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The connection between time-local and time-nonlocal perturbation expansions
Quantum Physics · Submitted 19 Jul 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Here we show that this implies a recursive relation between their perturbative expansions, allowing a series for the kernel $\mathcal{K}$ to be translated directly into a corresponding series for the more complicated generator $\mathcal{G}$.
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The abstract, sentence by sentence
There exist two canonical approaches to describe open quantum systems by a time-evolution equation: the Nakajima-Zwanzig quantum master equation, featuring a time-nonlocal memory kernel $\mathcal{K}$, and the time-convolutionless equation with a time-local generator $\mathcal{G}$. These key quantities have recently been shown to be connected by an exact fixed-point relation [Phys. Rev. X 11, 021041 (2021)]. Here we show that this implies a recursive relation between their perturbative expansions, allowing a series for the kernel $\mathcal{K}$ to be translated directly into a corresponding series for the more complicated generator $\mathcal{G}$. This leads to an elegant way of computing the generator using well-developed, standard memory-kernel techniques for strongly interacting open systems. Moreover, it allows for an unbiased comparison of time-local and time-nonlocal approaches independent of the particular technique chosen to calculate expansions of $\mathcal{K}$ and $\mathcal{G}$ (Nakajima-Zwanzig projections, real-time diagrams, etc.). We illustrate this for leading and next-to-leading order calculations of $\mathcal{K}$ and $\mathcal{G}$ for the single impurity Anderson model using both the bare expansion in the system-environment coupling and a more advanced renormalized series. We compare the different expansions obtained, quantify the legitimacy of the generated dynamics (complete positivity) and benchmark with the exact result in the non-interacting limit.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | There exist two canonical approaches to describe open quantum systems by a time-evolution equation: the Nakajima-Zwanzig quantum master equation, featuring a time-nonlocal memory kernel $\mathcal{K}$, and the time-convolutionless equation with a time-local generator $\mathcal{G}$. | background 1.00 | 0.05 | 0.04 | asserted 1.00 |
| s1 | These key quantities have recently been shown to be connected by an exact fixed-point relation [Phys. Rev. X 11, 021041 (2021)]. | background 1.00 | 0.05 | 0.07 | asserted 1.00 |
| s2 | Here we show that this implies a recursive relation between their perturbative expansions, allowing a series for the kernel $\mathcal{K}$ to be translated directly into a corresponding series for the more complicated generator $\mathcal{G}$. | contribution 0.70 | 0.92 | 0.77 | asserted 1.00 |
| s3 | This leads to an elegant way of computing the generator using well-developed, standard memory-kernel techniques for strongly interacting open systems. | interpretation 0.90 | 0.69 | 0.24 | asserted 0.96 |
| s4 | Moreover, it allows for an unbiased comparison of time-local and time-nonlocal approaches independent of the particular technique chosen to calculate expansions of $\mathcal{K}$ and $\mathcal{G}$ (Nakajima-Zwanzig projections, real-time diagrams, etc.). | interpretation 0.79 | 0.62 | 0.13 | asserted 0.90 |
| s5 | We illustrate this for leading and next-to-leading order calculations of $\mathcal{K}$ and $\mathcal{G}$ for the single impurity Anderson model using both the bare expansion in the system-environment coupling and a more advanced renormalized series. | method 0.67 | 0.95 | 0.06 | asserted 1.00 |
| s6 | We compare the different expansions obtained, quantify the legitimacy of the generated dynamics (complete positivity) and benchmark with the exact result in the non-interacting limit. | contribution 0.48 | 0.95 | 0.07 | asserted 0.99 |
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Original abstract
There exist two canonical approaches to describe open quantum systems by a time-evolution equation: the Nakajima-Zwanzig quantum master equation, featuring a time-nonlocal memory kernel $\mathcal{K}$, and the time-convolutionless equation with a time-local generator $\mathcal{G}$. These key quantities have recently been shown to be connected by an exact fixed-point relation [Phys. Rev. X 11, 021041 (2021)]. Here we show that this implies a recursive relation between their perturbative expansions, allowing a series for the kernel $\mathcal{K}$ to be translated directly into a corresponding series for the more complicated generator $\mathcal{G}$. This leads to an elegant way of computing the generator using well-developed, standard memory-kernel techniques for strongly interacting open systems. Moreover, it allows for an unbiased comparison of time-local and time-nonlocal approaches independent of the particular technique chosen to calculate expansions of $\mathcal{K}$ and $\mathcal{G}$ (Nakajima-Zwanzig projections, real-time diagrams, etc.). We illustrate this for leading and next-to-leading order calculations of $\mathcal{K}$ and $\mathcal{G}$ for the single impurity Anderson model using both the bare expansion in the system-environment coupling and a more advanced renormalized series. We compare the different expansions obtained, quantify the legitimacy of the generated dynamics (complete positivity) and benchmark with the exact result in the non-interacting limit.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 3 sentences without a confident label
Uses the method or tool of 1 paper
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This is substantially simplified [ref] by combining diagrammatic [ref] and Liouville-Fock space techniques [ref] and exploiting the wideband limit from the very beginning.
Dynamics of open quantum systems · Bare perturbation theory for kernel $$ · 0.83
Cites as background 2 papers
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It was later studied on its own merits [ref] and revealed powerful exact relations [ref] .
Fermionic duality: General symmetry of open systems with strong dissipation and memory · Introduction · 0.81
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Several methods have been developed to compute $$, even for complicated models featuring strong interaction and memory effects [ref] .
Dynamics of open quantum systems · Introduction · 0.91
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