Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology

We study the entanglement spectrum of noninteracting band insulators, which can be computed from the two-point correlation function, when restricted to one part of the system. In particular, we analyze a type of partitioning of the system that maintains its full translational symmetry, by tracing over a subset of local degrees of freedom, such as sublattice sites or spin orientations. The corresponding single-particle entanglement spectrum is the band structure of an entanglement Hamiltonian in the Brillouin zone. We find that the hallmark of a nontrivial topological phase is a gapless entanglement spectrum with an “entanglement Fermi surface.” Furthermore, we derive a relation between the entanglement spectrum and the quantum geometry of Bloch states contained in the Fubini-Study metric. The results are illustrated with lattice models of Chern insulators and ${\mathbb{Z}}_2$ topological insulators.

Figure 1

Mapping from the BZ to the Bloch sphere for the bipartitioning defined by the projector ${\Pi }_1$ [Eq. (19)] in the case of a band with nonvanishing Chern number. The points $k_1^0$ and $k_2^0$, where the first and second components of the Bloch spinor vanish, are mapped to the north and south pole of the Bloch sphere, respectively. The entanglement Fermi surface is mapped to the equator. Any other choice of ${\Pi }_1$ corresponds to a rotation of the Bloch sphere, such that the mapping from the BZ to the Bloch sphere changes. However, the existence of the entanglement Fermi surface is guaranteed by the nonvanishing Chern number for any choice of ${\Pi }_1$.

Figure 2

Plots of the SLES of the $\pi$-flux model for $t_2=0.1$ and $\eta =-0.35$ (top; topological) and $\eta =-0.45$ (bottom; trivial). The entanglement Fermi sea ($\lambda <0$) is shown in purple/light gray color, regions with $\lambda >0$ are shown in orange/dark gray, the thick black line shows the entanglement Fermi surface. The entanglement spectrum is gapless (gapped) in the topological (trivial) case, with a phase transition at $\eta =-4t_2$. According to Eq. (41) this plot also shows the quantum distance, as $\lambda \left(k\right)=d_{{\Pi }_1}\left(k\right)-1/2$.

Figure 3

Single-particle entanglement spectrum $\lambda \left(k\right)$ (dots) and quantum distance $d^2\left(k\right)-n/2$ (continuous line) as a function of the wave vector parallel to the applied spatial cut for the $\pi$-flux model. The cut is applied in the middle of a strip of 100 lattice sites, i.e., $N=200$ and $n=m=100$. The parameters are the same as in Fig. 2, i.e., $t_2=0.1$ and $\eta =-0.35$ (top) and $\eta =-0.45$ (bottom). The quantum distance is calculated using Eq. (35).

Figure 4

Regions of gapless energy spectrum (left) and gapless SLES (right) in the Kane-Mele model for different values of the parameters ${\lambda }_{\nu }$ and ${\lambda }_{\mathrm{R}}$. The plots were obtained by numerically evaluating the (entanglement) spectrum at random points in the first BZ for each set of parameters. The color coding shows the maximal distance between eigenvalues; red and yellow colors show values below $0.05$. Additional precision with a larger sample of points was used in a range around the phase transition. The black contour shows the ellipse of Eq. (47a).