Geometric interpretation of the Hall conductivity as a directed area

The textbook expression for the Hall conductivity *s*_{xy} 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 *s*_{xy} 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 *f*_{0} = *h/e* threading the area *A*.
[This is true for arbitrary FS shape and *k*-dependence of the mfp **l**(**k**).]
We have

*s*_{xy}/[*e*^{2}/*h*]
= 2*BA*/*f*_{0}.

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 *s*_{xy}.
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).