Nonlinear Coherent States for Anisotropic 2D Dirac Materials
319
Fig. 1 Dirac cones for an isotropic (dashed gray contour) and anisotropic material (solid blue
contour). For the former, their cross-sections are circles and ellipses for latter with semi-major axis
along (a) the p x -axis (v xx < v yy ) or (b) the p y -axis (v xx > v yy )
vector potential which defines a magnetic field orthogonal to the sample. In Landau
gauge, such that
A(x, y) = B 0 x ˆ
j and
B = ∇ ×
A = B 0 ˆ
k, we can write
Ψ (x, y) = N exp(iky)
ψ + (x)
iψ − (x)
,
(4)
where N is for normalization. Substituting (4) into (3), and decoupling, we get
H
±
ζ ψ
± (x) =
−
d 2
dx 2 +
ω 2
ζ
4
x +
2k
ω B
2
±
1
2
ω ζ
ψ
± (x) = ((
± )
2 ψ
± (x),
(5)
with ± = E/v xx ¯
h, ω ζ = ω B /ζ , ω B = 2eB 0 / ¯
h and ζ = v xx /v yy being the
anisotropy parameter. The spectrum is
E
−
0 = 0, E
−
n = E
+
n−1 = ± ¯
h
v xx v yy ω B |n|, n = 0, ±1, ±2, . . . .
(6)
with ± standing for the conduction and valence bands. Eigenfunctions are
ψ n (x) =
1
2 n n!
ω ζ
2π
1/2
exp
−
ω ζ
4
x +
2k
ω B
2
H n
ω ζ
2
x +
2k
ω B
.
(7)
Finally, we take N 2 = 2 (δ 0n −1) , ψ −
n ≡ ψ n and ψ +
n ≡ ψ n−1 in (4).