Sommerfeld expansion

A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fermi–Dirac distribution.

When the inverse temperature ${\displaystyle \beta }$ is a large quantity, the integral can be expanded in terms of ${\displaystyle \beta }$ as

${\displaystyle \int _{-\infty }^{\infty }{\frac {H(\varepsilon )}{e^{\beta (\varepsilon -\mu )}+1}}\,\mathrm {d} \varepsilon =\int _{-\infty }^{\mu }H(\varepsilon )\,\mathrm {d} \varepsilon +{\frac {\pi ^{2}}{6}}\left({\frac {1}{\beta }}\right)^{2}H^{\prime }(\mu )+O\left({\frac {1}{\beta \mu }}\right)^{4}}$

where ${\displaystyle H^{\prime }(\mu )}$ is used to denote the derivative of ${\displaystyle H(\varepsilon )}$ evaluated at ${\displaystyle \varepsilon =\mu }$ and where the ${\displaystyle O(x^{n})}$ notation refers to limiting behavior of order ${\displaystyle x^{n}}$. The expansion is only valid if ${\displaystyle H(\varepsilon )}$ vanishes as ${\displaystyle \varepsilon \rightarrow -\infty }$ and goes no faster than polynomially in ${\displaystyle \varepsilon }$ as ${\displaystyle \varepsilon \rightarrow \infty }$. If the integral is from zero to infinity, then the integral in the first term of the expansion is from zero to ${\displaystyle \mu }$ and the second term is unchanged.