scattering (XRTS) measurements taken on warm dense matter combined with state-of-the-art ab initio path integral Monte Carlo (PIMC) simulations. As a practical example, we consider a an XRTS dataset taken at the National Ignition Facility (NIF) on strongly compressed beryllium. Interestingly, our new approach gives us a substantially lower density compared to previously used chemical models, which has potentially important implications for the integrated radiation hydrodynamics modelling of inertial fusion energy applications.

Matter under extreme densities, temperatures and pressures is ubiquitous throughout our universe
and naturally occurs in a variety of astrophysical objects, including giant planet interiors. On Earth,
such extreme states are important for technological applications such as inertial fusion energy
(IFE), where both the fuel capsule and the ablator material have to traverse this warm dense matter
regime in a controlled way to reach ignition. In the laboratory, warm dense matter is created in large
research facilities such as the European XFEL in Germany using a variety of techniques. Here, a key
challenge is given by the accurate diagnostics of the created samples due to the extreme conditions
and the ultrafast time scales. Over the last years, the X-ray Thomson scattering (XRTS) technique---
also known as inelastic X-ray scattering---has emerged as a promising method of diagnostics as it
is, in principle, capable of giving microscopic insights into the probed sample in the form of the
electronic dynamic structure factor [1]. In practice, however, the interpretation of XRTS
measurements has relied on theoretical models that are based on a number of de-facto
uncontrolled assumptions. Consequently, the quality of the thus inferred system parameters has
remained unclear. Here, I present an overview of a new approach that allows for the model-free
interpretation of XRTS spectra in the imaginary-time domain [2,3]. The latter naturally emerges in
Feynman’s celebrated path integral formulation of statistical mechanics and, by definition, contains
the same information as the usual spectral epresentation, only in an a-priori unfamiliar
representation. At the same time, working in the imaginary-time allows one to deconvolve the
physical nformation from effects due to the X-ray source and the detector. This, in turn, opens up
the way for the model-free extraction of important system parameters such as the temperature [2]
without the need for any
approximations or simulations.
[1] S. H. Glenzer and R. Redmer, X-ray Thomson scattering in high energy density plasmas, Rev. Mod.
Phys. 81, 1625 (2009)
[2] T. Dornheim et al., Accurate temperature diagnostics for matter under extreme conditions,
Nature Commun. 13, 7911 (2022)
[3] T. Dornheim et al., Physical insights from imaginary-time correlation functions, Matt. Radiat.
Extremes 8, 056601 (2023)

Many fields of physics use quantum Monte Carlo techniques, but struggle to estimate dynamic spectra via the analytic
continuation of imaginary-time quantum Monte Carlo data. One of the most ubiquitous approaches to analytic continuation is the maximum entropy method (MEM). We supply a dual Newton optimization algorithm to be used within
the MEM and provide analytic bounds for the algorithm’s error. The MEM is typically used with Bryan’s controversial algorithm. We present new theoretical issues that are not yet in the literature. Our algorithm has all the theoretical
benefits of Bryan’s algorithm without these theoretical issues. We compare the MEM with Bryan’s optimization to
the MEM with our dual Newton optimization on test problems from lattice quantum chromodynamics and plasma
physics. These comparisons show that in the presence of noise the dual Newton algorithm produces better estimates
and error bars; this indicates the limits of Bryan’s algorithm’s applicability. We use the MEM to investigate authentic
quantum Monte Carlo data for the uniform electron gas at warm dense matter conditions and further substantiate the
roton-type feature in the dispersion relation.

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., arXiv:2407.01044] 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 ∆μxc ≲ 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.

We explore the recently introduced η-ensemble approach to compute the free energy directly from \emph{ab initio} path integral Monte Carlo (PIMC) simulations [T.~Dornheim \emph{et al.}, arXiv:2407.01044] and apply it to the archetypal uniform electron gas model both in the warm dense matter and strongly coupled regimes. Specifically, we present an in-depth study of the relevant algorithmic details such as the choice of the free weighting parameter and the choice of the optimum number of intermediate η-steps to connect the real, non-ideal system (η=1) with the ideal limit (η=0). Moreover, we explore the inherent decomposition of the full free energy into its ideal bosonic, ideal-to-interacting, and bosonic-to-fermionic contributions for different parameter regimes. Finally, we compare our new free energy data with an existing free energy parametrization [Groth \emph{et al.}, Phys.~Rev.~Lett.~\textbf{119}, 135001 (2017)] obtained via adiabatic connection formula evaluations, and we find very good agreement in its range of applicability, i.e., for density parameters rs≤20; in addition, we present the first PIMC results for the free energy in the low density regime of 20

Warm dense matter (WDM) plays an important role in astrophysical objects and technological applications, but the rigorous diagnostics of corresponding experiments is notoriously difficult. In this work, we present a model-free analysis of x-ray Thomson scattering (XRTS) measurements at multiple scattering angles. Specifically, we analyze scattering data that have been collected for isochorically heated graphite at the Linac Coherent Light Source (LCLS). Overall, we find good consistency in the extracted temperature between small and large scattering angles, whereas possible signatures of non-equilibrium may be hidden by the source function, and by the available dynamic spectral range. The present proof-of-principle study directly points to improved experimental set-ups for equation-of-state measurements and for the model-free study of relaxation times.

The recently derived Fourier--Matsubara expansion of imaginary--time correlation functions comprises an exact result of linear response theory for finite-temperature quantum many-body systems. In its density--density version, the expansion facilitates systematic comparisons between quasi-exact \emph{ab initio} path integral Monte Carlo simulations and approximate dielectric formalism schemes at the level of the imaginary--time (density--density) correlation functions and the dynamic Matsubara local field corrections. On this theoretical basis, the dynamic properties of the quantum version of the Singwi--Tosi--Land--Sj\"olander scheme are analyzed for the paramagnetic warm dense uniform electron gas. The marginal improvement compared to the semi-classical version of the Singwi--Tosi--Land--Sj\"olander scheme is attributed to the weak Matsubara order dependence of the approximate dynamic Matsubara local field correction. The evaluation of the ideal density response function at the non-interacting occupation numbers is identified to constitute a general deficiency of the dielectric formalism, which calls for a reformulation in future works.

The density-density correlations of the non-interacting finite temperature electron gas are discussed in detail. Starting from the ideal linear density response function and utilizing general relations from linear response theory, known and novel expressions are derived for the pair correlation function, static structure factor, dynamic structure factor, thermal structure factor and imaginary time correlation function. Applications of these expressions in the classical mapping approach, self-consistent dielectric formalism and equation-of-state construction are analyzed in depth.

Research into Warm Dense Matter (WDM) has recently become more important with
advances in inertial confinement fusion and astrophysics. The complex nature of matter under
these conditions remains however, making it extremely difficult to describe theoretically, due
to the interplay between quantum degeneracy and Coulomb interactions, and the transition
of condensed and plasma phases. Several methods exist to describe matter at these
conditions, with recent extensions to Path Integral Monte Carlo (PIMC) allowing the
calculation of the Laplace of the dynamic structure factors. This allows quasi-exact calculations
for warm dense hydrogen, but does not give access to other important physical quantities
describing a plasma state, like the ionisation and ionisation potential depression (IPD). Since
both are difficult to measure experimentally, we instead compare the imaginary time
correlation function (ITCF) from PIMC simulations with synthetic X-Ray Thomson Scattering
(XRTS) spectra to obtain a best estimate for the ionisation state. The IPD is then directly
calculated using the Saha equation and compared against other commonly used models. We
expect this work to be relevant for future inertial confinement energy developments,
particularly in validating equation of state models.

Research into Warm Dense Matter (WDM) has become more important with recent advances in inertial confinement fusion and astrophysics. The interplay between quantum degeneracy and Coulomb interactions, and the transition of condensed and plasma phases occurring under these conditions, however, make WDM extremely difficult to describe theoretically. Several methods exist to describe matter at these conditions, with recent extensions to Path Integral Monte Carlo (PIMC) allowing the calculation of the Laplace transform of the dynamic structure factor, i.e. the imaginary time correlation function (ITCF), of warm dense hydrogen [1]. While PIMC is quasi-exact, it does not give access to other important physical quantities describing a plasma state, like the ionisation and ionisation potential depression (IPD). Moreover, both are difficult to measure experimentally. To remedy this, we instead compare the ITCF from PIMC simulations with synthetic X-Ray Thomson Scattering spectra [2], computed from a Chihara decomposition [3], to obtain a best estimate for the ionisation state. The IPD is then directly calculated using the Saha equation and compared against other commonly used models. We expect this work to be relevant for future inertial confinement energy developments, particularly in validating equation of state models, and for the refinement of astrophysical models.

[1] T. Dornheim et.al., arXiv preprint arXiv:2403.08570 (2024)

[2] T. Dornheim et.al., Phys. Plasmas 30, 042707 (2023)

[3] G. Gregori et.al., Phys. Rev. E 67, 026412 (2003)

Warm dense matter (WDM) is now routinely created and probed in laboratories around the world, providing unprecedented insights into conditions achieved in stellar atmospheres, planetary interiors, and inertial confinement fusion experiments. However, the interpretation of these experiments is often filtered through models with systematic errors that are difficult to quantify. Due to the simultaneous presence of quantum degeneracy and thermal excitation, processes in which free electrons are de-excited into thermally unoccupied bound states transferring momentum and energy to a scattered X-ray photon become viable. Here we show that such free-bound transitions are a particular feature of WDM and vanish in the limits of cold and hot temperatures. The inclusion of these processes into the analysis of recent X-ray Thomson Scattering experiments on WDM at the National Ignition Facility and the Linac Coherent Light Source significantly improves model fits, indicating that free-bound transitions have been observed without previously being identified. This interpretation is corroborated by agreement with a recently developed model-free thermometry technique and presents an important step for precisely characterizing and understanding the complex WDM state of matter.

