  The physics of interacting quantum wires has attracted a lot of attention
recently. When the density of electrons in the wire is very low, the strong
repulsion between electrons leads to the formation of a Wigner crystal. We
review the rich spin and orbital properties of the Wigner crystal, both in the
one-dimensional and quasi-one-dimensional regime. In the one-dimensional Wigner
crystal the electron spins form an antiferromagnetic Heisenberg chain with
exponentially small exchange coupling. In the presence of leads the resulting
inhomogeneity of the electron density causes a violation of spin-charge
separation. As a consequence the spin degrees of freedom affect the conductance
of the wire. Upon increasing the electron density, the Wigner crystal starts
deviating from the strictly one-dimensional geometry, forming a zigzag
structure instead. Spin interactions in this regime are dominated by ring
exchanges, and the phase diagram of the resulting zigzag spin chain has a
number of unpolarized phases as well as regions of complete and partial spin
polarization. Finally we address the orbital properties in the vicinity of the
transition from a one-dimensional to a quasi-one-dimensional state. Due to the
locking between chains in the zigzag Wigner crystal, only one gapless mode
exists. Manifestations of Wigner crystal physics at weak interactions are
explored by studying the fate of the additional gapped low-energy mode as a
function of interaction strength.
