  Disorder operators are a type of non-local observables for quantum many-body
systems, measuring the fluctuations of symmetry charges inside a region. It has
been shown that disorder operators can reveal global aspects of many-body
states that are otherwise difficult to access through local measurements. We
study disorder operator for U(1) (charge or spin) symmetry in 2D Fermi and
non-Fermi liquid states, using the multidimensional bosonization formalism. For
a region $A$, the logarithm of the charge disorder parameter in a Fermi liquid
with isotropic interactions scales asympototically as $l_A\ln l_A$, with $l_A$
being the linear size of the region $A$. We calculate the proportionality
coefficient in terms of Landau parameters of the Fermi liquid theory. We then
study models of Fermi surface coupled to gapless bosonic fields realizing
non-Fermi liquid states. In a simple spinless model, where the fermion density
is coupled to a critical scalar, we find that at the quantum critical point,
the scaling behavior of the charge disorder operators is drastically modified
to $l_A \ln^2 l_A$. We also consider the composite Fermi liquid state and argue
that the charge disorder operator scales as $l_A$.
