Functional renormalization group approach for inhomogeneous one-dimensional Fermi systems with finite-ranged interactions

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(N 2 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, focussing on obtaining the linear response conductance: First, we study a model of a quantum point contact (QPC) with a parabolic barrier top and onsite interactions. In our setup, where the characteristic length lx of the QPC ranges between approximately 4-10 sites, eCLA achieves convergence once L becomes comparable to lx. 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 onsite 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.

Functional renormalization group treatment of the 0.7 analog in quantum point contacts

“…This artefact was also found to varying extent in our previous fRG work on QPCs [1,4,9,10,19] and is an artefact of our method, presumably our truncation scheme. is consistent in the sense that we also already observed it in our static Matsubara implementation of the eCLA [1]. Together with the other inconsistencies, namely the violation of the Ward identity (37) and the associated issue that the two-particle contribution to the conductance is negative unless the Ward-correction (38) is used, this implies that in order to obtain quantitatively reliable results for the conductance one will have to go beyond the channel decomposition (10), and in general also beyond second-order truncated fRG.…”

“…where c iσ annihilates an electron at site i ∈ Z with spin σ and n iσ = c † iσ c iσ is the number operator. Instead of a quadratic onsite potential as used in [1], we use a quadratic modulation in the hopping, τ i = τ − ∆τ i , to model the QPC barrier. This approach was also used in [9].…”

“…In a previous work [1], we have devised an extended Coupled-Ladder Approximation (eCLA), an approximation scheme within the second-order truncated functional Renormalization Group (fRG) approach. The eCLA is capable of a controlled incorporation of the spatial extent of the one-particle irreducible two-particle vertex (hereafter simply called "vertex") into a channel-decomposed [2][3][4] fRG flow.…”

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

“…[29] called this flavor of the FRG "coupled ladder approximation" and in Ref. [19] "extended coupled ladder approximation" (eCLA). They were mostly interested in the transport through quantum point contacts, calculating the conductance and the susceptibility, with special focus on the so-called "0.7anomaly" [32].…”

Approximating the frequency dependence of the effective interaction in the functional renormalization group for many-fermion systems

“…Note that this approximation is exact in second order perturbation theory for the Hubbard interaction where the contributions only depend on either s, t or u. In fact, working with channel couplings P Λ , C Λ and D Λ that only depend on one frequency was shown to be a reasonable approximation in zero-dimensional 3 and one-dimensional models [14][15][16] . In two dimensions, a one-frequency parametrization (with adaptive boson-fermion vertices) was used by Husemann et al 6 , but discovered intriguing divergences at non-zero Matsubara frequencies in the density channel.…”