  Spontaneously symmetry-broken (SSB) phases are locally ordered states of
matter characterizing a large variety of physical systems. Because of their
specific ordering, their presence is usually witnessed by means of local order
parameters. Here, we propose an alternative approach based on statistical
correlations of noise after the ballistic expansion of an atomic cloud. We
indeed demonstrate that probing such noise correlators allows one to
discriminate among different SSB phases characterized by spin-charge
separation. As a particular example, we test our prediction on a 1D extended
Fermi-Hubbard model, where the competition between local and nonlocal couplings
gives rise to three different SSB phases: a charge density wave, a
bond-ordering wave, and an antiferromagnet. Our numerical analysis shows that
this approach can accurately capture the presence of these different SSB
phases, thus representing an alternative and powerful strategy to characterize
strongly interacting quantum matter.
