Adaptive time-dependent density-matrix renormalization-group technique for calculating the conductance of strongly correlated nanostructures

A procedure based on the recently developed “adaptive” time-dependent density-matrix-renormalization-group (DMRG) technique is presented to calculate the zero temperature conductance of nanostructures, such as quantum dots (QDs) or molecular conductors, when represented by a small number of active levels. The leads are modeled using noninteracting tight-binding Hamiltonians. The ground state at time zero is calculated at zero bias. Then, a small bias is applied between the two leads, the wave function is DMRG evolved in time, and currents are measured as a function of time. Typically, the current is expected to present periodicities over long times, involving intermediate well-defined plateaus that resemble steady states. The conductance can be obtained from those steady-state-like currents. To test this approach, several cases of interacting and noninteracting systems have been studied. Our results show excellent agreement with exact results in the noninteracting case. More importantly, the technique also reproduces quantitatively well-established results for the conductance and local density of states in both the cases of one and two coupled interacting QDs. The technique also works at finite bias voltages, and it can be extended to include interactions in the leads.

Figure 1

(Color online) Schematic representation of the geometry used in our study. The leads are modeled by tight-binding Hamiltonians. The ground state at time zero is calculated at zero bias. Then, a finite bias $\mathrm{\Delta }V$ is applied between the two leads (without ramping time for simplicity, but this can be changed in future studies) and the resulting current is measured.

Figure 2

(Color online) Exact results for $J\left(t\right)∕\mathrm{\Delta }V$ (in units of $e^2∕h$) vs time (in units of $\hslash$/$t_{\mathrm{leads}}$), for the noninteracting 1QD case obtained with clusters of different lengths $\left(L\right)$ and $\mathrm{\Delta }V=0.001$. (a) $J\left(t\right)∕\mathrm{\Delta }V$ obtained with a large cluster $(L=402)$. $J\left(t\right)∕\mathrm{\Delta }V$ shows clear steady-state plateaus at $\pm 2e^2∕h$. The periodic changes in the current direction are caused by its reflection at the open boundaries of the cluster. (b) $J\left(t\right)∕\mathrm{\Delta }V$ obtained with decreasing $L$. The steady-state plateau is obtained even with $L=32$. The current is quasiperiodic with a period proportional to $L$. The parameters used are $V_{\mathrm{g}}=U=0$ and $t^{\prime }=0.4$.

Figure 3

(Color online) This figure shows the propagation of the current in the cluster after the bias $\mathrm{\Delta }V=0.001$ is applied at $t=0$. The different panels show the current as a function of position $x$ along the chain, at different times $t$. The size is $L=402$ and the dot is at the site 201. The results were obtained exactly, since the Hubbard couplings are zero.

Figure 4

(Color online) DMRG results compared to the exact results for $J\left(t\right)∕\mathrm{\Delta }V$ obtained using different clusters $L$ and number of states $M$, with $\mathrm{\Delta }V=0.001$. (a) $L=96$, (b) $L=64$, (c) $L=32$, and (d) $L=16$. Note that for $L=96$ and 64, $M=200$ shows good qualitative agreement and $M⩾300$ even shows good quantitative agreement with the exact results. For $L=32$ and 16, $M=200$ and 100 already show excellent quantitative agreement with the exact results.

Figure 5

(Color online) (a) DMRG and exact results for $G$ vs $V_{\mathrm{g}}$ in the case of one noninteracting QD, and $t^{\prime }=0.4$ (with $\mathrm{\Delta }V=0.001$). The DMRG results are obtained with $L=64$ and $M=300$. In this case, $G$ is obtained from the value of the steady-state current plateau. The exact results are for infinite leads. The results show a resonant tunneling peak of full width at half maximum (FWHM) equals $4{t^{\prime }}^2$ at $V_{\mathrm{g}}=0$. (b) $J\left(t\right)∕\mathrm{\Delta }V$ for $V_{\mathrm{g}}=\pm 0.3$ and $\mathrm{\Delta }V=0.01$, 0.001. Note that decreasing the bias voltage reduces the asymmetry between positive and negative $V_{\mathrm{g}}$ and gives a better steady-state plateau.

Figure 6

(Color online) (a) DMRG results compared to the exact results in the case of one noninteracting QD for several intermediate and large values of $\mathrm{\Delta }V$, namely, exploring the influence of a finite bias in the calculations. The parameters used are $V_{\mathrm{g}}=U=0$, and $t^{\prime }=0.4$. Both DMRG and exact results are obtained with a cluster $L=32$, using $M=200$ states, for the DMRG results.

Figure 7

(Color online) The results obtained with the Suzuki-Trotter approach and the static Runge-Kutta method compared to the exact results for (a) $\mathrm{\Delta }V=0.001$ and (b) $\mathrm{\Delta }V=0.5$. The parameters used are $V_{\mathrm{g}}=U=0.0$, $t^{\prime }=0.4$, and $L=64$.

Figure 8

(Color online) DMRG and exact results for $G$ vs $V_{\mathrm{g}}$ in the case of two coupled noninteracting QDs. The DMRG results are obtained with $L=64$ and $M=300$ $(\mathrm{\Delta }V=0.001)$. The exact results are for infinite leads. The cases (a) $t^{″}=0.5$ and (b) $t^{″}=0.2$ are investigated, with $t^{\prime }=0.4$ in both cases. The results present the bonding and anti-bonding resonant tunneling peaks at $\pm t^{″}$.

