Functional renormalization group approach for inhomogeneous interacting Fermi systems

Abstract

The functional renormalization group (fRG) approach has the property that, in general, the flow equation for the two-particle vertex generates O(N 4 ) independent variables, where N is the number of interacting states (e.g. sites of a real-space discretization). In order to include the flow equation for the two-particle vertex one needs to make further approximations if N becomes too large. We present such an approximation scheme, called the coupled-ladder approximation, for the special case of an onsite interaction. Like the generic third-order-truncated fRG, the coupled-ladder approximation is exact to second order and is closely related to a simultaneous treatment of the random phase approximation in all channels, i.e. summing up parquet-type diagrams. The scheme is applied to a one-dimensional model describing a quantum point contact.

Microscopic origin of the ‘0.7-anomaly’ in quantum point contacts

“…These features are reproduced by our calculations using fRG and SOPT, and their origin is explained in detail section 3.3.6 (see also Bauer et al, 2014) and chapter 5 (see also Bauer et al, 2013), using a simple density-density interaction argument: at finite magnetic field, spin up electrons flow into the QPC, such that spin down electrons not only have to overcome the Zeeman energy but also the Coulomb repulsion of the additional spin up electrons. Since the inflow of spin up electrons depends on the DOS, this effect is strongest in the sub-open regime.…”

“…In most cases the only practical approximation schemes are fRG1 and fRG2, and for models involving a large number N of interacting states, such as the chain model considered in this work, fRG2 runs into problems, since the two-particle vertex is represented by O(N 4 ) independent variables. This complication can be resolved by using further approximations for models with a local interaction, as explained in section 3.3.6 (see also Bauer et al, 2014). …”

“…Therefore we present a compact but complete and generic derivation of the fRG flow equations. At the end of the chapter we introduce the approximation scheme developed to treat interacting inhomogeneous one-dimensional models, in the form of a publication (see also Bauer et al, 2014).…”

Relation between the 0.7 anomaly and the Kondo effect: Geometric crossover between a quantum point contact and a Kondo quantum dot

“…We note that it is not straightforward to distinguish To theoretically study the effect of interactions on the properties of the CCR at zero temperature, we used fRG [30][31][32][33] , a renormalization-group-enhanced perturbative expansion in the interaction. We used it to calculate the linear conductance g of the CCR, and three local quantities, the occupation n j , magnetization m j and spin susceptibility χ j of site j, defined, respectively, as…”

“…14, but now tune the shape of the potential barrier in such a way that it smoothly crosses over between a single barrier, representing a QPC, and a double barrier, representing a KQD. We use the functional renormalization group (fRG) [30][31][32][33] to calculate how transport and thermodynamic properties at T = 0 change during this crossover. This allows us to track the extent to which features characteristic for Kondo correlations do or do not survive in the QPC regime.…”