Electric field response of strongly correlated one-dimensional metals: A Bethe ansatz density functional theory study

We present a theoretical study on the response properties to an external electric field of strongly correlated one-dimensional metals. Our investigation is based on the recently developed Bethe ansatz local-density approximation (BALDA) to the density functional theory formulation of the Hubbard model. This is capable of describing both Luttinger liquid and Mott-insulator correlations. The BALDA calculated values for the static linear polarizability are compared with those obtained by numerically accurate methods, such as exact (Lanczos) diagonalization and the density-matrix renormalization group, over a broad range of parameters. In general BALDA linear polarizabilities are in good agreement with the exact results. The response of the exact exchange and correlation potential is found to point in the same direction of the perturbing potential. This is well reproduced by the BALDA approach, although the fine details depend on the specific parameterization for the local approximation. Finally we provide a numerical proof for the nonlocality of the exact exchange and correlation functional.

Figure 1

(Color online) Exchange-correlation potential $v_{\text{XC}}^{\text{hom}}\left(n,t,U\right)$ of the 1D Hubbard model as a function of the electron filling, $n$, for different values of interaction strength, $U$. Here we report data for both BALDA/LSOC and BALDA/FN. Note that the agreement between the two schemes improves as $U$ increases.

Figure 2

(Color online) Linear polarizability, $\alpha$, as a function of the Coulomb repulsion $U/t$. Results are presented for BALDA/LSOC and BALDA/FN and they are compared with those obtained with either ED or DMRG calculations. In the various panels we show: (a) $L=12$ at quarter filling $\left(n=1/2\right)$, (b) $L=16$ at quarter filling, (c) $L=60$ and $N=20$, and (d) $L=60$ at quarter filling. In panel (a) we also show results for BALDA/FN [labeled with “BALDA/FN(b)”] where $\alpha$ is calculated from the site occupation and not from the total energy.

Figure 3

(Color online) Relative error between BALDA calculated polarizabilities and those obtained with exact methods (either ED or DMRG). In the panels we show: (a) $L=12$ at quarter filling $\left(n=1/2\right)$, (b) $L=16$ at quarter filling, (c) $L=60$ and $N=20$, and (d) $L=60$ at quarter filling.

Figure 4

(Color online) Polarizability as a function of $U/t$ for a chain of 60 sites and various filling factors, $n$. The figure legend reports the fitted values for the exponent $\xi$ [see Eq. (12)]. The symbols represents the calculated data while the solid lines are just to guide the eyes. In the inset we present the exponent $\xi$ as a function of the filling factor $n$.

Figure 5

(Color online) Scaling of the polarizability as a function of the chain length, $L$. Panel (a) and (b) are for $n=1/3$ while (c) and (d) for $n=1/2$. Note the linear dependence of the $\alpha \left(L\right)$ curve when plotted on a log-log scale, proving the relation $\alpha \left(L\right)={\alpha }_1{L}^{\gamma }$.

Figure 6

(Color online) The difference, $\Delta v_{\text{XC}}/t$, between the XC potential calculated at finite electric field and in absence of the field as a function of the site index. Results are presented for a 60 site chain with $N=10$ $\left(n=1/6\right)$. The dots are the calculated data while the lines are a guide to the eye. The external potential has a negative slope.

Figure 7

(Color online) The difference, $\Delta v_{\text{XC}}/t$, between the XC potential calculated at finite electric field and in absence of the field as a function of the site index. Results are presented for a 60 site chain with $N=30$ $\left(n=1/2\right)$. The dots are the calculated data while the lines are a guide to the eye. The external potential has a negative slope.

Figure 8

(Color online) Variation in the XC potential per site, $\delta v/L$, as a function of the variation in the total energy, $\delta E_0$, for a 60 site chain in which the first 30 sites have an on-site energy lower by $\delta$ with respect to the remaining 30. The variation are calculated with respect to the homogeneous case. Three methods have been adopted to evaluate $\delta v/L$: Case A, summing over all the site in the chains. Case B, summing only over the first three sites in the left-hand side of the chain and the last three in the right-hand side. Case C, summing over the first three sites on each side adjacent to the potential discontinuity. The inset shows a magnification of the data for small $\delta E_0$.