Topological entanglement entropy of interacting disordered zigzag graphene ribbons

Interacting disordered zigzag graphene nanoribbons have fractional charges, are quasi-one-dimensional, and display an exponentially small gap. Our numerical computations showed that the topological entanglement entropy of these systems has a small finite but universal value, independent of the strength of the interaction and the disorder. The result that was obtained for the topological entanglement entropy shows that the disorder-free phase is critical and becomes unstable in the presence of disorder. Our result for the entanglement spectrum in the presence of disorder is also consistent with the presence of a topologically ordered phase.

Figure 1

(a) Zigzag edge antiferromagnetism of an interacting zigzag graphene nanoribbon without disorder showing the two degenerate ground states. (b) Schematic band structures of interacting (solid curves) and noninteracting (dashed curves) zigzag graphene nanoribbons. Unoccupied and occupied states near the wave vectors $k=\pm \pi /a_0$: $R$ and $L$ represent the states confined to the zigzag edges on the right and left, respectively (the length of the unit cell of a ribbon is $a_0$). The small arrows indicate the spins. Spin-split energy levels of the spin-up (solid lines) and spin-down (dashed lines) gap-edge states of the interacting disordered interacting zigzag graphene nanoribbons. These figures are taken from Ref. [7].

Figure 2

Dashed line: The exponentially small soft gap with $\alpha \sim 1/\sqrt{{\mathrm{\Delta }}_s}$ near the Fermi energy of the tunneling density of states of an interacting disordered zigzag graphene nanoribbon. Solid line: The hard gap, $\mathrm{\Delta }$, of the tunneling density of states of an interacting zigzag graphene nanoribbon in the absence of disorder.

Figure 3

Different regions of the ribbon used in Eq. (2) to compute the TEE. Vertical lengths are measured in number of horizontal carbon lines and horizontal lengths in number of vertical carbon lines.

Figure 4

TEE $\beta$ of an interacting disordered zigzag nanoribbon is computed using a rectangular ring with different ring widths $w$. Each point represents a different value of $(U,\mathrm{\Gamma })$. In (a), the ring width, ribbon length, ribbon width are, respectively, $w=7$, $L^{\prime }=260$, $W=112$. The side lengths of the ring are $l_{\text{zig}}=221$ and $l_{\text{arm}}=46$. In (b), they are $w=4$, $L^{\prime }=200$ $W=64$, $l_{\text{zig}}=161$, $l_{\text{arm}}=28$. The number of disorder realizations $N_D$ are, respectively, 10 and 20 in (a) and (b).