Distinct quantum anomalous Hall ground states induced by magnetic disorders

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.


I. INTRODUCTION
The discovery of QAH effect in magnetic TIs [1][2][3] represents a significant progress in the field of topological phase of matters. The QAH effect is generally regarded as a zero-magneticfield QH effect, but a thorough examination reveals fundamental distinctions between them.
Firstly, the QH effect requires the formation of discrete Landau levels generated by strong magnetic field, whereas the QAH effect relies on the exchange gap at the Dirac point arising from the interplay between topology and magnetic order [4][5][6]. The Dirac fermion's spin degree of freedom plays a central role in the QAH effect, thus its ground state is highly sensitive to the 3 magnetic properties. Secondly, the charge carriers in the QH effect have high mobility due to the relatively weak potential disorder, whereas the QAH system is strongly disordered due to the random distribution of magnetic dopants, as demonstrated by a variety of scanning probe experiments [7][8][9]. The magnetic disorders in QAH samples cause strong spatial fluctuations of exchange gap size, making the system peculiar from the magnetic-disorder-free QH system [7]. Therefore, magnetic TI is highly unique for exploring disorder-related magnetic and topological quantum phenomena [10][11][12][13][14][15][16][17].
In the past few years, the QAH effect has been widely investigated in magnetic TIs with diverse experimental parameters, including the type and density of magnetic dopants, chemical compositions, film thicknesses, and device configurations [1][2][3][18][19][20][21][22][23][24][25][26][27]. However, it is still far from clear how the magnetic disorders affect the QAH ground states. Previous study on this direction mainly focused on the effect of on-site potential disorder on the transport properties [28], in analogy to the QH system, which leads to the finding of Anderson insulator phase in a QAH sample. Later spectroscopy experiments suggest that the magnetic disorder plays a more significant role on properties of magnetic TIs [8,29]. However, from transport perspective, the role of magnetic disorder has not been thoroughly investigated, mainly due to the experimental challenges in manipulating and detecting the influence of magnetic disorders. The debates regarding the chiral Majorana mode in QAH films coupled with a superconductor highlight the importance of understanding the role of strong magnetic disorders [30][31][32][33]. Therefore, it is interesting to explore if the QAH system can host any novel quantum phases and phase transitions that are unavailable elsewhere.
Before elucidating the ground states of the QAH effect, it is instructive to survey the global phase diagram of the QH effect, which has been established theoretically [34]. Based on the Chern-Simons theories, Kivelson, Lee & Zhang predicted that two distinct phases, namely the QH liquid and Hall insulator, exist in different disorder or magnetic field regimes and can be characterized by different temperature (T) dependent transport behaviors. For the ground state of a QH liquid, both the longitudinal conductivity σxx and resistivity ρxx become zero, whereas the Hall conductivity σxy takes the quantized value (e 2 /h)sxy, where e 2 /h represents the quantum conductance and sxy is a particular rational fraction. For the Hall insulator phase, both σxx and σxy vanish at zero T, whereas ρxx goes to infinity and ρyx takes the classical value ~ B/ne (B denotes the magnetic field and n is the carrier density). Later experiments not only confirmed the existence of the two phases and the universal quantum phase transition between them [35][36][37], but also discovered an unexpected quantized Hall insulator phase characterized by a 4 quantized ρyx rather than the classical value [38]. However, in the development of QAH effect so far, the zero-magnetic-field version of the QH liquid and Hall insulator has seldomly been investigated, especially in experiment.

A. Experimental methodology
Recently there have been more theoretical interests on the effect of magnetic disorder on the QAH effect [10][11][12][13][14], but a thorough experimental investigation has been lacking. The main reason lies in the fact that magnetic disorder is introduced by the random deposition of magnetic dopants during sample growth, probably with a Gaussian-like distribution [8,10], thus it is difficult to control the strength of magnetic disorder. Recently, we show that co-doping Cr and V magnetic elements during the epitaxial growth of TI thin films is an effective way to reduce the magnetic inhomogeneity [9,20]. The accurate control of Cr-V ratio leads to the achievement of QAH effect at much higher T than that of the Cr and V singly doped magnetic TIs.
