  Two powerful theoretical predictions, Anderson localization and the Imry-Ma
argument, impose significant restrictions on the phases of matter that can
exist in the presence of even the smallest amount of disorder in
one-dimensional systems. These predictions forbid conducting states and ordered
states respectively. It was thus remarkable that a mechanism to circumvent
Anderson localization relying on the presence of correlated disorder was found,
that is also realized in certain biomolecular systems. In a similar manner, we
show that the Imry-Ma argument can be circumvented resulting in the formation
of stable ordered states with discrete broken symmetries in disordered one
dimensional systems. Specifically, we simulate a family of Hamiltonians of
spinless fermions with correlated disorder and interactions, where we find that
a charge density wave is stable up to a finite critical disorder strength.
Having circumvented the Imry-Ma mechanism, we then investigate other mechanisms
by which disorder can destroy an ordered state.
