Finite-temperature linear conductance from the Matsubara Green’s function without analytic continuation to the real axis

We illustrate how to calculate the finite-temperature linear-response conductance of quantum impurity models from the Matsubara Green’s function. A continued fraction expansion of the Fermi distribution is employed which was recently introduced by Ozaki [Phys. Rev. B 75, 035123 (2007)] and converges much faster than the usual Matsubara representation. We give a simplified derivation of Ozaki’s idea using concepts from condensed matter theory and present results for the rate of convergence. In case that the Green’s function of some model of interest is only known numerically, interpolating between Matsubara frequencies is much more stable than carrying out an analytic continuation to the real axis. We demonstrate this explicitly by considering an infinite tight-binding chain with a single site impurity as an exactly solvable test system, showing that it is advantageous to calculate transport properties directly on the imaginary axis. The formalism is applied to the single impurity Anderson model and the linear conductance at finite temperatures is calculated reliably at small to intermediate Coulomb interactions by virtue of the Matsubara functional renormalization group. Thus, this quantum many-body method combined with the continued fraction expansion of the Fermi function constitutes a promising tool to address more complex quantum dot geometries at finite temperatures.

Figure 1

(Color online) Approximation $f_M\left(x\right)$ to the Fermi function obtained from the continued fraction representation of Eq. (12) truncated at different $N=2M$. Already for $M=3$, the agreement with the exact result $f\left(x\right)=1/\left[\text{exp}\left(x\right)+1\right]$ is excellent for not too large arguments $\left|x\right|\lesssim 10$. Likewise, the $M=6$—curve nicely approximates $f\left(x\right)$ up to $\left|x\right|\lesssim 60$. For larger arguments, all $f_M\left(x\right)$ tend to 1/2.

Figure 2

(Color online) Convergence of the continued fraction expansion of the Fermi function to $f_M=1$ for very large negative arguments $x$ as a function of the “chain length” $N=2M$. Inset: by rescaling with $1/n$, the four curves collapse, indicating that the truncation $M$ has to increase only as $\sqrt{\left|x\right|}$ in order to obtain the same convergence.

Figure 3

(Color online) Derivative of the Matsubara Green’s function associated with an infinite tight-binding chain with nearest-neighbor hopping $t$ and a single site impurity of strength $ϵ$. Solid lines show the exact result of Eq. (18), symbols data obtained from computing a Padé approximation to $\mathcal{G}\left(i{\omega }_n\right)$ at different temperatures [and thus different physical Matsubara frequencies ${\omega }_n=\left(2n+1\right)\pi T$] (Refs. 16, 17). The curves at $ϵ/t=0.1$ and $ϵ/t=4$ were both multiplied by a factor of 2 for clarity.

Figure 4

(Color online) Tight-binding chain but now comparing the exact spectral function of Eq. (20) with the one obtained from Padé approximation. The position and spectral weight of the bound state is indicated by an arrow. Upper and lower panels show results computed with different discretizations of the imaginary frequency axis (Ref. 16).

Figure 5

(Color online) Linear-response conductance [normalized to the unitary value $G_0=1/\left(2\pi \right)$] of the noninteracting single impurity Anderson model as a function of the level position $ϵ$ and at different temperatures $T$. Solid lines show the exact result calculated from the real axis [Eq. (22)], symbols were obtained from the imaginary axis using the continued fraction representation of the Fermi function [Eq. (23)]. The expansion was truncated with rather small values of $M$ but agrees nicely with the exact data nevertheless.

Figure 6

(Color online) Linear conductance of the single impurity Anderson model at finite Coulomb interactions and different temperatures $T=0$, $T/\Gamma =0.1$, $T/\Gamma =0.2$, and $T/\Gamma =0.4$ (from top to bottom at $ϵ=0$). Solid lines show numerically exact data obtained from the NRG framework (Ref. 20), symbols functional RG results computed using the Matsubara FRG scheme outlined in Ref. 4 as well as the continued fraction expansion of the Fermi function which underlies Eq. (23). Inset to (a): linear conductance as a function of temperature at particle-hole symmetry. Symbols denote FRG data at $U/\Gamma =2$, the solid line a quadratic fit to the latter. The dashed line displays the noninteracting curve.