# New Julia programming language combines the best of two worlds

CASUS researcher promotes the use of Julia in high-energy physics At the end of September, the…

Our group develops novel methods and concepts to tackle problems from quantum many-body theory. The focus of our research is given by the simulation and diagnostics of warm dense matter, which requires a rigorous treatment of the complex interplay of effects like Coulomb coupling, quantum degeneracy, and strong thermal excitations. In addition, we apply our methodologies to other many-body systems such as ultracold atoms and electrons in quantum dots.

**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.

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”).

**Density functional theory (DFT) simulation of warm dense matter**

Linear-response time-dependent density functional theory constitutes a potentially powerful tool for the prediction of XRTS measurements. In addition to the usual XC-functional that is required for any type of DFT application, the computation of such dynamic properties needs the so-called XC-kernel as an additional input. In practice, computing an XC-kernel consistently was limited to relatively simple XC-functionals.

**Recently, we have developed a novel framework that allows the direct computation of the static XC-kernel for arbitrarily complex XC-functionals. This includes nonlocal hybrid functionals, which constitute the gold standard in the field of DFT simulations.**

Comparing these results with exact PIMC reference data where they are available has given new insights into the behavior of different XC-functionals. Moreover, the presented framework constitutes the basis for a number of DFT applications such as the prediction of XRTS experiments with warm dense matter.

The key strength of the DFT method is its balance between good accuracy and a manageable computational effort. In our group, we use exact PIMC reference data to rigorously assess the accuracy of different exchange—correlation functionals in DFT for the simulation of WDM. Moreover, we use DFT to predict XRTS experiments, and to study the linear and nonlinear response properties of a variety of systems.

Over the last two decades, the DFT method has become the de-facto work horse of WDM theory as it allows one to simulate complex systems with a manageable computation cost. In practice, the accuracy of a DFT simulation decisively depends on the employed exchange—correlation (XC) functional; it cannot be obtained within DFT itself, and has to be supplied as an empirical input. While the applicability of the available zoo of approximations for the XC-functional is reasonably well understood at ambient conditions, the development of thermal XC-functionals that are specifically designed for WDM simulations has started only recently. In our group, we use our quasi-exact PIMC simulation results to unambiguously benchmark existing functionals in the WDM regime. Moreover, we explore new concepts for the construction of novel functionals that consistently take into account the effect of the temperature.

A second field of application are linear-response time-dependent DFT (LR-TDDFT) simulations of various WDM systems such as hydrogen, beryllium, and carbon to predict the outcome of X-ray Thomson scattering (XRTS) experiments. To this end, we have developed a new framework that allows us to evaluate the static XC-kernel for arbitrary XC-functionals on any level of Jacob’s ladder of functional approximations. These efforts are further aided by extensive DFT+MD (Molecular Dynamics) simulations that give us access to important structural properties of the studied system such as the Rayleigh weight, a key property for the interpretation of XRTS experiments.

**X-ray Thomson scattering (XRTS) diagnostics**

While XRTS has emerged as a standard method of diagnostics for experiments with warm dense matter, the accuracy of the extracted system parameters such as the temperature has remained unclear.

**To break the vicious cycle between approximate theoretical models and a-priori uninterpreted experimental observations, we have introduced a novel model-free framework that allows us to directly extract the temperature from the XRTS signal. Our approach is free of any empirical parameters and works for arbitrarily complex systems.**

Due to its straightforward implementation and negligible computation cost, this new imaginary-time thermometry method has already been adapted by researchers around the globe, including world leading experts at the National Ignition Facility in California.

The XRTS approach has emerged as a key method of diagnostics for matter under extreme temperatures, densities and pressures. While in principle giving one insights into the microphysics of the probed sample, the interpretation of the measured intensity has generally relied on a number of model assumptions. In our group, we develop new concepts for the model-free interpretation of XRTS measurements such as the recent imaginary-time thermometry method that allows us to directly extract the temperature from the experimental data. Moreover, we use both PIMC and DFT simulations to predict XRTS signals, thereby guiding the development of new experimental set-ups.