*This work was partly funded by the Center for Advanced Systems Understanding (CASUS) which is financed by Germany's Federal Ministry of Education and Research (BMBF) and by the Saxon Ministry for Science, Culture and Tourism (SMWK) with tax funds on the basis of the budget approved by the Saxon State Parliament.Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-NA0003525.The work of Ti.~D., M.~J.~M, and F.R.G.~was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.

Matter under extreme densities, temperatures and pressures is ubiquitous throughout our universe
and naturally occurs in a variety of astrophysical objects, including giant planet interiors (e.g.
Jupiter, but also exoplanets), brown dwarfs, white dwarf atmospheres, in the outer layer of neutron
stars and during meteor impacts. On Earth, such extreme states are important for technological
applications such as the discovery and synthesis of novel materials. A particularly important
application is given by inertial fusion energy (IFE), where both the fuel capsule and the ablator
material have to traverse this warm dense matter regime in a controlled way to reach ignition.
Indeed, the recent spectacular news from the National Ignition Facility (NIF) at the Lawrence
Livermore National Laboratory in California, USA, who have reported a net energy gain of the
burning plasma with respect to the compression energy [1], opens up the intriguing possibility to
develop IFE into a clean, safe and nigh abundant source of energy in the future.
In the laboratory, warm dense matter is created in large research facilities such as the European
XFEL in Germany, SACLA in Japan, and the NIF, SLAC, and the OMEGA laser in the USA using a
variety of techniques. Here, a key challenge is given by the accurate diagnostics of the created
samples due to the extreme conditions and the ultrafast time scales. Over the last years, the X-ray
Thomson scattering (XRTS) technique---also known as inelastic X-ray scattering---has emerged as
a promising method of diagnostics as it is, in principle, capable of giving microscopic insights into
the probed sample in the form of the electronic dynamic structure factor [2]. In practice, however,
the interpretation of XRTS measurements has relied on theoretical models that are based on a
number of de-facto uncontrolled assumptions. Consequently, the quality of the thus inferred system
parameters has remained unclear.
Here, I present an overview of a new approach that allows for the model-free interpretation of
XRTS spectra in the imaginary-time domain [3-5]. The latter naturally emerges in Feynman’s
celebrated path integral formulation of statistical mechanics and, by definition, contains the same
information as the usual spectral representation, only in an a-priori unfamiliar representation. At the
same time, working in the imaginary-time allows one to deconvolve the physical information from
effects due to the X-ray source and the detector. This, in turn, opens up the way for the model-free
extraction of important system parameters such as the temperature [3] without the need for any
approximations or simulations.

[1] The Indirect Drive ICF Collaboration, Achievement of Target Gain Larger than Unity in an
Inertial Fusion Experiment, Phys. Rev. Lett. 132, 065102 (2024)
[2] S. H. Glenzer and R. Redmer, X-ray Thomson scattering in high energy density plasmas, Rev.
Mod. Phys. 81, 1625 (2009)
[3] T. Dornheim et al., Accurate temperature diagnostics for matter under extreme conditions,
Nature Commun. 13, 7911 (2022)
[4] T. Dornheim et al., Physical insights from imaginary-time correlation functions, Matt. Radiat.
Extremes 8, 056601 (2023)
[5] T. Dornheim et al., X-ray Thomson scattering absolute intensity from the f-sum rule in the
imaginary-time domain, Sci. Reports 14, 14377 (2024)

Understanding the properties of warm dense hydrogen [1] is of paramount importance for the modeling of astrophysical objects (giant planets, brown dwarfs, etc) and for the development of technological applications such as inertial fusion energy. Yet, the simultaneous presence of Coulomb correlations, partial ionization, quantum degeneracy and strong thermal excitations renders its accurate theoretical description challenging: a holistic approach that takes into account all of these effects without uncontrolled approximations is needed.

Here, I present new ab initio path integral Monte Carlo (PIMC) simulations of warm dense hydrogen [2,3], which have been obtained without the usual fixed-node approximation. While being computationally costly, these simulations give us access to a host of observables, most notably the linear density response and the related local field factors [3]. Finally, I discuss the direct connection between our simulations and upcoming x-ray Thomson scattering (XRTS) experiments with hydrogen, and the potential utility of the static density response function as a physical observable to quantify electronic localization around the ions.

[1] M. Bonitz et al., arXiv:2405.10627

[2] T. Dornheim et al., Journal of Chemical Physics 160, 164111 (2024)

[3] T. Dornheim et al., arXiv:2403.08570

*This work has received funding from the European Union's Just Transition Fund (JTF) within the project "Roentgenlaser-Optimierung der Laserfusion" (ROLF), contract number 5086999001, co-financed by the Saxon state government out of the State budget approved by the Saxon State Parliament.This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2022 research and innovation programme(Grant agreement No. 101076233, "PREXTREME").Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them

We derive equations of motion for higher order density response functions using the theory of thermodynamic Green's functions. We also derive expressions for the higher order generalized dielectric functions and polarization functions. Moreover, we relate higher order response functions and higher order collision integrals within the Martin-Schwinger hierarchy. We expect our results to be highly relevant to the study of a variety of quantum many-body systems such as matter under extreme temperatures, densities, and pressures.

Mosaic crystals, with their high integrated reflectivities, are widely employed in spectrometers used to diagnose high energy density systems. X-ray Thomson scattering (XRTS) has emerged as a powerful diagnostic tool of these systems, providing in principle direct access to important properties such as the temperature via detailed balance. However, the measured XRTS spectrum is broadened by the spectrometer instrument function (IF), and without careful consideration of the IF one risks misdiagnosing system conditions. Here, we consider in detail the IF of 40 and 100 μm mosaic Highly Annealed Pyrolytic Graphite crystals, and how the broadening varies across the spectrometer in an energy range of 6.7–8.6 keV. Notably, we find a strong asymmetry in the shape of the IF toward higher energies. As an example, we consider the effect of the asymmetry in the IF on the temperature inferred via XRTS for simulated 80 eV CH plasmas and find that the temperature can be overestimated if an approximate symmetric IF is used. We, therefore, expect a detailed consideration of the full IF will have an important impact on system properties inferred via XRTS in both forward modeling and model-free approaches.

I present an overview of current ab initio path integral Monte Carlo (PIMC) capabilities to simulate warm dense matter and related extreme states. In the first part, I introduce the PIMC method and summarize recent developments for the uniform electron gas. In the second part, I show how emerging PIMC simulations of real systems such as warm dense hydrogen and beryllium allow for novel ways to interpret x-ray Thomson scattering (XRTS) measurements. This is demonstrated for an experimental dataset for strongly compressed beryllium measured at the National Ignition Facility (NIF).

The kinetic energy (KE) kernel, which is defined as the second order functional derivative of the KE functional with respect to density, is the key ingredient to the construction of KE models for orbital free density functional theory (OFDFT) applications. For solids, the KE kernel is usually approximated using the uniform electron gas (UEG) model or the UEG-with-gap model. These kernels do not have information about the effects from the core electrons since there are no orbitals for the projection on nonlocal pseudopotentials. To illuminate this aspect, we provide a methodology for computing the KE kernel from Kohn-Sham DFT and apply it to the valence electrons in bulk aluminum (Al) with a face-centered cubic lattice and in bulk silicon (Si) in a semiconducting crystal diamond state. We find that bulk-derived local pseudopotentials provide accurate results for the KE kernel in the interstitial region. The effect of using nonlocal pseudopotentials manifests at short wavelengths, defined by the diameter of an ion surrounded by its core electrons. Specifically, we find that the utilization of nonlocal pseudopotentials leads to significant deviations in the KE kernel from the von Weizsacker result in this region, which is, as a rule, explicitly enforced in most widely used KE functional approximations for OFDFT simulations.

X-ray Thomson scattering (XRTS) has emerged as a powerful tool for the diagnostics of matter under extreme conditions. In principle, it gives one access to important system parameters such as the temperature, density, and ionization state, but the interpretation of the measured XRTS intensity usually relies on theoretical models and approximations. In this work, we show that it is possible to extract the Rayleigh weight—a key property that describes the electronic localization around the ions—directly from the experimental data without the need for any model calculations or simulations. As a practical application, we consider an experimental measurement of strongly compressed Be at the National Ignition Facility (NIF) [D¨oppner et al., Nature 618, 270-275 (2023)]. In addition to being interesting in their own right, our results will open up new avenues for diagnostics from ab initio simulations, help to further constrain existing chemical models, and constitute a rigorous benchmark for theory and simulations.

