Nonequilibrium pseudogap Anderson impurity model: A master equation tensor network approach

We study equilibrium and nonequilibrium properties of the single-impurity Anderson model with a power-law pseudogap in the density of states. In equilibrium, the model is known to display a quantum phase transition from a generalized Kondo to a local moment phase. In the present work, we focus on the extension of these phases beyond equilibrium, i.e., under the influence of a bias voltage. Within the auxiliary master equation approach combined with a scheme based on matrix product states (MPS) we are able to directly address the current-carrying steady state. Starting with the equilibrium situation, we first corroborate our results by comparing to a direct numerical evaluation of ground-state spectral properties of the system by MPS. Here, a scheme to locate the phase boundary by extrapolating the power-law exponent of the self energy produces a very good agreement with previous results obtained by the numerical renormalization group. Our nonequilibrium study as a function of the applied bias voltage is then carried out for two points on either side of the phase boundary. In the Kondo regime the resonance in the spectral function is split as a function of the increasing bias voltage. The local moment regime, instead, displays a dip in the spectrum near the position of the chemical potentials. Similar features are observed in the corresponding self energies. The Kondo split peaks approximately obey a power-law behavior as a function of frequency whose exponents depend only slightly on voltage. Finally, the differential conductance in the Kondo regime shows a peculiar maximum at finite voltages, whose height, however, is below the accuracy level.

Figure 1

Construction of effective sites for the MPS time evolution. The impurity sites are displayed as red circles, the full and empty bath sites as blue and white ones. As discussed in the text, by “full” and “empty” we mean sites for which ${\mathit{\Gamma }}^{\left(1\right)}=0$ or ${\mathit{\Gamma }}^{\left(2\right)}=0$, respectively, for details, see Appendix pp1-s1. Each site is labeled with an index and its spin and tilde degrees of freedom. The upper panel of this figure shows the sites and their couplings occurring in the Lindblad equation in the augmented state space. Here the upper (lower) part of this ladder structure is formed by nontilde (tilde) sites. Lines connecting these two sets of sites represent $\mathrm{\Gamma }$ terms, while lines within the same set are hoppings. The central panel shows the effective sites used in [56] that result from combining nontilde and tilde sites with the same index. Finally, the lower panel shows the effective sites used in this work that result from the combined sites by separating the spin degrees of freedom. The advantage of this representation is that the local Hilbert space has a dimension of 4, instead of 16 as in our previous work. On the other hand, it introduces two long-range hopping terms.

Figure 2

Single step in the MPS time evolution of the (PSG) SIAM with separated spin degrees of freedom. The impurity sites are represented as red circles, the full and empty bath sites as blue and white ones. The same coloring also classifies the time evolution gates that are represented as boxes. A time evolution step $\mathrm{\Delta }t$ consists of five layers, labeled “odd,” “even,” and “swap.” In each layer, a site $i\sigma$, with index $i$ and spin $\sigma$, is touched only by one gate. In the swap layer, swap gates displayed as crossing time lines are employed to cope with the long-range couplings between the empty bath sites and the impurity sites.

Figure 3

Equilibrium ($\varphi =0$) fit results. (a) Retarded hybridization function $-\text{Im}{\mathrm{\Delta }}^R\left(\omega \right)$ in units of the hybridization strength $\mathrm{\Gamma }$ and (b) distribution function $f$ determined from $\text{Im}{\mathrm{\Delta }}^R$ and $\text{Im}{\mathrm{\Delta }}^K$ via Eq. (7). The power-law exponent $r^{\prime }$ is obtained by fitting the AMEA hybridization function with Eq. (27) on the interval $\mathrm{\Omega }$. The same procedure applied to the HMPS result yields quite the same exponent (up to a deviation of $\approx 0.01$). ${\left|\omega \right|}^r$ is plotted for comparison, see Eq. (27). These curves are hardly distinguishable (black vs. red dots).

Figure 4

Equilibrium ($\varphi =0$) (a) spectral function $A\left(\omega \right)$ and (b) retarded self energy $-\text{Im}{\mathrm{\Sigma }}^R\left(\omega \right)$ in the GK phase. The power-law exponents $s^{\prime }$ and ${\kappa }^{\prime }$ are obtained by fitting the AMEA results with Eqs. (29) and (30) on the interval $\mathrm{\Omega }$. The same procedure applied to the HMPS results yields quite the same exponents (up to a deviation of $\approx 0.01$). A power-law $\propto {\left|\omega \right|}^{-s}$ is plotted for comparison, see Eq. (29). The error shadings, hardly to be seen in this figure, are estimates of the PH symmetry errors, see Appendix pp2.

Figure 5

Determination of the phase boundary by linear extrapolation of the power-law exponent ${\kappa }^{\prime }$ of the HMPS self energy in the GK phase. First, (a) ${\kappa }^{\prime }$ is extrapolated to vanishing values of the broadening $\eta$ to extract ${\kappa }_0^{\prime }={\kappa }^{\prime }(\eta \to 0)$ for various values of the interaction strength $U$. Second, (b) the critical interaction strength is determined from a second extrapolation, $U_c=U({\kappa }_0^{\prime }\to r)$.

Figure 6

Phase diagram adapted from [24] (with kind permission) displaying different NRG results. On top of this we present our HMPS results for the phase boundary obtained via the extrapolation scheme discussed in the text. We also indicate the two points we consider in AMEA, i.e. $r=0.25$, $U=6$ and $\mathrm{\Gamma }=0.25$ and $\mathrm{\Gamma }=1$. If $U$ is much smaller than the bandwidth, the phase boundary for a given $r$ is expected to depend on $\mathrm{\Gamma }{U}^{r-1}$ only [24].

Figure 7

Nonequilibrium ($\varphi >0$) quantities in the Kondo regime, (a) spectral function, (b) retarded self energy, and (c) differential conductance. The solid lines are the physical quantities and the dotted lines the auxiliary ones, see Sec. 2c4. Notice that the two curves are often indistinguishable. The error shadings and error bars are estimates based on symmetry considerations, see Appendix pp2.

Figure 8

Nonequilibrium ($\varphi >0$) quantities in the LM regime. Conventions are as in Fig. 7.

Figure 9

Nonequilibrium ($\varphi >0$) effective power-law exponents as a function of the bias voltage $\varphi$. The three pairs of exponents are extracted from a fit of the auxiliary retarded hybridization function ($r^{\prime },r^{\prime \prime }$), the spectral function ($s^{\prime },s^{\prime \prime }$) and the retarded self energy (${\kappa }^{\prime },{\kappa }^{\prime \prime }$) with Eqs. (31) to (33). The single and double primes correspond to different fitting intervals $\mathrm{\Omega }\left(\varphi \right)$ and ${\mathrm{\Omega }}_1\left(\varphi \right)$, see text.

Figure 10

(a) Impurity (red sphere) coupled to a partially filled bath (semicircle) at chemical potential $\mu$. (b) The same hybridization function can be obtained by coupling the impurity to a full and empty bath with appropriate DOS.

Figure 11

(a) Impurity (red sphere) coupled to a partially filled left bath and a partially filled right bath (semicircles), whose chemical potentials differ by $2\mathrm{\Delta }\varepsilon$. (b) The same situation with two full (blue) and two empty (white) baths.

Figure 12

Auxiliary spectral functions $A\left(\omega \right)$ obtained from different raw data for symmetry considerations and error estimation, see text.