  The presence of ubiquitous magnetic fields in the universe is suggested from
observations of radiation and cosmic ray from galaxies or the intergalactic
medium (IGM). One possible origin of cosmic magnetic fields is the
magnetogenesis in the primordial universe. Such magnetic fields are called
primordial magnetic fields (PMFs), and are considered to affect the evolution
of matter density fluctuations and the thermal history of the IGM gas. Hence
the information of PMFs is expected to be imprinted on the anisotropies of the
cosmic microwave background (CMB) through the thermal Sunyaev-Zel'dovich (tSZ)
effect in the IGM. In this study, given an initial power spectrum of PMFs as
$P(k)\propto B_{\rm 1Mpc}^2 k^{n_{B}}$, we calculate dynamical and thermal
evolutions of the IGM under the influence of PMFs, and compute the resultant
angular power spectrum of the Compton $y$-parameter on the sky. As a result, we
find that two physical processes driven by PMFs dominantly determine the power
spectrum of the Compton $y$-parameter; (i) the heating due to the ambipolar
diffusion effectively works to increase the temperature and the ionization
fraction, and (ii) the Lorentz force drastically enhances the density contrast
just after the recombination epoch. These facts result in making the tSZ
angular power spectrum induced by the PMFs more remarkable at $\ell >10^4$ than
that by galaxy clusters even with $B_{\rm 1Mpc}=0.1$ nG and $n_{B}=-1.0$
because the contribution from galaxy clusters decreases with increasing $\ell$.
The measurement of the tSZ angular power spectrum on high $\ell$ modes can
provide the stringent constraint on PMFs.