Energy functionals serve as the basis for different models and methods in quantum and classical many-particle physics. Arguably, one of the most successful and widely used approaches in material science at both ambient and extreme conditions is density functional theory (DFT). Various flavors of DFT methods are being actively used to study material properties at extreme conditions, such as in warm dense matter, dense plasmas, and nuclear physics applications. In this review, we focus on the warm dense matter regime, which occurs in the core of giant planets and stellar atmospheres, and as a transient state in inertial confinement fusion experiments. We discuss the connection between linear density response functions and free energy functionals as well as the utility of the linear response formalism for the construction of advanced functionals. As a new result, we derive the stiffness theorem linking the change in the intrinsic free energy to the density response properties of electrons. We review and summarize recent works that assess various exchange-correlation (XC) functionals for an inhomogeneous electron gas that is perturbed by a harmonic external field and for warm dense hydrogen using exact path integral quantum Monte Carlo data as an unassailable benchmark. This constitutes a valuable guide for selecting an appropriate XC functional for DFT calculations in the context of investigating the inhomogeneous electronic structure of warm dense matter. We stress that correctly simulating the strongly perturbed electron gas necessitates the correct UEG limit of the XC and non-interacting free-energy functionals.

Hydrogen at extreme temperatures and pressures has significant relevance for cutting-edge technological applications, and its natural occurrence in astrophysical objects further underscores its importance. In this work, we develop a new framework to identify the breaking of bound states due to pressure ionization in bulk hydrogen [1]. Firstly, we show that the dimensionless reduced density gradient (RDG) is a valuable tool for detecting pressure-induced ionization in the medium. Secondly, the generalized RDG is proposed as an effective means for examining the interstitial electronic structure. Finally, our rigorous assessment of a variety of exchange-correlation (XC) functionals in density functional theory calculations for different density regions reveals the crucial role of thermal XC effects in accurately describing density gradients in high-energy density systems. Our exact path integral Monte-Carlo (PIMC) test set generated for this project is freely available online [2]. The insights gained from this research could also have implications for our understanding of astrophysical phenomena, such as white dwarf stars. Furthermore, this study is an addition to our current research on thermal effects in XC functionals, which we are exploring at various complexity levels [3-6].

References:

[1] Z. Moldabekov, S. Schwalbe, M. P. Böhme, J. Vorberger, X. Shao, M. Pavanello, F.
Graziani, T. Dornheim, Journal of Chemical Theory and Computation (in print) (2023).
DOI: 10.1021/acs.jctc.3c00934 ; arXiv:2308.07916.
[2] The data is available according to the FAIR principles on the bound state breaking BSB
GitLab Repository. 2023; https://gitlab.com/theonov13/bsb
[3] Z. Moldabekov, M. Lokamani, J. Vorberger, A. Cangi, and T. Dornheim, The Journal of
Physical Chemistry Letters 14 (5), 1326-1333 (2023).
[4] Z. Moldabekov, M. Lokamani, J. Vorberger, A. Cangi, T. Dornheim, J. Chem. Phys.
158, 094105 (2023).
[5] Z. Moldabekov, T.Dornheim, M. Böhme, J. Vorberger, A. Cangi, J. Chem. Phys. 155,
124116 (2021).
[6] Z. Moldabekov, M. Böhme, J. Vorberger, D. Blaschke, and T. Dornheim, Journal of
Chemical Theory and Computation 19, 1286-1299 (2023)

The combination of isochoric heating of solids by free-electron lasers (FELs) and in situ diagnostics by X-ray Thomson scattering (XRTS) allows for measurements of material properties at warm dense matter (WDM) conditions relevant for astrophysics, inertial confinement fusion, and materials science. In the case of metals, the FEL beam pumps energy directly into electrons with the lattice structure of ions being nearly unaffected. This leads to a unique transient state that gives rise to a set of interesting physical effects, which can serve as a reliable testing platform for WDM theories. In this work, we present extensive linear-response time-dependent density functional theory (TDDFT) results for the electronic dynamic structure factor of isochorically heated copper with a face-centered cubic lattice. At ambient conditions, the plasmon is heavily damped due to the presence of d-band excitations, and its position is independent of the wavenumber. In contrast, the plasmon feature starts to dominate the excitation spectrum and has a Bohm–Gross-type plasmon dispersion for temperatures T ≥ 4 eV, where the quasi-free electrons in the interstitial region are in the WDM regime. In addition, we analyze the thermal changes in the d-band excitations and outline the possibility to use future XRTS measurements of isochorically heated copper as a controlled testbed for WDM theories.

Ultrafast heating of solids with modern x-ray free electron lasers (XFELs) leads to a unique set of conditions characterized by the simultaneous presence of heated electrons in a cold ionic lattice. In this work, we analyze the effect of electronic heating on the dynamic structure factor (DSF) in bulk aluminum (Al) with a face-centered cubic lattice and in silicon (Si) with a crystal diamond structure using first-principles linear-response time-dependent density functional theory simulations. We find a thermally induced red shift of the collective plasmon excitation in both materials. In addition, we show that the heating of the electrons in Al can lead to the formation of a double-plasmon peak due to the extension of the Landau damping region to smaller wave numbers. Finally, we demonstrate that thermal effects generate a measurable and distinct signature (peak-valley structure) in the DSF of Si at small frequencies. Our simulations indicate a variety of new features in the spectrum of x-ray-driven solids, specifically at finite momentum transfer, which can be probed in upcoming x-ray Thomson scattering experiments at various XFEL facilities.

We investigate the short wavelength limit of the dynamic Matsubara local field correction $\widetilde{G}(\mathbf{q},z_l)$ of the uniform electron gas based on direct \emph{ab initio} path integral Monte Carlo (PIMC) results over an unprecedented range of wavenumbers, $q\lesssim20q_\textnormal{F}$, where $q_\textnormal{F}$ is the Fermi wavenumber. We find excellent agreement with the analytically derived asymptotic limit by Hou \emph{et al.}~[\textit{Phys.~Rev.~B}~\textbf{106}, L081126 (2022)] for the static local field correction and empirically confirm the independence of the short wavelength limit with respect to the Matsubara frequency $z_l$. In the warm dense matter regime, we find that the onset of the quantum tail in the static local field correction closely coincides with the onset of the algebraic tail in the momentum distribution function and the corresponding empirical criterion reported by Hunger \emph{et al.}~[\textit{Phys.~Rev.~E} \textbf{103}, 053204 (2021)]. In the strongly coupled electron liquid regime, our calculations reveal a more complicated non-monotonic convergence towards the $q\to\infty$ limit that is shaped by the spatial structure in the system.
We expect our results to be of broad interest for a number of fields including the study of matter under extreme conditions, the development of improved dielectric theories, and the construction of advanced exchange--correlation functionals for thermal density functional theory.

We carry out highly accurate ab initio path integral Monte Carlo (PIMC) simulations to directly
estimate the free energy of various warm dense matter systems including the uniform electron gas
and hydrogen without any nodal restrictions or other approximations. Since our approach is based
on an effective ensemble in a bosonic configuration space, it does not increase the computational
complexity beyond the usual fermion sign problem. Its application to inhomogeneous cases such as
an electronic system in a fixed external ion potential is straightforward and opens up the enticing
possibility to benchmark density functional theory and other existing methods. Finally, it is not
limited to warm dense matter, and can be applied to a gamut of other systems such ultracold atoms
and electrons in quantum dots.

We present two methods for computing the dynamic structure factor for warm dense hydrogen
without invoking either the Born-Oppenheimer approximation or the Chihara decomposition, by
employing a wave packet description which resolves the electron dynamics during ion evolution.
Firstly, a semiclassical method is discussed which is corrected based on known quantum constraints,
and secondly, a direct computation of the density-density response function within the molecular
dynamics. The wave packet models are compared to PIMC and DFT-MD for the static and low-
frequency behaviour. For the high-frequency behaviour the models recover the expected behaviour
in the limits of small and large momentum transfers and show the characteristic flattening of the
plasmon dispersion for intermediate momentum transfers due to interactions, in agreement with
commonly used models for X-ray Thomson scattering. By modelling the electrons and ions on
an equal footing, both the ion and free electron part of the spectrum can now be treated within
a single framework where we simultaneously resolve the ion-acoustic and plasmon mode, with a
self-consistent description of collisions and screening.

Matter at extreme densities and temperatures displays a complex quantum behavior that is characterized by Coulomb interactions, thermal excitations, and partial ionization. Such warm dense matter (WDM) is ubiquitous throughout the universe and occurs in a host of astrophysical objects such as giant planet interiors and white dwarf atmospheres. A particularly intriguing application is given by inertial confinement fusion, where both the fuel capsule and the ablator have to traverse the WDM regime in a controlled way to reach ignition.

In practice, rigorously understanding WDM is highly challenging both from experimental measurements and numerical simulations [1]. On the one hand, interpreting and diagnosing experiments with WDM requires a suitable theoretical description. One the other hand, there is no single method that is capable of accurately describing the full range of relevant densities and temperatures, and the interpretation of experiments is, therefore, usually based on a number of de-facto uncontrolled approximations. The result is the vicious cycle of WDM diagnostics: making sense of experimental observations requires theoretical modeling, whereas theoretical models must be benchmarked against experiments to verify their inherent assumptions.

