In this final thesis we investigate the transition between the ergodic and many-body localized (MBL) phase in the $t$-$J$ model in the presence of spin or potential disorder. We concentrate on the two distinct cases, namely the case of one-hole doping and the case of one-third doping, where the transition between phases is investigated as a function of the magnitude of the appropriate type of disorder. Our numerical approach is based on full numerical diagonalization of the model Hamiltonians, where three different indicators are used in order to test the systems for their ergodic or MBL properties. The calculations of the mean ratio of the adjacent level-spacings $\langle \tilde{r}\rangle$ and of the spectral form-factor (SFF) are based on the analysis of the statistical properties of the investigated Hamiltonians' energy spectra. The third indicator we use is the calculation of the entanglement entropy of all system's eigenstates. In the ergodic phase, the calculated values of $\langle\tilde{r}\rangle$ and SFF match the ones obtained in the so-called gaussian orthogonal ensemble (GOE) of random matrices. The results are completely different in the MBL phase where they match those for systems in which the energy levels are distributed independently according to the Poisson probability distribution. The entanglement entropy of the highly excited eigenstates in the ergodic regime obeys the so-called volume-law scaling while in the MBL phase all the states in the spectrum are weakly entangled and thus obey the area law scaling of the entanglement entropy. All of our indicators imply that an increase in system's disorder leads towards a transition between an ergodic and the MBL phase in the case of one-third doping for both spin and potential disorder.
The same holds true in the case of one-hole doping in the presence of spin disorder while the case of only one hole and potential disorder is somewhat different from the remaining ones. As our indicators show, an increase in disorder does not lead towards a transition in the MBL phase in this case. As far as we know, we are among the very first to use the calculation of SFF as an idicator of system's ergodicity or the presence of MBL. Due to its rather simple implementation, the calculation of $\langle\tilde{r}\rangle$ is commonly used in order to test for ergodicity or the presence of MBL in a given system. In calculating $\langle\tilde{r}\rangle$, we only consider the correlations between the nearest levels in an energy spectrum. The calculation of SFF, on the other hand, accounts for correlations between all the energy levels in the spectrum. It thus allows for a more comprehensive insight in the system's properties, such as its behaviour at different time scales, at the expense of a more involved numerical implementation. In terms of determining the systems' ergodicity or the presence of MBL, the results of our SFF calculations match the predictions of our $\langle\tilde{r}\rangle$ calculations.
|
