In ideal Fermi gases, the Fermi surface at p = p_{F} = √2μm is the boundary in p-space between the occupied states (n_{p} = 1) at p^{2}/2m < μ and empty states (n_{p} = 0) at p^{2}/2m > μ. At this boundary (the surface in 3D momentum space) the energy is zero. What happens when the interaction between particles is introduced? Due to interaction the distribution function n_{p} of particles in the ground state is no longer exactly 1 or 0. However, it appears that the Fermi surface survives as the singularity in n_{p}. Such stability of the Fermi surface comes from a topological property of the one-particle Green’s function at imaginary frequency:

G^{-1} = iω – p^{2}/2m + μ —– (1)

Let us for simplicity skip one spatial dimension p_{z} so that the Fermi surface becomes the line in 2D momentum space (p_{x},p_{y}); this does not change the co-dimension of zeroes which remains 1 = 3−2 = 2−1. The Green’s function has singularities lying on a closed line ω = 0, p^{2}_{x} + p^{2}_{y} = p^{2}_{F} in the 3D momentum-frequency space (ω,p_{x},p_{y}). This is the line of the quantized vortex in the momentum space, since the phase Φ of the Green’s function G = |G|e^{iΦ} changes by 2πN_{1} around the path embracing any element of this vortex line. In the considered case the phase winding number is N_{1} = 1. If we add the third momentum dimension p_{z} the vortex line becomes the surface in the 4D momentum-frequency space (ω,p_{x},p_{y},p_{z}) – the Fermi surface – but again the phase changes by 2π along any closed loop empracing the element of the 2D surface in the 4D momentum-frequency space.

Fermi surface is a topological object in momentum space – a vortex loop. When the chemical potential μ decreases the loop shrinks and disappears at μ < 0. The point μ = T = 0 marks the Lifshitz transition between the gapless ground state at μ > 0 to the fully gapped vacuum at μ < 0.

The winding number cannot change by continuous deformation of the Green’s function: the momentum-space vortex is robust toward any perturbation. Thus the singularity of the Green’s function on the Fermi surface is preserved, even when interaction between fermions is introduced. The invariant is the same for any space dimension, since the co-dimension remains 1.

The Green function is generally a matrix with spin indices. In addition, it may have the band indices (in the case of electrons in the periodic potential of crystals). In such a case the phase of the Green’s function becomes meaning-less; however, the topological property of the Green’s function remains robust. The general analysis demonstrates that topologically stable Fermi surfaces are described by the group Z of integers. The winding number N_{1} is expressed analytically in terms of the Green’s function:

N_{1 }= tr ∮_{C} dl/2πi G(μ,p) ∂_{l}G^{-1}(μ,p) —– (2)

Here the integral is taken over an arbitrary contour C around the momentum- space vortex, and tr is the trace over the spin, band and/or other indices.

The Fermi surface cannot be destroyed by small perturbations, since it is protected by topology and thus is robust to perturbations. But the Fermi surface can be removed by large perturbations in the processes which reproduces the processes occurring for the real-space counterpart of the Fermi surface – the loop of quantized vortex in superfluids and superconductors. The vortex ring can continuously shrink to a point and then disappear, or continuously expand and leave the momentum space. The first scenario occurs when one continuously changes the chemical potential from the positive to the negative value: at μ < 0 there is no vortex loop in momentum space and the ground state (vacuum) is fully gapped. The point μ = 0 marks the quantum phase transition – the Lifshitz transition – at which the topology of the energy spectrum changes. At this transition the symmetry of the ground state does not changes. The second scenario of the quantum phase transition to the fully gapped states occurs when the inverse mass 1/m in (1) crosses zero.

Similar Lifshitz transitions from the fully gapped state to the state with the Fermi surface may occur in superfluids and superconductors. This happens, for example, when the superfluid velocity crosses the Landau critical velocity. The symmetry of the order parameter does not change across such a quantum phase transition. In the non-superconduting states, the transition from the gapless to gapped state is the metal-insulator transition.

The Lifshitz transitions involving the vortex lines in p-space may occur be- tween the gapless states. They are accompanied by the change of the topology of the Fermi surface itself. The simplest example of such a phase transition discussed in terms of the vortex lines is provided by the reconnection of the vortex lines.