Integration of these kind of functions

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Luqman Saleem 2024-1-16

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评论： Luqman Saleem 2024-1-17

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I want to integrate

is a 2 by 2 matrix (given below),

is identity matrix of order 2, and Eand η and positive real numbers. I want to calculate this integration (for each component of the matrix) with

Note that the integrand does not diverges as long as η is non-zero, however when η is significantly small, the integrand can be very large for some points in

plane. As an example, I have plot the imaginary part of (1,2) componet of the integrand matrix for

and

below. (the other components also follow the same behavior):

It can be seen that when η is small the integrand is fairly smooth and can be integrated easily even with integrate2 function. But when η is small, the integrand has very sharp peaks due to which integral2 fails. To calculate the correct integration with small η by trapz, we need a lot of grid points which makes everything very slow.

Is there a way to calculate this integration with

I have analytically calculated the condition for which the integrand shows sharp peaks, that condition is:

where

are components of matrix H. We could also find the exact

points on which the condition is satisfied.

where x are the roots of equation

and y is all values between -1 and +1. And

So, for each y there will be 4 roots, x, which will give eight

pionts (four

and four

. From these eight

pionts, we will take only the real

; imaginary

must be neglected. Also for each y there will be two

points (one

and one

Is there any way to tell MATLAB to take a lot of gird pionts near

points and then calculate the integration? Or is there any way to calculate this type of integration?

Code:

clear; clc;
% parameters
E = 4.4;
eta = 0.1;5e-4; 
% kx and ky limits and points
dkx = 0.006;
dky = dkx;
xmin = -2*pi/(3*sqrt(3));
xmax = 4*pi/(3*sqrt(3));
ymin = -2*pi/3;
ymax = 2*pi/3;
kxs = xmin:dkx:xmax;
kys = ymin:dky:ymax;
NBZx = length(kxs);
NBZy = length(kys);
% H matrix:
J = 1;
Dz = 0.5;
S = 1;
s0 = eye(2);
sx = [0, 1; 1, 0];
sy = [0, -1i; 1i, 0];
sz = [1, 0; 0, -1];
h0 = 3 * J * S;
hx = @(kx, ky) -J * S * (cos(ky / 2 - (3^(1/2) * kx) / 2) + cos(ky / 2 + (3^(1/2) * kx) / 2) + cos(ky));
hy = @(kx, ky) -J * S * (sin(ky / 2 - (3^(1/2) * kx) / 2) + sin(ky / 2 + (3^(1/2) * kx) / 2) - sin(ky));
hz = @(kx, ky) -2 * Dz * S * (sin(3^(1/2) * kx) + sin((3 * ky) / 2 - (3^(1/2) * kx) / 2) - sin((3 * ky) / 2 + (3^(1/2) * kx) / 2));
H = @(kx, ky) s0 * h0 + sx * hx(kx, ky) + sy * hy(kx, ky) + sz * hz(kx, ky);
%integrand:
G00 = @(kx, ky) inv(E*eye(2) - H(kx,ky) + 1i*eye(2)*eta); %integrand