MATLAB: Eigenvalues of a Matrix for a mesh of values and 3D plot.

3d plotsdiagonalisationeigenvaluesMATLABmeshgrid

Problem: Here is what i want to do. I need to construct an Hamiltonian matrix of dimension 2 by 2, then diagonalise it and find the eigenvalues for set of input values. I can do the problem for input values on a line, but i want a 3D plot and hence want to use a meshgrid.
The following working code plots for values on a line.
clear all
clc
a0 = 1.42; alpha0=1;
%-------------------------------------Define k range----------------------------------

kx = -2*pi/(3*a0):2*pi/(3*49.5*a0):2*pi/(3*a0);
ky = -2*pi/(3*a0):2*pi/(3*49.5*a0):2*pi/(3*a0);
a= 3*a0/2; b= sqrt(3)*a0/2;
N=2;
% % % % Hamiltonian generation
H=zeros(N,N);
for c = 1:100
    hk(c)=exp(1i*ky(c)*a0)+exp(1i*(a*kx(c)- b*ky(c)))+exp(-1i*(a*kx(c)+ b*ky(c)));%h(k) for kx, ky ne 0.

    
    %*****Hamiltoninan generation starts here**************
 for i=1:N
     for j=1:N
         if (j-i)==1
             H(i,j)= -alpha0*hk(c);
         elseif (j-i)==-1
             H(i,j)= -alpha0*conj(hk(c));
         end 
     end 
 end 
 E(:,c) = eig(H); %Eigenvalues
 
end
plot(kx,E);
Now for the above eigenvalues, i can't use surf to 3D plot and hence i tried the following code to generate eigenvalues in a meshgrid.
This code has an error.
a0 = 1.42; alpha0=1;
%-------------------------------------Define k range----------------------------------
kx = -2*pi/(3*a0):2*pi/(3*49.5*a0):2*pi/(3*a0);
ky = -2*pi/(3*a0):2*pi/(3*49.5*a0):2*pi/(3*a0);
[x,y]=meshgrid(kx,ky);
a= 3*a0/2; b= sqrt(3)*a0/2;
N=2;
H=zeros(N,N);
for l=1:100
    for m=1:100
   hk(l,m)=exp(1i*y(l,m)*a0)+exp(1i*(a*x(l,m)- b*y(l,m)))+exp(-1i*(a*x(l,m)+ b*y(l,m)));%h(k) for kx, ky ne 0.
    
   %*****Hamiltoninan**************
 for i=1:N
     for j=1:N
         if (j-i)==1
             H(i,j)= -alpha0*hk(l,m);
         elseif (j-i)==-1
             H(i,j)= -alpha0*conj(hk(l,m));
         end 
     end 
 end 
 E(l,m) = eig(H); %Here is the problem because it can't store two values in one grid
H=zeros(N,N);
    end 
 
end
 figure
 meshgrid(kx,ky,100);
 surf(kx,ky,E);
So the core of the problem is that, i want to solve the Hamiltonian matrix for the values scattered in a mesh and obtain the eigenvalues and plot those to obtain a 3D plot.
Any help or suggestion will be helpful.
Thank you.

Best Answer

It seems you already understand the problem
E(l,m) = eig(H); %Here is the problem because it can't store two values in one grid
and have solved it in your 1D case
E(:,c) = eig(H); %Eigenvalues
If you want to extend your solution, just augment E by one more dimension
E(l,m,:) = eig(H);
As for plotting, I don't think surf/mesh/contour can handle a multi-dimensional Z variable, so you'll probably have to split
figure(1)
meshgrid(kx,ky);
surf(kx,ky,E(:,:,1));
So a full solution is (after some simplification of your code0
clear;
close all;
clc
a0 = 1.42;
alpha0 = 1;
% Define k range
k = -2*pi/(3*a0):2*pi/(3*49.5*a0):2*pi/(3*a0);
% it was redundant to do this twice
% don't hard-code the length of k
M = length(k);
[x,y] = meshgrid(k);
a = 3*a0/2;
b = sqrt(3)*a0/2;
% compute hk for all pairs of kx and ky
hk = exp(1i*y*a0) + exp(1i*(a*x- b*y)) + exp(-1i*(a*x+ b*y));
% by the way, the need for a0 is unclear, as it seems to divide out
% pre-allocate your results matrix
E = nan(M,M,2);
for i = 1:M
	for j = 1:M
		
		% your H computation appears needlessly complex
		H = -alpha0*[0,hk(i,j);conj(hk(i,j)),0];
		
		E(i,j,:) = eig(H);
		
	end
end
figure; hold on;
surf(x,y,E(:,:,1));
surf(x,y,E(:,:,2));
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