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Röpke G, Dornheim T, Vorberger J, Blaschke D, Mahato B. Virial coefficients of the uniform electron gas from path-integral Monte Carlo simulations. Phys Rev E 2024; 109:025202. [PMID: 38491663 DOI: 10.1103/physreve.109.025202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2023] [Accepted: 12/22/2023] [Indexed: 03/18/2024]
Abstract
The properties of plasmas in the low-density limit are described by virial expansions. Analytical expressions are known from Green's function approaches only for the first three virial coefficients. Accurate path-integral Monte Carlo (PIMC) simulations have recently been performed for the uniform electron gas, allowing the virial expansions to be analyzed and interpolation formulas to be derived. The exact expression for the second virial coefficient is used to test the accuracy of the PIMC simulations and the range of validity of the interpolation formula of Groth et al. [Phys. Rev. Lett. 119, 135001 (2017)0031-900710.1103/PhysRevLett.119.135001], and we discuss the fourth virial coefficient, which is not exactly known yet. Combining PIMC simulations with benchmarks from exact virial expansion results would allow us to obtain more accurate representations of the equation of state for parameter ranges of conditions which are of interest, e.g., for helioseismology.
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Affiliation(s)
- G Röpke
- Institute of Physics, University of Rostock, Albert-Einstein-Str. 23-24, D-18059 Rostock, Germany
| | - T Dornheim
- Center for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-02826 Görlitz, Germany
| | - J Vorberger
- Institute for Radiation Physics, Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - D Blaschke
- Center for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-02826 Görlitz, Germany
- Institute of Theoretical Physics, University of Wroclaw, 50-204 Wroclaw, Poland
| | - B Mahato
- Institute of Theoretical Physics, University of Wroclaw, 50-204 Wroclaw, Poland
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2
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Dornheim T, Tolias P, Groth S, Moldabekov ZA, Vorberger J, Hirshberg B. Fermionic physics from ab initio path integral Monte Carlo simulations of fictitious identical particles. J Chem Phys 2023; 159:164113. [PMID: 37888764 DOI: 10.1063/5.0171930] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2023] [Accepted: 10/06/2023] [Indexed: 10/28/2023] Open
Abstract
The ab initio path integral Monte Carlo (PIMC) method is one of the most successful methods in statistical physics, quantum chemistry and related fields, but its application to quantum degenerate Fermi systems is severely hampered by an exponential computational bottleneck: the notorious fermion sign problem. Very recently, Xiong and Xiong [J. Chem. Phys. 157, 094112 (2022)] have suggested to partially circumvent the sign problem by carrying out simulations of fictitious systems guided by an interpolating continuous variable ξ ∈ [-1, 1], with the physical Fermi- and Bose-statistics corresponding to ξ = -1 and ξ = 1. It has been proposed that information about the fermionic limit might be obtained by calculations within the bosonic sector ξ > 0 combined with an extrapolation throughout the fermionic sector ξ < 0, essentially bypassing the sign problem. Here, we show how the inclusion of the artificial parameter ξ can be interpreted as an effective penalty on the formation of permutation cycles in the PIMC simulation. We demonstrate that the proposed extrapolation method breaks down for moderate to high quantum degeneracy. Instead, the method constitutes a valuable tool for the description of large Fermi-systems of weak quantum degeneracy. This is demonstrated for electrons in a 2D harmonic trap and for the uniform electron gas (UEG), where we find excellent agreement (∼0.5%) with exact configuration PIMC results in the high-density regime while attaining a speed-up exceeding 11 orders of magnitude. Finally, we extend the idea beyond the energy and analyze the radial density distribution (2D trap), as well as the static structure factor and imaginary-time density-density correlation function (UEG).
