Abstract
In low-temperature high-density plasmas quantum effects of the electrons are becoming increasingly important. This requires the development of new theoretical and computational tools. Quantum Monte Carlo methods are among the most successful approaches to first-principle simulations of many-body quantum systems. In this chapter we present a recently developed method—the configuration path integral Monte Carlo (CPIMC) method for moderately coupled, highly degenerate fermions at finite temperatures. It is based on the second quantization representation of the \(N\)-particle density operator in a basis of (anti-)symmetrized \(N\)-particle states (configurations of occupation numbers) and allows to tread arbitrary pair interactions in a continuous space. We give a detailed description of the method and discuss the application to electrons or, more generally, Coulomb-interacting fermions. As a test case we consider a few quantum particles in a one-dimensional harmonic trap. Depending on the coupling parameter (ratio of the interaction energy to kinetic energy), the method strongly reduces the sign problem as compared to direct path integral Monte Carlo (DPIMC) simulations in the regime of strong degeneracy which is of particular importance for dense matter in laser plasmas or compact stars. In order to provide a self-contained introduction, the chapter includes a short introduction to Metropolis Monte Carlo methods and the second quantization of quantum mechanics.
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Notes
- 1.
In fact, there are weaker conditions on the transition probabilities, but in practice, one uses the detailed balance.
- 2.
Efficient with respect to a certain system with fixed parameters to be simulated, i.e. currently, there is no method suitable for all systems.
- 3.
In this notation, operators in the interaction and the Schrödinger picture differ only concerning the presence of a time argument.
- 4.
Due to the finite number of one-particle states in a practical simulation, the sums will not go up to infinity but to \(N_B\), the number of orbitals used in the simulation.
- 5.
The one- and two-particle integrals used in these calculations have been calculated by K. Balzer with Mathematica [39].
- 6.
These integrals have been calculated with a program by K. Balzer.
- 7.
The finite temperature HF and CI programs were written by D. Hochstuhl.
- 8.
Shown is the standard error, see (5.15). Within two times the standard error the results can be considered to coincide.
- 9.
The data was produced by A. Filinov.
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Acknowledgments
We are grateful to K. Balzer, A. Filinov and D. Hochstuhl for providing supplementary data shown in this chapter. We acknowledge financial support by the Deutsche Forschungsgemeinschaft via grant BO1366-10 and a grant for CPU time at the HLRN.
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Schoof, T., Groth, S., Bonitz, M. (2014). Introduction to Configuration Path Integral Monte Carlo. In: Bonitz, M., Lopez, J., Becker, K., Thomsen, H. (eds) Complex Plasmas. Springer Series on Atomic, Optical, and Plasma Physics, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-319-05437-7_5
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