  We explore a possible phase where chiral $SU(2)_L \times SU(2)_R$ symmetry is
spontaneously broken while its center $Z_2$ symmetry remains unbroken and its
consequence on thermal quantities. In this phase, chiral symmetry breaking is
driven by a quartic quark condensate although a bilinear quark condensate
vanishes. A Ginzburg-Landau free energy leads to a new tricritical point (TCP)
between the $Z_2$ broken and unbroken phases. Furthermore, a critical point can
appear even in the chiral limit where explicit breaking is turned off, instead
of a TCP at which restoration of chiral and its center symmetries takes place
simultaneously. The net quark number density exhibits an abrupt change near the
restoration of the center symmetry rather than that of the chiral symmetry.
Hadron masses in possible phases are also studied in a linear sigma model. We
show that, in the $Z_2$ symmetric phase, the $\bar{q}q$-type scalar meson with
zero isospin I=0 splits from the $\bar{q}q$-type pseudoscalar meson with I=1.