In this work, we outline a strategy to break this vicious cycle by combining the X-ray Thomson scattering (XRTS) technique [2] with new ab initio path integral Monte Carlo (PIMC) capabilities [3,4,5]. As a first step, we have proposed to interpret XRTS experiments in the imaginary-time (Laplace) domain, which allows for the model-free diagnostics of the temperature [6] and normalization [7]. Moreover, by switching to the imaginary-time, we can directly compare our quasi-exact PIMC calculations with the experimental measurement [5]. This opens up novel ways to diagnose the experimental conditions, as we have recently demonstrated for the case of strongly compressed beryllium at the National Ignition Facility.

Our results open up new possibilities for improved XRTS set-ups that are specifically designed to be sensitive to particular parameters of interest [8]. Moreover, the presented PIMC capabilities are important in their own right and will allow for a gamut of applications, including equation-of-state calculations and the estimation of structural properties and linear response functions.

[1] T. Dornheim et al., Phys. Plasmas 30, 032705 (2023) [2] S. Glenzer and R. Redmer, Rev. Mod. Phys. 81, 1625 (2009) [3] T. Dornheim et al., J. Phys. Chem. Lett. 15, 1305-1313 (2024) [4] T. Dornheim et al., arXiv:2403.01979 [5] T. Dornheim et al., arXiv:2402.19113 [6] T. Dornheim et al., Nature Commun. 13, 7911 (2022) [7] T. Dornheim et al., arXiv:2305.15305 [8] Th. Gawne et al., arXiv:2403.02776

We investigate the impact of electronic correlations and quantum delocalization onto the static structure factor and static density response function of the strongly coupled electron liquid. In contrast to a classical system, the density response of the electron liquid vanishes on small length scales due to quantum delocalization effects, which we rigorously quantify in terms of imaginary-time correlation functions and dynamic Matsubara response functions. This allows us to analyze the interplay of structural order and dynamic quantum effects as it manifests in the dynamic Matsubara local field correction. Finally, we identify an effective electronic attraction in the spin-offdiagonal static density response when the wavelength of the perturbation is commensurate with the average interparticle distance.

This repository contains the PIMC data shown in the publication "Static linear density response from X-ray Thomson scattering measurements: a case study of warm dense beryllium".

We present density response estimators for Monte Carlo simulations that are based on a reweighting procedure, where the samples of an unperturbed system are used to estimate the properties of a system perturbed by an external harmonic potential. This allows the linear and nonlinear static density response to be estimated purely from simulations of the unperturbed system. The method is demonstrated for the uniform electron gas under warm dense matter and strongly coupled conditions using ab initio path integral Monte Carlo simulations. The performance of the method with respect to the number of particles and the number of imaginary time slices is investigated. The scheme is generalised to consider multiple external perturbations, acting on different species and with different wavenumbers, giving one access to additional cross-species density response functions and the complete quadratic response function resolved for both wave number arguments through mode coupling. The flexibility of the methodology opens the possibility to investigate numerous new density response properties to further advance our understanding of interacting quantum many-body systems across a broad range of applications.

Conditional probability density functional theory has recently been used to derive the temperature dependence of the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) for the exchange-correlation (XC) free energy. We implement and systematically benchmark thermal PBE within Kohn-Sham density functional theory calculations of warm dense matter. Comparisons with the local density approximation (LDA) and PBE functionals, as well as thermal LDA, show that thermal PBE significantly improves the description of warm dense matter properties, including energies, forces, pressures, and electronic charge densities. In particular, thermal PBE exhibits close agreement with path integral Monte Carlo (PIMC) reference data at negligible additional computational cost. This work demonstrates the practical utility of thermal PBE as an accurate semilocal functional for simulations in the warm dense regime.

We develop a wavepacket molecular dynamics framework for modeling the structural properties of partially ionized dense plasmas, based on a chemical model that explicitly includes bound state wave functions. Using hydrogen as a representative system, we compute self-consistent charge state distributions through free-energy minimization, following the approach of Plummer et al. [Phys. Rev. E 111, 015204 (2025)]. This enables a direct comparison of static equilibrium properties with path-integral Monte Carlo data, facilitating an evaluation of the model’s underlying approximations and its ability to capture the complex interplay between ionization and structure in dense plasma environments.

Presentation at DPG Spring Meeting SMUK - Erlangen, March 17, 2026 (German Physical Society)

This repository contains the PIMC raw data for the publication "Taylor series perspective on ab initio path integral Monte Carlo simulations with Fermi-Dirac statistics" using the same units and formatting as in the plots.

Heating and ionization are among the most fundamental processes in relativistic laser--solid interactions; however, their spatiotemporal evolution remains challenging to capture experimentally. Here we present detailed diagnosis of high-intensity laser interactions with wire targets, leveraging the extreme spectral brightness of an X-ray free-electron laser in sub-picosecond time-resolved resonant X-ray emission spectroscopy and absorption imaging. Experimental results are compared with comprehensive simulations using atomic collisional--radiative models, particle-in-cell, and magnetohydrodynamics codes to elucidate the underlying physics. These multi-scale simulations reveal extreme sensitivity of basic plasma parameters with widely used models, such as temperature and ionization depth, which are able to be constrained by incorporating a detailed accounting of laser spatial profiles, pre-plasma conditions, and collisional processes. These results provide new insights into heating and ionization dynamics in the high-energy-density regime relevant to inertial fusion energy research, both as an experimental platform for accessing theoretically challenging conditions and as a benchmark for improving models of high-power laser--plasma interactions.

Raw data for publication of "Microscopy X-ray Imaging enriched with Small Angle X-ray Scattering for few nanometer resolution reveals shock waves and compression in intense short pulse laser irradiation of solids" in Frontiers of Physics (2026)

The data is contained as pkl files. Use python module pickle to read.

Understanding how laser pulses compress solids into high-energy-density states requires diagnostics that simultaneously resolve macroscopic geometry and nanometer-scale structure. ere we present a combined X-ray imaging (XRM) and small-angle X-ray scattering (SAXS) approach that bridges this diagnostic gap. Using the Matter in Extreme Conditions end station at LCLS, we irradiated 25 μm copper wires with 45 fs, 0.9 J, 800 nm pulses at 3.5 × 1019 W/cm2 while probing with 8.2 keV XFEL pulses. XRM visualizes the evolution of ablation, compression, and inward-propagating fronts with ∼ 200 nm resolution, while SAXS quantifies their nanometer-scale sharpness via the time-resolved evolution of scattering streaks. The joint analysis reveals that an initially smooth compression steepens into a nanometer-sharp shock front after tsh ≈ (18 ± 3) ps, consistent with an analytical steepening model and hydrodynamic simulations. The front reaches a velocity of csh ≈ 25 km/s and a lateral width of several tens of microns, demonstrating direct observation of shock formation and decay at
solid density for the first time with few-nanometer precision. This integrated XRM–SAXS method establishes a quantitative, multi-scale diagnostic of laserdriven
shocks in dense plasmas relevant to inertial confinement fusion, warm dense matter, and planetary physics.

Matter under extreme densities, temperatures, and pressures is abundant throughout our universe and occurs in a variety of celestial objects such as giant planets, brown dwarfs, and the outer layer of neutron stars. On Earth, such warm dense matter (WDM) [1] plays an important role for cutting-edge technological applications such as material science and discovery. However, the holy grail of contemporary high energy density science is given by inertial confinement fusion, where both the fusion fuel and the ablator material have to traverse the WDM regime in a controlled way during the initial phase of the compression. Unfortunately, the rigorous theoretical description of WDM is notoriously difficult. Ab initio density functional theory (DFT) simulations have emerged as the workhorse in the field, but their accuracy decisively depends on the utilized exchange—correlation (XC) free energy functional, which has to be supplied as an external input. The gold standard is given by quasi-exact path integral Monte Carlo (PIMC) simulations, which are, in principle, capable of giving exact results for a great variety of material properties.
Here, I will give an overview of recent developments in the direct PIMC estimation of the free energy [2,3]---the key
quantity for thermal DFT simulations and equation-of-state tables. This includes novel technical developments to deal efficiently with the fermion sign problem [4,5], as well as new opportunities to directly estimate the chemical potential [6]. From a physics perspective, I will show recent results for the uniform electron gas as well as for warm dense hydrogen. Finally, I will give an outlook on upcoming developments and opportunities for improved equation-of-state tables and XC functionals.
[1] J. Vorberger et al., Roadmap for warm dense matter physics, arXiv:2505.02494
[2] T. Dornheim et al. Phys. Rev. B 111, L041114 (2025)
[3] T. Dornheim et al., Phys. Rev. Research 7, 023250 (2025)
[4] T. Dornheim et al., J. Chem. Theory. Comput. 21, 7290–7303 (2025)
[5] P. Svensson et al., J. Phys. Chem. Lett. 16, 10639–10646 (2025)
[6] T. Dornheim et al., Phys. Rev. B 111, 115149 (2025)

The imaging platform developed at the High Energy Density - Helmholtz International Beamline for Extreme Fields (HED-HiBEF) instrument at the European XFEL and its applications to high energy density and fusion related research are presented. The platform combines the XFEL beam with the high-intensity short-pulse laser ReLaX and the high-energy nanosecond-pulse laser DiPOLE-100X. The spatial resolution is better than 500 nm and the temporal resolution of the order of 50 fs. The influence of the XFEL source in the x-ray imaging method is discussed. Free-propagation x-ray phase contrast imaging and Talbot-Lau imaging setups are shown. We show examples of blast waves and converging cylindrical shocks in aluminium, resonant absorption measurements of specific charged states in copper with ReLaX and planar shocks in polystyrene material generated by DiPOLE-100X. For the first time, we show the application of Talbot-Lau interferometry to convergent cylindrical shocks as well as resonant absorption processes. We also discuss the possibilities introduced by combining this imaging platform with a kJ-class laser.

X-ray Thomson scattering (XRTS) constitutes an essential technique for diagnosing material properties under extreme conditions, such as high pressures and intense laser heating. Time-dependent density functional theory (TDDFT) is one of the most accurate available ab initio methods for modeling XRTS spectra, as well as a host of other dynamic material properties. However, strong thermal excitations, along with the need to account for variations in temperature and density as well as the finite size of the detector significantly increase the computational cost of TDDFT simulations compared to ambient conditions. In this work, we present a broadly applicable method for optimizing and enhancing the efficiency of TDDFT calculations. Our approach is based on a one-to-one mapping between the dynamic structure factor and the imaginary time density--density correlation function, which naturally emerges in Feynman’s path integral formulation of quantum many-body theory. Specifically, we combine rigorous convergence tests in the imaginary time domain with a constraints-based attenuation of narrow-band fluctuations to improve the efficiency of TDDFT modeling without the introduction of any significant bias. As a result, we can report a speed-up by up to an order of magnitude, thus substantially reducing the burden of computational cost required for XRTS analysis.

I present recent result for ab initio path integral Monte Carlo simulations of warm dense matter.

Understanding warm dense matter: from theory to experiment

Warm dense matter (WDM) [1], an extreme state that simultaneously features high densities, temperatures and pressures, is ubiquitous throughout our universe and occurs in a variety of celestial objects such as giant planet interiors, brown dwarfs and white dwarf atmospheres. In addition, WDM is of high current interest due to its relevance for cutting-edge technological applications such as inertial confinement fusion (ICF), where both the fusion fuel and the ablator material have to traverse the WDM regime in a controlled way to reach ignition. As a direct consequence, there has been a remarkable surge of activity in the field and extreme states of matter are nowadays frequently realized in large research facilities such as the National Ignition Facility (NIF) in the US and the European XFEL in Germany.
Despite these advances, a rigorous and comprehensive theoretical description of WDM has remained elusive as it must feature a holistic treatment of a plethora of physical effects including Coulomb coupling, quantum degeneracy and delocalization, strong thermal excitations, and often also partial ionization. This lack of reliable theory limits our ability to model astrophysical objects and practical applications alike. Moreover, it even limits our ability to diagnose experiments with WDM and the interpretation of experimental measurements is often based on a number of model assumptions and de-facto uncontrolled approximations.
To remedy this unfortunate situation, we have recently introduced a new framework for the model-free interpretation of X-ray Thomson scattering (XRTS) experiments, which allows us to infer key parameters such as the temperature directly from the experimental data [2]. In combination with new ab initio path integral Monte Carlo simulations, these capabilities have allowed us to thoroughly re-evaluate an XRTS measurement on strongly compressed beryllium taken at the NIF [3], leading to a substantially lower mass density [4] than it had been assumed based on previously used chemical models.
In this talk, I will give a comprehensive overview of these developments in both theory and experiment, including a discussion of current capabilities and limitations. In addition, I will outline promising opportunities for future experiments harnessing the unique capabilities for high repetition XRTS experiments with ultrahigh resolution [5] using HIBEF-HED at the European XFEL, and also at the NIF using the new colliding planar shock platform [6].

[1] J. Vorberger et al., Roadmap for warm dense matter physics, arXiv:2505.02494
[2] T. Dornheim et al., Accurate temperature diagnostics for matter under extreme conditions, Nature Comm. 13, 7911 (2022)
[3] T. Döppner et al., Observing the onset of pressure-driven K-shell delocalization, Nature 618, 270-275 (2023)
[4] T. Dornheim et al., Unraveling electronic correlations in warm dense quantum plasmas, Nature Communications 16, 5103 (2025)
[5] Th. Gawne et al., Ultrahigh resolution x-ray Thomson scattering measurements at the European X-ray Free Electron Laser, Physical Review B 109, L241112 (2024)
[6] M.J. MacDonald et al., The colliding planar shocks platform to study warm dense matter at the National Ignition Facility, Physics of Plasmas 30, 062701 (2023)

QuantumElectrodynamics.jl is an open-source Julia framework to model and simulate interactions between particle and strong electromagnetic fields.

The study of superionic (SI) water has been a highly active research area since its theoretical prediction. Despite significant experimental and computational efforts, its melting curve and the stability of different oxygen lattices remain debated, impacting our understanding of SI ice’s peculiar transport properties. Experimental results at lower pressures show disagreement, whereas data at higher pressures are scarce due to the extreme challenges of such experiments. In this work, we present ultrafast X-ray diffraction results of water compressed by multiple shocks to pressures up to ~ 180 GPa. At pressures exceeding 150 GPa and temperatures around 2500 K, our diffraction patterns challenge the pure FCC-SI phase model, providing experimental evidence of the mixed close-packed superionic phase predicted by advanced ab initio calculations. At lower pressures, we observe simultaneous signatures of BCC and FCC structures within a pressure-temperature range consistent with some static-compression experiments, helping to resolve contradictory results in literature. These insights offer new constraints on the stability domains of SI phases and reveal detailed structural features, such as stacking faults. Our results advance the structural understanding of high-pressure SI ice to a level approaching that of ice I polymorphs, with potential implications for water-rich interiors of giant planets.

We present the first systematic experimental validation of return-current-driven cylindrical implosion scaling in micrometer-sized Cu and Al wires irradiated by J-class femtosecond laser pulses. Employing XFEL-based imaging with sub-micrometer spatial and femtosecond temporal resolution, supported by hydrodynamic and particle-in-cell simulations, we reveal how return current density depends precisely on wire diameter, material properties, and incident laser energy. We identify deviations from simple theoretical predictions due to geometrically influenced electron escape dynamics. These results refine and confirm the scaling laws essential for predictive modeling in high-energy-density physics and inertial fusion research.

Understanding the complex plasma dynamics in ultra-intense relativistic laser–solid interactions is of fundamental importance for applications of laser–plasma-based particle accelerators, the creation of high-energy-density matter, understanding planetary science, and laser-driven fusion energy. However, experimental efforts in this regime have been limited by the lack of accessibility of over-critical densities and the poor spatiotemporal resolution of conventional diagnostics. Over the last decade, the advent of femtosecond brilliant hard X-ray free-electron lasers (XFELs) has opened new horizons to overcome these limitations. Here, for the first time, we present full-scale spatiotemporal measurements of solid-density plasma dynamics, including preplasma generation with tens of nanometer scale length driven by the leading edge of a relativistic laser pulse, ultrafast heating and ionization at the main pulse arrival, the laser-driven blast wave, and transient surface return current-induced compression dynamics up to hundreds of picoseconds after interaction. These observations are enabled by utilizing a novel combination of advanced X-ray diagnostics including small-angle X-ray scattering, resonant X-ray emission spectroscopy, and propagation-based X-ray phase-contrast imaging simultaneously at the European XFEL-HED beamline station.

A density functional theory (DFT) framework is presented that links functional derivatives of free-energy functionals to non-linear static density response functions in quantum many-body systems. Within this framework, explicit expressions are derived for various higher-order response functions of systems that are homogeneous on average, including the first theoretical result for the cubic response at the first harmonic χ(1,3) 0 (q). Specifically, our framework includes hitherto neglected mode-coupling effects that are important for the non-linear density response even in the presence of a single harmonic perturbation. We compare these predictions for χ(1,3) 0 (q) to new Kohn-Sham DFT simulations, leading to excellent agreement between theory and numerical results. Exact analytical expressions are also obtained for the long-wavelength limits of the ideal quadratic and cubic response functions. Particular emphasis is placed on the connections between the third- and fourth-order functional derivatives of the non-interacting free-energy functional Fs[n] and the ideal quadratic and cubic response functions of the uniform electron gas, respectively. These relations provide exact constraints that may prove useful for the future construction of improved approximations to Fs[n], in particular for warm dense matter applications at finite temperatures. Here, we use
this framework to assess several commonly employed approximations to Fs[n] through orbital-free DFT simulations of the harmonically perturbed ideal electron gas. The results are compared with Kohn-Sham DFT calculations across temperatures ranging from the ground state to the warm dense regime. Additionally, we analyze in detail the temperature- and wavenumber-dependent nonmonotonic behavior of the ideal quadratic and cubic response functions.

X-ray Thomson scattering (XRTS) has emerged as a valuable diagnostic for matter under extreme conditions, as it captures the intricate many-body physics of the probed sample. Recent advances, such as the model-free temperature diagnostic of Dornheim et al. [this http URL. 13, 7911 (2022)], have demonstrated how much information can be extracted directly within the imaginary-time formalism. However, since the imaginary-time formalism is a concept often difficult to grasp, we provide here a systematic overview of its theoretical foundations and explicitly demonstrate its practical applications to temperature inference, including relevant subtleties. Furthermore, we present recent developments that enable the determination of the absolute normalization, Rayleigh weight, and density from XRTS measurements without reliance on uncontrolled model assumptions. Finally, we outline a unified workflow that guides the extraction of these key observables, offering a practical framework for applying the method to interpret experimental measurements.

