Fermionic versus Spin Baths in non-Interacting Transport Models

TL;DR

This work interrogates whether a spin-1/2 bath can substitute for a fermionic reservoir in noninteracting transport by studying a resonant-level model. Using exact diagonalization and a perturbative master equation expanded to second and fourth order, the authors show that fermionic and spin baths yield identical reduced dynamics at second order, but diverge at fourth order due to exchange signs from fermionic anticommutation and non-Gaussian bath correlations in spins. As the bath size increases, higher-order corrections become negligible and both baths converge to the same Markovian dynamics predicted by the second-order theory, with spin baths effectively approximating Gaussian fermionic statistics in this regime. Practically, spin baths are reliable for large, weakly coupled environments, whereas accurately capturing exchange-statistics effects requires a fully fermionic treatment for small or strongly coupled baths.

Abstract

We investigate how fermionic anticommutation shapes transport in a noninteracting resonant-level model where a single central site is coupled to an environment. To this end, we compare a fermionic reservoir with a bath of spin-half modes using exact diagonalization and a perturbative expansion of the master equation to identify the differences. Notably, we find different reduced dynamics for the spin and fermionic baths even though the particles remain noninteracting and both models enforce local Pauli blocking. These differences originate from higher-order terms in the system-bath coupling, where fermionic anticommutation introduces exchange signs in higher-order correlations. By contrast, all second-order contributions, set solely by two-point correlators, coincide. Deviations are largest for small to intermediate bath sizes, fading in the effectively Markovian regime where higher-order corrections are negligible. These results identify when spin baths can be a substitute for fermionic reservoirs and vice versa.

Figures (4)

• Figure 1: Schematic representation of the resonant--level model: A system mode at energy $\omega_0$ is coupled (with coupling strengths $V_k$) to an environment consisting of discrete modes within a bandwidth $\omega_{\mathrm{BW}}$, partially filled up to the chemical potential $\mu$.
• Figure 2: Population dynamics from exact diagonalization for $N_E=3$, $n_{\mathrm{exc}}=2$, uniform coupling $V=1$, and $\omega_{\mathrm{BW}}=2$ with mode frequencies $\{-1,0,1\}$. Fermionic results (red dashed) vs. spins (blue).
• Figure 3: System population $n_S(t)$ with bandwidth $\omega_{\mathrm{BW}}=4\gamma$. Black solid: second-order master equation (ME2) red dashed: fermionic bath; blue dotted: spin bath. As $N_E$ increases, the fermionic and spin results converge and approach the ME2 prediction, reflecting the Markovian limit.
• Figure 4: Maximum population difference $\Delta_{\mathrm{max}}$ between fermions and spins with $\omega_{\mathrm{BW}}=4\gamma$, plotted as a function of $N_E$ (x-axis) and the number of occupied bath modes $n_{\mathrm{exc}}$ (y-axis). Differences peak at moderate bath sizes and near half filling, while vanishing for very small or large $N_E$.