Holographic duality between $(2+1)$-d quantum anomalous Hall state and   $(3+1)$-d topological insulators
  In this paper, we study $(2+1)$-dimensional quantum anomalous Hall states, i.e. band insulators with quantized Hall conductance, using the exact holographic mapping. The exact holographic mapping is an approach to holographic duality which maps the quantum anomalous Hall state to a different state living in $(3+1)$-dimensional hyperbolic space. By studying topological response properties and the entanglement spectrum, we demonstrate that the holographic dual theory of a quantum anomalous Hall state is a $(3+1)$-dimensional topological insulator. The dual description enables a new characterization of topological properties of a system by the quantum entanglement between degrees of freedom at different length scales. 
Yingfei Gu, Ching Hua Lee, Xueda Wen, Gil Young Cho, Shinsei Ryu,   Xiao-Liang Qi
cond-mat.mes-hall cond-mat.str-el hep-th quant-ph