Manipulating a multidimensional array.

PROBLEM: I have managed to create a multidimensional array storing eigenvectors of a hamiltonian. Now I want to manipulate the array to find the expectation value of a matrix and make a quiver plot.

EXPLANATION : I have the following Hamiltonian with . I want to find the eigenvectors for a set of values on a meshgrid and then use it to find and and then to do a quiver plot.

It can be done by solving analyticaly as well and the results are,

The above can be easily coded as follows,

kx = linspace(-0.5,0.5,15);
ky = linspace(-0.5,0.5,15);
[KX,KY] = meshgrid(kx,kx);

X = -sin(atan2(KY,KX));
Y = cos(atan2(KY,KX));

quiver(KX,KY,X,Y)

My Attempt

kx = linspace(-0.5,0.5,15);
ky = linspace(-0.5,0.5,15);
[KX,KY] = meshgrid(kx,kx);

alpha_R = 0.18851;
M = length(kx);

% SPIN MATRICES 

sigma_x = [0,1;1,0]; % sigma_x
sigma_y = [0,-1j;1j,0]; %sigma_y
I = [1,0;0,1]; % Identity

%Pre-allocation
E = nan(M,M,2);
Psi = nan(M,M,2);
Psi_prime = nan(M,M,2);



for i = 1: length(kx)
    for j = 1: length(ky)
        
        kx_t = kx(i);
        ky_t = ky(j);
        
        eps_k = kx_t^2 + ky_t^2; % epsilon_k
        
        %hamiltonian
        
        H = eps_k * I + alpha_R * (sigma_x .* ky_t - sigma_y .* kx_t);
        
        E(i,j,:) = eig(H);
        [V,D] = eig(H);
        Psi(i,j,:) = V(:,1);
        
    end 
end 
        
Psi_prime = pagectranspose(Psi);

%%%ERROR HERE
expx = Psi_prime .* sigma_x .* Psi ;
% expy
% quiver(KX,KY,expx,expy)

I want to find the expectation value of x and y and make a quiver plot of that to get the same texture as above.

Any help or guidance would be helpful. I just am unable to visualize a multidimensional array.

Thank You.

Answer 1

[Bruno Luong]

Try this:

kx = linspace(-0.5,0.5,15);
ky = linspace(-0.5,0.5,15);
[KX,KY] = meshgrid(kx,kx);

alpha_R = 0.18851;
M = length(kx);

% SPIN MATRICES 

sigma_x = [0,1;1,0]; % sigma_x
sigma_y = [0,-1j;1j,0]; %sigma_y
I = [1,0;0,1]; % Identity

%Pre-allocation
E = nan(M,M,2);
Psi = nan(2,1,M,M); % BL's change here

for i = 1: length(kx)
    for j = 1: length(ky)
        
        kx_t = kx(i);
        ky_t = ky(j);
        
        eps_k = kx_t^2 + ky_t^2; % epsilon_k
        
        %hamiltonian
        
        H = eps_k * I + alpha_R * (sigma_x .* ky_t - sigma_y .* kx_t);
        
        E(i,j,:) = eig(H);
        [V,D] = eig(H);
        Psi(:,:,i,j) = V(:,1); % BL's change here
        
    end 
end 
        
% Psi_prime = pagectranspose(Psi); % BL's change here

%%%NO LONGER ERROR HERE
expx = pagemtimes(pagemtimes(Psi,'ctranspose',sigma_x,'none'),Psi); % BL's change here
expx = reshape(expx,[M M]); % BL's change here
expy = pagemtimes(pagemtimes(Psi,'ctranspose',sigma_y,'none'),Psi); % BL's change here
expy = reshape(expy,[M M]); % BL's change here
% expy
quiver(KX,KY,expx,expy)

[Vira Roy]

Thanks for your reply Bruno Luong. I am really inclined to accept the answer although, I have a little query here. Since it is the same Hamiltonian as the one I could solve analytically, I was expecting the same spin texture(quiver plot) as I have shown in the question. Could it be due to how the array has been reshaped before plotting?

Thanks a lot.

[Bruno Luong]

Yes, but the error occurs in your original code as well since you did use i for first dimension and j for second index - which is opposite to the meshgrid convention.

You have 2 options to fix

[KX,KY] = ndgrid(kx,kx);

OR (exclusive)

expx = reshape(expx,[M M]).';
expy = reshape(expy,[M M]).';

[Vira Roy]

Thanks for your reply Bruno and forbeing so nice with the reply. Thanks a lot.

[SHUBHAM PATEL]

Hello @Bruno Luong,

Could you please tell how can we add a third feature (sigma_z, and expz) which is represented by the background color in the quiver plot?

In the code above if we want to add another two line.

% before the 'for' loop
sigma_z = [1,0;0,-1]; %sigma_z

%After the for loop
expz = pagemtimes(pagemtimes(Psi,'ctranspose',sigma_z,'none'),Psi); % BL's change here
expz = reshape(expz,[M M]); % BL's change here

And if we want to reflect the magnitude of espectation value of z by background color of the quiver plot, is it possible in MATLAB?

PS: The magnitude of expz is vecry small (zero), still I want to know the way.

[Bruno Luong]

It sounds lie a graphical/plotting related question. I suggest you to open a new question so other participants can give you answers.