Path Integral Monte Carlo simulation twist helps decipher ‘warm dense matter’

Simulating “warm dense matter” was a roadblock for researchers pursuing a better understanding of this state of matter, but a new approach to Path Integral Monte Carlo (PIMC) simulations by researchers from the Center for Advanced Systems Understanding (CASUS) at the Helmholtz Zentrum Dresden-Rossendorf in Germany and Lawrence Livermore National Laboratory (LLNL) is finally revealing secrets about this extreme state of matter.

Warm dense matter temperatures range from several thousand to hundreds of millions of Kelvin, with densities that can exceed solids, so the term “warm” is perhaps a bit of a misnomer for something created via meteorite impacts or experiments using high-power lasers.

“It’s an extreme state that occurs within giant planet interiors, brown dwarfs, and other compact astrophysical objects, but also plays a role in the compression path of fusion fuel and the surrounding ablator material for inertial fusion energy experiments,” explains Tobias Dornheim, a group leader at CASUS. “The rigorous theoretical description of warm dense matter is notoriously difficult because it must deal with the complex interplay of effects such as Coulomb coupling, quantum degeneracy, and strong thermal excitations.”

But it’s essential to understand its role in laser-driven inertial confinement fusion research at LLNL’s National Ignition Facility because when researchers fire at a capsule of fusion fuel with lasers, the fuel’s hydrogen passes through a state of warm dense matter.

Imaginary-time domain

Dornheim’s passion for quantum mechanics began in high school. “And then I encountered quantum many-body systems at extreme conditions and the advanced theoretical concepts that are used to describe them during my time as a Ph.D. student,” he says. “The main inspiration for my current work with experimental measurements stems from this background. I love to take abstract concepts such as imaginary-time path integrals and apply them to actual real-world measurements.”

The interpretation of x-ray Thomson scattering (XRTS) diagnostics of warm dense matter states within the laboratory previously tended to rely on de facto uncontrolled approximations and model assumptions.

This necessitated the team’s new approach, which involves fictitious particle physics (to mitigate the fermion sign problem)—it’s essentially a computational trick that allows the researchers to apply the exact PIMC method to beryllium.

“In our recent project, we transformed an XRTS measurement of strongly compressed beryllium that was taken at the U.S. National Ignition Facility (NIF) to the imaginary-time domain, which gives us model-free access to a number of important properties such as the temperature of the probed sample,” says Dornheim. “To get even further insight into the generated extreme state of matter, we developed a new setup for highly accurate path integral Monte Carlo simulations, which allows us to interpret the XRTS data set in novel ways. Interestingly, this analysis on a true ab initio level gave us a substantially lower density of the beryllium (~22 g/cm³) compared to previously used chemical models (~34 g/cm³).”

The team’s ab initio PIMC method is based on Richard Feynman’s imaginary-time path integral representation of statistical mechanics. “In this formulation, the original quantum many-body problem of interest is mapped onto an effectively classical system of interacting ring polymers,” explains Dornheim. “The corresponding polymer positions are then sampled stochastically—using random numbers—via the celebrated Metropolis algorithm, a.k.a. ‘Monte Carlo.’”

One of PIMC’s biggest strengths is it’s exact by design, without the need for any empirical external input such as the exchange-correlation functional in density functional theory. But this advantage comes at the cost of an exponential computational bottleneck: the notorious fermion sign problem.

“In our recent project, we developed a new setup that efficiently deals with the sign problem using a clever idea presented recently for path integral molecular dynamics simulations of electrons in quantum dots,” says Dornheim. “This was decisive to simulate sufficiently large systems to facilitate a meaningful comparison with experimental XRTS measurements.”

PIMC simulations

What does this work mean for materials science and the future of laser-driven inertial confinement fusion energy? “For both fields, integrated multiscale simulations such as radiation hydrodynamics require high input about the microphysics of extreme states of matter in the form of an equation of state that connects key parameters such as density, temperature, and pressure,” says Dornheim. “Our PIMC simulations can provide such an equation of state for at least light elements—hydrogen to beryllium have already been demonstrated. It’s very useful to benchmark existing data tables, but also to construct improved future parametrizations without previous approximations.”

Beyond this, PIMC is, in principle, also capable of providing exact information about the dynamic properties of the electrons in warm dense matter, which is something the team is actively working on.

The coolest moment of this project for Dornheim was the realization that highly accurate simulations of warm dense beryllium (as achieved at NIF) are actually feasible, thanks to their new PIMC setup. “This first direct comparison between ab initio PIMC simulations and a warm dense matter experiment is an important milestone for us personally and the broader community,” he says.

Technical challenges still “abound and range from the characterization of the experimental setup—in the form of the source and instrument function, for example—to the realization of large-scale simulation campaigns on supercomputers with the associated challenges in ensuring convergence and handling large data sets,” says Dornheim. “Nevertheless, the present project gave us a mountain of extremely valuable practical experience and we’re confident we can introduce improvements both to experimental and simulation aspects of our work.”

Novel experimental setups

The team’s new theoretical concepts for XRTS diagnostics of warm dense matter inspired the development of novel experimental setups.

“A few months ago, we carried out a corresponding successful experiment at Helmholtz International Beamline for Extreme Fields (HIBEF) at the European XFEL in Germany, using its unique capabilities for high-resolution and high-repetition rate XRTS measurements,” says Dornheim. “It gave us a wealth of high-quality data, which is currently being analyzed. And we’ve been awarded beamtime at NIF within the Discovery Science program to further develop the combination of XRTS diagnostics with cutting-edge warm dense matter theory.”

Ultimately, the goal “is to develop XRTS into a reliable and approximation-free diagnostic of extreme states of matter—with our ab initio PIMC simulations providing an unassailable theoretical foundation,” Dornheim adds.

FURTHER READING

T. Dornheim et al., Nat. Commun., 16, 5103 (2025); https://doi.org/10.1038/s41467-025-60278-3.