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

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 $\mathcal{O}\left(N^4\right)$ 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 $\mathcal{O}\left(N^2\right)$ 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 $\mathcal{O}\left(N^2{L}^2\right)$ 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.

Figure 1

The linear conductance $g=G/G_Q$ of a QPC as a function of gate voltage, plotted for the cases with and without feedback of $D^{\uparrow \downarrow }$ in an intermediate parameter regime for four equidistant magnetic fields. Note that the difference between the two cases is suppressed with increasing the magnetic field.

Figure 2

Typical QPC and QD barrier shapes, controlled via the parameters, ${\varepsilon }_F, V_g, N^{\prime }$, and, for the QD, $V_s$ and $j_s$. For these plots, both $\mu$ and the barrier top lie were chosen to lie below the center of the bulk band, which we take as reference energy where $\omega =0$. The case of half-filled leads, used for most of our calculations, corresponds to choosing $\mu =0$.

Figure 3

Linear conductance $g$ calculated using the static eCLA for five equidistantly chosen magnetic fields $B$ between 0 and ${\mathrm{\Omega }}_x/2$. (a)–(c) Conductance at fixed $U/\sqrt{{\mathrm{\Omega }}_x\tau }=3.0$ and four values of $L$. (d)–(f) Conductance at fixed $L=5$, for three values of $U/\sqrt{{\mathrm{\Omega }}_x\tau }$.

Figure 4

Convergence behavior of the conductance for different values of $l_x/a$, where $\mathrm{\Delta }{g}_L$ is defined in Eq. (49). The parameters for the $l_x/a=3.8$ data are the same as in Fig. 3. For the larger $l_x$ values the chemical potential was chosen as $\mu =-1.7$ and the parameter $\gamma$ was varied. The inset shows the dependence of $L_C$ on $l_x$.

Figure 5

Conductance and vertex quantities calculated for the two feedback lengths $L=0$ (left column) and $L=5$ (right column) with three different effective interaction strengths $U/\sqrt{{\mathrm{\Omega }}_x\tau }$, at zero magnetic field.

Figure 6

The crossover from a QPC to a QD. (a) The conductance as a function of gate-voltage $V_g$, calculated for several magnetic fields (black solid lines: $B=0,1,2,3\times {10}^{-4}$, black dashed lines: $B=6,9,12\times {10}^{-4}$) with feedback length $L=20$. Colored symbols indicate the conductance values obtained with smaller feedback lengths. (b) and (c) Noninteracting LDOS (color scale) and barrier shape (solid white curve) for the two gate voltages marked by the left and right vertical arrows in (a), respectively. Horizontal white dashed lines indicate the chemical potential $\mu$. (d), (e) The electron density per site $n_j$ again computed for the two gate voltages indicated in (a). Summing $n_j$ over all sites between the two density minima yields $n_{\mathrm{dot}}=1.01$ and 2.98.

Figure 7

(a) Distance dependence of the bare interaction $\tilde{U}(0,\tilde{x})$ between an electron located at the QPC center and one at $\tilde{x}$, plotted on a logarithmic scale, for three values of ${\tilde{l}}_s$. The dashed black line shows the limit of ${\tilde{l}}_s\to \infty$ and the dots on the lowest curve (red) illustrate the chosen discretization points for the case $N=61$. (b) $\tilde{U}(0,\tilde{x})$ (central peak) and $\tilde{U}({\tilde{x}}_s=4.5,\tilde{x})$ (side peak), plotted for ${\tilde{l}}_s=2.15$ on a linear scale for both negative and positive $\tilde{x}$ values.

Figure 8

QPC conductance step shape for three choices of the number of discretization points $N$ (with maximal feedback length $L=N-1$), for two QPCs with different curvatures. We used the following parameters, in absolute units [cf. Eqs. (38) and (61)]. In (a), $\gamma =0.85, {\varepsilon }_F=13.89\mathrm{meV}, {\mathrm{\Omega }}_y=2.35\mathrm{meV}, L_{\mathrm{bar}}=146.11\mathrm{nm}$, and $l_s=46.17\mathrm{nm}$; and in (b), $\gamma =0.85, {\varepsilon }_F=11.00\mathrm{meV}, {\mathrm{\Omega }}_y=2.00\mathrm{meV}, L_{\mathrm{bar}}=158.24\mathrm{nm}$, and $l_s=50.00\mathrm{nm}$. The insets zoom into the range $g\in [0.8,1.05]$ and plot $g$ as a function of $V_g-V_g^{\mathrm{po}}$ to align the pinchoffs. When expressed in terms of the dimensionless parameters of Eq. (65), the parameter choices in (a) and (b) differ only in ${\tilde{\mathrm{\Omega }}}_x$. For example, for the middle $N=61$ curves (green), we obtain for panel (a) $\mathfrak{A}=\{{\tilde{\mathrm{\Omega }}}_x=1.23,{\tilde{\mathrm{\Omega }}}_y=1.91,{\tilde{L}}_{\mathrm{bar}}=6.79,{\tilde{\mathrm{\Omega }}}_y^{\prime \prime }=-0.060,{\tilde{l}}_s=2.15\}$, and for panel (b), $\mathfrak{B}=\{{\tilde{\mathrm{\Omega }}}_x=1.05,{\tilde{\mathrm{\Omega }}}_y=1.91,{\tilde{L}}_{\mathrm{bar}}=6.79,{\tilde{\mathrm{\Omega }}}_y^{\prime \prime }=-0.060,{\tilde{l}}_s=2.15\}$.

