Holographic duality between $(2+1)$-dimensional quantum anomalous Hall state and $(3+1)$-dimensional topological insulators

In this paper, we study $(2+1)$-dimensional quantum anomalous Hall states, i.e., band insulators with quantized Hall conductance, using exact holographic mapping. 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 characterization of topological properties of a system by the quantum entanglement between degrees of freedom at different length scales.

Figure 1

Illustration of the duality between a topological insulator and a quantum anomalous Hall system.

Figure 2

A (simplest) example of the tensor network used to define the EHM. Each vertex is a two-site unitary transformation that transforms (in the upward direction) two input sites (below) to two output sites (above). One of the output sites (blue dot) is considered as a bulk site, which contains the high-energy degrees of freedom of the two input sites. The other site, containing the low-energy degrees of freedom, becomes the input of the next layer (level in the tree). The whole network is a composition of these layers of unitary operators, which defines the unitary mapping $M$. The number of bulk sites (blue dots and the red dot at the top) is equal to the number of boundary sites (black dots). Bulk sites closer (farther) to the boundary correspond to higher-energy (lower-energy) degrees of freedom.

Figure 3

(Left panel) An illustration of a transformation $V_n$ in two-dimensional EHM [see Eq. (4)]. The mapping is constructed by combining two $(1+1)$-dimensional EHM steps, one in the $x$ direction and the other in the $y$ direction. The blue dots are the output UV modes and the red dot is the IR mode at the present level, which will become the input for the next level. (Right panel) An illustration of the wave functions of the IR and UV modes. The IR mode is a uniform superposition of the four modes involved, while the UV modes are the three orthogonal oscillatory suppositions.

Figure 4

Numerical results on the Chern number density for the Dirac model with different masses. We notice there are two peaks, the UV peak on the left and the IR peak on the right, each contributing a Chern number of approximately $\frac{1}{2}$. Generically, the IR peak arises from the Berry curvature contributions of the massive Dirac cone, while the UV peak arises from states near the Brillouin-zone boundary, which provide UV regularization of the massive Dirac cone.

Figure 5

Numerical results of the $\theta$ angle for Dirac models with different masses. One can observe that plateaus emerge at $\theta =\pi$ for small masses.

Figure 6

An illustration of holographic topological insulators mapped from Dirac models. (a) The two band model [Eq. (5)] which only has one mass scale. (b) The four band model [Eq. (11)] which has two mass scales $m_1$ and $m_2$. (a) Dirac model 1. (b) Dirac model 2.

Figure 7

An entanglement cut in the bulk of the holographic TI (dashed rectangle) divides the TI into two parts $A$ and $A^C$. The reduced density matrix of $A$ is obtained by a partial trace over region $A^C$.

Figure 8

(a) The low-energy entanglement spectra $\epsilon \left(\mathbf{q}\right)$ for different positions of entanglement cuts at layers $n=2,14$, and 30 of the EHM bulk [Eq. (5)], with the small mass set to $m=-2^{-20}$. The spectra are gapped for cuts in deep UV $\left(n=2\right)$ and deep IR $\left(n=30\right)$ regimes, but are gapless when the cut is made in the bulk TI region $\left(n=14\right)$. The entanglement gap closes at three massless Dirac points $(\pi ,0),(0,\pi )$, and $(\pi ,\pi )$ in the reduced Brillouin zone. (b) Plot of the minimal gap of the correlator $C_A\left(\mathbf{q}\right)$ spectrum. The background color density represents the Chern number density $c\left(n\right)$ profile. Indeed, the entanglement gap vanishes only in the TI bulk region, where $c\left(n\right)$ is exponentially small between its two peaks in the UV and the IR. (a) Entanglement spectrum for cut position $n=2$, 14, and 30. (b) Minimal gap of the correlator as function of $n$.

Figure 9

Plot for functions $C\left(k\right),D\left(k\right)$, and $Y(k,n)$ for $k_y=0$. The label $k$ in the figure represents $k_x$ in our EHM. Shown are the plots for layers $n=1,2,3$.