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arXiv:2110.07527 · Molecular Physics

Jellium at finite temperature

Riccardo Fantoni

Published Oct 28, 2021• 8 citations
arXivHTMLPDF

Abstract

We adopt the fixed node restricted path integral Monte Carlo method within the ‘Worm algorithm’ to simulate Wigner's Jellium model at finite, non zero, temperatures using free-particle nodes of the density matrix. The new element is that we incorporate the Worm algorithm paradigm of Prokof'ev and Svistunov in the grand canonical ensemble in order to more efficiently handle the fermionic exchanges. We present results for the structure and thermodynamic properties of the ideal Fermi gas and three points for the interacting electron gas. We treat explicitly the case of the partially polarized electron gas.

DOI
10.1080/00268976.2021.1996648

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We adopt the fixed node restricted path integral Monte Carlo method within the ‘Worm algorithm’ to simulate Wigner's Jellium model at finite, non zero, temperatures using free-particle nodes of the density matrix. The new element is that we incorporate the Worm algorithm paradigm of Prokof'ev and Svistunov in the grand canonical ensemble in order to more efficiently handle the fermionic exchanges. We present results for the structure and thermodynamic properties of the ideal Fermi gas and three points for the interacting electron gas. We treat explicitly the case of the partially polarized electron gas.
Over the last ten years the `fixed-node approximation', well known in the literature, has been widely used for a numerical treatment of thermodynamic properties of strongly correlated Fermi systems. Results of the direct path integral Monte Carlo simulation performed here show that the `fixed-node approximation' describes the thermodynamic properties of the strongly coupling fermions rather well at `weak and moderate' degeneracy. To analyse the reasons for the increasing difference between the `fixed-node approximation' and the results of direct path integral Monte Carlo simulations for highly degenerate fermions, the correctness of the `fixed-node approximation' for ideal Fermi systems has been analysed by analytical methods. A rigorous proof has been given of the fact that the exact Fermi function with index 3 / 2 describing the logarithm of the grand partition function of ideal fermions cannot be reproduced in the `fixed-node approximation', which means that the `fixed-node approximation' does not give the correct ideal Fermi gas limit.
We introduce a quantum Monte Carlo method at finite temperature for interacting fermionic models in the canonical ensemble, where the conservation of the particle number is enforced. Although general thermodynamic arguments ensure the equivalence of the canonical and the grand-canonical ensembles in the thermodynamic limit, their approach to the infinite-volume limit is distinctively different. Observables computed in the canonical ensemble generically display a finite-size correction proportional to the inverse volume, whereas in the grand-canonical ensemble the approach is exponential in the ratio of the linear size over the correlation length. We verify these predictions by quantum Monte Carlo simulations of the Hubbard model in one and two dimensions in the grand-canonical and the canonical ensemble. We prove an exact formula for the finite-size part of the free energy density, energy density and other observables in the canonical ensemble and relate this correction to a susceptibility computed in the corresponding grand-canonical ensemble. This result is confirmed by an exact computation of the one-dimensional classical Ising model in the canonical ensemble, which for classical models corresponds to the so-called fixed-magnetization ensemble. Our method is useful for simulating finite systems which are not coupled to a particle bath, such as in nuclear or cold atom physics.
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The Fermi gas at unitarity is a particularly interesting system of cold atoms, being dilute and strongly interacting at the same time. It can be studied non-perturbatively with Monte Carlo methods, like the recently developed worm algorithm. We discuss our implementation and tests of this algorithm and suggest a modification that increases its efficiency by reducing autocorrelations. We then show how the worm algorithm can be applied to calculate the critical temperature of an imbalanced Fermi gas (unequal number of fermions in the two spin components). We finally present some results obtained with the modified algorithm, in the balanced as well as in the imbalanced case.
We study quantum wires and point contacts with a fixed-node path integral Monte Carlo technique. The fixed-node technique uses a variational principle to map fermionic problems into effective bosonic problems, which are then evaluated with standard quantum Monte Carlo techniques. While fixed-node is an approximation, it has the useful properties of being variational and being able to recover the exact answer when the exact nodes or phases of the density matrix are known. In these finite-temperature simulations we use the free particle density matrix as a fixed-node constraint to efficiently simulate hundreds of interacting electrons. We have calculated charge densities, pair correlation functions, and the currentcurrent Matsubura Green's functions for quantum wires and quantum point contacts.
High Resolution Image Download MS PowerPoint Slide We combine the recent η-ensemble path integral Monte Carlo approach to the free energy [Dornheim et al. Phys. Rev. B 2025 111, L041114] with a recent fictitious partition function technique based on inserting a continuous variable that interpolates between the bosonic and Fermionic limits [Xiong and Xiong J. Chem. Phys. 2022 157, 094112] to deal with the Fermion sign problem. As a practical example, we apply our setup to the warm, dense, uniform electron gas over a broad range of densities and temperatures. We obtain accurate results for the exchange–correlation free energy down to half the Fermi temperature and find excellent agreement with the state-of-the-art parametrization by Groth et al. [ Phys. Rev. Lett. 2017 119, 135001]. Our work opens up new avenues for the future study of a host of interacting Fermi systems, including warm dense matter, ultracold atoms, and electrons in quantum dots, and for Fermionic free energy calculations with unprecedented system size.