Contour calculus for many-particle functions

Hyrkäs, Markku; Karlsson, Daniel; Leeuwen, Robert van

doi:10.1088/1751-8121/ab165d

Cited by 5 publications

(14 citation statements)

References 17 publications

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“…However, unlike the zero-temperature case we find that at finite temperature it is in general difficult to derive approximate positive definite expressions without introducing unwanted vacuum diagrams in the expansion. Third, we then demonstrate how this issue can be resolved using an expansion in so-called retarded half-diagrams [13,11] with a clear physical interpretation as collective contributions of past scattering processes.…”

Section: Self-energy Cutting Rules At Finite Temperaturementioning

confidence: 96%

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Cutting rules and positivity in finite temperature many-body theory

Hyrkäs,

Karlsson,

van Leeuwen

2022

Preprint

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For a given diagrammatic approximation in many-body perturbation theory it is not guaranteed that positive observables, such as the density or the spectral function, retain their positivity. For zero-temperature systems we developed a method [Phys.Rev.B90,115134 (2014)] based on so-called cutting rules for Feynman diagrams that enforces these properties diagrammatically, thus solving the problem of negative spectral densities observed for various vertex approximations. In this work we extend this method to systems at finite temperature by formulating the cutting rules in terms of retarded N -point functions, thereby simplifying earlier approaches and simultaneously solving the issue of non-vanishing vacuum diagrams that has plagued finite temperature expansions. Our approach is moreover valid for nonequilibrium systems in initial equilibrium and allows us to show that important commonly used approximations, namely the GW , second Born and T -matrix approximation, retain positive spectral functions at finite temperature. Finally we derive an analytic continuation relation between the spectral forms of retarded N -point functions and their Matsubara counterparts and a set of Feynman rules to evaluate them.

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Section: Self-energy Cutting Rules At Finite Temperaturementioning

confidence: 96%

“…Internal loop integrals over a sub-diagram up to the external times representing the entry or exit point of the self-energy can be expressed in terms of real-time integrals over retarded functions [13,11]. Accordingly we define a multi-argument retarded component of a general Feynman diagram with integrand D(z N ) = D(z 1 , .…”

Section: Retarded Cutting Rulesmentioning

confidence: 99%

Cutting rules and positivity in finite temperature many-body theory

Hyrkäs,

Karlsson,

van Leeuwen

2022

Preprint

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“… the diagrams appearing in D are contour ordered rather than (anti)time ordered. To convert the contour expression into a real‐time expression the usual Langreth rules are inadequate due to the multi‐integral structure, and more generalized rules have to be used . Here we give a brief discussion of the real‐time conversion.…”

Section: Evaluation Of the Half‐diagramsmentioning

confidence: 99%

“…An integral over all but one variables of a contour‐diagram

\begin{array}{c} center & A^{'} (z_{i}) = \int_{γ} d z_{N / i} A (z_{scriptN}), & normali \in scriptN \end{array}

is a function symmetric with respect to the branch index, so that

A^{'} true(t true) = A^{'} true(t_{\pm} true)

, that for both branch‐indices is equal to the real‐time integral

A^{'} true(t_{i} true) = \int_{t_{0}}^{\infty} d t_{N / i} A^{R true(i, scriptN / i true)} true(t_{N} true)

…”

Section: Evaluation Of the Half‐diagramsmentioning

confidence: 99%

Diagrammatic Expansion for Positive Spectral Functions in the Steady‐State Limit

Hyrkäs¹,

Karlsson²,

Leeuwen³

2019

Physica Status Solidi (b)

Self Cite

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Recently, a method was presented for constructing self-energies within manybody perturbation theory that is guaranteed to produce a positive spectral function for equilibrium systems, by representing the self-energy as a product of half-diagrams on the forward and backward branches of the Keldysh contour [Phys. Rev. B 2014, 90, 115134]. An alternative half-diagram representation that is based on products of retarded diagrams is derived here. This approach extends the method to systems out of equilibrium. When a steady-state limit exists, it is shown that our approach yields a positive definite spectral function in the frequency domain.

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“…Taking advantage of appropriate analytical continuation, one can arrive at the evaluation of Green's functions 21,24 . Expressions of physical quantities in terms of KPM have been developed recently [25][26][27][28][29][30][31][32][33] , including the applications to superconductors 34 , topological materials 35,36 , quantum impurity problems [37][38][39] and ab initio calculations 40,41 . However, these methods are applicable to bulk or isolated systems 21,33 , not to scattering processes between leads in open systems, which corresponds to a realistic experimental setup 4 .…”

Section: Introductionmentioning

confidence: 99%

Chebyshev polynomial method to Landauer–Büttiker formula of quantum transport in nanostructures

Zhang

Liu

et al. 2020

AIP Advances

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Landauer-Büttiker formula describes the electronic quantum transports in nanostructures and molecules. It will be numerically demanding for simulations of complex or large size systems due to, for example, matrix inversion calculations. Recently, Chebyshev polynomial method has attracted intense interests in numerical simulations of quantum systems due to the high efficiency in parallelization, because the only matrix operation it involves is just the product of sparse matrices and vectors. Many progresses have been made on the Chebyshev polynomial representations of physical quantities for isolated or bulk quantum structures. Here we present the Chebyshev polynomial method to the typical electronic scattering problem, the Landauer-Büttiker formula for the conductance of quantum transports in nanostructures. We first describe the full algorithm based on the standard bath KPM. Then, we present two simple but efficient improvements. One of them has a time consumption remarkably less than the direct matrix calculation without KPM. Some typical examples are also presented to illustrate the numerical effectiveness.

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Contour calculus for many-particle functions

Cited by 5 publications

References 17 publications

Cutting rules and positivity in finite temperature many-body theory

Cutting rules and positivity in finite temperature many-body theory

Diagrammatic Expansion for Positive Spectral Functions in the Steady‐State Limit

Chebyshev polynomial method to Landauer–Büttiker formula of quantum transport in nanostructures

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