Pseudofermion method for the exact description of fermionic environments: From single-molecule electronics to the Kondo resonance

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Pseudofermion method for the exact description of fermionic environments: From single-molecule electronics to the Kondo resonance

Mauro Cirio, Neill Lambert, Pengfei Liang, Po-Chen Kuo, Yueh-Nan Chen, Paul Menczel, Ken Funo, and Franco Nori

Phys. Rev. Research 5, 033011 – Published 7 July 2023

Abstract

We develop a discrete fermion approach for modeling the strong interaction of an arbitrary system interacting with continuum electronic reservoirs. The approach is based on a pseudofermion decomposition of the continuum bath correlation functions and is only limited by the accuracy of this decomposition. We show that to obtain this decomposition, one can allow for imaginary pseudofermion parameters, and strong damping in individual pseudofermions, without introducing unwanted approximations. For a noninteracting single-resonant level, we benchmark our approach against an analytical solution and an exact hierarchical-equations-of-motion approach. We also show that, for the interacting case, this simple method can capture the strongly correlated low-temperature physics of Kondo resonance, even in the difficult scaling limit, by employing matrix product state techniques.

Received 30 July 2022
Revised 25 May 2023
Accepted 26 May 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033011

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Open quantum systems & decoherence Quantum correlations, foundations & formalism

Non-Markovian processes

General Physics

Authors & Affiliations

Mauro Cirio ^1,*, Neill Lambert ^2,†, Pengfei Liang¹, Po-Chen Kuo ^3,4, Yueh-Nan Chen ^3,4, Paul Menczel ², Ken Funo ², and Franco Nori ^2,5,6

¹Graduate School of China Academy of Engineering Physics, Haidian District, Beijing 100193, China
²Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan
³Department of Physics, National Cheng Kung University, 701 Tainan, Taiwan
⁴Center for Quantum Frontiers of Research & Technology, NCKU, 70101 Tainan, Taiwan
⁵Center for Quantum Computing, RIKEN, Wakoshi, Saitama 351-0198, Japan
⁶Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*cirio.mauro@gmail.com
^†nwlambert@gmail.com

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Vol. 5, Iss. 3 — July - September 2023

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Images

Figure 1
Illustration of the pseudofermion model. In (a) a fermionic system $S$ interacts with a continuum of environmental fermionic modes at different frequencies. The spectral density function in (b) describes the distribution of the system-environment coupling. As shown in (c) the dynamics of $S$ can be equivalently computed by considering the interaction with a discrete set of pseudofermions (red and blue circles) whose dissipative properties are described by a Lindblad master equation (wiggly arrows). This equivalence holds as the pseudofermions are designed to reproduce the correlation functions of the original bath. Specifically, for the spectral density in (b), the red (blue) pseudofermions in (c) reproduce the resonant (Matsubara) contribution to the correlation function plotted in (d); see Eq. (20).
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Figure 2
Current into the right reservoir from a single impurity (coupled to two reservoirs) as a function of chemical potential difference $Δ μ = μ_{L} - μ_{R}$ . For both reservoirs we choose $ε = Γ, W = 2.5 Γ$ , and $β = 1 / (0.2 Γ)$ . The truncation parameter for the HEOM results is $n_{\max} = 2$ .
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Figure 3
Spectral density $A (ω)$ for a single impurity containing two interacting fermions coupled to a spin-labeled reservoir. Here, $μ = 0, U = 3 π Γ, ε = - U / 2$ , and $β = 1 / (0.2 Γ)$ . (a) shows the results for width $W = 2.5 Γ$ , while (b) is for $W = 10 Γ$ . In (a) we see that the HEOM result with $N_{k} = 2$ has not converged, while all other results are (pseudofermion and HEOM, $N_{k} = 4$ results overlap). In (b), which uses the Matsubara series in Eq. (B49), the HEOM results converge for $N_{k} = 5$ , while the standard pseudofermion results, relying on $K^{fit} = 1$ terms in the series in Eq. (29) to fit the Matsubara contribution (shown in the inset), have not converged. To efficiently include more exponents, and hence more pseudofermions, we employ matrix product states, allowing us to include $K^{fit} = 3$ terms in the Matsubara series (yellow curve in inset) and achieve convergence (solid purple curve).
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Figure 4
Schematics of the superfermion representation. Blue circles correspond to physical fermions, and orange circles correspond to auxiliary ones. They are combined in pairs to give one MPS site, shown as dashed ellipses. The pair on the left represents the system, while remaining sites represent the bath. Solid curves between sites indicate the long-ranged hopping terms induced by the Hamiltonian.
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Figure 5
Tensor network simulation of the pseudofermion master equation (16) using the superfermion representation and the swap-gate technique. (a) The density matrix of the composite system is represented as a MPS (red square for the system and blue squares for the pseudofermions). The Trotterized propagator acts on the MPS sequentially in the form of two-site gates which can be long ranged. (b) Each long-ranged gate can be decomposed into a series of two-site, nearest-neighbor gates which combine the swap gate $S$ and the Trotterized propagator. Neighboring swap gates (in the dashed green box) can be combined to the identity operator, leading to the simplified version of the algorithm in (c).
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