Beyond quantum mean-field approximation: Phase-space formulation of many-body time-dependent density functional theory and efficient spectral approximations

TL;DR

This work develops a phase-space formulation of many-body TDDFT based on the 2-RDM, addressing the prohibitive dimensionality by employing a Wigner function representation. It introduces efficient spectral approximations: a pseudo-difference operator for spatially periodic problems and Chebyshev spectral elements for non-periodic settings, both integrated into a massively parallel CHASM framework. Numerical experiments reveal two-body corrections to quantum kinetics, including entropy growth and damping/modulation of plasma instabilities, with XC effects often secondary to explicit two-body interactions. The approach provides a scalable path toward accurate 2-RDM dynamics in many-body TDDFT and lays groundwork for practical applications and future refinements of dynamical XC functionals.

Abstract

As a universal quantum mechanical approach to the dynamical many-body problem, the time-dependent density functional theory (TDDFT) might be inadequate to describe crucial observables that rely on two-body evolution behavior, like the double-excitation probability and two-body dynamic correlation. One promising remedy is to utilize the time-dependent 2-reduced density matrix (2-RDM) that directly represents two-body observables in an N-particle system, and resort to the extended TDDFT for multibody densities to break the confines of spatial local on one-body density [Phys. Rev. Lett. 26(6) (2024) 263001]. However, the usage of 2-RDM is prohibitive due to the augmented dimensionality, e.g., 4-D space for unidimensional 2-RDM. This work addresses the high-dimensional numerical challenges by using an equivalent Wigner phase-space formulation of 2-RDM and seeking efficient spectral approximations to nonlocal quantum potentials. For spatial periodic case, a pseudo-difference operator approach is derived for both the Hartree-exchange-correlation term and two-body collision operator, while the discretization via the Chebyshev spectral element method is provided for non-periodic case. A massively parallel numerical scheme, which integrates these spectral approximations with a distributed characteristics method, allows us to make the first attempt for real simulations of 2-RDM dynamics. Numerical experiments demonstrate the two-body correction to the quantum kinetic theory, and show the increase in the system's entropy induced by the two-bdoy interaction. Thus it may pave the way for an accurate description of 2-RDM dynamics and advance to a practical application of many-body TDDFT.

Figures (12)

• Figure 1: The picture of the two-body truncation of the quantum BBGKY hierarchy.
• Figure 2: The non-diagonal elements $b_{ij}$ in $A^{-1}$ decay exponentially away from the main diagonal $b_{ii}$, and the coefficients ${\bm{\eta}} = A^{-1}(f(r_{-1}), \dots, f(r_{N_x+1}))^{T}$ can be recovered by $\widetilde{\bm{\eta}}^{(l)} = (A_M^{(l)})^{-1}(\phi_L^{(l)}, f(r_0^{(l)}), \dots, f(r_M^{(l)}), \phi_R^{(l)})^T$.
• Figure 3: Three-fold or four-fold zero-padding is necessary in evaluating the Hartree potential.
• Figure 4: Nonlinear quantum Landau damping up to $T =200$ (left: with XC potential, right: with two-body interaction). In the linear stage ($t = 0$-$15$), the purely local exchange–correlation effect does not alter the Landau damping rate of the plasma, whereas the two-body interactions significantly modify the linear Landau damping rate and can even change the nonlinear coupling processes.
• Figure 5: Nonlinear quantum Landau damping: Evolution of the reduced Wigner function. The cases from left side to right side are: Hartree, Hartree with XC correction, Hartree with two-body correction ($\epsilon = 1$ or $\epsilon = 0.01$).

Theorems & Definitions (6)

• Example 1: 3-D Coulomb potential
• Example 2: 1-D screened Coulomb potential
• Example 3: 1-D smoothed Coulomb potential
• Example 4: 1-D Coulomb potential under the neutralized ion background
• Remark 1
• Remark 2