Characterizing second-order topological insulators via entanglement topological invariant in two-dimensional systems

Abstract

Higher-order topological insulators have attracted significant interest in recent years. However, identifying a universal topological invariant capable of characterizing higher-order topology remains challenging. Here, we propose a entanglement topological invariant designed to characterize secondorder topological systems. This entanglement topological invariant captures the entanglement of topological corner states under open boundary conditions by employing a bipartite entanglement entropy method. In several representative models, the entanglement topological invariant assumes a nonzero value exclusively in the presence of second-order topology, with its magnitude exactly matching the number of topologically protected corner states. Consequently, the proposed entanglement topological invariant not only provides a clear criterion for detecting higher-order topology, but also offers a quantitative measure for the related corner states. Our study establishes a universal and precise method for characterizing higher-order topological phases, opening avenues for their fundamental understanding and future investigations.

Figures (4)

• Figure 1: Spatial partition for entanglement topological invariant computation. Schematic diagram of the nanoflake partition with width $W$ and length $L$. The lengths of partition $A$ and $B$ are $L_{A}$ and $L_{B}$, respectively. And both of them have the same width as the entire nanoflake.
• Figure 2: Topological corner states and entanglement topological invariant evolution. a, Energy levels of the square-shaped nanoflake for the coupled BHZ model with mass $\varepsilon=-1$. The inset shows the probability density distribution of the zero-energy states represented by blue dots. b, The energy $E$ near Fermi energy (black and red lines) and second-order ETI $S^{T}$ (blue star lines) as a function of mass $\varepsilon$. The alternating red and black lines represent the double degenerate states. The other parameters are set as system width $W=10a$, system length $L=40a$, subsystem lengths $L_{A}=L_{B}=8a$, intra-orbital hopping $t=1$, kinetic energy $\lambda_{x}=\lambda_{y}=1$, interlayer coupling $\eta=0.4$ and Zeeman field $B_{z}=0$.
• Figure 3: Topological corner states and entanglement topological invariant evolution under exchange Zeeman field. a, Same as Fig. \ref{['fig2']}a , except the exchange Zeeman field $B_{z} = 1.8$. b, The energy $E$ near Fermi energy (black and red lines) and second-order ETI $S^{T}$ (blue star lines) as a function of Zeeman field $B_{z}$. The alternating red and black lines represent the double degenerate states. The other parameters are set as system width $W=10a$, system length $L=40a$, subsystem lengths $L_{A}=L_{B}=8a$, intra-orbital hopping $t=1$, kinetic energy $\lambda_{x}=\lambda_{y}=1$, interlayer coupling $\eta=0.4$ and mass $\varepsilon = -1$.
• Figure 4: Dependence of entanglement topological invariant on subsystem and system size. Second-order ETI $S^{T}$ as a function of mass $\varepsilon$ with different subsystem lengths $L_{A}$ for a and different system lengths $L$ for b . The length of region $B$ is consistently equal to the length of region $A$. The parameters are system length $L=40a$ in a and subsystem length $L_{A}=12a$ in b , other parameters are the same as those in Fig. \ref{['fig2']}b except Zeeman field $B_{z} = 1.8$. The insets are the details of $S^{T}$ within the range of mass $\varepsilon \in [1, 2.1]$.