A Study On The Effect Of Disorder In 1-dimensional And 2-dimensional Topolgical System

In this thesis, I study the effect of randomness in 1-dimensional and 2-dimensional topo-logical system. In the 1-dimensional system, the topic is the effect of random partition on the entanglement spectrum of the spin-1 Afflect-Kennedy-Lieb-Tasaki chain. In the 2-dimensional system, I study the effect of random potential on the wave functions and tunneling amplitudes in the Laughlin fractional quantum Hall systems.In the first project, I extract the critical boundary theory of the spin-1 AKLT chain from its quantum entanglement information. The infinite randomness fixed point of the spin-1/2 degrees of freedom of the spin-1 AKLT chain can merge from extensive bipartition. The nested entanglement entropy of the ground state exhibits logarithmic scaling behavior, and its effective central charge c (?)ln2.In the second project, I study the tunneling amplitude of v= 1/3 Laughlin quasiholeon a disk at various disorder strength. From observing the overlap between ordered and disordered wave functions for both Laughlin ground state and quasihole state, we find it evolves to zero a critical disorder strength. Then we observe the ratio between tunneling amplitude of disordered and clean system for both Laughlin ground state and quasihole state, we find a scaling behavior of the ratio's asymptotic value in the thermodynamic limit. From this scaling relation, we extract the critical disorder strength that Laughlin FQH regime makes a phase transition into an insulator. Both of our results on the critical disorder strength value, from wave function overlap and tunneling amplitude,agree with each other and also with the result obtained from the topological Chern number approach.