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Panagiotis Tolias
- Space and Plasma Physics, Royal Institute of Technology (KTH), Stockholm SE-100 44, Sweden
| | - Simon Groth
- Christian-Albrechts-Universität zu Kiel, D-24098 Kiel, Germany
| | - Zhandos A Moldabekov
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Barak Hirshberg
- School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel
- The Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
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3
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Dornheim T, Vorberger J, Moldabekov ZA, Böhme M. Analysing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220217. [PMID: 37393936 DOI: 10.1098/rsta.2022.0217] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Accepted: 01/10/2023] [Indexed: 07/04/2023]
Abstract
Rigorous diagnostics of experiments with warm dense matter are notoriously difficult. A key method is X-ray Thomson scattering (XRTS), but the interpretation of XRTS measurements is usually based on theoretical models that entail various approximations. Recently, Dornheim et al. [Nat. Commun. 13, 7911 (2022)] introduced a new framework for temperature diagnostics of XRTS experiments that is based on imaginary-time correlation functions. On the one hand, switching from the frequency to the imaginary-time domain gives one direct access to a number of physical properties, which facilitates the extraction of the temperature of arbitrarily complex materials without relying on any models or approximations. On the other hand, the bulk of theoretical work in dynamic quantum many-body theory is devoted to the frequency domain, and, to the best of our knowledge, the manifestation of physics properties within the imaginary-time density-density correlation function (ITCF) remains poorly understood. In the present work, we aim to fill this gap by introducing a simple, semi-analytical model for the imaginary-time dependence of two-body correlations within the framework of imaginary-time path integrals. As a practical example, we compare our new model to extensive ab initio path integral Monte Carlo results for the ITCF of a uniform electron gas, and find excellent agreement over a broad range of wavenumbers, densities and temperatures. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Zhandos A Moldabekov
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Maximilian Böhme
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
- Technische Universität Dresden, D-01062 Dresden, Germany
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4
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Rygg JR, Celliers PM, Collins GW. Specific Heat of Electron Plasma Waves. PHYSICAL REVIEW LETTERS 2023; 130:225101. [PMID: 37327418 DOI: 10.1103/physrevlett.130.225101] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2022] [Revised: 03/31/2023] [Accepted: 04/07/2023] [Indexed: 06/18/2023]
Abstract
Collective modes in a plasma, like phonons in a solid, contribute to a material's equation of state and transport properties, but the long wavelengths of these modes are difficult to simulate with today's finite-size quantum simulation techniques. A simple Debye-type calculation of the specific heat of electron plasma waves is presented, yielding up to 0.05k/e^{-} for warm dense matter (WDM), where thermal and Fermi energies are near 1 Ry=13.6 eV. This overlooked energy reservoir is sufficient to explain reported compression differences between theoretical hydrogen models and shock experiments. Such an additional specific heat contribution refines our understanding of systems passing through the WDM regime, such as the convective threshold in low-mass main-sequence stars, white dwarf envelopes, and substellar objects; WDM x-ray scattering experiments; and the compression of inertial confinement fusion fuels.
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Affiliation(s)
- J R Rygg
- Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623, USA
- Department of Mechanical Engineering, University of Rochester, Rochester, New York 14623, USA
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14623, USA
| | - P M Celliers
- Lawrence Livermore National Laboratory, Livermore, California 94551, USA
| | - G W Collins
- Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623, USA
- Department of Mechanical Engineering, University of Rochester, Rochester, New York 14623, USA
- Department of Physics and Astronomy, University of Rochester, Rochester, New York 14623, USA
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Dornheim T, Tolias P, Moldabekov ZA, Cangi A, Vorberger J. Effective electronic forces and potentials from ab initio path integral Monte Carlo simulations. J Chem Phys 2022; 156:244113. [PMID: 35778089 DOI: 10.1063/5.0097768] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The rigorous description of correlated quantum many-body systems constitutes one of the most challenging tasks in contemporary physics and related disciplines. In this context, a particularly useful tool is the concept of effective pair potentials that take into account the effects of the complex many-body medium consistently. In this work, we present extensive, highly accurate ab initio path integral Monte Carlo (PIMC) results for the effective interaction and the effective force between two electrons in the presence of the uniform electron gas. This gives us a direct insight into finite-size effects, thereby, opening up the possibility for novel domain decompositions and methodological advances. In addition, we present unassailable numerical proof for an effective attraction between two electrons under moderate coupling conditions, without the mediation of an underlying ionic structure. Finally, we compare our exact PIMC results to effective potentials from linear-response theory, and we demonstrate their usefulness for the description of the dynamic structure factor. All PIMC results are made freely available online and can be used as a thorough benchmark for new developments and approximations.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
| | - Panagiotis Tolias
- Space and Plasma Physics, Royal Institute of Technology (KTH), Stockholm SE-100 44, Sweden
| | | | - Attila Cangi
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
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6
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Dornheim T, Vorberger J, Militzer B, Moldabekov ZA. Momentum distribution of the uniform electron gas at finite temperature: Effects of spin polarization. Phys Rev E 2021; 104:055206. [PMID: 34942706 DOI: 10.1103/physreve.104.055206] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2021] [Accepted: 10/25/2021] [Indexed: 11/07/2022]
