Second-order functional renormalization group approach to one-dimensional systems in real and momentum space

We devise a functional renormalization group treatment for a chain of interacting spinless fermions which is correct up to second order in interaction strength. We treat both inhomogeneous systems in real space as well as the translationally invariant case in a $k$-space formalism. The strengths and shortcomings of the different schemes as well as technical details of their implementation are discussed. We use the method to study two proof-of-principle problems in the realm of Luttinger liquid physics, namely, reflection at interfaces and power laws in the occupation number as a function of crystal momentum.

Figure 1

Top: Conductance $g$ across an interacting region of length $L=120$ of a spinless fermion model [Eq. (1) with $t_j=t$ and $U_j=U]$ connected to noninteracting leads characterized by hoppings $t_L$ as simulated with fRG-2. Various $U$, increasing from left to right, are depicted. Bottom: The conductance maxima for different $U$ occur at values $t_{L,0}^{\mathrm{fRG}-2}$ (dots) that nicely agree with the analytical predictions based on Luttinger liquid theory from Eq. (27) (solid line). Analogous results from fRG-1 are shown as crosses.

Figure 2

Top: fRG results for the occupation numbers $n_k$ of a translationally invariant system in the vicinity of the Fermi momentum for $U/t=\pm 1$. The solid lines show DMRG results from Ref. [33], and the dash-dotted lines denote the results of second-order perturbation theory (which coincide for $U/t=\pm 1$). Bottom: To check if the fRG captures the power-law nature of $n_k$, we plot a log-log-derivative of the data from the top panel with asymptotic values of the Bethe ansatz exponent from Luttinger liquid theory denoted by horizontal dashed lines that are also approached by the fRG data.