  As perhaps the most studied paradigm for a quantum phase transition, the
periodic quantum Ising chain is exactly solvable via the Jordan-Wigner
transformation followed by a Fourier transform that diagonalizes the model in
the momentum space of spinless fermions. Although the above procedures are
well-known, there remain some subtle points to be clarified regarding the
correspondence between the real-space and momentum-space representations of the
quantum Ising ring, especially those related to fermion parities. In this work,
we establish the relationship between the two fully aligned ferromagnetic
states in real space and the two degenerate momentum-space ground states of the
classical Ising ring, with the former being a special case of the factorized
ground states of the more general XYZ model on the frustration-free
hypersurface. Based on this observation, we then provide a Pfaffian formula for
calculating real-time dynamics of the parity-breaking longitudinal
magnetization with the system initially prepared in one of the two
ferromagnetic states and under translationally invariant drivings. The
formalism is shown to be applicable to systems with the help of online programs
for the numerical computation of the Pfaffian, thus providing an efficient
method to numerically study, for example, the emergence of discrete time
crystals in related systems.
