  Quantum spin liquids are exotic states of matter that may be realized in
frustrated quantum magnets and feature fractionalized excitations and emergent
gauge fields. Here, we consider a gapless U(1) spin liquid with spinon Fermi
pockets in two spatial dimensions. Such a state appears to be the most
promising candidate to describe the exotic field-induced behavior observed in
numerical simulations of the antiferromagnetic Kitaev honeycomb model. A
similar such state may also be responsible for the recently-reported quantum
oscillations of the thermal conductivity in the field-induced quantum
paramagnetic phase of $\alpha$-RuCl$_3$. We consider the regime close to a
Lifshitz transition, at which the spinon Fermi pockets shrink to small circles
around high-symmetry points in the Brillouin zone. By employing renormalization
group and mean-field arguments, we demonstrate that interactions lead to a gap
opening in the spinon spectrum at low temperatures, which can be understood as
a nesting instability of the spinon Fermi surface. This leads to proliferation
of monopole operators of the emergent U(1) gauge field and confinement of
spinons. While signatures of fractionalization may be observable at finite
temperatures, the gapless U(1) spin liquid state with nested spinon Fermi
pockets is ultimately unstable at low temperatures towards a conventional
long-range-ordered ground state, such as a valence bond solid. Implications for
Kitaev materials in external magnetic fields are discussed.
