It can be found inside gas giants such as Jupiter and is briefly created during meteorite impacts or in laser fusion experiments: warm dense matter. This exotic state of matter combines features of solid, liquid and gaseous phases. Until now, simulating warm dense matter accurately has been considered a major challenge. An international team led by researchers from the Center for Advanced Systems Understanding (CASUS) at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR) in Germany and Lawrence Livermore National Laboratory (LLNL) has succeeded in describing this state of matter much more accurately than before using a new computational method. The approach could advance laser fusion and help in the synthesis of new high-tech materials. The team presents its results in the journal Nature Communications (DOI: 10.1038/s41467-025-60278-3).
Warm dense matter (WDM) is characterized by temperatures ranging from several thousand to hundreds of millions of Kelvin and densities that sometimes exceed those of solids. “Such conditions can be found, for example, inside gas planets, in brown dwarfs, or in the atmospheres of white dwarfs,” explains Dr. Tobias Dornheim, junior group leader at CASUS and first author of the publication. “On Earth, it can be created during meteorite impacts or, for example, in experiments with powerful lasers.”
WDM is of particular interest for materials research. For example, tiny diamonds can be produced by compressing and heating plastics. WDM also plays a central role in fusion research, especially in laser-driven inertial confinement fusion that is studied at LLNL’s National Ignition Facility (NIF). Here, a capsule containing fusion fuel — typically the hydrogen isotopes deuterium and tritium — is heated and compressed so intensely by laser bombardment that the atomic nuclei fuse, releasing energy. “When the fusion capsule is fired at with lasers, the hydrogen passes through the state of warm dense matter,” explains Dr. Tilo Döppner, a scientist at LLNL who has played a key role in numerous fusion experiments at the NIF. “In order to achieve energy gain in fusion experiments, we need to understand the WDM state as well as possible.”
A solution to the sign problem
Computer simulations can help describe WDM. However, conventional simulation techniques have their limitations. “The problem is that WDM is an intermediate state — neither solid, liquid, nor fully ionized plasma,” explains Dr. Maximilian Böhme, who earned his doctorate at CASUS in 2024 and then continued his scientific career as a Lawrence Fellow at LLNL. “Most existing models involve a series of approximations and therefore often fail to achieve the necessary accuracy,” says Böhme.
Path integral Monte Carlo simulation (PIMC) would be a precise method. In principle, it allows a complete quantum mechanical description of WDM, but it usually fails due to the so-called sign problem: in order to calculate material properties without approximations, the respective contributions of all electrons within a material must be added together. However, while electrons are negatively charged, the wavefunction used to describe their quantum state oscillates between positive and negative. These opposite contributions to the PIMC simulation can cancel each other out. With each additional particle in the system, the number of combinations of these “sign-affected” contributions relevant for an accurate calculation increases exponentially. Even the world's most powerful supercomputers can therefore often only calculate PIMC simulations for a few particles.
This is where Dornheim and his team stepped in. “We introduced imaginary particle statistics, which are not physically real, but help to mitigate the sign problem,” explains Dornheim. “This computational trick enabled us to apply the exact PIMC method to a realistic material for the first time, in this case beryllium.”
Simulations meet experiments
This is where experiments at LLNL, led by Döppner, come into play. In these experiments, beryllium capsules were compressed beyond 10 times solid density and heated using the 192 laser beams at NIF. Simultaneously, powerful X-rays were used to examine the tiny sample. Scattered X-rays revealed how dense and hot the material became during laser compression. “In the past, relatively simple models were used to analyze the X-ray scattering data,” says Dornheim. “With our new method, we can determine important parameters such as density and temperature from the scattering signal now without approximation.”
In fact, the analysis revealed that the density of the sample was lower than inferred with previously used models. “Our findings are crucial for future modeling of the hydrogen fusion process,” emphasizes Dr. Jan Vorberger from the Institute of Radiation Physics at HZDR. “Previous simulations of fusion capsule compression may be based on incorrect assumptions. Our method provides a precise diagnostic tool for analyzing the processes more accurately.” In addition to diagnostics, the new method could also be used to obtain equations of state — i.e., the relationships between pressure, temperature and energy. Such data is relevant for the development of fusion power plants, but also for understanding exoplanets.
Additional experiment planned at NIF
In the fall of 2025, the team plans to conduct a new series of experiments at NIF. “We want to further refine the diagnostics and find out how sensitive our method is to small changes,” explains Dornheim. In the future, the calculations should not only explain existing data, but also actively help to plan and optimize new experiments — for example, for the development of more efficient fusion capsules.
Researchers from several institutions participated in the study. In addition to HZDR and LLNL, these included the Royal Institute of Technology (KTH) in Stockholm (Sweden), the University of Rostock, the Technical University of Dresden (both Germany), the University of Warwick (UK) and the SLAC National Accelerator Laboratory (USA).
END