Half-Quantized Hall Metal and Marginal Metal in Disordered Magnetic Topological Insulators

TL;DR

The study addresses whether a half-quantized Hall effect can survive in disordered semimagnetic topological insulators. It combines a lattice model with real-space Kubo formalism, effective medium theory, transfer-matrix localization, and density-of-states analysis to map the phase diagram and characterize transport. Key findings include the robust HQHM phase in weak disorder, a distinct marginal metal with scale-invariant conductance at intermediate disorder, and a transition to an Anderson insulator at strong disorder, with a nontrivial scaling structure reminiscent of BKT-like criticality. These results illuminate disorder-driven topological transitions in magnetic TIs and suggest pathways for experimental observation and device applications relying on robust topological transport under realistic imperfections.

Abstract

A semimagnetic topological insulator -- a heterostructure combining a topological insulator with a ferromagnet -- exhibits a half-quantized Hall effect, characterized by a quantized Hall conductance of $\frac{1}{2}\frac{e^{2}}{h}$ (where $e$ is the elementary charge and $h$ is the Planck constant), which reinforces the established understanding of topological phenomena in condensed matter physics. However, its stability in realistic, disordered systems remains poorly understood. Here, we demonstrate the robustness of the half-quantized Hall effect in weakly disordered systems, stemming from a single gapless Dirac cone of fermions and coexisting with weak antilocalization due to the $π$ Berry phase that suppresses backscattering. Furthermore, we uncover a marginal metallic phase emerging between weak antilocalization and Anderson insulation -- a transition that defies conventional metal-insulator transitions by lacking an isolated critical point -- where both conductance and normalized localization length exhibit scale invariance, independent of system size. The half-quantized Hall metal and the marginal metallic phase challenge existing localization theories and provide insights into disorder-driven topological phase transitions in magnetic topological insulators, opening avenues for exploring quantum materials and next-generation electronic devices.

Figures (6)

• Figure 1: The phase diagram in disordered semimagnetic topological insulator films. (a). Schematic of a semimagnetic topological insulator film. The acronyms HQHM (Half-Quantized Hall Metal), MM (Marginal Metal), and AI (Anderson insulator) label distinct phases in the diagram. The red solid line indicates that the clean system is a metal rather than MM. (b). Evolution of the spectral function $A(\epsilon,\mathbf{k})$ with the disorder strength $W$. (c). The phase diagram of the semimagnetic topological insulator in the $W-E_{\rm F}$ plane: The metallic phases of the scaling function $\alpha=1$ with the half-quantized anomalous Hall conductivity; the marginal metallic phase of $\alpha=0$ with non-quantized Hall conductivity, and the Anderson insulating phase. Here we take the system size $L$ large enough.
• Figure 2: Robustness of the Half-quantized Hall effect in Weak Disorder. (a). The phase diagram of the Hall conductivity in the $W-E_{\rm F}$ plane. We set the lattice size $L_{x}=L_{y}=L=20$ in the simulation. The red solid line indicates that the clean system is metallic. The bright yellow areas highlight the half-quantized Hall metal (HQHM) phase, whereas the chartreuse region indicates marginal metal (MM) with non-quantized Hall conductivity. The white solid line marks the phase boundary as determined by the effective medium theory. (b) The calculated Hall conductivity and disorder renormalized Dirac mass $\widetilde{m}_{0}$ and energy broadening $\eta_{\mathrm{top}}$ at $E_{\rm F}=0.01$ eV as a function of $W$ in the effective medium theory. The black dashed line denotes the critical threshold $W_{\mathrm{c}}=2.6$ eV. The finite-size scaling analysis of the quantization error, $\sigma_{xy}^{0}-\sigma_{xy}$, is presented as the function of $1/L$ in (c-1) along the vertical line of $W=1.0$ eV for varying $E_{\rm F}$, and (d-1) along the horizontal line of $E_{\rm F}=0.01$ eV for varying $W$. The corresponding $\sigma_{xy}^{0}$ and $l_{e}$ are displayed in (c-2) and (d-2), respectively. The error bars reflect the uncertainties arising from the numerical fitting. We have used the set of parameters $L_{z}=10$, $L_{z}^{\mathrm{Mag}}=3$, $\lambda_{\parallel}=0.41$ eV, $\lambda_{z}=0.44$ eV, $t_{\parallel}=0.566$ eV, $t_{z}=0.40$ eV, $V_{0}=0.1$ eV, and lattice constants $a=b=1$ nm and $c=0.5$ nm unless otherwise specified. The raw data points are averaged over 50 random samples.
• Figure 3: The normalized localization length and the emergence of the marginal metal region in moderate disorder. (a) The normalized localization length $\Lambda_{x}$ along $x$ direction, where the error bars show standard deviations of $\Lambda_{x}$Yamakage2013:PRB. The used longitudinal size is $L_{x}=2\times10^{6}$. The thickness is $L_{z}=10$, and we set Fermi energy at $E_{\rm F}=0.01$ eV. In the marginal metallic phase, the normalized localization length exhibits a unique behavior by collapsing onto a single line and showing no size dependence, demonstrating its exotic criticality. (b) The normalized delocalization length near the transition point. (c) The fit of the numerical data near the transition point via a universal scaling function $\Lambda_{x}=F\left(L_{y}/\xi\right)$. The transition points and the critical exponents are enumerated in the figure.
• Figure 4: Results of quantum interference corrections. Dependence of $\alpha_{L}=\frac{\pi h}{e^{2}}\frac{\partial\sigma}{\partial\ln L}$ on $E_{\rm F}/V_{0}$ for different inter-surface scattering strength (a) $\zeta=0$ and (b) $\zeta=0.1$ with $L/l_{m}$ varied and $l_{m}=\frac{\lambda_{\parallel}^{2}}{V_{0}U_{0}}$. The characteristic length scales $(l_{e}^{s}/l^{s})^{2}$ for two Cooperon channels ($s=t,b$) as a function of $V_{0}/E_{\rm F}$ for (c) $\zeta=0$ and (d) $\zeta=0.1$. Here, $l_{e}^{s}$ represents the mean free paths for the respective channels.
• Figure 5: The arithmetic and geometric mean DOS$\rho_{\mathrm{a}}(E)$and$\rho_{\mathrm{t}}(E)$. (a) The DOS as the function of disorder strength $W$ at the Fermi energy $E_{\rm F}=0.01$ eV, revealing the phase transitions between half-quantized Hall metal (HQHM), marginal metal (MM), and Anderson insulator (AI). The cyan dashed line fitted to $\rho_{\mathrm{t}}$ intersects the horizontal axis at the hexagonal marker ($W=13.47$ eV), thereby demarcating the phase boundary between MM and AI. (b) The DOS as the function of the Fermi energy $E_{\rm F}$ at $W=1.0$ eV. The insets show an enlarged view of the band center and band edge.