  Electronic, magnetic or structural inhomogeneities ranging in size from
nanoscopic to mesoscopic scales seem endemic, and are possibly generic, to
colossal magnetoresistance manganites and other transition metal oxides. We
show here that an extension, to include long range Coulomb interactions, of a
quantum two-fluid $\ell-b$ model proposed recently for manganites [Phys. Rev.
Lett., {\bf 92}, 157203 (2004)] leads to an excellent description of such
inhomogeneities. In the $\ell-b$ model two very different kinds of electronic
states, one localized and polaronic ($\ell$), and the other extended or broad
band ($b$) co-exist. For model parameters appropriate to manganites, and even
within a simple dynamical mean-filed theory (DMFT) framework, it describes many
of the unusual phenomena seen in manganites, including colossal
magnetoresistance (CMR), qualitatively and quantitatively. However, in the
absence of long ranged Coulomb interaction, a system described by such a model
would actually phase separate, into macroscopic regions of $l$ and $b$
electrons respectively. As we show in this paper, in the presence of Coulomb
interactions, the {\em macroscopic} phase separation gets suppressed, and
instead nanometer scale regions of polarons interspersed with band electron
puddles appear, constituting a new kind of quantum Coulomb glass. Our work
points to an interplay of strong correlations, long range Coulomb interaction
and dopant ion disorder as the origin of nanoscale inhomogeneities, rather than
disorder frustrated phase competition as is generally believed. Based on this,
we argue that the observed micrometer(meso)-scale inhomogeneities owe their
existence to extrinsic causes, eg. strain due to cracks and defects. We suggest
possible experiments to validate our speculation.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics