Density-matrix renormalization group method for the conductance of one-dimensional correlated systems using the Kubo formula

We improve the density-matrix renormalization group (DMRG) evaluation of the Kubo formula for the zero-temperature linear conductance of one-dimensional correlated systems. The dynamical DMRG is used to compute the linear response of a finite system to an applied ac source-drain voltage; then the low-frequency finite-system response is extrapolated to the thermodynamic limit to obtain the dc conductance of an infinite system. The method is demonstrated on the one-dimensional spinless fermion model at half filling. Our method is able to replicate several predictions of the Luttinger liquid theory such as the renormalization of the conductance in a homogeneous conductor, the universal effects of a single barrier, and the resonant tunneling through a double barrier.

Figure 1

Scheme of the system. A one-dimensional lattice of $M$ sites is divided into three segments by an external potential with a profile determined by coefficients $C\left(j\right)$ in Eq. (3): left and right leads, where the potential is constant, and quantum wire of $M_{\text{W}}$ sites, where the potential decreases linearly.

Figure 2

Alternating-current conductance $G\left(\omega \right)$ of the noninteracting chain for (a) $M=100$, with broadening $\eta =0.48t/M$, and (b) three different system sizes $M$, with broadening $\eta =48t/M$. In all cases, $M_{\text{W}}=10$.

Figure 3

DMRG results for the conductance $G\left(M\right)$ of the noninteracting chain as a function of the ratio $M_{\text{W}}/M$ between system size and wire length for different wire lengths $M_{\text{W}}$. For all cases, $\eta =48t/M$. Solid lines are polynomial fits. The dashed line shows the exact result for $M_{\text{W}}=10$.

Figure 4

DMRG results for the conductance $G\left(M\right)$ as a function of the inverse system size for various nearest-neighbor interaction strengths $V$. Solid lines are polynomial fits. For all cases, $\eta =48t/M$ and $M_{\text{W}}=10$. Pentagons on the vertical axis ($1/M=0$) show the exact values predicted by the Luttinger liquid theory (12) combined with the Bethe ansatz solution (13).

Figure 5

DMRG results for the conductance $G\left(M\right)$ of a noninteracting chain with a single on-site impurity as a function of the inverse system size for various barrier strengths $\epsilon$. Solid lines are polynomial fits. For all cases, $\eta =48t/M$ and $M_{\text{W}}=10$. Pentagons on the vertical axis ($1/M=0$) show the exact values predicted by the Landauer formula (15) and (16).

Figure 6

DMRG results for the conductance $G\left(M\right)$ of a chain with a repulsive interaction $V=2t$ and a single barrier of strength $0\le \epsilon \le 3t$ as a function of the inverse system size. Solid lines are polynomial fits. For all cases, $\eta =48t/M$ and $M_{\text{W}}=10$.

Figure 7

DMRG results for the conductance $G\left(M\right)$ of a chain with an attractive interaction $V=-t$ and a single barrier of strength $0\le \epsilon \le 3t$ as a function of the inverse system size. Solid lines are polynomial fits. For all cases, $\eta =48t/M$ and $M_{\text{W}}=10$.

Figure 8

DMRG results for the conductance $G\left(M\right)$ of a repulsive Luttinger liquid ($V=2t$) with two nonresonant barriers of absolute strengths $\epsilon$ as a function of the inverse system size. The impurity sites are nearest neighbors and have the same on-site potential. Solid lines are polynomial fits. For all cases, $\eta =48t/M$ and $M_{\text{W}}=10$.

Figure 9

DMRG results for the conductance $G\left(M\right)$ of a repulsive Luttinger liquid ($V=2t$) with two resonant barriers of absolute strengths $\epsilon$ as a function of the inverse system size. The impurity sites are next-nearest neighbors and have on-site potentials of opposite signs. Solid lines are polynomial fits. For all cases, $\eta =48t/M$ and $M_{\text{W}}=10$.