Disorder and interactions in one-dimensional systems

We present a numerical approach to the study of disorder and interactions in quasi-one-dimensional (1D) systems which combines aspects of the transfer matrix method and the density matrix renormalization group which have been successfully applied to disorder and interacting problems, respectively. The method is applied to spinless fermions in 1D, and the existence of a conducting state is demonstrated in the presence of attractive interactions.

Figure 1

The recursive procedure adds new sites to both ends of a 1D chain at each iteration.

Figure 2

A flow diagram showing the central procedure of the algorithm. This whole procedure is repeated many times with different disorder realizations.

Figure 3

Results from a (clean) short chain of ten sites demonstrating phase separation for $U<-2$. For each particle number the (grand canonical) ground-state energy is plotted. The chemical potential is set to give half-filling as the overall ground state (i.e., $\mu =U$). Plotted energies are grand canonical.

Figure 4

The average of $\frac{1}{2}\ln \left(D^2\right)$ as a function of chain length $L$, where $D$ is the reduced density matrix (4). Averages are geometric, that is, the mean of $\ln \left(D^2\right)$ is found. The chain length is allowed to extend until numerical precision is lost. The $\mu =0$ case was stopped at 20 000 sites as it had saturated long before. System parameters are $W=2$, $U=0$, and the energy cutoff is set using $M=10$ (corresponding to a total of approximately 160 basis states per iteration). The averages were taken over 2000 systems.

Figure 5

The average of $\frac{1}{2}\ln \left(D^2\right)$ against chain length $L$. The parameters are the same as Fig. 4, but restricted to the middle of the band $(\mu =0)$. The only difference between the three curves is the procedure for discarding basis states. Note that the data around $-30$ are numerical noise as discussed in Sec. 3D1.

Figure 6

The dependence of inverse localization length on disorder when interactions are turned off. Results from the new method should correspond to known result for the middle of the band. Systems were allowed to extend up to 1000 sites, retaining an average of 480 basis states per iteration. Averages were taken over 1000 disorder realizations.

Figure 7

The dependence of inverse localization length on the average number of basis states retained per iteration for the noninteracting case with disorder $W=2$ (a) and $W=5$ (b). The insets show this quantity plotted against the actual energy cutoff used. Such plots can be used to test for convergence. Data were averaged over 100 systems with chains extending up to 1000 sites.

Figure 8

The dependence of localization length on interaction strength. The three lines correspond to different energy cutoff values. Each line is averaged over 1000 systems which are allowed to extend to a maximum of 1000 lattice sites. Disorder $W=2$.

Figure 9

Disorder-interaction phase space plot for the single-chain model at half-filling. The contours represent the inverse localization length in intervals of 0.01. The lowest interval corresponds to a localization length greater than 1000 sites. This plot was produced using over 1300 points. Each point was averaged over 250 systems in which chains were allowed to extend to 2000 sites and approximately 240 basis states were retained per iteration. Data for $W<0.6$ are not shown because the method is unreliable for low disorder.