We show how Oshikawa's theorem for the Fermi surface volume of the Kondo lattice can be extended to the SU$(N)$ symmetric case. By extending the theorem, we are able to show that the mechanism of Fermi surface expansion seen in the large $N$ mean-field theory is directly linked to the expansion of the Fermi surface in a spin-$1/2$ Kondo lattice. This linkage enables us to interpret the expansion of the Fermi surface in a Kondo lattice as a fractionalization of the local moments into heavy electrons. Our method allows extension to a pure U(1) spin liquid, where we find the volume of the spinon Fermi surface by applying a spin-twist, analogous to Oshikawa's flux insertion. Lastly, we discuss the possibility of interpreting the FL$^*$ phase characterised by a small Fermi surface in the absence of symmetry breaking, as a non-topological coexistence of such a U(1) spin liquid and an electronic Fermi liquid.
Categories: cond-mat.str-el cond-mat.supr-con