Effective narrow ladder model for two quantum wires on a semiconducting substrate

We present a theoretical study of two spinless fermion wires coupled to a three-dimensional semiconducting substrate. We develop a mapping of wires and substrate onto a system of two coupled two-dimensional ladder lattices using a block Lanczos algorithm. We then approximate the resulting system by narrow ladder models, which can be investigated using the density-matrix renormalization group method. In the absence of any direct wire-wire hopping we find that the substrate can mediate an effective wire-wire coupling so that the wires could form an effective two-leg ladder with a Mott charge-density-wave insulating ground state for arbitrarily small nearest-neighbor repulsion. In other cases the wires remain effectively uncoupled even for strong wire-substrate hybridizations leading to the possible stabilization of the Luttinger liquid phase at finite nearest-neighbor repulsion as found previously for single wires on substrates. These investigations show that it may be difficult to determine under which conditions the physics of correlated one-dimensional electrons can be realized in arrays of atomic wires on semiconducting substrates because they seem to depend on the model (and consequently material) particulars.

Figure 1

The upper panel display sketch of two atomic wires (red spheres) on a 3D substrate with four numbered shells. The lower panel ladder representation of the same system with the uppermost legs corresponding to the atomic wires (red circles) and the other legs (in green and light blue) representing the shells one to four.

Figure 2

Hopping parameters of the two-impurity subsystem after the Block-Lanczos transformation. The parameters are given in the text. (a) Intrarung hopping terms ${\left[e_l\left(k_x\right)\right]}_{n,n^{\prime }}$ for the system with NN impurities. (b) Intraleg hopping terms ${\left[{\tau }_l\left(k_x\right)\right]}_{n,n}$ as well as interleg diagonal hoppings ${\left[{\tau }_l\left(k_x\right)\right]}_{1,2}$ for the system with NN impurities. (c) Intrarung hopping terms ${\left[e_l\left(k_x\right)\right]}_{n,n^{\prime }}$ for the system with NNN impurities. (d) Intraleg hopping terms ${\left[{\tau }_l\left(k_x\right)\right]}_{n,n}$ as well as interleg diagonal hoppings ${\left[{\tau }_l\left(k_x\right)\right]}_{1,2}$ for the system with NNN impurities. The horizontal axis represents the number of Bloc-Lanczos shell. Note that the results are indistinguishable for $n=1$ and $n=2$ (square symbols) in (a), (b), and (c) but not in (d).

Figure 3

Spectral functions of two NN wires on a semiconducting substrate with $t_{\text{w}}=3, t_{\text{ws}}=8$, and $t_{\text{s}}=1$. (a) Full TWSS model. (b) NLM with $N_{\text{leg}}=2N_{\text{shell}}=54$. (c) NLM with $N_{\text{leg}}=2N_{\text{shell}}=6$.

Figure 4

Spectral functions for two NNN wires on a semiconducting substrate with $t_{\text{w}}=3, t_{\text{ws}}=8$, and $t_{\text{s}}=1$. (a) Full TWSS model. (b) NLM with $N_{\text{leg}}=6$.

Figure 5

Enlarged view of the spectral functions in Fig. 4. (a) Full TWSS model. (b) NLM with $N_{\text{leg}}=6$.

Figure 6

Single-particle gap $E_p$ in a six-leg NLM as a function of the intrawire interaction $V$. The upper plots (a) and (b) show results for two NN wires and two NNN wires, respectively. The lower figures (c) and (d) show an enlarged view of the same results. The upper and lower triangles are results for a three-leg NLM representing a single wire on a substrate. The diamonds show results for the two-leg ladder model with a rung hopping $t_{\perp }=0.5$ and a leg hopping $t_{\parallel }=3$ but no substrate. The ladder length is $L_x=128$.

Figure 7

Charge-density-wave order parameters ${\delta }_{l,n}$ for various legs $(l,n)$ (see the text) in the six-leg NLM for a pair of NN and NNN wires on a substrate. The parameters for the original TWSS models are $t_{\text{w}}=3, t_{\text{ab}}=0, t_{\text{s}}=1$ and (a) $t_{\text{ws}}=0.5$ (b) $t_{\text{ws}}=8$. The CDW order parameters are also shown for the first leg of a three-leg NLM representing a single wire on a substrate as well as for the two-leg ladder model with a rung hopping $t_{\perp }=0.5$ and a leg hopping $t_{\parallel }=3$ but no substrate. The ladder length is $L_x=128$.

Figure 8

Distribution of the single-particle excitation density $N_{l,n}$ defined in Eq. (34) between the two wires of a six-leg NLM with $t_{\text{ws}}=3, t_{ab}=0$, and $t_{\text{s}}=1$: (a) NN wires with $t_{\text{ws}}=0.5$, (b) NNN wires with $t_{\text{ws}}=0.5$, (c) NN wires with $t_{\text{ws}}=8$, and (d) NNN wires with $t_{\text{ws}}=8$. The upper and lower triangles show results for the two-leg ladder with rung hopping $t_{\perp }=0.5$ and a leg hopping $t_{\parallel }=3$ but no substrate. The ladder length is $L_x=128$.