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.”