Exact description of fermionic reservoirs via purified damped ancillary fermions

TL;DR

The paper tackles accurate modeling of fermionic reservoirs in nonequilibrium open quantum systems by introducing purified pseudofermions, a four-type purification that decouples positive- and negative-time reservoir contributions. The method uses analytic continuation and adiabatic elimination to derive Lindblad-like generators for purified degrees of freedom, with reservoir correlations C^ extsigma(t) decomposed as $C^ extsigma(t) \approx \sum_l w_l^ extsigma e^{i\textsigma\epsilon_l^ extsigma t - \gamma_l^ extsigma t}$ via AAA or similar decompositions. It provides explicit mappings for four types (I–IV), constructs a ρ_ppf whose tracing yields ρ_S, and demonstrates high-accuracy numerical results across a single impurity, a three-site engine, Kondo resonance, and chain transport, agreeing with Landauer–Büttiker theory and resolving band-edge features. The approach offers a scalable, numerically stable route to modeling fermionic reservoirs in quantum transport, quantum thermodynamics, and nonequilibrium phase transitions, with potential applicability to extended, strongly correlated systems.

Abstract

We present a method for the modeling of fermionic reservoirs using a new class of ancillary damped fermions, dubbed purified pseudofermions, which exhibit unusual free correlations. We show that this key feature, when combined with existing efficient decomposition algorithms for the reservoir correlation functions, enables the development of an easily implementable and accurate scheme for constructing effective models of fermionic reservoirs. We numerically demonstrate the validity, accuracy, efficiency and potential use of our method by studying the particle transport of spinless fermions in a one-dimensional chain. Beyond its utility as a quantum impurity solver, our method holds promise for addressing a wide range of problems involving extended systems in fields like quantum transport, quantum thermodynamics, thermal engines and nonequilibrium phase transitions.

Figures (7)

• Figure 1: (a) Sketch of a fermionic open quantum system and its modeling in terms of purified pseudofermions. Darkness of the blue color corresponds to the occupation of reservoir fermionic states at the initial time. As listed in Table. \ref{['tab:ppf']}, the type I and II (III and IV) purified pseudofermions are initially occupied (empty). (b) The $2\times2$ correlation matrix $\mathbf{C}_\text{X}$ with the free correlations $C_\text{X}^{\pm1}(t)$ ($\text{X}\in\{\text{I},\text{II},\text{III},\text{IV}\}$) as entries. In this representation, $\mathbf{C}_\text{X}$ has only one nonvanishing entry for each type of purified pseudofermions.
• Figure 2: In (a) the spectrum $S^{+1}(\epsilon)=2J_\text{rsq}(\epsilon)f(\epsilon)$ and the reservoir correlation $C^{+1}(t)$ (inset) of the left lead are plotted. Black dashed lines correspond to their approximations obtained with the AAA algorithm. In (b) the steady-state particle current $I_{ss}$ from the purified pseudofermion model (labeled as "PPF") is plotted as a function of the impurity energy $\epsilon_0$. Results of the LB formula is shown in black dashed line. Simulation parameters are: $W=8\Gamma$, $A=200$, $\mu_L=-\mu_R=W/16$, and $T_L=T_R=0$.
• Figure 3: The steady-state particle current $I_{ss}$ is plotted as a function of the average chemical potential $\mu=(\mu_L+\mu_R)/2$ and the bias $V=\mu_L-\mu_R$, using (a) the LB formula and (b) the purified pseudofermion model. In the region with $VI_{ss}<0$, we set $I_{ss}=0$ by hand. Simulation parameters are the same as those in Fig. 8 in Ref. PhysRevX.10.031040.
• Figure 4: The steady-state particle current $I_{ss}$ as a function of the hybridization strength $\Gamma$ for a three-site thermal engine in the absence $U=0$ (solid line and blue dots) and presence $U=1.2t_S$ (orange crosses) of the nearest-neighbour interaction. The relative error of the results for $U=0$ is below $2\%$, and the number of purified pseudofermions used in the simulations are shown in the inset. Other simulation parameters are $T_L=10t_S$, $T_R=t_S$, $\mu_L=-\mu_R=-t_S/2$, $W=8\Gamma$ and $A=200$.
• Figure 5: (a) Steady-state impurity correlation function $\text{Re}\langle f_\uparrow(t)f_\uparrow^\dagger(0)\rangle_\text{ss}$ for simulation times $\Gamma T_\text{sim} = 100, 200, 300, 400$, corresponding to interaction strengths $U/\pi\Gamma= 1,2,3,4$, respectively. The reservoir temperature is set to half the Kondo temperature, $T=T_K/2$. The left inset shows the corresponding computation time (blue dots) along with an exponential fitting (dashed line). The right inset displays displays the Kondo temperature $T_K$ as a function of $U$, with dots marking the values of $U$ used in the main figure. (b) Corresponding impurity spectral function $A_\uparrow(\epsilon)$. The inset provides a magnified view of the Kondo peaks. Other simulation parameters are $W=20\Gamma$, $A=20$ and $\mu=0$.