Phase transitions in the one-dimensional transverse Ising model in a longitudinal magnetic field

The phase transitions in the one-dimensional transverse Ising model in the presence of a longitudinal magnetic field were studied by the quantum fidelity method. We used exact diagonalization to obtain the ground-state energies and corresponding eigenvectors for lattice sizes up to 24 spins.The maximum of the fidelity susceptibility is used to locate the various phase boundaries present in the system. The type of dominant spin ordering for each phase was identified by examining the corresponding ground-state eigenvector. For a given antiferromagnetic nearest-neighbor interaction ($J_2$), we calculated the fidelity susceptibility as a function of the transverse field ($B_x$) and the strength of the longitudinal field ($B_z$). The phase diagram in the ($B_x,B_z$)-plane shows three phases. These findings are in contrast with the published literature that claims that the system has only two phases. For $B_x < 1$, we observed an antiferromagnetic phase for small values of $B_z $ and a paramagnetic phase for large values of $B_z$. For $B_x > 1$ and low $B_z$, we found a disordered phase that undergoes a phase transition to a paramagnetic phase for large values of $B_z$.