TABLE OF CONTENTS In this chapter we shall remind the reader of some basic ideas of thermodynamics and statistical physics of interacting electron and spin systems. We shall show how thermal fluctuations destroy long-range order in low- dimensional quantum interacting systems at any nonzero temperature if only short-range interactions are present.
Low-dimensional electron systems (insulating magnets and conductors) have been an active topic of scientific research long before their experimental realization in organic conductors, polymers, Peierls insulators and nanoscale and mesoscopic systems, e.g., quantum wires and edge states of the fractional quantum Hall effect devices. There are several principal differences between one space dimension and higher dimensions, most of which can be traced back to the reduced phase space in one dimension. Key properties distinguishing one-dimensional systems used to be connected with thermal fluctuations destroying long-range order at any nonzero temperature if only short-range interactions are present and quantum fluctuations tending to suppress a broken continuous symmetry, the spin-charge separation (the charge and spin content of wave functions of electrons move with different speeds), the breakdown of the Fermi liquid description, i.e.,absence of Fermi liquid quasiparticle pole in the Green's function (it becomes a marginal Fermi liquid or Tomonaga Luttinger liquid with collective excitations due to global conformal symmetry), and the localization of electrons with even a small amount of disorder.
During recent years the interest in the strongly correlated electron and quantum spin systems has grown...