In this work, we employ a statistical methodology to investigate the effect of the degree of magnetic disorders on the QAH ground state. We have performed transport measurements on 82 magnetic TIs, with the results of 16 samples displayed in the main text. All the data presented in this work are taken when the gate voltage (Vg) is tuned to the charge neutral point (CNP) of each sample if not specified. All the samples were grown under parameters similar to that in previous reports for achieving the QAH effect [1,20,23,27]. The large quantity of samples, in conjunction with the co-doping method, allows us to cover a wide range with different strength of magnetic disorders. We first classify these samples by their zero-field ρxx values (ρxx 0 ), which reflect the degree of magnetic disorder of a sample without the influence of external magnetic field. Because ρxx takes all kinds of scattering processes into account, this criterion was commonly adopted in the studies of quantum phase transitions in different disordered electronic systems and AH effect in ferromagnets [34,36,37,[39][40][41]. By thoroughly analyzing the magnetic field dependent ρxx and ρyx at varied Ts and Vgs, we discover two novel quantum phases: the QAH insulator and AH insulator. We propose that the transmission between chiral edge states at domain boundaries, tunable by the strength of magnetic disorder and magnetic fields, is the key for determining the QAH ground state in magnetic TIs. Our results also shed new lights on some longstanding issues regarding the peculiar transport phenomena in QAH effect. 5 We start by presenting the transport data of five representative magnetic TI samples (denoted as #S1 to #S5) with different strength of magnetic disorder. The lowest disordered sample #S1 was grown by the Cr-V co-doping method [20], whereas the other four samples were Cr singly doped magnetic TIs with relatively higher level of disorder [8]. The details of all the samples studied in this work are documented in the supplementary information. h/e 2 as T is lowered from 30 K to 1.6 K. To better visualize such distinct behaviors, Fig. 1(f) summarizes the T dependent ρxx 0 for the five samples extracted from Figs. 1(a) to 1(e). From the slope of the ρxx 0 vs. T curves, we clearly observe a phase transition from metallic to insulating behavior with the increase of disorder from #1 to #5. The observed trend is highly analogous to the classification of QH liquid and Hall insulator in conventional QH system [34][35][36][37][38]. Inspired by such analogy, we divide the five samples into two distinct ground states, dubbed as the QAH insulator (#S1, #S2, and #S3) and AH insulator (#S4 and #S5) phases, respectively, as the zero-magnetic-field counterparts of the QH liquid and Hall insulator phases. The different topological properties of the QAH insulator and the AH insulator are also reflected in the distinct behaviors of the Hall conductivity σxy, which is directly linked to the Chern number (C) in the form of σxy = Ce 2 /h. In the good QAH insulator sample #S1, σxy 0 is quantized at 1.004 e 2 /h, whereas in the typical AH insulator sample #S5, it is only as small as 0.004 e 2 /h. More details about the behaviors of conductivity in the two phases are documented in the supplementary Figs. S2 and S3. The horizontal black broken line in Fig. 1(f) at ρxx 0 ~ 2 h/e 2 roughly marks the boundary between the QAH insulator and AH insulator phases. 6 Another highly unexpected feature of the QAH ground state concerns the ρxx value at HC (ρxx Hc ), in which the magnetic configuration is the most random and complex. The T dependent ρxx Hc values for the five samples are shown in Fig. 1(g). In the most disordered sample #S5, ρxx Hc displays highly insulating behavior, reaching as high as 20 h/e 2 at T = 1.6 K. With the decrease of disorder, the insulating behavior becomes weaker. In the lowest disordered sample #S1, interestingly, ρxx Hc retains the quantum resistance h/e 2 over a wide T range. The appearance of h/e 2 implies that a topologically nontrivial mechanism dominates the transport behavior near the HC.

