A compilation of studies on random systems: measurements of correlation functions and localization properties - Getelina, João Carlos de Andrade (2021)
We present a collection of studies about several properties of two random systems, namely (i) the random one-dimensional spin-1/2 chain, and (ii) the random two-dimensional Bose-Hubbard model. In study (i), we consider two variants of random spin chains: the usual case with uncorrelated random couplings, and the correlated case, in which the even and odd sublattices are identical to each other. Using the Jordan-Wigner transformation, we map the spin-1/2 chain Hamiltonian into a noninteracting fermionic model. On this new basis, we perform an exact diagonalization routine, which allows us to compute spin- spin correlation functions and related observables. Our results are presented here as a reproduction of two published papers. In the first one, we measure entanglement properties and the violation of Bell inequalities. We show that the correlated case does not violate the Bell inequality up to a small degree of randomness, thus contradicting the prior belief that any amount of disorder should lead to a violation of Bell inequalities and, equivalently, the existence of nonlocal states. In the second paper, we confirm the strong-disorder renormalization group predictions about the scaling of spin-spin correlation functions for the uncorrelated disorder case. We show that results suggesting a possible correction to the scaling function may be consequence of either a lack of numerical precision or a relatively large crossover length. In addition, we show that much of the nonuniversal properties of the spin-spin correlation functions can be understood from a single parameter scaling perspective. In study (ii), we consider the two-dimensional Bose-Hubbard model with disorder introduced either as random site dilution or as onsite interactions generated from a uniform probability distribution. We investigate the localization properties of collective modes by employing a multifractal analysis and a recursive Green´s function method. Using a variational mean-field approach, we obtain noninteracting Hamiltonians for the Goldstone (phase) and Higgs (amplitude) modes. Our results show that only the lowest-excitation Goldstone mode undergoes a localization-delocalization transition close to the superfluid-Mott insulator phase transition; higher-excitation phase modes and all amplitude modes remain localized. This behavior is observed for the two types of disorder investigated.