Geometric interpretation of the Hall conductivity as a directed area

The textbook expression for the Hall conductivity sxy of a metal (in weak fields) is incredibly unwieldy (it is displayed, for e.g., with a typographical error, as the last equation in Ch. 12 of Kittel’s Quantum Theory of Solids).  For a general Fermi Surface with arbitrary relaxation time anisotropy, it involves the integral of a complicated kernel over the Fermi Surface FS.

The unwieldy expression belies a very simple, geometric interpretation of sxy that has been realized only recently.  Let us trace a closed path in k-space on a FS of arbitrary shape.  The tip of the mean-free-path l(k), a vector of arbitrary length normal to the surface, will describe a closed path G in l-space (the map from k to l is a slight generalization of the Gauss map familiar in differential geometry).

The interpretation is clearest for a 2D Fermi Surface, for which the curve G traced out by l is planar (the mapping is shown for a hypothetical FS in 2D in the figures above).  Let the directed (Stokes) area of the curve shown on the right be A
(unlike the simple situation shown, G may self-intersect if l(k) changes rapidly when the local curvature of the FS changes sign). Segments traversed in a clockwise sense make a positive contribution to A, while anti-clockwise segments make a negative contribution.)  The weak-field Hall conductivity of a 2D metal, expressed in units of universal conductance, just equals the number of flux quanta f0 = h/e threading the area A.  [This is true for arbitrary FS shape and k-dependence of the mfp l(k).]   We have

sxy/[e2/h]  =  2BA/f0.

This simple expression is equivalent to the complicated expression given in standard texts.  Its prescription is purely geometric:  Take the point k around the FS.  The vector l(k) will trace the companion curve G in l-space.  The flux threading A is positive (negative) if the local circulation is clockwise (anti-clockwise).  Count the net number of flux quanta captured.  This gives the Hall conductivity (in units of universal conductance).  For the trivial case of a 2D isotropic, circular FS, it is easy to see that the expression above reduces to the standard Drude sxy.  The Hall conductivity captures a mathematical property that combines the global curvature of the Fermi Surface and its mfp variation.

A derivation with several examples is given in

1) N.P. Ong, Phys. Rev. B 43, 193 (1991).