A novel path integral Monte Carlo (PIMC) approach for correlated many-particle systems with arbitrary pair interaction in continuous space at low temperatures is presented. It is based on a representation of the N -particle density operator in a basis of (anti-)symmetrized N -particle states (configurations of occupation numbers). The path integral is transformed into a sum over trajectories with the same topology and, finally, the limit of M →∞ , where M is the number of high-temperature factors, is analytically performed. This yields exact expressions for the thermodynamic quantities and allows to perform efficient simulations for fermions at low temperature and weak to moderate coupling. Our method is expected to be applicable to dense quantum plasmas in the regime of strong degeneracy where conventional PIMC fails due to the fermion sign problem. c
Recent advances in warm dense matter physics, e.g. laser compressed matter, lead to an increasing interest in the description of correlated, degenerate electrons at finite temperatures. Path integral Monte Carlo (PIMC) methods cannot correctly describe weakly to moderately coupled and strongly degenerate Fermi systems due to the so-called fermion sign problem. The Configuration Path Integral Monte Carlo (CPIMC) approach greatly reduces the sign problem and allows for the exact computation of thermodynamic properties in this regime. In addition, the first successful implementation of the diagrammatic worm algorithm for a general Hamiltonian in Fock space with arbitrary pair interactions gives direct access to the Matsubara Green function.
This thesis demonstrates the capabilities of the CPIMC approach for a model system of Coulomb interacting fermions in a two-dimensional harmonic trap. The correctness of the CPIMC implementation is verified by rigorous comparisons with an exact diagonalization method. Benchmark results are presented, which reveal large errors of the Hartree-Fock approximation in open shell configurations even for weak coupling and a significant deviation of multi-level blocking PIMC data in the complete basis set limit.
The application of the CPIMC method to the warm dense homogeneous electron gas (HEG) quantifies the accuracy of recently published restricted PIMC (RPIMC) results, which have been the basis for the construction of exchange-correlation free energy functionals to be used in finite-temperature density functional theory calculations of warm dense matter. It is shown that the errors of the RPIMC data exceed 10 % at intermediate densities. Additionally, highly accurate data for the exchange-correlation energy at high densities, which are inaccessible by the RPIMC method, are provided. These results are needed to significantly increase the quality of future exchange-correlation functionals to be used in finite-temperature applications.
Correlated fermions are of high interest in condensed matter (Fermi liquids, Wigner molecules), cold atomic gases and dense plasmas. Here we propose a novel approach to path integral Monte Carlo (PIMC) simulations of strongly degenerate non-ideal fermions at finite temperature by combining a fourth-order factorization of the density matrix with antisymmetric propagators, i.e., determinants, between all imaginary time slices. To efficiently run through the modified configuration space, we introduce a modification of the widely used continuous space worm algorithm, which allows for an efficient sampling at arbitrary system parameters. We demonstrate how the application of determinants achieves an effective blocking of permutations with opposite signs, leading to a significant relieve of the fermion sign problem. To benchmark the capability of our method regarding the simulation of degenerate fermions, we consider multiple electrons in a quantum dot and compare our results with other ab initio techniques, where they are available. The present permutation blocking PIMC approach allows us to obtain accurate results even for N = 20 electrons at low temperature and arbitrary coupling, where no other ab initio results have been reported, so far.
Recent advances in warm dense matter physics, e.g. laser compressed matter, lead to an increasing interest in the description of correlated, degenerate electrons at finite temperatures. Path integral Monte Carlo (PIMC) methods cannot correctly describe weakly to moderately coupled and strongly degenerate Fermi systems due to the so-called fermion sign problem. The Configuration Path Integral Monte Carlo (CPIMC) approach greatly reduces the sign problem and allows for the exact computation of thermodynamic properties in this regime. In addition, the first successful implementation of the diagrammatic worm algorithm for a general Hamiltonian in Fock space with arbitrary pair interactions gives direct access to the Matsubara Green function. This thesis demonstrates the capabilities of the CPIMC approach for a model system of Coulomb interacting fermions in a two-dimensional harmonic trap. The correctness of the CPIMC implementation is verified by rigorous comparisons with an exact diagonalization method. Benchmark results are presented, which reveal large errors of the Hartree-Fock approximation in open shell configurations even for weak coupling and a significant deviation of multi-level blocking PIMC data in the complete basis set limit. The application of the CPIMC method to the warm dense homogeneous electron gas (HEG) quantifies the accuracy of recently published restricted PIMC (RPIMC) results, which have been the basis for the construction of exchange-correlation free energy functionals to be used in finite-temperature density functional theory calculations of warm dense matter. It is shown that the errors of the RPIMC data exceed 10 % at intermediate densities. Additionally, highly accurate data for the exchange-correlation energy at high densities, which are inaccessible by the RPIMC method, are provided. These results are needed to significantly increase the quality of future exchange-correlation functionals to be used in finite-temperature applications.
ABSTRACT Ab initio path integral Monte Carlo (PIMC) simulations constitute the gold standard for the estimation of a broad range of equilibrium properties of a host of interacting quantum many‐body systems spanning a broad range of conditions from ultracold atoms to warm dense quantum plasmas. A key practical limitation is given by the notorious fermion sign problem, which manifests as an exponential computational bottleneck with respect to system size and inverse temperature. In practice, the sign problem is particularly severe in the grand‐canonical ensemble, where the bosonic and fermionic configuration spaces differ not only with respect to the symmetry of the thermal density matrix but, crucially, also with respect to the particle number distribution for a given chemical potential μ [T. Dornheim, J. Phys. A 54 , 335001 (2021)]. Here, we present a simple reweighting scheme that basically allows one to retain access to grand‐canonical expectation values at the cost of fermionic PIMC simulations in the canonical ensemble for the largest significant particle number in the fermionic sector. As a practical example, we consider the warm dense electron gas, which has attracted considerable recent attention due to its relevance, e.g., for the modeling of compact astrophysical objects and inertial fusion energy applications.