Does spatial homogeneity break down in strongly correlated electron systems?

In considering the electronic properties of the metallic phase of solids one almost always assumes spatial homogeneity of the the underlying electron fluid.
This is convenient and powerful. But, that does not mean that it is always correct!
Occasionally one might entertain the possibility of some sort of charge order and symmetry breaking such as a charge density wave or stripes.

Over the past two decades it has been found that the two-dimensional electron gases (2DEGs) that occur at interfaces in semiconductor heterostructures exhibit a metal-insulator transition that has confounded definitive theoretical understanding. Furthermore, there has been considerable debate about the relative importance and interplay of disorder (due to impurities) and strong electronic correlations.

A fundamental challenge is to describe the dependence of the resistance on the temperature, density, and magnetic field (parallel to the 2DEG). Furthermore, double layer systems exhibit extremely large and unanticipated "drag" resistance: current is passed through one layer and the

A nice review of the experiments is in a 2010 Rev. Mod. Phys. Colloquium article by Boris Spivak, Sergei Kravchenko, Steve Kivelson, and X.P.A. Gao.

The authors discuss the shortcomings of different theories and make the novel proposal that the key relevant physics may be the existence of new phases ["microemulsion phases"] between the Fermi liquid metal and Wigner crystal. The relevant [speculative and schematic] phase diagram is shown below. The horizontal axis is density. The vertical axis is 1/d where d is the distance between the 2DEG and a metallic ground plane in a ideal MOSFET.

A key to understanding the temperature dependent properties is the Pomeranchuk effect [analogous to that in 3He] where the Wigner solid has a larger entropy than the Fermi liquid due to spin entropy.

Spivak and Kivelson have proposed that the large drag resistance arises from a phase consisting of mobile bubbles of Wigner crystal embedded in a Fermi liquid.

A key to testing these novel explanations experimentally is the development of new probes for the direct imaging of the spatial inhomogenities associated with the microemulsion phases.

A recent post Is hydrodynamics ever relevant in metals? considered a more recent theory of Andreev, Kivelson, and Spivak to describe transport properties above the Fermi liquid degeneracy temperature.

A previous post Deconstructing the metal-insulator transition in 2DEGs considered an alternative theory of the temperature dependence of the electrical resistance, based on a Dynamical Mean-Field Theory treatment of an extended Hubbard model.