  Recent theoretical studies have found quantum spin liquid states with spinon
Fermi surfaces upon the application of a magnetic field on a gapped state with
topological order. We investigate the thermal Hall conductivity across this
transition, describing how the quantized thermal Hall conductivity of the
gapped state changes to an unquantized thermal Hall conductivity in the gapless
spinon Fermi surface state. We consider two cases, both of potential
experimental interest: the state with non-Abelian Ising topological order on
the honeycomb lattice, and the state with Abelian chiral spin liquid
topological order on the triangular lattice.