Figure 9

(Color online) DMRG local density-of-states (LDOS) results for (a) 1QD and (b) 2QDs compared to the exact (bulk) results. The parameters used are $V_{\mathrm{g}}=U=0$, $t^{\prime }=0.4$ in both cases and $t^{″}=0.5$ in (b). In (a), a broadening imaginary component $\eta =0.1$ was introduced. In (b), $\eta =0.15$. Smaller values of $\eta$ would reveal the discrete nature of the LDOS obtained with DMRG on a finite $L=64$ system.

Figure 10

(Color online) $J\left(t\right)∕\mathrm{\Delta }V$ in the case of one quantum dot for different cluster lengths $L$. The parameters used are $U=1.0$, $t^{\prime }=0.4$, $\mathrm{\Delta }V=0.005$, and $V_{\mathrm{g}}=-0.5$. As $L$ increases, the conductance approaches the unitarity limit $(2e^2∕h)$ due to the Kondo screening effect.

Figure 11

(Color online) $J\left(t\right)∕\mathrm{\Delta }V$ in the case of one interacting QD for different values of $V_{\mathrm{g}}$, and with $\mathrm{\Delta }V=0.005$. The value of the conductance is obtained by averaging the current over an interval of time, corresponding to the steady state. The solid horizontal lines represent this time interval over which the average of the current is taken, and the value of the average. The parameters used are $U=1.0$, $t^{\prime }=0.4$, $L=128$, and $M=300$.

Figure 12

(Color online) $J\left(t\right)∕\mathrm{\Delta }V$ in the case of one interacting QD for $V_{\mathrm{g}}∕U=-1.5$, $U=1.0$, $t^{\prime }=0.4$, $L=128$, and different values of $M$. Note that as $M$ increases, the oscillations at long times tend to decrease.

Figure 13

(Color online) Conductance $G$ and the dot occupation $⟨n_d⟩$ for one interacting QD. The circles show $G$ obtained by averaging the current over an interval of time corresponding to the steady state, as shown in Fig. 11. The squares show $G$ obtained from $⟨n_d⟩$ using the Friedel sum rule (FSR). $G$ has the shape of the expected Kondo or mixed-valence plateau centered at $V_{\mathrm{g}}=-U∕2$. The feature would be sharper reducing $t^{\prime }$. Results shown are: (a) $U=1.0$, $t^{\prime }=0.4$, and (b) $U=2.0$, $t^{\prime }=0.5$. In both cases, $L=128$ and $M=300$. Note that the DMRG conductance results in (a) show a slightly better agreement with the FSR results. This is expected since the finite-size effects are stronger for larger $U$.

Figure 14

(Color online) Conductance $G(\mathrm{\Delta }V=0.001)$ and charge at the dot $⟨n_d⟩$, for one interacting QD in the case of large $U$. The circles show $G$ obtained from DMRG current procedure outlined in this paper, while the squares show $G$ obtained from $⟨n_d⟩$ using the Friedel sum-rule. The finite-size effects are obvious here, since the results are halfway between the expected Kondo plateau (properly reproduced by the FSR procedure) and the Coulomb blockade peaks. This case is shown as an illustration of important size effects in some limits. The parameters used are $U=4.0$, $t^{\prime }=0.4$, $L=128$, and $M=300$.

Figure 15

(Color online) Finite-size scaling of the conductance $G$ at $V_{\mathrm{g}}=-U∕2$ for the odd-1QD-even (circles) and even-1QD-even clusters (triangles) setups. Note that in both cases $G$ converges to 1 in the bulk limit. However, the odd-1QD-even cluster converges faster, which makes it the most useful for practical calculations. The parameters used are $U=1.0$ and $t^{\prime }=0.4$.

Figure 16

(Color online) Finite-size scaling of $G$ for odd-QD-even clusters using OBC (circles) and DBC (triangles). $d=0.5$ was used. Note that $L=64$ cluster with DBC gives better results than a $L=128$ cluster with OBC. The same parameters are used as in Fig. 15.

Figure 17

(Color online) Conductance $G$ vs gate voltage $V_{\mathrm{g}}$ in the case of one interacting QD ($U=2.0$, $t^{\prime }=0.5$) for different values of the magnetic field $B$ (and $\mathrm{\Delta }V=0.001$). For $B=0$, a Kondo plateau is obtained, centered at $V_{\mathrm{g}}=-U∕2$. As $B$ increases, the Kondo effect is suppressed, and for moderate $B$, two Coulomb blockade peaks are observed at $V_{\mathrm{g}}=-U$ and $V_{\mathrm{g}}=0$, as expected.

Figure 18

(Color online) $J\left(t\right)∕\mathrm{\Delta }V$ for two coupled QD’s at $V_{\mathrm{g}}=-U∕2$ for different values of $t^{″}∕{t^{\prime }}^2$ (and $\mathrm{\Delta }V=0.005$). As in the case of one interacting QD, the conductance is obtained by averaging the steady-state current over the indicated intervals. The parameters used are $U=1.0$, $t^{\prime }=0.5$, $L=127$, and $M=300$.

Figure 19

(Color online) Conductance $G$ as a function of $t^{″}∕{t^{\prime }}^2$, at $V_{\mathrm{g}}=-U∕2$ $(\mathrm{\Delta }V=0.005)$. In this regime, $G$ is determined by the competition between the Kondo correlation of each dot with the neighboring leads and the antiferromagnetic correlation between the two dots. The circles represent our DMRG results obtained with $L=127$ and $M=300$. The solid line is the plot of the functional form obtained by Georges and Meir using SBMFT.[40]