  An unusual correlation function is conjectured by M. Campostrini et al.
(Phys. Rev. E 91, 042123 (2015)) for the ground state of a transverse Ising
chain with geometrical frustration in one of the translationally invariant
cases. Later, we demonstrated the correlation function and showed its non-local
nature in the thermodynamic limit based on the rigorous evaluation of a
Toeplitz determinant (J. Stat. Mech. 113102 (2016)). In this paper, we prove
rigorously that all the states that forming the lowest gapless spectrum
(including the ground state) in the kink phase exhibit the same asymptotic
correlation function. So, in a point of view of cannonical ensemble, the
thermal correlation function is inert to temperature within the energy range of
the lowest gapless spectrum.
