  Many body localization (MBL) is a phenomena that allows for the preservation
of quantum information for long times. We study a variation of the
disordered-Heisenberg model, which is known to exhibit an MBL phase [5][6],
known as the quasi-Heisenberg model. Our model is a variation of the
quasi-Heisenberg model with fields in both x and y directions. We found that
while our model shares some characteristics for MBl, as seen by other
quasi-Heisenberg models, the adjacent gap ratio for our system falls well below
the expected Poisson limit when it transitions to an MBL phase. A similar model
to ours has been experimentally realized in [11], and so we are motivated
further to study MBL characteristics. To determine the characteristics of the
system when transitioning from the ergodic to MBL phase, we calculate the
quantities: entanglement entropy, adjacent gap ratio, and the probability
distributions for both. We calculate the aforementioned quantities through the
eigenvalues and eigenvectors of the system's Hamiltonian, which are obtained
through exact diagonalization. We find the entanglement entropy of the system
behaves as expected for MBL, with the spectral average of the entanglement
entropy dropping close to zero as it enters the MBL phase. However, the
spectral average for the adjacent gap ratio falls below the Poisson limit,
0.386, which corresponds to uncorrelated energy levels This drop in adjacent
gap ratio can be explained through some level of attraction between energy
levels, and is ultimately a consequence of the separability of the quasirandom
field. The probability distribution for the adjacent gap ratio is fitted from a
derived distribution which was found using the Brody function for energy level
spacing. The Brody function accounts for energy level attraction when the Brody
parameter, $\omega$, becomes negative...
