I am trying to calculate the Berry curvature of a topological photonic crystal and to solve this I need to write the formula of Berry curvature in MATLAB

Hello,

i am trying to get the plot of Berry Curvature for topological photonic crystal where the formula given below needs to be implemented in MATLAB.

u_nk data have been extracted from COMSOL, k varies from 0 to 3.

I am finding it difficult to implement this formula.

Any help would be appreciated.

Thanks

Answer 1

[SANKALP DEV]

Hi Dibaskar,

I understand that you want assistance in implementing the equation,’ Ωn(k) = ∇k x ∫unitcelld2runk*(r) ∇runk(r)’, which represents the Berry curvature for a topological photonic crystal.

Kindly refer to the following steps to implement the equation:

1. Define the necessary variables:

• ‘k’ (wavevector): Create an array of values from 0 to 3, depending on the desired range.
• ‘u_nk’ (eigenstate data): Import or generate the eigenstate data from ‘COMSOL’. This should be a matrix with dimensions ‘n x k’, where ‘n’ is the number of eigenstates and ‘k’ is the number of wavevectors.

2.Initialize the Berry curvature array:

• Create an empty matrix ‘omega’ with dimensions ‘n x k’.

3.Perform the calculations:

• Loop through each wavevector ‘k’ and eigen state ‘n’.
• For each combination of ‘k’ and ‘n’ calculate the berry curvature as shown in following code snippet.

for n = 1:size(u_nk, 1) 
    for ki = 1:length(k) 
        % Calculate the gradient of eigenstate with respect to k 
        gradk = gradient(u_nk(n, ki), k(ki));
        % Calculate the integral over the unit cell 
        integral = calculateIntegral(u_nk(n, ki));  % Implement this function 
        % Calculate the Berry curvature 
        Omega(n, ki) = cross(gradk, integral); 
    end 
end 

Note that the function, ‘calculateIntegral’, mentioned here is a user defined function which can be implemented by the following code.

function integral = calculateIntegral(u_nk) 
    x = linspace(x_min, x_max, num_points); 
    y = linspace(y_min, y_max, num_points); 
    % Initialize the integral 
    integral = 0;
    % Perform the integration using the trapezoidal rule 
    for i = 1:length(x)-1 
        for j = 1:length(y)-1 
            % Evaluate the integrand at each point 
            integrand = u_nk(x(i), y(j)) * gradient(u_nk(x(i), y(j)), x(i), y(j));
            % Accumulate the results using the trapezoidal rule 
            integral = integral + trapz(y, integrand); 
        end 
    end 
end 

Replace ‘x_min’, ‘x_max’, ‘y_min’, ‘y_max’, and ‘num_points’ with the appropriate values specific to your problem.

To know more about MATLAB functions used in above code, refer to following documentations.

• ‘trapz’- Trapezoidal numerical integration - MATLAB trapz (mathworks.com)
• ‘cross’-Cross product - MATLAB cross (mathworks.com)
• ‘gradient’-Numerical gradient - MATLAB gradient (mathworks.com)
• Hope it helps.

Regards,

Sankalp

[Dibaskar]

Thank you so much for such a detailed answer to my query.

I am indeed grateful to your reply.

I am trying to implement that code.

Also one last thing, can you tell me after calculating the berry curvature matrix, how to get the intensity plot?

The plot is between kx and ky vectors.

I am attaching a picture of the plot(taken from a literature) below.

Regards,

Dibaskar

Answer 2

[Guifeng]

Dear Dibaskar,

I am also working on similar problems recently. May I know how do you export un_k data from COMSOL. Moreover, as I understand, the un_k data of any single band at any single k vector should be a matrix for 2D phononic crtsyals, which means the un_k should be a 4 dimension matrix, how do you calculate gradient abour k for such matrix?

Wish to here from you, thank you!

Best,

Guifeng