The fermion sign problem constitutes a fundamental computational bottleneck across a plethora of research fields in physics, quantum chemistry and related disciplines. Recently, it has been suggested to alleviate the sign problem in \emph{ab initio} path integral Molecular Dynamics and path integral Monte Carlo (PIMC) calculations based on the simulation of fictitious identical particles that are represented by a continuous quantum statistics variable [\textit{J.~Chem.~Phys.}~\textbf{157}, 094112 (2022)]. This idea facilitated a host of applications including the interpretation of an x-ray scattering experiment with strongly compressed beryllium at the National Ignition Facility [\textit{Nature Commun.}~\textbf{16}, 5103 (2025)]. In the present work, we express the original isothermal -extrapolation method as a special case of a truncated Taylor series expansion around the limit of distinguishable particles. We derive new PIMC estimators that allow us to evaluate the Taylor coefficients up to arbitrary order and we carry out extensive new PIMC simulations of the warm dense electron gas to systematically analyze the sign problem from this new perspective. This gives us important insights into the applicability of the -extrapolation method for different levels of quantum degeneracy in terms of the Taylor series radius of convergence. Moreover, the direct PIMC evaluation of the -derivatives, in principle, removes the necessity for simulations at different values of and can facilitate more efficient simulations that are designed to maximize compute time in those regions of the full permutation space that contribute most to the final Taylor estimate of the fermionic expectation value of interest.

An exact expression is derived for the kinetic contribution to the odd (arbitrary order) frequency moments of the dynamic structure factor via a finite summation that features averages of even (all lower orders) powers of the momentum over the exact momentum distribution. The derivation is carried out for the non-interacting Fermi gas and generalized to the interacting case based on the conjecture that averages over the Fermi distribution can be substituted with averages over the exact distribution. The expression is validated against known results (first, third frequency moments) and new explicit calculations (fifth, seventh frequency moments).

Ab initio path integral Monte Carlo (PIMC) simulations constitute the gold standard for the estimation of a broad range of equilibrium properties of a host of interacting quantum many-body systems spanning conditions from ultracold atoms to warm dense quantum plasmas. A key practical limitation is given by the notorious fermion sign problem, which manifests as an exponential computational bottleneck with respect to system size and inverse temperature. In practice, the sign problem is particularly severe in the grandcanonical ensemble, where the bosonic and fermionic configuration spaces differ not only with respect to the symmetry of the thermal density matrix but, crucially, also with respect to the particle number distribution for a given chemical potential [T. Dornheim, J. Phys. A 54, 335001 (2021)]. Here, we present a simple reweighting scheme that basically allows one to retain access to grandcanonical expectation values at the cost of fermionic PIMC simulations in the canonical ensemble for the largest significant particle number in the fermionic sector. As a practical example, we consider the warm dense electron gas, which has attracted considerable recent attention due to its relevance, e.g., for the modeling of compact astrophysical objects and inertial fusion energy applications

We compare the predictions of the dynamic structure factor (DSF) of ambient polycrystalline aluminium from time-dependent density functional theory (TDDFT) in the pair continuum regime to recent ultrahigh resolution x-ray Thomson scattering measurements, collected at the European XFEL. TDDFT predicts strong anisotropy in the DSF at the wavenumber examined here, even with $q$-blurring accounted for. The experimental spectrum has more than sufficient resolution and signal-to-noise levels to resolve these orientation dependencies, and therefore the orientational averaging of the polycrystalline sample is observed rigorously. Once the orientation averaging is accounted for, TDDFT is able to reproduce the experimental spectrum adequately. Finally, comparisons of predicted DSFs from jellium to experiment demonstrates the importance of accounting for lattice effects in modelling the spectrum from a polycrystal.

Fast neutron-induced nuclear reactions are crucial for advancing our understanding of fundamental nuclear processes, stellar nucleosynthesis, and applications, including reactor safety, medical isotope production, and materials research. With many research reactors being phased out, compact accelerator-based neutron sources are becoming increasingly important. Laser-driven neutron sources (LDNSs) offer unique advantages—ultrashort neutron pulsees for superior energy resolution, high per-pulse flux, and a drastically reduced footprint. However, their use in single-event fast neutron spectroscopy remains unproven, requiring stable multi-shot operation and detectors capable of functioning in the extreme environment of petawatt-class laser-plasma interactions. Here, we present a proof-of-concept experiment at the DRACO PW laser in a pitcher-catcher configuration, stably producing 6–7×107 neutrons/shot with energies above 1 MeV, over more than 200 shots delivered at a shot-per-minute rate. Neutron time-of-flight measurements were performed using a single-crystal diamond detector, which is located only 1.5 m away from the source and capable of resolving individual neutron-induced reactions. Observed reaction rates are consistent with Monte Carlo simulations inferred by real-time diagnostics of accompanying gamma, ion, and electron fluxes. With the recent
advances in repetition rate, targetry, and ion acceleration efficiency, this work establishes LDNSs as a promising, scalable platform for future fast neutron-induced reaction studies, particularly for measurements involving short-lived isotopes or requiring high instantaneous neutron flux.

Understanding the properties of warm dense hydrogen is of key importance for the modeling of compact astrophysical objects and to understand and further optimize inertial confinement fusion (ICF) applications. The work horse of warm dense matter theory is given by thermal density functional theory (DFT), which, however, suffers from two limitations: (i) its accuracy can depend on the utilized exchange–correlation (XC) functional, which has to be approximated and (ii) it is generally limited to single-electron properties such as the density distribution. Here, we present a new ansatz combining time-dependent DFT results for the dynamic structure factor See(q, ω) with static DFT results for the density response. This allows us to estimate the electron–electron static structure factor See(q) of warm dense hydrogen with high accuracy over a broad range of densities and temperatures. In addition to its value for the study of warm dense matter, our work opens up new avenues for the future study of electronic correlations exclusively within the framework of DFT for a host of applications.

Dense plasma environment affects the electronic structure of ions via variations of the microscopic
electrical fields, also known as plasma screening. This effect can be either estimated by simplified
analytical models, or by computationally expensive and to date unverified numerical calculations.
We have experimentally quantified plasma screening from the energy shifts of the bound-bound
transitions in matter driven by the x-ray free electron laser (XFEL). This was enabled by identifi-
cation of detailed electronic configurations of the observed Kα, Kβ and Kγ lines. This work paves
the way for improving plasma screening models including connected effects like ionization poten-
tial depression and continuum lowering, which will advance the understanding of atomic physics in
Warm Dense Matter regime.

Warm dense matter (WDM) is now routinely created and probed in laboratories around the world,
providing unprecedented insights into conditions achieved in stellar atmospheres, planetary interiors,
and inertial confinement fusion experiments. However, the interpretation of these experiments is often
filtered through models with systematic errors that are difficult to quantify. Due to the simultaneous presence of quantum degeneracy and thermal excitation, processes in which free electrons are
de-excited into thermally unoccupied bound states transferring momentum and energy to a scattered
x-ray photon become viable. Here we show that such free-bound transitions are a particular feature
of WDM and vanish in the limits of cold and hot temperatures. The inclusion of these processes
into the analysis of recent X-ray Thomson Scattering experiments on WDM at the National Ignition Facility and the Linac Coherent Light Source significantly improves model fits, indicating that
free-bound transitions have been observed without previously being identified. This interpretation is
corroborated by agreement with a recently developed model-free thermometry technique and presents
an important step for precisely characterizing and understanding the complex WDM state of matter.

First information on the timelike electromagnetic structure of baryons in the second resonance region has been obtained from measurements of invariant mass and angular distributions in the quasi-free reaction π−p→nee at sπ−p−−−−√ = 1.49 GeV with the High Acceptance Di-Electron Spectrometer (HADES) detector at GSI using the pion beam impinging on a CH2 target. We find a total cross section σ(π−p→nee)=2.97±0.07data±0.21acc±0.31Zeffμb. Combined with the Partial Wave Analysis of the concurrently measured two-pion channel, these data sets provide a crucial test of Vector Meson Dominance (VMD) inspired models. The commonly used "strict VMD" approach strongly overestimates the e+e− yield. Instead, approaches based on a VMD amplitude vanishing at small e+e− invariant masses supplemented coherently by a direct photon amplitude provide a better agreement. A good description of the data is also obtained using a calculation of electromagnetic timelike baryon transition form factors in a covariant spectator-quark model, demonstrating the dominance of meson cloud effects. The angular distributions of e+e− pairs demonstrate the contributions of virtual photons with longitudinal polarization, in contrast to real photons. The virtual photon angular dependence supports the dominance of J=3/2, I=1/2 contributions observed in both the γ⋆n and the ππn channels.

