  Dislocation pileups directly impact the material properties of crystalline
solids through the arrangement and collective motion of interacting
dislocations. We study the statistical mechanics of these ordered defect
structures embedded in two dimensional crystals, where the dislocations
themselves form one-dimensional lattices. In particular, pileups exemplify a
new class of inhomogeneous crystals characterized by spatially varying lattice
spacings. By analytically formulating key statistical quantities and comparing
our theory to numerical experiments using an intriguing mapping of dislocation
positions onto the eigenvalues of recently studied random matrix ensembles, we
uncover two types of one-dimensional phase transitions in dislocation pileups:
a thermal depinning transition out of long-range translational order from the
pinned-defect phase, due to a periodic Peierls potential, to a floating-defect
state, and finally the melting out of a quasi-long range ordered floating
defect-solid phase to a defect-liquid. We also find the set of transition
temperatures at which these transitions can be directly observed through the
one-dimensional structure factor, where the delta function Bragg peaks, at the
pinned-defect to floating-defect transition, broaden into algebraically
diverging Bragg peaks, which then sequentially disappear as one approaches the
two-dimensional melting transition of the host crystal. We calculate a set of
temperature-dependent critical exponents for the structure factor and radial
distribution function, and obtain their exact forms for both uniform and
inhomogeneous pileups using random matrix theory.