Figure 9

QPC conductance curves at fixed $N$, calculated with feedback length $L=N-1$ for several values of the interaction cutoff $L_U$ (solid lines), and with $L=15$ for $L_U=10$ (dashed line). The QPC parameters were chosen as in Fig. 8. Note that while convergence in $L$ is rapid, the conductance becomes independent of the cutoff length only for $L_U>40$. Furthermore, for $L_U\lesssim l_x/a\approx 4.4$, we recover the conductance shape of short-ranged interactions.

Figure 10

(a) and (b) The conductance curves corresponding to the interactions depicted in Fig. 7, for two different QPC mean curvatures ${\tilde{\mathrm{\Omega }}}_x=1.2$ and 1.0, respectively. The arrows at the right (red) ${\tilde{l}}_s=0.86$ and the left (blue) ${\tilde{l}}_s=2.15$ curve in (b) indicate the gate voltages ${\tilde{V}}_g=-1.43$ and ${\tilde{V}}_g=3.73$ at which the density profiles in Figs. 11 and 11 were calculated, respectively.

Figure 11

Density profiles (thin lines) calculated for two fixed parameter choices from Fig. 10, indicated for panels (a) and (b) by the right and left arrows in Fig. 10, respectively. For comparison, the thick lines depict (a vertically rescaled version of) the imaginary part of the interacting single-particle propagator at the chemical potential, ${\mathcal{A}}_{0,\tilde{x}}=-\frac{1}{\pi {\overline{l}}_x}Im{G}_{0,\tilde{x}}^R(\omega =0)$. Horizontal dashed lines indicate where ${\mathcal{A}}_{0,\tilde{x}}=0$. In (b), the distance between the two density maxima (marked by the dashed vertical lines) is $\lambda =3.62{\overline{l}}_x$. This agrees well with two estimates of ${\lambda }_F/2$, either from the distance between the two central zeros of ${\mathcal{A}}_{0,\tilde{x}}$ finding ${\lambda }_F/2=3.82{\overline{l}}_x$ or from the mean density $\overline{n}$ in the center of the QPC (shaded region) finding ${\lambda }_F/2=3.55{\overline{l}}_x$.

Figure 12

Study of two QPCs with different ${\tilde{L}}_{\mathrm{bar}}$, for three choices of ${\tilde{\mathrm{\Omega }}}_x$. The other dimensionless parameters were chosen the same as in $\mathfrak{B}$ [cf. caption of Fig. 8]. (a) and (b) Conductance as function of gate voltage and (c)–(h) density as function of position and gate voltage. While the conductance changes its shape for both QPCs, the shorter one (b) shows stronger features, preeminently a shoulder in the conductance step. In the density, both QPCs show the development of oscillations with approximate wavelength ${\lambda }_F/2$, which is determined by the Green's function as in Fig. 11 and indicated by the distance between the black lines. In the last plots (g) and (h), the density oscillations transition at smaller gate voltages from two to three maxima. The cut along the dashed white line in (f) is precisely the density profile plotted in Fig. 11.

Figure 13

(a)–(c) Barrier shapes (dashed lines) and corresponding noninteracting densities (solid lines) for almost open QPCs with (a) a parabolic barrier top, (b) a flat barrier top with wide flanks, (c) and a flat barrier top with steep flanks. (d)–(f) Density profiles corresponding to these three barrier shapes, plotted as functions of position and gate voltage. In these plots, ${\lambda }_F/2$ is again indicated by the distance between the black lines. The flat barrier top with steep flanks of panel (c) yields pronounced Friedel oscillations in the density profile shown in (f), which resemble the density oscillations caused by the long-range interaction in the open regime of the QPCs of Figs. 12–12. This suggests that for the latter, the renormalized barriers have a rather flat tops with steep flanks.

Figure 14

Interacting LDOS in the static approximation [Eq. (67)], shown as function of position and energy (color scale), for three different values of the screening length ${\tilde{l}}_s$. Solid white lines show the bare potential $V_j$ and dashed white lines $V_j+{\mathrm{\Sigma }}_{jj}$, as functions of position. The physical parameters used for this plot correspond to those of Fig. 10, with the gate voltage was set to ${\tilde{V}}_g=-1.91$ in (a), ${\tilde{V}}_g=-1.43$ in (b), and ${\tilde{V}}_g=3.73$ in (c). (The latter two correspond to the red and blue markers in Fig. 10.) The shape of the band bottom reflects that of the renormalized barrier. (The fact that the renormalized barrier top lies below the bare barrier top in (c) is due to the artifact of static fRG discussed in Sec. 4b.)