Diagonal Padé approximant of the one-body Green's function: A study on Hubbard rings

Tarantino, Walter; Sabatino, S. Di

doi:10.1103/physrevb.99.075149

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…where the coefficients of the polynomials A(λ) and B(λ) are determined by collecting and comparing terms for each power of λ with the low-order terms in the Taylor series expansion. Padé approximants are extremely useful in many areas of physics and chemistry [138][139][140][141] as they can model poles, which appear at the roots of B(λ). However, they are unable to model functions with square-root branch points (which are ubiquitous in the singularity structure of perturbative methods) and more complicated functional forms appearing at critical points (where the nature of the solution undergoes a sudden transition).…”

Section: Padé Approximantmentioning

confidence: 99%

Perturbation theory in the complex plane: exceptional points and where to find them

Marie

Burton

Loos

2021

J. Phys.: Condens. Matter

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We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree–Fock approximation and Rayleigh–Schrödinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller–Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller–Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

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Section: Padé Approximantmentioning

confidence: 99%

Perturbation theory in the complex plane: exceptional points and where to find them

Marie

Burton

Loos

2021

J. Phys.: Condens. Matter

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“…where the coefficients of the polynomials A(λ) and B(λ) are determined by collecting terms for each power of λ. Padé approximants are extremely useful in many areas of physics and chemistry [136][137][138][139] as they can model poles, which appear at the roots of B(λ). However, they are unable to model functions with square-root branch points (which are ubiquitous in the singularity structure of perturbative methods) and more complicated functional forms appearing at critical points (where the nature of the solution undergoes a sudden transition).…”

Section: Resummation Methodsmentioning

confidence: 99%

Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them

Marie,

Burton,

Loos

2020

Preprint

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We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree-Fock approximation and Rayleigh-Schrödinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller-Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller-Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

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“…Many different approaches were studied by researchers, such as the Padé approximation by Hunt, the Taylor expansion approach by Nielsen, and two different rootfinding methods by You [12]. Padé approximation is often used, for example, to find an approximation to Green's function [13]. So, in this research, we try to modify the dispersion relation with other transcendental functions before applying the Padé approximation.…”

Section: Introductionmentioning

confidence: 99%

Modeling a Wave on Mild Sloping Bottom Topography and Its Dispersion Relation Approximation

Abdullah

Erianto²

2022

KUBIK

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Linear wave theory is a simple theory that researchers and engineers often use to study a wave in deep, intermediate, and shallow water regions. Many researchers mostly used it over the horizontal flat seabed, but in actual conditions, sloping seabed always exists, although mild. In this research, we try to model a wave over a mild sloping seabed by linear wave theory and analyze the influence of the seabed’s slope on the solution of the model. The model is constructed from Laplace and Bernoulli equations together with kinematic and dynamic boundary conditions. We used the result of the analytical solution to find the relation between propagation speed, wavelength, and bed slope through the dispersion relation. Because of the difference in fluid dispersive character for each water region, we also determined dispersion relation approximation by modifying the hyperbolic tangent form into hyperbolic sine-cosine and exponential form, then approximated it with Padé approximant. As the final result, exponential form modification with Padé approximant had the best agreement to exact dispersion relation equation then direct hyperbolic tangent form.

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Diagonal Padé approximant of the one-body Green's function: A study on Hubbard rings

Cited by 3 publications

References 15 publications

Perturbation theory in the complex plane: exceptional points and where to find them

Perturbation theory in the complex plane: exceptional points and where to find them

Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them

Modeling a Wave on Mild Sloping Bottom Topography and Its Dispersion Relation Approximation

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