Entanglement, excitations, and correlation effects in narrow zigzag graphene nanoribbons

We investigate the low-lying excitation spectrum and ground-state properties of narrow graphene nanoribbons with zigzag edge configurations. Nanoribbons of comparable widths have been synthesized very recently [P. Ruffieux et al., Nature (London) 531, 489 (2016)], and their descriptions require more sophisticated methods since in this regime conventional methods, like mean-field or density-functional theory with local-density approximation, fail to capture the enhanced quantum fluctuations. Using the unbiased density-matrix renormalization-group algorithm, we calculate the charge gaps with high accuracy for different widths and interaction strengths and compare them with mean-field results. It turns out that the gaps are much smaller in the former case due to the proper treatment of quantum fluctuations. Applying the elements of quantum information theory, we also reveal the entanglement structure inside a ribbon and examine the spectrum of subsystem density matrices to understand the origin of entanglement. We examine the possibility of magnetic ordering and the effect of magnetic field. Our findings are relevant for understanding the gap values in different recent experiments and the deviations between them.

Figure 1

The applied notations for the length and width of a zigzag ribbon and the mapping used in the DMRG calculation.

Figure 2

Finite-size scaling of charge gaps for different values of $U$ and $L_y=4$ as indicated by the inset figure. The dotted lines denote the exponential fit to the data described in the main text.

Figure 3

DMRG results for the charge gaps as a function of $U$ for two different widths ($L_y=2$ and 4) as indicated in the legend. The dotted lines are guides to the eye. The inset shows the band gaps obtained with the mean-field approximation of the Hubbard model (solid and dotted lines) together with the DMRG data.

Figure 4

Entanglement patterns in a zigzag ribbon for $L_x=6,L_y=4$, and $U=0$. The various types of lines correspond to different magnitudes as indicated in the sidebar. The numbers indicate the positions of sites along the one-dimensional DMRG topology. The blue dashed lines connect only nearest-neighbor sites (for example, sites $i=1,2$ or $i=1,5$) or opposite sites within a hexagon (for example, sites $i=9,10$ or $i=22,23$). The red dash-dotted lines connect opposite zigzag sites, for example, $i=13,16$.

Figure 5

Similar to Fig. 4, but for $U/t=2$; furthermore, the red dash-dotted lines at the zigzag edges connect neighboring zigzag sites, for example, $i=13,21$ or $i=24,32$.

Figure 6

Finite-size scaling of spin correlations between the opposite edges (red squares) and between neighboring edge atoms for $U/t=2$ (black circles), calculated in the middle of the ribbon as they are indicated in the inset.

Figure 7

(a)–(c) The local magnetic moments $S_i^z$ of the ground state in the presence of a pinning magnetic field at the bottom zigzag sites, $h/t=0.01$, for various values of $U$. (d)–(f) The distribution of local magnetic moments in the $S_{\mathrm{TOT}}^z=1,S=1$ sector for different values of $U$. The magnitudes of up and down moments are proportional to the area of the circles.