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Weidinger, Lukas; Bauer, Florian; Delft, Jan von (2017): Functional renormalization group approach for inhomogeneous one-dimensional Fermi systems with finite-ranged interactions. In: Physical Review B, Vol. 95, No. 3, 35122
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Abstract

We introduce an equilibrium formulation of the functional renormalization group (fRG) for inhomogeneous systems capable of dealing with spatially finite-ranged interactions. In the general third-order truncated form of fRG, the dependence of the two-particle vertex is described by O(N-4) independent variables, where N is the dimension of the single-particle system. In a previous paper [Bauer et al., Phys. Rev. B 89, 045128 (2014)], the so-called coupled-ladder approximation (CLA) was introduced and shown to admit a consistent treatment for models with a purely onsite interaction, reducing the vertex to O(N-2) independent variables. In this work, we introduce an extended version of this scheme, called the extended coupled ladder approximation (eCLA), which includes a spatially extended feedback between the individual channels, measured by a feedback length L, using O((NL2)-L-2) independent variables for the vertex. We apply the eCLA in a static approximation and at zero temperature to three types of one-dimensional model systems, focusing on obtaining the linear response conductance. First, we study a model of a quantum point contact (QPC) with a parabolic barrier top and on-site interactions. In our setup, where the characteristic length l(x) of the QPC ranges between approximately 4-10 sites, eCLA achieves convergence once L becomes comparable to l(x). It also turns out that the additional feedback stabilizes the fRG flow. This enables us, second, to study the geometric crossover between a QPC and a quantum dot, again for a one-dimensional model with on-site interactions. Third, the enlarged feedback also enables the treatment of a finite-ranged interaction extending over up to L sites. Using a simple estimate for the form of such a finite-ranged interaction in a QPC with a parabolic barrier top, we study its effects on the conductance and the density. We find that for low densities and sufficiently large interaction ranges the conductance develops additional features, and the corresponding density shows some fluctuations that can be interpreted as Friedel oscillations arising from a renormalized barrier shape with a rather flat top and steep flanks.