Imaginary-time formulation of steady-state nonequilibrium in quantum dot models

We examine the recently proposed imaginary-time formulation for strongly correlated steady-state nonequilibrium for its range of validity and discuss significant improvements in the analytic continuation of the Matsubara voltage as well as the fermionic Matsubara frequency. The discretization error in the conventional Hirsch-Fye algorithm has been compensated in the Fourier transformation with reliable small frequency behavior of self-energy. Here we give detailed discussions for generalized spectral representation ansatz by including high-order vertex corrections and its numerical analytic continuation procedures. The differential conductance calculations agree accurately with available data from other nonequilibrium transport theories. It is verified that, at finite source-drain voltage, the Kondo resonance is destroyed at bias comparable to the Kondo temperature. Calculated scaling coefficients of differential conductance for zero-bias anomaly fall within the range of experimental estimates.

Figure 1

(a) Source and drain reservoirs for electron and a quantum dot level. Chemical potentials of the reservoirs are the same in equilibrium. (b) Voltage bias $\Phi$ in steady-state nonequilibrium divided between chemical potentials of the reservoirs. The quantum dot energy level can be arbitrarily positioned with respect to the chemical potentials.

Figure 2

(a) Keldysh contour for greater Green’s function with the first-order scattering (marked by X) happening between time $t=0$ and $t$ [corresponding to the case (i) considered in the text] (b) Time ordering of case (ii). (c) Case (iii).

Figure 3

(a) Interaction $\widehat{V}$ mapping a state $|n⟩$ to $|m⟩$. (b) Initial and final states without changes in the $Y_0$ number of scattering states. (c) Initial and final states with different $Y_0$ numbers. (d) Contraction with the $d$ creation and annihilation operators. Each line represents the contraction of scattering state basis ${⟨{\psi }_{\alpha k\sigma }^{†}{\psi }_{\alpha k\sigma }⟩}_0$ and contributes the factor $t_{\alpha }^2{\left|g\left({ϵ}_{\alpha k}\right)\right|}^2$. Two diagrams are with $|n⟩$ and $|m⟩$ states interchanged.

Figure 4

Nonlinear Transport through (a) single QD, (b) side-coupled QDs, and (c) parallel QDs can be studied within the imaginary-time formalism. (d) Green’s functions in serially coupled QDs may not be correctly continued to lesser Green’s functions.

Figure 5

Imaginary-time electron self-energy at $U=10$ and $\beta =24$ for Matsubara voltages ${\phi }_m$ with $m=0\left(\text{filled}\text{circle}\right),\dots ,5\left(\text{open}\text{circle}\right)$. (a) Conventional Hirsch-Fye algorithm without the discretization correction shows spurious structures at small ${\omega }_n$ and finite ${\phi }_m$. (b) With smaller discretization $\Delta \tau =1/10$, the spurious structures become weaker. (c) The discretization correction produced smooth self-energy at small ${\omega }_n$. (d) At higher bias $\Phi =1$, the curvature at high ${\phi }_m$ becomes weaker, suggesting reduced correlation effects. The unit of energy is given by the noninteracting level width of QD, $\Gamma =1$.

Figure 6

Self-energy diagram of second-order perturbation in the Coulomb parameter $U$ of the Anderson model.

Figure 7

(Color online) (a) Comparison of differential conductance in weakly interacting limit at $U/\Gamma =3$ from the imaginary-time QMC (ITQMC) at $\beta =25$, functional renormalization group [fRG (Refs. 14, 15, 17) data taken from Ref. 17], iterative summation of path integral [ISPI (Refs. 16, 17)], time-dependent density-matrix renormalization group [tDMRG (Ref. 11)]. (b) Conductance in the intermediate coupling regime at $U/\Gamma =5$. Curves are from ITQMC $\left(\beta =24\right)$ and the time-dependent numerical renormalization group [tNRG (Ref. 9) at $\beta =25$].

Figure 8

(a) Spectral functions for the ansatz Eqs. (52, 53) at $U=10$, $\beta =36$, and $\Phi =0.5$. Dashed lines are for negative $\gamma$’s, e.g., ${\sigma }_{-1}\left(\omega \right)$, ${\sigma }_{-3}\left(\omega \right)$, etc. As expected, ${\sigma }_{\gamma }\left(ϵ\right)={\sigma }_{-\gamma }\left(-ϵ\right)$ is satisfied in the particle-hole symmetric limit. Short vertical lines indicate $\omega =\Phi /2$ (single) and $\omega =3\Phi /2$ (double line). (b) Intralead single-particle excitations dressed by particle-hole excitations. They contribute to ${\sigma }_1\left(\omega \right)$ at excitation energy of $\Phi /2$. (c) Interlead excitations for ${\sigma }_3\left(\omega \right)$ at excitation energy of $3\Phi /2$.

Figure 9

(a) Spectral functions of one-particle Green’s function at $U=10$, $\beta =36$, and bias voltage $\Phi =0,0.2,0.6,1.0,1.6,2.0$. The Kondo resonance becomes quenched quickly as the bias $\Phi$ is applied. The spectral functions are shifted by 0.05 for clarity. Short vertical lines mark spectral features at $\omega =\Phi /2$. Inset: spectral function for the whole frequency range. (b) Spectral functions at $U=14$, $\beta =48$. Double vertical lines denote the spectral contributions at $\omega =3\Phi /2$. At larger $U$, the Kondo peaks get quenched faster to lower intensity.

Figure 10

(Color online) Conductance for $U=10$ at inverse temperatures $\beta =16$, 24, 36, and 60. (a) Conductance curves calculated using equilibrium spectral function are shown with dashed lines. The narrower zero-bias anomaly width (about $50–60\mathrm{\%}$ from the equilibrium Kondo scale) indicates reduction in the Kondo effect at finite bias. (b) Conductance without the Padé correction. The difference is insignificant.

Figure 11

(a) Fourth-order vertex correction to the self-energy with the imaginary-frequency labels. (b) The same diagram in the real-time theory. Numerical labels denote scattering states, i.e., $1\equiv \left({\alpha }_1,k_1,{\sigma }_1\right)$. (c) Time ordering for the greater Green’s function with the interaction times $\left(s_1,s_2\right)$ with $s_1$ extending to $t=T$ on the upper Keldysh branch. (d) The same diagram with $s_1$ on the lower branch.