Efficient computation of Hopf invariant

Question

I am trying to check using Mathematica the following result: $$-\frac{1}{4\pi^2}\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{2\pi} \mathbf{F}\cdot\mathbf{A}\;dk_x dk_y dk_z=1,\quad\text{where}$$ $$z_{\uparrow}=\sin k_x+i\sin k_y,\quad z_{\downarrow}=\sin k_z+i(\cos k_x+\cos k_y+\cos k_z-3/2),$$ $$\mathbf{A}=i(z_{\uparrow} \; z_{\downarrow})^{*}\mathbf{\nabla} (z_{\uparrow} \; z_{\downarrow})^{\intercal},$$ $$\mathbf{F}=\mathbf{\nabla}\times\mathbf{A}.$$

So far I wrote

zup0 = Sin[kx] + I Sin[ky];
zdo0 = Sin[kz] + I (Cos[kx] + Cos[ky] + Cos[kz] - 3/2);

zup = zup0/Norm[{zup0, zdo0}] // ComplexExpand // Simplify;
zdo = zdo0/Norm[{zup0, zdo0}] // ComplexExpand // Simplify;

ax = I Conjugate[{zup, zdo}]. {D[zup,kx], D[zdo,kx]};
ay = I Conjugate[{zup, zdo}]. {D[zup,ky], D[zdo,ky]};
az = I Conjugate[{zup, zdo}]. {D[zup,kz], D[zdo,kz]};

fx = D[az,ky]-D[ay,kz];
fy = D[ax,kz]-D[az,kx];
fz = D[ay,kx]-D[ax,ky];

integrand = ax fx + ay fy + az fz;

result = NIntegrate[integrand, {kx,0,2Pi}, {ky,0,2Pi}, {kz,0,2Pi}]

This code returns the non-numerical value error and I suspect that this happens because of the clumsy use of Conjugate and D. How can I fix the problem? I will also appreciate further suggestions to improve my code, including how to use ComplexExpand and Simplify to enhance computational efficiency.

Answer 1

There are multiple improvements for your code, the most obvious ones being you could use Grad and Curl to compute {ax,ay,az} and {fx,fy,fz} in one line each, and also use Normalize for {zup0,zdo0}. Combining all of this, your code can be "vectorized" to be much more concise and mathematical-looking.

You also may want to use Simplify to simplify intermediate expressions assuming 0<kx,ky,kz<2Pi, instead of using ComplexExpand, to avoid having Conjugate's and Arg's popping up everywhere.

With all of the above, Mathematica can check the identity albeit after some hard numerical integration work:

z = {Sin[kx] + I Sin[ky], Sin[kz] + I (Cos[kx] + Cos[ky] + Cos[kz] - 3/2)}//Normalize//ComplexExpand//Simplify;

a = I Conjugate[z] . Grad[z, {kx, ky, kz}]//Simplify[#, 0 < kx < 2 \[Pi] && 0 < ky < 2 \[Pi] && 0 < kz < 2 \[Pi]] &;

f = Curl[a, {kx, ky, kz}]//Simplify[#, 0 < kx < 2 \[Pi] && 0 < ky < 2 \[Pi] && 0 < kz < 2 \[Pi]] &;

integrand = a . f // Simplify;

NIntegrate[integrand, {kx, 0, 2 Pi}, {ky, 0, 2 Pi}, {kz, 0, 2 Pi}]
(*-39.4784*)
```

Answer 2

Integrate localizes its variables. Therefore the variables kx,ky,kz in integrand and the integration variables of NIntegrate are not the same. What you can do is to write integrand as a function like:

integrand[kx_, ky_, kz_] = ax fx + ay fy + az fz;
result = NIntegrate[
  integrand[kx, ky, kz], {kx, 0, 2 Pi}, {ky, 0, 2 Pi}, {kz, 0, 2 Pi}]

However, there is a more serious second problem. The derivative of Conjugate is not defined. You need to rewrite this without using Conjugate.