FIG. 5: Results for a dot with size 50 × 50 nm $^{2}$. Dependence of N$_{T}$ / 2 on B for δ = 0 . 01eV. FIG. 3: (a) Eigenenergies ϵ$_{n}$ at φ = 0. Quasi-degenerate states are present near zero energy. Size of the dot is 74 × 71 ˚ A $^{2}$. A quantization energy of order γk$_{min}$ ∼ 0 . 03eV can be seen as an excitation gap near zero energy. (b) Eigenenergies ϵ$_{n}$ at φ = 0 . 01. 3 GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s50>GLYPH<s48> GLYPH<s51>GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s53>GLYPH<s48> GLYPH<s54>GLYPH<s48> GLYPH<s55>GLYPH<s48> GLYPH<s56>GLYPH<s48> GLYPH<s57>GLYPH<s48> GLYPH<s78> GLYPH<s84> GLYPH<s47> GLYPH<s50> GLYPH<s66>GLYPH<s91>GLYPH<s84>GLYPH<s93> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s48>GLYPH<s46>GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s50> GLYPH<s47>GLYPH<s76> GLYPH<s120> FIG. 6: Size of the dot is 74 × 71 ˚ A $^{2}$. (a) The probability wavefunction of the state with ϵ$_{1027}$ = − 0 . 07eV at φ =0. The length unit is a . (b) Profile of z-component of pseudospin: sizes of red (blue) dots represent probabilities of occupying A (B) carbon atoms. Note that blue (red) dots are dominant in the upper (lower) of dot. When the probabilities are less than 0.00001, the radius of the dots is set to the smallest value. The upper and lower horizontal edges represent zigzag edges. (c) The probability wavefunction of the state with ϵ$_{1027}$ = − 2 . 0 × 10 − $^{3}$eV at ℓ/L$_{x}$ = 0 . 12 ( φ = 0 . 01). (d) Profile of z-component of pseudospin for n = 1027 at φ = 0 . 01. FIG. 4: (a) Some energy levels ϵ$_{n}$ change as φ increases while some do not. (b) The total number of states within the energy interval δ around zero energy at a finite value of φ . Size of the dot is 74 × 71 ˚ A $^{2}$. tem length 2 ℓ ∼ L$_{x,y}$ . Note that in this regime many cyclotron orbits get affected by the presence of the edges and corners of the rectangular dot. Since we must also assume that Hofstadter effect is negligible the validity regime of Eq.(3) is a$_{0}$ ≪ 2 ℓ < L$_{x,y}$ . Note also that the scaling function f ( x ) should be different for each δ . The ratio between the number of nearly zero energy states induced by the magnetic field and the number of hexagons, N$_{D}$ ( φ ) N$_{h}$ , should depend on a dimensionless quantity consisting of a combination of ℓ , a$_{0}$ , L$_{x}$ and L$_{y}$ , which are the important parameters of rectangu- lar graphene dots. The lengths L$_{x}$ and L$_{y}$ should ap- pear as L$_{x}$ + L$_{y}$ so that for rectangular-shaped graphene sheets N$_{D}$/N$_{h}$ remains the same when L$_{x}$ and L$_{y}$ are ex- changed. These considerations lead us to the dimension- less variable ℓ 2 a 0 ( L$_{x}$ + L$_{y}$ ) = 3 4 π 1 φ ( M $_{+}$√$_{3}$$_{N}$$_{+2}$$_{/}$ √$_{3)}$ , see Eq.(3). We are especially interested in the regime where the di- ameter of the cyclotron orbit is comparable to the sys- We first show how the wavefunction of a non-zero en- ergy state at φ = 0 changes into a state with nearly zero energy as φ increases. Consider the probability wave- function for n = 1027 at a finite φ = 0 . 01, as shown in III. WAVEFUNCTIONS OF QUASI-DEGENERATE STATES IN A MAGNETIC FIELD