According to the conventional definition the Fermi surface in momentum space surrounds the positions of occupied particle states at zero temperature. This definition has the clear meaning  for the system of noninteracting Fermi-particles in the absence of inhomogeneities. 

In the presence of interactions the Fermi surface may be defined as the position of the poles of the Green function in momentum space. This definition is more general but it still cannot be applied in the presence of disorder or other sources of inhomogeneities. Then instead of the Green's function depending on momentum we can consider the Wigner transformed Green's function depending both on momentum and on coordinates. Its singularities in momentum space (or singularities of certain quantities composed of it) may   be considered as the space dependent Fermi surface. 
In the homogeneous systems the topological invariant responsible for the stability of the Fermi surface (or Fermi point) is composed of the one-particle Green function $G(p)$. It has the form of an integral along the three – dimensional hypersurface embracing the Fermi surface/Fermi point in momentum space. This expression may be extended to the case of the inhomogeneous systems. Then the form of the given expression is similar to that of the homogeneous system. There the Green's function $G(p)$ should be replaced by the Wigner transformed Green's function $G_W(p,x)$, and the ordinary products are to be replaced by the Moyal products $\star$. The integration over the coordinate space is to be added, while integration in momentum space should be performed along the hypersurface surrounding the positions of Fermi surfaces. Unlike the case of homogeneous system, in general case of the inhomogeneous system we cannot restrict integration to the compact hypersurface. It may have the form of the two hyperplanes of opposite orientations with constant Matsubara frequency $p_4= ±\epsilon$  ($\epsilon \to 0$, it is supposed that the Fermi surfaces are situated between these two hyperplanes):
\begin{eqnarray}
    {\cal N}&&
    =  {-} \frac{\epsilon_{ijk4}}{ |{\bf V}| \,3!\,4\pi^2}\, \int d^3x \int  d^3p
    \, {\rm tr}\, \hat{T} {G}_{W}(x,p )\nonumber\\&&\star \frac{\partial {Q}_{W}(x,p )}{\partial p_i} \star \frac{\partial  {G}_{W}(x,p )}{\partial p_j} \star \frac{\partial  {Q}_{W}(x,p )}{\partial p_k}
    \label{calM2d230I}
\end{eqnarray}
By $Q_W$ we denote the Weyl symbol of Dirac operator that is a solution of equation $Q_W  \star G_W=1$. Here $\bf V$  is the volume of the given system,  $\hat T$ is a symmetry matrix that commutes with both $Q_W$ and $G_W$. The above expression may also be rewritten in the form of an integral over infinite hypersurface $\Sigma$ embracing the Fermi surface that appears as deformation of the two hyperplanes mentioned above. 

There are certain questions concerning the above defined quantity. First of all, for the Dirac fermions with $\hat{T} = \gamma^5$  it enters expression for the conductivity of the chiral separation effect. Next, if the system may be continuously deformed to the uniform one, this topological invariant is reduced to the previously known quantity composed of the one-particle Green's function $G(p)$. In this case the nontrivial value of $\cal N$  is obtained due to the matrix structure of the Green's function and reveals analogy with the degree of mapping. An important question is whether exist the systems, in which the nontrivial value of $\cal N$  is not due to the matrix structure, but due to the spatial dependence of $G_W (p,x)$. We give examples of such systems in the present paper. Besides, there are certain (still unsolved) pure mathematical questions concerning the above mentioned expression of $\cal N$ itself. We are trying to address some of these questions as well. 

It is worth mentioning that the present paper does not represent the exhaustive description of all issues related to the given topological invariant. These issues still await their investigation. 


 

M. Zubkov
JETP Letters 122, issue 8 (2025)