  The Zeeman effect lowers the energy of electrons with spin states which are
anti-parallel to the applied magnetic field but lifts that of spin parallel
states. In quantum spin liquids where the spin and charge degrees of freedom
are fractionalized, anomalous Zeeman response may be expected. In the case of
spin liquids with spinon Fermi surface, the threshold energy to excite an
electronic state is found to exhibit no Zeeman shift. This is specific to the
spinon Fermi surface case. In contrast, other gapped spin liquids are expected
to exhibit the standard Zeeman shift at the band edge even though they also
exhibit spin-charge fractionalization. When gauge field fluctuations are
included, we find that the Zeeman shift of the electronic states gets affected
by the gauge field induced binding. In the electronic density of states
spectra, weak gauge binding induces band edge resonance peaks which exhibit the
Zeeman shift in the same direction as that in the standard Zeeman effect, but
the shift is reduced as the binding potential increases. With further increase
in the binding potential the resonance becomes true in-gap bound states and
eventually the shift direction reverses so it is opposite to the standard
Zeeman effect. We propose that one can perform spin polarized scanning
tunneling microscope measurements as a test of the spinon Fermi sea ground
state in quantum spin liquid candidate materials.