B. Classification of the QAH ground state
An instructive way to summarize the evolution of transport behavior with disorder is to plot ρxx Hc as a function of ρxx 0 for different Ts, as shown in Fig. 1(h). It clearly shows that for the lowest disordered sample #S1, ρxx Hc remains a constant at h/e 2 for the entire T range. With the increase of disorder (samples #S2 and #S3), the ρxx Hc vs. ρxx 0 traces have a negative slope that becomes steeper as ρxx 0 approaches 2 h/e 2 , which is the boundary between the QAH insulator and AH insulator phases. When the disorder is further increased (samples #S4 and #S5), the slope changes sign to positive. The correlation between ρxx Hc and ρxx 0 can be summarized by a power law ρxx Hc ~ (ρxx 0 ) α , and the α values for different samples are listed in the figure. For α = 0, the sample is in the low disorder limit of the QAH insulator phase with ρxx Hc ~ h/e 2 . For α = 1, the sample is in the high disorder limit of the AH insulator phase when the peak near HC is so broad that ρxx Hc ~ ρxx 0 . In fact, the ρxx Hc ~ h/e 2 behavior was present in some previous data on QAH samples with varied parameters [18,20,22,24,26,28,33], although without being noticed.

C. More transport data on typical QAH insulator and AH insulator samples
To demonstrate the universality of the distinct QAH ground states, in Figs. 2(a) and 2(b) we display transport data on more magnetic TI samples in the typical QAH insulator and AH insulator phases. All the data shown here are measured at T = 1.6 K. Figure 2

D. Magnetic-field-induced quantum phase transition
In addition to disorder, magnetic field also plays a crucial role in determining the ground 8 state of QAH system. Figure 4(a) displays the magnetic field dependent ρxx from T = 100 mK to 800 mK for sample #13, which is an AH insulator in zero magnetic field. The blue arrow marks the direction of field sweep, which ensures that the initial state is always fully magnetized by strong out-of-plane magnetic field. The overall trend is that ρxx increases with the decrease of magnetic field, and the crossing at 0.25 T indicates the existence of a quantum critical point (QCP). In Fig. 4(b) we extract the ρxx value of different magnetic fields and replot them as a function of T. It clearly shows that the value of ρxx vs. T curve at the transition field (HT) is nearly a constant around 2.6 h/e 2 , which sets a phase boundary between the metallic (high field) and insulating (low field) ground states. This value is close to the phase boundary ρxx 0 ~ 2 h/e 2 obtained in Fig. 1(f). Therefore, increasing magnetic field drives a quantum phase transition from AH insulator to QAH insulator phase, in a similar manner as reducing disorder.
In conventional QH effect, a universal scaling law governs the behavior of both the liquidinsulator transition [36,37] and plateau-plateau transition [42][43][44][45]. To check the validity of the so-called "super-universality" in the QAH effect [34,46], following the criteria based on the crossing point in resistivity curves [47], we perform the same scaling analysis of the quantum phase transition between the QAH insulator and AH insulator phases. Figure 4(c) shows that all the ρxx vs (H-HT)/T κ curves, where (H-HT)/T κ is the single parameter scaling function, collapse into a universal curve for a critical parameter κ ~ 0.31±0.01. The critical ρxx value at the QCP is found to be 2.6 h/e 2 , which sets a phase boundary between the metallic (high field) and insulating (low field) ground states. The value of κ ~ 0.31 is about 26% smaller than κ ~ 0.42 in the QH phase transition [36,37]. Meanwhile, the ρxx value at the QCP deviates significantly from the theoretical expectation of h/e 2 for the QH phase transition [34][35][36][37]48].
Consistent result is obtained in another sample, in which κ and critical ρxx are found to be 0.29 and 1.7 h/e 2 (see supplementary Fig. S9 for details). These results strongly suggest that the quantum critical behavior in the QAH system is different from that in conventional QH systems, probably due to the existence of strong magnetic disorders.

