[Paper Review] Reduced density matrix functional theory at finite temperature. II. Application to the electron gas: Exchange only

Tim Baldsiefen, F. G. Eich+1 more|arXiv (Cornell University)|Aug 23, 2012

Advanced Chemical Physics Studies3 citations

TL;DR

This paper applies finite-temperature reduced density matrix functional theory (FT-RDMFT) in the first-order approximation—equivalent to finite-temperature Hartree-Fock (FT-HF)—to the homogeneous electron gas (HEG), analyzing both collinear and planar spin spiral (PSS) phases. It reveals that temperature suppresses the characteristic kink in the dispersion relation more rapidly than convergence to the noninteracting limit, and identifies an instantaneous phase transition in the PSS amplitude due to a nonanalytical jump, contrasting with Overhauser’s proposal.

ABSTRACT

Using the newly introduced theory of finite-temperature reduced density matrix functional theory, we apply the first-order approximation to the homogeneous electron gas. We consider both collinear spin states as well as symmetry broken states describing planar spin spirals and investigate the magnetic phase diagram as well as the temperature-dependence of the single particle spectra.

Motivation & Objective

• To develop and apply finite-temperature reduced density matrix functional theory (FT-RDMFT) to the homogeneous electron gas (HEG) in the first-order approximation.
• To investigate the magnetic phase diagram of the HEG, including collinear and planar spin spiral (PSS) states, under thermal effects.
• To analyze the temperature dependence of single-particle spectra and grand potential contributions in the Hartree-Fock approximation.
• To establish a foundation for future correlation functional development in Part III of this series.
• To compare FT-RDMFT with finite-temperature many-body perturbation theory (FT-MBPT), highlighting improved thermodynamic consistency.

Proposed method

• Utilizes the first-order functional in FT-RDMFT, which corresponds to the finite-temperature Hartree-Fock (FT-HF) solution, minimizing the grand potential via the 1-reduced density matrix (1RDM).
• Applies the variational principle to the grand potential functional Ω[𝒟̂] = tr{𝒟̂(Ĥ − μÂN − 1/β ln 𝒟̂)} with the SDO 𝒟̂ = e^−β(Ĥ−μÂN)/tr{e^−β(Ĥ−μÂN)}.
• Constructs the 1RDM and natural orbitals (NOs) for both collinear and PSS configurations, using the PSS state with wavevector q = 1.6k_F at r_s = 5.0 a.u.
• Computes single-particle energies and dispersion relations using the HF eigenvalue equation E_b = ∫ d𝐫 d𝐫′ γ(𝐫,𝐫′) [t(𝐫,𝐫′) + v(𝐫,𝐫′)] with the exchange term explicitly included.
• Evaluates the grand potential contributions and entropy, ensuring thermodynamic consistency through the 1RDM minimization procedure.
• Analyzes the temperature evolution of the PSS order parameter, identifying a nonanalytical jump indicating an instantaneous phase transition.

Experimental results

Research questions

• RQ1 How does finite temperature affect the magnetic phase diagram of the homogeneous electron gas in the FT-HF approximation?
• RQ2 What is the temperature dependence of the single-particle dispersion relation in both collinear and planar spin spiral (PSS) phases?
• RQ3 Does the PSS order parameter exhibit a continuous or discontinuous transition with increasing temperature?
• RQ4 How does FT-RDMFT compare to finite-temperature MBPT in terms of thermodynamic consistency, particularly regarding the free energy's temperature dependence?
• RQ5 To what extent does thermal smearing suppress the characteristic kink in the HF dispersion relation compared to the noninteracting limit?

Key findings

• The temperature-induced suppression of the kink in the HF dispersion relation occurs more rapidly than the convergence to the noninteracting dispersion, suggesting potential feasibility of an effective mass approximation at intermediate temperatures.
• For the PSS phase at r_s = 5.0 a.u. and q = 1.6k_F, the amplitude of the spin spiral exhibits a nonanalytical jump at a critical temperature, indicating an instantaneous phase transition, contrary to Overhauser’s prediction of a continuous transition.
• The unoccupied band in the PSS phase does not follow the noninteracting dispersion at zero temperature due to explicit coupling of both PSS channels in the exchange term.
• The grand potential decreases monotonically with increasing temperature in FT-RDMFT, in contrast to FT-MBPT, which can show unphysical increases, demonstrating improved thermodynamic consistency.
• Thermal smearing reduces the influence of the exchange energy relative to the kinetic energy due to reduced overlap between NOs of different quantum numbers, leading to a gradual closing of the HF gap.
• The Hartree-Fock dispersion for the collinear phase shows a vanishing
• valley
• in the occupied band with increasing temperature, followed by convergence to the noninteracting k²/2 relation.