\documentclass{article}
\usepackage{amssymb,amsmath}
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{eso-pic}
\usepackage{epstopdf}
\usepackage{type1cm}
\usepackage[utf8x]{inputenc}
\newcommand{\TeXForWeb}{\TeX$^4$Web}
\newcommand{\cref}[2][\relax]{\href{http://sciencewise.info/ontology/#2}{\ifx#1\relax#2\else#1\fi}}
\newcommand{\dref}[2][\relax]{\href{http://sciencewise.info/definitions/#2}{\ifx#1\relax#2\else#1\fi}}
\newcommand{\fileref}[2][\relax]{\href{http://sciencewise.info//definitions/Fermi_surface_by_Vadim_Cheianov/#2}{\ifx#1\relax#2\else#1\fi}}
\title{Fermi surface by Vadim Cheianov}
\begin{document}
\maketitle
The Fermi surface is a surface in the momentum space characterizing the ground state of a \cref{Fermi liquid}. It is defined as the surface of discontinuity of the momentum distribution function of the constituent particles.
In the \cref{Fermi gas} model, describing a homogeneous system of non-interacting
fermions, the ground state distribution of particles is given by the Fermi-Dirac
function
$$
n(\mathbf k) = \left\{
\begin{array}{ll}
1, & E(\mathbf k)E_F \end{array}
\right.
$$
where $E(\mathbf k)$ is the energy of a particle as a function
of its momentum and $E_F$ is the Fermi energy.
The Fermi surface is found as a solution to the algebraic equation
$$
E_F= E(\mathbf k).
$$
In a system of interacting fermions the momentum distribution function
is defined as the vacuum expectation value
$$
n(\mathbf k) = \langle c^\dagger_{s \mathbf k} c_{s \mathbf k}\rangle ,
$$
where $c^\dagger_{s\mathbf k} ( c_{s \mathbf k} )$
is the creation (annihilation) operator of a fermion with
spin $s$ and momentum $\mathbf k$.
Under conditions defining
the \cref[Landau Fermi liquid]{Fermi liquid theory} $n(\mathbf k)$
experiences a jump at a surface in the reciprocal space defining
the Fermi surface of the interacting system.
Of particular importance is the case of a spherical Fermi surface, which
is called the Fermi sphere. The radius of the Fermi sphere in the reciprocal
space is called the Fermi momentum $p_F$.
The Kohn-Luttinger theorem states that
in the absence of spontaneously broken symmetries the radius of the Fermi
sphere does not depend on interactions between particles. For a system of
spin $s$ fermions at particle density $\rho$ the
Fermi momentum is
$$
p_F= \hbar \left(\frac{6\pi^2 \rho}{2s+1} \right)^{1/3}.
$$
\end{document}