E. Edge state transmission model
Because a rigorous theory about the QAH phase transition is still lacking, here we take a phenomenological approach to give an intuitive explanation for the classification of phases and the transition between them. The basic idea is similar to the phase incoherent network model invented to describe the quantized Hall insulator phase in QH systems [49], in which electron transport proceeds via a network of edge states surrounding QH puddles. As long as the puddle size is larger than the phase coherent length, ρyx is always quantized regardless of the ρxx behavior [50,51]. The most unique feature of QAH systems is the existence of magnetic domains, and the chiral edge states at domain boundaries [8]. We propose that the transmission between the edge states of neighboring domains plays the most crucial role in the quantum transport process. In a macroscopic QAH sample, magnetic disorders lead to spatially separated and randomly distributed magnetic domains [7][8][9][10][11][12][13][14][15]. Under such situation, charge transport is mainly determined by two quantities, the average distance λ between neighboring domains and the decay length ξ of the chiral edge state at domain boundaries.
Without loss of generality, we start from the simplest situation consisting of two domains.
MFM measurements in Cr-V codoped and Cr singly doped TI samples show that the value of λ in a good QAH insulator is estimated to be around 100 nm. Whereas in AH insulator samples with relatively high degree of magnetic disorder, its value can be up to 1 μm [20]. The value of ξ is in the same range, from a few hundred nanometers to 1 μm, as revealed by the visualization of spatial conductivity on a magnetic TI by MIM measurements [52]. Figure 5(a) schematically illustrates the magnetic domain distribution in an AH insulator, in which λ is larger than ξ. The transmission between chiral edge states is by quantum mechanical hopping across the barrier, similar to variable range hopping in strongly disordered systems, thus becomes insulating at low T. With the reduction of disorder strength or increase of magnetic field, the average size of magnetic domain grows so that the distance λ between chiral edge states becomes smaller, as shown in Fig. 5(b). When λ becomes comparable to ξ, the chiral edge states overlap substantially and their chemical potentials equilibrate. Consequently, the transmission has an ohmic contact behavior, which corresponds to the metallic ρxx behaviors in the QAH insulator regime. The quantum phase transition from the AH insulator to QAH insulator, achieved either by reducing disorder or increasing magnetic field, can be explained naturally in this picture.
The universal ρxx H c ~ h/e 2 observed in the low disorder limit of the QAH insulator phase can also be understood in terms of edge state transmission. As shown in Fig. 5(c), at the HC of a QAH insulator, the edge states of two neighboring domains with opposite magnetization achieve chemical potential equilibration due to substantial wavefunction overlap. When perfect transmission occurs at the domain boundary, the chemical potential of the two upstream edge states get fully equalized, so that the two upper edges have the same potential. Following the Landauer-Buttiker formalism [53], the ρxx at different edges obtain different quantized values, 2 h/e 2 on the lower edge and 0 on the upper edge [numbers in Fig. 5(c) indicate the chemical 10 potential distribution on different sections of the edge states]. This effect has been directly demonstrated by transport measurement in a two-domain device fabricated by the tip of a magnetic force microscope [53]. The macroscopic samples studied in our experiment consist of large numbers of up and down magnetic domains at HC, and the average of the two quantized values gives rise to the observed quantum resistance h/e 2 . The universal ρxx H c ~ h/e 2 behavior is thus a manifestation of topological edge state transport that obeys the Landauer-Buttiker formalism in the low disorder limit of QAH insulator phase [3,21,22,24] (see Supplementary session J and K for details).

III. DISCUSSIONS AND CONCLUSIONS
The proposed phenomenological picture not only explains the formation and evolution of the QAH insulator and AH insulator phases, but also sheds important new lights on the peculiar transport phenomena in magnetic TIs. For example, the QAH state is theoretically expected to show quantized ρyx and vanishing ρxx simultaneously due to the dissipationless edge states.
