This is another one of those examples where textbooks always just gloss over it with the remark that it "can be done" and then just state the result.
I want to compute the general form of a van Hove singularity if the dispersion relation expanded to second order has a saddle point. As a simple example, consider
$$E = E_0 + a_x k_x^2 – a_y k_y^2$$
where the coefficients $a_x$ and $a_y$ are both positive.
I understand how the derivation works when all coefficients are negative or all coefficients are positive, because then the surface of constant energy is actually finite: It's an ellipsoid and we can estimate its volume.
But how to proceed for saddle point?
In the 2D-example above, I have
$$D(E) \propto \int_{E(k_x,k_y) = E} \frac{1}{\sqrt{a_x^2 k_x^2 + a_y^2 k_y^2}}$$
and I'm not sure how to go from there. Do I actually find a parametrization for that path and compute that integral or is there a better way to arrive at the approximate form?
Best Answer
The complete expression for the integral is $$ D(E) = \frac{1}{4\pi^2}\int_{E(k_x,k_y)=E} \frac{d\mathbf{l}}{\left|\nabla_k E(k_x,k_y)\right|} $$ This is the integration over a curve which in your case is described by the following equation: $$ E = E_0 + a_x k_x^2 - a_y k_y^2 $$
Depending on what energy value you substitute, you should select either $k_x$ or $k_y$ as the integration variable. The variable selected should change continuously from $-\infty$ to $+\infty$ on the curve.
This is just for convenience. One variable is convenient on the one side of the singularity, another is on the other side.
If you select say $k_x$ the integral will look as follows: $$ D(E) = 4\cdot\frac{1}{4\pi^2}\int_0^\infty \frac{1}{\left|\nabla_k E\bigl(k_x,k_y(k_x)\bigr)\right|} \left|\frac{d\mathbf{l}(k_x)}{dk_x}\right| dk_x $$ This is a usual one-dimensional integral.
The factor $4$ is here because there are four identical parts of the curve while we are integrating over only one.