Hi everyone. Suppose we consider an electron in a two dimensional lattice, whose dispersion relation is given by:(adsbygoogle = window.adsbygoogle || []).push({});

$$

\epsilon(k_x,k_y)=-J(\cos(k_x a)+\cos(k_y a)),

$$ and where the wave vectors belong to the first Brillouin zone ([itex]k_i\in [-\pi/a,\pi/a][/itex]).

In this case it turns out that the Fermi energy is zero and the Fermi surface is a square defined by:

\begin{align}

&k_y=-k_x+\pi/a\;\;\;\;\text{ for the first quarter of the k-space} \\

&k_y=k_x-\pi/a\;\;\;\;\text{for the second quarter} \\

&k_y=-k_x+\pi/a\;\;\;\;\text{for the third quarter} \\

&k_y=k_x+\pi/a\;\;\;\;\text{for the fourth quarter}

\end{align}

I would like to compute the density of states near the Fermi surface. The density of states is given by:

\begin{align}

\nu (\epsilon)=\int\frac{d\vec l}{4\pi^2}\frac{1}{|\nabla \epsilon|}.

\end{align}

In the situation I am describing the we have, along the Fermi surface:

\begin{align}

|\nabla \epsilon|=|\sin(k_x a)|.

\end{align}

Since I should integrate from [itex]-\pi/a[/itex] to [itex]\pi/a[/itex] the integral is clearly divergent. This is the so called Van Hove divergency. How do I deal with it?

I am expecting a logarithmic divergency around [itex]\epsilon\simeq0[/itex], how do I find it?

Thank you all

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# Van Hove singularity for a two dimensional lattice

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