TABLE OF CONTENTS In this chapter we shall study the next order corrections in L ?1 to thermodynamic characteristics of correlated quantum chains. These corrections are related to quantum topological effects in these chains (like persistent currents) and to the asymptotic behaviour of correlation functions in the conformal limit.
Let us start to consider finite size corrections with the simplest model of a spin- Heisenberg-Ising chain with periodic boundary conditions. Bethe ansatz equations Eq. (3.19) for the state with the z-projection of the total spin S z =( L/2) ? M can be re-written for J z/ J =cos ? as
where we introduced the so-called counting function
with
and quantum numbers J j =( M +1)/2 (mod 1). The energy and the total momentum of this state are defined as
where E 0 is the energy of the ferromagnetic, spin-polarized state (with M = 0). Let us specify the set of quantum numbers J j. Let us choose two numbers J = M/2 (mod 1), so that
For J j we take all the numbers equal to ( M +1)/2 (mod 1) between J + and J ?. This pertains to a Dirac sea of M particles with D particles moved from the left Fermi point to the right one. We can also introduce the function ? L( x)=
