Frontiers of Computational Quantum Many-Body Theory

Path integral Monte Carlo (PIMC) simulations


We have demonstrated the capability to carry out highly accurate PIMC simulations of warm dense matter without the exponential bottleneck with respect to the number of simulated electrons. This has been achieved using the controlled xi-extrapolation method that has been suggested by Xiong and Xiong [JCP 2022].

In a recent breakthrough, we have simulated strongly compressed beryllium as it has been realized in experiments at the National Ignition Facility in California. This has allowed us to rigorously analyze different XRTS measurements with an unprecedented level of consistency, and without the need for any empirical parameters. The unambiguous predictive capability of our simulations constitutes a game changer for the understanding of warm dense matter, and will give new insights into a wealth of properties of light elements such as hydrogen and beryllium, and potentially even material mixtures such as lithium hydrate.

The PIMC method constitutes the gold standard in quantum many-body theory at finite temperatures as it is in principle capable of providing exact results without the need for any empirical parameters. In practice, the PIMC simulation of quantum degenerate Fermi systems (such as the electrons in WDM) is afflicted with an exponential computational bottleneck: the notorious fermion sign problem. In our group, we develop different strategies to deal with the sign problem, which allows us to compute highly accurate results for a variety of observables for light elements over a broad range of densities and temperatures.

Figure 1: left: snapshot of a PIMC simulation of N=50 Be atoms in the warm dense matter regime, with the green orbs and blue paths representing the semi-classical nuclei and quantum degenerate electrons; right: schematic illustration of the PIMC method, where a configuration of four electrons is depicted along the imaginary-time domain. Note the pair exchange involving the two electrons in the center that takes into account fermionic anti-symmetry and leads to the notorious sign problem.

The ab initio PIMC method is based on Feynman’s celebrated imaginary-time path integral representation of statistical quantum mechanics. Since its introduction in the 1960s, PIMC has given important insights into a variety of physics phenomena including superfluidity and Bose-Einstein-condensation. Unfortunately, the application of PIMC to quantum degenerate Fermi systems is severely hampered by a notorious computational bottleneck: the fermion sign problem. It leads to an exponential increase in the required compute time e.g. with increasing system size or decreasing temperature.

In our group, we develop new methodologies to deal with the sign problem. This allows us to carry out highly accurate PIMC simulations of electronic systems and light elements over substantial parts of the warm dense matter regime. First and foremost, these simulation campaigns give us new insights into the behavior of matter under extreme conditions. A particular strength of the PIMC method is its straightforward access to many-body correlation functions (including electronic pair correlation properties), which is not directly possible with less accurate methods such as density functional theory. Moreover, the direct PIMC method that is employed in our group allows us to estimate dynamic many-body properties in the imaginary-time domain. Such imaginary-time correlation functions can be estimated in X-ray Thomson scattering (XRTS) experiments, and give one access to a variety of linear and nonlinear response properties.

The further development of novel PIMC methodologies is supported by the European Union’s Horizon 2022 research and innovation programme (Grant agreement No. 101076233, “Predicting the extreme: PREXTREME”).