Topologically ordered zigzag nanoribbon: $e/2$ fractional edge charge, spin-charge separation, and ground-state degeneracy

We numerically compute the density of states (DOS) of interacting disordered zigzag graphene nanoribbon (ZGNR) having midgap states showing $e/2$ fractional edge charges. The computed Hartree-Fock DOS is linear at the critical disorder strength where the gap vanishes. This implies an $I\text{−}V$ curve of $I\propto V^2$. Thus, $I\text{−}V$ curve measurement may yield evidence of fractional charges in interacting disordered ZGNR. We show that even a weak disorder potential acts as a singular perturbation on zigzag edge electronic states, producing drastic changes in the energy spectrum. Spin-charge separation and fractional charges play a key role in the reconstruction of edge antiferromagnetism. Our results show that an interacting disordered ZGNR is a topologically ordered Mott-Anderson insulator.

Figure 1

(a) Zigzag edge antiferromagnetism of interacting ZGNR without disorder. (b) Schematic of interacting (solid curves) and noninteracting (dashed curves) ZGNR band structures. The unoccupied and occupied states near the wave vectors $k=\pm \pi /a_0$ are shown: $R$ and $L$ represent states confined on the right and left zigzag edges, respectively ($a_0=1.73a$ is the unit cell length of the ZGNR and $a=1.42$ Å is the C-C distance). The small arrows indicate spins. The spin-split energy levels of the spin-up (solid lines) and spin-down (dashed lines) gap-edge states of the interacting disordered ZGNRs are shown. These states decay exponentially from the zigzag edges. In the limit of an infinitely long ribbon the gap ${\mathrm{\Delta }}_s$ may vanish and a soft gap can develop. Another degenerate ground state can be obtained by exchanging $\uparrow$ and $\downarrow$ spins.

Figure 2

Two $e/2$ fractional zigzag edge charges of an interacting disordered ZGNR. Note that red (blue) probability density means that the wave function has A (B) chirality, i.e., it is finite only on A (B) carbon atoms. Since there is negligible tunneling between these sites we will call this type of state a mixed chiral state.

Figure 3

A tunneling electron is fractionalized when it enters an interacting disordered ZGNR.

Figure 4

Off-diagonal disorder: Random network of hexagons consisting of A and B carbon atoms. A zigzag edge site is connected to two other carbon atoms while a site away from the edges is connected to three other carbon atoms. The hopping parameter $t$ is not the same for all sites. The zigzag edges have definite chirality, consisting A or B carbon atoms. In contrast, armchair edges have mixed chirality.

Figure 5

Diagonal disorder: Site energies are varied randomly. Colors represent strength of on-site disorder potential. Again a zigzag edge site is connected to two other carbon atoms while a site away from the edges is connected to three other carbon atoms. But the hopping parameter $t$ is the same for all sites.

Figure 6

States localized on the right and left zigzag edges are represented, respectively, by $R$ and $L$. The long arrows indicate the coupling, induced by a short-ranged disorder potential, between states $R\uparrow$ and $L\uparrow$ or $R\downarrow$ and $L\downarrow$.

Figure 7

Schematic drawing of the site probability distribution ${\left|\psi \right|}^2$ of two degenerate edge states with wave functions ${\psi }_L$ and ${\psi }_R$ is shown. They are localized on the (a) left and (b) right edges, respectively. (c) Disorder couples these states and can generate antibonding and bonding edge states with the wave functions $\psi =\frac{1}{\sqrt{2}}({\varphi }_L-{\varphi }_R)$ and $\psi =\frac{1}{\sqrt{2}}({\varphi }_L+{\varphi }_R)$, respectively.

Figure 8

(a) Plot of $q_A$ for disorder potential strength $\mathrm{\Gamma }=0.1t$, where each point represents the probability of finding an electron of a gap-edge state on $A$ carbon sites, $q_A$. The disorder realization number is $N_D=300$, and the ribbon length is $\ell =196.8$ Å. Here the impurity or defect–to–carbon atom ratio is $n_{\mathrm{imp}}=0.1$. The gap size is ${\mathrm{\Delta }}_s\approx 0.12\frac{\mathrm{\Delta }}{2}$. (b) Plot of $q_A$ for $U=t$ in the absence of disorder, where the on-site electron repulsion and hopping parameter are indicated by $U$ and $t$, respectively.

