  Heavy fermion systems represent a prototypical setting to study magnetic
quantum phase transitions. A particular focus has been on the physics of Kondo
destruction, which captures quantum criticality beyond the Landau framework of
order-parameter fluctuations. In this context, we study the spin one-half
Kondo-Heisenberg model on a honeycomb lattice at half filling. The problem is
approached from the Kondo destroyed, antiferromagnetically ordered insulating
phase. We describe the local moments in terms of a coarse grained quantum
non-linear sigma model, and show that the skyrmion defects of the
antiferromagnetic order parameter host a number of competing order parameters.
In addition to the spin Peierls, charge and current density wave order
parameters, we identify for the first time Kondo singlets as the competing
orders of the antiferromagnetism. We show that the antiferromagnetism and
various competing singlet orders can be related to each other via generalized
chiral transformations of the underlying fermions. We also show that the
conduction electrons acquire a Berry phase through their coupling to the
hedgehog configurations of the N\'eel order, which cancels the Berry phase of
the local moments. Our results demonstrate the competition between the
Kondo-singlet formation and spin-Peierls order when the antiferromagnetic order
is suppressed, thereby shedding new light on the global phase diagram of heavy
fermion systems at zero temperature.
