Many Body Localization in 2D Systems with Quasi-Random fields in X and Y Directions

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, ω, becomes negative...