A Functional Generalization of the Field-Theoretical Renormalization Group Approach for the Single-Impurity Anderson Model

Abstract

We apply a functional implementation of the field-theoretical renormalization group (RG) method up to two loops to the single-impurity Anderson model. To achieve this, we follow a RG strategy similar to that proposed by Vojta et al. [Phys. Rev. Lett. 85, 4940 (2000)], which consists of defining a soft ultraviolet regulator in the space of Matsubara frequencies for the renormalized Green's function. Then we proceed to derive analytically and solve numerically integro-differential flow equations for the effective couplings and the quasiparticle weight of the present model, which fully treat the interplay of particle-particle and particle-hole parquet diagrams and the effect of the two-loop self-energy feedback into them. We show that our results correctly reproduce accurate numerical renormalization group data for weak to slightly moderate interactions. These results are in excellent agreement with other functional Wilsonian RG works available in the literature. Since the field-theoretical RG method turns out to be easier to implement at higher loops than the Wilsonian approach, higher-order calculations within the present approach could improve further the results for this model at stronger couplings. We argue that the present RG scheme could thus offer a possible alternative to other functional RG methods to describe electronic correlations within this model.

Effective models for Anderson impurity and Kondo problems from continuous unitary transformations

“…The functional RG approach yielded good results up to intermediate interactions for the Anderson impurity model 12 . Many other studies [13][14][15][16][17] improved the functional RG approach recently but did not capture the strong coupling regime. Only in 2013, Streib and co-workers succeeded 18 exploiting a magnetic field as regulatory cutoff and conserved Ward identities similar to a renormalized perturbation theory developed by Hewson and his co-workers [19][20][21] .…”

“…The functional RG approach was applied to the Anderson impurity model 12 yielding good results for small and intermediate interactions, but failing to reproduce the exponentially small Kondo energy scale in the strong coupling regime. Subsequently, a series of papers [13][14][15][16][17] tried different variants of the functional RG approach to reproduce this small energy scale. While the successes in the regime of small to intermediate couplings were very interesting, the strong coupling regime eluded a description by functional RG.…”

Functional renormalization group approach to the singlet-triplet transition in quantum dots

“…[15], which, despite its simplicity, is accurate not just at weak coupling but also at moderate and even at rather strong coupling. The advantage of this approach, which we refer to as the static approximation since the frequency dependence of all vertex functions is neglected, is that it is considerably faster than NRG, and also simpler than other alternative approaches such as Fluctuation Exchange Approximation (FLEX) [16] or field theoretical RG approaches [17] and can be easily applied also to multidot geometries. [15] The approach in Ref.…”

Solution of the Anderson impurity model via the functional renormalization group

“…In this Rapid Communication we show that this can be achieved by means of the functional renormalization group (FRG) method. [7][8][9] In the past few years, many authors have studied the AIM using different versions of the FRG, [10][11][12][13][14][15][16][17] but failed in reproducing the correct strong-coupling behavior of the AIM. We show that this problem can be solved using a simple truncation of the FRG hierarchy involving only frequency-independent interaction vertices which are fixed by Ward identities (WIs), provided that the transverse spin fluctuations are properly bosonized and that the corresponding bosonic self-energy, obtained from a skeleton equation, fulfills the WI.…”