  The free energy per lattice site of a quantum spin chain in the thermodynamic
limit is determined by a single `dominant' Eigenvalue of an associated quantum
transfer matrix in the infinite Trotter number limit. For integrable quantum
spin chains, related with solutions of the Yang-Baxter equation, an appropriate
choice of the quantum transfer matrix enables to study its spectrum, e.g.\ by
means of the algebraic Bethe Ansatz. In its turn, the knowledge of the full
spectrum allows one to study its universality properties such as the appearance
of a conformal spectrum in the low-temperature regime. More generally,
accessing the full spectrum is a necessary step for deriving thermal form
factor series representations of the correlation functions of local operators
for the spin chain under consideration. These are statements that have been
established by physicists on a heuristic level and that are calling for a
rigorous mathematical justification. In this work we implement certain aspects
of this programme with the example of the XXZ quantum spin chain in the
antiferromagnetic massless regime and in the low-temperature limit. We
rigorously establish the existence, uniqueness and characterise the form of the
solutions to the non-linear integral equations that are equivalent to the Bethe
Ansatz equations for the quantum transfer matrix of this model. This allows us
to describe that part of the quantum transfer matrix spectrum that is related
to the Bethe Ansatz and that does not collapse to zero in the infinite Trotter
number limit. Within the considered part of the spectrum we rigorously identify
the dominant Eigenvalue and show that those correlations lengths that diverge
in the low-temperature limit are given, to the leading order, by the spectrum
of the free Boson $c=1$ conformal field theory. This rigorously establishes a
long-standing conjecture present in the physics literature.