The direct investigation of the specific solidification phenomena (morphological transitions, local branch evolution, defect accumulation, etc.) is rather complex and requires high spatial and temporal resolution and sensitivity of the detector. In this work, in-situ X-ray synchrotron imaging (e.g. radiography or tomography) and (local) diffraction methods are combined to study the dendritic microstructures of solidifying Ga - In alloys. The X-ray diffraction techniques are well suited for the analysis of lattice constant variation, lattice orientation and misorientation defects. The combination techniques allow precise control of the position of the X-ray beam on a selected dendrite or microstructure feature. The tomography/diffraction experiments are performed using a 400 µm diameter capillary cell at the ID19 beamline (ESRF, France) with a spatial resolution of ~1 µm and a diffraction spot size of 250 µm. Our measurements show that the diffractograms obtained from a single indium dendrite are equivalent to the TEM (Transmission Electron Microscopy) pattern, but at a different size scale. The majority of the indium dendrites grow along the <110> orientation typically observed in body-centered metals. Alternatively, the indium crystal lattice could be considered close to a slightly distorted FCC structure, with the major growth direction being the <100> orientation. Not all groups of diffraction reflections appear on the indium diffractogram, which may be related to the mosaicism of a single dendrite crystal. The present data demonstrate that the combination of these X-ray techniques can provide experimental data on the orientation distribution, lattice spacing and mosaicity of a dendrite crystal and could help to validate microstructural solidification models.

Previous studies of the solidification of a low-melting temperature ternary Ga-In-Bi alloy with a laboratory microfocus X-ray system demonstrated that the morphology of the growing indium dendrites was significantly altered by the presence of a small amount of bismuth (2.5wt%) [ICASP 2021 and TMS 2023 conference contributions]. Our observations revealed the formation of curved dendrites, multiple splitting events, and the development of branched or seaweed-like structures. We explained this phenomenon by the fact that the addition of bismuth as a third alloying element reduces the anisotropy properties of the phase boundaries, making them more susceptible to disturbances.
This work uses X-ray synchrotron radiography to study the formation of a variety of solid phase morphologies in the same ternary system. The synchrotron imaging experiments were performed at the ID19 beamline (ESRF, France) with a spatial resolution of ~1 µm. The high spatial resolution allows us to analyse the shape of the dendrites' tips and local dynamics of splitting or branching in great detail. The preliminary analysis of the data clearly shows that the direction of the dendrite growth can deviate because of different mechanisms, e.g. branching or unstable tip growth. The ongoing data processing will provide further insights into the physics of the transition to branched growth in ternary alloys.

Plasma screening was experimentally measured by observing energy shifts in bound-bound
transitions in solid density, T ~ 100 eV plasma. Ultra-fast melting was studied via the time
evolution of ion and electron temperatures from x-ray absorption. These studies provide
data for refining plasma screening models and related effects, enhancing our understanding
of atomic physics in Warm Dense Matter.

We present a model of x-ray thermal diffuse scattering (TDS) from a cubic polycrystal with an arbitrary crystallographic texture, based on
the classic approach of Warren [B. E. Warren, Acta Crystallogr. 6, 803 (1953)]. We compare the predictions of our model with femtosecond
x-ray diffraction patterns gathered from ambient and dynamically compressed rolled copper foils obtained at the High Energy Density instrument of the European X-Ray Free-Electron Laser facility and find that the texture-aware TDS model yields more accurate results than does the
conventional powder model owed to Warren. Nevertheless, we further show: with sufficient angular detector coverage, the TDS signal is largely
unchanged by sample orientation and in all cases strongly resembles the signal from a perfectly random powder; shot-to-shot fluctuations in
the TDS signal resulting from grain-sampling statistics are at the percent level, in stark contrast to the fluctuations in the Bragg-peak intensities
(which are over an order of magnitude greater); and TDS is largely unchanged even following texture evolution caused by compressioninduced plastic deformation. We conclude that TDS is robust against texture variation, making it a flexible temperature diagnostic applicable
just as well to off-the-shelf commercial foils as to ideal powders.

In dynamic-compression experiments, the line-imaging Velocity Interferometer System for Any Reflector (VISAR) is a well-established diagnostic used to probe the velocity history, including wave profiles derived from dynamically compressed interfaces and wavefronts, depending on material optical properties. Knowledge of the velocity history allows for the determination of the pressure achieved during compression. Such a VISAR analysis is often based on Fourier transform techniques and assumes that the recorded interferograms are free from image distortions. In this paper, we describe the VISAR diagnostic installed at the HED-HIBEF instrument located at the European XFEL along with its calibration and characterization. It comprises a two-color (532, 1064 nm), three-arm (with three velocity sensitivities) line imaging system. We provide a procedure to correct VISAR images for geometric distortions and evaluate the performance of the system using Fourier analysis. We finally discuss the spatial and temporal calibrations of the diagnostic. As an example, we compare the pressure extracted from the VISAR analysis of shock-compressed polyimide and silicon.

Laser plasma-based particle accelerators attract great interest in fields where conventional accelerators reach limits based on size, cost or beam parameters. However, laser accelerators have not yet reached their full potential in producing high-radiation doses at high particle energies. The quest to fully leverage the available laser pulse energies is guided by first principles simulations predicting efficient ion acceleration mechanisms at near critical plasma densities. The most stringent limitation for accessing this regime is the lack of a system that provides a high degree of control of the plasma density conditions at high-repetition rates.
In this talk I will outline our approach for overcoming these challenges using a novel cryogenic hydrogen target in combination with petwawatt-class lasers. Controlled pre-expansion of the initially solid target by low intensity pre-pulses allows for tailored density scans from the overdense to the underdense regime transitioning between different acceleration mechanisms. Under ideal conditions, the near-critical density produces proton energies of 80 MeV representing a boost in maximum energy by a factor of more than two compared to the solid jet case. Furthermore, recent investigations provide the basis for transferring the high single-shot performance into a reproducible, robust and, above all, highly repetitive operation mode.

Laser plasma based particle accelerators have attracted great interest in fields where conventional accelerators reach limits based on size, cost or beam parameters. However, laser accelerators have not yet reached their full potential in producing simultaneous high-radiation doses at high particle energies. To overcome limitations a high degree of control of the plasma conditions is needed, making methods for optimizing and stabilizing the performance of the accelerator essential.
In this talk, we will present techniques for the optimization of laser-driven proton beams using the DRACO laser system of the Helmholtz-Zentrum Dresden-Rossendorf. With its ultra-short laser pulses of up to 23J energy on target yielding intensities on the order of 5*10e21W/cm², DRACO enables exciting research on ion acceleration, as it has been shown in recent publications [Ziegler2024, Rehwald2023, Kroll2022]. This talk will provide an overview of our workflows to optimize and stabilize the acceleration process, including manipulating the laser intensity contrast (e.g. by changing the spectral laser phase) or tailoring the target evolution during the interaction (e.g. by varying the initial target thickness or by introducing an intentional pre-expansion). Our view on the experiment automation and automated laser-plasma systems in this context will conclude the presentation.

[Ziegler2024] T. Ziegler et al. "Laser-driven high-energy proton beams from cascaded acceleration regime" Nature Physics, 20, pages 1211–1216 (2024)
[Rehwald2023] M. Rehwald et al. ”Ultra-short pulse laser-driven acceleration of protons to 80 MeV from density tailored cryogenic hydrogen jets” Nature Communications 14, 4009 (2023).
[Kroll2022] F. Kroll et al. ”Tumor irradiation in mice with a laser-accelerated proton beam” Nature Physics 18, 316-322 (2022)

We report on a single-shot longitudinal phase-space reconstruction diagnostic for electron beams in a laser
wakefield accelerator via the experimental observation of distinct periodic modulations in the angularly
resolved spectra. Such modulated angular spectra arise as a result of the direct interaction between the
ultrarelativistic electron beam and the laser driver in the presence of the wakefield. A constrained theoretical
model for the coupled oscillator, assisted by a genetic algorithm, can recreate the experimental electron
spectra and, thus, fully reconstructs the longitudinal phase-space distribution of the electron beam with a
temporal resolution of approximately 1.3 fs. In particular, it reveals the slice energy spread of the electron
beam, which is important to measure for applications such as x-ray free electron lasers. In our experiment, the
root-mean-square slice energy spread retrieved is bounded at 9.9 MeV, corresponding to a 0.9%–3.0%
relative spread, despite the overall GeV energy beam having approximately 100% relative energy spread.

Vacuum fluctuations give rise to effective nonlinear interactions between electromagnetic fields. These generically modify the characteristics of light traversing a strong-field region. X-ray free-electron lasers (XFELs) constitute a particularly promising probe, due to their brilliance, the possibility of precise control and favorable frequency scaling. However, the nonlinear vacuum response is very small even when probing a tightly focused high-intensity laser field with XFEL radiation and direct measurement of light-by-light scattering of real photons and the associated fundamental physics constants of the quantum vacuum has not been possible to date. Achieving a sufficiently good signal-to-background separation is key to a successful quantum vacuum experiment. To master this challenge, a dark-field detection concept has recently been proposed. Here we present the results of a proof-of-principle experiment validating this approach by demonstrating that using real-world x-ray optics the background signal can be suppressed sufficiently to measure the weak nonlinear response of the vacuum

Presentation about the work: Phys. Rev. Research 7, 033016

Presentation on obtaining ionization potential depression and charge state for hydrogen as published in Phys. Rev. Research 7, 033016.

Presentation about the published work Phys. Rev. Research 7, 033016 on determining ionization potential depression and ionization from ab initio simulations of Hydrogen.

Presentation about the published work "Toward model-free temperature diagnostics of warm dense matter from multiple scattering angles", Appl. Phys. Lett. 126, 044104 (2025).

I provide an overview of recent developments in the ab initio quantum Monte Carlo (QMC) simulation of warm dense matter.

