Etydhv

I am trying to Plot Eigenvalues of a Hamiltonian, but I am getting noisy plot, which is incorrect. Here is the code.

A1 = 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0;
 A2 = 0, -I, 0, 0, I, 0, 0, 0, 0, 0, 0, -I, 0, 0, I, 0;
 A3 = 0, 0, 0, -1, 0, 0, 1, 0, 0, 1, 0, 0, -1, 0, 0, 0;
 A4 = 0, -I, 0, 0, I, 0, 0, 0, 0, 0, 0, I, 0, 0, -I, 0;
 A5 = 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1;
 A6 = 0, 0, 0, -I, 0, 0, I, 0, 0, -I, 0, 0, I, 0, 0, 0;
 A7 = 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0;
 A8 = 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1;
H[d_, λ_, β_, m_] := 
 a (Sin[x] A1 + Sin[ky] A2) + A3 β + 
 d A4 + (t Cos[z] + 2 b (2 - Cos[x] - Cos[ky])) A5 + α*
 Sin[ky] A6 + λ Sin[z] A7+m*A8;
 ky = 0;
 a = 1;
 b = 1;
 t = 1.5;
 α = 0.3;
 Plot3D[Eigenvalues[H[0.1, 0.5, 0.7, 0]][[4]], x, -π, π, z, 0, 2 π]

Any help will be highly appreciated.

By default, the eigenvalues are ordered by absolute value. All the eigenvalues of this particular matrix have the same absolute value plus some rounding errors. Thus, it can easily happen, that the fourth eigenvalue is positive or negative, depending on the parameters.

You can use Max to plot the largest eigenvalue:

Plot3D[Max@Eigenvalues[H[0.1, 0.5, 0.7, 0.]], x, -Pi, Pi, z, 0, 2 Pi]

Alternatively, you may use the "Criteria" suboption of the Method "Arnoldi":

Plot3D[
 Eigenvalues[
 H[0.1, 0.5, 0.7, 0], -1, 
 Method -> "Arnoldi", "Criteria" -> "RealPart"
 ], 
 x, - Pi, Pi, z, 0, 2 Pi]

Not sure why you pick the 4th element, but maybe this will help:

ev4 = Eigenvalues[H[p, q, r, s]][[4]] /. 
 Thread[p, q, r, s -> 0.1, 0.5, 0.7, 0];
Plot3D[ev4, x, -π, π, z, 0, 2 π]