However, numerous previous experiments show that ρyx can be well quantized at zero magnetic field, but ρxx is always finite (up to a few h/e 2 ) [1][2][3][18][19][20][21][22][23][24][25][26][27][28]. Several theoretical models have been proposed to explain this discrepancy [10,15,54], but there is no consensus yet. Our results provide a new physical picture to resolve this controversy. In an ideal QAH insulator, the quantization of ρyx and vanishing ρxx should occur simultaneously, whereas for an AH insulator, a reduced ρyx and insulating ρxx are expected. However, when the system just enters the AH insulator regime, such as sample #S13, the ρyx value is quite close to h/e 2 but ρxx is very large.
This behavior is reminiscent of the quantized Hall insulator phase with quantized ρyx and insulating ρxx in the QH system [38], which has been found to lie between the QH liquid and Hall insulator phases. Due to the unavoidable inhomogeneities in magnetic TIs, the QAH insulator sample may contain a small fraction of quantized AH insulator phase, which will give rise to a sizable ρxx and nearly quantized ρyx.
Notably, recent studies in some stoichiometric topological magnets also reveal large but unquantized AH or topological Hall effect [55][56][57][58][59][60], which arises from enhanced Berry curvature near the Fermi surface similar to magnetically doped TIs. However, the physics for the absence of quantization is different from the AH insulator found here. In the former cases, it is mainly due to the contributions of bulk conduction channels. Whereas in the AH insulator phase, it is attributed to the magnetic-disorder-induced exchange gap fluctuations [8].
Besides the different scaling parameter κ and critical resistivity value compared to the QH phase transition, magnetic field also plays a drastically different role in the QAH phenomena.
In QH effect, magnetic field directly couples to itinerant electrons through the orbital effect, which alters the kinetic energy in the form of Landau levels. But for the QAH effect, magnetic field mainly couples with the local moments through the Zeeman energy, which in turn affects the magnetic domains and chiral edge state transport. As a consequence, in the magnetic-fieldinduced QAH insulator to AH insulator transition the insulating phase exists in the low field side of the QCP, which is opposite to that in the QH liquid to Hall insulator transition. These differences strongly suggest that the QAH phase is not a simple zero-field version of conventional QH effect. We note that κ itself is not a universal exponent but is instead expressed as κ ~ 1/νz, where ν is the critical correlation length term and z is the dynamic term. Because both ν and z determine the class of quantum criticality for a quantum phase transition, the exact universality class of the QAH phase transition awaits further investigation to disentangle the critical and dynamic exponents.
Although the zero-field version of Hall insulator, the AH insulator, discovered here looks superficially similar to the Anderson insulator reported previously [28], the physics between them are highly distinct in several ascpects. The Anderson insulator phase is induced by the onsite potential disorder [61,62], and the exchange gap is a constant in space. In contrast, the AH insulator phase originates from the magnetic disorder, and the fluctuation of exchange gap plays a dominant role. Because the two types of disorders exhibit different responses to external parameters, it is natural to expect distinct transport signatures in the Anderson insulator and AH insulator phases. For example, electrostatic potential fluatuation usually does not change with temperature, whereas the exchange gap fluctuation due to magnetic disorder decreases with lower temperatures because of better magnetic order. This fundamental distinction may explain the distinct ground state properties of ρyx between the two phases, both in the vast discrepancy in the absolute values and the opposite temperature dependent behaviors. The QH related edge state physics in the AH insulator phase and the almost quantized ρyx at the ground state also make it different from the conventional magnetic semiconductors.
In conclusion, we find that magnetic TIs with different degrees of magnetic disorder can be classified into two novel phases denotated as the QAH insulator and AH insualtor phases. In the low-disorder limit of the QAH insulator phase, a universal quantum resistance h/e 2 is observed at HC. In the AH insulator samples, magnetic field can drive it to the QAH insulator phase through a quantum phase transition. Both the scaling parameters and the critical resistivity, as well as the role of magnetic field, exhibit apparent discrepancies from that in the QH phase 12 transition. We propose a phenomenological model involving the transmission between chiral edge states at domain boundaries, which explains the novel ground states and peculiar quantum transport phenomena in the QAH effect.     10 # S12 # S13 # S14 # S15 # S16