A Gentle Introduction to the Functional Renormalization Group: The Kondo Effect in Quantum Dots

Andergassen, Sabine; Enss, Tilman; Karrasch, Christoph; Meden, V.

doi:10.1007/978-1-4020-8512-3_1

Cited by 6 publications

(5 citation statements)

References 29 publications

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“…It was first applied to quantum dot systems, where electronic correlations lead to interesting strong-coupling effects, roughly five years ago. The employed approximation scheme, which can be viewed as a kind of RG enhanced Hartree Fock theory not suffering from typical mean-field artifacts, succeeds in accurately describing linear transport properties (such as the conductance) of various single-as well as multi-level spinful and spin-polarised quantum dot geometries at zero temperature and even captures aspects of Kondo physics [1][2][3][4][5].…”

Section: Short Summarymentioning

confidence: 99%

Functional renormalization group study of the interacting resonant level model in and out of equilibrium

et al. 2010

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We investigate equilibrium and steady-state non-equilibrium transport properties of a spinless resonant level locally coupled to two conduction bands of width ∼ Γ via a Coulomb interaction U and a hybridization t ′ . In order to study the effects of finite bias voltages beyond linear response, a generalization of the functional renormalization group to Keldysh frequency space is employed. Being mostly unexplored in the context of quantum impurity systems out of equilibrium, we benchmark this method against recently-published time-dependent density matrix renormalization group data. We thoroughly investigate the scaling limit Γ → ∞ characterized by the appearance of power laws. Most importantly, at the particle-hole symmetric point the steady-state current decays like J ∼ V −α J as a function of the bias voltage V ≫ t ′ , with an exponent αJ (U ) that we calculate to leading order in the Coulomb interaction strength. In contrast, we do not observe a pure power-law (but more complex) current-voltage-relation if the energy ǫ of the resonant level is pinned close to either one of the chemical potentials ±V /2.

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Section: Short Summarymentioning

confidence: 99%

Functional renormalization group study of the interacting resonant level model in and out of equilibrium

et al. 2010

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“…The issue of obtaining an exponential scale but not the correct exponent for the functional dependence on U is common to various approximate methods (for example variational wave functions where the issue was cured by introducing an extended Ansatz by Schönhammer 72 , saddle-point approximations of a functional integral approach 73 or FRG 74 ). A faint analogy may be drawn here to Gutzwiller approximation, where an exponential energy scale in U arises by a renormalized hybridization parameter V 75 , which is also the case for VCA Ω .…”

Section: E Low Energy Properties Kondo Temperaturementioning

confidence: 99%

Variational cluster approach to the single-impurity Anderson model

Nuss

Arrigoni²,

Aichhorn³

et al. 2012

Phys. Rev. B

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We study the single impurity Anderson model by means of cluster perturbation theory and the variational cluster approach (VCA). An expression for the VCA grand potential for a system in a non interacting bath is presented. Results for the single-particle dynamics in different parameter regimes are shown to be in good agreement with established renormalization group results. We aim at a broad and comprehensive overview of the capabilities and shortcomings of the methods. We address the question to what extent the elusive low energy properties of the model are reproducible within the framework of VCA. These are furthermore benchmarked against continuous time quantum Monte Carlo calculations. We also discuss results obtained by an alternative, i.e. self consistent formulation of VCA, which was introduced recently in the context of non equilibrium systems.

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“…The functional renormalization group (FRG) is a versatile tool to treat many-body systems with diverse energy scales and competing ordering tendencies [1][2][3]. In this particular flavor of the RG concept, a flow parameter Λ is introduced in such a fashion that at an initial Λ i the system can be solved exactly.…”

Section: Introductionmentioning

confidence: 99%

Detecting phases in one-dimensional many-fermion systems with the functional renormalization group

Markhof¹,

Sbierski²,

Meden³

et al. 2018

Phys. Rev. B

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The functional renormalization group (FRG) has been used widely to investigate phase diagrams, in particular the one of the two-dimensional Hubbard model. So far, the study of one-dimensional models has not attracted as much attention. We use the FRG to investigate the phases of a onedimensional spinless tight-binding chain with nearest and next-nearest neighbor interactions at half filling. The phase diagram of this model has already been established with other methods, and phase transitions from a metallic phase to ordered phases take place at intermediate to strong interactions. The model is thus well suited to analyze the potential and the limitations of the FRG in this regime of interactions. We employ flow equations that are exact up to second order in the interaction, which implies that we take into account the frequency dependence of the two-particle vertex as well as the feedback of the dynamic self-energy. For intermediate nearest neighbor interactions, our scheme captures the phase transition from a metallic phase to a charge density wave with alternating occupation. The critical interaction, at which this transition occurs, is underestimated due to our approximations. Similarly, for intermediate next-nearest neighbor interactions, we observe a transition to a charge density wave with occupation pattern ..00110011... We show that taking into account a feedback of the two-particle vertex in the flow equation is essential for the detection of those phases. arXiv:1803.00272v2 [cond-mat.str-el]

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A Gentle Introduction to the Functional Renormalization Group: The Kondo Effect in Quantum Dots

Cited by 6 publications

References 29 publications

Functional renormalization group study of the interacting resonant level model in and out of equilibrium

Functional renormalization group study of the interacting resonant level model in and out of equilibrium

Variational cluster approach to the single-impurity Anderson model

Detecting phases in one-dimensional many-fermion systems with the functional renormalization group

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