  The quantum anomalous Hall (QAH) effect in magnetic topological insulator
(TI) represents a new state of matter originated from the interplay between
topology and magnetism. The defining characteristics of the QAH ground state
are the quantized Hall resistivity ($\rho_{yx}$) and vanishing longitudinal
resistivity ($\rho_{xx}$) in the absence of external magnetic field. A
fundamental question concerning the QAH effect is whether it is merely a
zero-magnetic-field quantum Hall (QH) effect, or if it can host unique quantum
phases and phase transitions that are unavailable elsewhere. The most dramatic
departure of the QAH systems from other QH systems lies in the strong magnetic
disorders that induce spatially random magnetization. Because disorder and
magnetism play pivotal roles in the phase diagram of two-dimensional electron
systems, the high degree of magnetic disorders in QAH systems may create novel
phases and quantum critical phenomena. In this work, we perform systematic
transport studies of a series of magnetic TIs with varied strength of magnetic
disorders. We find that the ground state of QAH effect can be categorized into
two distinct classes: the QAH insulator and anomalous Hall (AH) insulator
phases, as the zero-magnetic-field counterparts of the QH liquid and Hall
insulator in the QH systems. In the low disorder limit of the QAH insulator
regime, we observe a universal quantized longitudinal resistance $\rho_{xx} =
h/e^{2}$ at the coercive field. In the AH insulator regime, we find that a
magnetic field can drive it to the QAH insulator phase through a quantum
critical point with distinct scaling behaviors from that in the QH phase
transition. We propose that the transmission between chiral edge states at
domain boundaries, tunable by disorder and magnetic fields, is the key for
determining the QAH ground state.