We present the first observation of 3ΛH and 4ΛH in Ag+Ag collisions at √sNN = 2.55 GeV, emitted around mid-rapidity. The hypernuclei are reconstructed via their two-body decay channels and identified through their weak-decay topology, employing an artificial neural network for enhanced discrimination. The analysis methodology is validated using Λ hyperons. The resulting rapidity distributions, dN/dy, exhibit a bell shape centered at mid-rapidity. The yield of 4 ΛH is equal to or exceeds that of 3 ΛH, which contrasts the measurement from the STAR collaboration at √sNN = 3 GeV and is consistent with a scenario in which hypernuclei receive feed-down from excited states. The data enable a high-precision measurement of the hypernuclei lifetimes. For the 3
ΛH, a lifetime of τ3 ΛH = (239 ± 23(stat) ± 18(sys)) ps, is extracted, consistent on the 1σ level with that of the free Λ. In contrast, the 4ΛH lifetime of τ4 ΛH = (209 ± 7(stat) ± 10(sys)) ps, shows a 4.5 σ deviation from the free Λ lifetime. The results consolidate the available world data.

I present a topical overview of current warm dense matter research with a focus on x-ray Thomson scattering, experiments and theory.

Micro-structured targets represent a well-researched approach for optimizing the laser-target interaction towards specific parameters of particle acceleration or radiation generation processes, e.g. enhanced production of Kalpha xray radiation or proton yield. However, due to the complexity of target production, experiments usually focus on a very small set of target geometries and a very concrete enhancement goal.

In our work, we investigate micro-scale pillar and tube targets with different geometry parameters, irradiated in a grazing incidence geometry at the Draco PW facility (30 fs pulse duration, intensity up to ~ 5 x 1021 W/cm2, plasma mirror enhanced contrast). A diverse diagnostic suite combines angularly-resolved electron and proton spectrometry with 1D-imaging x-ray line spectrometry, providing data for all major emissions from the targets.

Our measurements show that strongly enhanced electron temperatures as predicted for grazing incidence schemes can be observed for the micro-tube geometry which offers optimal laser confinement during irradiation. Additionally, we observe that micro-structures on the target surface can lead to enhanced yields for protons and electrons. X-ray line emission yields insights into the laser-target interaction for optimal grazing conditions.

I present an overview of recent model-free and path integral Monte Carlo based frameworks for the interpretation of x-ray Thomson scattering experiments with warm dense matter.

I present DMA highlights related to the ab initio description of warm dense matter and corresponding x-ray scattering experiments.

Warm dense matter (WDM) is an extreme state that is characterized by the complex
interplay of Coulomb correlations, strong thermal excitations, quantum delocalization and
degeneracy effects, and often also partial ionization and overlapping bound-state
wavefunctions [1]. These conditions are fairly ubiquitous throughout nature and occur in a
variety of compact astrophysical objects such as giant planet interiors, brown dwarfs and
white dwarf atmospheres. In addition, matter under extreme conditions is starting to play an
increasingly important role for cutting-edge technological applications, such as the
discovery and synthesis of novel and exotic materials. The holy grail of contemporary high
energy density science is given by inertial confinement fusion (ICF), which holds the
tantalizing promise of a potentially unlimited source of clean energy in the future. In ICF
experiments, both the fusion fuel and the ablator have to traverse the WDM regime in a
controlled way during the early segments of the compression path while avoiding the
formation of any notorious instabilities. The further optimization that is required to realize
an actual ICF power plant thus makes a more rigorous understanding of WDM
indispensable.
Ab initio path integral Monte Carlo (PIMC) methods are, in principle, uniquely suited to
capture the full complexity of WDM, but their application is severely limited by the
notorious fermion sign problem [2]; it constitutes an exponential computational bottleneck
with respect to increasing numbers of electrons and decreasing temperature. In a seminal
recent work, Xiong and Xiong [3] have suggested to partially avoid the sign problem by
carrying out path integral simulations of fictitious identical particles that are guided by a
continuous spin-statistics parameter ξ.
In this presentation, I give a broad overview about a variety of advances in PIMC
simulations of WDM using this ξ-extrapolation technique, including the estimation of
different observables [4], large-scale simulations of up to 1000 electrons [5], and free
energy calculations [6]. Moreover, I discuss recent methodological developments such as re-
weighting [7] and a generalized Taylor series perspective onto the simulation of fictitious
identical particles [8]. As the capstone of our work, I demonstrate how the ξ-extrapolation
method can be used to compute the properties of real WDM systems [9,10], facilitating for
the first time the direct comparison with x-ray scattering experiments with strongly
compressed beryllium taken at the National Ignition Facility in the USA [11,12,13].
[1] J. Vorberger et al., Roadmap for warm dense matter physics, arXiv:2505.02494
[2] T. Dornheim, Physical Review E 100, 023307 (2019)
[3] Y. Xiong and H. Xiong, The Journal of Chemical Physics 157, 094112 (2022)
[4] T. Dornheim et al., The Journal of Chemical Physics 159, 164113 (2023)
[5] T. Dornheim et al., The Journal of Physical Chemistry Letters 15, 1305 (2024)
[6] T. Dornheim et al., Journal of Chemical Theory and Computation 21, 7290 (2025)
[7] T. Dornheim et al., arXiv:2508.12323 (under review)
[8] T. Dornheim et al., arXiv:2509.11317 (under review)
[9] T. Dornheim et al., The Journal of Chemical Physics 160, 164111 (2024)
[10] T. Dornheim et al., Matter and Radiation at Extremes 9, 057401 (2024)
[11] T. Dornheim et al., Nature Communications 16, 5103 (2025)
[12] T. Dornheim et al., Physics of Plasmas 32, 052712 (2025)
[13] S. Schwalbe et al., arXiv:2504.13611 (under review)

This repository contains the PIMC raw data presented in the publication "Re-weighting estimator for ab initio path integral Monte Carlo simulations of fictitious identical particles" in the same format and units as in the main text.

The fermion sign problem constitutes one of the most fundamental obstacles in quantum many-body theory. Recently, it has been suggested to circumvent the sign problem by carrying out path integral simulations with a fictitious quantum statistics variable , which allows for a smooth interpolation between the bosonic and fermionic limits [\textit{J.~Chem.~Phys.}~\textbf{157}, 094112 (2022)]. This -extrapolation method has subsequently been applied to a variety of systems and has facilitated the analysis of an x-ray scattering measurement taken at the National Ignition Facility with unprecedented accuracy [\textit{Nature Commun.}~\textbf{16}, 5103 (2025)]. Yet, it comes at the cost of performing an additional simulations, which, in combination with the required small error bars, can pose a serious practical limitation. Here, we remove this bottleneck by presenting a new re-weighting estimator, which allows the study of the full -dependence from a single path integral Monte Carlo (PIMC) simulation. This is demonstrated for various observables of the uniform electron gas and also warm dense beryllium. We expect our work to be useful for future PIMC simulations of Fermi systems, including ultracold atoms, electrons in quantum dots, and warm dense quantum plasmas.

We compare the predictions of the dynamic structure factor (DSF) of ambient polycrystalline aluminium from time-dependent density functional theory (TDDFT) in the pair continuum regime to recent ultrahigh resolution x-ray Thomson scattering measurements, collected at the European XFEL. TDDFT predicts strong anisotropy in the DSF at the wavenumber examined here, even with $q$-blurring accounted for. The experimental spectrum has more than sufficient resolution and signal-to-noise levels to resolve these orientation dependencies, and therefore the orientational averaging of the polycrystalline sample is observed rigorously. Once the orientation averaging is accounted for, TDDFT is able to reproduce the experimental spectrum adequately. Finally, comparisons of predicted DSFs from jellium to experiment demonstrates the importance of accounting for lattice effects in modelling the spectrum from a polycrystal.

We present a methodology for accelerating the estimation of the free energy from path integral Monte Carlo simulations by considering an intermediate artificial reference system where interactions are inexpensive to evaluate numerically. Using the spherically averaged Ewald interaction as this intermediate reference system for the uniform electron gas, the interaction contribution for the free energy was evaluated up to 18 times faster than the Ewald-only method. Furthermore, a ξ-extrapolation technique was tested and applied to alleviate the fermion sign problem and to resolve the sign for large particle numbers. Combining these two techniques enabled the evaluation of the free energy for a system of 1000 electrons, where both finite-size and statistical errors are below chemical accuracy. The general procedure can be applied to systems relevant for planetary and inertial confinement fusion modeling with low to moderate levels of quantum degeneracy.

A core–corona decomposition of compact (neutron) star models was compared with the current mass–radius data of the outlier XTE J1814-338. The corona (which may also be dubbed the envelope, halo or outer crust) is assumed to be of Standard Model matter, with an equation of state that is supposed to be faithfully known and accommodates nearly all other neutron star data. The core, solely parameterized by its mass, radius and transition pressure, presents a challenge regarding its composition. We derived a range of core parameters needed to describe the current data of XTE J1814-338.

An exact series representation of the even frequency moments of the dynamic structure factor is
derived. Truncations are proposed that allow to evaluate the explicitly unknown second, fourth and
fifth frequency moments for the finite temperature uniform electron gas. Their applicability range
in terms of degeneracy parameter and wavenumber is determined by exploiting the non-interacting
limit and by comparing with the quasi-exact results of path integral Monte Carlo simulations.