  In quantum spin systems, singlet phases often develop in the vicinity of an
antiferromagnetic order. Typical settings for such problems arise when
itinerant fermions are also present. In this work, we develop a theoretical
framework for addressing such competing orders in an itinerant system,
described by Dirac fermions strongly coupled to an O(3) nonlinear sigma model.
We focus on two spatial dimensions, where upon disordering the
antiferromagnetic order by quantum fluctuations the singular tunneling events
also known as (anti)hedgehogs can nucleate competing singlet orders in the
paramagnetic phase. In the presence of an isolated hedgehog configuration of
the nonlinear sigma model field, we show that the fermion determinant vanishes
as the dynamic Euclidean Dirac operator supports fermion zero modes of definite
chirality. This provides a topological mechanism for suppressing the tunneling
events. Using the methodology of quantum chromodynamics, we evaluate the
fermion determinant in the close proximity of magnetic quantum phase
transition, when the antiferromagnetic order parameter field can be described
by a dilute gas of hedgehogs and antihedgehogs. We show how the precise nature
of emergent singlet order is determined by the overlap between dynamic fermion
zero modes of opposite chirality, localized on the hedgehogs and antihedgehogs.
For a Kondo-Heisenberg model on the honeycomb lattice, we demonstrate the
competition between spin Peierls order and Kondo singlet formation, thereby
elucidating its global phase diagram. We also discuss other physical problems
that can be addressed within this general framework.
