4 FIG. 9: (a) The probability wavefunction of the state with ϵ$_{1053}$ = 0 . 144eV at ℓ/L$_{x}$ = 0 . 12 ( φ = 0 . 01). Size of the dot is 74 × 71 ˚ A $^{2}$. (b) Profile of z-component of pseudospin of the same state. FIG. 7: Size of the dot is 74 × 71 ˚ A $^{2}$. (a) The probability wavefunction of the state with ϵ$_{1045}$ = 5 . 38 × 10 − $^{6}$eV at φ = 0. (b) Profile of z-component of pseudospin for n = 1045 at φ = 0. (c) The probability wavefunction for n = 1045 at ℓ/L$_{x}$ = 0 . 12 ( φ = 0 . 01). (d) Profile of z-component of pseudospin for n = 1045 at φ = 0 . 01. changed from 5 . 38 × 10 − $^{6}$eV to 4 . 5 × 10 − $^{6}$eV when φ is changed to 0.01 from zero. The pseudospin profiles are shown in Figs.7(b) and (d). Fig.8 shows a probability wavefunction at a smaller value of ℓ/L$_{x,y}$ = 0 . 09 (corre- sponding to φ = 0 . 02), and we see that the wavefunction is less localized on the zigzag edges and LLL character is more pronounced in comparison to result of ℓ/L$_{x}$ = 0 . 12 (Fig.6(c)). All N$_{D}$ ( φ ) states have similar properties men- tioned above with finite probabilities of finding an elec- tron inside the dot. There are also N$_{ℓ}$ zigzag edges states that are more strongly localized on the edges with local- ization lengths comparable to a . When an electron is in one of these states the probability of find the electron away from the edges is practically zero. The energies of these states are less then 10 − $^{10}$eV. We can summarize our results as follows: all nearly zero energy states are localized on the zigzag edges with or without some weight inside the dot. When the magnetic length is much smaller than the system size magnetic edge states can be formed, see Fig.9. The probability wavefunction of a magnetic edge state with ϵ$_{1053}$ = 0 . 144eV is shown in Fig.9(a). It is a mixture of ordinary magnetic and zigzag edge states. The proba- bility wavefunction decays from the armchair edges while it is strongly peaked on the zigzag edges. Pseudospin expectation values on each zigzag edge display opposite chiral behavior. On armchair edges the chiralities are more or less evenly mixed, see Fig.9(b). Fig.6(c). It is localized on the zigzag edges with a fi- nite probability inside the dot. On the armchair edges the wavefunction is vanishingly small. The wavefunc- tion has changed significantly from the φ = 0 result, see Fig.6(a), and also its energy has changed from - 0.07eV to − 2 . 0 × 10 − $^{3}$eV. The values of the z-component of the pseudospin, Fig.6(b) and Fig.6(d), are larger on the zigzag edges at φ = 0 . 01 compared to the result at φ = 0. The probability wavefunction of another state with nearly zero energy is shown in Fig.7(c) at φ = 0 . 01. We see that the result is somewhat different from the zero field result of Fig.7(a), which displays a localized state with the localization length comparable to the unit cell length a . Now there is a finite probability to find an elec- tron inside the dot while the probabilities on the zigzag edges are reduced. Note that the energy of this state has We now explain qualitatively how mixed states of Fig.6(c) and Fig.7(c) can arise. An infinitely long zigzag nanoribbon in a magnetic field has nearly zero energy surface states that are localized on the edges in addition to ordinary lowest Landau level states, see IV. DISCUSSIONS AND CONCLUSIONS FIG. 8: Size of the dot is 74 × 71 ˚ A 2 and ℓ/L$_{x}$ = 0 . 09 ( φ = 0 . 02) (a) The probability wavefunction of the state with ϵ$_{1027}$ = − 2 . 04 × 10 − $^{5}$eV. (b) Profile of z-component of pseudospin of the same state. These results should compared with those in Fig.6.