  For the one-dimensional Ising chain with spin-$1/2$ and exchange couple $J$
in a steady transverse field(TF), an analytical theory has well been developed
in terms of some topological order parameters such as Berry phase(BP). For a TF
Ising chain, the nonzero BP which depends on the exchange couple and the field
strength characterizes the corresponding symmetry breaking of parity and time
reversal(PT). However, there seems to exist a topological phase transition for
the one-dimensional Ising chain in a longitudinal field(LF) with the reduced
field strength $\epsilon$. If the LF is added at zero temperature, researchers
believe that the LF also could influence the PT-symmetry and there exists the
discontinuous BP. But the theoretic characterization has not been well founded.
This paper tries to aim at this problem. With the Jordan-Wigner transformation,
we give the four-fermion interaction form of the Hamiltonian in the
one-dimensional Ising chain with a LF. Further by the method of Wick's theorem
and the mean-field theory, the four-fermion interaction is well dealt with. We
solve the ground state energy and the ground wave function in the momentum
space. We discuss the BP and suggest that there exist nonzero BPs when
$\epsilon=0$ in the paramagnetic case where $J<0$ and when $-1<\epsilon<1$, in
the diamagnetic case where $J>0$.