While warm dense matter is routinely realized in the laboratory using different experimental techniques, the extreme conditions and short time-scales render the rigorous diagnostics of these states a formidable challenge. Over the last years, the X-ray Thomson scattering (XRTS) approach has emerged as a key technique. In principle, it gives one access to the dynamic structure factor of the probed system, which contains a wealth of information about its microphysics. Of particular importance is the determination of system parameters such as the temperature, density, and ionization state, which are important elements of the equation-of-state of the probed material.

In the past, the interpretation of XRTS measurements was generally based on a number of uncontrolled model assumptions such as the decomposition of the total population of electrons into effectively bound and free states. Such chemical models are always based on approximate models for the individual terms, and are expected to break down alltogether at high densities where even the electronic orbitals of supposedly bound electrons begin to overlap. This unsatisfactory situation can be understood as a classical chicken and egg problem: on the one hand, we need theoretical models to interpret experimental data. On the other hand, the theoretical models have been de-facto uncontrolled and require experimental observations as a benchmark.

To break this vicious cycle, we develop novel concepts in our group for the model-free diagnostics of XRTS measurements in the imaginary-time domain. As a first application, we have shown that it is possible to directly extract the temperature of a given system from the XRTS signal without the need for models and approximations. The model-free inference of other properties of the system is the subject of intense ongoing work. These efforts are complemented by the development of a more rigorous theoretical description of XRTS experiments based on ab initio path integral Monte Carlo (PIMC) and time-dependent density functional theory (TD-DFT) simulations.

**Linear and nonlinear density response theory**

The estimation of nonlinear response properties of interacting quantum many-body systems has remained limited to a few model cases due to the required very high computational effort.

**To overcome this bottleneck, we have developed a new framework for the computation of a variety of nonlinear response functions, which is based on the estimation of imaginary-time many-body correlation functions. It gives one access to the full wavenumber dependence of the nonlinear response from a single simulation of the unperturbed system.**

In practice, this results in a reduction of compute time by several orders of magnitude, enabling the systematic study of nonlinear properties of complex systems for the first time. In addition, the derived hierarchy that relates many-body correlations to nonlinear responses gives new insights into a number of effects such as the nonlinear coupling of several external perturbations.

Linear response theory is one of the most powerful concepts in quantum many-body theory, in general, and WDM theory, in particular. It is at the heart of modeling XRTS measurements, and gives one access to a wealth of interesting material properties such as electrical and thermal conductivities, opacity, and stopping power. In addition, our PIMC and DFT simulations also give us access to a wide class of nonlinear response properties, which play an important role for experiments with high-intensity laser beams and for the description of strongly coupled systems.

The central role of linear-response theory (LRT) in the description of nonideal quantum many-body systems such as warm dense matter or ultracold atoms can hardly be overstated. For example, the dynamic linear density response function gives one direct access to the dynamic structure factor, which is the key property in XRTS experiments. In addition, LRT is connected to a multitude of transport properties such as the dynamic dielectric function, conductivity, and stopping power. In our group, we use cutting-edge path integral Monte Carlo (PIMC) and density functional theory (DFT) simulations to estimate linear-response properties of warm dense matter, and other applications.

A related topic that has attracted considerable interest over the last years is the study of nonlinear response properties. These become important when a system is subject to a strong external perturbation such as a high-intensity laser beam, and also enter the theoretical description of material properties such as effective potentials and the stopping power. The most direct way to study nonlinear response properties is indeed the dedicated simulation of a strongly perturbed system to explicitly measure its response. While being formally exact, this approach requires multiple independent simulations to estimate a nonlinear response function for a single wavenumber at a given combination of density and temperature. A more elegant alternative is given by the estimation of higher-order imaginary-time correlation functions, which give one access to the full wavenumber dependence of the nonlinear response from a single simulation of the unperturbed system. The investigation of a variety of nonlinear response properties of real WDM systems, and their impact on the modeling of material properties, will be pursued in future works.