  We study the effect of disorder on frustrated dimerized spin-1/2 chains at
the Majumdar-Ghosh point. Using variational methods and density-matrix
renormalization group approaches, we identify two localization mechanisms for
spinons which are the deconfined fractional elementary excitations of these
chains. The first one belongs to the Anderson localization class and dominates
at the random Majumdar-Ghosh (RMG) point. There, spinons are almost
independent, remain gapped, and localize in Lifshitz states whose localization
length is analytically obtained. The RMG point then displays a quantum phase
transition to phase of localized spinons at large disorder. The other mechanism
is a random confinement mechanism which induces an effective interaction
between spinons and brings the chain into a gapless and partially polarized
phase for arbitrarily small disorder.

Strongly Correlated Electrons