*Phys. Rev. B* **100**, 214201 (2019)

**Metal to marginal-metal transition in two-dimensional ferromagnetic electron gases**

Weiwei Chen^{1}, **C. Wang**^{2,3,*}, Qinwei Shi^{1}, Qunxiang Li^{1}, and X. R. Wang^{4}^{,5,*}

*1. Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China*

*2. Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China*

*3. School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China*

*4. Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong*

*5. Shenzhen Research Institute, The Hong Kong University of Science and Technology, Shenzhen 518057, China*

*Corresponding author: physcwang@tju.edu.cn phxwan@ust.hk

**Abstract**

Two-dimensional ferromagnetic electron gases subject to random scalar potentials and Rashba spin-orbit interactions exhibit a striking quantum criticality. As disorder strength W increases, such a system undergoes two transitions. It first changes from a normal diffusive metal consisting of extended states to a marginal metal consisting of critical states at a critical disorder Wc,1. Another transition from the marginal metal to an insulator occurs at a stronger disorder Wc,2. Through highly accurate numerical procedures based on the recursive Green's function and the exact diagonalization methods, we elucidate the nature of the quantum criticality and the properties of the pertinent states. The conductance is described by an unorthodox one-parameter scaling law: Conductance of various system sizes and disorders collapse into two branches of a scaling curve corresponding to diffusive metal and insulating phases with an exponentially diverging correlation length, ξ∝exp[α/√|W−Wc|], near transition points. Finite-size scaling analysis of the inverse participation ratio reveals that the states of the marginal metal are fractals of a universal fractal dimension D=1.90±0.02, while those of the diffusive metal spread over the whole system with D=2 and states in the insulating phase are localized with D=0. The phase diagram for diffusive metals, marginal metals, and the Anderson insulators is plotted in the disorder-magnetic-coupling plane.