Figure 9

Plot of $q_A$ for $\mathrm{\Gamma }=0.03t$, $U=t$, $n_{\mathrm{imp}}=0.1$, $l=1232.5$ Å, and $N_D=2166$. Zero-energy states with $q_A$ rather close to $1/2$ are indicated by an arrow. Blue (red) dots are for spin-up (-down) states.

Figure 10

Plot of the probability density of a fractionalized gap-edge state in the presence of off–diagonal disorder. It is a mixed chiral state; green and blue represent different chiralities. Its energy is $E=-0.019t$ with $q_A=0.495$. The range of hopping parameters is $0.94<t_{ij}/t<1.06$ and the ribbon length is $\ell =199.3$ Å. A zigzag ribbon consists of zigzag lines. In this ribbon they are labeled from $j=1$ to 8.

Figure 11

Same as in Fig. 10 but with a longer ribbon length $\ell =494.5$ Å. The gap-edge-state energy is $E=0.055t$ with $q_A=0.51$. The range of hopping parameters is $0.94<t_{ij}/t<1.06$.

Figure 12

Plot of DOS $\rho \left(E\right)$ for disorder potential strength $\mathrm{\Gamma }=0.18t$. Here the disorder realization number $N_D=4409$, the impurity or defect–to–carbon atom ratio $n_{\mathrm{imp}}=0.1$, and ribbon length $\ell =740.46$ Å. The histogram interval is $\delta E=0.02\frac{\mathrm{\Delta }}{2}$. The solid line represents a linear fit to $\rho \left(E\right)$.

Figure 13

The DOS displays an exponential soft gap near $E=0$. Parameters are $U=t$, $\mathrm{\Gamma }=0.03t$, $n_{\mathrm{imp}}=0.1$, $l=1232.5$ Å, and $N_D=2166$. Histogram interval is $0.014\frac{\mathrm{\Delta }}{2}$.

Figure 14

Net site spin values $S_i=S_{i\uparrow }+S_{i\downarrow }=n_{i\uparrow }-n_{i\downarrow }$ are plotted, where $n_{i\sigma }$ is the site occupation number for spin $\sigma$. A zigzag ribbon consists of zigzag lines. In this ribbon they are labeled from $j=1$ to 8. Blue (red) lines indicate positive (negative) spin values. This result is for one disorder realization with diagonal disorder. The parameters are $U=t$.

Figure 15

(a) Site-spin $z$ components, $S_{i\uparrow }=n_{i\uparrow }$ (blue) and $S_{i\downarrow }=-n_{i\downarrow }$ (red), plotted along edges. Here $n_{i\sigma }$ is the site occupation number. The left and right figures correspond to the left and right zigzag edges, respectively. The number $1/2$ indicates a removed or added electron occupation number. (b) Net site spin values $S_i=S_{i\uparrow }+S_{i\downarrow }$ are plotted.

Figure 16

Plot of total $z$ component of ground-state site spin $S_i$. Process of spin-charge separation is displayed in (a) and (b).

Figure 17

(a) Finite-length dimer chain with unit cell containing two carbon atoms connected by single bond. The intracell hopping $t^{\prime }$ is smaller than the intercell hopping $t$. (b) Tight-binding energy spectrum. Two nearly degenerate gap states exist. (c) Probability density of a gap state as function of site index $i$. A peak is apparent at the red (blue) site at the left (right) end. The probability densities of the bonding and antibonding states are almost identical.

Figure 18

Kitaev's chain has two degenerate zero-energy states. They can be combined to give one bonding and one antibonding states, see Fig. 7. For each of these states the probability to find an electron (hole) at site $i=1,...,15$ is $|u_i{|}^2$ ($|v_i{|}^2$). We have set the hopping parameter equal to the value of the gap $t=\mathrm{\Delta }$ and the chemical potential $\mu =0$. A Majorana zero mode (half a real fermion mode) at each end of the chain is displayed.