Preserving fermionic statistics for single-particle approximations in microscopic quantum master equations

TL;DR

The paper addresses unphysical fermionic evolution caused by single-particle approximations in microscopic quantum master equations. It derives a unitality-based constraint on the dissipative dynamics to guarantee N-representability of the 1-electron reduced density matrix, and applies this to the Redfield, Unified, and Universal Lindblad equations. Pauli blocking is proposed as a practical remedy for operators that violate the constraint, preserving positivity and N-representability in many cases. Benchmark and benzene case studies illustrate when standard microscopic MEs fail and how the proposed constraint and Pauli-blocked corrections restore physically meaningful dynamics, guiding the use of reduced-fermion models in chemistry and materials science.

Abstract

Microscopic master equations have gained traction for the dissipative treatment of molecular spin and solid-state systems for quantum technologies. Single particle approximations are often invoked to treat these systems, which can lead to unphysical evolution when combined with master equation approaches. We present a mathematical constraint on the system-environment parameters to ensure that microscopically-derived Markovian master equations preserve fermionic, $N$-representable statistics when applied to reduced systems. We demonstrate these constraints for the recently derived unified master equation and universal Lindblad equation, along with the Redfield master equation for cases when positivity issues are not present. For operators that break the constraint, we explore the addition of Pauli factors to recover $N$-representability. This work promotes feasible applications of novel microscopic master equations for realistic chemical systems.

Figures (5)

• Figure 1: Example of a clustering approach prescribed by the UME with five distinct Bohr frequencies. Note that if all clusters resemble $\mathcal{F}_{\overline{\omega}_3}$, the unified master equation reduces to the secular equation.
• Figure 2: Dynamics of the 1-hole RDM for a 3-level system at 50 K. (a) Application of the ULE, which results in a non-unital map. (b) Application of the Pauli-blocked ULE, which results in a unital map. Both (a) and (b) compare the explicit solution for $\mathds{1} - \leftindex^{1}\rho$, obtained from propagating $\leftindex^{1}\rho$(lines), against the dynamics for $\leftindex^{1}q$(circles).
• Figure 3: Unconstrained dynamics of benzene at $50$ K by the UME, RME, and ULE that violate the Pauli-exclusion principle. (a) No clustering is invoked in the UME, reducing this equation to the secular or Davies form. (b) A clustering threshold of $0.091$ a.u. is invoked to cluster Bohr frequencies $0.169$ a.u. and $0.260$ a.u.
• Figure 4: Use of Pauli blocking to recover fermionic statistics for benzene with the RME, UME, and ULE at 50 K.
• Figure 5: Spectral function $\Gamma(\omega)$ for a bosonic bath with a Drude-Lorentz spectral density with $\lambda = 0.01$ a.u. and $\hbar$ in atomic units. The color map denotes the temperature range.