  We continue the study of the Tomita-Takesaki modular conjugation for a
massless Dirac field in a generic multicomponent region in $1+1$ spacetime
dimensions. In this paper we focus on the computations for a thermal state on a
circle, namely on the euclidean torus. By analytic continuation from the
modular flow we arrive at an explicit expression for the modular conjugation in
this scenario and derive its relevant limits. In contrast to the case of the
vacuum on the line, this new result has a non-local behaviour even for
connected regions. It also presents a novel contribution coming from the
purification one has to introduce in order to deal with a mixed state: a term
that maps the algebra of operators of the region to a copy of the global one,
the so called 'second world' algebra.
