The Fermi surface is a surface in the momentum space characterizing the ground state of a Fermi liquid. It is defined as the surface of discontinuity of the momentum distribution function of the constituent particles.
In the Fermi gas model, describing a homogeneous system of non-interacting fermions, the ground state distribution of particles is given by the Fermi-Dirac function 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 In a system of interacting fermions the momentum distribution function is defined as the vacuum expectation value 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 Landau Fermi liquid $n(\mathbf k)$ 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