Abstract
We carry out extensive direct path integral Monte Carlo (PIMC) simulations of the uniform electron gas (UEG) at finite temperature for different values of the spin-polarization ξ. This allows us to unambiguously quantify the impact of spin effects on the momentum distribution function n(k) and related properties. We find that interesting physical effects like the interaction-induced increase in the occupation of the zero-momentum state n(0) substantially depend on ξ. Our results further advance the current understanding of the UEG as a fundamental model system, and are of practical relevance for the description of transport properties of warm dense matter in an external magnetic field. All PIMC results are freely available online and can be used as a benchmark for the development of methods and applications.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany.,Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Burkhard Militzer
- Department of Earth and Planetary Science, University of California, Berkeley, California 94720, USA.,Department of Astronomy, University of California, Berkeley, California 94720, USA
| | - Zhandos A Moldabekov
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany.,Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
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7
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Röpke G, Schörner M, Redmer R, Bethkenhagen M. Virial expansion of the electrical conductivity of hydrogen plasmas. Phys Rev E 2021; 104:045204. [PMID: 34781483 DOI: 10.1103/physreve.104.045204] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2021] [Accepted: 09/22/2021] [Indexed: 11/07/2022]
Abstract
The low-density limit of the electrical conductivity σ(n,T) of hydrogen as the simplest ionic plasma is presented as a function of the temperature T and mass density n in the form of a virial expansion of the resistivity. Quantum statistical methods yield exact values for the lowest virial coefficients which serve as a benchmark for analytical approaches to the electrical conductivity as well as for numerical results obtained from density functional theory-based molecular dynamics simulations (DFT-MD) or path-integral Monte Carlo simulations. While these simulations are well suited to calculate σ(n,T) in a wide range of density and temperature, in particular, for the warm dense matter region, they become computationally expensive in the low-density limit, and virial expansions can be utilized to balance this drawback. We present new results of DFT-MD simulations in that regime and discuss the account of electron-electron collisions by comparison with the virial expansion.
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Affiliation(s)
- G Röpke
- Institut für Physik, Universität Rostock, D-18051 Rostock, Germany
| | - M Schörner
- Institut für Physik, Universität Rostock, D-18051 Rostock, Germany
| | - R Redmer
- Institut für Physik, Universität Rostock, D-18051 Rostock, Germany
| | - M Bethkenhagen
- École Normale Supérieure de Lyon, Laboratoire de Géologie de Lyon, LGLTPE UMR 5276, Centre Blaise Pascal, 46 allée d'Italie, Lyon 69364, France
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8
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Dornheim T, Moldabekov ZA, Vorberger J. Nonlinear density response from imaginary-time correlation functions: Ab initio path integral Monte Carlo simulations of the warm dense electron gas. J Chem Phys 2021; 155:054110. [PMID: 34364322 DOI: 10.1063/5.0058988] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The ab initio path integral Monte Carlo (PIMC) approach is one of the most successful methods in quantum many-body theory. A particular strength of this method is its straightforward access to imaginary-time correlation functions (ITCFs). For example, the well-known density-density ITCF F(q, τ) allows one to estimate the linear response of a given system for all wave vectors q from a single simulation of the unperturbed system. Moreover, it constitutes the basis for the reconstruction of the dynamic structure factor S(q, ω)-a key quantity in state-of-the-art scattering experiments. In this work, we present analogous relations between the nonlinear density response in the quadratic and cubic order of the perturbation strength and generalized ITCFs measuring correlations between up to four imaginary-time arguments. As a practical demonstration of our new approach, we carry out simulations of the warm dense electron gas and find excellent agreement with previous PIMC results that had been obtained with substantially larger computational effort. In addition, we give a relation between a cubic ITCF and the triple dynamic structure factor S(q1, ω1; q2, ω2), which evokes the enticing possibility to study dynamic three-body effects on an ab initio level.
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Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
| | | | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
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9
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Dornheim T, Vorberger J. Overcoming finite-size effects in electronic structure simulations at extreme conditions. J Chem Phys 2021; 154:144103. [DOI: 10.1063/5.0045634] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
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10
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Dornheim T, Invernizzi M, Vorberger J, Hirshberg B. Attenuating the fermion sign problem in path integral Monte Carlo simulations using the Bogoliubov inequality and thermodynamic integration. J Chem Phys 2020; 153:234104. [DOI: 10.1063/5.0030760] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Tobias Dornheim
- Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
| | - Michele Invernizzi
- Institute of Computational Sciences, Università Della Svizzera Italiana, 6900 Lugano, Switzerland
- National Centre for Computational Design and Discovery of Novel Materials MARVEL, Università Della Svizzera Italiana, 6900 Lugano, Switzerland
- Department of Physics, ETH Zurich, 8092 Zurich, Switzerland
| | - Jan Vorberger
- Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
| | - Barak Hirshberg
- Institute of Computational Sciences, Università Della Svizzera Italiana, 6900 Lugano, Switzerland
- Department of Chemistry and Applied Biosciences, ETH Zurich, 8092 Zurich, Switzerland
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